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

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generating the aggregate demand: 5. Empirical issues for the collective model 215 piqi = pici+ £ α a ¡ ¢ ¡ ¡ ¢¢¤ 0 x z b 0 x z i μ + μ x + μ z + αi 1 − μ + μ x + μ z Y (5.20) and for public goods: PjQj = PjCj + £ α a ¡ ¢ ¡ ¡ ¢¢¤ 0 x z b 0 x z j μ + μ x + μ z + αj 1 − μ + μ x + μ z Y Here, ci = ca i + cb ³ i and Y = x − P i,s pics P i − j PjCj ´ . The household demand is thus a direct generalization of the standard LES, with additional quadratic terms in x2 and cross terms in xpi and xPj, plus terms involving the distribution factor z; one can readily check that it does not satisfy Slutsky symmetry in general, although it does satisfy SNR1. A first remark is that ca i and cbi cannot be individually identified from group demand, since the latter only involves their sum ci. As discussed above, this indeterminacy is however welfare irrelevant, because the collective indirect utilities of the wife and the husband are, up to an additive constant: W a (p, P, x, z) = logY +log ¡ μ 0 + μ x x + μ z z ¢ − X α a i log pi − X i j α a j log Pj W b (p, P, x, z) = logY +log ¡ 1 − ¡ μ 0 + μ x x + μ z z ¢¢ − X α b i log pi − X which does not depend on the cs i . Secondly, the form of aggregate demands is such that private and public goods have exactly the same structure. We therefore simplify our notations by defining ξi = qi for i ≤ n, ξi = Qi for n
216 5. Empirical issues for the collective model ante knowledge of the public versus private nature of each good is necessary for the identifiability result to hold in general, for many parametric forms it is actually not needed. Identifiability The general case The question, now, is whether the empirical estimation of the form (5.21) allows to recover the relevant parameters - namely, the α s i ,theγi ,andthe μα . We start by rewriting (5.21) as: πiξ i = πiγ i+ ¡ α b i + ¡ α a i − α b¢ ¡ 0 a i μ + αi − α b¢ ¢ x z i (μ x + μ z) Ã x − X πmγ m ! (5.22) The right hand side of (5.22) can in principle be econometrically identified; we can thus recover the coefficients of the variables, namely x, x2 ,xz, the πm and the products xπm and zπm. For any i and any m 6= i, theratio of the coefficient of x by that of πm gives γm ;theγmare therefore vastly overidentified. However, the remaining coefficients are identifiable only up to an arbitrary choice of two of them. Indeed, an empirical estimation of the right hand side of (E0 i ) can only recover for each j the respective coefficients of x, x2 and xz, that is the three expressions Kj x = αb j +¡ αa j − αb ¢ 0 j μ , Kj xx = ¡ αa j − αb ¢ x j j μ and Kxz = ¡ αa j − αb ¢ z j μ .Now,pickuptwoarbitrary ¢ values for μ 0 and μ x ,withμ x 6=0.Thelasttwoexpressionsgive ¡ α a j − αb j and μz ;thefirst gives αb j therefore αaj . As expected, a continuum of different models generate the same aggregate demand. Moreover, these differences are welfare relevant, in the sense that the individual welfare gains of a given reform (say, a change in prices and incomes) will be evaluated differently by different models; in practice, the collective indirect utilities recovered above are not invariant across the various structural models compatible with a given aggregate demand. A unitary version of the model obtains when the Pareto weights are constant: μx = μz =0.ThenKj xz =0for all j (since distribution factors cannot matter9 ), and Kj xx = 0 for all j (demand must be linear in x, since a quadratic term would violate Slutsky). We are left with Kj x = αb j + ¡ αa j − αb ¢ 0 j μ , and it is obviously impossible to identify independently α a j ,αb j and μ0 ; as expected, the unitary framework is not identifiable. Identification under exclusion We now show that in the non-unitary version of the collective framework, an exclusion assumption per member is sufficient to exactly recover all the 9 For a discussion of the role of distribution factor in a unitary context, see Browning, <strong>Chiappori</strong> and Lechene (2006). m
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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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266 6. Uncertainty and Dynamics in
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294 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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