Pierre André Chiappori (Columbia) "Family Economics" - Cemmap

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10. An equilibrium model of marriage, fertility and divorce 441 expectation. That is, p = U(θm(p),θc(p)). (10.15) The function U(., .), viewed as a function of p, is a non decreasing function from [0, 1] to [0, 1]. Therefore, by the Tarski fixed point theorem (see Mas- Colell et al (1995), section MI), there is at least one equilibrium point in the interval [0, 1] at which expectations are realized. We may narrow down the range of possible equilibria, based on some a priori information. Because of the advantages of joint consumption and the zero mean and symmetry assumptions on G(θ) and F (ε), more than half of the population will choose to marry, and those who subsequently received asufficiently favorable shock to the quality of match will remain married even if the probability of finding a new mate is 1, implying that p 0. Because of the positive feedback, whereby an increase in the expected number of singles induces more people to become single, there may be multiple equilibria. Having assumed that all individuals are ex ante identical, we can rank the different equilibria based on their common expected value of life time utility: W1 = E max(W1,1(θ),W1,0(θ),V1). (10.16) The expectation is taken at the beginning of the first period prior to any meeting, when the quality of prospective matches is yet unknown. An equilibrium with a higher number of unattached individuals at the beginning of the second period will generally have less marriages, more divorces and fewer couples with children. Despite these apparently negative features, equilibria with higher p are in fact Pareto superior, because of the better option for couples who suffered a bad shock to their first marriage to recover by forming a new marriage. To see this, note that by (10.11), ∂W1,n ∂p and ∂V1 ∂p are positive, implying that an increase in p causes an increase in the expected welfare of all members of society, irrespective of the value of θ that they draw. In other words, the search frictions, represented here by random meetings with members of the opposite sex, irrespective of whether or not they are already attached, imply that those who choose to divorce or remain single exert a positive externality on other members of society who find it easier to find a mate for remarriage. This externality dominates the welfare comparisons because all other factors, such as the damage to children, are internalized by the partners. The presence of children implies that married couples are more reluctant to divorce, which yields a lower
442 10. An equilibrium model of marriage, fertility and divorce equilibrium value for p. However, it is still true that all couples, including couples with children, will be better off in an equilibrium with a higher p if multiple equilibria exist. 10.2 An Example We now introduce a simple example with multiple equilibria and discuss their properties. Assume that ε takes only two values, −a and +a with equal probability, while θ is distributed uniformly on [−b, b]. For this example, we assume that 2a > ¡ q ∗ − q 0¢ ; that is, the variance of the match quality shock is large relative to the loss for children from divorce, so that even couples with children may divorce if the revised quality of their match is low enough. The expected utility profile if marriage takes place, conditional on having children or not (n =0, 1), is: ⎧ ⎪⎨ W1,n(θ) = ⎪⎩ 7Y 2 3Y + θ + p(Y + β)+n(q 0 − c) if − b ≤ θ0). 3 In this figure we have h1 + a > h0− a. This follows from the assumption that 2a > � q∗ − q0� .
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Family Economics Martin Browning Pi
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iv 3.2 Householdproduction ........
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vi 6.4 IntertemporalBehavior ......
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viii 11 Marriage, Divorce, Children
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List of Figures 1.1 Marriage Rates
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xiii 4.2 Apotentiallycompensatingva
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Introduction The existence of a nuc
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two distinct solution concepts; one
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were affectedbytherisinginvestments
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Murat Iyigun, Guy Lacroix, Valérie
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Part I Models of Household Behavior
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14 1. Facts there appear to be age,
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16 1. Facts 1.1.4 Transitions The m
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18 1. Facts sumption goods within h
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20 1. Facts may not capture the ful
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22 1. Facts This process is driven
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24 1. Facts 1.4 Children Children a
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26 1. Facts a major policy concern
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28 1. Facts have means for, say, hi
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30 1. Facts [21] Michael, Robert,
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32 1. Facts Age group USA Denmark 1
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34 1. Facts Birth cohort 1931-36 19
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36 1. Facts USA Canada UK Norway Ye
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38 1. Facts Country Male Participat
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40 1. Facts Mother’s Age 20-30 31
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42 1. Facts rate per 1000, 20+ 18 1
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44 1. Facts Marriage rate per 1000
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46 1. Facts percent entering first
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48 1. Facts percent entering divorc
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50 1. Facts 100% 90% 80% 70% 60% 50
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52 1. Facts 1 0.8 0.6 0.4 0.2 0 23
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54 1. Facts 0.8 0.6 0.4 0.2 0 23 25
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56 1. Facts 0.8 0.6 0.4 0.2 0 1968
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58 1. Facts 6.8 6.6 6.4 6.2 6 5.8 2
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60 1. Facts 0.4 0.2 0 -0.2 -0.4 196
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62 1. Facts 0.6 0.4 0.2 0 1968 1970
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64 1. Facts FIGURE 1.23. Age Pyrami
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66 1. Facts 0.6 0.4 0.2 0 1930 1933
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68 1. Facts 100% 80% 60% 40% 20% 0%
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70 1. Facts 0.35 0.3 0.25 0.2 0.15
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72 1. Facts 100% 80% 60% 40% 20% 0%
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74 1. Facts Average number of child
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76 1. Facts Age 28 27 26 25 24 23 1
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78 1. Facts FIGURE 1.37. Consumptio
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80 1. Facts
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82 2. The gains from marriage for o
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84 2. The gains from marriage Q 2 y
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86 2. The gains from marriage this
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88 2. The gains from marriage as wh
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90 2. The gains from marriage ing a
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92 2. The gains from marriage absen
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94 2. The gains from marriage 2.5 C
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96 2. The gains from marriage 2.5.3
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98 2. The gains from marriage This
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100 2. The gains from marriage [14]
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102 2. The gains from marriage
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104 3. Preferences and decision mak
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158 4. The collective model: a form
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200 5. Empirical issues for the col
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244 6. Uncertainty and Dynamics in
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294 7. Matching on the Marriage Mar
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492 11. Marriage, Divorce, Children
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Pierre André Chiappori (Columbia) "Family Economics" - Cemmap

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