Fertility and Time Inconsistency

1. Fertility and Time Inconsistency Matthias Wrede RWTH Aachen University (presented by Jessica Schuring)

2. Introduction • Fertility-related behavior has changed dramatically in OECD countries over the last 30 years: • Average Total Fertility Rate (average number of children per woman of childbearing age) is declining: from 2.7 in 1970 to 1.6 in 2002 • Mean age at first childbirth is increasing: from 23.8 in 1970 to 27.2 in 2000 • The number of families with four or more children has declined significantly from 1970 to 2000

3. Introduction: Motivation for Studying Fertility Behaviors • Changes in fertility behaviors have lasting effects on many policy issues: • Social security, tax policy, public spending, public infrastructure, society’s productivity and potential to innovate • Due to the effects of changing fertility behaviors, many countries have incentive to change family policies • In order to best shape family policy, it’s important to thoroughly understand fertility behavior

4. Fertility Behavior:Perfect Rationality? • Standard models of fertility assume perfectly rational agents • However, recent experimental studies have observed bounded rationality • Since having children has long-lasting consequences, theories of bounded rationality focusing on time discounting are useful in analyzing fertility behaviors of women who are not perfectly rational

5. Time Discounting • There is experimental evidence to suggest that the discount rate declines over time • It has been shown that a declining discount rate affects many behaviors including savings, addiction, and retirement • Analyzing its effect on fertility is a natural extension • Note: any discounting raises the issue of time inconsistency since changes in the discount rate may lead to previously optimal decisions becoming suboptimal

6. Focus of this Paper • This paper analyzes whether declining discount rates have an effect on: • The total number of children • The timing of childbirths • The model utilizes quasi-hyperbolic discounting, which simplifies the analysis • This paper also briefly addresses an empirical study on the effect of time inconsistency on the total number of births

7. Model • Three period model • The woman is only fertile in periods one and two • In those periods, she chooses the number of children (treated as a continuous variable) • The woman derives utility both from consumption and from children • Children increase utility as consumption goods when they are young, w(nt), and as investment goods when the mother is old, v(n1 + n2)

8. Model • Lifetime Utility in period s (call this (*)): where ci = consumption in period i ni = number of children in period i for i = 1,2 δ = normal discount factor γ > 1 captures quasi-hyperbolic discounting u(·), v(·), w(·) have the usual properties Note: ci = yi – kiniwhere yi = income in period i and kini = childcare costs (including opportunity costs)

9. Agent’s Problem:(with an Imperfect Capital Market) • First, we will assume the woman has no access to a capital market – no means to save or borrow • The woman’s lifetime problem: • In Period 1, the woman chooses n1and makes plans forn2, maximizing (*) for s = 1 • In Period 2, the woman reconsiders her plans for this period, choosing n2 and maximizing (*) for s = 2 • The F.O.C. in this period deviates from the woman’s previous plans – the woman is time inconsistent • In Period 3, there are no more decisions to be made,and the woman consumes her entire income (bequest motives are excluded by assumption)

10. Results of Time Discounting:with an Imperfect Capital Market • Proposition 1:Introducing quasi-hyperbolic preferences on a perfectly imperfect capital market leads to postponement of births and to less children in the second period than intended. • When a woman can neither save nor borrow, an increase in γ yields: • Period 1: the woman shifts motherhood into the future, as the present becomes more valuable • Period 2: the mother reduces the number of births, in reaction to investment in children becoming less attractive (the future is discounted more heavily)

11. Results of Time Discounting:with an Imperfect Capital Market • Is there a different result if the woman is a “sophisticated” as opposed to a “naïve”? • Proposition 2: Sophisticated women with quasi-hyperbolic preferences postpone births even more than naive women. • For the sophisticated woman (one who anticipates her preference shift) who can neither save nor borrow, an increase in γ yields: • Additional incentive to postpone births in Period 1, in order to encourage her second period self to give birth to more children

12. Results of Time Discounting:with a Perfect Capital Market • Now assume the woman has unrestricted access to a perfect capital market, with interest rate r • The woman can now have savings, si, in period i, so her income becomes: y1 = ω and yj= (1+ r) sj-1, for j=2, 3 • We find that the effect of quasi-hyperbolic preferences on the number and timing of births depends on how those births affect utility in the presence of a perfect capital market • Consider two cases: “pure investment utility” and “pure consumption utility” (since general results are ambiguous)

13. Results of Time Discounting: Pure Investment Benefit • In this case, children provide no consumption utility • Thus, women postpone births until Period 2 • Proposition 3: Introducing quasi-hyperbolic preferences on a perfect capital market leads to lesschildren than intended when the mother only benefits from children when she is old. • An increase in γ yields: • A desire for more children in Period 1  reduced savings, s1 = ω – c1 reduced income in Period 2  higher MU of income in Period 2 (children become relatively more expensive)  less children in the second period • Additionally, in Period 2, women give birth to less children because second period consumption becomes more valuable

14. Results of Time Discounting: Pure Consumption Benefit • In this case, children provide no investment utility • Women usually give births in both periods • Proposition 4: Introducing quasi-hyperbolic preferences on a perfect capital market leads to morechildren than intendedin the second period when the mother only benefits from children while they are young. • An increase in γ yields: • A desire for more children in Period 1  higher child care costs and first period consumption  reduced income in Period 2  less children in the second period • But, since Period 2 consumption utility of children is more valuable than expected, the woman gives birth to more children than expected for this period

15. Results of Time Discounting:with a Perfect Capital Market • What about in this case – Are there different results for “sophisticated” versus “naïve” women? • Note: n2 > 0 is assumed • Proposition 5: A sophisticated woman consumes more in the first period than a naive woman. If opportunity costs decline sufficiently sharply over the life cycle, even sophisticated women postpone births. • A sophisticated woman disagrees with her second period self on the allocation of resources • Thus she is motivated to increase first period consumption in order to decrease second period income • If opportunity costs are declining enough over the woman’s life, she may be motivated to delay births

16. Conclusions • The paper also gives extensions and an empirical model • In the empirical model, the author creates an interesting proxy for time inconsistency: • He uses a variable measuring the readiness and ability of smokers to quit smoking • Based on the idea that naïve agents overestimate their ability to stop smoking when they smoke their first cigarette • The empirical model finds that continued smoking and giving birth to children are substitutes • Time inconsistency reduces the number of births

17. Conclusions • In summary: • Without the ability to save and borrow, naïve hyperbolic discounters postpone births and give births to less children than previously intended • Sophisticated mothers also postpone births • With perfect capital markets, we require additional restrictions to get the same results: • We must have opportunity costs of children declining over the life cycle and the investment motive prevalent • Otherwise, declining discount rates may have an opposite effect on births

18. Questions???