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Education Mismatch and Earnings Inequality

Rongsheng TangShanghai University of Finance and Economics

Gaowang WangShandong University

February 2020 ∗

Abstract

To better understand education mismatch, we build a model featuring three un-derlying channels—preference, promotion, and search friction—and quantify theirimpacts on earnings inequality for the highly educated. Statistically, these three chan-nels account for 70% of education mismatch, measured by the relatedness betweena worker’s field of study and his or her current occupation. In addition, educationmismatch is known to have a negative wage effect and a negative correlation withwage inequality across occupations and majors. In a stationary equilibrium, thesethree channels affect wage inequality through occupational choice. In particular, ahigh promotion level, low job amenity, or a low match premium motive results in alow likelihood to accept a matched job. Search friction also affects employment distri-bution. Quantitatively, we find that the preference channel negatively contributes toinequality increase and the match premium contributes by 23.4%. The contributionsof search friction and promotion are 5.2% and 2%, respectively.

Keywords: education mismatch, earnings inequality, highly educated, field of study

JEL Code: E24, I24, J24, J31, O15

∗Tang: Institute for Advanced Research, Shanghai University of Finance and Economics; Key Labora-tory of Mathematical Economics (SUFE), Ministry of Education, 777 Guoding Road, Shanghai, China 200433(e-mail: [email protected]). Wang: Center for Economic Research, Shandong University, Jinan,China. E-mail: [email protected]. Rongsheng Tang is grateful for the financial support from theNational Natural Science Foundation of China (Grant No.71803112). Gaowang Wang thanks Young ScholarProgram of Shandong University, China (No. 2018WLJH09) for their financial supports.

1 Introduction

Education mismatch happens when a worker’s skill type does not match the job require-

ments. Although education level mismatch has received much attention in the literature

(e.g., Lee et al. (2015)), research on education type mismatch is scarce. It has been docu-

mented that this type of mismatch has a substantial effect on both earnings and inequal-

ity (e.g., Altonji et al. (2014)); therefore, an understanding of the fundamental reasons for

education mismatch is important. In this paper, we model education mismatch by in-

corporating three underlying channels—preference, promotion, and search friction—and

quantify their impacts on earnings inequality for the highly educated. Although the effect

of education mismatch on wage inequality has been discussed, no paper has specified the

underlying reasons; this paper addresses that gap.

In the National Survey of College Graduates (NSCG), people were asked about the

relatedness between their current occupation and their field of study. We use this infor-

mation to measure education mismatch. Although the field of study may not be a perfect

proxy of skill type, it represents the knowledge type, which is an important skill com-

ponent for workers with college degrees, particularly for workers with graduate school

experience. In addition, the survey also asks about the main reasons for the mismatch.

The data show that 70% of education mismatch can be attributed to the three channels

mentioned above. For example, as documented in section 2, in 1990, the proportions are

32%, 20%, and 16% for promotion, preference, and search friction, respectively. Thus, this

paper will only focus on these three channels.

Consistent with the fact in Ritter and West (2014) that there is a negative wage effect

of education mismatch, the wage ratio of unmatched to matched workers was 0.91, 0.94,

and 0.87 in 1990, 2000, and 2010, respectively. This ratio also varies with reasons, thereby

implying that wages vary within the unmatched group. For example, in 1990, the ra-

tios were 0.98, 0.85, and 0.66 for promotion, preference, and search friction, respectively.

Therefore, it would be meaningful to distinguish the reasons and study the wage inequal-

ity. Moreover, the empirical fact of a positive correlation between education mismatch and

wage inequality across both majors and occupations suggests the potential importance of

education mismatch in wage inequality.

We build a model and quantify the contribution of each channel on wage inequality.

In the model, workers vary by skill type, and firms vary by productivity type. Given a

worker’s skill type, there are two types of jobs: matched and unmatched. Working in the

matched job requires workers that have related skill types and draw a match premium.

The total output is a product of the firm’s productivity, the worker’s skill, the match pre-

mium, and the worker’s effort. A worker acquires a share of the output, and promotion

is defined as an increase of this share. The job amenity on a matched job is random and

affects the worker’s effort. Therefore, the match degree between a worker and a job af-

1

fects the human capital level and the worker’s occupational choice. In the quantitative

analysis, we calibrate the model to target the US economy and subsequently quantify

the contribution of each channel to the increase in wage inequality from the 1990 to 2000

period.

In a stationary equilibrium, the underlying channels affect wage inequality through

occupational choice. In particular, a high promotion level in an unmatched job, a low job

amenity, or a low match premium in a matched job motives result in a low likelihood of ac-

cepting the matched job. Search friction on a matched job will also affect the employment

distribution. Quantitatively, we found that the preference channel negatively contributes

to the inequality increase and the match premium contributes 23.4%; for search friction

and promotion, the contributions are 5.2% and 2%, respectively.

The policy implications of this paper are as follows. First, improving the degree of

education match itself will shrink wage inequality, which could improve the education

signal and lower market friction. Second, since the match premium channel explains a

significant part of the increase, improving match quality would prove fruitful.

Related Literature

The related literature primarily includes studies on wage inequality, education mismatch,

and skill mismatch. There is a growing collection of papers on wage inequality for the

highly educated. It is commonly documented that wage inequality has increased rapidly

since the 1980s (e.g., Autor et al. (2008), Piketty and Saez (2014)). Skill premium has been

studied intensively (e.g., Acemoglu and Autor (2011), Burstein et al. (2015)), and the study

of wage inequality within educational groups has also been plentiful (e.g., Violante (2002),

Lee et al. (2015)).

Studies on within-group or residual wage inequality emphasize the impact of unob-

served skills (e.g., Lemieux (2006)). The findings of Violante (2002) and Kambourov and

Manovskii (2009) are closely related to this paper. The former paper emphasizes the role

of skill transferability across machines of different vintages in explaining wage differ-

ences among ex-ante workers; the author finds that this channel could explain one third

of the residual wage inequality. The latter paper connects occupational mobility and wage

inequality by emphasizing occupation-specific human capital. Although the authors con-

clude that occupational mobility would explain most of the residual wage inequality, they

attribute occupational mobility to idiosyncratic productivity shocks. Since the aforemen-

tioned paper does not identify the fundamental reasons for shocks, this study serves as a

complement in providing some specific reasons for occupational choice.

Recent studies on high-skills groups have focused on the match between skills or edu-

cation and jobs. For example, Altonji et al. (2014) argue that the earnings difference across

college majors can be larger than the skill premium between college and high school.

Furthermore, while Altonji et al. (2014) argue that the substantial widening or earnings

2

inequality between majors is related to the task composition of occupations associated

to majors, Ritter and West (2014) found that the changing distribution of majors causes

slight, if any, shift in earnings distribution. Based on the results of these two papers,

my study goes one step further and asks why, given the wage difference between occu-

pations for people with the same major, does education mismatch still happen and also

asks to what extent mismatch accounts for the increase in wage inequality for the highly

educated.

Studies on education mismatch usually investigate the wage effect of mismatch be-

tween college majors and occupations. Robst (2007) is one of pioneers in employing NSCG

data to measure education mismatch. Lemieux (2015) shows that the return on education

varies greatly depending on occupation, the field of study, and the match between these

two factors. It is observed that the college major match-related channel accounts for close

to half of the conventionally measured return to education. Other papers have also shown

empirical results between field studies and earnings difference (e.g., Arcidiacono (2004),

Kirkebøen et al. (2014)).

The present study is also related to the literature on skill mismatch (e.g., Guvenen et al.

(2015), Lise and Postel-Vinay (2015)). In these papers, skills level is measured using a test

score such as the ASVAB and the skill requirements in each occupation. Skill mismatch is

measured as the distance between a worker’s skill acquirement and the job’s skill require-

ment. In a recent paper, Cooper and Liu (2019) measured the mismatch between skills

and education attainment.

Organization of the Paper The remainder of the present paper is organized as follows.

Section 2 is the description of statistical and empirical facts. In Section 3, we build a

benchmark model on education mismatch incorporating underlying reasons. Section 4 is

an equilibrium analysis, followed by a quantitative analysis in section 5. Section 6 is the

conclusion.

2 Facts

Some main features related to education mismatch are documented in this section. Data

are collected from the NSCG, which tracks a sample of people who reported having col-

lege degrees in a census survey. Every 10 years, it provides information about the relat-

edness between the field of study and occupation. There are three possible responses in

the survey, namely, close, somewhat close, and not at all, which are treated as the proxy

of the degree of education mismatch. The sample includes NSCG (1993, 2003, and 2013)

data based on the censuses of 1990, 2000, and 2010, respectively.

The trimming strategy is standard and as follows: it only includes workers who are

employed full time and are between 16 and 65 years of age. The top annual earnings are

3

four million, and earnings less than 2,800 are excluded. For race, only the categories of

white, black, and Hispanic are used. The major code is regrouped under the categories

listed in the Department of Education, following the methodology of Altonji et al. (2014),

which includes around 50 majors in total. Subsequently, occupation is regrouped under

the method proposed by Dorn (2009), who makes a consistent three-digit occupation code.

In the data, there are four levels of schooling year: 16, 18, 19, and 21. These levels are

regrouped into three categories: Bachelor (16), Masters (18, 19), and PhD (21). In addition,

tenure is calculated as a potential experience, that is, max(age− schooling− 6, 0). Wage

residue is based on the following regression, and wage is the exponential of the residue

as in Kambourov and Manovskii (2009),

ln(wage) = constant + β1edu + β2exp + i.gender + i.race + ε

where edu is education, exp is the potential experience in the labor market, and gender

and race are also controlled.

2.1 Statistical Description

Table 1 through Table 3 present the observations, average tenure, earnings, inequality, and

other factors for different demographic groups from 1990 to 2010. In the sample, the over-

all inequality increases from 0.24 in 1990 to 0.39 in 2010. Men have higher earnings and

wage inequality than women. As the education level increases, annual earnings increase;

however, the Masters level workers have the lowest wage inequality. White people earn

higher wages and have higher inequality than other racial groups. The proportion of job

relatedness does not change substantially: the proportion of closely related jobs is around

0.56. However, this does not mean that education mismatch does not contribute to in-

creases in earnings inequality. If the match premium increases, education mismatch still

matters for earnings inequality. In addition, if the main reason for this mismatch changes,

it also matters for earnings inequality. Table 4 lists several inequalities; it shows that in

statistical terms, education mismatch generally contributes for 15% of the inequality.1

Table 5 lists the proportion of match degree in different demographic groups. There is

not much difference in match degree among the gender and racial groups. However, job

relatedness increases as the education level increases; for example, in 1990 the proportion

increases from 0.46 for a Bachelor degree to 0.88 for a PhD. Table 6 shows the reasons for

worker mismatch. In the survey, people were asked about the most important reason for

working outside their fields; the table lists the seven potential reasons, and the proportion

for each reason is calculated for each year. The data show that there are three main reasons

1In this table, var1 is the residual inequality after further controlling the major dummy, and var2 is theresidual inequality after further controlling the variables of major, occupation, and match status; thus, thecontribution of major-occupation match is computed as var1−var2

var1 .

4

for mismatch: pay or promotion opportunities; seven interests; and the unavailability of

a job in their highest degree field. These three factors in total constitute around 70% of the

mismatch reported for each year. 2

Table 8 presents the wage ratio between the matched and the unmatched group. In

the raw data, the average wage in the matched group is higher than the average wage

in the unmatched group regardless of the reason. However, for the residual wage, the

group with the reason “Pay, promotion opportunity” has a higher or almost the same

wage. The wage varies within the unmatched group. Therefore, it would be meaningful

to distinguish the different reasons for education mismatch.3

2.2 Wage Effect

This subsection documents the wage effect of education mismatch. As in Ritter and West

(2014), I regress log annual earning on the demographic and match variables as follows,

ln(earnings)ijm = βDi + αZj + θMm + δ1closejm + δ2somejm + γXi + εijm,

where Di includes a vector of demographic variables (tenure, age, gender, race, etc.) for

individual i. Zj denotes the job j, and Mm denotes the major m. The terms closejm and

somejm denote that the job j and the major m are closely related and somewhat related,

respectively. Xi includes all of the other factors for individual i: parents’ education, degree

location, work location, etc. Finally, εijm is the residual term. Therefore, δ1 and δ2 capture

the wage effect of mismatch. In particular, δ1(δ2) represents the percentage of change in

earnings when unmatched workers become closely (or somewhat) related.

Table 9 presents a part of the regression results. It shows that δ1 = 0.171, δ2 = 0.118

in 1990; δ1 = 0.229, δ2 = 0.170 in 2000; and δ1 = 0.265, δ2 = 0.160 in 2010. The result that

δ1 > δ2 > 0 suggests mismatch has a significant negative effect on earnings (e.g., 17.1%

in 1990). Moreover, the results show that the effect of education mismatch is becoming

larger over time. Compared to the mismatch case (not related at all), the matched group

(closely related) has 17.1%, 22.9%, and 26.5% higher annual earnings in 1990, 2000, and

2010, respectively.

2.3 Wage Inequality Effect

To discern the relationship between education mismatch and wage inequality, we plot

education mismatch and wage inequality across both majors and occupations. In Figure

2Table 7 presents the mismatch reasons for a worker who has a potential experience of less than 10 years. InKambourov and Manovskii (2009), it is documented that it typically takes 10 years to become an experiencedworker. This rule is also applied in Ritter and West (2014).

3Table 10 shows the wage ratio between the matched and the unmatched group for inexperienced workers.

5

1, each point represents one major and the relatedness is calculated as the proportion

of matched workers (job closely related) of this major.4 Wage inequality is the residual

variance of log annual earning controlling for demographic characteristics in each major.

Figure 1 shows that there is a significant negative correlation between job relatedness and

wage inequality.

Figure 3 plots the mismatch and wage inequality across occupations.5 It shows a nega-

tive correlation between mismatch and wage inequality. As documented in the literature,

the within-occupation factor has a significant contribution to the increase of residual wage

inequality; this negative correlation suggests the potential importance of education mis-

match on wage inequality.

3 The Model

3.1 Model environment

In a general model, a worker has skills in two dimensions (a, k), where a represents the

skills level and k represents the skill type. A firm has productivity in two dimensions

(A, k′), where A represents the productivity level and k

′represents firm’s type. To mea-

sure the mismatch, following Berliant et al. (2006), I assume that k and k′

are in a cycle

of [0, 1]. The distance reflects the degree of education mismatch. Moreover, a worker ac-

quires a share α of the output jointly produced with the firm, and promotion is modeled

as an increase of α.

Production

The joint output between a worker and a firm is linear in a job’s productivity A, worker’s

skills level a, work effort e, and the match premium between the work and job h(k, k′),

that is, y(a, k, k′) = Aah(k, k

′)e. A worker takes α share of the joint output as a wage,

or w(a, k, k′) = αy(a, k, k

′). In addition, the match premium function decreases in the

distance between job and worker’s type d(k, k′). It will be effective only if the worker

and the job’s type are close enough or in the match-specific knowledge spread as in

Berliant et al. (2006). In particular, there is a cutoff δk such that the match premium is

the lowest level hL if d(k, k′) ≥ δk, and it decreases in distance if d(k, k

′) < δk, formally,

h(k, k′)

= hL d(k, k′) ≥ δk

> hL d(k, k′) < δk

, where δk could be specific to the worker’s skill type; in other

4Figure 2 presents the case of measuring job relatedness as a proportion of the closely related group andthe somewhat related group.

5Figure 2 presents the case of measuring job relatedness as a proportion of the closely related group andthe somewhat related group.

6

words, a different skill type may have a different knowledge spread.

Worker

A worker’s utility is a function of consumption level c, work effort e, and job amenity τ,

in particular,

u(c, e) =c1−θ

1− θ− 1

τ

e1−ρ

1− ρ

where θ captures risk aversion, ρ captures elasticity of effort, and a large τ implies high

job amenity or a low disutility of effort. In addition, τ is fixed in a non-matched job (τM)

and is random in a matched job; namely, τ =

τM d(k, k′) ≥ δk

τ(k, k′) d(k, k

′) < δk

, where τ(k, k′) fol-

lows a distribution that may also depend on worker and job type. After τ is realized,

worker’s utility maximization implies the following wage function: w(α, A, a, h, τ) =

[(αAah)1−ρτ]1

θ−ρ , and effort function e(α, A, a, h, τ) = [(αAah)−(θ−1)τ]1

θ−ρ . Subsequently,

the indirect utility function is as follows:

U(α, A, a, h, τ) = − θ − ρ

(θ − 1)(1− ρ)[(αAah)1−ρτ]−

θ−1θ−ρ .

Given the assumption that ρ < 1 < θ, wages and utility increase in α, A, a, h, τ while effort

increases in τ but decreases in α, A, a, h.

Value function

In every period, an unemployed worker with skill (a, k) receives a job offer (A, k′) with

probability Pf . The worker decides whether to take the job or wait for another period.

For an employed worker, there is a separating rate Ps; when the separation happens, the

worker will be unemployed for one period. If there is no separation, the worker also

receives a job offer. Depending on the current position, the promotion level might be

different. Specifically, for a worker who is currently not matched (d(k, k′) > δk), there is

no promotion. For a worker who is currently in a matched job (d(k, k′) < δk), there is

a probability of Pα of promotion if the offer is from a non-matched job; however, if the

offer is from a matched job, there is no promotion. In addition, the promotion level in a

matched job is always α0, and the preference level in an unmatched job is always τM.

In each period, a worker (a, k) is in one of the following three status: unemployed,

working in an unmatched job (A, k′) with promotion level α, or working in matched job

with amenity τ. The lifetime expected utilities are denoted as VU(a, k), Vg(A, a, k, k′, α, τM),

and Vs(A, a, k, k′, α0, τ), respectively. In the rest of the present paper, the analyses will be

only conducted based on a benchmark model, and the Appendix B describes the general

model.

7

3.2 Benchmark model

In this benchmark model, we proceed with further assumptions to make the model tractable.

Workers are assumed to have the same skills level a, which is normalized to be 1; firms

are homogeneous in productivity (A = 1); skills have the same knowledge spread (δk =

δ); And the match premium has two values, hH for a matched job and hL for an un-

matched job, that is, h(k, k′)

= hL d(k, k′) ≥ δ

= hH d(k, k′) < δ

. The distribution of preference level for

a matched job is across three values τL, τM, τH, where τL < τM < τH. In addition, the

promotion level in a matched job is always α0, and the promoted value is α1(> α0). These

simplifications are made for two reasons. First, this paper is only studying inequality for

the highly educated workers for whom the skills difference could be small. Second, the

focus of this model is type mismatch; hence, productivity and skill differences are omitted.

For a worker who is working in a matched job (S) with preference τ, the instant utility

is U(α0, hH, τ), where we simplify the notation by denoting U(α, h, τ) = U(α, 1, 1, h, τ).

For the next period, there is a probability of Ps that the job and worker will be separated.

If they are not separated, there is a probability of Pδ that the worker will be offered a

matched job outside the firm. The job amenity in any matched job is τ′, which is drawn

from τL, τM, τH. With a probability of (1− Pδ), the worker will receive an offer of an

unmatched job (g). For the unmatched job, with a probability of Pα, the promotion level

is α1. There is a probability of (1− Pα) that the promotion is α0. In both cases, the worker

chooses to take the job offer or not. Hence, the value function can be written as

Vs(τ) = U(α0, hH, τ) + βEτ′ PsVU + (1− Ps)[PδVs(τ

′)

+ (1− Pδ)Pαmax[Vg(α1), Vs(τ′)] + (1− Pα)max[Vg(α0), Vs(τ

′)]],

where β is the time discount rate in the utility function.

For the worker in an unmatched job with promotion α, since the match premium is hL

and the preference is τM, the indirect utility function is U(α, hL, τM). For the next period,

there is a probability of Ps that the job and worker will be separated. If they are not sepa-

rated, there is a probability of Pδ that the worker will be offered a matched job outside the

firm with a promotion level of α0, of which the preference is τ′

drawn from τL, τM, τH.There is a probability of (1− Pδ) that the worker will receive an offer of an unmatched job

with the same promotion level. Subsequently, the value function is as follows:

Vg(α) = U(α, hL, τM) + βEτ′ PsVU + (1− Ps)[(1− Pδ)Vg(α) + Pδmax[Vg(α), Vs(τ

′)]].

An unemployed worker will have an unemployment benefit V in the current period. In

the next period, there is a probability of Pf that this worker will receive an offer. There

is a probability of Pδ that it is a matched job with preference from a random draw and a

8

probability of (1− Pδ) that it is an unmatched job with a promotion level of α0. The value

function can then be written as follows:

VU = V + βEτ′ (1− Pf )VU + Pf [PδVs(τ

′) + (1− Pδ)Vg(α0)]

In this benchmark model, mismatch happens when Vg(α) > Vs(τ), for some (α, τ).

Let D(α) be the set of preferences on a matched job that a mismatch happens, that is,

D(α) = τ : Vg(α) > Vs(τ). In this case, the profile for mismatch is (α1, τ) : τ ∈ D(α1)and (α0, τ) : τ ∈ D(α0). A mismatch through occupational choice is differentiated into

two parts based on two underlying reasons: preference and promotion. Subsequently, the

set of promotion is PM = (α1, τ) : τ ∈ D(α1), andτ /∈ D(α0) and others are due to

preference.

4 Equilibrium

There are potential multiple equilibria listed in Appendix A.1. For illustration, in this

section we only discuss one (Eq1), which is characterized as D(α0) = τL and D(α1) =

τL, τM, τH. The profile of a mismatched worker is (α0, τL), (α1, τL), (α1, τM), (α1, τH).6

Furthermore, a mismatch due to promotion is described as the following case. When

given a promotion level of α1, people will choose an unmatched task, but if the promotion

level is downgraded to α0, the worker will choose a matched task. By this rule, the set for

a promotion-driven unmatched worker is (α1, τM), (α1, τH), and the set for preference is

(α0, τL), (α1, τL).

Stationary equilibrium

In the stationary equilibrium, let Ng0, Ng1 be the number of employed people that are

unmatched with a promotion level of α0, α1, respectively, and NU the total number of un-

employed people. If NS is the number of matched workers, the employment distribution

requires following conditions.

1. The unemployed workers include unlucky job seekers and unlucky employed work-

ers, as follows.

NU = NU(1− Pf ) + (Ng0 + Ng1 + Ns)Ps

2. The workers in unmatched jobs with promotion α0 come from lucky job seekers,

stayers, and switchers from matched jobs due to low amenity.

Ng0 = NU Pf (1− Pδ) + Ng0(1− Ps)[(1− Pδ) + PδPL] + Ns(1− Ps)(1− Pδ)(1− Pα)PL

6In the quantitative analysis, I will allow the model to choose an equilibrium to match the data.

9

3. The workers in unmatched job with promotion α1 are stayers and switchers from

matched jobs due to promotion.

Ng1 = Ng1(1− Ps) + Ns(1− Ps)(1− Pδ)Pα

4. The workers in matched job are lucky job seekers, switchers from unmatched jobs,

and stayers.

Ns = NU Pf Pδ + Ng0(1− Ps)Pδ(PH + PM)+ Ns(1− Ps)[Pδ +(1− Pδ)(1− Pα)(PH + PM)]

5. The total number of labor force is normalized to 1, leading to the following.

1 = NU + Ng0 + Ng1 + Ns

The employment and wages are computed in Appendix A.2 and are summarized as fol-

lows.

Employment Let NPF, NPM, NSF be the number of total employment unmatched due to

preference, promotion, and search friction, respectively; NsL, NsM, NsH are the number

of workers matched with preference τL, τM, τH, respectively. Then they are computed in

Equation (1) through Equation (6).

Wages Let wPF, wPM, wSF be the wages of unmatched workers due to preference, pro-

motion, and search friction, respectively; wsL, wsM, wsH are the wages of matched work-

ers with the preferences τL, τM, τH, respectively. Further, wg, ws are the average wages

of unmatched and matched workers, respectively. Given the wage function w(α, h, τ) =

[(αh)1−ρτ]1

θ−ρ , they are computed in Equation (7) through Equation (14).

Earnings inequality The total inequality can be decomposed into within-group inequal-

ity (Varj) and between-group inequality (( ¯lnwj − ¯lnw)2,

Var(lnw) = ∑j=g,s

Nj

Ng + Ns[Varj + ( ¯lnwj − ¯lnw)2]

where Varg is the inequality within an unmatched job,

Varg = ∑j=0,1

Ngj

Ng(lnw(αj, hL, τM)− ¯lnwg)

2

and Vars is the inequality within a matched job.

Vars = ∑j=L,M,H

Nsj

Ns(lnw(α0, hH, τj)− ¯lnws)

2

10

Given all the functions in Appendix A.2, the next section will quantify the impact of edu-

cation mismatch on earnings inequality.

5 Quantitative analysis

In the quantitative analysis, we first calibrate the model with data from 1990 as the bench-

mark. Subsequently, to quantify the contribution of each factor, we re-calibrate the channel-

specific parameters in the model by targeting the economy in 2000. Finally, given the pa-

rameters in 1990 and 2000, we conduct several counterfactual experiments to examine the

impact of each factor on inequality.

5.1 Calibration

The following parameters required calibration: the preference parameters τH, τM, τL, and

PH, PM, PL; the promotion parameters α0, α1, Pα; the search friction parameters Pδ, Pf , Ps;

and the skill premium parameters hL, hH as well as A, V, ρ, θ, and β. First, we normalize

the following parameters: job productivity, match premium in an unmatched job, prefer-

ence in an unmatched job, that is, A = 1, hL = 1, τM = 1. Following the literature, we

set θ = 2 and β = 0.95. Other parameters are calibrated by jointly targeting several main

characteristics.

Table 11 presents the calibration results by targeting the data in 1990. The terms

(α0, α1) are promotion level (low, high). Pα is the promotion probability, hH is the match

premium, (τH, τL) are the job amenity (high, low), and (PM, PL) are the probability dis-

tributions on job amenity with PH + PM + PL = 1. The term Pδ is the knowledge spread,

and the terms (Pf , Ps) are the job receive rate and separation rates. The terms (V, ρ) are

the unemployment disutility and elasticity of effort. The data are from NSCG (1993),

which collected the information in 1990, and the targets are the following: the labor

share (ls); matched to the unmatched employment ratio ( NsNg

) and the wage ratio (WgWs

);

employment ratio between matched to the unmatched due to promotion ( NPMNg

), prefer-

ence ( NPFNg

), and search friction ( NSFNg

); wage ratios between unmatched and matched work-

ers due to promotion (WPMWs

), preference (WPFWs

), and search friction (WSFWs

); wage inequality

within the matched group (Vars) and the total wage inequality (Var); and unemploy-

ment (Nu). Overall, the model matches the targets quite well, particularly for inequalities

Var, Vars, Varg and employment ratios NsNg

, NPMNg

, NPFNg

, and NSFNg

. The fitness of these two is

important since the model studies inequality through occupational choice.

11

5.2 Counterfactual Analysis

To quantify the contribution of each channel, we re-calibrate the channel-specific param-

eters by targeting the economy in 2000. The result is shown in Table 12. Similar to the

result in 1990, overall, the model matches the targets well, particularly for inequalities

and employment ratios.

Given the parameters in 1990 and 2000, several counterfactual experiments are con-

ducted to examine the impact of each factor on inequality. First, parameters on preference

(τL, τH, PL, PM, PH) in 1990 are replaced with the values from 2000. The impact of this

channel is computed as the ratio of the difference between this counterfactual value and

the baseline value in 1990 to the inequality change from 1990 to 2000. In Table 13, the first

row shows the inequality level when replacing the parameter in 1990 with that of 2000 in

the counterfactual cases. The second row shows the difference between inequality in the

counterfactual case and that of the benchmark. The third row shows the ratio of the value

in the second row to the wage change from 1990 to 2000.

As shown in Table 13, the probability distribution plays an important role in explain-

ing the increase in wage inequality. In particular, if (PL, PM, PH) are replaced with the

values from 2000, the wage inequality is 35.9% higher than the baseline value. We con-

clude that the amenity probability distribution contributes 35.9% to the increase in wage

inequality, although it may also incorporate other indirect impacts. However, if the job

amenity level (τL, τH) is replaced, the wage inequality is 93.5% lower than the baseline

value. This result is consistent with the calibration result in 1990 in that the difference

between τH and τL is considerably larger than in 2000. In addition, if both (τL, τH) and

(PL, PM, PH) are replaced, the inequality is 72.6% lower than the baseline value. Therefore,

overall, the preference channel contributes negatively to the increase of inequality.

A similar experiment is conducted on match premium (hH). As shown in Table 14,

if hH is replaced, the inequality is 23.4% higher than the baseline model. Given that the

match premium increase from 2.09 to 3.65, this exercise suggests the an increase in match

premium contributes to a wage inequality increase of 23.4%.

For search friction, three parameters are considered: the job finding rate (Pf ), the sep-

aration rate (Ps), and the knowledge spread (Pδ). As shown in Table 15, the contribution

of each channel is 2.7%, 0%, 3.1%, respectively. The overall contribution of search friction

is 5.2%.

The counterfactual exercise on promotion channel involves the following parameters:

the promotion offer probability (Pα) and the promotion level (α0, α1). As shown in Ta-

ble 16, the probability of promotion negatively contributes to the increase of inequality

(−9%), the promotion level distribution contributes 9.7%, and the overall contribution of

the promotion channel is 2%.

The change of inequality may be caused by factors that were missed in the model.

To check this, we performed similar counterfactual exercises for (V,ρ) to account for the

12

contribution from residues. As shown in Table 17, the remaining part contributes to in-

equality by 275.8%; hence, the model still misses a significant portion of the explanation

for the inequality increase. To make more sense of the magnitude of each channel, we

summed all the contributions from the five components: preference (PF), match premium

(PE), search friction (SF), promotion (PM), and residue (RS) and computed the ratio of

each component to the sum. As reported in Table 18, the first row is the inequality level

when replacing the parameters in 1990 with those of 2000 in the counterfactual cases. The

second row is the difference between the inequality in a counterfactual case and inequal-

ity in the benchmark. The third row is the ratio of the value in the second row to the wage

change from 1990 to 2000. The fourth row is the standardized value, that is, the ratio of the

value in the third row to the total sum. Subsequently, in this standardized measurement,

the contribution of preference, match premium, search friction, and promotion are −31%,

10%, 2.2%, and 0.9%, respectively. Thus, the contribution is still significant given that the

inequality is residual inequality.

6 Conclusion

In the present study, we investigated education mismatch and earnings inequality and

quantified the contribution of match premium and three other underlying reasons: pro-

motion, preference, and search friction. These channels affect the wage inequality through

occupational choice. In particular, when the promotion level in an unmatched job is high,

the job amenity in a matched job is low, or the match premium is low, people are more

likely to accept an unmatched job. A knowledge spread will also affect the employment

distribution in both matched and unmatched jobs. Quantitatively, we found that the pref-

erence channel negatively contributes to an increase in inequality and a match premium

contributes 23.4%. For search friction and promotion, the contributions amount to 5.2%

and 2%, respectively.

The policy implications of this paper are as follows. First, improving education in

terms of matching degrees to desired occupations will shrink wage inequality. This could

be accomplished by improving the education signal or lowering market friction, among

other solutions. Second, since the match premium channel explains a significant part of

the increase, improving the match quality would be a fruitful task.

The model could be extended to incorporate dynamic changes. A worker may update

his or her preference based on the working experience and on-the-job learning may in-

crease the skill match. The model could also be extended to include skill and productivity

heterogeneity. Under these two extensions, both preference and search friction may have

a higher quantitative importance. A third extension of this model would be to make the

heterogeneity of preference and promotion level into a continuum wherein people have

continuous attitudes or promotion levels.

13

Table 1: Statistical description: 1990

groups observations tenure earning inequality proportion

genderFemale 34467 18.17 53814.12 0.20 0.39

male 59893 19.73 76401.97 0.26 0.61

educationBachelor 58063 18.77 61039.09 0.23 0.64Master 25757 20.17 68416.25 0.20 0.25

PhD 10540 18.68 104475.18 0.32 0.11

raceWhite 79175 19.16 68444.02 0.24 0.91Black 9478 19.20 55785.14 0.19 0.06

Hispanic 5707 17.27 62247.22 0.20 0.03

relatednessClose 55613 18.96 70803.82 0.22 0.56Some 24066 19.04 67695.83 0.23 0.26Not 14681 19.69 57021.46 0.28 0.18

all sample 94360 19.11 67514.19 0.24 1

Source: National Survey of College Graduates (1993)

Table 2: Statistical description:2000


genderFemale 21180 20.13 60732.11 0.28 0.43

male 34285 21.51 90929.36 0.38 0.57


PhD 7348 19.72 131856.72 0.49 0.10

raceWhite 47212 21.07 79947.82 0.35 0.89Black 4411 20.70 61798.49 0.25 0.07

Hispanic 3842 18.50 65085.11 0.33 0.05


all sample 55465 20.92 78042.82 0.34 1


14

Table 3: Statistical description: 2010


genderFemale 26871 19.21 74392.51 0.34 0.47

male 36619 21.22 111361.32 0.43 0.53


PhD 7844 19.16 170399.52 0.53 0.09

raceWhite 51949 20.44 96302.70 0.38 0.86Black 4873 20.54 80507.39 0.43 0.07

Hispanic 6668 18.16 79348.42 0.37 0.08


all sample 63490 20.28 93939.57 0.39 1


Table 4: Earnings inequality

year Var_raw Var_res Var1 Var21993 0.29 0.24 0.21 0.182003 0.40 0.34 0.31 0.262013 0.47 0.39 0.35 0.29

Note: The second column is the earnings inequality with raw data; the third column is the residual inequal-ity; the fourth column is residual inequality after further controlling major dummy; and the fifth column isresidual inequality after further controlling major, occupation, and match status.

Table 5: Proportion of match

1990 2000 2010groups close some not close some not close some not

genderFemale 0.61 0.21 0.18 0.61 0.22 0.17 0.59 0.23 0.17Male 0.53 0.29 0.18 0.52 0.29 0.19 0.53 0.29 0.18

educationBachelor 0.46 0.30 0.24 0.46 0.30 0.24 0.46 0.31 0.23Master 0.68 0.22 0.10 0.68 0.22 0.10 0.69 0.22 0.09PhD 0.88 0.08 0.04 0.87 0.09 0.04 0.88 0.09 0.03

raceWhite 0.59 0.26 0.15 0.60 0.25 0.15 0.56 0.27 0.17Black 0.60 0.22 0.18 0.59 0.24 0.17 0.54 0.26 0.20Hispanic 0.61 0.23 0.15 0.66 0.21 0.13 0.58 0.23 0.19

all sample 0.56 0.26 0.18 0.56 0.26 0.19 0.56 0.26 0.19

Source: National Survey of College Graduates (1993,2003,2013)

15

Table 6: Reasons for Mismatch

Reason 1990 2000 2010Pay, promotion opportunities 0.32 0.32 0.29

Working conditions [hours, equip., working envir.] 0.08 0.10 0.09Job location 0.04 0.06 0.07

Change in career or professional interests 0.20 0.20 0.19Family-related reasons 0.08 0.10 0.10

Job in highest degree field not available 0.16 0.14 0.18Others 0.12 0.07 0.08

Note: The columns “1990, 2000, and 2010” present the proportion of workers who answered “my major andoccupation are not related at all.”

Table 7: Reason of mismatch: exp<=10

Reasons 1990 2000 2010Pay, promotion opportunities 0.30 0.35 0.27

Working conditions [hours, equip., working envir.] 0.07 0.09 0.09Job location 0.04 0.05 0.05

Change in career or professional interests 0.18 0.22 0.15Family-related reasons 0.07 0.08 0.07

Job in highest degree field not available 0.21 0.17 0.28Others 0.12 0.05 0.09

Note: This table only includes workers with an experience of no more than 10 years.

Table 8: Wage ratio to matched group

Reasonsraw data residue

1990 2000 2010 1990 2000 2010Pay, promotion opportunities 0.98 0.94 0.89 1.02 1.00 0.99

Working conditions [hours, equip., working envir.] 0.74 0.66 0.59 0.78 0.73 0.66Job location 0.68 0.64 0.57 0.72 0.69 0.65

Change in career or professional interests 0.85 0.76 0.74 0.89 0.81 0.81Family-related reasons 0.70 0.59 0.52 0.78 0.66 0.59

Job in highest degree field not available 0.66 0.60 0.49 0.72 0.67 0.59Others 0.74 0.71 0.57 0.79 0.75 0.65

Note: The column “Raw Data” presents the raw wage ratio of the unmatched group (under different reasons)to the matched group for 1990, 2000, and 2010. The column “Residue” presents the residual wage ratio (aftercontrolling for demographic characteristics) of the unmatched group to the matched group for 1990, 2000,and 2010.

16

Table 9: Wage effect of education mismatch: NSCG(1993,2003,2013)

VARIABLES 1990 2000 2010closely related 0.171*** 0.229*** 0.265***

(0.00450) (0.00687) (0.00681)some related 0.118*** 0.170*** 0.160***

(0.00450) (0.00690) (0.00688)exp 0.0361*** 0.0386*** 0.0441***

(0.000651) (0.00101) (0.000796)male 0.158*** 0.206*** 0.165***

(0.00332) (0.00498) (0.00479)hgc 0.0681*** 0.0666*** 0.0858***

(0.00134) (0.00212) (0.00203)black -0.0381*** -0.0519*** -0.107***

(0.00620) (0.00924) (0.00900)Constant 9.395*** 9.711*** 9.348***

(0.0256) (0.103) (0.108)Observations 92,802 55,039 62,452R-squared 0.354 0.334 0.380Standard errors in parentheses*** p<0.01, ** p<0.05, * p<0.1

Note: This is regression result of

ln(earnings)ijm = βDi + αZj + θMm + δ1closejm + δ2somejm + γXi + εijm,

where Di includes a vector of demographic variables (tenure, age, gender, race, etc.) for individual i. Zjdenotes the job j, Mm denotes the major m, and closejm and somejm denote that the job and the major areclosely related and somewhat related, respectively. Xi includes all other factors for individual i: parents’education, degree location, work location, etc., and εijm is the residue term. Data source: National Survey ofCollege Graduates (1993, 2003, 2013).

Table 10: Wage ratio to matched group: exp<=10

Reasonsraw data residue

1990 2000 2010 1990 2000 2010Pay, promotion opportunities 0.92 0.94 0.82 0.98 1.01 0.92

Working conditions [hours, equip., working envir.] 0.74 0.80 0.61 0.80 0.88 0.67Job location 0.70 0.72 0.59 0.73 0.81 0.72

Change in career or professional interests 0.78 0.77 0.69 0.84 0.83 0.80Family-related reasons 0.76 0.56 0.70 0.81 0.64 0.81

Job in highest degree field not available 0.64 0.58 0.52 0.71 0.66 0.63Others 0.66 0.71 0.60 0.75 0.75 0.67

Note: This table only includes workers with an experience of no more than 10 years.

17

Note: This figure shows the correlation between major relatedness and within-major inequality. The left,middle, and right panels represent the results for 1990, 2000, and 2010, respectively. In each panel, a dotrepresents a major, the x-axis is the relatedness in that major, and the y-axis is the inequality within thatmajor.

Figure 1: Job Relatedness (Major) and Wage Inequality

Note: This figure shows the correlation between major relatedness and within-major inequality. The relat-edness herein includes the answer of both “some close” and “very close.” The left, middle, and right panelsrepresent the results for 1990, 2000, and 2010, respectively. In each panel, a dot represents a major, the x-axisis the relatedness in that major, and the y-axis is the inequality within that major.

Figure 2: Job relatedness (major) and wage inequality

Note: This figure shows the correlation between job (occupation) relatedness and within-job inequality. Theleft, middle, and right panels represent the results for 1990, 2000, and 2010, respectively. In each panel, a dotrepresents a job, the x-axis is the job relatedness in that job, and the y-axis is the inequality within that job.

Figure 3: Job relatedness (occupation) and wage inequality

18

Note: This figure shows the correlation between job (occupation) relatedness and within-job inequality. Therelatedness herein includes the answer of both “some close” and “very close.” The left, middle, and rightpanels represent the results for 1990, 2000, and 2010, respectively. In each panel, a dot represents a job, thex-axis is the job relatedness in that job, and the y-axis is the inequality within that job.

Figure 4: Job relatedness (occupation) and wage inequality

Table 11: Parameters in 1990

Parameters Descriptions Value target modelα0 promotion level (low) 0.31 ls 0.60 0.37α1 promotion level (high) 0.99 Varg 0.27 0.23Pα promotion probability 0.03 NPM

Ng0.36 0.36

hH match premium 2.09 WgWs

0.88 0.87τH job amenity(high) 14.06 Vars 0.22 0.23τL job amenity(low) 0.01 WPF

Ws0.83 0.67

PM preference probability(M) 0.09 WPMWs

1.02 1.17PL preference probability(L) 0.29 NPF

Ng0.46 0.46

Pδ knowledge spread 0.84 NsNg

5.13 5.13Pf job receive rate 0.73 Nu 0.05 0.05Ps job separation rate 0.04 WSF

Ws0.72 0.79

V unemployment disutility -551.03 NSFNg

0.19 0.19ρ elasticity on effort -4.15 Var 0.23 0.23

Note: The data is from NSCG (1993), which collects the information of 1990, and the targets are the following:

labor share (ls); matched to unmatched employment ratio ( NsNg

) and wage ratio (WgWs

); matched to unmatched

employment ratio due to promotion ( NPMNg

), preference ( NPFNg


); wage ratios

between unmatched and matched due to promotion ( WPMWs

), preference ( WPFWs

), and search friction ( WSFWs

);wage inequality within matched group (Vars) and total wage inequality (Var); and unemployment rate (NU).

19

Table 12: Parameters in 2000

Parameters Descriptions Value target modelα0 promotion level (low) 0.27 ls 0.60 0.33α1 promotion level (high) 0.99 Varg 0.40 0.22Pα promotion probability 0.05 NPM

Ng0.35 0.35

hH match premium 3.65 WgWs

0.82 0.78τH job amenity(high) 2.63 Vars 0.31 0.36τL job amenity(low) 0.02 WPF

Ws0.75 0.64

PM preference probability(M) 0.01 WPMWs

1.00 1.03PL preference probability(L) 0.36 NPF

Ng0.50 0.50

Pδ knowledge spread 0.88 NsNg

4.72 4.72Pf job finding rate 0.90 Nu 0.05 0.05Ps job separation rate 0.05 WSF

Ws0.67 0.72

V unemployment disutility -331.59 NSFNg

0.15 0.15ρ elasticity on effort -1.87 Var 0.34 0.34

Note: The data is from NSCG (2003), which collects the information of 2000, and the targets are the following:

labor share (ls); matched to unmatched employment ratio ( NsNg

) and wage ratio (WgWs

); matched to unmatched

employment ratio due to promotion ( NPMNg

), preference ( NPFNg


); wage ratios

between unmatched and matched due to promotion ( WPMWs

), preference ( WPFWs

), and search friction ( WSFWs

);wage inequality within matched group (Vars) and total wage inequality (Var); and unemployment rate (NU).

Table 13: Counterfactual analysis: preference

Wage inequality Counterfactual analysis

data data modelτL τH (τL, τH) (PL, PM, PH) Preference(PF)

(1990) (2000) (1990)

0.232 0.336 0.2320.214 0.148 0.135 0.269 0.156-0.018 -0.084 -0.097 0.037 -0.075-0.173 -0.809 -0.935 0.359 -0.726

Note: The columns under “Wage inequality” list the inequality from 1990, 2000, and from the benchmarkthat is calibrated with data from 1990. The columns under “Counterfactual analysis” list the wage inequalityunder different counterfactual cases. The column of τL represents the result that replaces τL in 1990 withthe value in 2000 and keeps others with the benchmark values. The same is true for the columns of “τH”,“(τL, τH)”, and “(PL, PM, PH)”, and the column of “Preference(PF)” is the result when replacing (τL, τH) and(PL, PM, PH).

20

Table 14: Counterfactual analysis: match premium


data data model(hH)match premium (PE)

(1990) (2000) (1990)

0.232 0.336 0.2320.2560.0240.234

Note: The columns under “Wage inequality” list the inequality from 1990, 2000, and from the benchmarkthat is calibrated with data from 1990. The column of “(hH)match premium(PE)” represents the result thatreplaces hH in 1990 with the value in 2000 and keeps others with the benchmark values.

Table 15: Counterfactual analysis: search friction


data data model Pδ Pf Ps search_friction(SF)(1990) (2000) (1990)

0.232 0.336 0.2320.235 0.232 0.235 0.2370.003 0.000 0.003 0.0050.027 0.000 0.031 0.052

Note: The columns under “Wage inequality” list the inequality from 1990, 2000, and from the benchmark thatis calibrated with data from 1990. The column of “Pδ” represents the result that replaces Pδ in 1990 with thevalue in 2000 and keeps others with benchmark values. The same is true for the columns of “Pf ” and “Ps”,and the column of “search_friction(SF)” is the result when replacing (Pδ, Pf , Ps).

Table 16: Counterfactual analysis: promotion


data data model Pα α0 α1 (α0, α1) promotion(PM)(1990) (2000) (1990)

0.232 0.336 0.2320.223 0.242 0.235 0.242 0.234-0.009 0.010 0.003 0.010 0.002-0.090 0.096 0.026 0.097 0.020

Note:The columns under “Wage inequality” list the inequality from 1990, 2000, and from the benchmark thatis calibrated with data from 1990. The column of Pα represents the result that replaces Pα in 1990 with thevalue in 2000 and keeps others with the benchmark values. The same is true for the columns of “α0”, “α1”,and “(α0,α1)”, and the column of “promotion(PM)” is the result when replacing Pαand (α0,α1).

21

Table 17: Counterfactual analysis: residues


data data model V ρ residue(RS)(1990) (2000) (1990)

0.232 0.336 0.2320.232 0.519 0.5190.000 0.287 0.2870.000 2.758 2.758

Note The columns under “Wage inequality” list the inequality from 1990, 2000, and from the benchmark thatis calibrated with data from 1990. The column of V represents the result that replaces V in 1990 with the valuein 2000 and keeps others with the benchmark values. The same is true for the column of “ρ”, and the columnof “residue(RS)” is the result when replacing V and ρ.

Table 18: Counterfactual analysis: decomposition


data data modelPreference(PF) match premium (PE) search friction(SF) promotion(PM) residue(RS)

(1990) (2000) (1990)

0.232 0.336 0.232

0.156 0.256 0.237 0.234 0.519-0.075 0.024 0.005 0.002 0.287-0.726 0.234 0.052 0.020 2.758-0.31 0.10 0.022 0.009 1.18

Note The columns under “Wage inequality” list the inequality from 1990, 2000, and from the benchmark thatis calibrated in 1990. The columns under “Counterfactual analysis” list the wage inequality under differentcounterfactual cases.

22

References

Acemoglu, D., Autor, D., 2011. Skills, tasks and technologies: Implications for employ-

ment and earnings. Handbook of Labor Economics 4, 1043–1171.

Altonji, J. G., Kahn, L. B., Speer, J. D., 2014. Trends in earnings differentials across college

majors and the changing task composition of jobs. The American Economic Review:

Papers & Proceedings 104 (5), 387–393.

Arcidiacono, P., 2004. Ability sorting and the returns to college major. Journal of Econo-

metrics 121 (1), 343–375.

Autor, D. H., Katz, L. F., Kearney, M. S., 2008. Trends in us wage inequality: Revising the

revisionists. The Review of Economics and Statistics 90 (2), 300–323.

Berliant, M., Reed III, R. R., Wang, P., 2006. Knowledge exchange, matching, and agglom-

eration. Journal of Urban Economics 60 (1), 69–95.

Burstein, A., Morales, E., Vogel, J., 2015. Accounting for changes in between-group in-

equality.

Cooper, R., Liu, H., 2019. Mismatch in human capital accumulation. International Eco-

nomic Review, Forthcoming.

Dorn, D., 2009. Essays on inequality, spatial interaction, and the demand for skills. Ph.D.

thesis, Verlag nicht ermittelbar.

Guvenen, F., Kuruscu, B., Tanaka, S., Wiczer, D., 2015. Multidimensional skill mismatch.

Kambourov, G., Manovskii, I., 2009. Occupational mobility and wage inequality. The Re-

view of Economic Studies 76 (2), 731–759.

Kirkebøen, L., Leuven, E., Mogstad, M., 2014. Field of study, earnings, and self-selection.

Lee, S. Y., Shin, Y., Lee, D., November 2015. The option value of human capital: Higher

education and wage inequality (21724).

Lemieux, T., 2006. Increasing residual wage inequality: Composition effects, noisy data,

or rising demand for skill? The American Economic Review, 461–498.

Lemieux, T., 2015. Occupations,fields of study and returns to education. Canadian Journal

of Economics.

Lise, J., Postel-Vinay, F., 2015. Multidimensional skills, sorting, and human capital accu-

mulation.

Piketty, T., Saez, E., 2014. Inequality in the long run. Science 344 (6186), 838–843.

23

Ritter, J. A., West, K. L., 2014. Field of study and earnings inequality among the highly

educated: 1993-2010.

Robst, J., 2007. Education and job match: The relatedness of college major and work.

Economics of Education Review 26 (4), 397–407.

Violante, G. L., 2002. Technological acceleration, skill transferability, and the rise in resid-

ual inequality. The Quarterly Journal of Economics, 297–338.

24

Appendix

A Appendix of math detail on benchmark model

A.1 Multiple Equilibria

There are seven potential equilibria (labeled Eq1 to Eq7) as in Table A.1

Table A.1: Multiple equilibriums

Equilibrium D(α0) D(α1) PM PFEq1 τL τL, τM, τH (α1, τM), (α1, τH) (α0, τL), (α1, τL)Eq2 τL τL, τM (α1, τM) (α0, τL), (α1, τL)Eq3 τL τL (α0, τL), (α1, τL)Eq4 τL, τM, τH (α1, τL), (α1, τM), (α1, τH) Eq5 τL, τM (α1, τL), (α1, τM) Eq6 τL (α1, τL) Eq7

Note This table lists all seven equilibriums. Columns “D(α0)” and “D(α1)” represent the set of job amenityof a job switcher when the promotion level is α0, and α1, respectively. The columns “PM” and “PF” list thecombination of promotion level and amenity due to promotion and preference, respectively.

A.2 Employment distribution and wages in equilibrium 1 (Eq1)

Employment Let NPF, NPM, andNSF be the number of total unmatched employees due to

preference, promotion, and search friction, respectively. NsL, NsM, andNsH are the number

of workers matched with their preference of τL, τM, andτH, respectively. Employment in

an unmatched job that comes from stayers with different promotion levels and switchers

from matched jobs due to preference is as follows.

NPF = Ng0(1− Ps)PδPL + Ns(1− Ps)(1− Pδ)PL + Ng1(1− Ps)PδPL. (1)

Employment in an unmatched job that comes from stayers and switchers from matched

jobs due to promotion is as follows.

NPM = Ns(1− Ps)(1− Pδ)Pα(PM + PH) + Ng1(1− Ps)Pδ(PM + PH) (2)

Employment in an unmatched job that comes from lucky job seekers receiving offers of

unmatched jobs and stayers with different promotion levels who did not receive an offer

25

of a matched job due to search friction is as follows.

NSF = NU Pf (1− Pδ) + Ng0(1− Ps)(1− Pδ) + Ng1(1− Ps)(1− Pδ) (3)

Employment in a matched job with low job amenity that comes from lucky job seekers

and stayers is as follows.

NsL = NU Pf PδPL + Ns(1− Ps)PδPL (4)

Employment in matched jobs with medium job amenity that comes from lucky job seek-

ers, switchers from unmatched jobs with promotions of α0, and stayers is as follows.

NsM = NU Pf PδPM + Ng0(1− Ps)PδPM + Ns(1− Ps)(1− Pδ)(1− Pα)PM + Ns(1− Ps)PδPM

(5)

Employment in matched jobs with high job amenity that comes from lucky job seekers,

switchers from unmatched jobs, and stayers is as follows.

NsH = NU Pf PδPH + Ng0(1− Ps)PδPH + Ns(1− Ps)[(1− Pδ)(1− Pα) + Pα]PH (6)

Wages Let wPF, wPM, andwSF be the wage of unmatched workers due to preference, pro-

motion, and search friction, respectively; wsL, wsM, andwsH are the wages of matched

workers with the preference τL, τM, andτH, respectively; wgandws are the average wage

of unmatched and matched workers, respectively. Given the wage function w(α, h, τ) =

[(αh)1−ρτ]1

θ−ρ , the wage in unmatched job due to preference is the average wage among

workers with different promotion levels.

wPF = w(α0, hL, τM)[Ng0(1− Ps)PδPL + Ns(1− Ps)(1− Pδ)(1− Pα)PL]/NPF (7)

+ w(α1, hL, τM)[Ns(1− Ps)(1− Pδ)PαPL + Ng1(1− Ps)PδPL]/NPF

Similarly, the wage in unmatched jobs due to search friction is the average wage among

workers of different promotion levels.

wSF = w(α0, hL, τM)[NU Pf (1− Pδ) + Ng0(1− Ps)(1− Pδ)]/NSF (8)

+ w(α1, hL, τM)[Ng1(1− Ps)(1− Pδ)]/NSF

Given that other wages are the same for workers in the same group, the following holds.

wPM = w(α1, hL, τM) (9)

wsL = w(α0, hH, τL) (10)

wsM = w(α0, hH, τM) (11)

26

wsH = w(α0, hH, τH) (12)

Finally, the average wage in unmatched and matched jobs is as follows.

wg = ∑j=0,1

Ngj

Ngw(αj, hL, τM) (13)

ws = ∑j=L,M,H

Nsj

Nsw(α0, hH, τj) (14)

B Appendix of math detail of the model with skills and produc-

tivity heterogeneity

B.1 Value function in dynamics

The value function of an unemployed worker (a) is as follows.

Vut (a) = V + β(1− Pf )Vu

t+1(a)

+ βPf ∫

A′ ,τ′PδVs

t+1(A′, a, τ

′)dGt+1(A

′, τ′ |a)

+∫

A′(1− Pδ)V

g0t+1(A

′, a)dGt+1(A

′ |a)

The value function of worker (a) in an unmatched job (A) with promotion (α0) and pref-

erence (τM) is as follows.

Vg0t (A, a) = ug(α0, A, a) + βPsVu

t+1(a)

+ β(1− Ps)∫

A′ ,τ′Pδmax[Vg0

t+1(A, a), Vst+1(A

′, a, τ

′)]dGt+1(A

′, τ′ |a)

+∫

A′(1− Pδ)max[Vg0

t+1(A, a), Vg0t+1(A

′, a)]dGt+1(A

′ |a)



Vg1t (A, a) = ug(α1, A, a) + βPsVu

t+1(a)

+ β(1− Ps)∫

A′ ,τ′Pδmax[Vg1

t+1(A, a), Vst+1(A

′, a, τ

′)]dGt+1(A

′, τ′ |a)

+∫

A′(1− Pδ)max[Vg1

t+1(A, a), Vg0t+1(A

′, a)]dGt+1(A

′ |a)

27

The value function of worker (a) in a matched job (A) with promotion (α0) and preference

(τ) is as follows.

Vst (A, a, τ) = us(A, a, τ) + βPsVu

t+1(a)

+ β(1− Ps)∫

A′ ,τ′Pδmax[Vs

t+1(A, a, τ′), Vs

t+1(A′, a, τ

′)]dGt+1(A

′, τ′ |a)

+∫

A′ ,τ′(1− Pδ)(1− Pα)max[Vs

t+1(A, a, τ′), Vg0

t+1(A′, a)]dGt+1(A

′, τ′ |a)

+∫

A′ ,τ′(1− Pδ)Pαmax[Vs

t+1(A, a, τ′), Vg1

t+1(A′, a)]dGt+1(A

′, τ′ |a)

Density function of conditional distribution in dynamics

Assume that the density function of conditional distribution in unemployment, a gen-

eral task without promotion, a general task with promotion, and specific task are f ut (a),

f g0t (A, a), f g1

t (A, a), and f st (A, a, τ), respectively. Subsequently, given a distribution of

G(A, a, τ) and a density function of g(A, a, τ), the density functions can be written as

follows.

Nut f u

t (a) = Nut−1(1− Pf ) f u

t−1(a) + Ng0t−1Ps

∫A′

f g0t−1(A

′, a)dA

′

+ Ng1t−1Ps

∫A′

f g1t−1(A

′, a)dA

′+ Ns

t−1Ps

∫A′ ,τ′

f st−1(A

′, a, τ

′)dA

′dτ′

Ng0t f g0

t (A, a) = Nut−1Pf f u

t−1(a)(1− Pδ)g(A|a)

+ Ng0t−1(1− Ps)(1− Pδ)[

∫Ω1(A,a)

f g0t−1(A

′, a)g(A|a)dA

′]

+ Ng0t−1(1− Ps)(1− Pδ)

∫Ω2(A,a)

f g0t−1(A, a)g(A|a)dA

+ Ng0t−1(1− Ps)Pδ

∫Ω3(A,a)

f g0t−1(A, a)g(A, τ|a)dAdτ

+ Ng1t−1(1− Ps)(1− Pδ)[

∫Ω4(A,a)

f g1t−1(A

′, a)g(A|a)dA

′]

+ Nst−1(1− Ps)(1− Pδ)(1− Pα)[

∫Ω5(A,a)

f st−1(A

′, a)g(τ)g(A|a)dA

′dτ]

where

Ω1(A, a) = A′

: Vg0t (A, a) > Vg0

t (A′, a),

Ω2(A, a) = A : Vg0t (A, a) ≥ Vg0

t (A, a) = Ω1(A, a),

Ω3(A, a) = (A, τ) : Vg0t (A, a) ≥ Vs

t (A, a, τ),

Ω4(A, a) = A′

: Vg0t (A, a) > Vg1

t (A′, a),

Ω5(A, a) = (A′, τ) : Vg0

t (A, a) > Vst (A

′, a, τ) = Ω3(A, a),

28

and

Ng1t f g1

t (A, a) = Ng1t−1(1− Ps)(1− Pδ)

∫Ω6(A,a)

f g1t−1(A, a)g(A|a)dA


∫Ω7(A,a)

f g1t−1(A, a)g(A, τ|a)dAdτ

+ Nst−1(1− Ps)(1− Pδ)Pα[

∫Ω8(A,a)

f st−1(A

′, a)g(τ)g(A|a)dA

′dτ]

where

Ω6(A, a) = A : Vg1t (A, a) ≥ Vg0

t (A, a),

Ω7(A, a) = (A, τ) : Vg1t (A, a) > Vs

t (A, a, τ),

Ω8(A, a) = (A′, τ) : Vg1

t (A, a) ≥ Vst (A

′, a, τ) = Ω7(A, a),

and

Nst f s

t (A, a, τ) = Nut−1Pf f u

t−1(a)Pδg(A, τ|a)

+ Ng0t−1(1− Ps)Pδ[

∫Ω9(A,a,τ)

f g0t−1(A

′, a)g(A, τ|a)dA

′]

+ Ng1t−1(1− Ps)Pδ[

∫Ω10(A,a,τ)

f g1t−1(A

′, a)g(A, τ|a)dA

′]

+ Nst−1(1− Ps)Pδ[

∫Ω11(A,a,τ)

f st−1(A

′, a)g(A, τ|a)dA

′]

+ Nst−1(1− Ps)(1− Pδ)(1− Pα)

∫Ω12(A,a,τ)

f st−1(A, a)g(A, τ|a)dA

+ (1− Pδ)Pα

∫Ω13(A,a,τ)

f st−1(A, a)g(A, τ|a)dA + Pδ

∫Ω14(A,a)

f st−1(A, a)g(A, τ|a)dA

where

Ω9(A, a, τ) = A′

: Vst (A, a, τ) ≥ Vg0

t (A′, a),

Ω10(A, a, τ) = A′

: Vst (A, a, τ) ≥ Vg1

t (A′, a),

Ω11(A, a, τ) = A′

: Vst (A, a, τ) > Vs

t (A′, a, τ),

Ω12(A, a, τ) = A : Vst (A, a, τ) ≥ Vg0

t (A, a) = Ω9(A, a, τ),

Ω13(A, a, τ) = A : Vst (A, a, τ) ≥ Vg1

t (A, a) = Ω10(A, a, τ),

Ω14(A, a, τ) = A : Vst (A, a, τ) ≥ Vs

t (A, a, τ) = Ω11(A, a, τ),

29

Employment in dynamics

Assume that the density function of unemployment is as follows.

Nut =

∫a

Nut f u

t (a)d(a)

= Nut−1(1− Pf ) + Ng0

t−1Ps + Ng1t−1Ps + Ns

t−1Ps

Ng0t =

∫A,a

Ng0t f g0

t (A, a)d(A, a)

= Nut−1Pf (1− Pδ)

+ Ng0t−1(1− Ps)(1− Pδ)

∫A,a

[∫

Ω1(A,a)f g0t−1(A

′, a)dA

′]g(A|a)d(A)da

+ Ng0t−1(1− Ps)(1− Pδ)

∫A,a

[∫

Ω2(A,a)g(A|a)dA] f g0

t−1(A, a)dad(A)

+ Pδ

∫A,a

[∫

Ω3(A,a)g(A, τ|a)dAdτ] f g0

t−1(A, a)dad(A)

+ Ng1t−1(1− Ps)(1− Pδ)

∫A,a

[∫

Ω4(A,a)f g1t−1(A

′, a)dA

′]g(A|a)dad(A)

+ Nst−1(1− Ps)(1− Pδ)(1− Pα)[

∫A,a

[∫

Ω5(A,a)f st−1(A

′, a)g(τ)dA

′dτ]g(A|a)dad(A)]

where∫A,a

[∫

Ω1(A,a)f g0t−1(A

′, a)dA

′]g(A|a)d(A)da +

∫A,a

[∫

Ω2(A,a)g(A|a)dA] f g0

t−1(A, a)dad(A) = 1

and

Ng1t =

∫A,a

Ng1t f g1

t (A, a)d(A, a)

= Ng1t−1(1− Ps)

∫A,a

[(1− Pδ)∫

Ω6(A,a)g(A|a)dA + Pδ

∫Ω7(A,a)

g(A, τ|a)dAdτ] f g1t−1(A, a)dad(A)

+ Nst−1(1− Ps)(1− Pδ)Pα

∫A,a

[∫

Ω8(A,a)f st−1(A

′, a)g(τ)dA

′dτ]g(A|a)dad(A)

30

Nst =

∫A,a,τ

Nst f s

t (A, a, τ)d(A, a, τ)

= Nut−1Pf Pδ


∫A,a,τ

[∫

Ω9(A,a,τ)f g0t−1(A

′, a)dA

′]g(A, τ|a)d(A, a, τ)


∫A,a,τ

[∫

Ω10(A,a,τ)f g1t−1(A

′, a)dA

′]g(A, τ|a)d(A, a, τ)

+ Nst−1(1− Ps)Pδ

∫A,a,τ

[∫

Ω11(A,a,τ)f st−1(A

′, a)dA

′]g(A, τ|a)d(A, a, τ)

+ Nst−1(1− Ps)

∫A,a,τ

[(1− Pδ)(1− Pα)∫

Ω12(A,a,τ)g(A, τ|a)dA

+ (1− Pδ)Pα

∫Ω13(A,a,τ)

g(A, τ|a)dA + Pδ

∫Ω14(A,a)

g(A, τ|a)dA] f st−1(A, a)d(A, a, τ)

where∫A,a,τ[∫

Ω11(A,a,τ)f st−1(A

′, a)dA

′]g(A, τ|a)+ [

∫Ω14(A,a)

g(A, τ|a)dA] f st−1(A, a)d(A, a, τ) = 1

B.2 Stationary Equilibrium

Value functions in stationary equilibrium

In stationary equilibrium, f u(a), f g0(A, a), f g1(A, a), and f s(A, a, τ) are the density func-

tions of conditional distribution; Vu(a), Vg0(A, a), Vg1(A, a), and Vs(A, a, τ) are the value

functions; and Nu, Ng0 , Ng1 , and Ns are the employment levels.

The value function of an unemployed worker (a) is as follows.

Vu(a) = V + β(1− Pf )Vu(a)

+ βPf ∫

A′ ,τ′PδVs(A

′, a, τ

′)g(A

′, τ′ |a)d(A

′, τ′)

+∫

A′(1− Pδ)Vg0(A

′, a)g(A

′ |a)d(A′)



Vg0(A, a) = ug(α0, A, a) + βPsVu(a)

+ β(1− Ps)∫

A′ ,τ′Pδmax[Vg0(A, a), Vs(A

′, a, τ

′)]g(A

′, τ′ |a)d(A

′, τ′)

+∫

A′(1− Pδ)max[Vg0(A, a), Vg0(A

′, a)]g(A

′ |a)d(A′)


31


Vg1(A, a) = ug(α1, A, a) + βPsVu(a)

+ β(1− Ps)∫

A′ ,τ′Pδmax[Vg1(A, a), Vs(A

′, a, τ

′)]g(A

′, τ′ |a)d(A

′, τ′)

+∫

A′(1− Pδ)max[Vg1(A, a), Vg0(A

′, a)]g(A

′ |a)d(A′)

The value function of worker (a) in a matched job (A) with promotion (α0) and preference

(τ) is as follows.

Vs(A, a, τ) = us(A, a, τ) + βPsVu(a)

+ β(1− Ps)∫

A′ ,τ′Pδmax[Vs(A, a, τ

′), Vs(A

′, a, τ

′)]g(A

′, τ′ |a)d(A

′, τ′)

+∫

A′ ,τ′(1− Pδ)(1− Pα)max[Vs(A, a, τ

′), Vg0(A

′, a)]g(A

′, τ′ |a)d(A

′, τ′)

+∫

A′ ,τ′(1− Pδ)Pαmax[Vs(A, a, τ

′), Vg1(A

′, a)]g(A

′, τ′ |a)d(A

′, τ′)

Density function in stationary equilibrium

f u(a) =1

Nu Nu(1− Pf ) f u(a) + Ng0 Ps f g0(a)

+ Ng1 Ps f g1(a)d + NsPs f st−1(a)

f g0(A, a) =1

Ng0NuPf f u(a)(1− Pδ)g(A|a)

+ Ng0(1− Ps)[∫

Ω1(A,a)f g0(A

′, a)dA

′](1− Pδ)g(A|a)

+ Ng0(1− Ps) f g0(A, a)[(1− Pδ)∫

Ω2(A,a)g(A|a)dA + Pδ

∫Ω3(A,a)

g(A, τ|a)dAdτ]

+ Ng1(1− Ps)[∫

Ω4(A,a)f g1(A

′, a)dA

′](1− Pδ)g(A|a)

+ Ns(1− Ps)[∫

Ω5(A,a)f s(A

′, a)g(τ)dA

′dτ](1− Pδ)(1− Pα)g(A|a)

and

f g1(A, a) =1

Ng1Ng1(1− Ps) f g1(A, a)[(1− Pδ)

∫Ω6(A,a)

g(A|a)dA + Pδ

∫Ω7(A,a)

g(A, τ|a)dAdτ]

+ Ns(1− Ps)[∫

Ω8(A,a)f s(A

′, a)g(τ)dA

′dτ](1− Pδ)Pαg(A|a)

32

and

f s(A, a, τ) =1

Ns NuPf f u(a)Pδg(A, τ|a)

+ Ng0(1− Ps)[∫

Ω9(A,a,τ)f g0(A

′, a)dA

′]Pδg(A, τ|a)

+ Ng1(1− Ps)[∫

Ω10(A,a,τ)f g1(A

′, a)dA

′]Pδg(A, τ|a)

+ Ns(1− Ps)[∫

Ω11(A,a,τ)f s(A

′, a, τ)dA

′]Pδg(A|a, τ)

+ Ns(1− Ps)[∫

τ′

f s(A, a, τ′)dτ

′][(1− Pδ)(1− Pα)

∫Ω12(A,a,τ)

g(A, τ|a)dA

+ (1− Pδ)Pα

∫Ω13(A,a,τ)

g(A, τ|a)dA + Pδ

∫Ω14(A,a,τ)

g(A, τ|a)dA]

Employment in stationary equilibrium

Nu = Nu(1− Pf ) + Ng0 Ps + Ng1 Ps + NsPs

Ng0 = NuPf (1− Pδ)

+ Ng0(1− Ps)(1− Pδ)

+ Ng0(1− Ps)Pδ

∫A,a

[∫

Ω3(A,a)g(A, τ|a)dAdτ] f g0(A, a)dad(A)

+ Ng1(1− Ps)(1− Pδ)∫

A,a[∫

Ω4(A,a)f g1(A

′, a)dA

′]g(A|a)dad(A)

+ Ns(1− Ps)(1− Pδ)(1− Pα)∫

A,a[∫

Ω5(A,a)f s(A

′, a)g(τ)dA

′dτ]g(A|a)dad(A)

Ng1 = Ng1(1− Ps)(1− Pδ)∫

A,a[∫

Ω6(A,a)g(A|a)dA] f g1(A, a)dad(A)

+ Ng1(1− Ps)Pδ

∫A,a

[∫


+ Ns(1− Ps)(1− Pδ)Pα

∫A,a

[∫

Ω8(A,a)f s(A

′, a)g(τ)dA

′dτ]g(A|a)dad(A)

33

Ns = NuPf Pδ

+ Ng0(1− Ps)Pδ

∫A,a,τ

[∫

Ω9(A,a,τ)f g0(A

′, a)dA

′]g(A, τ|a)d(A, a, τ)

+ Ng1(1− Ps)Pδ

∫A,a,τ

[∫

Ω10(A,a,τ)f g1(A

′, a)dA

′]g(A, τ|a)d(A, a, τ)

+ Ns(1− Ps)Pδ

+ Ns(1− Ps)(1− Pδ)(1− Pα)∫

A,a,τ[∫

Ω12(A,a,τ)g(A, τ|a)dA] f s(A, a)d(A, a, τ)


∫A,a,τ

[∫

Ω13(A,a,τ)g(A, τ|a)dA] f s(A, a)d(A, a, τ)

Solving the equation

Ng1 = P1Ns

where

P1 =(1− Ps)(1− Pδ)PαPr8

1− (1− Ps)(1− Pδ)Pr6 − (1− Ps)PδPr7

Pr6 =∫

A,a[∫


Pr7 =∫

A,a[∫


Pr8 =∫

A,a[∫

Ω8(A,a)f s(A

′, a)g(τ)dA

′dτ]g(A|a)dad(A)

then

Nu =Ps

Pf(Ng0 + (1 + P1)Ns)

Ng0 =NuPf (1− Pδ) + Ng1(1− Ps)(1− Pδ)Pr4 + Ns(1− Ps)(1− Pδ)(1− Pα)Pr5

1− (1− Ps)(1− Pδ)− (1− Ps)PδPr3

where

Pr3 =∫

A,a[∫


Pr4 =∫

A,a[∫

Ω4(A,a)f g1(A

′, a)dA

′]g(A|a)dad(A)

Pr5 = ∫

A,a[∫

Ω5(A,a)f s(A

′, a)g(τ)dA

′dτ]g(A|a)dad(A)

substituting

Ng0 =

PsPf(Ng0 + (1 + P1)Ns)Pf (1− Pδ) + P1Ns(1− Ps)(1− Pδ)Pr4 + Ns(1− Ps)(1− Pδ)(1− Pα)Pr5

1− (1− Ps)(1− Pδ)− (1− Ps)PδPr3

or

Ng0 =(1 + P1)Ps(1− Pδ) + P1(1− Ps)(1− Pδ)Pr4 + (1− Ps)(1− Pδ)(1− Pα)Pr5

1− (1− Ps)(1− Pδ)− (1− Ps)PδPr3 − Ps(1− Pδ)Ns

34

denote

P0 =(1 + P1)Ps(1− Pδ) + P1(1− Ps)(1− Pδ)Pr4 + (1− Ps)(1− Pδ)(1− Pα)Pr5

1− (1− Ps)(1− Pδ)− (1− Ps)PδPr3 − Ps(1− Pδ)

then

Nu =Ps

Pf[P0 + (1 + P1)]Ns = PuNs

where

Pu =Ps

Pf[P0 + (1 + P1)]

then the market clear condition implies the following.

Ns =1

1 + Pu + P0 + P1

Mismatch in stationary equilibrium

The mismatched workers could come from task g0 and g1. In each group, they could

come from search friction (SF), preference (PF), and productivity (A). The total number is

Ng = Ng0 + Ng1 , which could be divided into the following groups. In task g0, mismatch

from search friction comes from new job finders, stayers in g0, and switchers from task g1.

Ng0SF = NuPf (1− Pδ)

+ Ng0(1− Ps)(1− Pδ)

+ Ng1(1− Ps)(1− Pδ)∫

A,a[∫

Ω4(A,a)f g1(A

′, a)dA

′]g(A|a)dad(A)

In task g0, a mismatch from preference comes from stayers and from switchers from task

s.

Ng0PF = Ng0(1− Ps)Pδ

∫A,a

[∫

Ω3(A,a)&τ<τMg(A, τ|a)dAdτ] f g0(A, a)dad(A)

+ Ns(1− Ps)(1− Pδ)(1− Pα)∫

A,a[∫

Ω5(A,a)&τ<τMf s(A

′, a)g(τ)dA

′dτ]g(A|a)dad(A)

In task g0, mismatch from productivity difference comes from stayers and from switchers

from task s.

Ng0A = Ng0(1− Ps)Pδ

∫A,a

[∫

Ω3(A,a)&τ>τMg(A, τ|a)dAdτ] f g0(A, a)dad(A)

+ Ns(1− Ps)(1− Pδ)(1− Pα)∫

A,a[∫

Ω5(A,a)&τ>τMf s(A

′, a)g(τ)dA

′dτ]g(A|a)dad(A)

35

In task g1, mismatch from promotion comes from mismatched workers who are willing

to switch to task g1 but not to task g0 and would leave from task g1but not from task g0.

Ng1PM = Ng1(1− Ps)Pδ

∫A,a

[∫

Ω7(A,a)&Ωc3(A,a)

g(A, τ|a)dAdτ] f g1(A, a)dad(A)


∫A,a

[∫

Ω8(A,a)&Ωc5(A,a)

f s(A′, a)g(τ)dA

′dτ]g(A|a)dad(A)

In task g1, a mismatch from search friction comes from workers in task g1 who stay in

their task when facing choice of task g0.

Ng1SF = Ng1(1− Ps)(1− Pδ)

∫A,a

[∫


In task g1, a mismatch from preference comes from stayers and from switchers from task

s.

Ng1PF = Ng1(1− Ps)Pδ

∫A,a

[∫

Ω3(A,a)&τ<τMg(A, τ|a)dAdτ] f g1(A, a)dad(A)


∫A,a

[∫


′, a)g(τ)dA

′dτ]g(A|a)dad(A)

In task g1, a mismatch from productivity difference comes from stayers and from switch-

ers from task s.

Ng1A = Ng1(1− Ps)Pδ

∫A,a

[∫

Ω3(A,a)&τ>τMg(A, τ|a)dAdτ] f g1(A, a)dad(A)


∫A,a

[∫


′, a)g(τ)dA

′dτ]g(A|a)dad(A)

Wages in stationary equilibrium

The wage functions are as follows.

w0(A, a) = w(α0, A, a, hL, τM),

w1(A, a) = w(α1, A, a, hL, τM),

ws(A, a, τ) = w(α0, A, a, hH, τ),

36

wg0SF =

1Ng0

SFNuPf (1− Pδ)

∫A,a

f u(a)g(A|a)w0(A, a)d(A, a)

+ Ng0(1− Ps)(1− Pδ)∫

A,a[∫

Ω1(A,a)f g0(A

′, a)dA

′]g(A|a)w0(A, a)d(A, a)

+ Ng0(1− Ps)(1− Pδ)∫

A,af g0(A, a)[

∫Ω2(A,a)

g(A|a)dA]w0(A, a)d(A, a)

+ Ng1(1− Ps)(1− Pδ)∫

A,a[∫

Ω4(A,a)f g1(A

′, a)dA

′]g(A|a)w0(A, a)d(A, a)

wg0PF =

1Ng0

PFNg0(1− Ps)Pδ

∫A,a

f g0(A, a)[∫

Ω3(A,a)&τ<τMg(A, τ|a)dAdτ]w0(A, a)d(A, a)

+ Ns(1− Ps)(1− Pδ)(1− Pα)∫

A,a[∫


′, a)g(τ)dA

′dτ]g(A|a)w0(A, a)d(A, a)

wg0A =

1Ng0

ANg0(1− Ps)Pδ

∫A,a

f g0(A, a)[∫

Ω3(A,a)&τ>τMg(A, τ|a)dAdτ]w0(A, a)d(A, a)

+ Ns(1− Ps)(1− Pδ)(1− Pα)∫

A,a[∫


′, a)g(τ)dA


and

wg1PM =

1Ng1

PMNg1(1− Ps)Pδ

∫A,a

f g1(A, a)[∫

Ω7(A,a)&Ωc3(A,a)

g(A, τ|a)dAdτ]w1(A, a)d(A, a)


∫A,a

[∫

Ω8(A,a)&&Ωc5(A,a)

f s(A′, a)g(τ)dA


wg1SF =

1Ng1

SFNg1(1− Ps)(1− Pδ)

∫A,a

f g1(A, a)[∫

Ω6(A,a)g(A|a)dA]w1(A, a)d(A, a)

wg1PF =

1Ng1

PFNg1(1− Ps)Pδ

∫A,a

f g1(A, a)[∫

Ω3(A,a)&τ<τMg(A, τ|a)dAdτ]w1(A, a)d(A, a)


∫A,a

[∫


′, a)g(τ)dA


wg1A =

1Ng1

ANg1(1− Ps)Pδ

∫A,a

f g1(A, a)[∫

Ω3(A,a)&τ>τMg(A, τ|a)dAdτ]w1(A, a)d(A, a)


∫A,a

[∫


′, a)g(τ)dA


and

ws =1

Ns

∫A,a,τ

ws(A, a, τ) f s(A, a, τ)d(A, a, τ)

37

gaowangwang.weebly.com · 2020-02-11 · Education Mismatch and Earnings Inequality Rongsheng Tang...

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