- Email: [email protected]
Incentives, insurance, and the variability of consumption and leisure
Journal
of Economic
Dynamics
and Control
18 (1994) 581-599.
North-Holland
Incentives, insurance, and the variability of consumption and leisure* Christopher
Phelan
University of Wisconsin, Madison, WI 53706, USA Received
May 1992, final version
received
March
1993
This paper considers to what extent dynamic incentive models such as Green (1987), Phelan and Townsend (1991), and Atkeson and Lucas (1992) can quantitatively explain why some individuals consume (or work) more than others and why a typical individual’s consumption (or leisure) varies over time. A simple repeated agency model is shown to be able to better match several population moments concerning life-cycle labor supply and consumption variation than the same model without incentive constraints. On the other hand, I show for the standard stylized models of repeated agency, incentive-induced variability is an insufficient explanation of the variability in the data. Key words: Incentives; Consumption JEL classification: D31; D82; D91
variability
1. Introduction Models where individuals can take private ductivity or receive privately observed taste
Correspondence to: Christopher Phelan, Department Observatory Drive, Madison, WI 53706, USA.
actions which affect their proor wealth shocks give rise to
of Economics,
University
of Wisconsin,
1180
*This work is drawn from my University of Chicago Ph.D. dissertation of the same title. I would like to thank Andrew Atkeson, William Carrington, John Heaton, Robert Lucas, John Matsusaka, Karl Snow, the members of the Money and Banking Workshop at the University of Chicago, and seminar participants at Cornell University, Princeton University, the University of Pennsylvania, Northwestern University, the University of Wisconsin, the University of Illinois, the University of Rochester, Boston University, the University of Virginia, the 1990 meetings of the Society for Economic Dynamics and Control, and the associate editor for helpful comments. My committee members John Cochrane and Lars Hansen were especially helpful. Finally I want to thank my committee chairman Robert Townsend for his comments and constant support throughout my graduate education. Computer support from the National Center for Supercomputing Applications, Champaign, Illinois, and financial support from the Earhart Foundation, Bradley Foundation, National Science Foundation grant SES-911 l-926, and my wife, Elizabeth M. Bloomer, is appreciated.
0165-1889/94/$07.00
0
1994-Elsevier
Science B.V. All rights reserved
582
C. Phelan. Incentives, insurance, and consumption and leisure
inequality. In such incentive models, full insurance against productivity, wealth, or taste shocks is impossible since every individual will minimize effort if his consumption is the same regardless of his productivity, and every individual will claim hardship or urgency to consume if by doing so he can raise his consumption while bearing no other costs. This lack of full insurance allows for an incentive-based theory of distribution. The dynamic, general equilibrium, incentive models of Green (1987), Phelan and Townsend (199 l), and Atkeson and Lucas (1992) each generate a theory of why some individuals consume (or work) more than others and why a typical individual’s consumption (or leisure) varies over time. In each paper the incentive problem can be the sole reason why any individual consumes more than any other individual, and the sole reason why any individual’s consumption varies over time’. The purpose of this paper is to see to what extent such incentive-induced variability can account for the variability we actually see. To make specific quantitative predictions, this paper focuses on the dynamic, general equilibrium, agency model in Phelan and Townsend (1991). This is essentially the standard principal-agent model. In each period, each agent takes an unobserved action (an effort level) which affects the probability of various output levels. In incentive-compatible plans, an agent’s consumption must depend on his output to get the agent to work, which leaves the agent less than fully insured against idiosyncratic shocks to his output. This incomplete insurance causes consumption and leisure to vary. This type of incentive-induced variability takes a specific dynamic form that differs from the kind of variability produced by more traditional means such as heterogeneity of initial endowments (of either wealth or human capital) or transient shocks such as preference shocks and measurement error. Thus, by looking at variability in the data we should be able to see to what extent actual variability is consistent with the incentive story. I take a simple approach here. First, I derive moments summarizing both cross-sectional and intertemporal consumption and leisure variability from data from the Consumer Expenditure Survey. Second, I parameterize and numerically solve the incentive model attempting to match these moments. The parameterization used allows three methods of creating variability. The first
‘While the ability to formalize these dynamic incentive problems is recent, the notion that incentive problems may be crucial in understanding why some people get to consume more than others is a common notion, In fact, in their survey of attitudes toward free markets, Shiller, Boycko. and Korobov (1991) include the question: Some have expressed thefollowing: ‘It’s too bad that some people are poor while others are rich. But we can’t jx that: if the government were to make sure that eoeryone had the same income, we would all be poor, since no one would have any material incentive to work hard.’ Have you heard such a theory or not? If yes, then how often? 11% of Moscow residents and 45% of New York residents sampled had heard of the theory, although 59% of Moscow residents and 62% of New York residents disagreed with it.
C. Phelan, Inceritives. insurance, and consumption
and leisure
583
method is to create endogenous incentive-induced variability. The second method is to simply assume exogenous initial inequality in expected lifetime utility levels. This stands in for any permanent differences across agents which would cause permanent consumption or leisure differences, including differences in inmate human capital. The third method is to assume exogenous transient (or i.i.d.) consumption variability. While this is consistent with preference shocks, the paper refers to such variability as measurement error. The premise of the paper is then as follows: The extent to which one must rely on the exogenous methods of generating variability to match the moments in the data is the extent to which a particular incentive model is incapable of accounting for the variability we see. The principal quantitative result is that for the simple repeated-agency model considered here, in order to get close to the moments in the data, most variability must be attributed to the exogenous factors. The dynamics of the endogenous incentive-produced variability does not square well with the variability in the data. A secondary (and not as robust) quantitative result is that the dynamic-agency model constitutes an improvement on the full-information framework on several dimensions. It allows for endogenous early retirement and predicts that cross-sectional consumption variability should move with age and leisure in directions consistent with the data. It is important to note that this project makes no attempt to explain the variation in the data, except in regard to the narrow question of whether the variability in the data could have been generated by the standard prototype dynamic agency model. When it is shown that the incentive model needs large amounts of exogenous variability to match the moments in the data, I do not dwell on identifying the correct sources of this exogenous variation other than discussing possibilities in the conclusion. While ultimate purpose of any research line is to explain the data, a valid intermediate goal is to understand where our standard theoretical models quantitatively fail and succeed. This is in line with the recent works of Mace (1991), Cochrane (1991), and Townsend (1989) which test the full-consumption-insurance implications of full-information/complete-market frameworks. It is hoped that such an exercise will help determine how simple models should be enriched in order to match data.’ The structure of the paper is as follows. Section 2 presents the unobserved-effort (or agency) model along with a benchmark full-information/ complete-markets framework. Section 3 discusses the data from the Consumer
2The research on consumption insurance relates to this one in another respect as well. The fact that these papers generally reject full consumption insurance makes the standard incentive model a natural candidate for quantitative study. Papers by Altonji, Hayashi, and Kotlikoff (1989) and Abel and Kothkoff (1988) also point to incomplete insurance. Townsend (1989) finds a sururisinalv high amount of insurance in Indian villages, but nevertheless rejects full insurance. Mace (1991) aiso finds a surprisingly high level of insurance, but does reject full insurance for important consumption categories such as nondurable consumption.
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C. Phelan. Incentives, insurance, and consumption and leisure
Expenditure Survey and the methods used to select a sample of 898 white, non-Hispanic males with exactly a high school education, and the assumptions made to derive statistics on individual consumption and labor supply. That section also presents a summary of the moments associated with this sample. Section 4 examines how and to what extent the model accounts for the consumption and labor supply variability found in the sample, and to what extent the agency model improves on the characteristics of the full-information/complete-markets framework. The paper concludes (Section 5) by discussing the implications of these results on models with incentive problems.
2. The model This section describes the common physical structure between the full-information model and the unobserved-effort (or agency) model. This structure consists of identical overlapping generations of agents who live for exactly T periods, die, and are then replaced by the same number of newborns. Each generation contains an infinite number of members, but at any time a generation with a given birth date represents l/T fraction of the population. The total number in the population is assumed constant over time. There is a single consumption good and each agent is matched for life with a probabilistic production technology which produces this good. At the beginning of each date, an agent can exert one of two effort levels a E {0, l> if he is still of working age. After some retirement age, R, the agent is assumed to lose his ability to take the higher effort level. For generality then let A, denote the set of possible actions (or effort levels) for an agent with t periods of life left. After the agent takes his action a, the agent’s technology produces one of two output amounts q E Q = {q, ij} c R, according to known nonzero probabilities Wa). This probability structure is assumed independent and identical across agents and time, and thus P(q/a) also represents the fraction of agents taking action a who will realize output q.3 After the outputs are realized, society can store the outputs or distribute them for consumption. The period then ends. It is assumed that society has access to a productive linear storage technology. For each unit of the consumption good stored (where the total amount stored is allowed to be negative), l/6 units (6 < 1) are assumed available for consumption or storage the next period. This can be considered equivalent to society as a whole having access to world credit markets at the constant interest rate (l/6 - 1).
3More generally, all aggregates for each generation can be known the model while there is still uncertainty at the individual level.
with certainty
in a solution
to
C. Phelan, Incentives, insurance, and consumption and leisure
585
Each agent’s common point-in-time utility function is defined as U(a, c)
E
tC_t2 - La, a
a < 1, 1 > 0,
where c is an element of a finite set C c R and c is the lowest point in set C. Agent’s have preferences over their expected lifetime utility discounted by the common discount factor p < 1. The difference between the full-information model and the unobserved-effort model is solely that in the latter individual actions are private to the agent. Solutions to each of the models can be computed using similar recursive techniques discussed in Phelan and Townsend (1991). To find Pareto-efficient allocations, I first assume that each agent is born with the right to receive some ex ante expected discounted utility w E W,, where W, is the set of possible expected discounted utilities for agents with T periods of life left. The cost of delivering a given utility to an agent under a given allocation is defined as the expected discounted sum of his consumption under that allocation minus his expected discounted output. Since this cost does not depend on how the allocation treats other agents, we can consider an efficient allocation as a collection of separate efficient contracts between society and each of its individual members. An efficient contract minimizes the cost (or maximizes the surplus) of providing the ex ante utility of the agent facing the contract. Where effort is assumed unobservable, I add incentive constraints regarding obedience to the programming problem of minimizing the cost of a contract subject to the constraint that it provides the required ex ante expected utility to the agent. To insure aggregate feasibility and optimality, I require that the distribution of initial promised utilities be such that the aggregate cost of the efficient contracts given these promises equals zero. That is, the discounted stream of aggregate consumption must equal the discounted stream of aggregate output.
3. Data from the consumer expenditure survey This section discusses the data from the Consumer Expenditure Survey concerning the variability of consumption and leisure. The first subsection simply defines the moments I use to compare the data to solutions to the models. The second subsection discusses the derivation of these moments from the data, while the third discusses the moments themselves.
3. I. Moment dejinitions Since the model predicts the fraction of agents that experience all possible realizations of lifetime effort, output, and consumption streams, one can derive
586
C. Phelan,
Incentives.
insurance,
and consumption
and leisure
population statistics on the variability of consumption and leisure both crosssectionally at a point in time and for individuals over time. I derive the following moments to describe variability. These moments were chosen because I believe they essentially capture the characteristics of consumption and leisure variability which can be gleaned from the data. What I am interested in is capturing how variable the typical individual’s consumption and leisure is over time and how much one agent’s consumption and leisure correlates with an otherwise identical agent. For a given two-date snapshot of an economy, let the fraction of agents of a given age group who go through a given consumption and labor supply transition be denoted by F(a, c, a’, ~‘1age). The population moments I consider are derived from F(a,c,a’c’/age) and defined as follows: The fraction of individuals by age who go through a given labor supply transition, n(a, a’lage), is defined by n(u, ~‘1age) E
C F(a, c, a’ c’( age) . CXC’
(1)
Cross-sectional variation in labor is the extent to which these fractions vary from zero to unity, and intertemporal variation is the extent to which n(u = 0,~ = 1 Iage) and n(u = 1, a’ = Olage) are greater than zero. For each age and work category, the cross-sectional mean and variance of time-averaged consumption (the average amount the agent eats over both periods), mc(u, u’(age) and ~(a, a’lage), are defined by
mc(u,u’Iage)
uc(a,a’)age) =
=
1
c (c + c’) __ F(u, c, a’ c’ Iage) , n(a,a’lage) CrCr 2 1
+4~‘lage)c.c~IL
(c + c’)
1
(2)
2
- mc(u, a’[ age)
2
F(a, c, a’ c’I age). (3)
The mean difference in consumption, &(a, a’] age), or the average amount that consumption differs between periods, and its variance, &(a, a’ 1age), or how much these differences vary across individuals are defined by
md(u, a’\ age) E n(a,J,age),Z
(c - cY%c,a’c’Iage), XC’
vd(u, a’/ age) 3 n(u ,t ,ageJ .Fc,CCC - 4
- m&4a’Iw)12 W4 c,a’ c’Iage).
(5)
C. Phelan. Incentives, insurance, and consumption and leisure
587
The cross-sectional variance of consumption, ~(a, a’/ age), essentially captures how much consumption levels differ across same-age individuals at a point in time, and the variance of time-differenced consumption, vd(a, a’1age), essentially captures how variable consumption is for given individuals over time.
3.2. Description
of the data
The data come from the Bureau of Labor Statistics’ Consumer Expenditure Survey for nine quarters, from 1984-1 to 1986-1. The CES interviews households for four consecutive quarters. Each calendar quarter, one quarter of the families complete their participation and are replaced. Thus, I have a full year of data on five cohorts of families starting with those observed from 1984-1 to 1984-4 and ending with thsoe observed from 19852 to 1986-1. The model is kept simple to allow for computation, so I have instead manipulated the data to fit the theoretical framework of the model. The model assumes equally skilled agents with identical preferences and ignores home production considerations such as raising children. Thus I have extracted from the CES a set of 898 white, non-Hispanic males with exactly a high school education. This was the largest group of males with identical ethnic and educational backgrounds. Each of these males is listed as either the CES’s reference person (their main contact) or the spouse of the reference person. The CES records consumption expenditures for a large number of categories and subcategories. Consumption is nondurable in the model, so I only consider consumption expenditures on food, alcoholic beverages, clothing, transportation expenses, entertainment, personal care products, reading materials, and tobacco. Durables such as household appliances are excluded. The household’s consumption from the first two quarters are added to form one observation and the last two quarters are similarly added. This forms two consecutive obervations for each household. If I had assumed the shorter period length of a quarter, with four observations, then agents in the model with reasonable life spans live twice as many periods. This makes computation take longer. In the other direction, one wants to have at least two observations per household to observe consumption changes over time. The model makes predictions concerning the magnitudes of individual changes in consumption and leisure from one period to the next. The model also considers individuals while the data is reported for households. This requires some method for taking household consumption from the data and deriving a number which best corresponds to the object c in the model. Simply dividing household consumption by the number of household members does not take into account that children consume less than adults and that adults derive satisfaction from the consumption of their children. For these
588
C. Phelan, Incentives, insurance, and consumption and leisure
reasons and for simplicity, I treat children as consumption goods and impute c by dividing household consumption by the number of adult houselhold members.4 The models also have constant aggregates, and consumption is in real units. The data set reflects inflation, seasonality, and aggregate changes (including growth). Thus an individual’s consumption for each six-month period is defined as the individual’s log consumption minus the average log consumption of those observed during the same time period. Since labor efforts are assumed equal to zero or one in the model, I record those reporting to have worked an average of 20 or more hours a week for a given six-month period as having worked one that period and all others as having worked zero.5 Lastly, I do not attempt to measure output q. There is nothing in the data which reports a household’s contribution to societal wealth (the role of q in the models). A tempting, but inappropriate measure of q is reported income. However, because of the existence of employeremployee relationships and government insurance programs, reported income could just as well be a measure of the result of insurance schemes smoothing over shocks to q as a measure of q itself. For this reason I assume that q is unobserved to the econometrician.
3.3. Data summary These assumptions make the testable implications of the full-information and unobserved-effort models as stated, the fraction of agents of any age group which go through a given work and consumption transition, F(a, c, a’ c’lage). I focus on the same moments used to discuss variability in the earlier sections: the fraction of agents in each quadrant representing the four work states, n(a, a’ )age), and the cross-sectional and intertemporal variances of consumption
4Lazear and Michael (1989) use CES data to estimate the proportion of family consumption which goes to children versus adults, and thus I could have used their estimates to impute the consumption of the male reference person (or male spouse of the reference person). Nevertheless, this would have ignored the fact that in the model all consumption utility comes from the agent’s own consumption, but in reality people care about their children. Treating a child’s consumption as equivalent to the parent’s (in the parent’s eyes) is equivalent to a simple altruism framework. ‘It is important to note that since the incentive constraint is satisfied, we can ex post assume agents actually work the recommended amount, and thus the data reflects true work levels. 61t is important to note that incentive theory only makes predictions regarding the mapping from outcomes (q’s in the model) to consumption and leisure levels, The theory makes no predictions regarding the mechanisms by which such mappings are implemented. Many compensation and insurance schemes (or mechanism) can achieve the same consumption and leisure mapping, thus no compensation (or earnings) predictions are implied by the model.
C. Phelan, Incentives, insurance, and consumption and leisure
589
for each work group, UC@,a’1age) and ud(a, a’lage). Table 1 shows the estimated moments7 The first section of table 1 shows the percentage of agents in each work category - n(u, u’l age). The first column of this section shows that at higher ages a larger fraction of agents are observed not working in either period. The next two columns show that very few people are observed to switch from working in the first period to not working in the next, or vice versa. Moments conditional on switching actions are thus poorly estimated. The next section displays the variability of consumption over time ~ ud(u,u’lage). One fact that will prove important is that for all ages grouped together the variance of time-differenced consumption is only slightly greater for those who work than those who don’t. Otherwise both workers and nonworkers tend to show a greater variance as age increases, but not monotonically. The last section of table 1 shows that the cross-sectional variance of consumption for those who do not work either period tends to rise with age, and that the variance of consumption for those who work both periods is less than those who don’t work, and may tend to rise with age except for the 18-28 year olds. I will focus heavily on these profiles in the next section.
4. Implications
of the data on the incentive method
This section discusses the extent to which the unobserved effort model is capable of producing variation consistent with the observed variation from the Consumer Expenditure Survey and to what extent it improves upon the ‘The standard errors I report assume that the observations are drawn independently from the population of white, non-Hispanic, high-school-educated males. The standard error of the estimated n(a, a’1age) is defined by
C& a’Iage)U- 0, ~‘IwW%w)1°~5 , where N(age) (the sample size) is the number of uc(a, a’( age) is defined as
(c + c’) CXC
~
2
of observations
for each age group. The standard
2
- mc(u, a’/ age)
(6)
1 2
- ~(a, a’( age)
N(age) n(a, a’ I333
F(u, c, a’ c’ 1age)
error
I. 0.5
(7)
The standard
error of ud(a, a’1 age) is defined as
J$X’c[{(c
- c’) -md(~,a’jage))~
- ud(u,u’lage)]2F(u,c,u’c’~age)
N(age) n (a, a’ Iage)
1(8) . O3
590
C. Phelan, Incentives, insurance. and consumption and leisure Table 1 Estimated
Age group
moments,
where a and a’ are the work levels in the first and second six-month period, respectively. a = 0, a’ = 0 Work
transition
a = 0, a’ = 1 fractions
a=l,a’=O
a = 1, a’ = 1
by age group
18-28 29-39
2.5% (1.0%)
1.2 (0.7%)
(0%)
100% (-) 96.3% (1.2%)
40-50
3.5% (1.3%)
3.0% (1.2%)
0.5% (0.5%)
92.9% (1.8%)
51-61
12.4% (2.5%)
(0%)
(O-“)
87.6% (2.5%)
62-90
66.3% (3.6%)
2.8% (1.3%)
4.0% (1.5%)
26.9% (3.4%)
18-90
16.7% (1.2%)
1.6% (0.4%)
0.9% (0.3 %)
80.8% (1.3%)
p_”
Intertemporal
variance
p-”
of log consumption
(Y)
by age group
18-28 (I)
(I) 29-39 40-50 51-61
(I)
0.105 (0.023)
0.013 (0.004)
0.084 (0.034)
(I)
0.120 (0.032)
0.042 (0.012)
0.035 (0.024)
(1)
0.088 (0.014)
(7)
0.138 (0.03 1)
0.161 (0.054)
(1)
62-90
0.095 (0.014)
0.236 (0.088)
0.135 (0.044)
0.201 (0.061)
18-90
0.101 (0.014)
0.179 (0.83)
0.127 (0.044)
0.120 (0.014)
Cross-sectional
variance
of log consumption
by age group (I)
0.202 (0.027)
18-28 (I) 29-39 40-50 51-61
(1)
0.072 (0.029)
0.221 (0.090)
(I)
0.162 (0.0 19)
0.180 (0.078)
0.090 (0.034)
(I)
0.174 (0.02 1)
(I)
0.184 (0.021)
0.320 (0.113)
(I)
62-90
0.257 (0.029)
0.049 (0.032)
0.059 (0.03 1)
0.201 (0.044)
18-90
0.255 (0.028)
0.225 (0.83)
0.061 (0.024)
0.184 (0.011)
C. Phelan, Incentives, insurance, and consumption and leisure
591
characteristics of the full-information/complete-markets framework. The first subsection generally discusses how the unobserved-effort model generates both cross-sectional and intertemporal variability. The following subsections then discuss the ability of the unobserved-effort model to match the moments of the data through the use of computed examples. The principal quantitative result is that although the unobserved-effort model generates endogenous variability, most variability must nevertheless be attributed exogenous variability such as initial inequality and measurement error. The kind of endogenous variation the unobserved-effort model produces is inconsistent with a majority of the observed variation. A secondary result is that the unobserved-effort model constitutes an improvement on several dimensions. It allows for early retirement and for the cross-sectional variances to depend on age and work class. Both of these characteristics are present in the data.
4.1. Solution characteristics In the full-information model, cross-sectional variation in work levels and consumption can only be caused by initial cross-sectional variation in the initial expected utilities, w. Those required to receive high expected discounted utilities are given high consumption levels. Agents with very high w’s are given high leisure (a = 0) as well as high consumption. Because agents are insured against variation in their own output, they receive the same consumption and leisure amounts for each period in life up to the retirement age R. Upon reaching R, the agent’s labor supply level may fall, but his consumption will remain constant if utility is additively separable as assumed here. The cross-sectional distribution of consumption is thus the same over all age groups and the same for retired as well as working people. Intertemporal variability in the full-information model is nonexistent except for the intertemporal labor supply variability associated with an agent retiring. The unobserved-effort model generates endogenous variability over and above the exogenous variability present in its full-information counterpart. In the unobserved-effort model, as in the full-information model, cross-sectional variation in expected discounted utilities causes cross-sectional variation in work levels and consumption. However, in the unobserved-effort model there is endogenously generated variability as well. This is best understood by considering the recursive nature of solutions to the unobserved-effort model. As shown in Phelan and Townsend (1991) and Spear and Srivastava (1987), the expected discounted utility of an agent from any point in time on sufficiently describes the history of that agent. Thus, the punishments and rewards for first-period outputs consist entirely of making first-period consumption contingent on first-period output and making promised expected
592
C. Phelan, Incentives, insurance, and consumption and leisure
discounted utility w for the next period contingent on first-period output. These promises then generate the new distribution of required expected discounted utilities for the environment where agents have T - 1 periods of life left. In the full-information model, if one standardizes for the number of periods left, the distribution of expected discounted utilities stays constant over the life cycle. In the unobserved-effort model, variation in expected discounted utilities grows as agents are punished and rewarded based on output levels, again standardizing for the number of periods left. High outputs are rewarded not only with high current consumption amounts but also by high expected future utilities, w’. As variability in expected discounted utilities grows, variability in consumption grows as well. It is important to note that in solutions to the unobserved-effort model, changes in log consumption tend to be nearly permanent. That is, the typical agent’s log consumption follows a near random walk. For a wide variety of parameter choices, the autocorrelation of consumption changes was never observed below -0.18 and was usually observed much closer to zero. This autocorrelation was never observed below -0.08 for parameters that produce nontrivial amounts of variation. This persistence can also be seen in the fact that in computed examples the intertemporal variances are much smaller than the cross-sectional variances. Differences across individuals are much greater than differences for the same individual over time. (Again, this characteristic was observed for every parameter configuration tested.) If consumption levels for individuals were i.i.d. across time, then the intertemporal variances would be exactly four times greater than the cross-sectional variances.*
4.2. Implications for the unobserved-effort
model
Figs. 1 through 5 present the moments of an example designed to illustrate the extent to which the unobserved-effort model can produce moments near to some of those in the data, and why one must assume high amount of initial variation and measurement error. The example is in some sense an informal ‘best fit’ of the model. The parameters for the example are the result of the author testing scores of parameter specifications and choosing one specification that allowed the illustration of where the model fails and succeeds. The specification of this example is as follows. Preferences are specified by the curvature parameter for consumption a = 0.5, the disutility of effort J. = 9.9, and the discount parameter fl = 0.97. Generally, as agents become less risk-averse and discount less, the incentive model produces more endogenous consumption variation. The parameters CI= 0.5 and ‘This comes from the 2 in the denominator of the definition and the fact that var(X/constant) = var(X)/constant’).
of the cross-sectional
variance,
eq. (3),
C. Phelan, Incentives. insurance, and consumption and leisure
593
/? = 0.95 represented the smallest amount of risk aversion and discounting (over a six-month time horizon) compatible with other studies. The utility loss associated with working, I = 9.9, has no real justification. As workers dislike work more, the incentive constraints tighten, causing more endogenous variation. The number used was arrived at because it helped produce moments that best matched the data. The technological parameters are specified by the probability of the low output given each effort level, P(dg) = 0.97 and P(q/ci) = 0.845,9 and the output levels themselves, q = 650 and q = 4000. One result of the large number of parameter specifications tested was that technologies where even those who take the high effort level usually fail tend to produce high rewards for those who succeed. Thus these parameters were chosen for thier ability to produce significant quantities of endogenous consumption variability. A more technical specification is that the set of consumption levels C is the even grid (in logs), e5, e5.1, e5.‘, . . . , e9. For each model, the reported moments are sensitive to the fineness of the consumption grid, however, this sensitivity becomes smaller as the grid is made finer. The above grid of 41 consumption points was chosen because the moments of the unobserved-effort model did not change much (three significant digits) to further refinements of the grid. Lastly, the technological rate of discount, 6, is set equal to the preference rate of discount, /I Agents are assumed to live for 105 six-month periods (to match the observation period in the data), the first 90 of which they can work. Age 18 in the data is assumed to equal age one in the model. Lastly, the example assumes initial variation in expected utilities and measurement error of consumption. These levels will be explained shortly. Fig. 1 shows the fraction of agents who work both periods. This example produces moments near the data not only for all groups together, but also improves on the full-information model by allowing for a decreasing fraction of agents working as age increases. Under all parameter choices, the full-information model allows only for an initial fraction of agents to not work due to initial diversity, and for this fraction to remain constant until the age where everyone is forced to retire.” Fig. 2 shows the intertemporal variances of consumption for agents who work neither period, adjusting for assumed measurement error in consumption which ’ P(~/cJ) = 0.03 and P(g/ici) = 0.155 are implied by P(q/g) and P(q/ti).
“‘It should be noted here that it is only this simple version of the full-information model which does not allow for early retirement. For instance, I do not allow for sickness. There is a large body of theoretical and empirical work [see Lazear (1986) for a survey] on the causes of retirement. It is difficult to compare this work with most theoretical work on retirement since most often the problem is posed as finding the optimal retirement age given an exogenous wage and benefits profile. Empirical papers tend to focus on the effects of changes in Social Security and other pension benefits. This work allows for endogenous retirement and endogenous retirement benefits.
594
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Fig
1. Fraction
working
both periods.
+
0
+ 0
d 16-28
29-39
40-50 age
Fig. 2. Intertemporal
variance:
+
+
d +
q
+
+
51-61
62-90
18-90
group
Working
neither
period
is explained shortly. Measurement error must be assumed for the following reason. For those who have stopped working (including those going from working to not working) there is again full insurance against variations in output, and thus the intertemporal variances are zero. If agents aren’t working, there is no incentive problem and thus no reason to make consumption contingent on output.” Since the observed intertemporal variance for non workers is
r’ For those going from working to not working, full insurance exists because the consumption in both periods occurs after the output from the first period is realized. This first output is the only one that depends on effort and thus any punishment or reward based on this output will affect the consumption in both periods equally.
C. Phelan, Incentives, insurance, and consumption and leisure
595
1
I
-I
16-28
29-39
40-50 age
Fig. 3. Intertemporal
51-61
62-90
18-90
group
variance:
Working
both periods.
quite significantly different from zero (as can be seen in table l), the unobservedeffort mod.el (in this simple form) is simply incompatible with a basic fact of the data. A possible correction is to assume that consumption is measured with error, which, by the nature of panel data, is certainly true. If this error term is independent and log-normally distributed with a variance of D’, this will add 2~* to the predicted intertemporal variance for each age and work group and a*/2 to each predicted cross-sectional variance.‘* However, when all ages are grouped together, the intertemporal variance of consumption for nonworkers is not very different (and not significantly different) than the intertemporal variance for workers. Thus, if one sets O* high enough to account for the intertemporal variability of nonworkers and chooses any set of parameters which causes the model to match the intertemporal variance of workers, then most of the intertemporal variance for both work groups will be due to measurement error. This can be seen in figs. 2 and 3. Here 2a2 is set equal to the observed intertemporal variance of consumption for nonworkers as a whole. However, the intertemporal variance for workers (fig. 3) is only slightly higher than what it would be without the incentive problem. What it is possible to choose different parameters to increase this difference, one is still constrained to match the observed intertemporal variance for workers, and thus most of the intertemporal variance will still be due to the assumed measurement error. Figs. 4 and 5 display the cross-sectional variance of consumption by age group. Cross-sectional variation can be endogenously generated in the unobserved-effort model through the output-based punishing and rewarding of agents, “These follow from var(A + B) = var(A) + var(B) for independent A and B and var(A/constant) = var(A)/constant’). Again, the difference in the effects comes from the 2 in the denominator of eq. (3).
596
C. Phelan. Incentives, insurance, and consumption and leisure
18-28
29-39
40-50 age
Fig. 4. Cross-section
51-61
62-90
18-90
group
variance:
Working
neither
period.
_I
18-28
29-39
40-50 age
Fig. 5. Cross-section
51-61
62-90
18-90
group
variance:
Working
both periods.
or exogenously by assuming measurement error or initial inequality in expected discounted utilities. Each method produces different results. As shown above, increases in assumed measurement error increase the crosssectional variances of all age and work groups equally. An increase in initial inequality works similarly. It raises the level of cross-sectional variability that would have existed even without the incentive problem, and it raises this level equally over all age and work groups. On the other hand, changing a preference or technology parameter does not affect all age and work groups equally. Increases in punishments and rewards have disproportionately greater effects on the cross-sectional variances of older groups since punishments and rewards are nearly permanent. As agents age, punishments and rewards accumulate.
C. Phelan, Incentives, insurance, and consumption and leisure
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Put differently, the unobserved-effort model predicts an upsloping age-variance profile for workers as in the solid line on fig. 5. Increasing initial inequality or measurement error raises the base level, or intercept, of this profile, while increasing the severity of punishments and rewards tends chiefly to increase the slope. Increasing initial inequality or measurement error also equally increases the crosssectional variances of both work groups as a whole (the 18-90 age group), while increasing the severity of punishments and rewards disproportionately increases the cross-sectional variance of nonworkers since they are older than workers. While it is possible to choose different parameters to increase or decrease the slope of the age-variance profile in fig. 5, one is still constrained to match the leftmost point, the observed cross-sectional variance for young workers. Increasing the severity of punishments and rewards will increase the difference across models of the cross-sectional variance of 18-28 year olds (the difference between the two lines at the leftmost point). However, the main effect of this is to increase the slope of the age-variance profile for the unobserved-effort problem, causing one to overshoot the observed variances for the older age groups and for nonworkers.13 Thus one must assume at least enough initial inequality to raise the intercept to an acceptable level. In this example, this causes the population cross-sectional variance of consumption for the unobserved-effort model to be only one-third higher than its full-information counterpart. (Equivalently, 25% of the cross-sectional variances in this example can be attributed to the incentive problem.) One cannot choose parameters to greatly increase this difference without contradicting the data points in figs. 4 and 5. In the other direction, one cannot simply raise the level of initial inequality and cause the full-information model to match all the cross-sectional moments. While one cannot reject that the profile in fig. 5 is flat (as the full-information model predicts), the model also predicts that the cross-sectional variance of consumption for old retired workers equals the cross-sectional variance of young workers. In the full-information model, cross-sectional variance is caused only by variability in initial utility promises and measurement error - both assumed unrelated to age or work group. Nevertheless, in the data the crosssectional variance of consumption for workers of all ages is significantly lower than that of nonworkers. The unobserved-effort model correctly predicts a higher cross-sectional variance for nonworkers since nonworkers are older (in both the theory and the data) and thus have accumulated a lifetime of punishments and rewards.14 This and the fact that the unobserved-effort model allows 13Again, it is important to note that this is a general solution characteristic, not dependent on the parameters of this example. It follows directly from the near permanence of consumption changes in solutions to the unobserved-effort model. 14This study is not the first to look at these type of age-variance profiles. Nevertheless, most of the labor literature, starting with Mincer (1974), focuses on the variability of wages. Mincer argues that if life-cycle wage profiles for different careers have varying slopes and intercepts, then life-cycle variance profiles can be increasing or U-shaped.
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for early retirement allows for the claim that improves upon the ability of the full-information
the unobserved-effort model model to match the data.
5. Conclusions The inability of this particular dynamic agency model to account for the majority of the variability we see is more robust than one would first guess given the extreme assumptions of the particular model I consider. In my unpublished dissertation of the same title [Phelan (1990)] I show that an unobservedendowment model similar to Green (1987) suffers from a nearly identical inability to match the data, and for the same reasons. In Green’s model, as in the one presented here (but unlike the data), each agent’s consumption should start out equal and consumption changes should be permanent. Further, I show in the dissertation that simply assuming closed insurance markets over outputs 4, and thus having each agent solving a permanent-income consumption-smoothing problem, also has similar difficulty in matching the moments of the data without large amounts of initial inequality. Again, that model had difficulty generating sufficient amounts of endogenous early-life consumption inequality. Simply put, any time-separable moral-hazard or limited-insurance model where all periods of life are identical will have a very difficult time generating sufficient levels of early-life consumption inequality, and thus early-life inequality will have to be brought about by dispersion in initial required utility levels. Given this, what conclusions should be drawn? One possibility is that the majority of the cross-sectional variation we see can be attributed to inequality of initial wealth levels. However, results from Altonji, Hayashi, and Kotlikoff (1989) suggest that inequality in initial wealth levels is probably not a sufficient explanation for cross-sectional consumption inequality. Specifically, they find significant dispersion in the consumption of yuoung adult siblings, even though one would expect that siblings would start life with equal wealth levels. There are other possibilities as well. The conclusion in section 4 that one must assume large amounts of dispersion in initial expected discounted utilities does not necessarily mean one must assume dispersion in initial wealth levels. Dispersion in expected utilities can also be interpreted as dispersion in innate skill levels if society allows agents to keep most of the fruits of their production. In fact, this can be considered a form of initial wealth dispersion in that those with higher innate skill levels ‘own’ a higher level of human capital and thus are wealthier, although not by conventional measures. Nevertheless, this is problematic. If skill levels are observable, there is no incentive-based reason why those with higher skill levels should consume more than those less skilled. Note that differing skill levels can be brought into the model by having heterogeneity in the possible output levels. High skilled people could be represented as having possible output levels @ and q* and low skilled
C. Phelan, Incentives. insurance, and consumption and leisure
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as having possible output levels 4’ and ql, where gh > 4’ and q” > ql. If agents of both skill levels have the same function P mapping effort levels toprobabilities of their respective output levels q and 4, then both types will be treated exactly the same, holding expected discounted utility constant. The differing q’s will only affect aggregate output and thus the ability to support any distribution of expected discounted utilities. What one needs to get a model of incentive-based variability to look like the variability we see in data is an incentive-based theory of why expected discounted utilities differ across agents at the beginning of adulthood. An incentivebased theory requiring those with higher innate skills to consume more than those less skilled may suffice. Nevertheless, such a model would entail incentive problems markedly different than the kinds of incentive problems posed in standard dynamic agency models.
References Abel, A. and L.J. KotlikolT, 1988, Does the consumption of different age groups move together? A new nonparametric test of intergenerational altruism, NBER working paper no. 2490. Altonji, J.G. and A. Siow, 1987, Testing the response of consumption to income changes with (noisy) panel data, Quarterly Journal of Economics 102, 293-328. Altonji, J.G., F. Hayashi, and L.J. Kotlikoff, 1989, Is the extended family altruistically linked? Direct tests using micro data, NBER working paper no. 3046. Altug, S. and R. Miller, 1990, Household choice in equilibrium, Econometrica 58, 543-570. Atkeson, A. and R. Lucas, 1992, On efficient distribution with private information, Review of Economic Studies 59, 427-453. Cochrane, J., 1991, A simple test of consumption insurance, Journal of Political Economy 99,957-976. Green, E., 1987, Lending and the smoothing of uninsurable income, in: E. Prescott and N. Wallace, eds., Contractual arrangements of intertemporal trade (University of Minnesota Press, Minneapolis, MN) 3-25. Hayashi, F., 1985a, The effect of liquidity constraints on consumption: A cross sectional analysis, Quarterly Journal of Economics 100, 183-206. Hayashi, F., 1985b, The permanent income hypothesis and consumption durability: Analysis based on Japanese panel data, Quarterly Journal of Economics 100, 1083-1113. Hayashi, F., 1987, Tests for liquidity of constraints, in: T. Bewley, ed., Advances in econometrics: Fifth world congress, Vol. 2 (Cambridge University Press, Cambridge) 91-120. Lazear, E.P., 1986, Retirement from the labor force, in: 0. Ashenfelter and R. Layard, eds., Handbook of labor economics, Vol. 1 (Elsevier Science Publishers, Amsterdam) 305-355. Lazear, E.P. and R.T. Michael, 1988, Allocation of income within the household (University of Chicago Press, Chicago, IL). Mace, B., 1991, Full insurance in the presence of aggregate uncertainty, Journal of Political Economy 99,928-956. Mincer, J., 1974, Schooling, experience and earnings (NBER, Cambridge, MA). Phelan, C., 1990, Incentives, insurance, and the variability of consumption and leisure, Ph.D. dissertation (University of Chicago, Chicago, IL). Phelan, C. and R.M. Townsend, 1991, Computing multiperiod, information-constrained optima, Review of Economic Studies 58, 853-881. Shiller, S., M. Boycko, and V. Korobov, 1991, Popular attitudes toward free markets: The Soviet Union and thk United States compared, American Economic Review 81, 385-400. Spear, S. and S. Srivastava, 1987, On repeated hazard with discounting., Review of Economic Studies, 180, 599-618. Townsend, R.M., 1989, Risk and insurance in village India, Manuscript.
of Economic
Dynamics
and Control
18 (1994) 581-599.
North-Holland
Incentives, insurance, and the variability of consumption and leisure* Christopher
Phelan
University of Wisconsin, Madison, WI 53706, USA Received
May 1992, final version
received
March
1993
This paper considers to what extent dynamic incentive models such as Green (1987), Phelan and Townsend (1991), and Atkeson and Lucas (1992) can quantitatively explain why some individuals consume (or work) more than others and why a typical individual’s consumption (or leisure) varies over time. A simple repeated agency model is shown to be able to better match several population moments concerning life-cycle labor supply and consumption variation than the same model without incentive constraints. On the other hand, I show for the standard stylized models of repeated agency, incentive-induced variability is an insufficient explanation of the variability in the data. Key words: Incentives; Consumption JEL classification: D31; D82; D91
variability
1. Introduction Models where individuals can take private ductivity or receive privately observed taste
Correspondence to: Christopher Phelan, Department Observatory Drive, Madison, WI 53706, USA.
actions which affect their proor wealth shocks give rise to
of Economics,
University
of Wisconsin,
1180
*This work is drawn from my University of Chicago Ph.D. dissertation of the same title. I would like to thank Andrew Atkeson, William Carrington, John Heaton, Robert Lucas, John Matsusaka, Karl Snow, the members of the Money and Banking Workshop at the University of Chicago, and seminar participants at Cornell University, Princeton University, the University of Pennsylvania, Northwestern University, the University of Wisconsin, the University of Illinois, the University of Rochester, Boston University, the University of Virginia, the 1990 meetings of the Society for Economic Dynamics and Control, and the associate editor for helpful comments. My committee members John Cochrane and Lars Hansen were especially helpful. Finally I want to thank my committee chairman Robert Townsend for his comments and constant support throughout my graduate education. Computer support from the National Center for Supercomputing Applications, Champaign, Illinois, and financial support from the Earhart Foundation, Bradley Foundation, National Science Foundation grant SES-911 l-926, and my wife, Elizabeth M. Bloomer, is appreciated.
0165-1889/94/$07.00
0
1994-Elsevier
Science B.V. All rights reserved
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C. Phelan. Incentives, insurance, and consumption and leisure
inequality. In such incentive models, full insurance against productivity, wealth, or taste shocks is impossible since every individual will minimize effort if his consumption is the same regardless of his productivity, and every individual will claim hardship or urgency to consume if by doing so he can raise his consumption while bearing no other costs. This lack of full insurance allows for an incentive-based theory of distribution. The dynamic, general equilibrium, incentive models of Green (1987), Phelan and Townsend (199 l), and Atkeson and Lucas (1992) each generate a theory of why some individuals consume (or work) more than others and why a typical individual’s consumption (or leisure) varies over time. In each paper the incentive problem can be the sole reason why any individual consumes more than any other individual, and the sole reason why any individual’s consumption varies over time’. The purpose of this paper is to see to what extent such incentive-induced variability can account for the variability we actually see. To make specific quantitative predictions, this paper focuses on the dynamic, general equilibrium, agency model in Phelan and Townsend (1991). This is essentially the standard principal-agent model. In each period, each agent takes an unobserved action (an effort level) which affects the probability of various output levels. In incentive-compatible plans, an agent’s consumption must depend on his output to get the agent to work, which leaves the agent less than fully insured against idiosyncratic shocks to his output. This incomplete insurance causes consumption and leisure to vary. This type of incentive-induced variability takes a specific dynamic form that differs from the kind of variability produced by more traditional means such as heterogeneity of initial endowments (of either wealth or human capital) or transient shocks such as preference shocks and measurement error. Thus, by looking at variability in the data we should be able to see to what extent actual variability is consistent with the incentive story. I take a simple approach here. First, I derive moments summarizing both cross-sectional and intertemporal consumption and leisure variability from data from the Consumer Expenditure Survey. Second, I parameterize and numerically solve the incentive model attempting to match these moments. The parameterization used allows three methods of creating variability. The first
‘While the ability to formalize these dynamic incentive problems is recent, the notion that incentive problems may be crucial in understanding why some people get to consume more than others is a common notion, In fact, in their survey of attitudes toward free markets, Shiller, Boycko. and Korobov (1991) include the question: Some have expressed thefollowing: ‘It’s too bad that some people are poor while others are rich. But we can’t jx that: if the government were to make sure that eoeryone had the same income, we would all be poor, since no one would have any material incentive to work hard.’ Have you heard such a theory or not? If yes, then how often? 11% of Moscow residents and 45% of New York residents sampled had heard of the theory, although 59% of Moscow residents and 62% of New York residents disagreed with it.
C. Phelan, Inceritives. insurance, and consumption
and leisure
583
method is to create endogenous incentive-induced variability. The second method is to simply assume exogenous initial inequality in expected lifetime utility levels. This stands in for any permanent differences across agents which would cause permanent consumption or leisure differences, including differences in inmate human capital. The third method is to assume exogenous transient (or i.i.d.) consumption variability. While this is consistent with preference shocks, the paper refers to such variability as measurement error. The premise of the paper is then as follows: The extent to which one must rely on the exogenous methods of generating variability to match the moments in the data is the extent to which a particular incentive model is incapable of accounting for the variability we see. The principal quantitative result is that for the simple repeated-agency model considered here, in order to get close to the moments in the data, most variability must be attributed to the exogenous factors. The dynamics of the endogenous incentive-produced variability does not square well with the variability in the data. A secondary (and not as robust) quantitative result is that the dynamic-agency model constitutes an improvement on the full-information framework on several dimensions. It allows for endogenous early retirement and predicts that cross-sectional consumption variability should move with age and leisure in directions consistent with the data. It is important to note that this project makes no attempt to explain the variation in the data, except in regard to the narrow question of whether the variability in the data could have been generated by the standard prototype dynamic agency model. When it is shown that the incentive model needs large amounts of exogenous variability to match the moments in the data, I do not dwell on identifying the correct sources of this exogenous variation other than discussing possibilities in the conclusion. While ultimate purpose of any research line is to explain the data, a valid intermediate goal is to understand where our standard theoretical models quantitatively fail and succeed. This is in line with the recent works of Mace (1991), Cochrane (1991), and Townsend (1989) which test the full-consumption-insurance implications of full-information/complete-market frameworks. It is hoped that such an exercise will help determine how simple models should be enriched in order to match data.’ The structure of the paper is as follows. Section 2 presents the unobserved-effort (or agency) model along with a benchmark full-information/ complete-markets framework. Section 3 discusses the data from the Consumer
2The research on consumption insurance relates to this one in another respect as well. The fact that these papers generally reject full consumption insurance makes the standard incentive model a natural candidate for quantitative study. Papers by Altonji, Hayashi, and Kotlikoff (1989) and Abel and Kothkoff (1988) also point to incomplete insurance. Townsend (1989) finds a sururisinalv high amount of insurance in Indian villages, but nevertheless rejects full insurance. Mace (1991) aiso finds a surprisingly high level of insurance, but does reject full insurance for important consumption categories such as nondurable consumption.
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Expenditure Survey and the methods used to select a sample of 898 white, non-Hispanic males with exactly a high school education, and the assumptions made to derive statistics on individual consumption and labor supply. That section also presents a summary of the moments associated with this sample. Section 4 examines how and to what extent the model accounts for the consumption and labor supply variability found in the sample, and to what extent the agency model improves on the characteristics of the full-information/complete-markets framework. The paper concludes (Section 5) by discussing the implications of these results on models with incentive problems.
2. The model This section describes the common physical structure between the full-information model and the unobserved-effort (or agency) model. This structure consists of identical overlapping generations of agents who live for exactly T periods, die, and are then replaced by the same number of newborns. Each generation contains an infinite number of members, but at any time a generation with a given birth date represents l/T fraction of the population. The total number in the population is assumed constant over time. There is a single consumption good and each agent is matched for life with a probabilistic production technology which produces this good. At the beginning of each date, an agent can exert one of two effort levels a E {0, l> if he is still of working age. After some retirement age, R, the agent is assumed to lose his ability to take the higher effort level. For generality then let A, denote the set of possible actions (or effort levels) for an agent with t periods of life left. After the agent takes his action a, the agent’s technology produces one of two output amounts q E Q = {q, ij} c R, according to known nonzero probabilities Wa). This probability structure is assumed independent and identical across agents and time, and thus P(q/a) also represents the fraction of agents taking action a who will realize output q.3 After the outputs are realized, society can store the outputs or distribute them for consumption. The period then ends. It is assumed that society has access to a productive linear storage technology. For each unit of the consumption good stored (where the total amount stored is allowed to be negative), l/6 units (6 < 1) are assumed available for consumption or storage the next period. This can be considered equivalent to society as a whole having access to world credit markets at the constant interest rate (l/6 - 1).
3More generally, all aggregates for each generation can be known the model while there is still uncertainty at the individual level.
with certainty
in a solution
to
C. Phelan, Incentives, insurance, and consumption and leisure
585
Each agent’s common point-in-time utility function is defined as U(a, c)
E
tC_t2 - La, a
a < 1, 1 > 0,
where c is an element of a finite set C c R and c is the lowest point in set C. Agent’s have preferences over their expected lifetime utility discounted by the common discount factor p < 1. The difference between the full-information model and the unobserved-effort model is solely that in the latter individual actions are private to the agent. Solutions to each of the models can be computed using similar recursive techniques discussed in Phelan and Townsend (1991). To find Pareto-efficient allocations, I first assume that each agent is born with the right to receive some ex ante expected discounted utility w E W,, where W, is the set of possible expected discounted utilities for agents with T periods of life left. The cost of delivering a given utility to an agent under a given allocation is defined as the expected discounted sum of his consumption under that allocation minus his expected discounted output. Since this cost does not depend on how the allocation treats other agents, we can consider an efficient allocation as a collection of separate efficient contracts between society and each of its individual members. An efficient contract minimizes the cost (or maximizes the surplus) of providing the ex ante utility of the agent facing the contract. Where effort is assumed unobservable, I add incentive constraints regarding obedience to the programming problem of minimizing the cost of a contract subject to the constraint that it provides the required ex ante expected utility to the agent. To insure aggregate feasibility and optimality, I require that the distribution of initial promised utilities be such that the aggregate cost of the efficient contracts given these promises equals zero. That is, the discounted stream of aggregate consumption must equal the discounted stream of aggregate output.
3. Data from the consumer expenditure survey This section discusses the data from the Consumer Expenditure Survey concerning the variability of consumption and leisure. The first subsection simply defines the moments I use to compare the data to solutions to the models. The second subsection discusses the derivation of these moments from the data, while the third discusses the moments themselves.
3. I. Moment dejinitions Since the model predicts the fraction of agents that experience all possible realizations of lifetime effort, output, and consumption streams, one can derive
586
C. Phelan,
Incentives.
insurance,
and consumption
and leisure
population statistics on the variability of consumption and leisure both crosssectionally at a point in time and for individuals over time. I derive the following moments to describe variability. These moments were chosen because I believe they essentially capture the characteristics of consumption and leisure variability which can be gleaned from the data. What I am interested in is capturing how variable the typical individual’s consumption and leisure is over time and how much one agent’s consumption and leisure correlates with an otherwise identical agent. For a given two-date snapshot of an economy, let the fraction of agents of a given age group who go through a given consumption and labor supply transition be denoted by F(a, c, a’, ~‘1age). The population moments I consider are derived from F(a,c,a’c’/age) and defined as follows: The fraction of individuals by age who go through a given labor supply transition, n(a, a’lage), is defined by n(u, ~‘1age) E
C F(a, c, a’ c’( age) . CXC’
(1)
Cross-sectional variation in labor is the extent to which these fractions vary from zero to unity, and intertemporal variation is the extent to which n(u = 0,~ = 1 Iage) and n(u = 1, a’ = Olage) are greater than zero. For each age and work category, the cross-sectional mean and variance of time-averaged consumption (the average amount the agent eats over both periods), mc(u, u’(age) and ~(a, a’lage), are defined by
mc(u,u’Iage)
uc(a,a’)age) =
=
1
c (c + c’) __ F(u, c, a’ c’ Iage) , n(a,a’lage) CrCr 2 1
+4~‘lage)c.c~IL
(c + c’)
1
(2)
2
- mc(u, a’[ age)
2
F(a, c, a’ c’I age). (3)
The mean difference in consumption, &(a, a’] age), or the average amount that consumption differs between periods, and its variance, &(a, a’ 1age), or how much these differences vary across individuals are defined by
md(u, a’\ age) E n(a,J,age),Z
(c - cY%c,a’c’Iage), XC’
vd(u, a’/ age) 3 n(u ,t ,ageJ .Fc,CCC - 4
- m&4a’Iw)12 W4 c,a’ c’Iage).
(5)
C. Phelan. Incentives, insurance, and consumption and leisure
587
The cross-sectional variance of consumption, ~(a, a’/ age), essentially captures how much consumption levels differ across same-age individuals at a point in time, and the variance of time-differenced consumption, vd(a, a’1age), essentially captures how variable consumption is for given individuals over time.
3.2. Description
of the data
The data come from the Bureau of Labor Statistics’ Consumer Expenditure Survey for nine quarters, from 1984-1 to 1986-1. The CES interviews households for four consecutive quarters. Each calendar quarter, one quarter of the families complete their participation and are replaced. Thus, I have a full year of data on five cohorts of families starting with those observed from 1984-1 to 1984-4 and ending with thsoe observed from 19852 to 1986-1. The model is kept simple to allow for computation, so I have instead manipulated the data to fit the theoretical framework of the model. The model assumes equally skilled agents with identical preferences and ignores home production considerations such as raising children. Thus I have extracted from the CES a set of 898 white, non-Hispanic males with exactly a high school education. This was the largest group of males with identical ethnic and educational backgrounds. Each of these males is listed as either the CES’s reference person (their main contact) or the spouse of the reference person. The CES records consumption expenditures for a large number of categories and subcategories. Consumption is nondurable in the model, so I only consider consumption expenditures on food, alcoholic beverages, clothing, transportation expenses, entertainment, personal care products, reading materials, and tobacco. Durables such as household appliances are excluded. The household’s consumption from the first two quarters are added to form one observation and the last two quarters are similarly added. This forms two consecutive obervations for each household. If I had assumed the shorter period length of a quarter, with four observations, then agents in the model with reasonable life spans live twice as many periods. This makes computation take longer. In the other direction, one wants to have at least two observations per household to observe consumption changes over time. The model makes predictions concerning the magnitudes of individual changes in consumption and leisure from one period to the next. The model also considers individuals while the data is reported for households. This requires some method for taking household consumption from the data and deriving a number which best corresponds to the object c in the model. Simply dividing household consumption by the number of household members does not take into account that children consume less than adults and that adults derive satisfaction from the consumption of their children. For these
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C. Phelan, Incentives, insurance, and consumption and leisure
reasons and for simplicity, I treat children as consumption goods and impute c by dividing household consumption by the number of adult houselhold members.4 The models also have constant aggregates, and consumption is in real units. The data set reflects inflation, seasonality, and aggregate changes (including growth). Thus an individual’s consumption for each six-month period is defined as the individual’s log consumption minus the average log consumption of those observed during the same time period. Since labor efforts are assumed equal to zero or one in the model, I record those reporting to have worked an average of 20 or more hours a week for a given six-month period as having worked one that period and all others as having worked zero.5 Lastly, I do not attempt to measure output q. There is nothing in the data which reports a household’s contribution to societal wealth (the role of q in the models). A tempting, but inappropriate measure of q is reported income. However, because of the existence of employeremployee relationships and government insurance programs, reported income could just as well be a measure of the result of insurance schemes smoothing over shocks to q as a measure of q itself. For this reason I assume that q is unobserved to the econometrician.
3.3. Data summary These assumptions make the testable implications of the full-information and unobserved-effort models as stated, the fraction of agents of any age group which go through a given work and consumption transition, F(a, c, a’ c’lage). I focus on the same moments used to discuss variability in the earlier sections: the fraction of agents in each quadrant representing the four work states, n(a, a’ )age), and the cross-sectional and intertemporal variances of consumption
4Lazear and Michael (1989) use CES data to estimate the proportion of family consumption which goes to children versus adults, and thus I could have used their estimates to impute the consumption of the male reference person (or male spouse of the reference person). Nevertheless, this would have ignored the fact that in the model all consumption utility comes from the agent’s own consumption, but in reality people care about their children. Treating a child’s consumption as equivalent to the parent’s (in the parent’s eyes) is equivalent to a simple altruism framework. ‘It is important to note that since the incentive constraint is satisfied, we can ex post assume agents actually work the recommended amount, and thus the data reflects true work levels. 61t is important to note that incentive theory only makes predictions regarding the mapping from outcomes (q’s in the model) to consumption and leisure levels, The theory makes no predictions regarding the mechanisms by which such mappings are implemented. Many compensation and insurance schemes (or mechanism) can achieve the same consumption and leisure mapping, thus no compensation (or earnings) predictions are implied by the model.
C. Phelan, Incentives, insurance, and consumption and leisure
589
for each work group, UC@,a’1age) and ud(a, a’lage). Table 1 shows the estimated moments7 The first section of table 1 shows the percentage of agents in each work category - n(u, u’l age). The first column of this section shows that at higher ages a larger fraction of agents are observed not working in either period. The next two columns show that very few people are observed to switch from working in the first period to not working in the next, or vice versa. Moments conditional on switching actions are thus poorly estimated. The next section displays the variability of consumption over time ~ ud(u,u’lage). One fact that will prove important is that for all ages grouped together the variance of time-differenced consumption is only slightly greater for those who work than those who don’t. Otherwise both workers and nonworkers tend to show a greater variance as age increases, but not monotonically. The last section of table 1 shows that the cross-sectional variance of consumption for those who do not work either period tends to rise with age, and that the variance of consumption for those who work both periods is less than those who don’t work, and may tend to rise with age except for the 18-28 year olds. I will focus heavily on these profiles in the next section.
4. Implications
of the data on the incentive method
This section discusses the extent to which the unobserved effort model is capable of producing variation consistent with the observed variation from the Consumer Expenditure Survey and to what extent it improves upon the ‘The standard errors I report assume that the observations are drawn independently from the population of white, non-Hispanic, high-school-educated males. The standard error of the estimated n(a, a’1age) is defined by
C& a’Iage)U- 0, ~‘IwW%w)1°~5 , where N(age) (the sample size) is the number of uc(a, a’( age) is defined as
(c + c’) CXC
~
2
of observations
for each age group. The standard
2
- mc(u, a’/ age)
(6)
1 2
- ~(a, a’( age)
N(age) n(a, a’ I333
F(u, c, a’ c’ 1age)
error
I. 0.5
(7)
The standard
error of ud(a, a’1 age) is defined as
J$X’c[{(c
- c’) -md(~,a’jage))~
- ud(u,u’lage)]2F(u,c,u’c’~age)
N(age) n (a, a’ Iage)
1(8) . O3
590
C. Phelan, Incentives, insurance. and consumption and leisure Table 1 Estimated
Age group
moments,
where a and a’ are the work levels in the first and second six-month period, respectively. a = 0, a’ = 0 Work
transition
a = 0, a’ = 1 fractions
a=l,a’=O
a = 1, a’ = 1
by age group
18-28 29-39
2.5% (1.0%)
1.2 (0.7%)
(0%)
100% (-) 96.3% (1.2%)
40-50
3.5% (1.3%)
3.0% (1.2%)
0.5% (0.5%)
92.9% (1.8%)
51-61
12.4% (2.5%)
(0%)
(O-“)
87.6% (2.5%)
62-90
66.3% (3.6%)
2.8% (1.3%)
4.0% (1.5%)
26.9% (3.4%)
18-90
16.7% (1.2%)
1.6% (0.4%)
0.9% (0.3 %)
80.8% (1.3%)
p_”
Intertemporal
variance
p-”
of log consumption
(Y)
by age group
18-28 (I)
(I) 29-39 40-50 51-61
(I)
0.105 (0.023)
0.013 (0.004)
0.084 (0.034)
(I)
0.120 (0.032)
0.042 (0.012)
0.035 (0.024)
(1)
0.088 (0.014)
(7)
0.138 (0.03 1)
0.161 (0.054)
(1)
62-90
0.095 (0.014)
0.236 (0.088)
0.135 (0.044)
0.201 (0.061)
18-90
0.101 (0.014)
0.179 (0.83)
0.127 (0.044)
0.120 (0.014)
Cross-sectional
variance
of log consumption
by age group (I)
0.202 (0.027)
18-28 (I) 29-39 40-50 51-61
(1)
0.072 (0.029)
0.221 (0.090)
(I)
0.162 (0.0 19)
0.180 (0.078)
0.090 (0.034)
(I)
0.174 (0.02 1)
(I)
0.184 (0.021)
0.320 (0.113)
(I)
62-90
0.257 (0.029)
0.049 (0.032)
0.059 (0.03 1)
0.201 (0.044)
18-90
0.255 (0.028)
0.225 (0.83)
0.061 (0.024)
0.184 (0.011)
C. Phelan, Incentives, insurance, and consumption and leisure
591
characteristics of the full-information/complete-markets framework. The first subsection generally discusses how the unobserved-effort model generates both cross-sectional and intertemporal variability. The following subsections then discuss the ability of the unobserved-effort model to match the moments of the data through the use of computed examples. The principal quantitative result is that although the unobserved-effort model generates endogenous variability, most variability must nevertheless be attributed exogenous variability such as initial inequality and measurement error. The kind of endogenous variation the unobserved-effort model produces is inconsistent with a majority of the observed variation. A secondary result is that the unobserved-effort model constitutes an improvement on several dimensions. It allows for early retirement and for the cross-sectional variances to depend on age and work class. Both of these characteristics are present in the data.
4.1. Solution characteristics In the full-information model, cross-sectional variation in work levels and consumption can only be caused by initial cross-sectional variation in the initial expected utilities, w. Those required to receive high expected discounted utilities are given high consumption levels. Agents with very high w’s are given high leisure (a = 0) as well as high consumption. Because agents are insured against variation in their own output, they receive the same consumption and leisure amounts for each period in life up to the retirement age R. Upon reaching R, the agent’s labor supply level may fall, but his consumption will remain constant if utility is additively separable as assumed here. The cross-sectional distribution of consumption is thus the same over all age groups and the same for retired as well as working people. Intertemporal variability in the full-information model is nonexistent except for the intertemporal labor supply variability associated with an agent retiring. The unobserved-effort model generates endogenous variability over and above the exogenous variability present in its full-information counterpart. In the unobserved-effort model, as in the full-information model, cross-sectional variation in expected discounted utilities causes cross-sectional variation in work levels and consumption. However, in the unobserved-effort model there is endogenously generated variability as well. This is best understood by considering the recursive nature of solutions to the unobserved-effort model. As shown in Phelan and Townsend (1991) and Spear and Srivastava (1987), the expected discounted utility of an agent from any point in time on sufficiently describes the history of that agent. Thus, the punishments and rewards for first-period outputs consist entirely of making first-period consumption contingent on first-period output and making promised expected
592
C. Phelan, Incentives, insurance, and consumption and leisure
discounted utility w for the next period contingent on first-period output. These promises then generate the new distribution of required expected discounted utilities for the environment where agents have T - 1 periods of life left. In the full-information model, if one standardizes for the number of periods left, the distribution of expected discounted utilities stays constant over the life cycle. In the unobserved-effort model, variation in expected discounted utilities grows as agents are punished and rewarded based on output levels, again standardizing for the number of periods left. High outputs are rewarded not only with high current consumption amounts but also by high expected future utilities, w’. As variability in expected discounted utilities grows, variability in consumption grows as well. It is important to note that in solutions to the unobserved-effort model, changes in log consumption tend to be nearly permanent. That is, the typical agent’s log consumption follows a near random walk. For a wide variety of parameter choices, the autocorrelation of consumption changes was never observed below -0.18 and was usually observed much closer to zero. This autocorrelation was never observed below -0.08 for parameters that produce nontrivial amounts of variation. This persistence can also be seen in the fact that in computed examples the intertemporal variances are much smaller than the cross-sectional variances. Differences across individuals are much greater than differences for the same individual over time. (Again, this characteristic was observed for every parameter configuration tested.) If consumption levels for individuals were i.i.d. across time, then the intertemporal variances would be exactly four times greater than the cross-sectional variances.*
4.2. Implications for the unobserved-effort
model
Figs. 1 through 5 present the moments of an example designed to illustrate the extent to which the unobserved-effort model can produce moments near to some of those in the data, and why one must assume high amount of initial variation and measurement error. The example is in some sense an informal ‘best fit’ of the model. The parameters for the example are the result of the author testing scores of parameter specifications and choosing one specification that allowed the illustration of where the model fails and succeeds. The specification of this example is as follows. Preferences are specified by the curvature parameter for consumption a = 0.5, the disutility of effort J. = 9.9, and the discount parameter fl = 0.97. Generally, as agents become less risk-averse and discount less, the incentive model produces more endogenous consumption variation. The parameters CI= 0.5 and ‘This comes from the 2 in the denominator of the definition and the fact that var(X/constant) = var(X)/constant’).
of the cross-sectional
variance,
eq. (3),
C. Phelan, Incentives. insurance, and consumption and leisure
593
/? = 0.95 represented the smallest amount of risk aversion and discounting (over a six-month time horizon) compatible with other studies. The utility loss associated with working, I = 9.9, has no real justification. As workers dislike work more, the incentive constraints tighten, causing more endogenous variation. The number used was arrived at because it helped produce moments that best matched the data. The technological parameters are specified by the probability of the low output given each effort level, P(dg) = 0.97 and P(q/ci) = 0.845,9 and the output levels themselves, q = 650 and q = 4000. One result of the large number of parameter specifications tested was that technologies where even those who take the high effort level usually fail tend to produce high rewards for those who succeed. Thus these parameters were chosen for thier ability to produce significant quantities of endogenous consumption variability. A more technical specification is that the set of consumption levels C is the even grid (in logs), e5, e5.1, e5.‘, . . . , e9. For each model, the reported moments are sensitive to the fineness of the consumption grid, however, this sensitivity becomes smaller as the grid is made finer. The above grid of 41 consumption points was chosen because the moments of the unobserved-effort model did not change much (three significant digits) to further refinements of the grid. Lastly, the technological rate of discount, 6, is set equal to the preference rate of discount, /I Agents are assumed to live for 105 six-month periods (to match the observation period in the data), the first 90 of which they can work. Age 18 in the data is assumed to equal age one in the model. Lastly, the example assumes initial variation in expected utilities and measurement error of consumption. These levels will be explained shortly. Fig. 1 shows the fraction of agents who work both periods. This example produces moments near the data not only for all groups together, but also improves on the full-information model by allowing for a decreasing fraction of agents working as age increases. Under all parameter choices, the full-information model allows only for an initial fraction of agents to not work due to initial diversity, and for this fraction to remain constant until the age where everyone is forced to retire.” Fig. 2 shows the intertemporal variances of consumption for agents who work neither period, adjusting for assumed measurement error in consumption which ’ P(~/cJ) = 0.03 and P(g/ici) = 0.155 are implied by P(q/g) and P(q/ti).
“‘It should be noted here that it is only this simple version of the full-information model which does not allow for early retirement. For instance, I do not allow for sickness. There is a large body of theoretical and empirical work [see Lazear (1986) for a survey] on the causes of retirement. It is difficult to compare this work with most theoretical work on retirement since most often the problem is posed as finding the optimal retirement age given an exogenous wage and benefits profile. Empirical papers tend to focus on the effects of changes in Social Security and other pension benefits. This work allows for endogenous retirement and endogenous retirement benefits.
594
C. Phelan, Incentives, insurance, and consumption and leisure
Fig
1. Fraction
working
both periods.
+
0
+ 0
d 16-28
29-39
40-50 age
Fig. 2. Intertemporal
variance:
+
+
d +
q
+
+
51-61
62-90
18-90
group
Working
neither
period
is explained shortly. Measurement error must be assumed for the following reason. For those who have stopped working (including those going from working to not working) there is again full insurance against variations in output, and thus the intertemporal variances are zero. If agents aren’t working, there is no incentive problem and thus no reason to make consumption contingent on output.” Since the observed intertemporal variance for non workers is
r’ For those going from working to not working, full insurance exists because the consumption in both periods occurs after the output from the first period is realized. This first output is the only one that depends on effort and thus any punishment or reward based on this output will affect the consumption in both periods equally.
C. Phelan, Incentives, insurance, and consumption and leisure
595
1
I
-I
16-28
29-39
40-50 age
Fig. 3. Intertemporal
51-61
62-90
18-90
group
variance:
Working
both periods.
quite significantly different from zero (as can be seen in table l), the unobservedeffort mod.el (in this simple form) is simply incompatible with a basic fact of the data. A possible correction is to assume that consumption is measured with error, which, by the nature of panel data, is certainly true. If this error term is independent and log-normally distributed with a variance of D’, this will add 2~* to the predicted intertemporal variance for each age and work group and a*/2 to each predicted cross-sectional variance.‘* However, when all ages are grouped together, the intertemporal variance of consumption for nonworkers is not very different (and not significantly different) than the intertemporal variance for workers. Thus, if one sets O* high enough to account for the intertemporal variability of nonworkers and chooses any set of parameters which causes the model to match the intertemporal variance of workers, then most of the intertemporal variance for both work groups will be due to measurement error. This can be seen in figs. 2 and 3. Here 2a2 is set equal to the observed intertemporal variance of consumption for nonworkers as a whole. However, the intertemporal variance for workers (fig. 3) is only slightly higher than what it would be without the incentive problem. What it is possible to choose different parameters to increase this difference, one is still constrained to match the observed intertemporal variance for workers, and thus most of the intertemporal variance will still be due to the assumed measurement error. Figs. 4 and 5 display the cross-sectional variance of consumption by age group. Cross-sectional variation can be endogenously generated in the unobserved-effort model through the output-based punishing and rewarding of agents, “These follow from var(A + B) = var(A) + var(B) for independent A and B and var(A/constant) = var(A)/constant’). Again, the difference in the effects comes from the 2 in the denominator of eq. (3).
596
C. Phelan. Incentives, insurance, and consumption and leisure
18-28
29-39
40-50 age
Fig. 4. Cross-section
51-61
62-90
18-90
group
variance:
Working
neither
period.
_I
18-28
29-39
40-50 age
Fig. 5. Cross-section
51-61
62-90
18-90
group
variance:
Working
both periods.
or exogenously by assuming measurement error or initial inequality in expected discounted utilities. Each method produces different results. As shown above, increases in assumed measurement error increase the crosssectional variances of all age and work groups equally. An increase in initial inequality works similarly. It raises the level of cross-sectional variability that would have existed even without the incentive problem, and it raises this level equally over all age and work groups. On the other hand, changing a preference or technology parameter does not affect all age and work groups equally. Increases in punishments and rewards have disproportionately greater effects on the cross-sectional variances of older groups since punishments and rewards are nearly permanent. As agents age, punishments and rewards accumulate.
C. Phelan, Incentives, insurance, and consumption and leisure
597
Put differently, the unobserved-effort model predicts an upsloping age-variance profile for workers as in the solid line on fig. 5. Increasing initial inequality or measurement error raises the base level, or intercept, of this profile, while increasing the severity of punishments and rewards tends chiefly to increase the slope. Increasing initial inequality or measurement error also equally increases the crosssectional variances of both work groups as a whole (the 18-90 age group), while increasing the severity of punishments and rewards disproportionately increases the cross-sectional variance of nonworkers since they are older than workers. While it is possible to choose different parameters to increase or decrease the slope of the age-variance profile in fig. 5, one is still constrained to match the leftmost point, the observed cross-sectional variance for young workers. Increasing the severity of punishments and rewards will increase the difference across models of the cross-sectional variance of 18-28 year olds (the difference between the two lines at the leftmost point). However, the main effect of this is to increase the slope of the age-variance profile for the unobserved-effort problem, causing one to overshoot the observed variances for the older age groups and for nonworkers.13 Thus one must assume at least enough initial inequality to raise the intercept to an acceptable level. In this example, this causes the population cross-sectional variance of consumption for the unobserved-effort model to be only one-third higher than its full-information counterpart. (Equivalently, 25% of the cross-sectional variances in this example can be attributed to the incentive problem.) One cannot choose parameters to greatly increase this difference without contradicting the data points in figs. 4 and 5. In the other direction, one cannot simply raise the level of initial inequality and cause the full-information model to match all the cross-sectional moments. While one cannot reject that the profile in fig. 5 is flat (as the full-information model predicts), the model also predicts that the cross-sectional variance of consumption for old retired workers equals the cross-sectional variance of young workers. In the full-information model, cross-sectional variance is caused only by variability in initial utility promises and measurement error - both assumed unrelated to age or work group. Nevertheless, in the data the crosssectional variance of consumption for workers of all ages is significantly lower than that of nonworkers. The unobserved-effort model correctly predicts a higher cross-sectional variance for nonworkers since nonworkers are older (in both the theory and the data) and thus have accumulated a lifetime of punishments and rewards.14 This and the fact that the unobserved-effort model allows 13Again, it is important to note that this is a general solution characteristic, not dependent on the parameters of this example. It follows directly from the near permanence of consumption changes in solutions to the unobserved-effort model. 14This study is not the first to look at these type of age-variance profiles. Nevertheless, most of the labor literature, starting with Mincer (1974), focuses on the variability of wages. Mincer argues that if life-cycle wage profiles for different careers have varying slopes and intercepts, then life-cycle variance profiles can be increasing or U-shaped.
598
C. Phelan, Incentives. insurance, and consumption and leisure
for early retirement allows for the claim that improves upon the ability of the full-information
the unobserved-effort model model to match the data.
5. Conclusions The inability of this particular dynamic agency model to account for the majority of the variability we see is more robust than one would first guess given the extreme assumptions of the particular model I consider. In my unpublished dissertation of the same title [Phelan (1990)] I show that an unobservedendowment model similar to Green (1987) suffers from a nearly identical inability to match the data, and for the same reasons. In Green’s model, as in the one presented here (but unlike the data), each agent’s consumption should start out equal and consumption changes should be permanent. Further, I show in the dissertation that simply assuming closed insurance markets over outputs 4, and thus having each agent solving a permanent-income consumption-smoothing problem, also has similar difficulty in matching the moments of the data without large amounts of initial inequality. Again, that model had difficulty generating sufficient amounts of endogenous early-life consumption inequality. Simply put, any time-separable moral-hazard or limited-insurance model where all periods of life are identical will have a very difficult time generating sufficient levels of early-life consumption inequality, and thus early-life inequality will have to be brought about by dispersion in initial required utility levels. Given this, what conclusions should be drawn? One possibility is that the majority of the cross-sectional variation we see can be attributed to inequality of initial wealth levels. However, results from Altonji, Hayashi, and Kotlikoff (1989) suggest that inequality in initial wealth levels is probably not a sufficient explanation for cross-sectional consumption inequality. Specifically, they find significant dispersion in the consumption of yuoung adult siblings, even though one would expect that siblings would start life with equal wealth levels. There are other possibilities as well. The conclusion in section 4 that one must assume large amounts of dispersion in initial expected discounted utilities does not necessarily mean one must assume dispersion in initial wealth levels. Dispersion in expected utilities can also be interpreted as dispersion in innate skill levels if society allows agents to keep most of the fruits of their production. In fact, this can be considered a form of initial wealth dispersion in that those with higher innate skill levels ‘own’ a higher level of human capital and thus are wealthier, although not by conventional measures. Nevertheless, this is problematic. If skill levels are observable, there is no incentive-based reason why those with higher skill levels should consume more than those less skilled. Note that differing skill levels can be brought into the model by having heterogeneity in the possible output levels. High skilled people could be represented as having possible output levels @ and q* and low skilled
C. Phelan, Incentives. insurance, and consumption and leisure
599
as having possible output levels 4’ and ql, where gh > 4’ and q” > ql. If agents of both skill levels have the same function P mapping effort levels toprobabilities of their respective output levels q and 4, then both types will be treated exactly the same, holding expected discounted utility constant. The differing q’s will only affect aggregate output and thus the ability to support any distribution of expected discounted utilities. What one needs to get a model of incentive-based variability to look like the variability we see in data is an incentive-based theory of why expected discounted utilities differ across agents at the beginning of adulthood. An incentivebased theory requiring those with higher innate skills to consume more than those less skilled may suffice. Nevertheless, such a model would entail incentive problems markedly different than the kinds of incentive problems posed in standard dynamic agency models.
References Abel, A. and L.J. KotlikolT, 1988, Does the consumption of different age groups move together? A new nonparametric test of intergenerational altruism, NBER working paper no. 2490. Altonji, J.G. and A. Siow, 1987, Testing the response of consumption to income changes with (noisy) panel data, Quarterly Journal of Economics 102, 293-328. Altonji, J.G., F. Hayashi, and L.J. Kotlikoff, 1989, Is the extended family altruistically linked? Direct tests using micro data, NBER working paper no. 3046. Altug, S. and R. Miller, 1990, Household choice in equilibrium, Econometrica 58, 543-570. Atkeson, A. and R. Lucas, 1992, On efficient distribution with private information, Review of Economic Studies 59, 427-453. Cochrane, J., 1991, A simple test of consumption insurance, Journal of Political Economy 99,957-976. Green, E., 1987, Lending and the smoothing of uninsurable income, in: E. Prescott and N. Wallace, eds., Contractual arrangements of intertemporal trade (University of Minnesota Press, Minneapolis, MN) 3-25. Hayashi, F., 1985a, The effect of liquidity constraints on consumption: A cross sectional analysis, Quarterly Journal of Economics 100, 183-206. Hayashi, F., 1985b, The permanent income hypothesis and consumption durability: Analysis based on Japanese panel data, Quarterly Journal of Economics 100, 1083-1113. Hayashi, F., 1987, Tests for liquidity of constraints, in: T. Bewley, ed., Advances in econometrics: Fifth world congress, Vol. 2 (Cambridge University Press, Cambridge) 91-120. Lazear, E.P., 1986, Retirement from the labor force, in: 0. Ashenfelter and R. Layard, eds., Handbook of labor economics, Vol. 1 (Elsevier Science Publishers, Amsterdam) 305-355. Lazear, E.P. and R.T. Michael, 1988, Allocation of income within the household (University of Chicago Press, Chicago, IL). Mace, B., 1991, Full insurance in the presence of aggregate uncertainty, Journal of Political Economy 99,928-956. Mincer, J., 1974, Schooling, experience and earnings (NBER, Cambridge, MA). Phelan, C., 1990, Incentives, insurance, and the variability of consumption and leisure, Ph.D. dissertation (University of Chicago, Chicago, IL). Phelan, C. and R.M. Townsend, 1991, Computing multiperiod, information-constrained optima, Review of Economic Studies 58, 853-881. Shiller, S., M. Boycko, and V. Korobov, 1991, Popular attitudes toward free markets: The Soviet Union and thk United States compared, American Economic Review 81, 385-400. Spear, S. and S. Srivastava, 1987, On repeated hazard with discounting., Review of Economic Studies, 180, 599-618. Townsend, R.M., 1989, Risk and insurance in village India, Manuscript.