Discussion Papers In Economics
And Business

Note on the Determinants of Wage Dispersion
Yosuke Oda
Discussion Paper 08-37

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANNote on the Determinants of Wage Dispersion
Yosuke Oda
Discussion Paper 08-37

December 2008

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANNote on the Determinants of Wage Dispersion
Yosuke Oda ∗Graduate school of Economics, Osaka university †December 2008

Abstract
The purpose of this note is to explain theoretically the observed entire densityof wages which is hump-shaped and right-skewed. I extend the model brought up
by Halko et al . (2008) to introduce heterogeneity of ﬁrm’s productivity. It causes
a diﬀerence in the support of wage oﬀers, a wider (narrower) range for high (low)productivity ﬁrms. The diﬀerent support roughly results in the observed wage
dispersion because low wage oﬀers are made by all ﬁrms (right-skewed), whereas
high wage oﬀers are made by only high producivity ﬁrms.
Keywords: search; matching; wage posting; wage dispersion
JEL classiﬁcation: J31; J64∗

I would like to thank my supervisor Masaru Sasaki for his helpful suggestions and comments. Allremaining errors are, of course, my own.
†

E-mail:JAPAN.

dg011oy@mail2.econ.osaka-u.ac.jp, 1-7 Machikaneyama, Toyonaka, Osaka 560-0043,

1

Introduction

Many studies have attempted to explain the shape of empirically observed wage density
which is hump-shaped and right-skewed. The aim of this note is also to constructa model which can explain empirically observed wage density which is hump-shaped
and right-skewed, and suggest an intuitive explanation why such a shape is formed.
Bontemps et al . (2000) assumes a continuous distribution of ﬁrm productivity types
and derives hump-shaped wage density. Julien et al . (2006) makes ﬁrm’s productivity
endogenous by selecting either high or low levels. In the latter model, workers’ wage
can diﬀer even if they are employed by the same job according to the number of oﬀer
they are made. Halko et al . (2008), which is closed to this note, considers a model that
there is a market in which only ﬁrms make a wage oﬀer (wage oﬀer market), in which
only workers make a wage oﬀer (wage demand market), and in which the two markets
coexist. In their model, which situation occurs depends on the vacancy-unemployment
ratio. In particular, hump-shaped is derived due to wage oﬀer market whereas rightskewed is mainly due to wage demand market. In the actual economy, however, mostwage level would be determined by ﬁrms (with a few exceptions, for example, labor
union’s wage negotiation).
The main contribution of this note is that I exhibit a quite simple model that shows
the observed wage distribution and intuitive explanation using by “ wage oﬀer market”
as in Halko et al . (2008), so the wage level in this note is determined by ﬁrms only. In
the market ﬁrms randomize over the workers and oﬀer a wage using a mixed strategy
with a cumulative distribution function. Unlike Halko et al . (2008), I assume that, as
in Julien et al . (2006), ﬁrm’s productivity is two types but it is exogenous in this note.
This productivity diﬀerence leads to long-tail in the wage oﬀer market. Introducting
the diﬀerence in productivity to the model constructed by Halko et al . (2008), the2variant support of wage oﬀer between high and low productivity ﬁrms is led since highproductivity ﬁrms can make a higher wage oﬀer than low productivity ﬁrms. Given
the fact, both high and low productivity ﬁrms can make a low wage oﬀer but only highproductivity ﬁrms can oﬀer a high wage. Hence the upper bound of the support for highproductivity ﬁrms is wider than for low productivity ﬁrms, which leads to high density
within low wage oﬀers and low density within high wage oﬀers. By using the theoreticalmodel, I show numerical results and intuitive exlpanations of empirically observed wage
density. The model predicts that, as the fraction of high productivity ﬁrm increases, theskewedness is enhanced whereas as the productivity diﬀerence decreases, the skewedness
is weakened.
The rest of note is organized as follows. Section 2 constructs the theoretical model.
Numerical analysis of the model and intuitive explanations are shown in section 3 andsection 4 concludes.

2

The Model

Consider a static labor market in which ﬁrms with a vacancy measured by v send a wageoﬀer and unemployed workers measured by u receive the oﬀer(s). Each vacancy canhire at most one worker and oﬀers a wage using a mixed strategy. Each worker chooses
the highest oﬀer and can not make any counter oﬀer. Firm’s productivity is either
high pH or low pL , (pH > pL > 0), and the fraction of high productivity in aggregate
vacancy is indicated by α ∈ (0, 1). As in Halko et al . (2008), the Poisson rate underwhich ﬁrms randomly make an oﬀer to workers is θ ≡ v/u, the vacancy-unemployment
rate. Throughout this note, I take this value as given.
Let Gi (·) be a cumulative distribution function of wage oﬀer by type i ﬁrm withsupport wi ∈ [wi , wi ] and Vi an expected value of type i vacancy. Since workers accept
the highest wage oﬀer, the expected value of type L vacancy that makes a wage oﬀer
3

w is given by
VL = e−(1−α)θ (pL − w) + (1 − α)θe−(1−α)θ (pL − w)GL (w) + · · ·[(1 − α)θ]k e−(1−α)θ
+(pL − w)[GL (w)]k + · · ·k!
∞∑[(1 − α)θ]k e−(1−α)θ
= (pL − w)
[GL (w)]kk!
k=0= (pL − w)e−(1−α)θ(1−GL (w)) .

(1)

Interpretation of this expression is as follows: when a ﬁrm oﬀering w meets a certain
worker and no other ﬁrms make an oﬀer to the worker, then this ﬁrm hires this worker
and obtains the gain pL − w. When the ﬁrm competes against another ﬁrm for a certain
worker, this ﬁrm hires the worker and obtain the gain if the competitor bids a smaller
wage oﬀer than w (GL (w)). In general, when the ﬁrm competes with k ﬁrms, this ﬁrm
obtains the worker if it oﬀers the highest wage ([GL (w)]k ). Analogously, the expectedvalue of type H vacancy that bids a wage w is
VH = (pH − w)e−αθ(1−GH (w)) .

(2)

Since ﬁrm’s oﬀer is determined by a mixed strategy, it must be indiﬀerence between
the highest oﬀer and the lowset oﬀer. For type H, the expected value oﬀering the lowest
oﬀer wH , (1 − wH )e−αθ(1−GH (wH )) , equals to the expected value with the highest oﬀer
wH , (1 − wH )e−αθ(1−GH (wH )) . Note that wH = 0 since if no other ﬁrms make an oﬀer toa certain worker, this ﬁrm can hire the worker with no payment. Given GH (wH ) = 0,
GH (wH ) = 1 and wH = 0, the highest oﬀer for type H ﬁrm is wH = pH (1 − e−αθ ).
By similar procedure, I obtain the highest oﬀer for type L ﬁrm, wL = pL [1 − e−(1−α)θ ].
Note that the support of type H wage oﬀer, wH ∈ [0, wH ], is wider than that of type−(1−α)θ

L, wL ∈ [0, wL ], if the diﬀerence of productivity is suﬃciently large, ppHL > 1−e1−e−αθ
1. Hereafter, I focus on this case. From (1) and (2), the wage distribution function−(1−α)θ

It can be easily shown that limα→0 1−e1−e−αθ
is satisﬁed for almost all values of pH /pL .
1−(1−α)θ

→ +∞ and limα→1 1−e1−e−αθ

4→ 0. So the condition

1
Lof type L is GL (wL ) = (1−α)θ
ln pLp−wwith support wL ∈ [0, wL ] and that of type H is1
H
GH (wH ) = αθln pHp−wwith support wH ∈ [0, wH ], respectively. Note that, as in HalkoH

et al . (2008), the density functions are increasing in w (G′i (wi ) > 0 for i = H, L).
Next I will derive the aggregate wage oﬀer distribution. Note that both type L andtype H ﬁrms make an oﬀer in the range of w ∈ [0, wL ]. Let β be the fraction in overall
type H. By deﬁnition, β = GH (wL ) can be easily derived as follows:β = (αθ)−1 P, where P ≡ ln

pH.
pH − pL [1 − e−(1−α)θ ]Since the fraction of type H ﬁrm is α, in aggregate, 1 − α type L ﬁrm and αβ type H
ﬁrm make an wage oﬀer in the range of w ∈ [0, wL ] and α(1 − β) type H ﬁrm does in
the range of w ∈ [wL , wH ]. Hence the entire wage oﬀer distribution, indexed by G(w),is summarized as

G(w) =

 [ (
)
(
)]
1
p
p

L
H
 ln
+ β ln, for w ∈ [0, wL ],
pL − w
pH − w
θ

(
)


1−β
pH


ln
,

θ
pH − w

for w ∈ [wL , wH ].

Note that, in general, this distribution does not consist with the “realized” wage
distribution since not all the oﬀers are accepted: a worker who receives multiple oﬀers
chooses only the highest one. So an oﬀered wage is rejected if a competitor bids ahigher wage. Letting G̃(·) be the realized wage distribution, it is given by G̃(w) =∑∞ θk e−θ1k
k=1 k! [G(w)] , which is the probability that the highest oﬀer, conditional on
1−e−θ
the worker receiving at least one oﬀer, is at most w. Substituting G(w) for G̃(w), I

5

obtain the realized wage distribution 2

[(
]
)(
)P/αθ
−θ

p
p
e
L
H

− 1 , for w ∈ [0, wL ],


pL −w
pH −w
1−e−θ


G̃(w) =
[(])1−P/αθ

 e−θ
pH

−1 ,
for w ∈ [wL , wH ].
 1−e−θ
pH −w(3)

As noted above, since not all oﬀers are accepted, it can be easily conﬁrmed that G̃(0) = 0
but G̃(wH ) < 1 3 .

3Numerical Analysis

In this section I show numerical results of the realized wage distribution (3). Notethat parameters must satisfy the condition that the support of high productivity ﬁrm−(1−α)θ

. As a benchmark, I set theis wider than that of low productivity ﬁrm, ppHL > 1−e1−e−αθ
parameters as follows: pH = 1.0 (normalized) and pL = 0.1, θ = 1.2, and α = 0.1. Thus
there is suﬃcient productivity diﬀerence ( ppHL > 10.0) and vacancy-unemployment ratio
is larger than one (θ = 1.2), which is empirically observed (e.g., Hall, 2005). Figure 1
shows the realized wage distribution under benchmark parameters. As mentioned above,since both type H and type L ﬁrms make an wage oﬀer in the range of w ∈ [0, wL ],the density is substantially high than in the range of w ∈ [wL , wH ] where only type H
ﬁrms oﬀer. Note also since each of the wage density functions are increasing in w, theaggregate density function is increasing within w ∈ [0, wL ] and w ∈ [wL , wH ].
One may claim that degeneration of the aggregate density function is questionable(since empirically observed distribution does not degenerate) and that it does not have

2

1
1−e−θ

∑∞

θ k e−θkk=1
k! [G(w)]

e−θ
1−e−θ

[∑
∞

θ0
θk
k
0
k=0 k! [G(w)] − 0! [G(w)]

]
.

Since
{
[ (
)
(
)]
}
pL
pH
m
e
m
− 1 for w ∈ [0, wL ]. The arrangex=0 x! = e , G̃(w) = 1−e−θ exp ln pL −w + β ln pH −w
ment of this expression and analogous calculation for w ∈ [wL , wH ] yields the result.
Note that G̃(w) =

∑∞

x

=

−θ

Substitution wH = pH (1 − e−αθ ) into G̃(w) for w ∈ [wL , wH ] yields G̃(wH ) = e
Since −[(1 − α)θ + P ] < 0, G̃(wH ) is deﬁnitely smaller than one.
3

6−[(1−α)θ+P ]

1−e−θ

−e−θ

.

a long-tail (since the function is increasing in right-hand edge), however, they would
not be problematic. Consider a general case in which there are k diﬀerent types of
productivity 0 < p1 < p2 < · · · < pk < +∞ and the fraction of type i is denoted by αi∑( ki=1 αi = 1). From discussion in the previous section, a ﬁrm with higher productivity
has wider support of wage oﬀer. Let si be the support of ﬁrm with productivity i (s1 <s2 < · · · < sk ) and ni be the number of ﬁrms which make an oﬀer within the support
si . Recalling that the total number of ﬁrms (vacancies) is v, ni is determined by ni =∑v kj=i αj and n1 > n2 > · · · > nk . Assuming that the fraction of high productivity ﬁrms
is relatively less than low roductivity ﬁrms, a similar shape of wage density function
would be obtained in the model with k diﬀerent types of productivity and degeneration
would be moderately improved. In the extreme case in which productivity heterogeneity
is continuous as in Bontemps et al . (2000), the analogous discussion is applied anddegeneration would not arise. Analogous discussions justify an increasing tail. The
aggregate wage density is weighted with the measure of ﬁrm with productivity i, αi .
Consequently, in the k types model, the measure is generally small in right-hand edgeof the entire wage density, which would lead to non-increasing (or decreasing) tail inaggregate.
density
30
25
20
15
10
5
0.05

0.1

0.15

0.2

wage

Figure 1: benchmark case(α = 0.2, θ = 1.2, pH = 1.0, pL = 0.1)
Next I examine the eﬀect of change in α and productivity diﬀerence. Figure 2 shows7the results under various values of α (the other parameters are the same as benchmarkcase). It suggests that as the fraction of type H ﬁrm increases, the skewedness isenhanced: that is, the density in low wage oﬀers is decreased and the long tail islengthened. The intuition is straightforward. Given the productivity diﬀerence pH /pL ,increase in α makes type H ﬁrms more apt to oﬀer a high wage because competitors forthem increase. To make sure that they hires a worker, they must make a high wage oﬀer.
Justiﬁcation of degeneration and increasing tail is same as the above discussion. Finallythe eﬀect of change in productivity diﬀerence, under which pH is held constant and pL
is increased, is in Figure 3. It shows that as the productivity diﬀerence decreases, theskewedness is weakened. The intuition is also straightforward. Given the component
ratio of productivity α, this is because increase in pL widens the support of wage oﬀer
for type L ﬁrms whereas the support for type H ﬁrms. Brieﬂy speaking, increase(decrease) in α enhances (weakens) the right skewedness whereas decrease (increase) in
productivity diﬀerence weakens (enhances) the skewedness 4 .
From the numerical results, some predictions are obtained regarding the real economy: (i) there is substantial diﬀerence in productivity among ﬁrms; (ii) the fraction ofhigh productivity is very small; (iii) many high productivity ﬁrms oﬀer a low wage inspite of they can aﬀord to make a high wage due to randomizing over workers.

densitydensityΑ=0.3

density

Α=0.4

20
15
10
5
0.05 0.1 0.15 0.2 0.25 0.3

wage

Α=0.5

14
12
10
8
6
4
2

17.5
15
12.5
10
7.5
5
2.5
0.1

0.2

0.3

wage

0.1

0.2

0.3

0.4

wage

Figure 2: change in α−(1−α)θ

holds. If the condition does not satisfy, a situation
These results are conditional on ppHL > 1−e1−e−αθ
in which type L ﬁrms have wider support than type H ﬁrms may arise, so the skewedness is not deﬁned(for instance, α is close to 0).
48(θ = 1.2, pH = 1.0, pL = 0.1)

densitydensitypL=0.2

densitypL=0.25

14
12
10
8
6
4
2

15
12.5
10
7.5
5
2.5
0.05

0.1

0.15

0.2

wage

pL=0.3

12
10
8
6
4
2

0.05

0.1

0.15

0.2

wage

0.05

0.1

0.15

0.2

wage

Figure 3: change in productivity diﬀerence(α = 0.2, θ = 1.2, pH = 1.0)

4Conclusion

This note have explored the theoretical model explains empirically observed wage distribution. It shows that the right skewedness and long tail can be explained by two
reasons: First, both high and low productivity make a low wage oﬀer due to randomizing over workers. Second, only high productivity (can) make a high wage oﬀer. The
model predicts that, as the fraction of high productivity ﬁrm increases, the skewedness isenhanced whereas as the productivity diﬀerence decreases, the skewedness is weakened.
In this note, I have shown a theoretical model that can explain the actual wage
distribution but not shown the empirically observed wage distributions. As further
tasks, I shall cite the empirical distributions and explain why such a distribution occurs.
For instance, following my theoretical model, a contributing factor of a high degree
hump-shaped is predicted as a large ratio of low productivity ﬁrms. As another example,
a longer tail seems to be derived by a large productivity diﬀerence. To prove right ofmy theoretical model, I shall show some empirical facts that exhibit the ratio of highand low productivity ﬁrms, the size of productivity diﬀerence, and so on.

9References[1] Acemoglu, D., and R.Shimer, “Wage and Technology Dispersion”, Review of Economic Studies 67 (2000), 585-607.
[2] Bontemps, C., J.M. Robin, and G.van den Berg, “Equilibrium Search with Continuous Productivity Dispersion: Theory and Nonparametric Estimation”, International Economic Review 41 (2000), 305-57.
[3] Halko, Marja-Liisa, K.Kultti, and J.Virrankoski, “Search Direction and Wage Dispersion”, International Economic Review 49 (2008), 111-34.
[4] Hall, R.E., “Employment Fluctuations with Equilibrium Wage Stickness”, American Economic Review 95 (2005), 50-65.
[5] Julien, B., J.Kennes, and I.King, “Residual Wage Disparity and Coodination Unemployment”, International Economic Review 47 (2006), 961-89.
[6] Montgomery, J.K., “Equilibrium Wage Dispersion and Interindustry Wage Diﬀerentials”, Quarterly Journal of Economics 106 (1991), 163-79.
[7] Pissarides, C.A., Equilibrium Unemployment Theory (Cambridge, MA: MIT Press,
2000).

10