数理解析研究所講究録 1215 巻 2001 年 112-126

112

On the Fate of an Education Obsessed Society
Koichi Futagami and Akiomi Kitagawa\dag er*

February 22, 2001

Abstract.This paper constructs an economic growth model with overlapping
generations. Agents’ ability in the model can be high or low. Agentswith high ability incur low costs to obtain education. On the other
hand, agents with low abflity incur high costs to obtain an education.
With physical capital accumulation, the wage becomes high enough,and then the low-ability agents want to be thought as ahigh-ability
agent. In order to separate from the low-ability agents, the high ability
agents must send asignal to firms by obtaining high level of education.
This incurs unnecessarily high costs to the high ability agents andabsorbs their saving. This reduces physical capital accumulation andcan bring up trade cycles.
JEL Classification Numbers: J24, 041
Keywords: Signal, Human Capital, Cycles

1Introduction
In almost advanced countries, people eager to have high levels of education.
In order to get agood position in acompany, people tries to enter highly
ranked universities and get degrees. However, abilities are quite different
among individuals. Some people have high abilities and can easily get humancapital with little effort. On the other hand, another people needs much effort
Faculty of Economics, Osaka University, Machikaneyama, Toyonaka, Osaka 560-0043.
(phone): 81-6-8515266, (fax): 81-6-850-5274, (e–mail):futagami@econ.osaka-u.ac.jp\dag erFaculty of Economics, Osaka Prefecture University, Sakai, Osaka, (e-nuail):kitagawa@eco.osakafu-u.ac.jp*

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to get human capital and further can get only alow level of human capitalcompared to individuals with high abilities. Spence’s seminal paper (1973,1974) investigates this situation and shows that the high-ability agents havean incentive to send signals to firms in order to discriminate them ffom the
low-ability individuals by getting ahigher level of educations than the lowability individuals.
Now let’s consider this issue in amacroeconomic framework. Gettinghigh levels of educations of course needs much cost. Ordinary people sometimes must borrow money from banks to enter universities. In particular,
in Japan, primary schoolchildren or junior high school students often attend
cram schools (called Jyuku) in order to enter famous private schools. It costmuch. This may reduce the saving which was once invested into physical capital. Consequently, this may reduce output level of the education obsessedsociety..
This paper constructs an economic growth model with overlapping generations in order to examine this issue. Agents with high ability incur low
costs to obtain education. On the other hand, agents with low ability incur high costs to obtain an education. With physical capital accumulation,
the wage becomes high enough, and then the low-ability agents become towant to be thought as ahigh-ability agent. In order to separate ffom the
low-ability agents, the high ability agents must send asignal to firms byobtaining ahigher level of education. This incurs unnecessarily high costs
to the high ability agents and absorbs their saving. This reduces physicalcapital accumulation and can bring up cycles.
The rest of this paper is structured as follows: Section 2builds up themodel to be considered. Section 3examines the signaling game among theagents. Section 4defines equilibrium and dynamics of the model. Section 5gives some concluding remarks.

2ModelThe model of this paper is an overlapping generations economy ofnous growth with physical and human capital. The consumption
is produced by using physical, $K$ and human capital, $H$ . The production
function takes the following Cobb-Douglas form, $Y=AK^{\alpha}H^{1-\alpha},$ $A>0$ ,$0<\alpha<1$ . Both physical capital and human capital depreciate completely
after the production.
$\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{o}_{\mathrm{o}}^{\sigma}\mathrm{e}-$$b\sigma \mathrm{o}\mathrm{o}\mathrm{d}_{\dot{l}}Y$

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Each agent lives three periods (young, adult, old). We assume population
is normalized to be one. There are be two types of agents. One is an agent
with an ability to accumulate human capital cheaply. The other is an agent
without such ability. The population ratio of the agents with the ability is ,. These ratios remain
and the ratio of the agents without the ability isconstant. When young, the agent with the ability (the agent without theand get human capital as follows:ability) receives an education,$\theta$$1-\theta$$e_{t}^{a}(e_{t}^{n})$$h_{t+1}^{a}=e_{t}^{a}+(1-\delta)H_{t},$$h_{t}^{a}\leq\gamma H_{t},$$h_{t+1}^{n}=e_{t}^{n}+(1-\delta)H_{t},$$\gamma>1$$h_{t}^{n}\leq H_{t}$

(1)
(2)

means aggregate human capital of the economy. Each agent can
wherehave apart of human capital of their parents’ generation without any cost.
The agent with the ability can advance the human capital level than theirparents’ level, however, the agents without the ability at most get the samelevel of human capital as their parents’ level. The agents incur costs to gethuman capital, but different level. The high-ability agent can accumulate
human capital more cheaply than the low-ability agent. The high-ability, on the other hand, the low-ability agent must pay
agent must pay. and $0<\beta^{a}<\beta^{n}$ . In order to finance this cost, they borrow from.
adult agents at the asset
When adult, they work by selling their human capital to firms and get
wage income according to their human capital level which firms believe they
have. They save this income all for the old period. There are two saving
methods, one is to invest it into physical assets, the other is to lend it toyoung agents who want to get educations. By arbitrage, the rates of returnsof these savings become the same.
When old, they consume all their wealth, both principal and interest.
This is the only source of their utility. Consequently, their objective becomesthe maximization of their wage income minus their repayment.
At the first period, there are only adult agents and old agents.
$H_{t}$$\beta^{a}e^{a}$$\beta^{n}e^{n}$$\mathrm{m}\pi \mathrm{k}\dot{\mathrm{e}}\mathrm{t}$3Job Market Signaling
Because there are two types of agents, we have to consider the signaling gamesituation at each period. The low-ability agents may have an incentive to
mimic the high-ability agents. The timing of the game is the following:115
1. Agents choose alevel of education with knowing their ability.

2. Firms observe agents’ education level, but not knowing theirability (therefore, their borrowing levels), and make wage offers
to the agentsagents
3. The agents accept or reject the wage offer.
The objective of the agents is given by:$\max_{ei}w_{t+1}h_{t+1}^{i}-r_{t+1}\beta^{i}e_{t}^{i},$$i=a,$ $n$ .

Taking account of (1) and (2), we must distinguish the following fivecases:(I) Both of the agents do not have an education.
If $r_{t+1}\beta^{n}/w_{t+1}>r_{t+1}\beta^{a}/w_{t+1}>1$ , then both types choose not be educated (see Figure 1).
$e_{t}^{a}=e_{t}^{n}=0$.

(II) The high-ability agents begin to have an education.
YVhen $r_{t+1}\beta^{n}/w_{t+1}>r_{t+1}\beta^{a}/w_{t+1}=1$ , then the high-ability agents are
indifferent between getting an education and not getting an education. On
the other hand, the low-ability agents have no incentive to have an education(see Figure 2).
$e_{t}^{a}\in[0, (\gamma+\delta-1)H_{t}]$$e_{t}^{n}=0$(III) Only the high-ability agents have an education.
’VVhen $r_{t+1}\beta^{n}/w_{t+1}>1>r_{t+1}\beta^{a}/w_{t+1}$ , then the high-ability agents havean incentive to be educated up to the maximum level. However, the lowability agents still have no incentive to have an education (see Figure 3).
$e_{t}^{a}=(\gamma+\delta-1)H_{t}$$e_{t}^{n}=0$

116(IV) The low-ability agents begin to have an education.
When $1=r_{t+1}\beta^{n}/w_{t+1}>r_{t+1}\beta^{a}/w_{t+1}$ , then the low-ability agents are
indifferent between getting an education and not getting an education. The
high-ability agents have an incentive to have an education up to their maximum level. If the low ability agents invest up to the maximum level ofthe high-abilty agents, then firms want to distinguish the low-ability agent
who mimic the high-ability agents. But, firms cannot separate them fromthe high-ability agents because they cannot observe the actual ability of theagents. The firms can observe only the education level. Therefore, the highability agents have an incentive to invest human capital over their maximum
level in order to separate them from the low-ability agents (see Figure )Consequently, we obtain the following:$4$

e7$=(\gamma+\delta-1)H_{t}+\epsilon H_{t},$$.1$$\epsilon>0$$e_{t}^{n}\in[0, \delta H_{t}]$We assume that the high-ability agents need to overinvest
in order todiscriminate them from the low-ability agents.
(V) Both of the agents invest up to their maximum levels.
When $1>r_{t+1}\beta^{n}/w_{t+1}>r_{t+1}\beta^{a}/w_{t+1}$ , then both of the agents havean incentive to have an education up to their maximum levels. The same
situation as case (IV) occurs. Therefore, the high-ability agents must send
asignal to firms to separate them from the low-ability agents. Accordingly
there is an unnecessary overinvestment in human capital (see Figure 5).
$\epsilon H_{t}$$e_{t}^{a}=[ \frac{w_{t+1}}{r_{t+1}\beta^{n}}(\gamma-1)+\delta]H_{t}+\epsilon H_{t},$ $\epsilon>0$$e_{t}^{n}=\delta H_{t}$Summarizing the preceding arguments and noting (1) and (2), we canobtain the following human capital accumulation expressions:lThere can be pooling equilibria other than the separating equilibrium. However, byresorting to the Intuitive Criteria of Cho and Kreps (1987), we can refine the perfect
Bayesian equilibrium. We can show that all pooling equilibria cannot survive through theIntuitive Criteria (see Gibbons (1992).

117$H_{t+1}$$=$$\theta h_{t+1}^{a}+(1-\theta)h_{t+1}^{n}$when$(1-\delta)H_{t}$$\frac{r_{t+1P}}{w_{\ell+1}}>1$,$[0, \{\theta\gamma+(1-\theta)(1-\delta)\}H_{t}]$$=$$\{\theta\gamma+(1-\theta)(1-\delta)\}H_{t}$$\{$’$when \frac{\frac{\tau_{\ell+}\beta^{n}n_{1}}{\mathrm{r}e+1\Psi w_{t+1}}}{w_{t+}n}\frac{}{w_{t+1}}when>\frac{\mathrm{r}t+1\beta^{\alpha}1}{rt+1\beta^{a}w_{t+1}}whe\frac{\mathrm{r}\iota+1\beta^{\mathrm{n}}11>\sigma--}{w_{t+1}}whe\frac{\tau e+1\beta}{1>1w_{t+1}-->}.’,$$[\{\theta\gamma+(1-\theta)(1-\delta)\}H_{t}, \{\theta(\gamma-\mathrm{I})+1\}H_{t}]$$\{\theta(\gamma-1)+1\}H_{t}$4Market Equilibrium
Bertrand competition among the firms drives the profit of the firms down to
zero. Hence, the following conditions must hold:$w_{t}=(1-\alpha)Ak_{t}^{\alpha}$$r_{t}=\alpha Ak_{t}^{\alpha-1}$

,

(3)
(4)

,

where
and stands for the wage rate and the (gross) interest rate respec.
tively, andAsset market equilibrium condition becomes$w_{t}$$r_{t}$$k_{t}\equiv K_{t}/H_{t}$$K_{t+1}+\theta\beta^{a}e_{t}^{a}+(1-\theta)\beta^{n}e_{t}^{n}=w_{t}H_{t}-r_{t}[\theta\beta^{a}e_{t-1}^{a}+(1-\theta)\beta^{n}e_{t-1}^{n}]$.

(5)The left hand side means the demand for funds for physical and humancapital investment. On the contrary, the right hand side means supply forthe funds. Dividing the both side of (5) byand taking account of (3) and(4), we get the following:$H_{t}$$k_{t+1} \frac{H_{t+1}}{H_{t}}+\theta\beta^{a}\frac{e_{t}^{a}}{H_{t}}+(\mathrm{I}-\theta)\beta^{n}\frac{e_{t}^{n}}{H_{t}}$(6)$=A(1- \alpha)k_{t}^{\alpha}\wedge-A\alpha k_{t}^{\alpha-1}[\theta\beta^{a}\frac{e_{t-1}^{a}}{H_{t-1}}+(1-\theta)\beta^{n_{\frac{e_{t-1}^{n}}{H_{t-1}}}}]\frac{H_{t-1}}{H_{t}}$We first examine the demand for the funds. Let’s denoting the demand. There are five cases which is described above. The
for the funds as$D_{t+1}$

118’$\frac{\alpha\beta^{a}}{1-\alpha,\mathrm{m}}<\mathrm{a}\mathrm{r}\mathrm{k}\mathrm{e}\mathrm{t}$$\mathrm{f}\mathrm{i}\mathrm{v}\mathrm{e}\mathrm{c}\mathrm{c}\mathrm{a}s\mathrm{a}\mathrm{e}\mathrm{a}\mathrm{r}\mathrm{e}.\cdot(1)k_{t+1}<\frac{\alpha\beta^{a}}{1-\alpha,k},(\mathrm{I}\mathrm{I}k_{t+1}=\frac{\alpha\beta^{a}}{1-\alpha,\mathrm{t}\mathrm{h}\mathrm{e}’}(1\mathrm{V})k_{t+1}=\frac{\alpha\beta^{n}}{1-\alpha},(\mathrm{V})\frac{\alpha\beta^{n}}{1-\alpha}<_{t+1}.\mathrm{B}\mathrm{y}\mathrm{u}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{o}\mathrm{r}(\mathrm{I}\mathrm{l}\mathrm{I})$$k_{t+1}< \frac{\alpha\beta^{n}}{1-\alpha,\mathrm{u}}\mathrm{e}\mathrm{q}\mathrm{u}\mathrm{i}1\mathrm{i}\mathrm{b}\mathrm{r}\mathrm{i}\mathrm{m}$conditions, (3) and (4), the five cases become as follows:(I)$k_{t+1}(1-\delta)$$[(1- \delta)\frac{\alpha\beta^{a}}{1-a,\{\theta},\theta(\gamma+\delta-1)(\beta^{a}+\frac{\alpha\beta^{a}}{1-\alpha,\beta^{a}})+\frac{\alpha\beta^{a}}{1-\alpha,\delta-}(1-\theta)]k_{t+1}\gamma+(1-\theta)(1-\delta)\}+\theta(\gamma+1)$$(\mathrm{I}\mathrm{I}\mathrm{I})(\mathrm{I}\mathrm{I})$$D_{t+1}=\{$

(IV)
(V)

$[D_{\mathrm{I}\mathrm{V}},\overline{D}_{1\mathrm{V}}]$$k_{t+1} \{\theta(\gamma-1)+1\}+\theta\beta^{a}[\frac{1-\alpha}{\alpha\beta^{\mathfrak{n}}}(\gamma-1)k_{t+1}+\delta+\epsilon]+(1-\theta)\beta^{n}\delta$

where

and
.
is acorrespondence which assigns anonempty
compact subset to every. We denote this correspondence as $D(k_{t+1})$ .
Next, let’s examine the supply side. By denoting the supply for the funds
as , we similarly get the following for $t\geq 2$ :$D_{\mathrm{I}\mathrm{V}}= \theta(\gamma+\delta-1)(\beta^{a}+\frac{\alpha\beta^{n}}{1-\alpha})+\frac{\alpha\beta^{n}}{1-\alpha}(1-\theta)$$\delta)_{\frac{\alpha\beta^{n}-\alpha}{}}+(1-\theta)\beta^{n}\delta+\theta\beta^{a}\epsilon$$\overline{D}_{\mathrm{I}\mathrm{V}}=D_{\mathrm{I}\mathrm{V}}+\theta(1-$$D_{t+1}$$k_{t+1}$$S_{t}$

(I)
(II)
(III)
(IV)
(V)

$S_{t}=\{$$A(1-$$\mathrm{w}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{e}k_{t}<$$\frac{\alpha\Psi \mathrm{f}\mathrm{i}\mathrm{v}}{1-\alpha},(\mathrm{l}\mathrm{V})\mathrm{e}\mathrm{c}\mathrm{a}\mathrm{e}\mathrm{e}\mathrm{s}\mathrm{a}\mathrm{r}\mathrm{e}k_{t}=\frac{\alpha\Psi \mathrm{f}\mathrm{o}}{1-\alpha}(\mathrm{V}^{\cdot})\frac{\alpha\beta^{n}(1)}{1-\alpha}<k_{t}.S_{11}=\mathrm{a}\mathrm{s}11\mathrm{o}\mathrm{w}\mathrm{s}.k_{t}<\frac{\alpha\beta^{a}}{1-\alpha},(\mathrm{I}\mathrm{I})A(1-[\frac{\alpha\beta^{a}(\mathrm{I}\mathrm{I}}{\mathrm{I}-\alpha}]^{\alpha_{11}},=k_{t}=\frac{\alpha\beta^{\Phi}}{1-\alpha,\alpha)},\mathrm{I})\frac{\alpha\beta^{a}}{1-\alpha S},<$$A(1- \alpha)[\frac{\alpha\beta^{a}}{1-\alpha}]^{\alpha}\frac{1}{\theta(\gamma+\delta-1)+1-\theta},$,$S_{\mathrm{I}\mathrm{V}}=A(1- \alpha)[\frac{\alpha\beta^{n}}{1-\alpha}]^{\alpha}\frac{(\beta^{f*}-\beta^{a})\theta(\gamma+\delta-1)+\beta^{n}(1-\theta)}{\theta(\gamma+\delta-1)+1-\theta}$

and. As for the first period,
we get $S_{1}=w_{1}H_{1}=A(1-\alpha)k_{1}^{\alpha}$ . Similar to the demand correspondence, we
can define the supply correspondence as.
Consequently, we get the following dynamics from the asset market equilibrium condition:$S_{1\mathrm{V}}=A(1- \alpha)[\frac{\alpha\beta^{n}}{\mathrm{i}-\alpha}]^{\alpha}\frac{\underline{(\beta^{n}}-\beta^{a})\theta(\gamma+\delta-1)+\beta^{f*}(1-\delta)-\theta\beta^{a}\epsilon}{\theta(\gamma+\delta-1)+1-\theta-}$$S(k_{\mathrm{t}})$$k_{\mathrm{t}+1}\in\Phi(k_{t})\equiv\{k_{\mathrm{t}+1}^{\wedge}|D(k_{t+1})\cap S(k_{t})\neq\emptyset f\sigma rk_{t}\}$This define the dynamic path of$k_{t}^{\wedge}.\mathrm{S}\mathrm{t}\mathrm{e}\mathrm{a}\mathrm{d}\mathrm{y}$Definition 1Steady State
Steady states of the (7) is defined by

State:$k^{-*}\in\Phi(k^{*})$

.

(7)

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There can be many patterns of the dynamics. So, let’s pick up an interesting case. Figure 6depicts the demand and supply correspondences..
There is asteady state between point Aand point B. As can be seen fromenters into this
this figure, this steady state happens to be case (IV). If
this case, we haveregion, thenmust be in this region because of (by , taking accountto examine the following dynamics of . Denoting, we can express (6) as follows:of $e_{t}^{a}=\gamma+\delta-1,$$k_{t}$$k_{t+1}$$7J\cdot\ln$$\frac{e_{t}^{n}}{H_{t}}$$\frac{e_{t}}{H_{t}}$$x_{t}^{n}$$k^{*}= \frac{\alpha\beta^{n}}{1-\alpha}$$\frac{\alpha\beta^{n}}{1-\alpha}[\{\theta\gamma+(1-\theta)(1-\delta)\}+(1-\theta)x_{t}]+\theta\beta^{a}(\gamma+\delta-1+\epsilon)+(1-\theta)\beta\chi\emptyset$$=$$(1- \alpha)A[\frac{\alpha\beta^{n}}{1-\alpha}]^{\alpha}\frac{(\beta^{n}-\beta^{a})\theta(\gamma+\delta-1+\epsilon)+\beta^{n}(1-\theta)}{\beta^{n}[\{\theta\gamma+(1-\theta)(1-\delta)\}+(1-\theta)x_{t-1}^{n}]}$This defines the dynamics of case (IV). Consequently, the steady state of thisdynamics is defined by:$\frac{\alpha\sqrt{}^{n}}{1-\alpha}\{\theta\gamma+(1-\theta)(1-\delta)\}+\theta\beta^{a}(\gamma+\delta-1+\epsilon)+\frac{(1-\theta)\beta^{n}}{1-\alpha}x^{n*}(9)$$=$$(1- \alpha)A[\frac{\alpha\beta^{n}}{1-\alpha}]^{\alpha}\frac{(\beta^{n}-\beta^{a})\theta(\gamma+\delta-1)+\mathcal{B}^{n}(1-\theta)}{\beta^{n}[\{\theta\gamma+(1-\theta)(1-\delta)\}+(1-\theta)x^{n*}]}$

We first examine the stability of the steady state of the dynamics of .
and dividing
By differentiating the right hand side of (8) with respect to, we getthis by$x_{t}^{n}$$x_{t-1}^{n}$$\frac{(1-\theta)\beta^{n}}{1-\alpha}$$-(1- \alpha)^{2}A[\frac{\alpha\beta^{n}}{1-\alpha}]^{\alpha}\frac{(\beta^{n}-\beta^{a})\theta(\gamma+\delta-1+\epsilon)+\beta^{n}(1-\theta)}{(\beta^{n})^{2}[\{\theta\gamma+(1-\theta)(1-\delta)\}+(1-\theta)x^{n*}]^{2}}$This is the slope of the graph of the dynamics of . Therefore, when this is, then the steady state is unstable (see Figure 7). By making
smalleruse of (9), we can rearrange this as follows:$x_{t}^{n}$$\mathrm{t}\mathrm{h}\mathrm{a}\mathrm{n}-1$$-[1+(1- \alpha)\frac{\beta^{a}\theta(\gamma+\delta-1+\epsilon)-\beta^{n}\{\theta\gamma+(1-\theta)(1-\delta)]}{\beta^{n}[\{\theta\gamma+(1-\theta)(1-\delta)\}+(1-\theta)x^{n*}]}]$

Hence, the numerator takes apositive value, then the steady state becomes
unstable. We get the following condition under which the steady state is
unstable:120$\epsilon>(1-\frac{\beta^{a}}{\beta^{n}})(\gamma+\delta-1)+\frac{1-\delta}{\theta}$

This inequality can be consistent with Figure 6because this inequality does
not contain the productivity parameter .
When this condition holds, even if enters into case (IV), leave case
(IV). Then, enters case (III) or case (IV).
$A$$k_{t}$$k_{t}$$k_{t}$

5Concluding Remarks
We have shown that overinvestment to human capital may absorb funds forinvestment for physical capital. As mentioned in the introduction, the saving
is absorbed by expense to getting educations. This reduces income in theadult period and thus saving volume of the adult individuals. Consequently,
this leads to adecrease in physical capital and can produce permanent cycles.

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References[1] Cho I.-K. and D. Kreps, 1987, ”Signaling Games and Stable Equilibria,”
Quarterly Journal of Econornics, 102, pp179-222.
[2] Gibbons, R., 1992, Game Theory for Applied Economists, Princeton UP.
[3] Spence, A. M., 1973, ”Job Market Signaling,” Quarterly Journal of EcOnomics, 87, pp355-374.
[4] Spence, A. M., 1974, ”Competitive and Optimal Responses to Signaling:
An Analysis of Efficiency and Distribution,” Journal of Economic Theory,8, 296-332.

122

6 H,

,

$(_{\gamma+\delta- 1)}H$

Figure 1,$\delta^{H}$,$(_{\gamma+\delta- 1)^{H}}$Figure 2

123

$a$

,

$\delta^{H}$,$a$$\beta e$,$(_{\gamma+\delta- 1)}H$

Figure 3,$\mathrm{f}1^{aa}e$,$\delta^{H}$,$(_{\gamma+\delta- 1)}H$

Figure 4124

6 $H$,,$(_{\gamma+\delta- 1)}H$

Figure 5

125

,

$k$

Figure 6126$x_{\mathrm{t}+1}$

Figure 7