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Essays on understanding consumer contribution behaviors in the context of crowdfunding
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Essays on understanding consumer contribution behaviors in the context of crowdfunding
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Essays on Understanding Consumer Contribution Behaviors in the Context of Crowdfunding by Xiaoqian Yu A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (BUSINESS ADMINISTRATION) August 2017 Copyright © 2017 by Xiaoqian Yu i Abstract My dissertation focuses on understanding consumers’ contribution behaviors in the context of crowdfunding. Crowdfunding is an emerging method for charitable contribution and entrepreneurial finance, in which small amounts of capital are obtained from a large number of individuals who share common interests. In 2015, nearly $34 billion was raised through crowdfunding platforms worldwide. Although the crowdfunding market is booming rapidly, the research on crowdfunding is still at its relatively nascent stage. In my first essay, I examine the effort-based incentives in the context of crowdfunding, a highly important and relevant context to marketing promotion. Effort-based incentives, wherein consumers are required to complete specific tasks to obtain monetary rewards, is a popular promotional tool in marketing. Unlike in a standard marketing context where sales promotions often trigger a non-negative effect on consumer demand, incentives may have a complex effect on consumer contribution behavior in the crowdfunding context when involving raising money for projects that can benefit the general public. On the one hand, effort-based incentives allow an individual to turn effort into a monetary reward, which can relax the budget and increase the individual’s contribution amount. On the other hand, monetary incentives may reduce people’s contribution amount by crowding out individuals’ intrinsic prosocial motivation. Therefore, I anticipate an overall positive or negative effect of effort-based incentives on individuals’ contribution amount, depending on the relative magnitude of the positive budget expansion effect and the negative psychological effect. I propose a structural model to uncover the underlying behavioral mechanisms for the impact of effort-based incentives on people’s prosocial contribution behavior. I develop a unified utility maximization framework to simultaneously model two interdependent consumer decisions: ii contribution decision (i.e. how much to contribute) and incentive participation decision (i.e. how many incentives tasks to participate in). The proposed model allows incentive participation to relax individuals’ budget constraint, which has not been considered in the literature. Since incentive participation directly affects budget, the optimal contribution and incentive participation amount involve simultaneity for some data observations. I adopt a two-step estimation approach to cope with the simultaneity issue. I use the Bayesian method to estimate model parameters to account for the multiple-constraint requirement and unobserved heterogeneity. Using data from a pioneering crowdfunding platform for journalism, I find that incentive availability may reduce individuals’ marginal utility for contribution, leading to either positive or negative overall effect of effort-based incentives on an individual’s contribution. Based on individuals’ heterogeneous reactions to the effort-based incentives, I can customize the offering of the incentives in a way to improve the return on investment for the incentive sponsors, increase the revenue for the crowdfunding platform, and also maximize individual donors contribution benefit (i.e. make them happier). Hence the policy simulation analysis in this study leads to important managerial implications on the effective use of the effort-based incentives. On crowdfunding platforms, contributors usually need to make contributions in the present and then wait to see if they can obtain the project reward at the end of the funding drive. Because the project reward delivery decision depends on the final funding performance, for instance, whether the funding goal is accomplished or not. Given this funding structure, individuals have to make contributions decisions in a forward looking manner, because they need to take into account of the future funding success probability. So for my second essay, I develop a dynamic structural model that explicitly captures individuals’ inter-temporal trade-offs and dynamic concerns when they make contribution decisions to a crowdfunding project. Understanding the factors that iii influence crowdfunders contribution behavior over time is a key to the success of funding projects and the crowdfunding platform. A structural model that characterizes the dynamic response of crowdfunders contribution behavior is therefore critical. In general, a crowdfunding project has a predetermined funding goal and deadline, and during the funding drive, people can observe the time varying funding status such as current funding amount raised, funding time left and the number of contributors who have previously supported the project. Thus, an individual can potentially make contribution decisions based on prior others behaviors. This is especially true for the current project, because the crowdfunding platform that I’m studying specializes in raising money to help project initiators publish a digital report that will be accessible to everyone online for free. This means that this crowdfunding platform is actually raising money for public goods, as it’s possible for people to obtain the project reward (i.e. digital report) without bearing any costs (i.e., money or time investment). Thus, given this empirical context, and also based on the economic theory on public goods provision, it’s essential for us to account for individuals’ dynamic behaviors in this setting by accounting for their motivations to manipulate the timing and amount of their contributions conditional on others behaviors. In this essay, I use data from a novel journalistic crowdfunding platform. On this platform, a project can receive funding from two types of contributors: individuals and organizations (e.g., New York Times). Typically, organizations make a large contribution in a single donation. And crowdfunders can observe the detailed funding status information from those two different types of contributors. This particular funding context leads to some interesting questions such as: how organization’s contribution would affect individual’s contribution? Would it crowd in or crowd out individual’s contribution? Is it a good idea for this crowdfunding platform to disclose iv organization’s contribution? To answer these questions, I build up a dynamic structural model which allows individuals to be forward looking, and which also accounts for individuals’ inter- temporal tradeoffs. The proposed model is flexible enough to handle funding deadlines and project reward schemes based on the final funding result (i.e., whether a project achieves the funding goal or not), all of which are ubiquitous in crowdfunding setting. In addition, it can handle the dynamic crowding in/out issue induced by organizations’ contributions, which is a unique feature in our crowdfunding setting. The estimation of our dynamic structural model involves several computational challenges, given that we have both continuous and constrained states and action space. To overcome the computational and estimation-related challenges, I adopt the modified Bayesian IJC algorithm, form the kernel density estimator to simulate the likelihood, use parallel computing (OpenMP and MPI) to speed up our estimation process, and solve the model using numerical dynamic programming techniques. Finally, I use the estimation result to evaluate the impact of changes to the extant funding result and the impact of changes to the extant organization’s contributions on individual contributions. v ACKNOWLEDGEMENTS Foremost, I would like to express my deepest gratitude to my advisor Dr. Sha Yang for her continuous support of my research, for her patience, motivation, and immense knowledge. Her guidance and encouragement helped me in all the time of both my research and personal life. I could not have imagined having a better advisor and mentor for my Ph.D. study. Besides my advisor, I would like to thank the rest of my thesis committee: Dr. Gerard J. Tellis, Dr. Anthony Dukes, Dr. Botao Yang, and Dr. Yu-Wei Hsieh, for their insightful comments and suggestions on my dissertation. My sincere gratitude also goes to my coauthor Dr. Yi Zhao, for the stimulating discussions, insightful suggestions, kindness encouragement and full support at every stage of my research. I’m also grateful to my coauthor Dr. Lian Jian, for offering me great research opportunities, inspirations and support. I also feel very fortunate to have been studying at Marshall School of Business, where I learned a lot and received generous help from the faculty members and fellow students. vi DEDICATIONS I would like to dedicate my dissertation to my dearest family. I’m deeply indebted to my parents, Jianbin Yu and Aiping Li, for their unconditional love, care and support. A special dedication goes to my husband, Guixi Zou, for all his love, understanding and support. I earnestly feel that without his support and dedication, I would not be able to go through this whole process. vii Contents Abstract ........................................................................................................................................... i Acknowledgements ........................................................................................................................v Dedications.................................................................................................................................... vi Contents ....................................................................................................................................... vii List of Tables ................................................................................................................................ ix List of Figures .................................................................................................................................x Chapter 1. Modeling Consumer Crowdfunding Behaviors under Effort-Based Incentives ..1 1.1 Introduction ...........................................................................................................................1 1.2 Related Literature ..................................................................................................................6 1.3 Empirical Context .................................................................................................................9 1.3.1 The Crowdfunding Platform ..........................................................................................9 1.3.2 The Data ......................................................................................................................10 1.3.3 Data Patterns and Insights ..........................................................................................13 1.4 Proposed Model ..................................................................................................................15 1.4.1 Modeling Individual Contribution Amount and Incentive Participation Amount .......15 1.4.2 Model Identification and Parameter Estimation .........................................................21 1.4.3 Model Derivation ........................................................................................................24 1.5 Results and Counterfactual Analysis ..................................................................................27 1.5.1 Model Comparison ......................................................................................................27 1.5.2 Findings .......................................................................................................................28 1.5.3 Counterfactual Analysis ..............................................................................................33 1.6 Conclusion ..........................................................................................................................40 Chapter 2. A Dynamic Model of Crowdfunders’ Contributions to Public Goods .................43 2.1 Introduction .........................................................................................................................43 2.2 Empirical Context ...............................................................................................................47 2.3 Related Literature ................................................................................................................49 2.4 Data .....................................................................................................................................57 2.4.1 Model-Free Data Patterns ..........................................................................................59 2.5 Modeling Framework ..........................................................................................................63 viii 2.5.1 Motivation for Structural Modeling and Sources of Dynamics ..................................63 2.5.2 Model Setup .................................................................................................................65 2.5.2.1 Model Timeline ....................................................................................................65 2.5.2.2 Utility Specification .............................................................................................67 2.5.2.3 Period Utility during the Funding Stage .............................................................67 2.5.2.4 Period Utility during the Final Delivery Stage ...................................................69 2.5.2.5 Donor’s Belief on Final Report Delivery Probability .........................................69 2.5.3 Evolution of State Variables ........................................................................................70 2.5.4 Dynamic Structural Model ..........................................................................................73 2.5.4.1 Modified Bayesian Algorithm ..............................................................................75 2.5.4.1.1 The Outer Loop of MCMC Algorithm .........................................................76 2.5.4.1.2 The Inner Loop of MCMC Algorithm ..........................................................77 2.5.4.1.3 The M-H Steps .............................................................................................78 2.6 Estimation Result ................................................................................................................81 2.7 Conclusion .........................................................................................................................82 Bibliography for Chapter One ....................................................................................................85 Bibliography for Chapter Two ...................................................................................................88 Appendices ....................................................................................................................................90 Appendices for Chapter One .....................................................................................................90 Appendix A: Model Derivation and Likelihood Construction .............................................90 Appendix B: Predicting Contribution Amount 𝑚̂ ................................................................99 Appendix C: Predicting Number of Incentive Participation 𝑛̂ ..........................................102 Appendix D: Bayesian MCMC Algorithm ........................................................................105 Appendix E: Model Identification and Parameter Estimation ...........................................107 Appendices for Chapter Two ..................................................................................................113 Appendix A: A Funding Project Example .........................................................................113 Appendix B: Estimation Result for State Transition ..........................................................114 ix List of Tables Table 1. Summary Statistics .....................................................................................................13 Table 2. Seven Possible Cases of Observed Outcomes ............................................................21 Table 3. Estimation Result .......................................................................................................30 Table 4. Heterogeneity Estimates from the Proposed Model ...................................................31 Table 5. Characteristics of the Eight Segments and Counterfactual Analysis I .......................36 Table 6. Counterfactual Analysis II: Customized Incentives ...................................................39 Table 7. Empirical Evidence on Crowding Effect ...................................................................54 Table 8. Experimental Evidence on Crowding Effect ..............................................................55 Table 9. Summary of the Overall Dataset ................................................................................58 Table 10. Summary Statistics for Funding Project Information...............................................58 Table 11. Summary Statistics for Individual’s Weekly Contribution ......................................59 Table 12. Impact of Organization’s Contribution on Individual’s Contribution......................62 Table 13. State Variables and the Law of Motion ....................................................................73 Table 14. Estimation Result .....................................................................................................82 Table 15. Predicting m using Censored Regression ...............................................................100 Table 16. Variance-Covariance for m using Censored Regression ........................................101 Table 17. Estimation Result for n using Truncated Poisson Regression ................................103 Table 18. Variance-Covariance for n using Truncated Poisson Regression ..........................104 Table 19. Estimation Result for Weekly Donation Amount ..................................................116 Table 20. Estimation Result for Weekly Number of Donor ...................................................117 Table 21. Estimation Result for Weekly Organizational Amount .........................................118 x List of Figures Figure 1. Contribution and Incentive Participation Incidences Over Time .............................14 Figure 2. Heterogeneous Impact of Incentives on Contribution ..............................................15 Figure 3. Histogram of the Impact of Incentive Availability on Contribution Preference ......32 Figure 4. Histogram of the Baseline Budget ............................................................................32 Figure 5. Histogram of the Reward Saving Benefit .................................................................33 Figure 6. Funding Trend along the Funding Drive ..................................................................60 Figure 7. Funding Percentage Raised over Funding Time Passed for Unsuccessful Projects .61 Figure 8. Funding Percentage Raised over Funding Time Passed for Successful Projects .....61 Figure 9. Model Timeline ........................................................................................................66 1 Chapter 1. Modeling Consumer Crowdfunding Behaviors under Effort-Based Incentives 1.1 Introduction One central topic in marketing is to understand how promotion affects consumer behavior. Numerous papers have documented various promotional effects on consumer product awareness (e.g. Terui, Ban and Allenby 2011), purchase (e.g. Kamakura and Russell 1989), and consumption (e.g. Neslin, Henderson and Quelch 1985). In a typical sales promotion such as price discount, getting a reward often requires little effort from the consumer. However, there is a large body of literature in marketing on promotions such as coupons (e.g. Bawa and Shoemaker 1987) and social causes (e.g. Arora and Henderson 2007), which require consumer effort for getting the promotion. More specifically, effort-based incentives are those that require individuals to engage in a specific task and expend effort in exchange for a reward. For instance, Google Consumer Surveys allow businesses to pay to get survey responses from consumers on the Internet who wants to earn micropayments to gain access to premium content (e.g., news articles). 1 Some mobile applications, such as iBotta and Receipt Hog, allow users to earn monetary rewards by performing small tasks (e.g. answering a survey question or scanning one’s grocery store receipt), and users can donate the reward to charities of their choices. Elance, a popular online freelance marketplace, awards an existing customer $10 or 10 connects for making a qualified referral. In this paper, we examine the effort-based incentives in crowdfunding, a highly important and relevant context in marketing. In recent years, crowdfunding markets have emerged to provide entrepreneurs a new channel for raising capital in support of creative projects or new business ideas 2 . Promotion is an important marketing tool to crowdfunders as well as crowdfunding 1 See details at https://www.google.com/insights/consumersurveys/home, retrieved on Oct 24, 2014. 2 See details at “The New Thundering Herd,” http://www.economist.com/node/21556973 retrieved on Oct 24, 2014. 2 platforms. Unlike consumers in other market transactions such as retailing, crowdfunding participants often contribute without receiving a direct tangible benefit 3 , and as such, need to be better motivated. The empirical context of this study is a pioneering crowdfunding platform for journalism. On this journalism-based crowdfunding platform, freelance journalists propose topics or stories for funding with a set of specified deliverables, and contributors can support a fund requester to research a topic and produce the deliverables by donating money. Third-party company sponsors of this platform offer effort-based incentives so that contributors can participate in an incentive task (i.e. filling out a survey provided by the sponsors) to obtain rewards that can be credited towards their contribution amount. Both the platform and the incentive sponsors can benefit from such a promotion. On the one hand, if this type of promotion helps increase user contribution amount, the platform can anticipate a revenue gain as the platform keeps a percentage of every contribution dollar as its revenue. On the other hand, if the promotion incentive attracts more participants, it helps build more goodwill and gather more survey feedbacks for the incentive sponsors. Unlike in a standard marketing context where sales promotions often trigger a non-negative effect on consumer demand, incentives may have a complex effect on consumer contribution behavior in the crowdfunding context when involving raising money for projects that can benefit the general public. This is because when crowdfunding projects are raising money for public goods or causes, individuals are behaving prosocially by making contributions to support such projects. And monetary incentives may have dual effects on individuals’ crowdfunding contribution in such 3 For example, DonorsChoose.org is a U.S. non-profit crowdfunding platform that allows individuals to donate directly to public school classroom projects. GoFundMe.com is a crowdfunding platform that allows people to raise money for events ranging from life events such as celebrations and graduations to challenging circumstances like accidents and illnesses. On these kind of crowdfunding platforms, crowdfunders are purely making prosocial contributions, as they do not receive tangible reward for making contributions. 3 case. On the one hand, incentive rewards can relax one’s budget and thus help increase the individual’s contribution amount, based on the basic economic principle. On the other hand, incentives can have detrimental effects on prosocial behaviors because they could mitigate participants’ intrinsic joy of giving or spoil their social image value (e.g. Ariely, Bracha and Meier 2009). Therefore, we may well anticipate an overall positive or negative impact of effort-based incentives on individuals’ contribution amount, depending on the relative magnitudes of the positive budget expansion effect and the negative psychological effect. This is quite different from a standard marketing context where sales promotions often trigger a non-negative effect on consumer demand. A structural model is therefore needed to help better understand the underlying behavioral mechanisms that would explain the complex relationship between effort-based incentives and consumer contribution to prosocial crowdfunding projects. We develop a structural model to study consumer crowdfunding contribution behavior under effort-based incentives. The proposed model has two unique features. First, we jointly model consumer demand (i.e. how much to contribute) and incentive participation (i.e. how many incentive tasks to participate) based on a unified utility maximization framework. This is important because incentive participation incurs cost but brings reward, which could relax a consumer’s budget constraint, and as such, the two decisions on contribution amount and incentive participation may be dependent on each other. In literature, incentive participation is relatively under-researched, possibly due to lack of data in the traditional retailing context. For example, researchers often observe consumer coupon redemption but do not observe the actual coupon collection behavior. Second, we model the dual effects of the effort-based incentive on consumer contribution. In particular, our model allows incentive participation to relax individuals’ budget constraint, 4 which has not been considered in the previous literature, but is an important nature of promotion that requires non-trivial effort from consumers to receive the reward. We also capture the potential hidden cost of monetary rewards (Deci 1971, Lepper and Greene 1978) by allowing whether an incentive is available to affect an individual’s marginal contribution utility. Modeling such a dual- effect process helps us understand why the effort-based incentives may increase or decrease individual contribution amount, depending on the magnitude of these two competing forces. The proposed model imposes several challenges. On model derivation, solving the Kuhn- Tucker optimality conditions involves additional complexity due to the joint modeling of contribution amount and incentive participation level, and allowing incentive participation decision to directly affect the budget constraint. Our proposed model also gives rise to several econometric challenges: (i) Simultaneity, that is, contribution and incentive participation affect each other in some data cases. We adopt a two-step estimation approach to cope with the simultaneity issue. (ii) Implicitness, that is, the specification of a decision variable is not explicit. We use changes-in-variable method to obtain the likelihood. (iii) Unobserved budget, that is, we treat budget as unknown and make inference. As a result, some model parameters need to be estimated with constraints, and classical estimation methods like MLE are difficult to implement in this case to account for a large number of constraints. We develop a Bayesian method to estimate model parameters while conveniently ensuring model parameters to satisfy multiple constraints. Our paper provides useful managerial insights on the effective use of effort-based incentives. From the platform’s perspective, we show that effort-based incentives have a positive effect on some individuals but a negative effect on others, and a targeted strategy would help the platform increase revenue. We segment consumers into eight groups based on mean split of estimated baseline budget, contribution preference, and incentive participation cost. We find that, 5 making incentives available can increase consumer contribution by up to 109.6% for segments with relatively low detrimental effect from the incentives. For those segments with relatively large negative impact from the incentives, effort-based incentives tend to decrease their contribution amount, as the positive budget expansion effect is dominated by the negative psychological effect. From an incentive sponsor’s perspective, we show that a customized strategy can help the sponsor improve the incentive participation rate at a lower cost (i.e. reward amount). We design a policy experiment to compare incentive participation and cost in two scenarios: (i) uniform incentive strategy where all groups are provided with incentives with the same reward amount; and (ii) customized incentive strategy where groups will be exposed to incentives with different reward amount. We simulate the total contribution amount and the total incentive participation frequency under these two settings. We find that the reward amount customization can improve the return on investment for incentive sponsors, raise more contribution for the crowdfunding platform, and more importantly, maximize individual’s contribution utility. In other words, our customization creates a win-win-win scenario for the crowdfunding platform, incentive sponsors, and also the crowdfunders. Thus, our counterfactual analysis underscores the importance of targeting and customization for the effective use of effort-based incentives. The remainder of this paper is organized as follows. In the next two sections, we review the relevant literature and provide background information about our empirical context. We then develop a model of consumer contribution and incentive participation under effort-based incentives in section 4. In section 5, we discuss our empirical findings and derive managerial implications via counterfactual simulations. Section 6 concludes the paper with implications for future research. 6 1.2 Related Literature Our study is built upon a large body of marketing literature on coupon promotion. Although coupon does not require consumers to complete a particular task as in our case, it sometimes demand search effort. One stream of research has examined factors affecting consumer coupon redemption. For instance, Reibstein (1982) found that coupon redemption is explained by distribution methods along with several other factors. Mittal (1994) found that consumer characteristics are important correlates to coupon redemption. Bawa, Srinivasan and Srivastava (1997) developed a model to understand the interaction effect of coupon attractiveness and coupon proneness on coupon redemption. Another stream of research has sought to quantify the behavioral and managerial implications of coupon promotions. For example, Neslin (1990) developed a market response model for coupon promotions. Inman and McAlister (1994) investigated whether and how coupon expiration dates affect consumer behavior. Chiang (1995) studied how coupons from competing brands affect individual consumers’ category demand. Our research differentiates from these two streams of coupon related research in several ways. First, we study a different type of effort-based promotion requiring participants to complete a task for a reward, in a context of making contributions to crowdfunding projects for public goods. In coupon promotions, researchers often do not observe whether individual consumers collect and save coupons, and for that reason, can only study coupon redemption as a measure for effort. Here, we study consumers’ responses to incentives with varying observable reward amounts and effort requirements. Second, we jointly model consumer participation in effort-based incentives and the impact of such incentives on their consumption behavior, i.e., contributing to crowd-funded journalistic projects. Such a modeling approach distinguishes the current study from most previous research focusing on either coupon redemption or sales effect of coupon promotion, with a well- 7 noted exception by Chiang (1995) where a selection type of econometric model is put forth to simultaneously study consumer coupon use and demand. Third, we model the contribution participation and amount as outcomes from a joint utility maximization framework. This allows us to closely examine the alternative behavioral mechanisms of the influence from effort-based incentives. This structural formulation has an advantage over the reduced-form specification as adopted in Chiang (1995) in exploring the underlying behavioral mechanisms. In addition, we incorporate dynamics by modeling consumer budget evolvement over time and reward accumulation as done in the literature of loyalty programs (e.g. Kopalle et al 2012), which were not considered in Chiang (1995). Since the products of crowd-funded journalism are published online with free access, they are public goods and contributing to public goods is therefore deemed as a prosocial behavior. Our empirical investigation into the effect of effort-based incentives on individual contribution behavior also connects to the broad research on consumer prosocial behavior. Some prior research in marketing has sought to understand consumer charity donation behavior and how consumers respond to cause-related promotion. For the former, attention has been devoted to the effect of asking amount on consumer donation interest. Recently, Lee and Feinberg (2014) empirically investigated how “appeal scales” (i.e. the suggested donation amount levels) affect consumer charitable donation incidence and amount and suggested the use of customized scales based on individual donation history. For the latter, Strahilevitz and Myers (1998) found that charity incentives are more effective in promoting frivolous products than in promoting practical products. Arora and Henderson (2007) found that embedded premium (i.e. a sales promotion with a social cause) works more effectively than an equivalent price discount in generating sales. 8 Another stream of research has examined how monetary incentive affects individual’s prosocial behavior. From a theoretical perspective, incentives can affect contribution behavior to public goods through complex interactions of multiple effects. On one hand, standard economics theory predicts a direct price effect, that is, incentives can positively influence prosocial behavior by offsetting one’s cost of contribution or relaxing one’s budget constraint (Arrow 1972). On the other hand, incentives can have detrimental psychological effects on prosocial behaviors which can reduce consumer intention to contribute to public goods. Several explanations are offered in the literature to address such detrimental effect of monetary reward on prosocial behavior. First, incentives can crowd out intrinsic joy from a prosocial action (Deci 1971). Second, incentives can spoil the social image value of doing good deeds (Ariely, Bracha and Meier 2009). Third, incentives can shift one’s perception of the interaction norm from a social one to a market transactional one, making free-riding more acceptable or less shameful (Heyman and Ariely 2004). Overall, prior research shows that explicit incentives have mixed effects on prosocial behavior. For example, explicit incentives have worked in encouraging garbage recycling (Kinnaman 2006) and blood donation (Lacetera, Macis, and Slonim 2012), but have backfired in other cases such as in motivating volunteering activities (Gneezy and Rustichini 2000a) and in encouraging parents picking up their children from daycare on time (Gneezy and Rustichini 2000b). These studies do not separately identify individuals’ decisions in participating in an incentive task and their decisions in their prosocial contribution, as they are essentially one decision, e.g., donate blood in exchange for rewards. In the context of effort-based incentives, these are two separate decisions and need to be distinguished and then modeled jointly. This new context calls for a more nuanced modeling approach, which is taken in the present study. 9 Our research is also built upon a vast literature in marketing on modeling consumer discrete-continuous demand (Arora, Allenby and Ginter 1998, Kim, Allenby and Rossi 2002, Bhat 2005, and Satomura, Kim and Allenby 2011). Our model makes two important extensions in order to incorporate effort-based incentives into consumer demand. First, the standard discrete- continuous demand model assumes the budget not to be affected by the decision outcome, and consequently there is no need to estimate the budget parameter. In our model, number of participated incentive tasks and hence the reward obtained will affect the budget. Since the decision variables affect the budget, we encounter the simultaneity issue, that is, the optimal outcomes of contribution amount and incentive participation will affect each other. On top of that, we now need to infer model parameters that have to satisfy multiple constraints. Both add a substantial complexity to model inference. Second, unlike a standard discrete-continuous model where there is only one key behavioral decision variable (i.e. the demand), we have two decision variables: contribution and incentive participation decisions. This adds another layer of complexity to the model derivation and estimation as shown in the Appendix A. 1.3 Empirical Context 1.3.1 The Crowdfunding Platform We obtain data from a leading crowdfunding platform for community-funded online journalism established in November 2008. As of April 2013, the platform has received support from more than 20,900 contributors. Although this crowdfunding platform is smaller than other popular crowdfunding platforms like Kickstarter, it is one of the earliest and pioneering crowdfunding platforms specially designed for journalistic reporting. It allows freelance journalists to raise money in support of their work by pitching journalism ideas to the online community, just as an 10 entrepreneur would pitch venture capitalists. The platform keeps a small percentage of every contribution dollar as its revenue. People can contribute to support the projects they are interested in. If a project creator manages to raise enough money, she will deliver the final report on the funding website, and the public has free access to the report. In other words, people are contributing money for public goods (i.e., free digital reports) on this crowdfunding website. The platform has been experimenting with an effort-based incentive mechanism that allows users to earn credit dollars by participating in a survey designed by third-party sponsors of this platform. An example of a survey questions is “Considering all aspects of your daily life (school, work, etc.), in total, your printing usage is light-personal use, medium or heavy?” Potential contributors are not obliged to take a survey, and if participating, they can allocate the credit dollars obtained from their survey participation to any projects they are interested in in any increment, or they can save some or all credits for future use on this website. As third party sponsors, they will pay for the credit dollars that compensate users for completing the incentive tasks. In return, they can collect survey responses from individuals’ incentive participation. In addition, offering such incentives may help sponsors promote their brand and build up goodwill. 1.3.2 The Data In total, we observe contribution behavior from 5,311 users of this platform. We observe each individual’s contribution amount and number of incentive tasks participated in each month from November 2008 to May 2011. We expand the dataset by treating the non-contribution and non- incentive participation as zero observations 4 . The final re-constructed data have 55,305 observations on those individuals’ monthly contribution and monthly incentive participation. 4 The original dataset has 5,541 users, but 230 of them are excluded due to either no activity or missing information. 11 These 5,311 individuals have made 7,474 positive monthly contributions and have 5,885 non-zero monthly incentive participation during the time window. For each consumer and during each time period (i.e., month), we model how much money she contributes and how many incentive tasks she participates in. We choose not to model the incentive participation as a 0-1 dummy variable, because in a given time period, there may be multiple number of incentive tasks (i.e., surveys) available, and individuals are allowed to participate in multiple incentive tasks (i.e., take more than one survey). Given this empirical context, modeling one single 0-1 dummy for incentive participation is not sufficient, and using multiple 0-1 dummy variables will make it much more complicated to model the contribution and incentive participation decision in a unified structural way in order to answer our main research question. So we decide to model the incentive participation by using the number of incentive tasks taken. In this way, we are able to parsimoniously model the incentive participation trade-offs. Table 1 reports the summary statistics of the variables used in our analysis. The average monthly individual contribution amount (𝑚 𝑖𝑡 ) is $2.245 with a wide dispersion ranging from $0 to $2,500. The monthly average number of surveys a consumer takes (𝑛 𝑖𝑡 ) is 0.133, ranging from 0 to 4. The variable Lag_CumContri_Dummy measures the contribution status in the last period, and this variable allows us to capture contribution state dependence. In a given time period, there are on average 31 projects available for funding. The monthly average reward amount from available surveys ( it R ) is $0.560. The monthly average number of questions per survey is 9.666. The effort-based incentives are available for 83.4% of time. The average number of available incentives is 2.505 with a standard deviation of 1.374, and the maximum number of available surveys is 5. The variable Lag_CumSurvey_Dummy measures whether an individual has taken any incentive tasks in the last period. We use this 12 variable to measure individual state dependence in incentive participation. Variable it L measures the reward amount left from the last period (or at the beginning of period t). Individuals can save the earned reward for future use. We find that consumers have positive reward credit balance for 20.5% of the time periods and its monthly average is $0.372, which is much smaller than the average monthly contribution amount ($2.245 or $16.614). Our crowdfunding website is a pioneering funding platform for journalism. In our dataset, we observe contributions not only from general individuals, but also from news reporters. Thus we create a Reporter dummy which is an individual specific variable, taking the value of 1 if the person is a news reporter, and 0 otherwise. About 15.6% of the people in the analysis sample are reporters. On this crowdfunding platform, the funding projects are classified by sixteen story types. For example, a funding project that pitches a story related to public health and environment will be assigned with a tag of News_pub_health as well as a tag of News_environment. Moreover, a funding project is usually assigned with multiple news type tags (i.e., the types are not mutually exclusive). Using this information, we are able to calculate the share of different types of funding projects available in a given time period 5 . We see that stories related to poverty, race, government & employment issues, and diversity are more popular on this crowdfunding platform. 5 For simplicity, we re-classify those sixteen story types into five groups. 13 Variables Mean Std. Dev. Min Max Contribution Amount 𝑚 𝑖𝑡 2.245 21.463 0 2500.000 Contribution Amount 𝑚 𝑖𝑡 >0 16.614 56.307 0 2500.000 Number of Incentive Participation 𝑛 𝑖𝑡 0.133 0.422 0 4 Number of Incentive Participation 𝑛 𝑖𝑡 >0 1.252 0.525 0 4 Lag_CumContri_Dummy 0.852 0.355 0 1 Number of Available Funding Projects 31.160 10.350 5 42 Incentives Availability Dummy 0.834 0.372 0 1 Number of Available Incentives 2.505 1.374 0 5 Average Reward Available 𝑅̅ 𝑖𝑡 0.560 1.675 0 10 Average Number of Questions 9.666 9.576 0 32 Lag_CumSurvey_Dummy 0.547 0.498 0 1 Reward left from last period 𝐿 𝑖𝑡 0.319 1.196 0 25 Reward Saving Indicator 𝑆 𝑖𝑡 0.205 0.403 0 1 Reward Saving Amount (𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 )≥0 0.372 1.296 0 25.000 Reporter 0.156 0.362 0 1 Share_Project_Type I 0.469 0.113 0.167 0.800 Share_Project_Type II 0.821 0.084 0.556 1.000 Share_Project_Type III 0.351 0.126 0.000 0.750 Share_Project_Type IV 0.425 0.123 0.000 0.688 Share_Project_Type V 0.446 0.118 0.300 0.750 Note: Project Type I: Public Health, Environmental Issues Project Type II: race, diversity, poverty, government, employment issues Project Type III: education justice issues Project Type IV: Local infrastructural issues (LA, Vietnam) Project Type V: Others Table 1. Summary Statistics 1.3.3 Data Patterns and Insights First, we find that, when incentives are available, contribution amount and number of incentive tasks participated are significantly correlated (ρ=0.14, p-value<0.001). We also plotted the contribution and incentive participation incidences over time (see Figure 1), and found that these two variables are highly correspondent to each other. These data patterns suggest that when incentive tasks are available, contribution amount and incentive participation may be two interdependent decisions, and such interdependence justifies a unified utility maximization framework and possibly simultaneity between the two decisions. 14 Figure 1. Contribution (solid line) and Incentive Participation (dotted line) Incidences over Time Second, we compare the average contribution amount between two scenarios: when incentives are available (mean = 1.79) vs. when incentives are not available (mean=4.53). Surprisingly, we find that the average contribution amount is significantly lower when incentives are available (t-stats=9.27, p-value<0.001). Our regression analysis shows that the impact of incentive availability on contribution amount varies across individuals. Figure 2 plots the histogram of the reduced-form overall effect of incentive availability on contribution amount across individuals. We see that while incentives increase contribution amount for some individuals, they lower contribution amount for others. 15 Figure 2. Heterogeneous Impact of Incentives on Contribution The overall effect could be an end result from two opposing forces: (i) availability of incentive tasks allow individuals to expand budget and therefore to increase contribution amount; (ii) availability of incentive tasks provides monetary rewards to participants, and monetary rewards haven been shown to undermine people’s prosocial behavior, which will lower the individual’s contribution preference and the contribution amount. This explains that why incentive availability may have a negative impact on contribution amount for some individuals. 1.4 Proposed Model 1.4.1 Modeling Individual Contribution Amount and Incentive Participation Amount We first introduce our notations used in the model. 𝑚 𝑖𝑡 is the observed contribution amount and 𝑛 𝑖𝑡 is the observed number of effort-based incentives participated by individual i during time period t (month). These are the two decision variables that we model. 𝜑 𝑖𝑡 captures the baseline marginal utility from contribution. 𝑚 𝑖𝑡 𝑜𝑢𝑡𝑠𝑖𝑑𝑒 denotes person i’s outside good allocation. 𝐶 𝑖𝑡 denotes the average incentive participation disutility or cost per incentive task, and therefore 𝐶 𝑖𝑡 ∙ 16 𝑛 𝑖𝑡 measures the total incentive participation cost through taking 𝑛 𝑖𝑡 number of incentive tasks. 𝐿 𝑖𝑡 denotes the reward saving amount the individual has accumulated at the beginning of t. 𝑅̅ 𝑖𝑡 represents the average reward amount available, thus, 𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 calculates the total reward amount obtained through taking 𝑛 𝑖𝑡 number of incentive tasks. (𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 ) measures the reward saving amount at the end of period t 6 , and 𝛾 𝑖 captures the marginal benefit from reward saving. We also create a reward saving indicator 𝑆 𝑖𝑡 =𝐼 (𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 >0) . If 𝑆 𝑖𝑡 =1, it suggests that the individual has made the contribution entirely by using her reward credits. If 𝑆 𝑖𝑡 =0, it suggests that the individual contributed out of her own pocket in addition to using all of her available credits. We make this distinction to account for the fact that incentive reward can only be used on this website, and cannot be redeemed as real cash for consumers to use on other consumptions at their will. We employ a flexible utility function specification that can produce either corner or interior solutions (Arora, Allenby and Ginter 1998, Kim, Allenby and Rossi 2002, Bhat 2005, and Satomura, Kim and Allenby 2011). We assume that an individual i makes the contribution amount decision (𝑚 𝑖𝑡 ) and the number of incentive participation decision (𝑛 𝑖𝑡 ) jointly in order to maximize a unified utility that consists of the following four components: (1) benefit from making a contribution 𝑚 𝑖𝑡 ≥0; (2) benefit from allocating the remaining budget on outside good (𝑚 𝑖𝑡 𝑜𝑢𝑡𝑠𝑖𝑑𝑒 ) ; (3) disutility or cost from incentive participation; (4) benefit from saving the reward for future use to account for the possibility of finding a better project to contribute in the future. Thus, consumer i (i = 1, 2…N) maximizes her utility in time period t (t = 1, 2… i T ): 6 𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 can be negative, and we use a reward saving dummy 𝑆 𝑖𝑡 to make sure that the reward amount saved is non-negative. 17 ln(m 1) ln(m 1) m it outside it it it it it it i it it it it U C n S L R n (1) 7 subject to the budget constraint: 𝑚 𝑖𝑡 +𝑚 𝑖𝑡 𝑜𝑢𝑡𝑠𝑖𝑑𝑒 +𝑆 𝑖𝑡 (𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 )≤𝑀 𝑖𝑡 (2) where 𝑀 𝑖𝑡 stands for individual i’s monetary donation related budget in period t, and 𝑆 𝑖𝑡 (𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 ) denotes the reward amount left for future use. Note that the left-hand side of the equation (2) involves the reward saving 𝑆 𝑖𝑡 (𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 ) . This is because reward credits saved are not transferrable to outside good consumption, since reward credits earned can be only used to contribute to projects on this website. In order to capture the unique aspect of effort-based incentives in relaxing consumer budget, we specify 𝑀 𝑖𝑡 as follows, 𝑀 𝑖𝑡 =exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 (3) where exp(𝑀̅ 𝑖 ) represents person i’s baseline budget, and the budget, 𝑀 𝑖𝑡 , can be expanded by the reward amount accumulated (𝐿 𝑖𝑡 ) and the reward obtained (𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 ). 7 We include +1 to allow for corner solution. (1) Contribution Benefit (4) Benefit from Reward Saving (3) Incentive Participation Disutility or Cost (2) Outside Good Benefit 18 When𝑆 𝑖𝑡 =1 , an individual makes a contribution it m entirely from the reward she accumulated, but the remaining credits cannot be used on outside good given the policy. In that case, equation (2) and (3) suggest that she can allocate all her remaining budget on outside good, that is, 𝑚 𝑖𝑡 𝑜𝑢𝑡𝑠𝑖𝑑𝑒 =exp(𝑀̅ 𝑖 ). In that case, the person still needs to decide on how much to contribute now, how much credits to save for future contribution, and how many incentive tasks to participate. However when 𝑆 𝑖𝑡 =0, an individual contributes 𝑚 𝑖𝑡 by using all her reward credits accumulated as well as some of her baseline budget. So the remaining budget she left on outside good becomes 𝑚 𝑖𝑡 𝑜𝑢𝑡𝑠𝑖𝑑𝑒 =exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 . By substituting in the outside good allocation as derived above, we can rewrite the utility function as follows under two cases. When, 𝑆 𝑖𝑡 =1, 𝑈 𝑖𝑡 =𝜑 𝑖𝑡 ln(𝑚 𝑖𝑡 +1)+ln(exp (𝑀̅ 𝑖 )+1)−𝐶 𝑖𝑡 ∙𝑛 𝑖𝑡 +𝛾 𝑖 ∙(𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 ) (4a) And when there is no reward saving for future, i.e., 𝑆 𝑖𝑡 =0, the utility function can be written as 𝑈 𝑖𝑡 = 𝜑 𝑖𝑡 ln(𝑚 𝑖𝑡 +1)+ln(exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1)−𝐶 𝑖𝑡 ∙𝑛 𝑖𝑡 (4b) To ensure the positivity of the incentive participation cost 𝐶 𝑖𝑡 , we adopt the following censored normal specification: 𝐶 𝑖𝑡 ={ 𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′𝜏 +𝜐 𝑖𝑡 , 𝑖𝑓 𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖 𝑛 ′𝜏 +𝜐 𝑖𝑡 ≥0 0, 𝑜𝑡 ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (5) 19 where 𝑥 𝑖𝑡 𝑛 and 𝑧 𝑖𝑡 𝑛 stand for the vector of covariates that affect the incentive participation cost (e.g., Lag_Survey_Dummy, number of available surveys, number of survey questions, reporter dummy), and 𝜏 𝑖 𝑎𝑛𝑑 𝜏 denote the corresponding vector of response coefficients 8 . 𝜐 𝑖𝑡 captures the random shocks that affect the incentive participation cost. For identification purpose, we assume that 𝜐 𝑖𝑡 ~𝑖 .𝑖 .𝑑 ,𝑁 (0,1) . Following the literature, we introduce a multiplicative random element to the marginal baseline utility. The exponential function guarantees the marginal utility to be positive: 𝜑 𝑖𝑡 =exp (𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 +𝜀 𝑖𝑡 ) (6) where 𝑥 𝑖𝑡 𝑚 ′and 𝑧 𝑖𝑡 𝑚 ′ stand for the vector of covariates that affect the baseline marginal benefit from contributing 𝑚 𝑖𝑡 (e.g., Lag_Contri_Dummy, number of funding projects available, incentive availability dummy, share of different types of funding project, reporter dummy), and 𝛽 𝑖 and 𝛽 denote the corresponding vector of response coefficients. Based on the literature, the presence of the incentives may deter people’s prosocial behavior. In order to test the potential detrimental psychological effect from the effort-based incentives, we allow the incentive availability dummy to directly influence the baseline marginal contribution benefit, so that we can test whether the presence of the incentives would have an influence on individuals prosocial contribution behavior. 𝜀 𝑖𝑡 captures the random shocks that affect the baseline marginal contribution utility. For identification purpose, we assume that 𝜀 𝑖𝑡 ~𝑖 .𝑖 .𝑑 ,𝑁 (0,1) . 8 Some variables in 𝑧 𝑖𝑡 𝑛 are individual specific, so we use fixed coefficient, and other variables in 𝑧 𝑖𝑡 𝑛 do not have enough variation over time, so we use fixed coefficient instead of random coefficient. 20 Finally, we also adopt the exponential transformation to ensure the positivity of the marginal benefit from reward saving, 𝛾 𝑖 =exp (𝛾 𝑖 ∗ ) (7) Given that in our data, the contribution amount (𝑚 𝑖𝑡 ), incentive participation (𝑛 𝑖𝑡 ), and the reward saving indicator ( 𝑆 𝑖𝑡 =𝐼 (𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 >0) ) can be either zero or non-zero, we have eight different data combinations depending on the value of these three variables: Case (1). 𝑚 𝑖𝑡 =0,𝑛 𝑖𝑡 =0,𝑆 𝑖𝑡 =0; Case (2). 𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 =0,𝑆 𝑖𝑡 =0; Case (3). 𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 >0,𝑆 𝑖𝑡 =0; Case (4). 𝑚 𝑖𝑡 =0,𝑛 𝑖𝑡 =0,𝑆 𝑖𝑡 =1; Case (5). 𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 =0,𝑆 𝑖𝑡 =1; Case (6). 𝑚 𝑖𝑡 =0,𝑛 𝑖𝑡 >0,𝑆 𝑖𝑡 =1; Case (7). 𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 >0,𝑆 𝑖𝑡 =1; Case (8). 𝑚 𝑖𝑡 =0,𝑛 𝑖𝑡 >0,𝑆 𝑖𝑡 =0. Table 2 reports the data frequency for the eight cases. 70.63% of the observations belong to case 1 (i.e., no contribution, no incentive participation and no reward accumulation). Case 5 is the least likely because the average reward saving amount is much smaller than average contribution amount. We have a total of 7 possible cases, since Case 8 is infeasible because when 𝑚 𝑖𝑡 =0 and 𝑛 𝑖𝑡 >0, 𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 would always be greater than zero, and therefore 𝑆 𝑖𝑡 21 has to take the value of 1 instead of 0. We provide the complete derivation for the Kuhn-Tucker first-order conditions for these seven cases in Appendix A. Seven Possible Cases of Observed Outcomes Count Percentage Case 1: 𝑚 𝑖𝑡 =0,𝑛 𝑖𝑡 =0, 𝑆 𝑖𝑡 =0 39063 70.63% Case 2: 𝑚 𝑖𝑡 . >0,𝑛 𝑖𝑡 =0, 𝑆 𝑖𝑡 =0 1831 3.31% Case 3: 𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 >0, 𝑆 𝑖𝑡 =0 3101 5.61% Case 4: 𝑚 𝑖𝑡 =0,𝑛 𝑖𝑡 =0, 𝑆 𝑖𝑡 =1 8405 15.20% Case 5: 𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 =0, 𝑆 𝑖𝑡 =1 121 0.22% Case 6: 𝑚 𝑖𝑡 =0,𝑛 𝑖𝑡 >0, 𝑆 𝑖𝑡 =1 363 0.66% Case 7: 𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 >0, 𝑆 𝑖𝑡 =1 2421 4.38% Table 2. Seven Possible Cases of Observed Outcomes 1.4.2 Model Identification and Parameter Estimation In our model setup, we assume that individuals maximize their utilities subject to budget constraint by jointly choosing the amount of contribution (mit) and number of incentive tasks to participate (nit). Solving this constrained utility maximization problem leads to seven different Kuhn-Tucker optimality conditions. As shown in Appendix A, the Kuhn-Tucker optimality conditions for cases 4-7 (i.e., 20.5% of the entire data) do not involve simultaneity, that is, 𝑚 𝑖𝑡 and 𝑛 𝑖𝑡 are determined by exogenous covariates and do not affect each other. And for 20.9% of data in cases 1-3 (i.e., 16.6% of the entire data), since incentives are not available, individuals only decide on how much to contribute. Our model directly follows the standard utility maximization demand model (Satomura, Kim and Allenby 2011). And there is no simultaneity involved between contribution and incentive participation. As such, there is no model identification issue due to simultaneity for 37.1% (i.e. 20.5% + 16.6%) of all our data observations. However, for the remaining 79.1% of the data in cases 1-3 (i.e. 62.9% of total data), incentives are available, and the Kuhn-Tucker optimality conditions imply simultaneous equation 22 model, that is the contribution amount and incentive participation decisions are interdependent (see Appendix A for details on derivation). We adopt the two-stage approach to take care of the simultaneity issue involved in 62.9% of the total observations from case 1-3 (Greene 2000, Maddala 1984, Amemiya 1974, Heckman 1976, Lee, L., F. 1978, 1982, Nelson and Olsen 1978, Lee 1982, Bajari et al. 2010,Yang, Narayan and Assael 2006, Yang and Ghose 2010). More specifically, in the first stage, using all covariates 9 , we adopt censored regression to predict 𝑚 𝑖𝑡 (≥ 0), and truncated Poisson to predict 𝑛 𝑖𝑡 (0≤𝑛 𝑖 𝑡 ≤𝑁 𝑖𝑡 ) where 𝑁 𝑖𝑡 stands for the maximum number of available surveys. Using the estimation result in the first stage, we form the predicted 𝑚̂ 𝑖𝑡 and 𝑛̂ 𝑖𝑡 . Then in the second stage, when we estimate the Kuhn-Tucker first order condition for 𝑚 𝑖𝑡 , we substitute 𝑛̂ 𝑖𝑡 for 𝑛 𝑖𝑡 . And similarly, we replace 𝑚 𝑖𝑡 by 𝑚̂ 𝑖𝑡 when estimating the Kuhn-Tucker first order condition for 𝑛 𝑖𝑡 . By doing so, we can estimate all the structural parameters. For identification purpose, we impose exclusion restrictions. That is, when we estimate 𝑚 𝑖𝑡 in the first stage, we also include incentive participation related covariates such as number of available incentive tasks at t, average number of questions for the available tasks, and lag incentive participation dummy (Lag_Survey_Dummy), and those covariates only influence the incentive participation cost in the proposed structural model. And when we estimate 𝑛 𝑖𝑡 in the first stage, we also include contribution related covariates such as number of available funding projects at t, and lag contribution dummy (Lag_Contri_Dummy), and these covariates only affect the marginal baseline contribution utility in the structural model. The complete estimation results for the first stage can be found in Appendix B and C. In addition to the exclusion restriction, there are also several other identification strategies imposed in our model. First, the simultaneity issue does not apply to cases 4-7 and 20.9% of the 9 Both contribution related and incentive participation related covariates. 23 observations from cases 1-3, this data setting naturally helps our model identification because for those cases, 𝑚 𝑖𝑡 and 𝑛 𝑖𝑡 are exogenously determined (i.e., the endogenous variables are excluded). Second, in deriving the Kuhn-Tucker optimality conditions, we have automatically normalized coefficients in front of the endogenous variables. For example, for Case 1, the first order derivative equation of 𝑚 𝑖𝑡 (equation A1a in Appendix A) suggests that the coefficient of 𝑛 𝑖𝑡 is the average reward amount 𝑅̅ 𝑖𝑡 (which is directly observed from data). Furthermore, the first order derivative equation of 𝑛 𝑖𝑡 (equation A1b in Appendix A) suggests that the coefficient of 𝑚 𝑖𝑡 is -1. This parameter normalization, naturally imposed through our structural modeling, has greatly helped our model identification. Third, following Greene (2000), we have constrained the error terms in the structural model, 𝜀 𝑖𝑡 and 𝜐 𝑖𝑡 , to be uncorrelated to further aid in model identification. Fourth, nonlinearities in our model specification can also help model identification. For model parameters, we are able to estimate the marginal contribution benefit (𝜑 𝑖𝑡 ), average cost of incentive participation (𝐶 𝑖𝑡 ), and reward saving benefit because we have data variations in contribution amount, number of incentive participation and reward saving amount across individuals and within individuals. More specifically, the covariation between contribution amount and the contribution related covariates (𝑥 𝑖𝑡 𝑚 and 𝑧 𝑖𝑡 𝑚 ) enables us to empirically identify the parameters (𝛽 𝑖 and 𝛽 ) specified in the marginal contribution benefit. And similarly, the covariation between incentive participation and the incentive participation related covariates (𝑥 𝑖𝑡 𝑛 and 𝑧 𝑖𝑡 𝑛 ) allows us to identify the parameters (𝜏 𝑖 and 𝜏 ) involved in the average incentive participation cost. And we can identify 𝛾 𝑖 through data variation on reward saving 𝑆 𝑖𝑡 ∙(𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 ) . Finally, we can estimate budget related parameters mainly based on the data variation patterns between contribution and incentive participation. Since in our empirical context, incentives are not restricted to any specific individuals or funding projects, we observe data 24 variations both within and across individuals, and data variations with and without the presence of effort-based incentives. These data variations allow us to identify the budget related parameters, because we expect that effort-based incentives would affect different individuals in different ways. Specifically, for individuals with low baseline budget, the presence of the effort-based incentives can make a big difference. For instance, without the effort-based incentives, even if this individual has a high contribution preference, she may not be able to contribute due to the budget constraint. But with the presence of the incentives, the reward can increase her budget, which allows her to be able to make a contribution. For individual with high baseline budget, the presence of the effort- based incentives may not change her contribution pattern that much because her behavior is less constrained by the budget. More details on model identification and parameter estimation are reported in Appendix E. 1.4.3 Model Derivation We next provide an illustration for deriving the likelihood function for case (1) where 𝑚 𝑖𝑡 =0, 𝑛 𝑖𝑡 =0,𝑎𝑛𝑑 𝑆 𝑖𝑡 =0, our utility function is specified as: 𝑈 𝑖𝑡 = 𝜑 𝑖𝑡 ln(𝑚 𝑖𝑡 +1)+ln(exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1)−𝐶 𝑖𝑡 ∙𝑛 𝑖𝑡 And the following Kuhn-Tucker first-order conditions must be satisfied: 𝜕𝑢 𝑖𝑡 𝜕𝑚 𝑖𝑡 = 𝜑 𝑖𝑡 𝑚 𝑖𝑡 +1 − 1 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1 <0 (8a) 𝜕𝑢 𝑖𝑡 𝜕𝑛 𝑖𝑡 = 𝑅̅ 𝑖𝑡 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1 −𝐶 𝑖𝑡 <0 (8b) Since 𝑚 𝑖𝑡 =0, Equation (8a) becomes 25 𝜕𝑢 𝑖𝑡 𝜕𝑚 𝑖𝑡 =𝜑 𝑖𝑡 − 1 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 +1 <0 plugging in Equation (6), we have exp(𝑥 𝑖𝑡 𝑚 ′ 𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 +𝜀 𝑖𝑡 )< 1 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 +1 Taking the logarithm we obtain the first inequality condition: 𝜀 𝑖𝑡 <𝑙𝑜𝑔 ( 1 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 +1 )−(𝑥 𝑖𝑡 𝑚 ′ 𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 ) (9a) Similarly, because 𝑛 𝑖𝑡 =0, and 𝑅̅ 𝑖𝑡 ≥0 , plugging the specification for incentive participation cost, Equation (5), into Equation (8b), we have 𝐶 𝑖𝑡 =𝑥 𝑖𝑡 𝑛 ′ 𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′ 𝜏 +𝜐 𝑖𝑡 > 𝑅̅ 𝑖𝑡 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 −𝑚 𝑖𝑡 +1 ≥0 hence we obtain the following second inequality condition: 𝜐 𝑖𝑡 > 𝑅̅ 𝑖𝑡 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 −𝑚 𝑖𝑡 +1 −(𝑥 𝑖𝑡 𝑛 ′ 𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′ 𝜏 ) (9b) 26 To overcome the endogeneity issue induced by the simultaneous decision making process, we use a two-stage estimation process (Nelson and Olsen 1978). In the first stage, reduced form estimates are used to construct instruments 𝑚̂ 𝑖𝑡 and 𝑛̂ 𝑖𝑡 . Then in the second stage, we will substitute the predicted 𝑚̂ 𝑖𝑡 for 𝑚 𝑖𝑡 in the optimality condition for incentive participation, and predicted 𝑛̂ 𝑖𝑡 for 𝑛 𝑖𝑡 in the optimality condition for contribution. Therefore, the probability of observing zero contribution, zero incentive participation and no reward saving can be written as: f ̂ 1 (𝑚 𝑖𝑡 =0,𝑛 𝑖𝑡 =0,𝑆 𝑖𝑡 =0) =Φ 𝜀 (𝑙𝑜𝑔 ( 1 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛̂ 𝑖𝑡 +1 )−(𝑥 𝑖𝑡 𝑚 ′ 𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 )) ∙[1−Φ 𝑣 ( 𝑅̅ 𝑖𝑡 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 −𝑚̂ 𝑖𝑡 +1 −(𝑥 𝑖𝑡 𝑛 ′ 𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′ 𝜏 ))] (10) To complete the econometric specification, we model the unobserved heterogeneity across individuals by adopting the standard random coefficients specification, that is, 𝜃 𝑖 ~𝑀𝑉𝑁 (𝜃 ̅ ,𝑉 𝜃 ) (11a) 𝜏 𝑖 ~𝑀𝑉𝑁 (𝜏 ̅ ,𝑉 𝜏 ) (11b) where 𝜃 𝑖 =(𝑀̅ 𝑖 ,𝛽 𝑖 ,𝛾 𝑖 ) . Another estimation challenge is to ensure that model parameters satisfy the multiple- constraint requirement. Since we need to estimate the budget, we need to ensure that exp (𝑀̅ 𝑖 )+ 𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 +1≥max(𝑚 𝑖𝑡 ) for all i across t. This makes classical estimation methods almost impossible to implement in order to account for a large number of constraints. We develop a Bayesian estimation approach to address the multiple-constraint problem and unobserved heterogeneity in parameters. Specifically, we use Markov Chain Monte Carlo methods 27 implemented in the Gibbs Sampler and an improved Metropolis-Hastings algorithm (Rossi, Allenby and McCulloch 2005). Algorithms for model estimation are provided in Appendix D. We run the chain for 30,000 iterations and use the last 15,000 draws for calculating the conditional posterior means and standard deviations of model parameters. Convergence is assessed by inspecting the time series of model parameters. 1.5 Results and Counterfactual Analysis 1.5.1 Model Comparison We estimated two models including the proposed model (Model II) and one benchmark model (Model I). Model I is a special case of the proposed model without accounting for unobserved heterogeneity across individuals. At the bottom of Table 3, we report the fit statistics (Log Marginal Density) for these two models. As shown, our proposed model fits the data substantially better than the benchmark model, suggesting that there is a large amount of unobserved variation in consumer contribution and incentive participation preferences. And for our proposed model, the mean absolute error for contribution amount and number of incentive participation are 2.201 and 0.143 respectively 10 . 1.5.2 Findings Individual Contribution Behavior. A core issue we investigate in this paper is how effort-based incentives affect individual contribution. Table 3 reports the model parameter estimates and Table 4 reports the variance-covariance estimates. We find that the baseline marginal utility diminishes if an individual has made any contribution in the last period. This negative state dependence could be due to our empirical context involving contributing to public goods (i.e. making prosocial 10 In comparison to the standard derivation of the contribution amount (21.46) and of the number of incentive participation (0.422), our model fits the data reasonably well. 28 contribution). A contributor does not get tangible rewards as in other market transactions such as buying products. Contribution once may have already satisfied the person’s need for being prosocial. And people get a higher contribution benefit when there are more funding projects available. This is as expected, because when there are more funding projects available, it’s more likely for people to find a funding project they’re interested in. And our estimation result indicates that on average, the existence of effort-based incentives can have a negative effect (-3.50) on the marginal contribution utility, and Figure 3 plots the histogram of the impact of incentive availability on marginal contribution utility for all individuals. We can see that incentive availability increases the marginal contribution preference for a small fraction of individuals, but at the mean level it can decrease the marginal contribution benefit. Thus, in the context of crowdfunding that involves public goods and prosocial contribution behavior, we find an empirical support for the theory on hidden cost of monetary incentives. In other words, we find that the offering of the effort-based incentives can mitigate individual’s prosocial contribution. On other covariates, we do not see significant difference between news reporters and general individuals in their contribution preferences. We find some significant effects of various features of the funding projects. Specifically, consumers tend to have a higher baseline utility for projects related to public health, environmental, race, diversity or poverty issues etc. (i.e. project type I, II, IV, and V). In addition, the average baseline budget across people is $33.12 (i.e. exp(3.50)), and Figure 4 provides the histogram of the baseline budget for all those 5,311 individuals 11 . Our findings suggest that, the average preference for saving some reward for future 11 When estimating the proposed model, we did not incorporate individual demographic information as covariates. This is because, only for a fraction of the sample (i.e. 208 individuals), we have information such as their income level. For these 208 individuals, we do have their baseline budget estimates obtained from the proposed model without using income level data. Ex ante, we examine the relationship between the baseline budget estimates and the income 29 use is 0.018(i.e. exp(-4.00)), and Figure 5 provides the histogram of the reward saving benefit for all individuals. We can see that, at the mean level, the benefit from saving one dollar reward for future use is quite small. This is consistent with our data, which show most people would use the reward to contribute soon after they obtain it, and the average reward saving amount is much smaller than the average contribution amount. Incentive Participation. We find that the incentive task participation cost decreases as the number of available incentive tasks increases, As more surveys are available, participants may be able to take more, and then enjoy more efficiency in completing surveys. But the incentive participation cost increases if a person has taken any incentive tasks in the last period. We find that reporters tend to have a lower participation cost compared to non-reporters. In addition, we find a U-shape relationship between incentive participation cost and the number of survey questions. That is, the participation cost first decreases with the number of questions, but then increases after the number of questions reaches a certain threshold. Our result indicates a negative threshold, meaning that the incentive participation cost only increases with the number of questions, because the number of questions can only be positive. level for these 208 individuals. We find that their correlation is significantly positive (ρ=0.31, p-value<0.001), suggesting that our model can reasonably predict consumer contribution budget. 30 Model I Benchmark Model II Proposed Model Baseline Marginal Utility Intercept -7.359 (0.017) -1.700 (0.102) Lag_Contri_Dummy -0.271 (0.016) -3.700 (0.097) Number of Funding Projects 0.544 (0.010) 1.600 (0.045) Incentive Availability Dummy -1.454 (0.020) -3.500 (0.103) Share of Project Type I 0.044 (0.008) 0.108 (0.013) Share of Project Type II 0.019 (0.012) 0.218 (0.017) Share of Project Type III -0.333 (0.009) -0.712 (0.016) Share of Project Type IV 0.408 (0.015) 0.893 (0.022) Share of Project Type V 0.235 (0.007) 0.176 (0.011) Reporter 0.010 (0.016) -0.022 (0.045) Budget Baseline i M 7.826 (0.001) 3.500 (0.032) Future Saving Benefit * i Intercept -8.438 (0.014) -4.000 (0.053) Reporter Incentive Participation Cost Intercept 1.640 (0.015) 1.866 (0.022) Lag_Survey_Dummy 0.060 (0.025) 0.334 (0.033) Number of Available Surveys -0.370 (0.010) -0.313 (0.016) Question 0.280 (0.017) 0.314 (0.027) Question^2 -0.150 (0.011) 0.021 (0.024) Reporter -0.100 (0.018) -0.095 (0.027) Log(Marginal Density) -123914.7 22996.380 MAE_Amount 2.552 2.201 MAE_Incentive 0.143 0.142 Note: Posterior means and posterior standard deviations are reported, and significant estimates at 95% level are bolded in tables. Table 3. Estimation Results 31 Variance-Covariance i M 2.680 (0.097) Intercept -1.540 (0.149) 21.16 (0.596) Lag_Contri_Dummy -0.120 (0.122) -1.830 (0.336) 5.690 (0.293) Number of Funding Projects -0.250 (0.070) 10.150 (0.286) -1.030 (0.169) 5.280 (0.162) Incentive Availability Dummy -1.350 (0.164) -17.360 (0.560) 2.850 (0.348) -8.860 (0.297) 18.560 (0.618) i -2.540 (0.102) 6.290 (0.222) 0.150 (0.160) 2.730 (0.113) -1.840 (0.195) 4.84 (0.174) Participation Cost Intercept 0.074 (0.007) Lag_CumSurvey 0.018 (0.007) 0.163 (0.019) Number of Available Surveys -0.018 (0.004) -0.008 (0.006) 0.045 (0.004) Question 0.024 (0.006) 0.018 (0.009) -0.002 (0.004) 0.083 (0.009) Question^2 -0.017 (0.004) -0.002 (0.006) 0.009 (0.003) -0.024 (0.005) 0.051 (0.004) Table 4. Heterogeneity Estimates from the Proposed Model 32 Figure 3. Histogram of the Impact of Incentive Availability on Contribution Preference Figure 4. Histogram of the Baseline Budget 33 Figure 5. Histogram of the Reward Saving Benefit 1.5.3 Counterfactual Analysis Based on the four utility components, we know that individual baseline budget (𝑀̅ 𝑖 ), contribution preference (𝜑 𝑖𝑡 ), incentive participation cost (𝐶 𝑖𝑡 ) and reward saving preference (𝛾 𝑖 ) all play important roles in explaining the impact of incentives on contribution. In order to better understand the complex effect of incentives on individual contribution behavior and the underlying mechanism, we simulate individual level contribution preference (𝜑 𝑖 ) and incentive participation cost (𝐶 𝑖 ), and classify contributors into eight groups using the mean 12 split of 𝑀̅ 𝑖 , 𝜑 𝑖 and 𝐶 𝑖 13 . 12 Using mean split or median split gives us similar result. 13 Based on our individual level analysis, adding 𝛾 𝑖 would not change our simulation result, so for simplicity, we choose to classify consumers into eight groups, instead of 16 groups. 34 Based on this classification, we have 663 individuals for each group, and Table 5 reports the detailed characteristics for these eight classified segments. Counterfactual Analysis I. The relationship between incentive and contribution behavior is complex given that effort-based incentives have a dual competing effect (i.e., positive budget expansion and detrimental psychological effect) on individual’s prosocial contribution behavior. To answer our main research question on how effort-based incentives affect individual’s contribution, we simulate each individual’s optimal contribution amount and incentive participation amount simultaneously under two scenarios where incentives are not available and are always available, and in Table 5 we report the predicted segment level contribution amount for each of the two scenarios (𝑚 0 for without incentives, and 𝑚 1 for with incentives). As shown, for six segments (except for Segment 3 and 4), effort-based incentives tend to increase contribution amount (e.g., the increase ranges from 5.18% to 109.61%). To further explore the underlying mechanism, we also calculate the average impact of the incentives availability on marginal contribution utility (see column 5 in Table 5), some interesting patterns emerge after getting this negative psychological effect from incentives for those eight segments. We notice that effort-based incentives tend to decrease contribution amount when the detrimental effect from the incentives is relatively large (i.e., Segment 3 and 4). So for those two segments, the detrimental psychological effect tend to dominate the positive budget expansion effect. However, for other segments (e.g., Segment 1, 2 and 5 to 8), the average detrimental effect from incentives is relatively small, and the incentives can increase contribution amount as the budget expansion effect can overwrite the negative effect from the incentives. In addition, we can see that segment 5 has the lowest negative impact (-1.810) from incentives among the eight segments. Without the presence of the effort-based incentives, segment 35 5 makes the least amount of contribution (i.e., $1,279 accounts for only 2.86% of total contribution). With the presence of incentives, more people in this segment are willing to make a positive contribution, because the contribution incidence (i.e. number of positive contribution) without incentives is 588, but contribution incidence with incentives is 853. So the effort-based incentives create a 45% increase in contribution incidence for segment 5. Furthermore, people in segment 5 also seem to be willing to contribution more, because the total contribution amount increase is 109%, which is higher than the 45% increase in contribution incidence. Although the effort-based incentives can increase contribution for segment 1, 2, 5, 6, 7, and 8, the increase mechanism seems to differ. For instance, for segment 5, 6, 7 and 8, the contribution incidence all increases. But for segment 1 and 2, the contribution incidence actually decreases, even though the total contribution amount for those segments all increases. A careful examination on the characteristics of those segments, we notice that segment 5, 6, 7 and 8 tend to have relatively lower baseline budget 𝑀̅ 𝑖 , hence they’re more likely to be constrained by the donation related budget. So for those people, the presence of the effort-based incentives can significantly increase the contribution incidence as it lowers the contribution cost. But for segment 1, 2, 3 and 4, the budget constraint issue may not hold anymore as they have relatively higher baseline budget. For them, whether incentives increase or decrease contribution would mainly depend on the relative magnitude of the positive budget expansion effect and the negative psychological effect. In summary, the overall effect of effort-based incentives on contribution differs across different types of consumers. Using our structural model, we are able to capture such complex and heterogeneous effect from effort-based incentives, and based on our estimates, we can create consumer segments and identify which segments may respond favorably to effort-based incentives 36 and which segments may respond unfavorably. Using our segment level analysis, we can help the platform increase revenue by targeting the effort-based incentives to the right type of consumers. 36 Mean 𝑀̅ 𝑖 Mean 𝜑 𝑖 Mean 𝐶 𝑖 Mean 𝛾 𝑖 Mean 𝑑 𝑖 Total contribution amount (incidence) without Incentives 𝑚 0 Total contribution amount (incidence) with 100% Incentives 𝑚 1 𝑚 1 − 𝑚 0 𝑚 1 − 𝑚 0 𝑚 0 Segment 1 low 𝜑 𝑖 , high 𝐶 𝑖 , high 𝑀̅ 𝑖 5.167 0.003 2.442 0.004 -2.851 6168 (1358) 8810 (1303) 2642 42.83% Segment 2 low 𝜑 𝑖 , low 𝐶 𝑖 , high 𝑀̅ 𝑖 5.115 0.003 0.885 0.005 -2.793 8306 (1515) 9119 (1496) 813 9.79% Segment 3 high 𝜑 𝑖 , low 𝐶 𝑖 , high 𝑀̅ 𝑖 3.730 0.141 0.855 0.061 -5.871 12358 (1461) 10758 (1301) -1600 -12.95% Segment 4 high 𝜑 𝑖 , high 𝐶 𝑖 , high 𝑀̅ 𝑖 3.693 0.175 2.336 0.064 -5.973 9650 (1395) 8955 (1284) -695 -7.20% Segment 5 low 𝜑 𝑖 , high 𝐶 𝑖 , low 𝑀̅ 𝑖 3.137 0.008 2.454 0.023 -1.810 1279 (588) 2681 (853) 1402 109.61% Segment 6 low 𝜑 𝑖 , low 𝐶 𝑖 , low 𝑀̅ 𝑖 3.191 0.009 0.877 0.025 -2.355 1430 (602) 2405 (825) 975 68.18% Segment 7 high 𝜑 𝑖 , low 𝐶 𝑖 , low 𝑀̅ 𝑖 2.147 0.332 0.782 0.108 -3.343 3052 (1053) 4632 (1223) 1580 5.18% Segment 8 high 𝜑 𝑖 , high 𝐶 𝑖 , low 𝑀̅ 𝑖 2.101 0.268 2.387 0.097 -3.192 2493 (938) 3444 (1156) 951 38.15% Total 44,736 50,804 6,068 13.56% Notes * Contribution preference (𝜑 𝑖 ), Baseline budget (𝑀̅ 𝑖 ), Incentive participation cost (𝐶 𝑖 ), Reward saving preference (𝛾 𝑖 ), impact of incentive availability on contribution preference ( 𝑑 𝑖 ). * Each segment has 663 individuals. * Contribution incidence (reported in parentheses): number of positive contribution Table 5. Characteristics of the Eight Segments and Counterfactual Analysis I 37 Counterfactual Analysis II. In our context, the effort-based incentives are provided by third party sponsors, and this means that, sponsors are paying for the incentive credits or money (i.e. incurring a cost). From sponsors’ perspective, they are interested in getting more survey responses at a lower cost. Therefore, we run this second counterfactual analysis to see whether we can increase the total number of survey participation at a lower cost while keeping or increasing the total funding amount for the platform. In particular, we compare two incentive strategies. In the uniform incentive strategy, we jointly simulate individuals’ contribution and incentive participation where all eight segments are given the same reward amount from a survey provided by the sponsor. In the customized incentive strategy, we construct an eight dimensional grid where each grid represents the reward amount (ranges from $0 to $15) offered to each of the eight segments. Then, for each of the uniform incentive setting (e.g., $3, $4…$10), we simulate the optimal contribution and incentive participation under each of the possible reward settings for these eight segments. For each of the uniform incentive setting, we can find an optimized reward setting where we can obtain a larger number of survey responses at a lower cost for the sponsors, and get a higher contribution amount for the crowdfunding platform. Results are reported in Table 6 for eight uniform reward cases (i.e. in eight rows). We see that, for example, in the $7 uniform reward case, by customizing the reward (e.g., $0 survey to segment 1, $1 to segment 2, $2 to segment 3, $0 to segment 4 and 5, $2 to segment 6, $7 to segment 7, and $9 to segment 8), the survey responses can increase by 1.2% at a lower cost (a decrease of 1.13%) for the incentive sponsor, and the total contribution amount gets a 5.21% raise for the crowdfunding platform. Overall, this counterfactual analysis demonstrates that we can customize incentive reward to selectively target to different segments based on their behavioral reactions to incentives. Customization and targeting enable us to improve the return on investment for incentive sponsors, 38 raise more contribution for the crowdfunding platform, and more importantly, maximize individuals’ contribution utilities which makes them happier. In other words, our customization creates a win-win-win scenario for the crowdfunding platform, incentive sponsors, and also the crowdfunders. 39 Segment 1 low 𝜑 𝑖 , high 𝐶 𝑖 , high 𝑀̅ 𝑖 Segment 2 low 𝜑 𝑖 , low 𝐶 𝑖 , high 𝑀̅ 𝑖 Segment 3 high 𝜑 𝑖 , low 𝐶 𝑖 , high 𝑀̅ 𝑖 Segment 4 high 𝜑 𝑖 , high 𝐶 𝑖 , high 𝑀̅ 𝑖 Segment 5 low 𝜑 𝑖 , high 𝐶 𝑖 , low 𝑀̅ 𝑖 Segment 6 low 𝜑 𝑖 , low 𝐶 𝑖 , low 𝑀̅ 𝑖 Segment 7 high 𝜑 𝑖 , low 𝐶 𝑖 , low 𝑀̅ 𝑖 Segment 8 high 𝜑 𝑖 , high 𝐶 𝑖 , low 𝑀̅ 𝑖 Change from m Change from n Change in Cost Uniform Incentive Strategy $3 $0 $1 $1 $0 $0 $1 $2 $8 +4.04% +0.31% -2.02% Uniform Incentive Strategy $4 $0 $1 $1 $0 $0 $1 $4 $7 +5.09% +0.55% -1.41% Uniform Incentive Strategy $5 $0 $1 $1 $10 $0 $1 $5 $7 +7.41% +0.24% -1.11% Uniform Incentive Strategy $6 $0 $1 $1 $10 $0 $1 $6 $8 +8.92% +0.60% -0.40% Uniform Incentive Strategy $7 $0 $1 $2 $0 $0 $2 $7 $9 +5.21% +1.20% -1.13% Uniform Incentive Strategy $8 $0 $1 $2 $0 $0 $2 $8 $9 +5.80% +1.18% -1.02% Uniform Incentive Strategy $9 $0 $1 $2 $0 $0 $2 $9 $11 +3.75% +1.19% -0.66% Uniform Incentive Strategy $10 $0 $1 $7 $0 $0 $1 $10 $11 +2.52% 0.59% -0.14% Note: We have constructed eight uniform reward cases (e.g., $3, $4 …$10). For each (row) of the uniform incentive setting, we can find an optimized reward setting. The customized reward amount for each segment and the change in m, n and cost are listed in each row. Table 6. Counterfactual Analysis II: Customized Incentives 40 1.6 Conclusion Our study focuses on the effort-based incentive, a marketing promotion that requires non-trivial effort from consumers by completing a specific task in order to obtain the reward. There is a large body of literature in marketing on promotions such as coupons and cause marketing incentives, which require consumer effort of getting the promotion. Our research builds upon this foundation and contributes to the existing work in the following two important ways. First, we examine effort- based incentives in the context of crowdfunding, a highly important and relevant context to marketing promotion for several reasons. Promotion is a very important marketing tool to crowdfunders as well as crowdfunding platforms, because crowdfunders sometimes contribute without receiving a direct tangible benefit and therefore need to be better incentivized. Furthermore, effort-based incentives may have a complicated impact on crowdfunders’ contribution behaviors when they are making contributions to support prosocial projects. More specifically, effort-based incentives allow individuals to turn effort into a reward, and such reward can relax one’s budget and hence increase individual’s contribution level. However, the incentives may undermine people’s prosocial contribution behavior as the psychological theory would predict. Thus, we develop a structural model to capture these two underlying behavioral mechanisms by jointly modeling the micro-level process of incentive participation, reward accumulation, and contribution decisions. Second, we jointly model consumer demand and incentive participation based on a unified utility maximization framework. This is important because incentive participation incurs cost but brings reward, which could relax a consumer’s budget constraint, and as such, the two decisions of consumer demand (i.e., contribution) and incentive participation may depend on each other. Incentive participation is relatively under-researched, possibly due to lack of data in the traditional 41 retailing context. For example, researchers often do not observe the actual coupon collection behavior. In our empirical context, we observe the incentive participation information, and we are able to connect the contribution decision and the incentive participation decision in a structural manner by allowing the incentive participation to affect the budget. This is an important modeling contribution and has not been done in the literature. Using data from a pioneering crowdfunding platform for journalism, we find that individuals may react favorably or unfavorably to the effort-based incentives, depending on the relative magnitude of the positive budget expansion effect and the negative psychological effect from incentives. In other words, we find empirical support for the theory on the hidden cost of monetary incentives, suggesting that effort-based incentives may have either positive or negative overall effect on individual’s contribution. Based on individual’s heterogeneous reactions to the effort-based incentives, we are able to customize the offering of the incentives to improve the return on investment for the incentive sponsors, increase the revenue for the crowdfunding platform, and also maximize individual donor’s contribution benefit. Thus, our structural modeling approach allows us to understand how effort-based incentives affect crowdfunders prosocial contribution on a micro-foundation level, and our counterfactual analysis allows us to offer important managerial implications on the effective use of the effort-based incentives. In particular, our customization and targeting strategy enables us to create a win-win-win scenario for the crowdfunding platform, the incentive sponsors, and also the crowdfunders. Our study focuses on a context in which consumption is in the form of making contributions to support a funding project for public goods. It provides us a clean context to study but may limit the generalizability of the findings to other contexts such as direct private consumptions. With the increasing popularity of Google Consumer Surveys as a form of earning 42 and paying for premium content on the Internet, different effects of effort-based incentives might manifest. Nevertheless, to the extent that Google Consumer Surveys is a platform and its service can be applied to other contexts such as charity contribution, our results can be helpful in evaluating the best strategies in that domain. More importantly, the modeling approach that connects two interdependent decisions through budget relaxation is directly applicable across various contexts. In future research, our study can be extended in several ways. First, one may incorporate forward-looking in modeling individuals’ contribution decisions over time. Due to the large number of infrequent users and complexity of the model, we have only partially considered forward looking by capturing the benefit of saving credit reward for future use. Second, if individuals’ contribution history on each funding project is observed, one can develop a micro- level model to quantify how effort-based incentives affect an individual’s contribution decision in the funding process of a project. Without this detailed history information, we jointly model individual’s total contribution amount and number of incentive tasks participated in each month. Third, the generalizability of the proposed conceptual framework needs to be tested. In our empirical context, we examine one specific type of effort-based incentive, that is, individuals fill out surveys in exchange for credit dollars to be contributed to fundraising projects. It would be interesting to study other effort-based incentives such as referrals, product improvement suggestions, etc. Fourth, with the rise of social media, it is also interesting to examine how social interactions among contributors and between fundraisers and contributors affect the effectiveness of such effort-based incentives. Notwithstanding these limitations of our study, we hope this research can spawn more future work on effort-based incentives, a relatively under-researched yet important promotional tool. 43 Chapter 2. A Dynamic Model of Crowdfunders’ Contributions to Public Goods 2.1 Introduction Over the past few years, crowdfunding has become an increasingly popular practice of funding a project by raising monetary contributions from a large number of people. In general, it works in the following way: first a crowdfunding project creator pitches an idea on the crowdfunding platform, and then crowdfunders who are interested in the pitched idea can make monetary contributions to support the project. If the project creator manages to raise enough money, then she will deliver a project reward to crowdfunders who have supported the project previously. Typically, the project reward won’t be delivered until the terminal of the funding drive, and the reward delivery decision is conditional on the final funding performance (e.g., whether the funding goal is accomplished or not). Given this funding reward structure, individual donors need to make contribution decisions in a forward looking way, because they need to take into account of the future funding success probability. In addition, a crowdfunding project usually has a predetermined funding goal and funding deadline, and during the funding drive, people can observe the time varying funding status such as current funding amount raised, funding time left, and the number of contributors who have previously supported the project. Hence, under this transparent funding process, it is possible for crowdfunders to make contribution decisions based on prior others behaviors. This is especially true for our empirical context, because the crowdfunding platform that we’re studying specializes in raising money to help project initiators to publish a digital report that can be read by anyone online for free. This means that the crowdfunding platform is actually raising money for public goods. Because the project reward (i.e., report) will not only be delivered to the donors who have directly supported the project, but also other people who have not made any monetary 44 contributions to the funding project. In other words, it’s possible for people to obtain the project reward without bearing any costs (i.e., money or time investment). Thus, given our empirical context and the crowdfunding reward structure, and also based on the economic theory on public goods provision, it’s essential for us to account for individual’s dynamic concerns when setting up the framework to model individual donor’s contribution decision on crowdfunding platform. In this paper, we collect data from a novel journalistic crowdfunding platform that raises money for public goods (i.e., digital reports with free access). When making the contribution decision, crowdfunders on this platform can observe the current funding status such as the total funding amount accomplished, time left, and number of donors attracted etc. And whether individuals can receive the final reward or not depends on the final funding performance. So just like how it works on other crowdfunding platforms, crowdfunders face a situation in which they must donate in the present and then wait to see if they can obtain the final reward in the future. However, there are two unique and interesting features about this journalistic crowdfunding platform. First, successfully funded projects will deliver a digital report as the funding reward. However, unlike the general crowdfunding platform where the final reward will only be delivered to crowdfunders who have previously supported the funding project, on this platform, the report will be published online with free access to anyone, not just to the crowdfunders. So basically, this journalistic crowdfunding platform is raising money for public goods. Second, unlike a typical crowdfunding platform where projects only receive contributions from individual donors, this platform allows projects to receive funding from two types of contributors: individuals and organizations (/firms like New York Times). And organizations usually make a much larger contribution in a single time donation. This particular funding context leads to some interesting research questions such as: how organization’s contribution would affect 45 individual contribution? Does it positively attract more individual contribution (i.e. crowding-in)? Or does it negatively deter some individual contribution (i.e. crowding-out)? The crowding out/in research question has been widely studied in Economics, but the conclusion is mixed and most studies are either theoretical or experimental. Among the limited amount of the empirical studies, to the best of our knowledge, none of them have directly taken care of individual’s dynamic concerns over the funding provision process. The main reason is due to either lack of proper data or the computational challenge. Thus, to answer these questions, we will build a dynamic structural model which can handle the funding deadlines and project reward schemes based on the final funding result (i.e., whether a project achieves the funding goal or not), all of which are ubiquitous in crowdfunding setting. In addition, our model can handle the dynamic free riding issue (i.e. crowding in/out), which is a unique feature in our crowdfunding setting. So our main objective of this paper is to propose a dynamic structural framework to model individual’s contribution decision (i.e. how much to contribute) on a crowdfunding platform that is raising money for public goods. In particular, we develop a framework to characterize the dynamic behaviors of individual crowdfunders in this empirical setting, accounting for their motivations to manipulate the timing and/or amount of their contributions conditional on others behaviors. Even though individuals cannot obtain the final project reward until the terminal of the funding process, given the public goods feature, individuals can obtain period utilities from their contribution behaviors. Because individuals are behaving prosocially by making contributions to support a project that can be potentially beneficial to the general public. In economic literature, researchers call this utility as warm-glow effect, which is defined by Andreoni (1989, 1990) as an abstract feeling of personal gratification obtained from the act of giving itself. Note that individual’s contribution not only brings direct period utility, it also increases the final funding 46 success probability, and therefore it also enhances the chance to obtain the final project reward benefit. But of course, it’s not costless to make contributions. Thus, we model individuals to be forward-looking, in order to trade off the cost of making a contribution, with the risk/cost of not being able to receive the final report from a funding project, as well as the good feeling generated from the act of helping a project that can be beneficial to others. During the funding drive, individuals cannot control when or how much others would support a funding project, thus during each funding period, they form expectations over how much funding a project will receive, and how many donors would come to support that project. The estimation of our dynamic structural model involves several computational challenges, given that the state space has both continuous (funding amount raised, funding amount from organizational donors, individual’s cumulative contribution amount) and discrete dimensions (time periods left, and number of previous donors). In addition, individual’s period choice (i.e. nonnegative contribution amount) is continuous, and it is both lower and upper constrained, this complicates the modeling and estimation process. And to overcome the computational and estimation-related challenges, we use a conjunction of different approaches. First, we adopt a Bayesian estimation approach, using a modified Bayesian IJC algorithm (Ishihara et al. 2013 and 2016, Imai et al. 2009) that helps deal with finite-horizon DP models. Second, since we do not have a close form solution for the likelihood, we use the kernel density estimator to simulate the likelihood (Yao et al. 2012, Liu et al 2014). Third, to deal with the constrained continuous decision, we use golden section search method to find the optimal action when solving the value function. Fourth, to overcome the computational challenge, we use parallel computing (OpenMP and MPI) to speed up our estimation process. 47 The remainder of this paper is organized as follows. In the next section, we introduce the empirical context for this study. In section 3, we review three related research streamlines to better understand our research objective and potential contribution. In section 4, we describe the data and show some model free data patterns to understand our identification strategy. In section 5, we develop a dynamic structural framework to model individual’s contribution decisions by taking into account of individuals’ inter-temporal tradeoffs. Finally, we discuss our empirical findings and the potential counterfactual analysis to be conducted. 2.2 Empirical Context Crowdfunding is an emerging method for charitable contribution and entrepreneurial finance, in which small amounts of capital are obtained from a large number of individuals who share common interests. The crowdfunding market is booming rapidly as nearly $34 billion was raised through crowdfunding platform worldwide in 2015, significantly up from $ 530 million raised in 2009 (The Economist 2012). For this study, we collect data from a crowdfunding platform for online journalism. As of April 2013, the platform has received supports from more than 20,900 contributors and 110 organizational donors such as Los Angeles Times and New York Times. It allows freelance journalists to raise money in support of their work by pitching journalism ideas to the online community, just like an entrepreneur would pitch venture capitalists (see Appendix A for an example of a funding project on this platform). Since this platform is a pioneer journalistic crowdfunding platform, people would come to this online place to search for interesting but under covered report ideas. If they find the pitched journalistic idea worth investigating, then people would make donations to help the project raise 48 enough money to cover the cost for producing the report. And firms or organizations, like Los Angeles Times, New York Times, AAPR etc., who are interested in the pitched ideas can also make contributions to support the funding projects on this platform. Each funding project pitched on this platform has a predetermined funding goal (i.e., how much money to raise), funding deadline (i.e., funding termination date), a description of the report idea or topic, how the investigation on the report idea will be helpful to the public, the report delivery format (e.g., text, video, audio, photo), and the project initiator’s qualifications such as professional writing experience. Any member of this platform can act upon those funding requests by contributing funds in any increment. The contribution process continues in this manner until the project expires. Every project has an estimate for the cost of providing the deliverable. Examples of such cost include travel, locating and copying documents, and other production related expenses. The amount of money required to cover these costs is set as the funding goal of the project. Once the project funding process is completed successfully, the project creator will deliver the predetermined project deliverables, most of the time in terms of articles (with text or photo), and sometimes using videos or audios. Each project creator needs to choose a funding deadline when creating a funding pledge. Before the deadline, interested donors can make any contributions at any time in any incremental amount. However, after the deadline, donors can no longer make any contributions. And the only thing they can do is to wait for the decision on the delivery of the final project reward (i.e., report articles or interview videos). If the predetermined project funding goal is met, then the project creator is obligated to deliver the final project deliverable (i.e. report). Otherwise, the project creator needs to make a choice among two options: (1) she can choose either to accept the raised 49 funding (less than the goal) and deliver the project report, or (2) she can return all the raised money back to the donors and she will not cover any report on the pitched idea. Therefore, for each funding project we define two different stages: the funding stage (t=1,…,T) and the project delivery stage (t=T+1), where T represents the funding deadline. One important and interesting feature of this journalistic crowdfunding platform is that the project deliverables will be published on this funding platform and everybody will have free access to those published reports. This means that the funding projects are raising money for public goods (Kaye and Quinn 2010). And individual’s monetary support to the production of those free reports ultimately can provide benefit to the masses, thus individuals are behaving prosocially by making contribution to support the funding projects on this platform. According to a recent case study on that crowdfunding platform (Aitamurto 2011), contributors reported a number of reasons for supporting projects including that a pitch is relevant to their lives, a topic can affect lives of their relatives, friends or neighborhood, etc. Therefore, supporting the funding projects on this journalistic crowdfunding platform can provide crowdfunders with both direct and indirect benefit, because crowdfunders can obtain information or knowledge they seek to know directly from the report, and since the published report is publicly available to others, their contribution behaviors can be beneficial to others as well. 2.3 Related Literature Our study intersects multiple domains of literature from a substantive viewpoint: crowdfunding, dynamic provision to public goods, dynamic free riding or crowding in/out issue. Although crowdfunding is not a new concept, it is a relatively new phenomenon, and the research on crowdfunding is still at its nascent but rapidly increasing stage. Past works have sought 50 to understand the motivations and mechanisms underlying the donor’s contribution behavior (Liu et al 2009, Aitomato 2011). Some studies have also investigated the dynamics of crowdfunding and social influence among crowdfunders. For example, Agrawal et al. (2015) explored how geography, social networks, and the timing affect crowdfunders investment decisions on a crowdfunding platform for artists. Burtch et al. (2012) examined the impact the observable indicators of prior others’ contributions on subsequent individuals’ contribution decisions using aggregate daily data from a crowdfunding platform for journalism. Their results indicate that donors tend to contribute less when prior others contribute more frequently or with greater amount. Later, using data from Kickstarter, one of the most influential U.S. based crowdfunding website, Kuppuswamy and Bayus (2013) found a similar negative effect of prior support on current contribution, and they also identified a deadline effect--- contributions to relatively successful projects tend to go up right before the projects close. In spite of such obvious dynamic patterns, almost all the previous empirical work on crowdfunding has adopted the reduced form approach. Another related literature is the dynamic provision to public goods. Individual’s contribution to public goods not only affects her own utility, but also that of other members in the collectivity. This interdependence creates a strong incentive for donors to take advantage of the public good without bearing a suitable cost. In static setting, a donor can only make one contribution decision and that decision must be made without any knowledge of decisions made by others. By contrast, in the dynamic setting, a donor can make contribution decisions in multiple rounds, and she can condition each decision on the level of contribution in the previous round, a state variable that is periodically updated. A paper by Fischerter and Garcher (2009) investigates the role of social preference and beliefs in voluntary contribution to public goods. They are interested in one type of social preferences, that is, people’s propensity to cooperate provided that 51 others cooperate as well. Such “conditional cooperation” depends directly on how others behave or are believed to behave. Conditional cooperators who observe or believe others free ride will reduce their contributions correspondingly. Fischerter and Garcher design two experiments to examine whether contributions decline because of contribution preferences or/and because of the way people form their beliefs on how others will behave. Their results indicate that people condition their contributions based not only on the contribution preference, but also on their beliefs. They argue that one explanation for why beliefs matter, in addition to the contribution preference, is subject’s willingness to cooperate in order to induce high belief and cooperation in the population. However, given the strong evidence of the potential dynamic concerns involved in the provision process to public goods, probably due to the lack of empirical data and the computational challenge, to the best of our knowledge, there’s no empirical work that directly model donors inter- temporal tradeoffs using a dynamic programming model which accounts for donor’s dynamic concerns such as their belief on others contribution. Given our empirical context, our study is also related to the economic literature on examining how organization’s contribution affect individual’s contribution to public goods. Researchers use the crowding effect to describe the impact of organization’s giving on individual’s giving to public goods. More specifically, the crowding effect can be described on a number of dimensions: direction and intensity. In terms of direction a crowding in occurs when organization’s support stimulates individual’s donations, while crowding out corresponds to the situation where organization’s support prohibits individual’s donations. The literature further distinguishes between total and partial (i.e., complete and incomplete) crowding effects to indicate the intensity of the relationship (Clotfelter, 1985). In the presence of a total effect changes in organization’s support produce equal changes in individual’s donation, while in the case of a partial effect such 52 changes produce a less than equal response in individual’s donation. The empirical evidence on the crowding effect of organization’s provision on individual’s contribution is somewhat mixed. Many studies indicate that organization’s contribution would crowd out individual’s contributions, but the crowding out is incomplete and usually small. We summarize the previous findings on crowding in and crowding out effect from both the empirical and experimental research in Table 7 and Table 8. 53 Authors (year) Data Findings on crowding effect Assumptions/Treatments/Explanations Abrams and Schmitz (1978, 1984) Aggregate tax return data Governmental social welfare transfers lowered private charitable contributions by about 28% Assume that others’ contributions and government contributions are exogenously determined Clotfelter (1985) Similar data as Abrams and Schmitz Crowding-out effect is just about 5% Kingma (1989) Contribution data to Public Radio (household-level data) Crowding-out is about 13% Using IV to derive others’ contributions and government contributions Khanna et al. (1995) UK health, religion, social welfare organizations Partial crowding-in effect of government support on UK health organizations and independence in the case of religion and social welfare organizations Payne (1998) A panel of charities drawn from IRS 990 forms 50% crowding-out Use aggregate government transfers to individuals in the state as an IV for government grant Khanna and Sandler (2000) Accounting data from British charities Substantial crowding-in effect, between 13% and 89% Taking into account of the endogeneity of government grants Okten and Weisbrod (2000) Panel data of IRS from individual nonprofit organizations Government grants have a crowding-in effect on private donations in six of the seven industries, significantly in four — libraries, hospitals, scientific research, and higher education. Only take care of the endogeneity issue for fundraising and price (2SLS, IV) Brooks (2000,2003) American symphony orchestras and public radio stations An inverted U shape: initially crowding in occurs and the higher the level of public support awarded to non- profit institutions, the higher the level of private donations. However after a certain point crowding out dominates and the higher the level of public support the lower the level of private donation Ribar and Wilhelm (2002) Panel of donations and government funding from 125 international relief and development organizations -Extended Andreoni’s impure altruism model to an economy with an infinitely large number of donors. -When public good provision is large and preferences are concave, increased public provision has negligible effect on individual choice. They argue that these results reconcile the different crowding-out estimates from large- scale empirical studies and small-scale experiments. Specifically, empirical analyses of data from large populations indicate partial, but usually very low, crowding-out by governmental contribution, whereas experimental investigations provide evidence of sizable crowding-out effects. Borgonovi (2006) Panel of American non-profit theatres -The crowding effect induced by the level of public support takes an inverted U shape: at low levels public support crowds-in private donations while at higher levels it displaces them. 54 -The change in total public support in the past year produces a constant crowding-in effect on the level of private donations - Federal and state support have a crowding-in effect at all levels, while local support has a similar impact to total public support (inverted U). Andreoni (2010) Panel of tax returns from charitable organizations Significant crowding out of about 73%, and most of the crowding out is the result of reduced fund-raising. No evidence for classic crowding out, and found a slight crowding in of donors by government grants. 1. Donors who count their contributions through taxation as part of their total contribution will reduce voluntary contributions to offset the grant. 2. Indirect crowding out due to reduced fund- raising effort. Garth Heutel (2010) A large panel data set gathered from nonprofit organizations' tax returns. -Find evidence that government grants crowd in private donations, with a larger effect for younger charities, consistent with signaling. -private donations crowd out government grants, but they are not statistically significant Note: First empirical paper test crowding out in the opposite direction (private on public contribution) Table 7. Empirical Evidence on Crowding Effect 55 Authors (year) Treatments Findings on crowding effect Notes Andreoni (1993) An involuntary transfer resembling a tax is levied on individuals and the resulting revenue is transferred to the provision of the public good. No tax treatment: each player was allocated seven tokens, and had to decide how many to contribute to a public good. Tax treatment: each player was now constrained to contribute at least two tokens (as tax) Crowding-out is 71 percent which is incomplete -His results may be “taken as evidence for alternative models that assume people experience some private benefit from contributing to public goods.” -Use Cobb-Douglas utility function to decide the payoff structure 1 [( ) ] ii u A w g G Bolton & Katok (1998) -Two treatments: (1) In the 15-5.a treatment, $15 was distributed to the subject in role A (the donor role) and $5 was distributed to the subject in role B (the recipient role). A was then given the opportunity to redistribute some of A's money to B. (2) The 18-2 treatment was identical, now $18 to A and $2 to B - Incomplete crowding out (74%) - Provide direct evidence that donor preferences have an impure altruism component which results in partial rather than complete crowding-out. -Modified version of Andreoni’s design: Test impure altruism using the dictator game which provides a test of pure versus impure altruism in a context where the strategic aspect of the public goods model is stripped away. Chan et al. (2002) -Individuals were asked to allocate their endowments between a private good and a public good. -tax group (15% to 25%) -no tax group -Reject complete crowding-out, but crowding-out increases as the involuntary transfer increases and sufficiently large involuntary transfers may offset the benefits of warm-glow giving. (non-linear crowding-out) Extends Andreoni’s between-subjects design by considering two levels of the involuntary transfer and by using a design in which all subjects see all transfer treatments (with-in subject). Eckel et al. (2005) - two initial allocations and two frames. (1) Initial allocations are either US$18 for the subject and US$2 for the charity, or US$15 and US$5 (2) In one frame, subjects are simply informed of the initial allocations between themselves and their chosen charity. In the other, subjects are told that their US$20 allocation has been taxed, and the amount allocated to their chosen charity. (3) same payoff structure across conditions Note: -The no-tax frame design mimics fiscal illusion: subjects are not told where the initial contribution to their charity comes from. -Subjects play a single dictator game In the first frame, crowding out that is close to zero; In the second frame, nearly 100% crowding out. (1) Forced contributions (Government transfers) do not crowd out private giving when the source of transfers is not apparent to the subjects, and the recipient is a charitable organization. (2) Forced contributions crowd out private giving when the source of the funding of the forced transfers is apparent to the subjects (3) By substituting a charity of choice for the previously anonymous subject, potential donors’ altruistic preferences can be activated. The satisfaction from giving, and the incentive to give, is greater if the donor has reason to believe that the recipient is deserving of assistance. Table 8. Experimental Evidence on Crowding Effect 56 The crowding out literature goes back to Warr (1982), Roberts (1984) and BBV (1986) who theoretically show that an exogenous increase in government grant could crowd out individual’s giving dollar-for-dollar (i.e., complete crowding out). From the donor’s point of view, the government is perceived just as another individual donor ready to make a contribution to the public good. So when government decides to take part in the provision and finances its contingent by taxing individual donors, the latter will decrease their contribution exactly by the amount of their individual tax load. Some studies including Coltfelter (1985) and Kingma (1989) find support for crowding out, but typically at a rate less than dollar-for-dollar (i.e., incomplete crowding out). One explanation for incomplete crowding out, provided by Andreoni (1989, 1990), is that individuals are “impure altruistic” in that they obtain warm glow from their contribution, independent of the level of public good. Andreoni describes warm glow as an abstract feeling of personal gratification obtained from the act of giving itself. In contrast, some other studies find crowding in (Kahanne and Sandler 1989; Payne 2001). Crowding in may occur if the public giving provides a signal on charity quality. A signaling model is presented by Payne (2001), Vesterlund (2003) and Andreoni (2006) where “seed money” by large donors or announcement of previous contributions increase subsequent contribution by acting as a signal of charity quality. In summary, the past theoretical, experimental and empirical work on public good provision provide compelling but inconsistent findings on crowding effect (i.e., complete vs. incomplete crowding out, crowding out vs. crowding in). So to understand how organization’s contribution would affect individual donor’s contribution on our crowdfunding platform, we need to empirically test that. It’s important to understand whether organization’s contribution 57 can crowd in or crowd out individual’s contribution, because currently all the contribution information including organization’s contribution are publicly available to all the potential donors. If organization’s contribution can potentially negatively crowd out individual’s contribution, then it’s not wise for the platform to disclose such information. However, if organization’s contribution can spur more individual’s contribution, then the platform can offer “seed money” (just like the organization’s contribution) to increase a funding project’s success probability. 2.4 Data In this section, we will show the characteristics of our data set and describe the model free data patterns that will help us set up and identify the structural model that will be described in the next section. We obtained a sample of 43,697 weekly contributions from 1,143 individual donors for 115 funding projects. Among these 115 funding projects, 80 (70%) projects received funding from organizational donors, and 49 (43%) funding projects successfully raised enough funding within the funding deadline (see Table 9). On average, the funding goal is $1,695 and funding duration is 18 weeks. The average number of donors per project is 57 and on average a project has 2 organizational donors. The mean funding percentage contributed by organizational donors is 18.4%. Thus, organizational donors typically make a much larger contribution amount compared to individual donors. And the average total funding percentage raised is 81.4%. Text, Video, Photo, and Audio are four dummies representing the project report delivery format. And we also have a year dummy to control for the time trend of the platform (see Table 10). 58 For individual’s weekly contribution, the mean contribution amount is $0.981, and an individual’s own cumulative contribution amount over time is on average about $21.24. When making the weekly contribution amount decision, individuals can observe funding status such as the number of previous donors, the total funding amount raised, and the funding amount donated by organizational donors previously. As we can see from Table 11, on average, the funding percentage donated by organizations is about 9.59%, and the number of previous number of donors is 94. Variables Value/Mean Number of Funding Projects 115 Number of Donors 1143 Number of Weekly Donations 43697 Projects with Organizational Donors 70% Successfully Funded Projects 43% Table 9. Summary of the Overall Dataset Variables Mean Std. Dev. Minimum Maximum Funding Goal 1695.010 4840.02 75 40000 Funding Duration (Week) 17.970 19.39 2 87 Number of Donor 56.630 52.72 6 294 Number of Organization 2.035 2.60 0 16 Total Funding Percentage Raised 81.36% 25.37% 5.90% 100% Total Organizational Funding Percentage 18.40% 24.58% 0 88% Text 0.887 0.318 0 1 Video 0.348 0.478 0 1 Photo 0.757 0.431 0 1 Audio 0.313 0.466 0 1 Log (Funding Goal) 6.561 1.034 4.317 10.597 Year Dummy 0.774 0.420 0 1 Table 10. Summary Statistics for Funding Project Information 59 Variables Mean Std. Dev. Minimum Maximum Contribution Amount 0.981 17.684 0 2500.00 Lag_cum_own_contribution 21.243 111.809 0 2500.00 Funding Time Left (%) 50.52% 28.76% 0 98.95% Funding Raised (%) 55.76% 31.57% 0 99.99% Prior Cumulative Number of Donor 94.902 81.122 0 293 Prior Organization Contribution 9.59% 16.25% 0 88.00% Table 11. Summary Statistics for Individual’s Weekly Contribution 2.4.1 Model-Free Data Patterns We begin with checking whether dynamics play an important role in understanding crowdfunders’ contribution behavior under this context. We start by looking in the data to see whether there exists patterns consistent with agent’s shifting the allocation of contributions within the funding drive. First, Kuppuswamy and Bayus (2013) identified a deadline effect--- contributions to relatively successful projects tend to go up right before projects close. When incentives exist for donors to manipulate timing, output, for example the cumulative funding, should look lumpy over the course of the funding-drive. In particular, we expect to see spikes in output when agents are close to the funding deadline especially for projects that are close to the funding goal. In Figure 6, we plot the funding trends along the funding drive for several projects, and we indeed see a lumpy funding process and some spikes for projects that are relatively successful in raising funds. 60 Figure 6. Funding Trend along the Funding Drive In Figure 7 and 8, we plotted out the funding percentage raised over time passed for projects that are unsuccessful and successful respectively. We can notice that for unsuccessful projects (i.e. funding goal not accomplished), overall, the funding drive looks relatively flat. And in comparison, the funding drive looks much steeper, suggesting that donors indeed take into account of the funding status over time, and potentially they use those information to form belief on how likely it is for a project to be able successfully raise enough money within the timeline. From the plot for unsuccessfully funded projects, we can see that a project (in blue line on the very left) managed to raise over 90% of the funding amount in a very short time period, but it still failed. The reason for that is it only attracts 3 donors, so we believe that, in addition to the funding percentage raised, individuals also care about the number of donors who have supported the project previously. Because this information can be viewed as how 61 many people are directly interested in the project, and therefore individuals can have an idea on how many people their contribution can have an influence on. Figure 7. Funding Percentage Raised over Funding Time Passed for Unsuccessful Projects Figure 8. Funding Percentage Raised over Funding Time Passed for Successful Projects 62 To directly test whether organization’s contribution would influence individual donor’s contribution, we estimate individual i’s weekly contribution 𝑦 𝑖𝑡 using the following censored regression specification: 𝑦 𝑖𝑡 ={ 𝑠 𝑖𝑡 𝐴𝑚𝑡 _𝐿𝑒𝑓𝑡 , 𝑖𝑓 𝑦 𝑖𝑡 ∗ ≥𝑠 𝑖𝑡 𝐴𝑚𝑡 _𝐿𝑒𝑓𝑡 𝑦 𝑖𝑡 ∗ , 𝑖𝑓 0<𝑦 𝑖𝑡 ∗ <𝑠 𝑖𝑡 𝐴𝑚𝑡 _𝐿𝑒𝑓𝑡 0, 𝑖𝑓 𝑦 𝑖𝑡 ∗ ≤0 (1) with 𝑦 𝑖𝑡 ∗ =𝜃 𝑖 0 +𝜃 𝑖 1 𝑥 𝑖𝑡 𝑇𝑖𝑚𝑒𝐿𝑒𝑓𝑡 +𝜃 𝑖 2 𝑥 𝑖𝑡 𝐹𝑢𝑛𝑑 _𝑝𝑒𝑟𝑐 +𝜃 𝑖 3 𝑥 𝑖𝑡 𝑇𝑖𝑚𝑒𝐿𝑒𝑓𝑡 𝑥 𝑖𝑡 𝐹𝑢𝑛𝑑 _𝑝𝑒𝑟𝑐 +𝜃 𝑖 4 𝑥 𝑖𝑡 𝐷𝑜𝑛𝑜𝑟 +𝜃 𝑖 5 𝑥 𝑖𝑡 𝑂𝑟𝑔 _𝐴𝑚𝑡 _𝑝𝑒𝑟𝑐 +𝜃 6 𝑤 𝑥 𝑗 𝑓𝑒𝑎𝑡𝑢𝑟𝑒 +𝑒 𝑖𝑡 where 𝑥 𝑖𝑡 𝑂𝑟𝑔 _𝐴𝑚𝑡 _𝑝𝑒𝑟𝑐 represents the funding percentage contributed by organizational donors. We assume 𝑒 𝑖𝑡 ~𝑁 (0,𝜎 2 ) , and 𝜃 𝑗 𝑤 ={𝜃 𝑖 0 ,𝜃 𝑖 1 ,𝜃 𝑖 2 ,𝜃 𝑖 3 ,𝜃 𝑖 4 ,𝜃 𝑖 5 }~𝑀𝑉𝑁 (𝜃 ̅ ,Ω 𝜃 ). The estimation result for this reduced form model is reported in the Table 12. The result suggests that, on average, organization’s contribution has a positive impact on individual’s contribution. Although the number of donors does not significantly affect individual’s contribution on the population level, it has a lot heterogeneity, suggesting that different people may react differently to the observed funding status like number of donors. Posterior Mean (Std.) Posterior Variance Intercept 0.110 (0.040) 0.016 Funding Time Left (𝒙 𝒊𝒕 𝑻𝒊𝒎𝒆𝑳𝒆𝒇𝒕 ) 0.012 (0.043) 0.019 Funding Percentage Raised (𝒙 𝒊𝒕 𝑭𝒖𝒏𝒅 _𝒑𝒆𝒓𝒄 ) -0.282 (0.054) 0.026 Funding Percentage *Time (𝒙 𝒊𝒕 𝑻𝒊𝒎𝒆𝑳𝒆𝒇𝒕 𝒙 𝒊𝒕 𝑭𝒖𝒏𝒅 _𝒑𝒆𝒓𝒄 ) -0.066 (0.054) 0.024 Number of Donor (𝒙 𝒊𝒕 𝑫𝒐𝒏𝒐𝒓 ) 0.005 (0.029) 0.009 Organization’s Funding Percentage (𝒙 𝒊𝒕 𝑶𝒓𝒈 _𝑨𝒎𝒕 _𝒑𝒆𝒓𝒄 ) 0.260 (0.061) 0.032 Table 12. Impact of Organization’s Contribution on Individual’s Contribution 63 2.5 Modeling Framework 2.5.1 Motivation for Structural Modeling and Sources of Dynamics Understanding the factors that influence crowdfunders contribution behavior over time is a key to the success of funding projects and the crowdfunding platform. A structural model that characterizes the dynamic response of crowdfunders contribution behavior is therefore critical for the following reasons. First, we need to account for impact of the dynamic funding structure on crowdfunders contribution behavior. Individuals are allowed to contribute at any time, in any amount as long as it’s before the funding deadline or the funding goal hasn’t yet been fulfilled, and they can observe others contribution history and therefore can potentially make contributions conditional on others behavior (Tucker and Zhang 2009; Zhang and Liu 2011; Burtch et.al, 2012). And since the final project reward won’t be realized until the terminal of the funding process, individuals have uncertainties on whether they can obtain the final reward from supporting a project. However, if they do not make contributions to support the funding project, then there’s a risk that the funding project would fail in raising enough money within the funding deadline, because their contributions now may spur others to donate in the following periods. In addition, the final report will be published online with free access to everyone, so this public goods feature automatically brings in the dynamic free riding, or crowding in/out issue. So we need a model that can account for those dynamic concerns. Second, we need a structural model because we wish to conduct counterfactual policy simulations to understand the effects of changing platform policies on crowdfunders contribution behavior. A model based on micro-foundations of consumer behavior uses theory about consumer behavior to recover primitives of consumer preferences. These preference parameters can then be used to evaluate how individuals would make choices in a 64 counterfactual scenario, allowing us to provide recommendations that have managerial implications for the platform. A fundamental process we need to account for in our model is the inter-temporal tradeoff in making contributions. The dynamics in individual’s contribution decisions stem from (1) the inter-temporal tradeoff of the individual’s current benefit versus the future benefits, (2) the dynamic funding provision process itself, (3) and the dynamic free riding or crowding in/out issue inherent in the provision process to public goods. First, given the funding and reward giving structure, an individual will evaluate the utility of making a contribution today versus tomorrow. Because the final reward will only be delivered in the terminal period conditional on the final funding performance. So an individual is uncertain about whether she can get the final reward or not during the decision making periods. As the funding deadline approaches, the less uncertain she is about the funding success probability. However the good thing about contributing early lies in that her action space is constrained by the distance to the funding goal, so the earlier she contributes, the larger amount she is allowed to contribute as the action space is less constrained. And the more she contributes, the larger the contribution benefit (e.g., warm-glow) she can obtain. Second, at the time deciding when to make a contribution, an individual would think about how likely it is for the project to successfully raise enough money within the funding drive, in the meanwhile, she also takes into account of the impact of her own action on the funding performance in the subsequent funding period. The more and the earlier she contributes, the more likely it is to attract more potential donors or funding, and hence the higher the funding success probability becomes. 65 2.5.2 Model Setup We model an individual donor’s weekly contribution decision to support a crowdfunding project that raises money for public goods (i.e., digital reports). We model donors as maximizing their intertemporal utilities, conditional on periodically updated funding status, their belief on how much funding a project can potentially raise within the funding drive, and the funding success probability (i.e., the project reward delivery probability). We’ll describe the model setup by first presenting the contribution timeline, and then explaining the period utility and the DP problem for an individual decision maker, and finally describing the proposed estimation algorithm to solve the DP problem. 2.5.2.1 Model Timeline The timing of the model is as follows (see Figure 9). In each week: 1. Individuals check two things before making a contribution decision. First, an individual checks whether it is the funding stage or the reward delivery stage. Second, she checks whether the funding project has achieved the funding goal or not. a. If it’s the reward delivery stage (i.e., t=T+1), then an individual can no longer make any contributions. Otherwise, b. If the project has already raised enough funding, then an individual donor is not allowed to make any contributions neither. Otherwise, c. If the funding goal has not been achieved and it’s still during the funding stage, then an individual donor observes the current states (e.g., funding status) and makes a contribution decision (can either be zero or nonzero) in a dynamically optimal manner. 66 2. An idiosyncratic shock is realized. The shock plus the donor’s contribution amount decision together determine the individual’s realized period payoff. 3. Next funding week comes. Figure 9. Model Timeline No Yes (no action is allowed) No Yes New Period Starts Check t=T+1 Check Funding=Goal Makes a Contribution Decision (𝐴 ≥0) Reward Delivery Stage contributions are not allowed Project Funding Stage 67 2.5.2.2 Utility Specification We will specify the period utility based on the model timeline. On this crowdfunding platform, each funding project has a predetermined funding deadline and funding goal. And individuals can make contributions to a project they’re interested in at any time, as long as the project’s funding goal hasn’t yet been fulfilled and the funding deadline hasn’t yet been met. Since we collect data from a journalistic crowdfunding platform that is raising money for public goods, we assume that an individual’s overall utility for making contribution A = {𝐴 1 ,….,𝐴 𝑇 } 14 can be decomposed into three components: First, if a funding project successfully raises enough money, then a donor can obtain a reward benefit because the project creator will deliver a digital report to the public. However this reward benefit can only be realized at the end of the funding drive, because the report will only be published in the final delivery stage. Second, a donor can derive direct benefit from making a contribution even during the funding period, because she is behaving prosocially by donating money to support a project that can be beneficial to the public. An individual’s contribution behavior can generate warm glow effect or make one feel good about herself. Third, the contribution amount is the cost that an individual needs to bear as she will have less money to spend on outside goods under budget constraint if she decides to make a positive contribution to support a project in a given time period. 2.5.2.3 Period utility during the funding stage First, during the funding stage (i.e., t=1,…T), in each time period t, conditional on the observed funding status, an individual i makes a decision on how much to contribute, 𝐴 𝑖𝑡 ≥0, to a funding project in order to maximize her payoff. Since the reward benefit will only be realized in the final delivery stage, an individual i’s period utility from contributing 𝐴 𝑖𝑡 dollars in the funding stage 14 Hereafter, we will use letter A to denote Action, s to denote State, and x to denote covariates. 68 contains two components: the warm glow effect from her act of giving, and the corresponding cost she has to bear. Following the literature, we adopt the log transformed Cobb-Douglas utility specification (Andreoni 1993; Satomura, Kim and Allenby 2011) to model individual’s monetary contribution behavior. Thus, the period utility for individual donor i, contributing 𝐴 𝑖𝑡 ≥0 can be specified as, 𝑢 𝑖𝑡 =𝜑 𝑖𝑡 1 [log( 𝑠 𝑖𝑡 𝐼𝑛𝑑 _𝑐𝑢𝑚𝑎𝑚𝑡 +𝐴 𝑖𝑡 +1 𝑠 𝑖𝑡 𝐼𝑛𝑑 _𝑐𝑢𝑚𝑎𝑚𝑡 +1 )]−𝜑 2 𝐴 𝑖𝑡 (2) where 𝑠 𝑖𝑡 𝑖𝑛𝑑 _𝑐𝑢𝑚𝑎𝑚𝑡 denotes individual i’s cumulative contribution amount to a funding project at the beginning of period t. We assume that when there is no discount, warm glow 15 only depends on individual donor’s own cumulative contribution amount. In particular, if an individual decides to make zero contribution in period t (i.e., 𝐴 𝑖𝑗 ,𝑡 =0), then she will not get any extra warm glow benefit for the current period. −𝐴 𝑖𝑡 represents the monetary contribution cost. For identification purpose, we set 𝜑 2 to 1. 𝜑 𝑖𝑡 1 captures the marginal contribution benefit, and we use censored normal to specify 𝜑 𝑖𝑡 1 𝜑 𝑖𝑡 1 ={ 0 𝑖𝑓 𝜑 𝑖𝑡 1∗ ≤0 𝜑 𝑖𝑡 1∗ 𝑖𝑓 𝜑 𝑖𝑡 1∗ >0 𝜑 𝑖𝑡 1∗ =𝜓 1𝑖 +𝜓 2𝑖 ∗𝑠 𝑖𝑡 𝑑𝑜𝑛𝑜𝑟 +𝜓 3𝑖 ∗𝑠 𝑖𝑡 𝑂𝑟𝑔 _𝐶𝑢𝑚𝑎𝑚𝑡 +𝜀 𝑖𝑡 𝜑 (3) where 𝜀 𝑖𝑡 𝜑 captures the random action shock and 𝜀 𝑖𝑡 𝜑 ~𝑖 .𝑖 .𝑑 ,𝑁 (0,𝜎 𝜑 2 ). Each time period, when an individual makes a contribution, her benefit (e.g., good feeling) for making any contribution can be adjusted by the number of people who are also interested in the project. Even though she cannot directly observe the number of people who may potentially benefit from the project, she can observe the number of donor who have made contributions previously (𝑠 𝑖𝑡 𝑑𝑜𝑛𝑜𝑟 ). She can use this 15 We adopt the logarithm function to capture the warm glow effect so that it increases with contribution, but at a decreasing rate. 69 number as an approximation. As suggested by the public goods provision literature, individual’s contribution can be crowded out or crowded in by organization’s contribution. So we believe that individual’s marginal contribution benefit 𝜑 𝑖𝑡 1 can be influenced by organization’s contribution (𝑠 𝑖𝑡 𝑂𝑟𝑔 _𝐶 𝑢 𝑚𝑎𝑚𝑡 ) as well. 2.5.2.4 Period utility for final delivery state In the final report delivery stage (t=T+1), no more contributions can be made, and the final reward delivery benefit can be realized depending on the final funding status. If the funding goal is met (i.e., 𝑠 𝑇 +1 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 =𝑥 𝑗 𝐺𝑜𝑎𝑙 ), then the reward benefit can be realized for sure. However, if the raised funding falls below the goal (i.e., 𝑠 𝑇 +1 𝑐𝑢𝑚𝑎𝑚𝑡 <𝑥 𝑗 𝐺𝑜𝑎𝑙 ), then the reward benefit will be realized with certain probability. Therefore, the final period utility can be specified as, 𝑢 𝑇 +1,1 =𝜑 3 , 𝑖𝑓 𝑠 𝑖 ,𝑇 +1 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 =𝑥 𝑔𝑜𝑎𝑙 (4a) 𝑢 𝑇 +1,2 =𝜑 3 , 𝑖𝑓 𝑠 𝑇 +1 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 <𝑥 𝑔𝑜𝑎𝑙 & 𝐼 (𝑑𝑒𝑙𝑖𝑣𝑒𝑟 )=1 (4b) 𝑢 𝑇 +1,3 = (𝑠 𝑇 +1 𝐼𝑛𝑑 _𝐶𝑢𝑚𝑎𝑚𝑡 ), 𝑖𝑓 𝑠 𝑇 +1 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 <𝑥 𝑔𝑜𝑎𝑙 & 𝐼 (𝑑𝑒𝑙𝑖𝑣𝑒𝑟 )=0 (4c) where 𝑠 𝑇 +1 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 denotes the total contribution amount raised by all donors at the beginning of period T+1, and 𝑥 𝑔𝑜𝑎𝑙 represents the funding goal. And in order to guarantee the positivity, we let 𝜑 3 =exp (𝜓 3 ∗ ). When the project is not delivered, then there’s no reward benefit, but all individual’s previous contribution, 𝑠 𝑖 ,𝑇 +1 𝑖𝑛𝑑 _𝑐𝑢𝑚𝑎𝑚𝑡 , will be returned back. 2.5.2.5 Donor’s Belief on Final Report Delivery Probability In our dataset, we do not observe the detailed contribution history for the 43 abandoned projects which did not accomplish the funding goal and chose not to deliver the final report. The only information we have for those abandoned projects is the total funding amount raised at the end of the funding drive. However, for other 73 projects that did not achieve the funding goal but chose 70 to deliver the final report, we observe complete contribution history. In order to form donor’s belief on final report delivery probability, we use the 43 abandoned projects and 73 delivered projects to estimate the delivery probability using the following Logit model, 01 01 exp( * ) ( 1| 1) 1 exp( * ) Raise j Raise j Raise j s Prob deliver s s (5) Our reduced form result shows that 𝛼 0 =−2.92 and 𝛼 1 =14.34. Therefore, the probability of delivering the final report conditional on the final observed funding status at T+1 can be written as, 𝑃 𝑑𝑒𝑙𝑖𝑣𝑒𝑟 = { 1, 𝑖𝑓 𝑠 𝑖 ,𝑇 +1 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 =𝑥 𝐺𝑜𝑎𝑙 exp (𝛼 0 +𝛼 1 𝑠 𝑖 ,𝑇 +1 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 𝑥 𝐺𝑜𝑎𝑙 ) 1+exp (𝛼 0 +𝛼 1 𝑠 𝑖 ,𝑇 +1 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 𝑥 𝐺𝑜𝑎𝑙 ) 𝑜𝑡 ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (6) We assume that when making the contribution decision, a donor has a belief on how much funding percentage a project is able to raise. And based on that belief, she calculates final report delivery probability according to equation (6). 2.5.3 Evolution of State Variables There are two important sources of dynamics in the model. First, the reward benefit won’t be realized until the final delivery stage, just like the nonlinear compensation scheme (e.g., annual bonus) in the sales force literature (Misra and Nair 2011), this fundraising structure itself generates a dynamic into donor’s problem. In particular, a reduction in contribution now may reduce the chance to obtain the direct reward benefit in the future. A second source of dynamic is introduced by the dynamic crowding out or crowding in as we discussed in the previous sections. These dynamic aspects are embedded in the transitions of the state variables in the model. In the following, we will first explain how the state variables evolve, and then present the value function 71 that encapsulates the optimal inter-temporal decisions of the agents. In this study, we have five observed state variables: 𝑠 𝑡 ={𝐼𝐶𝐴 𝑡 ,𝑇𝐿 𝑡 ,𝐹𝑈 𝑡 ,𝐷𝑂𝑅 𝑡 ,𝑂𝐴 𝑡 }={𝑠 𝑡 𝐼𝑛𝑑 _𝐶𝑢𝑚𝑎𝑚𝑡 ,𝑠 𝑡 𝑇𝑖𝑚𝑒𝐿𝑒𝑓𝑡 ,𝑠 𝑡 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 ,𝑠 𝑡 𝐷𝑜𝑛𝑜𝑟 ,𝑠 𝑡 𝑂𝑟𝑔 _𝐴𝑚𝑡 } 16 Next, we describe how these five state variables evolve. We assume that donors have rational belief on state evolution in the sense that it follows the actual transition observed from the empirical data. The state s t Ind_Cumamt captures individual donor’s cumulative contribution at the beginning of period t. Its law of motion, 𝑠 𝑡 +1 Ind_Cumamt =𝑠 𝑡 Ind_Cumamt +𝐴 𝑡 , is deterministic as individual’s own cumulative contribution amount in period t+1 (𝑠 𝑡 +1 Ind_Cumamt ) is equal to the sum of her cumulative contribution at the beginning of period t (𝑠 𝑡 Ind_Cumamt ) and her contribution amount in the current period t (𝐴 𝑡 ). The second state s t TimeLeft tracks the funding time left, it evolves deterministically as s t TimeLeft decreases by 1 for each period. The third state variable s it Funding keeps track of the total funding amount raised. This is updated via the linear law of motion 𝑠 𝑖𝑡 +1 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 =𝑠 𝑖𝑡 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 +𝐴 𝑖𝑡 +𝑤 𝑖𝑡 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 , which is simply the sum of the funding raised at the beginning of period t, individual’s own contribution amount, and the contribution amount donated by others in this period (𝑤 𝑖𝑡 Funding ). Based on this law of motion, the evolution of 𝑠 𝑖𝑡 Funding is fully determined by the evolvement of the weekly contribution amount 𝑤 𝑖𝑡 Funding . Since individual i does not have control on others contribution, she forms expectation on how much funding others would contribute in that period of time. And we use the empirical 16 For simplicity, we will only use the t index in the following sections. 72 data on weekly contribution to estimate its transition probability using lower and upper censored normal regression: 𝑤 𝑗𝑡 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 ={ 𝑠 𝑗𝑡 𝐴𝑚𝑡 _𝐿𝑒𝑓𝑡 , 𝑖𝑓 𝑤 𝑗𝑡 ∗ ≥𝑠 𝑗𝑡 𝐴𝑚𝑡 _𝐿𝑒𝑓𝑡 𝑤 𝑗𝑡 ∗ , 𝑖𝑓 0<𝑤 𝑗𝑡 ∗ <𝑠 𝑗𝑡 𝐴𝑚𝑡 _𝐿𝑒𝑓𝑡 0, 𝑖𝑓 𝑤 𝑗𝑡 ∗ ≤0 (7) where 𝑠 𝑗𝑡 𝐴𝑚𝑡 _𝐿𝑒𝑓𝑡 =𝑥 𝑗 𝐺𝑜𝑎𝑙 −𝑠 𝑗𝑡 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 . The fourth state 𝑠 𝑡 Donor keeps track of the number of donors at the beginning of period t. It is also updated via a linear law of motion: 𝑠 𝑡 +1 𝐷𝑜𝑛𝑜𝑟 =𝑠 𝑡 𝐷𝑜𝑛𝑜𝑟 +𝐼 (𝐴 𝑡 >0 & 𝑠 𝑡 𝐼𝑛 𝑑 𝐶𝑢𝑚𝑎𝑚𝑡 =0)+ℎ 𝑡 𝑑𝑜𝑛𝑜𝑟 (8) That is, the number of donors at beginning of time period t+1 is equal to the number of donors at t, plus one or zero depending on individual’s contribution decision, plus the number of other donors coming within this time period (ℎ 𝑡 𝑑𝑜𝑛𝑜𝑟 ). If an individual decides to make a positive contribution and in addition, she has not made any contributions previously, then her positive contribution action leads to an increase of one for the state variable 𝑠 𝑡 Donor . We use the empirical data on weekly number of new donors attracted to a project to form an individual’s belief on new donors ℎ 𝑡 𝑑𝑜𝑛𝑜𝑟 . And we assume that the arrival of new donors follows a Poisson process: ℎ 𝑡 𝑑𝑜𝑛𝑜𝑟 ~𝑃𝑜𝑖𝑠𝑠𝑜𝑛 (𝜆 𝑡 ) , Pr(𝑁 𝑗𝑡 =𝑛 )=𝑒 𝜆 𝑗𝑡 𝜆 𝑗𝑡 𝑛 𝑛 ! , (n=0,1,2,…) (9) The fifth state s ijt Org_Amt keeps track of the funding amount raised by organizational donors. Similar to the total funding amount state variable, the organizational funding at the beginning of period t+1 is the sum of the organizational funding 𝑠 𝑖𝑗𝑡 𝑂𝑟𝑔 _𝐴𝑚𝑡 at the beginning of period t and the new contribution, 𝑔 𝑖𝑗𝑡 𝑂𝑟𝑔 _𝐴𝑚𝑡 , made by organizational donors within period t. 𝑠 𝑖𝑗 ,𝑡 +1 𝑂𝑟𝑔 _𝐴𝑚𝑡 =𝑠 𝑖𝑗𝑡 𝑂𝑟𝑔 _𝐴𝑚𝑡 +𝑔 𝑖𝑗𝑡 𝑂𝑟𝑔 _𝐴𝑚𝑡 (10) 73 And we use the empirical data on weekly contribution from organizations to estimate its transition probability using lower and upper censored normal regression 𝑔 𝑗𝑡 𝑂𝑟𝑔 _𝐴𝑚𝑡 (𝑠 𝑗𝑡 ;𝑘 𝑔 )={ 𝑠 𝑗𝑡 𝐴𝑚𝑡 _𝐿𝑒𝑓𝑡 𝑖𝑓 𝑔 𝑗𝑡 ∗ ≥𝑠 𝑗𝑡 𝐴𝑚𝑡 _𝐿𝑒𝑓𝑡 𝑔 𝑗𝑡 ∗ 𝑖𝑓 0< 𝑔 𝑖𝑗𝑡 ∗ <𝑠 𝑗𝑡 𝐴𝑚𝑡 _𝐿𝑒𝑓𝑡 0 𝑖𝑓 𝑔 𝑗𝑡 ∗ ≤0 (11) Where 𝑠 𝑗𝑡 𝐴𝑚𝑡 _𝐿𝑒𝑓𝑡 =𝑥 𝑗 𝐺𝑜𝑎𝑙 −𝑠 𝑗𝑡 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 . The following Table 13 summarizes the observed and unobserved state variables and the corresponding laws of motion. And the detailed estimation for the evolvement of the stochastic states (e.g., 𝑠 𝑡 +1 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 (𝑤 𝑡 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 ),𝑠 𝑡 +1 𝐷𝑜𝑛𝑜𝑟 (ℎ 𝑡 𝑑𝑜𝑛𝑜𝑟 ),𝑠 𝑡 +1 𝑂𝑟 𝑔 𝐴𝑚𝑡 (𝑔 𝑡 𝑂𝑟𝑔 _𝐴𝑚𝑡 ) ) are presented in Appendix B. State Variables Description Type Law of Motion 𝑠 𝑡 +1 𝐼𝑛𝑑 _𝐶𝑢𝑚𝑎𝑚𝑡 Donor’s own cumulative contribution Observed, deterministic 𝑠 𝑡 +1 𝐼𝑛𝑑 _𝐶𝑢𝑚𝑎𝑚𝑡 =𝑠 𝑡 𝐼𝑛𝑑 _𝐶𝑢𝑚𝑎𝑚𝑡 +𝐴 𝑡 𝑠 𝑡 +1 𝑇𝑖𝑚𝑒𝐿𝑒𝑓𝑡 Periods left for funding Observed, deterministic 𝑠 𝑡 +1 𝑇𝑖𝑚𝑒𝐿𝑒𝑓𝑡 =𝑠 𝑡 𝑇𝑖𝑚𝑒𝐿𝑒𝑓𝑡 −1 𝑠 𝑡 +1 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 Funding received Observed, continuous, stochastic 𝑠 𝑡 +1 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 =𝑠 𝑡 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 +𝐴 𝑡 +𝑤 𝑡 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 (𝑠 𝑡 ) 𝑠 𝑡 +1 𝐷𝑜𝑛𝑜𝑟 Cumulative number of donors Observed, Count, Stochastic 𝑠 𝑡 +1 𝐷𝑜𝑛𝑜𝑟 =𝑠 𝑡 𝐷𝑜𝑛𝑜𝑟 +𝐼 (𝐴 𝑡 >0 & 𝑠 𝑡 𝐼𝑛𝑑 _𝐶𝑢𝑚𝑎𝑚𝑡 =0)+ℎ 𝑡 𝑑𝑜𝑛𝑜𝑟 (𝑠 𝑡 ) 𝑠 𝑡 +1 𝑂𝑟𝑔 _𝐴𝑚𝑡 Cumulative organizational contribution Observed, continuous, stochastic 𝑠 𝑡 +1 𝑂𝑟𝑔 _𝐴𝑚𝑡 =𝑠 𝑡 𝑂𝑟𝑔 _𝐴𝑚𝑡 +𝑔 𝑡 𝑂𝑟𝑔 _𝐴𝑚𝑡 (𝑠 𝑡 ) 𝜀 𝑡 𝝋 Contribution Decision Shock Unobserved 2 ~ i.i.d, (0, ) t N Table 13. State Variables and the Law of Motion 2.5.4 Dynamic Structural Model This section we will propose Bayesian MCMC algorithm to be applied to our finite-horizon continuous choice dynamic programming model. We’ll first provide the model framework. Let t index time (discrete), t=0,…,T+1, where (T+1)<∞ is the terminal period during which the project 74 creator delivers the final report. Let 𝐴 𝑡 ∊𝐶 𝑡 be an action (continuous) and 𝐶 𝑡 be the action space at time t. 𝑠 𝑡 =(𝑥 𝑡 ,𝜀 𝑡 ) be a vector of state variables where 𝑥 𝑡 and 𝜀 𝑡 represent the observed and unobserved state vectors, respectively. And 𝜃 ∊Θ is the vector of model parameters. Let 𝑢 𝑡 = (𝑠 𝑡 ,𝐴 𝑡 ;𝜃 ) be per period utility function at time t given(𝑠 𝑡 ,𝐴 𝑡 ) . The expected discounted sum of current and future utilities at time t given 𝑠 𝑡 and (𝐴 𝑡 ,…,𝐴 𝑇 ) is given by 𝐄 [∑ 𝛽 𝜏 −𝑡 𝑈 𝜏 (𝑠 𝜏 ,𝐴 𝜏 ;𝜃 ) 𝑇 +1 𝜏 =𝑡 |𝑠 𝑡 ] (12) where 𝛽𝜖 (0,1) represents the discount factor, and the expectation is taken over the evolution of state variables given (𝐴 𝑡 ,…,𝐴 𝑇 ) . Let 𝑓 𝑡 +1 (∙|𝑠 𝑡 ,𝐴 𝑡 ;𝜃 ) be the transition probability given(𝑠 𝑡 ,𝐴 𝑡 ) . A forward-looking individual’s optimization problem is to choose a sequence of actions 𝐴 =(𝐴 𝑡 ,…,𝐴 𝑇 ) to maximize the expression (12). max 𝐴 𝐄 [∑ 𝛽 𝜏 −𝑡 𝑈 𝜏 (𝑠 𝜏 ,𝐴 𝜏 ;𝜃 ) 𝑇 +1 𝜏 =𝑡 |𝑠 𝑡 ] (13) The solution to the above dynamic programming problem is the same as the solution to the Bellman equation, which is referred to as the value function. For finite-horizon DP problems, the value function at time t, 𝜐 𝑡 (𝑠 𝑡 ;𝜃 ) can be recursively computed by backward induction (Bellman 1957). More specifically, at t=T+1, 𝜐 𝑇 +1 (𝑠 𝑇 +1 ;𝜃 )=𝐸 [𝑢 𝑇 +1 (𝑠 𝑇 +1 ;𝜃 )] (14) During this terminal period, no more action is allowed, and the expectation is taken over the final report delivery probability following equation (6). And 𝑢 𝑇 +1 is given by equation (4a-4c). During the funding period t=T,…,1, the value function can be written as, 𝜐 𝑡 (𝑠 𝑡 ;𝜃 )=max 𝐴 𝑡 ∈𝐶 𝑡 (𝑉 𝑡 (𝑠 𝑡 ,𝐴 𝑡 ;𝜃 )) (15) where the action space is constrained by the funding amount left, 𝐴 𝑡 ∈𝐶 𝑡 =[0,𝑠 𝑡 𝐴𝑚𝑡𝐿 𝑒 𝑓𝑡 ]. And the alternative-specific value function, 𝑉 𝑡 (𝑠 𝑡 ,𝐴 𝑡 ;𝜃 ) , is given by: 𝑉 𝑡 (𝑠 𝑡 ,𝐴 𝑡 ;𝜃 )=𝑢 𝑡 (𝑠 𝑡 ,𝐴 𝑡 ;𝜃 )+𝛽 𝐸 𝑆 𝑡 +1 [𝜐 𝑡 +1 (𝑠 𝑡 +1 ;𝜃 )|𝑠 𝑡 ,𝐴 𝑡 ] (16) 75 The optimal decision rule is 𝐴 𝑡 ∗ =argmax 𝐴 𝑡 ∈𝐶 𝑡 {𝑉 𝑡 (𝑠 𝑡 ,𝐴 𝑡 ;𝜃 )}. And the expected value function is given by 𝐸 𝑆 𝑡 +1 [𝜐 𝑡 +1 (𝑠 𝑡 +1 ;𝜃 )|𝑠 𝑡 ,𝐴 𝑡 ]=∫𝜐 𝑡 +1 (𝑠 𝑡 +1 ;𝜃 )𝑓 𝑡 +1 (𝑠 𝑡 +1 |𝑠 𝑡 ,𝐴 𝑡 ;𝜃 𝑠 )𝑑 𝑠 𝑡 +1 (17) In our setting, the state space includes three stochastic continuous or discrete state variables (e.g., total funding amount, organizational funding amount, number of donors), and the expected value function in equation (17) does not have a closed form solution. Thus we need to numerically approximate the high-dimensional integral. Inspired by the MCMC algorithm proposed by Imai, Jain, and Ching (2009, IJC) for infinite-horizon discrete choice dynamic programming models, and the modified MCMC algorithm proposed by Ishihara and Ching (2013, 2016) for finite- horizon discrete choice dynamic programming models, we develop the following MCMC algorithm for our finite-horizon continuous choice dynamic programming model. 2.5.4.1 Modified Bayesian Algorithm The key innovation of the IJC algorithm (Imai, Jain and Ching 2009) is that instead of fully solving for the fixed point of the Bellman operator in each outer loop, it gradually solves it by applying the Bellman operator only once per outer loop iteration, and uses the set of partially solved value functions from the past outer loop iterations for approximating the expected value functions. In other words, the IJC algorithm solves for the fixed point and estimates parameters simultaneously, and therefore, the computational burden per outer loop reduces to that of a static model. However, the IJC computational advantage does not directly apply in the finite-horizon problem, where value functions are recursively computed by backward induction rather than by the method of successive approximation. Ishihara and Ching (2013) extended the IJC algorithm for solving finite-horizon DP problems. Similar to the IJC algorithm, the expected value functions are approximated by the weighted average of partially solved value functions in the past outer loop iterations. However, the 76 weighted average, does not only allow us to non-parametrically approximate expected value functions with respect to the parameter space, it also allows us to approximate the high- dimensional integration with respect to the state variables. Ishihara and Ching’s modified algorithm evaluates value functions only at a small subset of state points in each outer loop iteration. In other words, we do not need to apply the Bellman operator at each state point in each outer loop iteration. The modified algorithm consists of (a) the outer-loop, in which we draw parameters from the posterior distribution using the Metropolis-Hastings algorithm, and (b) the inner-loop, in which we first approximate expected value functions using the past history of pseudo-value functions and then update the history by computing pseudo value functions by the modified backward induction procedure. Since we approximate the value function, we will use the terms, pseudo value function(𝜐̂ 𝑡 𝑟 (𝑠 𝑡 ;𝜃 )) , pseudo expected value function(𝐸̂ [𝜐 𝑡 +1 (𝑠 𝑡 +1 ;𝜃 )|𝑠 𝑡 ,𝐴 𝑡 ]) , pseudo alternative specific value function (𝑉̂ 𝑡 𝑟 (𝑠 𝑡 ,𝐴 𝑡 ;𝜃 )), and pseudo likelihood(L ̂ 𝑟 (𝐴 |𝑥 ;𝜃 )). The superscript r indicates that those values are the ones derived at outer loop iteration r. 2.5.4.1.1 The Outer Loop of MCMC Algorithm The outer loop is a Metropolis-Hastings (M-H) algorithm. Suppose that we are at the beginning of iteration r. we first draw a candidate parameter vector from a proposal density. Then we decide whether or not to accept the candidate parameter vector. We will denote the candidate parameter vector in iteration r by θ ∗𝑟 and the accepted parameter vector in iteration r by θ 𝑟 Let q(θ ∗𝑟 |θ 𝑟 −1 ) denote the proposal density for the candidateθ ∗𝑟 given θ 𝑟 −1 (e.g. θ ∗𝑟 ~𝑁 (θ 𝑟 −1 ,𝜎 2 ). First drawθ ∗𝑟 ~q(θ ∗𝑟 |θ 𝑟 −1 ) . Then accept θ ∗𝑟 with probabilityΛ. That is, θ 𝑟 ={ θ ∗𝑟 𝑤𝑖𝑡 ℎ 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 Λ θ 𝑟 −1 𝑤𝑖𝑡 ℎ 𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 1−Λ 77 where Λ is the acceptance probability calculated as follows, Λ=min( 𝜋 (𝜃 ∗𝑟 )𝐿̂ 𝑟 (𝐴 |𝑥 ;𝜃 ∗𝑟 )𝑞 (𝜃 𝑟 −1 |𝜃 ∗𝑟 ) 𝜋 (𝜃 𝑟 −1 )𝐿̂ 𝑟 (𝐴 |𝑥 ;𝜃 𝑟 −1 )𝑞 (𝜃 ∗𝑟 |𝜃 𝑟 −1 ) ,1) (18) where 𝜋 denotes the prior distribution, 𝐿̂ 𝑟 is the pseudo-likelihood function based on pseudo alternative specific value function which is computed in the inner loop. 2.5.4.1.2 The Inner Loop of MCMC Algorithm The inner loop does two jobs. First, it computes the pseudo action/alternative specific value function, which is then used to construct the pseudo likelihood in the outer loop. Second, it updates pseudo value functions by applying the modified backward induction procedure. In these procedures, we need to compute the expected value function first. In order to alleviate the computational burden of fully solving the value function by backward induction at each state point in each time period, we approximate the expected value function by using the information from earlier iterations of the MCMC estimation algorithm. We store the following information: H 𝑟 ≡ {𝜃 ∗𝑘 ,{𝑠 𝑡 𝑘 ,𝜐̂ 𝑡 𝑘 (𝑠 𝑡 𝑘 ;𝜃 ∗𝑘 )} 𝑡 =1 𝑇 +1 } 𝑘 =𝑟 −𝑁 𝑟 −1 . N is the number of past iterations used for computing the pseudo value function. 𝜃 ∗𝑘 is a vector of candidate parameter values in outer loop iteration k, {𝑠 𝑡 𝑘 } 𝑡 is a random sequence of state points generated in outer loop iteration k, and {𝜐̂ 𝑡 𝑘 (𝑠 𝑡 𝑘 ;𝜃 ∗𝑘 )} 𝑡 is a sequence of pseudo value functions in outer loop iteration k evaluated at (𝑠 𝑡 𝑘 ;𝜃 ∗𝑘 ) . The pseudo expected value function at(𝑠 𝑡 ,𝐴 𝑡 ) can be approximated as 𝐸̂ 𝑟 [𝜐 𝑡 +1 (𝑠 𝑡 +1 ;𝜃 ∗𝑟 )|𝑠 𝑡 ,𝐴 𝑡 ] = ∑ 𝜐̂ 𝑡 +1 𝑘 (𝑠 𝑡 +1 𝑘 ;𝜃 ∗𝑘 ) K ℎ𝜃 (𝜃 ∗𝑟 −𝜃 ∗𝑘 )𝑓 𝑡 +1 (𝑠 𝑡 +1 𝑘 |𝑠 𝑡 ,𝐴 𝑡 ;𝜃 𝑠 ) ∑ K ℎ𝜃 (𝜃 ∗𝑟 −𝜃 ∗𝑙 )𝑓 𝑡 +1 (𝑠 𝑡 +1 𝑙 |𝑠 𝑡 ,𝐴 𝑡 ;𝜃 𝑠 ) 𝑟 −1 𝑙 =𝑟 −𝑁 𝑟 −1 𝑘 =𝑟 −𝑁 =∑ 𝜐̂ 𝑡 +1 𝑘 (𝑠 𝑡 +1 𝑘 ;𝜃 ∗𝑘 ) 𝑟 −1 𝑘 =𝑟 −𝑁 𝑊 𝑘 (𝜃 ∗𝑟 ,𝜃 ∗𝑘 ,𝑠 𝑡 +1 𝑘 ,𝑠 𝑡 ,𝐴 𝑡 ) (19) 78 where K ℎ𝜃 is the Gaussian kernel density with bandwidth h. With this pseudo expected value function, we can construct the pseudo alternative specific value function conditional on 𝑠 𝑡 and 𝐴 𝑡 . 𝑉̂ 𝑡 𝑟 (𝑠 𝑡 ,𝐴 𝑡 ;𝜃 ∗𝑟 )=𝑢 𝑡 (𝑠 𝑡 ,𝐴 𝑡 ;𝜃 ∗𝑟 )+𝛽 𝐸̂ 𝑟 [𝜐 𝑡 +1 (𝑠 𝑡 +1 ;𝜃 ∗𝑟 )|𝑠 𝑡 ,𝐴 𝑡 ] (20) By Bellman’s principle of optimality, the pseudo value function can be written as 𝜐̂ 𝑡 𝑟 (𝑠 𝑡 ;𝜃 ∗𝑟 )=max 𝐴 𝑡 𝜖𝐶 𝑡 𝑉̂ 𝑡 𝑟 (𝑠 𝑡 ,𝐴 𝑡 ;𝜃 ∗𝑟 ) (21) And the optimal contribution is chosen to maximize the above bellman equation, 𝐴 𝑡 ∗ (𝑠 𝑡 ;𝜃 ∗𝑟 )=argmax 𝐴 𝑡 𝜖𝐶 𝑡 𝑉̂ 𝑡 𝑟 (𝑠 𝑡 ,𝐴 𝑡 ;𝜃 ∗𝑟 ) =argmax 𝐴 𝑡 𝜖𝐶 𝑡 {𝑢 𝑡 (𝑠 𝑡 ,𝐴 𝑡 ;𝜃 ∗𝑟 )+𝛽 𝐸̂ 𝑟 [𝜐 𝑡 +1 (𝑠 𝑡 +1 ;𝜃 ∗𝑟 )|𝑠 𝑡 ,𝐴 𝑡 ]} (22) 2.5.4.1.3 The M-H steps: 1. Suppose that we are at iteration r. we start with H 𝑟 ≡{𝜃 ∗𝑘 ,{𝑠 𝑡 𝑘 ,𝜐̂ 𝑡 𝑘 (𝑠 𝑡 𝑘 ;𝜃 ∗𝑘 )} 𝑡 =1 𝑇 +1 } 𝑘 =𝑟 −𝑁 𝑟 −1 , where N is the number of past iterations used for the expected future value approximation. 2. Draw parameter vector 𝜃 from the posterior distribution 𝑓 𝑡 (𝜃 |𝐴 𝑡 )∝𝑓 𝑡 (𝐴 𝑡 |𝜃 )𝜋 (𝜃 ) Since there’s no easy way to draw from this posterior, we use the M-H algorithm. (a) Draw 𝜃 𝑟 from the proposal distribution 𝑞 (𝜃 𝑟 −1 ,𝜃 ∗𝑟 ) (e.g., 𝜃 ∗𝑟 ~𝑁 (𝜃 𝑟 −1 ,𝜎 2 ) ) where 𝜃 ∗𝑟 is the candidate parameter. (b) Compute the pseudo-likelihood for 𝐴 𝑡 at 𝜃 ∗𝑟 . Because there’s no closed form solution to the optimal action policy, a likelihood function based on observed 𝐴 𝑡 becomes infeasible. Instead, we implement a numerical approximation method to establish a simulated likelihood function for the estimation. For each 𝐴 𝑡 observed in the data, and its corresponding state point 𝑆 𝑡 , we use the following steps to simulate its density: (i) Draw nr=250 random shocks {ε 𝑡 } 𝑘 =1 𝑛𝑟 from 𝜀 𝑖𝑡 𝐵 ~𝑖 .𝑖 .𝑑 ,𝑁 (0,𝜎 𝐵 2 ) 79 (ii) For each random draw of shocks and the observed state point, calculate the optimal contribution action by solving the following equation 𝐴 𝑡 ∗ =argmax 𝐴 𝑡 𝜖𝐶 𝑡 𝑣̂ 𝑡 𝑟 (𝑠 𝑡 ,𝐴 𝑡 ;𝜃 ∗𝑟 )=argmax 𝐴 𝑡 𝜖𝐶 𝑡 (𝑢 𝑡 (𝑠 𝑡 ,𝐴 𝑡 ;𝜃 ∗𝑟 )+𝛽 𝐸̂ 𝑟 [𝜐 𝑡 +1 (𝑠 𝑡 +1 ;𝜃 ∗𝑟 )|𝑠 𝑡 ,𝐴 𝑡 ]) (iii) Using the calculated nr=250 optimal 𝐴 𝑡 ∗ (𝑠 𝑡 ) , simulate 𝑓̃ 𝑡 (𝐴 𝑡 |𝜃 ) , the density of observed 𝐴 𝑡 using a Gaussian kernel density estimator (Following Yao, Mela, Chiang and Chen (2012), Xiao Liu, et al. (2015)). We repeat the same step and obtain the pseudo-likelihood 𝑓̃ 𝑡 (𝐴 𝑡 |𝜃 𝑟 −1 ) conditional on 𝜃 𝑟 −1 . Then we determine whether or not to accept 𝜃 ∗𝑟 . The acceptance probability Λ is given by Λ=min( 𝜋 (𝜃 ∗𝑟 )𝑓̃ 𝑡 (𝐴 𝑡 |𝜃 ∗𝑟 )𝑞 (𝜃 𝑟 −1 ,𝜃 ∗𝑟 ) 𝜋 (𝜃 𝑟 −1 )𝑓̃ 𝑡 (𝐴 𝑡 |𝜃 𝑟 −1 )𝑞 (𝜃 ∗𝑟 ,𝜃 𝑟 −1 ) ,1) where 𝜋 () denotes the prior distribution. (c) repeat step (a) and (b) for all i (if we have random coefficient). To obtain the pseudo value function 𝑣̂ 𝑡 𝑟 (𝑠 𝑡 ;𝜃 ∗𝑟 ) , we need 𝐸̂ 𝑟 [𝜐 𝑡 +1 (𝑠 𝑡 +1 ;𝜃 ∗𝑟 )|𝑠 𝑡 ,𝐴 𝑡 ], which is obtained by a weighted average of {𝜐̂ 𝑡 𝑘 (𝑠 𝑡 𝑘 ;𝜃 ∗𝑘 )} 𝑘 =𝑟 −𝑁 𝑟 −1 , treating 𝜃 ∗ as one of the parameters when computing the weights. In the context, we have five observed state variables 𝑠 𝑡 = {𝐼𝐶𝐴 𝑡 ,𝑇𝐿 𝑡 ,𝐹𝑈 𝑡 ,𝐷𝑂𝑅 𝑡 ,𝑂𝐴 𝑡 }={𝑠 𝑡 𝐼𝑛𝑑 _𝐶𝑢𝑚𝑎𝑚𝑡 ,𝑠 𝑡 𝑇𝑖𝑚𝑒𝐿𝑒𝑓𝑡 ,𝑠 𝑡 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 ,𝑠 𝑡 𝐷𝑜𝑛𝑜𝑟 ,𝑠 𝑡 𝑂𝑟𝑔 _𝐴𝑚𝑡 } , because 𝐼𝐶𝐴 𝑡 is continuous and evolves deterministically, 𝐹𝑈 𝑡 ,𝐷𝑂𝑅 𝑡 ,𝑂𝐴 𝑡 are continuous and evolve stochastically, and 𝑇𝐿 𝑡 is discrete, we have 𝐸̂ 𝑟 [𝜐 𝑡 +1 (𝐼𝐶𝐴 𝑡 +1 ,𝑇𝐿 𝑡 +1 ,𝐹𝑈 𝑡 +1 ,𝐷𝑂𝑅 𝑡 +1 ,𝑂𝐴 𝑡 +1 ,ε 𝑡 +1 ;𝜃 ∗𝑟 )|(𝐼𝐶𝐴 𝑡 ,𝑇𝐿 𝑡 ,𝐹𝑈 𝑡 ,𝐷𝑂𝑅 𝑡 ,𝑂𝐴 𝑡 ),𝐴 𝑡 ] =∑ {𝜐̂ 𝑡 +1 𝑘 [(𝐼𝐶𝐴 𝑡 +1 ,𝑇𝐿 𝑡 +1 ,𝐹𝑈 𝑡 +1 𝑘 ,𝐷𝑂𝑅 𝑡 +1 𝑘 ,𝑂 𝐴 𝑡 +1 𝑘 ,ε 𝑡 +1 𝑘 ;𝜃 ∗𝑘 )] 𝑟 =1 𝑘 =𝑟 −𝑁 K ℎ𝜃 (𝜃 ∗𝑟 −𝜃 ∗𝑘 )𝑓 (𝐹𝑈 𝑡 𝑘 |𝜇 𝑡 𝐹𝑈 ,𝜎 𝑤 2 )𝑓 (𝐷𝑂𝑅 𝑡 𝑘 |𝜆 𝑡 )𝑓 (𝑂𝐴 𝑡 𝑘 |𝜇 𝑡 𝑂𝐴 ,𝜎 𝑡 2 ) ∑ K ℎ𝜃 (𝜃 ∗𝑟 −𝜃 ∗𝑙 ) 𝑟 −1 𝑙 =𝑟 −𝑁 𝑓 (𝐹𝑈 𝑡 𝑙 |𝜇 𝑡 𝐹𝑈 ,𝜎 𝑤 2 )𝑓 (𝐷𝑂𝑅 𝑡 𝑙 |𝜆 𝑡 )𝑓 (𝑂𝐴 𝑡 𝑙 |𝜇 𝑡 𝑂𝐴 ,𝜎 𝑡 2 ) } (23) Detailed steps are as follows: 80 (a) Make one draw of the unobserved state variable from 𝜀 𝑖𝑡 𝑟 ~N(0, 𝜎 𝜀 2 ) (b) Compute the pseudo expected future value at 𝜃 ∗𝑟 𝐸̂ 𝑟 [𝜐 𝑡 +1 (𝐼𝐶𝐴 𝑡 +1 ,𝑇𝐿 𝑡 +1 ,𝐹𝑈 𝑡 +1 ,𝐷𝑂𝑅 𝑡 +1 ,𝑂𝐴 𝑡 +1 ,ε 𝑡 +1 ;𝜃 ∗𝑟 )|𝑠 𝑡 ,𝐴 𝑡 ] = K ℎ𝜃 (𝜃 ∗𝑟 −𝜃 ∗𝑘 )𝑓 (𝐹𝑈 𝑡 +1 𝑘 |𝜇 𝑡 𝐹𝑈 ,𝜎 𝑤 2 )𝑓 (𝐷 𝑂𝑅 𝑡 +1 𝑘 |𝜆 𝑡 )𝑓 (𝑂𝐴 𝑡 +1 𝑘 |𝜇 𝑡 𝑂𝐴 ,𝜎 𝑡 2 ) ∑ K ℎ𝜃 (𝜃 ∗𝑟 −𝜃 ∗𝑙 ) 𝑟 −1 𝑙 =𝑟 −𝑁 𝑓 (𝐹𝑈 𝑡 𝑙 |𝜇 𝑡 𝐹𝑈 ,𝜎 𝑤 2 )𝑓 (𝐷𝑂𝑅 𝑡 𝑙 |𝜆 𝑡 )𝑓 (𝑂𝐴 𝑡 𝑙 |𝜇 𝑡 𝑂𝐴 ,𝜎 𝑡 2 ) where 𝑠 𝑡 +1 𝑘 is a set of draws from a uniform distribution covering the support of S. (c) Compute the pseudo value 𝜐̂ 𝑡 𝑟 (𝑠 𝑡 ;𝜃 ∗𝑟 ) using the pseudo expected future values computed in (b) and the optimal choice 𝐴 𝑡 ∗ . 𝜐̂ 𝑡 𝑟 (𝑠 𝑡 ;𝜃 ∗𝑟 )=𝑢 𝑡 (𝑠 𝑡 ,𝐴 𝑡 ∗ ;𝜃 ∗𝑟 )+𝛽 𝐸̂ 𝑟 [𝜐 𝑡 +1 (𝑠 𝑡 +1 ;𝜃 ∗𝑟 )|𝑠 𝑡 ,𝐴 𝑡 ∗ ] Where the optimal action 𝐴 𝑡 ∗ satisfies 𝐴 𝑡 ∗ =argmax 𝐴 𝑡 𝜖𝐶 𝑡 𝑣̂ 𝑡 𝑟 (𝑠 𝑡 ,𝐴 𝑡 ;𝜃 ∗𝑟 ) =argmax 𝐴 𝑡 𝜖𝐶 𝑡 (𝑢 𝑡 (𝑠 𝑡 ,𝐴 𝑡 ∗ ;𝜃 ∗𝑟 )+𝛽 𝐸̂ 𝑟 [𝜐 𝑡 +1 (𝑠 𝑡 +1 ;𝜃 ∗𝑟 )|𝑠 𝑡 ,𝐴 𝑡 ∗ ]) (d) Repeat (a-c) for all i and t. To update the history, we first simulate on sequence of state points {𝑠 𝑡 𝑟 } 𝑡 =1 𝑇 +1 from a uniform distribution that covers 𝑠 𝑡 . We then apply the modified backward induction procedure, starting from t=T+1. The key here is that we compute pseudo value functions at only one state point (𝑠 𝑡 𝑟 ) per time period (Ishihara and Ching 2013, 2016). At t=T+1, given a random draw state point 𝑠 𝑇 +1 𝑟 , we compute and store 𝜐̂ 𝑇 +1 𝑟 (𝑠 𝑇 +1 𝑟 ;𝜃 ∗𝑟 )=𝐸 [𝑢 𝑇 +1 (𝑠 𝑇 +1 𝑟 ;𝜃 ∗𝑟 )] Move backward to t=T, T-1,…1, given a state point 𝑠 𝑡 𝑟 , we compute and store 𝜐̂ 𝑡 𝑟 (𝑠 𝑡 𝑟 ;𝜃 ∗𝑟 )=max 𝐴 𝑡 𝜖𝐶 𝑡 𝑉̂ 𝑡 𝑟 (𝑠 𝑡 𝑟 ,𝐴 𝑡 ;𝜃 ∗𝑟 )=max { 𝐴 𝑡 𝜖𝐶 𝑡 𝑢 𝑡 (𝑠 𝑡 ,𝐴 𝑡 ;𝜃 ∗𝑟 )+𝛽 𝐸̂ 𝑟 [𝜐 𝑡 +1 (𝑠 𝑡 +1 ;𝜃 ∗𝑟 )|𝑠 𝑡 ,𝐴 𝑡 ]} 81 where the computation of the pseudo action specific value functions requires the computation of the pseudo expected value function. This is done using equation (23) evaluated at (𝑠 𝑡 𝑟 ,𝐴 𝑡 ) . We continue this process until t=1. The entire modified backward induction procedure generates a sequence of pseudo value functions evaluated at {𝑠 𝑡 𝑟 ,𝜐̂ 𝑡 𝑟 (𝑠 𝑡 𝑟 ;𝜃 ∗𝑟 )} 𝑡 =1 𝑇 +1 . We then store {𝜃 ∗𝑟 ,{𝑠 𝑡 𝑟 ,𝜐̂ 𝑡 𝑟 (𝑠 𝑡 𝑟 ;𝜃 ∗𝑟 )} 𝑡 =1 𝑇 +1 } and update H 𝑟 to H 𝑟 +1 . 2.6 Estimation Result In Table 14, our estimation result indicates that, on average, funding status on the number of previous donors may decrease the marginal contribution period utility when keeping everything else constant. This is possible because when a project has received funding from a relatively large number of donors, individual may actually think that it’s likely that she can obtain the final reward without paying the cost. But individuals are heterogeneous in reacting to this funding status. Because some donors may think their contribution can potentially benefit more people when there are more donors attracted to the project. With regard to the funding status on organization’s funding, it enhances individual’s marginal contribution period utility, suggesting that disclosing organization’s contribution may encourage individual’s contribution keep anything else constant. But in order to fully evaluate the impact of organization’s contribution on individual’s contribution, we need to conduct counterfactual simulations. Because organization’s contribution not only can affect individual’s marginal contribution period utility, but also influences individual’s belief on how much funding the project can receive from all the donors and also from organizational donors, how many donors the project can attract, how likely the project can successfully accomplish the funding goal in the future. And all of these are captured in the evolvement of the state variables. 82 Posterior Mean (Std. Dev) Intercept 𝝋 𝟏 ̅̅̅̅ -1.130 (1.057) Number of Donors 𝝋 𝟐 ̅̅̅̅ -3.820 (0.635) Organization’s funding 𝝋 𝟑 ̅̅̅̅ 3.910 (0.631) 𝝈 𝝋 𝟏 𝟐 0.770 (0.388) 𝝈 𝝋 𝟐 𝟐 0.320 (0.051) 𝝈 𝝋 𝟑 𝟐 0.310 (0.053) Final reward benefit (𝝍 𝟑 ∗ ) -5.470 (0.452) Table 14. Estimation Result 2.7 Conclusion This study focuses on understanding crowdfunders contribution behavior on a prosocial crowdfunding platform that receives funding from two types of donors (i.e., individual and organizational donors). Understanding the factors that influence crowdfunders contribution behavior over time is a key to the success of funding projects and the crowdfunding platform. A structural model that characterizes the dynamic response of crowdfunders contribution behavior is therefore critical. Using data from a journalistic crowdfunding platform that raises money for public goods, we develop a dynamic structural framework to model individual’s contribution decision (i.e. how much to contribute), and we characterize the dynamic behaviors of individual crowdfunders in this empirical setting by accounting for their motivations to manipulate the timing and amount of their contributions conditional on others behaviors. In particular, we model individuals to be forward- 83 looking, in order to trade off the cost of making a contribution, with the risk/cost of not being able to receive the final report from a funding project, as well as the good feeling (e.g., warm glow effect) generated from the act of helping a project that can be beneficial to others. Furthermore, our empirical setting allows us examine how organization’s contribution would affect individual’s contribution in a transparent dynamic funding provision process. In literature, this is a widely researched and debated question. However, due to the lack of proper data and the involved computational challenges, to the best of our knowledge, there’s no empirical study that has explicitly taken care of the individual’s dynamic concerns over the funding provision process. In sum, our dynamic structural model can handle the funding deadlines and project reward schemes based on the final funding result (i.e., whether a project achieves the funding goal or not), all of which are ubiquitous in crowdfunding setting. In addition, our model can handle the dynamic crowding in/out issue induced by organizations’ contributions, which is a unique feature in our crowdfunding setting. The estimation of our dynamic structural model involves several computational challenges, given that we have both continuous and constrained states and action space. To overcome the computational and estimation-related challenges, we use a conjunction of different approaches. First, we adopt a Bayesian estimation approach, using a modified Bayesian IJC algorithm that helps deal with finite-horizon DP models. Second, since we do not have a close form solution for the likelihood, we use the kernel density estimator to simulate the likelihood. Third, to deal with the constrained continuous decision, we use golden section search method to find the optimal action when solving the value function. Fourth, to overcome the computational challenge, we use parallel computing (OpenMP and MPI) to speed up our estimation process. 84 Our modeling framework can be potentially modified to be applied to other general crowdfunding platforms (e.g., platforms that are not raising money for public goods). Because our model setup can handle individual’s forward looking behavior, the dynamic issues introduced by the funding reward scheme and the funding deadline. In addition, over the years, there’s an increasing number of crowdfunding platforms that seek to cooperate with organizations using the matching or other types of mechanism, for example DonorsChoose.org allows firms (like Google, Starbucks etc.) to match individual’s donation or match 50% of the funding goal conditional on the funding performance. 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Failure to Launch in Two-Sided Markets: A Study of the US Video Game Market, Working paper, SUNY at Stony Brook. 90 Appendices Appendices for Chapter One Appendix A. Model Derivation and Likelihood Construction We employ a flexible utility function specification that can produce either corner or interior solutions. When there’s no reward saving for future, i.e., 𝑆 𝑖𝑡 =𝐼 𝑖𝑡 (𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 >0) =0, our utility and FOC condition are specified as, 𝑢 𝑖𝑡 = 𝜑 𝑖𝑡 ln(𝑚 𝑖𝑡 +1)+ln(exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1)−𝐶 𝑖𝑡 ∙𝑛 𝑖𝑡 𝜕𝑢 𝑖𝑡 𝜕𝑚 𝑖𝑡 = 𝜑 𝑖𝑡 𝑚 𝑖𝑡 +1 − 1 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1 𝜕𝑢 𝑖𝑡 𝜕𝑛 𝑖𝑡 = 𝑅̅ 𝑖𝑡 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1 −𝐶 𝑖𝑡 When there’s reward saving for future, that is, 𝑆 𝑖𝑡 =1, we have 𝑢 𝑖𝑡 = 𝜑 𝑖𝑡 ln(𝑚 𝑖𝑡 +1)+ln(exp (𝑀̅ 𝑖 )+1)−𝐶 𝑖𝑡 ∙𝑛 𝑖𝑡 +𝛾 𝑖 ∙(𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 ) 𝜕𝑢 𝑖𝑡 𝜕𝑚 𝑖𝑡 = 𝜑 𝑖𝑡 𝑚 𝑖𝑡 +1 −𝛾 𝑖 𝜕𝑢 𝑖𝑡 𝜕𝑛 𝑖𝑡 =−𝐶 𝑖𝑡 +𝛾 𝑖 𝑅̅ 𝑖𝑡 Depending on the value of 𝑚 𝑖𝑡 and 𝑛 𝑖𝑡 , we have eight permutations: Case (1): 𝑚 𝑖𝑡 =0, 𝑛 𝑖𝑡 =0,𝑎𝑛𝑑 𝑆 𝑖𝑡 =0 Case (2): 𝑚 𝑖𝑡 >0, 𝑛 𝑖𝑡 =0,𝑎𝑛𝑑 𝑆 𝑖𝑡 =0 Case (3): 𝑚 𝑖𝑡 >0, 𝑛 𝑖𝑡 >0,𝑎𝑛𝑑 𝑆 𝑖𝑡 =0 Case (4): 𝑚 𝑖𝑡 =0, 𝑛 𝑖𝑡 =0,𝑎𝑛𝑑 𝑆 𝑖𝑡 =1 Case (5): 𝑚 𝑖𝑡 >0, 𝑛 𝑖𝑡 =0,𝑎𝑛𝑑 𝑆 𝑖𝑡 =1 Case (6): 𝑚 𝑖𝑡 =0, 𝑛 𝑖𝑡 >0,𝑎𝑛𝑑 𝑆 𝑖𝑡 =1 Case (7): 𝑚 𝑖𝑡 >0, 𝑛 𝑖𝑡 >0,𝑎𝑛𝑑 𝑆 𝑖𝑡 =1 91 Case (8): 𝑚 𝑖𝑡 =0, 𝑛 𝑖𝑡 >0,𝑎𝑛𝑑 𝑆 𝑖𝑡 =0 where case (8) is impossible because when 𝑚 𝑖𝑡 =0 𝑎𝑛𝑑 𝑛 𝑖𝑡 >0, the reward saving indicator, 𝑆 𝑖𝑡 =𝐼 (𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 >0), must be one. Therefore, the probability of observing 𝑚 𝑖𝑡 𝑎𝑛𝑑 𝑛 𝑖𝑡 is: 𝑓 (𝑚 𝑖𝑡 ,𝑛 𝑖𝑡 )=𝑓 1 ̂ (𝑚 𝑖𝑡 =0,𝑛 𝑖𝑡 =0,𝑆 𝑖𝑡 =0) 𝐼 (𝑐𝑎𝑠𝑒 =1) ∗ 𝑓 ̂ 2 (𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 =0,𝑆 𝑖𝑡 =0) 𝐼 (𝑐𝑎𝑠𝑒 =2) ∗𝑓 ̂ 3 (𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 >0,𝑆 𝑖𝑡 =0) 𝐼 (𝑐𝑎𝑠𝑒 =3) ∗𝑓 ̂ 4 (𝑚 𝑖𝑡 =0,𝑛 𝑖𝑡 =0,𝑆 𝑖𝑡 =1) 𝐼 (𝑐𝑎𝑠𝑒 =4) ∗𝑓 ̂ 5 (𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 =0,𝑆 𝑖𝑡 =1) 𝐼 (𝑐𝑎𝑠𝑒 =5) ∗𝑓 ̂ 6 (𝑚 𝑖𝑡 =0,𝑛 𝑖𝑡 >0,𝑆 𝑖𝑡 =1) 𝐼 (𝑐𝑎𝑠𝑒 =6) ∗𝑓 ̂ 7 (𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 >0,𝑆 𝑖𝑡 =1) 𝐼 (𝑐𝑎𝑠𝑒 =7) We next derive the Kuhn-Tucker first-order condition for each case step by step. Case (1): 𝒎 𝒊𝒕 =𝟎 , 𝒏 𝒊𝒕 =𝟎 ,𝒂𝒏𝒅 𝑺 𝒊𝒕 =𝟎 𝜕𝑢 𝑖𝑡 𝜕𝑚 𝑖𝑡 = 𝜑 𝑖𝑡 𝑚 𝑖𝑡 +1 − 1 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1 <0 (A1a) 𝜕𝑢 𝑖𝑡 𝜕𝑛 𝑖𝑡 = 𝑅̅ 𝑖𝑡 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1 −𝐶 𝑖𝑡 <0 (A1b) For (A1a) because 𝑚 𝑖𝑡 =0, we have 𝜕𝑢 𝑖𝑡 𝜕𝑚 𝑖 𝑡 =𝜑 𝑖𝑡 − 1 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 +1 <0 Plug in the specification for 𝜑 𝑖𝑡 , we have 𝑒𝑥𝑝 (𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 +𝜀 𝑖𝑡 )< 1 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 +1 Therefore, 𝜀 𝑖𝑡 <𝑙𝑜𝑔 ( 1 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 +1 )−(𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 ) For (A1b) because 𝑛 𝑖𝑡 =0, we have 𝜕𝑢 𝑖𝑡 𝜕𝑛 𝑖𝑡 = 𝑅̅ 𝑖𝑡 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 −𝑚 𝑖𝑡 +1 −𝐶 𝑖𝑡 <0 where 𝐶 𝑖𝑡 is specified as 𝐶 𝑖𝑡 =(𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′𝜏 +𝜐 𝑖𝑡 ) , therefore, 92 𝜐 𝑖𝑡 > 𝑅̅ 𝑖𝑡 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 −𝑚 𝑖𝑡 +1 −(𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′𝜏 ) To overcome the endogeneity issue induced by the simultaneous decision making process, we use a two-stage estimation process (Nelson and Olsen 1978). In the first stage, reduced form estimates are used to construct instruments 𝑚̂ 𝑖𝑡 and 𝑛̂ 𝑖𝑡 . Then in the second stage, we will substitute the predicted 𝑚̂ 𝑖𝑡 for 𝑚 𝑖𝑡 in the optimality condition for incentive participation, and predicted 𝑛̂ 𝑖𝑡 for 𝑛 𝑖𝑡 in the optimality condition for contribution. Therefore, the probability of observing zero contribution, zero incentive participation and no reward saving can be written as, 𝑓 ̂ 1 (𝑚 𝑖𝑡 =0,𝑛 𝑖𝑡 =0,𝑆 𝑖𝑡 =0) =Φ 𝜀 (𝑙𝑜𝑔 ( 1 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛̂ 𝑖𝑡 +1 )−(𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 )) ∗[1−Φ 𝜐 ( 𝑅̅ 𝑖𝑡 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 −𝑚̂ 𝑖𝑡 +1 −(𝑥 𝑖 𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′𝜏 ))] (𝐴 1𝑐 ) Case (2): 𝒎 𝒊𝒕 >𝟎 , 𝒏 𝒊𝒕 =𝟎 ,𝒂𝒏𝒅 𝑺 𝒊𝒕 =𝟎 𝜕𝑢 𝑖𝑡 𝜕𝑚 𝑖𝑡 = 𝜑 𝑖𝑡 𝑚 𝑖𝑡 +1 − 1 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1 =0 (A2a) 𝜕𝑢 𝑖𝑡 𝜕𝑛 𝑖𝑡 = 𝑅̅ 𝑖𝑡 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1 −𝐶 𝑖𝑡 <0 (A2b) For (A2a), we have 𝑒𝑥𝑝 (𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 +𝜀 𝑖𝑡 )= 𝑚 𝑖𝑡 +1 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1 Hence 𝜀 𝑖𝑡 =𝑙𝑜𝑔 ( 𝑚 𝑖𝑡 +1 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1 )−(𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 ) = 𝑙𝑜𝑔 (𝑚 𝑖𝑡 +1)−𝑙𝑜𝑔 (exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1)−(𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 ) Jacobian: 𝐽 𝑖𝑡 = 𝜕𝜀 𝑖𝑡 𝜕𝑚 𝑖𝑡 =| exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 +2 (𝑚 𝑖𝑡 +1)(exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1) | For (A2b) because 𝑛 𝑖𝑡 =0, we have 𝜕𝑢 𝑖𝑡 𝜕𝑛 𝑖𝑡 = 𝑅̅ 𝑖𝑡 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 −𝑚 𝑖𝑡 +1 −𝐶 𝑖𝑡 <0 93 where 𝐶 𝑖𝑡 is specified as 𝐶 𝑖𝑡 =(𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖 𝑛 ′𝜏 +𝜐 𝑖𝑡 ) , therefore, 𝜐 𝑖𝑡 > 𝑅̅ 𝑖𝑡 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 −𝑚 𝑖𝑡 +1 −(𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′𝜏 ) Thus using the two-stage estimation process, the probability of observing a positive contribution, zero incentive participation and no reward saving should be 𝑓 ̂ 2 (𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 =0,𝑆 𝑖𝑡 =0) =𝜙 𝜀 (𝑙𝑜𝑔 ( 𝑚 𝑖𝑡 +1 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛̂ 𝑖𝑡 −𝑚 𝑖𝑡 +1 )−(𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 )) ∗| exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛̂ 𝑖𝑡 +2 (𝑚 𝑖𝑡 +1)(exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛̂ 𝑖𝑡 −𝑚 𝑖𝑡 +1) | ∗[1−Φ 𝜐 ( 𝑅̅ 𝑖𝑡 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 −𝑚̂ 𝑖𝑡 +1 −(𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′𝜏 ))] (𝐴 2𝑐 ) Case (3): 𝒎 𝒊𝒕 >𝟎 , 𝒏 𝒊𝒕 >𝟎 ,𝒂𝒏𝒅 𝑺 𝒊𝒕 =𝟎 𝜕𝑢 𝑖𝑡 𝜕𝑚 𝑖𝑡 = 𝜑 𝑖𝑡 𝑚 𝑖𝑡 +1 − 1 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1 =0 (A3a) 𝜕𝑢 𝑖𝑡 𝜕𝑛 𝑖𝑡 = 𝑅̅ 𝑖𝑡 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1 −𝐶 𝑖𝑡 =0 (A3b) For (A3a), we have 𝑒𝑥𝑝 (𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 +𝜀 𝑖𝑡 )= 𝑚 𝑖𝑡 +1 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1 Hence 𝜀 𝑖𝑡 =𝑙𝑜𝑔 ( 𝑚 𝑖𝑡 +1 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1 )−(𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 ) = 𝑙𝑜𝑔 (𝑚 𝑖𝑡 +1)−𝑙𝑜𝑔 (exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1)−(𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 ) For (A3b), we have 𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′𝜏 +𝜐 𝑖𝑡 = 𝑅̅ 𝑖𝑡 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1 Hence, 𝜐 𝑖𝑡 = 𝑅̅ 𝑖𝑡 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1 −(𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′𝜏 ) 94 Jacobian Matrix: it it it it it it it it it mn J vv mn With 𝐽 1 = 𝜕𝜀 𝑖𝑡 𝜕𝑚 𝑖𝑡 = exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 +2 (𝑚 𝑖𝑡 +1)(exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1) 𝐽 2 = 𝜕𝜀 𝑖𝑡 𝜕𝑛 𝑖𝑡 = −𝑅̅ 𝑖𝑡 (exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1) 𝐽 3 = 𝜕𝜐 𝑖𝑡 𝜕𝑚 𝑖𝑡 = 𝑅̅ 𝑖𝑡 (exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1) 2 𝐽 4 = 𝜕𝜐 𝑖𝑡 𝜕𝑛 𝑖𝑡 = −𝑅̅ 𝑖𝑡 2 (exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1) 2 Jacobian is the absolute value of the determination of 𝐽 𝑖𝑡 𝐽 𝑖𝑡 =|𝐽 1 𝐽 4 −𝐽 2 𝐽 3 | =| −𝑅̅ 𝑖𝑡 2 ∙(exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 +2) (𝑚 𝑖𝑡 +1)(exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1) 3 + 𝑅̅ 𝑖𝑡 2 (exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1) 3 | =| −𝑅̅ 𝑖𝑡 2 (exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1) 3 ( exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 +2 𝑚 𝑖𝑡 +1 −1)| =| −𝑅̅ 𝑖𝑡 2 (exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1) 3 ( exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1 𝑚 𝑖𝑡 +1 )| =| −𝑅̅ 𝑖𝑡 2 (exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚 𝑖𝑡 +1) 2 (𝑚 𝑖𝑡 +1) | Thus, using the two-stage estimation process, the probability of observing a positive contribution, positive incentive participation and no reward saving should be 𝑓 ̂ 3 (𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 >0,𝑆 𝑖𝑡 =0) =𝜙 𝜀 (𝑙𝑜𝑔 ( 𝑚 𝑖𝑡 +1 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛̂ 𝑖𝑡 −𝑚 𝑖𝑡 +1 )−(𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 )) ∗| −𝑅̅ 𝑖𝑡 2 (exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛̂ 𝑖𝑡 −𝑚 𝑖𝑡 +1) 2 (𝑚 𝑖𝑡 +1) | ∗𝜙 𝜐 ( 𝑅̅ 𝑖𝑡 exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚̂ 𝑖𝑡 +1 −(𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′𝜏 )) 95 ∗| −𝑅̅ 𝑖𝑡 2 (exp (𝑀̅ 𝑖 )+𝐿 𝑖𝑡 +𝑅̅ 𝑖𝑡 ∙𝑛 𝑖𝑡 −𝑚̂ 𝑖𝑡 +1) 2 (𝑚̂ 𝑖𝑡 +1) | (𝐴 3𝑐 ) Case (4): 𝒎 𝒊𝒕 =𝟎 , 𝒏 𝒊𝒕 =𝟎 ,𝒂𝒏𝒅 𝑺 𝒊𝒕 =𝟏 𝜕𝑢 𝑖𝑡 𝜕𝑚 𝑖𝑡 = 𝜑 𝑖𝑡 𝑚 𝑖𝑡 +1 −𝛾 𝑖 <0 (A4a) 𝜕𝑢 𝑖𝑡 𝜕𝑛 𝑖𝑡 =−𝐶 𝑖𝑡 +𝛾 𝑖 𝑅̅ 𝑖𝑡 <0 (A4b) For (A4a) because 𝑚 𝑖𝑡 =0, we have 𝜕𝑢 𝑖𝑡 𝜕𝑚 𝑖𝑡 =𝜑 𝑖𝑡 −𝛾 𝑖 <0 Plug in the specification for 𝜑 𝑖𝑡 , we have 𝑒𝑥𝑝 (𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 +𝜀 𝑖𝑡 )<𝛾 𝑖 Therefore, 𝜀 𝑖𝑡 <log(𝛾 𝑖 )−(𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 ) For (A4b) we have 𝜕𝑢 𝑖𝑡 𝜕𝑛 𝑖𝑡 =−𝐶 𝑖𝑡 +𝛾 𝑖 𝑅̅ 𝑖𝑡 <0 where 𝐶 𝑖𝑡 is specified as 𝐶 𝑖𝑡 =(𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′𝜏 +𝜐 𝑖𝑡 )>𝛾 𝑖 𝑅̅ 𝑖𝑡 ≥0, therefore, 𝜐 𝑖𝑡 >𝛾 𝑖 𝑅̅ 𝑖𝑡 −(𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′𝜏 ) Likelihood for (m it =0, n it =0,and S it =1) can be written as, 𝑓 4 (𝑚 𝑖𝑡 =0,𝑛 𝑖𝑡 =0,𝑆 𝑖𝑡 =1) =Φ 𝜀 (log(𝛾 𝑖 )−(𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 ))∗[1−Φ 𝜐 (𝛾 𝑖 𝑅̅ 𝑖𝑡 −(𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′𝜏 ))] (𝐴 4𝑐 ) Case (5): 𝒎 𝒊𝒕 >𝟎 , 𝒏 𝒊𝒕 =𝟎 ,𝒂𝒏𝒅 𝑺 𝒊𝒕 =𝟏 𝜕𝑢 𝑖𝑡 𝜕𝑚 𝑖𝑡 = 𝜑 𝑖𝑡 𝑚 𝑖𝑡 +1 −𝛾 𝑖 =0 (A5a) 𝜕𝑢 𝑖𝑡 𝜕𝑛 𝑖𝑡 =−𝐶 𝑖𝑡 +𝛾 𝑖 𝑅̅ 𝑖𝑡 <0 (A5b) For (A5a), we have 𝜑 𝑖𝑡 𝑚 𝑖𝑡 +1 =𝛾 𝑖 . Plugging in the definition for 𝜑 𝑖𝑡 , we have 𝑒𝑥𝑝 (𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 +𝜀 𝑖𝑡 ) 𝑚 𝑖𝑡 +1 =𝛾 𝑖 96 Therefore, 𝜀 𝑖𝑡 =log((𝑚 𝑖𝑡 +1)∗𝛾 𝑖 )−(𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 ) =log(𝑚 𝑖𝑡 +1)+log (𝛾 𝑖 )−(𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 ) Jacobian: 𝐽 𝑖𝑡 𝜀 =| 𝜕 𝜀 𝑖𝑡 𝜕𝑚 𝑖𝑡 |=| 1 𝑚 𝑖𝑡 +1 | For (A5b) we have 𝜕𝑢 𝑖𝑡 𝜕𝑛 𝑖𝑡 =−𝐶 𝑖𝑡 +𝛾 𝑖 𝑅̅ 𝑖𝑡 <0 where 𝐶 𝑖𝑡 is specified as 𝐶 𝑖𝑡 =(𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′𝜏 +𝜐 𝑖𝑡 )>𝛾 𝑖 𝑅̅ 𝑖𝑡 ≥0, therefore, 𝜐 𝑖𝑡 >𝛾 𝑖 𝑅̅ 𝑖𝑡 −(𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′𝜏 ) Likelihood for (m it >0, n it =0,and S it =1) can be written as, 𝑓 5 (𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 =0,𝑆 𝑖𝑡 =1) =𝜙 𝜀 (log((𝑚 𝑖𝑡 +1)∗𝛾 𝑖 )−(𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 ))∗| 1 𝑚 𝑖𝑡 +1 | ∗[1−Φ 𝜐 (𝛾 𝑖 𝑅̅ 𝑖𝑡 −(𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′𝜏 ))] (𝐴 5𝑐 ) Case (6): 𝒎 𝒊𝒕 =𝟎 , 𝒏 𝒊𝒕 >𝟎 ,𝒂𝒏𝒅 𝑺 𝒊𝒕 =𝟏 𝜕𝑢 𝑖𝑡 𝜕𝑚 𝑖𝑡 = 𝜑 𝑖𝑡 𝑚 𝑖𝑡 +1 −𝛾 𝑖 <0 (A6a) 𝜕𝑢 𝑖𝑡 𝜕𝑛 𝑖𝑡 =−𝐶 𝑖𝑡 +𝛾 𝑖 𝑅̅ 𝑖𝑡 =0 (A6b) For (A6a) because 𝑚 𝑖𝑡 =0, we have 𝜕𝑢 𝑖𝑡 𝜕𝑚 𝑖𝑡 =𝜑 𝑖𝑡 −𝛾 𝑖 <0 Plug in the specification for 𝜑 𝑖𝑡 , we have 𝑒𝑥𝑝 (𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 +𝜀 𝑖𝑡 )<𝛾 𝑖 Therefore, 𝜀 𝑖𝑡 <log(𝛾 𝑖 )−(𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 ) For (A6b) we have 𝜕𝑢 𝑖𝑡 𝜕𝑛 𝑖𝑡 =−𝐶 𝑖𝑡 +𝛾 𝑖 𝑅̅ 𝑖𝑡 =0 where 𝐶 𝑖𝑡 is specified as 𝐶 𝑖𝑡 =(𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′𝜏 +𝜐 𝑖𝑡 )=𝛾 𝑖 𝑅̅ 𝑖𝑡 , therefore, 𝜐 𝑖𝑡 =𝛾 𝑖 𝑅̅ 𝑖𝑡 −(𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′𝜏 ) Likelihood for (m it =0, n it >0,and S it =1) can be written as, 97 𝑓 6 (𝑚 𝑖𝑡 =0,𝑛 𝑖𝑡 >0,𝑆 𝑖𝑡 =1) =Φ 𝜀 (log(𝛾 𝑖 )−(𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 ))∗[𝜙 𝜐 (𝛾 𝑖 𝑅̅ 𝑖𝑡 −(𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′𝜏 ))] (𝐴 6𝑐 ) Case (7): 𝒎 𝒊𝒕 >𝟎 , 𝒏 𝒊𝒕 >𝟎 ,𝒂𝒏𝒅 𝑺 𝒊𝒕 =𝟏 𝜕𝑢 𝑖𝑡 𝜕𝑚 𝑖𝑡 = 𝜑 𝑖𝑡 𝑚 𝑖𝑡 +1 −𝛾 𝑖 =0 (A7a) 𝜕𝑢 𝑖𝑡 𝜕𝑛 𝑖𝑡 =−𝐶 𝑖𝑡 +𝛾 𝑖 𝑅̅ 𝑖𝑡 =0 (A7b) From (A7a), we have 1 it i it m . Plugging in the definition for 𝜑 𝑖𝑡 , we have 𝑒𝑥𝑝 (𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 +𝜀 𝑖𝑡 ) 𝑚 𝑖𝑡 +1 =𝛾 𝑖 Therefore, 𝜀 𝑖𝑡 =log((𝑚 𝑖𝑡 +1)∗𝛾 𝑖 )−(𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 ) =log(𝑚 𝑖𝑡 +1)+log (𝛾 𝑖 )−(𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 ) Jacobian 𝐽 𝑖𝑡 =| 𝜕 𝜀 𝑖𝑡 𝜕𝑚 𝑖𝑡 |=| 1 𝑚 𝑖𝑡 +1 | For (A7b) we have 0 it it it i it u CR n where 𝐶 𝑖𝑡 is specified as 𝐶 𝑖𝑡 =(𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′𝜏 +𝜐 𝑖𝑡 )=𝛾 𝑖 𝑅̅ 𝑖𝑡 , therefore, 𝜐 𝑖𝑡 =𝛾 𝑖 𝑅̅ 𝑖𝑡 −(𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′𝜏 ) Likelihood for (m it >0, n it >0,and S it =1) can be written as, 𝑓 7 (𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 >0,𝑆 𝑖𝑡 =1) =𝜙 𝜀 (log((𝑚 𝑖𝑡 +1)∗𝛾 𝑖 )−(𝑥 𝑖𝑡 𝑚 ′𝛽 𝑖 +𝑧 𝑖𝑡 𝑚 ′ 𝛽 ))∗| 1 𝑚 𝑖𝑡 +1 | ∗[𝜙 𝜐 (𝛾 𝑖 𝑅̅ 𝑖𝑡 −(𝑥 𝑖𝑡 𝑛 ′𝜏 𝑖 +𝑧 𝑖𝑡 𝑛 ′𝜏 ))] (𝐴 7𝑐 ) Therefore, the likelihood of observing (𝑚 𝑖𝑡 ,𝑛 𝑖𝑡 ) can be written as: 𝑓 (𝑚 𝑖𝑡 ,𝑛 𝑖𝑡 )=𝑓 1 ̂ (𝑚 𝑖𝑡 =0,𝑛 𝑖𝑡 =0,𝑆 𝑖𝑡 =0) 𝐼 (𝑐𝑎𝑠𝑒 =1) ∗ 𝑓 ̂ 2 (𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 =0,𝑆 𝑖𝑡 =0) 𝐼 (𝑐𝑎𝑠𝑒 =2) ∗𝑓 ̂ 3 (𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 >0,𝑆 𝑖𝑡 =0) 𝐼 (𝑐𝑎𝑠𝑒 =3) 98 ∗𝑓 4 (𝑚 𝑖𝑡 =0,𝑛 𝑖𝑡 =0,𝑆 𝑖𝑡 =1) 𝐼 (𝑐𝑎𝑠𝑒 =4) ∗𝑓 5 (𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 =0,𝑆 𝑖𝑡 =1) 𝐼 (𝑐𝑎𝑠𝑒 =5) ∗𝑓 6 (𝑚 𝑖𝑡 =0,𝑛 𝑖𝑡 >0,𝑆 𝑖𝑡 =1) 𝐼 (𝑐𝑎𝑠𝑒 =6) ∗𝑓 7 (𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 >0,𝑆 𝑖𝑡 =1) 𝐼 (𝑐𝑎𝑠𝑒 =7) (A8) 𝐿 =∏∏𝑓 (𝑚 𝑖𝑡 ,𝑛 𝑖𝑡 ) 𝑇 𝑖 𝑡 =1 𝑁 𝑖 =1 99 Appendix B. Predicting Contribution Amount 𝒎̂ Since 0 m , we adopt the following Censored Regression (Type I Tobit) Model for estimating contribution amount m : (m 0) (m 0) 1 ( ) 1 I I m x x prob m 1 log ( ) (m 0)*log (m 0)*log 1 m x x prob m I I where and represent the pdf and cdf for a standard normal distribution. The following Table is the estimation result. The last row (MAE) report the model performance in predicting m. 100 Posterior Mean (SD) Intercept -0.334 (0.063) Lag_Contri_Dummy -0.794 (0.030) Number of Funding Projects -0.009 (0.016) Incentive Availability Dummy -0.816 (0.060) Reporter -0.050 (0.018) Average Incentive Reward 0.415 (0.005) Reward Saved L_it 0.044 (0.005) Share of Project Type I 0.027 (0.006) Share of Project Type II -0.043 (0.008) Share of Project Type III -0.210 (0.010) Share of Project Type IV 0.075 (0.009) Share of Project Type V -0.069 (0.009) Lag_Survey_Dummy 0.639 (0.028) Number of Available Surveys -0.024 (0.011) Number of Question -0.321 (0.021) Question^2 -0.066 (0.015) 2 0.210 (0.005) Log (Marginal Density) - 8568.146 Mean Absolute Error 3.855 Table 15. Predicting m using Censored Regression 101 Variance-Covariance Intercept 2.867 (0.110) Lag_Contri_Dummy -0.012 (0.067) 0.188 (0.014) Number of Funding Projects 0.600 (0.034) 0.008 (0.013) 0.212 (0.011) Incentive Availability Dummy -3.015 (0.124) 0.013 (0.071) -0.668 (0.038) 3.330 (0.146) Lag_Survey_Dummy -0.097 (0.069) -0.097 (0.012) -0.021 (0.016) 0.057 (0.071) 0.226 (0.023) Number of Available Surveys 0.252 (0.018) -0.002 (0.009) 0.041 (0.005) -0.319 (0.020) 0.002 (0.009) 0.099 (0.006) Number of Question 0.891 (0.032) 0.009 (0.019) 0.193 (0.011) -0.905 (0.033) -0.057 (0.021) 0.078 (0.007) 0.375 (0.015) Question^2 -0.278 (0.018) -0.001 (0.008) -0.050 (0.005) 0.240 (0.020) 0.029 (0.009) -0.005 (0.004) -0.121 (0.007) 0.083 (0.006) Table 16. Variance-Covariance for m using Censored Regression 102 Appendix C. Predicting Number of Incentive Participation 𝒏̂ Since the incentive participation n is right truncated by the available number of incentives, we adopt the following Truncated Poisson Regression model to estimate n : (s) (s) 00 (N n) n! n! (N | N ) (N ( )) !! nn ii floor floor ii e prob prob n s cdf floor s e ii (s) 0 (s) 0 log( (N | N s)) log log n! ! log( ) log(n!) log ! ni floor i i floor i prob n i n i where s denotes the number of available surveys. The following Table is the estimation result. The last row (MAE) report the model performance in predicting n. 103 Posterior Mean (SD) Intercept -2.018 0.059) Lag_Contri_Dummy -0.030 (0.032) Number of Funding Projects -0.591(0.035) Incentive Availability Dummy -1.700 (0.057) Reporter 0.086 (0.040) Average Incentive Reward 1.481 (0.011) Reward Saved L_it 0.049 (0.018) Share of Project Type I 0.112 (0.050) Share of Project Type II -0.035 (0.052) Share of Project Type III 0.460 (0.100) Share of Project Type IV -0.116 (0.065) Share of Project Type V -0.593 (0.046) Lag_Survey_Dummy -0.273 (0.092) Number of Available Surveys -0.276 (0.026) Number of Question -0.459 (0.047) Question^2 -0.041 (0.037) Log (Marginal Density) -8758.847 Mean Absolute Error 0.182 Table 17. Estimation Result for n using Truncated Poisson Regression 104 Variance-Covariance Intercept 0.654 (0.058) Lag_Survey_Dummy -0.070 (0.053) 0.455 (0.114) Number of Available Surveys -0.016 (0.026) 0.032 (0.023) 0.211 (0.020) Number of Question -0.002 (0.042) 0.011 (0.043) -0.007 (0.020) 0.223 (0.026) Question^2 -0.006 (0.026) -0.005 (0.025) 0.023 (0.014) -0.083 (0.016) 0.121 (0.010) Lag_Contri_Dummy 0.029 (0.028) -0.110 (0.039) -0.013 (0.010) -0.010 (0.014) 0.002 (0.009) 0.129 (0.018) Number of Funding Projects -0.050 (0.040) -0.034 (0.027) -0.088 (0.019) 0.029 (0.015) -0.014 (0.013) 0.015 (0.013) 0.235 (0.024) Incentive Availability Dummy -0.543 (0.052) 0.029 (0.039) -0.017 (0.025) 0.060 (0.036) -0.047 (0.023) -0.013 (0.024) -0.008 (0.034) 0.631 (0.053) Table 18. Variance-Covariance for n using Truncated Poisson Regression 105 Appendix D. Bayesian MCMC Algorithm Estimation is carried out by sequentially generating draws from the following distributions: Step 1. Generate individual specific coefficients {M , , } i i i i for i 1,..., N . ( |*) Pr(m ,n | , , , ) ( | , ) i t it it i i i V Pr(m ,n | , , , ) it it i i is the likelihood given by equation (8) in Appendix A, where t indexes the contribution period. ( | , ) i V is the prior distribution of heterogeneity. An improved Metropolis-Hastings algorithm with a random-walk chain is used to generate draws of each parameter in i . More specifically, let (j) i be the jth draw for i . The next draw (j+1) for i is given by (j 1) (j) i ii , where i is a draw from the candidate generating density from 11 (0,sbeta*(H (V ) ) ) i i Normal , we choose tuning parameter sbeta =2.93. H i is the Hessian of the ith unit likelihood evaluated at the MLE for the fraction likelihood defined by multiplying the unit likelihood by the pooled likelihood. The probability of accepting the new draw, (j 1) i , is given by 1 (j 1) (j 1) (j 1) 1 (j) (j) (j) 1 exp * Pr(m ,n | , , , ) 2 min ,1 1 exp * * Pr(m ,n | , , , ) 2 i i i i i i i i i i i i V V If the new draw is rejected, then (j 1) (j) ii .And (j) Pr(m ,n | , , , ) i i i i is the likelihood that is evaluated by using (j) i . Step 2. Generate { ,i 1,..., N} i for incentive participation. ( |*) Pr(m ,n | , , , ) ( | , ) i t it it i i i V Pr(m ,n | , , , ) it it i i is the likelihood given by equation (8) in Appendix A. ( | , ) i V is the prior distribution of heterogeneity. An improved Metropolis-Hastings algorithm with a random- walk chain is used to generate draws of i . Let (j) i be the jth draw for 0 i . The next draw (j+1) for 0 i is given by (j 1) (j) i ii , where i is a draw from the candidate generating density from 11 (0,sbeta*( (V ) ) ) i i Normal H , we choose tuning parameter sbeta =2.93. i i H is the Hessian of the ith unit likelihood evaluated at the MLE for the fraction likelihood defined by multiplying the unit likelihood by the pooled likelihood. The probability of accepting the new draw, ( 1) j i , is given by ( 1) 1 ( 1) (j 1) ( ) 1 ( ) (j) 1 exp * Pr(m ,n | , , , ) 2 min ,1 1 exp * Pr(m ,n | , , , ) 2 jj i i i i i i jj i i i i i i V V 106 If the new draw is rejected, then (j 1) (j) ii . And (j) Pr(m ,n | , , , ) i i i i is the likelihood that is evaluated by using (j) i . Step 3. Generate ,V . For parameters in individual coefficient matrix i , we draw from the posterior of a multivariate regression with a nature conjugate prior specified as follows, i U with cov( ) i uV 1 | ~ ( , A ) ~ (nu_ ,V_ ) V Normal V V IW where is the prior mean of , 1 A is a prior precision matrix (i.e., k dimensional diagonal matrix), nu_ is the degree of freedom parameter for V , and we set it to be 3, and V_ is the location parameter for V ( V_ is a k dimensional diagonal matrix, and k is the number of column in matrix i ). Step 4. Generate , V . Same procedure as in Step 3. Step 5. Generate . ( |*) Pr(m ,n | , , , ) ( | 0,100) i t it it i i Pr(m ,n | , , , ) it it i i is the likelihood given by equation (8), where i indexes an individual, and t indexes contribution month. ( | 0,100) is a diffuse prior distribution. A Metropolis-Hastings algorithm with a random-walk chain is used to generate draw of . Let (j) be the jth draw for . The next draw (j+1) for is given by (j 1) (j) , where is a draw from the candidate generating density from 1 (0,sbeta* V ) mle Normal , we choose tuning parameter sbeta =2.93, and V mle is the variance at MLE. The probability of accepting the new draw, (j 1) , is given by (j 1) 2 1 (j 1) (j) 2 1 (j) 1 exp ( 0) *100 * Pr(m, ) 2 min ,1 1 exp ( 0) *100 * Pr(m, ) 2 n n If the new draw is rejected, then (j 1) (j) . And (j) Pr(m,n) is the full likelihood that is evaluated by using (j) . Step 6. Generate . Same procedure as in Step 5. 107 Appendix E. Model Identification and Parameter Estimation (1) How the model identification issue arises. Note that individuals maximize their utility subject to budget constraint by jointly choosing the amount of contribution (mit) and number of incentive tasks to participate (nit). The Kuhn-Tucker optimality conditions derived in Appendix A lead to 7 possible cases based on the observed data of m, n and S, where i indexes for individual and t stands for time. Case 1: 𝑚 𝑖𝑡 =0,𝑛 𝑖𝑡 =0, 𝑆 𝑖𝑡 =0 Case 2: 𝑚 𝑖𝑡 . >0,𝑛 𝑖𝑡 =0, 𝑆 𝑖𝑡 =0 Case 3: 𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 >0, 𝑆 𝑖𝑡 =0 Case 4: 𝑚 𝑖𝑡 =0,𝑛 𝑖𝑡 =0, 𝑆 𝑖𝑡 =1 Case 5: 𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 =0, 𝑆 𝑖𝑡 =1 Case 6: 𝑚 𝑖𝑡 =0,𝑛 𝑖𝑡 >0, 𝑆 𝑖𝑡 =1 Case 7: 𝑚 𝑖𝑡 >0,𝑛 𝑖𝑡 >0, 𝑆 𝑖𝑡 =1 Cases 1-3 account for 79.5% of total data observations We have two types data observations for cases 1-3: Type I situation: when incentives are not available (20.9% of data observations in cases 1-3, which account for 16.6% of the total data observations) In this situation, since incentives are not available, individuals only decide on how much to contribute. Our model directly follows the standard utility maximization (Satomura, Kim and Allenby 2011). The optimality condition leads to the following estimable equation for contribution amount (mit). Here i indexes for individual and t stands for time (i.e. month), 𝑥 1,𝑖𝑡 are covariates which will be defined specifically later. 𝑚 𝑖𝑡 = 𝑓 1 (𝑥 1,𝑖𝑡 , 𝜃 1,𝑖 ) + 𝜀 1,𝑖𝑡 (E0) Since incentives are not available, we only model contribution amount, and there is no simultaneity involved between contribution and incentive participation. As such, there is no model identification issue due to simultaneity. Simultaneity may arise in some cases of Type II situation, which we discuss next. Type II situation: when incentives are available (79.1% of data observations in cases 1-3, which account for 62.9% of the total data observations) When the incentives are available, the Kuhn-Tucker optimality conditions for type II data observations lead to a simultaneous equation model (detailed derivation is reported in the Appendix A in the paper). Although the optimality conditions for these three cases suggest different functional forms for f1 and f2, they share the same model structure as follows: 𝑚 𝑖𝑡 = 𝑓 1 (𝑛 𝑖𝑡 , 𝑥 1,𝑖𝑡 , 𝑥 3,𝑖𝑡 , 𝜃 1,𝑖 ) + 𝜀 1,𝑖𝑡 (E1) 108 𝑛 𝑖𝑡 = 𝑓 2 (𝑚 𝑖𝑡 , 𝑥 2,𝑖𝑡 , 𝑥 3,𝑖𝑡 , 𝜃 2,𝑖 ) + 𝜀 2,𝑖𝑡 (E2) 𝑥 1,𝑖𝑡 is a vector of covariates that uniquely affect individuals’ contribution preference after controlling for the effect of 𝑛 𝑖𝑡 and 𝑥 3,𝑖𝑡 : number of available funding projects at t, shares of types of available funding projects at t, if incentive is available at t, and contribution status at t-1. 𝑥 2,𝑖𝑡 is a vector of covariates that uniquely affect individuals’ incentive participation preference after controlling for the effect of 𝑚 𝑖𝑡 and 𝑥 3,𝑖𝑡 : number of available incentive tasks at t, average number of questions for an available task at t, and incentive participation at t-1. 𝑥 3,𝑖𝑡 is a vector of covariates that affect individuals’ contribution preference and incentive participation preference after controlling for all the other effects: average reward amount for an available task at t, reward amount carried over to t, and whether individual being a reporter (which does not vary over time). Average reward amount and reward amount carried over affect both contribution amount and incentive participation after controlling for 𝑥 1,𝑖𝑡 and 𝑥 2,𝑖𝑡 , due to our model setup and the subsequent Kuhn-Tucker optimality conditions. “Reporter” dummy appears in both (E1) and (E2) because we use it to capture observed heterogeneity across individuals. This is commonly recognized as a non-recursive or non-triangular simultaneous equation model in econometrics (e.g. Greene 2000), and widely used in economics and marketing to model interdependent decisions. The term “non-recursive” means that two decisions affect each other, as compared to a “recursive” situation where only one decision affects the other but not the other way around. For a simple example of a “recursive” situation involving two decisions of y1 and y2, y1 will be affected only by exogenous covariates x1, and y2 is affected by both y1 and exogenous covariates x2. Non-recursive simultaneous equations lead to endogeneity, which needs to be accounted for to achieve consistent model estimates (e.g. Greene 2000). To see this, we use a simple example. Let us assume the following two-equation non-recursive simultaneous system that mimics our model setup: 1 1 1 1 2 1 y x y (E3) 2 2 2 2 1 2 y x y (E4) 1 0 , 2 0 (A3) and (A4) can be re-written as a matrix form, that is, 1 1 1 1 1 1 2 2 2 2 2 2 00 00 y y x y y x (E5) or Y Y XB (E6) 109 Then the implicit or structural form of equation (A6) can be written as an explicit or reduced-form equation, that is, 11 ( ) ( ) Y I XB I (E7) It can be approved that 1 () I is a 2x2 matrix with all 4 elements not equal to zero. In other words, (E7) implies that y1 is a function of 1 and 2 , and similarly, y2 is a function of 1 and 2 . This suggests that y2 is endogenous in equation (E3) and y1 is endogenous in equation (E4). As such, in the first three cases in our modeling/empirical setting, simultaneity of 𝑚 𝑖𝑡 and 𝑛 𝑖𝑡 will imply that 𝑛 𝑖𝑡 is endogenous in equation (E1) and 𝑚 𝑖𝑡 will be endogenous in equation (E2). Cases 4-7 do not involve simultaneity (20.5% of total observations): As shown in our model derivation in the Appendix A of our paper, the remaining four cases (Case 4 – Case 7) do not involve simultaneity. Although these four cases suggest different functional forms for f1 and f2, they share the same model structure as follows: 𝑚 𝑖𝑡 = 𝑓 1 (𝑥 1,𝑖𝑡 ,𝑥 3,𝑖𝑡 , 𝜃 1,𝑖 ) + 𝜀 1,𝑖𝑡 (E8) 𝑛 𝑖𝑡 = 𝑓 2 (𝑥 2,𝑖𝑡 , 𝑥 3,𝑖𝑡 , 𝜃 2,𝑖 ) + 𝜀 2,𝑖𝑡 (E9) (E8) and (E9) lead to a standard model of two dependent variables, which does not involve any simultaneity or endogeneity. (2) Two-stage method for estimating simultaneous equations Since 79.1% of the data in the first three cases (i.e. 62.9% of the total observations) need a model with simultaneity, we have followed the two-stage method to account for the simultaneity arising from the interdependency between incentive participation and contribution, for these data observations. Two-stage method has been proposed as a popular method to estimate simultaneous equation models in econometric textbooks (e.g. Greene 2000, page 683), econometric books (e.g. Maddala 1984, page 243), and literature (e.g. Amemiya 1974b, Nelson and Olsen 1978, Lee 1982, Bajari et al. 2010, Yang and Ghose 2010) Step 1: estimate the reduced-form model by using all the exogenous covariates to predict the two outcomes and obtain 𝑚 𝑖𝑡 ̂ and 𝑛 𝑖𝑡 ̂ 𝑚 𝑖𝑡 = 𝑔 1 (𝑥 1,𝑖𝑡 , 𝑥 2,𝑖𝑡 , 𝑥 3,𝑖𝑡 , 𝛾 1,𝑖 ) + 𝑣 1,𝑖𝑡 (E10) 𝑛 𝑖𝑡 = 𝑔 2 (𝑥 1,𝑖𝑡 ,𝑥 2,𝑖𝑡 , 𝑥 3,𝑖𝑡 , 𝛾 2,𝑖 ) + 𝑣 2,𝑖𝑡 (E11) We use censored regression to predict 𝑚 𝑖𝑡 , and truncated Poisson to predict 𝑛 𝑖𝑡 , and the estimation results for step 1 are reported in the Appendix B and C. Step 2: Insert 𝑚 𝑖𝑡 ̂ and 𝑛 𝑖𝑡 ̂ back to the right hand side of the structural model (E1) and (E2) and estimate the parameters of interest, that is, 𝑚 𝑖𝑡 = 𝑓 1 (𝑛 𝑖𝑡 ̂ , 𝑥 1,𝑖𝑡 , 𝑥 3,𝑖𝑡 , 𝜃 1,𝑖 ) + 𝜀 1,𝑖𝑡 (E12) 𝑛 𝑖𝑡 = 𝑓 2 (𝑚 𝑖𝑡 ̂ , 𝑥 2,𝑖𝑡 , 𝑥 3,𝑖𝑡 , 𝜃 2,𝑖 ) + 𝜀 2,𝑖𝑡 (E13) 110 (3) Identify the model involving simultaneity through imposing exclusion restriction. Econometricians have proposed alternative ways to help identification in models involving simultaneity. A commonly used approach is by imposing exclusion restriction (e.g. Greene 2000, page 668, #3). This is done by removing from the equation some variables that significantly affect the endogenous regressor but do not directly affect the outcome, to help identify the parameters in that equation. Recall that in our structural model as specified in (E1) and (E2) applicable to Cases 1-3, 𝑚 𝑖𝑡 = 𝑓 1 (𝑛 𝑖𝑡 , 𝑥 1,𝑖𝑡 , 𝑥 3,𝑖𝑡 , 𝜃 1,𝑖 ) + 𝜀 1,𝑖𝑡 (E1) 𝑛 𝑖𝑡 = 𝑓 2 (𝑚 𝑖𝑡 , 𝑥 2,𝑖𝑡 , 𝑥 3,𝑖𝑡 , 𝜃 2,𝑖 ) + 𝜀 2,𝑖𝑡 (E2) We impose the exclusion restriction for Cases 1-3 in the following way: In identifying the contribution amount model in (E1), we exclude 𝑥 2,𝑖𝑡 . In particular, using a simple reduced-form analysis, we find that some covariates in 𝑥 2,𝑖𝑡 (i.e. number of available incentive tasks at t, average number of questions for an available task at t, and incentive participation at t-1) significantly affect nit, but do not significantly affect mit after controlling for nit. It is intuitive that these covariates affect incentive participation, because they reflect incentive-related information. However, these covariates may not directly affect contribution amount for two reasons. First, credits earned do not have an expiration date, and individuals can use these earned credits at any time to contribute to projects they like. Second, based on our discussion with the management team of the crowdfunding platform, we know that contributors usually have a strong preference for the topics of funding projects, and funding project topic is listed as the most important reason for individuals to consider a contribution. In identifying the model of number of incentives participated in (E2), we exclude 𝑥 1,𝑖𝑡 . In particular, using a simple reduced-form analysis, we find that some covariates in 𝑥 1,𝑖𝑡 (i.e. number of available funding projects at t, and contribution status at t-1) significantly affect mit, but do not significantly affect nit after controlling for mit. It is intuitive that these covariates affect contribution, because they capture information related to contribution. However, these covariates may not directly affect incentive participation for the same two reasons as mentioned above. (4) Identify the model involving simultaneity (79.1% of the observations from Cases 1-3) through other solutions. There are several other econometric solutions proposed to identify models involving simultaneity. For example, Greene (2000) suggested six solutions where “exclusion” is listed as #3. We have already discussed how we impose the exclusion restriction in our context. We next discuss how we apply additional rules to further aid in our model identification, by following solutions proposed in Greene (2000). “Exclusion”. 111 In addition to applying “exclusion” to covariates, one can exclude endogenous variables from equations to achieve better identification. Our structural model has a unique situation, that is, although the seven cases share common parameters, simultaneity does not apply to Cases 4-7 and 20.9% of the observations from case 1-3 where m and n are exogenously determined in (E8) and (E9) (i.e. removing endogenous variables from the two equations for some data observations). This natural setting helps our model identification to address the simultaneity issue involved in equations (E1) and (E2) for Cases 1-3. “Normalization”. A close examination of the model implied from Cases 1-3 suggests that we have naturally normalized coefficients in front of the endogenous variables m and n (See Equation (A1d), (A2d) and (A3d) in the Appendix A). For example, for Case 1, the contribution amount equation of m (equation (A1d)) suggests that the coefficient of n is average reward amount (which is directly observed from data). Furthermore, the incentive participation equation of n (equation (A1d)) suggests that the coefficient of m is -1. The parameter normalization naturally imposed through our structural modeling has greatly helped on model identification, an insight drawn from the econometric principle. “Restriction on the covariance matrix of the error terms” To further aid in model identification, we have constrained the error terms in the structural model of equations (E1) and (E2) to be uncorrelated, following Greene (2000). “Nonlinearities” Finally, note that equations (E1) and (E2) involve a model that his nonlinear, and as suggested by the econometric principle, nonlinearities can aid in model identification. (5). Parameter estimation (1) At the high level, we can estimate the marginal contribution utility, cost of incentive participation, and preference for reward saving, through variation in three pieces of data: contribution amount, incentive participation, and amount of reward saving. (2) Empirical identification of parameters in the marginal contribution utility is achieved through the co-variation between contribution amount and the covariates. (3) Similarly, empirical identification of parameters in the cost is achieved through the co- variation between incentive participation and the covariates. (4) We can estimate budget related parameters based on the data variation patterns between contribution and incentive participation. Since in our data, the incentives are available periodically and the available incentives are not restricted to any specific individuals or funding projects, we observe data variations both within and across individuals, and variations with and without the presence of effort-based incentives. Those data variations allow us to identify 112 the budget related parameters (e.g., high and low baseline budget level). We expect that effort- based incentives would affect individuals in different ways. Specifically, for individuals with low baseline budget, the presence of the effort-based incentives can make a big difference. Without the effort-based incentives, even if this individual has a high contribution preference, she may not be able to contribute too much due to the budget constraint. But with the presence of the incentives, the reward can increase her budget, and she would be able to contribute more. For individual with high baseline budget, the presence of the effort-based incentives would not change her contribution pattern that much because she has a high budget level. (5) We constrain the variance of the error term in the contribution amount equation (E1) as 1, in order to achieve better empirical identification. This directly follows the previous literature on structural models of demand (Kim, Allenby and Rossi 2002, Satomura, Kim and Allenby 2011). Here is the rationale. In Case 1, Case 4 and Case 6, over 80% of our data, the likelihood of contribution amount is a function of (x ) it i which is the cumulative distribution function of normal. Imagine if we do not fix the variance to be 1, then the function will change to be (x ) it i . We can easily see that as long as the ratio i stays the same, the probability will not change, and as such, we cannot separately identify parameter and variance. Similar rationale applies to variance of the error term in the incentive participation equation (E2), which is also fixed to be 1 for better empirical identification. (6) Since marginal utility for outside good and the baseline budget cannot be statistically identified at the same time, we set the marginal utility from outside good to 1, following Satomura, Kim and Allenby (2011). 113 Appendices for Chapter Two Appendix A. A Funding Project Example Short Description of the funding project: Cara Jones is the project creator. She is an award winning writer, reporter, and producer with vast experience in using video to tell compelling, inspiring human-interest stories. Cara and her team members created Storytellers for Good in 2010. Storytellers for Good is a non-profit team of journalists and photographers who aim to tell and promote compelling stories of people and organizations making a positive difference. They aim to tell the inspiring stories of local nonprofits in Bay Area, so they pledged a funding project on this journalistic crowdfunding platform. Funding project final delivery description: Storytellers for Good will produce five high quality promotional videos for non-profit organizations in need. The stories will be 2-5 minutes long and include interviews, graphics, text and music. 114 Appendix B. Estimation Result for State Transition 1. The stochastic evolution of cumulative funding amount (𝒔 𝒊𝒕 𝑭𝒖𝒏𝒅𝒊𝒏𝒈 ) 𝑠 𝑖𝑡 +1 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 =𝑠 𝑖𝑡 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 +𝐴 𝑖𝑡 +𝑤 𝑖𝑡 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 (B1) We use the empirical data on weekly contribution amount funded to a project to form the belief on new weekly funding 𝑤 𝑖𝑡 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 . Since the weekly contribution is both lower and upper bounded, we use the following censored normal regression to estimate it. 𝑤 𝑗𝑡 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 ={ 𝑠 𝑗𝑡 𝐴𝑚𝑡 _𝐿𝑒𝑓𝑡 , 𝑖𝑓 𝑤 𝑗𝑡 ∗ ≥𝑠 𝑗𝑡 𝐴𝑚𝑡 _𝐿𝑒𝑓𝑡 𝑤 𝑗𝑡 ∗ , 𝑖𝑓 0<𝑤 𝑗𝑡 ∗ <𝑠 𝑗𝑡 𝐴𝑚𝑡 _𝐿𝑒𝑓𝑡 0, 𝑖𝑓 𝑤 𝑗𝑡 ∗ ≤0 (B2) where 𝑠 𝑗𝑡 𝐴𝑚𝑡 _𝐿𝑒𝑓𝑡 =𝑥 𝑗 𝐺𝑜𝑎𝑙 −𝑠 𝑗𝑡 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 , and we allow the latent 𝑤 𝑗𝑡 ∗ to be influence by the observed states and also some project specific covariates such as funding goal, reward delivery format, year dummy etc. 𝑤 𝑗𝑡 ∗ =𝑤 (𝑠 𝑗𝑡 ,𝜈 𝑗𝑡 𝑤 ;𝜃 𝑤 )+𝜈 𝑗𝑡 𝑤 =𝜃 𝑗 0 𝑤 +𝜃 𝑗 1 𝑤 𝑠 𝑗𝑡 𝑇𝑖𝑚𝑒𝐿𝑒𝑓𝑡 +𝜃 𝑗 2 𝑤 𝑠 𝑗𝑡 𝐹𝑢𝑛𝑑 _𝑝𝑒𝑟𝑐 +𝜃 𝑗 3 𝑤 𝑠 𝑗𝑡 𝑇𝑖𝑚𝑒𝐿𝑒𝑓𝑡 𝑠 𝑗𝑡 𝐹𝑢𝑛𝑑 _𝑝𝑒𝑟𝑐 +𝜃 4 𝑤 𝑠 𝑗𝑡 𝐷𝑜𝑛𝑜𝑟 +𝜃 5 𝑤 𝑠 𝑗𝑡 𝑂𝑟𝑔 _𝐴𝑚𝑡 𝑥 𝑗 𝐺𝑜𝑎𝑙 + 𝜃 6 𝑤 𝑥 𝑗𝑡 𝑓𝑒𝑎𝑡𝑢𝑟𝑒 +𝜈 𝑗𝑡 𝑤 (B3) where 𝑠 𝑗𝑡 𝐹𝑢𝑛𝑑 _𝑝𝑒𝑟𝑐 = 𝑠 𝑗𝑡 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 𝑥 𝐺𝑜𝑎𝑙 , 𝜈 𝑗𝑡 𝑤 ~𝑁 (0,𝜎 𝑤 2 ) , and 𝜃 𝑗 𝑤 ={𝜃 𝑗 0 𝑤 ,𝜃 𝑗 1 𝑤 ,𝜃 𝑗 2 𝑤 ,𝜃 𝑗 3 𝑤 }~𝑀𝑉𝑁 (𝜃 ̅ ,Ω 𝜃 ). And 𝑥 𝑗 ,𝑡 𝑓𝑒𝑎𝑡𝑢𝑟𝑒 includes the following project specific covariates: 𝑥 𝑗 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 𝑤𝑒𝑒𝑘 : Total funding week (to the deadline) 𝑥 𝑗 𝐺𝑜𝑎𝑙 : Logarithm funding goal for project j 𝑥 𝑗 𝑌𝑒𝑎𝑟 : Year dummy for project creation 𝑥 𝑗 𝑓𝑜𝑟𝑚𝑎𝑡 : Four format dummies indicating the project delivery format: video, audio, photo, and text. These four format types are not mutually exclusive. 115 2. The stochastic evolution of cumulative number of donors (𝒔 𝒋 ,𝒕 𝒅𝒐𝒏𝒐𝒓 ) 𝑠 𝑗𝑡 +1 𝐷𝑜𝑛𝑜𝑟 =𝑠 𝑗𝑡 𝐷𝑜𝑛𝑜𝑟 +𝐼 (𝐴 𝑡 >0 & 𝑠 𝑡 𝐼𝑛𝑑 _𝐶𝑢𝑚𝑎𝑚𝑡 =0)+ℎ 𝑗𝑡 𝑑𝑜𝑛𝑜𝑟 (B4) We use the empirical data on weekly number of new donors attracted to a project to obtain the belief on new donors ℎ 𝑡 𝑑𝑜𝑛𝑜𝑟 . And we assume that the arrival of new donors follows a Poisson process: ℎ 𝑡 𝑑𝑜𝑛𝑜𝑟 ~𝑃𝑜𝑖𝑠𝑠𝑜𝑛 (𝜆 𝑡 ) , Pr(𝑁 𝑗𝑡 =𝑛 )=𝑒 𝜆 𝑗𝑡 𝜆 𝑗𝑡 𝑛 𝑛 ! , (n=0,1,2,…) (B5) 𝜆 𝑡 =exp(𝛾 𝑗 0 +𝛾 𝑗 1 𝑠 𝑗𝑡 𝑇𝑖𝑚𝑒𝐿𝑒𝑓𝑡 +𝛾 𝑗 2 𝑠 𝑗𝑡 𝐹𝑢𝑛𝑑 _𝑝𝑒𝑟𝑐 +𝛾 𝑗 3 𝑠 𝑗𝑡 𝑇𝑖𝑚𝑒𝐿𝑒𝑓𝑡 𝑠 𝑗𝑡 𝐹𝑢𝑛𝑑 _𝑝𝑒𝑟𝑐 +𝛾 4 𝑠 𝑗𝑡 𝐷𝑜𝑛𝑜𝑟 + 𝛾 5 𝑠 𝑗𝑡 𝑂𝑟𝑔 _𝐴𝑚𝑡 𝑥 𝑗 𝐺𝑜𝑎𝑙 +𝛾 6 𝑥 𝑗𝑡 𝑓𝑒𝑎𝑡𝑢𝑟𝑒 ) (B6) 3. The stochastic evolution of organizational funding (𝒔 𝒊𝒋 ,𝒕 𝑶𝒓𝒈 _𝑨𝒎𝒕 ) 𝑠 𝑗 ,𝑡 +1 𝑂𝑟𝑔 _𝐴𝑚𝑡 =𝑠 𝑗𝑡 𝑂𝑟𝑔 _𝐴𝑚𝑡 +𝑔 𝑗𝑡 𝑂𝑟𝑔 _𝐴𝑚𝑡 (B7) 𝑔 𝑗𝑡 𝑂𝑟𝑔 _𝐴𝑚𝑡 ={ 𝑠 𝑗𝑡 𝐴𝑚𝑡 _𝐿𝑒𝑓𝑡 𝑖𝑓 𝑔 𝑗𝑡 ∗ ≥𝑠 𝑗𝑡 𝐴𝑚𝑡 _𝐿𝑒𝑓𝑡 𝑔 𝑗𝑡 ∗ 𝑖𝑓 0< 𝑔 𝑗𝑡 ∗ <𝑠 𝑗𝑡 𝐴𝑚𝑡 _𝐿𝑒𝑓𝑡 0 𝑖𝑓 𝑔 𝑗𝑡 ∗ ≤0 (B8) Where 𝑠 𝑗𝑡 𝐴𝑚𝑡 _𝐿𝑒𝑓𝑡 =𝑥 𝑗 𝐺𝑜𝑎𝑙 −𝑠 𝑗𝑡 𝐹𝑢𝑛𝑑𝑖𝑛𝑔 . 𝑔 𝑗𝑡 ∗ =𝑔 (𝑠 𝑗𝑡 ,𝜈 𝑗𝑡 𝑔 ;𝑘 𝑔 )+𝜈 𝑡 𝑔 =𝜅 𝑗 0 𝑔 +𝜅 𝑗 1 𝑔 𝑠 𝑗𝑡 𝑇𝑖𝑚𝑒𝐿𝑒𝑓𝑡 +𝜅 𝑗 2 𝑔 𝑠 𝑗𝑡 𝐹𝑢𝑛𝑑 _𝑝𝑒𝑟𝑐 +𝜅 𝑗 3 𝑔 𝑠 𝑗𝑡 𝑇𝑖𝑚𝑒𝐿𝑒𝑓𝑡 𝑠 𝑗𝑡 𝐹𝑢𝑛𝑑 _𝑝𝑒𝑟𝑐 +𝜅 4 𝑔 𝑠 𝑗𝑡 𝐷𝑜𝑛𝑜𝑟 +𝜅 5 𝑔 𝑠 𝑗𝑡 𝑂𝑟𝑔 _𝐴𝑚𝑡 𝑥 𝑗 𝐺𝑜𝑎𝑙 +𝜅 6 𝑔 𝑥 𝑗𝑡 𝑓𝑒𝑎𝑡𝑢𝑟𝑒 +𝜈 𝑗𝑡 𝑔 (B9) where 𝜈 𝑗𝑡 𝑔 ~𝑁 (0,𝜎 𝑔 2 ) . 116 Posterior Mean (Posterior Std) Intercept 1.130 (0.120) TimeLeft -0.950 (0.150) FundingPerc -1.190 (0.160) Time Left *Fundingperc 0.750 (0.160) CumDonor 0.0004 (0.0003) CumOrgAmtPerc -0.314 (0.046) TotalFundingWeek -0.004 (0.001) Log_Goal -0.013 (0.002) Year Dummy -0.005 (0.003) Text 0.150 (0.001) Video -0.172 (0.039) Photo -0.025 (0.001) Audio 0.064 (0.047) 2 0.001 (0.0006) Log (Marginal Density) 424.710 Percent of exact match* 70.7% Percent of zeros** 58.9% Mean Absolute Error 0.042 Mean Square Error 0.007 Note: *Exact match means the absolute difference between predict and actual y is less than 0.05. **In the original data, there are 49.8% of zero observations. *** For estimation, we transform the dependent variable to be a percentage value ([0,1]). Table 19. Estimation Result for Weekly Donation Amount 117 Posterior Mean (Posterior Std) Intercept 11.000 (1.000) TimeLeft -12.000 (1.100) FundingPerc -18.000 (1.700) Time Left *Fundingperc 15.000 (2.200) CumDonor -0.020 (0.001) CumOrgAmtPerc 8.740 (0.420) TotalFundingWeek 0.041 (0.005) Log_Goal 0.080 (0.035) Year Dummy -0.446 (0.135) Text -1.008 (0.243) Video -0.594 (0.216) Photo 0.294 (0.235) Audio 0.657 (0.264) Log (Marginal Density) -3279.473 Percent of exact match* 85.9% Percent of zeros** 43.2% Mean Absolute Error 2.180 Mean Square Error 20.703 Note: *Exact match means the absolute difference between predict and actual y is less than 5. **In the original data, there are 49.8% of zero observations. Table 20. Estimation Result for Weekly Number of Donor 118 Posterior Mean (Posterior Std) Intercept -0.640 (0.180) TimeLeft -0.430 (0.140) FundingPerc -0.840 (0.240) Time Left *Fundingperc 0.860 (0.190) CumDonor -0.0007 (0.0006) CumOrgAmtPerc -1.250 (0.260) TotalFundingWeek -0.006 (0.002) Log_Goal 0.089 (0.011) Year Dummy 0.068 (0.072) Text 0.025 (0.110) Video -0.039 (0.133) Photo -0.303 (0.082) Audio -0.055 (0.112) 2 0.033 (0.006) Log (Marginal Density) -131.647 Percent of exact match* 94.1% Percent of zeros** 94.3% Mean Absolute Error 0.011 Mean Square Error 0.003 Note: *Exact match means the absolute difference between predict and actual y is less than 0.05. **In the original data, there are 89.7% of zero observations. *** For estimation, we transform the dependent variable to be a percentage value ([0,1]). Table 21. Estimation Result for Weekly Organizational Amount
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Creator
Yu, Xiaoqian
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Core Title
Essays on understanding consumer contribution behaviors in the context of crowdfunding
School
Marshall School of Business
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Doctor of Philosophy
Degree Program
Business Administration
Publication Date
07/15/2019
Defense Date
05/09/2017
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University of Southern California
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consumer contribution behaviors,crowdfunding,effort-based incentives,OAI-PMH Harvest,structural model
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Yang, Sha (
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), Hsieh, Yu-Wei (
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uxiaoqian@gmail.com,xiaoqiay@usc.edu
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consumer contribution behaviors
crowdfunding
effort-based incentives
structural model