DBJ Discussion Paper Series, No.1307

Income Disparity, Uneven Economic Opportunities, andVerifiability
Masayuki Otaki(Institute of Social Science, University of Tokyo)

January

2014

Discussion Papers are a series of preliminary materials in their draft form. No quotations,reproductions or circulations should be made without the written consent of the authors in orderto protect the tentative characters of these papers. Any opinions, findings, conclusions orrecommendations expressed in these papers are those of the authors and do not reflect the viewsof the Institute.

Income Disparity, Uneven Economic Opportunities, and Verifiability
Masayuki Otaki
Institute of Social Sciences, University of Tokyo
ohtaki@iss.u-tokyo.ac.jp
7-3-1 Hongo Bunkyo
Tokyo 113-0033, JapanAbstract
The function of income in educational investment is considered under asymmetric
information on individuals’ effort. High income strengthens deep commitment toinvestment, and hence facilitates access to the capital market. Low income individuals
tend to be excluded for the same reason even though they have the same abilities as
wealthy individuals. Thus, disparity of income produces uneven economic opportunities.
Some income redistribution policy is necessary to cure this social inefficiency. The
expenditure of transferred income should, however, be limited to educational
investment to avoid wasteful consumption.

1. Introduction
Human capital investment, especially educational investment, heavily depends on an
individual’s own income and/or wealth, and thus, non-wealthy individuals are excluded
from economic opportunities that not only enhance their intelligence but also increase
their future incomes. This article explores why income disparity hinders equality ofeconomic opportunities.
Verifiability of one’s effort to succeed plays a key role. Wealthy people, who can invest
sufficient money in education, are naturally incentivized to engage in high level efforts
regardless of whether it is verifiable or not. This is because they would lose much money
when their educational investment fails. Thus, high income and/or wealth signal thesoundness of the investment plan to financial intermediaries. Hence, high income
and/or wealth facilitate lending towards wealthy individuals’ investment.
Since non-wealthy individuals cannot be incentivized by the cost incurred when theireducational investment fails, and efforts towards cultivation via education are generally
not verifiable, financial intermediaries quote a high interest rate to compensate for sucha high risk. Therefore, financial intermediaries’ lending to non-wealthy individuals
cannot be accomplished easily. Thus, uneven economic opportunities coexist withdisparity of income as such.
This article is organized as follows. In Section 2, we construct a model that exhibits thecoexistence of uneven economic opportunities and income disparity. Section 3 explains
how this problem resolved. Section 4 presents some concluding remarks.
2. The Model
2.1 Structure of the model
The model is an application of the theory of moral hazard to limited liability infinancial deals. It originates from Arrow (1963) and Stiglitz and Weiss (1981).
We assume lenders (financial intermediaries) and borrowers (individuals who intend
to invest in education) are both risk neutral, and their concerns are confined to expected
return. The investment initially requires a unit of money. The probability of success in
the investment, which is controllable by the borrower’s effort, is p . The investment
generates X amounts of goods when it succeeds, and nothing when it fails.
The cost function of the effort to ensure the success probability, p , in terms of money,c  p  ,has the following properties.
c  0   c '  0   0, c '  0, c ''  0, if c  0 .

1(1)

Furthermore, we denote the lending interest rate as 1  r , and the deposit rate is equal
to unity. Finally, we assume the following inequality.

X  1 r .

(2)This inequality is necessary for all educational investments to be meaningful.
Based on this setting, the borrower’s payoff function  B can be defined as B  p  X  1  r 1  m  1  p  m  c  p  .

(3)

The first term in the left-hand side of Equation (3) is the net expected revenue from theinvestment whose income is m . The second term corresponds to the expected loss when
the investment fails. The third term is the disutility of the effort to ensure the successprobability, p , which is measured in terms of money.
Since an individual maximizes his or her payoff, we obtain the following bydifferentiating Equation (2) with respect to p .

c '  p   X  1  r 1  m  m,

(4)

The first term in the right-hand side of Equation (4) represents the gain from lightening
his or her redemption. The second term is the effect that reduces the loss frominvestment failure. Equation (4) is illustrated by Figure 1, where the success probability

p is clearly a monotonously increasing function of m . Let us denote this relationship asfollows.

p    m  , '  0 .

(5)

On the other hand, the payoff function of a lender is L  m     m  1  r  m   1 .

(6)

The first term in the right-hand side of Equation (6) is the expected interest revenue
from lending. The second term is the redemption of a deposit. We assume that thedeposit market is competitive, and the equilibrium profits from a loan are zero. That is,  m  1  r  m   1  0(7)holds.
2.2 Comparative statics and welfare implications
Presuming the relationship in Equation (7), and employing the envelope theorem, wecan show that2d B  B   m  1  r  m   1   m     m   0dm
m(8)To summarize, we obtain the following theorem.
Theorem 1

The success probability of an education investment is an increasing function of theindividual’s income, m . Furthermore, the expected net revenue from the investment isalso an increasing function of m .
Combining Equations (2) and (7), we obtain  m X    m  1  r   1, m   0 X  1.

(9)

From Theorem 1 and Inequality (9), we can ascertain that all potential educational
investments are socially desirable, because they bring about positive surpluses topotential borrowers.
However, some not wealthy strata cannot access the capital market because surpluses
for the investment are too small relatively to the cost incurred by the effort. Henceforth,
we assume that the following relationship holds. That is,
Assumption 1

  0   X  1  r   c   0    0(10)holds.1

11 2
p . Equations (4) and (7) imply that2
1   0     X  1  r  0      X  1  .
0 


The following is an example. Let c  p  Thus,   0  satisfies the following quadratic equation.

 2  0    X  1  0     0 .
The solution is  X  1   2  X  1  42

  0 
Consequently,2.

 2  0   X  1  0   3.
  0     0  X  1 
2
2Since,  0  is a monotonous decreasing function of  , as long as  is sufficiently
B

3This assumption, in conjunction with Theorem 1, implies that the net expected return

for borrowers whose income is located within the interval 0, m* become negative, 

where m* satisfies  B m*  0 (see Figure 2). Consequently, they have to give up theinvestment even though all individuals have the same innate abilities. Thus, income
disparity hinders equalizing economic opportunities, and we have the following
theorem.
Theorem 2

The uneven educational opportunity owing to the disparity of income is sociallyinefficient.
2.3 Verifiability and the law of large numbers
The above theory, similar to other economic theories under uncertainty, is based on thelaw of large numbers. Lenders can know borrowers’ effort only statistically. That is,while the average revenue of lenders can be calculated accurately from large samples,
lenders cannot anticipate the success or failure of an each individual borrower’seducational investment.
This property, which is intrinsic to stochastic phenomena, causes a problem concerning
the verifiability of borrowers’ efforts. Lenders are unable to identify the cause ofinvestment failure; because of misfortune or their laziness. That is, the cause of failure
is not verifiable by its stochastic nature. Hence lenders depend on observable
information such as incomes to infer borrowers’ efforts.
This implies that even though a talented but non-wealthy individual, who has a low
cost function c  p  , applies for a loan, his or her request is rejected because of the low
income. Theorem 2 proves that such an uneven opportunity reflects the inefficiency ofthe society, and suggesting the acute need of that some measures to provide educational 0  0 holds.
On the other hand,  1    X  1 , and thus,small, B22 X  1   X  1 X  1  0 .
2
2
B
*
*
Therefore, there is a value m such that   m   0 is satisfied.
 B 1   X  X  1 4

opportunities to non-wealthy individuals for educational opportunity are acutelydesirable.
3. Supplementary Policies
The society faces a problem of shortage of funds for non- wealthy individuals. Hence, it
is sufficient that the government transfers incomes to them up to the critical value m* .
However, the government should restrict the usage of money. It should limit the usage
to educational investment.
Without this restriction, those who receive the subsidy consume wasteful items. Thiscan be easily proved as follows. From Equation (8), the marginal utility from additional 

investment is  m*  1 . On the other hand, the marginal utility of wasteful
consumption is unity from the definition of the utility function (3). Accordingly, any
additional money transfers to non-wealthy individuals with no condition attached arealways used for wasteful expenditure. Thus, we obtain the following theorem.
Theorem 3

An optimal income redistribution policy involves income transfers from those whose
income exceeds unity to non-wealthy individuals up to m* . However, expenditure must
be restricted to educational investment.
4. Concluding Remarks
We have analyzed the role of income in education investment. The results obtained are
as follows. First, high income eases investment because it interests lenders. Excessively
low income individuals are deprived of educational opportunities even though they havethe same abilities as high income individuals. Such deprivation indicates socialinefficiency.
Second, some income redistribution policy is unavoidable to cure the inefficiency of thesociety. Nevertheless, the usage of the transferred money should be confined toeducational purposes.

5References
K.J. Arrow (1963). Uncertainty and the welfare economics of medical care, American
Economic Review 53, pp. 941-973.
J. E. Stiglitz and A. Weiss (1981). Credit rationing in markets with imperfect
information, American Economic Review 71, pp. 393-410.

6

c′ (p)X − [1 + r][1 − m] + mψ(m)OFigure 1: Optimal Eﬀort1pπ B (m)Omm∗

Figure 2: Borrower’s Payoﬀ Function2