The institution of marriage

Abstract

The institution of marriage is both old and ubiquitous. Yet, little work has been done by economists on why this social institution exists and why throughout history it has been intimately linked to fertility. We explain the institution of marriage as a societal consensus on the need to curb cuckoldry for the purpose of paternity certainty and biparental investment in offspring. By raising the costs of mating to individuals, marriage reduces cheating in society, a source of mating market failure, and makes paternity more certain. Men, in consequence, invest more in their putative offspring, a fact that also benefits mothers.

Appendix: Continuous choice

For expository and analytical reasons, we have changed to a continuous choice of mating partners in Section 5. Separable utility of women and men is isoelastic in nonreproductive consumption and Cobb–Douglas in the quantity and quality of own offspring (Eq. 19). Hence, the optimization problem of females is given by:

(20)

This change implies the following first-order necessary conditions for the representative female. Her optimal investment in child quality Q _Female has to meet the condition:

(21)

and her optimal continuous choice of male mating partners N _♀ has to satisfy:

(22)

A man’s expected utility from reproduction is determined by the same expression in child quality as in the female case times the (fractional) ν-th moment of the binomial distribution with N _♂ draws (female partners) and probability of success (fatherhood) δ:

(23)

According to Hoffmann-Jørgensen (1994, p. 303), the fractional moment of a nonnegative random variable can be calculated using the following formular:

(24)

where Γ is Euler’s Gamma function¹⁶ and M (τ) denotes the respective moment generating function with the property that the n-th moment about the origin is given by the n-th derivative evaluated at zero. The formular (24) can be obtained by using techniques of fractional calculus and taking the derivative of the $\nu^{\text{th}}$ order of M(τ) (for details see Wolfe 1975). For our purposes, it is important to note that K _♂ is indeed a nonnegative random variable and that the moment generating function of the underlying binomial distribution is given by:

(25)

Substituting the moment generating function (25) into formular (24) and applying the rule of integration by substitution with $z=t^{\frac{1}{\nu}}$ leads to the following result:

(26)

We can now address the man’s continuous choice optimization problem. Treating the probability of biological fatherhood δ as exogenously given, a male individual chooses q _♂ and N _♂ to maximize his expected utility:

(27)

This leads us to the following first-order necessary condition with respect to q _♂:

(28)

The first-order necessary condition with respect to N _♂, in turn, is as follows:

(29)

Note that given the probability of fatherhood δ, a man’s expected utility with respect to the number of own offspring, i.e., the fractional moment determined by Eq. 26, is increasing at a diminishing rate in the number of female partners N _♂:

(30)

In other words, the fractional moment is concave in N _♂.

The solution to the model is described by the first-order necessary conditions (Eqs. 21 and 22) for females, their counterparts for males (Eq. 28 and 29), the mating market clearing condition (Eq. 11), and the mating cost constraint (Eq. 12). We used the Mathematica function “FindRoot” to solve this system of equations for the optimal values of Q _♀, N _♀, q _♂, N _♂, s _♀, and s _♂.