Wage Volatility and Changing Patterns of

Labor Supply∗

Jay H. Hong†, Byoung Hoon Seok‡, and Hye Mi You§

May 21, 2014

Abstract

For the past four decades in the U.S., hours worked of skilled men relative to un-skilled men have increased while the skill premium rose. This fact is in contrast withprevious studies suggesting a value of labor supply elasticity close to zero. This paperexplores wage volatility to resolve this discrepancy. As wages become more volatile,individuals not only accumulate more precautionary savings but also increase workhours. Using the Panel Study of Income Dynamics (PSID), we document that skilledmen experienced a greater rise in wage volatility accompanied by increasing shares ofpersistent wage shocks than unskilled men. We build a general equilibrium incompletemarkets model and feed the estimated wage processes into the model. The model canreplicate the increasing trends in hours worked of skilled men relative to unskilled menin the data. We also find that the variances of persistent wage shocks are important inexplaining the evolution of hours worked, whereas the quantitative impact of a transi-tory component is fairly modest.

Keywords: Skill Premium, Wage Volatility, Labor Supply by Skill Group, Precau-tionary SavingsJEL Classifications: E24, J22, J24, J31

∗ We are grateful to Mark Aguiar, Mark Bils, Yongsung Chang, Dirk Krueger, and participants at theWegmans conference held at the U of Rochester in 2009, Midwest Macro Meetings 2011, annual confer-ence of the Canadian Economics Association 2011, Midwest Economics Association Annual Meeting 2013,NY/Philadelphia Workshop on Quantitative Macroeconomics 2013, Yonsei U, Cleveland Fed, Seoul NationalU, Sogang U, Korea U, and Sungkyunkwan Univ. for their valuable comments and suggestions. KyoohoKwon and Jinhee Woo provide excellent research assistance. This work was partially supported by ResearchResettlement Fund for the new faculty of Seoul National University. All errors remain ours.

† Department of Economics, Seoul National University, Email: jayhong@snu.ac.kr

‡ Department of Economics, Ohio State University, Email: seok.19@osu.edu

§ Corresponding Author: Department of Economics, State University of New York at Buffalo, Email:hyemiyou@buffalo.edu

1

1 Introduction

For the past four decades in the U.S., male hours worked of college graduates (skilled,

hereafter) relative to non-college graduates (unskilled, hereafter) increased. This pattern is

in line with a recent literature on the trends in hours worked by skill group in the U.S.1 Given

that the relative wages of skilled to unskilled men rose substantially during the period,2 the

increased skilled-unskilled hours differential is in contrast with empirical evidence on wage

elasticity of labor supply. A recent survey by Saez et al. (2012) writes “With some notable

exceptions, the profession has settled on a value for this elasticity close to zero for prime-age

males.”3. This suggests that the hours difference between the two skill groups should stay

roughly constant with the rising skill premium, contrary to what we observe in the data.

This paper explores the second moment of wages, i.e. wage volatility to resolve this

discrepancy. As wages become more volatile under the incomplete market, individuals work

longer hours as well as accumulate more precautionary savings for self-insurance. Flodén

(2006) shows that an increase in future wage uncertainty raises current labor supply due to

a precautionary motive in a simple two-period model. Pijoan-Mas (2006) also emphasizes

the impact of wage volatility on labor supply decisions, by showing that both the level of

1Aguiar and Hurst (2007) show that less-educated men reduced their time spent on total market work

more than those with a college or more over the period 1965-2003, based on five waves of American Time Use

Surveys (ATUS). Costa (2000) also finds a similar pattern using wage as a measure of skill. Costa (2000)

shows that the length of a work day by higher-wage earners has increased relative to that of lower-wage

earners between 1973 and 1991 in the Current Population Survey (CPS). Santos (2014) confirms that this

finding holds for the CPS data extended to 2006.

2For the evolution of relative wages of the skilled to the unskilled, i.e., skill premium in the U.S., see Katz

and Murphy (1992), Krusell et al. (1994), Acemoglu (2002), and Autor et al. (2008).

3Estimating this elasticity is a challenging task, given that it is diffi cult to measure purely exogenous

wage changes in the data and it is often the case that workers are not able to freely adjust their work hours.

A recent paper by Ashenfelter et al. (2010) exploits a panel data set of taxi drivers (considered to be free

from the two problems stated above) and estimates a negative elasticity of male labor supply. They view

that this finding suggests that the income effect is larger than the substitution effect in the long run. Given

their result, the increase in skilled-unskilled hours differential is more puzzling. For a survey of the literature

on wage elasticity of labor supply, see Keane (2011) and Saez et al. (2012).

2

precautionary savings and labor supply are higher in an economy with incomplete markets

than in one with complete markets. If skilled men were exposed to greater increases in wage

volatility than unskilled men, then this channel can potentially explain the observed rise in

skilled-unskilled hours differential.

Using the Panel Study of Income Dynamics (PSID), we document that there are marked

differences in the trends of the second moment—as well as the first moment—of wages between

skilled and unskilled men in recent decades.4 Our estimation shows that i) wage volatility

(measured by the variance of residual wages) has risen for both skill groups between 1967

and 2006 and ii) skilled men have experienced much greater increases in their wage volatility

compared to the unskilled. We also find that the rise in wage volatility for skilled men was

largely due to persistent wage shocks, while unskilled men experienced increasing shares of

transitory wage shocks in their residual wages for the period.5

This paper develops a general equilibrium incomplete markets model with heterogeneous

agents of different skill levels. In each period, agents face uninsurable idiosyncratic pro-

ductivity shocks, which consist of a persistent and a transitory component drawn from a

skill-specific distribution. We calibrate the initial steady state of the model to the U.S.

economy in 1967 and then feed the estimated evolution of wage processes from the PSID

into the model. We then evaluate the effect of changes in wage volatility by comparing the

predicted evolution of relative hours of skilled to unskilled men from the model with its data

counterpart along the transition.

The model can replicate the increasing patterns of skilled-unskilled hours differential of

U.S. men observed in the data. In our model, hours worked increase in the short run when

4There are many studies on increasing volatility of residual wages and earnings for overall workers in

the U.S. for the past few decades. See Gottschalk and Moffi tt (1994), Gottschalk and Moffi tt (2011), and

Heathcote et al. (2010).

5Although examining the causes of the rising wage volatility is outside the scope of this study, we view

deunionization and increasing adoption of performance-pay contracts in the U.S. labor market as important

factor behind the increases in wage volatility. Particularly, performance-pay wage contracts is likely to be

responsible for a greater increase in wage volatility among skilled men than unskilled men, as Lemieux et al.

(2009) suggest.

3

they face higher wage volatility. This effect is much stronger for skilled workers because they

are exposed to relatively greater increases in wage volatility than the unskilled. However, in

the long run, as skilled men accumulate a buffer stock of precautionary savings over time,

they reduce their work hours relative to unskilled men due to wealth effects in the new

steady state. This implies that hours adjustment is important for self-insurance in the short

run, whereas precautionary savings play a dominant role in the long run. We also find that

not only the level of wage volatility but also the relative composition of persistent versus

transitory wage shocks matter for individual labor supply decisions. Persistent wage shocks

are crucial in explaining the trends in hours worked, whereas the quantitative impact of a

transitory wage component is fairly small.

Our study is closely related to Heathcote et al. (2010), who explore the implications of

observed changes in the wage structure including a rising skill premium, a declining gender

gap, and the increasing overall residual wage dispersion for cross-sectional inequality in

hours, earnings, and consumption. Our work differs from theirs in that we focus on the

differences in the evolution of wage volatility across different skill groups. We explore the

quantitative effects of the differences in increased wage volatility on the evolution of the

relative hours worked of skilled to unskilled men. One of the unique features in our study

is that it investigates the impacts on individual labor supply of not only the level of total

wage volatility but also the changes in its composition of persistent versus transitory shocks.

Our result show that the shock composition is quantitatively important in individual labor

supply decisions.

We can also relate our paper to Erosa et al. (2011) and Castro and Coen-Pirani (2008)

who examine differences in male labor supply by skill groups. However, Erosa et al. (2011)

focus on their life-cycle features to study aggregate labor supply responses to tax changes.

Our work also differs from Castro and Coen-Pirani (2008) in that they have used demand-side

factors to explain changes in business cycle fluctuations of hours by skill groups,6 whereas

6Castro and Coen-Pirani (2008) attempt to explain increased cyclicality of skilled hours since the mid-

1980s and propose changes in capital-skill complementarity as potential explanations.

4

we consider supply-side factors to address the trends in hours by skill levels.

Several recent studies, including Santos (2014), Elsby and Shapiro (2012), and Michelacci

and Pijoan-Mas (2008), have tried to explain the trends or cross-country differences in hours

worked. These three studies all explore the effects of current experience/hours on future labor

market outcomes as potential explanations for the phenomenon, although the details of the

mechanism vary. Santos (2014) considers a model in which current work hours affect future

productivity in order to explain the increase in the correlation between hours and wages for

the last quarter of the 20th century in the U.S. He shows that in the CPS data, this dynamic

effect has become stronger for higher wage quintiles, whereas it has weakened for lower wage

quintiles, leading to the recent rise in hours-wages correlation. Elsby and Shapiro (2012)

focus on the extensive margin of labor supply and explore changes in the return to experience

as driving forces behind the changing patterns of employment/nonemployment rates by

education. They find that the return to experience for high-school dropouts has fallen

significantly since the 1970s, which contributed to a downward trend in their employment

rate. Michelacci and Pijoan-Mas (2008) exploit a search-matching framework to explain

the divergence in hours worked between the U.S. and Continental Europe. They assume

that current hours of work affect the future probability of getting outside job offers. In this

framework, greater wage inequality makes outside offers more attractive, inducing agents to

work longer hours. Relative these studies, we propose an alternative mechanism based on

self-insurance motive. Our work shows that due to the insurance motive, the evolution of

the second moment of wages plays a dominant role over the first moment in explaining the

recent changes in U.S. male hours worked across skill groups.

The remainder of this paper is organized as follows. Section 2 describes the stylized facts

on changes in the U.S. wage structure and the trends in hours worked by skill group. In

section 3, we describe a general equilibrium model with heterogeneous agents and incomplete

markets. We then describe the calibration procedure in section 4. Section 5 presents main

quantitative results and Section 6 concludes the paper.

5

2 Data

This section documents the changes in the wage structure and the hours worked by skill

groups for the past few decades. Our data on the first moment of wages and hours worked

are drawn from the Current Population Survey (CPS) March Supplements over the period

1967-2006. Since a panel dimension is required to estimate the time-varying variances of per-

sistent and transitory wage components, we also exploit the Panel Study of Income Dynamics

(PSID). In order to avoid issues regarding changing selection into labor force participation

among women, we restrict our attention to males aged between 25 and 59. Detailed selection

criteria are provided in the Appendix I.

2.1 Changes in the Wage Structure

We measure skill (e) based on one’s educational attainment: define a skilled worker (s) as

one with a college degree and an unskilled worker (u) as one without a college degree. We

assume that individual wages follow a skill-specific process, which depends on a time-varying

skill price per effi ciency unit of labor, experience, and a history of persistent and transitory

labor productivity shocks. Specifically, the hourly wage of individual i of age a at time t of

skill type e ∈ {u, s} is assumed to follow

lnweiat = βet + f e(Xit) + yeiat,

where βet is a skill-specific time dummy, Xit = a−Si−5 is potential experience, f e is a skill-

specific return to experience (assumed to be a cubic polynomial of Xit), and Si is years of

schooling. We assume that the log wage residual yeiat consists of a persistent and a transitory

component:

yeiat = µeiat + υeiat + θeiat,

where µeiat is a persistent component, υeiat ∼ (0, λυt (e)) is a transitory component, and θ

eiat ∼

(0, λθ) is a measurement error. The variance λυt (e) of the transitory component υeiat is allowed

to vary over time, whereas that of the measurement error is time-invariant and independent

6

of skill type. The persistent component µeiat is modelled as an AR(1) process:

µeiat = ρeµeiat−1 + ηeiat,

where ρe is the persistence, ηeiat ∼ (0, ληt (e)), and µei1t ∼ (0, λµ1(e)). The persistent wage shock

ηeiat has a time-varying variance of ληt (e).We assume that all four variables, υ

eiat, θ

eiat, η

eiat, and

µei1t are orthogonal and i.i.d. across individuals. We focus on time effects in this specification

because time effects rather than cohort effects have been important in explaining rising wage

inequality in the U.S. for recent decades as suggested by Heathcote et al. (2005).

With this statistical model, we estimate the wage processes of skilled and unskilled work-

ers separately in two stages. In the first stage, we run an Ordinary Least Squares (OLS)

regression of log hourly wages on a time dummy, a cubic polynomial of potential experience,

for each skill type. We then exploit the log wage residuals from these first-stage OLS regres-

sions to estimate the processes for persistent and transitory components. Since a transitory

shock in wages disappears after one period, autocovariances of individual log wage residuals

identify a persistent component.7 In implementing the estimation procedure, we compute

sample autocovariances of the log residual wages yeiat from the first-stage regressions of all

possible orders for each age group in every survey year. The parameters are then estimated by

minimizing the equally weighted distance between these sample autocovariances and their

model counterparts.8 To identify a transitory component separately from a measurement

error, we use an estimate of 0.02 for the measurement error from French (2004).

7A large literature in labor and macroeconomics has been devoted to estimating labor income processes.

As Krueger et al. (2010) summarize, these studies often found that the estimates of persistent and transitory

variances based on the covariances of the levels of log earnings are significantly different from those based on

their first differences. A recent paper by Daly et al. (2013) attempts to reconcile these different estimates and

finds that “outlying”earnings observations either prior to or following a missing observation are misspecified

in standard labor income processes used in this literature. However, they find that without dropping such

“outlying”observations, one can use the estimates for persistent variances based on the levels of log earnings

because the biases are quantitatively small. In section 5, we show that a time-varying persistent wage

component drives most of important quantitative results in our model while the impacts of a transitory wage

component is quantitatively small.

8Details of the estimation procedure are provided in the Appendix II.

7

We define the log skill premium as the difference between the mean log hourly wages of

male college graduates to those of men without a college degree, based on the predicted wages

from the first-stage OLS regressions. Figure 1 plots the trends in the male skill premium

for the period of 1967-2006 from both CPS and PSID. Both data show a drop in the skill

premium during the 1970s and a sharp rise since the late 1970s, as is consistent with the

literature.

Figure 2 presents the time series of the variances of persistent and transitory wage shocks

of each skill type, estimated in the second stage. The top panel of Figure 2 indicates that the

estimated variance (ληt (s)) of a persistent wage shock for skilled men rose rapidly beginning

in the late 1970s and continued to increase until it began to decline around 2000. To the

contrary, the estimated variance (ληt (u)) of a persistent wage shock for unskilled men declined

during the 1980s. This led to a substantial divergence in the level of wage volatility between

skilled and unskilled men during the 1980s and in the early 1990s. The estimated variances

(λυt (e)) of a transitory wage shock for both groups in the bottom panel of Figure 2 also show

different patterns [[accross]] skill groups. Unskilled men experienced a continued increase

in the variance of a transitory wage shock at a constant pace over the measured period.

However, skilled men experienced a slow-down in the increasing pace of the variance of a

transitory wage shock during the 1980s and faced a sudden increase in its variance since the

late 1990s.

Figure 3 presents the variance decomposition of log wage residuals for both skill groups

based on the estimated parameters. The left panel shows that total wage variance for

unskilled men was 0.22 in 1967 and increased by 77% to reach its peak at 0.39 in 2003 . The

right panel indicates, total wage variance of skilled men in 1967 was 0.21, similar to that of

unskilled men. However, it more than doubled for the same period and peaked in 2003 at

0.49. Comparison between the left and right panel indicates that skilled men experienced a

much larger increase in their residual wage volatility (measured as the estimated variance of

log wage residuals) for the past four decades than unskilled men.

Figure 4 reports the shares of persistent wage shocks in the variance of residual wages

8

for both skill groups. Among unskilled men, the fraction of wage volatility accounted for by

the persistent component was 0.78 in 1967. It declined steadily since then and reached 0.58

in the early 2000. The trend in the share for skilled men was not monotonic; it decreased

during the 1970s, but rose sharply during the 1980s and stayed roughly constant at about

0.75 during the 1990s. These patterns imply that unskilled men experienced the rise in

residual volatility with a rising share of the transitory wage shock during the sample period.

In contrast, skilled men went through a substantial increase in residual wage volatility during

the 1980s due to the accumulated effects of the rising variance of the persistent wage shock.

In summary, skilled men experienced a greater increase in the total wage volatility with a

larger fraction due to a persistent wage component than unskilled men between 1967 and

2000.

One might think that the greater increase in wage volatility for skilled men than for

unskilled men is largely due to a rising share of college graduates in the sample.9 Suppose

that recent college graduates have different characteristics which expose them to larger wage

volatility than older college graduates. In this case, wage volatility of skilled men may be

estimated to have risen due to the composition changes, although wage volatility actually

stayed constant for every agent. We claim that the estimation results are not mainly driven

by these composition changes. The fraction of college graduates increased most rapidly

during the 1970s and grew only slowly thereafter. However, wage volatility among skilled

men changed little during the 1970s and rose sharply since the early 1980s. This suggests

that the composition changes are not the main driving force behind the trends in wage

volatility of skilled men.

We also examine a possibility that wage volatility appears to have risen due to increased

unemployment risks. Considering that wages tend to decline after a spell of unemployment,

frequent unemployment spells may increase variability of observed wages. In the PSID

data, we did not find any distinct trends in the transition probabilities from employment

9According to the CPS data, the share was about 15% in 1967, whereas it rose to 31% in 2006.

9

to unemployment for either skill group, particularly during the 1980s and 1990s. Thus, the

estimated rises in residual wage volatility for either skill group are not due to changes in

unemployment risks.

2.2 Trends in Hours Worked

In this section, we document the trends in hours worked by skill group using CPS March

Supplements for the last 40 years.

The upper panel of Figure 5 plots the trends in the average weekly hours (annual hours

worked divided by 52 weeks) of employed men by skill type for the period 1967-2006. The

average weekly hours of both skilled and unskilled men were steady during the 1970 and

began to increase in the early 1980s. Even though the upward trends discontinued in the

early 2000s, both skill groups worked more in 2006 than they did in the beginning of the

sample period.

The lower panel presents the differences in the average weekly hours between skilled and

unskilled men over the same period. In the beginning of the sample period, skilled men on

average worked about 2 hours more than unskilled men. The hours differential rose to about

3.5 hours in the mid 1990s and declined afterwards. Most of the rise in skilled-unskilled

hours differentials took place during the 1980s and the 1990s during which skill premium

rose rapidly.10

We claim that the rise in relative hours of skilled to unskilled men is mainly due to

time effects rather than cohort effects. To illustrate this point, consider a statistical model

in which skilled-unskilled hours differential (hd) depends on age (a), time (t), and cohort

(t− a), where these three components are additively separable:

hd = g1(a) + g2(t) + g3(t− a).

Tracking the hours differential over time within cohorts removes cohort effects and captures

10Male work hours vary by age: prime-age male work more than the young and the old. However, the

trends in hours by skill type are not mainly driven by changing age composition. Holding the age distribution

of the sample the same as that in 1967 changes the trends (based on changing age distribution) little.

10

both time (∆g1(a)) and age effects (∆g2(t)). Instead, following the same age group over

time sorts out age effects, reflecting the difference between time and cohort effects (∆g2(t)−

∆g3(t−a)). Lastly, between-age variations in cross-sectional data indicate the sum (∆g1(a)+

∆g3(t− a)) of age and cohort effects, leaving out time effects.

Table 1 presents the average annual changes in the hours differential within cohort,

within age, and between age groups. Note that the age effects are time-invariant, whereas

both time and cohort effects are time-varying. The results show that within-cohort changes

in the relative labor supply are statistically different from zero for most of the sub-periods.

However, between-age groups variations are statistically insignificant except for the first

decade in the sample. Figure 6 also confirms this point. The top panel presents within-

cohort variations in hours differential across age groups, the slopes of which represent both

age and time effects. The slopes vary significantly by cohort, which is attributable to strong

time effects. On the other hand, the bottom panel suggests weak cohort effects. It indicates

between-age groups variations for different years, the slopes of which are determined by age

and cohort effects. Although the levels of the age profiles vary by period, the slopes are

similar across different years.

In addition, since both within-cohort and within-age changes in the skilled-unskilled

hours differential have the time effects in common, the correlation between the two should

be strong if the time effects are strong. The correlation is very close to one as shown in

Table 1. To the contrary, the correlation between within-age and between-age variations

is only 0.2, implying that the cohort effects have been weak. These results suggest that

time effects rather than cohort effect have been important in the increasing trends in the

skilled-unskilled hours differential during the 1980s and the 1990s.

3 The Model

We present a general equilibrium heterogeneous agent model with incomplete markets. We

consider two skill types of workers that face idiosyncratic productivity shock drawn from

11

skill-specific distributions.

3.1 Households

There are two types of single-person households—skilled and unskilled—in this economy. We

denote the skill type by e where e ∈ {u, s}11. Households are endowed with one unit

of time in each period. Each household faces a skill-specific process for his idiosyncratic

productivity. In the beginning of a period, a household of skill type e draws persistent and

transitory components (µ and υ, respectively) of his productivity, x, from her type-specific

distribution. After observing her productivity draw, the household decides whether to be

employed or non-employed. If employed, he provides hx effi ciency units and gets paid we per

effi ciency unit of labor in the labor market, where h is hours worked chosen by this household.

There is a minimum requirement for the number of hours such that each household should

work at least h when employed. If this household is non-employed, then his hours of work h

is zero, and he receives no wage income.

In this economy, only a risk-free asset is available for saving and no borrowing is allowed,

which make markets incomplete. Households insure themselves against low productivity

shocks through both precautionary savings and labor supply decisions.

At the end of the period, households face a survival probability of γ: a fixed fraction 1−γ

of the population dies and is replaced by newborn households in the next period. Unclaimed

assets of deceased households are collected and redistributed to the newborn household in a

lump-sum transfer, T .

Each household maximizes the expected lifetime utility from a stream of consumption ct

11We abstract from educational choice by households to focus on differences in labor supply by skill level.

According to the CPS March supplements, the fraction of individuals with a college degree has increased

rapidly until 1970s and slowed down afterwards. The extent that this change in acquiring a college education

affects labor supply decisions is unaccounted for in our analysis, thus, our quantitative results may be biased.

12

and non-working hours or leisure 1− ht:

E∞∑t=0

(βγ)tu(ct, ht)

u(c, h) =c1−σc

1− σc+ ψe

(1− h)1−σh

1− σhand β, γ ∈ (0, 1)

where β is the discount factor, ψe is a skill-specific constant, σc is the inverse of elasticity

of intertemporal substitution with respect to consumption, and σh is the same but with

respect to leisure. Consumption and leisure are assumed to be additively separable in a

CRRA utility function, which is common for both skill types. In period t, households choose

consumption ct, hours of work ht, and asset holdings at+1 for the next period subject to

ct + at+1 ≤ at(1 + rt) + wethtxt + Tt,

ct ≥ 0, at+1 ≥ 0, ht ∈ {0} ∪ [h, h]

where rt is real interest rate on the risk-free asset, wet is real wage per effi ciency unit of

labor for each skill type. In order to reflect the presence of a skill premium, we consider two

different real wages for skill types. The logarithm of the idiosyncratic productivity xet of a

skill type is specified as the sum of a persistent and a purely transitory component (µet and

υet , respectively), given by

lnxet = µet + υet

where υet ∼ N(0, λυt (e)). The persistent component is specified as an AR(1) process:

µet = ρeµet−1 + ηet

where ηet ∼ N(0, ληt (e)). We consider separate productivity processes for the two skill types

to account for differences in persistence of the shocks and in the variances of both persistent

and transitory productivity. Variances of both persistent and transitory productivity shocks

are also allowed to be time-varying to incorporate the changes in wage volatility in the PSID

data. We assume that a household born at period τ starts his lifespan with a persistent

productivity shock drawn from a type-specific distribution, which is time-invariant, defined

as µeτ ∼ N(0, λµ(e)).

13

The recursive fomulation of a household problem of skill type e is given by:

V e(a, µe, υe) = maxc≥0,a′≥0,h∈H

{u(c, h) + βγEV e(a′, µe′, υe′)}

subject to

c+ a′ ≤ a(1 + r) + wehxe + T

lnxe = µe + υe

µe′ = ρeµe + ηe, ηe ∼ N(0, λη (e))

υe ∼ N(0, λυ (e))

H = {0} ∪ [h, h]

3.2 Production

We consider a representative firm that produces output using a standard Cobb-Douglas

technology with competitive factor markets. The firm’s production function is given by

F (Kt, Ht, zt) = ztKαt H

1−αt

where Kt is aggregate capital and Ht is aggregate hours in effi ciency units, which is a CES

aggregator of hours worked by skilled St and unskilled Ut households:

Ht ={χtU

φt + (1− χt)S

φt

} 1φ,−∞ < φ ≤ 1

where 1 − 1φdenotes the elasticity of substitution between the two types of labor inputs.

Parameter χt is introduced to capture skill-biased technical change over time. A decrease

in χt makes effi ciency units of hours worked by skilled workers more productive relative to

those of unskilled workers, leading to a rise in skill premium.

The assumption of competitive factor markets imply

rt + δ = FK(Kt, Ht, zt) = ztα(Kt/Ht)α−1

wut = FU(Kt, Ht, zt) = zt(1− α)χt(Kt/Ht)αH1−φ

t Uφ−1t

wst = FS(Kt, Ht, zt) = zt(1− α)(1− χt)(Kt/Ht)αH1−φ

t Sφ−1t

14

where δ is the depreciation rate of physical capital. The skill premium, defined by the relative

wage of skilled men to unskilled men, is given by:

wst/wut =

1− χtχt

(St/Ut)φ−1

3.3 Competitive Equilibrium

A competitive equilibrium consists of value functions, optimal policy functions, aggregate

inputs, prices, and a law of motion for the distribution Gt+1 = Tt (Gt) such that:

i) Givenwe and r, households of skill type e optimally choose ct(e, at, µt, υt;Gt), at+1(e, at, µt, υt;Gt),

and ht(e, at, µt, υt;Gt) that are consistent with V e,

ii) A firm chooses Kt, Ut, and St that maximize profits,

iii) Labor market for skill type clears:

Ut =

∫ht(u, at, µt, υt;Gt) exp(µt + υt) dGt(u, at, µt, υt)

St =

∫ht(s, at, µt, υt;Gt) exp(µt + υt) dGt(s, at, µt, υt)

iv) Capital market clears:

Kt =∑e

∫at dGt(e, at, µt, υt)

v) Goods market clears:

∑e

∫ct(e, at, µt, υt;Gt) dGt(e, at, µt, υt) = F (Kt, Ht, zt) + (1− δ)Kt −Kt+1

vi) Individual and aggregate behaviors are consistent.

4 Calibration

We select a set of parameters using standard values used in related literature. This includes

the capital share α in the production function, the depreciation rate δ of capital, and the

elasticity φ of substitution between the two types of skills in production. We assign 0.33

15

and 0.08 to α and δ, respectively, following Aiyagari (1994) and Prescott (1986). There

is a large literature on the estimate of the elasticity of substitution between skilled and

unskilled workers and most estimates fall between 1.4 and 2. Katz and Murphy (1992) find

that this elasticity is about 1.4 based on CPS March Supplements for the period 1967-1973.

Autor et al. (2008) show that the elasticity is estimated to be about 1.6 using the CPS data

extended to 2005. Heckman et al. (1998) and Card and Lemieux (2001) obtain the estimates

of 1.5 and 2, respectively. We use 1.5 as the elasticity of substitution between skilled and

unskilled men, which sets the parameter φ to 13. We choose the survival probability γ = 0.972

to match the average life span of workers to be 35 years. The maximum hours of work h is

set at 5840 hours per year (16 hours per day times 365 days). The minimum hours of work

h is set to match 800 hours per year, which was used as a cutoff for a moderate attachment

to the labor force in Prescott et al. (2009). We set the preference parameter σc to 1.5,

following the common estimate of risk aversion parameter between 1 and 2 in the literature

from Attanasio (1999). The parameter σh is set to match the Frisch elasticity of hours of

0.4 as in Chang and Kim (2006).

Another set of time-invariant parameters is calibrated so that the model can replicate

target moments observed in the data in 1967, which is the initial period we assume the U.S.

economy to be in the steady state. The discount factor, β, is picked so that the equilibrium

real interest rate in the initial steady state matches an annual real interest rate of 0.04. The

parameters ψu and ψs in the utility function are chosen to match the average weekly hours

worked of unskilled and skilled men in the initial steady state, which are 41.3 and 43.5 hours,

respectively.

The last set of parameters are allowed to vary over time. This includes the parameter χt,

which governs skill-biased technical change, and is chosen to match the time series of skill

premium in the CPS data over the period we consider. As for the parameters that determine

the two components of idiosyncratic productivity x, we take direct estimates from the PSID

data, described in section 2.1. Table 2 summarizes the values of parameters used for our

model economy.

16

5 Results

Our model economy is assumed to be in the initial steady state, which is calibrated to 1967.

Then the economy experiences over-time changes in volatility of shocks and skill premium as

estimated and observed in the data until 2000. The economy gradually converges to the new

steady state. In this section, we first report the initial steady state results. We then explore

the transitional dynamics of labor supply by skill group in response to increased volatility

through persistent and transitory wage shocks. This is followed by some counterfactual

experiments. The computation algorithm to solve the model is described in the Appendix

III.

5.1 Initial Steady State

We solve for the steady state allocations of the model economy using parameters chosen to

match the wage structure in 1967. The variances of persistent and transitory wage com-

ponents for both skill types are set to their estimates in year 1967. The parameter for a

skill-biased technology, χ, is set to match the U.S. skill premium observed in 1967. These

values are presented in Table 3.

As reported in Table 4, skilled men face both higher wage per effi ciency unit of labor and

larger volatility in residual wages than unskilled men. The first two rows of Table 4 show

the variances of persistent and transitory shocks for each skill group in the initial steady

state. Skilled men face greater variances in both persistent and transitory wage shocks than

unskilled men. The third row shows that skilled men earn about 46% more per effi ciency

unit of labor than unskilled men in the initial steady state, consistent with what we observe

in the data. In the fourth row, we report the average weekly hours worked by skilled and

unskilled men. We pick preference parameters ψu and ψs so that they replicate what we

observe in the U.S. data in 1967. The remaining rows present labor income, assets, and

consumption net transfers for both skill groups.

17

5.2 Transitional Dynamics

5.2.1 Hours of Work

Our main exercise is to find transition dynamics of the economy in response to changes in the

wage structure. We assume that in 1967, households face unexpected changes in the variances

of persistent and transitory productivity shocks, skill premium, and labor composition (in

terms of skill type) and that these changes stop in 2000. After 2000, the economy converges

to a new steady state.

Figure 7 presents the transition path of hours worked of each skill group to the new

steady state. During the periods when wage volatility has risen, skilled men continue to

raise their hours worked. They increase their average weekly hours work by about 2 hours

between 1967 and 2000. However, as they accumulate enough stock of precautionary savings

over time, they are able to afford to reduce their work hours due to wealth effects in the

long run. They begin to [[decrease their hours worked starting in 2000 when]] wage volatility

stops increasing and work less in the new steady state than in the initial steady state.

Unskilled men initially increase their hours worked as they experience a rise in the vari-

ance of persistent wage shocks. However, they begin to reduce their labor supply in the

late 1970s as the share of transitory wage shocks in total wage volatility rises and that of

persistent wage shocks declines steadily. This pattern of changing labor supply hints that

the relative composition of persistent and transitory wage components in residual wages may

have significant effects on individual labor supply decisions. After 2000 when there is no ad-

ditional changes in the wage structure, unskilled men increase their labor supply, converging

to the new steady state. This is due to a general equilibrium effect. The substantial amount

of savings accumulated by skilled men lowers the real interest rate, providing unskilled men

disincentives to save. As a result, unskilled men reduce their savings, raising their work

hours in the new steady state, compared to the initial steady state.

Since skilled men increase their work hours more than unskilled men in the short-run,

skilled-unskilled hours differential expands for the period. As Figure 8 shows, differences in

18

work hours between skilled and unskilled men rises from 2.2 hours per week in 1967 to 4.3

hours in 2000. This model prediction on the hours differential is consistent with what we

observe in the data.12 An intriguing implication of the model is that this increased hours

differential between the two skill types is a short-run phenomenon. As skilled men reduce

their work hours due to wealth effects in the long run, the hours differential begins to decline

after 2000, converging to an even lower level in the new steady state than in the initial steady

state.

5.3 Decomposing the Forces

This subsection decomposes the forces that contributed to changes in hours worked and

savings by skill type. Specifically, short-run effects of the changes in the wage structure

between 1967 and 1999 are analyzed on hours and wealth for each skill type and their

differences. The changes in the skill premium and the variances of persistent and transitory

wage components of both skill types are summarized in Table 5. We proceed by implementing

the following four counterfactual experiments: (i) SP: agents experience the rise in skill

premium without a change in the variances of residual wages; (ii) SR1: agents anticipate the

variances in residual wages to rise in the next period; (iii) SR2: agents face increased wage

volatility and anticipate the wage volatility to stay constant at their new higher levels; and

iv) SS2: the economy reaches a new steady state with a wage structure corresponding to

year 2000.

Table 6 shows that in response to a rise in skill premium, skilled men reduce their hours

worked due to a dominant income effect. This causes hours differential between skilled and

unskilled men to decline from 2.2 hours per week in the initial steady state to 1.5 hours per

12The model slightly overstates the rise in the hours differential observed in the data. This overstatement

might be due to some missing factors contributing to a stronger income effect for skilled men, such as

increasing fractions of skilled men with working spouses. Data show that in recent decades, female labor

force participation has increased substantially among more educated women and that there is more positive

assortative matching with respect to education in the marriage market today than in the past (see Greenwood

et al. (2014)). This implies that there is a stronger income effect at work for skilled men through working

spouses than what we consider in this paper.

19

week. Savings do not respond much to a rise in skill premium for both skill groups. When

wage volatility increases, both skilled and unskilled men increase their hours worked in the

short run (SR2), as we expect. Since skilled men experience a greater rise in wage volatility

and derives less disutility from working than unskilled men (ψu > ψs), they increase their

hours and savings more than unskilled men. The hours differentials between the two skill

groups rise from 2.2 hours to 4.3 hours in SR2. The effect of anticipated increase in future

wage volatility (SR1) on hours worked is even greater than SR2 effect. However, in the long

run, skilled men accumulate substantial amount of precautionary savings as wages become

more volatile, resulting in a reduction of their work hours due to the income effect. This

significant increase in savings by skilled men lowers the real interest rate in the new steady

state, providing unskilled men with disincentives to save. As a result, unskilled men increase

their hours worked while reducing their savings in the long run.

Figure 9 shows how skilled men adjust their labor supply and savings by asset level in

expectation of an increase in the variance of their persistent productivity shock in the next

period (SR1), holding the transitory productivity fixed at the mean level. The first thing to

note is that in the initial steady state, the correlation between persistent productivity and

hours is positive for the rich and negative for the poor. When one faces a higher persistent

productivity, there are two effects at work: the substitution effect and the income effect.

A higher persistent productivity means a higher opportunity cost of taking leisure in the

current period, increasing labor supply. On the other hand, a higher persistent productivity

has a positive income effect, inducing individuals to reduce their hours worked. The greater

amount of assets one has, the smaller the income effect is, making the substitution effect more

significant than the income effect. This explains the changing correlation between persistent

productivity and hours depending on asset level. Secondly, we note that individuals with

higher persistent productivity reduce both their work hours and savings in response to a

future increase in volatility in persistent productivity. On the other hand, those with lower

persistent productivity increase both labor supply and savings. This is because an individual

with a good persistent productivity in the current period expects an even better productivity

20

in the next period, given the high probability of his receiving a good persistent productivity

again. Exactly the opposite applies to an individual with a bad persistent productivity

today. As a results, the dashed lines (that indicate hours worked and savings of skilled

men expecting an increase in future volatility through their persistent productivity shock)

cross the decision rules in the initial steady state in Figure 9. However, individuals with

the mean level of the persistent productivity increase both their work hours and savings for

self-insurance.

When the variance of the persistent productivity rises in the current period, skilled men

further adjust their work hours and savings as shown in Figure 10 (SR2). Entering the

world with higher volatility associated with persistent productivity, they increase their work

hours and savings for all asset levels—compared to those in the initial steady state—due to

the precautionary savings motive.

We present skilled men’s response of labor supply and savings to a future increase in

the variance of their transitory productivity shock by asset level, holding the persistent pro-

ductivity fixed at the mean level, in Figure 11 (SR1). Note that transitory productivity is

positively correlated with hours for all asset levels. Since transitory productivity only affects

current wages, the income effect is less than that associated with persistent productivity

and therefore the substitution effect is always dominant. As individuals expect an increase

in future volatility through transitory productivity, they react by working longer hours and

accumulating more savings for all asset levels. Regardless of the current transitory produc-

tivity, they expect greater volatility, adjusting their work hours and savings so that they

can insure themselves against this change. These effect are stronger for the poor than the

rich because wage income is more important for consumption for those households with little

wealth..

Given effects of an expected rise in future volatility through transitory productivity, indi-

viduals further adjust their work hours and savings, entering the world with higher volatility

associated with transitory productivity (SR2). This effect is shown in Figure 12. For all lev-

els of assets, individuals increase their work hours and savings relative to what they were in

21

the initial steady state, with the greatest adjustment taking place for those with lower levels

of assets. However, the quantitative effect of the increase in volatility through transitory

productivity is smaller than what we obtain when the variance of persistent productivity

increases by the same factor.

In summary, we confirm that unskilled men show similar response patterns of work hours

and savings as skilled men when faced with increases in the variances of persistent and

transitory productivity shocks. When they expect the persistent productivity to become

more uncertain in the next period, households with a good persistent productivity reduce

their labor supply whereas those with a bad persistent productivity increase hours worked.

When a household enters the period with more volatile persistent productivity, they increase

hours worked and savings, compared to initial steady state levels. The responses of unskilled

men in hours worked and savings to an increase in the variance of transitory productivity

are also similar what we find for skilled men.

5.4 Counterfactual Experiments

We run a few counterfactual experiments to examine the role of not only the level of total

wage volatility but also its composition of persistent versus transitory wage shocks, in ex-

plaining the trends in hours.13 For each experiment, both short-run effects and transition

dynamics are presented.

The first two counterfactuals investigate how skilled men adjust their hours worked when

they were exposed to the same changes in either level of total wage volatility or the shock

composition as unskilled men. In the first experiment (CF1), we have skilled men experience

the same magnitude of increases in wage volatility as unskilled men, that is, a smaller increase

in wage volatility than observed. With the same increase in wage volatility as that of unskilled

13Note that skilled and unskilled men differ in the degree of disutility from working and in labor income

fraction of wealth. These differences between the two skill groups affect the quantitative impacts of the level

of wage uncertainty, and its composition, on the trend of skilled-unskilled hours differentials we present using

these counterfactual experiments. Therefore, we suggest the readers focus on the relative importance of the

two channels instead of the absolute quantitative impact of each channel.

22

men, skilled men increase their work hours less in the short run than in the benchmark model

as Table 7 shows. The rise in the hours difference between the two skill groups is reduced

by two-thirds (SR2), compared with that in the benchmark model. This implies that the

greater rise in wage volatility for skilled men contributes significantly to the rise in skilled-

unskilled hours differentials for the past forty years. The effect of expected change in future

shock volatility captured by SR1 declines more dramatically in this counterfactual exercise,

compared with the SR2 effect. In expectation of greater wage volatility in future, skilled

men increase their work hours to 50.9 hours in the benchmark model whereas they work

46.2 hours in this counterfactual exercise. More than three quarters of the rise in skilled-

unskilled hours differential in SR1 can be ascribed to the rise in total volatility instead of

shock composition.

During the transition path to the new steady state shown in Figure 13, skilled men

increase their work hours less with the smaller rise in wage volatility than in the benchmark

case. This smaller increase in wage volatility for skilled men also weakens their precautionary

saving motive, compared with the benchmark model. The general equilibrium effect is also

weaker, reducing the real interest rate less than in the benchmark case. As a result, in the new

steady state, unskilled men accumulate more savings and work less than in the benchmark

case. This causes the skilled-unskilled hours differential to be higher in the new steady state

than in the benchmark model.

The second counterfactual experiment (CF2) assumes that skilled men experienced the

same changes in wage composition of persistent and transitory shocks as unskilled men. With

increased wage volatility more through transitory wage shocks as unskilled men, skilled men

increase their work hours less than in the benchmark model. As Table 7 presents, this reduces

the rise in skilled-unskilled hours differential by half (SR2), compared with the benchmark

case. This magnitude is similar to that in CF1. It implies that the changes in the level of

total wage volatility and in the relative composition of shocks are almost equally important

in increasing skilled-unskilled hours differential in the short run. However, the SR1 effect

is only slightly affected by the change in shock composition as Table 7 indicates. With

23

the same change in shock composition for skilled men as for unskilled men, the increase in

relative hours of skilled men to unskilled men in SR1 is slightly smaller in CF2 than in the

benchmark.

The transition dynamics in CF2 confirm that the composition of shocks is fairly important

in hours decision of skilled men. With increased wage volatility more through transitory

shocks, skilled men’s hours worked decline earlier and converge to even lower levels in the

long run than in the benchmark model. The precautionary savings motive associated with

transitory wage shock is weaker than with persistent wage shock, leading to smaller increases

in both work hours and savings of skilled men in CF2, compared with the benchmark model.

As a result, the long-run labor supply of unskilled men is slightly lower due to a weaker

general equilibrium effect in CF2 than in the benchmark model. This leads to a greater

decline in skilled-unskilled hours differential in the long run than in the benchmark case.

[need to point figure cf1234]

The remaining two counterfactuals are to examine the importance of composition of

the shocks on labor supply further. The third counterfactual (CF3) computes the average

weekly hours of each skill type when wage volatility rose through persistent wage shocks

only. In this case, both skill groups increase their work hours more in the short run than

in the benchmark model because of a stronger precautionary saving motive. However, the

difference in their hours changes little as shown in Table 8. This pattern also arises in the

transition dynamics. Although the time path of hours differentials during the transition to

the new steady state stays almost unchanged, both skill groups work longer hours than in

the benchmark economy. The general equilibrium effect is greater, inducing unskilled men

to work longer hours in the new steady state than in the benchmark model.

The last counterfactual experiment (CF4) considers a case where wage volatility increases

through transitory shocks only. With these changes, both skill groups work shorter hours

than in the initial steady state (SR2) as shown in Table 8. Recall that the value of a transitory

productivity shock is positively associated with hours worked. Also note that the reduction

in work hours of those with a bad productivity is greater than the increase in work hours

24

of those with a good productivity. Consequently, the average hours worked decline. Since

skilled men own larger assets than unskilled men, this reduction in hours is more pronounced

for them than for unskilled men. Transition dynamics also show similar patterns. Both skill

groups reduce their hours worked in response to increased wage volatility from transitory

shocks only. This effect is stronger for skilled men than for unskilled men. In addition, as

both skill groups accumulate more savings over time, they both reduce their hours worked

further, with this effect being greater for skilled men. As a result, their hours difference

decline substantially moving towards the new steady state.

6 Conclusion

In the past few decades in the U.S., relative labor supply of skilled men to unskilled men

increased while the skill premium rose sharply. This fact contrasts with the predictions of

previous literature suggesting a zero labor supply elasticity with respect to wage. This paper

attempts to explain this discrepancy using the second moment of wages, i.e. wage volatility.

Using the PSID., we document that wage volatility increased for both skill groups with

greater increases for skilled men than for unskilled men in recent decades. In addition, we

show that skilled men experienced the rise in wage volatility with a rising share of persistent

wage shocks, relative to unskilled men.

This paper develops a general equilibrium incomplete markets model where agents re-

ceive idiosyncratic labor productivity shocks drawn from skill-specific distributions. Feeding

the estimated wage processes from the PSID data into the model, we study the model’s

implications on the trends in hours by skill group. The model can replicate the observed

trend in hours worked of skilled men relative to unskilled men during the transitions to the

new steady state. However, as skilled men accumulate precautionary savings over time, they

can afford to reduce their labor supply relative to unskilled men in the new steady state.

This implies that hours adjustment is crucial for self-insurance in the short run, whereas

precautionary savings play more important role in the long run. Our study also finds that

25

the variances of persistent wage shocks mainly determines the trends in hours, while the

quantitative impact of transitory wage shocks is fairly small.

What caused wage volatility of skilled men to rise more with an increasing fraction

of a persistent wage component than that of unskilled men is outside the scope of this

study. Exploring potential explanations behind the phenomenon may help improve our

understanding of male labor supply and their macroeconomic implications.

26

Appendix I. Data - Wages and Hours Worked

We use data from Current Population Survey (CPS) March Supplements and Panel

Study of Income Dynamics (PSID) over the period 1967-2006 for analysis. Since PSID was

conducted biennially beginning 1997, we use a total of 35 surveys from PSID.

CPS and PSID Sample selection: We include only men aged between 25 and 59, not

a student and not disabled, with reported educational attainment in the sample. In CPS, we

classify individual employment status based on their annual hours worked and hourly wages

in the previous calendar year. Annual hours worked are a product of weeks worked in the

previous calendar year and usual hours per week in the previous year individuals reported.

Hourly wage is obtained by dividing annual labor income by annual hours worked. We

classify those who worked more than 800 hours for the past calendar year as employed and

others as non-employed. Conditional on being employed, one should earn more than half the

federal minimum wage per hour and not be self-employed to be included in the sample. We

apply the former restriction to exclude extreme outliers and the latter one due to it being

diffi cult to distinguish between labor and capital shares out of an individual’s annual income.

From PSID, we restrict the sample to male heads of household who participated in the labor

market last year (i.e., worked at least 260 hours). We exclude males in oversamples as in

many previous literature. The resulting sample is an unbalanced panel.

CPS and PSID Variables: In CPS, we obtain annual hours worked as a product of

weeks worked in the past year and usual hours worked per week in the past year. Since

usual hours worked per week in the past year are not available and weeks worked in the past

year are coded in intervals for surveys before 1976, we impute both variables for previous

surveys using the average weeks worked and the average usual hours worked per week in the

same education and weeks worked interval cells in the 1976 survey. Annual earnings in CPS

are income from wages and salaries. In PSID, we use head of household’s total annual work

hours and labor income of head to obtain annual hours worked and hourly wages.

27

Appendix II. Estimation of Wage Processes

As described in section 2.2, we model the log wage residual yeiat as the sum of a persistent

and a transitory component with time-varying variances, following Heathcote et al. (2010):

yeiat = µeiat + υeiat + θeiat

where µeiat is a persistent component, υeiat ∼ (0, λυt (e)) is a transitory component, and θ

eiat ∼

(0, λθ) is a measurement error. The persistent component µeiat is assumed to follow an AR(1)

process:

µeiat = ρeµeiat−1 + ηeiat

where ρe is the persistence, ηeiat ∼ (0, ληt (e)), and µei1t ∼ (0, λµ1(e)). We assume that all four

variables, υeiat, υeiat, η

eiat, and µ

ei1t are orthogonal and i.i.d. across individuals.

As we mention in 2.2, we take the estimate of 0.02 from French (2004) for the variance, λθ,

of measurement error. Then, we estimate a parameter vector Φe, which includes two time-

invariant parameters ρe and λµ1(e) and a set of time-varying parameters {ληt (e), λυt (e)}2006t=1967

for each skill group e ∈ {u, s}. Since the PSID are available biannually beginning in 1997,

we do not have empirical moments for transitory shocks for years 1997, 1999, 2001, 2003,

and 2005. In order to resolve this issue, we assume that the cross-sectional variance of log

residual wages in these missing years is the average of that in the previous year and in the

subsequent year and identify the variances of transitory wage shock for missing years as is

done in Heathcote et al. (2010).

For each sample year t, we construct 10-year adjacent age cells from ages 29 to 54 such

that, for instance, the age group 29 consists of those aged 25 to 34 years. We then compute

the empirical autocovariance, gea,t,n, of all possible orders for each age/year (a, t) cell in our

PSID sample using log wage residuals yeiat from the first-stage regressions:

gea,t,n =1

Iea,t,n

Ia,t,n∑i=1

yei,a,t, yei,a+n,t+n, n > 0,

where Iea,t,n is the number of observations for nth order autocovariance for age/year (a, t)

cell for skill group e. We then pick the parameters Φe that minimize the equally weighted

28

distance between this empirical autocovariance matrix and its theoretical counterpart:

Φe = arg minΦe

[Ge −Ge(Φe)

]′I[Ge −Ge(Φe)

],

where Ge is a stacked vector of empirical autocovariances as well as cross-sectional variances

for missing years, Ge(Φe) is a theoretical counterpart, and I is an identity matrix.

Appendix III. Algorithm

In this paper, we calibrate the parameters β, ψu, and ψs by targeting the initial steady

state. We allow the parameters λt = {χ, ληs , λυs , ληu, λυu}, governing persistent and transitory

idiosyncratic productivities and the skill premium, and the share of the skilled and the

unskilled in the laborforce, to vary over time.

Solving for the Steady State

Given sets of parameters (including λ0),

1. Guess price r. Given this guess, compute wu and ws (given λ0).

2. Solve the value function and get he(a, µ, υ), a′e(a, µ, υ), ce(a, µ, υ).

3. Generate a sample of population N over (e, a, µ, υ) space. (Or NS sample over (a, µ, υ)

and NU sample over (a, µ, υ). )

4. Compute aggregate Statistics:

K =∑e

∫adG(e, a, µ, υ)

U =

∫hu(a, µ, υ) dG(u, a, µ, υ)

S =

∫hs(a, µ, υ) dG(s, a, µ, υ)

H ={χUφ + (1− χ)Sφ

} 1φ

And define

r = zα (K/H)α−1 − δ

29

5. Check if |r − r| < ε. If not, update r and go back to step 1.

Solving transition economy (Changing parameters overtime)

The economy was originally at the initial steady state (∗), and there is a gradual change

for λ from λ∗ to λ∗∗ for periods t = t1, · · · , tτ and then the economy converges to the new

steady state (∗∗) at tT (> tτ ).

1. We know the parameters {ληs , λυs , ληu, λυu, π}tτt=t1 .

2. Solve two (initial and final) steady states: We needG∗(e, a, µ, υ), V e∗∗(a, µ, υ), r∗, χ∗, r∗∗, χ∗∗.

3. Guess the sequences of {r, χ, z}tT−1t=t1 . Then, the other prices are given by

wut = wu(zt, πt, rt, χt;α, δ, φ)

wst = ws(zt, πt, rt, χt;α, δ, φ)

wut = wtχt

(χt + (1− χt)

(χt

1− χt· πt) φ

φ−1) 1−φ

φ

wst = wut · πt

wt = zt(1− α)

(rt + δ

ztα

) αα−1

4. Solve the value function and get the decision rules during the transition periods:

(a) Using the value function at the new steady state: V e∗∗(a, µ, υ;λ∗∗),

(b) In each period during the transition (t = t1, · · · , tT ), we solve

V et (a, µ, υ) = max

c,a′,h{u(c, h) + βγEtV e

t+1(a′, µ′, υ′)}

subject to constraints with pt = {rt, wut , wst}, λt = {χt, ληs , λυs , ληu, λυu}.

30

(c) Solving the problem above backward from t = tT−1, · · · , t1. Notice that we do this

T − 1 times (also, this process doesn’t require convergence of the value function).

In each period, we obtain the decision rules: ct(e, a, µ, υ), ht(e, a, µ, υ), a′t(e, a, µ, υ)

and the value function, V et (a, µ, υ).

5. Given a distribution at t, Gt(e, a, µ, υ), simulate Gt+1(· · · ) by applying ct(·), ht(·), a′t(·).

Start from t = t1. Note that we already know Gt1(· · · ) from the distribution of the

initial steady state G∗(· · · )

(a) First calculate the aggregate statistics:

Kt =∑e

∫adGt(e, a, µ, υ)

Ut =

∫ht(u, a, µ, υ) exp(µ+ υ)dGt(u, a, µ, υ)

St =

∫ht(s, a, µ, υ) exp(µ+ υ)dGt(s, a, µ, υ)

(b) Using these aggregate statistics, compute

• an updated share χt (which implies the observed skill premium πt) such that

χt =(St/Ut)

φ−1

πt + (St/Ut)φ−1

• and an implied price rt such that

rt = ztα(Kt/Ht

)α−1

− δ

where

Ht ={χtU

φt + (1− χt)S

φt

} 1φ

(c) Get an updated distribution Gt+1 and go back to (5a) until we get GtT−1.

6. Check if the guessed sequences {rt}∗∗t=∗, {χt}∗∗t=∗ are close enough to the model-implied

sequences of {rt}∗∗t=∗, {χt}∗∗t=∗. If not, update the guesses and go back to step 3.

31

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34

Figure 1: Trends in Skill Premium for U.S. Men between 1967 and 2006

Note: Wages of employed men who worked more than 800 hours in the past year.

35

Table 1: Average Annual Changes in Skilled-Unskilled Hours Differential

Within Cohort Within Age Between Age

1967− 19710.0754∗

(0.0307)

0.0269

(0.0306)

0.0540∗∗

(0.0132)

1972− 19760.0338

(0.0496)

−0.0078

(0.0495)

0.0480∗∗

(0.0186)

1977− 19810.1731∗∗

(0.0381)

0.1614∗

(0.0402)

0.0099

(0.0230)

1982− 1986−0.0754∗

(0.0356)

−0.0847∗∗

(0.0378)

0.0068

(0.0166)

1987− 19910.1298∗

(0.0575)

0.1103∗

(0.0560)

0.0275

(0.0212)

1992− 1996−0.0437

(0.0448)

−0.0635

(0.0456)

0.0022

(0.0208)

1997− 20010.0382

(0.0456)

0.0247

(0.0469)

0.0214

(0.0209)

2002− 2006−0.1404∗∗

(0.0519)

−0.1648∗∗

(0.0484)

0.0260

(0.0191)

Correlation with Within Age 0.9970 −− ..

Correlation with Between Age .. 0.1661 ..

Note: Numbers in parentheses represent standard errors. Coeffi cients with

a * and a ** are statistically significant at a 5% and a 1% level, respectively.

36

Table 2: Parameters

Parameter Target

Parameters taken from the literature

σc= 1.5 Relative risk aversion of 1.5

σh= 4.26 Frisch elasticity of hours of 0.4

α = 0.36 Capital share (e.g. Aiyagari (1994))

δ = 0.08 Capital depreciation rate (e.g. Prescott (1986))

φ =13

Substitution elasticity of 1.5 b/w skill groups

(e.g. Heckman et. al. (1998))

h= 0.60 5, 840 hours per year

h= 0.14 800 hours per year

Parameters specific to model economy

β = 0.9859 Real interest rate of 0.04

ψu= 0.6171 Avg. weekly hours of employed unskilled men: 41.3

ψs= 0.4488 Avg. weekly hours of employed skilled men: 43.5

γ = 0.972 Average work life of 35 years

χt Time series of skill premium in CPS data

zt Normalize output to 1 in every period

ρu= 0.9838 Own estimate for the unskilled from PSID

ρs= 0.9859 Own estimate for the skilled from PSID

λµ1(u) = 0.1467 Own estimate for the unskilled from PSID

λµ1(s) = 0.1179 Own estimate for the skilled from PSID

{ληt (u), λυt (u),

λυt (s), ληt (s)}

Own estimates from PSID

37

Table 3: Parameters for the Initial Steady State

Parameter Target

χ = 0.6718 Skill Premium of 1.4681

ληSS1(s) = 0.0040 Variance of the innovation in Persistent Shock: Skilled

ληSS1(u) = 0.0038 Variance of the innovation in Persistent Shock: Unskilled

λυSS1(s) = 0.0137 Variance of Transitory Shock: Skilled

λυSS1(u) = 0.0132 Variance of Transitory Shock: Unskilled

Table 4: Initial Steady State Results

Skilled Unskilled

σ2P/ (1− ρ2) 0.1429 0.1182

σ2T 0.0137 0.0132

w 2.2024 1.5040

h 0.3863 0.3667

wxh 0.9113 0.5906

a 3.3051 2.9439

c− T 1.0338 0.7078

Table 5: Changes in the Wage Structure

Year 1967 1999

Skill Premium (SP) 1.4681 1.6471

ληt (s) (PS) 0.0040 0.0247

ληt (u) (PU) 0.0038 0.0096

λυt (s) (TS) 0.0137 0.1032

λυt (u) (TU) 0.0132 0.0938

38

Table 6: Short-Run Analysis

Hours Worked

Skilled Unskilled Differential

SS1 43.4 41.2 2.2

SP 42.8 41.3 1.5

SR1 50.9 43.5 7.4

SR2 45.8 41.5 4.3

SS2 42.0 41.6 0.4

Savings

Skilled Unskilled Differential

SS1 3.31 2.94 0.37

SP 3.32 2.94 0.38

SR1 3.60 2.99 0.61

SR2 3.54 3.00 0.54

SS2 6.19 1.99 4.20

Table 7: Counterfactual 1 and 2

S U Difference

SS1 43.4 41.2 2.2

SR1 50.9 43.5 7.4

CF1 46.2 43.5 2.7

CF2 50.0 43.5 6.5

SR2 45.8 41.5 4.3

CF1 44.5 41.5 2.9

CF2 44.6 41.5 3.0

39

Table 8: Counterfactual 3 and 4

S U Difference

SS1 43.4 41.2 2.2

SR1 50.9 43.5 7.4

CF3 51.6 44.1 7.4

CF4 46.9 42.1 4.7

SR2 45.8 41.5 4.3

CF3 46.7 42.4 4.3

CF4 40.0 39.7 0.3

Figure 2: Estimated Variances of Persistent and Transitory Wage Shocks

40

Figure 3: Variance Decomposition of Log Wage Residuals

Figure 4: Trends in the Share of Persistent Wage Shocks

.55

.6.6

5.7

.75

.8

1970 1980 1990 2000 2010Year

Uskilled Skilled

Trends in the share of persistent wage shocks

Figure 5: Trends in the Male Hours Worked in the U.S.

4041

4243

4445

1970 1980 1990 2000 2010Year

Uskilled Skilled

Average Weekly Hours Worked

1.5

22.

53

3.5

1970 1980 1990 2000 2010Year

Average Weekly Hours Difference between Skill Groups

41

Figure 6: Time Effects vs. Cohort Effects in Skilled-Unskilled Hours Differential

2

3

4

1930s

1940s

1950s

1960s

Within‐Cohort Variations

1

2

25 30 35 40 45 50 55

1970s

Age

2

3

4

1970

1980

1990

Between‐Age Variations

1

2

25 30 35 40 45 50 55

2000

Age

Figure 7: Transition Dynamics: Hours Worked by Skill Level

1950 2000 2050 210040.5

41

41.5

42

42.5

43

43.5

44

44.5

45

45.5

year

wk.

Hou

rs

Transition (1967−2000+)

S (Model)U (Model)

42

Figure 8: Transition Dynamics: Hours Differences between Skilled and Unskilled Men

1950 2000 2050 21001

1.5

2

2.5

3

3.5

4

4.5

year

Hou

rs D

iff

Transition (1967−2000+)

Modeldata

1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 20101

1.5

2

2.5

3

3.5

4

4.5

year

Hou

rs D

iff

Transition (1967−2000+)

Modeldata

Figure 9: The SR1 Effect of an Increase in Persistent Wage Shock of Skilled Men

43

Figure 10: The SR2 Effect of an Increase in Persistent Wage Shock of Skilled Men

Figure 11: The SR1 Effect of an Increase in Transitory Wage Shock of Skilled Men

44

Figure 12: The SR2 Effect of an Increase in Transitory Wage Shock of Skilled Men

Figure 13: Counterfactual 1: Same Increase in Wage Uncertainty as the Unskilled

1950 2000 2050 210040

41

42

43

44

45

46

year

wk.

Hou

rs

Transition (1967−2000+)

S (Bench)U (Bench)S (CF)U (CF)

1950 2000 2050 21001

1.5

2

2.5

3

3.5

4

4.5

5

year

Hou

rs D

iff

Transition (1967−2000+)

BenchCF

45

Figure 14: Counterfactual 2: Same Change in Shock Composition as the Unskilled

1950 2000 2050 210040.5

41

41.5

42

42.5

43

43.5

44

44.5

45

45.5

year

wk.

Hou

rsTransition (1967−2000+)

S (Bench)U (Bench)S (CF)U (CF)

1950 2000 2050 21000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

year

Hou

rs D

iff

Transition (1967−2000+)

BenchCF

Figure 15: Counterfactual 3: Increased Wage Uncertainty through Persistent Shock only

1950 2000 2050 210040

41

42

43

44

45

46

47

year

wk.

Hou

rs

Transition (1967−2000+)

S (Bench)U (Bench)S (CF)U (CF)

1950 2000 2050 21001

1.5

2

2.5

3

3.5

4

4.5

year

Hou

rs D

iff

Transition (1967−2000+)

BenchCF

Figure 16: Counterfactual 4: Increased Wage Uncertainty through Transitory Shock only

1950 2000 2050 210034

36

38

40

42

44

46

year

wk.

Hou

rs

Transition (1967−2000+)

S (Bench)U (Bench)S (CF)U (CF)

1950 2000 2050 2100−2

−1

0

1

2

3

4

5

year

Hou

rs D

iff

Transition (1967−2000+)

BenchCF

46