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Essays on consumption behavior, economic growth and public policy

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Content ESSAYS ON CONSUMPTION BEHAVIOR, ECONOMIC GROWTH AND PUBLIC POLICY Copyright 2005 by Chengyu Yang 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) May 2005 Chengyu Yang Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. UMI Number: 3180483 Copyright 2005 by Yang, Chengyu All rights reserved. INFORMATION TO USERS The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleed-through, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. ® UMI UMI Microform 3180483 Copyright 2005 by ProQuest Information and Learning Company. All rights reserved. This microform edition is protected against unauthorized copying under Title 17, United States Code. ProQuest Information and Learning Company 300 North Zeeb Road P.O. Box 1346 Ann Arbor, Ml 48106-1346 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Acknowledgements I am grateful to Professor Fukang Fang from Beijing Normal University for opening the door of the wonderful world of Economics for me when I was a graduate student of Physics. I am particularly grateful to Professor Richard H. Day from the University of Southern California for all the encouragements, guidance and comments on the research in this dissertation. I would like to thank Professor W. Bentley MacLeod and Professor Ayse Imrohoroglu from the University of Southern California for accepting to be in my dissertation committee and giving me valuable comments on my dissertation. I would also like to thank Professor Caroline Betts, Professor Robert Dekle, and Professor Joines Douglas from the University of Southern California for being in my guidance committee. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. iii Table of Contents Acknowledgements ii List of Tables iv List of Figures v Abstract vi Preface viii Chapter 1 Overlapping Altruistic Generations: An Analysis of Consumption and Saving Behavior in the U.S. Economy 1 Chapter 2 Economic Growth and Revealed Social Preference 47 Chapter 3 A Dynamic Analysis of Public Fiscal Policy under Different Consumption Patterns 69 References 8 7 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. iv List of Tables 1.1 Correlation between the saving rate and the growth rate of TFP 20 1.2 Benchmark calibration and estimation 32 1.3 Saving rate calibration and estimation: benchmark 35 1.4 Saving rate estimation: with social security 39 2.1 GMM estimates of production function parameters 58 2.2 GMM estimates of the other model parameters 58 2.3 A comparison of econometric estimates and calibrations of parameters 59 2.4 A comparison of percentage factor contributions to GDP growth by econometric estimation and calibration (%) 63 3.1 Parameter estimates for the linear model 79 3.2 Parameter estimates for the piecewise linear model 80 3.3 Key coefficients 82 3.4 The stability and KRB tax policy conditions 83 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. V List of Figures 1.1 The personal saving rate in the U.S. 2 1.2 U.S. net worth-disposable income ratio 4 1.3 Conditional survival rates for the U.S. 18 1.4 Saving rate calibration: benchmark (C.E.) 35 1.5 Saving rate calibration: benchmark (Phase) 36 1.6 Saving rate estimation: benchmark (C.E.) 36 1.7 Saving rate estimation: benchmark (Phase) 37 1.8 Saving rate calibration: with social security (C.E.) 38 1.9 Saving rate estimation: with social security (C.E.) 40 1.10 Saving rate estimation: with social security (Phase) 40 1.11 Simulation of the shares of bequest wealth: benchmark 42 1.12 Simulation of the shares of bequest wealth: bequest motives 43 1.13 Fertility rate estimation 45 2.1 Production function parameter estimates: b 59 2.2 Production function parameter estimates: ft 60 2.3 Utility function parameter estimates: 6 60 2.4 Utility function parameter estimates: y 61 2.5 Steady state output trajectories based on GMM estimates 64 2.6 Steady state output trajectories based on calibrated estimates 64 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Abstract My dissertation investigates consumption and saving behavior, their relationship to economic growth, and the effectiveness of public fiscal policy under different consumption patterns. In the first chapter “Overlapping Altruistic Generations: An Analysis of Consumption and Saving Behavior in the U.S. Economy”, I develop a three period overlapping generation model of altruistic household consumption, labor, fertility and bequest decisions to explain changes in personal saving rate over the last half century in the U.S. We find that wealth accumulated through life cycle saving and bequest transfers have different effects on consumption. TFP growth and demographic structure change are the two major driving forces for the first ascending, then declining trend of saving rate. In particular, TFP growth together with the decreasing share of bequest wealth and the increasing wealth-income ratio can account for most of the saving decline in 1990s. Calibration and estimation results show theory’s consistency with data. In the second chapter “Economic Growth and Revealed Social Preference”, the representative agent growth model is estimated econometrically using the GMM method for three separate growth eras from 1929 to 2002 and the results are compared to those obtained using the' standard Prescott calibration approach. The estimated parameters differ substantially in the three cases, which imply changing social preferences for present versus future income and work-leisure tradeoffs. These in turn imply switching among alternative balanced growth paths and differences in the contributions of capital, labor, and labor augmenting productivity among the three eras. In the third and last chapter “A Dynamic Analysis of Public Fiscal Policy under Different Consumption Patterns”, a dynamic aggregate demand and supply model is used to explain the public policy paradox of tax rate decreases being able to increase tax revenue. We find that under the consumption pattern of high marginal propensity to consume and invest, tax rate reduction can Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. vii stimulate both economic growth and tax revenue in the long run. The consumption boom since late 1980s implies the tax cut policy may be effective in practice during the same period. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Preface Consumption behavior is among the key issues of economic study. At the micro level, consumption/saving choice is one of the most important decisions any economic agent faces, which directly determines its welfare and affects almost all its other choices like work, leisure and fertility. At the macro level, consumption is the biggest component of GDP, whose change has crucial influence on saving, the accumulation of private capital and hence economic growth. It then becomes an important task of explaining the determination and evolution of consumption behaviors on both individual and aggregate levels for the understanding of economic growth and the effectiveness of public policy when individuals may change their decisions in reaction to the implementation of policy. The evolution of consumption behavior has been widely documented and studied. As an alternative measurement of consumption, the personal saving rate in the U.S. has experienced three different phases during the last half century. From 1930s to 1960s, it had risen from 4 percent to 10 percent; from 1970s to early 1980s, it fluctuated around a stable level of 10 percent; during the last two decades, it declined dramatically from 10.6 percent in 1984 to only 2 percent in 2000. The reasons for this change, its impact on economic growth, and its inconsistency with most formulations in intertemporal optimization models have attracted many discussions in the recent literature (for example, Slesnick, 1992, Browning and Lusardi, 1996, Mehra, 2001). Among the various explanations like wealth effect, demographic structure change, and intergenerational transfer, no single reason can account for most of the change in saving rate (Parker, 1999). A micro theoretical framework incorporating most important factors and being consistent with intertemporal optimization should be used to interpret the mechanism of household consumption and saving behavior and its linkage to the aggregate consumption performance. Since work and leisure, consumption and saving, and fertility and bequest are all jointly determined within the household at Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ix different age, an overlapping generation model of household with all the above choices endogenized might be a necessary and appropriate framework of analysis. Moreover, the endogenous wealth accumulation, demographic structure change and intergenerational transfer through bequest are all incorporated into one model, both single effect and cross effect can help to improve the model’s explaining power. The change of consumption and saving behavior also reflects growth and evolution of the whole economy. The shifting trend of saving rate shown in data may suggest the possibility of changes either in the economy’s balanced growth path, or in the society’s overall “preference” for growth. Using a standard representative agent model, time and leisure preference parameters can be calibrated and estimated. By separating the whole period from 1929 to 2002 into different phases in corresponding to the rising and declining trends of saving rate, we can examine if the estimation results show significant difference in the preference estimates for each separate phase, which implies accompanying the change of consumption behavior, the economy may rests on different balanced growth path and/or social preferences for time and leisure may have changed. In economic practice, public fiscal policy is an important tool to improve economic growth and equity. As many other policy instruments, its effectiveness depends on the conditions and environments of the economy, especially on the patterns of consumption and investment, and the situation of technological progress whenever the policy is implemented. It is interesting to study how the exactly same policy results in different outcomes under different economic environments. The widely presumed tax policy paradox that a reduction in tax rate can both stimulate economic growth and increase tax revenue provides a good example for examining the relationship between policy efficiency and economic conditions in terms of consumption and investment patterns. The tax cut policy was argued explicitly in the early 1960s by President Kennedy, subsequently by Ronald Reagan, and most recently by George W. Bush, which attracted a lot of skepticism and Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. debates among economists and politicians. Once one realizes that consumption and investment behavior, as well as productivity growth have been changed significantly in the U.S. during the last half century, it becomes easier to understand that the same tax policy faced by the three administrations may lead to different results for the long run economic growth and tax revenue. My dissertation consists of three essays and aims to investigate consumption and saving behavior, their relationship to economic growth, and the effectiveness of public fiscal policy under different consumption and growth patterns. In my first essay, I investigate the mechanism of household consumption at different age, and the consequent aggregate performance. An altruistic OLG model of household is developed and fitted to data to explain the change in personal consumption, labor, birth and bequest decisions over the last half century in the U.S. We find that wealth accumulated through life cycle saving and bequest transfers have different effects on consumption. TFP growth and demographic structure change are the two major factors for understanding the first ascending, then declining behavior of saving rate. In particular, TFP growth together with the decreasing share of bequest wealth and the increasing wealth-income ratio can account for most of the saving decline in 1990s. The effects of bequest motives, technological innovation and social security on consumption, labor force participation and fertility are then analyzed by the corresponding functions derived explicitly from the model. In my second essay, I explore the possibility of different economic growth stages and social preference change accompanying the evolution of consumption behavior in the long run. The standard representative agent growth model is estimated econometrically using the Generalized Method of Moments for the U.S. economy for three separate growth eras from 1929 to 2002. The parameter estimates differ substantially in the three cases, which imply social preference changes in time discount and work-leisure tradeoffs. These in turn imply switching among alternative balanced growth paths and differences in the contributions of capital, labor, and labor augmenting Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. xi productivity among the three Eras, which are also reflected in the evolution of consumption and saving behavior as shown in chapter one. In my third and last essay, I analyze the relationship between aggregate consumption behavior and the effectiveness of public fiscal policy. A dynamic aggregate demand and supply (DADS) model with different consumption patterns is used to explain the tax policy paradox that tax rate decreases could in principle - and might in practice - increase tax revenue and national income. We find that with the consumption pattern of high marginal propensity to consume and invest, the tax rate reduction can stimulate economic growth, tax revenue, and government budget surplus in the long run. The dramatic change of consumption behavior characterized by the consumption boom since late 1980s implies that the tax cut policy may be effective in practice for the U.S. during the same period. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 Chapter 1: Overlapping Altruistic Generations: An Analysis of Consumption and Saving Behavior in the U.S. Economy Abstract: This chapter develops a three period OLG model of altruistic household consumption, labor, fertility and bequest decisions to explain the change in personal saving rate over the last half century in the United States. We find that wealth accumulated through life cycle saving and intergenerational transfers have different effects on consumption. TFP growth, demographic structure change, together with the decreasing share of bequest wealth and the increasing wealth-income ratio can account for most of the saving decline in 1990s. The effects of bequest motives, technological innovation and social security on saving, labor force participation and fertility are then analyzed by the corresponding functions derived explicitly from the model. Calibration and estimation of the model show theory’s consistency with the data. A new way of estimating bequest wealth is also obtained. 1 Introduction This chapter develops a micro theoretical framework for long run consumption and saving behavior using an overlapping generation model with endogenous fertility, intentional bequests, and social security system. It studies a representative household’s consumption behavior over different income and wealth levels and at different ages, and the interactions between saving, bequest transfer, fertility rate, labor force participation, and social security at the macro level. It aims to answer the following questions: What is the difference between effects of the wealth accumulated through market saving and intergenerational bequest transfer and its implications? Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 2 How is household’s birth decision associated with its consumption and bequest behavior? How do productivity growth and demographic structure change affect aggregate consumption? What is the effect of social security taxes on saving? How do individual consumption behaviors aggregate to generate the rising and declining trends in personal saving rate? The personal saving rate in the United States has experienced three different phases during the last five decades as shown in Figure 1.1. 25% 22% Adjusted 19% r o 0 1 o> 16 % c o ? 13% 7 5 § 10% m d ) Q - 7% NIP A 4% 1% in o o t - c o c o r - - 0)0)0) o o Year Figure 1.1 The personal saving rate in the U.S. From early 1950s to 1960s, the saving rate had been rising from 4 percent to 10 percent, then from 1970s to early 1980s, it fluctuated around a steady level of 10 percent. During the last two decades, it declined dramatically from 10.6 percent in 1984 to below 2 percent in 20001 . The reasons for this change, its implications on economic growth, and its inconsistency with most formulations in optimization models have attracted many discussions in the literature. 1 The first saving rate series in Figure 1.1 is from National Income and Products Account (NIPA). The second series has been adjusted by excluding consumer durables from consumption and shows a similar trend over the same period. Empirical evidence indicates that consumption grows faster than income, or equivalently, that the personal saving rate declines over time after 1980s. Both the U.S. and international empirical studies show this fact (Slesnick, 1992, Attanasio and Batiks, 1998). Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 3 Among the various explanations for the recent saving decline, most are based on statistical estimations and focused on the effect of one or two factors. For example, Mehra (2001) estimates consumption equations using aggregate data and shows the presence of a significant wealth effect on consumption. Davis and Palumbo (2001) get similar results using household level data. Many other factors like demographic structure, social security system, intergenerational transfer and the development of financial markets have been studied specifically (Cox et al., 2000, Evans, 2001, Hurd, 1989); yet no single effect can successfully explain most of the variation in saving rate (Parker, 1999). In the meantime it is common for empirical studies to ignore the linkage between the optimal household decision mechanism and the macro level saving performance, so the consistence of statistical estimates with economic theories becomes another concern2. This paper explains the changes in personal saving rate from a theoretical point of view. It incorporates four of the most important factors in the literature that determine individual’s consumption decision, that is, altruistic motives and the intergenerational linkage (Barro, 1974), wealth and its decomposition through different ways of accumulation (Kotlikoff and Summers, 1981, Modigliani, 1988), birth decision and demographic structure (Becker and Barro, 1988), and social security system (Evans, 2001). The framework enables us to study the effect of each separate factor, as well as the cross effect of factor interactions on saving behavior at the aggregate level. A three period overlapping generation model with altruistic motives toward raising children and leaving bequests intentionally is developed; the optimal household savings of different 2 The importance of income and wealth effects has been widely verified and estimated. A common econometric consumption model estimating both effects takes the following form: c < W , m where Ct denotes consumption, Y t denotes disposable income, and W t denotes household net worth at the beginning of time t. The underlying microeconomic mechanism of (1) is still unclear. This is not a trivial issue since econometric regression of other ad hoc forms can also yield same or better result. Finding the answers to questions why consumption equation takes such a simple form and yields not bad result, what economic factors determine the coefficient values, and if there exists any other functional form both consistent with micro behavior and better in fitting the macro data the major motivations of this chapter. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 generations are aggregated to get the macro saving function, it is then tested and estimated with the data of the United States. The analyses lead to the following results. First, wealth accumulated through investment and intergenerational transfers have different effects on consumption, the shares and the relative importance of the two components change over time. Second, relatively small effect of bequest wealth and its decreasing share, as well as the increasing net worth-income ratio (see Figure 1.2) account for most of the saving decline in 1990s. o T O c c a 5.5 8 C d ) 5 IB to w 0 c l 4.5 C O Q 1 4 g a > 2 3.5 Y ear Figure 1.2 U.S. net worth-disposable income ratio Third, the calibration and estimation results show that the model fits the historical data quite well and reproduces the rising and declining trend in saving rate. Last, an alternative way of estimating bequest wealth is obtained from the model; the impacts of altruistic motives and time preference on the share of bequest wealth are also simulated. In section 2 a three-period OLG model of representative households is developed with double altruism, endogenous fertility and labor force participation decisions, and social security system. Section 3 solves the model. The aggregate consumption function will be derived analytically from Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 5 micro optimal decisions. Section 4 provides comparative static analyses of various causes for the change in saving rate. Section 5 calibrates the model. Section 6 gives the estimation and simulation results of the model, and finally, section 7 summarizes the main results and findings. 2 The Model Consider agents who are double altruistic: they voluntarily feed their children and leave bequests to descendants. There also exists a social security system which makes transfers in the opposite direction: from young to old. 2.1 Demographics Suppose the economy is populated with even number of males and females of each age. Each individual lives for three periods, childhood, adulthood (middle), and retirement (old), with age denoted by 1, 2, and 3 respectively and generation denoted by birth period. At the beginning of age 2 each pair of adult male and female marry to make a household and have children. The children live in parents’ household until they reach age 2 and make households of their own. We refer to the household as a generation t, or simply, t household, if the couple is bom at time t. The total population of generation s at time t is denoted as N *, which is also double the number of s households, if s<t. At the beginning of time t, each middle aged couple produces n, children. Individuals of cohort s face random survival from age t-1 to f, denoted by stochastic variable n* , t=s+l, s+2, 1 with probability p x 0 with probability \-px 1 with probability p 2 ^ 0 with probability 1 -p2 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 6 At the end of age 3, each agent dies off. Survival probabilities pl and p2 are exogenously given. Total population is N, = N‘ t + N 1 '1 + A '-2 = N ‘~lpl(pin,nt_l/4 + pin,_i/2 + p 2) . The demographic structure depends on both the endogenous fertility and the exogenous mortality rates. Throughout the paper, superscripts denote birth dates, subscripts denote time periods, uppercase letters denote aggregate variables, and lowercase letters denote household variables, so c'~j denotes the consumption of a t-j couple (born at t-j,j = 1,2) at time t. 2.2 Technologies There are two kinds of production in the economy, the market production of consumption goods and the household production of children. 2.2.1 Production of Consumption Goods Suppose the economy produces one output that can be either consumed or accumulated as capital stock. The good market is perfectly competitive, and a representative firm’s production function (normalized to be the total output) takes the standard Cobb-Douglas form Z t = F {K „ A tH ,) = K°{AtH t Y e (3) where A( + 1 = (1+ g)At is the labor-augmenting productivity and g is the exogenous productivity growth rate. Z, is real output, K t is the total capital stock, and H, is the total labor input. The aggregate stock of capital evolves according to Kl+ 1 = (1 - S)K t + /,, where 11 is net investment and S is the depreciation factor. A representative firm’s problem is to maximize its profits for each period, max Z t - (rt + S ) K t - w ,H t (4) K, ,H, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 7 where w, is the real wage and r, is the real interest rate. Rental markets for capital and labor are perfectly competitive, so factor prices are equal to marginal productivities: W, = a - 8)A}-eK?H;e (5) rt = dA)-eK?~lH)-e - S 2.2.2 Household Production of Children Raising children costs both money and time. Suppose there is no labor market for babysitting, so parents need to allocate part of their time endowment for taking care of children. A middle aged couple has access to a common technology of producing nt children nl ={et t-xY { B tm';l )x -p (6) where Bl+ l = (1 + g h )B, > 0 is the labor-augmenting technology of childbearing, e‘~ l denotes the total consumption expenditure on children, m '”1 is household’s labor input for children production. 2.3 Households Household is the basic unit of decision making in our model. A t-1 household, by definition, consists of one couple born at t-1, and their age 1 children at time t; and consists of only the couple when they get old at time t+1. Therefore, there exist two types of households at any time, one with young children, and the other without. Children of age 1 do not work and do not make any economic decision. Instead, they just consume whatever good their parents give them. Each household has two units of time endowment that can be allocated among leisure and working in the market or within family. Parents are assumed to be altruistic with both the motive of rearing descendants and the motive of leaving bequests. Bequests are not only left for immediate Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. descendents, but also for grandchildren. The couple of each household makes joint decisions on own current and future consumptions, expenditure on children, bequests, and labor inputs for market and for children production. 2.4 Bequest Motives and Wealth Decomposition Most people do not spend all their income and wealth on themselves; instead, they spend a substantial fraction on descendents. Consequently, the number of children, the provision for children’s necessities and education, and the bequests they will receive significantly affects parents’ consumption and welfare. In order to give rise to endogenous fertility rate and wealth transfer between generations, we incorporate altruistic motives explicitly into household’s utility function. Under the consideration that children’s preferences are uncertain or even unknown at their birth, we do not include children’s utilities in parents’ utility function directly like Barro (1974), Becker and Barro (1988) and many other altruistic models; instead, we choose the number of children and, following Blinder (1973) with modifications, the levels of bequest to children and grandchildren as reflections of parents’ altruism. This presentation of the utility function, due to its finite horizon feature, yields different results than the alternatives of introducing future generation’s utility directly into the utility of current household. Empirical studies show that intergenerational wealth transfer constitutes a substantial part of the total wealth (Kotlikoff and Summers, 1981, Modgliani, 1988). Despite of the big difference in estimates of the amount of bequests, the fact that bequests play a significant role in wealth accumulation is widely accepted. Therefore, we decompose wealth into two components by their accumulation approaches, i.e., wealth accumulated by investment (life-cycle saving) and by intergenerational transfer, like bequests. We will show later that the two wealth have different effects on consumption, and their shares in total wealth are changing. The combination of these two factors provides more explanatory power for saving behavior than the existing research limited to total wealth effect. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 9 2.5 Household Utility A generation t-1 household at time t maximizes its expected lifetime joint utility Etu(c'~x,c'~', , n,,b'~ 'x, gb‘~ x), which is assumed to take the logarithm form E,U =(1 - X)Inc/”5 + /In n , +A\n(2-h'~l -m ,'-1) + Pp2 ^ - 7 ) ^ c‘~ \ +v, lnZj,'"1 + v 2 ln g ^ ,1] where c / denotes the consumption of a generation j couple at time s, s = j+1, j+2, respectively; l ’~x is the leisure taken by each t-1 household, h'~x and m '-1 are the labor inputs for market production and children production, and l'~l - 2 - h'~x - m '~x; nt denotes the number of children to be raised; fc'*1 and gb‘~l denote household’s bequest for children and grandchildren; ft is the subjective time discount factor; y denotes the couple’s willingness to raise children; Vj and V 2 are another set of altruism parameters that measure the motives of leaving bequest to children and to grandchildren. To keep the constancy of altruism over time, we set v x + v 2 = y ■ In the extreme case of / = 0 and vx =V2 -0 , i.e., individuals have no motive of rearing descendants or leaving bequests, our model will be reduced to a standard OLG model of representative households. 2.6 Social Security System There exists an unfunded social security system reinforced by a benevolent government. The government levies a tax rate Tt on the current working generation’s labor income, and uses the tax contribution to pay benefits to each current retired individual in the amount of a fixed fraction, (p, of lifetime working earnings. It is a pay-as-you-go (PAYG) system and the payroll tax rate depends not only on the replacement rate (the benefits), but also on the relative size of the working generation compared to the retired generation. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 10 Assumption 1. (PAYG Social Security System) The social security system is unfunded; the entire social security pensions paid to the retirees come from the social security contributions of the current working generation = N ‘ , \ w th'-X (8) 2.7 Budget Constraints and the Optimization Problem Each household formed at t makes following decisions, labor input for market production and middle aged own consumption; number of children to raise thus expenditure and time spent on them; old aged own consumption, bequests for children and grandchildren. A couple of generation t-1 at time t (as parents) has initial wealth endowment inherited from their grandparents, 4 n ‘ tZl gb‘ tZl / pln,-int_2 , plus inheritance from accidentally deceased parents, 2(1 / / W i > where s[_2 is the savings of a t-2 household. They obtain income from two sources: 1. labor income (1 - r ^ h ^ w ,, and 2. capital gains from the appreciation of bequest wealth, pln^n,^ + 2(1 - ^ “2 )sj~? / plnt_ i) ■ The couple uses part of the total income and wealth on consumption for themselves and for children, and save the rest for next period. The couple at the beginning of time t+1 (as grandparents) has wealth consisting of its last period savings, s‘~ l , and inheritance from parents, 27Zi~ 2b‘~ 1 / plp2n,_l ■ Their income also has two sources: capital income, which includes net gains from savings rt+ls ‘~l and gains from the inherited bequest 2rl7r‘~2bt~ 2 / plp 2n ,^ , and social security pensions, < ph'~lwt . Combining the above conditions, each household of generation t-1 maximizes the expected lifetime utility (7) subject to the following budget constraints Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 1 1 cJ + J + ^+i + £#+’ - (i+ r( + i K 1 + — — - — t+ 1 ^ ' + < & > , PiP2« < -i (9) ( 10) C J >o bs s~ 2>o g^r2 > o i>/j;-',to,m >o j = 1,2, 5 = t-eo,-",oo The optimal household behavior is described by the Euler equations and Proposition 1. i = * £ - v> P i^ d + 'i+i) ! - r s K i v 2 e! ' = YP m‘~l ^ r(l - P ) c,'-1 1 - y > 1 [l + <9/(1 + rl+ 1 )]w, V,-1 _ i [l + <p/( 1 + r,+ 1 )V, W ,'-1 ^ 1 - p < x 1-r e p Proposition 1. (Optimal Household Behavior) 1. The ratio of household’s time spent on children to leisure is fixed, i.e., the more a couple likes to have children, the more they will spend time taking care of them. 2. The optimal leisure decision is made such that the market value of leisure has the same marginal utility with current consumption. 3. The optimal decision in children production is to make the market value of the time spent on children have the same marginal product with the expenditure spent on them. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 12 2.8 Competitive Equilibrium Now we define the competitive equilibrium of the economy. Definition 1. A competitive equilibrium is a set of household allocations, {Kt,H t ; real prices for the factors of production, {rt ,wt , all subject to the following: 1. All households maximize (7) subject to (9); 2. Firms maximize profits for each period, so prices for the factors of production are the marginal productivities as shown in (5). 3. The market for labor clears 3 Equation (12) states that current working generation’s market savings and current old generation’s bequests for descendants constitute the capital stock or total wealth of the next period. It’s obtained by using the law of motion of capital, any given initial capital stock, K0, and the fact that total net savings I, is the sum of three components: 1. total net savings of t-1 households at t ; 2. total net savings of t-2 households at t\ and 3. firm’s reserved depreciation 5Kt. y'=l,2; asset holdings (savings), {.s' 1} H, = N r ih r i /2 a n 4. The market for capital clears’ Kt+ 1 = + N ‘ t-\b[-2 + gb‘ t-2)/2 ( 12) 5. The market for consumption goods clears 6. The social security is unfunded and its account is in balance (14) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 13 2.9 Optimal Household Choices The model can be analytically solved in closed forms using the Euler equations and the equilibrium conditions. Given real wage wt , real interest rate rt , social security tax rate r , , and inheritances b'~? and gb‘~_l, the optimal interior working, leisure, consumption, saving, and bequest plans for a generation t-1 household are solved as follow 2(1 - y) C - / 1+A+P 2P 1+ - 1+ ^4- 7 1 - 2 l +r, w, ----— + L (P 2~V > )< P . PAt-l PA- (1 - p2+vl)C‘ + ,7-2 . 2 p2 v2 + v l u, 2 PA (15) i-r (16) (17) ,.,- 1 _ VlP2fi(l + rl+1) ,+1“ 1 - r ' (18) ,-1 _ V iP 20(l + r,+1) = 1 -y (19) _ 2 l + y ( l - p ) ( i - r ) [ i + ^/(i + r,+ 1 ) K ' (20) ^ = r± z£ l c' - ' (1- y ) [ l + 97/(1+ r,+i)]w, (21) n, =J £ L l z £ « 1+ — — \ w , ■\+p (22) The first term in brackets in the right hand side of (15) denotes the total labor and social security income, the second term denotes the total lifetime capital income. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 14 3 Aggregate Consumption and Saving Behavior 3.1 Aggregate Consumption Function In order to explain the aggregate consumption and saving behavior by the model, we need to sum up all households’ optimal choices. Total wealth level at time t is the sum of market savings and bequest savings K, M denotes the wealth accumulated by investment, and K f denotes the wealth accumulated by intentional bequest transfer, Total disposable income is the sum of all labor, social security, and capital incomes (including reserved depreciation) of each generation. K, = K f + Kf (23) (24) (25) (26) where Y,L = N ' - ' w X ' 1 /2 (27) denotes the income due to market labor participation, and YtK ={r,+S)Kt (28) denotes the net capital income. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15 The age distribution of cohort s is n 'm = p x n : n :+ 2 = p 2n ;+ 1 N ; = n sN;-l/2 s = - ° v , ° ° (29) Total consumption is the sum of consumptions of each household at time t C, = N'-1 (c( M + e'-1 )/2 + N ‘-2c‘-2 12 (30) Plugging into (30) the solution of c' ,e‘ \ and c't 2 given in (15)-(22) yields C , 1-(1 -p)y+p2 /3 , 1 ( PiP0--Y)'\ < P 1+— v.-p2 Y+-— — — k + ----- ---- p \ l ™ 1-(1 -p)rj' 1 + E,(r,J .0+0 PA- 1 i+vl-p2 y+ pffli-r) 1-(1 -p)Y pm- n i — a - p)y f t (31) Plugging into (31) the equilibrium conditions of real interest rate and wage (5) and social security tax rate (14), we get aggregate consumption as a function of total income and wealth C, = e 1 - s ( e B e M ) f K f A y~K,j l + £,(r,+ i) Y,+ey Yt_l+ eMK ^ e BK f (32) 1 - (1 - p)y i - (i - p)r + PiP 1 - (1 -p)Y y-' \ - { \ - p ) Y + p 2P 1 - (1 ~P)Y \ + e V\ - P ir + p IPQ- - r) 1 - (1 - P)Y (1 ~6)-P2Y + p \ m - Y) 1 - (1 - p)Y - i - ( i - p)y + p2P i - q - p )y 1 - (1 ~P)Y+ PiP 1 - (1 - p)Y (1 -6)<p (!-<?) l + ^i - P 2Y + 1 - (1 - p)Y + p2P ( 1 -S) 1 + \ P i Y j pIP(\ - r) 1 - (1 ~P)Y plPO- - Y) vx- p2y + 1 - (1 - p)Y (33) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 16 Dividing both sides of (32) by disposable income and subtract it from one gives the saving rate function s. = l - e f Y X it-1 y l+£,(rI + 1 ) K - ' e i-s (eB eM ) U J eM K €m ~eM^~rr K, 'K ,' (34) In general, eM^eB, er ^ 0 and er ^0. Equation (32) implies that in addition to current income and total wealth levels, the past period income level, future interest rate expectation and the composition of wealth all have effects on aggregate consumption. Wealth accumulated through market investment or life-cycle saving and wealth accumulated through bequest transfers (bequest wealth) have different effects on consumption. The difference of the two wealth effects is given by 1 - (1 - P ) Y i - ( i - p ) r + Pij3 1 - s P i Y ) V. ■ PlY) v, ~ P2Y + p I P Q - y) 1 - (1 - P)Y (35) It’s easy to verify that under some special situations, eB -eM =0 and (32) degenerates to the standard form consumption function that is determined solely by income and total wealth. Proposition 2. (Benchmark Consumption Function of Standard Form) Under the condition of no social security system, i.e., (p=0, and either one of the following conditions on bequest motives and survival uncertainties: Vi = p 2y or 1 /j = p 2y - p I P Q - y ) 1 — (1 - p )y (36) we have eY = er = 0 and eM = eB, aggregate consumption function then takes the standard form commonly used in econometric studies: Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 17 if v, = p2 r (37) c , = i- ( i-p)y+p2 p f-(l-p)r 3.2 Relative Size of Two Wealth Effects In general, (36) does not hold, bequest wealth, market wealth, and past income all have separate effect on saving rate. Proposition 3. (Comparison of Wealth Effects and Income Growth Effect) Suppose there exists a social security system, (p > 0. 1. If 0<p2<vl/y, then eB > eu , bequest wealth has a larger effect on consumption than market wealth; and eY > 0, the gross aggregate income growth rate Yt/Yt_x has a positive effect on saving rate. 2. If v, > j'2[l-( l-p )j']/4 /? (l-y ), when \> p2 > v jy , we have eB 0 3. If vl <y2[l-(l-p)y]/4/3(l-y), 3a. when p\ >p2> vjy, we have eB 0; 3b. when 1>p2>p*2, we have eB >eM and eY <0\ Survival probabilities are not constant in the long run; instead, they are increasing with living standard, environment, and health care level, etc., which are all positively correlated to aggregate 4 Data of mortality rates and life expectancies are from National Vital Statistical Reports, vol. 52. 2004. where income. Figure 1.3 shows how the two survival rates of the U.S. evolve in the 20th century4. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 18 1 — ------------------------------------------------------------------------------------------------------------------------------------------- □ □ ° " 0 . 9 6 - □ □ (l o 0 • • 0 - 9 2 - o ( 0 a > | 0.88 - 1 5 I 0 . 8 4 - m * 0.8 - • D o from 0 to 25 0 . 7 6 - a • from 25 to 50 q 7 2 '______ 1 ___1 _____ 1 _____1 _____1 _____1 _____1 _____1 _____1 _____ o o o o o o o o o o o O r - C N J C O r f i n < D N - O O a > 0 o > o > o > o > C D O > a > c n a > o > o Year Figure 1.3 Conditional survival rates for the U.S. The rate of surviving to 25 years old (an approximation of pi) increases by 31%, and the rate of surviving from 25 to 50 (an approximation of p2) increases by 22% during the last 100 years. While both survival rates grow over time, the growth rates are different. px grows faster than p2 at the beginning and then slows down after World War II, then p2 starts to catch up and the ratio of two rates is getting closer to unity. Note that although in general p2 < pv there are exceptions. The two situations of p 2(1901)> /?1 (1901) and /?2(19IQ ) > p,(1910) are due to the high infant mortality at the early stage of economic development. At the very early stage of economic development, survival rate is low so p2 < v j y , then bequest wealth has bigger effect on consumption than market wealth, and income growth rate has a positive effect on consumption ( eK ] > 0 )• As the economy grows p 2 becomes larger, or equivalently, parents are willing to leave more bequests to grandchildren when they have the ability to do so, the above result will change. When p2 >vx/y , the effect of bequest wealth on consumption becomes less than that of market wealth. Under the condition of preference □ ■ o from 0 to 25 • from 25 to 50 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 19 parameters given in case 3 of Proposition 3, the signs of eB — eM and eY | may change again when p 2 gets even greater than p*. As a result, the saving rate will change non-monotonically in the long run. To evaluate the discounted value of future income when making lifetime decision, households need to make expectation of future interest rate. We have the following assumption, Assumption 3. (Expectation of Future Interest Rate) Households make naive expectation of future real interest rate, i.e., E, (rl+ l ) = rt- 4 Qualitative Analysis of the Model 4.1 Effects of Productivity Growth In the existence of social security system, the current working generation pay social security pensions to the retired generation in the amount of a fixed fraction of the latter’s working income. Since the system is unfunded, opposite impacts on consumption of the two generations will result. The final effect on aggregate consumption is determined jointly by the relative cohort .size N r2/ N r = 2 p 2//?[«,_, , and the relative income size of the social security beneficiaries, Yt_jYt ■ At steady state, Yt_jYt = 2 /ptn(l+ g ) , where n is the steady state fertility rate. The growth rate of productivity of market production g and therefore the growth rate of total factor of productivity (TFP), (1 + -1 , could have different effects on saving rate depending on the sign of eY t ■ ds, _ 2eY i > 0 when eY _ t > 0 dg pjn(l + g)2 <0 when eY[ <0 From Proposition 3 we know that if vx > y2[l-(l-p)y]/4/3(l-y) , then >0 , an productivity growth rate shock will always cause saving rate to rise. But in an economy with Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 20 v, <y2[\-{\-p)y}/AP{\-y), which means the bequest preference toward grandchildren is very high, eY j will become negative as at high level of p 2, and leads to the decline in saving rate in response to a positive shock to TFP growth rate. At the earlier stage of economic development when both the portion of retired people and the income level is low, economic growth has a positive effect on saving rate as households want to smooth consumption over their lifetime. The empirical facts of most developing countries and developed countries at their earlier stage show this phenomenon. As the economy grows, the survival rate p 2 also increases. By Proposition 3, when p 2 > p 2, we get ey < 0 , the higher rate of TFP growth may result in a decline in saving rate. Possible economic explanations could include consumption preference and behavioral change and less incentive for precautionary saving as people become much wealthier. As an approximation of the above relation, Table 1.1 shows the correlation between the personal saving rate and the growth rate of TFP in the U.S. between 1947 and 2002 corresponding to the different stages of saving trends. Saving rate is positively correlated with TFP during the period 1947 to 1970 and the period 1971 to 1984 with a decreasing magnitude, and is negatively correlated with TFP during 1985 to 2002. It suggests that the productivity growth effect may have been reversed in late 1980s. Table 1.1 Correlation between the saving rate and the growth rate of TFP Phase Correlation Average of the saving rates Average of TFP growth rates 1947-1970 1971-1984 1985-2002 0.472 0.074 - 0.222 0.079 0.099 0.060 0.0216 0.0105 0.0162 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 21 4.2 Wealth Effects 4.2.1 Effects of Market Wealth and Bequests Wealth Total wealth consists of two components accumulated through different channels. In general, their effects on consumption and savings are also different. Taking the derivatives of (34) with respect to K™ and K? and then taking the difference yields ds. ds, dKM dK ~ ieB eM ) 9 1 y, a - s ) k , (39) Saving rate’s responses to the same unit market wealth shock and bequest wealth shock are different. By Proposition 3, if vl >y2[l-(l-p)y]/4]3(l-y), es p2>vjy)- If v, <y2[l-(l-p)y]/4/3(l-y ) , eB-eM may change signs from negative to positive as p 2 increases. In either case, a unit shock to wealth may have different results depending on the allocation of shock between its two components. The size of bequest wealth can be expressed as a dynamic function of total wealth and income with initial condition on past bequest wealth. k b = l r ( l + r(_i) 1 - (1 - p)y \ l + 0(v, - p2) + f t 1 - — )]T„ 2 + < p( 1 - eyyx - p2)Y,_ 3 1 + r(_ , + (1 - S)( 1 - p2 + v, )K,_2 + (1 + rt_2 )(v, - p 2 -1 )K?_2 p% y (40) (40) implies that bequest wealth is jointly determined by past income, total wealth and bequest wealth up to three period lags, which can not be a constant proportion of K t in general. 4.2.2 Effect of Relative Wealth to Income Although wealth accumulation is usually associated with income growth, the relative growth rate may vary over time. One of the most significant economic changes in the United States that Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 22 accompany the rapid decline in saving rate in the later half of 1990s is the dramatic increase of household net worth-disposable income ratio. From 1995 to 2000, that ratio rises almost 30 percent. Maki and Palumbo (2001)’s estimation shows that wealthy Americans’ consumption response to the stock-market boom can account for most of the decline in saving rate last decade. In this paper, we can get similar result using aggregate data, and what is more important, unlike Maki and Palumbo whose work is solely based on statistical estimation, our model provides an optimal micro foundation for the aggregate level estimation, and the consumption and saving rate equations are explicitly derived to be compared with the empirical data. Let fit - Kl/Yl be the wealth-income ratio, the derivation of (34) to wealth-income ratio gives rise to the following result d*, dfi, (k ” ) eB ~eM ______(u + J - \ e -e )d(K' /K'} <0 (41) [{\-s)n, +ef r i - J ' M m When the economy deviates from its steady state wealth-income ratio, saving rate will move to the opposite direction. The magnitude of the response will be strengthened if the wealth component with bigger consumption effect rises its share in total wealth, and will be weakened if that share drops. This allows the same percentage growth in wealth-income ratio to lead to different levels of decline in saving rate. For example, wealth increase mainly due to a rise in stock prices may raise market wealth more than bequest wealth; so the effect on saving rate is even bigger if market wealth has a greater consumption effect than bequest wealth. 4.2 Role of Social Security System The U.S. empirical data show that the average social security taxes over total disposable income increases steadily from zero in late 1920s to about 8 percent in 20005. To check if this 5 The current social security tax rates for employees and employers are both 6.2%, the upper limit for taxable income is $87,900. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 23 increase in tax rate can partially account for the declining trend in personal saving rate, we start with expressing replacement rate 0 in terms of social security tax. Keeping everything else constant, and taking the derivative of (34) with respect to Z, yields Social security payment serves as a kind of forced savings for the old aged, so it will crowd out part of the private precautionary savings, thereby an increase in Tt will make saving rate to decline. Although eY i might become negative when the income associating survival rate p 2 is greater than p* as indicated in Proposition 3, the overall effect is negative. An increase in replacement rate (p has the similar effect. However, the existence of bequest motive hence positive bequest will affect the magnitude of social security’s effect on saving. We have ds, \ d ds, \ <0 and — ) dtp ) (43) shows that the higher the motive for leaving bequest, the smaller (in absolute level) the negative effect that social security has on saving. This theoretical result extends Laitner (1988) on the interaction between bequest and social security, which claims that only negative net bequest wealth can do the job. 4.2 Bequests, Technology and Fertility 4.4.1 Bequest Motive’s Effect on Fertility Rate Parents’ altruism toward children and grandchildren plays an important role in their consumption and birth decisions. In the long run, the incentives of raising children and leaving Y, J [l+E,(rM) h e, <0 (42) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 24 bequests to descendents may change over generations. One of the most significant evidence is the declining fertility rate and the shrinking household size. U.S. birth rate drops about 40% since 1950s and the average household size declines from 3.11 in 1970 to only 2.67 in 2000. It will not be enough to explain these huge changes without considering the shift of values on family and altruism over generations. To test the impact of family preference changes on fertility, first derive the fertility rate function from (22) in terms of income and wealth per working household n m i P \ - p f e\ y1 [l - (l - p)y + p2P]a)~ p .P0--8). U ' J -f) \ - e (1+ Q{vx - Pi))y; + (1 - S)( 1+ V , - p2)*; + 0'i -Pi) PtY (44) where superscripts “e” on income and wealth variables denote per working household. Here we set q>- 0 to simplify the analysis and concentrate on the effect of altruism motive. Note that At = A0(l + g)‘ ■ Then the effect of altruism motive change on fertility is given by dnt dy l+Pifi v2 jvx f ix e d }{l-{\-p)y+p2 p} ' pvfiSPi-v2)kf fl/p-l y[i-(i-p)r+p2 /3]^fi{ \-e i -p m -p) M \ - e e £ + i- s \ ke \ K t ) I K J. p2 r K . >o (45) (45) implies that a downward shift in the motive of raising children will cause the fertility rate to drop, which is quite obvious. The impact of bequest preference shift between children and grandchildren is given by dn, Vp - i i-p g (i-p) e ^ + l - d K K 1 2v ke B 1- ( 1+------- ~)“7 7 " y p2y k‘ (46) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 25 The effect depends on the proportion of bequest wealth that parents and grandparents receive. If k f / k‘ < p2y/(p2y + Pi _ 2Vj), fertility rate will decrease in response to a preference shift from children to grandchildren, otherwise, the equilibrium fertility rate will rise. 4.4.2 Household Production and Fertility As the technology and structure of children production change, the relative cost of raising children also changes, which in turn affects household’s birth decision from the economic aspect. The effects of technology progress and structure shift of household production are dn, n, —L = -!->0 dB, B, (47) dn, dp ln (l-0) + ln Q 1 - p ) 1-(1 -p)y+p2j3 1-0 In k, X, <0 if p<p0, p0 >max(0,-^) The effect of the technology progress of children production in (47) is simple and obvious, if the technological growth makes it possible to have more children at no additional cost in money and time, then fertility will rise for certain. Introducing positive shocks to Bt thereby can help to explain the sudden rise in fertility rate that lasts a short period of time, like the baby boom in 1950s (Greenwood et. al., 2002). The elasticity of fertility’s response to technology shock is one, larger families will raise more children than smaller households in response to the same positive shock. The effect of the change in the household production structure is much more complicated, it also depends on the production structure of the market good, current capital-output ratio, and household’s time and altruistic preferences. In practice, the structure parameter 6 of market production usually takes the value of one third, as for children production, p maybe little higher or lower, but should be less than one half due to the time demanding nature of raising children. So in Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 26 general we have dnt/dp< 0, the effect of production structure shift on fertility rate is negative6. The economic intuition is, any household production structure change that makes raising children less time intensive but more money demanding will always lower fertility rate and push more women to the market, since the household now has to provide children more consumption goods, and the amount of labor released from household by the structure shift is not enough to earn extra income in the market for the increased demand of expenditure on children. As a result, the household either chooses to have fewer children, or to work more for the market, which also leads to the decline in fertility rate. This theoretical result provides an alternative and interesting explanation of the common decline in fertility rate in most developed countries during the last 30 years. With the progress of economic development and civilization, the meaning of raising children has been changed fundamentally. It is now far from enough to only provide children basic living necessities and play with them; better and higher level education and trainings become important part of the raising process, which makes parents to spend more money on children. In another word, raising children becomes more and more materialized, as a reflection in the production function, p is getting bigger. In order to have children, young couples both need to work to earn enough money. Now the problem is not parents’ not having enough time to raise children, but in the contrary, they do not have enough money. The order of cause and effect seems to be reversed here compared to the common understanding: it is not because women work, so they do not want to have children; it is because they want to have children, so they have to work. 6 We can prove that dn,/dp(p=0)<O, dn,/dp(p=0.5)<0 , d2n,Jdp2 > 0, and limdnt/dp = +°°- Then p - A dnjdp is a continuous, monotonically increasing function of p , there must exist some max(9,1 / 2) <pQ<\ such that <0 if P<Pa = 0 if P = Po >0 if P>Po Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 27 4.4.3 The Number of Children and the Size of Bequests Few theoretical studies discuss specifically the effect of the existing number of children in a family on the size of bequest each child will receive in the future, and its implications on children’s earnings and parent’s savings. Our model provides a simple and reasonable analysis. The average bequest each living individual of generation t will receive from parents at t+2 is Both bq‘ and nt are multi-variable functions with the same set of independent factors. To find out the effect of nt on bq‘ , we must study the separate effect of each factor on both of them. Take the logarithm of bq‘ and nt , then check each factor separately given all the others constant. We get the following results If the increase in the number of children is solely due to the rise in parent’s altruism, then the bequest each child will get is exactly the same as before. Parents raise the total amount of bequests proportionally and there is no trade-off between quantity and quality. Total bequests will increase proportionally to the increase in the number of children, the rise of share of bequest wealth may result, which has further influence on the economy as we discussed earlier in the paper. bq N, (50) N',+2 P i n , Substituting expression (18) and (22) into (50) yields = 2 v ,p 2/?(l+ r,+1) (51) YPPxB, 1 - p A. 9 In bq‘ 9 y 9 y 9 In nt (52) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28 B. d ln V dvx _ _ 1 Idlnn, (53) dvx dlnn, vlf dlnv. The effect of the number of children on bequest per child is the same as that of the bequest preference allocation on number of children. If parents increase the number of children because they prefer to leave more bequests to them rather than to grandchildren, then they will also increase each child’s bequest. Since b'^/gb’ ^ =vx/v2, the extra bequests are taken from the bequests for grandchildren. When 9 Inn,/9 In v, <0, increasing the number of children will decrease the bequest each of them can get. This trade-off between quantity and quality is natural because the total altruism toward descendants does not change. C. dlnbq1 dp _ 9lnbq' dB, (54) dp d Inn, dBt d Inn, If the change of household production structure makes raising children less time costing but more money costing, or the opposite, and it makes parents want fewer children as discussed in section 4.4.2, then the bequest each child should get will increase proportionally, and the total amount of bequests to children keeps unchanged. The same quantity-quality trade-off in the demand of children also happens when the advance of household production technology B, encourages young couples to have more children, because raising children become less expensive from an economic point of view. However, the amount of total bequests to be left won’t change; as a result, people who were born in the baby boom period may have to compete with their siblings for bequests left by parents. Our analytical results in (54) are consistent with most empirical findings (Adams, 1980, Wilhem, 1996) using U.S. micro data that there exists a significant negative correlation between the amount of inheritance received by child and the number of siblings. In addition to this, situations with zero or positive correlation between the two are also explained in (52) and (53), since many Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 29 economic, cultural and social considerations are associated with the choice of family size, and they may have different effects on the amount of bequests to be left. 4.5 Women’s Labor Force Participation In our model, household as a whole is assumed to take the responsibility of taking care of young children. To study women’s labor force participation, let’ s consider the extreme case in which only the mother spends time to raise children and provides labor input in household production. Modify the household utility function accordingly as E'U = (1- y)\nc'~' +y\nn, + ^ [ln (l-/i,m 'M) + ln ( l- h /'M + PPi [(1 - r) In c'~l + v, In b‘~l + v2 In gb‘~ \ ] (55) where h™’ 1 1 and h f’ ’ 1 denote the market labor inputs of male and female adults, and m f ’ 1 1 denotes women’s labor input in household production. Replace household market labor input h ‘~x in budget constraints and keep everything else the same. (Married) women’s market labor force participation, h f " 1-1, is derived from household utility optimization hf-"1 =1- A+2y(l-p) {\-{\-p)Y+p2 m + < P 1+ £ ,(^ 1 ) +(yx- p 2) P2V A K (56) where the superscript e on income and wealth terms denotes per labor efficiency unit. For simplicity, consider the benchmark case under Proposition 2, at steady state, we have hf = 1— A+2y(l-p) [i-a-p)y+ p2m i+ -r - ] 1 + r 1+06^ - p 2)+{i-8){\+vx- p 2)— y . (57) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 30 The difference between male and female labor force participation rates is -hf'1 '1 = 2y(l~p) (l- h f ’ 1 -1) (58) A+2y(l-p) and the relation between fertility rate and women’s labor force participation rate is - i A + 2y(l-p)( p V " 2yM!'B:-p l i - p ) Taking the derivatives of (57) to labor augmenting productivity growth rate g and household production parameter p both gives positive results, which implies that female labor force participation rate will rise in response to technological progress in market production and household production structure change which makes children production more money (expenditure) intensive and less time (labor) intensive. The influence on the participation rate difference is opposite. As women’s market labor participation rate increases, the difference between the two genders decreases as shown in (58). This result is consistent with the historical data of the U.S. which shows that female labor force participation rate has been increasing continuously over the last century. Married women’s labor force participation rate rises from 5.7% in 1900 to 15.4% in 1940, 38.50% in 1970, 49.4% in 1980, till 59.5% in 2003. In the mean time, the participation difference between male and female declines from about 50 percentage points to only 15 percentage points. / It’s also an interesting topic to analyze the relationship between fertility rate and married women labor force participation rate. As shown in (59), usually a higher fertility rate means a lower labor force participation rate, since all terms in brackets before nt in the right hand side of (59) are positive. But how to explain the fact that in the baby boom period of 1950s both fertility rate and women’ s labor force participation, rate increase? Equation (59) indicates that the answer is the * /(l-0 ) / (1-0) K, ..e ^yt , v 1+ - (59) l + E,(rM)> Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 31 combination growth of labor augmenting productivity of market production A, and household production technology Bt ■ If the increases in the two technologies are big enough to dominate the negative influence of increased fertility rate, then women can both work more and have more children. The wide application of technological innovation in household, like washing machine and kitchen appliances makes it much easier to take care of children and gives mothers more flexibility and availability in time, while the high economic growth rate provide them enough rewards to go out for work than stay in home. 5 Calibration and Estimation We calibrate the model economy to the U.S. economy for the period 1947 to 2002. All data used in the following part of the chapter, if not separately specified, are from National Income and Products Account of U.S. Department of Commerce, Flow of Funds Accounts of Federal Reserved Bank, and National Center for Health Statistics, from 1947 to 2002. In the period we study, substantial changes have taken place in many aspects of the economy, from production technology, men and women’s labor force participation rates, to fertility rate and life expectancies. 1947 to 1980 is the period that many changes in technology during the war years are incorporated into the civilian production, and the period after 1980 are another era of rapid, qualitative change in the nature of capital that is characteristic of personal computers and electronic telecommunications. We thereby consider two distinct eras, i.e. phase I, the post world war II period from 1947 to 1980, and phase II, the advanced technology period from 1981 to 2002. We refer the whole period 1947 to 2002 as consolidated era (C.E.). All calibrations and estimations are made for each of these phases separately. Benchmark calibration and estimation values of parameters are listed in Table 1.2. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 32 Table 1.2 Benchmark calibration and estimation Phase Parameter 1947-1980 1981-2002 1947-2002 Demographics Survival rate 0-25 p 1 0.958 0.980 0.969 Survival rate 25-50 p 2 0.927 0.946 0.934 Market Production Capital share 6 0.319 0.303 0.305 Productivity growth rate g 0.0241 0.0176 0.0192 Annual depreciation rate S 0.0375 0.0433 0.0398 Household Expenditure share p 0.18 0.23 0.20 production Household technology Bt 2.80 2.80 2.80 Altruism y 0.32 0.29 0.3 Leisure A 1.4 1.4 1.4 Preferences Bequest motive V, 0.30 0.27 0.28 Bequest motive V2 0.02 0.02 0.02 Annual time discount /? 0.985 0.985 0.985 Social security Replacement rate (p 0.14 0.14 0.14 5.1 Demographics We take the length of each period in the model to be 25 years. Age 1 in the model corresponds to real age 0 to 25, age 2 corresponds to real age 25 to 50, and age 3 corresponds to real age 50 to 75. For such a long time span, the two survival rates pl and p 2 are not constant as shown in Figure 1.3. We use the average of actual rates for each phase as the values of p{ and p 2 in benchmark model calibration and the real time path data for model simulation and estimation. 5.2 Technologies For market production, the labor augmenting productivity growth rate g and capital share 0 of production function (3) are estimated by GMM method after taking into account the actual capacity utilization rate. Annual depreciation rate 8 is taken as the average of the actual depreciation rates for the period 1947 to 2002, which is 3.98% and is equivalent to 63.8% for the 25 year period. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 33 For household production, Bt is set to be 2.80 and p to be 0.20 for the C.E. period as taken by Greenwood (2002). Considering the fact that the development of household production makes raising children less time consuming, we set p = 0.18 for phase I and 0.23 for phase II. 5.3 Preferences The annual subject time discount parameter ft is set to be 0.985 to target the steady state capital output ratio of 3 for the whole period, /is set to be 0.3. The leisure preference X is then set to be X = 2(1 — y) = 1.4 to reflect a leisure preference that allocates 1/3 of discretionary time to work. We choose Vx = p 2y, V2 = y — vx for the benchmark consumption function (37). 5.4 Social Security The setup of our model makes social security beneficiaries about the same size of the contributors, while in practice the latter is more than three times of the former; we need to make adjustment to replacement rate - 0.14 for being consistent with the average replacement rate of 45% and annual payroll tax rate of 12%. 6 Estimation and Simulation Results As a brief theoretical framework, our model has only three periods and each period lasts for 25 years. Although there are a lot of benefits to select such a setup, including exact analytical solutions and clear theoretical predictions of various effects, directly applying the model to data is not practical in lack of enough samples. The solution is straight forward. At the aggregate level, the identity in function form of the benchmark consumption function (37) with the widely used statistical annual function (1) is not just a coincidence. It implies that the equation derived from the model may also be able to explain the economic relationships when variables are measured in other Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 34 model may also be able to explain the economic relationships when variables are measured in other time units. In another word, (37) is a general description of the connection between consumption, income and wealth that is invariant to measurement. The only thing that could be changed when applying (37) to empirical data of different frequencies is the size of the coefficients, but the signs should be the same. The same property holds for other equations and comparative static predictions derived from the model. In fact, this is an easy and efficient way widely used in scientific research to test the consistency of theory with reality. In the following section, saving, bequest wealth and fertility rate equations that derived from the three period model are taken to be general descriptions of the relations between economic variables. They are calibrated, simulated and then estimated by OLS regression using the annual time series data, the signs and relative size of each pair of parameters in calibration and estimation model are compared to check if the theory is consistent with the data and how well it predicts the economy. To be consistent with the theoretical definition that consumer durables are taken as part of investment, saving rate data have been adjusted and are higher than the personal saving rates from NIPA, but the two are highly correlated. 6.1 Benchmark Saving Rate Model As a benchmark model we consider only income and total wealth effects on consumption. Transform consumption function (37) into saving rate form and assume V{ = p 2y to satisfy the benchmark model condition. The saving rate function is then s, = l-er - e M (60) where 1 — (1 ~P)Y 6y — r 1 -(l-p )y+ p 20 i — (i-p)r eM =(!-<?) 1-(1 ~ p)Y + P zP Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 35 Coefficient calibration and estimation are shown in Table 1.3. Figure 1.4 and 1.5 show the calibration results and Figure 1.6 and 1.7 show the estimation results for consolidated era and for phase I and II, respectively. Both models exhibit the rising and declining trends in the saving rate. Table 1.3 Saving rate calibration and estimation: benchmark Phase Coefficient 1947-1980 1981-2002 1947-2002 eY 0.451 0.466 0.459 Calibration l - e Y 0.549 0.534 0.541 e M 0.071 0.069 0.069 e Y 0.655 0.582 0.590 l - e Y 0.345 (0.036) 0.418 (0.023) 0.410 (0.022) Estimation e M 0.0260 (0.0078) 0.0444 (0.0047) 0.0412 (0.0047) R2 0.234 0.804 0.575 s td 0.0136 0.0111 0.0142 2 8 % 2 6 % 2 4 % 2 2 % 2 0 % 1 8 % 1 6 % A ctual E stim ated 1 4 % 12% Y ear Figure 1.4 Saving rate calibration: benchmark (C.E.) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 28% | 22% ■ a to 16% A ctual E stim ated < 14% h-OCOCOOCMLOCO^-Ti-h-OCOCOOCMir) Tl-lOlOLOlOCDCDCDN-N-h-COCOCOOOOaJ 0) 0) 0) 030 ) 0) 00 ) 00) 0) 0) 00 00 ) 0) Y ear Figure 1.5 Saving rate calibration: benchmark (phase) 28% 26% oc 24% O ) c ’ § 22% = 20% Q - 18% T 3 16% A ctual 14% E stim ated 12% o o CM Y ear Figure 1.6 Saving rate estimation: benchmark (C.E) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 37 28% 3 2 6 % T O » 24% c I 22% | 20% £ a 1 8 % * D 0 W 1 6 % D < 1 4 % A ctual E stim ated 12% Y ear Figure 1.7 Saving rate estimation: benchmark (phase) Note that the estimates for phase II (1981 to 2002) are very close to the result of Davis and Palumbo (2001) using micro data. The estimation indicates an approximately 4 percent of wealth effect on marginal consumption. So when wealth-income ratio rises, we’ll expect to see a consumption boom and decline in saving rate. From 1992 to 1999, personal saving rate declines by 4 percentage points, while in the same period household net worth-disposable income ratio increases by 20 percent as shown in Figure 1.3, which accounts for most of the saving rate decline. 6.2 Saving Rate Model with Social Security When 0 , there exists a unfunded social security system. After considering the actual relative size of working generation and old generation we set $ > = 0.14 so the annual social security tax rate is about 12%, which is the current total tax rate for employees and employers. Keep the condition V, - p 2y so there is no composition wealth effect but productivity growth effect and interest rate expectation effect emerge. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 38 The saving rate function now becomes s, = l - e r ------ — — C v f y ^ It-1 'K , ' -e* where \-(\-p)y + OplP(\-y) 1-(1 -p)y+p2/3 pl(S{\-y) (l -gyp y~ ' 1 - 0 ~ P )Y + PlP e 1-(1 -p)y \ - { \ - p ) Y + p 2@ p)Y+ pIP<\-y) (i -exp 1-(1 -p)y+p2P (1 -S) = eB (61) Several revisions to the benchmark calibration are made to reflect the possible changes due to the introduction of social security system, y is changed to 0.4, since household’s social security benefits also depend on the size of the young generation, to raise more children could means either to receive more pension in the future, or to pay less social security taxes today. All coefficients are positive under the current setting. The calibration result for phase C.E. is shown in Figure 1.8. 28% 26% a. 24% w 22% = 20% 18% A ctual E stim ated < 14% 12% 5 “ ^ I'- h- 05 05 Y ear Figure 1.8 Saving rate calibration: with social security (C.E) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 39 The OLS estimates of (61) are in Table 1.4, regression coefficients have the right signs for all three phases with a insignificant exception for the interest rate expectation term, er in phase C.E. Table 1.4 Saving rate estimation: with social security Phase Coefficient 1947-1980 1981-2002 1947-2002 eY 0.176 0.146 0.261 0.824 0.854 0.739 ~ ey (0.095) (0.245) (0.092) 0.0293 0.0414 0.0414 eM (0.0059) (0.0059) (0.0041) 0.199 0.319 -0.066* er (0.092) (0.337) (0.062) £ > 0.277 0.153 0.405 eY _ t (0.094) (0.190) (0.094) R 2 0.575 0.820 0.689 std 0.0102 0.0106 0.0122 Figure 1.9 and 1.10 show how well the estimation of (61) fits the actual data. The separation of phases makes it possible to set different assumptions on exogenous factors that are not directly controlled by the model and yields substantial improvement in fitness. The wrong sign of er in phase C.E. simply indicates that some basic economic conditions has changed across phases so it maybe impropriate to put them together under the same set of assumptions or parameter setting. It should be noted that the improvement in fitness of model (61) over the standard function (60) is not a coincidence, as (61) is not a statistical equation. The two new factors in (61) are not added into the standard form in an ad hoc way solely for improving the estimation performance; in fact, (61) is derived exactly from household’s optimal behavior, the two terms 1/(1 + Et (r(+ 1 )) and (Tm /K,) are part of the function but have been dropped in (60) under certain special conditions. Any change in coefficient is not only due to statistical errors, but also means Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 40 that there maybe something happens at the micro level on household’s economic environment; or in the contrary, if there is anything that affecting household’s economic decisions take places, then the impact will eventually show itself off somewhere at the micro level. 28% o > 26% o> 24% % 22% c 20% 18% 16% A ctual E stim ated < 14% 12% (O 05 C M If) 00 t- 00 00 05 O) 05 O 05 05 05 05 05 O •t- •«- t - C N T — ^ h- h - 0 5 0 5 Y ear Figure 1.9 Saving rate estimation: with social security (C.E) 28% S 26% O ) 24% c 20% 18% 16% A ctual < 14% E stim ated 12% I s - O C O < £ 5 0 5 C N I lO ^ io io w io <o to O ) 0 5 0 5 0 5 0 5 0 5 0 5 C O t - co r> - 0 5 0 5 Year Figure 1.10 Saving rate estimation: with social security (phase) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 41 6.3 Simulation of Bequest Wealth Now release the restriction on Vx to allow for both bequest wealth effect and market wealth effect. The saving rate function has the form as shown in (34). Due to the unavailability of bequest wealth data in any major statistical sources, it is very difficult to test and estimate the equation directly. Although there are many studies on the role of bequests in wealth accumulation, the empirical measurements are very limited, since most surveys do not collect direct information on the amount of bequests received or plans to leave. Different methods have been used in the literature to estimate the amount of bequests, some get contradictory results. For example, Kotlikoff (1988) claims that the share of inherited wealth in the United States is about 80%, but Modigliani (1988) estimates it as being below 20%. Finding a theory-consistent approach to estimate bequest wealth itself becomes an important task. The bequest wealth function (40) derived from the model serves as an alternative way to estimate bequest wealth. Carefully choose initial conditions on K f so that the shares of bequest wealth in the first two or three years starting from 1948 are not far from the initial given shares. The nature of (40) determines that the initial conditions on bequest wealth are not totally free to choose since they are also linked to wealth and income of previous periods, which are fixed. Only when we have the chance to change environment parameters like preference and technology, can we set up the starting point of bequest freely. Next we’ll try several different parameter settings to see various possible evolutionary paths of bequest wealth. Figure 1.11 shows the two trajectories of bequest wealth share simulated in phase C.E. and in phase I and II separately. The parameter settings are the same as the benchmark model except for < p ^ 0 and Vx ^ p 2y ■ The trajectory by C.E simulation has a smooth trend that reaches its summit in early 1980s. In 1984 the bequest wealth takes about 20% of total wealth, which is the estimate of that year by Laitner and Ohlsson (2001). The trajectory by phase simulation is more volatile and shows a significant declining trend from earlyl980s. In the meantime, household net worth keeps Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 42 growing, which implies that market saving or investment is increasing very fast and can account for most of the increased wealth (mostly due to the rapid development of stock market and the huge increase in stock prices). The dropping of the bequest wealth share stopped at 2000, exactly the same time when the stock market crashed and market wealth shrank. This is also consistent with the decline in the saving rates for the same period. From saving rate function (34) it is easy to see that the decreasing share of bequest wealth together with eB < eM which is verified by our calibration provides a clear and reasonable explanation for the decline in saving rate since 1980s. 28% E 25% Phase 1 22% 19% C.E. 13% oo t t Y ear Figure 1.11 Simulation of the shares of bequest wealth: benchmark The last two terms in (34) are our concerns here. Firstly, since eB~eM < 0 and Kf /K, is decreasing, - [ 0/(l-<y)](eB )(AT®/a:() is also decreasing with Kt or time t. Secondly, eM+(eB- e M)(K?/K,) is positive and is increasing with total wealth; in addition, the wealth-income ratio is also increasing for this period as shown in Figure 1.3. This fact, together with the increasing coefficient and a minus sign in front, makes the last term in (34) decreasing. Then the two factors in the right hand side of (34) together can account for most of the decline in saving rate in late 1980s and the whole 1990s. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 43 Figure 1.12 shows the two cases with /? = 0.985 and y = 0.5 when people are very altruistic toward descendents. As much as 70% of the total wealth could have been accumulated through intergenerational wealth transfer. Depending on how a couple allocates the planed bequests between children and grandchildren, the resulting share of bequest wealth can be very different. When vx = 0.3, 60% of a couple’s bequests are left to children and 40% to grandchildren, then the share of bequest wealth can grow from 50% to about 57% in approximately 30 years or 1.2 generations. When vx =0.1, 80% of the bequests are to be left to grandchildren, then bequest wealth can be accumulated to a very high level of 70% of the total wealth in about the same time. 75% £ 70% 75 | 65% ® 60% D C T m 55% 50% 40% COr^NQ(9(D O)C\|lO fl3r^NQCO (DO)W 3l0'l0 1 0 © { 0 ® © S S N C 0 C 0 C 0 0 ) 0 ) 0 ) 0 ) O 0 ) 0 ) 0 ) 0 ) 0 i 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 ) 0 ’ 0 * 1 Y ear Figure 1.12 Simulation of shares of bequest wealth: bequest motives This result is interesting since the parents do not need to leave more bequests to increase the share of bequest wealth. The key point is who is going to inherit at when. An old-aged inheritor (the child) does not work when receives bequests from parents so has fewer income sources. A larger part of the bequests may be consumed by herself rather than to be passed through to the next generation. A middle-aged inheritor (the grandchild) is different. Since she works she may not consume the inheritance right away, she could invest it into the market, and then her descendents Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 44 will have more chance to get it from her when she dies. Although this may also increase market wealth as well, in a society of highly altruistic people, as long as the bequests are not totally consumed, they will become new bequests again sometime. The above statement can be summarized by eB < eM, which means bequest wealth has smaller effect on consumption than market wealth, as a result, bequest wealth is less likely to be consumed than market wealth, and will be accumulated much faster to reach a high share level. From (35) we learn that if vx 1- P 2 Pi (62) When v 2 , individual’s preference for leaving bequests to grandchildren, is relative higher compared to the preference for leaving bequest to children, bequest wealth will have smaller effect on consumption than market wealth. This is quite common for the rich families, and bequests transferred within rich families constitute most bequest wealth of the economy. Time preference also plays an important role in the accumulation of bequest wealth. We find that a 5% decline in household’s time discount factor (y3 is dropped from 0.985 to 0.94) can result in more than 8 percent of decrease in the highest level that bequest wealth share can reach 6.4 Estimating Fertility Rate As stated in (22), fertility rate can be expressed as a function of income and wealth. Plug (15) into (22) and replace wage and interest rate with their equilibrium expressions given in equation (5), then JV', the number of newborns at time t, can be expressed as a function of aggregate income and wealth. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Although it looks very complicated, it only consists of Yt,K t, K f , Yt i , A, and Bt ■ (63) can be estimated given information on current and past aggregate income, current wealth and its composition, and current state of technologies of market and household production. Again we need information on share of bequest wealth, which can be estimated through various ways including the one discussed in section 6.3. In order to check if (63) works we make a very rough approximation by assuming that the two wealth composition effects are equal, therefore no composition wealth effect appears in the equation. Using fertility rate data of the United States for the last half century we estimate (63) and the result is shown in Figure 1.13. 3.9 3.6 3.3 .■ S ' 2.7 2.1 A ctu al E s tim a te d Y ear Figure 1.13 Fertility rate estimation Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 46 7 Conclusions In this paper we develop a three period altruistic OLG model to explain the nonlinear trend in the personal saving rate shown in the data of the United States and many other developed countries for the last half century. TFP growth, demographic structure change, the effect difference between bequest wealth and market wealth, together with their changing shares, turn out to be the most important reasons that cause saving rate to fluctuate. In the case of the U.S. in 1990s, the increasing wealth-disposable income ratio, weaker effect of bequest wealth on consumption and its declining share explain most decline in saving rate. One important feature of this paper is that the model includes both motives for raising children and leaving bequests, so birth behavior is linked with wealth transfer, which at the aggregate level determines the interaction between wealth accumulation, demographic structure and other major economic variables. As bequest wealth attracts ever growing attentions yet its measurement still unclear and on debates, a simple alternative method of estimating bequest wealth resulted from our model seems promising and is waiting for more data verification. Although the research in this paper was first inspired by the saving rate puzzle, its simple structure and the very nature of incorporating four major household decisions endogenously enable it to be extended to a broader field of interests. The interaction between micro behaviors and its sequences at the macro level can be studied analytically by the model, thus public policy issues like the role of social security and the effect of changing technology and cost of raising children on women’s labor force participation can be discussed in a common framework which allows for cross effect study. The model should be extended to many-periods (more than 20) for better verification with data and more explaining power. Heterogeneity in household’s work efficiency is another direction of extension to analyze the role of household expenditure on children on the accumulation of human capital and long run economic growth. Changes in labor force participation rate and its relationship with fertility decline and economic growth will also be studied in depth. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 47 Chapter 2: Economic Growth and Revealed Social Preference (Joint with Richard H. Day) Abstract: The Representative Agent Growth Model is estimated econometrically using the Generalized Method of Moments for the U.S. economy for three separate Growth Eras and the results compared to those obtained using the Kydland-Prescott calibration approach. The estimated parameters differ substantially in the three cases, which imply changing social preferences for present versus future income and work-leisure tradeoffs. These in turn imply switching among alternative balanced growth paths and differences in the contributions of capital, labor, and labor augmenting productivity among the three Eras. Using the GMM method yields very high productivity and capital elasticity parameters and a very low time preference parameter for Era I (1929-1941) compared to Eras III (1948-1980) and IV (1981-2002). While both GMM and the calibration method yield much smaller leisure parameters for Era IV than for Eras I and III. 1 Introduction The characterization of a national economy as a representative agent can reveal something interesting about a society’s “preference” for growth. This point is illustrated by calibrating time and leisure preference parameters based on econometric estimates of the U.S. economy, using the standard NIPA data, for three Growth Eras through the years 1929-2002. The results reveal distinct differences over time that suggest substantial changes in technology, consumption, and savings behavior. Robert Solow was perhaps the first to take macroeconomic growth theory seriously as an empirical tool. In the meantime, most of the literature has focused on theoretical properties and generalizations to which many noteworthy contributions have been made, culminating in Stokey Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 48 and Lucas’ comprehensive treatment from Bellman’s dynamic programming point of view. But it was Kydland and Prescott’s 1994 paper that restored interest in empirical applications by ingeniously applying properties of an optimal strategy along a balanced growth path to calibrate the parameters of a social utility function and following Cobb and Douglas’ original use of factor shares to estimate the elasticity of production with respect to capital and labor. In this study we use econometric methods to estimate the basic parameters on the basis of which the social utility function parameters can be calibrated and inferences drawn about the separate Eras. In section 2 the theoretical properties that are exploited in the calibration process are reprised. Section 3 describes the econometric estimation procedures. The results are given in section 4 and used to derive econometrically based estimates of the utility function parameters. Also presented are comparable parameter estimates using the Cooley-Prescott techniques. Section 5 then considers the implications. The separate Eras turn out to be strikingly different in several regards. In particular the economy is seen to approximate not one but three somewhat different steady states based on distinct parameter values for each. These imply factor contributions quite different than those obtained using the calibration approach on comparable data. They also suggest a need to consider the meaning of optimal growth in this context. Briefly, rather than thinking of the macroeconomy as exhibiting some kind of optimality, we suggest that the estimated utility function parameters provide a statistical characterization of the separate Eras in terms of the “social” time preference, consumption-leisure-work tradeoffs, and productivity characteristics. 2 Theoretical Background The RBC approach characterizes aggregate consumption, investment, and capital accumulation as the solution of an infinite horizon, dynamic program whose trajectories can be viewed as competitive intertemporal equilibria so that involuntary unemployment and excess Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 49 capacity cannot occur. To estimate parameters using the standard NIP A data, care must be taken in specifying the units of the various variables involved. We interpret preferences as those of a representative adult, a non-institutional person over the age of 16 who allocates income to consumption or saving and time between employment and non-employment. In the NIPA data the population of adults, L, so defined consists of the employed, E, (measured in full time equivalents), those formally unemployed, those looking for employment, and those not looking for employment. The term L - E consists of all adults not employed (again measured in full time equivalents). According to the model, those not employed are assumed to have chosen to work or not to work at all. It is not our purpose here to argue for or against the relevance of this assumption but to consider the implications of the model as a whole. Given the above interpretation, the description below follows Cooley-Prescott (1995) with minor variations that facilitate the use of the NIPA data and that uses per adult rather than per capita units. A preference ordering of alternative consumption and work streams, {c,,/it } “ 0, is described by the utility function, c, is per adult consumption; h„ the fraction of discretionary time devoted to work; 1 - h„ the fraction of the period devoted to non-work; a, the time preference or discount parameter. The parameter y governs the tradeoff between consumption and non-work. The discount parameter, a = 6 * (1 + v ) , where v is the average rate of population growth and 9 a measure of time preference. Aggregate production per employed adult, q“ = Q ,/E ,, is given by the production function, ( 1) where u(c„ ht) = [(1 - y) Inc, + y ln (l - h,)] Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 50 where k “ is social capital (private plus government) utilized per employed adult. If K, is the total social capital and v, is the reported capital utilization rate, then utilized capital stock per employed adult is k* =vtK ,/E l = v,Kl/(hlLt) ■ The variable B, is the index of labor augmenting technological progress where Bl+ l = (\+b)B, . (2) and the parameter b is the rate of productivity improvement. The term 7 7 '' represents the influence of outside forces. It is assumed to be generated by the auto-regressive process, where £f+ 1 ~ N (0,a) (3) Thus, s i is thought of as an exogenous, random shock and 7 7 ’ an endogenous variable representing the hysteresis effect of past shocks. A steady state in labor efficiency units can be obtained by dividing per adult variables by the term h,Bt. Because qt = htqu t , the production function expressed in adult units is qt — {h,B,)l~^ {vtktY ev’ or, in labor efficiency units q e t - {ke t Y en' (4) where ql = q j (h,Bt) and k f = v , k j (h,B, ). In adult efficiency terms the utility function becomes u (cl ,h,) = ( l - y ) l n ce , + (1 - y) In B, + (1 - y) In h, + y ln(l - h, ) (5) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 51 Capital per adult accumulates according to the equation where j, = q,~ c, is investment per adult, < 5 is the physical depreciation rate of capital stock, and v is the growth rate of the non-institutional adult population. In labor efficiency units (dividing both sides by Bt+ \ht+ \) k e = h<lh‘ + 1 ---- [ (1 - S)k‘ ; +jf] ( 6 ) , + 1 (l + v)(l + fc)L The choice among alternative consumption-leisure streams in labor efficiency units is subject to the constraints, < + r ; < qe t , < > 0, rt > 0, t = 0, 1 , 2, 3,- • • (7) and 0 < h, < 1 (8) The consumption-leisure-work-investment streams calibrated in labor efficiency units are assumed to satisfy the intertemporal optimization, = m ax£,{M{c( ‘ \/i(}“ 0| subject to ( 1) - ( 8) } (9) Icf.h,} where E(-) is the statistical expectation. The indirect utility function, V (k^ , T j ' t') , satisfies Bellman’s equation, V{k%i7 , ’ ) = max { , h, )^Q + aE[V(k:+l, ^ )]} (10) 0<,ce , < > q , Q<,h,<A Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 52 As the expectation operator integrates out the influence of future random shocks, the optimal consumption-employment strategy that satisfies this functional equation depends only on the current situation: h = h { k % n ,*) In the absence of shocks, a steady state must satisfy the equation (K ce <,qe 0<h<i where u(ce , h) is given by (5) and where q e V + S + b( 1 + V) (11) V(ifc*,0) = max u(ce,h) + a V (k e,0) (12) k e =-------- ------- - [ ( * ' / - c e] v + b + v b + S From this we infer that the steady state investment output ratio is $ = ~jel q e = [ v + S + b(l + v ) ] k e/ q e , (13) so the steady state capital output ratio is (14) Dividing both sides by ft and rearranging shows that the steady state marginal rate of return on capital satisfies r /3[v + S + b( 1 + r)] (15) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 53 The first order conditions for maximizing (10) with respect to Ce and h yield the ratios „ (l + y)(l + b) ( 16) [/3qe/k e+(l-S)] r 1 - P (17) l - y h 1-0 Finally, evaluating the derivatives, dV (kl+l) dV (k,) dkni dkt at the steady state, the rate of return on capital is found to satisfy the condition r = (l + m + V ,l + S - l (18) a Rearranging, ~ (l+fr)(l+v) r - S + l or, as a = 0(1 + v), (19) 9 = 1+b (20) 7 - S + 1 Equations (14)-(20) provide the basis for estimating the utility function parameters a or 6 and y by each of the two methods under consideration. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 54 3 Parameter Estimates The Bellman equation (10) is nonlinear. Its explicit solution to get an optimal consumption strategy has not been derived. In order to proceed econometrically, one approach is to follow Kydland, Prescott, et al., by assuming the economy to be fluctuating around a steady equilibrium state during the time periods under investigation with the fluctuations generated by auto regressive shocks. If such an assumption were to give a good approximation, then a discrete time Solow model, which has a simpler form, could be used as a basis for the parameter estimates because the two steady state investment ratios and the implied marginal rates of return on capital are the same. That is the approach we have taken. 3.1 Statistical Assumptions The Solow model consists of the same production function and capital accumulation equation which we retain without modification but instead of an optimal investment strategy defined by (9)-(10), a constant investment ratio < j) is hypothesized. In view of the fact that the NIPA data for the investment/output ratio, population growth rate and depreciation rate all vary considerably over time, we allow for shock terms as follows. For the depreciation ratio we assume (21) where the shock term is assumed to follow an auto-regressive process (22) Likewise, the observed population growth rate is assumed to be generated by Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 55 (23) where T j^+1, (24) Finally, we assume investment is generated by the process it =<pevlql (25) where r]l t satisfies, (26) where ej+ 1 can be thought of as a “expectational shock,” and where the “hysteresis term,” A 't j I , is the accumulated influence of past Expectational shocks on current plans. 3.2 Econometric Estimation At the macro level the goods and services produced by and for the government are part of GDP and government capital stock plays a role in the productivity of private capital (think of public highways and private vehicles). Accordingly, the production function is based on social capital the sum of private and public capital) and on the total adult employment including both private and public employees. The full employment assumption underlying the Solow and RBC approach forces one to choose between, on the one hand, using the data on employment and utilized capital, or, on the other hand, the estimate of the available labor supply (employed plus unemployed workers in full time equivalents) and total capital stock. Because unemployed resources do not contribute to output, the former would seem to be most appropriate, and that is what we do here. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 56 Given these assumptions, the following system of equations are derived for joint estimation using the GMM methodology. The production function relationships (2)— (4) imply In qM = (1- A) In A + (1 - /?) ln(l + b)+A9 In q ,+ (\-A q){\- p) ln(l + b)t + p \n kl+ l - Aq p in kt + e q + l where A = B . The depreciation of population and investment equations (21)-(26) imply, respectively, the equations Innt+ l = AN ln n ,+ { l- A N)lnv + ef+ l (28) Ind M - As Ind t + ( l - A s )\n S + £?+ ] (29) Jt+iItft+i = ^ lnO ',/4, ) + (1 - ^ )In^ + f /+ 1 (30) Because the shock terms cannot be independent and may be correlated, we treat the four equations as a system and derive estimates using the Generalized Method of Moments. 3.3 Utility Function Parameters Based on Econometric Parameter Estimation There are still the time preference parameter, 0, and the consumption-leisure tradeoff parameter, y, to be derived. Given the econometric estimates of < p , n, 3, b and /?, the implied steady state rate of return is given by equation (15) which, substituted in equation (18), yields an estimate of a. Instead of the survey data estimates used by Cooley and Prescott, we use the average workforce participation rate as a proxy for the discretionary work ratio, h. Substituting into equation (17) and solving gives an estimate for y. 3.4 Utility Function Parameters Based on Calibrated Parameter Estimation Our calibration estimates are not directly comparable to those obtained by Cooley and Prescott who constructed modified data for capital stock and rates of return on capital. With the exception Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 57 of the capital utilization for the years 1929-1941, our data are based solely on the NIPA tables. Also, our estimates are based on annual data while Cooley-Prescott use monthly figures. However, for methodological comparison, we also used their calibration procedure in so far as possible but using the NIPA data. They compute average values of the investment/capital ratio, j/k, the capital/output ratio, q/k, the average output consumption ratio q/c, the population growth rate, n, and the average rate of growth of per capita income to get b. Solving the steady state condition (13), yields their estimate of the depreciation rate, 8 = (j/k) - n - (1 + tijb (31) Using the average employment/workforce ratio as an estimate of h and substituting the afore mentioned ratios in (17) yields an estimate of the work/leisure choice parameter, y. (l- m ~ n ) ( z /c ) (32) h + (l-P)Q.-n)(z/c) The steady state rate of return given by (15) substituted in (20) yields the estimate of 6. 4 Empirical Results To assume that parameters are constant throughout the 1929-2002 Era may be misleading for several reasons. The Great Depression, 1929-1941, ended with the onset of World War II with demobilization and conversion to peace time conditions occurring during late 1945 through 1947. Because the period 1942-1947 involved such extreme changes in the composition of the workforce, in working hours, and in the type of things produced, we have (as in usual practice) excluded it from the analysis. Many changes to technology occurred during the war years based on innovations of the 1930s (multi-engine and jet plane design, large scale computers and electronics, communications, factory layout, and product coordination). These were subsequently incorporated Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 58 in civilian production in the postwar years, 1948-1980. The computer chip, personal computers, and related computational and electronic innovations introduced in the early 1980s lead to another period of rapid, qualitative change in the nature of capital. Based on these considerations, we considered four distinct Eras. Phase I: 1929-1941 The Depression Era Phase II: 1942-1947 World War II and Recovery Phase III: 1948-1980 Post World War II Phase IV: 1981-2002 Advanced Technology We refer to the years 1929-1941 and 1948-2002 with Phase II deleted as the “Consolidated Era.” The empirical estimates for the model parameters are given in Table 2.1 through 2.3. Table 2.1 GMM estimates of production function parameters Phase P Xq Phase I 0.0321 0.402 0.226 (1929-1941) (0.011) (0.057) (0.751) Phase III 0.0241 0.319 0.798 (1948-1980) (0.011) (0.134) (0.253) Phase IV 0.0176 0.303 0.482 (1981-2002) (0.005) (0.283) (0.438) Consolidated 0.0186 0.358 0.759 Era (0.003) (0.091) (0.180) 0.911 0.0236 0.993 0.0168 0.991 0.0101 0.999 0.0152 Table 2.2 GMM estimates of the other model parameters Phase < t > X1 S X5 v XN I 0.155 0.395 0.0285 0.721 0.00676 0.396 (0.044) (0.153) (0.013) (0.135) (0.0014) (0.121) III 0.206 0.334 0.0368 0.632 0.0110 0.922 (0.107) (0.348) (0.021) (0.213) (0.0065) (0.049) IV 0.194 0.560 0.0476 0.893 0.0105 0.740 (0.135) (0.258) (0.048) (0.104) (0.0047) (0.120) C.E. 0.204 0.622 0.0363 0.941 0.0104 0.932 (0.019) (0.032) (0.029) (0.048) (0.0043) (0.029) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 59 Table 2.3 A comparison of econometric estimates and calibrations of parameters Phase I III IV C.E. b GMM 0.0321 0.0241 0.0176 0.0186 Calibration 0.0253 0.0227 0.0201 0.0223 P GMM 0.402 0.319 0.303 0.358 Calibration 0.353 0.310 0.289 0.311 e GMM 0.900 0.953 0.950 0.944 Calibration 0.947 0.952 0.958 0.952 y GMM 0.568 0.579 0.495 0.539 Calibration 0.587 0.582 0.500 0.556 The values for b and p are shown in Figure 2.1 and 2.2. Econometric and calibrated parameter estimates are shown for each phase and for the Consolidated Era. Also shown (as a benchmark) is the Cooley-Prescott original estimate for the years those authors considered. In Figure 2.3 and 2.4 comparable comparisons of the time and leisure preference parameter estimates are shown. 0.04 0.03 0.02 0.01 Figure 2.1 Production function parameter estimate: b - II - - C & P C alib ratio n P h a s e ------------G M M P h a s e - - - - GM M C E - - - - C alib ratio n CE :_________________________________________ 1 - 1 III r . . . . . . . . . . . . . . . . . . . . . . . I V C O y — to 0 5 CO 1 " - kO 0 5 C O h- LO 0 5 CO h - T— C O CO LO LO CO CO CO N - h - CO CO C O 0 5 0 5 O o > 05 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 0 5 o T— Y ear T“ T*“ O J Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 ■ C&P ■C alibration P h a s e -G M M P h a s e ■GMM CE ■ C alibration CE I V , I ■ I I ....... I.J o > C M 0 3 C O C O C O 0 3 0 5 h- 0 5 0 5 C O I T 5 0 5 h- L O 0 5 t — L O 0 3 C O C O C O 0 5 0 5 0 5 Y ear C O h - 0 5 I " - t - h- co 0 5 0 5 L O 0 5 co oo 0 5 0 5 O O C M Figure 2.2 Production function parameter estimate: /? 0.97 0.96 0.95 0.94 0.93 0.92 0.91 0.90 0.89 ■ - C& P ^ “ C alibration P h a s e GMM P h a s e - -G M M C E - - C alibration CE I V 0 5 C M 0 3 C O C O 0 3 f s ~ - T “ L Q C O T j - Tt 0 3 0 3 0 3 0 5 C O h « . ^ in io 0 5 0 3 0 3 0 3 0 3 0 3 0 3 0 3 lO 0 5 C O h - 0 0 C O 0 5 0 5 0 3 0 3 0 3 0 3 O o C M Y ear Figure 2.3 Utility function parameter estimate: 6 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 61 0.65 0.60 0.55 0.50 0.45 0.40 ■C&P ■ C alibration P h a s e ■GMM P h a s e ■GMM C E ■ C alibration C E I V 0 5 C N 0 5 co r-- C O C O L O 0 5 C O LO LO CO CO CO C O h- L O 0 5 C O C O C O C O O ) 0 5 05 05 05 050 50 50 50 505 05 05 05 050 50 5 05 05 O O C N Y ear Figure 2.4 Utility function parameter estimate: y There are quite noticeable differences among the several estimates. First, those computed separately by phase are quite different than those for the Consolidated Era. That is especially true for the Depression Phase I. Moreover, technology parameters show systematic shifts through the phases. Both the capital elasticity of production, /?, and the labor augmenting productivity parameter, b, have relatively large values in Phase I and decline successively in Phases III and IV, especially between Phases I and III. In contrast to these pronounced distinctions, the time preference estimates by phase differ substantially only for Phase I; the Phase III and IV parameter values are relatively close, while the leisure parameter estimates are similar in Phases I and III but differ substantially in Phase IV. As for the distinction between estimation methods, we note that the greatest discrepancies in the phase estimates for /? and b are in Phase I. For Phase III and IV the econometric and calibration estimates differ, but not by much. The same is true for the time and leisure preference parameters. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 62 Finally, with the exception of the time preference parameter, our estimates are very different than the original Cooley-Prescott results. Putting these distinctions and similarities together, we find the parameter estimates very sensitive to the breakdown by phase, but not to the calibration versus the GMM method. We also conclude from the comparison with the original Cooley-Prescott parameters that the estimates (except for time preference) are very sensitive to differences in data used. 5 Implications 5.1 Factor Contributions to Economic Growth by Phase Using a variant of Solow’s original growth accounting procedure, we can compare factor contributions to economic growth among the phases and between econometric and calibrated estimates of capital elasticity and growth rate parameters /? and b. Express the production function (4) in the equivalent way, Q, = F (B t, K : ,E t) = B)-p {K: f E ^ e 1 1 ' (33) Taking the first order expansion and rearranging, we get Q, B, K, t, where AB/B, = b. The terms on the right partition the rate of growth of output into the three factor contributions, the autoregressive shock process, and the high-order terms in the Tyler expansion,/?,. The contribution of labor augmenting productivity to the change in GDP is (1 -p)b. Using annual rates of change for utilized capital and employment yields the annual contributions of capital and labor can be calculated. The first order approximation does not measure the change Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 63 caused by the decline in marginal productivity of capital and work between two periods. This total will differ from the average change in output. Taking the averages and dividing by the GDP rate of growth, the relative contributions are obtained. This can be done for the econometric and calibrated parameter estimates for each phase and for the Consolidated Era. The results are given in Table 2.4. Table 2.4 A comparison of percentage factor contributions to GDP growth by econometric estimation and calibration (%) Phase Methods Productivity Capital Work Shocks Error I GMM Calibration 58.08 49.60 -1.52 -1.34 10.67 11.55 38.21 43.42 -5.41 -3.24 III GMM Calibration 44.50 42.45 31.38 30.50 31.56 31.98 -8.63 -6.18 1.20 1.26 IV GMM Calibration 39.38 45.95 23.77 22.67 32.72 33.38 3.46 -2.57 0.66 0.57 C.E. GMM Calibration 34.74 44.80 26.82 23.30 26.77 28.73 11.52 2.85 0.16 0.32 Noticeably, and probably surprisingly, the GMM phase estimates attribute a greater contribution of labor augmenting productivity to growth in Phase I and lower in Phases III and IV, compared to the Calibrated Phase Estimates. The relative contribution of capital for the entire Consolidated Era is a little less than in Phase IV. 5.2 Steady State Per Capita Output Trajectories Except for Phase I, we have found what seem like relatively minor differences in some of the parameter estimates for the Solow model. However, small parameter differences are compounded in exponential paths and big differences are compounded hugely. Figure 2.5 and 2.6 show the steady state trajectories over the entire period based on the parameter estimates obtained separately for each phase. In each case the trajectory was anchored (or forced through) the least squares estimate of the initial period of the given phase using the estimated parameter values by the econometric or calibration procedure for that phase, respectively. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 4 0 0 0 0 e 3 5 0 0 0 Q 3 0 0 0 0 C D ® 2 5 0 0 0 s 20000 Q. O 1 5 0 0 0 A ctual C .E . 29-41 48-80 81-02 ,°o 10000 C L 5 0 0 0 0 0 ( 0 5 C O N - T - m O > C O I ^ - T - l 0 0 3 C O C Mc o c O ' t ' ^ - ' a - L o m e o c D C D i ^ L O 0 5 C O C O co r- t - 05 05 O T “ T - V T - V V T “ T — T - T-T- C M Y ear Figure 2.5 Steady state output trajectories based on GMM estimates 3 5 0 0 0 £ 3 0 0 0 0 £ 2 5 0 0 0 0 5 0 5 c 20000 g - 1 5 0 0 0 A ctual C .E . 29-41 48-80 81-02 ,°o C L 1 0 0 0 0 = 5 0 0 0 ocr 0 5 C O h- v- L O 0 5 c m co n j t s C O h- T - L O L O C O lO 0 5 C O h * C O C O h- h- r in O ) C O C O C O C O 0 5 Y ear Figure 2.6 Steady state output trajectories based on calibrated estimates Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Why should the GMM and calibration methods differ markedly in so many instances? The answer must lie in the differing treatments of the auto-regressive shock mechanisms. The Cooley-Prescott approach is aimed at a time series analysis of residuals, assuming that the entire burden of explanation lies with the productivity term, given stability over time in the underlying structural parameters. Among other practices, their estimates of the structural parameters use the average rates of change in the critical capital/ labor, consumption/output, and investment/output ratios together with average factor share estimates for the elasticity of production. Contrastingly, the econometric approach estimates the auto-regressive time series variables endogenously and simultaneously with the parameter estimates. Thus, estimates of /?, b, and , for example, depend on the data, not just for per capita output and capital, but on depreciation, investment, and population growth rate data. Especially critical from this dynamic perspective are differences in the labor augmented productivity parameter, b. Because this is the compounding parameter, small differences in that parameter compound quickly to give large differences in estimated GDP. 6 Conclusions and Reflections If it were not for the systematically changing parameter estimates among the three phases, we might be warranted in concluding that the steady state estimates are solely due to the auto-regressive shock components of the econometric Solow model. But these systematic differences are pronounced! The labor augmenting productivity and the capital elasticity parameters are much higher in Phase I than later and both are lowest in the most recent phase! Significantly, the leisure preference parameter is much lower in Phase IV than in the preceding phases! Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 66 As Table 2.4 shows, the average contribution of productivity in Phase I is far greater than for capital and labor though the estimated auto-regressive shocks also played a bigger role than later. Movie buffs, who have seen the films of the 30s, will be familiar with the trolleys, the narrow roads, steam locomotives, the mule powered sharecropper farms of the south, and what seemed then to be the fabulous Douglas DC3 and Pan American flying clippers. These were being replaced rapidly by concrete highways, narrow by present standards but far more efficient on a year-round basis than the gravel and dirt roads they were replacing. Trucks and tractors were rapidly replacing the remaining horses on farms and in cities. By the end of the phase they were all but eliminated, having been replaced by faster and bigger tractors, trucks, automated machinery. Four engine planes and automobiles were already replacing trains as preferred long distance travel. The shock term, which is also significantly higher in Phase I, is associated with the great decline in output and investment between 1929 and 1930. As the economy is assumed to be in a steady state equilibrium, these events necessarily have to be attributed to exogenous shocks and their hysteresis effects. Equally dramatic in terms of phase differences is the increase in the time preference parameter after Phase I and the drastic decline in the leisure preference parameter in Phase IV. The former may reflect the extremely high rate of unemployment and poverty: thoughts about the future no doubt have a diminished influence on society when many of its members are concerned about finding shelter and the next meal. An equilibrium model which does not account for involuntary unemployment inevitably must interpret a decline in market work as a voluntary increase in leisure. Just as changes in the production function parameters appear to reflect different technological phases, so the utility function parameters seem to reveal changing preferences prevailing throughout society. The fact that household production has never been properly accounted for in GDP data, however, suggests that apparent changing attitudes about work implied by the drastic increase in workforce participation by women may really be the result of changing technology. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 67 While we believe something is learned from this exercise, inferences drawn from the results should be interpreted as qualitative not quantitative for four fundamental reasons. First, parameter estimates are sensitive both to estimation method and to the specific data used. Second, a macroeconomic model by its nature is a radically simplified characterization of an economy with heterogeneous goods, production and marketing processes, and governed by a complex panoply of heterogeneous private and governmental organizations. But, of course, that is precisely why such macro models are needed. Third, the intertemporal equilibrium format of the theoretical model is used. The basis for the revealed preferences argument forces departures from the steady state trends to be interpreted as the result of outside forces and not to adjustments out of equilibrium. Fourth, the phases, which seem to be justified on the basis of a survey of the general features of the economy throughout the Consolidated Era in the analysis presented here, constitute discrete events, in the sense that the economy is seen as “jumping” from one steady state path to another. This is a sort of theoretical self-contradiction, for any given phase is governed by a steady state which is non -optimal with respect to earlier or later phases. To account for such structural phase transitions in a more satisfactory theoretical and empirical manner, and still remain operational for macroeconomic purposes, poses a daunting challenge. The construction of an empirical RBC model, estimated econometrically or by pure calibration, involves an inverse optimality exercise. Parameters of a dynamic problem are estimated which imply, or are consistent with, the behavior observed in the data. They say nothing about the data’s optimality. Obviously, economies do not think. Economies do not have preferences, they have cultures. Cultures have rules, laws, organizations, patterns of behavior. None of these derive directly from single individuals as such, but from the interact ion of all the individuals and all the organizations in which they participate. The macroeconomic data that emerge from this complexity are statistical artifacts that characterize the economy as a whole. The rationalization of these characteristics using an optimal growth model leads to parameters that reflect the society’s Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 68 behavior as a whole in terms of averages. The observed path of the economy that the model rationalizes is not the one chosen or wanted by any individual unless the economy is controlled by an autocrat or if everyone were identical and production technology were as simple as the macroeconomic representation of it. Giving up the idea of optimality and thinking of the technology, time preference, and consumption/leisure tradeoff parameters as social attributes seems to us, nonetheless, to have definite interest. They reveal properties of the aggregate economy that ultimately demand deeper explanations and suggest possible directions in which to look for them. Similar econometrically based inter -country estimates could be made and compared to the calibration approach followed by Kehoe (2002). One would expect those comparisons to change if the countries involved progressed through different apparent growth phases as seems to have been the case in the U.S. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 69 Chapter 3: A Dynamic Analysis of the Public Fiscal Policy under Different Consumption Patterns (Joint with Richard H. Day) Abstract: A dynamic aggregate demand and supply model is used to explain the policy paradox associated with the Kennedy, Reagan and Bush II administrations: that is, the possibility that tax rate decreases could in principle - and might in practice - increase tax revenue. We find that with high enough marginal propensity to consume and invest, the tax rate reduction can stimulate economic growth, tax revenue, and government budget surplus in the long run. A nonlinear elaboration is introduced to explain some aspects of monetary policy. Several factors explaining why employment lags behind the recovery of output after a recession are identified. 1 Introduction The Bush administration’s recent tax-cut policy attracts a lot of concerns and debates, many economists suspect its effectiveness in stimulating economic growth and employment, as well as in reducing government budget deficit in the long run. When looking back into the history, we find that the same argument exists from early 1960s for Kennedy and subsequently by Reagan administration in 1980s. The question we are going to answer is a widely presumed fiscal policy paradox which we refer to as the KRB tax policy paradox: Can a reduction in tax rates both stimulate economic growth and increase tax revenue? In this chapter we uses a discrete dynamic, aggregate demand and supply (DADS) growth model to examine the tax policy paradox and get somewhat surprising result, that is, under certain conditions on marginal propensity to consume and invest, and productivity growth, a permanent Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 70 decrease in tax rate will increase the entire future balanced growth path of income and tax revenue, hence the government budget surplus. Calibration and estimation with data show that our economy is currently in the period satisfying these conditions. That is to say, the policy position in common by Presidents Kennedy, Reagan, and Bush rests on a logical and theoretical basis, and the Bush administration’s tax legislation may also be effective in practice. The critical factor that enables us to get this result is the dramatic booming in personal consumption since late 1980s. Although the model presented cannot accommodate all the complications of economic growth and fluctuations, it clearly provides a useful approximation and a clear cut explanation of the KRB paradox7. 2 The Basic DADS Growth Model 2.1 Aggregate Demand The basic linear dynamic aggregate demand and supply (DADS) model incorporates aggregate consumption and investment functions, which are, c t+l ~ + Q ;i ( l — 7r+i)Tr+i (1) where C t+ 1 and y (+ 1 are per capita private consumption and income, respectively, and j,+i=</>(yt - y ') (2) where j t+ l is per capita private investment. The coefficients a 0,a v < p , and y' are parameters and the exogenous variable r (+ 1 is the average per capita net tax rate. The parameter a x is the marginal propensity to consume (MPC), o = I —0 £ x is the marginal propensity to save (MPS), and (f) is the marginal propensity to invest (MPI). We assume throughout that o < ax < 1 and < / > > o • 7 An early theoretical analysis of the Kennedy tax policy will be found in Day (1970). The present discussion introduces critical nonlinearities, distinguishes between short and long run growth effects, and presents empirical evidence relevant to the period 1929-2002. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 71 Fitting these functions to the data for 1929-2002 (excluding the war 1942-1947) by ordinary least squares method yields the highly significant parameter estimates The explained variation in the data for the consumption function is 98.5% and 95.6% for the investment function. Per capita aggregate demand z(+ 1 is given by where g (+ 1 equals government spending (public investment + government services), and xt+l is the exogenous net exports (exports - imports). Government is exogenously determined to allow for government budget surplus or deficit. It is assumed that aggregate supply y( + 1 (production ql+i) equals aggregate demand z t+1, and that the aggregate income generated by production is equal to the real value of production, a 0 = 0, ^ = 0 .6 9 , 0 = 0.28, y'= 3673 ^ t+ l ^ t+ l J l+ l S t+ l •*)+! (3) (4) As a result, y,+i = a 0 + 0 ^(1 - Tt+ 1 )y l+ 1 + 0 ( y ,- y ') + g t+ 1 + xl+ 1 (5 ) or, solving for y (+1, we get the non-autonomous first difference equation, = (6) where y = a o + 0y'. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 72 2.2 Production and Productivity The supply side of the macro economy is represented by the aggregate production function, Q ,= F {K ';,B tEl) = (K ';Y{BtE y (7) where Qt is GDP, K “ is utilized aggregate social (private plus public) capital, Bt is labor augmenting productivity, Et is employment. It is here assumed for simplicity that private and public capital are perfect substitutes. The productivity term is defined by an autonomous difference equation, Bl+l= (l+ b)B t (8) where b is productivity growth rate, a form that says that productivity feeds on itself. The productivity term cannot be observed directly but must be inferred from the other data. This can be done by using the implied trend Bt = (\ + b)‘B0 (9) Given this, production per employed worker can be expressed by q: = f ( k : , t ) = ( i + b f - ^ A ( k : Y a o ) where cft = GDP per employed worker, fc" is utilized social capital per employed worker or capital/labor ratio, and A = . The parameter estimates using data for the entire period are b = 0.0186 /? = 0.358 Both are highly significant and the percentage of data variability explained is 0.9989. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 73 2.3 Capital Accumulation Social capital consists of private and government components, say K p and K g or in per capita terms, k p and k g . The equation of government capital accumulation is 1+n, where j s is the public component of investment and S g the depreciation rate on government capital. Public investment is exogenous. Private investment is estimated by the private investment function, so the equation of private capital accumulation equation is * ,'♦ ! = 7 ^ 1 (1 + * U - / ) ] (12) 1 + nt where S p is the depreciation rate for private capital. Social capital per capita is 2.4 Employment By definition, Qt = N,q, = q e ,E t (14) where N t is the total population in period t. Given (4), the level of employment required for production is E ,+ X = ( r r j l ' t \ e(y^S,+ l+Xt+l+<h,) (15) (<T+ax Tt+x)q, Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 74 2.5 Steady State Growth The average tax rate, government spending, net exports, and wages, capital value, interest, and depreciation change somewhat from period to period, but the model does not explain their movement. It merely explains how GDP per capita changes given whatever values those exogenous variables assume. It is of considerable interest, however, to investigate a situation in which the average tax rate is constant and exogenous demand grows at the same rate as productivity, (g ( + 1 + xM) = (1 + b)( + 1 (g + x) • Given these assumptions, aggregate demand (6) becomes where A - y / ( a + a 1 T ) ,B - ( g + x ) /( a + a lT) and C = (Z > /[(1 + b){< 7 + «jT)]. Here we take the average tax rate T as f ( . Generating the left side recursively, the first term is a transient that vanishes as time passes so the behavior of efficiency income should be approximated with increasing accuracy by the difference equation (16) Divide both sides through by (1 + b)‘+ 1 and define per capita efficiency income in productivity units y* - y j{ [ + b) ' . Then 1 — '+ l < 7 + a,r,+ 1 (1+A)'+ 1 (17) or (18) xt+ l = B + Cx, (19) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 75 Recursively generate its solution to get the series x t = 5(1 + C + C 2 H h C ,_1) + C 'x 0 If C > 1, this series is unbounded. If 0 < C < 1, it converges. If x 0 — y ^ , then \\m x‘ = r = — = --------- i ± £ ^— (20) < - » “ 1 -C (cr+alT)-<j>/(l + b) Assume y is negative, so A is also. Let {xs . }q be a series generated recursively by equation (19), and a series generated by equation (18). Assume x 0 = y l < y e . The former is a positive, eventually monotonically increasing series converging to y e. The latter is also a positive, eventually monotonically increasing series, each element of which, y t < x t < y , t > 1 , so it converges also. It is always eventually increasing, however, so it has the same upperbound y e given by (20). If x 0 = y% > y e , the argument is identical except the series are eventually decreasing and the inequalities reversed. An analogous argument is followed for y positive. Consequently, we have proposition: Proposition A. Suppose exogenous demand (g + xM ) = (1 + b),+ 1 (g + x) where g, x and b are positive constants for an arbitrary base year. Assume the average tax rate is a constant, X . If (a) y + g + x> 0 and (b) ^ < (1 + b)((7 + CtjT) then there exists a unique positive, asymptotically stable, efficiency income y e, £±^L a + ayu - 0/(1 + b) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. The implication is that per capita and aggregate income converge to exponential growth paths given, respectively, by y t - * ( \ + b ) 'y e and Yt -> [(1 + b)(l + n)] y'’ 3 Fiscal Policy: Short and Long Run Effects 3.1 Short Run Effects of Changed Government Spending or Tax Rate Let us suppose government spending or tax changes could be implemented at the beginning of a new year. From (6) dy (+i _ 1 dgl+ i C f + (X 1 Tl >1 1‘itl (21) and dy, t+ i - ■a, d rM {a + a x zt+ \) <0 (22) Tax revenue in a given year will be d't+l ~ T t+ l^ t+ l ~ ^ r+ l^ + lJ V + l Given that 0 < (X , < 1, dT^ dr„ t a ir i+i v a + aiTi+ i j N,+ x y ,+ i >° (23) Government budget surplus will be Tt+ i G,+ l — N l+ 1 (Tl+ lyt+ l g l+]) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 77 where Gl+ [ - N l+ x g t+ [ is aggregate government spending. Then From these inequalities we find: Proposition B. (i) A government spending increase (decrease) increases (decreases) income by a multiplier greater than unity. (ii) An increase (decrease) in the average tax rate increases (decreases) income and increases (decreases) tax revenue and government budget surplus. 3.2 Growth Effects To consider the longer run influence of fiscal policy, we look at the effects of government spending and taxation on steady state growth. In such a situation the influence of a given tax or spending change continues throughout the future. Given Proposition A, HV - M ( l + fc)(l + n)f (25) dYt _ ax{g + x) yc[(l + h)(l + n)f < 0 (26) dr \a + ax r - 0/(1 + b)f and dTt _______ a£______ dr a + ax r - 0/(1 + b) y[(l + b)(l + n)l (27) and d (T ,-G ,)_ j cr + ax r -0 /( l + b) ax r F [(l + W + n)]' (28) dr Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 78 These inferences imply: Proposition C. If (gt+ 1 + xl+ 1 ) = (1 + b),+ 1 (g + x) where b, g and x are positive constants and if (a) y+g+ x >0 and (b) < /> < (l + b)(a + a 1 r) then (i) a permanent increase (decrease) in government spending increases (decreases) the steady state growth path; (ii) a permanent increase (decrease) in the tax rate decreases (increases) the growth path of income; (iii) in addition, if then a permanent increase (decrease) in the tax rate will decrease (increase) tax revenue and government budget surplus. (iv) Standard Tax Condition. Finally, if then an increase (decrease) in the tax rate will increase (decrease) tax revenue and government budget surplus. Putting together the stability condition and Proposition C (iii) we have The Two-Fold KRB Tax Policy Paradox: If (1 + b )a < < j > < (1 + b)(a + a x r) i.e., if (1+6)MPS<MPI<( 1 +6)(MPS+ T MPC), a permanent decrease in the tax rate will increase the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 79 entire future balanced growth path of income, tax revenue, and government budget surplus. Thus, the policy position taken in common by Kennedy, Reagan, and Bush rests on a theoretical basis. 3.3 Empirical Results for the Consolidated Era: 1929-1941,1948-2002 The aggregate demand parameter estimates and the average net tax rate obtained for the Consolidated Era together with the terms that enter the stability and KRB tax policy conditions are given in Table 3.1. Table 3.1 Parameter estimates for the linear model < p a Oj T ax r o + a x T 0.26 0.31 0.69 0.17 0.12 0.43 Using these estimates, we have = 0.26 < (l+ b)a = 0.32 so the Standard Conditions, C (i), (ii), and (iv) apply. The KRB Tax Condition of Proposition C (iii) does not appear to hold. The anti-recession policy of the current administration and that of Kennedy, G.W. Bush, and Reagan, who all took the same position, appears to be supported in that tax decreases should be expected and seem, indeed, to have stimulated output in the short run and growth in the longer run. On the other hand, their argument that tax revenues should increase as a result of tax reductions is not supported by these parameter estimates. But let us not jump to conclusions prematurely. Further empirical evidence on the issue gives a very different inference. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 80 4 A Nonlinear Complication 4.1 Piecewise Linear Consumption Function A close inspection shows that in spite of the overall correlation, the data exhibit systematic departures from linearity. These suggest structural changes or possibly a smooth nonlinearity in the consumption and savings relationships. For the purpose of testing this possibility we divided the period 1939- 2002 into the following growth Eras. Excluding Era II, we estimated the piecewise linear function separately for periods I, III, and IV, All coefficients are highly significant. The total amount of variation in the data explained is close to unity in Eras III and IV. Note that the marginal propensity to consume increased and the marginal propensity to save decreased successively through periods I, III, and IV. These changes I 1929-1941 The Great Depression Era II 1942-1947 World War II and Post War Recovery III 1948-1980 The Cold War Era IV 1981-2002 The Computer Global Information Era (29) where y d t = (1 — Tt )y t , obtaining the following parameter estimates in Table 3.2. Table 3.2 Parameter estimates for the piecewise linear model Piece I a'0 = 1827 a\ = 0.4378 y" =6777 R2 =0.966 Piece III a \ = 492 a\ = 0.6348 / " =20047 R2 =0.998 Piece IV a* = -3871 a* = 0.8674 R1 =0.995 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 81 are substantial. Indeed, the increase in the MPC and decline in the MPS has been drastic—the former almost double, the latter less than a fourth—in Era IV compared to Era I. At this point one may ask, “Why did these parametric changes occur?” A possible answer comes from the microeconomic theory of consumption and savings. Using a utility representation of people’s preferences, one can derive nonlinear consumption and savings fractions that can be approximated by the piecewise linear functions used here. Of course, the national economy is made up of people of great diversity in age, race, and various social characteristics. The data suggest that their aggregate behavior per capita behaves like a representative “average” individual who at low income levels spend a high fraction of income increases on housing, durable goods, and savings but who spend an increasing fraction of income on luxury consumables, such as eating out, vacations, fashionable clothing, etc., as income increases, a phenomenon found in the data by Hildenbrand (1995). 4.2 A Nonlinear DADS Model Except for two exceptional years at the bottom of the Great Depression, we have not found evidence of a similar nonlinearity in the investment income relationship, so we retain for the time being equation (2) and its parameter estimates. But aggregate demand using the nonlinear consumption function is also nonlinear and also has three pieces, a \ + a \{ \ - T l+ 1 )y l+ 1 + - y') + g M + x,+ 1 y,+ 1 = • a \ + c c\ (1 - Tl+i)y t+ l + < j){yt - / ) + g t+ i + x M a o + - y") + + *,+ i (30) which, solving for y t , gives the reduced form non— autonomous difference equation. y,+ i = -j \ k - < +SM + xM + ^ y , l t e S',i = I,III,IV (29) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 82 where s ;= ( i +t,)s ' S"‘ =(1 + T,)Sm Slv =(l + T,)S,V As before, we consider only incomes y t > y ' . The steady state income level for period i is determined by y “ = —. ____8>+X' ------- i = /,///, IV < jl + a,‘r ‘ - 0 ‘/(l + b‘) if the conditions in Proposition A (a) and (b) are satisfied for the period, where superscript i denote the period. Depending on the values of a l 0 and a\ , the economy may has one, two, or three stable steady states respectively. 4.3 Stability and Tax Implications The qualitative stability and KRB conditions must now be applied separately to each Era. Using the new parameter estimates for the piecewise segments, the picture changes drastically as shown in Table 3.3. Table 3.3 Key coefficients Phase (a) (b) (c) (d) (e) (f) 0 a, T (1 + b) (l + b)<7 (l + b)(a+ (C.E) 1929-1941,1948-2002 0.26 0.69 0.16 1.0186 0.32 0.43 (I) 1929-1941 0.44 0.44 0.12 1.0321 0.58 0.64 (III) 1948-1980 0.28 0.63 0.19 1.0241 0.38 0.50 (IV) 1981-2002 0.25 0.87 0.16 1.0176 0.13 0.31 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Comparing column (a) with (e) and (f) to determine which of Propositions B or C apply, we get the results summarized in Table 3.4. Table 3.4 The stability and KRB tax policy conditions Era Stability KRB Condition (C.E.) 1929-1941,1948-2002 Satisfied Not Satisfied (I) 1929-1941 Satisfied Not Satisfied (III) 1948-1980 Satisfied Not Satisfied (IV) 1981-2002 Satisfied Satisfied All three Eras were stable and the KRB condition was not satisfied in Eras I and III. Having incorporated the evident nonlinearity in the consumption and savings relationships, however, the KRB condition is satisfied during Era IV. Indeed, if the drastic change in consumer behavior is incorporated, it is clear Bush’s arguments rest on a sound theoretical and empirical basis. But Kennedy’s hopes were apparently not justified. He died too soon to see the vindication that emerged during the Reagan years due to the substantial increase in the MPC and corollary decline in the MPS. 4.4 Productivity and Employment We now consider the implications of the DADS model for the employment paradox: a rising rate of GDP with a lagging increase in employment. For this purpose we draw on the following definitions. N population E employment Ls labor supply (employed and unemployed workers) L work force (working age — institutional population) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 84 £l+ l = (E/L?)m = the employment rate Al+ I = (Ls/L )l+ i = the labor participation rate co t+ 1 = (L/N )l+ l = the work force fraction of population. (Bear in mind that the unemployment rate is u = 1 - £ ) For GDP we have the identity and since by assumption, qt = z t , we have for the ratio of employment to population, which depends on per capita demand divided by productivity per employed worker. Evidently, if the latter were to grow at a rate faster than that of demand, the employment/population ratio would decline. That ratio, however, is not the crucial one for macroeconomic policy. Rather, it is the ratio of employment to the labor supply, Et / L] , and how it is influenced by the labor participation rate, Ls t fL t , and the work force/population ratio, L j N t ■ Multiply the left side of (30) by E a ..h « A +i A + i and do some re-arranging to get the employment rate, From this equation we infer the following model estimates. Proposition D. The employment rate increases (decreases) if (i) aggregate demand increases (decreases); (ii) lab or productivity decreases (increases); (iii) the labor participation rate decreases (increases). E j N ^ z J q : (32) 1 1 ( r + g t+i+-*r+l+(for) < cr+ ay T„i (33) Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 85 In terms of the instrumental variables of fiscal policy we get Proposition E. The employment rate increases (decreases) if (i) government spending increases (decreases); (ii) the average tax rate decreases (increases). Fiscal policy is often discussed vis-a-vis a target unemployment rate, say u . The employment rate equation can be used to estimate the government spending and taxation policy that would be required to achieve such a rate. Set the left side at £( + 1 = 1 — u . and adjust T and g to find a desired combination. Fixing T , for example, and solving for g, we get c ? ,+ 1 = eAM0)l+,q e M (a + OyT) ~ ( y + x t+ 1 + fy, ) (34) Proposition F. The level of government spending required to achieve a given target employment rate increases if (i) the labor participation rate increases; (ii) the work force/population ratio increases; (iii) lab or productivity increases. 5 Discussion The model presented here cannot be taken as a perfect tool for policy repercussion analysis. Important variables have not all been taken into account. However, as far as empirical macroeconomics is concerned, it is evidently a good first approximation. The results would seem to compel serious attention. Of special interest is the KRB tax policy paradox. Evidently, tax reductions should be expected to stimulate the economy and—given the drastic reduction in the Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 86 economy’s per capita marginal propensity to consume—a rise in tax revenues should accompany the economic growth that follows - with approximately a year’s delay. The work force and employment data also support the inferences obtained in the preceding section. Increases in the labor participation rate were substantial in much of the post war period, just as with the work force as a percentage of the population. Together with a continued upward trend in productivity, these facts explain why unemployment lags behind economic recovery after a recession and constitute a continuing problem. The lesson of equation (32)—given fixed labor participation rates and the work force/population ratio—is basically that government must expect to grow along with population if a reasonable level of employment is to be sustained. But options also exist: labor participation can be discouraged by delaying entry into the labor force (military or other service, extended education) by reducing the full employment fraction of the year (shorter hours, longer vacations). A reversion to the traditional family structure of one bread winner, one family manager would have a similar effect. Eventually, in the U.S. as well as other developed countries, the need for one or more of these possibilities is being reduced by, and maybe eliminated by, the graying of the population which will lower the work force/population ratio. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 87 References 1. Adams J.D., 1980, Personal Wealth Transfers, Quarterly Journal o f Economics, 95, 159-179. 2. Attanasio, O. and J. Banks, 1998, Trends in Household Saving: A Tale of Two Countries, IFS Working Paper 98/15. 3. Barro, R.J., 1974, Are Government Bonds Net Wealth? Journal o f Political Economy, 82(6), 1095-1117. 4. Becker G and R. Barro, 1988, A Reformulation of the Economic Theory of Fertility, Quarterly Journal o f Economics, 103(1), 1-25. 5. Blinder, A.S., 1973, A Model of Inherited Wealth, Quarterly Journal o f Economics, Nov. 1973, 608-626. 6. Browning, M. and A. Lusardi, 1996, Household Saving: Micro Theories and Micro Facts, Journal o f Economic Literature, 34,1797-1855. 7. Cobb, C.W. and P.H. Douglas, 1928, A Theory of Production, American Economic Review, Supplements March 1928. 8. Cooley, T.F. and E.C. Prescott, 1995, Economic Growth and Business Cycles, in T.F. Cooley (ed.), Frontiers o f Business Cycle Research, Princeton: Princeton University Press. 9. Cox, D., S. Ng, and A. Waldkirch, 2000, Intergenerational Linkages in Consumption Behavior, Boston College Working Papers in Economics 482. 10. Davis, M. and M. Palumbo, 2001, A Primer on the Economics and Time Series Econometrics of Wealth Effects, FEDS Working Paper 2001-09. 11. Day, R. H., 1970, An Elementary Analysis of the Kennedy Tax Program, in W.L. Johnson and D.S. Kammerschen (ed.), Macroeconomics'. Selected Readings, Boston: Houghton Mifflin. 12. Evans, P., 2001, Consumer Behavior in the United States: Implication for Social Security Reform, Economic inquiry, 39(4), 568-582. 13. Greenwood J., A. Seshadri, and G Vandenbroucke, 2002, The Baby Boom and Baby Bust: Some Macroeconomics for Population Economics, University o f Rochester Economie D'avant Garde, Research Report No. 1. 14. Hurd, M., 1989, Mortality Risk and Bequests, Econometrica, 57(4), 779-813. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. 15. Kehoe, T., 2002, Decades Lost and Found: Mexico and Chile Since 1980, Federal Reserve Bank o f Minneapolis Quarterly Review, 26(1), 3 -30. 16. Kotlikoff, L., 1988, Intergenerational Transfers and Savings, Journal o f Economic Perspectives, Spring 1988, 41-58. 17. Kotlikoff, L. and L. Summers, 1981, The Role of Intergenerational Transfers in Aggregate Capital Accumulation, Journal o f Political Economy, 89, 706-732 18. Kydland, F. and E.C. Prescott, 1982, Time to Build and Aggregate Fluctuations, Econometrica 50, 1345 -1370. 19. Lainter, J., 1988, Bequests, Gifts, and Social Security, Review o f Economic Studies, 55(2), 275-299. 20. Laitner, J. and H. Ohlsson, 2001, Bequest Motives: A Comparison of Sweden and the United States, Journal o f Public Economics, 79,205-236. 21. Maki, M. and M. G. Palumbo, 2001, Distengling the Wealth Effect: A Cohort Analysis of Household Saving in the 1990s, FEDS Working Paper 2001-21. 22. Mehra, Y., 2001, The Wealth Effect in Empirical Life-cycle Aggregate Consumption Equations, Economic Quarterly, 87(2), 45-68. 23. Modigliani, F., 1988, The Role of Intergenerational Transfers and Life Cycle Saving in the Accumulation of Wealth, Journal o f Economic Perspectives, 2, 15-40. 24. Parker, J., 1999, Spendthrift in America? On Two Decades of Decline in the U.S. Saving Rate, NBER Working Paper 7328. 25. Slesnick, D., 1992, Aggregate Consumption and Saving in the Postwar United States, The Review o f Economics and Statistics, 74(4), 585-597. 26. Stokey, N.L., R.E. Lucas, Jr., and E.C. Prescott, 1989, Recursive Methods in Economic Dynamics, Cambridge: Harvard University Press. 27. Wilhem, M.O., 1996, Bequest Behavior and the Effect of Heirs’ Earnings: Testing the Altruistic Model of Bequests, American Economic Review, 86, 874-92. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

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Creator Yang, Chengyu (author)

Core Title Essays on consumption behavior, economic growth and public policy

School Graduate School

Degree Doctor of Philosophy

Degree Program Economics

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag Economics, General,Economics, Theory,OAI-PMH Harvest

Language English

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Advisor Day, Richard H. (committee chair), Imrohoroglu, Ayse (committee member), MacLeod, W. Bentley (committee member)

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c16-386546

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