A Dynamic Equilibrium Model of the U.S. Wage Structure, 1968-1996

by

Matthew Johnson CRA International, Oakland California

and

Michael P. Keane

ARC Federation Fellow, University of Technology Sydney Research Professor, Arizona State University

March, 2007 Abstract: We develop an overlapping generations equilibrium model of the U.S. labor market, and fit it to PSID data from 1968-1996. Our main innovation is a much finer distinction among types of labor than in prior work – i.e., labor is differentiated by 1-digit occupation, education, gender and age, giving 160 types. This is important, as prior work has shown that: (i) wage and employment patterns over the sample period differ across occupation-education-gender-age groups in complex ways not captured by simpler models, and (ii) conclusions about effects of policies to increase human capital are very sensitive to the assumed substitution patterns across types. In our model the fraction of each youth cohort that attends college is determined in part by the college wage premium (as well as by parental background and tastes). The model provides a rather good fit to college attendance rates and occupational choices for narrowly defined types over the whole 29 year period. Hence, we use it to simulate the effect of tuition subsidies aimed at youth whose parents did not attend college. Our model implies that roughly 2/3 of the gain in their lifetime earnings resulting from increased college attendance is negated by equilibrium feedback effects, whereby increased supply of college labor lowers college wages. This is intermediate between earlier estimates by Heckman, Lochner and Taber (1998), who find that equilibrium effects almost eliminate earnings gains from tuition subsides, and by Lee (2005), who finds equilibrium effects are negligible. We also find that labor supply elasticities can be very misleading as guides to tax responses, due to equilibrium feedback effects on wages.

Acknowledgements: This paper is heavily based on the first chapter of Mathew Johnson’s dissertation at Yale University. Conversations with John Roemer were instrumental in developing this project. We also thank Lance Lochner for providing very useful comments. Keane’s work on the project was supported by Australian Research Council grant FF0561843.

I. Introduction

There is a vast literature on the evolution of the U.S. wage structure over the past 40

years. By “wage structure,” we refer to patterns of wage differentials across demographic groups,

characterized by education, gender, occupation and age. Perhaps the most widely noted change

over this period was the growth – since roughly 1976 – of the College/High-School wage

premium, despite growth in the relative supply of college graduates. In the early 90s, the

consensus view of the profession attributed this growth in relative wages of more highly skilled

labor to “skill biased technical change” (SBTC) that drove up the rental price of skill in general.1

But the consensus began to crack in the early 2000s. Card and DiNardo (2002) and

Eckstein and Nagypal (2004) look at the wage structure in more detail, examining not just the

college premium, but also: (i) the college premium by age, (ii) gender and race gaps, (iii) relative

wages across occupations, (iv) wage/experience profiles by education and gender, etc.. In doing

so they find many patterns that SBTC alone cannot explain. Other factors must also be at work.

For example, Card and Lemiuex (2001) noted that the increase in the college premium is

concentrated among young workers, and that this could be explained by reduced relative supply

of young college graduates. But, as they note, it is unclear why the supply of young graduates

would stagnate even as the college premium soared in the 80s. Clearly, to explain changes in

wages and in education/employment choices simultaneously, one needs an equilibrium model.

With this as our motivation, we build an equilibrium model with labor differentiated

along 4 key dimensions (education, occupation, gender and age), and with 5 key sources of

change in the wage structure (SBTC, capital-skill complementarity,2 occupational demand shifts,

demographic changes, changing tastes for work and college). We seek to explain changes in

wages and employment over the 1968-1996 period at a detailed level using this model. If

successful, we will use the model to examine effects of tuition subsidies on the wage structure.

Our work differs from (most) earlier work in this area in three key ways: First is the fine

distinction among types of labor, and the attempt to fit data patterns by narrowly defined type.

Second is the inclusion of several factors that may drive changes in the wage structure (i.e., most

earlier work looks at one or two factors at a time – most often SBTC and supply shifts). Third is

the equilibrium nature of the analysis, accounting for both labor supply and college attendance.

1 See, e.g., Bound and Johnson (1992), Katz and Murphy (1992), Murphy and Welch (1992), Juhn, Murphy and Pierce (1993), Berman, Bound and Griliches (1994), Gottschalk and Moffitt (1994). 2 Krusell, Ohanian, Rios-Rull and Violante (2000) and Fallon and Layard (1975) favor this story over SBTC.

1

With few exceptions, the prior literature on the evolution of the wage structure has been

descriptive and/or partial equilibrium in nature. It has not attempted to build equilibrium models

that can explain the evolution of the wage structure, while simultaneously explaining educational

and employment/occupational choices. The notable exceptions are Heckman, Lochner and Taber

(1998a, b) – henceforth HLT – and Lee (2005), Lee and Wolpin (2006a, b) – henceforth LW.

Both groups of authors highlight the usefulness of an equilibrium approach by using their models

to simulate how college tuition subsidies might influence the wage structure (in particular, how

they might influence earnings inequality). Interestingly, they reach very different conclusions.

In HLT, there are two types of labor - college and high school graduates – and the

estimated elasticity of substitution between them is 1.44.3 Given this value, a college tuition

subsidy that raises supply of college graduates would substantially reduce the college wage

premium.4 Hence, in equilibrium, the HLT model implies that such a subsidy would do little to

increase college attendance, or to benefit those who do not attend college under the status quo.

In contrast, Lee (2005) differentiates types of labor by both education and occupation

(e.g., white vs. blue-collar). Unlike HLT, he predicts equilibrium and partial equilibrium effects

of a college tuition subsidy would be very similar. Thus, adding the occupational choice margin

makes college and high school labor more substitutable (in part because, within each occupation,

they are assumed to be perfect substitutes). As a result, only a small decline in the college wage

premium is needed to absorb the increased supply of college educated workers.5

The contrast between the HLT and Lee results illustrates how model structure – i.e., how

types of labor are defined and entered into the aggregate production function – has a large impact

on substitution elasticities. These, in turn, greatly influence answers to questions like (1) How do

tuition subsidies affect college attendance and the college premium? or (2) To what extent do

supply factors like changing cohort sizes vs. demand factors like SBTC explain the evolution of

the wage structure, the occupational distribution of employment, and educational choices?

Even prior work on the wage structure that is partial equilibrium or descriptive in nature

has, for the most part, looked at only a few types of labor, often differentiated only by education, 3 There is a clear consensus in the literature on an elasticity of substitution between college and High-School labor in the vicinity of 1.4 (see also Katz and Murphy (1992)). HLT get 1.26 when they instrument using cohort sizes. 4 It implies that a 5% increase in college labor, leading to roughly a 6.25% increase in the ratio of college to high-school labor, would reduce the wage ratio by 4.3%. Thus, if the ratio starts at 1.50, it falls to 1.43. This substantial drop in the college premium wipes out 90% of the effect of a tuition subsidy on college attendance in their model. 5 Lee and Wolpin (2006a) extend Lee (2005) to allow additional occupations and gender, but do not discuss tuition subsidies. They estimate that technical change accounts for most of the growth of the service sector in 1968-2000.

2

or by very broad skill groupings. But as Card and Lemieux (2001), Card and DiNardo (2002) and

Eckstein and Nagypal (2004) show, this broad perspective may lead one to miss more subtle

patterns that are key to understanding what drives changes in the wage structure. While prior

work has usually focused on education and age (and sometimes gender), we will argue it is also

important to look at occupation. Specifically, we find that the increase in the relative wage of

“skilled” labor in the 80s-90s was driven by increased wages of professionals, while wages of

managers stayed essentially flat. Like the singular increase in the college premium at young ages,

this is another awkward pattern for the SBTC hypothesis that our model needs to rationalize.

The important recent papers by Kambourov and Manovskii (2004a, b, 2005), Moscarini

and Vella (2002) and Kranz (1999, 2006) argue that occupation is a better measure of skill than

education, and that occupation specific demand shifts may be crucial for understanding changes

in the wage structure.6 Thus, we differentiate labor by occupation and allow for occupational

demand shifts in our model. To anticipate a key finding, another awkward fact the pure SBTC

hypothesis is that, while wages and employment of high-school males both began to fall in the

mid-70s, high-school females did well on both dimensions. Eckstein and Nagypal (2004) also

note the importance of this pattern. Our model is able to explain this via demand shifts towards

output of service occupations (also found to be important in Lee and Wolpin (2006a)).

Thus, our main goal is to estimate an equilibrium model of the labor market with many

more types of labor than in prior work. Specifically, we differentiate labor by education (college

vs. high-school), gender, age (four ten year intervals from 25 to 64), and by ten occupations

(roughly the 1-digit level). This gives 2·2·4·10=160 types of labor, that enter a multi-level nested

CES aggregate production technology. The greater richness of the model may lead to more

reliable predictions of how policy interventions like tuition subsidies, or exogenous factors like

technical change, affect wages, occupational and employment choices, and educational choices.

The much greater detail of our model in terms of types of labor is feasible because we

make a simpler assumption about expectations than do HLT and LW. That is, we assume each

entering cohort of youth uses the wage structure it sees at the time of entry to predict the present

value of earnings with and without a college degree. In contrast, HLT and LW assume youth 6 Kambourov-Manovskii and Kranz both present evidence that returns to occupation (given education) grew much more than returns to education (given occupation) during the 80s and 90s. Kambourov and Manovskii present evidence that occupational mobility increased in the PSID during the 80s and 90s. They infer that the variance of occupation specific labor demand shocks increased, and argue this can explain most of the increase in wage inequality. Moscarini and Vella (2002), however, find that occupational mobility did not increase in the CPS.

3

forecast the future evolution of equilibrium wages in the economy. This makes solution of their

models much more difficult, yet: (1) we see no philosophical reason to prefer that assumption,7

and (2) we suspect the expectations assumption is of second order importance to the behavior of

the model economy relative to assumptions about substitutability of different types of labor (as

the large divergence in results between HLT and Lee illustrates).

We fit our model to PSID data from 1968-1996 on wages and employment of each of the

16 labor types (college vs. high-school, gender, four age groups) in each of the 10 occupations,

as well as the fraction in the home sector in each year, giving 29·2·16·10 = 9280 data elements.8

In addition, for each entering cohort of 19 year-old youth from 1968 to 1990, we fit the fraction

of males and females from each of four parental background types9 who choose to attend college

and emerge as college types in 1974 to 1996.10 This gives 2·4·23 = 184 additional data elements.

In one version of the model we take the capital stock as exogenous, while in another we

endogenize it and fit the annual capital stock data as well. In each case, identification of model

parameters is achieved through a combination of factors assumed exogenous (i.e., the size and

parental background distribution of each entering cohort) and restrictions on preferences and

technology (i.e., the CES share parameters are constrained to vary smoothly over the 29 years).

Since we fit educational choice data along with wage/employment data, a key challenge

for our model is to explain why the fraction of each cohort that attends college actually fell from

over 60% to about 45% for men from 1974 to 1996, while staying flat at about 45% for women

during the whole period, despite the rise in the college wage premium. As we see in Figures 4-6,

the model does a good job of tracking these patterns. And, in nearly all cases, the model also

does a good job of tracking the wage paths over the 29 years of our data, for all 10 occupations

and for all 16 types of labor. The same is true of the employment shares (see Figures 7-18). 7 Indeed, even the assumption that youth know the current wage structure exactly is best viewed as an as if assumption, since it is implausible that this is literally true (see Manski (1993), Dominitz and Manski (1996) and Betts (1996) for some evidence on what youth know about wages). This applies a fortiori to the assumption that youth can forecast future equilibrium wages. Rational expectations (as in LW) or perfect foresight (as in HLT) assumptions are often invoked not because we literally believe them but rather because they make model solution simpler. But that is certainly not the case here, where assuming that youth forecast future wages would increase computational difficulty by an order of magnitude. Some recent work by Martins (2006), using surveys of Portuguese college students, suggests that: “students have a relatively good understanding of [current] market [wage] rates.” He argues for the same assumption we make here. 8 Note that the fraction of each type at home is not included in this count, since shares must sum to one. 9 This is similar to HLT’s treatment of heterogeneity: they split entering cohorts of white males into AFQT quartiles. 10 We assume a youth decides whether to attend college at age 19, and then enters the labor market as a College or High-School type at age 25. This means, e.g., the 1991 entering cohort doesn’t emerge into the labor market until 1997, after the end of our data. This is why we only fit college choices for the 1968 through 1990 cohorts.

4

Given the acceptable fit of the model, it seems reasonable to use it for counterfactual

policy experiments. Hence, we predict how tuition subsidies (or attendance bonuses) for youth

from relatively under-privileged backgrounds would affect college attendance rates and labor

market outcomes. For example, consider a bonus large enough to bring the college attendance

rate of youth whose parents had a high school (HS) or less than high school (

types, while leaving the overall supply of college educated labor almost unchanged. Thus, tuition

subsidies financed in this way have essentially no equilibrium feedback effects.

Finally, we also use our model to simulate the effect of proportional taxes on labor supply

(given lump sum redistribution). We find that labor supply elasticities can be very misleading as

guides to tax responses, due to equilibrium feedback effects on wages. The labor supply response

of a group of workers depends crucially on its substitutability with other groups.

The remainder of the paper is organized as follows: Section II presents our model, and

Section III describes the PSID data we use for estimation. Section IV discusses the estimation

results, including our estimates of the elasticity of substitution between different types of labor.

Section V discusses the fit of the model, and in so doing describes many interesting patterns in

the data. Section VI describes our simulations of tuition subsidies (financed using lump sum or

proportional taxes or higher gross tuition levels) and of a tax increase. Section VII concludes.

II. An Overlapping Generations Equilibrium Model of the Labor Market

In each period (year) the population of the model economy consists of overlapping

generations of individuals aged 19 to 64. These individuals are differentiated into imperfectly

substitutable types of labor, distinguished by education level (college vs. high school), gender,

age (25-34, 35-44, 45-54, 55-64) and occupation (10 categories), giving a total of 160 labor

types. These different types of labor, along with the aggregate capital stock, are combined, via a

nested CES production function to be described in section II.1, into aggregate output.

The supply of each type of labor in each period is determined endogenously, first via

education decisions of entering cohorts, and then via decisions on labor force participation and

choice of occupation by the older cohorts. Individuals enter the economy at age 19 and decide

whether or not to pursue college training. At that point, there are 8 types of people, differentiated

by gender and four levels of parental education, i.e., whether the best educated parent’s level of

completed education was less than high-school (

All individuals (regardless of their schooling decision) are assumed to enter the labor

force at age 25.12 Henceforth, a worker’s type is determined solely by gender, age and education

(i.e., parental background no longer matters once education is determined).13 In each period

(through age 64) workers of each type choose between remaining home or working in one of ten

occupations.14 Occupations are defined at roughly the 1-digit level, as we detail in Section III.

Workers make occupational choices based on current wages.15 Each gender-education-age type

faces a type-specific vector of competitively determined occupational wages. Individuals also

have type-specific non-pecuniary payoffs to each choice (e.g., female workers may have a

greater distaste for work as laborers than males). We describe the labor force participation and

occupational choice decision rule in detail in Section II.2.

In each period (year), the equilibrium of the model economy is a 160-element vector of

occupation-education-age-gender specific wages such that the market for each skill type clears.

Let Xg,e,a,t denote the type (g, e, a)-specific aggregate supply of labor to the economy at time t,

where g denotes gender, e denotes education and a denotes age. This is governed by the sizes,

gender composition and educational choices of past cohorts. Further, let Lg,e,a,k,t, denote the type

(g, e, a)-specific supply of labor to occupation k at time t. Occupation specific supplies of labor

are governed by a decision rule Dk(·,·) for occupational choice based on aggregate supplies,

wages {wg,e,a,k,t}k=1,10 and group specific tastes for occupations (and the home sector):

{ }10, , , , , , , , , , ,1 ,g e a k t k g e a k t g e a tkL D w X=⎡= ⎢⎣⎤⎥⎦

=

(1)

Equilibrium wages are given by partials of the aggregate production function, evaluated at the

aggregate capital stock At and the type specific labor aggregates {Lg,e,a,k,t}:

(2) ( ), , , , 10 10 2 2 4, , , , , , , , 1 , , , 1 1 1 1,{{{{ [{ } , ]} } } }g k e a tg e a k t L t k g e a k t k g e a t k g e aw f A D w X= = = ==Thus, at the equilibrium wage vector, individual labor supply decisions give type specific labor

supplies to occupations that equate wages to marginal products for each type in each occupation.

12 We assume work starts at age 25 because, as Geweke and Keane (2000) describe, the sporadic collection of schooling data in the PSID requires one to wait until a person is roughly age 25 to get an accurate read on their completed schooling level. If non-college types work more than college types at ages earlier than 25 this will cause us to somewhat overstate the college lifetime earnings premium. But this should be largely subsumed in the intercept of the college attendance decision rule, and so should have little impact on our results. 13 This is consistent with results in Geweke and Keane (2000) who find, using a sub-sample of these same data, that parental background is insignificant in earnings functions that control for education. 14 Attrition from the economy is also incorporated, using life table hazard rates. Hazard rates vary by age, and are not included in this outline of the model for the sake of notation simplicity. 15 That is, there is no learning-by-doing or on-the-job human capital investment in the model.

7

Fitting the model to the PSID data for 1968-1996 requires an iterative process. At each

trial parameter value, we must solve for the equilibrium of the economy in each of the 29 years

(i.e., the 160-vector of equilibrium wages, the associated occupational employment shares by

type, and the percentage of each entering cohort predicted to attend college, by type). In this

process, the initial 1968 distribution of types of labor Xg,e,a,1968 is taken as exogenously given.

Then, X evolves as a result of the demographics and educational choices of incoming cohorts.16

Solution of the model for equilibrium wages and employment shares in each occupation

in each year from 1968-1996 proceeds as follows. Starting with the first period, a vector of initial

candidate wages w0 is postulated. Labor then allocates across occupations given w0, and this

allocation determines marginal products w1. The initial wage vector w0 is then updated to w1 and

a new allocation is calculated. We iterate this process until the wage vector converges to the

equilibrium we. The 1st entering youth cohort then makes college attendance decisions based on

the wage vector we. These decisions determine the supply of labor of each skill type (g, e, a) that

enters the next period. Given this, we determine the equilibrium wage vector for the next period,

and so on. The details of this process are given in Appendix A on computation and estimation.

The solution of the model, including wages, occupation choice probabilities and college

attendance rates for each cohort, serves as input to the estimation procedure. Specifically, having

solved for the sequence of 29 annual equilibria from 1968-1996, the model generates 160·29·2

predicted values of type (g, e, a)-specific wages and labor supply to each occupation over the 29

years, as well as 8·23 predicted college attendance frequencies for the 8 background types

(gender crossed with parental education) over 23 years.17 Model parameters are chosen to best fit

these 9464 data elements using the method of moments estimator described in Appendix A.

The remainder of this section describes in detail: (1) the aggregate production

technology, (2) the occupation choice decision rule, and (3) the college attendance decision rule. II.1 Form of Production Technology

The model incorporates imperfect substitution among labor inputs by relying on a nested

CES production technology with several levels. Of course, one might order CES nests of labor

differentiated by occupation, education, gender and age in many ways, but we view our ordering

16 Indeed, because of our assumption that the cohort of youth that makes college choices at age 19 in 1968 will not enter the labor market until age 25, we actually take X to be exogenously given up until 1974. 17 We fit college attendance choice for only 23 years rather than 29 because the cohort aged 19 in 1991 is not assumed to enter the market until 1997, and we only fit data through 1996.

8

as quite natural and intuitive. First, we place occupations in the upper level nests – not education,

gender or age – because labor of multiple occupations must typically be combined to produce

products or services – not necessarily labor of different education, gender or age levels. We felt

education should come next, as it seems intuitive that education is the primary determinant of the

quality of an occupation’s workers, not their gender or age composition. Finally, the ordering of

gender vs. age is perhaps not so obvious, but this appears to be immaterial to our final results,

because, in the event, we find that different age groups are highly substitutable.

At the top level, aggregate output depends on capital and two CES aggregates of

“skilled” and “unskilled” labor. These aggregates are, in turn, composed of occupation groups:

“skilled” labor consists of professionals and managers, while “unskilled” labor consists of the

blue collar and service occupations.18 Thus, we have:

( ) ( )( ) ( )1

1 1u u

s s ss A A s s uY f A EMP EMP

ρu

ρρ ρ ρβ λ λ λ λ

⎡ ⎤= ⋅ = + − + −⎢ ⎥

⎣ ⎦

ρ (3)

where: β = (exp(b(0) + b(1) · t) = a scale and technical progress parameter

A = capital equipment and structures

,s uEMP EMP = skilled and unskilled labor aggregates In (3), the λ are “share” or “weight” weight parameters (sometimes known as relative efficiency

parameters) that govern income shares for each input, while the ρ (ρ < 1) govern the elasticity of

substitution between inputs.19 Note the λ only translate literally into income shares in the Cobb-

Douglas special case that arises if ρ→0 (see Arrow, Chenery, Minhas, and Solow (1961)).

In (3), the elasticity of substitution between capital and skilled labor, which are nested

together, is σs = 1/(1- ρs), while that between the capital-skill composite and unskilled labor

(EMPu) is σu = 1/(1- ρu). The function implies capital-skill complementarity, meaning growth in

the capital stock increases the marginal product of skilled relative to unskilled labor, if ρu > ρs,

which implies σu > σs. Our top-level setup is similar to that in Fallon and Layard (1975) and

Krusell et al (2000), except we define skill along occupational rather then educational lines.20

18 Service occupations are sometimes referred to as “pink collar” (see, e.g., Lee and Wolpin (2006b)) as they have a heavy representation of women. LW use a CES function like (1), except they enter blue and pink collar separately. 19 Note that this CES functional form imposes symmetry restrictions on substitution elasticities. Here, the elasticity of substitution between EMPu and A is restricted to be the same as that between EMPu and EMPs. 20 Our treatment of capital differs as well, as we aggregate equipment and structures, while Krusell et al do not.

9

At the second level nest we have skilled and unskilled labor aggregates, defined as:

( )( ) ( )( )1 1

1 1 1 2 1 21 1H H L LH Ls u u uEMP EMP EMP EMP EMP EMPρ ρ ρ

u uρρ ρλ λ λ λ= + − = + − (4)

Thus, skilled labor combines labor in occupation 1 (professionals) and occupation 2 (managers).

Unskilled labor combines labor in the service and blue-collar occupations, denoted EMPu1 and

EMPu2, respectively. The service and blue-collar labor aggregates are, in turn, combinations of

labor inputs to several other 1-digit level occupations. Specifically, the service sector input is a

CES aggregate of employment in occupations 3-6 (Technicians, Sales, Clerical and Service

workers), while the blue-collar input is a CES aggregate of employment in occupations 7-10

(Craftsmen, Operatives, Transport Operatives, and Laborers). These are constructed as follows:

( )( )

( )( )

1 1 1 1 1

2 2 2 2

1

1 3 3 4 4 5 5 3 4 5 6

1

2 7 7 8 8 9 9 7 8 9 10

1

1

u u u u u

u u u u u

u

u

EMP EMP EMP EMP EMP

EMP EMP EMP EMP EMP

ρ ρ ρ ρ ρ

ρ ρ ρ ρ 2 ρ

λ λ λ λ λ λ

λ λ λ λ λ λ

= + + + − − −

= + + + − − −

(5)

Thus, the parameter ρu1 governs substitution among the four service sector occupations, while ρu2

governs substitution among the four blue-collar occupations.

It is notable that, while substitution between inputs within a nest is governed solely by the

nest-specific substitution parameter (ρ), that between inputs across nests is far more complex. It

depends on: (1) the nest-specific ρ’s of both inputs, (2) the ρ parameters at each higher nest up to

the first common nest, (3) the λ parameters up to the first common nest, and (4) factor supplies.

For example, the elasticity of substitution between blue collar (EMPu2) and managers (EMP1) is:

( )2,1 1 2 3 41 1u u s Ha a a a Lσ ρ ρ ρ ρ= − − − − , where 1ia =∑ , while that between service sector labor (EMPu1) inputs and managers (EMP1) is:

( )1,1 1 2 3 41 1u u s Ha a a a Lσ ρ ρ ρ ρ′ ′ ′ ′= − − − − , where 1ia′ =∑ , where the parameters and are functions of the income share parameters (i.e., the λs) in the

different nests.

a a′21 Hence, substitutability of inputs from one nest (e.g., blue collar and services)

with one from another nest (e.g., managers) will converge when their income shares are close.22

21 The precise form of the mapping from the λ’s to the ’s is available from the authors on request. a22 The elasticities of substitution between mangers and blue collar or service sector inputs are equal only if blue collar and services have equal income shares. Relative income shares do not matter in the special case where all ρ’s are equal. Then the technology reduces to a single nest CES where all inputs share the same substitution parameter.

10

Next we consider occupation level effective labor inputs. Each occupational labor input is

assumed to consist of an aggregate of college and high school labor, as follows:

( )( )1

1e ek k k k kEMP HS COLρ eρ ρμ μ= + − for k=1,10, (6)

where HSk and COLk denote high school and college type labor in occupation k, respectively.

Note that, while the “share” or “weight” parameters μk are allowed to vary across the ten

occupations, the education substitution parameter ρe is assumed to be common to all occupations.

At the next nesting level, the amounts of college and high school labor supplied to each

occupation are assumed to consist of male and female workers. Thus we have:

( ) ( )( )( )( ) ( )( )( )

1

1

1

1

g g g

g g g

male femalek k k k k

male femalek k k k k

HS HS HS

COL COL COL

ρ ρ ρ

ρ ρ ρ

ξ ξ

γ γ

= + −

= + −

for k=1,10. (7)

Note that the degree of substitutability between genders is assumed to be the same across all

occupations and at both education levels, and is governed by the single parameter ρg. But the

share parameters ξk and γk are allowed to differ by both occupation and by education level.

The final CES nest aggregates labor in four different age groups: 25-34, 35-44, 45-54 and

55-64. These aggregate up to the four gender/education level inputs as follows:

( ) ( ) ( )( )( )

1

1 ,25 34 2 ,35 44 3 ,45 54

1 2 3 ,55 641

a a a

a

male male malek k k k k kmale

kmale

k k k k

HS HS HSHS

HS

ρ ρ aρ ρ

ρ

τ τ τ

τ τ τ

− − −

−

⎛ ⎞+ +⎜ ⎟= ⎜ ⎟⎜ ⎟− − −⎝ ⎠

+ (8a)

( ) ( ) ( )( )( )

1

1 ,25 34 2 ,35 44 3 ,45 54

1 2 3 ,55 641

a a a

a

female female femalek k k k k kfemale

kfemale

k k k k

HS HS HSHS

HS

ρ ρ aρ ρ

ρ

σ σ σ

σ σ σ

− − −

−

⎛ ⎞+ +⎜ ⎟= ⎜ ⎟⎜ ⎟− − −⎝ ⎠

+ (8b)

( ) ( ) ( )( )( )

1

1 ,25 34 2 ,35 44 3 ,45 54

1 2 3 ,55 641

a a a

a

male male malek k k k k kmale

kmale

k k k k

COL COL COLCOL

COL

ρ ρ aρ ρ

ρ

ν ν ν

ν ν ν

− − −

−

⎛ ⎞+ + +⎜ ⎟= ⎜ ⎟⎜ ⎟− − −⎝ ⎠

(8c)

( ) ( ) ( )( )( )

1

1 ,25 34 2 ,35 44 3 ,45 54

1 2 3 ,55 641

a a a

a

female female femalek k k k k kfemale

kfemale

k k k k

COL COL COLCOL

COL

ρ ρ aρ ρ

ρ

ϖ ϖ ϖ

ϖ ϖ ϖ

− − −

−

⎛ ⎞+ + +⎜ ⎟= ⎜ ⎟⎜ ⎟− − −⎝ ⎠

(8d)

11

Note that the degree of substitutability between age groups is assumed to be the same for all

occupation/gender/education groups, and is governed by the single parameter ρa. But the share

parameters τk, σk, νk and kϖ are allowed to differ by occupation/gender/education level.

However, allowing these share parameters to differ freely would result in a severe

proliferation of parameters. Specifically, 4 gender/education groups x 10 occupations x 3 age

share parameters per group gives 120 share parameters. Furthermore, we will allow the share

parameters to change over time to account for SBTC and/or demand shifts. Thus, we will need to

constrain the share parameters as describe below. First, we describe how they vary over time.

As is standard in this literature, we let the share parameters vary over time to capture

biased technical change/demand shifts. Specifically, we let them follow low order polynomials in

time, using logistic transformations to constrain them to lie in the (0, 1) interval, as follows:

( )( )

2 3 40 1 2 3 4

2 3 40 1 2 3 4

exp,

1 expi i i i i

ii i i i i

t t t ti s A

t t t t

λ λ λ λ λλ

λ λ λ λ λ

+ + + += =

+ + + + + (9)

Here, we are looking at the top-level nest (see equation 3), where the capital/skilled labor

aggregate is combined with unskilled labor to produce output. The capital share parameter λA and

the capital/skilled labor aggregate share λs are each allowed to follow 4th order polynomials in

time. Similar expressions apply to the other share parameters. As we’ll see below, 4th order

polynomials provided an adequate fit to the data (and often only 3rd order terms were needed).

Note that the polynomial order we choose for the share parameters plays a key role in

identification. We assume that supplies of various types of labor vary exogenously over time for

several reasons, including changing cohort sizes – HLT refer to cohort size as the “standard

instrument” in this literature – and different “quality” of entering cohorts (i.e., the distribution of

parental education). This leads to different educational choices and hence different supplies of

college vs. high-school labor. And, in one version of the model, we have an exogenously varying

capital stock. Elasticities of substitution between inputs are identified by how such exogenous

shifts in supplies affect income shares (e.g., shares are constant in the Cobb-Douglas case, but

respond to supply changes if substitution departs from unit elastic).

Now, the model is fit to 29 years of data, meaning it attempts to match (among other

things) the income shares of the 160 types of labor (defined by gender, education, age and

occupation) in each of those 29 years. If the share parameters were allowed to vary across years

12

in an unconstrained way, the model could fit those shares perfectly, regardless of how the

substitution parameters ρs, ρu, ρH, ρL, ρu1, ρu1, ρe, ρg and ρa are set. Hence, the assumption that the

share parameters vary slowly over time is crucial to identification of substitution elasticities. In

our view such an assumption is reasonable – if technical change or occupational demand shifts

occur, it seems plausible they would occur gradually rather than suddenly.23

Now we return to the issue of the share parameters for the different age groups within the

4 gender/education groups, contained in equations (8a)-(8d).24 As noted, there are 120 such share

parameters. If each were allowed to follow a 4th order polynomial, this gives 600 parameters. To

avoid such proliferation of parameters, we impose the following restrictions:

(1) Each occupational skill group has a common intercept in its time polynomial. Recall

that there are 3 such skill groups: occupations 1 and 2 comprising professionals, managers and

administrators, occupations 3-6 which comprise the service sector, and occupations 7-10 which

comprise blue collar. This reduces the number of polynomial intercepts from 120 to 36.

(2) The linear and higher order terms in the time polynomial are assumed to be common

across all ten occupations within each of the 12 education/gender/age groups. This reduces the

number of such terms from 480 to 48.

(3) In many cases we found that 4th order polynomials were not necessary to obtain an

adequate fit to the data. In these cases we stopped at the 3rd order.

We emphasize that these restrictions only apply to (8), which aggregates age groups.

Thus, assumption (2) essentially means that if the income share of an age group within one

gender/education category rises in one occupation it will tend to rise in all occupations. This

certainly does not mean that income shares of gender/education groups will move similarly

across occupations. For instance, how the gender shares move is crucially influenced by the

share parameters in (7), while how education shares move is crucially influenced by the share

parameters in (6). More subtly, share parameters in (3)-(5) come into play as well: Suppose the

share parameters increase in occupations where educated labor is relatively productive, and/or

where women, given their preferences, particularly like to locate. This will drive up income

shares of college-educated women, even given no changes in the share parameters in (6)-(7).

23 This steady trend assumption has been adopted in the prior literature, at least back to Katz and Murphy (1992)). 24 Equations (8a)-(8d) each contain four share parameters that must sum to one. Given the adding up constraint, this is equivalent to saying they contain three share parameters that must lie between 0 and 1. To impose this constant, we use a multinomial logit transformation. The same is true of equation (5).

13

II.2 Occupational Choice

In each period, workers aged 25 to 64 choose among 11 alternatives (10 paid occupations

and home work). Recall there are 16 types of workers, distinguished by age-gender-education,

and each type faces a different 10-vector of occupational wages. Types also differ in tastes for

working in each occupation (or equivalently, the non-pecuniary rewards they receive). Workers

choose among the 11 alternatives based on the 10-vector of occupational wages,25 as well as

their tastes. Specifically, the utility to a worker conditional on choice of occupation k is given by:

( )2, 0, , , 1 , , , , 2 , 3 , 1( , , ) Pr 1k t g e k g e a k t k t k t k tU g e q w e e d ,α α α α −⎡ ⎤= + + − + = +⎣ ⎦ ε (10) Here, α0,g,e,k is a non-pecuniary reward that a worker of gender g and education level e receives

from working in occupation k. Notice that this is assumed to be age and calendar time invariant.

In the second term, α1 is a parameter and wg,e,a,k,t is the competitively determined wage for a

worker of type (g, e, a) in occupation k at time t.

The third term involves the difference between the worker’s education level and the

average education level of workers in occupation k. This term is meant to capture two things: (i)

workers may receive a psychic benefit (cost) from working with other workers who are similar

(different) from themselves, and (ii) it will be more difficult (in terms of effort) for a worker with

a high school education to successfully enter a high-skilled occupation.

In the fourth term, dk,t-1 is a 1/0 indicator for whether occupation k was chosen in the

previous period. Thus, Pr[dk,t-1=1] is the proportion of workers who chose occupation k in the

previous year. This term is included to capture persistence in occupational choices.26

Finally, the error term εk,t is assumed to be type I extreme value and independent across

alternatives. This gives simple multinomial logit (MNL) forms for the choice probabilities.

The utility associated with the home or out-of-the-labor-force alternative is allowed to

differ by age, gender and calendar time as follows:

( )0, 0 1 0, 1, , ,( , , ) ( )t h h h f h f g a hU g e a t t I g f tα α α α α= + ⋅ + + ⋅ ⋅ = + +ε

(11)

Here αh0 is the overall constant and αht an overall time trend, which capture the value of home

time and changes in this value over time. The next term, αh0 + αh1 · t, captures deviations in the

25 As wages depend only on gender, education and age – and not on accumulated occupation-specific or general work experience – workers do not need to consider any impact of current choices on future wage offers. 26 Persistence may arise due to habit persistence, or labor adjustment costs at the individual or aggregate level.

14

value of home time for women (both in level and trend). We expect to see both level and trend

differences by gender, given the initially much lower but rapidly rising level of female

participation in the labor force over the sample period. [Of course, the trend in the value of home

time for women presumably arises not just from changes in tastes, but perhaps also from changes

in fertility behavior and marriage market opportunities, which we do not model explicitly.]

It is notable that the model has multiple means to explain increasing participation of

women – not just changing tastes in (11). Part of the explanation may also be increasing wages in

(10). And part of the explanation may be technical change leading to increased income shares for

the service sector occupations (see Lee and Wolpin (2006a)), provided the estimates imply that

women receive high non-pecuniary rewards from working in these occupations.

The term ag,~α is a vector of gender/age specific deviations in the value of home time.

These are meant to capture the lower market participation rate of older workers (as they retire).

Finally, εk,t is again an independent type I extreme value error term.

Given the setup in (10)-(11), the proportions of workers of type (g, e, a) who choose to

work in occupation k, or who choose the home sector 0, are given by the MNL expressions:

( ) ( ) ( )10

, ,0

P 1| , , exp ( , , ) exp ( , , )kt k t k tk

d g e a U g e a U g e a=

= = ∑ for k = 0,10 (12) We are now in a position to state the specific form of the decision rule in equation (1). Given the

type (g, e, a) specific choice probabilities, and the type specific aggregate supplies of labor to the

economy at time t, Xg,e,a,t, the type specific factor inputs are determined by:

Lg,e,a,k,t = = P(dkt=1|g,e,a)·Xg,e,a,t. (1’) , , , , 1 , , ,[{ } , ]K

k g e a k t k g e a tD w X= The equilibrium of the economy is a wage vector that, when substituted into the decision rule

(1’), gives labor supplies that, when plugged into the production technology of equations (3)-(8),

give rise to a 160-vector of marginal products that match the wage vector, as in equation (2). II.3 Educational Choice

At age 19, the members of each entering cohort, beginning with the 1968 cohort, decide

whether or not to attend college. Six years later, at age 25, they enter the labor market, as either

College or High-school type labor. [The assumption that careers begin at age 25 is motivated by

data considerations that we discuss in detail in Section III.2]. Recall that youth differ in two

15

dimensions: gender g and parental background, denoted by b∈{

It is worth commenting further on the “cost” terms φ2b for b∈{

decision, workers enter the labor market six years later, at age 25. The choice probability in (15)

governs the evolution of the stock of 25-34 year old college workers as follows:

25 34, 25 34, 1 25, 35,g g g

t t tCOL COL COL COL− − −= + −g

t for g=male, female (16)

where:

( )4

25 19, , , 6 6

1P ,g t g b t t

bCOL N e COL g b− −

=

= ⋅ =∑ and 19, , 6g b tN − is the number of 19 year olds of gender g and parental background type b who enter

the economy at calendar time t-6. We take these entering cohort sizes as exogenous. Ignoring

mortality to simplify notation, the stock of 35-44 year old college workers evolves as follows:

35 44, 35 44, 1 35, 45,g g g

t t tCOL COL COL COL− − −= + −g

t for g=male, female Analogous equations hold for the evolution of the stocks of 45-54 and 55-64 year old workers.

Of course, the number of high school type workers at any age is simply obtained as the residual.

We take the initial (1968) stocks of gender/education/age types as exogenously given.

Individuals who make education decisions at age 19 in 1968 enter the model labor market at age

25 in 1974. So it is only at that point that the supplies of different types of labor to the model

economy become endogenously determined. By 1996, the 1968 cohort has reached age 47.

Finally, note that in taking the model to data we will have to make a compromise in

defining educational types (as we have only two). We will assume high-school labor consists of

workers with any level of education at or below high school graduate, while college type labor

will be defined as any workers who attend college (even if they do not complete). II.4 Equilibrium Determination of the Capital Stock

We estimate two versions of the model, one where the capital stock is exogenous, and

another where it is determined by the equilibrium in each period. In the endogenous capital

model, the capital stock at time t is determined by equating the marginal product of capital

evaluated at current factor input levels to the exogenous rental price of capital . Formally, Ctr

tkaegtkaegttCt ALAYr ∂∂= ==== )}}}}{{{{,(

101

41

21

21,,,, (17)

The rental price can be thought of as an outside world price of capital not determined in the

model (that is, does not depend on model factor supplies). It is assumed to evolve over time Ctr

18

according to the polynomial:

2 30 1 2 3 4

Ctr c c t c t c t c t= + + + +

4 (18) Identification is again aided by the assumption that the price of capital evolves smoothly over

time. In estimating this version of the model, we also fit the capital stock data from 1968-1996.

Only the endogenous capital version of the model can be used for policy experiments that

involve forecasts beyond 1996. An example is how a tuition subsidy affects lifetime earnings of

the 1990 cohort of 19 year olds. They will be in the labor market until 2035, so we must project

the capital stock to that date. We do this by (i) assuming the price of capital determined by (18)

stays fixed at the 1996 level, and (ii) using (17) to determine the capital stock in 1997-2035. II.5 Summary

The exogenous capital model given by (1)-(16) has 306 parameters, and the endogenous

capital model that adds (17)-(18) has 311. Of these, 237 are in the share equation polynomials, 9

are elasticities of substitution, and 2 are scale/TFP parameters, giving 248 technology parameters

in all. The occupational choice model contains 46 parameters, of which 33 are gender/education

specific tastes for occupations. The college choice equations contain 12 parameters.

The model parameters are clearly over-identified. As we show in the next section, the

model is fit to 9,464 data elements (9,493 with endogenous capital). To perfectly match observed

income shares (by type/occupation/year), the production function share parameters (for each

gender/education/age/occupation) must vary freely across the 29 years of data. This leaves the

substitution parameters unidentified. Instead, we constrain the share parameters to (i) lie along

low order polynomials and (ii) move in common across certain types of labor (see Section II.1).

Furthermore, to also match the observed occupational employment shares (which, if we

fit income shares, gives a match to earnings as well), requires that group specific tastes for each

occupation vary freely over time. This would leave effects of wages on occupational choice

unidentified. Instead, in equation (10), we constrain the taste parameters to be constant over time.

Within this constrained structure, the key thing that identifies elasticities of substitution

between different types of labor is how income shares respond to the variation in the supply of

workers of different types to the economy over time. Exogenous variation in these supplies is

generated by the variation across annual cohorts both in their size and in their distribution of

gender/parental education types, denoted { 19, , 6g b tN − } in Section II.3.

19

III. Data

The model is fit to occupation specific wages and employment shares for each gender-

education-age group from 1968 to 1996 (2·160·29 = 9280 data elements), and college attendance

rates for each gender-parental background group from 1968 to 1990 (8·23 = 184 data elements),

giving 9,464 data elements in total. The endogenous capital model adds 29 data elements (capital

stock over 29 years), for a total of 9,493. This section describes construction of these variables.

III.1 Occupational Choices and Earnings

The data on occupational choices and wages are constructed using the core representative

sample of the Panel Study of Income Dynamics (PSID). The PSID is a longitudinal study that

began collecting socio-economic data on 5000 families in 1968. We aggregate the individual

data on earnings and occupation up to the level of the 160 gender-education-age-occupation cells

used in our model, with each observation weighted using the PSID core sample weights.30, 31

Of course, the drawback of the PSID is that it is relatively small compared to the Current

Population Survey (CPS), leading to more noise in our estimates of occupation specific earnings

and employment shares. But the key advantage of the PSID is the consistency of its occupational

coding. Occupations are coded at one to three-digit precision using the 1970 Census taxonomy

throughout the whole 1968-1996 period. In contrast, the CPS changed its occupational codes at

several points. Consistency over time in occupational classifications is crucial to our analysis.

We screen any observations with unassigned education, or invalid or missing occupation,

employment, or hours worked.32 We also dropped several occupations with consistently low

employment: farmers, farm laborers and supervisors, private household workers and armed

forces personnel. The self-employed with no reported occupation were also dropped.

We group occupations into ten categories, based primarily on 1970 Census 1-digit codes.

But some 1-digit categories were further disaggregated using 3-digit codes. Most notably, we felt

the “Professional, technical, and kindred workers” category covered too wide a skill range. So

we split it into two occupations, “Professionals” and “Technicians,” based on the 3-digit codes

listed in Appendix B. The former is relatively high skilled compared to the latter. For instance, 30 We also utilize two PSID supplemental data files: the Retrospective Occupation-Industry Supplemental Data Files and the 1994-1996 Hours and Wages Supplemental Data Files. 31 The PSID only collects detailed occupational information on “household heads” and wives, and not on other household members. Hence, we can only utilize heads and wives in computing occupational employment frequency and wages. But the data to be matched in this paper is on individuals from age 25 to 64, which should mitigate the exclusion of other household members in computing occupational wages/employment. 32 We also drop observations with positive hours and no labor earnings, or missing/zero hours and positive earnings.

20

professors and dentists are classified as “professionals,” while secondary school teachers and

dental technicians are “technicians.” Thus, our ten occupations (codes in parenthesis) are:33

High Skilled Occupations:

1. Professionals (selected from 001-195, see Appendix B)

2. Managers and Administrators (201-245)

Service Occupations (Unskilled Group 1):

3. Technicians (selected from 001-195, see Appendix B)

4. Sales Workers (260-285)

5. Clerical and Kindred Workers (301-395)

6. Service Workers, except Private Household (901-965)

Blue Collar Occupations (Unskilled Group 2):

7. Craftsmen and Kindred Workers (401-600)

8. Operatives, Except Transport (601-695)

9. Transport Equipment Operators (701-715)

10. Laborers, Except Farm (740-785)

Labor income and hours of work are assigned to occupations using the respondent’s

reported main occupation. The data do not allow us to see how annual labor income and work

hours are divided among occupations. This introduces potential misclassification if a worker

works in multiple occupations within a single year. However, a problem only arises if the worker

switches between, not within, our ten occupational categories.

Ultimately, the employment data to be fit by the model are the number of full-time

equivalent workers in an occupation in a given year, and the earnings data to be fit are the

average annual earnings of such a full-time equivalent worker. Thus, in order to count full-time

equivalents, we must decide what number of hours defines full-time. Say, for example, we define

a full-time professional as working 2400 hours per year. Then, a worker whose main occupation

is in the professional category, and who reports 1600 hours of work, will contribute 0.75 units of

labor (of his/her gender/education/age type) to that sector in that year. Of course, how we define

full-time hours is merely a scale normalization, which has no bearing on the substantive results. 33 While the PSID uses 1970 Census occupation codes throughout, the level of detail in describing occupations varied over time. From 1968-1973 and 1975, only 1-digit codes were recorded. In 1974, 3-digit codes were recorded. 2-digit codes were recorded from 1976 to 1980, and 3-digit codes were recorded thereafter. In 1999 the PSID released the Retrospective Occupation-Industry Supplemental Data Files. This recode relied on the original written interview descriptions of respondents’ occupations to assign 3-digit occupation codes back to 1968.

21

We arrive at a measure of full-time hours for each occupation as follows: Figure 1 reports

the average level of hours for “full-time” male workers in each occupation in each year (where

full-time is defined as working at least 35 weeks and 30 hours per week). Clearly, average full-

time hours differ significantly across occupations, with managers and administrators averaging

the highest and laborers the lowest. Hours levels also vary within occupations over time. For

each occupation, we use the maximum of the full-time hours measure across all 29 years in

Figure 1 as the occupation specific full-time hours level. Denote this by FTEk for k=1,10.

Next, occupational choice frequencies for each gender/education/age group are calculated

using reported hours worked relative to our measures of full-time hours. Specifically, letting

i=1,…, N(t) index PSID core sample members in year t, for each group (g, e, a) we have:

( ) ( )( ) ( )

1 1P 1| , , ( 1) 1,

N t N t

kt it i ikt ii i

d g e a n I d n for kα= =

= = × = =∑ ∑ 10 (19)

where N(t) is the number of 25-64 year old individuals in the sample in year t, dikt is an indicator

for whether person i chose occupation k in year t (equal to one for at most one occupation) and:

PSID core sample weight, , 0 1i it itn hours k iFTEα α= = < ≤ where αit is the fraction of a full-time unit of labor that person i supplies to occupation k.

As we assign each person to one main occupation, αit may be non-zero for only one

occupation per person. Any residual time (1- αit)ni is assigned to the home sector. Thus, the

number of FTEs assigned to home consists not only of workers who are fully unemployed or out

of the labor force, but also includes some fraction of the time of part-time workers.

Analogous to (19), the “wage rate” in occupation k – i.e., the annual earnings of a full-

time equivalent worker who chooses that occupation – is given by:

( )( ) ( )

, , , ,1 1

( 1) ( 1) 1,1N t N t

g e a k t it k it i ikt it i ikti i

w wage FTE n I d n I d for kα α= =

= ⋅ ⋅ × = × = =∑ ∑ 0

(20)

where wageit = earningsit / hoursit. The earnings data are put in 1999 US dollars using the CPI-U.

Before applying (20), we screen outliers by dropping the top and bottom 0.5% of full-time wage

estimates in each year by gender.34

34 The hours data in the 1969, 1970 and 1974 PSID are often missing or unreliable. In such cases, we are unable to construct αit, the fraction of an FTE that i supplies to an occupation. Rather, we impute hours using group averages from the surrounding years. Specifically, we compute average hours worked in surrounding years for each gender,

22

III.2. College Attendance Decisions

A key simplifying assumption of our model is that there are only two educational types,

those with a high-school education or less (HS) and those who attend college (COL), regardless

of whether they complete it. The use of two broad educational categories is similar to HLT. We

assume the college attendance decision is a one time decision made at age 19, and we attempt to

fit college attendance frequencies by cohort (from 1968-1990), gender and parental background,

as measured by the highest level of education completed by either parent. Here there are four

categories,

level faced by those deciding on college attendance. The data is not adjusted for grant and other

financial aid, nor does it include indirect costs such as room and board or commuting (note that

HLT share the same limitation). We do not include data on aid because, unfortunately, such data

are only available for 1987 onward (from NCES National Post-Secondary Student Aid Study).35

III.3. Cohort Size and Attrition: 1968-1990 Census, US Decennial Life Tables

Our overlapping generations model requires as inputs (i) the age 25 to 64 distribution of

the population by gender and education in 1968, and (ii) the cohort sizes of age 19 entrants from

1969-1990 (by gender and parental background). While we use the PSID to determine

educational and occupational choice frequencies, we use the Census data to determine the age

distribution of cohorts in 1968, and the size of subsequent entering cohorts. Thus, to arrive at our

1968 inputs, the age/education choice proportions computed using the PSID are used to divide

each Census age cohort into HS and COL subgroups. Finally, hazard rates from the 1970-1990

U.S. Decennial Life Tables provide attrition rates for each age group. Inputs to the model are not

updated to account for immigration over the period. (HLT find immigration effects are small).

III.4. Capital Stock Data

As our capital stock measure we use Bureau of Economic Analysis (BEA) estimates of

nonresidential fixed investment, reported in June 2003.36 We add equipment and software (E&S)

and non-residential structures (omitting government and residential fixed assets). Several prior

studies, including Krusell et al (2000), attempt to improve upon standard BEA measures of

capital by incorporating quality-adjusted measures of equipment prices. But, comparing the BEA

measures to those used in Krusell et al (2000), it appears the gap between the two has narrowed

considerably, as a result of a 2003 revision of the methods used to construct the national income

and product accounts. Included in this revision were new methodologies for computing the real

stock of fixed assets.37 Thus we decided to use the standard BEA measures. Consistent with the

rest of our data, the capital stock is expressed in 1999 dollars (see Figure 2).38

35 A few studies, such as Leslie (1984), use aid data collected by the Cooperative Institutional Research Program (CIRP). CIRP collects institution-reported bracketed data on financial aid for full-time freshmen reaching as far back as 1967. However, this data is not publicly available, and it may not be directly comparable to the NCES data. 36 Specifically, BEA Table 1.1 “Current-Cost Net Stock of Fixed Assets and Consumer Durable Goods”, and BEA Table 1.2 “Chain-Type Quantity Indexes for Net Stock of Fixed Assets and Consumer Durable Goods”. 37In describing the overhaul, Moulton (2002) states: “BEA is pursuing research on developing quality-adjusted prices indexes for software, photocopy equipment, and nonresidential structures. The BEA is also evaluating the use of quality-adjusted price indexes for communications equipment developed by the Federal Reserve Board.” 38 We use separate BEA deflators for equipment and software (E&S) and Structures. This is important as the two can behave rather differently (e.g., from 1996 to 1999 the price level for E&S fell by roughly 3%, while that for

24

IV. Estimation Results

We fit the overlapping generations equilibrium model of Section II to the PSID data

described in Section III using the method of moments (MOM). Estimation is computationally

burdensome, for similar reasons to why estimation of stochastic dynamic programming models is

burdensome – i.e., it requires nesting a computationally intensive model solution algorithm

within a parameter search algorithm. Finding the equilibrium means solving for the 160-vector of

equilibrium wages (i.e., a 160-dimensional fixed point problem) in each of 29 years. And we

must re-solve for the equilibrium of the model economy at each of the many trial parameter

vectors that arise in the course of searching for the optimum of the statistical objective function.

As our focus is on substantive results and not computation, we describe the algorithm to

solve for equilibria, and the details of the MOM estimator, in Appendix A. We also describe a

number of “tricks” to speed up the solution/estimation algorithm so that estimation is feasible.

As noted earlier, we estimate two versions of our model, with physical capital treated as

exogenous or endogenous, respectively. Interestingly, parameter estimates and model fit are very

similar in each case. Thus, to conserve on space, we primarily report on the parameter estimates

and the evaluation of model fit for the exogenous capital model.39 Of course, we must use the

endogenous capital model for experiments that involve out of sample forecasts. IV.1. Parameter Estimates – Production Technology

IV.1.A. Production Function – Substitution and TFP Parameters

Table 1 and Appendix C report the parameter estimates for the production technology.

We begin with Table 1, which reports the substitution and TFP parameters, and later turn to

Appendix C, which reports the CES share parameters. At the highest level of the CES nesting

(see equation (3)), we estimate that the elasticity of substitution between physical capital and

skilled labor is 0.47, while that between the capital + skilled labor aggregate and unskilled labor

is 3.23. Thus, as 3.23>0.47, the estimates imply capital-skill complementarity, consistent with

the findings of Fallon and Layard (1975) and Krusell et al.40 It should be remembered, however,

that we define skilled labor based on occupation while they define it based on education.

structures grew 15%). Note also that the BEA readily provides quantity indexes rather than value deflators. We used the quantity indexes to get the implied deflators and use these to convert current values of capital to 1999 dollars. 39 The endogenous capital model, which has 5 additional parameters, also provides a slightly better fit. 40 Our estimated substitution elasticity between skilled labor and capital (0.47) is similar to Krusell et al’s estimate of 0.67. But our estimate for that between the capital-skill composite and unskilled labor (3.27) is much higher than

25

Moving down to the next level (equation (4)), we estimate that the skilled occupations

(professionals and mangers) are highly substitutable in production (i.e., σH=6.25). On the other

hand, for the two unskilled aggregates (services and blue collar), substitution is rather inelastic

(σL=0.56). This seems intuitive (e.g., it is easier to substitute an engineer for a manager than a

plumber for a nurse). But it is interesting that these elasticities are much less precisely estimated

than are the capital, skilled and unskilled labor elasticities at the top-level nest. Apparently, our

sources of identifying variation (i.e., changes in the size and quality – in terms of parental

background – of entering cohorts) induce more independent variation in the skilled vs. unskilled

labor inputs than they do in more narrowly defined occupations (which seems reasonable).

At the next level (equation (5)), we estimate substitution elasticities among the four blue-

collar occupations that form the blue collar aggregate, and for the four service occupations that

form the service sector labor aggregate. In each case, we cannot reject that substitution is unit

elastic (as ρu1 and ρu2 do not differ significantly from zero – either statistically or quantitatively).

The next level down in the nesting (equation (6)) describes substitution between college

and high school educated workers – within an occupation. We estimate this elasticity as 1.6,

which is quite close to estimates in the prior literature (which, as HLT note, are centered around

1.4 to 1.5). It should be remembered, however, that our figure is not really comparable to

estimates in the prior literature, which has not differentiated labor by both education and

occupation. We do not directly estimate the elasticity between college labor in toto and high

school labor in toto, but only that within occupations. To obtain the overall elasticity, we must

simulate an exogenous increase in the supply of college-educated labor.41

Furthermore, as we discussed in Section II.1, the elasticity of substitution between two

inputs (like high school and college labor), which do not appear in just one CES nest, but rather

enter several different nests, depends on the share parameters in all common higher nests. As

these share parameters vary over time, so will the substitution elasticities. Thus, for inputs like

high school and college labor, we cannot give a single elasticity figure, but a range of figures.

As we note in Table 2, for high school and college labor, simulations of our model imply

substitution elasticities in the range of 1.15 – 1.26. This is well below the elasticity of 1.6 that we their estimate of 1.67 – implying even stronger capital-skill complementarity. This may reflect the different aggregation of skill inputs here, with skill defined along occupational rather than educational lines. On the other hand, when we endogenize the capital stock, we obtain an elasticity of 1.75, which is nearly identical to theirs. 41 That is, we simulate a “helicopter drop” of college workers, increasing the number of such workers at each age by the same proportion. These new workers choose occupations in the same way as workers do normally in the model.

26

estimate for high school and college labor within a one-digit occupation, but it is still well within

the range of the prior literature.42 It seems intuitive that different education levels are more easily

substitutable within occupations (e.g., a high school educated manager can more easily substitute

for a college educated manger than could a high school educated laborer).

The next lower nest (equation (7)) describes substitution between male and female

workers (within education and occupation cells). We estimate that this elasticity of substitution is

very high (5.26). Thus, male and female workers appear to have very similar skills, conditional

on education/occupation. On the other hand, we can also simulate the elasticity of substitution

between male and female workers in toto, just as we did for high school and college workers. As

we see in Table 2, when we do this, we obtain a range of elasticities of 1.85-2.20. Thus, not

surprisingly, male and female workers appear much less similar unconditionally.

The bottom level of the nest (see equation (8)) describes substitution between workers in

different age groups. In each case, we condition on the gender/education/occupation cell. Our

estimates of this substitution elasticity were very high, and the estimation algorithm had a hard

time pinning down a final estimate. So we decided to peg the elasticity at 10.

Finally, the estimate of b(1) at the bottom of Table 1 implies neutral technical progress of

0.4% per year (Note: the BEA’s estimate of TFP growth over the 1968-1996 period is 0.55% per

year).43 Remaining output growth is due to growth of capital and labor, including growth in labor

quality due to increasing education and shifts toward more skilled occupations.

IV.1.B. Production Function – Share/Productivity Parameters

The CES production function of equations (3)-(8) contains 235 share parameters, reduced

to 160 because shares within nests must sum to one. The large number of share parameters arises

because our model contains 160 distinct types of labor. As noted in Section II.1 (equation (9)),

we let the 160 independent share parameters vary over time, according to cubic or quartic time

trends. This leads to severe proliferation of parameters unless we impose certain restrictions. The

restrictions described in Section II.1, leave us with 237 polynomial parameters to be estimated.44

The CES share parameter estimates are reported in Appendix C, Tables C1 to C4. Since

polynomial coefficients are not very informative, we also report the implied values of the share 42 Recall that HLT obtain 1.26 when they instrument for the college/high-school supply ratio using cohort sizes. 43 See “Multifactor Productivity Trends, 1997,” at www.bls.gov/schedule/archives/all_nr.htm#PROD3. We may obtain a smaller estimate of TFP growth because our model better captures skill upgrading of the labor force. 44 Recall that in (8), which combines age groups, intercepts are not allowed to differ across all ten occupations, but only across the three broad skill groups, and higher order terms are restricted to be equal across all ten occupations.

27

parameters for three selected years: 1968, 1982 and 1992. Of course, to the extent that the CES

substitution parameters depart from zero (i.e., from Cobb-Douglas), the “share” parameters are

not exactly equal to income shares. However, their trends over time tell us how demands for (and

income shares of) the various groups move. Some interesting patterns emerge from the estimates:

First, in Table C1, while the share of the capital+skilled labor composite rises from 1968

to 1982, it falls back to its original level by 1996. Thus, this factor cannot explain the rise of the

college wage premium over the sample period. This finding seems consistent with recent papers

like Card and DiNardo (2002) and Eckstein and Nagypal (2004) who argue that a simple trend in

the share of skilled labor cannot explain changes in the wage structure over the 70s-90s.

Second, the share of the service occupations (relative to blue collar) rises substantially,

consistent with findings in Lee and Wolpin (2006a). Third, within the Services sector, the share

of the technicians rose while that of clerical workers fell.

Fourth, in Table C2, which reports the shares of high school vs. college workers within

occupations, we see that the high school share falls dramatically in six of the ten occupations:

managers, sales workers, service workers, craft workers, operatives, and laborers. It falls more

modestly for professionals and technicians. Clearly this is a key factor (along with capital skill

complementarity) in explaining the rise of the college wage premium.

Fifth, in Table C3, which reports gender shares, we see that the share of high school

males (relative to high school females) falls in all ten occupations. The college male share falls

in the skilled occupations (1-2), and in the service sector occupations (3-6) but not in the blue-

collar occupations (7-10). Together, the second and fifth patterns – growing demand for labor in

“pink collar” occupations and shifting demand toward female labor within most occupations –

can explain both (i) the closing gender wage and employment gaps, and (ii) the pattern that high-

school females fared much better than high-school males over this period.

Finally, Table C4 reports age group shares. These seem very stable, although there is

evidence of a weak upward trend for the younger age group (25-34) among high school and

college females and college males.

IV.2. Parameter Estimates – Occupational Choice

Table 3 reports estimates of the occupational choice equation. The coefficient α1 on the

annual wage is .0000862. Thus, e.g., a ten thousand dollar increase in the annual wage in an

occupation raises the latent index associated with that occupation (see equation (10)) by .862. To

28

put this in perspective, suppose each occupation and the home sector originally had equal choice

probabilities of 1/11 = .091. A ten thousand dollar increase in the annual wage of one occupation

(relative to all others) would, other things equal, raise its choice frequency to .191, a substantial

109% increase. [This is obtained via simple MNL arithmetic: exp(.862)/(exp(.862)+10·exp(0)) =

2.368/12.368 = .191].

To put the ten thousand dollar earnings increase inducing this change in perspective, the

average annual wage for 35-44 year old male college graduates was roughly $58,000 in 1996,

while that for male high school graduates was roughly $36,000 (see Figure 17). Hence, we have

17% and 28% earnings increases for these groups, respectively. Thus, the elasticity of labor

supply to a single occupation is roughly 4 to 6, depending on the group.

Consider next the elasticity of labor supply to the whole economy. To determine this, we

calculate what happens to the home sector (and, conversely, total employment) if the wage in

every occupation rises 1% - implying a $580 ($360) increase for college (high school) males. As

an approximation, suppose the home shares are 15% and 24% for college and high school males,

respectively, and that the employed are spread evenly across occupations. Given our estimate of

α1, the 1% wage increase raises the latent index associated with all occupations (equation (11))

by .05 for college males and .031 for high school males. Using the MNL formula, it is simple to

calculate that this causes the home sector share to drop to .1437 and .2344 for these groups,

respectively. Thus, employment increases by 0.74% for each (the equality is a coincidence), and

we have an elasticity of labor supply to the economy as a whole of roughly .74 for males.45

Next, the estimate of α2 implies, as expected, a cost (psychic or effort) of entering an

occupation with an average education level very different from ones own. And the estimate of α3

implies, as expected, inertia in occupational choices, presumably due to monetary and/or psychic

mobility costs. As a result, the increase in labor supply to an occupation resulting from increased

labor demand would not occur immediately, but rather would play out over time.

The next panel of Table 3 reports the non-pecuniary payoffs from working in each of the

ten occupations for each of the four gender/education types. These fall into predictable patterns,

45 For women, we see in Figure 17 that the average annual wage rates in 1996 for 35-44 year old college and high school graduates were roughly $32,000 and $21,000, respectively. Thus, a 1% wage increase means $320 and $210 increases in income. This translates into .028 and .018 increases in the latent indices in equation (11). From Figures 13 and 15 we see that home sector shares in 1996 were about 38% and 47%, respectively. Thus, doing the same calculations as above, we obtain labor supply elasticities of 1.13 and 0.94 for college and high school educated women respectively. These figures are about 50% greater than the figure for males.

29

with, for instance, women (whether high school or college educated) having a preference for

clerical and service (i.e., “pink collar”) occupations and a distaste for being laborers. For males

these occupation specific taste parameters are generally smaller in magnitude.

Table 3 Part B reports parameters related to the value of the home option. As expected,

females have a positive intercept shift (αh0,f =2.02), meaning they have higher utility in the home

sector than males (presumably due to child rearing activities). But they also have a negative time

trend relative to men (αh1,f = -.028), so the value they assign to home work was declining. Thus,

tastes and/or declining fertility explain part of the increase in female labor force participation

over the ’68-’96 period. But, of course, part is also due to the increasing demand for female labor

captured by the trends in the gender share parameters (discussed in Section IV.1.B). Finally, the

table reports age effects in the value of home time (αg,a). As expected, these are increasing with

age for both males and females. This is in part how the model accounts for retirement.

IV.3. Parameter Estimates – Educational Choice

Table 4 reports the parameter estimates for the college choice equation. To interpret the

estimates, it is useful to consider Figure 3. This presents, for each cohort of 19 year olds from

1968-1996, the expectation, at the time of their college choice, of the PV of lifetime income for

college vs. high school graduates. Given our assumption that youth use the current wage

structure to infer future wages, these figures can be calculated using that wage structure in

conjunction with equation (12), which determines occupational and home choice probabilities.

For males, Figure 3 implies that that the (perceived) college lifetime earnings premium was

roughly $415,000 for the 1968 youth cohort, narrowed to roughly $330,000 in the 1974-83

period, widened to $400,000 in 1990, and widened further to about $480,000 in 1996.

The coefficient on the expected PV of lifetime income captures the extent to which the

difference in this PV between college and high school graduates influences college attendance.

For males, the coefficient φ1m is .0000134. As shown in Figure 3, in the 1990 cohort the expected

PV of lifetime income for male college and high school graduates were roughly $930,000 and

$530,000, respectively, a $400,000 difference (in 1999 CPI-U dollars). And, as shown in Figure

4, the college attendance rate for males in this cohort was roughly 42%. Suppose the gain to

college increased by 10%, or $40,000. This increases the latent index associated with college

(equation (13)) by .54, which increases the rate of college attendance to 55.4%. [Note: this refers

to any college attendance – not just completion – and rates as high as 60% are within the range

30

of our data – see Figure 4]. As college attendance increases by 32%, the elasticity of supply of

college labor with respect to the gain to college attendance is roughly 3.2 for males.

Interestingly, the coefficient on PV of lifetime income for women φ1f is roughly half that

for men. This is consistent with findings in Keane and Wolpin (2006) that about half the gain to

college for women comes not in the form of increased PV of lifetime earnings but rather in the

form of improved marriage market opportunities. It is hard to escape this conclusion since, as we

see in Figure 3, the PV of income gain for women is less than half that for men, yet their college

attendance rate is just as high. To explain this, we must assume that women simply get a (much)

higher utility from college itself, or that they recoup the gain through a different channel (i.e., the

marriage market). As we do not model the marriage market explicitly, the high rate of female

college attendance is explained by their intercept shift in the college choice equation (φ3=4.12),

suggesting they get a (much) greater non-monetary reward from college than do males.

As expected, fixed effects for parental background (i.e., education) figure prominently in

the value to college. Translating these parameters into monetary equivalents the male coefficient

on earnings implies the “cost” of college is greater by (6.04-3.82)/(.0000134) = $165,000 for

youth whose parents were high school drop outs (

suggest that liquidity constraints do little to explain relatively low college attendance rates by

low-income youth. Thus, we feel the most plausible interpretation of the “cost” parameters is in

terms of psychic and/or effort costs, as in scenarios (i) and (ii).

For males, the time trend for tastes is quantitatively small. It implies the non-pecuniary

value of college had an inverted U-shape, and was nearly the same in the 90s as in 1968. This is

important, as it means the model can explain the puzzle of declining male college attendance –

despite the rising college wage premium – without resorting to declining male tastes for college.

Rather, its explanation is technology/demand driven – shifts in demand towards “pink collar”

occupations, shifts in demand toward female labor within most occupations (see Section IV.1.B),

and the 6 to 9 point increase in the home share for males that we discuss below (Section V). As

we see in Figure 3, these factors counteract any increase in the college wage premium, so the

lifetime earnings premium for college graduates is about $415,000 for the first (1968) but only

$400,000 for the last (1990) entering youth cohort in the data. Given the elasticity of 3.2, the

model thus predicts a 12% drop in male college attendance, similar to what we see in the data.

The situation for women is the flip side of that for men. As we see in Figure 4, their

college atten