The Keio Economic Observatory Review No. 8, March 1996

An Equilibrium Model of Continually Heterogeneous Labor Market^{∗}

Keiichiro Obi Dokkyo University and Keio Economic Observatory Keio University

1. Introduction

In this paper a model of the labor market, where wage differentials among the firms of various scales exist, is presented. The term“firm of various scale” is used to indicate that the heights of marginal productivity curves for labor are different for different firms.

If the labor market is competitive, a unique wage rate prevails so long as the labor force is homogeneous from the firm’s point of view. If the labor force is heterogeneous, but can be split into three groups A, B and C, where firm “ a” exclusively recruits workers from group A and the members of group A exclusively apply to a, and so on, we have three independent labor markets and the notion of non-competing groups can be applied to determine wages within each market. However, if the firms a,b and c respectively recruit among all the members of the groups A,B and C simultaneously, then the notion of non-competing groups is not applicable to the labor market. Since the actual labor market 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 firm’point of view. The grades or ordering of applicants might be directly 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 differences in grades of applying workers. In fact, statistical data shows that there are wage differences among workers of firms of different sizes when controling for these characteristics of the workers.

This observed fact suggests that firms recongnize different grades among workers of the same age, sex, work experience, and/or educational backgrounds. Any reason for the paying of higher wages by large firms, whose labor productivities are higher than smaller ones, cannot be found as long as the grades of workers are the same across firms. In fact, large scale firms with higher productivities offer comparatively favorable work conditions (higher wages and shorter hours of work) and as a result attract many applicants of various grades.The firms recruit what they perceive as the most favorable ones among those who applied. Smaller firms with lower productivity can offer only less favorable terms and recruit among the residual applicants who fail to be employed by the large scale firms. 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 the notion of grades^{1} 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.

1The notion of this kind, that is, labor que, is used in L.C.Thurow [10]

The model is suitably simplified. Although the labor supply actually consists of members of 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 firms is not explicitly treated. These simplifications will not impair the basic characteristics of the model which remain sufficiently 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 firm (or a group of firms) have been developed elsewhere.^{2} In this kind of model, individual labor supply and labor demand functions for a firm are assumed; that is, the notion of a kind of local labor market is introduced in the models. However, the relation between the individual labor supply function for the specific firm considered and the supply function of the market as a 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 firm ′ s product in an oligopolistic market. The wage level of the firm considered and the average wage level of other firms appear as explanatory variables of the individual labor supply function(The ratio of the both variables is adopted insome cases).

Elasticities of labor supply with respect to those variables or coefficients of those variables for each firm change, reflecting change in the conditions of the labor market as a whole including changes in the degree of of competition, the labor suppliers’ conjecture with respect to the recruitment policy of firms other than the firms to which the suppliers are applying. However, the mechanism of such interdependent changes of elasticities or coefficients of individual supply function has not been clarified. In this sense,models using individual supply functions lack autonomy.

Individual labor supply functions for each firm are not used in the model presented below. Instead, two basic relations are introduced. Instead of an individual supply function for a firm, which describes the relation between the number of applicants for the specific firm and the wage rate the firm offers, we use the labor supply function for the whole market 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 the quantity of labor supplied among firms. The distribution itself is determined by the model including firm 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 G_{i}, where i = 1, 2, ..., m, and m is the total number of people of working age. The range of G_{i} is supposed to

be

ε ≤ G_{i} ≤ 1

where ε is some positive small number．

The cumulative distribution (cumulative from the top of G, where G = 1) function of G is designated by ν(G) and the density distribution by ν′(G).

2C.A.Pissarides [9]

2.2. Labor Supply Probability Function

Suppose among n persons, n′ persons accept the employment opportunity at wage rate w, and assigned hoursof work h, offered by firms. The ratio n′/n is the supply ratio with respect to the employment opportunity.

Plim_{n→∞} n′/n ≡ μ

is defined 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 labor^{3} is defined 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 \underline{w}. Any supplier’s level of MSPL depends on following three factors:

a) the shape of his/her income-leisure preference curve,

b) the level of his guaranteed income X_{g} 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 firms, h.

Hence, we have

\underline{w}^{i} = \underline{w}(x_{g}^{i}, h^{i}, α^{i})  i = 1, 2,…, n  (1.1)

where α^{i} stands for the set of preference parameters of the ith supplier. x_{g}^{i} and h^{i} can be regarded as exogenous variables for the ith supplier. The value of α^{i} is specific to ith supplier; that is, the value of α^{i} differs among each of the n suppliers. Hence, we have the density distribution function ϕ(α).

Now, suppose a group of persons have the same level of guaranteed income \overline{x}_{g} ; that is,

x_{g}^{i} = x_{g}^{i+1} = \overline{x}_{g} (1.2)

From(1.1), (1.2) and ϕ(α), we have

g_{fϕ}(\underline{w} | \overline{x}_{g}, h) (1.3) which is the density distribution function of MSPL, h for brevity being assumed a common value for all persons considered. Subscript f and ϕ denote the fact that the analytical form of the function g depends on f and ϕ. Integration of g,

3The notion of reservation wage(RW) is given in Heckman[2]. MSPL is another definition of a kind of RW, because MSPL is defined by assuming h is a parameter which is assiged by the firm. In the modern employee labor market (in contract to self employed work), hours of work are assigned by the employer(firm). 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]

μ = ∫_{\underline{w}=0}^{w} g(\underline{w} | \overline{x}_{g}, h)d\underline{w} = μ = μ(w | \overline{x}_{g}, h), (1.4)

gives the supply-probability function μ of the group of persons with \overline{x}_{g} and h.

Multiplying by n, the number of persons in the group, we have the number of suppliers L^{s} , namely,

L^{s} = nμ(w | \overline{x}_{g}, h), (1.5)

When x_{g} and h are destributed as a joint density distribution

ψ(x_{g}, h), (1.6)

we have

μ(w) = ∫_{\underline{w}=0}^{w} ∫_{x_{g}=c}^{d} ∫_{h=a}^{b} g(\underline{w}, x_{g}, h)ψ(x_{g}, h) · dh · dx_{g} · d\underline{w} (1.7)

where a, b, c, d and e are the values standing for regions of integration for the relevant variables, h, x_{g} and w.

3. The Outline of the Model

Let the production function of the ith sector (or firm) be

Q_{i} = F (L_{i}, \overline{G}_{i}, A_{i})  (i = 1, 2,…, n)  (2.1)

where A and \overline{G}, respectively, stand for the set of firm parameters and the index of the grade of workers employed in the ith sector. Further, \overline{G}_{i} can be written as

\overline{G}_{i} = \overline{G}_{i}(G_{i}^{min}, G_{i}^{max}) (2.1′)

where G_{i}^{max} and G_{i}^{min} are indicators of the highest grade of workers (most preferable workers among applicants from the firm’s point of view) and the lowest grade of workers. It is supposed that

\frac{∂F}{∂\overline{G}_{i}} > 0, \frac{∂F}{∂L_{i}} > 0.

Let the supply probability equation (1.7) be

μ = μ(w, \overline{λ}) (2.2) where \overline{λ} 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 x_{g} is supposed to be included in the set \overline{λ}.

The (cumulative) distribution function of G is denoted by

ν_{G} = ν(G) (2.3)

where

ε ≤ 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 ν is common to all the sectors under consideration. Hence, by letting the number of potential suppliers be N, the number of suppliers with grade G and over, N_{G}, is given by

N_{G} = N · ν_{G} = N · ν(G) (2.5)

The number of suppliers with grade G and over going to the ith sections, L_{s}^{i}, is written as

L_{s}^{i} = N · ν(G) · μ(w_{i}, \overline{λ}) (2.6)

where w_{i} stands for the wage rate offered by the ith sector.

3.1. Behavior of the Leader

Imagine a sector (or firm) which offers the most favorable wage in comparison to other sectors in order to attract a number of potential suppliers. This sector can recruit workers of higher grades comparing to other sectors which offer less favorable working conditions. We shall call this sector a leader sector (firm) or a leader for short. Residual sectors are followers. Among those residual sectors, we can distinguish leaders and followers in accordance with the wage differentials each sector is willing to pay. That is, if we have three sectors with wage rates w_{1}, w_{2} and w_{3} where w_{1} > w_{2} > w_{3}, sector 2 plays the role of 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 indefinitely, 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 of production Q_{i} (i = 1, 2) by varying G_{i} and L_{i} in the production function (2.1) .

The distribution function (2.5), N · ν(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 firm ℓ (we denote leader by ℓ ) wishes to recruit workers with grades higher than G_{ℓ}^{min}. In this case, the labor supply curve for firm ℓ can be depicted by curve S_{ℓ}S′_{ℓ} in the 1st quadrant. This curve stands for equation (2.6) where G^{min} is inserted for G. Now, G^{max}(= 1) and G^{min} being given for the firm ℓ, the demand curve for labor is derived from the production function (2.1) and (2.1′) by applying the condition of cost minimization. This is depicted by curve D_{ℓ} in the first quadrant. The intersection of the supply curve s_{ℓ} s′_{ℓ} and D_{ℓ} gives the wage rate w_{ℓ} and the demand for labor L_{ℓ} by firm ℓ necessary to attain the given level of production Q_{ℓ}.

If firm ℓ were content to recruit workers with lesser grades, e.g. [G^{min}] < G_{ℓ}^{min} , the curve s_{ℓ} s′_{ℓ} would be less steep and stretched to the right. Hence, the required grade of 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 attain production level Q_{ℓ} are detemined by the procedure of cost minimization.

From (2.5) and (2.2) the number of potential applicants with grade G_{ℓ}^{min} and over, L_{G_{ℓ}}^{min}, is given by

L_{G_{ℓ}}^{min} = N_{ℓ} · G_{ℓ}^{min} · μ = N · ν(G_{ℓ}^{min}) · μ(w_{ℓ}, \overline{λ}) (2.6′)

which is a function of w_{ℓ}. Eq.(2.6′) is depicted by the curve s_{ℓ}s′_{ℓ} in Figure 1.

Figure 1

We have, for the leader, G^{max} = 1 in (2.1′). Hence, (2.1′) is written as \overline{G}_{ℓ} = \overline{G}_{ℓ}(G_{ℓ}^{min}, 1). Substituting this function and (2.6′) into (2.1) gives the leader’s production function.

Q_{ℓ} = F [N · ν(G_{ℓ}^{min}) · μ(w_{ℓ}^{min}, \overline{λ}), \overline{G}_{ℓ}(G_{ℓ}^{min}, 1), A_{ℓ}], (2.1′′)

where the subscript i in (2.1) has been replaced by ℓ to denote that eq. (2.1′′) refers to the leader. Defnition of cost is given by

C_{ℓ} = C_{0}^{ℓ} + w_{ℓ} · L_{G_{ℓ}}^{min} = C_{0}^{ℓ} + w_{ℓ}N · ν(G_{ℓ}^{min}) · μ(w_{ℓ}, \overline{λ}) (2.7)

where C_{0}^{ℓ} stands for capital cost which is regareded as given.

We can obtain w_{ℓ} and G_{ℓ}^{min} by minimizing C_{ℓ} in (2.7) under the constraint (2.1′′), Q_{ℓ} being given:

Letting

ψ_{ℓ} = C_{ℓ} + k\{Q_{ℓ} − F [N · ν(G_{ℓ}^{min}) · μ(w_{ℓ}, \overline{λ}), \overline{G}_{ℓ}(G_{ℓ}^{min}, 1), A_{ℓ}]\} (2.8)

where k is Lagrangian multiplyer, and C_{ℓ} is given by (2 ‐ 7), we have

\frac{αψ_{ℓ}}{∂G_{ℓ}^{min}} = \frac{αψ_{ℓ}}{∂w_{ℓ}} = 0. (2.9)

Solving (2.1′′) and (2.9) simultaneously for G_{ℓ}^{min} and w_{ℓ} , we obtain,

G_{ℓ}^{∗} = G_{ℓ}^{∗}(\overline{ν}_{0}, \overline{λ}, A_{ℓ}, Q_{ℓ}) (2.10)

and

w_{ℓ}^{∗} = w_{ℓ}^{∗}(\overline{ν}_{0}, \overline{λ}, A_{ℓ}, Q_{ℓ}) (2.11)

where \overline{ν}_{0} is a set of parameters in the grade distribution function (2.3). Equations (2.10) and (2.11) give optimal values for G_{ℓ}^{min} and w_{ℓ} both minimizing cost G_{ℓ} for the given production level Q_{ℓ}. The solution for employment L_{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_{ℓ}^{∗} and w_{ℓ}^{∗} 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 G_{ℓ}^{min} which is the lowest grade for the leader. Let the lowest grade of people in the group of potential applicants for the follower be G_{f}^{min}. The number of people with grades between G_{f}^{min} and G_{ℓ}^{min}, which we denote by N_{G_{f}}^{min}, is given by

N_{G_{f}}^{min} = N · ν(G_{f}^{min}) − N · ν(G_{ℓ}^{min}), (2.12)

which is shown by the length of N_{G_{f}}^{min} ∼ N_{G_{ℓ}}^{min} in Figure..1. Hence, the number of suppliers to the follower L_{G_{f}}^{min} is written as

L_{G_{f}}^{min} = N_{G_{f}}^{min} · μ = N[ν(G_{f}^{min}) − ν(G_{ℓ}^{min})] · μ(w_{f}, \overline{λ}). (2.13)

Substituting (2.13) into (2.1), we have the ptoduction function of the follower;

Q_{f} = F [N · (ν(G_{f}^{min}) − ν(G_{ℓ}^{∗})) · μ(w_{f}, \overline{λ}), \overline{G}_{f}(G_{f}^{min}; G_{ℓ}^{∗}), A_{f}] (2.14)

where A_{f} is the set of parameters of the follower’s production function, and G_{ℓ}^{∗} is given by (2.10). The definition of follower’s cost C_{f} is given by

C_{f} = C_{0}^{f} + w_{f} · L_{f} = C_{0}^{f} + w_{f} · N[ν(G_{f}^{min}) − ν(G_{ℓ})] · μ(w_{f}, \overline{λ}). (2.15) where C_{0}^{f} is capital (fixed) cost and (2.13) is substtitued for L_{f} .

Let us minimize C_{f} in (2.15) under the constraint of (2.14) where the level of Q_{f} is given.

ψ_{f} = C_{f} + j[Q_{f} − F {·}]. (2.16)

where j is the Lagrangean multiplier.

The minimization condition is as follows.

\frac{αψ_{f}}{∂G_{f}^{min}} = \frac{αψ_{f}}{∂w_{f}} = 0. (2.17)

Solving (2.17) for G_{f}^{min} and w_{f} , we have

G_{f}^{∗} = G_{f}^{∗}(\overline{ν}_{0}, \overline{λ}_{1}, A_{f}, Q_{f}, G_{ℓ}^{∗}) (2.18)

w_{f}^{∗} = w_{f}^{∗}(\overline{ν}_{0}, \overline{λ}, A_{f}, Q_{f}, G_{ℓ}^{∗}). (2.19)

where G_{ℓ}^{∗} is already given by the leader’s solution (2.10). L_{f} 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 firms (sectors), we can successively apply the above leaderfollower relationship. We shall call the state of market shared by firms (sectors) playing the role of leaders and followers successively as succession equilibrium. Let us suppose two firms 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 the position 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 hold the position of leader without augmenting the marginal productivity of his workers and if he finds losing his leader position is not profitable, 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 definition we have w_{A} > w_{B} and G_{A}^{min} > G_{B}^{min} . We shall examine the condition that guarantees a stable structure of wage differentials. We use the term “succession eqilibrium” to characterize a labor market with stable wage differentials.

The necessary condition for succession equilibrium is that

w_{ℓ} > w, (2.20) where w_{ℓ} and w stand for leader A′s and follower B’s wage rate respectively. Necessary and sufficient 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 satisfied.

(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_{ℓ} . Compute the follower solution for A. Let the solution be w. Then suppose (2.20),

w_{ℓ} > w

does not hold. This is the necessary and sufficient 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 labor market, a stable wage differential between unit A and B is observed. Then it must be considered that the leader-follower relationship between A and B is sustained by 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 whenever those 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 is satisfied.

(b.1) Let B be a leader instead of A and compute the leader solution for B. Let the solution for the wage rate be denoted by w_{ℓ} . Compute the follower solutionfor A. Let the solution be denoted by w. Next suppose (2.20)w_{ℓ} > w, does not hold. In this case, it can be said that the set of estimated parameters of the model is not correct 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 sufficient condition for the stability of successive equilibrium.

3.5. Simple Model

We shall specify the analytical form of production functions (2.1) and (2.1′) as

Q_{i} = b_{i}L_{i}^{α_{i}}(\overline{G}_{i})^{γ_{i}},  \overline{α}_{i} > 0, γ_{i} > 0  (i = 1, 2,…),  (3.1)

\overline{G}_{i} = (G_{i+1} · G_{i})^{\frac{1}{2}} , G_{i+1} < G_{i}, (3.1′)

where G_{i} and G_{i+1} respectively stand for the highest and the lowest values of G among the workers the ith firm employs. Let i = ℓ, f, where ℓ and f respectively stand for leader and follower.

Simplifying the distribution function ν(G) without impairing the basic characteristics of the model,we use

ν(G) = ν_{0} + ν_{1}G, (3.2)

where ν_{0} and ν_{1} are parameters.

We specify the supply-probability equation (2.2) as a linear function of w

μ = λ_{0} + λ_{1}w (3.3)

where as shown later

λ_{0} < 0,  λ_{1} > 0  and  0 ≤ μ ≤ 1  (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 firms) be two, unit 1 and 2. Numerical values of the parameters are assinged as follows.

α_{1} = α_{2} = 1, γ_{1} = 0.4, γ_{2} = 0.9, b_{1} = b_{2} = 1,

ν_{0} = 1, ν_{1} = −1, λ_{0} = −0.5, λ_{1} = 0.01, N = 10, 000

Suffix 1 and 2 stand for unit 1 and 2 respectively. The elasticity of production with respect to grades for unit 2, γ_{2}, is larger than that for unit 1.

The levels of production of unit 1 and 2 are experimentally given as shown in the first and second columns of Table 1a through 1g. These are exogenous variables in the simple model under consideration.

(a) In Table 1a, Q_{2} is increased from 150 to 300, Q_{1} being constant. In this case the computation process revealed the succession-equilibrium was stable and a stable leadaer-follower relationship holds as is shown in the table ; i.e. unit 2 and 1 are theleader and follower respectively. The wage differential w_{2}/w_{1} increases.

(b) In the second case, Q_{1} and Q_{2} were increased with a common rate of growth starting from Q_{1} = Q_{2} = 160 , as shown in Table 1b. The leader-follower relationship does not alter. The wage differential decreases, unlike that of case (1). It can be seen that the increase in the wage differential in case (1) stems from the growth and stagnation of production of unit 2 and 1 respectively.

(c) In Table 1a, Q_{2}, the production of unit 2 which has a larger value for γ compared to unit 1, was increased. In contrast to this, production Q_{1} of unit 1 is increased, Q_{2} being held constant at 150, in Table 1c. For the values of Q_{1} = 160, ⋯, 190, unit 2 occupies the position of leader, while case (b.1) appears when Q_{1} exceeds 200; that is, we do not have a consistent solution for Q_{1} ≥ 200 and Q_{2} = 150.

(d) Next, in order to clarify the response of production unit 2 against production unit 1 with Q_{1} = 200, we tentatively assigned Q_{2} values in the range 38 ≤ Q_{2} ≤ 750. (See Table 1d). It was found that the leader position switches if Q_{2}¡47. The altered leader follower relationship is stable for Q_{1} = 200 and 38 < Q_{2} < 47.

(e) A test analogous to (d) is shown in Table 3e. Here, Q_{2} is held constant at 150, while Q_{1} is varied between 63 ≤ Q_{1} ≤ 1250. The leader role switches when Q_{1} reaches 1250.

(f) Analogous to (e), we take Q_{1} = 250 and 38 < Q_{2} < 750. For Q_{2} > 250, unit 2 and 1 play the leader and follower respectively. If Q_{2} ≤ 54, the relationship alters. Between Q_{2} = 54 and Q_{2} = 250, we do not have stable succession equilibrium (consistent solutions).

(g) Analogous to (f), we vary Q_{2} between 38 and 750, Q_{1} being 300. For Q_{2} > 250, unit 2 and 1 are the leader and follower respectively. However, for Q_{2} ≤ 63 this relationship 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 stand for 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.

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

Figure 2

Table 1: aaa

(no solution) (no solution) (no solution)

Table 1a D_{ℓ} Table 1b D_{f} Table 1c D′_{ℓ} Table 1d D′_{f} Table 1e Table 1f Table 1g

ν_{0} = 1 ν_{1} = −1 α_{1} = 1.0 α_{2} = 0.8

γ_{1} = 0.4 γ_{2} = 0.9 λ_{0} = −0.5 λ_{1} = 0.01 N = 10, 000

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}

Q_{i} = b_{i}L_{i}. (3.1a)

Let the cost function of ith firm (or sector) be ^{5}

C_{i} = ψ_{i}(\overline{G}_{i}, L_{i}) + w_{i}L_{i} (3.1′a)

where

\overline{G}_{i} = (G_{i+1} · G_{i})^{\frac{1}{2}} , G_{i+1} < G_{i}

where G_{i} and G_{i+1} respectively stand for the highest and the lowest values of G among the workers the ith firm employs.

4.1.2. The distribution function of grade indicator

Simplifying the distribution function ν(G) without impairing the basic characteristics of the model,we use

ν(G) = ν_{0} + ν_{1}G, (3.2)

where ν_{0} and ν_{1} are parameters. ν(G) is the ratio of the number of potential applicants with grade G and over to the total number of potential applicants (the number of the people of working age). The magnitueds of G′s the potential supplier with the highest and lowest grade among all potential suppliers are respectively defined to equal unity and ε, ε being some small positive number. Hence, we have

ν(G) = 1  if G = ε,  (3.3a)

and

ν(G) = \frac{1}{N} if G = 1,

4With regard to this specification we can give an interpretation that in (2.1) we assume the analytical form of F as _{\frac{∂F}{∂\overline{G}_{i}}} = 0. Another interpretaion would be that we assume Leontief type[3] (factor limitational)

production function.

5ψ_{i} stands for an additional cost which is affected by the value of the grade G_{i}. This is an alternative and easier way of analysing the effect of G or the production behavior of the ith firm (so to speak G.

Becker Version[1]) compared to the way in which G is included in the production function of ith firm, as is done in the previous section. By this alternative Sppcification of production function (3.1a) means that the demand curve for labor D_{i}D′_{i} (i = ℓ, f) in Figure 1 are strait lines perpendicular to the abscissas.

where N stands for the number of the total potential suppliers. By applying (3.3a) to (3.2) we have

ν_{1} = −\frac{(1 − \frac{1}{N})}{(1 − ε)} (3.4a)

ν_{0} = \frac{1 + ε(1 − \frac{1}{N})}{(1 − ε)} (3.5)

Hence, the distribution function (3.2) is written as

ν(G) = 1 + \frac{ε(1 − \frac{1}{N})}{(1 − ε)} − \frac{(1 − \frac{1}{N})}{(1 − ε)} · G (3.2′)

By adopting the magnitude of ε as ε = \frac{1}{N} (3.2′) is written as

ν(G) = 1 + \frac{1}{N} − G (3.2′′)

If N is sufficiently a large number we have

ν(G) ≅ 1 − G. (3.2′′′)

The number of persons with G higher than G_{j}, N(G ≥ G_{j}), is given by

N(G ≥ G_{j}) = N · ν(G_{j}) (3.6)

Hence, applying (3.2′), we obtain

N(G ≥ G_{j}) = N [1 + (1 − \frac{1}{N})(\frac{ε}{1 − ε}) − (1 − \frac{1}{N})(\frac{ε}{1 − ε}) · G_{j}] (3.7)

Making use of the relation ε = 1/N, (3.7) is written as

N(G ≥ G_{j}) = N + 1 − NG_{j} = N(1 − G_{j}) + 1 (3.7′)

From (3.2′′) we have, as a good approximation for (3.7′),

N(G ≥ G_{j}) ≅ N(1 − G_{j}) (3.7′′)

4.1.3. Indirect observation of G_{i}

In the simple model, variables Q_{i}^{t}, L_{i}^{t} and w_{i}^{t} 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 G_{i} directly and we must therefore indirectly measure it making use of the model itself. We shall discuss the procedure to measure G_{i} below.

Suppose we have data on Q_{i}^{t}, L_{i}^{t} and w_{i}^{t} (i = ℓ, f). With respect to the parameters of the model we have ν_{0} = 1, ν_{1} = −1. Further suppose we have already estimated the parameters, λ_{0} and λ_{1}, in the supply probability function. Labor supply curves for the leader and the follower respectivity pass through points A_{ℓ} and A_{f} in Figure.1. The values of the coordinates of those points A_{ℓ} and A_{f} are known from observed data on the wages and employment of the leader and the follower.(Production units (firms, sectors) are ordered by the observed wage rates. Hence, successively, leader-follower relationships can be identified by this ordering.) Therefore, we can obtain N_{G_{ℓ}}^{min} and N_{G_{f}}^{min} by solving the simultaneous equations,

N_{G_{ℓ}}^{min}(λ_{0} + λ_{1}w_{ℓ}) = L_{ℓ}

(N_{G_{ℓ}}^{min} − N_{G_{ℓ}}^{min})(λ_{0} + λ_{1}w_{f}) = L_{f}

where actual wages and employment w_{ℓ}, w_{f}, L_{ℓ} and L_{f} are directly obtained from the observed data and λ_{0} and λ_{1} , are supposed to be already estimated, as mentioned above.

Applying N_{G_{ℓ}}^{min} and N_{G_{f}}^{min} thus obtained to the left hand side of the grade distribution function (ν(G_{i}) = 1 − G_{i}) in [3.2”’], we can calculate the numerical values for G_{ℓ}^{min} and G_{f}^{min}. These are the “indirectly observed” values for G_{ℓ}^{min} and G_{f}^{min}.

4.1.4. Equation of Supply-probability

We specify the supply-probability equation (2.2) as a linear function of w

μ = λ_{0} + λ_{1}w (3.8)

where as shown later

λ_{0} < 0,  λ_{1} > 0  and  0 ≤ μ ≤ 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 simplification means that we implicitly employ a rectangular distribution for the minimum supply price of labor, \underline{w}. In equation (3.8), we have w = λ_{0}/λ_{1} when μ = 0, hence,

μ = 0  if w ≤ −\frac{λ_{0}}{λ_{1}}  (3.10)

μ = 1 if w ≥ −\frac{1 − λ_{0}}{λ_{1}}

and for the range of w,

−\frac{λ_{0}}{λ_{1}} < w < \frac{1 − λ_{0}}{λ} ,

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 −λ_{0}/λ_{1}, stands for the minimal value of the range of distributed values of \underline{w}. This minimal values of \underline{w} must be positive, and

−\frac{λ_{0}}{λ_{1}} > 0 (3.11)

must hold. On account of the nature of the distribution function, λ_{1} must be positive. Hence, from (3.11) we have

λ_{0} < 0 (3.12)

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 have

Q_{ℓ} = b_{ℓ}L_{ℓ} (L.1)

where the suffix i in (3.1a) is replaced by ℓ to show that the equation is that of the leader. THe cost function in (4.1.1) can be written as

C_{ℓ} = ψ_{ℓ}(\overline{G}_{ℓ}, L_{ℓ}) + w_{ℓ}L_{ℓ} (L.2)

where ^{7}

6If the true shape of μ function is linear as shown in (3.8) λ_{0} < 0 must be held in the estimated relation as well. However, if the true function is non-linear, linear supply probability function, λ_{0} +λ_{1}w, originally is an aproximation. Hence, a constant term in an estimated linear supply probability function could be negative.

7Here after, for the sake of brevity, we assume \overline{G}_{ℓ} ≃ G_{ℓ} where G_{ℓ} stands for G_{ℓ}^{min}. By this approximation the basic characteristtics of the model will not be impaired.

\overline{G}_{ℓ} = (G_{ℓ}^{max} · G_{ℓ}^{min})^{\frac{1}{2}} (L.2′)

The number of suppliers to the leader is given by (letting \overline{N} be population of working age)

L_{ℓ}^{s} = \overline{N}[ν(G_{ℓ}) − ν(G_{ℓ}^{max})] · μ(w_{ℓ})

or

L_{ℓ}^{s} = \overline{N}[ν(G_{ℓ}) − ν(G_{ℓ}^{max})] · (λ_{0} + λ_{1}w_{ℓ} + λ_{2}A) (L.3)

where A stands for the effect of changing \overline{x}_{g} in sec. 2.3. (L.3) corresponds to (2.6). This equation states that effective suppliers to the leader must be the ones with at least grade G_{ℓ}.

When the value of Q_{ℓ} is given (L.3) can be written as ( taking into account (L.1),)

\frac{1}{b_{ℓ}} Q_{ℓ} = \overline{N}[ν(G_{ℓ}) − ν(G_{ℓ}^{max})] · μ(w_{ℓ}) (L.3′)

We minimize C_{ℓ} in (L.2) under the constraint (L.3′). That is, defining F as

F = ψ_{ℓ}(\overline{G}_{ℓ}, L_{ℓ}) + w_{ℓ}L_{ℓ} + Λ [\frac{1}{b_{ℓ}} Q_{ℓ} − \overline{N}[ν(G_{ℓ}) − ν(G_{ℓ}^{max})] · μ(w_{ℓ})] (L.4)

where Λ is a Lagrangian multiplier,

\frac{∂F}{∂G_{ℓ}} = \frac{∂F}{∂w_{ℓ}} = 0 (L.5)

has to hold if C_{ℓ} is minimized. Hence, from \frac{∂F}{∂G_{ℓ}} = 0, we have

\frac{∂ψ_{ℓ}}{∂G_{ℓ}} + Λ [−\overline{N} \frac{dν(G_{ℓ})}{dG_{ℓ}} · μ(w_{ℓ})] = 0 (L.6)

From \frac{∂F}{∂w_{ℓ}} , we have

L_{ℓ} − Λ\overline{N}[ν(G_{ℓ}) − ν(G_{ℓ}^{max})] · \frac{dμ}{dw_{ℓ}} = 0 (L.7)

Taking into account (3.8) we have \frac{dμ}{dw_{ℓ}} = λ_{1}, Hence, from (L.6) and (L.7) we get the

minimization condition

\frac{−\frac{∂ψ_{ℓ}}{∂G_{ℓ}}}{μ(w_{ℓ})} = \frac{\frac{1}{b_{ℓ}} Q_{ℓ}}{(G_{ℓ}^{max} − G_{ℓ})λ_{1}} , (L.8)

where G_{ℓ}^{max} is given. We can solve (L.8) and (L.3′) simultaneously for G_{ℓ} and w_{ℓ}. From (L.8) and (L.3′)

−\frac{∂ψ_{ℓ}}{∂G_{ℓ}} = \frac{\overline{N}}{λ_{1}} · [μ(w_{ℓ})]^{2} (L.9)

To begin with the simplest case we assume \frac{∂ψ_{ℓ}}{∂G_{ℓ}} to be a linear function, that is,

\frac{∂ψ_{ℓ}}{∂G_{ℓ}} = δ_{0} + δ_{2}L_{ℓ} (L.10)

where δ_{0}, δ_{1} and δ_{2} are the paramenters, and,

\frac{∂ψ_{ℓ}}{∂G_{ℓ}} < 0 (L.11)

must hold.

Inserting (L.10) into (L.9) and making use of (L.3), we have

δ_{0} + δ_{2}L_{ℓ} = \frac{−[\frac{1}{b}Q_{ℓ}]^{2}}{\overline{N}(G_{ℓ}^{max} − G_{ℓ})^{2}λ_{1}} (L.12)

From this we have

G_{ℓ} = G_{ℓ}^{max} ± \sqrt{\frac{−(\frac{1}{b}Q_{ℓ})^{2}}{\overline{N}λ_{1}(δ_{0} + δ_{2}L_{ℓ})}} (L.13)

Taking into account G_{ℓ} < G_{ℓ}^{max} we adopt

G_{ℓ} = G_{ℓ}^{max} − \sqrt{\frac{−(\frac{1}{b}Q_{ℓ})^{2}}{\overline{N}λ_{1}(δ_{0} + δ_{2}L_{ℓ})}} (L.13′)

where

δ_{0} + δ_{2}L_{ℓ} < 0.

(L.13′) is the “leader solution” of the grade variable.

Taking into account ν(G_{ℓ}) − ν(G_{ℓ}^{max}) = G_{ℓ}^{max} − G_{ℓ}, (L.3) can be written as,

L_{ℓ} = \overline{N}(G_{ℓ}^{max} − G_{ℓ})(λ_{0} + λ_{1}w_{ℓ} + λ_{2}A) (L.3′′)

where A stands for the effect of changing \overline{x}_{g} in sec. 2.3.

From (L.3′′)

w_{ℓ} = \frac{L_{ℓ}}{λ_{1}\overline{N}(G_{ℓ}^{max} − G_{ℓ})} − \frac{λ_{0}}{λ_{1}} − \frac{λ_{2}}{λ_{1}} A (L.14)

Substituting G_{ℓ} in (L.14) by (L.13′) we obtain

w_{ℓ} = \frac{1}{λ_{1}} · \frac{L_{ℓ}}{\overline{N} · \sqrt{\frac{−L_{ℓ}^{2}}{Nλ_{1}(δ_{0} + δ_{2}L_{ℓ})}}} − \frac{λ_{0}}{λ_{1}} − \frac{λ_{2}}{λ_{1}} A (L.15)

or

w_{ℓ} = \frac{1}{λ_{1}} · \frac{\frac{1}{b_{ℓ}} Q_{ℓ}}{\overline{N} · \sqrt{\frac{−( \frac{1}{b_{ℓ}} Q_{ℓ})^{2}}{Nλ_{1}(δ_{0} + δ_{2} \frac{1}{b_{ℓ}} Q_{ℓ})}}} − \frac{λ_{0}}{λ_{1}} − \frac{λ_{2}}{λ_{1}} A (L.15′)

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,

Q_{f} = b_{f}L_{f} (F.1)

and

C_{f} = ψ_{f}(\overline{G}_{f}, L_{f}) + w_{f}L_{f} (F.2)

where ^{8}

\overline{G}_{f} = (G_{f}^{max} · G_{f}^{min})^{\frac{1}{2}} (F.2′)

The number of suppliers to the follower is given by

L_{f}^{s} = \overline{N}[ν(G_{f}) − ν(G_{f}^{max})] · μ(w_{f}) (F.3)

G_{f} stands for the minimum value of the grade indicator of labor the follower can accept. G_{f}^{max} is the maximum value of the grade indicator which the follower can attain. When Q_{f} is given (F.3) can be written as

8The same assumption as that made in footnote 6 is made, that is \overline{G}_{f} ≃ G_{f} , where G_{f} stands for G_{f}^{min} .

\frac{1}{b_{f}} Q_{f} = \overline{N}[ν(G_{f}) − ν(G_{f}^{max})] · μ(w_{f}) (F.3′)

We minimize C_{f} in (F.2) with constraint (F.3′). The values of w_{f} and G_{f} minimizing C_{f} can be given by,

\frac{∂F}{∂G_{f}} = \frac{∂F}{∂w_{f}} = 0, (F.4)

where

F = ψ_{f}(\overline{G}_{f}, L_{f}) + w_{f}L_{f} + Λ [\frac{1}{b_{f}} Q_{f} − \overline{N}[ν(G_{f}) − ν(G_{f}^{max})] · μ(w_{f})] (F.5)

From (F.4) we get

\frac{−\frac{∂ψ_{f}}{∂G_{f}}}{μ(w_{f})} = \frac{\frac{1}{b_{f}} Q_{f}}{(G_{f}^{max} − G_{f})λ_{1}} (F.6)

Making use of (F.6) and (F.3′) we have

−\frac{∂ψ_{f}}{∂G_{f}} = \frac{\overline{N}}{λ_{1}} · [μ(w_{f})]^{2} (F.7)

We assume \frac{∂ψ_{f}}{∂G_{f}} to be linear, that is

\frac{∂ψ_{f}}{∂G_{f}} = ε_{0} + ε_{2}L_{f} (F.8)

Using (F.3)

μ(w_{f}) = \frac{L_{f}}{\overline{N}[ν(G_{f}) − ν(G_{f}^{max})]} (F.9)

where

ν(G_{f}) = 1 − G_{f} (F.10)

and

ν(G_{f}^{max}) = 1 − G_{ℓ} (F.11)

because G_{f}^{max} = G_{ℓ} .

Hence (ν(G_{f}) − ν(G_{f}^{max})) = (1 − G_{f}) − (1 − G_{ℓ}) = G_{ℓ} − G_{f} (F.12)

Taking into account (F.8), (F.9) and (F.12), (F.7) is rewritten as

ε_{0} + ε_{2}L_{f} = −\frac{\overline{N}}{λ_{1}} [\frac{L_{f}^{2}}{\overline{N}^{2}(G_{ℓ} − G_{f})^{2}} ] (F.13)

where G_{ℓ} is given by the solution for the leader. Again we suppose ε_{1} = 0 for brevity. Hence (F.14) can be written as

ε_{0} + ε_{2}L_{f} = \frac{−(\frac{1}{b_{f}} Q_{f})^{2}}{\overline{N}(G_{ℓ} − G_{f})^{2}λ_{1}} (F.14)

From this we have

G_{f} = G_{ℓ} ± \sqrt{\frac{−(\frac{1}{b_{f}} Q_{f})^{2}}{\overline{N}λ_{1}(ε_{0} + ε_{2} \frac{1}{b_{f}} Q_{f})}} (F.15)

Because G_{f} < G_{ℓ} we adopt

G_{f} = G_{ℓ} − \sqrt{\frac{−(\frac{1}{b_{f}} Q_{f})^{2}}{\overline{N}λ_{1}(ε_{0} + ε_{2} \frac{1}{b_{f}} Q_{f})}} (F.16).

where

ε_{0} + ε_{2}L_{f} < 0.

(F.16) corresponds to (L.13′) in the leader’s case, and gives the “follower’s solution” of the grade variable. From (F.3) and (F.12) , we have

L_{f} = \overline{N}(G_{f}^{max} − G_{f})(λ_{0} + λ_{1}w_{f} + λ_{2}A) (F.17)

By solving (F.17) we get

w_{f} = \frac{L_{f}}{λ_{1}\overline{N}(G_{f}^{max} − G_{f})} − \frac{λ_{0}}{λ_{1}} − \frac{λ_{2}}{λ_{1}} A (F.18)

Substition of G_{f} in (F.18) by (F.15) gives

w_{f} = \frac{1}{λ_{1}} · \frac{L_{f}}{\overline{N} · \sqrt{\frac{−L_{f}^{2}}{\overline{N}λ_{1}(ε_{0} + ε_{2}L_{f})}}} − \frac{λ_{0}}{λ_{1}} − \frac{λ_{2}}{λ_{1}} A (F.19)

or

w_{f} = \frac{1}{λ_{1}} · \frac{\frac{1}{b_{f}} Q_{f}}{\overline{N} · \sqrt{\frac{−(\frac{1}{b_{f}} Q_{f})^{2}}{\overline{N}λ_{1}(ε_{0} + ε_{2} \frac{1}{b_{f}} Q_{f})}}} − \frac{λ_{0}}{λ_{1}} − \frac{λ_{2}}{λ_{1}} A (F.19′)

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 shown in this section.

It can be seen the wage of financial sector is always at the top of the wage differential among the sectors(industries) during the observational period, 1970 through 1991. Hence financial sector can be identified as “leader sector”. The other sectors are aggregated and can be identified as “follower sector” as is shown by Figure 4. Hence, application of the model of leader behavior and follower behavior is straightforward . ^{9}

The observed yearly values of w_{i}, L_{i} and Q_{i}(i = ℓ, 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 λ_{0}, λ_{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 δ_{0} and δ_{1} are also shown in Table.2. The estimated and observed values for w_{i}, G_{i} (i = ℓ, f) are shown in Table.3. The estimated and observed values for w_{i} and G_{i} (i = ℓ, f) are shown in Figure 4 and Figure 5.

Table 2: Estimated Parameters of Structural Equations

\begin{matrix}λ_{0} & 1.515465 \\ λ_{1} & 0.5524484E-06 \\ λ_{2} & -0.1202143E-01 \\ δ_{0} & -0.7462759E+10 \\ δ_{2} & -2385271. \\ ε_{0} & -0.9816546E+10 \\ ε_{2} & 564116.1 \end{matrix}

9In case where the order of wage differential(i.e, top. second, third etc.) change during observational period, the application of the models of leader and follower was discussed in (3.4).

10In this section μ is specified as μ = λ′_{0} + λ_{1}w + λ_{2}A where A affects the changes in \overline{x}_{g} in (2.3).

However, values of A to be observed were substitued by time trend, that is, μ = λ′_{0} + λ_{1}w + at where a stands for the coefficient of the time trend term. λ′_{0} could be negative as shown in footnote footnote 6

6. Conclusion

From these results obtained in Figure 4, we can conclude that the “alternative simple model of the continually heterogeneous labor market” seems to be applicable to the Japanese data.

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 25

Table 3: Observations and Simulated Values of Endogenous Variables

year  W_{ℓ}  W_{f}  G_{ℓ}  G_{f}
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_{ℓ},W_{f}: yen per month (constant price of 1985)

Table 4: Gross Domestic Product by kind of Economic Activity

year  XTI  XAF  XFI  XRE PXTI85 PXAF85 PXFI85 PXRE85
70  70387.9  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 (Deflator) PXAF85: Gross Domestic Product by Industry-Agriculture,Forestry and Fishery (Deflator) PXFI85: Gross Domestic Product by Industry-Finance and Insurance (Deflator) PXRE85: Gross Domestic Product by Industry-Real Estate (Deflator)

An Equilibrium Model of Continually Heterogeneous Labor Market 27

Table 5: Number of Employed Persons by kind of Economic Activity / Population 15 Years Old and Over-Total / Cash Earnings

year EWTI EWAF EWFI EWRE  PT  WSMR1 WSMFI  WSMRE
70  5052.9 1073.6  131.6  35.0  7886  75494  84958  98167
71  5081.4  995.8  139.9  39.6  7979  86726  98056  112486
72  5097.7  947.8  142.8  39.0  8070  100485  114372  128158
73  5205.6  903.3  145.8  43.2  8239  122041  134753  144619
74  5166.2  880.9  149.2  44.8  8341  154840  172061  171702
75  5139.8  861.8  153.4  45.9  8443  177272  206658  191010
76  5177.3  844.3  158.3  47.4  8540  200805  238408  209067
77  5236.7  836.6  164.6  51.2  8631  219608  266636  227793
78  5279.5  826.3  166.4  54.0  8726  235367  287393  245327
79  5331.3  797.6  171.0  57.6  8824  246872  302055  269788
80  5358.6  756.6  176.9  60.1  8932  263166  323773  291001
81  5393.2  732.9  182.4  64.9  9017  278846  346036  300952
82  5435.8  718.9  186.7  66.7  9117  288112  366614  309144
83  5521.1  695.4  192.5  70.0  9232  297137  384973  316523
84  5538.7  668.9  195.0  71.1  9347  310238  402622  330292
85  5578.3  659.8  194.7  73.1  9465  316914  407887  332516
86  5626.4  643.3  201.6  75.9  9587  326906  424815  350368
87  5676.3  633.4  208.7  78.6  9720  335860  435997  361355
88  5774.5  622.3  209.7  79.8  9849  340742  474191  381113
89  5899.0  613.4  217.0  81.5  9974  356716  485684  423026
90  6028.9  605.7  228.9  85.6 10090  369424  488887  440013
91  6157.8  587.1  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 Estate W_{ℓ} = (W SMF I · EW F I + W SMRE · EW RE)/(EW F I + EW RE) W_{f} = [W SMR1 · EW T I − W_{ℓ} · (EW F I + EW RE)]/(EW T I − EW F I − EW RE)

References

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[5] Obi,K.(1968), “An Analysis of household Supply of Labor in Terms of Principal Earner’s Critical Income” : Paper presented at the 3rd Far Eastern meeting of the Econometric Society,Reprinted in Household’s Labor supply function in terms of Numerical Income-Leisure preference Field:Keio Economic Obsevatory Occasional Paper,1991

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[10] Thurow,L.(1975),Genarating Inequality

(Keio University,Mita 2-15-45,Minato-ku, Tokyo, Japan)