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Investigation of various factors behind non-deaccummulation of housing and wealth with aging
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Investigation of various factors behind non-deaccummulation of housing and wealth with aging
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INVESTIGATION OF V ARIOUS FACTORS BEHIND NON-DEACCUMMULATION OF HOUSING AND WEALTH WITH AGING by Asiye Aydilek A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ECONOMICS) August 2009 Copyright 2009 Asiye Aydilek Dedication To my husband Harun, my parents and my sisters. ii Acknowledgements I would like to thank my advisor Ayse Imrohoroglu for guiding me in my research , for her numerous helpful comments and encouragement. She has been an excellent mentor to me. I am grateful to Selale Tuzel for accepting to become an outside member of my committee. I would like to thank her one more time for introducing me into the topic and the housing literature. I would like to thank Vincenzo Quadrini for helping me with the technical details and giving me a new idea. I appreciate the insightful and constructive suggestions that he provided me with. My members have been incredibly kind and patient. I also received helpful advice from my committee member Robert Dekle. I also want to thank Professor Selahattin Imrohoroglu, Sagiri Kitao and my friends Okan Eren, Murat Ungor and Huseyin Gunay for helpful discussions. I am grateful to my parents for always supporting me. It might not have been possible to complete any of my achievements, without feeling their love and their faith in me. Last but certainly not least, I want to thank my husband Harun for his love, con- tinuous support and valuable suggestions. Not only my thesis but also my whole life during PhD, might be much more difficult without him. iii TableofContents Dedication ii Acknowledgements iii List of Tables vi List of Figures vii Abstract xi 1 Introduction 1 1.1 Related Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2 General Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.1 Preferences and Labor Income . . . . . . . . . . . . . . . . . 12 1.2.2 Housing Contract . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.3 Borrowing and Saving with the Riskless Rate . . . . . . . . . 15 1.2.4 Budget Constraints . . . . . . . . . . . . . . . . . . . . . . . 16 1.2.5 The Optimization Problem of the Household . . . . . . . . . 17 2 The Model in a Uniformly Distributed Population by age 19 2.1 Model with Transitory Income Shocks and Deep Habits . . . . . . . . 19 2.2 Parameter Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3 Results with Uniform Population Distribution . . . . . . . . . . . . . 23 2.4 Results of the Model in Step 1 . . . . . . . . . . . . . . . . . . . . . 24 2.4.1 Effects of Bequest Motive . . . . . . . . . . . . . . . . . . . 24 2.5 Results of the Model in Step 2 . . . . . . . . . . . . . . . . . . . . . 25 2.5.1 Effects of Bequest Motive in the Model in Step 2 . . . . . . . 26 2.6 Results of the Model in Step 3 . . . . . . . . . . . . . . . . . . . . . 26 3 The Model in a Non-Uniformly Distributed Population by age 35 3.1 Model with Transitory Income Shocks, Permanent Income Shocks and Deep Habits under Real Population Distribution . . . . . . . . . . . . 35 3.2 Parameter Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 36 3.3 Results of the Model in Step 1 . . . . . . . . . . . . . . . . . . . . . 36 3.3.1 Optimal Life Cycle Decisions with Transitory Income Shocks 37 3.4 Results of the Model in Step 2 . . . . . . . . . . . . . . . . . . . . . 37 3.4.1 Optimal Life Cycle Decisions with Transitory Income Shocks and Permanent Income Shocks . . . . . . . . . . . . . . . . . 38 iv 3.5 Results of the Model in Step 3 . . . . . . . . . . . . . . . . . . . . . 39 3.5.1 Optimal Life Cycle Decisions with Transitory Income Shocks, Permanent Income Shocks and Habit Formation in Housing . 41 4 The Model with Transaction Costs in House Trading 47 4.1 The General Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 4.1.1 Budget Constraints . . . . . . . . . . . . . . . . . . . . . . . 47 4.2 Parameter Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.3 Results of the Model with Transaction costs, Step 1 . . . . . . . . . . 51 4.4 Results of the Model with Transaction costs, Step 2 . . . . . . . . . . 53 4.5 Results of the Model with Transaction costs, Step 3 . . . . . . . . . . 56 5 The Model with Leisure 61 5.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 5.2 Solution of the Model . . . . . . . . . . . . . . . . . . . . . . . . . . 63 5.3 Parameters and Calibration . . . . . . . . . . . . . . . . . . . . . . . 64 5.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 5.4.1 Optimal Life Cycle Decisions . . . . . . . . . . . . . . . . . 66 5.5 Results of the Model with Leisure and Habit Formation in Housing together . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.5.1 Optimal Life Cycle Decisions . . . . . . . . . . . . . . . . . 70 5.6 Model with a Multiplier in Utility . . . . . . . . . . . . . . . . . . . 71 5.7 Results of the Model in 5.6 . . . . . . . . . . . . . . . . . . . . . . . 74 5.7.1 Optimal Life Cycle Decisions . . . . . . . . . . . . . . . . . 74 6 Model with Health Shocks and Medical Costs 78 6.1 Health Status Uncertainty . . . . . . . . . . . . . . . . . . . . . . . 80 6.2 Medical Expense Uncertainty . . . . . . . . . . . . . . . . . . . . . 80 6.3 Survival Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . 80 6.4 Parameter Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.5 Optimal Life Cycle Decisions . . . . . . . . . . . . . . . . . . . . . 81 7 Conclusion 84 References 88 Appendix 93 v ListofTables 2.1 Statistics of the Model in 2.1, Step 1 . . . . . . . . . . . . . . . . . . 25 2.2 Statistics of the Model in 2.1, Step 3 . . . . . . . . . . . . . . . . . . 27 2.3 Statistics of the Model in 2.1, Step 3, Continued . . . . . . . . . . . . 30 3.1 Statistics of the Model in 3.1, Step 1 . . . . . . . . . . . . . . . . . . 37 3.2 Statistics of the Model in 3.1, Step 2 . . . . . . . . . . . . . . . . . . 38 3.3 Statistics of the Model in 3.1, Step 3 . . . . . . . . . . . . . . . . . . 40 4.1 Statistics of the Model in 4.1, Step 1 . . . . . . . . . . . . . . . . . . 51 4.2 Statistics of the Model in 4.1, Step 2 . . . . . . . . . . . . . . . . . . 54 5.1 Statistics of the Model in 5.1 . . . . . . . . . . . . . . . . . . . . . . 65 5.2 Statistics of the Model in 5.5 . . . . . . . . . . . . . . . . . . . . . . 69 5.3 Statistics of the Model in 5.6 . . . . . . . . . . . . . . . . . . . . . . 74 vi ListofFigures 2.1 Consumption with Deterministic Income and without Habit . . . . . . 26 2.2 Housing Value with Deterministic Income and without Habit . . . . . 27 2.3 Wealth Holdings with Deterministic Income and without Habit . . . . 28 2.4 Consumption and Housing in the Data . . . . . . . . . . . . . . . . . 28 2.5 Wealth and Home Ownership in the Data . . . . . . . . . . . . . . . 29 2.6 Housing with Transitory Income Shocks and without Habit . . . . . . 29 2.7 Consumption with Transitory Income Shocks and without Habit . . . 30 2.8 House Value with Transitory Income Shocks and without Habit . . . . 31 2.9 Home Ownership with Transitory Income Shocks and without Habit . 31 2.10 Wealth Holdings with Transitory Income Shocks and without Habit . 32 2.11 Home Ownership with Transitory Income Shocks with and without Bequest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.12 Wealth with Transitory Income Shocks with and without Bequest . . . 33 2.13 Consumption with Habit Formation in Housing and Transitory Income Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.14 House Value with Habit Formation in Housing and Transitory Income Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.15 Wealth Holdings with Habit Formation in Housing and Transitory Income Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.1 Consumption with two or three States Transitory Income Shocks . . . 38 3.2 Housing with two or three States Transitory Income Shocks . . . . . . 39 3.3 Wealth Holdings with two or three States Transitory Income Shocks . 40 3.4 Home Ownership with two or three States Transitory Income Shocks . 41 3.5 Consumption under Transitory Income Shocks and with or without Permanent Income Shocks . . . . . . . . . . . . . . . . . . . . . . . 42 vii 3.6 Housing under Transitory Income Shocks and with or without Perma- nent Income Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.7 Wealth under Transitory Income Shocks and with or without Perma- nent Income Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.8 Home Ownership under Transitory Income Shocks and with or without Permanent Income Shocks . . . . . . . . . . . . . . . . . . . . . . . 44 3.9 Consumption with Transitory Income Shocks, Permanent Income Shocks and Habit Formation in Housing . . . . . . . . . . . . . . . . . . . . 44 3.10 Housing with Transitory Income Shocks, Permanent Income Shocks and Habit Formation in Housing . . . . . . . . . . . . . . . . . . . . 45 3.11 House Value with Transitory Income Shocks, Permanent Income Shocks and Habit Formation in Housing . . . . . . . . . . . . . . . . . . . . 45 3.12 Home Ownership with Transitory Income Shocks, Permanent Income Shocks and Habit Formation in Housing . . . . . . . . . . . . . . . . 46 3.13 Wealth Holdings with Transitory Income Shocks, Permanent Income Shocks and Habit Formation in Housing . . . . . . . . . . . . . . . . 46 4.1 Consumption with Transitory Income Shocks, Permanent Income Shocks, without Habit Formation in Housing and with Different Maintenance Cost Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.2 Housing with Transitory Income Shocks, Permanent Income Shocks, without Habit Formation in Housing and with Different Maintenance Cost Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.3 House Value with Transitory Income Shocks, Permanent Income Shocks, without Habit Formation in Housing and with Different Maintenance Cost Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.4 Wealth Holdings with Transitory Income Shocks, Permanent Income Shocks, without Habit Formation in Housing and with Different Main- tenance Cost Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 4.5 Home Ownership with Transitory Income Shocks, Permanent Income Shocks, without Habit Formation in Housing and with Different Main- tenance Cost Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 4.6 Consumption with Transitory Income Shocks, Permanent Income Shocks, Different Maintenance Cost Rates and Habit Strengths . . . . . . . . 57 viii 4.7 Housing with Transitory Income Shocks, Permanent Income Shocks, Different Maintenance Cost Rates and Habit Strengths . . . . . . . . 57 4.8 House Value with Transitory Income Shocks, Permanent Income Shocks, Different Maintenance Cost Rates and Habit Strengths . . . . . . . . 58 4.9 Wealth Holdings with Transitory Income Shocks, Permanent Income Shocks, Different Maintenance Cost Rates and Habit Strengths . . . . 58 4.10 Home Ownership with Transitory Income Shocks, Permanent Income Shocks, Different Maintenance Cost Rates and Habit Strengths . . . . 59 4.11 Non-Housing Consumption with Transitory Income Shocks, with and without Transaction Costs . . . . . . . . . . . . . . . . . . . . . . . 59 4.12 Housing with Transitory Income Shocks, with and without Transaction Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.13 Wealth Holdings with Transitory Income Shocks, with and without Transaction Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.1 Consumption Expenditure with Transitory Income Shocks and Leisure but without Habit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 5.2 Housing with Transitory Income Shocks and Leisure but without Habit 67 5.3 House Value with Transitory Income Shocks and Leisure but without Habit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.4 Wealth Holdings with Transitory Income Shocks and Leisure but with- out Habit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 5.5 Home Ownership with Transitory Income Shocks and Leisure but with- out Habit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5.6 Consumption Index with Transitory Income Shocks and Leisure but without Habit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 5.7 Consumption Expenditure with Transitory Income Shocks, Leisure and Habit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 5.8 Housing with Transitory Income Shocks, Leisure and Habit . . . . . . 71 5.9 House Value with Transitory Income Shocks, Leisure and Habit . . . 72 5.10 Wealth Holdings with Transitory Income Shocks, Leisure and Habit . 72 ix 5.11 Home Ownership with Transitory Income Shocks, Leisure and Habit . 73 5.12 Consumption Index with Transitory Income Shocks, Leisure and Habit 74 5.13 Consumption Expenditure with Lower Price of Consumption after Retire- ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.14 Housing with Lower Price of Consumption after Retirement . . . . . 76 5.15 House Value with Lower Price of Consumption after Retirement . . . 76 5.16 Wealth Holdings with Lower Price of Consumption after Retirement . 77 5.17 Home Ownership with Lower Price of Consumption after Retirement 77 6.1 Housing with Stochastic Health Shocks . . . . . . . . . . . . . . . . 82 6.2 Ownership with Stochastic Health Shocks . . . . . . . . . . . . . . . 83 6.3 Wealth Holdings with Stochastic Health Shocks . . . . . . . . . . . . 83 x Abstract In this paper, we explore whether a standard life cycle model in which house- holds purchase nondurable consumption and housing and face idiosyncratic income and mortality risk as well as endogenous borrowing constraints, enriched with several new components, like habit formation in housing, increasing leisure with retirement, stochastic health shocks, can account for key patterns of consumption, housing and asset holdings over the life cycle. First, consumption expenditures on non-housing consumption are hump-shaped. Second, the housing profile first increases monotoni- cally and then flattens out. Third, the household accumulates wealth until age 70s and then does not liquidate his wealth much until he dies. Fourth, as shown by Aguiar and Hurst, neither the quality nor the quantity of food intake deteriorates with retire- ment status, although food expenditures decrease dramatically. We develop a life-cycle model that explicitly incorporates the dual feature of housing as both a consumption good and an investment asset. Our analysis indicates that transaction costs in house trading are essential to explain the home ownership patterns in the data. The increase in leisure due to retirement helps to explain the non-deaccumulation in housing and wealth for the elderly people. Moreover, the composite of consumption goods bought from the market and consumption goods prepared at home, does not decrease with retirement. Stochastic health shocks and medical costs also help to explain the non- deaccumulation of housing and wealth for the elderly. Habit formation in housing brings some of the statistics closer to the data. The household acquires less housing and more consumption at each age with habit formation. Habit formation helps to explain the flat housing profile for the elderly but it still does not explain the wealth profile of the elderly. We thus conclude that a standard model with transaction costs in xi house trading should be enhanced with key features like leisure and health shocks to explain the key patterns in the data. xii Chapter1 Introduction Household portfolios is an interesting area for research with still some unanswered questions and puzzles. There is a wide range of household portfolio models. Among this range, the avenue with housing is very exciting. It is because many more people own homes than own shares. For many investors, housing is the largest component in their portfolio.The house is the single largest expenditure made by households over their life time. According to the 2001 Survey of Consumer Finances, so called SCF, on average, home value constitutes 55% of a homeowner’s total assets. In contrast, stock investments account for only 12% of household assets. In U.S., around 68% of people are homeowners. Moreover, a household who earns an average income, holds six times as much wealth in residential property as in shares. Housing is interesting and important not just because it is a large component of household portfolio but also because it is a special type of good. It is different from other financial assets in that housing has the dual role. First, it is a durable consumption good that provides housing service flow. Second, it is an investment vehicle that allows the household to hold home equity. It is clear that housing is a major source of utility to all the households. The housing service flows are as much important as food or luxury goods. Considering all these, homeownership and its impacts over the life cycle are incorporated to this paper. The first contribution is incorporating the habit formation on a good-by-good basis, so called deep habits to the preferences. It is known that experimentation with preferences should be done with caution since it changes one 1 of the fundamental building blocks of an economic model. Even if the model with the specified preferences explains household portfolio behavior, the model needs to be validated with reference to other types of household behavior. However there is another view that finds some exploration of flexible preference forms as fruitful and useful. So we decided that it will be fruitful to explore the implications of deep habits in life-cycle model with housing. The motivation of embedding habit formation to the model comes from the rel- ative success of habit formation models in solving asset pricing puzzles, aggregate consumption dynamics and in accounting for key business-cycle regularities. Habit formation models can differ along two different dimensions. First, while some papers such as, Campbell-Cochrane (1999) and Chan and Kogan (2002) use an external habit specification where the habit depends on the consumption of a reference group such as aggregate consumption, others assume that the stock of habit depends on the individ- ual’s own past consumption like, Sundaresan (1989) or Constantinides (1990). Since this model is a life-cycle model that solves the household’s optimization problem and is not a general equilibrium model, the habit process is specified as a function of the household’s past consumption. Moreover, in the existing related literature, the assump- tion of internal habit formation is prominent. Second dimension is that, some models specify the argument in the utility function as the difference between consumption and habit which is called additive habit models as in Constantinides (1990) or Campbell- Cochrane (1999). In other models, utility depends on the ratio between consumption and habit which is called multiplicative habit models, as in Abel (1990) or Chan and Kogan (2002). We incorporate the ratio specification in our model. Another question related to the use of habit formation is at what level habits are formed. That is, are habits created at the level of each individual consumption good 2 separately or at the level of the aggregate consumption. Ravn, Schmitt-Grohe and Uribe (2006), hereafter RSU, are motivated to propose a model of habit formation on a good-by-good basis (they call this as ’deep habits’) after reading the available empirical literature on consumption behavior. There is some empirical support for the hypothesis of rational addictive behavior at the level of individual consumption goods. Thus, we introduced a special type of internal habit formation. It is internal habit formation at the level of nondurable consumption and housing separately. How- ever RSU assumed that habits for each individual variety are external, purely because of analytical reasons. We separated consumption into nondurable consumption and housing. First, we assume that there is habit formation on both consumption and hous- ing separately, possibly with different habit strength parameters. Flavin and Nakagawa (2008) made empirical tests using household level data and rejected the existence of habit persistence in nondurable consumption. So we set the habit strength parame- ter in nondurable consumption to zero and assume there is habit formation in only housing. Actually this assumption made the model solvable, otherwise the problem becomes very complex and hard to solve in a reasonable time. The reason is, with this assumption, we decrease the number of continuous state variables by one. RSU did not consider housing specifically and they used external habits because otherwise consumer problem becomes more complex. Another question related to the structure of the model is : Should one build a partial equilibrium or a general equilibrium model? The assumption of infinite investor hori- zon is not realistic and it may not yield a good simplifying approximation to the port- folio behavior of some part of the population. Tractable general equilibrium models with habit in utility have usually representative agent structure and their implications 3 for the behavior of individuals are not well understood. Life-cycle models, are tradi- tionally used to examine the determinants of individual portfolio allocation, consump- tion and wealth profiles and their dynamics with age. Understanding how households make decisions about their savings and exposure to risk is important in asset pricing, macroeconomics and public finance. Net worth, housing value (gross and net of mort- gages) and income vary substantially among households by age. Thus we put habit in utility in the context of a life-cycle portfolio choice problem. This paper discusses calibration and numerical solution of a wide range of household portfolio models and explores the implications of them, isolating the individual contribution of each major factor wherever possible and understanding the main mechanisms. Even with social insurance (Medicare and Medicaid), households may have to make face substantial out-of-pocket medical expenses as stated in French and Jones (2004), Palumbo (1999) and Feenberg and Skinner (1994). The risk of making such medical expenses might cause precautionary savings and affect the wealth profile as stated by De Nardi, French and Jones (2006). The second contribution of this paper that may contribute to the literature is adding health shocks to the model. The goal of adding the health shocks to the model is to explore the effect of health shocks and their medical costs on the life cycle consumption and saving in an environment with hous- ing. This addition may enable us to understand some of the peculiar feature of house- hold portfolio choice that is observed in the data. Many elderly keep large amounts of assets until very late in life. Also elderly people do not liquidate their housing wealth late in life cycle. Micro data shows that household holdings of the housing stock are not hump- shaped: lifetime profile of housing stock is monotonically increasing and then stays rather flat. Housing has a different pattern than consumption since consumption has a 4 hump-shaped pattern over the life-cycle. Fang Yang (2006) states that this contradicts a key prediction of the standard life cycle model without market frictions: the ratio of housing and non-housing consumption should not be age-dependent. That is to say, housing consumption should follow the same pattern as non-housing consumption. Feinstein and McFadden (1989) suggest that more than one-third of elderly households are consuming large housing services since they live in dwellings with at least three more rooms than the number of inhabitants. As stated in Venti and Wise (2004), the Survey of Income and Program Participation (SIPP) shows that, housing ownership for two-person households is small for young households and increases to 75% at the age of 30 and then stays flat, declining only slightly after age 75. They find that in case of no shocks such as death of a spouse or entry of a family member into a nursing home, families continue home ownership. Even in case of a shock, discontinuing ownership is an exceptional situation. Another contribution of this thesis is to analyze the effects of the increase in leisure time that comes with retirement on household decisions including housing. The leisure time increases with retirement so the household has more time available to prepare food at home and to search for cheaper shopping opportunities. The person spends more time at home. So the home becomes more important after retirement. It is assumed that the leisure time is substitute with consumption expenditure and com- plement with housing consumption. As a result, we find that although consump- tion expenditure decrease with retirement, consumption itself does not. Consumption expenditure is a poor proxy for consumption although it is a commonly used and easily found one. Thus we embedded leisure to the model with housing so that it is substitute with consumption expenditure and complement with housing consumption. According to the results of this model, the agent does not decumulate his housing and his wealth 5 after retirement. The standard models produce both housing and wealth as decreasing after retirement. Due to more time available, the person can keep his consumption and housing without decreasing his wealth too much. Another assumption of the model is stochastic labor income. We make this assumption because both the level and risk of labor income change over the life-cycle of the household, and markets to insure such idiosyncratic risks are missing. Indeed, there are many financial economists who study the optimal portfolio allocation behav- ior in the context of models with stochastic uninsurable labor income and borrow- ing constraints. Heaton and Lucas (1996, 1997, 2000), Koo (1998), Viceira (2001) and Haliassos and Michaelides (2003), Cocco et al. (1999), Campbell et al. (2001), Dammon et al. (2001), Cocco (2001) are examples of those economists. Some of the model’s assumptions are stochastic labor income, stochastic house prices, uncertain life span with the lack of annuity market to insure against uncer- tain lifetime, habit formation separately over housing and utility from bequeathed wealth. The model assumes that housing can be used as collateral in borrowing and non-collateralized debt is not allowed.The interaction between housing and borrowing constraints results in the accumulation of housing stock early in life. In short, our model has many components so we are faced with numerous modeling choices. The main conceptual, technical and computational issues that arise in the context of house- hold portfolio choice will be illustrated and the implications of alternative modeling choices will be explored. We will try to illustrate the individual contribution of each major factor wherever possible. We investigated various factors that may cause non-deaccumulation of housing and wealth with aging while also producing the patterns of consumption and home ownership as in the data. Mainly, we analyzed the effects of transitory income shocks, 6 permanent income shocks, maintenance cost of home ownership, buying and selling cost of house trading, habit formation in housing, the increase in leisure time with retirement and stochastic health shocks and medical costs. First of all, we find that transitory income shocks are essential to explain the grad- ual increase in home ownership rate in the young ages. However transitory income shocks alone do not help us to explain the non-deaccumulation of housing and wealth with aging. Permanent income shocks do not help us to explain the patterns of con- sumption, housing, wealth and home ownership. Habit formation in housing helps us to explain the non-deaccumulation of housing with aging but it does not explain non-deaccumulation of wealth with aging. With a higher habit strength in housing, home ownership in the early ages becomes more gradual as in the data. The models with components described so far, produce much higher home ownership rates than the data. So we added transaction costs to the model. First, we added only the maintenance cost of owning a house. Adding the maintenance cost decreases the deaccumulation of housing in the old ages. However with the reasonable values for maintenance cost from the literature, we still cannot decrease the deaccumulation of housing in the old ages to the values in the data. Thus, maintenance cost helps us to decrease the deaccumulation of housing in the old ages but it is not enough. If we have maintenance cost together with habit formation in housing, we can decrease the deaccumulation in housing in the old ages a lot. Also we get the statistics of house value/labor income, average value of hump in the consumption much closer to data. With only maintenance cost as a transaction cost, we still get a very high value of average home ownership rate. This is why we need to add the buying and selling costs of house trading. Adding buying and selling costs to a standard model produces a home ownership rate close to the data. However if we abstract from habit formation in housing, we get a high decumulation of 7 housing and ownership rate in the old ages even with all the transaction costs. So far, in the models described above, we get high deaccumulation of wealth in the old ages. When we add leisure time in a way that it is substitute with non-durable consumption expenditure and complementary with housing expenditure, we succeed to decrease the deaccumulation of wealth and deaccumulation of housing in the old ages. With only leisure in the model, if we do not put habit formation or transaction costs, we overesti- mate the statistics average home value/labor income and average value of the hump in the consumption and average home ownership rate. A model without leisure will have quick deaccumulation of wealth in the old ages. Buying and selling costs are necessary to get the average home ownership rate in the data. 1.1 RelatedLiterature There is a huge related literature. For instance, Heaton and Lucas (1997) study the effects of habit formation in an infinite horizon model with uninsurable labor income risk. Polkovnichenko (2002) also study impacts of a life-cycle model with internal, additive habit formation preferences on asset allocation. In Polkovnichenko’s additive difference habit model, risk aversion is now a function of consumption relative to the habit level. He states that, a stronger habit preference, besides making the investor more willing to smooth consumption intertemporally, also increases risk aversion and prudence. As a result, strengthening the habit motive generates even more wealth accu- mulation early in life than in the ratio habit model. Sundaresan (1989), Constantinides (1990), Detemple(1995), Chapman (1998) and Campbell and Cochrane (1999) also explore additive habit models. Some of the other related papers that are about optimal portfolio choice in the pres- ence of consumer durables, are Grossman and Laroque (1990), Cocco (2000), Flavin 8 and Yamashita (2002), Flavin and Nakagawa (2002), Campbell and Cocco (2003), and Yao and Zhang (2005) and Ortalo-Magne and Rady (2006). Hindy, Huang and Zhu (1997) used an infinite-horizon complete markets set up to explore the effects of habits and consumption durability on portfolio allocation. Rodriguez, Diaz-Gimenez, Quadrini and Rios-Rull (2002) gives useful informa- tion on age-wealth profile which can be used in calibration. They state that some of the inequality in earnings, income and wealth maybe due to age. They define earn- ings as income obtained from labor. They define wealth as net worth of the house- hold, in other words stock of unspent past income. Capital income is obtained from wealth. They partition the SCF sample into 10 cohorts according to the age. Then they compute the statistics for each cohort and compare them with those of the entire sample. Average labor income or earnings is monotonically increasing with the age of the household until age 55 and it decreases thereafter. Average cohort wealth also increases monotonically with the life-cycle. However it reaches its peak in the 61- 65 cohort. Moreover, the over-65 cohort is still significantly wealth-rich: It owns 33 percent more wealth then the sample average. In the presence of uncertainty on the date of death, saving is higher (Yaari (1965), Davies (1981). It is also so if one saves as a precaution against unexpected income or health shocks (Hubbard, Skinner, Zeldes (1995)). In U.S., more than a third of the elderly live with at least three more rooms than inhabitants and overconsume housing (Feinstein and McFadden (1989)). Venti and Wise (1990ab) or Hurd (1987) find lit- tle movement out of home ownership and that those who move buy more expensive homes. However the poor in income but rich in housing are more likely to reduce the value of their home when moving, and conversely the housing poor and income rich are more likely to increase it (see also Heiss et al.(2003)). Globally the decrease in 9 house value after age 75 comes from those who experience a shock such as the loss of a spouse. Intact families hardly reduce their house value. Heiss et al. (2003) also find a decrease in owner-occupation after 80, with no difference between cohorts: 50 to 60 percent of couples owning their home at 70, will own it at the death of the surviving spouse. According to Skinner (1996), housing acts as precautionary saving in a life cycle model with uncertainty. De Nardi, French and Jones(2006) explore why many elderly do not decumulate their assets until very late in life. They find that the risk of living long and facing high medical expenditures explains a big portion of elderly’s high savings. There are many other studies that considered medical expense risk as a source of large asset holdings even at advanced ages. They estimate their model using the method simulated methods of moments and AHEAD data. Their focus is only retired people. They say that the elderly need to keep a large amount of assets since they are almost sure to pay large out-of-pocket medical costs if they live to very advanced ages. Venti and Wise (2000) state that home equity is usually not liquidated to buy non- housing consumption goods.Their findings indicate that people buy homes to live as they age during retirement years. In other words, home acts like a reserve or buffer that can be used in catastrophic circumstances. These results match with the findings of a survey by American Association of Retired Persons. The respondents answered the question: Do you agree with the statement: What I would really like to do is stay in my current residence as much as possible?. 75% of people aged between 45 and 54, 83% of people aged between 55 and 64, 92% of people aged between 65 and 74, 95% of people aged at least 75 agreed with this statement. So under these results, it seems that housing equity should not be counted on while checking whether families have saved enough to maintain pre-retirement standard of living after retirement. 10 Previous studies have documented a dramatic decline in food expenditures at retire- ment. In the household consumption literature, usually, food expenditures are used as the measure of non-durable consumption. One reason for that is, PSID only report food expenditures out of nondurable goods. Another reason for that is, food is a nec- essary good with a small income elasticity. According to CSFII data, consumption expenditures fall by 17 percent at retirement. On the other hand, time spent in food production increases by 53 percent at retirement. Aguiar and Hurst (2005) state that the pattern of expenditures may be different from that of actual consumption, given the sharp increase in time spent shopping for and preparing food. They show that neither the quality nor the quantity of food intake worsens during retirement. They think that the elasticity of substitution between time and expenditures may be large in food production. Given home production, food expenditures may mask consumption smoothing of individuals. Aguiar and Hurst (2008) also state that consumption expenditure increases through middle age and then decreases fast thereafter. The entire decline in nondurable expen- diture late in life cycle is due to the declines in food, non-durable transportation and clothing. Expenditure on these three categories are positively correlated with market work hours. Food can be prepared at home. Transportation and clothing are inputs to the market work. The remaining components of nondurable expenditure including housing services, which are the other half, do not decrease over the second half of the life. Half of the components of total nondurable expenditure actually increase during retirement. These patterns are consistent with the view that some goods are comple- ments with time like entertainment and others being substitutes with time, like food via home production. Moreover, employment rates and market work hours begin to decline in the mid 40s and begin to decrease sharply in the early 50s. 11 1.2 GeneralModel The model is set in partial equilibrium and is concerned with life-cycle choices of consumption, housing and borrowing, lending for an investor with endogenous (inter- nal) habit formation preferences. We assume a typical economic environment for a life cycle model. The investor receives labor income, which is subject to uninsurable transitory income shocks and permanent income shocks during the working life and is constant during retirement periods. We assume that exogenous variables follow a finite state markov chain. Under these assumptions, we solve the problem numerically. The investor derives utility from consumption of housing and consumption of nondurable consumption goods. The decision to buy a house versus rent a house is also considered. So different from many existing papers in the literature, rental market exists. The price of nondurable goods is fixed and normalized to 1. Investors may borrow up to the market value of the house they own minus a downpayment, which is a proportion of current market value of the house. 1.2.1 PreferencesandLaborIncome There is one agent in the model who lives a finite life of at most length T. The person derives utility from consuming numeraire good consumptionC t , consuming housing servicesH t and from bequeathing wealthW t . The preferences are represented by the following modified Cobb-Douglas utility function. Cobb-Douglas utility function is used by many authors in the literature including Li and Yao (2006), Campbell and Cocco (2005), Yao and Zhang (2005), Chambers, Garriga, and Schlagenhauf (2004), Davis and Heathcote (2003), Gervais (2002). The parameter measures the share of housing services in the composite consumption. The parameter measures the 12 curvature of the utility function with respect to the composite good. The single-period utility function depends on the quasi ratio of current consumption to the habit stock of each variety (namely numeraire consumption good and housing services). Different from the relative deep habits of Ravn, Schmitt-Grohe and Uribe (2005), internal habit is used instead of external habit. The parameters 1 and 2 measure the degree of time nonseparability in consumption of numeraire good and housing services. When 1 = 0, 2 = 0; we have time separable preferences. Specifically, we also allow for the following case:. 1 = 0 and 2 7 0 where there is habit formation on only housing services consumption. Flavin and Nakagava(2004) also separate composite consumption into housing and consumption. They assumed that nondurable consumption has habit persistence but then they found that the coefficient of habit persistence is not significantly different from zero. That is why we also assume that the habit strength in nondurable consump- tion is equal to zero. U =E 8 > > < > > : T X t=0 t 0 B B @ F(t) 1 1 ( Ct C 1 t1 ) 1 ( Ht H 2 t1 ) (1 +(F(t1)F(t))B(W t ;LB) 1 C C A 9 > > = > > ; (1.1) where E denotes the mathematical expectations operator conditional on information available at time t and U is a period utility index assumed to be strictly increasing in its first argument, strictly decreasing in its second argument, twice continuously differentiable and strictly concave. F(t) denotes the probability of being alive at time t. C t denotes consumption at time t, H t denotes housing at time t and B(W t ;LB) denotes utility from leaving wealth ofW t to his kid as a bequest at time t. Y t =P y t e f(t;Zt) " t (1.2) 13 whereY t denotes labor income at t,e f(t;Zt) denotes age dependent deterministic income profile and" t denotes transitory income shock at t. P y t =P y t1 v t (1.3) wherev t denotes shock to the permanent income at time t. F(t) = t Y j=0 j (1.4) where j is the probability that household is alive at timej conditional on being alive at timej1: It is assumed that 0 = 1; T = 0 and0< j < 1 for0<j <T: Here is the subjective discount factor, is the curvature parameter, B(:) is the bequest function,W t is the bequeathed wealth and LB controls the strength of bequest motives. The household receives stochastic labor incomeY t which is the product of perma- nent incomeP y t , transitory shocks t and the deterministic age dependent component e f(t;Zt) . Permanent incomeP y t is the product of previous period’s permanent income and shock to permanent income of that period. It is assumed that log( t ) and log(v t ) are i.i.d normal with means such that their exponents has a mean of one. It will be assumed that during the retirement years, labor income,Y t becomes con- stant and equal to a fraction (replacement ratio) of the income without the transitory component just prior to retirement. Y t = P y 65 e f(65;Z 65 ) forall t> 65: (1.5) 14 The generated income profile mimics that of Attanasio (1995), Gourinchas and Parker(2002), Hubbard, Skinner, Zeldes (1995). 1.2.2 HousingContract A person can consume housing services either through renting or owning the house. D o t is an indicator variable. If the person is a renter,D o t = 0, otherwiseD o t = 1: To rent the house, the person has to pay a constant fraction ( r ) of the market value of the house. To buy the house, the person pays all of the market value of the house. The house price appreciation rate follows an i.i.d normal process with mean rh and standard deviation rh : It is assumed that the shock to house prices is exogenous and permanent, following many authors in the literature including Flavin and Yamashita (2002), Yao and Zhang (2005), Li and Yao (2006), Campbell and Cocco(2003), Cocco(2005). 1.2.3 BorrowingandSavingwiththeRisklessRate In addition to holding home equity, a household can save in liquid assets which earns the constant real risk free interest rate, r. The net savings (or borrowing if negative) is denoted asM t : It is assumed that households cannot borrow non-collateralized debt. ThusM t has to satisfy the following constraint: M t (1d)P h t H t D o t : (1.6) where d is called down payment ratio. Thus the person has to pay at least proportion d of the house’s value to buy the house. If the person does not own a house, he cannot borrow any money. 15 1.2.4 BudgetConstraints In this model, we abstract from housing transaction costs which simplifies the analysis. In the budget constraints, C t denotes consumption at time t, P h t denotes house price at t,M t denotes net saving at t, r denotes risk free interest rate,P y t denotes permanent labor income at t," t denotes the transitory income shock at t and r denotes the rental cost as a percentage of house value. D o t is an indicator variable which is equal to 1 if the person is a homeowner at t and 0 if renter. The intertemporal budget constraint for a household for the benchmark model, can be written as follows: 1) For a renter at period t-1 who decides to be a homeowner at the current period t (D o t1 = 0;D o t = 1 ) C t +P h t H t +M t =M t1 (1+r)+P y t e f(t;Zt) " t (1.7) 2)For a renter at period t-1 who decides to continue to be renter at the current period t (D o t1 = 0;D o t = 0) C t + r P h t H t +M t =M t1 (1+r)+P y t e f(t;Zt) " t (1.8) 3)For a homeowner at period t-1 who decides to be a renter at the current period t (D o t1 = 1;D o t = 0 ) C t + r P h t H t +M t =M t1 (1+r)+P h t H t1 +P y t e f(t;Zt) " t (1.9) 4)For a homeowner at period t-1 who decides to continue to be a homeowner at the current period t (D o t1 = 1;D o t = 1 ) C t +P h t H t +M t =M t1 (1+r)+P h t H t1 +P y t e f(t;Zt) " t (1.10) 16 If we combine these four cases in one constraint, we get the following equation: C t +[ r (1D o t )+D o t ]P h t H t +M t =M t1 (1+r)+D o t1 P h t H t1 +P y t e f(t;Zt) " t (1.11) 1.2.5 TheOptimizationProblemoftheHousehold The household’s problem at time t=0 is represented as follows: max fCt;Ht;Mt;D o t g E 8 > > < > > : T X t=1 t1 0 B B @ F(t) 1 1 ( Ct C 1 t1 ) 1 ( Ht H 2 t1 ) (1 ) +(F(t1)F(t))B(W t ;LB) 1 C C A 9 > > = > > ; subject to C t +[ r (1D o t )+D o t ]P h t H t +M t =M t1 (1+r)+D o t1 P h t H t1 +P y t e f(t;Zt) " t M t (1d)P h t H t D o t W t =M t1 (1+r)+P h t H t1 C t 0;H t 0 P h t =P h t1 (1+r h t ) P y t =P y t1 v t D o t 2f0;1g where d denotes the downpayment requirement ratio, is the time discount factor, F(t) denotes the probability of being alive at time t and LB denotes the bequest strength parameter. C t denotes consumption at time t, P h t denotes house prices at t, M t denotes net saving at t, r denotes risk free interest rate, P y t denotes permanent labor income at t," t denotes the transitory income shock at t and r denotes the rental 17 cost as a percentage of house value. D o t is an indicator variable which is equal to one if the person is a home owner at t and 0 if renter. 18 Chapter2 TheModelinaUniformlyDistributed Populationbyage 2.1 Model with Transitory Income Shocks and Deep Habits In this part, the general model explained in Chapter 1 is used. However we set the per- manent income shocks,v t to 1, at every period t. So in this part, the only income shock is the transitory income shock. There are no well known values for the habit formation parameters, 1 and 2 in the literature. There are empirical studies which estimate the habit persistence in nondurable consumption as zero. In the light of those studies and also for computational feasibility, we assigned the value zero to 1 and we assigned various values between 0 and 1 to 2 : In this way, we have consumption at time tC t as a choice variable and housing at time tH t as a state variable. If we assign 1 to a number greater than zero, consumption also becomes a state variable. This makes the already complicated problem, much more complicated. The higher the number of state variables, the longer time the computer needs to give results. By looking at var- ious values of 2 ; we are able to see the effects of different levels of habit formation on housing on household decisions. After the value function is calculated for all the points in the grid, simulations are performed. The results of the simulations need to be averaged to get the life time consumption, housing and all the other decision values. 19 In the model in this chapter, while we are taking the averages of all the simulations, we assume that the population is uniformly distributed across all the age groups. For example, the number of people who are 30 years old is equal to the number of people who are 80 years old and etc. This assumption is not a realistic assumption because in the data there are different number of people in different age groups. This is why we relax this assumption in Chapter 3. We abstract from transaction costs in house trading in this chapter for simplicity. 2.2 ParameterCalibration Two sets of parameters are distinguished in this model: those that can be estimated independently of the model or those that are based on estimations provided by other papers and those that are chosen so that the model-generated statistics match a given set of targets from the data. In our numerical analysis, the model is established with the following parameter values. It is assumed that individuals in our economy begin their economic life at age 20 (that is, t = 1 corresponds to age 20) and live until a maximum age of 90 (t = 71). The mandatory retirement age is 65 (t = 46). The conditional survival rates are taken from Anderson (2001). The 1998 life tables of the U.S. National Center for Health Statistics is used in Anderson (2001) for the calibration of the annual mortality rate. It is assumed that the household makes decisions annually. The age dependent deterministic labor income profile,f(t;Z t ) is chosen to reflect the hump shape of the labor income over the life cycle. f(t;Z t ) is taken from Cocco, Gomes, and Maenhout (2004). They fitted a third order polynomial to the labor income, using the PSID data. The function f(t;Z t ) is additively separable in t and Z t : Z t denotes personal characteristics other than age, including marital status and 20 household composition. The labor income in their estimation also include unemploy- ment compensation, welfare and transfers. They regressed the logarithm of the labor income to the age dummies and dummies for other characteristics. They fitted a third order polynomial to the age dummies and obtained labor income profiles. In the model, we use the polynomial fitted for high school graduates. The income profile is hump shaped over the life cycle before retirement. After the individual retires at age 65, he will start receiving pension income which is a constant fraction of his labor income at age 64. This constant fraction is called replacement ratio and is taken to be 0.6 for high school graduates. The replacement ratio is different for different education groups. The corresponding income replacement ratios are similar to the ones in the study of Cocco, Gomes, and Maenhout. Cocco, Gomes and Maenhout calibrated the replacement ratio by dividing the average labor income for retirees to the average labor income in the last working year prior to retirement. The parameters for the income shocks are taken from Li and Yao (2006). Specifically, the transitory shock’s standard deviation( ) is taken to be 0.27. These estimates are similar to the ones reported in Storesletten, Telmer and Yaron (2004). The utility function, Cobb-Douglas aggregator between housing services and non- housing consumption, is of the constant relative risk aversion class, which is standard in the wealth distribution literature. It is not unreasonable to assume unit elasticity between housing services and non-housing consumption, based on empirical evidence (see Fernandez-Villaverde and Krueger (2002)). The curvature parameter is set to 2.3. Standard values for this parameter in life-cycle models are between 1-6. The benchmark model has a habit parameter of 1 = 0: We set 2 to different values and analyze the effects. We also set both 1 and 2 to zero in one combination in order 21 to identify the effects of habit formation in housing on optimal consumption, housing and portfolio choice. In the U.S. economy, a typical value for downpayment ratios which are fractions of the house value that has to be paid at the time of the purchase, is 20% (see Fernandez- Villaverde and Krueger (2002)). For this reason the borrowing constraint is expressed so that agents can borrow up to 80% of the house they want to buy. The average post-WWII real return available on T-bills is approximately 3 percent so the real risk free rate is set to 0.03 in the model. In the model, since we abstract from transaction costs such as house selling, pur- chase, maintenance and refinancing costs, we only need to set the annual rental cost. In order to be consistent with the historical real costs of renting and owning, the annual rental cost is set at 4.5% of the market value of the rental property. This number is lower than the numbers used in the literature. I abstract from transaction costs of home ownership like maintenance cost in this part. So renting a home would be very expensive compared to home ownership with a value from literature. The house price appreciation rate ~ r h t follows an i.i.d normal process with mean rh and variance 2 rh : It is assumed that the housing appreciation rater h t is serially uncor- related and it has a mean zero. The shock to prices is thus permanent and exogenous. Based on 80 quarters of housing index data between March 1980 and March 1999, Goetzmann and Spiegel (2002) estimated the real housing returns for the 12 largest metropolitan statistical areas. They find that the returns vary from -1 percent to 3.46 percent. The assumption of zero mean falls within this range. The volatility of housing return is set at rh = 0:1. This value is between the aggregate index level of Goetz- mann and Spiegel (2000) and the upper bound of the empirical estimates of Flavin and 22 Yamashita (2002). It is assumed that there is no correlation between the housing return and the labor-income growth. We need to choose the subjective discount rate parameter and the bequest strength parameter LB. The targets used in the calibration of these two parameters are average wealth-labor income ratio and over 65 wealth to average wealth ratio. The bequest function is taken from Li and Yao (2006). Due to the Cobb-Douglas utility, the beneficiary’s expenditure on numeraire consumption and housing service consumption will occur in a fixed proportion of 1 . Then the bequest function will be defined by the following form: B(W t ;LB) = LB 1 [(1) (1)(1 1 ) (1 2 ) W (1)(1 1 )+(1 2 ) t ( r P h t ) (1 2 ) ] (1 ) (2.1) 2.3 ResultswithUniformPopulationDistribution The model described in 2.1 is developed in three steps in this part. Step1: The transitory income shocks are set to one at every period and both habit formation parameters are set to 0. So the model in this step is simple model without income shocks and without habit in housing and consumption. Since there are no income shocks in this model, labor income is deterministic. Step 2: The transitory income shocks are added to the model in Step 1. Labor income becomes stochastic in this step. Still, there is no habit formation. By compar- ing the results of the models in Step 1 and Step 2, we can see the effects of transitory income shocks. 23 Step3: Habit formation in housing is added to the model in Step 2. So by compar- ing the results of models in Step 2 and Step 3, we can see the effects of habit formation in housing. 2.4 ResultsoftheModelinStep1 Table 2.1 shows some of the results of the simple model with deterministic income and without habit formation. The numbers in parenthesis show the results when there is no bequest. The first column shows the average statistics for the U.S. data. The average statistics for the U.S. data are calculated in Li and Yao (2006) using data from SCF, 1995-2001. The parameters beta and bequest constant are calibrated so that Average Net Worth-Labor Income Ratio and Over 65 Wealth-to-Average Wealth Ratio match to the corresponding values in the data. From the first row, we see that Average Home Ownership Rate is higher than its data value. One of the reasons may be the absence of transaction costs in house trading. Also, with the existence of bequest motive home ownership rate is higher. This seems intuitive since the person buys home not just for consumption purpose only but also to leave the house to his kid. From the third row, we see that Average House Value to Labor Income ratio is higher than its data counterpart. We develop the model to make it closer to data in the following parts to remove these discrepancies. 2.4.1 EffectsofBequestMotive Whether there is bequest motive or not, Consumption ( Figure 2.1) and House Value (Figure 2.2) are hump shaped. Consumption and Housing of the elderly is lower with bequest motive. Since the elderly person wants to leave wealth to his kid, he will 24 Table 2.1: Statistics of the Model in 2.1, Step 1 Statistics Data BenchmarkModel Avg:HomeOwnershipRate 0:68 0:810(0:78) Avg:NetWorthLaborInc:Ratio 3:10 3:007(3:08) Avg:HouseValueLaborInc:Ratio 2:77 5:850(6:02) Over 65WealthAvg:WealthRatio 1:33 1:328(1:17) Avg:Sizeof theHumpinCons: 1:90 2:100(2:16) choose to consume less. From Figure 2.4 a, Consumption is also hump shaped in the data. The model generated consumption does not decrease as much as that in the data for the elderly. However Housing is not hump shaped in the data, it first increases and then stays almost flat in the old ages. So a standard model with deterministic income and without habit, cannot mimic the housing in the data. Moreover, as seen from Figure 2.3, without bequest motive, the elderly decumulates almost all of his wealth. With bequest motive, he does not spend all of his wealth. He dies with some wealth to leave to his kid. As it can be seen from Figure 2.5 , there is little wealth decumulation for the elderly in the data. 2.5 ResultsoftheModelinStep2 As seen from Figure 2.8, the shape of the house value becomes smoother with the addi- tion of transitory income shocks. From Figure 2.9, home ownership starts to increase earlier and more gradually with transitory income shocks. The elderly person does not become renter in this case. This is also the case in data.(see Figure 2.5) Wealth accumulation starts earlier with transitory income shocks. Wealth decumulation is less with stochastic labor income. These results show that adding transitory income shocks to the model is essential. 25 Figure 2.1: Consumption with Deterministic Income and without Habit 2.5.1 EffectsofBequestMotiveintheModelinStep2 Home ownership rate is lower in the middle ages when there is bequest motive. The person stays as a homeowner in the old ages only when there is bequest (see Figure 2.11). Wealth accumulation is lower with bequest. Wealth decumulation is faster and higher without bequest (see Figure 2.12). These results show that having bequest motive make the results of the model closer to the data. 2.6 ResultsoftheModelinStep3 Model in Step 3 is obtained by adding habit formation in housing is to the model in Step 2. As seen from Table 2.2 and 2.3, House Value over Labor Income decreases as 2 increases. Home Ownership rate is much higher than the value in the data for all 2 . We abstract from all kinds of transaction costs of house trading up to now. Transaction costs may help to decrease the home ownership rates. Without transaction costs of house trading, house becomes a liquid investment vehicle and this increases 26 Figure 2.2: Housing Value with Deterministic Income and without Habit the motivation to own a house. The statistics, Net Worth over Labor Income Ratio and Over 65 Wealth/ Average Wealth are very close to the values in the data since we chose the parameters beta and LB to hit these two values in the data. Average Hump in Consumption, which is found by dividing the peak value of consumption with the initial consumption, is close to the value in data for all habit formation parameters. Table 2.2: Statistics of the Model in 2.1, Step 3 Statistics Data 2 =0:1 2 = 0 2 = 0:2 2 = 0:5 HomeOwnership 0:68 0:85 0:86 0:88 0:93 N:Worth=L:Income 3:10 3:14 3:09 3:14 3:06 HouseVal:=L:Income 2:77 6:15 5:86 5:10 3:69 Over 65W:=Avg:W: 1:33 1:33 1:32 1:33 1:32 HumpinConsumption 1:90 2:11 1:99 1:90 1:93 27 Figure 2.3: Wealth Holdings with Deterministic Income and without Habit Figure 2.4: Consumption and Housing in the Data (a) Consumption (b) Housing OptimalLifeCycleHousingandConsumptionDecisions As seen from Figure 2.13, consumption becomes closer to the data as 2 increases. That is because, the difference between the levels of initial and final consumption decrease with the increase in 2 . The peak of the consumption is reached between ages 45 and 55. From Figure 2.14, we see that the shape of the house value is very hump shaped for low values of 2 . As 2 increases, the hump shape changes. For high values of 2 , the house value does not decrease much for the elderly. From Figure 2.15, we 28 Figure 2.5: Wealth and Home Ownership in the Data (a) Wealth Composition (b) Home Ownership Figure 2.6: Housing with Transitory Income Shocks and without Habit see that wealth holdings is similar for different habit formation parameters. So we can say that we need something different from habit formation to explain the slow wealth decumulation for the elderly. However habit formation is useful in explaining the slow decumulation of housing wealth for the elderly. 29 Figure 2.7: Consumption with Transitory Income Shocks and without Habit Table 2.3: Statistics of the Model in 2.1, Step 3, Continued Statistics Data 2 = 0:5 2 = 0:55 2 = 0:6 HomeOwnership 0:68 0:93 0:93 0:93 N:Worth=L:Income 3:10 3:06 3:13 3:02 HouseVal:=L:Income 2:77 3:69 3:47 3:17 Over 65W:=Avg:W: 1:33 1:31 1:32 1:32 HumpinConsumption 1:90 1:93 1:90 1:92 30 Figure 2.8: House Value with Transitory Income Shocks and without Habit Figure 2.9: Home Ownership with Transitory Income Shocks and without Habit 31 Figure 2.10: Wealth Holdings with Transitory Income Shocks and without Habit Figure 2.11: Home Ownership with Transitory Income Shocks with and without Bequest 32 Figure 2.12: Wealth with Transitory Income Shocks with and without Bequest Figure 2.13: Consumption with Habit Formation in Housing and Transitory Income Shocks 33 Figure 2.14: House Value with Habit Formation in Housing and Transitory Income Shocks Figure 2.15: Wealth Holdings with Habit Formation in Housing and Transitory Income Shocks 34 Chapter3 TheModelinaNon-Uniformly DistributedPopulationbyage 3.1 ModelwithTransitoryIncomeShocks,Permanent IncomeShocksandDeepHabitsunderRealPopu- lationDistribution We use the model described in 1.2. When we compare the model here with the model in the previous chapter, we are extending that model with permanent income shocks. The model in this part is also solved using value function. After the value function is calculated for all the points in the grid, simulations are performed. The results of the simulations need to be averaged to get the life time consumption, housing and all the other decision values. While we are taking the averages of all the simulations, we no longer assume that the population is uniformly distributed across all the age groups. We use the distribution of population from the U.S. data. For example, the number of people who are 30 years old is higher than the number of people who are 88 years old. Since we relax the uniform population distribution assumption, the model becomes closer to the data in this chapter. We still abstract from transaction costs in house trading. The model in this chapter is developed in three steps. 35 Step1: The permanent income shocks are set to one at every period and both habit formation parameters are set to zero. So the model in this step is a simple model with- out permanent income shocks and without habit in housing and consumption. Income is still stochastic but with only transitory income shocks. Step2: The permanent income shocks are added to the model in Step 1. Still, there is no habit formation in this step. Step3: Habit formation in housing is added to the model in Step 2. 3.2 ParameterCalibration Some of the parameters are the same with the parameters in 2.2. So we will mention only the different ones here. For instance the curvature parameter is set to 1.5 which is in the range of standard values. The transitory and permanent income shock’s standard deviation are taken from Cocco, Gomes, and Maenhout (2004). The real risk free rate is set to 0.02. The annual rental cost is set to 7.5% of the market value of the rental property. This value is widely used in the literature. 3.3 ResultsoftheModelinStep1 This is the model with only transitory income shocks. We approximated the transitory income shock first with a two states Markov Chain and then with a three states Markov Chain. Table 3.1 shows the statistics of the simulations. House Value over Labor Income, Home Ownership Rate and Average Hump of the Consumption are higher than their corresponding values in the data. The values of the statistics do not change much whether the transitory shock has two states or three states. 36 Table 3.1: Statistics of the Model in 3.1, Step 1 Statistics Data 2StatesTr. Shock 3StatesTr. Shock HomeOwnership 0:67 1.00 0.99 N:Worth=L:Income 3:10 3.08 3.10 HouseVal:=L:Income 2:77 7.44 7.44 Over 65W:=Avg:W: 1:33 1.32 1.32 HumpinCons: 1:90 3.43 3.30 3.3.1 OptimalLifeCycleDecisionswithTransitoryIncomeShocks As seen from figures 3.1, 3.2 and 3.3, the consumption, housing and wealth holdings decisions are not sensitive to the number of states in the Markov Chain that we use to approximate the transitory income shocks. However the home ownership decision becomes closer to data if the transitory income shock has three states. Ownership increases gradually and then stays flat with three 3 states. Ownership is always flat with two states. The graph of housing, Figure 3.2, is much smoother when compared with the Figure 2.6. So housing is still hump shaped with a non-uniformly distributed population but has a much smoother shape. 3.4 ResultsoftheModelinStep2 This is the model with only transitory income shocks and permanent income shocks. We approximated the transitory income shock with a three states Markov Chain. We looked at the results first when there is no permanent income shocks and then when there are permanent income shocks with three possible states. Table 3.2 shows the statistics of the simulations. House value over labor income, home ownership rate and average hump of the consumption are higher than their corresponding values in the data. The values of the statistics are sensitive to the addition of permanent income 37 Figure 3.1: Consumption with two or three States Transitory Income Shocks shocks. The amount of the hump in consumption increases a lot with the addition of permanent income shocks. Table 3.2: Statistics of the Model in 3.1, Step 2 Statistics Data Tr. Shock Tr.Sh. andPer. Sh. HomeOwnership 0:680 0.995 0.996 NetWorth=L:Income 3:100 3.102 3.215 HouseValue=L:Income 2:770 7.431 7.315 Over 65Wealth:=Avg:W: 1:330 1.324 1.332 HumpinConsumption 1:900 3.175 5.919 3.4.1 OptimalLifeCycleDecisionswithTransitoryIncomeShocks andPermanentIncomeShocks As seen from the Figure 3.5, the Consumption reaches a higher peak point with the addition of permanent income shocks. From Figure 3.6, Housing also reaches a higher peak point. The decumulation of Housing in the old ages is higher with permanent 38 Figure 3.2: Housing with two or three States Transitory Income Shocks income shocks. Figure 3.7 shows that the household accumulates more wealth before retirement with permanent income shocks. Also the person decumulates more wealth after retirement. As seen from Figure 3.8, Home Ownership is earlier with the addition of permanent income shocks. It is more gradual without permanent income shocks. 3.5 ResultsoftheModelinStep3 This is the model with transitory income shocks, permanent income shocks and habit formation in housing. We approximated the transitory income shock and permanent income shock with three states Markov Chains. Table 3.3 shows the statistics of the simulations. For all levels of habit formation, home ownership rate is higher than the value in the data because of the absence of transaction costs in house trading. House Value over Labor Income is higher than its value in the data except in the cases 2 = 0:85 and 2 = 1. It is very close to data when 2 = 0:7. House Value over Labor Income decreases as 2 increases from 0 to 1. So we can say that if we choose the 39 Figure 3.3: Wealth Holdings with two or three States Transitory Income Shocks level of habit formation on housing, 2 , to match the House Value over Labor Income in the data, 2 should be between 0.7 and 0.85 and closer to 0.7. Average Hump in the Consumption is higher than the data for all habit formation parameters, although it decreases generally as 2 increases. Since we chose the parameters beta and bequest strength parameter to match the Networth/Labor Income and Over 65 Wealth/ Average Wealth, those values are very close to their counterparts in the data for all habit strength parameters. Table 3.3: Statistics of the Model in 3.1, Step 3 Statistics Data 2 = 0 0:2 0:4 0:5 0:7 0:85 1 H:Ownership 0:680 0.996 0.995 0.998 0.994 0.992 0.989 0.998 N:Wo:=L:In: 3:100 3.215 3.112 3.202 3.136 3.112 3.137 3.113 H:Val:=L:In: 2:770 7.315 6.495 5.600 4.836 3.320 1.960 1.030 +65W:=Avg:W 1:330 1.332 1.343 1.332 1.342 1.349 1.354 1.326 HumpinCon: 1:900 5.919 5.694 5.556 4.949 4.269 4.233 4.46 40 Figure 3.4: Home Ownership with two or three States Transitory Income Shocks 3.5.1 Optimal Life Cycle Decisions with Transitory Income Shocks, Permanent Income Shocks and Habit Formation in Housing As we see from Figure 3.9, the person reaches a higher peak level in Consumption as the strength of habit formation in housing, 2 , increases. Also the decrease in Con- sumption in the old ages is higher as 2 increases. From Figure 3.10, for all habit strength values, Housing first increases until the retirement and then decreases. The initial increase in Housing decreases as the value of 2 gets higher. The deaccumu- lation in Housing in the old ages is less with higher values of 2 . From Figure 2.4, we do not see a quick decumulation in Housing in the old ages. So the increase in 2 makes the housing results closer to data qualitatively. As seen from Figure 3.13, in all cases, we see high wealth decumulation for the elderly people on the contrary to the data. Thus adding habit formation in Housing to the model does not help to get the puzzling low wealth decumulation of the elderly people. As seen from Figure 3.12, 41 Figure 3.5: Consumption under Transitory Income Shocks and with or without Perma- nent Income Shocks for the values of 2 between 0 and 0.85, the increase in the home ownership rate in the young ages becomes more gradual as 2 increases. Home Ownership stays flat in the old ages. 42 Figure 3.6: Housing under Transitory Income Shocks and with or without Permanent Income Shocks Figure 3.7: Wealth under Transitory Income Shocks and with or without Permanent Income Shocks 43 Figure 3.8: Home Ownership under Transitory Income Shocks and with or without Permanent Income Shocks Figure 3.9: Consumption with Transitory Income Shocks, Permanent Income Shocks and Habit Formation in Housing 44 Figure 3.10: Housing with Transitory Income Shocks, Permanent Income Shocks and Habit Formation in Housing Figure 3.11: House Value with Transitory Income Shocks, Permanent Income Shocks and Habit Formation in Housing 45 Figure 3.12: Home Ownership with Transitory Income Shocks, Permanent Income Shocks and Habit Formation in Housing Figure 3.13: Wealth Holdings with Transitory Income Shocks, Permanent Income Shocks and Habit Formation in Housing 46 Chapter4 TheModelwithTransactionCostsin HouseTrading 4.1 TheGeneralModel The model in this section is the same with the model described in 1.2 except the model here includes transaction costs in house trading. So the basic difference between the model in this section and the model in 1.2 is the budget constraints. In the model in 1.2, it is assumed that the person can sell and buy houses without paying any transaction costs. Also the person does not pay any maintenance or depreciation costs in that model. However in real life, there are costs related with home ownership. Thus, to make the model closer to real world by one more step, we added transaction costs. If the person is a home owner at any period, it is assumed that he has to pay a maintenance cost which is expressed as a percentage of the value of the house. If the person buys a new house, he has to pay buying cost. If a person sells his existing house, he has to pay selling cost. 4.1.1 BudgetConstraints In this section, mc denotes the maintenance and depreciation cost of owning a house, bc denotes the cost that is incurred when you buy a house and sc denotes the selling cost that is incurred when you sell a house. 47 The intertemporal budget constraints for a household, can be written as follows: There are 5 cases that should be considered. 1) For a renter who decides to continue to be a renter at the current period (D o t1 = 0;D o t = 0) (The person will not pay maintenance, buying or selling costs in this case.) C t + r P h t H t +M t =M t1 (1+r)+P y t e f(t;Zt) " t (4.1) whereC t denotes consumption at time t,P h t denotes house prices at t,M t denotes net saving at t, r denotes risk free interest rate,P y t denotes permanent labor income at t," t denotes the transitory income shock at t and r denotes the rental cost as a percentage of house value. D o t is an indicator variable which is equal to one if the person is a homeowner at t and 0 otherwise. 2) For a renter who decides to be a homeowner at the current period (D o t1 = 0;D o t = 1) (Since the person will buy a house, he will pay the buying cost and since he will be a owner, he will pay the maintenance cost of owning a house.) C t +P h t H t (1+bc+mc)+M t =M t1 (1+r)+P y t e f(t;Zt) " t (4.2) 3) For a homeowner who decides to be a renter at the current period (D o t1 = 1;D o t = 0) (Since the person will sell his house, he will pay the selling cost.) C t + r P h t H t +M t =M t1 (1+r)+P h t H t1 (1sc)+P y t e f(t;Zt) " t (4.3) 4) For a homeowner who decides to continue to be a homeowner at the current period (D o t1 = 1;D o t = 1) We have an indicator variableI sb t which is 0 if the person 48 keeps his house and 1 if the person sells his old house and buys a new one. (Since the person is still a homeowner, he will pay the maintenance cost.) C t +(1+mc)P h t H t +M t +bcI sb t P h t H t =P h t H t1 (1scI sb t )+M t1 (1+r)+P y t e f(t;Zt) " t (4.4) If we combine these four cases in one constraint, we get the following: C t +M t +P h t H t ( r (1D o t )+D o t (1+mc))+bcD o t P h t H t ((1D o t1 )+D o t1 I sb t ) =Inc t (4.5) Inc t =M t1 (1+r)+P y t e f(t;Zt) +D o t1 P h t H t1 (1sc(1D o t )D o t scI sb t ) (4.6) whereI sb t is an indicator variable which is equal to 1 if the person sell his house and buys a new one and 0 otherwise. To see the effect of each type of transaction cost separately, the transaction costs are added to the model step by step. Step1: Only the maintenance cost (mc) of owning a house is added to the model. The model where there is no habit formation with three states Transitory and Perma- nent Income Shocks is solved with different rates of maintenance costs. First we set mc to 0%, then to 1.5% and then to 3%. Step 2: Habit formation in Housing is added to the model in Step 1. The model where there is habit formation with three states Transitory and Permanent Income Shocks is solved with different rates of maintenance costs. First we set mc to 0%, then to 1.5% and then to 3% similarly. Step3: Buying and selling costs are added to the model in Step 1. However there are no permanent income shocks in this step for the ease of computation. 49 4.2 ParameterCalibration The parameters are taken to be same with the parameters in Chapter 3 as expressed in 3.2. Since our interest is to look at how transaction costs affect consumption, saving and housing decisions, choosing the type of transaction costs is very important. Smith, Rosen and Fallis (1988) estimate the transaction costs of changing houses. It includes searching costs, legal costs, costs of readjusting home furnishings to a new house and psychic cost from disruption. They estimated it to be between 8% and 10% the unit being changed. Martin (2002) estimates that the monetary costs of buying a new home, which include agent fee, transfer fee, appraisal and inspection fee range on between 7 and 11 percent of a purchase price of a home. Gruber and Martin (2003) find that the median household pays costs of the order of 7% to sell their houses and 2.5% to purchase. Li and Yao (2007) set house selling cost at 6% of the market value of the house. They set annual maintenance and depreciation cost at 3% of the house value. They set the house purchase cost at 3% of the house value to capture buyers; search cost and mortgage initiation costs. Yao and Zhang (2005) set annual maintenance and depreciation cost at 1.5% of the market value of the owned property. They set the cost of selling an existing house at 6% of the market value of the house, the conventional fee charged by the vast majority of real estate agents. Leigh (1980) estimates the annual depreciation rate of housing units in the United States to be between 0.0036 and 0.0136. Cocco (2004) use depreciation rate of 0.01 on an annual basis. It is assumed that the maintenance cost is equal to 1%, selling cost equal to 5% and buying cost equal to 2%. These values are in the ranges used in the literature. 50 4.3 ResultsoftheModelwithTransactioncosts,Step1 This is the model with transitory income shocks, permanent income shocks and with- out habit formation in housing. We set maintenance cost to first 0%, then to 1.5% and then to 3%. We approximated the transitory income shock and permanent income shock with three states Markov Chains. Table 4.1 shows the statistics of the simu- lations. First of all, for all maintenance cost values, home ownership rate is higher than the value in the data because of the absence of selling and buying costs in house trading. Home ownership rate decreases as maintenance cost increases. Changing the maintenance cost, changes house value over labor income a lot. House value over labor income decreases a lot as maintenance cost increases. The value of the hump in the consumption is higher than that in the data for all the cases, although it decreases as maintenance cost increases. Since we choose the parameters beta and bequest strength parameter to match the Networth/Labor Income and Over 65 Wealth/ Average Wealth, those values are very close to their counterparts in the data for all cases. To sum up, having maintenance cost makes the model statistics closer to data by decreasing the home ownership rate, house value over labor income and average hump in consump- tion. However these statistics are still higher than data so we can say that having only the maintenance cost is not enough to explain the data. Table 4.1: Statistics of the Model in 4.1, Step 1 Statistics Data mc = 0 mc = 1:5% mc = 3% HomeOwnership 0:670 0.996 0.994 0.985 NetWorth=L:Income 3:100 3.215 3.071 2.976 HouseValue=LaborIncome 2:770 7.315 5.317 3.893 +65W:=Avg:W 1:330 1.332 1.321 1.336 HumpinCon: 1:900 5.919 5.124 4.770 51 OptimalLifeCycleDecisions,Step1 As seen from Figure 4.1, Consumption gets lower as maintenance cost increases. The person reaches the peak in consumption earlier with a higher maintenance cost. As seen from Figure 4.2, the increase in Housing before retirement is lower with a higher maintenance cost. The decumulation in Housing after retirement is also lower with a higher maintenance cost. So having maintenance cost helps us to explain the low decumulation in housing in the old ages. As seen from Figure 4.5, the initial increase in the home ownership in the young ages is more gradual with higher maintenance cost which is also gradual in the data as seen in the Figure 2.5. As seen from Figure 4.4, there is not much difference in wealth profiles for different maintenance cost rates. We see high wealth decumulation in the old ages in all cases. So we need something else to explain the low wealth decumulation in the old ages in the data. Figure 4.1: Consumption with Transitory Income Shocks, Permanent Income Shocks, without Habit Formation in Housing and with Different Maintenance Cost Rates 52 Figure 4.2: Housing with Transitory Income Shocks, Permanent Income Shocks, with- out Habit Formation in Housing and with Different Maintenance Cost Rates 4.4 ResultsoftheModelwithTransactioncosts,Step2 This is the model with transitory income shocks, permanent income shocks and habit formation in housing. We set maintenance cost to first 0%, then 1.5% and then to 3%. We approximated the transitory income shock and permanent income shock with three states Markov Chains. We solved the model for the cases when 2 = 0;mc = 0% and 2 = 0;mc = 1:5% and 2 = 0:2;mc = 0% and 2 = 0:2;mc = 1:5% and 2 = 0:7;mc = 0% and 2 = 0:7;mc = 1:5%. Table 4.2 shows the statistics of the simulations. First of all, home ownership rate is higher than its value in the data for all the cases. However home ownership rate decreases with higher maintenance cost and higher level of habit formation in housing. House value over labor income ratio also decreases with both higher maintenance cost and higher level of habit formation in housing. Average value of hump in consumption also decreases with both higher maintenance cost and higher level of habit formation in housing. 53 Figure 4.3: House Value with Transitory Income Shocks, Permanent Income Shocks, without Habit Formation in Housing and with Different Maintenance Cost Rates Table 4.2: Statistics of the Model in 4.1, Step 2 Statistics Data 2 = 0 0;mc 0:2 0:2;mc 0:7 0:7;mc H:Ownership 0:68 0.99 0.99 0.99 0.99 0.99 0.98 NW=L:Inc: 3:10 3.21 3.07 3.11 2.93 3.11 3.15 H:Val:=L:Inc: 2:77 7.31 5.31 6.49 4.41 3.32 2.10 +65W:=Avg:W 1:33 1.33 1.32 1.34 1.32 1.34 1.36 HumpinCon: 1:90 5.91 5.12 5.69 5.19 4.26 4.08 OptimalLifeCycleDecisions,Step2 As seen from Figure 4.6, the peak of the consumption shifts to right as 2 increases and also as maintenance cost increases. The decrease in consumption in the old ages is higher with higher habit strength. The peak also increase with higher 2 . Con- sumption at the middle ages decrease with maintenance cost. So habit formation and maintenance cost affect consumption profile differently. As seen from Figure 4.7, the increase in housing before retirement and the decumulation in housing after retirement 54 Figure 4.4: Wealth Holdings with Transitory Income Shocks, Permanent Income Shocks, without Habit Formation in Housing and with Different Maintenance Cost Rates is less with maintenance cost. So the hump shape in housing is more pronounced with- out maintenance cost and Housing is flatter with maintenance cost. Housing is flatter with increasing habit strength. As seen from Figure 4.10, the initial increase in the home ownership in the young ages is more gradual with higher habit strength which is also gradual in the data as seen in the Figure 2.5. The ownership increases in the young ages and then stays flat in all cases which is close to the pattern in the data. The initial increase is most gradual when 2 = 0:7;mc = 1:5%. As seen from Figure 4.9, the person accumulates wealth before retirement and decumulates almost all of it after retirement in all cases. So we need something else to explain the low wealth decumulation in the old ages in the data. The person reaches a higher peak in wealth with higher habit strength. 55 Figure 4.5: Home Ownership with Transitory Income Shocks, Permanent Income Shocks, without Habit Formation in Housing and with Different Maintenance Cost Rates 4.5 ResultsoftheModelwithTransactioncosts,Step3 OptimalLifeCycleDecisionsoftheModel,Step3 As seen from Figure 4.11, non-housing consumption profile resembles more to that of data under transaction costs. Consumption decreases more in the old ages as in the data with transaction costs. This is intuitive since the person spends some of his income for transaction costs which leaves less money for consumption and housing. As seen from Figure 4.12, the person continues to accumulate housing until his 70s with transaction costs. After age 70s, the person does not decumulate his housing much with transaction costs. In the absence of transaction costs, the person accumulates housing until age 60 and then decumulates it a lot until he dies. As we see from Figure 4.13, under transaction costs the person reaches a higher peak point in wealth and the person decreases his wealth more in the old ages with transaction costs. As a result, we 56 Figure 4.6: Consumption with Transitory Income Shocks, Permanent Income Shocks, Different Maintenance Cost Rates and Habit Strengths Figure 4.7: Housing with Transitory Income Shocks, Permanent Income Shocks, Dif- ferent Maintenance Cost Rates and Habit Strengths can say that transaction costs alone are not enough to explain the non-deaccumulation in wealth for the elderly. 57 Figure 4.8: House Value with Transitory Income Shocks, Permanent Income Shocks, Different Maintenance Cost Rates and Habit Strengths Figure 4.9: Wealth Holdings with Transitory Income Shocks, Permanent Income Shocks, Different Maintenance Cost Rates and Habit Strengths 58 Figure 4.10: Home Ownership with Transitory Income Shocks, Permanent Income Shocks, Different Maintenance Cost Rates and Habit Strengths Figure 4.11: Non-Housing Consumption with Transitory Income Shocks, with and without Transaction Costs 59 Figure 4.12: Housing with Transitory Income Shocks, with and without Transaction Costs Figure 4.13: Wealth Holdings with Transitory Income Shocks, with and without Trans- action Costs 60 Chapter5 TheModelwithLeisure As discussed in the articles of Aguiar and Hurst (2005, 2008) and mentioned in 1.1, the pattern of actual consumption of the household may be different from that of the consumption goods bought with income from the market. Some of the goods are complements with time spent at home like housing services and entertainment. Some of the goods are substitutes with time spent outside the market work, like food. Market hours and time spent at home changes with retirement, starting from the mid 40s. In the light of these views and ideas, the model is altered in a way that consumption bought with income is substitute with leisure time and leisure time is complement with housing services. The rest of the model stays the same. While expenditure declines with retirement, time spent on food production increases dramatically with retirement, where food production is defined as shop- ping for food and preparing meals. For instance male household heads aged 66-68 spend 21 % more time on food production than those aged 60-62. The pattern persists when directly comparing retired to non-retired households. For individuals aged 57- 71, retired ones spend 27% more time on food production per day than the non-retired counterparts. Aguiar and Hurst state that although expenditure on food declines a lot in the peak retirement age, the consumption index remains essentially constant. They find no evidence that their consumption index varies across retirement status. They find that households do not suffer consumption declines, as opposed to expenditure declines at retirement. So the general model used in the previous chapters is altered in 61 a way that non-market time available to the household is substitute with consumption expenditure and complement with housing. Parameter x refers to productivity at home. L t refers to the time left after market work, sleeping and personal care. It is a kind of a measure of leisure.L t includes time spent in home production at home. For simplicity, L t is assumed to be equal to 1/3 before retirement and 2/3 after retirement. 5.1 Model The household’s problem at time t = 0 is represented as follows: max fCt;Ht;Mt;D o t g E 8 > > < > > : T X t=1 t1 0 B B @ F(t) 1 1 (C t +xL t ) 1 ( Ht H 2 t1 ) (1 ) +(F(t1)F(t))B(W t ;LB) 1 C C A 9 > > = > > ; (5.1) subject to (5.2) C t +[ r (1D o t )+D o t ]P h t H t +M t =M t1 (1+r)+D o t1 P h t H t1 +P y t e f(t;Zt) " t (5.3) M t (1d)P h t H t D o t (5.4) W t =M t1 (1+r)+P h t H t1 (5.5) C t 0;H t 0 (5.6) P h t =P h t1 (1+r h t ) (5.7) P y t =P y t1 v t (5.8) D o t 2f0;1g (5.9) where C t denotes consumption at time t, P h t denotes house prices at t, M t denotes net saving at t, r denotes risk free interest rate, P y t denotes permanent labor income at t, " t denotes the transitory income shock at t and r denotes the rental cost as a 62 percentage of house value.D o t is an indicator variable which is equal to 1 if the person is a homeowner at t and 0 if renter. Here is the subjective discount factor, is the curvature parameter, B(:) is the bequest function, W t is the bequeathed wealth, LB controls the strength of bequest motives and d denotes downpayment ratio. We abstract from transaction costs in house trading in this chapter. The value function can be expressed as below: V t (Z t ) = max fCt;Ht;Mt;D o t ;D s t g 8 > > < > > : t ( T P t=0 1 1 (C t +xL t ) 1 ( Ht H 2 t1 ) (1 ) +E t [V t+1 (Z t+1 )])+(1 t )B(W t ;LB) 9 > > = > > ; (5.10) C t 0;H t 0 Z t =fC t1 ;H t1 ;M t1 ;P y t ;P h t ; t ;D o t1 g: C t 0;H t 0 (5.11) where the state vectorZ t is defined as Z t =fH t1 ;M t1 ;P y t ;P h t ; t ;D o t1 g: (5.12) 5.2 SolutionoftheModel Due to the big number of continuous state variables, the variables are redefined in the following way. 63 c t = C t P y t e f(t;Zt) (5.13) h t = P h t H t P y t e f(t;Zt) (5.14) l t = M t P h t H t (5.15) k t = xL t P y t e f(t;Zt) (5.16) The values of x , L t and e f(t;Zt) are assumed to be deterministic. We solved the model under the assumption of absence of permanent income shocks. It is because in Chapter 3, we see that permanent income shocks are not necessary to make the model results closer to data. Moreover there are papers in the literature without permanent income shocks. So the value ofP y t is assumed to be one. 5.3 ParametersandCalibration Parameter x refers to productivity at home. We need to take the value of x from lit- erature or we should calibrate it. We chose the second one. For calibration, we need a target. From Figure 1 in Aguiar and Hurst, ”Consumption vs Expenditure”, we see that the average consumption index of people aged between 66-68 is the same as that of aged between 57-59. Also, the average consumption index of people aged between 60-62 is the same as that of aged between 57-59. These two imply the equations 5.17 and 5.18. So we will choose the value of x so that the model generated value of C 6668 +xL 6668 C 5759 xL 5759 is zero or close to zero. 64 C 6668 +xL 6668 C 5759 xL 5759 = 0 (5.17) C 6668 +xL 6668 = C 6062 +xL 6062 (5.18) x = C 5759 C 6668 L 6668 L 5759 (5.19) x = C 57 +C 58 +C 59 (C 66 +C 67 +C 68 ) (L 66 +L 67 +L 68 )(L 57 +L 58 +L 59 ) (5.20) x = C 60 +C 61 +C 62 (C 66 +C 67 +C 68 ) (L 66 +L 67 +L 68 )(L 60 +L 61 +L 62 ) (5.21) 5.4 Results We run the program with different values for x and among them we choose the x where C 6668 +xL 6668 C 5759 xL 5759 is zero or closest to zero. We will call the value ofC 6668 +xL 6668 C 5759 xL 5759 as LTarget. In each run, we calibrate beta and LB so that over65 wealth/average wealth and average net worth/labor income match the values in the data. As seen from the table 5.1, we can match LTarget best when x = 30. Table 5.1: Statistics of the Model in 5.1 Statistics Data x=0 x=10 x=20 x=30 NetWorth=L:Income 3:1 3:1 2:95 3:05 3:09 Over 65Wealth:=Avg:W: 1:33 1:32 1:36 1:34 1:33 LTarget 0 3:64 2:76 0:75 HomeOwnership 0:67 0.99 0.99 1.00 1.00 HouseValue=L:Income 2:77 7.44 8.14 9.06 9.87 HumpinConsumption 1:90 3.30 4.05 4.65 6.75 65 5.4.1 OptimalLifeCycleDecisions As seen from Figure 5.2, the decumulation in housing in the old ages decrease as the value of x increase. From Figure 5.1, it is evident that the decrease in the consumption expenditure in the old ages is higher with higher x. From Figure 5.3, it is seen that the decrease in the housing value in the old ages decrease with higher x. Housing value graph shifts upward as x increase. So we can say that the shape of housing value becomes flatter with higher x. This is intuitive. Housing is complementary with leisure time. The person has more time to spend at home after retirement so the house becomes more important. The person does not want to decrease his housing after retirement for this reason. The home ownership is very high compared to data in all cases. From Figure 5.4, it is seen that the person reaches a higher peak level in wealth with lower x. Also the decumulation in wealth after retirement is lower with higher x. When x = 30, wealth decreases little in the old ages as in the data. So far, we added and removed many components to the models and solved them. In all cases, we get quick wealth decumulation in the old ages. This is the first time we get low wealth decumulation in the old ages. This result is intuitive. The person has more time after retirement for preparing food at home. So with less consumption expenditure, the person can keep his consumption level. He does not need to use his wealth for consumption after retirement. From Figure 5.6, composite consumption does not decrease with retirement forx = 30 case. This figure supports the idea of Aguiar and Hurst. They say that although consumption expenditure decrease with retirement, the quantity and quality of actual consumption does not decrease with retirement. To sum up, when x = 30 , we get rid of the quick decumulation in housing and wealth in the old ages and we get flat consumption in the old ages. However we still need something else 66 for getting the home ownership. As mentioned in the previous chapters, we need transaction costs in house trading to get the home ownership patterns in the data. Figure 5.1: Consumption Expenditure with Transitory Income Shocks and Leisure but without Habit Figure 5.2: Housing with Transitory Income Shocks and Leisure but without Habit 67 Figure 5.3: House Value with Transitory Income Shocks and Leisure but without Habit Figure 5.4: Wealth Holdings with Transitory Income Shocks and Leisure but without Habit 5.5 Results of the Model with Leisure and Habit For- mationinHousingtogether We set the parameter x to 30 and then run the program with different values for habit strength, 2 . In each run, we calibrate beta and LB so that over65 wealth/average wealth, average net worth/labor income and LTarget match the values in the data. As 68 Figure 5.5: Home Ownership with Transitory Income Shocks and Leisure but without Habit seen from the table 5.2, home ownership rate is much higher than its value in the data in all cases. House value over labor income is also higher than the data in all cases. However it decreases as 2 increase. Hump in consumption is also higher than the its data value in all cases and it also decreases as 2 increase. The high value of home ownership rate is due to the absence of transaction costs. Table 5.2: Statistics of the Model in 5.5 Statistics Data T=0 T=0.2 T=0.4 NetWorth=L:Income 3:100 3:096 3:083 3:064 Over 65Wealth:=Avg:W: 1:33 1:331 1:336 1:311 LTarget 0:000 -0.756 0:805 0:139 HomeOwnership 0:67 1.00 1.00 1.00 HouseValue=L:Income 2:770 9.878 8.883 7.598 HumpinConsumption 1:900 6.755 5.009 4.355 69 Figure 5.6: Consumption Index with Transitory Income Shocks and Leisure but with- out Habit 5.5.1 OptimalLifeCycleDecisions As seen from Figure 5.7, there is a sudden decrease in consumption expenditure with retirement and consumption stays flat after retirement. The sudden decrease is less with higher habit strength. From Figure 5.8, housing increases until retirement and then stays flat as in the data. The initial increase is less with higher habit strength. Home ownership is constant at value one in all cases. So the model with leisure and habit formation in housing needs some addition to explain the home ownership pattern in the data. From Figure 5.10, the person accumulates more wealth before retirement with higher habit strength. The decumulation of wealth in the old ages is also higher with higher habit strength. The reason is, the person decreases his consumption less with higher habit strength and the person decumulates his housing less in the old ages with higher habit strength. From Figure 5.12, consumption index which is composed of consumption bought with income and consumption prepared at home, first increases and then stays flat. It decreases little after retirement for small habit strength values. 70 Figure 5.7: Consumption Expenditure with Transitory Income Shocks, Leisure and Habit Figure 5.8: Housing with Transitory Income Shocks, Leisure and Habit 5.6 ModelwithaMultiplierinUtility Since the person has more time for searching for coupons, bargains and cheaper places after retirement, we assume that the price of nondurable consumption is no longer one after retirement, it is less than one. It is mathematically the same to put a multiplier which is higher than one in front of nondurable consumption in the utility function in place of putting a multiplier which is less than one in front of nondurable consumption 71 Figure 5.9: House Value with Transitory Income Shocks, Leisure and Habit Figure 5.10: Wealth Holdings with Transitory Income Shocks, Leisure and Habit in the budget constraint. We preferred the first way and solved the model with different multipliers. When the multiplier is constant and equal to 1, this model is equivalent to the model with 3 states transitory income shocks only (no habit, no permanent income 72 Figure 5.11: Home Ownership with Transitory Income Shocks, Leisure and Habit shocks, no transaction costs). The household’s problem at time t = 0 is represented as follows: max fCt;Ht;Mt;D o t g E 8 > > < > > : T X t=1 t1 0 B B @ F(t) 1 1 (m t C t ) 1 ( Ht H 2 t1 ) (1 ) +(F(t1)F(t))B(W t ;LB) 1 C C A 9 > > = > > ; (5.22) (5.23) m t = 1 before retirement andm t = mul which is greater than 1 after retirement. The rest of the constraints are the same with the ones described in section 5.1. 73 Figure 5.12: Consumption Index with Transitory Income Shocks, Leisure and Habit Table 5.3: Statistics of the Model in 5.6 Statistics Data mul=1.5 mul=2 mul=3 NetWorth=L:Income 3:100 3:040 3:046 3:103 Over 65Wealth:=Avg:W: 1:330 1:321 1:331 1:336 HomeOwnership 0:670 0.996 0.996 0.996 HouseValue=L:Income 2:770 7.370 7.337 7.310 HumpinConsumption 1:900 3.163 3.161 3.270 5.7 ResultsoftheModelin5.6 5.7.1 OptimalLifeCycleDecisions As seen from Figure 5.14, when we have a multiplier higher than one, we see a sudden decrease in housing at the age of retirement and then we see slow decumulation. In total, the decumulation in housing of the elderly is less with higher multiplier. As seen from Figure 5.16, the accumulation of wealth before retirement and the decumulation of wealth after retirement is slower with a higher multiplier. However we still see a 74 sizable decumulation of wealth for the elderly. In short, making consumption cheaper after retirement does not help us a lot in getting closer to the data. Figure 5.13: Consumption Expenditure with Lower Price of Consumption after Retire- ment 75 Figure 5.14: Housing with Lower Price of Consumption after Retirement Figure 5.15: House Value with Lower Price of Consumption after Retirement 76 Figure 5.16: Wealth Holdings with Lower Price of Consumption after Retirement Figure 5.17: Home Ownership with Lower Price of Consumption after Retirement 77 Chapter6 ModelwithHealthShocksand MedicalCosts As stated in DeNardi, French and Jones (2006), many elderly people keep a lot of assets until very late in life. Moreover, there is large variation in life expectancy, conditional on health status. Even with health insurance, out-of-pocket medical and nursing home expenses can be big, and can generate significant net income risk for the elderly people. Average out-of-pocket medical expenditures increase very rapidly as the person gets older. For example, average medical expenditures for a woman in bad health increase from $1,200 at age 70 to $19,000 at age 100. The rising medical expenses generates a strong incentive to save to the elderly. Medical expenditures after age 85 becomes very much a luxury good. Kotlikoff (1988) finds that out-of-pocket medical expenditures are potentially an important driver of aggregate saving. Hubbard et al. (1994) and Palumbo (1999), find that medical expenses have fairly small effects. DeNardi, French and Jones (2006) find that their model with health shocks and medical costs can go a long way towards accounting for the observed lack of asset decumulation after retirement, at least for the elderly singles. Scholz et al. find that a life cycle model, augmented with realistic income and medical expense uncertainty, can do a good job of fitting the distribution of wealth at retirement. 78 We assume that agent is faced with health shocks after age 70 due to the availability of data and also for simplicity. The within period utility function is given by: u(C t ;H t ;m) = 1 1 (m)[( C t C 1 t1 ) 1 ( H t H 2 t1 ) ] (1 ) (6.1) U = Ef T X t=0 t (F(t) (m) 1 [( C t C 1 t1 ) 1 ( H t H 2 t1 ) ] (1 ) +(F(t1)F(t))B(W t ;LB))g The function(m) determines how a person’s utility depends on his or her health status. The person is assumed to be in good health if m = 2 and in bad health if m = 1. We define(m) by the following way: (m) = 1+(2m) (6.2) So when = 0; health status does not affect utility. The old person, aged 70 or more, is faced with health status uncertainty, survival uncertainty and medical expense uncertainty. These sources of risk are assumed to be exogenous. One of the reasons lying behind this assumption is, older people have already shaped their health and lifestyle. 79 6.1 HealthStatusUncertainty It is assumed that the transition probabilities for the health status depend on sex, current health and age. k;j;g;t = Pr(m t+1 = jjm t = k;g;t); where g denotes sex, t denotes age and k denotes the current health. 6.2 MedicalExpenseUncertainty Health costs,hc t are defined as out-of-pocket costs. lnhc t = hc(g;m;t) +(g;m;t)' t , where t denotes idiosyncratic component. Similar to Feenberg and Skinner (1994) and French and Jones (2004), we assume that ' t has the following form: ' t = t + t ; t ~N(0; 2 ) t = hc t1 + t ; t and t are serially and mutually independent. t and t will be made discrete using quadrature methods of Tauchen and Hussey. t is the transitory component of health cost uncertainty while t is the persistent component with the autocorrelation hc : 6.3 SurvivalUncertainty Mortality rates depend upon age and previous health status. s t+1 =s(m t ;t+1) 80 6.4 ParameterCalibration DeNardi, French and Jones (2006) find that the variance of log medical expenses is 2.15. French and Jones (2004) find that a suitably-constructed lognormal distribution can match average medical expenses. They used AHEAD data. The data for health shocks, medical costs, income and survival rates are kindly provided by them. The AHEAD is a sample of non-institutionalized individuals, aged 70 or older in 1993. They also find that medical expenses are highly correlated over time. They state that 66.5 percent of the cross sectional variance of medical expenses is due to the transitory component, and 33.5 percent due to the persistent component. The persistent compo- nent has an autocorrelation coefficient of 0.922, so that innovations to the the persistent component of medical expenses have long-lived effects. Medical expenses are the sum of what the individuals spend out of pocket on insurance premia, drug costs, and costs for hospital, nursing home care, doctor visits, dental visits, and outpatient care. It does not include expenses covered by insurance, either public or private. French and Jones find that most of a household’s lifetime medical expense risk comes from the persistent component. That is why we omitted the transitory component put only the persistent component into the model. It is assumed that the innovation variance of persistent component is equal to 0.0503. 6.5 OptimalLifeCycleDecisions From Figure 6.1, we see that with health shocks, housing becomes flatter for the very old people. However the housing profile is still different from the data. So stochastic permanent health shocks are not enough alone to mimic the housing profile. From Figure 6.2, we see that ownership pattern cannot be explained by only health shocks. 81 From Figure 6.3, it is seen that the decumulation in wealth is less under health shocks. However the wealth profile cannot be explained with the permanent health shocks after age 70 alone. Figure 6.1: Housing with Stochastic Health Shocks 82 Figure 6.2: Ownership with Stochastic Health Shocks Figure 6.3: Wealth Holdings with Stochastic Health Shocks 83 Chapter7 Conclusion Three basic patterns are observed in the data: the hump-shaped non-housing consump- tion profile, the non-hump-shaped, in other words first monotonically increasing and then flat housing profile, and wealth profile which is increasing until age 70s and then decreasing little. In U.S., young households own no liquid financial assets, but hold a major fraction of their wealth as housing. Later in life, households shift their portfolios to financial assets. In my thesis, a life-cycle model with several components is developed to investi- gate the reasons behind the non-deaccumulation of housing and wealth for the elderly people. A few modifications to the basic life-cycle model with housing are made to produce consumption, housing and wealth profiles that more closely resemble to U.S. data. Several key features distinguish the model from the existing ones in the literature. First, we add habit formation on a good by good basis, so called deep habits, to analyze the effects on the household decisions. Second, we add stochastic health shocks and medical costs on a model with housing to see the effects on housing and wealth accu- mulation decisions. Third, we add substitutability between leisure and non-housing consumption and complementarity between housing and leisure to see the effects on household decisions. We developed a quantitative and realistically calibrated dynamic partial equilib- rium model to solve numerically for the optimal housing, consumption and wealth accumulation decisions for a finitely-lived person. First, we model housing choices 84 along both the extensive margin of owning versus renting and the intensive margin of house value. Our model has many realistic features including stochastic labor income process and housing market frictions such as collateral borrowing requirement. House- holds begin their economic lives without any housing stock or any wealth. I also investigate the quantitative relevance of the transaction costs, income shocks, habit formation in housing, increase in leisure that comes with retirement and health shocks in determining this pattern. I add two types of income shocks, transitory and permanent. I find that adding permanent income shocks does not make the results closer to the data. So permanent income shocks are not essential to mimic the patterns in the data. I add habit formation in housing consumption. I run the program with var- ious habit strength parameters. It is seen that the person prefers to have less housing with higher habit strength in housing. In other words, the level of housing is lower at all ages as the habit strength parameter increases. The decumulation in housing for the elderly is less and the decumulation in wealth for the elderly is more with higher habit strength. The person accumulates less housing with a higher habit strength in the young and middle ages. So the habit strength in housing makes the housing results closer to the data but at the expense of making the decumulation in wealth higher for the elderly. The increase in the ownership for the young is more gradual with higher habit strength. The person consumes more under higher habit strength. This is a rather intuitive result. Consuming a higher housing today raises the person’s utility in the cur- rent period but lowers the utility in the next period. Thus the person optimally chooses a lower housing with habit formation in housing. The person chooses a higher non- housing consumption with the higher income due to the lower housing to compensate the loss in the utility. 85 One of the models include stochastic health shocks and medical costs with housing. As of my knowledge, this is the first paper which analyzes health shocks together with housing. It is found that the decumulation of wealth and housing for the elderly is less with stochastic health shocks and medical costs. It is assumed that being in bad health decreases the utility taken from consumption and housing. The more the bad health decreases utility, the less is the decumulation in wealth and housing of the elderly. One of the models include transaction costs in house trading. The existing studies in the literature used buying cost, maintenance cost and selling cost in house trading. First, only the maintenance cost is added to the model. Maintenance cost in housing has an effect on housing similar to the effect of habit formation. The increase in hous- ing before retirement and the decrease after retirement is less pronounced with mainte- nance cost when compared with the habit formation. Maintenance cost also decreases wealth decumulation in the old ages. Maintenance cost decreases consumption but habit formation increases consumption. So maintenance cost and habit formation have different effects on consumption. This is also intuitive since with maintenance cost, there is income loss due to the depreciation in house but with habit formation there is no such an income loss. The decrease in wealth decumulation in the old ages with maintenance cost is because of the decrease in consumption. As a next step, the other transaction costs are also added to the model. The decu- mulation of wealth for the elderly is higher with transaction costs in house trading. However the decumulation of housing for the elderly is less with transaction costs in house trading. The statistics average homeownership rate is very high relative to the data without transaction costs. The ownership for the young is less and increases more gradually with transaction costs. However the decrease in ownership rate for the elderly is sharper and earlier with transaction costs. The increase in housing for the 86 young is much slower with transaction costs. Consumption decreases more in the old ages under transaction costs. Thus from many ways, it is seen that putting transaction costs to the model is essential although it makes the programming more complicated. As far as I know, this paper is the first which analyzes the effect of increase in leisure at retirement on housing. The model with leisure explains the non- decumulation in housing and wealth for the elderly. Also with this model, although the retired people spends less on consumption, they do not decrease their consumption. This model is successful except explaining the ownership. As said before, transaction costs are necessary to explain the ownership. Since the elderly people have more time to deal with coupons and to search for the cheaper stores, consumption may become cheaper for them. Making consumption cheaper after retirement, decreases wealth decumulation of the elderly. However this model is not good at explaining the other choices. To sum up, the results of various models are checked. Among them, the models with health shocks and leisure helps to explain the non-decumulation of housing and wealth for the elderly. However transaction costs are still necessary to explain the home ownership pattern. Habit formation helps to explain the non-decumulation in housing but at the expense of more decumulation in wealth. The model with leisure makes the results closer to the data than the model with health shocks. 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Wise, Have IRAs Increased U.S. Saving?: Evidence from the Consumer Expenditure Survey, Quarterly Journal of Economics,105 (1990). [64] L.M. Viceira, Optimal Portfolio Choice for Long-Horizon Investors with Non- tradable Labor Income, Journal of Finance,56 (2001), 433-470. [65] M.E. Yaari, Uncertain lifetime, Life Insurance and the Theory of the Consumer, Review of Economic Studies,32 (1965), 137-150. [66] R. Yao, and H. Zhang, Optimal Consumption and Portfolio Choices with Risky Labor Income and Borrowing Constraints, Review of Financial Studies, 18 (2005), 197-239. 92 Appendix Due to the complex structure of the model, the analytical solution for the optimization problem does not exist. So numerical solutions are obtained using value function. The household problem is to find the sequences of consumptionfC t (Z t )g; net savings fM t (Z t )g and housingfH t (Z t )g as a function of the state (Z t ) for t = 1;2;:::;T subject to the constraints to maximize the utility. For numerical solution, we write the household problem as a sequence of stochastic Bellman equations. The problem has recursive nature, thus we can write the optimization problem in the following form: V t (Z t ) = max fCt;Ht;Mt;D o t g 8 > > < > > : t ( T P t=0 1 1 ( Ct C 1 t1 ) 1 ( Ht H 2 t1 ) (1 ) +E t [V t+1 (Z t+1 )])+(1 t )B(W t ;LB) 9 > > = > > ; (1) C t 0;H t 0 Z t =fC t1 ;H t1 ;M t1 ;P y t ;P h t ; t ;D o t1 g: C t 0;H t 0 (2) where the state vectorZ t is defined as Z t =fC t1 ;H t1 ;M t1 ;P y t ;P h t ; t ;D o t1 g: (3) 93 Among the state variables, only t is exogenous and the rest are endogenous. Among the endogenous state variables, only D o t is discrete with two possible val- ues, 0 and 1. The choice variables are represented by the following vector X t = fC t ;H t ;M t ;D o t g. In this form of the value function, there are 5 continuous endogenous state variables which will make the numerical analysis very difficult and slow. Thus we need to simplify this structure by a normalization, in other words by redefinition of some of the variables. We simplify the solution by exploiting the scale independence of the optimization problem. The vector of endogenous state variables is transformed to z t =fc t1 ;h t1 ;l t1 ; t ;v t ;r h t ;D o t1 g; where c t = C t P y t e f(t;Zt) (4) c t is the household’s consumption- labor income (without transitory shocks) ratio, h t = P h t H t P y t e f(t;Zt) (5) h t is the household’s house value-labor income (without transitory shocks) ratio, l t = M t P h t H t (6) l t is the savings(or mortgage loan)- house value ratio. The normalization allows us to get rid of two continuous state variables but we added two discrete state variablesv t andr h t . This will increase the speed of the computation a lot. The normalized budget constraint: 94 c t +([ r (1D o t )+D o t ]+l t )h t = e f(t1;Z t1 ) v t e f(t;Zt) h t1 ((1+r)l t1 +D o t1 (1+r h t ))+" t (7) l t (1d)D o t (8) Define v t (z t ) = V t (Z t ) ( (P y t e f(t;Z t ) ) (1 1 )(1)+(1 2 ) (P h t ) (1 2 ) ) 1 (9) Thus the recursive optimization problem can be written as the following form: v t (z t ) = max fct;ht;lt;D o t g 8 > > < > > : t ( T P t=0 1 1 ( ct c 1 t1 ) 1 ( ht h 2 t1 ) ( e f(t;Z t ) vt e f(t1;Z t1 ) ) 1 (1)+ 2 1 (1+r h t ) 2 (1 ) +E t [v t+1 (z t+1 )] ( v t+1 e f(t+1;Z t+1 ) e f(t;Z t ) ) con1 (1+r h t+1 ) (1 2 )(1 ) )+(1 t )b(W t ;LB) 9 > > = > > ; (10) con1 = ((1 1 )(1)+(1 2 ))(1 ) c t +([ r (1D o t )+D o t ]+l t )h t = e f(t1;Z t1 ) v t e f(t;Zt) h t1 ((1+r)l t1 +D o t1 (1+r h t ))+" t (11) l t (1d)D o t (12) b(w t ;LB) =LB 1 1 " (1) (1)(1 1 ) (1 2 ) w (1)(1 1 )+(1 2 ) t ( r ) (1 2 ) # (1 ) (13) c t > 0;h t > 0 (14) We will solve the problem by backward recursion. 95 At the terminal date T, T = 0 so the household’s value function is equal to the value of the bequest function. Thusv T (z T ) is constant and same for all the points in the state space. v T (z T ) = b(w T ;LB) v T (z T ) = =LB 1 1 " (1) (1)(1 1 ) (1 2 ) w (1)(1 1 )+(1 2 ) T ( r ) (1 2 ) # (1 ) (15) where w t =" t +h t1 e f(t1;Z t1 ) v t e f(t;Zt) (l t1 (1+r)+(1+r h t )) (16) The value function at date T,v T (z T ) is then used to compute the optimal decision rules for all the points on the state space at date T-1. For a household who comes to the current period t as a renter, we compute the value functions contingent on buying a house and becoming a homeowner or staying as a renter. The household’s optimal house tenure choice for the current period depends on which of the two value functions is bigger. To calculate the expected next period’s value function, we use discrete states as approximations for the exogenous state variables. Similarly for a household who comes to the current period t as a homeowner, we compute the value functions contin- gent on staying as a homeowner or becoming a renter. The household’s optimal house tenure choice for the current period depends on which of the two value functions is bigger. This procedure is repeated recursively for each period until the solution for date t=1 is found. We optimize over the different choices using grid search. 96 In short, for every age t before the last period T, and for each point in the state space, we optimize using grid search. Thus we will compute the value associated with each level of housing, level of net savings and home ownership status. From the Bellman equation, these values are equal to the sum of current utility and discounted expected continuation value, which can be computed since we have just obtainedV t+1 : All numerical integrations are performed using Gaussian quadrature to approximate the distributions of the shocks to the labor income and house price. Once we compute the value of all alternatives, we take the maximum to obtain the policy rules for the current period. We substitute these decision rules in the Bellman equation to obtain current period’s value function,V t which is used to solve the maximization problem of the previous period. This iteration continues until t=1. 97
Abstract (if available)
Abstract
In this paper, we explore whether a standard life cycle model in which households purchase nondurable consumption and housing and face idiosyncratic income and mortality risk as well as endogenous borrowing constraints, enriched with several new components, like habit formation in housing, increasing leisure with retirement, stochastic health shocks, can account for key patterns of consumption, housing and asset holdings over the life cycle. First, consumption expenditures on non-housing consumption are hump-shaped. Second, the housing profile first increases monotonically and then flattens out.
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Aydilek, Asiye
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Core Title
Investigation of various factors behind non-deaccummulation of housing and wealth with aging
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College of Letters, Arts and Sciences
Degree
Doctor of Philosophy
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Economics
Publication Date
07/12/2009
Defense Date
04/16/2009
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habit,health,Housing,Leisure,OAI-PMH Harvest,Wealth
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English
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Imrohoroglu, Ayse (
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), Dekle, Robert (
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), Quadrini, Vincenzo (
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aaydilek@usc.edu,asiye_koc@hotmail.com
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https://doi.org/10.25549/usctheses-m2351
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