The Keio Economic Observatory Review No. 8, March 1996

An Equilibrium Model of Continually Heterogeneous Labor
MarketKeiichiro Obi
Dokkyo University
andKeio Economic Observatory Keio University

1. Introduction
In this paper a model of the labor market, where wage di erentials among the rms ofvarious scales exist, is presented. The term\ rm of various scale" is used to indicate thatthe heights of marginal productivity curves for labor are di erent for di erent rms.
If the labor market is competitive, a unique wage rate prevails so long as the labor
force is homogeneous from the rm's point of view. If the labor force is heterogeneous,
but can be split into three groups A, B and C, where rm a" exclusively recruits workers
from group A and the members of group A exclusively apply to a, and so on, we havethree independent labor markets and the notion of non-competing groups can be applied
to determine wages within each market. However, if the rms a,b and c respectively
recruit among all the members of the groups A,B and C simultaneously, then the notionof non-competing groups is not applicable to the labor market. Since the actual labormarket we observe has such a nature, we need to construct a model which can describe
the performance of a competitive and heterogeneous labor market.
By heterogenity, we mean the existence of various grades (or labor ques) among applying workers from the rm'point of view. The grades or ordering of applicants might bedirectly or indirectly correlated with their work experience,educational background, age,
and/or sex. However, even if those characteristics or qualities are controlled, there may
yet exist some ordering or di erences in grades of applying workers. In fact, statistical
data shows that there are wage di erences among workers of rms of di erent sizes whencontroling for these characteristics of the workers.
This observed fact suggests that rms recongnize di erent grades among workers of
the same age, sex, work experience, and/or educational backgrounds. Any reason for the
paying of higher wages by large rms, whose labor productivities are higher than smaller
ones, cannot be found as long as the grades of workers are the same across rms. In fact,large scale rms with higher productivities o er comparatively favorable work conditions(higher wages and shorter hours of work) and as a result attract many applicants of various
grades.The rms recruit what they perceive as the most favorable ones among those who
applied. Smaller rms with lower productivity can o er only less favorable terms andrecruit among the residual applicants who fail to be employed by the large scale rms.
This is the common experience of high school and college graduates in Japan.
In the following section we present a model of the labor market making use of thenotion of grades1 of labor in order to realistically approximate the labor market in Japan.
 Reprinted and corrected from Keio Economic Observatory Occasional Paper E.No.14(January 1995).
This paper was presented at The International Symposium on Economic Modeling, Athens 1993.
1 The notion of this kind, that is, labor que, is used in L.C.Thurow [10]2

The Keio Economic Observatory Review No. 8, March 1996

The model is suitably simpli ed. Although the labor supply actually consists of membersof self‐employed households (e.g. farmers' households) and employee households whose
principal earners are employees, only the latter type of household is taken into account. As
well, the investment behavior of rms is not explicitly treated. These simpli cations will
not impair the basic characteristics of the model which remain suciently autonamous.
The performance of the model is tested by numerical examples and by the application to
Japanese data.
Models of wage and employment determination with respect to a rm (or a group ofrms) have been developed elsewhere.2 In this kind of model, individual labor supply andlabor demand functions for a rm are assumed; that is, the notion of a kind of local labormarket is introduced in the models. However, the relation between the individual labor
supply function for the speci c rm considered and the supply function of the market asa whole is not explicitly discussed. Such an individual supply function is, to some extent,
an ad hoc relation just as is the individual demand function for a rm ′s product in an
oligopolistic market. The wage level of the rm considered and the average wage level ofother rms appear as explanatory variables of the individual labor supply function(Theratio of the both variables is adopted insome cases).
Elasticities of labor supply with respect to those variables or coecients of those variables for each rm change, re ecting change in the conditions of the labor market as awhole including changes in the degree of of competition, the labor suppliers' conjecturewith respect to the recruitment policy of rms other than the rms to which the suppliers
are applying. However, the mechanism of such interdependent changes of elasticities orcoecients of individual supply function has not been clari ed. In this sense,models using
individual supply functions lack autonomy.
Individual labor supply functions for each rm are not used in the model presented
below. Instead, two basic relations are introduced. Instead of an individual supply function
for a rm, which describes the relation between the number of applicants for the speci crm and the wage rate the rm o ers, we use the labor supply function for the wholemarket describing the quantity of labor supplied, the wage rate being given. That is,
the supply function used in the following model does not specify the distribution of thequantity of labor supplied among rms. The distribution itself is determined by the model
including rm demand functions for labor.

2. Basic Equations of the Model
2.1. Distribution Function of Grades of Labor

Let the indicator of the grade of the worker be Gi , wherei = 1; 2; :::; m;and m is the total number of people of working age. The range of Gi is supposed tobe"  Gi  1
where " is some positive small number．
The cumulative distribution (cumulative from the top of G, where G = 1) function of
G is designated by  (G) and the density distribution by  0 (G).
2

C.A.Pissarides [9]An Equilibrium Model of Continually Heterogeneous Labor Market

3

2.2. Labor Supply Probability Function

Suppose among n persons, n0 persons accept the employment opportunity at wage ratew, and assigned hoursof work h, o ered by rms. The ratio n0 =n is the supply ratio withrespect to the employment opportunity.
Plim n0 =n  

n!1

is de ned as the supply probability, which is a function of w and h.

2.3. Distribution of Minimum Supply Price of Labor
The minimum supply price of labor3 is de ned as a critical wage rate below which suppliers
reject the employment opportunity, assigned hours of work h being given. The minimum
supply price of labor (MSPL) is denoted by w. Any supplier's level of MSPL depends onfollowing three factors:a) the shape of his/her income-leisure preference curve,b) the level of his guaranteed income Xg which he/she can obtain without working (e.g.
principal earner's income is a guaranteed income for non principal earners),c) hours of work assigned by rms, h.
Hence, we have

wi = w(xig ; hi ; i )i = 1; 2; . . . ; n(1:1)

where i stands for the set of preference parameters of the ith supplier. xig and hi canbe regarded as exogenous variables for the ith supplier. The value of i is speci c to ith
supplier; that is, the value of i di ers among each of the n suppliers. Hence, we have thedensity distribution function ( ).
Now, suppose a group of persons have the same level of guaranteed income xg ; thatis,xig = xig+1 = xg

(1:2)

From(1.1), (1.2) and ( ), we havegf (w j xg ; h)

(1:3)

3 The notion of reservation wage(RW) is given in Heckman[2]. MSPL is another de nition of a kindof RW, because MSPL is de ned by assuming h is a parameter which is assiged by the rm. In themodern employee labor market (in contract to self employed work), hours of work are assigned by theemployer( rm). The employees have some minimal leeway as they can reduce or increase the hours
worked to some extent. However, a complex array of social, psychological, and institutional factors usually
produces a situation where excessive absenteeism etc. will results or dismissal. The analogus situation
exists with respect to \overtime". Analysis of the MSPL of labor using an income{leisure preference
fuction assuming maximization behavior is shown in Keiichiro Obi[5],[6],[7]. and T.Miyauchi[4]4The Keio Economic Observatory Review No. 8, March 1996

which is the density distribution function of MSPL, h for brevity being assumed a common
value for all persons considered. Subscript f and  denote the fact that the analyticalform of the function g depends on f and . Integration of g,

=

Z w

w=0

g(w j xg ; h)dw =  = (w j xg ; h);(1:4)

gives the supply-probability function  of the group of persons with xg and h.
Multiplying by n, the number of persons in the group, we have the number of suppliers
Ls , namely,

Ls = n(w j xg ; h);When xg and h are destributed as a joint density distribution(xg ; h);

(1:5)

(1:6)

we have(w) =Z w Z dZ b

w=0 xg =c h=ag(w; xg ; h) (xg ; h)  dh  dxg  dw

(1:7)

where a; b; c; d and e are the values standing for regions of integration for the relevant
variables, h, xg and w.

3. The Outline of the Model
Let the production function of the ith sector (or rm) be

Qi = F (Li ; Gi ; Ai ) (i = 1; 2; . . . ; n)(2:1)
where A and G, respectively, stand for the set of rm parameters and the index of thegrade of workers employed in the ith sector. Further, Gi can be written as
max
Gi = Gi (Gmin(2:10 )
i ; Gi )where Gmax
and Gminii are indicators of the highest grade of workers (most preferable

workers among applicants from the rm's point of view) and the lowest grade of workers.
It is supposed that

@F > 0;@Gi@F@Li > 0:

Let the supply probability equation (1:7) be = (w; )(2:2)

An Equilibrium Model of Continually Heterogeneous Labor Market

5

where  is a set of parameters of individuals, and for the sake of brevity assigned hours
of work, h, is excluded and the guaranteed income level xg is supposed to be included in
the set .
The (cumulative) distribution function of G is denoted by

G =  (G)

(2:3)

where

"G1

(2:4)

An indirect method of observation of G is discussed in section 4.1.3.
Let us suppose that the analytical form of the function  is common to all the sectors
under consideration. Hence, by letting the number of potential suppliers be N , the numberof suppliers with grade G and over, NG , is given by

NG = N  G = N   (G)(2:5)

The number of suppliers with grade G and over going to the ith sections, Lis, is written as

Lis = N   (G)  (wi ; )
where wi stands for the wage rate o ered by the ith sector.

3.1. Behavior of the Leader(2:6)

Imagine a sector (or rm) which o ers the most favorable wage in comparison to othersectors in order to attract a number of potential suppliers. This sector can recruit workers
of higher grades comparing to other sectors which o er less favorable working conditions.
We shall call this sector a leader sector ( rm) or a leader for short. Residual sectors
are followers. Among those residual sectors, we can distinguish leaders and followers inaccordance with the wage di erentials each sector is willing to pay. That is, if we havethree sectors with wage rates w1 ; w2 and w3 where w1 > w2 > w3 , sector 2 plays the roleof the follower of sector 1, while sector 2 plays the role of leader of sector 3. Follower sector
2, against leader sector 1, recruits workers with relatively higher grades amongst residual
applicants which the leader has left for followers to employ because those applicants are
not fully suitable for employment from the leader's point of view. Sector 2 as a leader
against sector 3 will again leave undesirable labor suppliers. This pattern can be viewed
as continuing inde nitely, ith sector 3 acting as a leader to sector 4, and so on.
Let us imagine a labor market which consists of two sectors to simply present the basic
characteristics of the model, where one of the sectors is able to attain a given level ofproduction Qi (i = 1; 2) by varying Gi and Li in the production function (2:1) .
The distribution function (2:5), N   (G), is depicted in the fourth quadrant in Figure
1. The curve GN is the cumulative distribution curve from the top labor grade G(= 1).
Suppose rm ` (we denote leader by ` ) wishes to recruit workers with grades higher thanGmin
by curve S` S 0̀ in the` . In this case, the labor supply curve for rm ` can be depictedmin
1st quadrant. This curve stands for equation (2:6) where G is inserted for G. Now,
Gmax (= 1) and Gmin being given for the rm `, the demand curve for labor is derived from

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The Keio Economic Observatory Review No. 8, March 1996

the production function (2:1) and (2:10 ) by applying the condition of cost minimization.
This is depicted by curve D` in the rst quadrant. The intersection of the supply curves` s0̀ and D` gives the wage rate w` and the demand for labor L` by rm ` necessary toattain the given level of production Q` .
If rm ` were content to recruit workers with lesser grades, e.g. [Gmin ] < Gmin
` ,the curve s` s0̀ would be less steep and stretched to the right. Hence, the required gradeof workers would be less and the number of workers employd would increase. At any
rate, given the production function (2:1), the grade distribution function (2:3) , and the
supply probability function (2:2), the number of workers and the required grade to attainproduction level Q` are detemined by the procedure of cost minimization.
From (2:5) and (2:2) the number of potential applicants with grade Gmin
` and over,minLG` , is given by
min
min
Lmin
G` = N`  G`   = N   (G` )  (w` ; )

(2:60 )

which is a function of w` . Eq.(2:60) is depicted by the curve s` s0̀ in Figure 1.

Figure 1
We have, for the leader, Gmax = 1 in (2:10 ). Hence, (2:10 ) is written as G` =0G` (Gmin` ; 1). Substituting this function and (2:6 ) into (2:1) gives the leader's production function.
minminQ` = F [N   (Gmin` )  (w` ; ); G` (G` ; 1); A` ];(2:100 )`minC` = C0` + w`  LminG` = C0 + w` N   (G` )  (w` ; )

(2:7)

where the subscript i in (2:1) has been replaced by ` to denote that eq. (2:100 ) refers
to the leader. Defnition of cost is given by

where C0` stands for capital cost which is regareded as given.

An Equilibrium Model of Continually Heterogeneous Labor Market

700
We can obtain w` and Gmin
` by minimizing C` in (2:7) under the constraint (2:1 ), Q`
being given:
Lettingmin` = C` + k fQ` , F [N   (Gmin` )  (w` ; ); G` (G` ; 1); A` ]g(2:8)

where k is Lagrangian multiplyer, and C` is given by (2‐7), we have
` = ` = 0:
@w`
@Gmin
`
Solving (2:100 ) and (2:9) simultaneously for Gmin
` and w` , we obtain,G` = G` ( 0 ; ; A` ; Q`)(2:9)(2:10)andw` = w` ( 0 ; ; A` ; Q`)(2:11)
where  0 is a set of parameters in the grade distribution function (2:3). Equations (2:10)
and (2:11) give optimal values for Gmin
` and w` both minimizing cost G` for the given
production level Q`. The solution for employment L` can be calculated by substituting(2:10) and (2:11) into (2:6) for G and w respectively. We shall call the number of workers
thus obtained, and G` and w` given by (2:10) and (2:11) as the \leader solution".

3.2. Follower's Behavior

The highest grade of workers attainable to the follower is Gmin
` which is the lowest grade
for the leader. Let the lowest grade of people in the group of potential applicants for theminminfollower be Gmin
f . The number of people with grades between Gf and G` , which wemindenote by NGf , is given by
min
NGmin
= N   (Gminf ) , N   (G` );f(2:12)

which is shown by the length of NGmin NGminin Figure..1. Hence, the number of suppliersf`minto the follower LGf is written as
min
min
min
Lmin
Gf = NGf   = N [ (Gf ) ,  (G` )]  (wf ; ):(2:13)

Substituting (2:13) into (2:1), we have the ptoduction function of the follower;min 
Qf = F [N  ( (Gmin(2:14)f ) ,  (G` ))  (wf ; ); Gf (Gf ; G` ); Af ]where Af is the set of parameters of the follower's production function, and G` is givenby (2:10). The de nition of follower's cost Cf is given by

Cf = C0f + wf  Lf = C0f + wf  N [ (Gminf ) ,  (G` )]  (wf ; ):(2:15)

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The Keio Economic Observatory Review No. 8, March 1996

where C0f is capital ( xed) cost and (2:13) is substtitued for Lf .
Let us minimize Cf in (2:15) under the constraint of (2:14) where the level of Qf isgiven.
f = Cf + j [Qf , F fg]:(2:16)

where j is the Lagrangean multiplier.
The minimization condition is as follows.
ff= @w = 0:
@Gmin
f
f

(2:17)

Solving (2:17) for Gminf and wf ,we haveGf = Gf ( 0 ; 1 ; Af ; Qf ; G` )(2:18)

wf = wf ( 0 ; ; Af ; Qf ; G` ):(2:19)
where G` is already given by the leader's solution (2:10). Lf can be obtained from (2:13)
by inserting (2:18) and (2:19). We shall call this employment level and (2:18) and (2:19)
the \follower's solution".

3.3. Succession Equilibrium

When we have three or more rms (sectors), we can successively apply the above leaderfollower relationship. We shall call the state of market shared by rms (sectors) playing
the role of leaders and followers successively as succession equilibrium. Let us suppose two
rms are in a state of succession equilibrium. Now, suppose relative or absoluate changes
in the production level of the leader cause a \leader's solution" with a wage rate w lower
than the follower's. Then, of course, the initial state of the market cannot be sustained.
A new leader-follower relation has to be established. The former follower succeeds to theposition of leader and the former leader now becomes a follower. However, alternative
cases could be considered. If the initial leadaer expects that he will not be able to holdthe position of leader without augmenting the marginal productivity of his workers and if
he nds losing his leader position is not pro table, he might invest in capital to augment
his workers' productivity.

3.4. Conditions for Succession-Equilibrium in Labor Market

Let us concentrate on the leader unit A and the successive follower unit B . By de nitionmin
we have wA > wB and Gmin
A > GB . We shall examine the condition that guarantees a stable structure of wage di erentials. We use the term \succession eqilibrium" tocharacterize a labor market with stable wage di erentials.
The necessary condition for succession equilibrium is thatw` > w;(2:20)

An Equilibrium Model of Continually Heterogeneous Labor Market

9

where w` and w stand for leader A0 s and follower B 's wage rate respectively. Necessary
and sucient conditions read as follows. (Precise discussion is given in Obi[7])(a) Letting A and B be the leader and follower respectively, when (2:20) holds,the leaderfollower relationship is stable, if the following condition is satis ed.
(a.1) Let B be a leader instead of A, A being a follower, and compute the leader solution
for B . Let the solution for the wage rate be w` . Compute the follower solution for
A. Let the solution be w. Then suppose (2:20),

w` > w
does not hold. This is the necessary and sucient condition for stable successionequilibrium.
(a.2) When the leader-follower relationship between A and B is inverted in the computation procedure (a.1), if (2:20) holds in this case as well, then the leader-follower
relationship cannot be stable. Now suppose that the analytical forms of the production function and the grade-distribution function are true and the estimated parameters are correct. Further suppose that numerical values of the set of parameters
and the production levels of production unit A and B are such that they generate
the unstable case mentioned above. On the other hand suppose, in the real labormarket, a stable wage di erential between unit A and B is observed. Then it must
be considered that the leader-follower relationship between A and B is sustainedby fctors other than those already considered ; e.g. historical or random factors.
Hence, the observed leader-follower position of A and B will be inverted wheneverthose factors change.
(b) Letting A and B be the leader and follower respectively, when (2:20) does not hold,the inverse leader-follower relationship is stable so long as the following condition issatis ed.
(b.1) Let B be a leader instead of A and compute the leader solution for B . Let thesolution for the wage rate be denoted by w` . Compute the follower solutionfor A.
Let the solution be denoted by w. Next suppose (2:20)w` > w, does not hold. In
this case, it can be said that the set of estimated parameters of the model is notcorrect or the model itself is at fault.
(b.2) When the leader-follower relationship between A and B is inverted in the computation procedure, if (2:20) holds, then the leader-follower relationship is stable, B
and A being the leader and follower respectively. However, this case, (b.2), is substantially equivalent to case (a.1), and the indepent cases are (a.1), (a.2) and (b.1).
Hence, (a.1) is the necessary and sucient condition for the stability of successive
equilibrium.

3.5. Simple Model

We shall specify the analytical form of production functions (2:1) and (2:10 ) as

Qi = bi Li i (Gi ) i ;i > 0;

i>0(i = 1; 2; . . .);

(3:1)

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The Keio Economic Observatory Review No. 8, March 1996

Gi = (Gi+1  Gi ) 21 ; Gi+1 < Gi ;(3:10 )
where Gi and Gi+1 respectively stand for the highest and the lowest values of G among
the workers the ith rm employs. Let i = `; f; where ` and f respectively stand forleader and follower.
Simplifying the distribution function  (G) without impairing the basic characteristicsof the model,we use (G) = 0 + 1 G;(3:2)

where 0 and 1 are parameters.
We specify the supply-probability equation (2:2) as a linear function of w

 = 0 + 1 w

(3:3)

where as shown later0 < 0;1 > 0

01

and

(3:4)

3.6. Numerical Experiments
3.6.1. Simulation System

We shall present a few numerical experiments to examine the workability of the successive
equilibrium model. Let the number of production units (or rms) be two, unit 1 and 2.
Numerical values of the parameters are assinged as follows.
1= 2 = 1;1

= 0:4;

2= 0:9; b1 = b2 = 1;

0 = 1; 1 = ,1; 0 = ,0:5; 1 = 0:01; N = 10; 000
Sux 1 and 2 stand for unit 1 and 2 respectively. The elasticity of production with respect
to grades for unit 2, 2 , is larger than that for unit 1.
The levels of production of unit 1 and 2 are experimentally given as shown in the rst
and second columns of Table 1a through 1g. These are exogenous variables in the simple
model under consideration.
(a) In Table 1a, Q2 is increased from 150 to 300, Q1 being constant. In this case thecomputation process revealed the succession-equilibrium was stable and a stableleadaer-follower relationship holds as is shown in the table ; i.e. unit 2 and 1 aretheleader and follower respectively. The wage di erential w2 =w1 increases.
(b) In the second case, Q1 and Q2 were increased with a common rate of growth startingfrom Q1 = Q2 = 160 , as shown in Table 1b. The leader-follower relationship does
not alter. The wage di erential decreases, unlike that of case (1). It can be seen thatthe increase in the wage di erential in case (1) stems from the growth and stagnationof production of unit 2 and 1 respectively.

An Equilibrium Model of Continually Heterogeneous Labor Market

11(c) In Table 1a, Q2 , the production of unit 2 which has a larger value for compared tounit 1, was increased. In contrast to this, production Q1 of unit 1 is increased, Q2
being held constant at 150, in Table 1c. For the values of Q1 = 160;    ; 190, unit 2
occupies the position of leader, while case (b:1) appears when Q1 exceeds 200; thatis, we do not have a consistent solution for Q1  200 and Q2 = 150.
(d) Next, in order to clarify the response of production unit 2 against production unit
1 with Q1 = 200, we tentatively assigned Q2 values in the range 38  Q2  750.
(See Table 1d). It was found that the leader position switches if Q2 <47. The alteredleader follower relationship is stable for Q1 = 200 and 38 < Q2 < 47.
(e) A test analogous to (d) is shown in Table 3e. Here, Q2 is held constant at 150, while
Q1 is varied between 63  Q1  1250. The leader role switches when Q1 reaches
1250.
(f) Analogous to (e), we take Q1 = 250 and 38 < Q2 < 750. For Q2 > 250, unit 2 and 1
play the leader and follower respectively. If Q2  54, the relationship alters. Between
Q2 = 54 and Q2 = 250, we do not have stable succession equilibrium (consistentsolutions).
(g) Analogous to (f), we vary Q2 between 38 and 750, Q1 being 300. For Q2 > 250,
unit 2 and 1 are the leader and follower respectively. However, for Q2  63 thisrelationship alters.

3.6.2. The Ranges of Production which Guarantee Stable Succession Equilibria

The ranges for production of sectors 1 and 2, which guarantee stable succession equilibria,
are depicted in Figure 2. The thick lines and dotted lines or segments respectively standfor the ranges where succession equilibria are guaranteed and not guaranteed. Thus, it
can be seen that the hatched area represents (a part of) the unstable regions.

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The Keio Economic Observatory Review No. 8, March 1996

Thick lines and segments stand for the region where stable succession equilbrium holds. Attached numbers standed for leader follower relations, e.g.2{1states that unit 2 and 1, respectively, play the rule leader and follower.

Figure 2

An Equilibrium Model of Continually Heterogeneous Labor Market
Table 1: aaa(no solution)(no solution)(no solution)
Table 1aTable 1b
Table 1c
Table 1d
Table 1e
Table 1f
Table 1g

D`
Df
D0̀
Df0

0 = 1 1 = ,1
1

= 0:4

2

1

= 1:0

2

= 0:8

= 0:9 0 = ,0:5 1 = 0:01 N = 10; 000

13

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The Keio Economic Observatory Review No. 8, March 1996

4. An Alternative Simple Model
4.1. Basic Equations
4.1.1. Production function

We shall specify the analytical form of production functions (2:1) as 4

Qi = bi Li :(3:1a)

Let the cost function of ith rm (or sector) be 5

Ci = i (Gi ; Li ) + wi Li(3:10 a)where

Gi = (Gi+1  Gi ) 21 ; Gi+1 < Gi
where Gi and Gi+1 respectively stand for the highest and the lowest values of G among
the workers the ith rm employs.

4.1.2. The distribution function of grade indicator

Simplifying the distribution function  (G) without impairing the basic characteristics ofthe model,we use (G) = 0 + 1 G;(3:2)

where 0 and 1 are parameters.  (G) is the ratio of the number of potential applicantswith grade G and over to the total number of potential applicants (the number of thepeople of working age). The magnitueds of G0 s the potential supplier with the highest
and lowest grade among all potential suppliers are respectively de ned to equal unity and", " being some small positive number. Hence, we haveand

 (G) = 1

if G = "; (G) = N1

if G = 1;(3:3a)

4 With regard to this speci cation we can give an interpretation that in (2:1) we assume the analytical@F = 0. Another interpretaion would be that we assume Leontief type[3] (factor limitational)
form of F as @Giproduction function.
5
i stands for an additional cost which is a ected by the value of the grade Gi . This is an alternativeand easier way of analysing the e ect of G or the production behavior of the ith rm (so to speak G.
Becker Version[1]) compared to the way in which G is included in the production function of ith rm, as
is done in the previous section. By this alternative Sppci cation of production function (3:1a) means thatthe demand curve for labor Di Di0 (i = `; f ) in Figure 1 are strait lines perpendicular to the abscissas.

An Equilibrium Model of Continually Heterogeneous Labor Market

15

where N stands for the number of the total potential suppliers. By applying (3:3a) to(3:2) we have
(1 , 1 )1 = , (1 , N")(3:4a)

1 + "(1 , N1 )
(1 , ")
Hence, the distribution function (3:2) is written as

0 =

(3:5)

"(1 , 1 ) (1 , 1 ) (G) = 1 + (1 , "N) , (1 , N")  G(3:20 )

By adopting the magnitude of " as " = N1 (3:20 ) is written as

 (G) = 1 + N1 , G(3:200 ) (G) = 1 , G:

(3:2000 )If N is suciently a large number we have

The number of persons with G higher than Gj , N (G  Gj ), is given by

N (G  Gj ) = N   (Gj )Hence, applying (3:20), we obtain


N (G  Gj ) = N 1 + (1 , N1 )( 1 ," " ) , (1 , N1 )( 1 ," " )  Gj

(3:6)


(3:7)

Making use of the relation " = 1=N , (3:7) is written as

N (G  Gj ) = N + 1 , NGj = N (1 , Gj ) + 1
From (3:200 ) we have, as a good approximation for (3:70 ),N (G  Gj ) = N (1 , Gj )4.1.3. Indirect observation of Gi(3:70 )
(3:700 )In the simple model, variables Qti; Lti and wit where i and t stands for the production unit
and time respectively, are directly observable from the data. However, we cannot observe
the magnitude of Gi directly and we must therefore indirectly measure it making use of
the model itself. We shall discuss the procedure to measure Gi below.

16

The Keio Economic Observatory Review No. 8, March 1996

Suppose we have data on Qti ; Lti and wit (i = `; f ). With respect to the parametersof the model we have 0 = 1; 1 = ,1. Further suppose we have already estimated theparameters, 0 and 1 , in the supply probability function. Labor supply curves for theleader and the follower respectivity pass through points A` and Af in Figure.1. The
values of the coordinates of those points A` and Af are known from observed data on thewages and employment of the leader and the follower.(Production units ( rms, sectors)
are ordered by the observed wage rates. Hence, successively, leader-follower relationships
can be identi ed by this ordering.) Therefore, we can obtain NGminand NGminby solving`f
the simultaneous equations,NGmin(0 + 1 w` ) = L``
(NGmin, NGmin)(0 + 1 wf ) = Lf``
where actual wages and employment w` ; wf ; L` and Lf are directly obtained from theobserved data and 0 and 1 , are supposed to be already estimated, as mentioned above.
Applying NGminand NGmin
thus obtained to the left hand side of the grade distribution`ffunction ( (Gi ) = 1 , Gi ) in [3.2"'], we can calculate the numerical values for Gmin
` and
min and Gmin .
Gmin
.
These
are
the
\indirectly
observed"
valuesforGf`f

4.1.4. Equation of Supply-probability
We specify the supply-probability equation (2:2) as a linear function of w

 = 0 + 1 w

(3:8)

where as shown later0 < 0;1 > 0

and

01

(3:9)

are postulated. 6 In order to make our model simple without impairing its basic characteristics, we use a linear function as a supply probability function. This simpli cationmeans that we implicitly employ a rectangular distribution for the minimum supply priceof labor, w. In equation (3:8), we have w = 0 =1 when  = 0; hence,

=0

if w  , 0

=1

if w  , 1 , 01

and for the range of w,,  < w < 1 ,  ;0

1(3:10)

1

0

An Equilibrium Model of Continually Heterogeneous Labor Market

17

Figure 3(3:8) holds. The supply probability curve with the characteristics (3:8) and (3:10) is
depicted in Figure 3.
The numerical value of ,0 =1 , stands for the minimal value of the range of distributed
values of w. This minimal values of w must be positive, and,  > 0(3:11)

0 < 0(3:12)

0
1

must hold. On account of the nature of the distribution function, 1 must be positive.
Hence, from (3:11) we have

4.2. Behavior of the Leader in the alternative Simple Model
4.2.1. Basic Equations
Let the leader's production function be (3:1a), and we haveQ` = b` L`(L:1)
where the sux i in (3:1a) is replaced by ` to show that the equation is that of the leader.
THe cost function in (4:1:1) can be written asC` = ` (G` ; L`) + w` L`(L:2)

where 7
6 If the true shape of  function is linear as shown in (3:8)  < 0 must be held in the estimated relation0
as well. However, if the true function is non-linear, linear supply probability function, 0 + 1 w, originallyis an aproximation. Hence, a constant term in an estimated linear supply probability function could be
negative.
7 Here after, for the sake of brevity, we assume G ' G where G stands for Gmin . By this approxi````
mation the basic characteristtics of the model will not be impaired.

18

The Keio Economic Observatory Review No. 8, March 1996
(L:20 )min 1
G` = (Gmax`  G` ) 2

The number of suppliers to the leader is given by (letting N be population of working age)Ls` = N [ (G` ) ,  (Gmax` )]  (w` )orLs` = N [ (G` ) ,  (Gmax(L:3)` )]  (0 + 1 w` + 2 A)
where A stands for the e ect of changing xg in sec. 2.3. (L:3) corresponds to (2:6). Thisequation states that e ective suppliers to the leader must be the ones with at least gradeG` .
When the value of Q` is given (L:3) can be written as ( taking into account (L:1),)
1maxb` Q` = N [ (G` ) ,  (G` )]  (w` )We minimize C` in (L:2) under the constraint (L:30 ). That is, de ning F as


F = ` (G` ; L` ) + w` L` +  b1 Q` , N [ (G` ) ,  (Gmax` )]  (w` )`(L:30 )

(L:4)

where  is a Lagrangian multiplier,@F @F@G` = @w` = 0(L:5)@F = 0, we have
has to hold if C` is minimized. Hence, from @G`@ ` +  ,N d (G` )  (w ) = 0
`
@G`
dG`
@F , we have
From @w`dL` , N [ (G` ) ,  (Gmax` )]  dw = 0
`

(L:6)

(L:7)

d =  , Hence, from (L:6) and (L:7) we get theTaking into account (3:8) we have dw
1`
minimization condition
@ `
, @G

` =

(w` )

1Q

b` `
;
(Gmax
` , G` )1(L:8)

An Equilibrium Model of Continually Heterogeneous Labor Market

190
where Gmax
` is 0 given. We can solve (L:8) and (L:3 ) simultaneously for G` and w` . From(L:8) and (L:3 )@ ` = N  [(w )]
, @G
`


(L:9)

2

`

1

@ ` to be a linear function, that is,
To begin with the simplest case we assume @G`@ `@G` = 0 + 2 L`
where 0 ; 1 and 2 are the paramenters, and,

(L:10)@ `@G` < 0

(L:11)

must hold.
Inserting (L:10) into (L:9) and making use of (L:3), we have,[ 1b Q` ]
 +  L` =N (Gmax` , G` ) 
2

0

2

2

1

(L:12)

From this we have

v
u
u
u
max
G =G t``,( 1b Q`)

2N1 (0 + 2 L`)(L:13)Taking into account G` < Gmax
` we adoptvuuu
G = Gmax , t``,( 1b Q`)

2N1 (0 + 2 L`)(L:130)where0 + 2 L` < 0:
(L:130) is the \leader solution" of the grade variable.
max
Taking into account  (G` ) ,  (Gmax` ) = G` , G` , (L:3) can be written as,

L` = N (Gmax` , G` )(0 + 1 w` + 2 A)
where A stands for the e ect of changing xg in sec. 2.3.
From (L:300)(L:300 )20

The Keio Economic Observatory Review No. 8, March 1996

w` =

L`
, 0 , 2 A
1 N (Gmax
,
G
)
1
1
`
`

(L:14)

Substituting G` in (L:14) by (L:130 ) we obtainw` = 1 
1

s

L`

,L`

2

N  N ( +  L )1 0
2 `,  ,  A
0

2

1

1

(L:15)orw = 1 
`

1

1

v
u
u
u
N u
t

b` Q`

1
,( Q )2b` `N1 (0 + 2 b1 Q`),  ,  A
0

2

1

1

(L:150)`This is the \leader solution" of the wage variable .

4.3. Behavior of the Follower in the Simple Model
Let the production function and the cost function of the follower be respectively,

Qf = b f L f(F:1)

Cf = f (Gf ; Lf ) + wf Lf

(F:2)

and

where 8min 1
Gf = (Gmaxf  Gf ) 2(F:20 )The number of suppliers to the follower is given by

Lsf = N [ (Gf ) ,  (Gmaxf )]  (wf )(F:3)

Gf stands for the minimum value of the grade indicator of labor the follower can accept.
Gmax
f is the maximum value of the grade indicator which the follower can attain. When
Qf is given (F:3) can be written as
8

The same assumption as that made in footnote 6 is made, that is Gf ' Gf , where Gf stands for

Gmin
f .

An Equilibrium Model of Continually Heterogeneous Labor Market

21

1 Q = N [ (G ) ,  (Gmax )]  (w )

bf f

f

(F:30 )

f

f

We minimize Cf in (F:2) with constraint (F:30 ). The values of wf and Gf minimizing Cf
can be given by,@F = @F = 0;
@Gf @wf

(F:4)

where



F = f (Gf ; Lf ) + wf Lf +  b1 Qf , N [ (Gf ) ,  (Gmaxf )]  (wf )
f



(F:5)

From (F:4) we get

@ f
, @G

1

bf Qf
f =
(wf ) (Gmaxf , Gf )1(F:6)

Making use of (F:6) and (F:30 ) we have

@ f = N  [(w )]
, @G
f

f

2

1

(F:7)

@ f to be linear, that is
We assume @Gf@ f =" +" L
@Gf 0 2 f
Using (F:3)
(wf ) =

(F:8)

Lf

N [ (Gf ) ,  (Gmax
f )]

(F:9)

where

 (Gf ) = 1 , Gf

(F:10)

 (Gmaxf ) = 1 , G`

(F:11)

andbecause Gmax
f = G` .
Hence

22

The Keio Economic Observatory Review No. 8, March 1996
( (Gf ) ,  (Gmaxf )) = (1 , Gf ) , (1 , G` ) = G` , Gf

(F:12)Taking into account (F:8), (F:9) and (F:12), (F:7) is rewritten as"#

L2(F:13)"0 + "2 Lf = , N 2 f
1
N (G` , Gf )2
where G` is given by the solution for the leader. Again we suppose "1 = 0 for brevity.
Hence (F:14) can be written as,( b1 Qf )f" + " Lf =N (G` , Gf ) 
2

0

2

2

1

(F:14)

From this we have
v
u
u
u
u
Gf = G`  ut,( b1 Qf )2fN1 ("0 + "2 b1 Qf )(F:15)

f

Because Gf < G` we adopt
v
u
u
u
u
Gf = G` , ut,( b1 Qf )f2N1 ("0 + "2 b1 Qf )(F:16):fwhere"0 + "2 Lf < 0:
(F:16) corresponds to (L:130) in the leader's case, and gives the \follower's solution" of
the grade variable. From (F:3) and (F:12) , we have

Lf = N (Gmaxf , Gf )(0 + 1 wf + 2 A)
By solving (F:17) we get
wf =Lf0 , 2 A,
max
1 N (Gf , Gf ) 1 1

Substition of Gf in (F:18) by (F:15) gives(F:17)(F:18)

An Equilibrium Model of Continually Heterogeneous Labor Market

23Lf0 , 2 A
,
1 1
1
,L2f
N
N1 ("0 + "2 Lf )wf = 1 s(F:19)

or1Q

bv, 0 , 2 A
u
1
1
1
1
u
,( b Q )2uu
N tN1 ("0 + "2 1 Q )wf = 1 f

f

f(F:190 )f

bff

This is the \follower solution" of the wage variable.

5. Application of the alternative Simple Model to Japanese data
The result of application of the model in the previous section to Japanese data is shownin this section.
It can be seen the wage of nancial sector is always at the top of the wage di erential
among the sectors(industries) during the observational period, 1970 through 1991. Hencenancial sector can be identi ed as \leader sector". The other sectors are aggregated andcan be identi ed as \follower sector" as is shown by Figure 4. Hence, application of themodel of leader behavior and follower behavior is straightforward . 9
The observed yearly values of wi ; Li and Qi (i = `; f ) were obtained from SNA data
arranged by Economic Planning Agency. Observational period is 1970 through 1991.
These are shown in Table 3.
The estimated values of the parameters 0 ; 1 are shown in Table.2. 10
Indirectly observed values obtained by using the estimation method in 4.1.3 function
are shown in Table 3. Estimated values of 0 and 1 are also shown in Table.2. The
estimated and observed values for wi ; Gi (i = `; f ) are shown in Table.3. The estimatedand observed values for wi and Gi (i = `; f ) are shown in Figure 4 and Figure 5.
Table 2: Estimated Parameters of Structural Equations
0
1
2
0
2
"0
"2

1.515465
0.5524484E-06-0.1202143E-01-0.7462759E+10-2385271.
-0.9816546E+10564116.1

9 In case where the order of wage di erential(i.e, top. second, third etc.) change during observational
period, the application of the models of leader and follower was discussed in (3:4).
10 In this section  is speci ed as  =  +  w +  A where A a ects the changes in x in (2:3).
g1
20
However, values of A to be observed were substitued by time trend, that is,  = 0 + 1 w + at
where
a stands for the coecient of the time trend term. 0 could be negative as shown in footnote footnote 6
0

0

0

24

The Keio Economic Observatory Review No. 8, March 1996

6. Conclusion
From these results obtained in Figure 4, we can conclude that the \alternative simple modelof the continually heterogeneous labor market" seems to be applicable to the Japanesedata.
This model would also be applicable to the growth mechanism of developing economies
as was shown elsewhere[8].

An Equilibrium Model of Continually Heterogeneous Labor Market

Table 3: Observations and Simulated Values of Endogenous Variables
year
W`
Wf
G`
Gf
70 obs. 190723.9 164117.4 0.9686200 0.2366719
70 sml. 225210.9 207227.4 0.9693688 0.2595301
71 obs. 212242.0 181815.5 0.9670115 0.2333205
71 sml. 231979.4 206573.3 0.9674667 0.2467064
72 obs. 236075.2 202183.1 0.9672889 0.2357605
72 sml. 240207.1 211359.1 0.9673843 0.2406927
73 obs. 242492.2 216001.8 0.9666206 0.2261282
73 sml. 242717.1 205364.5 0.9666260 0.2203420
74 obs. 254782.4 229392.6 0.9660790 0.2373796
74 sml. 253821.1 216853.4 0.9660554 0.2305865
75 obs. 283595.2 247586.6 0.9658823 0.2491550
75 sml. 265557.8 228423.5 0.9654353 0.2384550
76 obs. 298899.2 259103.2 0.9651493 0.2452453
76 sml. 278351.2 237753.7 0.9646257 0.2331408
77 obs. 312024.5 266191.5 0.9636751 0.2351443
77 sml. 293480.8 248250.0 0.9631802 0.2247163
78 obs. 319961.3 271786.4 0.9630249 0.2269524
78 sml. 307068.5 258810.4 0.9626725 0.2192794
79 obs. 334005.4 280536.4 0.9620401 0.2178648
79 sml. 319750.1 266674.2 0.9616375 0.2094702
80 obs. 345523.0 288243.2 0.9610689 0.2122245
80 sml. 330872.6 273689.6 0.9606411 0.2032560
81 obs. 354781.7 296014.9 0.9595347 0.2042027
81 sml. 345882.0 284714.5 0.9592632 0.1971854
82 obs. 364612.8 298871.4 0.9587308 0.1937988
82 sml. 359798.1 295412.0 0.9585801 0.1915384
83 obs. 376508.9 305068.8 0.9576265 0.1786572
83 sml. 372576.0 302153.6 0.9574993 0.1766975
84 obs. 387559.1 313688.6 0.9573984 0.1760640
84 sml. 384304.5 311829.1 0.9572917 0.1747734
85 obs. 387313.4 316914.0 0.9570192 0.1671898
85 sml. 396973.9 322461.5 0.9573409 0.1711148
86 obs. 394973.4 319244.1 0.9556640 0.1560513
86 sml. 410530.0 332169.2 0.9562019 0.1651734
87 obs. 408629.6 330245.8 0.9545292 0.1518785
87 sml. 423995.9 342070.6 0.9550777 0.1603886
88 obs. 440171.0 334388.6 0.9552247 0.1347124
88 sml. 436453.7 349725.5 0.9550931 0.1452633
89 obs. 453168.6 344986.5 0.9541766 0.1166702
89 sml. 450787.6 357213.9 0.9540898 0.1253832
90 obs. 452937.7 351832.4 0.9515034 0.0936549
90 sml. 467234.0 365684.3 0.9520572 0.1045376
91 obs. 448169.7 358395.9 0.9505394 0.0684420
91 sml. 481306.3 373074.6 0.9518565 0.0811564
Unit of W` ,Wf : yen per month (constant price of 1985)

25

26

The Keio Economic Observatory Review No. 8, March 1996
Table 4: Gross Domestic Product by kind of Economic Activity

year
XTI
XAF
XFI
XRE PXTI85 PXAF85 PXFI85 PXRE85
7070387.9 4488.0 3120.5 5899.0
46.0
46.8
64.4
39.4
71
77039.0 4273.8 3766.4 6972.9
47.7
46.8
62.3
43.0
72
88648.7 5049.9 4550.5 8135.8
49.7
48.7
54.7
46.5
73
108763.0 6675.1 5560.9 9853.5
56.5
61.2
66.0
50.0
74
127727.1 7505.9 7001.1 10944.5
67.5
69.5
97.3
53.1
75
138707.7 8141.1 7795.8 12138.0
71.6
75.6
92.2
57.8
76
156191.4 8870.0 8348.7 14208.2
77.5
86.9
95.1
64.0
77
172864.1 9401.6 9050.5 16663.5
82.5
94.5
89.7
70.2
78
190517.4 9440.6 10294.0 19036.6
86.6
94.7
88.2
76.3
79
207251.8 9623.0 11413.0 20965.4
88.0
95.2
95.1
79.6
80
224266.2 8847.2 12440.4 22654.3
91.3
96.9
104.4
82.2
81
239883.1 9075.4 12307.3 24402.3
94.2
98.2
101.0
86.2
82
252930.0 9238.4 13990.5 25675.4
96.4
95.3
113.6
89.8
83
264260.5 9516.4 15370.2 27409.2
97.4
96.5
111.0
93.6
84
281948.5 9956.9 15843.5 29802.4
98.9
97.2
102.0
96.7
85
301175.2 10213.7 16971.9 32358.5
100.0
100.0
100.0
100.0
86
313154.4 9974.9 17714.3 34729.0
102.4
99.7
94.2
103.0
87
328761.1 9767.5 19228.1 37734.4
101.7
94.6
91.8
106.7
88
351749.3 9753.8 21015.0 40653.1
101.9
97.6
91.5
109.3
89
379150.4 10131.8 23436.1 43569.0
103.4
98.1
92.0
112.1
90
407334.4 10552.6 23021.5 46507.8
105.0
101.7
92.4
116.2
91
431061.2 10442.3 22896.0 49098.7
107.1
109.8
92.6
120.6
Notes
Unit: Billion Yen
Periodicity: Calendar Yearly Data
Source: Annual Report on National Accounts
XTI:Gross Domestic Product,Producers'Values-Industries
XAF:Gross Domestic Product by Industry-Agriculture,Forestry and Fishery
XFI:Gross Domestic Product by Industry-Finance and Insurance
XRE:Gross Domestic Product by Industry-Real Estate
PXTI85: Gross Domestic Product,Producers'Values-Industries (De ator)
PXAF85: Gross Domestic Product by Industry-Agriculture,Forestry and Fishery (De ator)
PXFI85: Gross Domestic Product by Industry-Finance and Insurance (De ator)
PXRE85: Gross Domestic Product by Industry-Real Estate (De ator)

An Equilibrium Model of Continually Heterogeneous Labor Market

Table 5: Number of Employed Persons by kind of Economic Activity/ Population 15 Years Old and Over-Total / Cash Earnings
year EWTI EWAF EWFI EWREPT
WSMR1 WSMFI
WSMRE
705052.9 1073.6 131.6
35.0 7886
75494
84958
98167
71
5081.4995.8 139.9
39.6 7979
86726
98056
112486725097.7947.8 142.8
39.0 8070
100485 114372
128158735205.6903.3 145.8
43.2 8239
122041 134753
14461974
5166.2880.9 149.2
44.8 8341
154840 172061
171702755139.8861.8 153.4
45.9 8443
177272 206658
191010765177.3844.3 158.3
47.4 8540
200805 238408
209067775236.7836.6 164.6
51.2 8631
219608 266636
227793785279.5826.3 166.4
54.0 8726
235367 287393
245327795331.3797.6 171.0
57.6 8824
246872 302055
269788805358.6756.6 176.9
60.1 8932
263166 323773
291001
815393.2732.9 182.4
64.9 9017
278846 346036
300952825435.8718.9 186.7
66.7 9117
288112 366614
30914483
5521.1
695.4 192.5
70.0 9232
297137 384973
316523845538.7668.9 195.0
71.1 9347
310238 402622
330292855578.3659.8 194.7
73.1 9465
316914 407887
332516865626.4643.3 201.6
75.9 9587
326906 424815
350368875676.3633.4 208.7
78.6 9720
335860 435997
361355885774.5622.3 209.7
79.8 9849
340742 474191
381113895899.0613.4 217.0
81.5 9974
356716 485684
423026906028.9605.7 228.9
85.6 10090
369424 488887
440013916157.8587.1 230.3
87.8 10199
383842 490963
451207
Notes on EWTI, EWAF, EWFI, EWRE
Source: Annual Report on National Accounts Unit: 10000 Persons
EWTI:Employed by kind of Economic Activity-Industries
EWAF:Employed by kind of Economic Activity-Agricultrue, Forestry and Fishery
EWFI:Employed by kind of Economic Activity-Finance and Insurance
EWRE:Employed by kind of Economic Activity-Real Estate
Notes on PT
Source: Monthly Reprot on the Labour Force Survey Unit: 10000 Persons
PT: Population 15 Years Old and Over-Total
Notes on WSMR1, WSMFI, WSMRE
Source: Monthly Labour Survey Unit: yen Periodicity: Monthly Data
WSMR1: Ave.Monthly Cash Earnings of Regular Workers(incl.Bonus) -All Industries
WSMFI: Ave.Monthly Cash Earnings of Regular Workers(incl.Bonus) -Finance and Insurance
WSMRE: Ave.Monthly Cash Earnings of Regular Workers (incl.Bonus)-Real EstateW` = (W SMF I  EW F I + W SMRE  EW RE )=(EW F I + EW RE )Wf = [W SMR1  EW T I , W`  (EW F I + EW RE )]=(EW T I , EW F I , EW RE )

27

28

The Keio Economic Observatory Review No. 8, March 1996

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