Testing Parental Altruism:Implications of a Dynamic Model Kathleen McGarry 1 Department of Economics University of California,Los Angeles and NBER original version,February 1998 this version,April 2006 1I am grateful to Michael Hurd,Janet Currie,Wei-Yin Hu,and Jean-Laurent Rosenthal for helpful comments.Financial support was received from the National Institute on Aging through grant number T32-AG00186 and from the Brookdale Foundation
Testing Parental Altruism: Implications of a Dynamic Model Kathleen McGarry 1 Department of Economics University of California, Los Angeles and NBER original version, February 1998 this version, April 2006 1I am grateful to Michael Hurd, Janet Currie, Wei-Yin Hu, and Jean-Laurent Rosenthal for helpful comments. Financial support was received from the National Institute on Aging through grant number T32-AG00186 and from the Brookdale Foundation
Abstract Each year parents transfer a great deal of money to their adult children.While intuition might suggest that these transfers are altruistic and made out of concern for the well-being of the children, the fundamental prediction of the altruistic model has been decisively rejected in empirical tests. Specifically,the required derivative restriction-that an increase of one dollar in the income of the recipient,accompanied by a decrease of one dollar in the income of the donor,leads to a one dollar reduction in transfers-fails to hold.I show in this paper that in fact,this prediction will not hold if parents use observations on the current incomes of children to update their expectations about future incomes.This result implies that many past studies have relied on too restrictive a test,and furthermore,that our ability to distinguish empirically between altruistic and exchange behavior is severely limited.The paper also analyzes the variation in transfer behavior over time and finds substantial change across periods in recipiency status as well as a strong correlation between inter vivos transfers and the transitory income of the recipient.This evidence suggests that dynamic models can provide insights into transfer behavior that are impossible to obtain in a static context
Abstract Each year parents transfer a great deal of money to their adult children. While intuition might suggest that these transfers are altruistic and made out of concern for the well-being of the children, the fundamental prediction of the altruistic model has been decisively rejected in empirical tests. Specifically, the required derivative restriction–that an increase of one dollar in the income of the recipient, accompanied by a decrease of one dollar in the income of the donor, leads to a one dollar reduction in transfers–fails to hold. I show in this paper that in fact, this prediction will not hold if parents use observations on the current incomes of children to update their expectations about future incomes. This result implies that many past studies have relied on too restrictive a test, and furthermore, that our ability to distinguish empirically between altruistic and exchange behavior is severely limited. The paper also analyzes the variation in transfer behavior over time and finds substantial change across periods in recipiency status as well as a strong correlation between inter vivos transfers and the transitory income of the recipient. This evidence suggests that dynamic models can provide insights into transfer behavior that are impossible to obtain in a static context
1 Introduction Intergenerational transfers between family members are an important economic phenomenon,par- ticularly those transfers from parents to children.Gale and Scholz (1994)estimate yearly flows between parents and their non-coresident children of $50 billion in 1999 dollars.Because of the magnitude of the funds involved such transfers are likely to have a substantial impact on other economic behaviors.Certainly they will affect the well-being of both donors and recipients and will have consequences for the distribution of wealth.In addition,however,familial transfers may interact with public transfers,and in doing so could alter the effectiveness of government assistance programs. The importance of the effects in any of these dimensions depends crucially on the motivation behind the private transfers.While the relationship between motive and impact has been long recognized,only recently have data of sufficient quality to test alternative hypotheses become available.Unfortunately,despite several efforts,a consensus has not yet been reached on the most appropriate model of behavior as none of the hypothesized models appears to be consistent with observed patterns giving. The two most prominant models of familial behavior,altruism and exchange,make very dif- ferent predictions about the redistributional aspects of private transfers,and about the interaction between public and private assistance,so distinguishing between the two is of both theoretical and practical interest.In the literature,attention has centered on testing the predictions of the altruism model.This model,wherein parents care about the welfare of their children,makes strong predic- tions about the signs and relative magnitudes of the effects of the donors'and recipients'incomes on transfer amounts,so testing is apparently straightforward.However,while the standard altruism model is written in a static context with the effects of interest being changes in lifetime transfers in response to changes in permanent incomes,the actual data used to test the model come from a dynamic world:Researchers observe current rather than permanent income,and single period rather than lifetime transfers.As I show in this paper this difference in measurement is likely to lead to incorrect conclusions in tests of the model's validity. Recent work testing the predictions of the altruism model in a dynamic context (Altonji, Hayashi,and Kotlikoff,1997)has concluded that the model is inconsistent with observed data
1 Introduction Intergenerational transfers between family members are an important economic phenomenon, particularly those transfers from parents to children. Gale and Scholz (1994) estimate yearly flows between parents and their non-coresident children of $50 billion in 1999 dollars. Because of the magnitude of the funds involved such transfers are likely to have a substantial impact on other economic behaviors. Certainly they will affect the well-being of both donors and recipients and will have consequences for the distribution of wealth. In addition, however, familial transfers may interact with public transfers, and in doing so could alter the effectiveness of government assistance programs. The importance of the effects in any of these dimensions depends crucially on the motivation behind the private transfers. While the relationship between motive and impact has been long recognized, only recently have data of sufficient quality to test alternative hypotheses become available. Unfortunately, despite several efforts, a consensus has not yet been reached on the most appropriate model of behavior as none of the hypothesized models appears to be consistent with observed patterns giving. The two most prominant models of familial behavior, altruism and exchange, make very different predictions about the redistributional aspects of private transfers, and about the interaction between public and private assistance, so distinguishing between the two is of both theoretical and practical interest. In the literature, attention has centered on testing the predictions of the altruism model. This model, wherein parents care about the welfare of their children, makes strong predictions about the signs and relative magnitudes of the effects of the donors’ and recipients’ incomes on transfer amounts, so testing is apparently straightforward. However, while the standard altruism model is written in a static context with the effects of interest being changes in lifetime transfers in response to changes in permanent incomes, the actual data used to test the model come from a dynamic world: Researchers observe current rather than permanent income, and single period rather than lifetime transfers. As I show in this paper this difference in measurement is likely to lead to incorrect conclusions in tests of the model’s validity. Recent work testing the predictions of the altruism model in a dynamic context (Altonji, Hayashi, and Kotlikoff, 1997) has concluded that the model is inconsistent with observed data. 1
Specifically,a fundamental prediction of the altruism model-that conditional on transfers being made,an increase of one dollar in the income of the recipient,accompanied by a decrease of one dollar in the income of the donor,will lead to a one dollar reduction in transfers-was decisively rejected.(I will term this relationship the "derivative restriction."1)However,under reasonable assumptions about the formation of expectations of future income,the derivative restriction will not hold.Thus,caution should be used when drawing conclusions from studies that have used the restriction as a test of altruism.Furthermore,because this derivative restriction has been the sole means of differentiating between altruism and exchange regimes,my result suggests that our ability to distinguish between the two explanations of behavior may be severely limited.2 The model developed here assumes that the future income of a child is not known,rather the parent knows only its distribution.Observations on the child's current income in each period provide new information that is used to update the distribution of future income.Changes in this distribution affect expected future transfers,and in a dynamic model feed back to affect current transfers.Through this mechanism,the responsiveness of current transfers to changes in current income is reduced,causing the derivative restriction to fail. Intuitively,consider an example wherein the arrival of a new observation on the child's income causes the parent to reduce her estimate of the child's future income and consumption.Not only will the parent wish to increase transfers in the current period,but she will also expect to increase future transfers as well.In this case the derivative of interest-the change in current transfers for a change in current income-differs from its value in the absence of updating,and the derivative restriction predicted by the static altruism model no longer holds. Using data from the Health and Retirement Survey (HRS),the paper provides an empirical test of this dynamic framework.Specifically,I examine the relationship between current transfers and both current and lagged income of the potential recipient and find that both variables have significant explanatory power.I interpret these results as evidence that a parent's assessment of her child's future income depends on past realizations of income.This result is consistent with Altonji et al.use the term "transfer income derivatives restriction." 2An alternative test of the two models relies on the sign of the effect of the recipient's income on the amount of the transfer.An altruism model predicts a negative relationship while either a positive or negative relationship is consistent with exchange.An estimated positive effect can therefore be used to discount the altruism model. However,recent empirical work has consistently found a negative relationship,and thus this test has provided no distinguishing power. 2
Specifically, a fundamental prediction of the altruism model—that conditional on transfers being made, an increase of one dollar in the income of the recipient, accompanied by a decrease of one dollar in the income of the donor, will lead to a one dollar reduction in transfers—was decisively rejected. (I will term this relationship the “derivative restriction.”1) However, under reasonable assumptions about the formation of expectations of future income, the derivative restriction will not hold. Thus, caution should be used when drawing conclusions from studies that have used the restriction as a test of altruism. Furthermore, because this derivative restriction has been the sole means of differentiating between altruism and exchange regimes, my result suggests that our ability to distinguish between the two explanations of behavior may be severely limited.2 The model developed here assumes that the future income of a child is not known, rather the parent knows only its distribution. Observations on the child’s current income in each period provide new information that is used to update the distribution of future income. Changes in this distribution affect expected future transfers, and in a dynamic model feed back to affect current transfers. Through this mechanism, the responsiveness of current transfers to changes in current income is reduced, causing the derivative restriction to fail. Intuitively, consider an example wherein the arrival of a new observation on the child’s income causes the parent to reduce her estimate of the child’s future income and consumption. Not only will the parent wish to increase transfers in the current period, but she will also expect to increase future transfers as well. In this case the derivative of interest–the change in current transfers for a change in current income–differs from its value in the absence of updating, and the derivative restriction predicted by the static altruism model no longer holds. Using data from the Health and Retirement Survey (HRS), the paper provides an empirical test of this dynamic framework. Specifically, I examine the relationship between current transfers and both current and lagged income of the potential recipient and find that both variables have significant explanatory power. I interpret these results as evidence that a parent’s assessment of her child’s future income depends on past realizations of income. This result is consistent with 1Altonji et al. use the term “transfer income derivatives restriction.” 2An alternative test of the two models relies on the sign of the effect of the recipient’s income on the amount of the transfer. An altruism model predicts a negative relationship while either a positive or negative relationship is consistent with exchange. An estimated positive effect can therefore be used to discount the altruism model. However, recent empirical work has consistently found a negative relationship, and thus this test has provided no distinguishing power. 2
the updating model introduced below.As a specification test I test the effect of future income on current transfers and find that it does not have significant explanatory power. In exploring the implications of this dynamic framework,I also examine variability of transfers over time.In the two years for which I have data I find a significant number of changes in recipiency. In each year approximately 13 percent of children are reported to be receiving transfers,yet only 5.5 percent of the sample receives transfers in both survey years.These dynamic aspects of behavior have frequently been ignored because of data limitations,3 yet from the empirical work presented here they appear to be an important phenomenon. The availability of repeated observations on each child allows me to examine transfer behavior net of permanent differences across individuals.If there exist unobserved characteristics of the parent or child that are correlated with transfer behavior and with the included measures of income, then estimated derivatives from equations that ignore these effects will be biased.In a fixed effects framework I find that the effect of current income on transfer behavior is large and significantly different from zero,but is approximately 60 percent smaller than its effect in specifications that do not control for unobservable differences across children.These results indicate a strong negative correlation between transfers and the transitory income of potential recipients and also demonstrate the necessity of adequate controls for permanent income in dynamic models of transfer behavior. The remainder of the paper is organized as follows:In section 2 I outline the standard altruism model and discuss the tests used to distinguish this model from an exchange regime.I then expand the static model to include two periods and note the conditions under which the predicted results differ.Section 3 describes the data I use in the empirical work and section 4 discusses the estimated effects of current income on transfers.A final section concludes and summarizes the results. 2 Background and Theory I divide the theoretical discussion into two subsections.The first presents the standard altruism model and the second extends this model to two periods. 3Dunn (1997),and Rosenzweig and Wolpin (1994)are exceptions.These papers both use multiple waves of the NLS surveys.However,information is not available on all siblings of the (potential)recipients,so a complete understanding of the allocation within families is not possible. 3
the updating model introduced below. As a specification test I test the effect of future income on current transfers and find that it does not have significant explanatory power. In exploring the implications of this dynamic framework, I also examine variability of transfers over time. In the two years for which I have data I find a significant number of changes in recipiency. In each year approximately 13 percent of children are reported to be receiving transfers, yet only 5.5 percent of the sample receives transfers in both survey years. These dynamic aspects of behavior have frequently been ignored because of data limitations,3 yet from the empirical work presented here they appear to be an important phenomenon. The availability of repeated observations on each child allows me to examine transfer behavior net of permanent differences across individuals. If there exist unobserved characteristics of the parent or child that are correlated with transfer behavior and with the included measures of income, then estimated derivatives from equations that ignore these effects will be biased. In a fixed effects framework I find that the effect of current income on transfer behavior is large and significantly different from zero, but is approximately 60 percent smaller than its effect in specifications that do not control for unobservable differences across children. These results indicate a strong negative correlation between transfers and the transitory income of potential recipients and also demonstrate the necessity of adequate controls for permanent income in dynamic models of transfer behavior. The remainder of the paper is organized as follows: In section 2 I outline the standard altruism model and discuss the tests used to distinguish this model from an exchange regime. I then expand the static model to include two periods and note the conditions under which the predicted results differ. Section 3 describes the data I use in the empirical work and section 4 discusses the estimated effects of current income on transfers. A final section concludes and summarizes the results. 2 Background and Theory I divide the theoretical discussion into two subsections. The first presents the standard altruism model and the second extends this model to two periods. 3Dunn (1997), and Rosenzweig and Wolpin (1994) are exceptions. These papers both use multiple waves of the NLS surveys. However, information is not available on all siblings of the (potential) recipients, so a complete understanding of the allocation within families is not possible. 3
2.1 The static altruism model In the standard altruism model parents care about the well-being of their children;they receive utility from their own consumption and from the utility of their children.Following the specification used in Cox(1987),the utility function of a parent is written as Up=Up(cp,V(ck))where cp and ck are the consumption of the parent and child,respectively.The consumption of the child is determined by his own income y and transfers from the parent T.Thus,ck=yk+T.Because this is a one-period model there is no saving. The comparative statics of the altruism model yield two testable predictions.First,the change in transfers for a change in a child's income is negative();as the child's income increases, the marginal utility of an additional dollar of consumption decreases,and the parent transfers less. This result implies that in families with more than one child,parents will make greater transfers to lower income children,in effect compensating the lower income children for their lack of resources. The second testable implication of the altruism model is that if transfers are positive,an increase of one dollar in the child's income along with a decrease of one dollar in the parent's income,will result in a decrease of one dollar in transfers to the child.That is,1where is the income of the parent.To see why the relationship holds,consider the following intuitive example. In a one-period model a parent has an income of $200 and her child has an income of $50.Given this initial allocation the parent chooses to transfer $50 to her child so that the consumption of the parent is $150 and the consumption of the child is $100.Now suppose the parent's income were $199 and child's income,$51.If the parent continues to transfer $50 to her child,the distribution of resources would be ($149,$101).If this allocation were preferred to a($150,$100)division then the parent would have initially chosen to transfer $51.Thus,by a revealed preference argument the parent prefers ($150,$100)to ($149,$101)and will reduce her transfer by one dollar in response to the change in incomes. Given these straightforward predictions,empirical tests of the model have centered on the estimates of andEarly work by Cox (1987)and Cox and Rank (192)found a positive relationship between a child's income and the amount of a transfer,a contradiction of the negative relationship predicted by the altruism model.However,these early studies were not able to control adequately for the income of the parent.Because well-off children tend to have well-off parents, 4
2.1 The static altruism model In the standard altruism model parents care about the well-being of their children; they receive utility from their own consumption and from the utility of their children. Following the specification used in Cox (1987), the utility function of a parent is written as Up = Up(cp, V (ck)) where cp and ck are the consumption of the parent and child, respectively. The consumption of the child is determined by his own income yk and transfers from the parent T. Thus, ck = yk +T. Because this is a one-period model there is no saving. The comparative statics of the altruism model yield two testable predictions. First, the change in transfers for a change in a child’s income is negative ( ∂T ∂yk < 0); as the child’s income increases, the marginal utility of an additional dollar of consumption decreases, and the parent transfers less. This result implies that in families with more than one child, parents will make greater transfers to lower income children, in effect compensating the lower income children for their lack of resources. The second testable implication of the altruism model is that if transfers are positive, an increase of one dollar in the child’s income along with a decrease of one dollar in the parent’s income, will result in a decrease of one dollar in transfers to the child. That is, ∂T ∂yk − ∂T ∂wp = −1 where wp is the income of the parent. To see why the relationship holds, consider the following intuitive example. In a one-period model a parent has an income of $200 and her child has an income of $50. Given this initial allocation the parent chooses to transfer $50 to her child so that the consumption of the parent is $150 and the consumption of the child is $100. Now suppose the parent’s income were $199 and child’s income, $51. If the parent continues to transfer $50 to her child, the distribution of resources would be ($149, $101). If this allocation were preferred to a ($150, $100) division then the parent would have initially chosen to transfer $51. Thus, by a revealed preference argument the parent prefers ($150, $100) to ($149, $101) and will reduce her transfer by one dollar in response to the change in incomes. Given these straightforward predictions, empirical tests of the model have centered on the estimates of ∂T ∂yk and ∂T ∂wp . Early work by Cox (1987) and Cox and Rank (1992) found a positive relationship between a child’s income and the amount of a transfer, a contradiction of the negative relationship predicted by the altruism model. However, these early studies were not able to control adequately for the income of the parent. Because well-off children tend to have well-off parents, 4
and well-off parents make greater transfers,the estimates obtained forwere positively biased. More recent efforts that better control for the incomes of both the parent and child have found a strong negative relationship between the child's income and the amount of the transfer (Cox and Jappelli 1990,Dunn 1997,McGarry and Schoeni,1995,1997),a result consistent with the altruism model,but also with alternative models.Although the sign offound by these studies is conistent with the altruism model,the magnitudes ofand(where estimated)fail to satisfy the derivative restriction,with estimatesofthat are closer tothan to-1. 2.2 Static versus dynamic outcomes The model outlined above is placed in a static framework.In the context of a single period model, parents know the lifetime earnings of their children,and the lifetime consumption of children is calculated directly as the sum of earnings and transfers.Parents make greater transfers to children with lower lifetime incomes and the timing of earnings and transfers is not an issue.However, in a more representative multiperiod framework,with an uncertain future,the timing of transfers becomes an important matter. As noted by Altonji et al.(1997),absent additional constraints,if the child's permanent income is uncertain,a parent will delay transfers in order to obtain additional information and more efficiently allocate resources.Furthermore,parents who are uncertain of their own date of death or future needs will be reluctant to part with resources they themselves might need some day and prefer to postpone transfers (Davies,1981).Acting against the postponement of transfers is the fact that children are unlikely to be able to borrow against future transfers and therefore may not be able to smooth consumption optimally across time.Even children with high lifetime incomes may be the recipients of transfers if they are temporarily liquidity constrained and unable to attain the level of consumption predicted by their permanent incomes(Cox,1990).Thus one would expect a negative relationship between transfers and current income,and a positive relationship between transfers and indicators of liquidity constraints.However,whereas the derivative restriction holds 4The most frequently cited alternative to the altruism model is an exchange model wherein observed transfers represent payment for services provided by the child.In the exchange model parents care about their own utility and the services(s)provided by the child.Formally,Up=U(cp,s).In contrast to the predictions of an altruism model,in an exchange regime the sign of the relationship between a child's income and the magnitude of a transfer is indeterminate.As a child's income increases,the price of his time increases and the quantity of time purchased therefore decreases.The net amount spent by the parent to purchase services(pricexquantity)can either increase or decrease depending on the elasticities of supply and demand for services. 5
and well-off parents make greater transfers, the estimates obtained for ∂T ∂yk were positively biased. More recent efforts that better control for the incomes of both the parent and child have found a strong negative relationship between the child’s income and the amount of the transfer (Cox and Jappelli 1990, Dunn 1997, McGarry and Schoeni, 1995, 1997), a result consistent with the altruism model, but also with alternative models.4 Although the sign of ∂T ∂yk found by these studies is consistent with the altruism model, the magnitudes of ∂T ∂yk and ∂T ∂wp (where estimated) fail to satisfy the derivative restriction, with estimates of ∂T ∂yk − ∂T ∂wp that are closer to 0 than to -1. 2.2 Static versus dynamic outcomes The model outlined above is placed in a static framework. In the context of a single period model, parents know the lifetime earnings of their children, and the lifetime consumption of children is calculated directly as the sum of earnings and transfers. Parents make greater transfers to children with lower lifetime incomes and the timing of earnings and transfers is not an issue. However, in a more representative multiperiod framework, with an uncertain future, the timing of transfers becomes an important matter. As noted by Altonji et al. (1997), absent additional constraints, if the child’s permanent income is uncertain, a parent will delay transfers in order to obtain additional information and more efficiently allocate resources. Furthermore, parents who are uncertain of their own date of death or future needs will be reluctant to part with resources they themselves might need some day and prefer to postpone transfers (Davies, 1981). Acting against the postponement of transfers is the fact that children are unlikely to be able to borrow against future transfers and therefore may not be able to smooth consumption optimally across time. Even children with high lifetime incomes may be the recipients of transfers if they are temporarily liquidity constrained and unable to attain the level of consumption predicted by their permanent incomes (Cox, 1990). Thus one would expect a negative relationship between transfers and current income, and a positive relationship between transfers and indicators of liquidity constraints. However, whereas the derivative restriction holds 4The most frequently cited alternative to the altruism model is an exchange model wherein observed transfers represent payment for services provided by the child. In the exchange model parents care about their own utility and the services (s) provided by the child. Formally, Up = U(cp, s). In contrast to the predictions of an altruism model, in an exchange regime the sign of the relationship between a child’s income and the magnitude of a transfer is indeterminate. As a child’s income increases, the price of his time increases and the quantity of time purchased therefore decreases. The net amount spent by the parent to purchase services (price×quantity) can either increase or decrease depending on the elasticities of supply and demand for services. 5
with respect to changes in permanent income in a static model,it is not clear that the same relationship must exist with respect to current income in this dynamic framework,even if children are liquidity constrained. To illustrate the problem formally consider a simplified version of the two period model from Altonji et al.(1997).5 Parents receive utility from their own consumption in each period cp and cp2 and from the utility of their children,V(ck),and V(ck2)where cke denotes the child's consumption level in period t.Ignoring interest rates and the time rate of discount,let the utility function of the parent be Up =U(cp)+nv(ck)+U(Cp2)+nV(ck2) where U and V are concave functions and the child's utility is weighted by n.Following the previous literature,I assume that the parent has income wp in period 1 and no second period income.She saves A1 in period 1 to finance period 2 consumption and transfers.The child has income ye in each period t,where t =1,2.Here I focus on the case in which children are liquidity constrained in period 1 and cannot borrow across periods:5 their consumption in each period is therefore the sum of their income,ye,and received transfers,Ti.The budget constraints can therefore be written as CpI Wp-T1-A Ck1=1十T1 Cp2 A1-T2 C2=2十T2. In the first period the parent does not know her child's period 2 income,but does know its distri- bution,conditional on information I available in period 1,f(I).The parent will maximize her utility using the expected value of second-period utility, Up=U(cp)+nv(ck)+[U(cpa)+nv(ckz)]f(uk2I)dykz. In the first period the parent observes y and wp and chooses Ti and cp.In period 2 the parent then observes y2 and divides remaining resources A1 between T2 and cp2. 5This discussion,and the notation used here draws directly on Altonji et al.(1997). 6As demonstrated by Altonji et al.(1997)if a child is not liquidity constrained the parent has no incentive to make transfers in the first period.The more interesting case is therefore the one in which the child does face these liquidity. 6
with respect to changes in permanent income in a static model, it is not clear that the same relationship must exist with respect to current income in this dynamic framework, even if children are liquidity constrained. To illustrate the problem formally consider a simplified version of the two period model from Altonji et al. (1997).5 Parents receive utility from their own consumption in each period cp1 and cp2 and from the utility of their children, V (ck1), and V (ck2) where ckt denotes the child’s consumption level in period t. Ignoring interest rates and the time rate of discount, let the utility function of the parent be Up = U(cp1) + η V (ck1) + U(cp2) + η V (ck2) where U and V are concave functions and the child’s utility is weighted by η. Following the previous literature, I assume that the parent has income wp in period 1 and no second period income. She saves A1 in period 1 to finance period 2 consumption and transfers. The child has income ykt in each period t, where t = 1, 2. Here I focus on the case in which children are liquidity constrained in period 1 and cannot borrow across periods:6 their consumption in each period is therefore the sum of their income, ykt and received transfers, Tt. The budget constraints can therefore be written as cp1 = wp − T1 − A1 ck1 = yk1 + T1 cp2 = A1 − T2 ck2 = yk2 + T2. In the first period the parent does not know her child’s period 2 income, but does know its distribution, conditional on information I available in period 1, f(yk2|I). The parent will maximize her utility using the expected value of second-period utility, Up = U(cp1) + η V (ck1) + [ U(cp2) + η V (ck2) ] f(yk2|I) dyk2. In the first period the parent observes yk1 and wp and chooses T1 and cp1. In period 2 the parent then observes yk2 and divides remaining resources A1 between T2 and cp2. 5This discussion, and the notation used here draws directly on Altonji et al. (1997). 6As demonstrated by Altonji et al. (1997) if a child is not liquidity constrained the parent has no incentive to make transfers in the first period. The more interesting case is therefore the one in which the child does face these liquidity. 6
The solution to this dynamic programming model can be obtained by first solving for the optimal allocation in period 2 as a function of A1 and y2.That is,the parent maximizes the function U2(A1-T2)+7V2(y2+T2) with respect to T2,yielding an optimal value for T2(and thus cp2)as a function of A1 and yk2, T2=T2(A1,k2) 2=A-T2. Using this result,the first period maximization problem is then to choose A1 and Ti to maximize U(ep)+nV(ck)+[U(Ai-T2(A1:a))+nV(k2+T2(A1:v))]f(alI)dukz subject to Cp Wp-T1-A1 C%1=1+T1. One should note that in the above maximization problem the variables wp,yk,and T1,ap- parently always enter in pairs as either wp-Ti or y+Ti.Altonji et al.note this result and concde from it that the derivative restriction1contimues to hold provided that I does not change.7 However,in a plausible multiperiod framework one could expect f()to be a function of y,rather than y+T1.If the distribution of second period income does depend on the observation of first period income then the derivative restriction is broken.In their theo- retical discussion,Altonji et al.note that "the determinants of expected future income"need be held constant when considering the partial derivative of transfers with respect to current income. Their empirical analysis attempts to hold constant these determinants by including measures of permanent income carefully constructed from repeated observations on income and other covari- ates.However,it is likely that changes in current income also change expected future income (and 7To understand how this conclusion is reached,consider the first order conditions of the above utility maximization problem.These equations will be functions of wp-Ti and y+Ti.Writing one such equation as H(wp-Ti,y+T), and differentiating first with respect to wp and then with respect to y yields a system of two equations such that (and+H0,where,is the derivative of the function H with respect to the ithargument.These two equations can be combined and the terms rearranged to yield the result that 恶兴=1 7
The solution to this dynamic programming model can be obtained by first solving for the optimal allocation in period 2 as a function of A1 and yk2 . That is, the parent maximizes the function U2(A1 − T2) + ηV2(yk2 + T2) with respect to T2, yielding an optimal value for T2 (and thus cp2) as a function of A1 and yk2 , T∗ 2 = T2(A1, yk2) c∗ p2 = A1 − T2. Using this result, the first period maximization problem is then to choose A1 and T1 to maximize U(cp1) + η V (ck1) + yk2 [ U(A1 − T2(A1, yk2)) + η V (yk2 + T2(A1, yk2)) ] f(yk2|I) dyk2 subject to cp1 = wp − T1 − A1 ck1 = yk1 + T1. One should note that in the above maximization problem the variables wp, yk1 , and T1, apparently always enter in pairs as either wp − T1 or yk1 + T1. Altonji et al. note this result and conclude from it that the derivative restriction ∂T ∂yk1 − ∂T ∂wp = −1 continues to hold provided that I does not change.7 However, in a plausible multiperiod framework one could expect f(yk2 |I) to be a function of yk1 , rather than yk1 + T1. If the distribution of second period income does depend on the observation of first period income then the derivative restriction is broken. In their theoretical discussion, Altonji et al. note that “the determinants of expected future income” need be held constant when considering the partial derivative of transfers with respect to current income. Their empirical analysis attempts to hold constant these determinants by including measures of permanent income carefully constructed from repeated observations on income and other covariates. However, it is likely that changes in current income also change expected future income (and 7To understand how this conclusion is reached, consider the first order conditions of the above utility maximization problem. These equations will be functions of wp−T1 and yk1 +T1. Writing one such equation as H(wp−T1, yk1 +T1), and differentiating first with respect to wp and then with respect to yk1 yields a system of two equations such that H1(1 − ∂T ∂wp ) + H2 ∂T ∂wp = 0 and H1(− ∂T ∂yk1 ) + H2(1 + ∂T ∂yk1 )=0, where Hi is the derivative of the function H with respect to the i th argument. These two equations can be combined and the terms rearranged to yield the result that ∂T ∂yk1 − ∂T ∂wp = −1. 7
thus expectations of permanent income).If this were the case,then an analysis of the effect of changes in current income on transfer behavior needs first to understand the relationship between changes in current and future incomes,and then to take into account this additional effect.The accuracy of our estimated effects and the correctness of the conclusions we draw from them will depend on our ability to do this. To understand this mechanism in the context of the above model,consider the specific case where yk and ykz have a bivariate normal distribution with means u and u2,variances of and 2,and a correlation coefficient p.The conditional distribution of y given y has an expected value equal topand variance (1-).Thus,if p is positive,a low value of 02 (y<)will reduce the child's period 1 consumption and increase the marginal utility of a transfer T1.At the same time,a low value of will shift the distribution ofy to the left.This shift will decrease expected second period consumption of the child,increase the marginal utility of a transfer in that period,and thus increase the marginal utility of A1.To equalize marginal utilities across arguments of the utility function,the parent will reduce cp and increase both Ti and A1.Because of the change in A1,the increase in Ti will be less than if the distribution of y were unaffected. The proof of this result for the general case is in the appendix.There I show that under reasonable assumptions about the relationship between income in the two periods,the value of lies betweener and negative,conistent with the resut of previous empirical studies.In the specific case of the bivariate normal distribution I show that the distance between and-1 depends directly on the magnitude of the correlation cfficient.Ifp,s that period 1 income is uninformative about period 2 income,the derivative restriction holds. In the empirical work that follows I use repeated observations on the income of a child to test the validity of this dynamic framework. 3 Data The data used in this study are from the Health and Retirement Survey(HRS).The HRS is a panel survey of the U.S.population born between 1931 and 1941 and their spouses.When appropriately SThe assumption of normality is made solely for expositional simplicityi Because the joint distribution of two normal random variables is so familiar to readers the conclusions should be clear.The appendix derives the result for the general case. 8
thus expectations of permanent income). If this were the case, then an analysis of the effect of changes in current income on transfer behavior needs first to understand the relationship between changes in current and future incomes, and then to take into account this additional effect. The accuracy of our estimated effects and the correctness of the conclusions we draw from them will depend on our ability to do this. To understand this mechanism in the context of the above model, consider the specific case where yk1 and yk2 have a bivariate normal distribution with means μ1 and μ2, variances σ2 1 and σ2 2, and a correlation coefficient ρ.8 The conditional distribution of yk2 given yk1 has an expected value equal to μ2 + ρσ1 (yk1 −μ1) σ2 and variance (1 − ρ2)σ2 2. Thus, if ρ is positive, a low value of yk1 ( yk1 < μ1) will reduce the child’s period 1 consumption and increase the marginal utility of a transfer T1. At the same time, a low value of yk1 will shift the distribution of yk2 to the left. This shift will decrease expected second period consumption of the child, increase the marginal utility of a transfer in that period, and thus increase the marginal utility of A1. To equalize marginal utilities across arguments of the utility function, the parent will reduce cp1 and increase both T1 and A1. Because of the change in A1, the increase in T1 will be less than if the distribution of yk2 were unaffected. The proof of this result for the general case is in the appendix. There I show that under reasonable assumptions about the relationship between income in the two periods, the value of ∂T ∂yk1 − ∂T ∂wp lies between zero and negative one, consistent with the results of previous empirical studies. In the specific case of the bivariate normal distribution I show that the distance between ∂T ∂yk1 − ∂T ∂wp and -1 depends directly on the magnitude of the correlation coefficient. If ρ = 0, so that period 1 income is uninformative about period 2 income, the derivative restriction holds. In the empirical work that follows I use repeated observations on the income of a child to test the validity of this dynamic framework. 3 Data The data used in this study are from the Health and Retirement Survey (HRS). The HRS is a panel survey of the U.S. population born between 1931 and 1941 and their spouses. When appropriately 8The assumption of normality is made solely for expositional simplicity¿ Because the joint distribution of two normal random variables is so familiar to readers the conclusions should be clear. The appendix derives the result for the general case. 8
