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

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6. Uncertainty and Dynamics in the Collective model 277 private goods other than good 1 are given by the conditions: ∂Gi ¡ ¢ i s Q, q−1 ∂Qj =1, 1 ≤ j ≤ N and ∂Gi ¡ ¢ i s Q, q−1 =1, 2 ≤ k ≤ n . ∂q i ki Neither these conditions nor the optimal levels of all private and public consumptions (except for good 1) dependonincome.Lettheseoptimal levels be denoted ¡ ¢ Q, ¯ i ¯q −1 . To simplify notations, we choose units such that Gi ¡ ¢ ¯Q, P i N s ¯q −1 = j=1 ¯ Qj + Pn k=2 ¯qi k ,i= a, b. Then, the indirect utility of a single person equals his or her income. Now, consider a man with income yb married with a woman with income ya . There is a unique efficient level for the consumption of each of the public goods and each of the private goods 2 to n. Moreover, these levels depend only on the total income of the partners, y = ya + yb .Ifwedefine η (y) = max (Q,qa −1 ,qb ( h F (Q) y − −1) PN j=1 Qj + Pn ¡ a k=2 qk + qb ¢ k i +Ga ) ¡ ¢ ¡ ¢ a b b m Q, q−1 + Gm Q, q−1 then the Pareto frontier is given by u a m + u b m = η (y)+θ a + θ b , (6.41) Here, u a m and u b m are the attainable utility levels that can be implemented by the allocations of the private good q1 between the two spouses, given the efficient consumption levels of all other goods. The Pareto frontier is a straight line with slope -1: utility is transferable between spouses (see chapter 3). Assuming, as is standard, that the optimal public consumptions are such that F (Q) is increasing in Q, weseethatη(y) is increasing and convex in y. 18 Moreover, η (0) = 0 and η 0 (0) = F (0) = 1. Sinceη is convex, this implies that η (y) >yand η 0 (y) > 1 for all y>0. Finally, if divorce takes place, the post-divorce utility of agent i is: ¡ i D ¡ y a ,y b¢¢ = D i ¡ y a ,y b¢ (6.42) V i s In particular, we see that V a s + V b s = Da ¡ y a ,y b¢ + D b ¡ y a ,y b¢ = y a + y b = y (6.43) In this framework, the divorce decision takes a particularly simple form. Indeed, agents divorce if and only if the point ¡ V a s ,Vb ¢ s is outside the Pareto set when married. Given (6.41), this occurs when the sum V a s +V b s is larger 18 By the envelope theorem, the derivative η 0 (y) is equal to F (Q). Therefore, η is increasing in y and, if F (Q) is increasing in y as well, then η is convex. Note that a sufficient (but by no means necessary) condition is that public consumptions are all normal.
278 6. Uncertainty and Dynamics in the Collective model than η (y) +θ a + θ b . Using (6.43), we conclude that divorce takes place whenever: η(y)+θ a + θ b
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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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328 7. Matching on the Marriage Mar
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334 8. Sharing the gains from marri
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392 9. Investment in Schooling and
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436 10. An equilibrium model of mar
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460 11. Marriage, Divorce, Children
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Pierre André Chiappori (Columbia) "Family Economics" - Cemmap

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