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A widely held perception among managers at Japanese firms is that “human capital investment in women will be wasted because women quit when marrying or when having children” and that “women are less productive and less ambitious than men.” On the surface, these statements might appear to be the case. However, this problem is not the fault of female workers but a consequence of Japanese firms’ choices that create a selffulfilling prophecy—for which this chapter presents the rationale. More specifically, the analysis relies on the gametheoretic model of Coate and Loury (CL) and provides two new solutions for breaking the selffulfilling prophecy equilibrium not considered under the CL theory. Furthermore, the analysis modifies the CL theory—a statistical discrimination theory that integrates incentive problems—by formally theorizing incentive problems pertaining to indirect discrimination and presents new theoretical findings that are consistent with empirical facts regarding the gender disparity in the attainment of managerial and professional positions in Japan. Moreover, these mathematical theoretical models on discrimination clarify the importance of incentive problems in the promotion of gender equality in economic activities.
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The paradox presented by French economist Maurice Allais (
1953). Examples are repeatedly shown in experiments on how people’s preferred choices contradict expected utility theory.
The instrumentalvariable method is a causal analytical method in econometrics to eliminate selection bias in the effect of the treatment variable when unobserved confounding variables are expected to exist.
The final assumption of a monotonically decreasing likelihood ratio, called the assumption of stochastic dominance, is a necessary and sufficient condition for
\( F_{q} (\theta ) \le F_{u} (\theta ) \). As is subsequently described, this assumption generates a single local maximum to the selfinvestment supply curve for employees.
Therefore, when companies maximize their utility, restrictions are added to allocate Job 1 proportionally by gender using the Lagrange method. Although a detailed explanation is omitted, the CL theory indicates that the equilibrium under AA is equal to the replacement of the ratio between benefit and cost,
\( x_{q} /x_{u} \), when restrictions on companies are not in place, by
\( (x_{q}  \gamma /\lambda )/(x_{u} + \gamma /\lambda ) \) for men and by
\( (x_{q} + \gamma /(1  \lambda ))/(x_{u}  \gamma /(1  \lambda )) \) for women. Here,
\( \lambda \) is the proportion of male employees and
\( \gamma \) is a positive Lagrange constant representing the conditions of the restrictions placed by AA. This outcome is equivalent to companies paying a tax of
\( \gamma /\lambda \) for every male assigned to Job 1 and receiving a subsidy of
\( \gamma /(1  \lambda ) \) for every female assigned to Job 1.
Here, the mean and variance of the signal when
\( a = 1 \) are expressed as
\( E(\theta ) = \int_{0}^{1} {\theta f(\theta )d\theta } \) and
\( V(\theta ) = \int_{0}^{1} {(\theta  E(\theta ))^{2} f(\theta )d\theta } \), respectively. When doing so, the mean value when
\( a \ne 1 \) is:
with a variance of:
$$ \begin{aligned} E(\theta ,a) & = \int_{0}^{1} {\theta (a\theta^{a  1} f(\theta^{a} ))d\theta = \int_{0}^{1} {a\theta^{a} f(\theta^{a} )d\theta = \int_{0}^{1} {a\theta^{a} f(\theta^{a} )\left( {\frac{d\theta }{{d\theta^{a} }}} \right)} } } d\theta^{a} \\ & = \int_{0}^{1} {\theta f(\theta^{a} )} d\theta^{a} = \int_{0}^{1} {\phi^{1/a} f(\phi )d\phi } = E(\theta^{1/a} ) \\ \end{aligned} $$
$$ \begin{aligned} V(\theta ,a) & = \int_{0}^{1} {(\theta  E(\theta ))^{2} (a\theta^{a  1} f(\theta^{a} ))d\theta = \int_{0}^{1} {(\theta  E(\theta ))^{2} (a\theta^{a  1} f(\theta^{a} ))\left( {\frac{d\theta }{{d\theta^{a} }}} \right)d\theta^{a} } } \\ & = \int_{0}^{1} {(\theta  E(\theta ))^{2} f(\theta^{a} )d\theta^{a} } = \int_{0}^{1} {(\phi^{1/a}  E(\phi^{1/a} ))^{2} f(\phi )d\phi } \\ & = E((\theta^{1/a}  E(\theta^{1/a} ))^{2} \\ \end{aligned} $$
Both are dependent on the value of
\( a \).
If we construct a model similar to the CL theory regarding employer behavior, for example, in place of likelihood ratio
\( \varphi (\theta ) = f_{u} (\theta )/f_{q} (\theta ) \), employers may employ, for a given proportion of men expressed as
\( \lambda \), likelihood ratio
\( \varphi (\theta ) = ((1  \lambda )f_{f,u} (\theta ) + \lambda f_{m,u} (\theta ))/((1  \lambda )f_{f,q} (\theta ) + \lambda f_{m,q} (\theta )) \) to apply to Eq. (
7.4), to estimate a threshold for a given uniform
\( \pi \) value for the proportion of qualified persons. Then, a demand curve that does not depend on gender results. In reality, a difference exists in the selfinvestment behavior between men and women. Thus, this decisionmaking is pseudorational because it ignores the fact that the expected proportion of qualified persons
\( \pi \) differs according to gender. However, it is consistent with the assumption that companies are genderblind with respect to their employees. Moreover, the interpretation from this model is relevant only for the discussion on the relative costs of mismatches subsequently described and is unrelated to the other results in this section.
Kawaguchi (
2008) states that the situation in Japan in which women’s economic activity has failed to advance was formed by mutually complementary systems both in the workplace and at home, premised on the traditional division of labor. He elucidates the difficulty in breaking the equilibrium through the use of game theoretical analysis. The limitation of discussion in this chapter of this book, as also pointed out by Daiji Kawaguchi, is that the mathematical model of statistical discrimination and indirect discrimination is not extended to cover the family system.
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 Title
 Statistical and Indirect Discrimination: Revisiting the Incentive Problem
 DOI
 https://doi.org/10.1007/9789811376818_7
 Author:

Kazuo Yamaguchi
 Publisher
 Springer Singapore
 Sequence number
 7
 Chapter number
 7