Fertility and Time Inconsistency

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Fertility and Time Inconsistency

• Matthias Wrede
• RWTH Aachen University
• (presented by Jessica Schuring)

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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

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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, societys 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, its
  important to thoroughly understand fertility
  behavior

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Fertility BehaviorPerfect 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

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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

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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

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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)

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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
• d normal discount factor
• ? gt 1 captures quasi-hyperbolic discounting
• u(), v(), w() have the usual properties
• Note ci yi kini where yi income in period
  i and
• kini childcare costs (including opportunity
  costs)

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Agents 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 womans lifetime problem
• In Period 1, the woman chooses n1 and makes plans
  for n2, 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
  womans 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)

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Results of Time Discountingwith 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 ? 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)

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Results of Time Discountingwith 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 ? yields
• Additional incentive to postpone births in Period
  1, in order to encourage her second period self
  to give birth to more children

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Results of Time Discountingwith 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 ? and yj (1 r) sj-1, for j2, 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)

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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 ? yields
• A desire for more children in Period 1 ? reduced
  savings, s1 ? 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

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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 ? 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

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Results of Time Discountingwith a Perfect
Capital Market

• What about in this case Are there different
  results for sophisticated versus naïve women?
• Note n2 gt 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 womans life, she may be motivated to delay
  births

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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

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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

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Questions???