Explaining Earnings Persistence: Does College Education Matter?

Transcription

1 Explaining Earnings Persistence: Does College Education Matter? Christoph Winter European University Institute Department of Economics Preliminary - Do not circulate March 9, 2007 Abstract Policy makers often consider higher education as the primary force for economic and social mobility. Designing e ective public policies requires measuring their e ect on the intergenerational persistence of earnings. We develop a quantitative theory of human capital transmission that allows us to formally evaluate the e ect of di erent college subsidy rules on earnings mobility. Our model is consistent with empirical ndings showing that only a fraction 10 percent of households is borrowing constrained for their college decision. We show that even when the number of constrained households is small, public policy can have a signi cant impact on economic mobility. Our analysis suggests that reducing college fees by 10 percent lowers the degree of intergenerational earnings persistence by 5 percent. We also nd (i) that existing college aid increases earnings mobility, that (ii) a switch to a at subsidy is more resource-e cient and that (iii) reducing social security bene ts curbs human capital investments. Keywords: Education, Intergenerational Mobility, College Subsidies JEL classi cation: I22, J62, E6, D91, C68 I would like to thank Morten Ravn and Salvador Ortigueira for their advice and encouragement. I also bene ted greatly from conversations with Ken Judd, Dirk Krueger, John Knowles, Víctor Ríos-Rull and Kenneth Wolpin. Address: Via della Piazzuola 43, I Firenze, Italy. christoph.winter@eui.eu

2 This paper develops a dynamic general equilibrium overlapping-generations model of household saving and human capital investment to analyze the role of parental transfers for college education in the transmission of earnings and education across generations. We also study how social policy and college subsidy rules in uence household savings and transfer behaviour and compare di erent tuition policies with respect to their e ect on the degree of social mobility. The research is motivated by recent empirical evidence indicating that the degree of intergenerational mobility of economic status, measured by the slope coe cient obtained by regressing o spring s log earnings (when adults) on their parent s log earnings, is around 0.4 in the United States (see Solon (1992) and Zimmerman (1992)). That is, around 40 percent of the parent s relative position in the earnings distribution is transmitted to the child. This degree of persistence is almost twice as high as in other developed countries (see Solon (2002)). Higher education has a prominent position in shaping labor market success, and it requires a substantial amount of private resources, which makes it a primary candidate for explaining the low degree of social mobility (see Haveman and Smeeding (2006)). Gale and Scholz (1994) nd that college expenses paid by parents on behalf of their children are sizable. According to their results based on data from the Survey of Consumer Finances (SCF), U.S. parents paid almost $ 100 billion for their o spring s college education during Savings for human capital investments thus accounts for a substantial share of total wealth. Keane and Wolpin (2001) present empirical evidence showing that even low-income families nance a considerable share of their higher education expenses out of their own resources, and that this share is rapidly increasing with income. Keane and Wolpin further nd evidence that borrowing constraints, i.e. restrictions on the availability of uncollateralized education loans, exist and are tight. This suggests that parental resources are an important source for education nancing. We construct a computable dynamic general equilibrium model in which agents go through a life-cycle. Households are organized into dynasties, with overlapping periods of parents and their children. Parent households are altruistic in the sense that they incorporate their o spring s utility, as in the Barro-Becker framework. Upon entering the labor market, child households differ with respect to their initial productivity types. Initial productivity types are correlated across generation. Conditional on their initial productivity, children decide on whether to attend college or not. College education is costly, and there is a certain probability that the child will become a dropout. The dropout probability is assumed to decrease as the level of initial productivity increases. This ensures that in equilibrium, high productivity households are more likely to attend college. Since education takes place at the very beginning of a household s economic life and borrowing against future human capital is not possible, the child household depends on parental support and college subsidies provided by the government. Parents are by no means restricted to human capital transfers if they want to give support to their children. They can also make inter-vivos gifts or leave bequests. This assumption is important because low ability children are 1

3 less likely to be successful at college. Therefore, their parents may prefer to transfer wealth rather than invest in education. In this paper, we extend the standard computable OLG framework with idiosyncratic productivity shocks and lifetime uncertainty by incorporating altruism and transfers to formally evaluate to what extend parental investment in college education can account for the transmission of earnings inequality across generations. In order to do so, we calibrate the parameter values in our model such that it replicates key features of the U.S. economy. In particular, we nd that our model matches the ow of intergenerational transfers observable in the U.S. very well. Using the procedure outlined in Carneiro and Heckman (2002), we nd that only a fraction of 10 percent of all households deciding about college education are constrained by the borrowing limit. This result is broadly consistent with empirical evidence provided by Carneiro and Heckman. Despite the small fraction of people who face binding borrowing constraints, we nd that there is a substantial role for public policy. Reducing the cost of college education by 10 percent lowers the degree of earnings persistence by about 5 percent. Hence, our setup suggests that even a small number of borrowing constrained households can account for a signi cant share of the degree of earnings persistence. Our second main result is that a switch from the current means-tested college aid system to a at system requires fewer resources for a given degree of social mobility. Intuitively, an asset based means-tested system implicitly imposes high marginal tax rates on a single year of parents resources (Feldstein 1995). This reduces wealth holdings among the poor, as they bene t the most from a progressive subsidy. A at subsidy removes these disincentives implicit in a means-tested aid system, thus encouraging wealth accumulation. However, to the extent that the bene ciaries of a at subsidy are those who were planning to go to college anyway, the at subsidy system may constitute a pure transfer program. Our results indicate that the resources saved from removing the high marginal tax rates more than o sets the amount that is implicitly used for transfers. We also show that this result depends on the degree of earnings uncertainty. The larger the dispersion of shocks, the smaller is the expected resource gain that can be achieved by this policy change. The reason is that households mainly save for precautionary reasons when facing higher uncertainty, and implicit tax rates on capital accumulation induced by a means-tested subsidy system are lower. This highlights the importance of analyzing the impact of college policies in a quantitative framework together with household savings decisions. Our model is related to several other papers that focus on education decisions and policy interventions in a general equilibrium context. Heckman et al. (1998) were the rst who highlighted the importance of general equilibrium effect induced by policy changes. Fernandez and Rogerson (2003) compare equityresource trade-o s between ve education systems. Our paper is probably the closest to Hanushek et al. (2004) and Restuccia and Urrutia (2004). Compared to Hanushek et al., who motivate human capital transfers by assuming a joy-of-giving motive, we argue that introducing altruistic parents who value 2

4 their children s consumption is more in the line with the data on human capital investment (see, among others, McGarry and Schoeni (1995) and Hochguertel and Ohlsson (2003)). Restuccia and Urrutia compare the impact of parental investment in early and college education. However, they abstract from the accumulation of physical wealth and idiosyncratic productivity shocks, which turn out to be important determinants of parental transfer behavior in our model. The remaining of our paper is structured as follows. We present the model in section 1. Section 2 introduces the equilibrium de nition, while the calibration is explained in section 3. We discuss our results in section 4 and conduct policy experiments in section 5. Finally, section 6 concludes. 1 Model The economy consists of three types of agents, households, rms and a government. There is a large number of nitely lived households, a large number of perfectly competitive rms and a government. We consider idiosyncratic shocks to households productivity, liftetime uncertainty and intergenerational links. Parent households make inter vivos transfers, invest in their o spring s education transmit part of their productivity type to their o spring. 1.1 The life cycle of a household There is a continuum of agents with total measure one. We assume that the size of the population is constant over time. Let j denote the age of an agent, j 2 J = f1; 2; :::J max g. Agents enter the economy when they turn 26 (model period j = 1). Before this age, they belong to their parent household and depend on its economic decisions. During the rst 35 years of their economic life, agents work. This implies that the agents work up to age 60 (model period J work = 35): Retirement takes place at the age of 61 (j = 36), which is mandatory. When agents turn 61, their children of age 26 form their own household. It is assumed that there is one child household for each parent household. We assume that parents make transfers at the age of 61 (j = 36), when their child household enters the economy. Transfers are made either in form of property transfers, investment in human capital or both. More speci cally, we assume that all agents have a high school degree. 1 Parents can allow its child household to go to college, in which case the parent household must bear the costs of tuition. In order to avoid any kind of strategic interaction between the parent and its child household, we make the following two timing assumptions. First, we assume that transfers are made at the age of 61 shortly after children have left home, but before they decide upon their consumption in the 1 This assumption is justi ed as the share of high school dropouts is small in the data. According to Díaz-Gimenez et al., 2002, only 17 percent of their sample has not nished high school. 3

5 Figure 1: The life-cycle rst period. And second, we assume that parents decide upon transfers and investment in college education simultaneously. We assume further that parents observe their children s productivity type before deciding on the transfer. Agents face a declining survival probability during retirement. Terminal age is 95 (J max = 70). Since annuity markets are closed by assumption, agents may leave some wealth upon the event of death. The remaining wealth of a deceased parent household is passed on to its child household. Our assumptions regarding the life-cycle are summarized in Figure Labor income and capital income The household receives capital income given by the market interest rate times its wealth. Agents cannot borrow against future income, and the wealth level of each household must be nonnegative. During each of the 35 periods of their working life, agents supply one unit of labor inelastically. The productivity of this labor unit of an j-year old agent is measured by " e j j;e, where " e J w j is a deterministic age pro le of average j=1 labor productivity of an agent with education level e, where e 2 E = fhighschool(hs); college( col )g (1) For retired agents, " e j = 0. j;e describes the stochastic labor productivity status of a j-year old agent with education level e. Given the level of education e, we assume that the stochastic process is identical and independent across agents and that it follows a nite-state Markov process with stationary transitions over 4

6 time. More speci cally, Q( hs ; N hs ) = Pr( j+1;hs 2 N hs j j;hs = hs ) (2) for high-school graduates. N hs = hs 1 ; hs 2 ; :::; hs n is the set of possible realizations of the productivity shock hs. Similarly, we express the stochastic shock process for college graudates as Q( col ; N col ) = Pr( j+1;col 2 N col j j;col = col ) (3) with N col = col 1 ; col 2 ; :::; col n. We assume that N hs and N col contain the same number of elements. Children inherit part their parents productivity process. We interpret this as transmission of productivity types across generations. In particular, it is assumed that the there is a positive correlation between the realization of the productivity shock in the rst working period of children and parents, where j = 26. Afterwards, the shocks evolves according to its stochastic process. In order to link the initial productivity levels across generations, we assume that the initial shock is governed by the following transition matrices: Q initial;hs (i; i 2 I = f1; 2; :::; ng) = Pr(i; i 2 f1; 2; :::; ng j 60;hs = hs ) (4) Q initial;col (i; i 2 I = f1; 2; :::; ng) = Pr(i; i 2 f1; 2; :::; ng j 60;col = col ) (5) That is, the transmission depends on the realization of the parents productivity shock in their last working period (j = 60) and on their education level. In our calibration, the transmission of productivity types depends de facto only on parent s education achievement. Notice that parents transmit only the productivity level. The speci c realization - that is, whether the shock is taken from N hs or N col - is determined by the education level, which is chosen by parents after they observe their o springs productivity type. College education College education is assumed to be costly in terms of time and tuition fees. Sending a child to college therefore requires a parent household to pay a xed cost in resources. There is evidence that the share of xed costs in total cost of college education are substantial (see Kane (1999) and Dynarski (2001)). It is assumed that this xed cost covers both opportunity costs and actual college expenses. Since borrowing against future income is not permitted, parents need to pay for college expenditure either out of their current income or out of their savings. 2 In accordance with the literature, we assume that there is a risk of no completion. In our framework, a higher productivity type is associated with a lower dropout probability. As parents observe their o spring s productivity 2 This assumption may seem strong at the rst glance. However, we will show later that the share of parents who cannot a ord education because of borrowing constraints is actually very small. 5

7 type, their decision on college investment also incorporates the risk that their children will not complete college education. If the child becomes a dropout, we assume that its productivity process follows the process of high school graduates. Taxes and social security bene ts Households during working life pay a proportional tax on their labor income; all households pay a proportional tax on their capital income. We abstract from gift and estate taxation. Tax revenue from labor income and capital income taxation is used by an in nitely lived government in order to nance pension bene ts (pen). We assume that pensions are independent of the employment history of a retiree. 2 The households recursive problem 2.1 Young households with deceased parents Consider a household during working age (j 2 J w = 1; :::; J work ) whose parent household is dead. At age j, this household consumes c y;d and has endof-period wealth holdings of a 0 y;d, where y,d means that we consider a young household with deceased parents. Given a discount factor ; a rate of return to capital r, a wage rate per e ciency unit of labor w, tax rates on labor income and capital income w and k, the optimization problem of this household reads as 8 9 < V y;d (s y;d ) = max c y;d ;a 0 : u(c y;d) + X = V y;d (s 0 y;d)q(; 0 ) ; 8j 2 1; :::J work 1 y;d 0 2N e (6) where V y;d (:) is the value function of a young household with deceased parents and s y;d is the vector of state variables in period j, which is given by s y;d = (a y;d ; e; j;e ; j) (7) Consequently, next period s state vector reads as follows: s 0 y;d = (a y;d ; e; j+1;e ; j + 1) (8) In each period j 2 1; :::J work 1 3, agents maximize (6) subject to the budget constraint a 0 y;d = (1 + r(1 k ))a y;d + (1 w )" e j j;e w c y;d (9) 3 Notice that for j = J work, the value function reads as 8 9 < V y;d (s y;d ) = max c y;d ;a 0 : u(c y;d) + X = V p;1 (s p;1 )Q initial;e (; di) ; y;d 0 2N e When j = J work, child households become parent household in j + 1. This implies that they observe their o spring s initial productivty level which becomes part of their state vector s p;1. As we will see below, compared to s y;d, s p;1 does not contain agents education level nor their productivity shock. 6

8 The state space S y;d of an household of type y; d thus includes four variables: own asset holdings, a y;d 2 R +, education level, e 2 E, stochastic productivity, j;e 2 N j;e, and age j 2 J w = 1; :::; J work. Notice that S y;d = R + E N e J w. Let P(E);P(N e ) and P(J w ) be the power sets of E; N e and J w, respectively, and let B(R + ) the Borel -algebra of R +. It follows that S y;d = B(R + ) P(E) P(N e ) P(J w ) is a -algebra on S y;d and that M y;d = (S y;d ; S y;d ) is a measurable space. We will assume that the value function V y;d :S y;d! R and the policy functions c y;d :S y;d! R + and a 0 y;d :S y;d! R + are measurable with respect to M y;d. 2.2 Young households whose parents are alive Let us now turn to a household whose parents are still alive. At any age j 2 J w, such households consume c y;a and have end-of-period wealth holdings of a 0 y;a, where a y;a denotes a young household whose parents are alive. Its parent household has wealth holdings of a p y;a. The optimization problem can be described by the following functional equation: u(cy;a ) + (1 V y;a (s y;a ) = max j+35 ) P 0 2N V y;d(s 0 e y;d )Q(; d0 ) c y;a;a 0 + y;a j+35 P 0 2N V y;a(s 0 y;a)q(; d 0 ) e (10) 8j 2 1; :::J work 1 4 where j+35 is the survival probability of the parent household, V y;a (s y;a ) denotes the value function given the state vector s y;a, where s y;a is described by s y;a = (a y;a ; a p y;a; e; j;e ; j) (11) Similarly, the state vector in the next period is given by s 0 y;a = (a y;a ; a p0 y;a; e; j+1;e ; j + 1) (12) The household maximizes (10) subject to its current period budget constraint a 0 y;a = (1 + r(1 k ))a y;a + (1 w )" e j j;e w c y;a (13) If the parent household dies in period j 1; the ow budget constraint becomes a 0 y;d = (1 + r(1 k ))(a y;a + a p y;a) + (1 w )" e j j;e w c y;a (14) In this case, the child household inherits the remaining parental assets in period j. Consequently, the value of the child household s states is a weighted sum 4 If j = J work, the child household knows that its parent household will die for sure in the current period. The Bellman equation thus reads as 8 9 < V y;a(s y;a) = max c y;a;a 0 : u(c y;d) + X = V p;1 (s p;1 )Q initial;e (; di) ; y;a 0 2N e Also see the previous footnote. 7

9 of the utility it receives if the parent household dies and the utility which is obtained if the parent continues to live for another period, where the parental survival probability j+35 serves as a weight. Because the child household inherits its parent household wealth upon the event of death, the child keeps track of it parent s wealth holdings and a p y;a becomes part of the child household s state vector. Notice that this may imply that children reduce their own savings if they observe that their parents are a uent. This is di erent to the model of De Nardi (2004) in which it is assumed that children do not observe their parents wealth holdings. We instead assume that the child has perfect knowledge about the law of motion of their parents asset holdings, which follow a p0 y;a = (1 + r(1 k ))a p y;a + pen c p (15) Notice that (15) is only valid for j 2. For j = 1, parents have a di erent budget constraint. Once the parent dies, the state vector reduces to s y;d and the child continues as a household with deceased parents, as described above. The state space S y;a of an household of type y; a includes parental asset holdings a p y;a 2 R +, which adds a continuous state variable to the state space. S y;a is thus given by S y;a = R + R + EN e J w. It follows from the de nitions made in section (2.2) that S y;a = B(R + )B(R + )P(E)P(N e )P(J w ) is a -algebra on S y;a and that M y;a = (S y;a ; S y;a ) is a measurable space. Further, let V y;a :S y;a! R, c y;a :S y;a! R + and a 0 y;a :S y;a! R + be measurable with respect to M y;a. 2.3 Parent households Consider now a parent household, 37 j J max. This household is retired and receives social security bene ts, pen. It also faces a declining survival probability, j < 1. This household chooses current consumption c p and its end-of-period wealth level a 0 p. The subscript p indicates that we consider a parent household. The optimization problem of this household can be written in recursive formulation as follows: V p (s p ) = max u(cp ) + j V p (s 0 p) 8j 2 f37; :::J max 1g (16) c p;a 0 p 8j 2 f37; :::J max where V p (s p ) is the value function, given the state vector s p. It follows that and The household maximizes (16) subject to 1g s p = (a p ; j) (17) s 0 p = (a 0 p; j) (18) a 0 p = (1 + r(1 k ))a p + pen c p (19) 8

10 In the terminal period J max, (16) reduces to subject to V p (s p ) = max c p fu(c p )g (20) c p (1 + r(1 k ))a p + pen (21) The Bellman equation for the terminal period re ects the fact that parent households do not receive any utility from assets which are left upon the event of death. In other words, all end-of-life bequests made are accidental. As parent household are out of the labor force and pension payments are independent from individual earnings histories, the state vector simpli es to s p. The state space is now given by S p = R + f37; :::; J max g, in a similar fashion to above, we can construct the -algebra on S p as S p = B(R + ) P(f37; :::; J max g) where P(f37; :::; J max g) is the power set of f37; :::; J max g. As a result, we obtain the measurable space M p = (S p ; S p ), with respect to which we de ne V p :S p! R, c p :S p! R + and a 0 p :S p! R + to be measurable. 2.4 Parent household, rst period Parent households in their rst period of parenthood (j = 36) play a crucial role in our economy. Not only do they decide on their consumption c p;1 and on endof-period asset holdings a 0 p;1, they also choose the amount of wealth they want to transfer (tra) to their children and they decide on investment in education (ed). Recall that the education level is a binary variable, that is, ed 2 f0; 1g, where ed = 0 if parents choose not to send their children to college and ed = 1 if parents send their children to college. Expressed in terms of a Bellman equation, the decision problem of a parent household at j = 36 reads as u(c V p;1 (s p;1 ) = max p ) + 36 V p (s 0 p) c p;1;a 0 p;1 ;tra;ed +& (E [V y;a (s y;a )jed = 1] + E [V y;a (s y;a )jed = 0]) (22) where & is the intergenerational discount factor, V p;1 (s p;1 ) is the value function for a given state vector s p;1, where s p;1 = (a p;1 ; i) (23) Notice that the initial productivity level i of the child becomes part of the parent household s state space. As investment in college education is risky, parents base their decisions on the expected values E [V y;a (s y;a )jed = 1] and E [V y;a (s y;a )jed = 0], which denotes the expected maximal lifetime utility of a child household if the parent household invests in college education (ed = 1) and if parents do not invest (ed = 0), respectively. More speci cally, the expected utility if ed = 1 is given by (i)v E [V y;a (s y;a )jed = 1] = y;a (tra; a p;1 ; col; 1;col ; 1) +(1 (i))v y;a (tra; a p;1 ; hs; 1;hs (24) ; 1) 9

11 where (i) is the probability that the child household completes college education will be successfully. We assume that the dropout probability depends on initial productivity level in a linear way. The expected utility is thus a weighted average of the lifetime utility if the child completes education and of the total utility the child household receives if it does not complete education. In the latter case, the utility is identical to the case where the parent did not invest (the education level in this case is high school hs) and the investment in education is waisted. If instead ed = 0, there is no risk involved and the expected utility becomes E [V y;a (s y;a )jed = 0] = V y;a (tra; a p;1 ; hs; 1;hs ; 1) (25) where the term on the left-hand side is the lifetime utility a child household with high school degree, initial capital stock tra, parentel wealth holdings a p;1 and initial productiviy 1;hs. Parent household maximize (22) subject to a 0 p;1 = (1 + r(1 k ))a p;1 + pen + jed=1 (a p;1 ) tra jed=1 c p;1 (26) where denotes the college subsidy the household receives if ed = 1, denotes the xed college expenses that the parent household pays. Note that the subsidy depends on the wealth level. We assume that parent households do not necessaritly weight their own utility in the same way as they weight their o spring s utility. This is re ected in the intergenerational discount factor &, where 0 & 1. If & = 0, parents are not altruistic towards their children at all and care only about their own utility. Using the fact that the state space of parents in their rst period is given by S p;1 = R + f1; 2; :::; ig, we construct a -algebra on S p;1 as S p;1 = B(R + ) P(f1; 2; :::; ig) where P(f1; 2; :::; ig) is the power set of f1; 2; :::; ig. M p;1 = (S p;1 ; S p;1 ) is then a measurable space, and we assume that V p;1 :S p;1! R, c p;1 :S p;1! R +, a 0 p;1 :S p! R +, tra :S p;1! R + and ed :S p;1! f0; 1g are measurable on M p; The Firm s Problem There are in nitely many rms with total measure one. All rms produce a single identical output good Y using a constant returns to scale production technology with physical capital K and labor measured in e ciency units L. The pro t-maximizing condition of a representative rm is r + = F K (K; L) (27) 2.6 The Government s Problem w = F L (K; L) (28) Let be probability measures de ned over the measurable spaces M y;d, M y;a, M p;1 and M p. 5 The government computes old-age pension bene ts pen as 5 Notice that the total population size is normalized to one. The probability measure thus de nes the number of people (or equivalently, the total population share) facing a speci c 10

12 the average lifetime income of a high-school graduate times a social security replacement ratio: R w R +fhsgn " hs pen = rep hs J w j j;hs d + R R +fhsgn " hs hs J w j j;hs d R R +fhsgn d + R hs J w R +fhsgn d hs J w (29) In order to keep the budget balanced, the government adjusts the tax rate on labor income w : R pen d + R d Sp;1 k rk Sp w = (30) wl where K and L are aggregate physical capital and aggregate e cient labor (human capital). 3 De nition of a Stationary Competitive Equilibrium We now de ne the equilibrium that we study: De nition 1 Given a replacement rate rep and a tax rate for capital income k, a Stationary Recursive Competitive Equilibrium in this economy is a set of functions V y;d (s y;d ); V y;a (s y;a ); V p;1 (s p;1 ); V p (s p ), c y;d (s y;d ), c y;a (s y;a ), c p;1 (s p;1 ), c p (s p ), a 0 y;d (s y;d), a 0 y;a(s y;a ), a 0 p;1(s p;1 ), a 0 p(s p ), tra(s p;1 ), ed(s p;1 ), prices for physical capital and e ective labor fr; wg, and set of probability measures on the state spaces of the respective household problem as de ned in sections (2.2)-(2.4) such that the following hold: 1. Given prices and policies, V y;d (s y;d ), V y;a (s y;a ), V p;1 (s p;1 ) and V p (s p ) are the solution to the household problem outlined in (2.2)-(2.4) with c y;d (s y;d ), c y;a (s y;a ), c p;1 (s p;1 ), c p (s p ), a 0 y;d (s y;d), a 0 y;a(s y;a ), a 0 p;1(s p;1 ), a 0 p(s p ), tra(s p;1 ), ed(s p;1 ) being the associated policy functions. 2. The prices r and w solve the rm s problem (27) and (28). 3. The government policies satisfy (29) and (30). 4. Markets for physical capital, labor in e ciency units and the consumption good clear: ( R a 0 S K = y;d y;d (s y;d)d + R ) S y;a a 0 y;a(s y;a )d + R S p;1 a 0 p;1(s p;1 )d + R S p a 0 (31) p(s p )d endowment with state variables. 11

13 Z Z L = " e j j;e d + S y;d " e j j;e d S y;a (32) C + [K (1 )K] + T + I = F (K; L) (33) where ( R c S y;d (s y;d )d + R ) C = y;d S y;a c y;a (s y;a )d + R S p;1 c p;1 (s p;1 )d + R (34) S p c p (s p )d Z T = tra(s p;1 )d (35) S p;1 Z I = ed(s p;1 )d (36) S p;1 5. The Aggregate Law of Motion is stationary: = H() (37) The function H is generated by the policy functions a 0 y;d (s y;d), a 0 y;a(s y;a ), a 0 p;1(s p;1 ), a 0 p(s p ), tra(s p;1 ), ed(s p;1 ), the Markov process Q( e ; N e ) and the transmission matrix Q initial;e (i; i 2 I = f1; 2; :::; ng) and can be written explicitly as (a) For all sets (A; A p ; E; N e ; J ) with J = f2; :::; 35g such that (A; A p ; E; N ; J ) 2 S y;a, the measure of agents whose parents are alive is given by Z a = P y;a (s y;a ; (A; A p ; E; N e ; J ))d a (38) S y;a where P y;a (s y;a ; (A; A p ; E; N e ; J )) ( P (j+35) Q(; 0 ) if a 0 y;a(s y;a ) 2 A; a 0 p(s p ) 2 A p ; e = e 0 2 E; j J = 0 2N e 0 otherwise P y;a (s y;a ; (A; A p ; E; N e ; J )) is the transition function. It gives the probability that an agent with endowment s y;a at age j ends up in j+1 with asset holdings a 0 y;a 2 A, productivity state 0 2 N e and parental asset holdings a 0 p 2 A p. The education level remains constant. (b) For all sets (A; E; N e ; J ) with J = f2; :::; 35g such that (A; E; N ; J ) 2 S y;d, the measure of agents with deceased parents is given by 8 R 9 < P S y;d (s y;d ; (A; E; N e ; J ))d d + = d = R y;d : S y;a 1 (j+35) P y;a (s y;a ; (A; A p ; E; N e ; J ))d a ; where y;d (s y;d ; (A; E; N e ; J )) ( P Q(; 0 ) if a 0 y;a(s y;a ) 2 A; e = e 0 2 E; j J = 0 2N e 0 otherwise (39) 12

14 As fraction 1 (j+35) of parents dies in period j, d incorporates the measure of young agents whose parents died in the previous period. (c) For all sets (A; I; J ) with J = f36g such that (A; I; J ) 2 S p;1, the measure of parent households in j = 36 is given by ( R ) P S p;1 (s y;d ; (A; I; J ))d d p;1 = y;d + R (40) S y;a P p;1 (s y;a ; (A; I; J ))d a where p;1 (s y;d ; (A; I; J )) ( P Q initial;e (i; i 0 ) if a 0 y;a(s y;a ) 2 A; j J = i 0 2I 0 otherwise and P p;1 (s y;d ; (A; I; J )) follows straightforwardly. P p;1 (:; (A; I; J )) shows the transition from child households to parent households. The measure of parent households collects all child households. (d) For all sets (A; A p ; E; N e ; J ) with J = f1g such that (A; A p ; E; N ; J ) 2 S y;a, the measure of agents in their rst period is given by Z initial = Pp;1 initial (s p;1 ; (A; A p ; E; N e ; J ))d p;1 (41) S p;1 where P initial p;1 (s p;1 ; (A; A p ; E; N e ; J )) is given by a 0 p;1(s p;1 ), tra(s p;1 ) and ed(s p;1 ) where a 0 p;1(s p;1 ) 2 A, tra(s p;1 ) 2 A p and ed(s p;1 ) 2 E. (e) The measure of agents during retirement is generated by the policy function a 0 p(s p ) where a 0 p(s p ) 2 A. A few remarks regarding the equilibrium conditions are in order. (31) and (32) state that aggregate physical and labor measured in e ciency units follow from aggregating the respective holdings of each agent and weighting them appropriately. (33) requires that the good market clears, i.e. that demand for goods, which is shown on the left-hand side, is equal to the supply of goods. The term [K (1 )K] on the left-hand side determines the amount of investment that is necessary to keep the aggregate capital stock constant, whereas I and T are aggregate college expenditures and transfers in stationary state, respectively. I can be viewed as an investment to keep the level of human capital constant. (37) requires stationarity of the probability measure. The function H is the transition function which determines the probability that an agent will end up with a certain combination of state variables tomorrow, given his endowment with state variables today. Notice that the stationarity condition requires that child households are (on average) identical to their parents in the sense that the reproduce their parent households distribution once they become parents themselves. This in turn implies that the distribution of transfers and inheritances that child households receive is consistent with the distribution of transfers that is actually left by parent households. 13

15 4 Parametrization and Calibration We assume that the utility from consumption in each period is given by u(c) = c 1 1. Production in the United States is assumed to follow the aggregate production function F (K; L) = K L 1. We calibrate parameter values of our benchmark economy to represent relevant features of the U.S. economy as close as possible. In our calibration procedure, we distinguish two sets of parameters. In the rst set, we collect parameters that are estimated in the empirical literature or that can be calibrated directly. Parameters in the second group are calibrated so that the model-generated data matches a given set of targets. With a length of a period in the model of 1 year, we set the capital share in income () equal to 0:36, as estimated by Prescott (1986). Following Imrohoroglu et. al. (1995) and Heer (2001), we assume that capital depreciates at an annual rate of 8 percent. The conditional survival probability j is taken from the National Vital Statistics Report, Vol. 53, No. 6 (2004) and refers to the conditional survival probabilities for the U.S. population. Only values between age 61 and age 95 are used. We assume that the survival probability is zero for agents at the age of 95. If an agent is younger than 61, he survives with a probability of 1. 6 The preference parameter determines the relative risk aversion and is the inverse relation to the intertemporal elasticity of substitution. We follow Attanasio et al. (1999) and Gourinchas and Parker (2002) who estimate using consumption data and nd a value of 1:5. This value is well in the interval of 1 to 3 commonly used in the literature. Regarding the log earnings process for high school graduates and college graduates, we assume that both processes follow an AR(1) process with persistence parameter hs for high school graduates and col for college graduates. The variance of the innovations are hs and col, respectively. These parameters are estimated by Hubbard, Skinner and Zeldes (1995) from the Panel Study of Income Dynamics (PSID). They nd that high school graduates have a lower earnings persistence and a higher variance ( hs = 0:946, hs = 0:025) compared to college graduates ( col = 0:955, col = 0:016). It should be noted that both estimates are rather conservative as Hubbard et al. use the combined labor income of the husband and wife (if married) plus unemployment insurance for their estimates. In the following, we will thus consider each household as a married couple that makes decisions jointly. As shown in the appendix, both earnings processes are quite similar. 7 We approximate both processes with a four-state Markov chain using the procedure outlined by Tauchen and Hussey (1991), which implies that the num- 6 The actual survival probability before 61 is close to 1. See the National Vital Statistics Report. 7 This is important because we implicitly assume that the di erence between the two earnings processes is solely due to skills acquired during college but not related to the level of pre-college ability (recall that the latter only in uences the realization of the rst productivity shock). This may overstate the e ect of college education on earnings. The fact that both earnings processes are almost the same tells us that the potential bias must be very small. 14

16 Parameter Description Value capital share of income 0.36 capital depreciation rate 0.08 risk aversion 1.5 hs earnings persistence high school hs variance shocks col earnings persistence college col variance shocks K capital income tax rate 0.2 rep replacement ratio pensions 0.4 Table 1: Parameters provided by other studies ber of productivity states is equal to 4. Further details regarding the calibration of the earnings processes and the underlying age-e ciency pro le are outlined in the Appendix. Following De Nardi (2004), we use a capital income tax rate K of 0:2 and a replacement rate for pension bene ts of 0:4. Table 1 summarizes the parameter that we take from the empirical literature. With respect to the parameters in the second group, we calibrate the discount factor such that our baseline economy targets a wealth-output ratio of 3 for the U.S. economy. This results in a of 0:96. The cost for college education is calibrated to target a ratio of high school graduates relative to college graduates of 1:66 as measured by Rodríguez, Díaz-Giménez, Quadrini and Ríos- Rull (2002). We obtain a of 1. Since the income per capita in our economy is 1 as well, our result implies that college expenses in our model amount to the equivalent of the average annual income. 8 We assume that the probability of college completion (i) is an increasing function of the initial productivity state i. In particular, we assume (i) = d + 0:2i (42) where d 2 [ 0:2; 0:2] such that (i) 2 [0; 1] if i 2 f1; :::; 4g. We calibrate the parameter d such that the model reproduces a college dropout rate of 50% that was measured by Restuccia and Urrutia (2004) for the U.S. economy. The fact that the probability of college completion is an increasing function of the initial shock i implies that the child household with a higher i is more likely to attain education. We use this relationship between i and probability of college attainment in order to calibrate the transmission of productivity types. More speci cally, we assume that the transmission of productivity types depend on the last realization of the parental productivity shock and the parental education level. The link between parental education attainment and productivity types of the child household ensures a positive correlation between the initial level of market skills of the parent and its o spring. In order to get this positive 8 Notice that our result is perfectly consistent with the notion that average college expenditures per year are about $10000 for a four-year college (see Collegeboard 2003) and that per-capital GDP is around $

17 Parameter Description Value discount factor 0.96 cost of college 1 d probability college completion transmission prod. type 0.83 w labor income tax 0.15 & intergen. disc. factor slope coll. subsidy 0.01 Table 2: Parameters calibrated to match certain features of the U.S. economyprovided by other studies link, we construct the transmission matrices Q initial;hs and Q initial;col as linear combinations of a matrix that gives equal probability to each potential productivity level on the one hand and a matrix where this probability is decreasing (Q initial;hs ) or increasing (Q initial;col ) with i on the other hand. We use a parameter as a weighting factor, which determines the relative weight between the equal-outcome matrix and the matrix which gives higher weight on either low or high productivity shocks. is calibrated such that our model reproduces the intergenerational correlation of earnings, which is 0:4 as measured by Solon (1992) and Zimmerman (1992). Parents who send their children to college receive a government subsidy (a) for each unit of expenditure in college education. We assume that (a) is a decreasing function parental wealth, with (0) 1 and (1) = 0, to represent a progressive subsidy. In particular, we assume that the subsidiy is linear (see Restuccia and Urrutia). Following Keane and Wolpin (2001) it is assumed that an individual s grant cannot exceed a certain fraction of college expenes. Keane and Wolpin set this maximum at 50 percent. Their estimate is likely to be biased downwards, given the sharp increase in tuition fees over the last decade, see Haveman and Smeeding (2006). Our assumptions result in the following functional form: (a) = max fmin [(1 0 a) ; 0:5] ; 0g (43) We calibrate the parameter 0 such that our model replicates the college expenditures made by private households relative to total wealth, as measured by Gale and Scholz (1994). We nd that this corresponds to a 0 of 0:01, which implies that the college subsidy is relatively at. Similarly, we calibrate the intergenerational discount factor & such that our model matches the monetary inter-vivos transfers (gifts etc.) relative to total household wealth, also taken from Gale and Scholz (1994). Finally, we adjust the tax rate on labor income w such that the government budget (30) is balanced. This results in a tax rate of 15 percent. The results are summarized in Table 2. 16

18 4.1 Measuring Earnings Persistence For a precise measurement of the persistence of earnings, Mulligan (1997) proposes to use the average earnings received by an individual over his lifetime. Due to data limitations, this is hardly possible in empirical studies. Instead, most studies measure the economic status of parents and children for several years rather than a single year (see Solon (1992), Zimmerman (1992) and Mulligan (1997)) and take averages. Compared to single year estimates, this has the additional advantage of correcting for transitory earnings uctuations, which can be quite high (Gottschalk and Mo tt, 1994). Most studies, such as Mulligan (1997), additionally correct for life-cycle in uences by controlling for age. In order to be consistent with the empirical literature, we measure average lifetime earnings as the average earnings of an individual over the course of his working life, conditional on his education level and his initial productivity. Because an individual s earnings process depends only on his education level, we need to distinguish at least two di erent levels of lifetime earnings. However, since we further assume that the earnings shock that a household receives in its rst period of working life is determined by the level of its productivity type average lifetime earnings also depend on the initial productivity type which is partly inherited from the parents. Consequently, we distinguish 8 di erent lifetime earnings groups (2 education groups, 4 initial productivity types). Following the empirical literature (see Mulligan 1997 for a summary), we measure the degree of intergenerational persistence as the estimated b 1 coe cient in the regression log (y c ) = log (y p ) + " (44) using the parent-children pairs properly weighted by the invariant distribution of productivity types and education groups 9. We nd that our model produces a standard deviation of lifetime earnings of Estimates of the standard deviation of log permanent earnings vary widely. Restuccia and Urrutia (2004) nd a standard deviation of the log of permanent earnings of 0.6, where permanent earnings refer to a 5 year earnings average. Using a 9 year average, Gottschalk and Mo t (1994) estimate a standard deviation between 0.4 and 0.49, depending the time period under consideration. Zimmerman (1992) measures a standard deviation of 0.4, using 4 year averages. 9 It is important to note that our de nition of lifetime earnings is also consistent with assumptions regarding the labor income process. While we follow the study of Hubbard, Skinner and Zeldes (1994,1995), assuming highly persistent earnings shocks and average age-earnings pro les that di er only with respect to education levels, a di erent strand of the literature, which was originated by Lillard and Weiss (1979), assumes that age-earnings pro les di er widely depending on initial ability, while earnings shocks are only of low persistence. See Guvenen (2006) for a comprehensive overview of the implications of both set of assumptions. Despite the fact that the earnings process only depends on the education level, our de nition of lifetime earnings is consistent with both speci cations, as initial ability determines the rst period s productivity shock and shocks are persistent. This implies that changing the assumptions regarding the earnings process would probably leave the degree of earnings persistent constant, but it would of course change the level of precautionary savings generated by our model. 17

19 Annual ows as a percentage of total wealth Gale and Scholz Benchmark economy Monetary inter-vivos transfers College expenses End-of-life bequests Table 3: Flows of Transfers Estimates for the standard deviation for the log of total (permanent and transitory) earnings vary widely as well. Mulligan (1997) nds a value of 0.8, while Gottschalk and Mo t s estimates suggest that the standard deviation of log earnings is Using our model, we nd a value of The fact that our benchmark economy generates a standard deviation of earnings which is relatively low compared to the literature may be due to the fact that our estimates for the earnings processes are rather conservative, as discussed above Results In this section, we analyze in detail the quantitative behaviour of our benchmark economy. 5.1 The relative size of intergenerational transfers. Table 3 shows the annual ow of college expenses, monetary inter-vivos transfers and end-of-life bequests as a share of total private wealth. The rst column refers to Table 4 in Gale and Scholz (1994), while the second column shows the ow of transfers generated by our benchmark model. The benchmark economy replicates the transfer ows (human and nonhuman) observable in the U.S. economy very well. In particular our results match the observed ow of end-of-life bequests remarkably well relative to previous studies using altruistic models in which parents incorporate their children s utility (see Davies and Shorrocks, p. 25). 5.2 Decomposing Earnings Persistence Our benchmark economy is calibrated such that it reproduces the degree of intergenerational earnings persistence of 0.4 observed in the U.S. economy. Restuccia and Urrutia (2004) use the di erence between the persistence before college education (persistence of productivity types in our model) and the persistence after college education (persistence of earnings) as a measure for the relative contribution of college education to the observable degree of earnings persistence. As Table 6 shows, persistence of productivity types in our baseline economy is 0.32, which suggests that around 20 percent of the total earnings persistence is 10 Notice that we also exclude high school dropouts from our analysis and assume that college dropouts have the same earnings process as high school graduates. 18

20 Ability/Wealth I II III I II III IV Table 4: Average enrolement rate for given ability and wealth tercile accounted for by college education. However, this measure does not incorporate the full impact of college education on earnings persistence as productivity types and college education are highly interdependent. Hence, we propose a di erent measure for the e ect of college education on earnings persistence. Instead of taking the di erence between the two di erent levels of persistence as a measure for the contribution of college education, we compute the elasticity of a decrease in tuition fees on the degree of the intergenerational persistence of earnings. We nd that the elasticity is approximately 0.5, which is quite high as it reveals a tight link between the college education and earnings persistence: reducing the college costs by 1 percent cuts the degree of persistence by about 0.5 percent. 5.3 Persistence of education attainment Our model generates an intergenerational correlation of education attainment of about In our model, education attainment is measured as the highest degree obtained, i.e. either high school or college. Mulligan (1997) nds an intergenerational persistence of schooling of about Mayer (2002) considers the correlation of college choices between father and sons and reports a correlation of Interestingly, the persistence of types and the persistence of education appear to be identical, suggesting a tight relationship between ability (in a broad sense) and education attainment 12. This result is in line with evidence found by Keane and Wolpin (2001) who conclude that borrowing constraints have only a small impact on school attendance decisions. In the next section, we take a closer look on the role borrowing constraints play in generating persistence. 5.4 Policy rules and ampli cation mechanisms Table 4 shows the average enrolment rate in college, given the wealth tercile and the initial productivity level. We observe that the enrolment rate increases 11 Notice that Mayer looks at whether father and sons attended some college whereas we compute the persistence of college graduation. Considering any college attendance is likely to increase the persistence of education in our model. 12 Note that the fact that the persistence of education attainment and productivity types coincides is simply by coincidence. As soon as we change parameters, the persistence parameters diverge. This can be seen from the third row of table (6), where we abstract from college subsidies. Note that the persistence of productivity types remains constant, whereas the persistence of education increases. 19