Collective Model with Children: Public Good and Household Production

Collective Model with Children: Public Good and Household Production Eleonora Matteazzi June 25, 2007 1

1 Introduction This paper aims at modeling household decisions concerning children in a collective approach. The original framework, initially developed by Chiappori [5], considers agents who individually maximize an egotistic utility function, defined over their private consumption of market goods and leisure, subject to the Pareto constraints that the partner s utility is not less than a given level of welfare, the household budget constraint, and individual time constraints. Alternatively, the household maximizes a weighted sum of individual utilities with respect to the household budget constraint and individual time constraints. Chiappori [6] shows 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 [6] shows that the model still holds in the case of caring (not paternalistic) preferences. Browning et al. [4] have generalized the collective approach by introducing public goods. Another generalization (Apps and Rees [1] and Chiappori [7]) of the collective approach considers household production of a (marketable or non marketable) good consumed privately by household members. Our objective 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. However, this seems to us the most relevant / SUIT- ABLE way of modeling household decisions concerning children. Although children are modeled in a similar way in Blundell et al. [2] or in Donni [10], to the best of our knowledge, the way we propose to model how the cost of children is shared among spouses has never been analyzed up to now. When household production is explicitly considered in the collective approach (Apps and Rees [1] and Chiappori [7]), the good produced is generally 2

assumed consumed privately. Here, on the opposite, we consider that household production is consumed publicly rather than individually within the household, which seems more relevant concerning children (see, for example, Bourguignon [3]). Husband and wife therefore consume the same quantity of public good. What adds up is their monetary contributions to the market costs of producing the public good. Whereas Blundell et al. [2] consider the cost of children globally paid by the household, our paper focuses on the way this cost is shared among spouses. When public goods are considered in a collective approach, it is usually assumed that the quantity of public good is chosen first and the sharing rule is defined conditional on the quantity of public good, namely on what is left after it has been financed by the household. Here, on the opposite, we first consider a sharing rule defined on the full income and then the public good is chosen conditional on this sharing rule and financed by each spouse. We show that it is equivalent to the previous approach. The household equilibrium condition gives the quantity of public good produced at home and the (endogenous) cost of producing it for each spouse. Since each spouse s cost is made both of time and money, one possible question is whether the Pareto-optimal allocation of inputs can be decentralized in the following sense. If individual total costs (aggregating time and money) were collectively chosen and each spouse were free to allocate his total cost between money and time, would he choose the Pareto-optimal quantities of inputs? Lindahl prices offer a positive answer to this question, but they represent the marginal prices of the public good for each spouse. Since we are interested in the share of costs between spouses, we do not limit the analysis to the degenerate cases of perfect substitutability between husband s and wife s time in producing children (see Donni [10]) or constant returns to scale (see Chiappori [7] or Apps and Rees [1]). Under weak assumptions concerning the technology of production for the public good (ei- 3

ther constant or decreasing returns to scale; quasi concavity), combined with the usual assumptions concerning preferences, we recover the total cost of children for each spouse. 2 The Pareto-Optimal Case 2.1 Definitions and program In what follows, we consider a two-person household which members i = 1, 2, respectively the husband and the wife, are characterized by his or her own rational preferences. Index 3 denotes what is outside the household. All the analysis is conducted under the hypothesis that each spouse is selfish toward the other and that, whatever the decision making process inside 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 the wage rate w 1 and w 2 ), enjoy leisure l 1 and l 2, consume a private composite 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 household children. The price of the private composite good is normalized to 1. As shown by Chiappori and Ekeland [8], 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 preferences of the form: U i (u i (l i, c i ; Z p ), y; Z p ) where Z p denotes the vector of individual characteristics that affect prefer- 4

ences 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 necessarily infinite when y = 0). In this model, y is a good that is both publicly consumed by household members, in the sense that the husband and the wife consume the same quantity of the public good, and produced within the household. In fact, each member shares his time T between leisure, paid 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 for their children. In order to produce y, the household also buys some input goods c 3 (such as clothing, school insurance, school meal, transport, education, etc.) which price is normalized to 1, and time inputs h 3 paid at the exogenous wage w 3, such as a nurse, a baby-sitter, etc. Then, let define the household monetary cost of producting y as c y w 3 h 3 + c 3. Notice that husband and wife have the same way of valuating money and time inputs. Then, the household total cost of producing the public good is T C = w 1 h 1 + w 2 h 2 + c y. The vector of individual or household characteristics affecting household productivity is denoted by Z h. The household production technology is given by 1 : y = Y (h 1, h 2, c y ; Z h ) Assumption 2. The function Y is increasing and concave in each argument and globally quasi-concave. 1 It corresponds to a more complete production function Ỹ (h1, h 2, h 3, c 3 ; Z h ) 5

Notice that we only make the assumption of decreasing (or constant) marginal productivities for each input but, unlike Chiappori [7] and Apps and Rees [1], we need not to assume constant returns to scale. In what follows, we consider the case of some complementarity between inputs. Finally, household non labor income is denoted by m 2. Any Pareto-Optimal solution solves the following constrained maximization program P1, namely it maximizes a linear social welfare function Max λu 1 (u 1 (l 1, c 1 ; Z p ), y; Z p ) + (1 λ)u 2 (u 2 (l 2, c 2 ; Z p ), y; Z p ) 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 m + w 1 T + w 2 T y Y (h 1, h 2, c y ; Z h ) l i + t i + h i T The function λ represents the Pareto-weight that depends on the exogenous variables entering the budget constraint, such as wages and non labor income, and on distributional factors s, that are variables that influence the decision process without affecting the budget set or preferences. An interpretation of this welfare weight is that it represent the bargaining power of the 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. The fact that the Pareto-weight is function of wages and non labor income implies that, as well, household preferences, as captured by the linear 2 As usuals in collective models, the decomposition of non labor income between spouses own non labor incomes m 1 and m 2 could be used as a distribution factor 6

household welfare function, depend on wages and non labor income, namely we have price dependent preferences. 2.2 The Pareto-optimal solution in the traditional approach Both egoistic preferences and caring preferences make an alternative resolution of the program P1 presented in section 2.1 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, the production level of y is collectively chosen, which gives optimal inputs levels. Optimality and interior solutions 3 imply following first order conditions: Y h i Y cy = w i, i = 1, 2 which gives unique inputs levels h i = H i (y, w 1, w 2 ) and c y = C y (y, w 1, w 2 ). For the sake of simplicity we omit 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 his 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 ) = w 1 H 1 (y, w 1, w 2 ) + w 2 H 2 (y, w 1, w 2 ) + C y (y, w 1, w 2 ). At the second stage, household income net of the cost of producing the public good m + w 1 T + w 2 T T C(y, w 1, w 2 ) is shared between spouses, according to a sharing rule Ψ: Ψ 1 = Ψ(m, w 1, w 2, y, H 1 (.), H 2 (.), C y (.), Z h, Z p, s) = Ψ(m, w 1, w 2, y, Z h, Z p, s) 3 For corner solutions, see Blundell et alii [2] and Donni [11] 7

and Ψ 2 = m + w 1 T + w 2 T 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 [6]). Given any λ we can find a sharing rule Ψ such that the outcomes of the two associated programs are the same and vice versa, 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 depends both on the level of y and also on the level of each input. 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 third stage, each spouse, separately, maximizes her own utility under her own budget constraint implied by previous steps, namely each spouse chooses how to allocate her own budget between composite consumption good and leisure. Optimality and interior solutions imply following first order conditions: u i l u i c = w i, i = 1, 2 Demand functions for consumption and leisure are c i = C i (Ψ i, w i, Z p ) and l i = L i (Ψ i, w i, Z p ). This relation represents the marginal value of leisure relative to consumption good. For the individual i, one hour spent enjoying leisure is marginally equivalent to w i $ spent for consumption good c i. The associated indirect utility functions are: v i (Ψ i ; w i, Z p ) = u i (C i (Ψ i ; w i, Z p ), L i (Ψ i ; w i, Z p ); Z p ) This approach does not allow to determine explicitly how much each spouse contributes to the monetary cost c y of the public good, even if the 8

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 tradistional approach. 2.3 An alternative approach to the Pareto-optimal solution In this section, we present a second program, named P2, totally equivalent to program P1, but with two advantages. First, it makes explicit the repartition of the monetary cost c y between spouses, namely c 1 y and c 2 y; second, it allows to define and measure the individual total cost of children for each parent. As we said in section 2.1, the household total cost of producing the public good is made by a monetary cost c y and the remuneration 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 borne individually by each parent. In other words, we want to determine T C 1 = w 1 h 1 + c 1 y and T C 2 = w 2 h 2 + c 2 y. The knowledge of time devoted by each spouse to take care for children, namely the domestic work h i in the production function, cannot give an idea of how much each parent is willing to spend 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 consume more of private good and leisure and, consequently, work less on the labor market. 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. On the other hand, if the mother spends 9

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. Then, let us consider the following three-stages program P2. At the first stage, household members agree on the repartition of family full income I = m + w 1 T + w 2 T according to a sharing rule Φ(w 1, w 2, m, s, Z h, Z p ), as if they still had to contribute to the production of the public good, then as if they would be without the public good. Then, let us define Φ 1 = Φ(m, w 1, w 2, Z h, Z p, s) and Φ 2 = m + w 1 T + w 2 T Φ 1 The value of the sharing rule Φ should maximize household welfare in the absence of public good: λu 1 (v 1 (Φ; w 1, Z p ), 0; Z p ) + (1 λ)u 2 (v 2 (m + w 1 T + w 2 T Φ; w 2, Z p ), 0; Z p ) which gives the first order condition: λuuv 1 1 = (1 λ)uuv 2 2 (1) Thus, marginal utilities of income are inversely proportional to the Paretoweights. Notice that Φ is not affected by the price of market time w 3, because household production decisions are taken in a following step. At the second stage, the production level of y is collectively chosen, which gives optimal inputs levels h i = H i (y, w 1, w 2 ) and c y = C y (y, w 1, w 2 ). Then, the household monetary cost for children c y is shared between spouses, which gives c i y = α i c y with α 1 +α 2 = 1. Let us consider the (collective) choice 10

of α 1 = α and then α 2 = (1 α). Pareto-optimality implies that c y, h 1 and h 2 maximize the household welfare function: λu 1 (v 1 (Φ 1 w 1 h 1 αc y ; w 1, Z p ), Y (h 1, h 2, c y ; Z h ); Z p ) +(1 λ)u 2 (v 2 (Φ 2 w 2 h 2 (1 α)c y ; w 2, Z p ), Y (h 1, h 2, c y ; Z h ); Z p ) which gives first order condition for c y : λu 1 uv 1 ( α) + λu 1 y Y cy + (1 λ)u 2 uv 2 (α 1) + (1 λ)u 2 y Y cy = 0 where (1 λ)u 2 uv 2 = λu 1 uv 1 because of (1). Then, we obtain: 1 = U y 1 Y cy Uuv + U y 2. (2) 1 1 Uuv 2 2 Similarly, for h 1 and h 2 : λu 1 uv 1 ( w 1 ) + λu 1 y Y h 1 + (1 λ)u 2 y Y h 1 = 0 λu 1 y Y h 2 + (1 λ)u 2 uv 2 ( w 2 ) + (1 λ)u 2 y Y h 2 = 0 then, we have: w i Y h i = U y 1 Uuv + U y 2 = 1 (3) 1 1 Uuv 2 2 Y cy because of (2). 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. For given levels or y, c y, h 1 and h 2, the repartition of costs affects spouses utilities, but not the production sphere because of the separability. The optimal repartition α should then maximize: λu 1 (v 1 (Φ 1 w 1 h 1 αc y ; w 1, Z p ), y ; Z p ) +(1 λ)u 2 (v 2 (Φ 2 w 2 h 2 (1 α)c y ; w 2, Z p ), y ; Z p ) 11

which gives the first order condition: λu 1 uv 1 = (1 λ)u 2 uv 2. (4) Once more, we obtain that the marginal utilities of income are inversely proportional to the Pareto-weight. This first order condition is sufficient, so there exists a unique solution α R, where: α = α(m, w 1, w 2, y, H 1 (.), H 2 (.), C y (.), Φ 1 (.), Φ 2 (.), Z h, Z p ) There is no reason why α should lie in the interval [0,1]. Indeed, it may well be the case that α 1 < 0 or α 2 < 0: this means that one spouse may spend much time for domestic production, but she is compensated by implicitly getting more money from the sharing rule, other ways spending less money (just to have a negative c i y because α i is negative) for child care. Indeed, since household net income of step 2 is less than full income of step 1, decreasing marginal utilities of income for both members imply that the loss of income has to be shared between spouses, namely both spouses will face a lower income at the second stage: Φ 1 w 1 h 1 αc y < Φ 1 and Φ 2 w 2 h 2 (1 α)c y < Φ 2. Since c i y > 0, this implies that: w 1h 1 c y < α < 1 + w 2h 2 The monetary cost of the public good may be negative for one member (because she is very productive in household production, then works more in the household), but the individual total cost (aggregating household working time and money) is always positive for both spouses. At the third stage, each spouse separately maximizes her own utility under her own budget constraint implied by the first two stages. Demand functions for consumption and leisure are c i = C i (Φ i w i h i α i c y, w i, Z p ) and l i = L i (Φ i w i h i α i c y, w i, Z p ). c y 12

Finally, let us consider the optimality condition for y. Given: 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 )] 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 = w 1 Hy 1 + w 2 Hy 2 + C yy = U y 1 Y cy Uuv + U y 2 (5) 1 1 Uuv 2 2 The optimal level of y is such that the marginal cost of producing y for the household is equals to the sum of the marginal benefits of consuming 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 (6) the marginal value of y for individual i, that is i is marginally indifferent between one unit more of y and p i dollars to share between her private consumption and leisure. The p i s simply correspond to Lindhal prices. Then, we can show that, 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 1 + p 2 p 1 = U 1 y U 1 u v1 + U 2 y U 2 u v2 = 1 + Uy 1 Uu 1v1 Uy 2 Uy 1 U 1 uv 1 U 2 uv 2 = 1 + λ (1 λ) where the third equality derives from relation (1). Note also that: p 2 p 1 = (1 λ) Uy 2. λ Uy 1 13 Uy 2 Uy 1

Moreover, that individual i s marginal cost of the public good is the derivative of individual total cost w i H i + Cy i = w i H i + α i C y with respect to y, that is w i Hy i + α i C yy + α iy C y. Pareto-optimality implies that this marginal cost equals the marginal benefit for individual i: w i H i y + α i C yy + α iy C y = p i = U i y U i uv i Let us now determine individual i s marginal value of h i relative to his own expense c i y = α i c y, starting from the marginal value of the time that individual i spends on domestic production. The first order condition (3) says that the marginal value of h i relative to c y is w i, namely, individual i is marginally indifferent between one hour spent in domestic production and w i dollars spent for c y. However, h i has an additional marginal value for spouse i implied by the fact that α i depends on h i : the more i gives domestic time h i, the less he has to pay for c y. Then, at the Pareto-optimal allocation, one hour spent by i in domestic production is also equivalent to α i w i dollars more given to each spouse to finance their consumptions of both private goods and leisure. So the direct marginal value of h i relative to c i y is α i w i. Finally, let us remark that the correspondence between the two programs implies that the net income available to each spouse after she has paid for his own costs for the public good is Φ i w i h i α i c y = Ψ i of program P1. Notice that the sharing rule α i for market cost and Ψ i, that is what is left for each individual s private consumption, are affected by the level of y and by the level of each input h i and c y. In the Pareto-optimal case, from the second theorem of welfare economics, the decisions collectively chosen can be transferred to individual choice without welfare loss, namely the centralized and decentralized program have the same solution, just as, for example, with consumption leisure/choice solved 14

at the third step. Finally, let us concentrate on the choice solved at the second stage. The Pareto-optimal decisions taken by household at this stage can be seen as individually optimal because, since spouse i anticipates the impact of his decision on the sharing rule, she has no incentive to deviate from the Pareto-optimal solution. 3 Further decentralization: Lindhal prices In this section, we solve a decentralized program, where the amounts of individual total cost T C i are collectively chosen and then each spouse freely allocates his own total cost between h i and c i y. Finally, the remaining of household resources is shared between spouses. Our question is: the solution of this decentralized program is the same as the Pareto-optimal solution? Are Pareto-optimal individual total cost T C 1 and T C 2 collectively chosen such that each spouse will take Pareto-optimal decisions, although they may have an egoistic behavior? 3.1 The program Let us consider the following three-stages program named P3. The first stage is similar to the first stage in program P2, where household members agree on the repartition of family full income according to a sharing rule Φ, as they would be without the public good: Φ 1 = Φ(m, w 1, w 2, Z h, Z p, s) and Φ 2 = m + w 1 T + w 2 T Φ 1. At the second stage, the amounts of individual total cost T C i devoted by each spouse to the production of the public good are collectively chosen 15

and then each spouse decides privately how allocate T C i between h i and c i y. This defines the quantity of public good produced. Finally, at the third stage, each individual i separately maximizes her utility under her own budget constraint deriving from previous steps. This step is similar to the third stage in program P3 and P2 and, as in program P2, individual budget constraint is given by Φ i w i h i c i y = Φ i T C i = Ψ i. Demand functions for consumption and leisure are c i = C i (Φ i T C i, w i, Z p ) and li = L i (Φ i T C i, w i, Z p ). As regards the second stage, this new program allows to separate the consumption sphere and the production sphere embedded in children. This is what we examine in next sections. 3.2 The production side Whereas in program P1 and P2, the quantity of public good produced y was collectively chosen and then the Pareto-optimal solution minimized household total cost to produce that quantity of public good, in program P3, individual total cost T C i are collectively chosen and, consequently, the amount of public good produced is defined. At this point, each spouse decides the repartition of individual total cost T C i between h i and c i y, considering h j and c j y as given, namely each individual i maximizes the quantity y to be produce given individual total cost T C i previously collectively chosen, the technology of production Y and input prices w i and w j. In program P1 and P2, the minimization of total cost, given the quantity of public good y collectively chosen, determined inputs demands as functions of y, that is h P O i = H i P O (y, w 1, w 2, Z h ) c P O y = C P O y (y, w 1, w 2, Z h ) 16

where P O is for Pareto-optimal. Instead, as regard program P3, the results of individual maximizations, given the individual total cost T C i collectively chosen, gives: h D i = H id (T C 1, T C 2, w 1, w 2, Z h ) c D y = C y D (T C 1, T C 2, w 1, w 2, Z h ) where D is for decentralized solution. These solutions are unique if we assume some complementarity between input. Notice that individually optimal allocation of T C i between h i and c i y is totally driven by production concerns and it is not affected by preferences. Moreover, it give a constrained production frontier: y = θ(t C 1, T C 2, Z h ) = Y ( H 1D, H 2D, C y1d + C y2d, Z h ) (7) represented by a strictly convex function in the space (T C 1, T C 2 ) in Figure 1 4. In the following, we report three lemmas that hold both in case T C 1 and T C 2 are Pareto-optimal and in the case they are not. Lemma 1. The decentralized solution gives the same first order condition as the Pareto-optimal one concerning the repartition of the individual total cost T C i between inputs h i and c i y. In fact, if in the second stage of program P3, spouse 2 choses the individually optimal levels h 2 = H 2D and c 2 2D y = C y, the allocation of T C 1 between h 1 and c 1 y is given by: Max (h 1,c 1 y)y (h 1, H 2D, c 1 y + C 2D y, Z h ) subject to: T C 1 = w 1 h 1 + c 1 y 4 Complementarity between h 1 and h 2 implies convexity of the function Y, resulting in the convexity of the function θ. 17

or equivalently: Max (h 1 )Y (h 1, H 2D, T C 1 + T C 2 w 1 h 1 + w 2 H2D ; w 3, Z h ) because: c 1 y = T C 1 w 1 h 1 C 2D y = T C 2 w 2 H2D Then, it follows that: Y h i Y cy = w i Notice that, if both spouses allocate their Pareto-optimal total costs T C ip O in step 2 of the decentralized program and if spouse 2 chooses the Pareto-optimal inputs H 2P O and Cy 2P O, then spouse 1 will have no incentive to deviate the Pareto-optimal solution. The same is true for spouse 2. This implies that if both spouses allocate their Pareto-optimal costs T C ip O in the second stage of the program P3, they will both choose the Pareto-optimal levels for their own inputs. This condition still holds in the case of non Pareto-optimal allocation of total costs T C 1 and T C 2 since, when the other spouse s behavior is fixed, minimizing individual cost is equivalent to maximizing household production. Lemma 2. The derivatives of the constrained production frontier with respect to each spouse total cost is equal to the derivative of the unconstrained production function Y with respect to the household monetary cost c y. [ θ T C i = Y h 1 H1 T C i + Y h 2 H2 T C i + Y cy C1 y T C i + C ] y 2 T C i According to Lemma 1, we have: θ T C i = Y cy [ w1 H1 T C i + C 1 y T C i + w 2 H 2 T C i + C 2 y T C i ] = Ycy 18

because: w 1 H1 T C i + C 1 y T C i + w 2 H 2 T C i + C 2 y T C T C1 T C2 i = + T Ci T C i where, in the right-hand side, one of the terms is 0 and the other is 1. The derivatives of the constrained production frontier with respect to each spouse total cost, namely θ T C i, corresponds to the marginal effect on the quantity of public good produced if spouse i contributes one dollar more in T C i, with T C j being fixed. Equivalently, the derivative of the unconstrained production function Y with respect to the household monetary cost c y, namely Y cy, corresponds to the marginal effect on the quantity of public good produced if household spends one dollar more in c y, with h 1 and h 2 being unchanged. If these two derivatives are equals, this means that the result of Lemma 2 corresponds to the marginal effect on the quantity of public good produced if spouse i contributes one dollar more in c i y, all other things equal. Lemma 3. Any quantity y of public good produced at home in this decentralized program lies on the production frontier θ. Assume that a quantity of y is produced using T C 1 and T C 2 and lies strictly above the production frontier θ (it cannot lie strictly below θ by definition of a production frontier). The same y could be produced at a lower cost for one spouse, say 1, leaving the other spouse s cost unchanged, which could be improved 1 s utility without affecting 2 s (except through the increase in y that would increase spouse 2 s utility). So the initial combination (y, T C 1, T C 2 ) is not optimal from the egoistic point of view of spouse 1. As a conclusion, the production side (production frontier) is not affected by the bargaining process. 19

3.3 The consumption side Let us now turn to preferences. The repartition of costs collectively decided in the second stage must maximize household welfare given the sharing rule, defined at the first step, and the desired level of y. Then, the second step of the program corresponds to the maximization of the household welfare subject to the technological constraint y = θ(t C 1, T C 2, Z h ), in other words to the maximization of λu 1 [v 1 (Φ 1 T C 1 ), θ(t C 1, T C 2, Z h ); Z p ]+(1 λ)u 2 [v 2 (Φ 2 T C 2 ), θ(t C 1, T C 2, Z h ); Z p ] which gives: λu 1 uv 1 + λu 1 y θ T C1 + (1 λ)u 2 y θ T C1 = 0 λuy 1 θ T C2 (1 λ)uuv 2 2 + (1 λ)uy 2 θ T C2 = 0 Using Lemma 2, we obtain λuuv 1 1 = (1 λ)uuv 2 2 namely, the same condition that defines the repartition α of monetary costs between spouses in program P2. This implies the following proposition: Proposition 1. In the Pareto-optimal formulation, if the level of household production is the same in the Pareto-optimal program P2 and in the decentralized one P3, then the repartition of the total cost between spouses is the same in the Pareto-optimal program P2 and in the decentralized program P3. Given Lemma 2, this also implies that if the level of y chosen in programs P2 and P3 is the same, then all the inputs (h 1, h 2, c 1 y, c 2 y) chosen in the two programs are the same. 20

We now have to check if the two programs lead to the same quantity of public good. Given that individual total cost T C i depends on w i, the desired level of public good y, and the sharing rule, namely T C i D = T C id (y, w i, Φ(m, w 1, w 2, Z h, Z p, s)) the optimal production level of y maximizes household welfare λu 1 [v 1 (Φ T C 1D (y, w 1, Φ); Z p ), y, Z p ] +(1 λ)u 2 [v 2 (m + w 1 T + w 2 T Φ T C 2D (y, w 2, Φ); Z p ), y, Z p ] that gives the following first order condition λu 1 uv 1 T C 1D y + λu 1 y (1 λ)u 2 uv 2 T C 2D y + (1 λ)u 2 y = 0 Dividing by λu 1 uv 1 = (1 λ)u 2 uv 2, and rearranging we obtain λu 1 y λu 1 uv 1 + (1 λ)u y 2 (1 λ)uuv 2 2 = λu uv 1 1 T C 1D y λuuv 1 1 + (1 λ)u uv 2 2 T C 2D y (1 λ)uuv 2 2 which gives Uy 1 Uuv 1 1 + U y 2 Uuv 2 2 = T Cy 1D + T Cy 2D = T C y Moreover, differentiating the constrained production frontier with respect to y, we find y = θ[t C 1 (y, w 1, Φ 1 ), T C 2 (y, w 2, Φ 2 ), Z h ] 1 = θ T C 1T C 1 y + θ T C 2T C 2 y and using Lemma 1 and previous results, we have: 1 = T Cy 1 + T Cy 2 = T C y = U y 1 Y cy Uuv 1 1 + U y 2 Uuv 2 2 21

This exactly corresponds to the Lindhal condition for the Pareto-optimal case, so optimal quantities are the same in programs P2 and P3. As a conclusion, we can state the following proposition: Proposition 2. In the Pareto-optimal case, the solutions of programs P2 and P3 are the same: Y P O = Y D, T C ip O = T C id, H ip O = H id, and C P O y = C D y. In other words, the Pareto-optimal solution can be decentralized. Of course, this result is not original since it simply corresponds to the second theorem of Welfare Economics, but the method allows us to recover not only marginal individual costs but also total costs. In fact, the repartition of costs collectively chosen in step 2 must maximize household welfare subject to the constraint that y level considered in the W curve can be produced with the constrained technology θ. According to Assumption 1, for a given level of public good, constant W curve 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 o, because lower T C i means higher wealth. Otherwise, if we assume some complementarity between production factors, the constrained production frontier θ will be represented by a strictly convex function in the space (T C 1, T C 2 ). If the W and the θ curves crossed, there would be a point on the same θ curve that would lie on a curve W strictly below the initial one: it would be possible to achieve a higher welfare level for the same quantity of public good. For this reason, the repartition of total costs at step 2 implies a tangency condition between W and the θ curves. In the Pareto-optimal case, the slope correspond to the opposite of the relative Lindhal prices. This would be enough to recover individual marginal prices for public good since the sum of spouses marginal prices simply correspond to the marginal cost of y for 22

the household. But we go a step further since the tangency condition gives not only the marginal cost, but also the total cost for each spouse: Lindhal prices are the derivatives of total cost T C i with respect to the quantity of public good y. Finally, 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. According to Assumption 1, for a given level of public good, constant W curve can be represented by decreasing strictly concave curve. Then, household welfare is maximized when the W curve goes down and left for a given value y o, because lower T C i means higher wealth. 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 the assumption of constant or increasing marginal costs, as we done, for a fixed y increase, the extent of the upwards shift of the θ curve is either constant or increasing in y. The higher 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. In fact, with y + y, each spouse could get the same utility U i with a lower wealth, that is with a higher cost T C i > T C. Under the assumption of decreasing marginal utilities, as we done, for a fixed y increase, the extent of the upwards shift of 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, 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 + y than with y, so the initial level was not optimal. For large values of y, 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. 23

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 y than with y, so the initial level was not optimal. 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 3. There exists a unique quantity y of public good, and hence a unique repartition of costs T C 1 and T C 2 that maximizes household welfare W (or any other W function in the non Pareto-optimal case). Then the following corollary is immediate: Corollary 1. Whatever the bargaining process, there exists a unique solution to the decentralized program P3. 24

25

4 Identification results As shown in section 3.1, the sharing rule Ψ i, namely the sharing rule in presence of the public good produced, is defined as difference between the sharing rule Φ i, namely the sharing rule defined over household full income, et the individual optimal cost T C i, in other words Φ i T C i = Ψ i. Then, we can define the individual optimal contribution to the public good as the difference between what individual i can obtain in the absence of public good and what he obtains when the public good is financed and produced. Our purpose is to identify T C i. Then, in order to do so, we have to identify the two sharing rules used. This is what we do in following sections. 4.1 Restrictions on total labor supplies and the sharing rule Φ In the absence of household production, and then in the absence of public good production, individual i s time endowment is shared only between market labor supply and leisure. At the first stage, household members agree on the repartition of household full income according to the sharing rule Φ(w 1, w 2, m, s, Z p ), and then, at the second stage, each one choses his consumption of market good and leisure maximizing his utility under his budget constraint. Then, from individual time constraint, we obtain individual market labor supply. The collective framework with egotistic preferences imposes certain restrictions on the labor supply functions. Given individual i market labor supply, t i (w 1, w 2, m, s) = T i (w i, Φ i (w 1, w 2, m, s, Z p ), Z p ) et following Chiappori s results [9], the partial derivatives of the sharing rule with respect to wages, nonlabor income, and the distribution factor are given 26

by where: Φ w2 = AD D C Φ w1 = BC D C Φ s = CD D C Φ m = D D C A = t 1 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 Then, Chiappori et al. [9] shows that, if some conditions hold, the sharing rule is defined up to an additive function depending only on the preference factors Z p. 4.2 Restrictions on total labor supplies and the sharing rule Ψ In the collective model taking into consideration household production, individual s i time is shared between market work, domestic work and leisure. Let us define f i individual i total labor supply, namely where: f 1 = t 1 + h 1 = F 1 (w 1, Ψ 1 (w 1, w 2, y, m, s, Z h, Z p ), Z h, Z p ) f 2 = t 2 + h 2 = F 2 (w 2, Ψ 2 (w 1, w 2, y, m, s, Z h, Z p ), Z h, Z p ) Ψ 2 = w 1 + w 2 + m T C(w 1, w 2, y, Z h ) Ψ 1 (w 1, w 2, m, y, s, Z h, Z p ) We differentiate total labor supply functions with respect to wages, non labor income, the level of public good, and the distribution factor: f 1 w 2 = F 1 Ψ 1 Ψ 1 w 2 27

f 1 s = F 1 Ψ 1 Ψ 1 s f 1 y = F 1 Ψ 1 Ψ 1 y f 1 m = F 1 Ψ 1 Ψ 1 m f 2 = F 2 ( 1 T C ) Ψ1 w 1 Ψ 2 w 1 w 1 f 2 s = F 2 ( ) Ψ1 Ψ 2 s f 2 y = F 2 ( T C ) Ψ 2 y Ψ1 y f 2 m = F 2 Ψ 2 ( 1 Ψ1 m ) Defining A = f 1 w 2 /f 1 m, B = f 1 s /f 1 m, C = f 1 y /f 1 m, D = f 2 w 1 /f 2 m, E = f 2 s /f 2 m, and F = f 2 y /f 2 m and solving the system, we obtain the derivatives of the sharing rule Ψ w1 = 1 T C w1 + BD E B Ψ w2 = AE E B Ψ s = BE E B Ψ y = CE E B Ψ m = E E B and the following restrictions: lw i i li m [Ψ i Ψ i w i l i ] 0 m 28

Rearranging we obtain: E E B = F + T C y F C T C y = BF EC E B 5 Data As we said previously, if we are able to identify the sharing rules φ and ρ, we can also identify individual total cost of producing the public good. Difficulties in identification rely on the fact that generally microdata does not have information about individual monetary contribution to public good. Then, there are some information that we absolutely need if we want to identify individual total cost. In what follows, we design the ideal survey for our purpose, in other word we explain which information we need if we want to identify individual total cost. As first thing, we need a time survey, that is a survey that collects time budgets from household member, at least for the husband and the wife. This means that, in this survey, we have to collect information about the time spent, by each household member, in the job market, said t i, and in domestic activities, said h i. Domestic activities include household activities like housework (interior cleaning, laundry, sewing, repairing, maintaining textiles, storing interior household items, food and drink preparation, kitchen and food clean-up, interior and exterior maintenance, repair, and decoration) and caring for and helping household members as children (physical care, reading to/with them, playing with them, and all activities related to household children s education such as homework, school conferences, etc.) and adults (physical care, medical care, organizing and planning for them, etc.). Moreover, we have to collect information about individual market wage. Then we need information on non labor income m, and thus we obtain information on total household consumption, namely 29

c 1 + c 2 + c y = m + w 1 t 1 + w 2 t 2. Let us now turn on information about household monetary cost of producing public good, namely c y. Here, we have to know the extra-household wage, said w 3, the input goods c 3 and the extrahousehold time inputs h 3, as, for example, baby-sitting or nursery schools. In this way, we can calculate the household total cost of producing public good, namely T C = w 1 h 1 + w 2 h 2 + c y. For each household member is better to have information regarding, for example, the age, the level of schooling, the origin, the country where he/she was born, the region who he/she lives, if he/she lives in a countryside or metropolitan area, the type of housing unit, and all other useful informations related to individual characteristics that affect preferences, said Z p, and individual or household characteristic affecting household production, said Z h. Finally, we take in consideration the public good. The public good, in our case, represents the quality and quantity of child. Then, we should have information of the number of children under 13 years old, for example, the type of school frequented (private or public), their participation to educational activities, like doing sport or playing musical instruments or dancing, spaciousness of child s sleeping room or spaciousness of garden (according to studies regarding the quality of child care, see Gotts [12] for example, a quality environment invites children to learn and grow. For this reason, most states require 36 square feet of room per child for indoor areas, while 100 square feet per child is recommended outside ), etc. Then, if we dispose of these information we are able to identify individual total cost. 30

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[11] O. Donni, Collective Household Labor Supply: Nonparticipation and Income Taxation, The Journal of Public Economics, 2003, 87, 1179-1198. [12] E.E. Gotts, The right to quality child care. Childhood Education, 1988, 64, 269-273. 32