Collective Model with Children: Public Good and Household Production

Transcription

1 Collective Model with Children: Public Good and Household Production Eleonora Matteazzi Nathalie Picard June 1, 21 Abstract The paper develops a theoretical model of labor supply with domestic production and public goods. The objective is to deal simultaneously with these two aspects and thus consider a good that is both produced and publicly consumed by household members. This good is the quality and quantity of children. We focus on the way the cost of children is shared between parents. The cost of children is made both of time and money, including all goods and time bought at market prices for children and the value of parental time devoted to children. The model is illustrated by an empirical application using French time use data. Key-words: collective model, market labor supply, domestic labor supply, household production, identification, children cost, public good, time use survey JEL-codes: D13, J21, J22 THEMA, Université de Cergy-Pontoise and Università degli Studi di Verona THEMA, Université de Cergy-Pontoise 1

2 1 Introduction This paper aims at modeling household decisions concerning children in a collective approach. The original framework, initially developed by Chiappori (1988, 1992), considers agents who individually maximize an egotistic utility function, defined over their private consumption of market goods and leisure, subject to the household budget constraint, to individual time constraints, and to the Pareto constraint that it would not be possible to increase one s utility without reducing the partner s utility. Alternatively, the household maximizes a weighted sum of individual utilities with respect to the household budget constraint and individual time constraints. Chiappori showed that an alternative interpretation is given by the existence of two stages in the household internal decision process: household members first share non labor income, according to a given sharing rule, and, then, each one chooses his or her own labor supply and consumption. Finally, Chiappori showed that the model still holds in the case of caring (not paternalistic) preferences. Browning et al. (1994) have generalized the collective approach by introducing public goods. Another generalization (Apps and Rees 1997, and Chiappori 1997) of the collective approach considers household production of a (marketable or non marketable) good consumed privately by household members. The objective of our paper is to deal simultaneously with these two aspects and thus consider a good that is both produced within the household and publicly consumed by household members. In our paper, this good is represented by the quality and quantity of household children. In particular, we are interested in the cost of children and in the way this cost is shared between parents. The total cost of children is made both of time and money (Apps and Rees 21, Bradbury 1994, and Perali and Caiumi 29) in the sense that it is defined as a market consumption cost plus the value of parental time devoted to take care of children. Although 2

3 children are modeled in a similar way in other works (Blundell et al.26, Donni 22), to the best of our knowledge, the way we propose to model and identify how the total cost of children is shared among spouses has never been analyzed up to now. Under weak assumptions concerning the technology of production for the public good (either constant or decreasing returns to scale; quasi concavity), combined with the usual assumptions concerning preferences (individual utilities strictly increasing and quasi concave in their arguments; separability in individual utilities betweeen the public good and the private sphere that involves consumption and leisure), we recover the total cost of children borne by each parent, and not only the derivatives of this cost. More precisely, assuming stable preferences of spouses, the individual contribution to the total cost of children is given by the difference between the amount that a spouse would get without children and the amount that the same individual gets with children. Equivalently, the individual contribution to the total cost of children is given by the difference between the amount that an individual would get if spouses share household non labor income and the amount that the same individual gets when spouses share the household exogenous income minus the total production cost. In other words, how much the husband (or the wife) gets if he (she) and his (her) partner share household non labor income? And, how much the husband (or the wife) gets if he (she) and his (her) partner share household non labor income minus total cost devoted to children? The difference between these two amounts gives the the husband (or the wife) constribution to the total cost of children. Namely, the individual contribution to the total cost of children is given by the difference between the sharing rule defined over household non labor income income and the sharing rule defined over household non labor income minus total cost of children. The paper is organised as follows. In section 2, we present two equivalent 3

4 Pareto programs that allow us to define the individual contribution to the total cost of children as difference between the two sharing rules mentioned above. In section 3, we present identification results of the sharing rule and the individual contribution to the total production cost. In section 4 we discuss the construction of the index that explains the quality and quantity of children. In sections 5 and 6, we present, respectively, empirical specification and data sources. Finally, in section 7, we discuss estimation results. 2 The Model 2.1 Definitions and centralized model P In what follows, we consider a two-person household. Each household member i = 1, 2, respectively the husband and the wife, are characterized by his or her own rational preferences. All the analysis is conducted under the hypothesis that each spouse is selfish toward the other and that, whatever the decision-making process within the household, the spouses will always exploit all consumption opportunities and they will come up to a Pareto efficient allocation. Household members work for money (t 1 and t 2 hours paid at wage rates w 1 and w 2 ), enjoy leisure l 1 and l 2, consume a private good c 1 and c 2 (what we empirically observe is C = c 1 + c 2 ) and a quantity of a public good y that represents both the quantity and quality of their children. When there is a public good, identification can be achieved only under the hypothesis of separability in the individual utilities between the public good y and the private sphere that involves consumption c i and leisure l i. Assumption 1. Individual utilities are characterized by egoistic prefer- 4

5 ences of the form: U i ( u i ( c i, l i ; Z p ), y; Zp ) where Z p denotes the vector of individual characteristics that affect preferences and U i and u i are strictly increasing in their arguments and strictly quasi concave and verify the Inada conditions (except that the marginal utility of y is not infinite when y = ). In this model, y is a good that is both publicly consumed by household members (in the sense that the husband and the wife enjoy the same quantity of the public good) and produced within the household. In fact, each spouse member shares her time endowment T between leisure, market work and domestic work h i, so that: T = l i + t i + h i where h i represents the time spent by each parent to take care of their children. In order to produce y, the household also buys some market goods and services (such as clothing, school insurance, school meals, transport, education, etc.), and time inputs such as a nurse, a baby-sitter, etc... Let us define c y the monetary cost for children. Then, the household total cost of producing the public good is T C = w 1 h 1 + w 2 h 2 + c y. The price of the composite good c 1 + c 2 + c y is normalized to 1. Finally, the household production technology is given by y = Y ( ) h 1, h 2, c y ; Z h (1) Assumption 2. The function Y is increasing and concave in each argument and globally quasi-concave. We exclude perfect substituability between inputs so that production frontiers are convex. Z h denotes the vector 5

6 of household and individual characteristics that affect production efficiency. Notice that we only make the assumption of decreasing (or constant) marginal productivities for each input but, unlike Chiappori (1997) and Apps and Rees (1997), we need not to assume constant returns to scale. Any Pareto-Optimal solution solves the following constrained maximization program (P), namely it maximizes a social welfare function W = λu ( ) 1 u 1 (c 1, l 1 ; Z p ), y; Z ( ) p + (1 λ)u 2 u 2 (c 2, l 2 ; Z p ), y; Z p (2) subject to the household budget constraint, the household production technology and individual time constraints: c 1 + c 2 + c y + w 1 l 1 + w 2 l 2 + w 1 h 1 + w 2 h 2 w 1 T + w 2 T + m y Y (h 1, h 2, c y ; Z h ) l i + t i + h i T, i = 1, 2 where m is household non labor income. The function λ represents the Pareto-weight that depends on the exogenous variables entering the budget constraint (wages and non labor income), on preferences (Z p ), on distributional factors s (variables which influence the decision process without affecting the budget set or preferences), on Z h and on y. An interpretation of this welfare weight is that it represents the bargaining power of individual 1 in the intra-household allocation process. Namely, λ determines the final position on the Pareto-frontier. Changes in wages or non labor income may shift bargaining power from one individual to the other, with consequences on observable household consumption and labor supply. 6

7 2.2 The two-stage program P1 Both egoistic preferences and caring preferences make an alternative resolution of the program P possible. In fact, an application of the second fundamental theorem of welfare economics allows us to solve the program in different stages. At the first stage of the program P1, the production level of the public good y is collectively chosen, and technically, it is a fuction of all exogenous variables (w 1, w 2, m, s, Z h, and Z p ). But, just as the value the level of expenditure for the publig good K in Blundell et al. (25), in what follows, y = y (w 1, w 2, m, s, Z h, Z p ) is taken as given, or more exactly predetermined. Based on the enveloppe theorem we don t have to derive y with respect to exogenous variables because it is the optimum. The minimization of household total cost of producing the domestic good, given the production function of the public good, gives optimal inputs levels. Optimality and interior solutions 1 imply following first order conditions 2 : Y h i Y c y = w i, i = 1, 2. (3) which gives unique inputs levels of the individual domestic labor supply h i = h i (y, w 1, w 2 ; Z h ) and of the monetary cost c y = c y (y, w 1, w 2 ; Z h ). This relation says that for individual i the marginal value of time spent in household production, namely h i, relative to the monetary cost c y is equal to her market wage. In other words, household member i is marginally indifferent between one hour spent in household production and w i $ spent for c y. Given the optimal input levels, the total cost of producing the public good is defined as: T C(y, w 1, w 2 ; Z h ) = w 1 h 1 (y, w 1, w 2 ; Z h )+w 2 h 2 (y, w 1, w 2 ; Z h )+c y (y, w 1, w 2 ; Z h ). 1 For corner solutions, see Blundell et alii (27) and Donni (23) 2 In what follows, the notation X k stands for the partial differential of the function X with respect to variable k. 7

8 At this point, household non labor income net of the cost of producing the public good, that is m T C(y, w 1, w 2 ; Z h ), is shared between spouses, according to a sharing rule Ψ: Ψ 1 = Ψ(w 1, w 2, m, s, y, h 1 (.), h 2 (.), c y (.); Z p, Z h ) = Ψ(w 1, w 2, m, s, y; Z p, Z h ) and, omitting the explanatory variables, Ψ 2 = m T C Ψ 1. The existence of a sharing rule implies no more (and not less) than the efficiency of the collective decision process (see Chiappori 1992). Given any Pareto weight λ we can find a corresponding sharing rule Ψ such that the outcomes of the two associated programs are the same, in other words there is a one to one correspondence between λ and Ψ. This means that bargaining power within the household can be measured alternatively by any of those functions since they are equivalent. The sharing rule Ψ depends both on the level of y and, also, on the level of each input, but the sharing rule Ψ does not depends on inputs since optimal levels of inputs are considered here implicitly (reduced form). Pareto-optimal decisions taken by spouses at the first stage can be seen as individually optimal in the sense that, because individual i anticipates the impact of her decisions on the sharing rule, she has no incentive to deviate from the Pareto-optimal solution. At the second stage, each spouse, separately, chooses how to allocate her own budget between composite private consumption good and leisure. In other words, each household member maximizes her own utility u ( ) i c i, l i ; Z p under her own budget constraint, implied by previous step, namely c i + w i l i w i T + Ψ i. 8

9 Optimality and interior solutions imply following first order conditions: u i l u i c = w i, i = 1, 2. (4) This relation represents the marginal value of leisure relative to the private consumption good. For individual i, one hour spent enjoying leisure is marginally equivalent to w i $ spent for the private consumption good c i. Demand functions for consumption are c i = c i (w i, Ψ i (w 1, w 2, m, s, y; Z p, Z h ) ; Z p ). Demand functions for leisure are l i = l i (w i, Ψ i (w 1, w 2, m, s, y; Z p, Z h ) ; Z p ). Moreover, from the time constraints, we can determine the Marshallian total labor supply funtion L i = T l i = t i + h i = L ( ) i w i, Ψ i (w 1, w 2, m, s, y; Z p, Z h ) ; Z p. Finally, the associated individual indirect utility functions are: v i (w i, Ψ i ) = u ( i c ( ) ) i w i, Ψ i ; Z p ), l i (w i, Ψ i ; Z p ; Zp. (5) This approach does not allow us to determine explicitly how much each spouse contributes to the monetary cost c y devoted to children, even if the first two stages implicitly define such a repartition. Our goal is to make it explicit, then, in the next section, we develop an alternative way of presenting the program P Sharing total cost of children between parents In this section, we present a second program, named P2, totally equivalent to program P1, but with two advantages. First, it makes explicit the implicit (because not actually paid) repartition of the monetary cost c y between spouses, namely c y1 and c y2 ; second, it allows to define and measure the total cost of children borne by each parent. As stated in section 2.1, the total cost of producing the public good is made of a monetary cost c y and the 9

10 value of parents time devoted to children, namely T C = w 1 h 1 + w 2 h 2 + c y. Our goal is to measure how much of that cost is implicitly borne individually by each parent. In other words, we want to determine T C 1 = w 1 h 1 + c y1 and T C 2 = w 2 h 2 + c y2, where c y1 = αc y and c y2 = (1 α)c y. The knowledge of time devoted by each spouse to take care of children, namely the domestic work h i in the production function, gives a misleading idea of how much each parent implicitly spends for children because children s cost may be compensated in the sharing rule. Let us consider the following example. Suppose that in a family the father cares more for children than the mother (namely, he is willing to spend more than the mother for child care) but he is less productive in household production. This means that the wife will spend a large amount of time with children but she will be compensated by the husband through the sharing rule. This allows her to increase her consumption of the market private good and leisure. At the same time, the husband cares more for children and then he undergoes a greater share of the children cost c y that reduces his share of the household income. In this example, the (implicit and unnobservable) total cost of children borne by the mother is low although it can be observed that she spends a lot of time for children. On the other hand, if the mother spends a large amount of time with children because she cares a lot for them (more than husband), this will not be compensated in the sharing rule. This means that she will have a lower share of household income that will force her to reduce her consumption of private composite good and leisure. In this example, both the observed time devoted to children and the implicit cost of children are large for the woman. Then, let us consider the following three-stage program, named P2. At the first stage, household members agree on the repartition of family non labor income according to a sharing rule Φ, from which each spouse will 1

11 contribute, at the second stage, to children cost. Then, let us define Φ 1 = Φ(w 1, w 2, m, s; Z p ) and, omitting the explanatory variables, Φ 2 = m Φ. Once the sharing rule Φ is defined, the production level of y is collectively chosen (just as in the first stage of program P1). Then the minimization of household cost of producing the domestic good gives optimal inputs levels h i = h i (y, w 1, w 2 ; Z h ) and c y = c y (y, w 1, w 2 ; Z h ). At the second stage, the household monetary cost for children c y is shared between spouses according to a sharing rule α, which gives c y1 = αc y and c y2 = (1 α)c y. The optimal repartition α should then maximize: λu ( 1 v 1 (w 1, Φ 1 w 1 h 1 αc y ), y ) +(1 λ)u ( 2 v 2 (w 2, Φ 2 w 2 h 2 (1 α)c y ), y ) which gives the first order condition: λuuv 1 1 = (1 λ)uuv 2 2. (6) We obtain that the marginal utilities of income are inversely proportional to the Pareto-weight. This first order condition is necessary and sufficient, so there exists a unique solution α R, where: α = α(w 1, w 2, m, y, h 1 (.), h 2 (.), c y (.), Φ(.), Z p ) There is no reason why α should lie in the interval [,1]. Indeed, it may well be the case that α < (or α > 1), for example if individual 1 (or individual 2) is very productive in taking care of children but does not like much children. In that case, individual 1 (or individual 2) will spend a lot of time in household production, but she will be compensated reducing her contribution to the monetary cost (just to have a negative c yi because α i is negative or greater than 1). In any case, even if the individual monetary cost may be negative for one member, individual total costs, T C 1 = w 1 h 1 + αc y 11

12 and T C 2 = w 2 h 2 +(1 α)c y, (aggregating remuneration of household working time and individual monetary cost) are always positive for both spouses because of the assumption that U i is increasing in y. This means that, at the second stage, both spouses will face a lower income: Φ 1 w 1 h 1 αc y < Φ 1 and Φ 2 w 2 h 2 (1 α)c y < Φ 2. Since T C i >, this implies that: utility w 1h 1 < α < 1 + w 2h 2 c y c y Finally, at the third stage, each spouse separately maximizes her own u i ( c i, l i ; Z p ) under her own budget constraint implied by previous steps, namely c i + w i l i w i T + Φ i T C i. Demand functions for consumption and leisure are c i = C i (w i, Φ i T C i ; Z p ) and l i = l i (w i, Φ i T C i ; Z p ). Moreover, from the individual time constraint, we can determine the Marshallian total labor supply funtion L i = T l i = t i + h i = L i (w i, Φ i T C i ; Z p ). Notice that both α and the income left to each individual s private consumption Φ i T C i, are affected by the level of y and by the level of each input h i and c y. Then, the Pareto-optimal decisions taken by household at the first stage can be seen as individually optimal because, since spouse i anticipates the impact of her decision on the following stage, she has no incentive to deviate from the Pareto-optimal solution. To conclude, let us remark that the equivalence between the programs P1 and P2 implies that the net income available to each spouse after she has contributed to household production, that is Φ i T C i, is equal to the sharing rule Ψ i defined in program P1. This allows us to determine the value 12

13 of individual total cost as difference between the sharing rule defined over non labor income and the sharing rule defined over non labor income minus total production cost, that is T C i (w 1, w 2, m, s, y; Z p, Z h ) = Φ i (w 1, w 2, m, s; Z p ) Ψ i (w 1, w 2, m, s, y; Z p, Z h ). More precisely, assuming stable preferences of household members, the individual contribution to the total cost of children is given by the difference between the amount that a household member (in our case, the husband or the wife) would get if there are no children and the amount that the same individual gets when there are household children. Or alternatively, the individual contribution to the total cost of children is given by the difference between the amount that a household member would get if spouses share household non labor income and the amount that the same individual gets once spouses share the household exogenous income minus the total production cost. In other words, how much the husband (or the wife) gets if he (she) and his (her) partner share household non labor income? And, how much the husband (or the wife) gets if he and his (her) partner share household non labor income minus the total production cost? The difference between these two amounts gives the the husband (or the wife) constribution to the total production cost. Note that, as usual, the constant term of both sharing rules cannot be identified, but those two household-specific constant terms are equal by definition and, therefore, they disappear in the difference of the two sharing rules. 2.4 How is the efficiency condition for the public good? In what follows, we show that the efficiency condition for the public good takes the standard Bowen-Lindhal-Samuelson form. Given the production function (1) y = Y [h 1 (y, w 1, w 2 ; Z h ), h 2 (y, w 1, w 2 ; Z h ), c y (y, w 1, w 2 ; Z h ); Z h ] 13

14 deriving it with respect to y we have: 1 = Y h 1 h 1 y + Y h 2 h 2 y + Y c y c y y and substituting relation (3), we obtain: 1 Y c y = w 1 h 1 y + w 2 h 2 y + c y y. (7) At the first stage of program P2, Pareto-optimality implies that c y, h 1 and h 2 maximize the household welfare function: λu ( 1 v 1 (w 1, Φ 1 w 1 h 1 αc y ), Y (h 1, h 2, c y ) ) +(1 λ)u ( 2 v 2 (w 2, Φ 2 w 2 h 2 (1 α)c y ), Y (h 1, h 2, c y ) ) which gives first order condition for c y : λuuv 1 1 ( α) + λuy 1 Y c y + (1 λ)uuv 2 2 (α 1) + (1 λ)uy 2 Y c y = where (1 λ)uuv 2 2 = λuuv 1 1 because of (6) 3. Then, we obtain: 1 Y c y = U y 1 Uuv + U y 2. (8) 1 1 Uuv 2 2 Finally, combining equations (7) and(8), we obtain that the optimal level of y is such that the marginal cost of producing y for the household is equal 3 Similarly, for h 1 and h 2. The first order condition, respectively, for h 1 and h 2 are Then, rearranging, we obtain: λu 1 uv 1 ( w 1 ) + λu 1 y Y h 1 + (1 λ)u 2 y Y h 1 = λu 1 y Y h 2 + (1 λ)u 2 uv 2 ( w 2 ) + (1 λ)u 2 y Y h 2 = w i Y h i = U y 1 Uuv U y 2 Uuv 2 2 = 1. Y c y We find again that the marginal value of h i relative to c y is equal to w i, as expected if labor market is competitive. 14

15 to the sum of the marginal benefits of enjoying y for the two spouses, other ways to the sum of the marginal amounts spouses are eager to pay in order to enjoy one unit more of y. Let us define: p i U i y U i uv i (9) the marginal value of y for individual i, that is i is marginally indifferent between one unit more of y and p i $ to share between her private consumption and leisure. The p i simply correspond to Lindhal prices. Then, at the optimum, the relative marginal value for y for the spouses only depends on their own relative preferences for the public good and on Pareto-weight, but not on the production side: p 2 p 1 = U 2 y U 2 uv 2 U 1 uv 1 U 1 y = (1 λ) Uy 2 λ Uy 1 where the last equality derives from relation (6). 2.5 Optimal level of public good y In this section, we show that there is a unique optimal y level, corresponding to the tangency point between the production frontier θ and the social welfare function W defined in the space (T C 1, T C 2 ), as illustrated in Figure 1. Let us define the production frontier θ as y = θ(t C 1, T C 2 ; w 1, w 2, Z h ) (1) It represents all possible combinations of (T C 1, T C 2 ) necessary to produce a certain y level. More precisely, a particular value of α is associated to each point of this frontier. Besides, according to Assumption 2, the production frontier θ is represented by a strictly convex function in the space (T C 1, T C 2 ). As stated in section 2.3, the optimal repartition α of the monetary cost has 15

16 to maximize W = λu ( 1 v 1 (w 1, Φ 1 T C 1 ), y ) + (1 λ)u ( 2 v 2 (w 2, Φ 2 T C 2 ), y ) (11) where T C 1 = w 1 h 1 +αc y, T C 2 = w 2 h 2 +(1 α)c y, and the y level considered in the W curve has to be produced with the technology θ. According to Assumption 1, for a given level of public good, the Pareto frontier (constant W ) can be represented by decreasing strictly concave curve, as you can see in Figure 1. Then, household welfare is maximized when the W curve goes down and left for a given value y, because lower T C i means higher wealth. Let us turn to the unicity of y. Start from one initial y o value of public good, defining optimal costs T C i as above and thus defining a welfare level W o = λu 1 + (1 λ)u 2. Consider now the effect of a marginal increase y. This will shift the θ curve upwards because producing more public good is more costly. Under our assumption of constant or increasing marginal costs, for a fixed y increase, the extent of the upwards shift of the θ curve is either constant or increasing in y. The larger quantity of public good y + y defines a new W curve corresponding to the new series of individual total costs (T C 1, T C 2 ) that would give the same welfare level W o with this higher quantity of public good. Namely, W (y o + y, T C 1, T C 2 ) = W o, which corresponds to U ( 1 v ( 1 w 1, Φ 1 T C 1 ), y o + y ) = U 1 and U 2 ( v 2 ( w 2, Φ 2 T C 2 ), y o + y ) = U 2. In other words, with y o + y, each spouse could get the same utility U i with a lower wealth, that is with a larger cost T C i > T C i. Obviously, based on Assumption 1, for a fixed y increase, the extent of the upwards shift of 16

17 the W curve is strictly decreasing in y. For this reason, two cases may occur depending on the value of y. For low values of y, see Figure 2a, the upwards shift of the W curve is large compared to the upwards shift of the θ curve, so the new W and θ curves cross. This means that there is another W curve lying below the second W curve and tangent to the second θ curve: spouses may get more utility with y o + y than with y o, so the initial level was not optimal (too small). For large values of y, see Figure 2b, we have to show that y level is not optimal because it is too large, so we consider a reduction of the quantity of the public good. The downwards shift of the W curve is small compared to the downwards shift of the θ curve, so the new W and θ curves cross: there is is another W curve lying below the second W curve and tangent to the second θ curve: spouses may get more utility with y o y than with y o, so the initial level was not optimal (too large). Assumption of constant or increasing marginal costs and of decreasing marginal utilities, together with the assumption that y can take continuous values, imply that there is a unique y level at the frontier between these two cases, leading to the following proposition: Proposition 1. There exists a unique quantity y of public good, and hence a unique sharing of costs T C 1 and T C 2 that maximizes household welfare W. As shown in Figures 2, this unique optimal y level corresponds to the tangency point E between the production frontier θ (convex) and the concave Pareto frontier W. 17

18 3 Identification results In following sections we show identification results concerning the sharing rule Ψ(w 1, w 2, m, s, y; Z p, Z h ) and the individual total cost T C i (w 1, w 2, m, s, y; Z p, Z h ). 3.1 Restrictions on total labor supplies and the sharing rule Chiappori (22) shows that a set of testable restrictions of the collective approach on observable market labor supplies can be derived and that the sharing rule Φ(w 1, w 2, m, s; Z p ) can be identified up to an additive constant. 4 Donni (23) showed that Chiappori s conclusions are still valid if either the husband or the wife (but not both) does not work 5. In this section we show that a set of testable restrictions of the collective approach on observable total labor supplies can be derived and that the sharing rule Ψ(w 1, w 2, m, s, y; Z p, Z h ) can be identified up to an additive constant. Then, following Chiappori, we differentiate total labor supply functions L 1 = t 1 + h 1 = L 1 ( w 1, Ψ 1 (w 1, w 2, m, s, y; Z p, Z h ); Z p ) L 2 = t 2 + h 2 = L 2 ( w 2, Ψ 2 (w 1, w 2, m, s, y; Z p, Z h ); Z p ) with respect to wages, non labor income, the level of public good, and the distribution factor. Recall that, although y is endogenous, based on the enveloppe theorem, it is not necessary to derive. Thus, we have: 4 Chiappori (22) identifies the partial derivatives of the sharing rule Φ in terms of observable labor supplies t 1 and t 2, namely: Φ w2 = AD D C, Φ w 1 = BC D C, Φ s = CD D C, Φ m = D D C where: A = t1 w 2 /t 1 m, B = t 2 w 1 /t 2 m, C = t 1 s/t 1 m, and D = t 2 s/t 2 m (with t i m and D C). 5 Let us to remark that wages are assumed to always be observed by the economist, even when one individual does not work. In that case, a potential wage is estimated by an auxiliary equation. 18

19 L 2 = L2 w 1 Ψ 2 L 1 w 2 = L1 Ψ 1 Ψ 1 w 2 L 1 w 2 L 1 m = L1 Ψ 1 Ψ 1 m L1 m L 1 s = L1 Ψ 1 Ψ 1 s L1 s L 1 y = L1 Ψ 1 Ψ 1 y L1 y L 2 m = L2 Ψ 2 L 2 s = L2 Ψ 2 ( L 2 y = L2 Ψ 2 ( T C ) Ψ1 L 2 w w 1 w 1 1 ( ) 1 Ψ1 L 2 m m ( ) Ψ1 L 2 s s T C y Ψ1 y ) L 2 y Defining A = L 1 w 2 /L 1 m, B = L 1 s/l 1 m, C = L 1 y/l 1 m, D = L 2 w 1 /L 2 m, E = L 2 s/l 2 m, and F = L 2 y/l 2 m, with L 1 m and L 2 m. Assuming that E B and F C, which will obviously be the case in the application below, and solving the system, we obtain the derivatives of the sharing rule Ψ w1 = T C w1 + Ψ w2 = AE E B, Ψ s = BE E B, Ψ y = CE E B, BD E B, 19

20 Ψ m = E E B. These partials are compatible if and only if they satisfy the usual crossderivative restrictions. Hence, the following conditions are necessary and sufficient : (a) Ψ ms = Ψ sm, (b) Ψ mw1 = Ψ w1 m, (c) Ψ mw2 = Ψ w2 m, (d) Ψ my = Ψ ym, (e) Ψ w1 w 2 = Ψ w2 w 1, (f) Ψ w1 s = Ψ sw1, (g) Ψ w1 y = Ψ yw1, (h) Ψ w2 s = Ψ sw2, (i) Ψ w2 y = Ψ yw2, (l) Ψ sy = Ψ ys, (m) lw 1 1 l1 m Ψm [T l 1 + Ψ w1 ], (m) lw 2 2 l2 m 1 Ψ m [T l i E F +T Cy Ψ w2 ], and (o) =. If these conditions E B F C are fulfilled, then the sharing rule Ψ(w 1, w 2, m, s, y; Z p, Z h ) can be identified up to an additive constant. 3.2 The identification of the individual total costs In this section, we show the identification result concerning the individual contribution to the total production cost. As previously, the sharing rule Φ 1 = Φ(w 1, w 2, m, s; Z p ) defines how household non labor income is shared between spouses. Indeed, the sharing rule Ψ 1 = Ψ(w 1, w 2, m, y, s; Z p, Z h ) defines how household non labor income minus total production cost is shared between spouses. For any given household, the two sharing rules have to be continuous in y =, namely if y = Ψ(w 1, w 2, m,, s; Z p, Z h ) = Φ(w 1, w 2, m, s; Z p ). Let us consider a numerical example. The sharing rule defined over the non labor income income is Φ 1 = η + η 1 w 1 + η 2 w 2 + η 3 w1 2 + η 4 w2 2 + η 5 w 1 w 2 + η 6 m + η 7 s, (12) while the sharing rule defined over the non labor income minus total production cost is Ψ 1 = κ +κ 1 w 1 +κ 2 w 2 +κ 3 w 2 1 +κ 4 w 2 2 +κ 5 w 1 w 2 +κ 6 m+κ 7 s κ 8 w 1 y κ 9 w 2 y+ 2

21 w κ 1y κ w2y κ 12w 1 w 2 y κ 13 my κ 14 sy κ 15 y κ 16 y 2. (13) As we previously stated, if y =, the two sharing rules are equal for any given household, that is η + η 1 w 1 + η 2 w 2 + η 3 w η 4 w η 5 w 1 w 2 + η 6 m + η 7 s = κ + κ 1 w 1 + κ 2 w 2 + κ 3 w κ 4 w κ 5 w 1 w 2 + κ 6 m + κ 7 s, and thus: η = κ, η 1 = κ 1, η 2 = κ 2, η 3 = κ 3, η 4 = κ 4, η 5 = κ 5, η 6 = κ 6, and η 7 = κ 7. Given that the sharing rule has to be continuous in y =, and that these equalities always hold, we can identify T C 1 as difference between equations (12) and (13): T C 1 = Φ 1 Ψ 1 w = κ 8 w 1 y + κ 9 w 2 y + κ 1y κ w2y κ 12w 1 w 2 y +κ 13 my + κ 14 sy + κ 15 y + κ 16 y 2 (14) where, obviously, T C 1 = if y =. Finally, given the equalities between parameters η j and κ j for j = 1,..., 6, we can write the sharing rule Φ 1 as Φ 1 = κ + κ 1 w 1 + κ 2 w 2 + κ 3 w κ 4 w κ 5 w 1 w 2 + κ 6 m + κ 7 s. (15) The identification and estimation of the sharing rule Ψ 1 allows us to recover information about Φ 1 and T C 1. The wife s total cost T C 2 is, then, obtaines as difference between total production cost and the husband total cost. 21

22 4 The Empirical Specification 4.1 Total Cost, Labor Supplies and Sharing Rule A. Total Production Cost. The total production cost is given by the value of spouses domestic work, h 1 and h 2, and by a market cost for children c y. This market cost for children includes demand for childcare and market goods and services bought for children. production cost has the following functional form: T C(w 1, w 2, w y, y) = A 1 w 1 y + A 2 w 2 y + A 3 2 Then, we suppose that the total w1y 2 + A 4 w2y 2 + w y 2 w y +A 5 w 1 w 2 y w y + A 6 w y y + A 7 w y y 2 + ɛy (16) where w y is the price of c y. The total cost function is homogeneous of degree one in w = (w 1, w 2, w y ) and non decreasing in y (see Mas-Colell, Whinston, and Green, 1995). As stated in section 2.1, the price of the composite good, that include c y, is normalized to 1. Then in what follows w y = 1. By applying Shepard s lemma, we obtain spouses domestic labor supplies and the demand of childcare and market goods for children. i. Domestic labor supplies. The spouses domestic labor supplies have the following linear form: h 1 (w 1, w 2, y) = A 1 y + A 3 w 1 y + A 5 w 2 y + ɛ 1 y (17) h 2 (w 1, w 2, y) = A 2 y + A 4 w 2 y + A 5 w 1 y + ɛ 2 y. (18) These equations satisfy a symmetry property. ii. Demand for childcare and market goods. The demand for childcare and market goods is obtained as difference between the total cost 22

23 of production and the remuneration of parents time devoted to children, c y (w 1, w 2, y) = A 3 2 w2 1y A 4 2 w2 2y A 5 w 1 w 2 + A 6 y + A 7 y 2 + ɛ 3 y (19) Note that: (a) if y = then spouses domestic labor supplies, market cost for children, and total production cost will be equal to zero; (b) the heteroskedastic error term in total cost function is ɛ = ɛ 1 w 1 + ɛ 2 w 2 + ɛ 3. B. Sharing Rule. The total cost function is a second-order polynomious, thus, for symmetrical reasons, the sharing rule must be a polynomious of the same order. The husband s share has the following form, as specified in equation (13): Ψ 1 (w 1, w 2, m, y, s) = κ + κ 1 w 1 + κ 2 w 2 + κ 3 w κ 4 w κ 5 w 1 w 2 + κ 6 m + w1 κ 7 s κ 8 w 1 y κ 9 w 2 y κ 2y 1 κ w2 2y 2 11 κ 2 12w 1 w 2 y κ 13 my κ 14 sy κ 15 y κ 16 y 2. The wife s share has the following form: Ψ 2 (w 1, w 2, m, y, s) = m T C(w 1, w 2, y) Ψ 1 (w 1, w 2, m, y, s). C. Total Labor Supply. The husband s total labor supply, given by the sum of market and domestic labor supplies, has the following linear structural form: L 1 (w 1, Ψ 1 ) = α + α 1 w 1 + α 2 Ψ 1 + ɛ 4 (2) with α 1 and α 2. Similarly for the wife: L 2 (w 2, Ψ 2 ) = β + β 1 w 2 + β 2 Ψ 2 + ɛ 5. (21) with β 1 and β 2. 23

24 D. Market Labor Supply. The husband s market labor supply is obtained as difference between equations (2) and (17): t 1 = α + α 1 w 1 + α 2 Ψ 1 A 1 y A 3 w 1 y A 5 w 2 y ɛ 2 y + ɛ 4. (22) Similarly for the wife, the arket labor supply is derived as difference between total and domesticc labor supplies: t 2 = β + β 1 w 2 + β 2 Ψ 2 A 2 y A 4 w 2 y A 5 w 1 y ɛ 1 y + ɛ 5. (23) 5 Data The data used in this work are the French Time-Use Survey (Enquête Emploi du Temps) conducted by INSEE in The survey was designed to provide estimates of time that French people spend in various activities. The survey includes 8,186 households, of which 7,46 are complete (i.e. in which all household members filled in a time use booklet and an individual questionnaire). We selected a subsample of married or living-together couples with at least one child under 18 years old. We also restrict the sample to couples in which the male reports a paid activity. This selection leaves a sample of 649 households. The data reveal that women participation rate in employment is about 66 %. Mean monthly market working hours for men are about 158, whereas they are 129 for women. Market wage is determined by the ratio between weekly labor income and hours worked per week. since wage may be endogenous, it was instrumented using educational level (from -no diploma- to 8 - Grandes Ecoles ) second-order polynomial, the age second-order polynomial, a dummy for self-employment status, a dummy for the country of origin of the worker (born in France or not), and the number of children by selected age group (-2 years of age, 3-6 years of age, 7-12 years of age, and years of 24

25 age). Other instruments are related to housing: geographical area (dummy variable for Paris region), the type of housing unit (house or flat) and the housing tenure type (owned or not). Potential wage was imputed to nonworking women. The male hourly wage is about 1 $ per hour, whereas the wage rate of females is about 9 $ per hour. Household non labor income was imputed using the French Family Budget Survey. Household non labor income was defined as the sum of income from savings and income from state support for families. The variables used are the level of education, the age, and the country of origin of both parents, the number of children under 18 years of age, geographical area (regional dummy variables), the housing tenure type (owned or not), and the number of rooms in the household. Finally, the sex ratio is computed at the regional level using tha data from the Population Survey (Recensement de la population 1999) conducted in This corresponds to the number of men in the relevant age range, based on the age of the household head (5-year intervals), divided by the number of the whole population whose age belongs to that range. Summary statistics are reported in Table Children Cost In the theoretical model, we suppose that each spouse shares her time between leisure, working in the job market and working inside the domestic walls. Domestic work is then defined as the time devoted to take care for and help children. This definition include physical care, reading to/with children, playing with them, and all activities related to household children s education such as homework, school conferences, transportation, etc. Our data reveal that mean monthly working hours for male are about 34, whereas they are 68 for females. We also observe that domestic time is very large when 25

26 children are younger and diminish as children grow up. As stated in section 2.1, in order to produce the public good, the household buys in the market some input goods - such as clothing, school insurance, school meal, transport, school fees, etc. - and time inputs - such as a nurse, a baby-sitter, etc. This defines the household monetary cost of production of the public good, c y. Since such information was not provided in the French Time-Use Survey, we imputed this monetary cost from the French Family Budget Survey (Enquête Budget de Familles) conducted in 2. The variables used are the level of education and the age of both parents, the employment status of the mother (employed or not employed), the number of children by selected age group (-2 years of age, 3-6 years of age, 7-12 years of age, and years of age), the household total income, geographical area (regional dummy variables), and the type of childcare away from home. Our data reveal that monetary cost is decreasing in children age. This is because the purchased childcare represents a large fraction of the total costs. Moreover, children monetary cost in increasing in parents wages, other things constant. Finally, as regards the total production cost (value of parents time plus monetary cost). It is decreasing in children s age till age 12 and then increase. Moreover, it is increasing in parents wages The Definition of the Quantity and Quality of Children We would expect that the total cost of children varies with factors such as the number and the age of children, the household income, the scale effect in household production and, finally, the parents preferences for children. Now, we let define an index measuring the quality and quantity of children, namely y. 6 Result available upon request. 26

27 What are, then, the variables that define the optimal level of the public good y? In other words, what are determinants of the quality and quantity of children? Surely, household income. Richer households may increase the quality of their children allowing them to attend better school, or music lessons, dance...and so on. Also the number of children has an impact on the value of y. It influences directly the quantity of children, and indirectly their quality, partly in relation to economies of scale and also because of budget constraints. A greater number of children may, in fact, lower their quality. Finally, the preferences. Let us consider two households with one child of 1 years old each and the same income. But the first household is eager to spend more for his child than the other one. In this case, the joint preferences of parents determines different values of y. Besides, we suppose that preferences are stable during the time. Then when parents decide, at the first stage of program P1, the optimal level of the public good, and then the optimal quantity and quality of children, this decision does not change during the time. In other words, the optimal level of y is decided once, and then it is spread across life cycle since children are more costly at some ages. For this reason, we can state that the value of y does not vary with children age. Then, we defined the following total cost function, at a given time in parents life cycle, depending on the number of children (n) and their age (age), on household income (I), and on an average weight of a child when the houshold has n children ( (n)) (in other words, a sort of measure of the scale effect in household production): where [ n ] T C(n) = f 1 (age i ) f 2 (I) (n) i=1 f 1 (age i ) = exp(γ 1 age i + γ 2 age 2 i + γ 3 age 3 i ) 27

28 f 2 (I) = exp (γ + γ 4 ln(i)) ( n ) (n) = exp δ n 1 n children i=2 Then, for example, (2) = exp (δ 2 ) is the average weight of a child when the household has two children. In Table 1 we report the estimates of (n). As we can see, the average weight of a child when the household has n children decreases as the number of children increases and the marginal cost of an additional children also decreases. Besides, let us to note that the function f 1 (age i ) sums across children. In other words, if f 1 (age i ) does not depend on children s age we would get T C(n) = n f 2 (I) (n) exp (µ) Finally, once estimated parameters γ and δ, we computed the value of y using this expression: ( n ) ln(y) = ln(n) + ˆγ + ˆγ 4 ln(i) + ˆδ n 1 n children. i=2 Note that the value of y is computed in a way which does not depend on children s age because of the assumption of stable household preferences over time. 28

29 Table 1. Number of children Average weight of a child Equivalence Marginal when the household has Scale Cost n n children: (n) (n) n The Estimation Method We estimate, by maximum likelihood, a system of five structural equations (the wife s and the husband s market labor supplies - y 1 and y 2 -, the wife s and the husband s domestic labor supplies - y 3 and y 4 - and the monetary cost for children - y 5 -), considering two different regimes: the wife participates to the labor market or not. The wife s labor force participation is based on whether her desired hours of work are greater or less than zero. The set of observations i is sorted so that the wife is working in observations 1 to k and she is not working in observations k + 1 to n. Using obvious notations, the system can be written as: t 2 = xβ 1 + υ 1, t 1 = xβ 2 + υ 2, h 2 = xβ 3 + υ 3, h 1 = xβ 4 + υ 4, c y = xβ 5 + υ 5, 29

30 The variable t 2 is latent; and the corresponding observable variables is given by: t 2 = t 2 if t 2 > t 2 = otherwise. The system can be written more compactly as: y i = x i β + υ i. Restrictions are imposed on β 3, β 4, β 5 based on equations (17), (18), and (19). Finally, Ω denotes the covariance matrix of υ, which covariances are set to zero. Let us consider the two different regimes. A. The wife participates in the labor market. If the latent representation of the wife s labor supply is greater than zero, that is, t 2 = xβ 1 + υ 1 >, then, t 2 = t 2. The contribution to the likelihood function for each observation i = 1,..., k such that the wife participates in the labor market is the following: [ L i 1 = (2π) 5 Ω 1/2 exp 1 ( yi x i β ) ( Ω 1 y i x i β )] 2 B. The wife does not participate in the labor market. If the latent representation of the wife s labor supply is not greater than zero, that is, t 2 = xβ 1 + υ 1, the wife s observed labor supply is equal to zero. The contribution to the likelihood for each observation such that the wife does not participate in the labor market is the following: 3

31 [ L i 2 = (2π) 4 Ω 1/2 exp 1 (ỹ ( x β) 2 Ω 1 ỹ x β )] ( ) Φ xβ 1 ω 11 where ỹ = (t 1, h 2, h 1, c y ), β = (β 2, β 3, β 4, β 5 ), Ω is the covariance matrix of (υ 2, υ 3, υ 4, υ 5 ), and ω 11 is the standard deviation of υ 1. 7 Estimation Results The parameter estimates of the total cost function (and domestic production inputs), according to equations (16) to (19), are reported in Table 2. Moreover, in Table 5 are reported the derivatives of the total cost function and of domestic production inputs with respect to wages and quantity of public good. The estimations show that an increase in wages and quantity of public good have a positive effect on total production cost. An increase in public good has a positive effect both on husband and wife domestic labor supply. In addition, estimations reveals that an increase in woman s wage reduces her domestic labor supply since h2 w 2 = A 4 y < for all y >. On the opposite, a change in man s wage has not a significant effect on his domestic labor supply. Finally, monetary cost is increasing with wages and the public good. The estimation of total labor supplies (see equations (2) and (21)) is reported in Table 3. An income increase has a negative effect on spouses total labor supply. Moreover, an increase in woman s wage increases her total labor supply. Instead, a change in husband s wage has not a significant impact on his total labor. In Table 4 is reported the estimation of the sharing rule Ψ 1. Once estimated the parameters of the sharing rule Ψ 1, we constructed individual total cost T C 1, according to equation (14), and individual total cost T C 2 as difference between total production cost and T C 1. Our estimates shows that 31

32 for 23 households the woman s total cost is negative. The rest of our analysis is then conducted on a sample of 626 households (from the initial sample of 649 households). Let us, now, turn to the impact of environmental variables on household ressources sharing. How does a change in husband s wage, or wife s wage, or non labor income, or sex ratio affect the way spouses share household non labor income? According to equation (15), these marginal effect, whose distribution is reported in Table 6, vary between households. We observe that a 1$ increase of non labor income increases man s share of household non labor income of.3 cents. On the opposite, the sharing rule is decreasing in sex ratio 7. On average, a change in husband s wage has a positive effect on the sharing rule, but significant for only 3.7% of households. On the opposite, the sharing rule is, on average, decreasing in wife s wage. In particular for 37.8% of households this negative effect is significant against the 1% of households where the effect is positive and significant. These results change if we concentrate our analysis on the way spouses share household non labor income minus total production cost (see Table 6). In this case, a 1$ increase in non labor income increases, on average, man s ressources of.127 cents, then the effect is smaller if compared to the case spouses share only household non labor income. For about 7% of households the positive effect is significant at the 5% level. For some households the effect is negative but never significant. On the opposite, for all woman in the sample the effect of a change in non labor income has a positive effect on their share of household ressources Ψ 2. We can explain these effects saying that there is a sort of compensation by the husband with respect to his wife because the latter is more productive in household production and spends more time than her husband to take care of their children, as Table 1 reveals. Finally, a change in non labor income does not affect significantly neither on 7 See Rapoport et al. (29) for a similar result. 32

33 individual monetary costs nor individual total costs (see Table 7 and Figure 3). The sex ratio has always a negative effect on Ψ 1 (and thus positive on Ψ 2 ) but significant for only 12.6% of households (see Table 6). It, also, has never a significant effct on the way sposes share monetary cost for children and, the, on individual monetary costs (see Table 7 and Figure 4). Let us now to analyse what happens if husband s wage increases (see Tables 6 and 7, and Figures 5 and 6). On average the effect is positive on Φ 1 and Ψ 1 but significant for very few households. The impact on the husband s monetary cost c y1 is, on average, negative and the negative impact is significant for about 5% of households (for only 2% of households the positive effect is significant at the 5% level). The effect on the value of husband s domestic work w 1 h 1 is always positive. Thus the final effect on the husband s total cost T C 1 is, on average, positive but not very significant. As regards the wife, a man s wage increase has, on average, a negative effect on Ψ 2 (the negative effect is significant for 33% of households). It increases, in fact, wife s contribution to the monetary cost of children, and, thus, to the total production cost. What happens, instead, if wife s wage increases (see Tables 6 and 7, and Figures 7 and 8)? On average the effect is negative on Φ 1 (the negative effect is significant for about 38% of households) and positive on Ψ 1 (the positive effect is significant for about 22% of households). In particular, the impact on the husband s monetary cost c y1 is, on average, negative. Also the effect on the value of husband s domestic work is always negative (but never significant). Obviously, the final effect on the husband s total cost is, on average, negative (the negative effect is significant for about 49% of households). As regards the wife, a woman s wage increase has, on average, a negative effect on Ψ 2 (the negative effect is significant for 8% of households). It 33