DBJ Discussion Paper Series, No.1308

Efficacy in Education and Intergenerational Affluence

Masayuki Otaki(Institute of Social Science, University of Tokyo)

March

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 or
recommendations expressed in these papers are those of the authors and do not reflect the viewsof the Institute.

Efficacy in Education and Intergenerational Affluence
Masayuki Otaki
Institute of Social Sciences
University of Tokyo
ohtaki@iss.u-tokyo.ac.jp
Abstract
We consider an intergenerational and educational investment stochastic model todetermine the effects of efficacy in education on affluence over time. Efficacy ineducation, defined by the success probability of intergenerational education, seriously
affects the affluence in the long run of families concerned with the prosperity ofdescendants. Since lenders of educational investment funds cannot take collateral forhumanitarian reasons, families without sufficient wealth are excluded from educational
opportunities.
Since the results of education investment are assumed to follow a random walk process,every family, independent of its wealth, faces bankruptcy risk with positive probability
in the long run, even if affluent at present. If education is more effective, theaccumulation of wealth releases more households from the threat of bankruptcy in the
future.
We show that low efficacy in education forces non-wealthy families to dissipate theirwealth even though they place their economic priority on their descendants’ wellbeing.
Keywords: efficacy in education; intergenerational wellbeing; lexicographic utility;
social stratification; pedagogical skills; random walk with one-sided absorbing barrier
JEL Classifications: I21, I22, I38,

1. Introduction
When parents are deeply devoted to their descendants’ wellbeing, they strive to save as
much money as possible. This is because their descendants might not be able to sustain
investment in education when their wealth is constrained and borrowing from outsideof the family is difficult.
This stems from the fact that educational investment is financed without collateral forhumanitarian reasons and that the effort that would be expensed by beneficiary
towards ensuring success in education is not verifiable. Financial intermediaries
estimate rationally that a wealthy and deeply committed family achieves highperformance in education because it loses more income when its education fails. Suchproperties of educational investment are analyzed by Otaki [1] based on Arrow [2] andStiglitz and Weiss [3].
This article analyzes how efficacy in education, which is defined by the ``success’’probability of a child in education, affects wealth accumulation. ``Success’’ means that
educated children acquire good natured and socialized characteristics and become able
to earn reasonable incomes, according to Dewy [4] who considers such success theultimate aim of education.1 It does not mean the acquisition of abilities for uncommonlyhigh incomes.
When education efficacy is sufficiently high, as wealth accumulation advances, mostfamilies are released from the fear of bankruptcy (i.e. constrained educational funds).
On the other hand, in the case of low efficacy, the accumulation of wealth is hindered,even though parents have a lexicographic preference for their descendants’ wellbeing
over their own.
This is because over time, success is canceled by low efficacy in education and, thus, itdoes not improve the sustainability of the family in being able to obtain educational
funding. Consequently, if efficacy is low, the fruits of education are dissipated rather
than being reinvested.
Efficacy depends not only on the contributions of a family’s own funds but also on skills
deployed by teachers since education comprises mutual communication. In other words,
when each family can finance educational costs from its own wealth, the decisive factor
that enables the sustenance of intergenerational affluence, is scholastic skills.

According to Dewy [4], ``To the one who is learned, subject matter is extensive,
accurately defined, and logically interrelated. To one who is learning, it is fluid, partial,
and connected through his personal occupations. The problem of teaching is to keep theexperience of the student moving in the direction of what the expert already knows.
Hence, the need that the teacher know both subject matter and the characteristic needs
and capacities of the student. ’’

11Nevertheless, educational economics has not given much importance to such skills.2
The rest of the paper is organized as follows. Section 2 constructs an intergenerationaland educational investment stochastic model under asymmetric information. In Section
3, we analyze the dynamic aspects of the model. Section 4 provides brief concluding
remarks.
2. The model
We regard an extended family as a kind of dynasty. Each individual has the following
lexicographic utility. He or she gives maximum priority to the sustainability of thefamily, and thereafter regards his private wellbeing. An extended family is assumed tobe sustained as long as its constituents can continue to be educated and earn decent
incomes to purchase various perishable goods. In other words, accumulated
non-perishable assets within the extended family serve as the capital from which to
produce perishable goods.
Furthermore, if parents succeed in educational investment, they obtain 1  mt units of
non-perishable goods (or consumable), and obtain nothing when they fail. mt is theamount of non-perishable goods that the family must outlay for educational investment
during period t .
Thus, the net return during period t , X t 1 ism , success
X t 1   t
1, failure .

(1)

The sum of accumulated non-perishable goods at the beginning of period t , St is

St  St 1  X t
if St i  0, St i  j  0, j .

(2)That is, St follows a random walk process with one-sided absorbing barrier St  0 . Thisimplies that a family cannot recover from an uncultivated (uneducated and with very
low income) situation once it is bankrupt, because a financial intermediary cannot lendmoney to a non-wealthy family as will be proved below.
Parents and their children cooperatively strive to succeed in educational investment.
The more they strive, the higher the success probability becomes. Let us denote therelationship between efforts, e , measured in terms of non-perishable goods and success
As Deer and Vesovic [5] summarize, the main concern of educational economists is
how educational investment affects one’s income and/or wealth. However, we must notethat quality of education symbolized by scholastic skills correlate with the fruits ofeducation, although such skills are hardly measurable.

2

2

probability, p as

et  e  pt |   , e  0 |    0,
e  2 ee, 2  0, | p 0  0, lim e  p |    ,
p 1
p p
p

(3)

e
 2e
 2e
 0, 2  0,
 0.


p
where  denotes pedagogical skills deployed by a teacher that strengthen efficacy ineducation.
A family that does not have sufficient educational funds possibly enters into a debt
contract with a financial intermediary. Then, the instantaneous payoff of a family,  B
becomes

 tB  p | mt ,1  r ,    p 1  mt  1  r 1  mt   mt   1  p  mt  e  p |   p 1  1  r 1  mt   1  p  mt  e  p |  ,(4)where r is the borrowing interest rate from the financial intermediary. It is clear from
Equation (4) that a non-wealthy family, whose wealth, mt is located in the vicinity of 0,
does not apply for the educational loan, because the return becomes negative. That is,
Lemma 1.

Some lump-sum of wealth is necessary for accessing educational investment. In turn,non-wealthy families are excluded from educational investment and are forced todissipate their own income.

A family maximizes  tB with respect to p . Assuming an interior solution, we obtain thefollowing first-order condition.

e '  pt |    1  1  r 1  mt   mt .

(5)

Furthermore, we assume that1 
e ' |    2 .
2 
Equation (6) guarantees that the optimal success probability is larger than

(6)1around2

the vicinity of mt  1 . This condition also implies that the expected return without3

borrowing,  t 1|  Bsatisfies

 tB 1|    0 .

(7)Equation (5) is illustrated in Figure 1. It is apparent that pt is a monotonously
increasing function of mt and a decreasing function of r . Let us denote this relationshipaspt    mt , r |   ,




 0,
 0,
 0.
mt
r


(8)

Equation (8) implies that a family, which can afford to provide sufficient funds foreducation, strives to succeed more because it has lost more wealth than a less wealthyfamily.
In addition, we assume that the loan market is competitive, and thus the profits of afinancial intermediary are equal to zero. When the deposit rate is zero, the zero-profit
condition of a financial intermediary is  mt , r  mt |   |   1  r  mt |      mt |   1  r  mt |    1 .

(9)Equation (8) defines implicitly the equilibrium loan rate, r  mt |   , which is applied to a
family whose own expenditure is mt .3
We assume that r  mt |   is a decreasing function of mt , or equivalently,   mt |   isan increasing function of mt . This assumption means that the marginal increment ofeffort for raising the success probability cannot be canceled by a lowered interest rate
offered under more favorable credit conditions. That is, the effective cost for raising the
success probability increases with the probability itself.4

Whenever  is the boundary such that  0 , any financial intermediary does notoffer a loan.
4 By differentiating Equation (5) with consideration for the relationship in Equation (9) ,we obtain3
1  mt 1
e '' 2  dpt  1   dmt .
pt  pt 
Thus, it is clear that Assumption 1 is validated for the affluent class, which can commit
more deeply to educational investment (i.e., mt  1 ).
4

Assumption 1.

The equilibrium loan interest rate, r  mt |   , is a monotonously decreasing function of

mt .
Employing the envelope theorem to Equation (4) and using the property of Equation(9) and Assumption 1, we obtain
dr  mt |   d tB   mt |   1  1  mt 
 0.
dmt
dm
t



(10)

Consequently,
Theorem 1.

The instantaneous utility of a family,  tB is a continuous and monotonously increasing
function of the education expenditure, mt . Furthermore, there is a unique criticalexpenditure level m that satisfies

 tB  m |    0,  tB  mt |    0 if mt  m , and  tB  mt |    0 if m  mt .

(11)

Proof.
The intermediate value theorem validates the assertion based on Lemma1 andEquations (7) and (10).
Theorem 1 implies that even though an extended family has a lexicographic preference
for its descendants’ wellbeing, no matter how wealthy it once was, its members areforever dispossessed of comfortable lives with modest income once its educational funds
are depleted. Nevertheless, we must still analyze the effect of wealth accumulation on
opportunities for enjoying a cultivated life. We consider this in the next section.
3. Intergenerational Wellbeing and Development in Pedagogical Skills
3.1 Wealth and Intergenerational Wellbeing
In this subsection, we analyze the relationship between a family’s wealth andintergenerational wellbeing (i.e., the sustainability of an educated extended family).
This sustainability is defined as the probability of an extended family continuing to
educate descendants and to enjoy cultivated lives forever.

5

To calculate this sustainability, let us denote p j  m0   as the probability with which an
extended family, which possesses j  m0 units of non-perishable goods, depletes its
wealth and is dispossessed from cultivated life someday. j is a non-negative integer
and m0  m  m0  1 is the initial endowment.
This problem is formulated as that of a random walk process, which is expressed by
Equation (2), with the absorbing barrier at m  0 .5 p j  m0   satisfies the following
difference equation. That is,

pm0    1   m0 |   
p j  m0     1|    p j 1 m0    1  1|     p j 1 m0   , j  1.

(12)

The solution of this difference equation isp j  m0    1   m0 |    j   ,    

1  1|   1|  

1,

(13)

where    is the eigen-value of difference Equation (12) and is less than unity.
Consequently, the sustainability of a family, which possesses j  m0 units of wealth,S  m0  j |   isS  m0  j |    1  p j  m0    1  1   m0 |    j   .

(14)

It is evident from Equation (14) that members of wealthier extended families can enjoy
cultivated lives throughout time, since S is a monotonously increasing function of m0
and j . That is,
Theorem 2.
The wealthier an extended family is, the higher the sustainability of its members

enjoying cultivated lives becomes.
Equation (14) also suggests a theory of social stratification with random social mobility.
Wealthier extended families rarely descend to the stratum that is bothered by daily life
and is excluded from educational opportunities. Nevertheless, those who belong to a
5

See Cox and Miller [6] for more detail.
6

stratum whose income is located below m  s 

are entirely dispossessed ofopportunities for ascending to the cultivated stratum, and consume income for
ephemeral purposes that belong to the second priority in its lexicographic preference,even though they wish to educate their children. We emphasize that such seemingly
selfish consumption of the non-wealthy stratum is not due to its shortage of love forchildren but due to its disillusionment owing to the shortage of educational funds.
Such stratification can be regarded spuriously as the result of relative deprivation(Veblen [7]; Merton [8]) or of differences in social good (Krause [9]). However, thesesociological theories underestimate the effect of economic affluence. Relative
deprivation emerges from income disparity. Many resources are poured into educationfor establishing a social good. That is, although relative deprivation and formation of an
intrinsic social good come from an extended family’s economic affluence, sociologicaltheories never succeed in explaining the cause of income disparity.
Our theory clarifies that imperfect information concerning the effort level of children
plays a key role in income disparity. Even though a child in a non-wealthy family would
strive to succeed, his/her efforts are unverifiable. The financial intermediary nevervalidates his/her efforts because the effort level is unobservable and the family’s default
cost is smaller under the principle of limited liability (we assume here that collateral forthe loan is zero owing to humanitarian reasons 6 ). Thus, non-wealthy families are
excluded entirely from educational loan markets. This is an acute origin of income
disparity.
3.2 Development of Pedagogical Skills and Intergenerational Wellbeing
In this subsection, we consider how development of pedagogical skills affects
intergenerational wellbeing.
First, we calculate the effect of development of the skills to the minimally required
fund m for education investment. By differentiating Equation (11) and applying theenvelope theorem in Equation (4), we obtain

ed tB dm e
dm

0
 B  0 .
d tdm d 
d
dm

(15)

As Johnson [10] argues, income contingent repayment (a kind of equity finance) might
be desirable. However, this is not a prevalent type of contract. Equity finance isgenerally an unsuitable measure under an asymmetric information structure. For moredetail, see Jensen [11].

67

Thus, the minimal funds required for educational investment are reduced bydevelopment of pedagogical skills. To summarize,
Theorem 3.

Less wealthy families can access educational loan markets by development ofpedagogical skills. This raises the intergenerational wellbeing of such families.
Such an egalitarian result is based on the fact that development of teachers’ skills
enhances children’s efforts through stimulating their curiosity. This heightens theefficacy in education, and hence, financial intermediaries may grant educational loanseven in cases in which a family is not so wealthy. Thus, development of pedagogicalskills contributes to equalizing educational opportunities and enriching less wealthyfamilies.
Second, we consider how the sustainability of an extended family is affected by thedevelopment of pedagogical skills. By differentiating Equation (5), it is evident that 2e 
 2e


m
|



 0.


p 2 
p

(16)Hence, from Equation (13),    is a decreasing function of  . Thus, Equation (14)implies that the sustainability of an extended family, S , is an increasing function of  .
This is because development in pedagogical skills makes education more effective byadvancing children’s intelligence abilities and raises the sustainability of extended
families (i.e., intergenerational wellbeing). That is,
Theorem 4.

Development of pedagogical skills heightens the sustainability of an extended family
defined by Equation (14). Such skills contribute to enhancing intergenerational
wellbeing.
4. Concluding Remarks
This article analyzed how efficacy in education affects intergenerational wellbeing. We
obtained the following results.
First, capital market imperfection, which comes from the inability to verify children’s
efforts, is an acute cause of stratification. This is because those who can invest muchmoney incur heavier losses when their investment fails and, thus, it is reasoned thatthey are diligent and have a higher success probability in education. For the same8

reason, a non-wealthy family is excluded from capital markets, and deprived ofopportunities for education.
Second, when education becomes more effective owing to progress in pedagogical skills,
less wealthy families are able to access capital markets. The development of teachers’
skills such that mutual communication between teacher and students is facilitatedmakes educational investment more effective, and this improves credit conditions forfamilies. Accordingly, pedagogical skills play a key role in equalizing educational
opportunities. In addition, high efficacy in education uniformly enhances thesustainability of extended families such that they can enjoy cultivated and affluent
lives.
For these two reasons, pedagogical skills are of profound social significance.

References[1]Otaki, M (2014) Income Disparity, Uneven Economic Opportunities, and Verifiability,
Advances in Social Sciences Research Journal, 1, pp. 44-49.
[2]Arrow, K. J. (1963) Uncertainty and the Welfare Economics of Medical Care,
American Economic Review, 53, pp. 941-873.
[3] Stiglitz, J. E., and Weiss, A. (1981) Credit Rationing in Markets with Imperfect
Information, American Economic Review, 71, pp. 393-410.
[4] Dewey, J. (1916) Democracy and Education, Macmillan, London, U.K.
[5]Deere, D. R., and Vosovic, J. (2006) Educational Wage Premiums and the U.S. Income
Distribution: a Survey, in Hanuschek, E. and Welch, F. (eds.) Hand Book of Educational
Economics Vol. 1, North-Holland, Amsterdam, Netherland, pp. 253-306.
[6]Cox, D. R., and Miller, H. D. (1965) The Theory of Stochastic Process, Chapman &Hall, London, U.K.
[7] Veblen, T. (1899). The Theory of the Leisure Class: An Economic Study of Institution,
Macmillan, London, U.K.

9

[8] Merton, R. K. (1949). On Theoretical Sociology, The Free Press, N.Y., USA.
[9] Krauss, I. (1976). Stratification, Class, and Conflict, The Free Press, N.Y., U.S.A.
[10] Johnston, D. B. (2001) The Economics and Politics of Income Contingent
Repayment Plans, The International Higher Education Finance and Accessibility
Project, State University of New York, Buffalo.
[11]Jensen, M. C. (1986). Agency Costs of Free Cash Flow Corporate Finance, andTakeovers, American Economic Review, 76, pp. 323-329.

10

e′ (pt |α)1 − [1 + r][1 − mt ] + mt

p∗tO

Figure 1: Optimal Eﬀort Level1pt