FERTILITY SHOCKS AND EQUILIBRIUM MARRIAGE‐RATE DYNAMICS

• DOI: http://doi.org/10.1111/iere.12395
• Published date: 01 November 2019
• Author: John Knowles,Guillaume Vandenbroucke
• Date: 01 November 2019

INTERNATIONAL ECONOMIC REVIEW

Vol. 60, No. 4, November 2019 DOI: 10.1111/iere.12395

FERTILITY SHOCKS AND EQUILIBRIUM MARRIAGE-RATE DYNAMICS∗

BYJOHN KNOWLES AND GUILLAUME VANDENBROUCKE1

Simon Fraser University, Canada; Federal Reserve Bank of St. Louis, U.S.A.

Female marriage probabilities were 50% higher in France in the years after World War 1, despite a large

drop in the sex ratio. We develop a model of marital matching in which composition effects in the singles

pool affect postdisruption matching rates. When calibrated to French data from World War 1, this mechanism

explains 2/3 of the postwar rise in female marriage probabilities as the result of better composition of the pool of

single men. We conclude that endogeneity issues make the sex ratio a potentially unreliable indicator of female

marriage prospects.

1. INTRODUCTION

Finding a marriage partner takes time; as in labor markets, the empirical observation that at

any point in time parties on both sides of the market remain unmatched is often explained as

the result of matching frictions. A central feature of frictional models of two-sided matching is

that the effect of a change in the number of unmatched agents on the speed of matching is a

function only of the ratio of the number of agents on each side of the market, and the effect on

each side has opposite sign. This feature, which arises from the assumption of constant returns

to scale (CRS) in the matching process, implies that workers are quicker to find jobs when the

vacancy unemployment ratio is high, and women quicker to marry when the “sex ratio,” the

ratio of men to women in the singles pool, is high. Exogenous shocks to these ratios are rare,

however, so the hypothesis is not easy to verify.

The population of young single men (ages 20–40) in European countries declined abruptly as

a result of World War 1; in France after the war, the sex ratio of singles in this age group was

28% lower than before the war. This decline constitutes an exogenous and unexpected shock to

the marriage market; because France and Germany had universal male military conscription,

the death rate of young males in these countries was essentially orthogonal to marital status.

What makes France unique, however, is the availability of annual age-specific population data,

by sex and marital status, starting from 1900. This combination of circumstances makes postwar

France a rare opportunity to learn about the aggregate dynamics of the marriage-matching

process. Previous papers, such as Vincent (1946), Henry (1966), and Abramitzky et al. (2011)

have stressed the impact of the postwar sex ratio on lifetime marital outcomes. If the sex-ratio

decline is indeed the main shock to the postwar marriage market, we should expect annual

postwar marriage probabilities (the “marriage hazard rates”) in France to be lower for women

and higher for men than before the war.

∗Manuscript received July 2016; revised May 2018.

1We thank the editor and two anonymous referees for useful comments. We are also grateful for constructive

comments from conference participants at REDg Barcelona 2013, LAEF-NYU 2013, Sciences Po Paris 2013, Essex

SAM 2013, OLG Days Paris 2014, ENSAI Rennes 2014, Iowa 2014, ZEW Mannheim 2014, Philadelphia Fed 2014,

and department seminars at USC, Southampton, Birmingham, SFU, IMF, Laval, UC Louvain, Tel Aviv, Be’er Sheva,

Dallas Fed, EIEF (Rome), EUI (Florence), and Rochester. Knowles thanks the Southampton CPC for encouragement

and financial support. The views expressed here are those of the authors and do not necessarily reflect the views of

the Federal Reserve Bank of St. Louis or the Federal Reserve System. Please address correspondence to: Guillaume

Vandenbroucke, Research Division, Federal Reserve Bank of St. Louis, P.O. Box 442, St. Louis, MO 63166-0442.

E-mail: guillaumevdb@gmail.com.

1505

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(2019) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

1506 KNOWLES AND VANDENBROUCKE

In this article, we show that the postwar marriage hazards do not fit the predicted pattern. We

document the time path, by age and sex, of the marriage hazards in France from 1905 to 1930.

We find that these increased dramatically for both sexes after the war, and that, contrary to the

standard theory, a striking rise in the female marriage hazards accounts for a large share of the

postwar rise in per capita marriage rates. We show that this was more than a transitory blip in

the data; five years after the war’s end, female marriage hazards were still above the prewar

average, so the contradiction between theory and data cannot be dismissed as an aberration

driven by conditions at the end of the war. We also find that the peak in female hazards cannot be

explained away by the postwar formalization of wartime matches, as the response of unmarried

births is implausibly small.

In principle, it would be possible to explain the rise in the postwar female hazard rates

by assuming increasing returns to scale (IRTS) in the matching process. The higher postwar

prevalence of singles would then generate higher matching hazards for both sexes, eliminating

the contradiction between theory and data. However, an important drawback of IRTS would

be the loss, due to multiple equilibria, of the predictive power of the CRS model. Furthermore,

the preponderance of empirical work on matching, as documented by Pissarides and Petrongolo

(2001, for labor markets) and Botticini and Siow (2008, for marriage markets) supports the CRS

assumption. Finally, the fact that the peace time marriage rate is very stable in France over the

period from 1800 to 1950, despite several large wartime shocks, would be difficult to reconcile

with the multiplicity of steady states implied by IRTS models.

Instead, we propose that a second war-related shock, the wartime “marriage bust,” accounts

for the postwar rise in the female marriage hazards. The per capita marriage rate over the 1914–

18 period fell abruptly to an average 51% of the peace time rate. Our idea is that this prolonged

interruption of the matching process resulted in a radical change in composition of the pool of

postwar singles. To simplify, suppose that singles differ in their marriage “propensity,” which

is public information. Normally, high-propensity singles would be comparatively rare in the

singles pool, because they would exit into marriage relatively quickly. Now consider an extreme

case: Suppose that marriage rates fall to zero during wartime. Then the high-propensity singles

would tend to accumulate and in the postwar singles pool would be unusually abundant, raising

the average marriage hazard. In the presence of matching frictions, it could take considerable

time for this composition effect to dissipate.

To formalize this mechanism in a more realistic context, our article proposes a matching

model in which heterogeneity in marriage propensity is derived from age-related differences

in the utility from single life, and, among women, in the propensity to bear children. Following

recent advances in the labor-matching literature, we use the competitive-search assumption

to incorporate this heterogeneity into the matching process. We use French Census and Vital

Statistics reports to calibrate the model’s steady state to the relevant age-specific statistics from

prewar France: the marriage hazards by sex, the sex ratio of singles, and the birth hazards of

married women. We then model the war as a pair of unanticipated shocks of known duration, to

male mortality and to fertility, which we calibrate so that the model matches the age distribution

of military deaths and the wartime decline in marriage hazards. The empirical foundations of

these shocks are well established; the age pattern of military deaths is based on the tabulations

of Vallin (1973), and the fertility shock reflects the wartime decline in per capita birth rates

based on the tabulations of Mitchell (1998), which Vandenbroucke (2014) shows is likely to

have been a response to the risk of paternal mortality leading to single motherhood.

We then compute the equilibrium time path of the marriage market from the start of the

war until the postwar return to the steady state. We find that the model generates a significant

postwar increase in female marriage hazards, because the negative effect of the sex-ratio decline

is offset by a larger, positive effect due to the wartime accumulation of single males who would

normally have married much earlier. In fact, our baseline calibration accounts for 66% of the

postwar peak in female marriage hazards. The model also generates a leisurely transition back to

the steady state, accounting for 38% of the half-life of the postwar peak for women. Although

the model peak occurs in 1919, a year before the peak in the data, the model successfully

EQUILIBRIUM MARRIAGE-RATE DYNAMICS 1507

predicts the sex ratio of singles throughout the 1920s. Thus our baseline calibration succeeds in

accounting for the basic features of the postwar transition dynamics in the marriage market.

Our results also suggest that the main force underlying the rise in the female marriage hazard

is the postwar abundance, per single woman, of men with a high propensity to marry. There

is no corresponding increase of high-propensity women, because the calibrated model implies

that women’s marriage propensity falls with age. Furthermore, we find that our results do not

rely on minimizing the impact of military mortality: We show that the male mortality shock, on

its own, delivers large postwar declines in the female marriage hazard.2

Finally, we ask whether the exceptional size of the female postwar peak might be due in

part to pronatalist sentiment in the postwar, as manifested by political events such as the

postwar repudiation of feminist policies and the adoption of legislation favoring large families

(see Roberts, 1994; Robert, 2005). We show that births per married woman increased sharply

relative to trend, a change that persisted well into the 1930s. When we augment the model

with a postwar fertility shock, recalibrated to match the postwar birth-rate bump, the resulting

transition path explains essentially all of the postwar peak in the female marriage hazards. Thus

fertility plays key roles in generating variation in marriage propensities, both in cross section

and in time series.

1.1. Related Literature. There is a long tradition in demography, dating from Groves and

Ogburn (1928), of studying the causal effect of the sex ratio on marriage rates. In economics, the

seminal work is the Becker (1973) model of frictionless matching. Our basic assumptions, includ-

ing the assumption of fully transferable utility within marriage, are derived from the canonical

Becker (1973) model, augmented with matching frictions, in the tradition of Shimer and Smith

(2000), but with repeated matching, so that the singles distribution evolves endogenously over

time. The problem of finding exogenous variation in sex ratios has been addressed by economists

such as Angrist (2002), who used immigration flows, and by Bronson and Mazzocco (2018), who

use variation in the size of birth cohorts.

The effect on eventual marital outcomes of war-induced declines in the sex ratio have been

studied by Henry (1966), Vincent (1946), and Abramitzky et al. (2011) for the case of World

War 1 in France and by Brainerd (2017) for World War 2 in the Soviet Union. Henry (1966) and

Abramitzky et al. (2011) both find that lower postwar sex ratios reduce the fraction of women

in the affected birth cohorts who ever married and change the mix of who marries whom.

However, Brainerd (2017) finds that both men and women are less likely to marry. Bethmann

and Kvasnicka (2014) find that low sex ratios increase births to unmarried mothers in Germany

after the Second World War.3None of these papers are directly concerned with the time it

takes to marry, so our analysis of hazard rates and dynamics do not speak to their main results.

This distinction is critical in our view, because the fraction of people who eventually marry is a

function not only of the hazard rate, but also of the length of time singles spend in matching.

If the heterogeneity in our model were permanent, the environment would be similar to that

of Burdett and Coles (1999), who show how composition effects generate multiple steady states

in a random-matching model. That sort of multiplicity does not occur in our model, however, due

to our reliance on the competitive-search mechanism, similar to the labor-matching model of

Shimer (2005), where endogenous surplus allocation ensures uniqueness and Pareto optimality

of the equilibrium. In this sense, our model is closely related to the matching and fertility model

of Kennes and Knowles (2015), which builds on elements of Regalia and R´

ıos-Rull (1999) and

Shimer (2005).

2We also show in Appendix A.1 that although the war increased the average age of single men, this on its own could

not have generated an increase in the female marriage hazard rates, because the number of single women increased

even more than the number of older single men.

3However, Henry (1966) and Abramitzky et al. (2011) both find that the lower sex ratio was associated with a smaller

postwar age gap between new spouses, a result that our model also generates. In the steady state, our model implies an

age difference of 1.79 years, whereas in the mid-1920s the age difference is on average 1.12 years.