2001 Maliar y Maliar

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2.3. Aggregation

In this section, we show how to employ aggregation theory in order tosimplify characterization of the equilibrium under the assumption of theCobb }Douglas and addilog preferences. Given that both of these prefer-ences are commonly used in macroeconomics, it is of interest to studytheir implications for the economy with heterogeneous consumers. Anotherreason for our choice is that these two types of preference are assumedin Kydland (1984) and GMV (1995), respectively. Considering bothCobb }Douglas and addilog preferences will enable us to evaluate whetherthe discrepancy in " ndings of these studies is explained by the preferencechoice.

2.3.1. Quasi-homothetic preferencesGorman 's (1953) theorem implies that if the consumers di ! er only in wealth

and have identical quasi-homothetic preferences, then demand is linear inwealth and, as a result, at the aggregate level, the economy behaves as if therewas a single representative consumer. Examples of Gorman 's (1953) aggregationin neoclassical economies with a single consumption commodity are discussedin Chatterjee (1994) and Caselli and Ventura (1996).

It turns out that with heterogeneity in both wealth and skills, the aggregation

result changes. Precisely, demand for physical hours worked nQ

R is not linear inwealth any longer; however, demand for e $ ciency hours worked nQReQQ is. Conse-quently, the preferences of a representative consumer depend on aggregatee$ ciency hours worked, hR, and not on aggregate physical hours worked,nR" 1 nQRd Q. Rather than elaborate a strict proof of this fact, we illustratethe aggregation results by using a particular example of quasi-homotheticpreferences.

Assume that the agent 's momentary utility is of the Cobb }Douglas type

u(cQR, nQR)"((cQR)I(1! nQR) \ I) \ E! 1

1!, 1' ' 0, ' 0. (7)

Then, the equilibrium in the model can be described by a single-agent utilitymaximization problem

max+AR FR I R> , RZ2

ER

R(cIR(1! hR) \ I) \ E! 11!

s.t. RC , (8)

a set of equations which relates individual and aggregate allocations

cQR" cR f Q, nQR" 1! (1! hR) f Q

eQ, (9)

1374 L. Maliar, S. Maliar / Journal of Economic Dynamics & Control 25 (2001) 1367 }1397


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Then, the equilibrium in the model can be characterized by a utility maximiza-tion problem of a single-agent

max+AR FR I R> , RZ2

ER

R c\ AR ! 11!

# XB(1! hR) \ N! 1

1!s.t. RC , (12)

a condition which identi " es the parameter X

X " 1 (eQ) \ N( f Q)ANd Q N, (13)a set of equations which relates individual and aggregate allocations

cQR" cR f Q, nQR" 1! (1! hR)X \ N(eQ)\ N( f Q)AN, (14)

and conditions which determine the agents ' shares of consumption + f Q, QZ1

ER

Ou (cO, hO)u (c , h )

[ cOf Q! wOe

Q(1! (1! hO)X\ N(eQ)\ N( f Q)AN)]" Q , (15)

where f Q" ( Q) A/ 1 ( Q) Ad Q.

Proposition 3. Under utility (11), (4)}(6) and (12)}(15) are equi valent.

Proof . See Appendix B.

Here, as in the case of quasi-homothetic preferences, the preferences of therepresentative consumer depend on aggregate e $ ciency hours worked and theequilibrium relation between individual and aggregate quantities is explicit.There are two di ! erences, however. First, the utility of the representativeconsumer depends on the joint distributions of wealth and skills through the

parameter X and, second, constraint (15) does not allow for a closed-formsolution with respect to the consumption share, unless " .As the utility (12) depends on the endogenous parameter X , a numerical

solution cannot be computed without iterations. An example of algorithm whichcalculates the equilibrium is as follows: " x X to some value, solve model (12),compute the shares from (15) and restore X according to (13); iterate on X untila " xed-point solution is found. Note that, unlike the algorithm iterating onwelfare weights, the above procedure iterates on a single parameter X indepen-dently of the number of heterogeneous agents. As a result, with any number of agents, the cost of calculating a numerical solution by using this algorithm is

roughly equal to that of " nding the equilibrium in the two-agent model by usingiterating on welfare weights.



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3. Model 's implications

This section is dedicated to qualitative analysis of the model 's predictions.Speci " cally, we study the relation between the implications of the model at theindividual and aggregate levels and identify the factors which play a determinantrole for the properties of the equilibrium.

3.1. Aggregate dynamics

Let us " rst discuss a potential e ! ect of the parameter X which appears in theutility of the representative consumer under the assumption of the addilogutility. According to (12) this parameter premultiplies the utility parameterB and, therefore, has the same e ! ect on the properties of the model as a variationin B. Normally, the parameter B is chosen to match average hours worked in thereal economy. The estimate of average hours worked, however, largely dependson whether the time endowment is measured in terms of &real ' or &discretionary 'time (discretionary time is total time minus time spent on personal care such assleeping, eating, etc.). This implies a substantial degree of freedom in the choiceof the parameter B. For instance, when calibrating identical models, Hansen(1985) assumes &discretionary ' time and obtains B" 2, while Christiano andEichenbaum (1992) match &real ' time and get B" 2.99, a value which is about

50% higher. This example suggests that the e ! ect associated with X is of littleinterest (at least, for the second moments of aggregate variables) unless it di ! ersfrom one by a very large order of magnitude. Therefore, the parameter X will bedisregarded in this section.

Given the supposition above, under both Cobb }Douglas and addilog typesof preferences, we would expect the following. First, the time series properties of variables +cR, kR, wR, rR, in the model are not a ! ected by the assumed types of heterogeneity. Second, the variable e $ ciency hours worked +hR, in the hetero-geneous model behaves as physical hours worked in the representative agentcase. Therefore, the only implication of heterogeneity which is of potential

interest is its e ! ect on time series properties of physical hours worked +nR, .Consequently, in the rest of the section, we concentrate on the model 's implica-tions regarding labor markets.

Integrating individual hours worked, we get that under both the Cobb }Douglas and addilog utilities, e $ ciency hours and &simple ' hours are related as

hR" 1! (1! nR) ) , (16)

where under Cobb }Douglas utility the parameter is computed from (9)

" 1 f Q

eQ dQ, (17)

L. Maliar, S. Maliar / Journal of Economic Dynamics & Control 25 (2001) 1367 }1397 1377


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while in the case of addilog utility it is obtained from (14)

" 1(eQ) \

N( f Q)

ANd Q

1 (eQ)\ N( f Q)ANd Q . (18)

We call the parameter labor input bias. We will denote by V and corr (x , y) thevolatility of a variable x and the correlation of variables x and y.

Consider two empirical regularities on labor market behavior:

(i) physical hours worked #uctuate more than e $ ciency hours worked;(ii) average productivity and hours worked are weakly correlated.

The " rst fact is reported, for example, by Kydland and Prescott (1993). Using the

PSID data, they construct aggregate labor-input series by summing individualhours worked weighted by average hourly wage and " nd that the resulting series#uctuate less than the unadjusted hours worked. Hansen (1993) obtains a sim-ilar result using monthly data on age-sex groups from the U.S. Bureau of LaborStatistics ' household survey. Note that in order for the model to be consistentwith this empirical observation, it is necessary that ( 1. Indeed, according to(16), we have F " L .

The second fact is well-known to the literature and is often referred to as theDunlop }Tarshis observation. Christiano and Eichenbaum (1992) document thisregularity for the U.S. economy. Also, they show that the representative agent

version of the model insistently predicts that this correlation is close to one and,therefore, does not reproduce fact (ii). Below we demonstrate that by taking intoaccount the e ! ect of heterogeneity on labor markets, we can improve themodel 's predictions in this dimension.

It follows from (16) that physical and e $ ciency hours worked are perfectlycorrelated in the model, corr (n, h)" 1, and, thus,

corr (y/n, n)" corr (y/h )h/n, h),

where y/n is average labor productivity. Given that in the heterogeneous model,hR behaves as nR in the representative agent model and also, that the representa-tive agent model generates almost perfect positive correlation between produc-tivity and hours worked, we have corr (y/h, h)K 1. Therefore, corr (y/n, n) in theheterogeneous case depends on how the variable hR/nR behaves over the businesscycle. Eq. (16) implies that whether this variable is procyclical or countercyclicaldepends on the value of the parameter

d(hR/nR)dhR

"1nR

1!dnR/nRdhR/hR

"!1!

nR. (19)

It follows by the last result that if ( 1, then the correlation between productiv-

ity and hours worked in the heterogeneous model will be smaller than thisstatistic in the representative agent setup.



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3.2. Indi vidual dynamics

In this section, we focus on the relation between aggregate and distributivepredictions of the model. For the purpose of subsequent analysis, we assumethat there is a continuum of agents in the economy so that decisions of a singleindividual have no e ! ect on aggregate allocation.

Kydland (1984) and R m Hos-Rull (1993) document two empirical regularities of the individual labor choice:

(iii) individual hours worked are increasing in skills;(iv) the volatility of individual hours worked is decreasing in skills.

In terms of our model, empirical facts (iii) and (iv) imply d nQ

R/d eQ'

0 andd QR/d eQ( 0 respectively, where

QR"

dnQRdwR

wRnQR

is the period elasticity of agent 's labor supply with respect to wage.First, we demonstrate that if regularity (iii) is satis " ed in our model, then (iv)

will also be satis " ed. Using the formulas for individual working hours given in(9) and (14) for the Cobb }Douglas and addilog utilities, one can show that

d QRdeQ

"! RhR

nQR(1! hR)dnQRdeQ

, (20)

where

R"dhRdwR

wRhR

' 0.

The latter follows because R is the same as the elasticity of labor with respectwage in the representative agent model.

Let us analyze how distributive facts (iii) and (iv) are related to previouslydiscussed aggregate regularities (i) and (ii). Using formula (19), the change insupply of physical and e $ ciency hours worked in response to wage increase canbe written as

dnR"dwRwR 1 nQR QR d Q, dhR" dwRwR 1 nQR QReQd Q.

Together with (19), these formulas imply

d(hR/nR)dhR" 1 " 1 n

Q

RnQY

R (eQ!

eQY

)(Q

R! QY

R ) dQ

dQY

nR ) 1 nQR QReQd Q. (21)



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Observe that if individual elasticities decrease in skills, d QR/d eQ( 0, then eachterm in the numerator of the above expression is negative and, therefore,

d(hR/nR)/dhR( 0. As shown in the previous section, the last result corresponds to( 1 which is consistent with aggregate facts (i) and (ii).

The elasticity considerations allows us to gain some intuition on why thecorrelation between productivity and hours worked in the heterogeneous modelmight be lower than in the representative agent model. In response to a positiveshock, all individuals supply more hours on the market. However, if theelasticity of labor supply decreases in skills, the increase in hours of low skilledworkers is larger than that of high skilled workers. As a result, a fraction of lowskilled hours in total hours worked is larger after the shock than it was beforethe shock and the average productivity of labor, yR/nR, goes down. If thedi! erence in the elasticities across agents is high enough, then it could be thecase that corr (y/n, n)( 0.

3.3. Working hours

The analysis of the previous section suggests that the model can produce theappropriate aggregate and distributive predictions only if it is capable of generating hours worked which increase in the level of skills (fact (iii)). Toevaluate the model 's ability to account for this observation, we will employ an

additional empirical regularity at the individual level, namely:

(v) wage di! erentials across agents do not exceed wealth di ! erentials.

In terms of our model, this implies d log Q /d log eQ5 1, since this ratio givesus the percentage change in wealth relative to one percent change in skills.GMV (1995) divide the PSID sample into two groups according to twoalternative criteria, the wage to wealth ratio and the level of wealth. Dependingon the criterion, they obtain that the di ! erence in the wages of two groupsamount to 2 while the di ! erences in wealth range from 5 to 30. We " ndthat a quantitative expression of fact (v) is highly sensitive to a criterion which isused for dividing the population into groups. For example, in the next section,we will show that if the PSID data are divided according to the educationallevel, then the distributions of wealth and wages across group do not di ! ersigni " cantly.

Let us analyze the relation between facts (iii) and (v). Formulas (10), (15) implythat under both Cobb }Douglas and addilog utility functions, the lifetime budgetconstraint can be written as follows:

EO

Ou (cO, nO)u (c , n ) (cOf Q! nQOeQwO) " Q . (22)



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Di ! erentiating (22) with respect to skills and subtracting from the resultingcondition Eq. (22), previously divided by skills, we get

1!dlog f Q

dlog eQ"

Q

eQ1!

dlog Q

dlog eQ! E

OOu (cO, nO)u (c , n )

eQwOdnQOdeQ

f Q

eQE

OOu (cO, nO)u (c , n )

cO. (23)

Under the assumption of Cobb }Douglas utility function, the condition forindividual hours provided in (9) yields

1!dlog f Q

dlog eQ"

dnQR

deQ1

1! nQR. (24)

Observe that if the model is to reproduce fact (iii) and fact (v), then theright-hand side of (23) must be negative. However, the condition (24) demon-strates that the model can account for fact (iii) only if the left-hand side of (23) ispositive. Thus, under the assumption of Cobb }Douglas utility, the modelcannot explain facts (iii) and (v) simultaneously. In other words, if such model iscalibrated to reproduce the distribution of wealth in the data, then it counter-factually predicts that rich (skilled) agents supply less labor on the market thanpoor (unskilled). Equivalently, if the model is calibrated to match the di ! erencein hours worked across productivity groups, then the implied distribution of wealth is such that regularity ( v) is violated.

For the addilog type of preferences, using (14), we obtain

1!dlog f Q

dlog eQ"

dnQRdeQ

eQ

(1! nQR)!

1# 1. (25)

Note that Eq. (25) together with (23) implies that, similar to the Cobb }Douglascase, the model will also fail with respect to facts (iii) and (v), if the agents 'preferences are of the addilog type with the standard elasticity of intertemporal

substitution for consumption, 1/ 4 1. The above results suggest that theassumptions about preferences which are standard for macroeconomic litera-ture are inconsistent with basic cross-sectional observations.

Kydland (1984) does not analyze the model 's implications with respect to the distribution of wealth and, therefore, does not pin down this failure of the model.

Precisely this feature of the model accounts for GMV 's (1995) puzzle discussed in the introduc-tion. This paper assumes the addilog utility with 1/ " 1, calibrates the model to reproduce thedistribution of wealth (fact (v)), and " nds that such model generates undesirable predictions at both

individual and aggregate levels. Indeed, our analysis implies that in this case regularities (i) }(iv) areviolated.



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There is another implication that follows from (25). Speci " cally, if the utility isof the addilog type and if the intertemporal elasticities of substitution for

consumption and leisure, 1/ and 1/ , are large enough, then the model might beable to reproduce facts (iii) and (v) simultaneously. Our previous analysissuggests that in this case, the model 's predictions will also be consistent with theremaining empirical regularities (i), (ii) and (iv). Given that the model canaccount for all empirical facts of interest only under the assumption of theaddilog preferences, we will study the quantitative implications of the modelonly under this type of preferences.

4. Calibration procedure

This section discusses the calibration procedure. To compute numericalpredictions, we use the model which is adjusted to growth as shown in AppendixC. The solution algorithm is described in Appendix D.

For studying quantitative implications of the model, speci " c values need to beassigned to the model parameters. We calibrate the model to reproduce thecapital to output ratio, I , the consumption to output ratio, A, and aggregatehours worked, n, in the U.S. economy. Output is produced according to theCobb }Douglas production function, f (kR, hR)" k?Rh \ ?R . The parameter choice issummarized in Table 1.

Here, I

is the discount rate in the model with growth, and g is the rate of technological progress. The parameters

I

, d, B are computed from the optimal-ity conditions evaluated in the steady state as it is shown in Appendix C. Weassume that initially the economy is in the steady state, i.e. we set " 1 andchoose aggregate capital, k , to be equal to the steady-state value. The remain-ing parameters are set to the values standard in the RBC literature. The ratios

I and A and the values of the parameters g and are adopted from Christianoand Eichenbaum (1992); following their paper, we de " ne consumption in A asa sum of private and government consumption. The value of n is taken from the

micro study by Juster and Sta ! ord (1991). Finally, the aggregate technologyshock R has the time series representation R" MR\ exp( R), 3 [0,1], whereR& N(0, C) for t3 . The parameters and Care calibrated as in Hansen(1985).

Table 1Model parameters

Parameter I A n g d CValue 10.62 0.727 0.31 1.004 0.993 0.339 0.0217 0.95 0.00712



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Table 2Heterogeneity parameters generated by the household data

Groups

Parameter 1 2 3 4 5 6 7 8

Q 0.268 0.730 0.419 0.600 0.796 1.270 1.853 1.904Q 0.123 0.272 0.624 0.771 0.946 1.129 1.643 2.120

d Q 0.173 0.640 1.274 1.781 0.806 1.624 1.188 0.513

Note: Each heterogeneity parameter is the average of corresponding individual variable in thesubsample. The share, d Q, is the relative weight of the subsample in the total population. The sharesand the heterogeneity parameters weighted by the shares are normalized to 8.

To calibrate the heterogeneity parameters, we use the PSID sample (1989).This data set contains the observations on 7114 U.S. households. FollowingKydland (1984), we split the population into the groups by the level of educationof the head of the household. We remove from the PSID sample 96 householdsfor which the level of education of the head is not available. As a proxy for skills,we use average hourly earnings of the head of the household. Initial endowmentis calibrated according to the total wealth of the household. To compute theaverages, we adjust the households ' observations to PSID sample weights.Exact labels of the cross sections used are < 17545, < 17536, < 17389, < 17612.Table 2 summarizes the estimates.

The groups 1 }8 distinguished in the table correspond to subsamples of thehouseholds, whose heads completed: (1) grades 0 }5; (2) grades 6}8; (3) grades9}11; (4) grade 12 grade (high school); (5) grade 12 plus non-academic training;(6) college but no degree; (7) college BA but no advanced degree; (8) college andadvanced or professional degree.

5. Results

The quantitative implications of the model to a large extent depend on thechoice of the parameters and . To illustrate the tendencies, we report theresults from simulations for several pairs of and .

5.1. Distributi ve predictions

This section reports the model 's predictions on consumption and workinghours of eight heterogeneous groups distinguished in Section 4 and compares

them with the corresponding quantities in the U.S. economy. As a proxy forconsumption, we use monetary income of households. Working hours in the



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U.S. economy are calibrated according to those worked by the head of thehousehold. The groups ' quantities in the U.S. economy are the averages of

individual variables in the subsamples. To compute the averages, we adjust thehouseholds ' observations to PSID sample weights. Exact labels of the crosssections used are < 16335, < 17533. The predictions of the model are computedusing Eqs. (14) and (16) which are evaluated in the steady state. Table 3 presentsthe results.

In Section 3.3, we show qualitatively that the values of the parameters andplay a decisive role in distributive predictions of the model. The results in

Table 3 allow us to evaluate their e ! ect quantitatively. As we can see, thetendency is that, if ' 1, the model " ts relatively well the empirical distributionof consumption; however, it counterfactually predicts that working hours de-crease with the level of skills. If ( 1, the model generates the appropriatepredictions for hours worked; however, it fails to produce the appropriatevariability of consumption across groups.

The e ! ect of the parameter on the groups ' consumption is determined by thevalue of . In such a way, the variability of consumption across groups increaseswith if ' 1, and it decreases with if ( 1. The variability of hours workeddecreases with under any value of ; this implies that if ' 1, an increment in

induces the low skilled agents to work less and high skilled agents to workmore, while if ( 1, the opposite is true. If K 1, the e! ect of the parameter onthe groups ' quantities is ambiguous and small.

Only in two cases from all those reported in Table 3, namely, " 0.6, " 1.0and " 0.6, " 0.2, does the model generate the distributional patterns forboth consumption and working hours as in the U.S. economy. The " t of themodel is somewhat better under " 1.0 than under " 0.2, although in bothcases the variability of consumption across groups is too large. In particular, themodel severely underpredicts the value of consumption for two bottom-skillgroups.

In fact, the model 's failure with respect to individual consumption of un-skilled individuals is not surprising. In the real economies, a large portion

of the expenditures of such individuals comes from government transfersand public services which are not included in the model. For example,D m Haz-Gime Hnez et al. (1996) divide the U.S. population into three groups accord-ing to the level of education; they " nd that the share of transfers in the totalincome of the bottom group (28%) is about six times as high as that of the topgroup (4.7%).

5.2. Aggregate predictions

Table 4 contains the estimates of the parameters X and , and selected second

moments computed using the series for e $ ciency hours, hR, and physical hoursworked, nR. With slight abuse of notation, in this section, V, corr (x , y) will



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T a b l e 3

G r o u

p s ' c o n s u m p t i o n a n d w o r k i n g h o u r s f o r U

. S .

a n d a r t i

" c i a l e c o n o m i e s

H e t e r o g e n e o u s m o d e l

"

1 . 5 ,

"

1 . 5 ,

"

1 . 0 ,

"

0 . 6 ,

"

0 . 6 ,

"

0 . 6 ,

" 1 0 . 0

" 1 . 0

" 1 . 0

"

1 . 0

" 0

. 2

" 0

. 1 5

U . S .

e c o n o m y

C

c Q

n Q

c Q

n Q

c Q

n Q

c Q

n Q

c Q

n Q

c Q

n Q

c Q

n

Q

1

0 . 1 5

0 . 3 5

0 . 2 1

0 . 4

8

0 . 1 3

0 . 2

9

0 . 0 5

0 . 0 8

0 . 0 3

0 . 0 3

0 . 0 3

0 . 0 2

0 . 3

5

0 . 0

5

2

0 . 3 4

0 . 3 3

0 . 3 9

0 . 3

8

0 . 2 9

0 . 2

8

0 . 1 6

0 . 1 4

0 . 1 2

0 . 0 7

0 . 1 1

0 . 1 2

0 . 5

1

0 . 1

2

3

0 . 6 4

0 . 3 2

0 . 7 1

0 . 3

4

0 . 6 2

0 . 3

2

0 . 4 9

0 . 2 8

0 . 4 3

0 . 2 5

0 . 4 3

0 . 2 5

0 . 6

4

0 . 2

6

4

0 . 7 8

0 . 3 2

0 . 8 4

0 . 3

2

0 . 7 7

0 . 3

2

0 . 6 6

0 . 3 0

0 . 6 0

0 . 2 8

0 . 6 0

0 . 2 8

0 . 7

8

0 . 3

0

5

0 . 9 5

0 . 3 1

0 . 9 8

0 . 3

1

0 . 9 4

0 . 3

1

0 . 8 8

0 . 3 2

0 . 8 3

0 . 3 2

0 . 8 3

0 . 3 2

0 . 9

5

0 . 3

4

6

1 . 1 4

0 . 3 0

1 . 1 2

0 . 2

8

1 . 1 3

0 . 3

1

1 . 1 3

0 . 3 4

1 . 1 1

0 . 3 5

1 . 1 0

0 . 3 5

1 . 1

0

0 . 3

6

7

1 . 6 1

0 . 2 9

1 . 4 8

0 . 2

5

1 . 6 5

0 . 3

1

1 . 8 6

0 . 3 9

1 . 9 7

0 . 4 4

1 . 9 8

0 . 4 4

1 . 5

5

0 . 3

9

8

2 . 0 0

0 . 2 9

1 . 7 8

0 . 2

4

2 . 1 1

0 . 3

1

2 . 5 6

0 . 4 3

2 . 8 8

0 . 5 1

2 . 9 2

0 . 5 2

2 . 0

0

0 . 3

9

N o t e : C o n s u m p t i o n

, c

Q

, a n d h o u r s w o r k e d

, n

Q

, w e i g h t e d b y t h e s h a r e s a r e n o r m a l i z e d t o 8 a n d t o 8 ;

0 . 3

1 r e s p e c t i v e l y

.



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Table 4Endogeneous parameters and selected labor statistics for U.S. and arti " cial economies

Heterogeneous model

" 1.5 " 1.0 " 1.0 " 0.6 " 0.6 " 0.6 " 0.6 U.S." 1.0 " 1.0 " 0.2 " 1.0 " 0.3 " 0.2 " 0.15 economy

X 1.145 1.000 0.992 0.894 0.897 0.898 0.898 *1.031 0.998 0.997 0.951 0.930 0.926 0.923 *

F 0.57 0.70 1.15 0.83 1.33 1.47 1.58 *(0.07) (0.09) (0.14) (0.11) (0.16) (0.18) (0.20)

L 0.51 0.70 1.16 0.97 1.65 1.85 2.01 1.66(0.06) (0.09) (0.14) (0.13) (0.20) (0.23) (0.26)

WF 0.73 0.69 0.55 0.64 0.50 0.45 0.43 *(0.09) (0.09) (0.08) (0.09) (0.07) (0.07) (0.07)

WL 0.78 0.69 0.54 0.51 0.25 0.23 0.25 1.18(0.10) (0.09) (0.08) (0.07) (0.05) (0.05) (0.05)

corr (y/h, h) 0.91 0.94 0.86 0.96 0.90 0.87 0.83 *(0.02) (0.02) (0.03) (0.01) (0.02) (0.03) (0.03)

corr (y/n, n) 0.92 0.94 0.86 0.93 0.48 0.05 ! 0.29 ! 0.20(0.02) (0.02) (0.03) (0.02) (0.05) (0.05) (0.09)

corr (y/h, y) 0.98 0.98 0.94 0.99 0.94 0.92 0.89 *(0.00) (0.00) (0.01) (0.00) (0.01) (0.01) (0.02)

corr (y/n, y) 0.99 0.98 0.99 0.97 0.59 0.17 ! 0.17 0.42(0.00) (0.00) (0.00) (0.01) (0.04) (0.06) (0.11)

The standard deviations and correlations are sample averages of statistics computed for each of 400 simulations; each simulation consists of 115 periods. Numbers in parentheses are samplestandard deviations of these statistics. All statistics are computed after " rst logging and thendetrending the simulated time series using the Hodrick }Prescott " lter.

Source: Hansen (1985, Table 1).Source: Christiano and Eichenbaum (1992, Table 4).

represent the volatility of a logged variable x and correlation between logged

variables x and y.As we see from Table 4, in all of the cases considered, the values of X di! erfrom one by less than 15%. In Section 3.1, we have argued that if the parameterX is not signi " cantly di ! erent from one, then e $ ciency hours in the heterogen-eous model behave in the same way as physical hours worked in the representa-tive agent case. To verify this conjecture, we computed the solutions underX " 1 and compared the resulting statistics to those reported in the table. We" nd that the e ! ect of this restriction on the model 's predictions is fairly small (afew percent, at most). Therefore, the statistics F , WF , corr (y/h, h) and corr (y/h, y)can be viewed as the volatilities of hours worked and productivity, and the

correlations between productivity and hours worked and between productivityand output in the associated representative agent model.



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Hansen (1985) and Christiano and Eichenbaum (1992) consider the represen-tative agent version of the model under " 1 and " 1. Hansen (1985) shows

that such a model underpredicts the volatility of hours worked and fails toaccount for the large #uctuations in hours compared to the relatively small#uctuations in productivity. Christiano and Eichenbaum (1992) point out an-other drawback of the model, precisely, its failure to produce the appropriatecorrelation between productivity and hours worked. Also, as follows from thetable, the correlation between productivity and output in such a model is farfrom the one observed in the data. The results reported in Table 4 indicate thata variation in the values of and improves on statistics F and F / WF ;however, it has only a minor e ! ect on corr (y/h, h) and corr (y/h, y). In particular,our " ndings suggest that the representative agent version of the model cannotaccount for the weak correlation between productivity and hours worked underany reasonable values of and .

As is argued in Section 3.1, accounting for labor input bias, , may bring themodel into closer conformity with the data provided that ( 1. From Table 4,we can observe the following tendency. If K 1, then K 1 and the e ! ect of on

is ambiguous and small. If ' 1, then ' 1 and it decreases with ; " nally, if ( 1, then ( 1 and it increases with . In particular, under the elasticities 1/

and 1/ , which are large enough, the value of is substantially smaller than one,and, consequently, the e ! ect of heterogeneity on aggregate dynamics is quantit-atively signi " cant.

Table 4 shows that under " 0.6 and 3 +0.3, 0.2, 0.15, , the model is capableof accounting for the weak correlation between productivity and hours worked;it also generates the correlation between productivity and output which is closeto that observed in the data. Finally, it reproduces the empirical observationdiscussed in Section 3 that the volatility of physical hours worked, L , is largerthan the volatility of e $ ciency hours, F . Unfortunately, under these and , themodel also has undesirable features. Speci " cally, the volatility of productivityis too low and the model 's predictions are not robust to small changes in theparameters.

To understand the origin of the shortcomings, consider the correlationbetween productivity and hours worked

corr (y/n, n)"corr (y, n) W! L

WL

"corr (y, n) W! L

( W# L ! 2 corr (y, n) W L. (26)

If the model is to generate zero correlation between productivity and hoursworked, it is necessary that Wcorr (y, n)K L . Given that in our model corr (y, n)is close to one (see Table 5), the preceding condition together with (26) impliesboth that WL is small and that corr (y/n, n) is highly sensitive to small changes in

statistics L and W. It can be reasonably expected that any modi " cation whichreduces corr (y, n) will improve the model 's performance. Two examples of such



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Observe that under and which are lower than one, the volatilities of consumption, capital, investment and output are closer to the corresponding

volatilities in the data.Summarizing, on the one hand, the heterogeneous agents model improves on

some labor market statistics which the representative agent model seriously failsto predict. On the other hand, the model with heterogeneity preserves thepositive features of the representative agent setup.

6. Conclusion

This paper incorporates into the neoclassical model with labor }leisure choiceheterogeneous agents who di ! er with respect to endowments and non-acquiredskills. We show that the qualitative and quantitative analysis of the equilibriumin the model can be simpli " ed substantially by using the results from aggrega-tion. We demonstrate that under the standard for the macroeconomic literatureassumptions on preferences, the model cannot account for cross-sectional obser-vations. Speci " cally, we " nd that if agents ' preferences are of the Cobb }Douglastype, the model fails to reproduce the simple empirical regularity that high-productivity agents work more compared to the low productivity. We reach thesame conclusion for the case of addilog utility with the standard (smaller thanone) intertemporal elasticity of substitution for consumption. We show, how-ever, that if such elasticity in the addilog utility is set to a value which is largeenough, then the model is capable of generating the appropriate pattern forhours worked. Moreover, such model can account for several time-series labormarket facts which cannot be reconciled within a similar representative agentsetup.

Appendix A

With an interior solution, the " rst-order conditions (FOCs) of agent 's s3 Sutility maximization problem (1), (2) with respect to insurance holdings, capitalholdings and the transversality conditions are as follows:

u (cQR, nQR)pR( )" u (cQR> ( ),nQR> ( ))Pr+ R> " " R" ,FFYZ , (A.1)u (cQR, nQR)" ER[ u (cQR> , nQR> )(1! d# rR> )], (A.2)

limR

E Ru (cQR, nQR) kQR> # pR( )mQR> ( ) d " 0, (A.3)where u , u are the " rst-order partial derivatives of the utility u with respect to

consumption and labor and cQR> ( ), n

QR> ( ) are equilibrium consumption andworking hours as functions of the realization of the aggregate shock.



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FOC (27) implies

ER\u (cQR, nQR)

u (cQR\ , nQR\ ) mQR( R) " m

QR( )pR\ ( ) d . (A.4)

Further, FOC (28) together with the fact that kQR is known at t! 1 yields

ER\u (cQR, nQR)

u (cQR\ , nQR\ )kQR(1! d# rR) " kQR. (A.5)

Multiplying each term of (2) by

u (cQR, nQR)u (cQR\ , nQR\ )

,

taking the expectation E R\ on both sides and using Eqs. (A.5) and (A.4), one canshow that for all t ' 0 the following condition holds:

kQR# mQR( )pR\ ( ) d" E R\

u (cQR, nQR)u (cQR\ , nQR\ )

(cQR! nQReQwR)

#ER\

u (cQR, nQR)

u (cQR\ , nQR\ ) kQ

R>#

mQ

R> ( )pR( ) d .

Applying forward recursion, using the law of iterative expectations and impos-ing transversality condition (A.3), we get

(1! d# r )kQ # mQ ( )

" cQ ! nQ eQw # kQ # mQ ( )p ( ) d" E

OR u (c

Q

O, nQ

O)u (c

Q

, nQ

)(cQ

O! nQ

OeQw

O)

# Eu (cQ , nQ )u (cQ , nQ )

kQ # mQ ( )p ( ) d"2" E

OOu (c

Q

O, nQ

O)u (cQ , nQ )

(cQO! nQ

OeQwO)

# limR

E Ru (cQR, nQR)u (cQ , nQ )

kQR> # mQR> ( )pR( ) d" E

OOu (c

Q

O, nQ

O)u (cQ , nQ ) (cQO! nQOeQwO) .



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The last result corresponds to expected lifetime budget constraint (6) used in themain text.

Appendix B

The FOCs of planner 's problem (4), (5) with respect to cQR, nQR, kR are

Qu (cQR, 1! nQR)" R, (B.1)

Qu (cQR, 1! nQR)" ReQwR, (B.2)

R" ER[ R> (1! d# rR> )], (B.3)

where R is the Lagrange multiplier associated with the economy 's resourceconstraint, wR, R f (kR, hR) is marginal product of labor input.

Proof of Proposition 2. Under the Cobb }Douglas utility, solving for cQR and(1! nQR) from FOCs (B.1), (B.2), we get

cQR"R

1!wR

\ I \ E E( Q) E(eQ)\ \ I \ EE, (B.4)

1! nQR"1!

R

1!wR

\ I \ E E( Q) E(eQ) I \ E\ E. (B.5)

Integration of (B.4) and (B.5) over the set of agents (both sides of the latter arepreviously premultiplied by eQ) yields

cR"R

1!wR

\ I \ E E 1 ( Q) E(eQ)\ \ I \ EEd Q, (B.6)1! hR" 1!

R1!

wR

\

I\

E E 1 (Q) E(eQ)\ \ I \ EEd Q. (B.7)

Dividing (B.4) by (B.6) and (B.5) by (B.7) and introducing f Q, we obtain (9) in themain text.

Rearranging the terms of Eqs. (B.6) and (B.7), we get

u (cR, 1! hR)" R 1 ( Q) E(eQ)\ \ I \ EEd Q\ E

, (B.8)

u (cR, 1! hR)" RwR 1 (Q) E(eQ)\ \ I \ EEd Q

\

E, (B.9)



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where u(cR, 1! hR)" [(cIR(1! hR) \ I) \ E! 1]/(1 ! ). Combining (B.8), (B.9)yields the FOC of (8) with respect to labor; substituting (B.8) into (B.3) gives us

the FOC with respect to capital. These are, respectively,u (cR, 1! hR)" wRu (cR, 1! hR),

u (cR, 1! hR)" ER[ u (cR> , 1! hR> )(1! d# rR> )].

This proves that aggregate dynamics of the heterogeneous economy is describedby single-agent utility maximization problem (8). Finally, condition (10) followsafter substituting into expected lifetime budget constraint (6) both conditionsgiven in (9).

Proof of Proposition 3. Under the addilog utility, conditions (B.1), (B.2) become

cQR" \ AR ( Q) A, (B.10)1! nQR" ( RwR)\ NB N( Q) N(eQ)\ N. (B.11)

Integrating them over the set of agents ( " rst, premultiplying the latter by eQ)yields

cR" \ AR 1 ( Q) Ad Q, (B.12)1! hR" ( RwR)\ NB N 1 ( Q) N(eQ) \ Nd Q. (B.13)

Taking the ratios of (B.10) to (B.12) and (B.11) to (B.13), introducing f Q andde" ning the parameter X as it is in (13) of the main text, we get the twoconditions in (14).

Rearranging the terms in (B.12) and (B.13), we have

u (cR, 1! hR)" R

1

( Q) Ad Q\ A

, (B.14)

u (cR, 1! hR)" RwR 1 ( Q) N(eQ) \ Nd Q\ N

, (B.15)

where u(cR, 1! hR)" (c \ AR ! 1)/(1! )# B((1! hR) \ N! 1)/(1! ). Eqs. (B.14),(B.15) combined together yield the FOC of single-agent problem (12) withrespect to hR. Further, substituting (B.14) in (B.3) we get the correspondingintertemporal condition of (12). They are, respectively,

Xu (cR, 1! hR)" wRu (cR, 1! hR),

u (cR, 1! hR)" ER[ u (cR> , 1! hR> )(1! d# rR> )].



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This veri " es that at the aggregate level the heterogeneous economy behaves assingle-agent economy (12). Finally, after substituting (14) into expected lifetime

budget constraint (6) we get (15).

Appendix C

We introduce growth as it is usually done in the RBC literature. In theeconomy with growth, the problem of an individual s3 S is

max+A

QR L

QR I

QR> K

QR>

F, F Z RZ2

ER

R (cQR) \ A! 11!

# B(1! nQR) \ N! 1

1!gR \ A

s.t.

cQR# kQR> # pR( )mQR> ( ) d " (1! d# rR)kQR# nQReQwRgR# mQR( R),where the parameter g is the rate of labor-augmenting technological progress.Finding the FOCs of the above problem, introducing new variablesc QR" cQRg\ R, k

I QR" kQRg\ R and m QR" mQRg\ R and following the procedure described

in Appendix B, we obtain the conditions that identify aggregate dynamics:

gc \ AR " I

E R[ c \ AR> (1! d# r R> )], (C.1)c \ AR w R" BX (1! hR)\ N, (C.2)c R# k

I

R> g" k I

R(1! d)# R f (k I

R, hR), (C.3)

where r R, R* f (k I

R, hR)/*k I

R, w R, R* f (k I

R, hR)/*hR, the parameter X is de" ned by(C.7), and

I

, g \ Ais the discount rate adjusted to growth. The expectedlifetime budget constraint takes the form

EO

I Oc \ AO

c \

A[ c

Of Q! w

OeQ(1! (1! h

O)X \ N(eQ)\ N( f Q)AN)]" Q . (C.4)

Eqs. (14), (16), (18) do not change after introducing growth except that theappropriate variables in (14) are c QR and c R.

Optimality conditions (C.1) }(C.3) provide a basis for calibrating the para-meters

I

, d and B. Evaluating (C.1) and (C.3) in the steady state yields

I

" Ig

I g# A# ! 1, d"

1! AI

# 1! g.

Condition (C.2) in the steady state can be written as

BX " (1! ) \ A? \ ?I \ AA (1! h)Nh\ A, (C.5)



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where h denotes steady-state e $ ciency labor. The parameter B is calibratedto the same value as in the representative agent case. Setting X " 1 and " 1,

we get

B" (1! ) \ A? \ ?I \ AA (1! n)Nn\ A. (C.6)Dividing (C.5) by (C.6) and substituting the resulting condition into (16) evalu-ated in the steady state, we obtain

X " NnA(1! (1! n) )\ A. (C.7)This formula relates the parameters X and .

Appendix D

To compute the solution, we use the following iterative algorithm:

Step 1. Fix to some level and compute X according to (C.7).Step 2. Use (C.1) }(C.3) to solve for aggregate equilibrium quantities.Step 3. Recover f Q's from (C.4) and recompute according to (18).

Iterate on these steps until the " xed point value of is found.

Once the equilibrium law of motion for the aggregate quantities and thecorresponding set of the agent-speci " c parameters + f Q, QZ1 is known, the remain-ing variables can be restored by direct calculations.

To complete Step 2 of the solution algorithm, we use parametrized expecta-tions algorithm, see, e.g., Marcet (1989). Under this algorithm, the conditionalexpectation in (C.1) is parameterized by a function of the state variables and,subsequently, iterations on the parameters of this function are performed untilthe equilibrium law of motion for the marginal utility is found. As a functionparametrizing (C.1), we use second-order degree exponentiated polynomial; thelength of simulations is 10,000.

To compute the expectations in expected lifetime budget constraints (C.4), wefollow the approach described in GMV (1995). Under this approach, theexpected in " nite sum is approximated by the average of N simulations of length

EO

c \ AOc \ A

I OzOK1N

,

L

2

Oc \ AOc \ A

I OzO, (D.1)

where zO3 +c O, w O, w O(1! hO), . To obtain a precise estimate, both N and haveto be large. Since it is computationally costly to set both N and large, theright-hand side of (D.1) is subdivided in two parts, the head and the tail.Subsequently, the head is computed as average of N short draws of length



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and the tail is approximated by a single long draw of the length

1N

,

L

2

Oc

\

AOc \ A I

OzOK 1N,

L

2 Y

Oc

\

AOc \ A I

OzO# I

2 Y

2

Oc

\

AOc \ A I

OzO.

We choose N " 400, " 115, " 10,000. As is argued in GMV (1995),a similar choice guarantees substantial accuracy of simulated solutions.

Finally, we show that a solution + f Q, QZ1 to expected lifetime budget con-straints (C.4) computed on Step 3 exists, is unique and such that f Q' 0 for s3 S.Indeed, let +c O, w O, hO,OZ2 be such that 0 ( c O, w O(R and 0 ( hO( 1 for any3 . Consider functions Q( f Q) and Q( f Q) such that

Q

( f Q

), X\

N(eQ

)\

N( f Q

)ANE O I

Oc \ AOc \ Aw O(1! hO),

Q( f Q), Q # eQEO

I Oc \ AO

c \ Aw O! f QE

O I Oc

\ AOc \ Ac O.

In terms of the above functions, the agent 's expected lifetime budget constraint(C.4) can be expressed as Q( f Q)" Q( f Q). For any X 3 R > , we have Q(0)" 0 and( Q) ' 0. Further, given that Q ' 0 for s3 S, we have Q(0)' 0 and ( Q) ( 0.Finally, the functions Q( f Q) and Q( f Q) are continuous on R > . Thus, there existsa unique value f Q which satisfy the expected lifetime budget constraint of eachagent s3 S. A solution + f Q, QZ1 and formula (18) determine uniquely the corre-sponding value of the parameter . Therefore, if the equilibrium exists, is interiorand unique, then the value of which is consistent with the equilibrium is alsounique.

General results about the existence of the equilibrium are hard to achieve.Whether the equilibrium in our economy exists will depend on a particularchoice of the model 's parameters. To see the point, consider, for example, thecase when all consumers have the preferences of Cobb }Douglas type andassume that there are some consumers whose endowment to skills ratio is veryhigh. According to formulas (9), (10) such consumers will choose to worka negative amount of hours, which implies that there is no interior equilibriumin the model. However, in all the numerical experiments which we reported, theequilibrium exists and is interior; also, the iterative procedure had no di $ cultiesto converge to a " xed point value of the parameter .

7. For further reading

Constantinides, 1982; Cooley and Ogaki, 1996; Danthine and Donaldson,

1995; Huang, 1987; King et al., 1988; Mas-Colell et al., 1995; Negishi, 1960;Ogaki, 1997; R m Hos-Rull, 1995; Rosen, 1968; Stokey and Lucas, 1989.



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