Male investment in schooling with frictional labour and marriage markets

Abstract

We present an equilibrium model with inter-linked frictional labour and marriage markets. Women’s flow value of being single is treated as given, and it captures returns from employment. Men can undertake a costly ex-ante investment in schooling. In the marriage market, women search sequentially for men characterised by wages, so they use a reservation value strategy. Single unemployed men conduct marital sequential search and, with an eye on the marriage market, also conduct a so-called constrained sequential job search. Given this setup, schooling enhances men’s marriage prospects as well as their labour market returns. In turn, women’s behaviour affects men’s schooling investment decision and their optimal job search strategy. We establish that for any given set of parameters, there exists a unique market equilibrium where a fraction of men get educated, and show that this fraction decreases if women’s labour market returns increase. We also examine the robustness of such an equilibrium.

1 Introduction

One of the most puzzling reported trends is that, while recently both female and male educational attainment have been increasing, the rate of increase in male schooling has lagged behind. In an important study, Goldin et al. (2006) found that if women made up only 39% of U.S. undergraduates in 1960, within four decades they constituted the majority of U.S. college students and of those graduating with a bachelor’s degree. This is also documented by Becker et al. (2010a, 2010b). The trend is by no means limited to the U.S. The same study reported that, while school enrollment rates of women in 17 OECD countries were, in the mid-80’s, below those of men, by 2002 women’s college enrollment rates exceeded those of men in 15 of these countries.

In this paper, we explore a particular mechanism that, taken in isolation, leads to a decrease in male schooling as a direct response to an increase in female educational attainment - an effect that could contribute to the above observed lag. In order to single out this mechanism, we provide an equilibrium model of inter-linked frictional labour and marriage markets, with focus on men’s choice of schooling investment. If women select male partners based on their wages, education enhances men’s prospects in the labour market as well as their marriage opportunities, through access to better paid jobs. We establish the existence and investigate the properties of a market equilibrium where a proportion of men choose to invest in schooling and we show that an increase in single women’s labour market options (viewed as a proxy for their educational attainment) leads to a decrease in the equilibrium fraction of men who invest in education.

To understand the economics behind this result, note that the story is essentially a simple Becker-type human capital model with frictions and inter-linked markets, where men can make a costly investment in education. Women select marriage partners based on men’s earnings, and therefore any expected returns to the schooling investment now stem not only from the labour market (access to a better wage distribution), but also the marriage market (better marital prospects). Consider an equilibrium where a proportion of men get educated, so expected returns equal the fixed cost of schooling. Women’s marital choice is affected by their outside options as well as the proportion of educated single men. Then, any increase in women’s opportunity cost of marriage will make them pickier, so the marriage prospects of men worsen. As the two markets are inter-linked, this leads to men having to adjust their labour market strategies, and returns to educations change. For a given fixed cost of schooling, only a lower fraction of men who invest in education can restore the equality between expected returns to education and cost of education, by restoring the original female pickiness. To see the effect on individual incentives of men, note that if the increase in women’s pickiness harms the marital prospects of educated men relatively more compared to those of uneducated men, the returns to schooling drop overall due to this negative effect on the marriage market, leading to a decrease in the proportion of educated men. In turn, with fewer educated men around, women become less picky, and the associated decrease in the female pickiness continues until it returns to its old level. Only then will men be once again indifferent between acquiring education or not.

Note two important modelling devices we employ. First, in order to obtain a meaningful female reservation strategy in the frictional marriage market, we include the flow utility of a woman from being single. Here, this parameter will capture female labour market options and returns (proxy for educational attainment). Upon marriage a woman loses this flow value and, if the man is employed, his wage becomes a public good for the couple. Second, we assume there are no exogenous differences across men, and concentrate on the case where women do not marry unemployed men. Then, as it turns out, the only male heterogeneity that will matter to women in the marriage market is their earned wages: single women only accept a single employed man if his wage is higher than a particular female reservation wage. A single unemployed man is therefore involved in a so-called constrained sequential job search problem, whereby his marriage market prospects (marriageability) depend on his earnings, so his labour market strategy is a best response to the female reservation wage.

Importantly, it is this partial search equilibrium in the inter-linked frictional markets that determines the returns to education for men, both in terms of wages and marriage prospects. Then, for any given cost of schooling, and with a binary education decision problem, a market equilibrium where a fraction of men choose to undertake the schooling investment requires that the return to education equals the cost of education. Crucially, what ensures that this market equilibrium condition holds is the proportion of educated men. To see this, consider an increase in female education (labour market returns). Women become pickier in the marriage market and increase their marital reservation wage. However, the only direct effect of such an increase is on the partial equilibrium in the joint frictional markets. With a fixed cost of schooling, the equilibrium returns from education for males need to remain unchanged - and this can only happen if the proportion of men who invest in schooling decreases. This is because a change in the fraction of educated men affects the female reservation wage in the same direction as a change in female education attainment. Therefore, the former must react in the opposite direction for the economy to remain in equilibrium.

Please note that although our comparative statics exercise results in decreased male schooling, it does not preclude extensions with outcomes where both sexes increase their education: the mechanism we highlight in this paper would still be at work. Having said that, apart from contributing to an explanation for the puzzling trend documented in the literature, our setup and results also seem to be in line with several other empirical findings. In terms of women’s attitudes in the marriage market, Gould and Paserman (2003) find that women are pickier if female wages (viewed by them as a proxy for flow value of being single) increase. Similarly, Blau et al. (2000) find that higher labour market returns for females lead to lower marriage rates for women between ages 16–24 and 25–34. Finally, Oppenheimer (1988) and Oppenheimer and Lew (1995) argue that an improvement in labour market gains for women results in them delaying the timing of marriage.

We also examine the robustness of our market equilibrium in terms of whether a change in women’s schooling preserves an outcome characterised by a fraction of educated men, or whether it leads to a corner solution scenario. As it turns out, the answer involves a very intuitive comparison between the marginal effect of a change in female reservation value on the proportion of unmarriageable wages across men. We characterise all possible scenarios for a given pair of wage distribution functions, and this robustness exercise also offers some clues as to what happens if there are changes in these distributions. However, the relevance of posing the latter question is ultimately an empirical matter. Greenwood et al. (2016) argue that the documented changes in wage distributions across educated/uneducated men in the U.S. are a main factor behind schooling patterns across sexes. In stark contrast, Blundell et al. (2016) report that no such wage distribution shifts occurred in the U.K. for those born on or after 1960.

Some comments about a couple of our explicit and implicit assumptions. First, it is clear that the fundamental logic of the model is underpinned by the well known fact that couples tend to sort according to various traits - see Becker (1991). In our paper, the only relevant differentiating endogenous trait men bring to the marriage market is their wage, and this in turn is affected by an ex-ante educational choice. However, as the study of equilibrium class formation is not one of the objectives of the present work, here we restrict the sorting aspect in the frictional marriage market to women having a reservation strategy.¹

Second, note that our emphasis is on the “breadwinner effect” whereby direct selection into marriage is based on the male wage. Grossbard-Shechtman and Neuman (2003) argue for the importance of this effect. Third, as our focus is on the effect of an exogenous increase in female education on the fraction of men who invest in schooling, we choose to proxy the labour market returns of women with their flow utility outside marriage - just like in Blau et al. (2000). Furthermore, the empirical results in Gould and Paserman (2003) suggest that men do not seem to care much about their partner’s wage. In turn, Goussé et al. (2017) document that the labour supply of married women is significantly lower than that of single women, while the opposite is true for men. In addition, Gould and Paserman (2003) provide evidence that women build this into their expectations and behaviour in the marriage market. Using more recent data, Ludwig and Bruderl (2018) find a clear pattern of selection into marriage based on male wage (its level and growth), in line with the above results. The data we ourselves have used to test and corroborate other extensions of our core model (see the references at the end of Introduction) is also quite recent. All this empirical work confirms gender asymmetries concerning the link between marriage and labour markets, and therefore supports our modelling choice to simplify women’s labour market participation. Relaxing this in-built asymmetry would only distract our analysis from the specific transition mechanism we aim to study: having a symmetric setup would not affect our key results qualitatively, but would instead compromise the tractability of the model.²

As stressed by Autor et al. (2019), the focus on men’s educational choices given the marriage market prospects is a timely endeavour. In terms of the specific issue of male-female schooling gap, our findings complement the important contributions of Chiappori et al. (2009), Greenwood et al. (2016) and Zhang (2019), who all offer explanations that also emphasise the link between the marriage market and labour market.³

The approach in Chiappori et al. (2009) is completely different from ours. They investigate stable marriage assignments in a frictionless environment with transferable utility within couples, with the focus being on women’s choice of schooling and the factors that would lead to female educational attainment lagging behind or possibly overtaking that of men’s. In Chiappori et al. (2009) the effect of the link between the marriage and labour market works through the complementarity in the home production function, as well as the discrete effects on the marital surplus of both sexes (and hence incentives to invest in education) that is generated by a change in the side of the market whose educated individuals are in short supply.⁴ In our model, the results do not hinge on the latter, and this is because we explicitly consider search frictions. Furthermore, in order to isolate the market-level mechanism of interest, we disregard the above-mentioned complementarity so that we avoid the related within-marriage incentives it would generate. As we will argue later, including such complementarity in home production function does not destroy our key results.

Greenwood et al. (2016) develop and estimate a unified model of marriage, divorce, educational attainment, and married female labour force participation that can explain multiple empirical facts. In their model marriage is embedded in a search theoretic framework, and while everyone individually makes a college investment decision, married couples decide jointly on women’s labour force participation. Of particular interest to us are their results concerning women’s education and labour market participation. Because men are assumed to work full time, while women’s work behaviour is endogenous and taken by the household as a whole, technological progress in home production coupled with shifts in the wage structure have a stronger impact on the education decision of women, with obvious consequences on female labour market participation.⁵

In turn, Zhang (2019) presents an investment and marriage model in which people have the option to make an investment in college education as well as a (post-college) career investment in education. Due to loss of fecundity, a post-college investment is more costly for women than for men, and as a result high income women are relatively scarce. In the unique equilibrium of the model, this marriage market effect provides an additional incentive for women to go to college, relative to men, thus generating an educational gap in favour of women. However, one perhaps should note that it is then not easy to see how this model could explain the education gap in favour of men observed pre-1990s.

Beyond the particular question tackled here, the present paper is part of a research agenda whose main message is that many observed outcomes in labour markets (including human capital accumulation) may very well be the result of individuals’ considerations and expectations in the marriage market - and vice-versa. As such, this work builds on Bonilla and Kiraly (2013) and Bonilla et al. (2019), where the concept of constrained sequential job search was first introduced and analysed in detail. For more recent papers that consider theoretical extensions as well as empirical tests of our core model, please see Bonilla et al. (2022a) and Bonilla et al. (2022b).

2 The model

We consider steady state equilibria of an economy that consists of a continuum of risk neutral men and women, where all agents discount the future at rate r.

Men enter the economy unemployed, single and of type L. The distribution of wages faced by them is F_L(. ) with continuous support $[{\underline{w}}_{L},{\overline{w}}_{L}]$. Men have a choice whether to enter the labour market immediately, or undertake an investment in education at a given cost c, same for all. An individual who undertakes the schooling investment becomes a type H man who now faces a wage distribution F_H(. ) with support $[{\underline{w}}_{H},{\overline{w}}_{H}]$. We assume that F_H(. ) first order stochastically dominates F_L(. ), so one can think of male types as capturing different educational attainments. Following the education decision, all men (H and L) become active in both the labour market and the marriage market. In the former, they look for wage offers using costless random sequential search, and job opportunities arrive at rate λ₀. If employed at wage w, a man receives the flow payoff w. We assume there is no on-the-job search. While active in the labour market, single men also conduct a costless sequential search in the frictional marriage market, looking for partners. Marriage requires mutual acceptance, and we assume that divorce is not possible. For a man, marriage confers a flow payoff y that captures the non-economic utility of the partnership. Overall therefore, a married man employed at wage w has a flow payoff w + y.

Women enter the economy single, and let x > 0 denote the flow payoff of a single woman. This parameter is crucial for our investigation, as it captures a woman’s options outside marriage. Here, we interpret this as her career opportunities, where an increase in x would mean higher labour market returns, possibly due to higher ex-ante schooling. Single women look for single males using costless sequential search, but (as we will show) they are not interested in marrying unemployed men. Hence, for them the relevant wage distribution is that of wages earned by single type i men (i = L, H), denoted by G_i(. ). Once a marriage partnership is formed, the woman loses x and enjoys a flow utility equal to her partner’s wage only.⁶

Given sequential search and the fact that utilities are monotonic in wages (for both sexes), the optimal strategies for women and men are characterised by the reservation property. Let R_i denote the reservation wage of unemployed type i men in the labour market. Similarly, let T_i denote the reservation wage of women in the marriage market, meaning an employed man of type i is marriageable only if his wage is no lower than T_i. As we will show, all equilibria we consider are characterised either by men matching the female reservation wage, or men setting a labour market reservation wage that falls short of female expectations in the marriage market.

Everyone (irrespective of employment and marital status) leaves the economy at rate δ. Let Γ denote the exogenous flow (measure) of new (unemployed and single) men who enter the economy at every instance, and let N_i denote the number of marriageable employed single men of type i. Similarly, let n denote the measure of single women; it is exogenous as we assume that a new single woman comes into the market every time a single woman gets married or exits the economy. Denote by ${\lambda }_{w}^{i}$ the rate at which a single woman meets an eligible bachelor of type i, and let λ_m denote the rate at which single men meet single women. We assume a quadratic matching function with parameter λ that measures the efficiency of the matching process. Then, we have ${\lambda }_{w}^{i}=\frac{\lambda ({N}_{H}+{N}_{L})n}{n}\frac{{N}_{i}}{({N}_{H}+{N}_{L})}=\lambda {N}_{i}$, and ${\lambda }_{m}=\frac{\lambda ({N}_{H}+{N}_{L})n}{({N}_{H}+{N}_{L})}=\lambda n$, where both N_i and ${\lambda }_{w}^{i}$ are of course endogenous. Crucially, let τ denote the (endogenous) proportion of male entrants who decide to invest in schooling. Our main objective is to investigate the effect of a change in x on the steady state equilibrium fraction of educated men.

3 Steady-state and optimal search

3.1 Unemployed men, marriageable men, and wages

Let u_i denote the number of unemployed men of type i. In steady state we require Γτ = u_H[δ + λ₀(1 − F_H(R_H))]. That is, the inflow of unemployed men who choose to invest in schooling needs to equal the outflow of these educated unemployed, either into employment (at an acceptable wage) or full exit. Consequently, in steady state we have:

$${u}_{H}=\frac{\tau \Gamma }{\delta +{\lambda }_{0}\left[1-{F}_{H}({R}_{H})\right]},$$

and

$${u}_{L}=\frac{(1-\tau )\Gamma }{\delta +{\lambda }_{0}\left[1-{F}_{L}({R}_{L})\right]}.$$

Using the same logic, the number of single marriageable men of type i is given by:

$${u}_{i}{\lambda }_{0}[1-{F}_{i}({T}_{i})]={N}_{i}(\lambda n+\delta ).$$

Then, using u_i as above, we obtain:

$${N}_{H}=\frac{\tau \Gamma }{\delta +{\lambda}_{0} \left[1-{F}_{H}({R}_{H})\right]}\frac{{\lambda }_{0}[1-{F}_{H}({T}_{H})]}{\lambda n+\delta }$$

and

$${N}_{L}=\frac{(1-\tau )\Gamma }{\delta +{\lambda}_{0} \left[1-{F}_{L}({R}_{L})\right]}\frac{{\lambda }_{0}[1-{F}_{L}({T}_{L})]}{\lambda n+\delta }.$$

Crucially, as we will show, the equilibria where the marriage market does affect men’s behaviour in the labour market are either characterised by either R_i = T_i or R_i < T_i, with obvious implications concerning marriage prospects.

Furthermore, since the female reservation policy depends on N_i, and the latter is clearly a function of R_i and T_i (see above), here we briefly examine this particular link in the two scenarios relevant for our equilibria:

(i) In an equilibrium with R_i < T_i the number of marriageable men N_i increases with R_i. Given that the wage distributions are exogenous for both types, if the reservation wages R_i increase, men of type i leave unemployment at a lower rate, so the steady state u_i increases. In addition, the rate at which unemployed men accept marriageable wages remains unchanged. Hence, all this leads to an increase in N_i when R_i increases. Furthermore, N_i increases when the female reservation wage T_i decreases as men now find marriageable wages at a higher rate. (ii) In contrast, when R_i is optimally set equal to T_i an increase in T_i (and thus R_i) results in a decrease in N_i.

Finally, the distribution of wages earned by marriageable men of type i is given by the steady state condition:

$${u}_{i}{\lambda }_{0}[{F}_{i}(w)-{F}_{i}({T}_{i})]={G}_{i}(w){N}_{i}(\lambda n+\delta ).$$

In the above, the number of marriageable men of type i with wages no higher than w is G_i(w)N_i, and they leave this stock if they get married or exit the economy altogether. The left-hand side captures the flow of unemployed men of type i who find and accept a job with a wage that confers marriageability but which is no higher than w.

From here, also using the solution for N_i previously obtained, we have:

$${G}_{i}(w)=\frac{{F}_{i}(w)-{F}_{i}({T}_{i})}{1-{F}_{i}({T}_{i})}.$$

3.2 Optimal search: women

In this section, we derive the female reservation wage, noting straight away that T_H = T_L( ≡ T).⁷

To start, we need to investigate women’s decision whether or not to marry unemployed men. Denote the pure labour market (i.e., without marriage market) reservation wage of type i men by ${\underline{R}}_{i}$ where:

$${\underline{R}}_{i}=\frac{{\lambda }_{0}}{r+\delta }\int\nolimits_{{\underline{R}}_{i}}^{{\overline{w}}_{i}}\left[w-{\underline{R}}_{i}\right]d{F}_{i}(w)$$

(1)

First, we establish that under certain natural conditions women only marry employed single men.⁸

Proposition 1

Women only marry unemployed single men of type i if the female reservation wage T is no higher than ${\underline{R}}_{i}$.

Proof.

See Appendix A. □

The proof above follows to a great extent the one carried out in Bonilla et al. (2019), with the crucial difference that in the present paper male types (here, education) are endogenous. It is perhaps worth noting that the intuition behind Proposition 1 hinges on the assumption that there is no divorce. Without the possibility of divorcing a man, a woman knows that if she is to marry an unemployed, his reservation wage will immediately drop to ${\underline{R}}_{i}$ since for him the marital market becomes irrelevant after marriage. A woman’s utility in marriage then mirrors the man’s utility in marriage: her expected flow utility in marriage is the man’s wage, and is thus determined by the man’s job acceptance strategy. It follows that marriage to an unemployed man is, for the woman, equivalent to marrying a man employed at a wage equal to ${\underline{R}}_{i}$ and she would only marry unemployed men if $T\le {\underline{R}}_{i}$.

Throughout this paper, we study equilibria characterised by $T > {\underline{R}}_{H}$. By doing so, we essentially eliminate the uninteresting equilibrium where the marriage market does not affect men’s job search, as they are always marriageable, even when unemployed.

Next, we turn to the derivation of women’s reservation wage. Use W^S to denote the value of being single for a woman. Standard derivations lead to the Bellman equation:

$$\begin{array}{l}(r+\delta ){W}^{S}=x+\lambda {N}_{H}\displaystyle\int\nolimits_{T}^{{\overline{w}}_{H}}\left[{W}_{H}^{M}(w)-{W}^{S}\right]d{G}_{H}(w)\\\qquad\qquad\quad\;\;\, +\,\lambda {N}_{L}\displaystyle\int\nolimits_{T}^{{\overline{w}}_{L}}\left[{W}_{L}^{M}(w)-{W}^{S}\right]d{G}_{L}(w).\end{array}$$

Making use of the solutions for N_i and G_i(w) previously obtained, we get:

$$\begin{array}{l}(r+\delta ){W}^{S}=x+\frac{\lambda \tau \Gamma {\lambda }_{0}}{[\delta +{\lambda}_{0} (1-{F}_{H}({R}_{H}))](\lambda n+\delta )}\displaystyle\int\limits_{T}^{{\overline{w}}_{H}}\left[{W}_{H}^{M}(w)-{W}^{S}\right]d{F}_{H}(w)\\\qquad\qquad\quad\;\; +\,\frac{\lambda (1-\tau )\Gamma {\lambda }_{0}}{[\delta +{\lambda}_{0} (1-{F}_{L}({R}_{L}))](\lambda n+\delta )}\displaystyle\int\limits_{T}^{{\overline{w}}_{L}}\left[{W}_{L}^{M}(w)-{W}^{S}\right]d{F}_{L}(w).\end{array}$$

Finally, using W^S = T/(r + δ) and applying standard integration by parts, we obtain:

$$\begin{array}{l}T=x+\frac{\lambda \tau \Gamma {\lambda }_{0}}{[\delta +{\lambda}_{0} (1-{F}_{H}({R}_{H}))](\lambda n+\delta )}\displaystyle\int\nolimits_{T}^{{\overline{w}}_{H}}[1-{F}_{H}(w)]dw\\\qquad +\,\frac{\lambda (1-\tau )\Gamma {\lambda }_{0}}{\left[\delta +{\lambda}_{0} (1-{F}_{L}({R}_{L}))\right](\lambda n+\delta )}\displaystyle\int\nolimits_{T}^{{\overline{w}}_{L}}[1-{F}_{L}(w)]dw.\end{array}$$

(2)

At this point, we make three observations that will be important in establishing how the fraction of educated men τ adjusts to an exogenous shock in x:

First, clearly ∂T/∂x > 0: as one would expect, women raise their reservation wage in the marriage market if their instantaneous utility while single increases. Second, ∂T/∂τ > 0: intuitively, a ceteris paribus increase in the fraction of educated men with better job prospects makes women pickier, since their marriage market prospects have also improved now. Third, please recall the discussion around N_i in Section 3.1 above: ceteris paribus, a higher reservation wage of type i men increases the number of marriageable men. Hence, ∂T/∂R_i > 0 reflecting that again women become pickier.

3.3 Optimal search: men

We are interested in equilibria in which the marriage market affects all men’s decisions in the labour market. For any $T \,>\, {\underline{R}}_{H}$ single unemployed men of both types undertake a so-called constrained search, knowing that by accepting a particular wage (for life), they either become marriageable or lose forever the prospect of marriage. Their optimal strategy is described by a function that assigns a labour market reservation wage for any female marriage market reservation wage: R_i(T).

In what follows, we fully characterise this function. Although the derivation steps are the same for both types,⁹ the actual reservation wage functions will be different across types, essentially due to the fact that men with different types (schooling choices) face different wage distributions. The main insight is that this function is non-monotonic in the female reservation wage, and has a unique maximum, attained at $T={\widehat{T}}_{i}$, where the latter is implicitly defined by the unique solution to:

$${\widehat{T}}_{i}=\frac{{\lambda }_{0}}{r+\delta }\left[\int\nolimits_{{\widehat{T}}_{i}}^{{\overline{w}}_{i}}\left[1-{F}_{i}(w)\right]dw+\frac{\lambda n[1-{F}_{i}({\widehat{T}}_{i})]}{r+\delta +\lambda n}y\right].$$

(3)

Clearly, for y > 0 and $0\le {F}_{i}({\widehat{T}}_{i}) < 1$, we have ${\widehat{T}}_{i} \,>\, {\underline{R}}_{i}$.

The formal derivation is as follows. Overall, a man (of either type) can be in one of three states: unemployed and single, employed at wage w and single (S), or employed at wage w and married (M). Denote a type i man’s value of being unemployed by U_i, and let ${V}_{i}^{S}(w)$ describe the value of being single and earning a wage w. Standard derivations lead to the Bellman equation for a type i unemployed man:

$$(r+\delta ){U}_{i}={\lambda }_{0}\displaystyle\int\nolimits_{{\underline{w}}_{i}}^{{\overline{w}}_{i}}\max \left[{V}_{i}^{S}(w)-{U}_{i},0\right]d{F}_{i}(w).$$

Anticipating that ${V}_{i}^{S}(w)$ is not a continuous function, we can define:

$${R}_{i}(T)\equiv \min \left\{w:{V}_{i}^{S}(w)\ge {U}_{i}\right\}.$$

With no divorce, the value of being married and earning a wage w is ${V}_{i}^{M}(w)=(w+y)/(r+\delta )$. Hence, for any T, we have:

$${V}^{S}(w)=\left\{\begin{array}{ll}\frac{w}{r+\delta }\qquad\qquad\qquad\qquad\quad{{\mbox{if}}}\,\,w \,<\, T\\ \frac{w}{r+\delta }+\frac{\lambda n}{(r+\delta +\lambda n)(r+\delta )}y\,\qquad{{\mbox{if}}}\,\,w\ge T\end{array}\right\}.$$

Please note that when λn = 0 (i.e., no marriage market), we have ${V}_{i}^{S}(w)=w/(r+\delta )$ for all w and, from ${U}_{i}={V}_{i}^{S}({R}_{i})$, standard manipulation yields ${R}_{i}={\underline{R}}_{i}$. As stated before, this is also the reservation wage that would be chosen by a hypothetical unemployed married man since, without divorce, this man is no longer involved in the marriage market.

It follows that the Bellman equation for unemployed men (also used in the proof of Proposition 2 below) is given by:

$$\begin{array}{l}(r+\delta ){U}_{i}={\lambda}_{0}\displaystyle\int\nolimits_{{R}_{i}}^{T}\left[\frac{w}{r+\delta}-{U}_{i}\right]d{F}_{i}(w)\\\qquad\qquad\quad +\,{\lambda}_{0}\displaystyle\int\nolimits_{T}^{{\overline{w}}_{i}}\left[\frac{w}{r+\delta}+\frac{\lambda n}{(r+\delta +\lambda n)(r+\delta )}y-{U}_{i}\right]d{F}_{i}(w),\end{array}$$

or

$$(r+\delta ){U}_{i}={\lambda}_{0}\int\nolimits_{{R}_{i}}^{{\overline{w}}_{i}}\left[\frac{w}{r+\delta}-{U}_{i}\right]d{F}_{i}(w)+\frac{{\lambda}_{0}\lambda n\left.\right(1-{F}_{i}(T)}{(r+\delta +\lambda n)(r+\delta )}y.$$

Using U_i = R_i/(r + δ), and after standard integration by parts, we obtain:

$${R}_{i}(T)=\frac{{\lambda }_{0}}{r+\delta }\int\nolimits_{{R}_{i}}^{{\overline{w}}_{i}}\left[1-{F}_{i}(w)\right]dw+\frac{{\lambda }_{0}\lambda n\left[1-{F}_{i}(T)\right]}{(r+\delta )(r+\lambda n+\delta )}y.$$

The Proposition below presents the full characterisation of the male reservation wage function.

Proposition 2

The reservation wage function R_i(T) is continuous, piece-wise differentiable, and:

• (a) ${R}_{i}={\underline{R}}_{i}$ for $T\le {\underline{R}}_{i}$ and $T \,>\, {\overline{w}}_{i}$;

• (b) R_i = T for $T\in ({\underline{R}}_{i},{\widehat{T}}_{i}]$;

• (c) ${R}_{i}(T)=\frac{{\lambda }_{0}}{r+\delta }\displaystyle\int\nolimits_{{R}_{i}}^{{\overline{w}}_{i}}\left[1-{F}_{i}(w)\right]dw+\frac{{\lambda }_{0}\lambda n\left[1-{F}_{i}(T)\right]}{(r+\delta )(r+\lambda n+\delta )}y$, with R_i < T and R_i decreasing for $T\in ({\widehat{T}}_{i},{\overline{w}}_{i}]$.

Furthermore, ${\widehat{T}}_{H} > {\widehat{T}}_{L}$ and ${\underline{R}}_{H} \,>\, {\underline{R}}_{L}$.

Proof.

See Appendix B. □

In essence, when the marriage market does affect men’s job search strategy, unemployed males can react in two ways. For relatively low values of female reservation wages, they hold out for such wages as R_i is equal to T.¹⁰ At the critical ${\widehat{T}}_{i}$ the labour market related cost of holding out for it equals the gains from the marriage market. For even higher female reservation wages men gradually give up on trying to match T, so they only get married if they are lucky and land a high enough wage. This is because higher and higher female reservation wages make it less and less likely to encounter a marriageable wage, so the male reservation wage decreases.

Two further observations follow. First, men’s value of unemployment U_i is not directly affected by x, since women’s flow utility of being single does not affect the male reservation wage functions, i.e., ∂R_i(T)/∂x = 0. However, note that x will of course affect the equilibrium male reservation wages through its direct effect on the female reservation wage function T. Second, the value of unemployment U_i is not directly affected by the proportion of educated men either, so ∂R_i/∂τ = 0.

4 Equilibrium

In this section, we investigate the existence and properties of a market equilibrium by first looking at the partial search equilibrium in the joint frictional markets, and then pinning down the steady state fraction of educated men that is consistent with optimal schooling investment choices. Intuitively, as they face a binary decision, men will choose to invest in schooling as long as the returns from education—as captured here by the difference in the values of educated and uneducated single unemployed men, is higher than the cost of schooling. A mixed market equilibrium with a fraction of educated men therefore requires ΔU( ≡ U_H − U_L) = c, meaning all males are indifferent between investing in schooling or not.

The exogenous parameter x plays a key role in the determination of the partial equilibrium in the labour and marriage markets. In addition, women also take τ as given, and it affects them as it determines the quality of the pool of employed men they face in the marriage market. Crucially, τ is the only endogenous variable that can adjust to ensure the equality between the returns to education and cost of schooling.

The central result of our paper concerns the nature of the interaction between these two variables x and τ. To get a flavour of the argument, consider a change in the flow utility of single women. This has a direct effect on T(R_H, R_L) only - recall that neither the value of unemployment nor the reservation wages of men are directly affected. However, the shift in T(R_H, R_L) itself has an immediate impact on U_H and U_L due to the change in the proportion of marriageable wages (different across male types). This leads to an adjustment in male reservation wages and with that, also a change in the returns to education. With comparative statics in mind recall that we have ∂U_i/∂x = 0 and ∂U_i/∂τ = 0, so τ affects T(R_H, R_L) only, with ∂T/∂τ > 0. Therefore, τ is indeed the only endogenous variable that can adjust in order to restore ΔU to its equilibrium level. After we establish this main result, we also investigate the adjustment of τ following a shock in x, in order to gain further insights about the robustness of our equilibrium.

4.1 Partial search equilibrium

First, note that the number of steady state educated single unemployed men (u_H) is essentially determined by the proportion of men who decide to invest in schooling, τ. Taking this as given, a search equilibrium for the inter-linked frictional markets is then the triplet $\{{R}_{L}^{* },{R}_{H}^{* },{T}^{* }\}$ together with steady state conditions, such that male reservation wages satisfy Proposition 2 and the female reservation wage satisfies (2). There are three types of potential equilibria: Type 1, characterised by ${R}_{i}^{* } < {T}^{* },{R}_{L}^{* } < {R}_{H}^{* }$ and ∂R_i/∂T < 0; Type 2, characterised by ${R}_{H}^{* }={T}^{* },{R}_{L}^{* } < {T}^{* }$ and ∂R_L/∂T < 0; and Type 3, with ${R}_{i}^{* }={T}^{* }$ for i = H, L.

Proposition 3

A partial search equilibrium exists and it is unique. In any such partial equilibrium ∂T^*/∂x > 0 and ∂T^*/∂τ > 0.

Proof.

To show existence, recall that R_i(T) is continuous and non-monotonic, with ${R}_{i}(T)={\underline{R}}_{i}$ for both $T={\underline{R}}_{i}$ and $T={\overline{w}}_{i}$, while T(R_H, R_L) is continuous and increasing in R_i for i = H, L.

The second statement is proved by contradiction. Consider a potential Type 1 equilibrium. Let x increase and assume a resulting new equilibrium with a lower T^*. Since ∂R_i/∂T < 0 while the reservation wage of a particular male type is not directly affected by the reservation wage of the other type, this would necessarily involve higher ${R}_{i}^{* }$. From (2), a higher x together with higher ${R}_{i}^{* }$ unambiguously results in a higher female reservation wage, which contradicts the initial assumption. Consider next a potential Type 3 equilibrium, and increase x. Imposing ${R}_{i}^{* }={T}^{* }$ in (2) we have ∂T/∂x > 0, and therefore a new equilibrium with a lower T^* would be a contradiction. Finally, consider a potential Type 2 equilibrium, and increase x. Given that ∂R_L/∂T < 0 and ${R}_{H}^{* }$ is optimally set equal to T^*, the above two arguments once again imply that a new equilibrium with a lower T^* would be a contradiction.

The reasoning which proves that ∂T^*/∂x > 0 also implies that the female reservation wage function implicitly given in (2) crosses the 45 degree line from below, and hence the uniqueness of equilibrium follows. Finally, an increase in τ shifts the female reaction function in (2) to the right. □

It is worth stressing the role played by N_i in the above results. Please recall the discussion in Section 3.1. Now, for the Type 1 equilibrium, an increase in R_i leads to an increase in N_i, which (ceteris paribus) makes women pickier. This, coupled with an increase in x must result in a higher female reservation wage. For an equilibrium of Type 2 with ${R}_{i}^{* }={T}^{* }$, a lower equilibrium female reservation wage would mean a higher N_i. But the combined effect of both a higher N_i and a higher x is that single women become pickier.

Panel (a) in Fig. 1 captures a Type 1 partial search equilibrium for the joint labour and marriage markets, with the reservation wage of educated men plotted against the female reservation wage. Panel (b) captures the same partial search equilibrium, but this time with the reservation wage of uneducated men plotted against the female reservation wage. Note that in Panel (b) the female reservation wage is positioned more to the right compared to the case depicted in Panel (a). This configuration always holds for R_L > R_H.

Fig. 1

Partial search equilibrium. The panels show the equilibrium triplet (${R}_{H}^{* },{R}_{L}^{* },{T}^{* }$). a depicts the reservation wage of educated men as a function of T (R_H(T)) from item (c) in Proposition 2, as well as women’s reservation wage as a function of this reservation wage (T(R_H)) from equation (2). b is analogous but involving R_L instead of R_H. Analysis of (2) shows that (T(R_L)) is to the right of (T(R_H))

We are now also in the position to describe the range of the parameter x for which these different types of partial search equilibria obtain. To this end, define x₁, x₂, x₃, and x₄ such that ${T}^{* }({x}_{1})={\underline{R}}_{H},{T}^{* }({x}_{2})={\widehat{T}}_{L},{T}^{* }({x}_{3})={\widehat{T}}_{H}$, and ${T}^{* }({x}_{4})={\overline{w}}_{L}$. Then, for $x\in \left({x}_{3},{x}_{4}\right]$ a Type 1 equilibrium exists; for $x\in \left({x}_{2},{x}_{3}\right]$ a Type 2 equilibrium exists, while a Type 3 equilibrium exists for $x\in \left({x}_{1},{x}_{2}\right]$.

Finally, Fig. 2 below illustrates the effect of an increase in x or an increase in τ on the Type 1 partial equilibrium:

Fig. 2

Comparative statics in partial search equilibrium. The shifts of T(R_i) to the right that result from either an increase in x or from an increase in τ. The new partial search equilibrium exhibits lower R_i. a T(R_H, R_L) against R_H, b T(R_H, R_L) against R_L

Please note that an increase in either x or τ leads to shifts of T(R_i) to the right in both panels, and the resulting partial search equilibrium is characterised by lower R_i.

4.2 Market equilibrium with schooling

Our focus is on a mixed market equilibrium (MME) characterised by a Type 1 partial search equilibrium in the joint frictional markets and a fraction of men choosing to get educated. Men will choose to invest in schooling as long as ΔU ≡ U_H − U_L > c, and hence a necessary condition for an interior equilibrium is that ΔU = c, so all men are indifferent between paying or not for education. Because in a Type 1 partial search equilibrium U_i = R_i/(r + δ) the above condition for such a mixed market equilibrium amounts to:

$$\Delta {R}^{* }\left(\equiv {R}_{H}^{* }-{R}_{L}^{* }\right)=(r+\delta )c.$$

Clearly, this equality pins down the value of returns to education required for an equilibrium. Since ∂R_i(T)/∂T < 0 for $T > {\widehat{T}}_{H}$, while ∂R_L(T)/∂R_H = ∂R_H(T)/∂R_L = 0, we can write ΔR = ΔR(T).

Assume (for now) that ΔR is monotonic in T.¹¹ Then, the above condition in fact pins down a unique market equilibrium value of T, and implicitly the associated partial search equilibrium values of R_H and R_L. In other words, a market equilibrium has $T=\widetilde{T}$ such that $\Delta R(\widetilde{T})=(r+\delta )c$.

Although the proportion of men who choose to invest in schooling is endogenous in the overall market equilibrium, it acts as a parameter in the determination of T^* in the partial search equilibrium. More specifically, we have T^*(x, τ). Then, the overall market equilibrium requires $\tau =\widetilde{\tau }$ such that:

$${T}^{* }(x,\widetilde{\tau })=\widetilde{T}$$

(4)

Crucially, note that τ is the variable that ensures that this market equilibrium condition holds: it will respond to changes in any parameters affecting T^* in the partial search equilibrium, such as parameter x highlighted in (4).

Our main objective is to establish the effect of an increase in single women’s flow utility on the equilibrium fraction of men who invest in schooling. First, we provide the formal definition of the mixed market equilibrium MME. To ensure that we differentiate in the correct manner, please observe that * refers to partial as well as overall market equilibrium variables, while ~ refers only to the latter.

Definition 1

A mixed market equilibrium (MME) consists of the quadruple $\{{R}_{i}^{* },{T}^{* },\widetilde{\tau }\}$ where:

• i. In the neighbourhood of $\widetilde{\tau }$ the partial search equilibrium is of Type 1 and thus ${R}_{i}^{* }$ is as in Proposition 2(c),

• ii. T^* solves (2),

• iii. $\widetilde{\tau }\in (0,1)$ solves (4).

The focus here is on the equilibrium with $x\in \left({x}_{3},{x}_{4}\right]$, and we are ready to state our main result:

Theorem 1

Consider a mixed market equilibrium (MME) in the neighbourhood of $\widetilde{\tau }$ where the partial search equilibrium is of Type 1. If x increases, the equilibrium will be characterised by the same T^* and a lower proportion of educated men.

Proof.

Condition (4) needs to hold for an MME. Recall that x and τ are both parameters in the partial search equilibrium that determines T^* and hence ΔR^*. For (4) to hold after an increase in x, there must be a change in τ such that dT^*(x, τ) = 0. Total differentiation of T^* yields $d{T}^{* }(x,\tau )=\frac{\partial {T}^{* }}{\partial x}dx+\frac{\partial {T}^{* }}{\partial \tau }d\tau$. Since ∂T^*/∂x > 0 (see Proposition 3), an increase in x leads to a higher T^*, so $\frac{\partial {T}^{* }}{\partial x}dx \,>\, 0$. Hence, dT^*(x, τ) = 0 only if $\frac{\partial {T}^{* }}{\partial \tau }d\tau \,<\, 0$. As ∂T^*/∂τ > 0 (again, see Proposition 3), this in turn requires dτ < 0, i.e., a decrease in τ. □

To see this, consider equilibrium condition (4), and an increase in x. Such an increase affects the triplet $\{{R}_{H}^{* },{R}_{L}^{* },{T}^{* }\}$ through its direct effect on T(R_i) only. Ceteris paribus, women become pickier in the marriage market, and hence men’s marriage prospects suffer (again, ceteris paribus). As a consequence, men re-optimise: as T(R_i) shifts to the right, the male reservation wages of both types decrease (see Fig. 2). Ceteris paribus (i.e., for a given τ), new returns to education have now been determined. Clearly, since a change in x does not alter the cost of schooling, equilibrium condition (4) does not hold anymore. But both x and τ affect women’s value of being single (and thus T) in the same direction, and the partial search equilibrium remains at T^*. As a consequence, only a lower fraction of men who invest in education can restore ${T}^{* }=\widetilde{T}$ so the equality between expected returns to education and cost of education holds again.

4.3 Robustness

We now address the question of robustness of our mixed market equilibrium MME.

4.3.1 MME with Type 1 partial search equilibrium

With ΔR monotonic in T, there is only one mixed strategy equilibrium, but the economy may also settle on corner equilibria with τ = 1 (all men educated) or τ = 0 (no educated men). Each of the latter two would correspond to a different partial search equilibrium, and would of course be consistent with our determination of T^* in equation (4).

We show below that the robustness of the overall economy equilibrium to a change in x depends on whether ΔR is, in the prevailing equilibrium, increasing or decreasing in T. Starting from the mixed strategy equilibrium, we show the conditions such that an increase in x results in a new mixed market equilibrium (with a different τ), but corresponding to the same partial search equilibrium. By doing this, we also establish the conditions under which an increase in x takes the economy to a corner solution. Which outcome prevails depends of course on how a change in x affects the incentives of individual men of each type.

First, note that ceteris paribus an increase in T (due to an increase in x) has a negative effect on all men in the sense that it leads to a decrease in the proportion of marriageable wages. The magnitude of this effect depends on the equilibrium value T^* as well as the respective wage distribution functions, so the decrease is generically different across male types. From Proposition 2(c), we have ∂ΔR/∂T < 0 if in equilibrium (at T^*) the negative effect of an increase in T is stronger for H type men than for L type men - that is:¹²

$$\frac{\partial {F}_{H}(T)}{\partial T}/\frac{\partial {F}_{L}(T)}{\partial T} > \frac{r+\delta +{\lambda }_{0}[1-{F}_{H}({R}_{H})]}{r+\delta +{\lambda }_{0}[1-{F}_{L}({R}_{L})]}$$

(5)

Intuitively, the returns to education diminish as the female reservation wage increases if the increase in the proportion of unmarriageable wages in the distribution F_H(. ) faced by educated men is high enough relative to that in the distribution faced by uneducated men F_L(. ), where “high enough” takes into account the fact that the female reservation wage affects employment probabilities through its effect on male reservation wages. Proposition 4, together with Fig. 3 and the analysis below address the consequences of this.

Fig. 3

Decreasing and increasing ΔR. In panel (a), the returns to education are decreasing in T, Condition (5) is satisfied. After an increase in x, the new equilibrium has the same returns to education as the original equilibrium, but it has a lower proportion of educated men. In panel (b), Condition (5) is not satisfied

Proposition 4

Consider a mixed market equilibrium (MME) in the neighbourhood of $\widetilde{\tau }$ where the partial search equilibrium is of Type 1.

• (a) If inequality (5) holds, an increase in x leads to either (i) an MME with same T^*and a lower proportion of educated men, or (ii) a corner solution with no educated men.

• (b) If inequality (5) holds in the opposite direction, an increase in x leads to a corner solution with no uneducated men.

Proof.

In the MME we have ΔR^*(T^*(x, τ)) = (r + δ)c. Ceteris paribus, an increase in x leads to an increase in T^*. When (5) holds we have ∂ΔR/∂T < 0, and therefore now ΔR(T) < (r + δ)c, so τ adjusts downwards (reversing the increase in T^*) until either the original partial equilibrium $\left\{{R}_{i}^{* },{T}^{* }\right\}$ is restored or a corner solution emerges, with τ = 0. When (5) holds in the opposite direction, ∂ΔR/∂T > 0, and hence ΔR(T) > (r + δ)c, so τ adjusts upwards, thus increasing T even further. As ΔR is monotonic, the cycle continues until τ = 1. □

The results in Proposition 4 are illustrated in Fig. 3 below.

In Panel (a), the returns to education are decreasing in T. As x increases, T(R_i) shifts to the right in the partial search market, and the new partial equilibrium exhibits a higher T (shown as ${T}^{{\prime} }$) and lower R_i. Consequently, ΔR is lower (shown as $\Delta {R}^{{\prime} }$). The number of educated men decreases until the original partial search equilibrium $({T}^{* },{R}_{i}^{* })$ is reached again, albeit with a lower proportion of educated men (τ). In Panel (b), the returns to education are increasing in T. As x increases, T(R_i) shifts to the right in the partial search market and the new partial equilibrium exhibits a higher T (shown as ${T}^{{\prime} }$) and lower R_i. Consequently, ΔR is higher (shown as ΔR^″). The number of educated men increases, and this generates a further increase in T. With ΔR monotonic in T, the cycle is repeated until τ = 1.

We can now spell out in detail and provide an intuitive interpretation of the chain of reactions that follow a positive shock in women’s options outside marriage. Recall that, in our model, such a shock is meant to capture changes in female labour market returns that are either partly or entirely due to enhanced schooling investment on their part. The immediate effect of any such change is that women become pickier in the marriage market. In turn, the resulting increase in the female reservation wage has an adverse effect on the men active in the marriage market, as it leads to a decrease in the proportion of marriageable wages, for both educated and uneducated single men. Unemployed men adjust their labour market strategy, with all males reducing their reservation wages. If the increase in x harms (through the subsequent increase in T) the marital prospects of educated men relatively more compared to those of uneducated men, the returns to schooling drop overall due to this negative effect on the marriage market. Now the returns to education are too low, and therefore the fraction of men who undertake investment in schooling decreases. At this point, the countervailing effect kicks in. With fewer educated men around, women become less picky, and the associated decrease in the female reservation wage continues until it returns to its old level. Only then will men be once again indifferent between acquiring education or not.

Interestingly, this transition mechanism suggests that if women experience a positive shock in the labour market (higher returns, possibly due to increased educational attainment), the overall benefits of this are exactly offset by a negative effect on the marriage market, where their prospects suffer as the pool of educated eligible men shrinks.¹³

Furthermore, the rate at which men of type i get married depends on the rate at which they find marriageable wages, and this in turn depends on T. With T back at its original equilibrium value (and an exogenous number of single women), men of type i marry at the same rate as in the old equilibrium, while their overall marriage rate is now lower - the latter follows because there are now more L type men in the economy, and these men marry at a lower rate than H types. Mirroring this of course, women also marry at a lower rate, and as a consequence overall marriage rates decrease due to the new composition of male types in the economy.

As stated in Proposition 4, it is also possible to end up with a corner solution that has no educated men. Women are now better off because the initial positive effect of an increase in their labour market returns is not fully eroded by the deterioration in their marriage market prospects.

In turn, if the increase in x harms the marriage market prospects of uneducated men relatively more compared to those of educated men, the overall returns to schooling increase. This makes women even more picky, and the aggregated response of all men further increases the proportion of educated men, in a virtuous cycle. Irrespective of whether this leads to a different type of equilibrium with no uneducated men, or stops at another mixed market equilibrium, women are always better off.

When is condition (5) likely to be satisfied? Consider for example Fig. 4: it depicts the density functions of F_i(w) with a long tail to the right (see for instance Christensen et al., 2005), and where F_H(. ) first-order stochastically dominates F_L(. ). Then it is clear that (5) would hold for high values of T, corresponding to high values of x, which in turn we would interpret as a high enough female rate of schooling.¹⁴ This would be consistent with the interpretation that the reversal of the education gap only kicked in when female schooling rates became high enough (in the USA that would be the 1970 birth cohort, according to Goldin et al., 2006), and that it has continued since.

Fig. 4

Wage distributions with long tail to the right

4.3.2 Other scenarios

What about the other two types of possible mixed market equilibria, characterised by Type 2 and Type 3 partial search equilibria? In terms of existence and uniqueness, the exact same arguments as before apply, suitably adjusted by substituting U_i for R_i/(r + δ). Furthermore, it can also be shown that ∂F_H(T)/∂T > ∂F_L(T)/∂T is now a sufficient condition for a decrease in returns to education following an increase in female reservation wage.

For completeness, consider now the scenario when ΔR is not monotonic in T and hence there are several partial search equilibria which satisfy MME equilibrium condition (4), all characterised by τ < 1. Say there are k such values of T, so that ΔR(T_j) = (r + δ)c, with j = 1, 2, . . . k (in the natural left to right direction, so the next higher j corresponds to a higher $\widetilde{\tau }$). That is, each j equilibrium corresponds to a different partial search equilibrium. Start at any j < k equilibrium, and assume that condition (5) does not hold. Then an increase in T (for example following an increase in x) would take the economy to equilibrium j + 1. If the starting equilibrium is the j = k one, then the economy would shift to the τ = 1 corner solution.

5 Discussion

In this section, we (re)examine several other key assumptions in our model, with particular focus on our asymmetric framework, the role of search frictions and the lack of complementarity in household production. As already mentioned in the Introduction, these three aspects make our theoretical setup markedly different from the one in Chiappori et al. (2009).

5.1 On our asymmetric framework

First, it is perhaps worth pointing out that in the frictionless matching model of Chiappori et al. (2009) women and men are ex-ante heterogenous, and it is this heterogeneity that allows for a symmetric treatment of both sexes in terms of their decisions. In contrast, in our search model all agents are homogenous ex-ante, but constrained job search on the part of men leads to an ex-post heterogeneity in terms of their earned wages. In other words, search frictions are needed to generate the male heterogeneity that in turn plays a role in the marriage market, and through it determines the optimal schooling investment decision of males. A more symmetric setup would of course include the explicit modelling of women’s education and labour market decisions as well. However, it is well known that in equilibrium models of frictional markets this involves a coordination problem that leads to potential inefficiencies.¹⁵ As we wanted to avoid this complication, and focus instead on isolating our own transmission mechanism, we have chosen to work with the asymmetric “breadwinner” model, as argued in the Introduction.

We have focused on single earner households with no intra-marital bargaining. Ignoring this aspect of surplus sharing makes our model more tractable and allows us to side-step the well-understood question of inefficiency in frictional markets characterised by the hold-up problem. Nonetheless, it is easy to see that any potential negotiation over the marriage surplus would not affect the nature of men’s constrained job search decision. Ceteris paribus, it would make women pickier in the marriage market, but our key comparative statics result would remain.

Alternatively, if men gave up a portion of their wage upon marriage, unemployed single men would become pickier in the labour market, leading to an upward shift in the reservation wage function. As women would also enjoy less than the full wage earned by their husbands, ceteris paribus they would become pickier in the marriage market, leading to a rightward shift of the T function. Again, our qualitative results would continue to hold.

Finally, since our aim was to investigate the consequences of the breadwinner effect, we modelled women as having to quit their job (give up x) after marriage. Relaxing this assumption would not change the main thrust of our result either: as long as women’s attachment to the labour market weakens after marriage, the effect we study goes through. For any given x, as the share of it that women must give up after marriage decreases, the value of marriage increases, thus decreasing the relative value of remaining single. This would make women less picky in the marriage market, leading to a leftward shift in the T function.

5.2 On the role of search frictions

In order to understand better the role of search frictions, consider our model but with a frictionless labour market instead. Now, no worker will accept a wage lower than the maximum wage in the market available to them. In our case, these would be ${\overline{w}}_{H}$ for educated men and ${\overline{w}}_{L}$ for uneducated men. Accordingly, the ex-post within-type heterogeneity which matters in the marriage market (i.e., the dispersion of earned wages) disappears.

There are only two possible scenarios: (i) women marry only educated men, and all of them; or (ii) women marry all men, regardless of education (that is, if both types exist in equilibrium). Indeed, the only way women will reject marriage to any educated man (and it would be all of them), is if $x \,>\, {\overline{w}}_{H}$. There are two important consequences of this:

First, an increase in x, which generates an increase in T, can never affect educated men negatively. A key feature in the determination of equilibrium in our model with frictions in the labour market is that when x (and thus T) increases, this harms both educated and uneducated men, as both men are less likely to encounter marriageable wages. Second (and closely related to the first), a continuous increase in x does not lead to continuous effects on women’s strategy: the only two increases in x that matter are the ones that either make women marry only educated men (a switch from marrying them all) or make them marry no one (a switch from marrying only educated men). A further important feature of our model is that as women’s pickiness increases, this has an effect on men’s strategy (the determination of their reservation wage). As stated above, without frictions in the labour market men will not accept any wage lower than the highest one available, regardless of women’s strategies. If $x\le {\overline{w}}_{L}$, women prefer marriage to being single, regardless of the education level of the partner. In turn, if ${\overline{w}}_{L} \,<\, x \,<\, {\overline{w}}_{H}$ then women prefer marriage to educated workers, but reject marriage to uneducated workers.

Having said that, it is interesting to determine the effect of an increase in x on the share of educated men. We show below that the only two possibilities are that (a) there is no effect, or (b) that an increase in x increases the share of educated men. This would of course be contrary to our main result, in which the share of educated men decreases for any given j equilibrium (i.e., one that corresponds to a particular search market equilibrium).

If $x \,<\, {\overline{w}}_{L}$, women prefer marriage to any man over being single. Consider a potential equilibrium with a proportion 0 < τ < 1 of men getting education. If there are more women than educated men, then all educated men will marry, and some uneducated men will also marry. The returns to education include the increase in wage (labour market returns) and the increase in the probability of getting married (from positive but lower than 1 if uneducated to equal to 1 if educated), i.e., the increase in expected marriage returns. In equilibrium, the share of uneducated men must be such that the cost of education equals the total returns to education described above. In that scenario, consider an increase in x. As long as $x \,<\, {\overline{w}}_{L}$, women will continue to prefer marriage to any man over staying single, and thus there is no effect on men’s education decision: even if the opportunity cost of marriage has increased, women still do not reject any wage that actually exists in equilibrium. As their increased pickiness does not affect men’s strategy in the labour market nor their marriage prospects, equilibrium τ remains unchanged.

If instead x increases above ${\overline{w}}_{L}$ to ${\overline{w}}_{L} \,<\, x \,<\, {\overline{w}}_{H}$, women now reject marriage to uneducated men. This increases the (marriage) returns to education as now education raises the probability of marriage from zero to 1 and, with the cost of education unchanged, men will find it optimal to invest in schooling. As long as the number of women is higher than the number of educated men, marriage probabilities remain unchanged: even if τ is higher, the probability of marriage is still zero if uneducated and still 1 if educated. When τ has increased such that the numbers are equal, the returns to education decrease as τ increases: with more educated men than women the probability of marriage is less than 1 and decreases with the number of educated men. Depending on the cost of education, the new equilibrium could exhibit τ < 1 or τ = 1, the latter occurring if the probability of marriage has not declined enough to make education too costly relative to the sum of the marriage and labour market returns to education. Alternatively, if equilibrium τ is such that there are more educated men than women, some educated men will not marry, and no uneducated man will marry. The marriage returns to education now involve an increase in the probability of marriage from zero to positive but lower than 1. Consider next an increase in x. Even if the increase is such that now ${\overline{w}}_{L} \,<\, x \,<\, {\overline{w}}_{H}$, there are no effects on the equilibrium τ. This is because it will remain true that all women will marry educated men, and the same number of educated men will not get married, while uneducated men will still not marry. With the probabilities of marriage unchanged after the increase in x, there is no effect on the marriage returns to education.

Further highlighting the effect of search frictions, Bonilla and Kiraly (2013) show that it is entirely possible (using noisy search) to generate an endogenous distribution of wages in a setup similar to the one discussed in this paper. Crucially, the comparative statics following an increase in x mimics the results we obtain here (in the partial search equilibrium with an exogenous distribution of wages). This is essentially because the distribution of wages, even if endogenous to the model, is regarded as exogenous by men and individual firms alike.

5.3 On complementarities

In order to isolate our mechanism, we have abstracted from any potential complementarity in home production. Relaxing this assumption and regarding w and x as complements would, following an increase in x, generate additional effects that would now stem from within-marriage incentives. However, we show below that our key results still obtain for given parameter values, and in doing so we also unpick the specific effects on the components of our mechanism.

Consider for example the household utility given by w(1 + kx), with k > 0. There are then two effects following an increase in x:

• (i) The effect through T. For a woman, the opportunity cost of accepting marriage to a man earning w = T is x − T(1 + kx), and as x increases this changes by 1 − Tk. It is then easy to show therefore that if Tk < 1, as x increases we still observe a shift to the right of T(R_i), and our comparative statics result follows. For Tk > 1 the effect on T(R_i) is the opposite, but even in this particular case (ii) below points to a rightward shift of T(R_i).

• (ii) The direct effect on R_i. In Appendix C we show that returns to schooling increase as x increases. This is because now x interacts with the wage and educated men face a wage distribution that first order stochastically dominates that of uneducated men: F_H(w) < F_L(w). Hence, a direct reaction to an increase in x would trigger a similar mechanism as the one studied in our paper. To see this, note that by increasing the returns to schooling (ceteris paribus), an increase in x leads to a higher share of educated men τ. However, such an increase in τ in turn generates a shift in T(R_i) to the right, mimicking the effect of an increase in x that we address in the paper.

Furthermore, Appendix C shows that an increase in x leads to a downward shift of both R_i(T), which follows because now the same utility can obtain from a lower wage compared to the case without complementarity (that is, with k = 0). Ceteris paribus, this points towards an equilibrium with lower R_i and lower T. Finally, Appendix C also confirms that, in terms of being a function of T and R_i, the structure of our inequality condition (5) does not change - the only effect on that condition comes from the shifts in T(R_i) and R_i(T) discussed above.

6 Conclusion

In this paper, we contribute to the discussion which surrounds the puzzling fact that the rate of increase in male schooling has recently lagged behind that of female education attainment. We consider a version of a Becker-type human capital model where the male educational decision is embedded in a framework of inter-linked frictional labour and marriage markets. We show that in the steady state market equilibrium the proportion of men who choose to undertake the (costly) schooling investment can decrease as a direct response to an increase in the female labour market returns (viewed as a proxy for female educational attainment). This could partially explain the above-mentioned lag.

If education improves men’s prospects both in the labour and the marriage market (where marital selection occurs based on male wages), then unemployed men react to the expectations set by women. An increase in women’s labour returns (education) increases the value of being a single woman and hence the female marriage market reservation wage. In turn, this affects the unemployment values of males, and consequently their returns to education. Now, with the equilibrium condition not fulfilled, only a decrease in the proportion of educated men can re-balance the equality between these returns and the cost of schooling, since only a lower fraction of such men would restore the original lower female reservation wage.

Interestingly, although the constrained sequential job search gives rise to a non-monotonic male reservation wage function, this mechanism holds for any equilibrium where the marriage market affects men’s labour market decisions. Therefore our analysis points towards a more general feature of models of inter-connected frictional markets where (i) access to one market is conditional on the outcome of search in the other market, and (ii) the prospects in both markets are influenced by an ex-ante costly investment. In such models, the proportion of agents who undertake the investment is crucial, and is determined by the equality between the expected returns from this investment and its cost. If there is a change in an exogenous variable that affects these expected returns (through the change in the requirements to access the other market), the fraction of these agents is the only endogenous variable left to restore the market equilibrium condition.

Appendices

Appendix A—Proof of Proposition 1

Consider a married and employed man of any type. With neither on-the-job search nor the possibility of divorce, standard considerations give the discounted expected lifetime utility of such a man:

$${V}^{M}(w)=\frac{w+y}{r+\delta }.$$

This shows that once employed and married, type is irrelevant for men. Next, consider a married but unemployed man of type i. As he can ignore the marriage market (there is no divorce), his reservation wage is simply the standard pure labour market one:

$${\underline{R}}_{i}=\frac{{\lambda }_{0}}{r+\delta }\int\nolimits_{{\underline{R}}_{i}}^{{\overline{w}}_{i}}\left[w-{\underline{R}}_{i}\right]d{F}_{i}(w)$$

Note that ${\underline{R}}_{H} \,>\, {\underline{R}}_{L}$ since F_H(. ) first order stochastically dominates F_L(. ).

In principle, women could of course marry unemployed men as well. Let us therefore examine the situation of a woman who is married to a type H jobless man. Her value function, denoted by ${W}_{H}^{U}$, is given by:

$$(r+\delta ){W}_{H}^{U}={\lambda }_{0}\int\nolimits_{{\underline{R}}_{H}}^{{\overline{w}}_{H}}\left[{W}_{H}^{M}(w)-{W}_{H}^{U}\right]d{F}_{H}(w),$$

where ${W}_{H}^{M}(w)=w/(r+\delta )(={W}_{L}^{M}(w))$ is the discounted lifetime utility of being married to a type H employed man who earns wage w. The above equation incorporates the fact that a married type H unemployed man has reservation wage ${\underline{R}}_{H}$.

For T to be a female reservation wage, it needs to satisfy the condition W^S = T/(r + δ), where W^S is the woman’s value of being single. Furthermore, the condition for women to accept unemployed men of type H is ${W}_{H}^{U}\ge {W}^{S}$. At the cutoff, we then have ${W}_{H}^{U}={W}^{S}=T/(r+\delta )$. Given the equation for ${W}_{H}^{U}$ and making use of ${W}_{H}^{M}(w)$ above, this means (using the standard integration by parts) that in this cut-off point T is the unique solution to:

$$T=\frac{{\lambda }_{0}}{r+\delta }\int\nolimits_{{\underline{R}}_{H}}^{{\overline{w}}_{H}}\left[w-T\right]d{F}_{H}(w).$$

It follows that $T={\underline{R}}_{H}$. Note that if this holds, the L men are not marriageable since ${\underline{R}}_{L} \,<\, {\underline{R}}_{H}$.

Now, if a woman’s value of being single W^S increases, her reservation wage also increases. In contrast, ${W}_{H}^{U}$ is independent of T. Therefore, ${W}^{S} \,\gtreqless\, {W}_{H}^{U}$ if and only if $T \,\gtreqless\, {\underline{R}}_{H}$ and hence we conclude that if $T \,>\, {\underline{R}}_{H}$ women will reject marriage to all unemployed men. Throughout this paper, we study equilibria characterised by $T \,>\, {\underline{R}}_{H}$.¹⁶

Appendix B—Proof of Proposition 2

First, consider $T\in ({\widehat{T}}_{i},{\overline{w}}_{i}]$. Assume for a moment that R_i(T) ≤ T. Then, using ${V}_{i}^{S}({R}_{i})={R}_{i}/(r+\delta )={U}_{i}$, the male reservation wage R_i(T) is given by the expression in Proposition 2(c). From here, ${R}_{i}({\widehat{T}}_{i})={\widehat{T}}_{i}$, where ${\widehat{T}}_{i}$ as defined in (3). Call this reservation wage ${\widehat{R}}_{i}$. Also, from the above, when $T\ge {\overline{w}}_{i}$ we have ${R}_{i}(T)={\underline{R}}_{i}$, since then F_i(T) = 1. It is easy to show that ${\widehat{T}}_{i} \,<\, {\overline{w}}_{i}$, and R_i(T) is decreasing in T. Hence, R_i(T) < T iff $T > {\widehat{T}}_{i}$. For $T\le {\widehat{T}}_{i}$, the reservation function derived above does not survive as an optimal strategy. Consider then $T\in ({\underline{R}}_{i},{\widehat{T}}_{i}]$. Unemployed men are still not marriageable, and the value of being a single unemployed with a reservation wage R_i > T is given by:

$$(r+\delta ){U}_{i}=\frac{{\lambda }_{0}}{r+\delta }\int\nolimits_{{R}_{i}}^{{\overline{w}}_{i}}[1-{F}_{i}(w)]dw+\frac{{\lambda }_{0}\lambda n}{(r+\delta )(r+\delta +\lambda n)}y.$$

On the other hand, when choosing T as the reservation wage, this value is given by:

$$(r+\delta ){U}_{i}=\frac{{\lambda }_{0}}{r+\delta }\int\nolimits_{T}^{{\overline{w}}_{i}}[1-{F}_{i}(w)]dw+\frac{{\lambda }_{0}\lambda n}{(r+\delta )(r+\delta +\lambda n)}y.$$

For R_i > T, the latter is higher than the former. Intuitively, the only reason to increase R_i above ${\underline{R}}_{i}$ would be in order to become marriageable. But R_i = T is already enough for that.

Next, consider $T\le {\underline{R}}_{i}$. If men believe they are marriageable irrespective of their employment status, they choose ${R}_{i}={\underline{R}}_{i}$ both when single and married. This is because ${V}_{i}^{S}(w)=\frac{w}{r+\delta }+\frac{\lambda n}{(r+\delta +\lambda n)(r+\delta )}y$ for all $w\ge {\underline{R}}_{i}$. For $T\le {\underline{R}}_{i}$, they are indeed always marriageable. Now consider the case with $T > {\overline{w}}_{i}$. We have 1 − F_i(T) = 0, and therefore a man of type i can never get married, as the highest available wage is ${\overline{w}}_{i}$. Men optimally set ${R}_{i}(T)={\underline{R}}_{i}$.

Finally, ${\widehat{T}}_{H} \,>\, {\widehat{T}}_{L}$ and ${\underline{R}}_{H} \,>\, {\underline{R}}_{L}$ follow from the fact that F_H(. ) first order stochastically dominates F_L(. ): the former is clear from equation (3), while ${\underline{R}}_{i}=\frac{{\lambda }_{0}}{r\,+\,\delta }\int\nolimits_{{\underline{R}}_{i}}^{{\overline{w}}_{i}}\left[1-{F}_{i}(w)\right]dw$ as discussed following equation (1).

Appendix C—Complementarity in home production

Consider a flow utility within marriage given by w(1 + kx), with k ≥ 0. Please note that the framework of our main model corresponds to the case with k = 0.

With no divorce, the value of being a married man earning a wage w is now ${V}_{i}^{M}(w)=[w(1+kx)+y]/(r+\delta )$. Hence, for any T, we have:

$${V}^{S}(w)=\left\{\begin{array}{ll}\frac{w(1+kx)}{r+\delta }\,\qquad\qquad\qquad\quad\;{{\mbox{if}}}\,\,w < T\\ \frac{w(1+kx)}{r+\delta }+\frac{\lambda n}{(r+\delta +\lambda n)(r+\delta )}y\;\,{{\mbox{if}}}\,\,\,w\ge T\end{array}\right\}.$$

It follows that the Bellman equation for unemployed men is given by:

$$\begin{array}{l}(r+\delta ){U}_{i}={\lambda }_{0}\displaystyle\int\nolimits_{{R}_{i}}^{T}\left[\frac{w(1+kx)}{r+\delta }-{U}_{i}\right]d{F}_{i}(w)\\\qquad\qquad\;\;\; +\,{\lambda }_{0}\displaystyle\int\nolimits_{T}^{{\overline{w}}_{i}}\left[\frac{w(1+kx)}{r+\delta }+\frac{\lambda n}{(r+\delta +\lambda n)(r+\delta )}y-{U}_{i}\right]d{F}_{i}(w),\end{array}$$

or

$$(r+\delta ){U}_{i}={\lambda }_{0}(1+kx)\int\nolimits_{{R}_{i}}^{{\overline{w}}_{i}}\left[\frac{w}{r+\delta }-{U}_{i}\right]d{F}_{i}(w)+\frac{{\lambda }_{0}\lambda n\left.\right(1-{F}_{i}(T)}{(r+\delta +\lambda n)(r+\delta )}y.$$

(A1)

Using U_i = R_i(1 + kx)/(r + δ), and after standard integration by parts, we obtain:

$${R}_{i}(T)=\frac{{\lambda }_{0}}{r+\delta }\int\nolimits_{{R}_{i}}^{{\overline{w}}_{i}}\left[1-{F}_{i}(w)\right]dw+\frac{{\lambda }_{0}\lambda n\left[1-{F}_{i}(T)\right]}{(1+kx)(r+\delta )(r+\lambda n+\delta )}y$$

(A2)

From here, ∂ΔR/∂T < 0 ⇔ ∂R_H(T)/∂T < ∂R_L(T)/∂T, where:

$$\frac{\partial {R}_{i}(T)}{\partial T}=\frac{\partial {F}_{i}(T)}{\partial T}\frac{{\lambda }_{0}\lambda ny}{\{r+\delta +{\lambda }_{0}[1-{F}_{i}({R}_{i})]\}(r+\lambda n+\delta )(1+kx)}$$

Hence, ∂R_H(T)/∂T < ∂R_L(T)/∂T if:

$$\frac{\partial {F}_{H}(T)}{\partial T}/\frac{\partial {F}_{L}(T)}{\partial T} > \frac{r+\delta +{\lambda }_{0}[1-{F}_{H}({R}_{H})]}{r+\delta +{\lambda }_{0}[1-{F}_{L}({R}_{L})]}$$

which is independent of k since (1 + kx) is a constant.

In addition, since F_H(w) < F_L(w) it follows from the expression for U_i above that an increase in x increases U_H more than U_L, thus increasing the returns to schooling. Finally, it is easy to see from the expression for R_i(T) above that for k > 0 an increase in x leads to a downward shift of R_i(T).