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(n t ), and as investment goods when the mother is old, v(n 1 + n 2 )

8. Model Lifetime Utility in period s (call this ( * )): where c i = consumption in period i n i = 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: c i = y i – k i n i where y i = income in period i and k i n i = 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 n 1 and makes plans for n 2, maximizing ( * ) for s = 1 In Period 2, the woman reconsiders her plans for this period, choosing n 2 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, s i, in period i, so her income becomes: y 1 = ω and y j = (1+ r) s j-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 less children 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, s 1 = ω – c 1  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 more children than intended in 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: n 2 > 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???