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We consider an intergenerational and educational investment stochastic model to determine the effects of efficacy in education on affluence over time. Efficacy in education, defined by the success probability of intergenerational education, seriously affects the affluence in the long run of families concerned with the prosperity of descendants. This is because since lenders of educational investment funds cannot take collateral for humanitarian reasons, families without sufficient wealth are excluded from educational opportunities. If education is more effective, the accumulation of wealth releases more households from the threat of bankruptcy in the future.

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 outside of the family is difficult.

This stems from the fact that educational investment is financed without collateral for humanitarian 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 high performance in education because it loses more income when its education fails. Such properties of educational investment are analyzed by Otaki [

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 [^{1}. It does not mean the acquisition of abilities for uncommonly high incomes.

When education efficacy is sufficiently high, as wealth accumulation advances, most families 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, it does 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. 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 intergenerational and educational investment stochastic model under asymmetric information. In Section 3, we analyze the dynamic aspects of the model. Section 4 provides brief concluding remarks.

It is a well-known statistical property that there is a robust positive correlation between enrollment rates and real GDP per capital. For example, Mankiw, Romer and Weil [

However, converse causality is also thinkable and emphasized mostly by researchers who are engaged in development economics. That is, lower income hinders educational opportunities. Ito [

We consider that such causalities coexist, and hence, we construct a model which enables to analyze both causalities simultaneously.

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 the family, and thereafter regards his private wellbeing. An extended family is assumed to be 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

Thus, the net return during period

The sum of accumulated non-perishable goods at the beginning of period

That is,

Parents and their children cooperatively strive to succeed in educational investment. The more they strive, the higher the success probability becomes. Let us denote the relationship between efforts,

where

Afamily that does not have sufficient educational funds possibly enters into a debt contract with a financial intermediary. Then, the instantaneous payoff of a family,

where

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

A family maximizes

Furthermore, we assume that there exits

Equation (6) guarantees that the optimal success probability is larger than

Equation (5) is illustrated in

Equation (8) implies that a family, which can afford to provide sufficient funds for education, strives to succeed more because it has lost more wealth than a less wealthy family.

In addition, we assume that the loan market is competitive, and thus the profits of a financial intermediary are equal to zero. When the deposit rate is zero, the zero-profit condition of a financial intermediary is

Equation (8) defines implicitly the equilibrium loan rate,

expenditure is ^{3}.

We assume that ^{4}.

The equilibrium loan interest rate,

Employing the envelope theorem to Equation (4) and using the property of Equation (9) and Assumption 1, we obtain

Consequently,

The instantaneous utility of a family,

Proof. The intermediate value theorem validates the assertion based on Lemma1 and Equations (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 are forever 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.

In this subsection, we analyze the relationship between a family’s wealth and intergenerational 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.

To calculate this sustainability, let us denote

This problem is formulated as that of a random walk process, which is expressed by Equation (2), with the absorbing barrier at ^{5}.

The solution of this difference equation is

where

It is evident from Equation (14) that members of wealthier extended families can enjoy cultivated lives throughout time, since

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 stratum whose income is located below

Such stratification can be regarded spuriously as the result of relative deprivation (Veblen [

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 never validates 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 for the 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.

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

Thus, the minimal funds required for educational investment are reduced by development of pedagogical skills. To summarize,

Less wealthy families can access educational loan markets by development of pedagogical 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 the efficacy in education, and hence, financial intermediaries may grant educational loans even in cases in which a family is not so wealthy. Thus, development of pedagogical skills contributes to equalizing educational opportunities and enriching less wealthy families.

Second, we consider how the sustainability of an extended family is affected by the development of pedagogical skills. By differentiating Equation (5), it is evident that

Hence, from Equation (13),

Development of pedagogical skills heightens the sustainability of an extended family defined by Equation (14). Such skills contribute to enhancing intergenerational wellbeing.

There are serious implications in Theorem 4 concerning desired nature of pedagogical skills and wellbeing. Generally, it is considered that education should delve into how students can earn higher income regardless of their displeasure and different kinds of abilities. Instead of such existing human capital theory, we emphasize that the acute necessity for education is inspiring their various intellectual curiosity, which facilitate to induce students into their talented jobs. This effect is expressed

Such a change in education will enhance social division of labor, and thus, surely equalizes income distribution compared with the economy in which every student is educated towards the same kind of job and cut-throat competition in labor market is inevitable.

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 much money incur heavier losses when their investment fails and, thus, it is reasoned that they are diligent and have a higher success probability in education. For the same reason, a non-wealthy family is excluded from capital markets, and deprived of opportunities 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 facilitated makes educational investment more effective, and this improves credit conditions for families. Accordingly, pedagogical skills play a key role in equalizing educational opportunities. In addition, high efficacy in education uniformly enhances the sustainability of extended families such that they can enjoy cultivated and affluent lives.

For these two reasons, pedagogical skills are of profound social significance.