A Cournot-Nash Model of Family Decision Making

Zhiqi Chen and Frances Woolley CARLETON UNIVERSITY

Revised: June 1999

Correspondence:

Frances Woolley Department of Economics Carleton University Ottawa, Canada K1S 5B6 tel: 613 520-2600 x 3756 fax: 613 520-3906 e-mail: frances_woolley@carleton.ca

The authors are indebted to Tony Atkinson, Andrew Clark, Fiona Coulter, David Long, David Ulph, Nick Rowe and Nick Stern, two anonymous referees and participants at a number of seminars for helpful comments on earlier drafts on this paper. The usual disclaimer applies.

ABSTRACT

This paper develops a model of two person family. Each family member attempts to maximize his or her own utility. Yet they are interdependent in two respects. Family members are interdependent, first of all, because they care about each other. Second, there are local public goods or household expenditures within the family, such as housing. The presence of household expenditures means that one family member's consumption choices affect the other family member's level of well-being. The two family members' interdependent utility maximization problems are first solved using a non-cooperative, or Cournot-Nash, game theoretic framework then the model is extended to take the Cournot-Nash equilibrium as a threat point in a bargaining game. The model's predictions differ substantially from the "unitary" framework usually used in economic analysis, in which households maximize a single household utility function. When the spouses are relatively equal in income, or when one spouse is much wealthier than the other and the wealthier spouse has all the bargaining power in the family, the equilibrium depends, as in the unitary model, on household income but not on the division of income between spouses. In the intermediate case between equality and substantial inequality or in the case where one spouse is much wealthier than the other but the wealthier spouse does not have all the bargaining power, the distribution of income does shape expenditure patterns, contrary to the predictions of the unitary model. The contribution of the paper is to provide a rigorous derivation of the properties of household demands in the Cournot-Nash setting, a full analysis of the determinants of intrahousehold resource allocation, including the effect of varying household bargaining power, and an explication of the models implications for policy analysis.

A Cournot-Nash Model of Family Decision Making Introduction

This paper models the decisions of two people living together. Each attempts to maximize his or her own utility. Yet they are interdependent in two respects. Family members are interdependent, first of all, because they care about each other. Second, there are local public goods within the family, such as housing. The presence of public goods means that one family member's consumption choices affect the other family member's level of well-being. The two family members' interdependent utility maximization problems are solved in two stages. In the first stage we find the non-cooperative Cournot-Nash equilibrium. In the second stage, we take the Cournot-Nash equilibrium as a threat point in a bargaining game. There are a number of approaches to modelling family decision making taken in the literature, which are surveyed in Bergstrom (1996, 1997) and Lundberg and Pollak (1997). Our paper ties together two strands of this literature: the non-cooperative and cooperative bargaining approaches. In non-cooperative models of the family, each family member maximizes his or her well-being (which may depend upon the consumption or utility of others) taking the behaviour of others as given. Influential early models which have this feature are Leuthold (1968) and Becker (1974), even though neither author uses the term "non-cooperative". Non-cooperative models have also been developed by Ulph (1988) and Konrad and Lommerud (1995), applied to the division of housework (Bragstad 1991), domestic violence (Tauchen, Witte and Long, 1991), and expenditures on children by divorced parents
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(Del Boca and Flinn, 1994). The great attraction of the non-cooperative framework is that, because each person's behaviour maximizes his or her well-being, the equilibrium is self-enforcing. In contrast, cooperative bargains generally require a mechanism for contract enforcement, as there is no internal incentive to move to the bargaining solution. But, because of legal obstacles and transaction costs, families rarely write explicit contracts governing their behaviour. It can be argued that, because families involve long-term, repeated interaction, cooperation will evolve over time, and therefore family decision making can be best understood as the outcome of a cooperative bargaining process. All cooperative models share the feature that, if cooperation does evolve, the particular cooperative allocation reached will depend crucially on what happens in the event of disagreement, variously known as the threat point, disagreement point, or status quo. The seminal contributions in the cooperative bargaining literature are McElroy and Horney (1981) and Manser and Brown (1980), which use divorce as the threat point. More recently, Lundberg and Pollak (1993), Haddad and Kanbur (1994) and Konrad and Lommerud (1996) have developed models where the threat point is some form of non-cooperative behaviour. We agree with these authors that non-cooperation within marriage is the best choice of threat point; our points of departure are, first, we expand the non-cooperative threat point to include household public goods and caring partners. Secondly, we consider bargaining over income transfers within the household. Finally, we use a generalized Nash bargaining framework, in which we are able to characterize fully the effects of changes in bargaining power on intra-household distribution. The first contribution of our paper is to provide a rigorous analysis of the relationship between spending patterns and a host of other variables such as relative income of the two spouses, relative
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bargaining power, and caring between spouses. A number of these results, for example the characterization of observed household demands, are new or different from those of the other models in the literature. While some results are the same as the existing ones, for example the replication of Beckers (1974) rotten kid theorem, they are also contributions in that we confirm the robustness of these results in a much more compelling model of family. The rigorous formulation developed in this paper allows us to predict the effect of male and female incomes, particularly government transfer income, on expenditure patterns, and on the properties of the household's demand function. Tax or transfer policy designed to assist families generally involves payments to specific members of the family. Child benefits in several countries are received by mothers (for example, in Canada or the U.K.), while support in the form of tax exemptions for dependants may reduce the fathers, the mothers, or both spouses tax liabilities, depending on the income tax act in question. The unitary model of the household gives no reason to believe that it makes any difference whether benefits are received by men or women, mothers or fathers.1 The cooperative bargaining approach developed by McElroy and Horney (1981) predicts only changes which alter the divorce positions of each partner affect expenditure patterns. However in policy debates it is generally agreed that it does matter who receives benefits when a couple is married, and not just in the event of divorce, even if there is no agreement who the appropriate recipient is (Parker and Sutherland, 1991). This paper gives a firm theoretical basis for the claim that the targeting of benefits within families matters. There are some things that our paper does not do. It does not follow Chiapporis (1988, 1992) "collective" approach to modeling the family. We believe that the collective approach has fruitful applications to empirical work on intra-household distribution, but is less useful in explaining why any
3

one of the many possible Pareto optimal allocations of resources in the households will be chosen. Our paper can be thought of as complementary to the collective approach, in that it can explain why particular variables influence intra-household allocations. Second, we do not consider dynamic aspects of household decision making, such as investment in children (Ott, 1995) or education (Konrad and Lommerud, 1996). While a valuable project, it is beyond the scope of this paper. The first part of the paper sets out and solves the basic Cournot-Nash model and describes the three types of equilibria. The remaining parts of the paper extend the model to use the Cournot-Nash equilibrium as a threat point in a cooperative game,

1. A Model of Family Decision Making In the model there are two family members, one male (m) and one female (f).2 The household resource allocation decision is made in two stages. In the first stage, transfers between spouses are determined, either through bargaining, or through a voluntary decision. In the second stage, each family member makes his or her consumption choices, conditional on the transfer received. We solve the familys decision making problem through backwards induction. First we describe consumption choices conditional on transfer received, and then consider the outcome of intra-household bargaining over transfers. In the resource allocation decision, each family member allocates the income at his or her disposal, denoted by Yi, between spending on private, or personal, goods, pixi, and household public goods, xih, subject to the budget constraint:

Yi ' x i h % px i, (i ' m, f ).

(1)

We assume that the partners face the same price for the private good purely for notational convenience. All of our results would be qualitatively similar if the partners faced different prices, however they would be more cumbersome to state. The price of the household good is normalized to one. There is no home produced good. The model is most valuable in addressing the policy of targeting income transfers within the family, hence we focus on transferring, and spending money incomes. One spouse may also transfer part of his income to the other. Let Ii denote the earned income of spouse i. Without any loss of generality, we assume that the male spouse has a higher income level than the female spouse (Im > If) and let p be the net transfer from the former to the latter. Then the disposable income of the male spouse is Ym = Im - p and the disposable income of the female spouse is Yf = If + p. Each spouse's level of well-being depends upon her own consumption and, because she cares about her spouse, on his well-being. A spouse's preferences over his or her own consumption are represented by the egocentric utility function, Ui. Utility depends upon the aggregate level of consumption of household goods (xh=xmh+xfh) and personal goods (xi):
Ui ' U(x i,x h) ' u(x i) % v(x h).
(2)

The functions u() and v() satisfy the standard assumptions that uN > 0, vN > 0, uO < 0, vO < 0, and uN(0) =4, vN(0) = 4. These assumptions imply that xi and xh are normal goods.

The form of the utility function (2) implies that household goods share with Samuelson public goods the characteristic that they are non-rival in the generation of utility. For example, a CD that generates utility for both partners is a household good. In contrast, private expenditures generate utility for only one person. The characterization of goods as household or private expenditures depends both on the nature of the good in question and the preferences of the individuals. Individual musical tastes determine whether a CD is a "private" or "household" expenditure. A person's actions are guided by her preferences about her own consumption and her caring for her spouse. Her objective is to maximize her welfare function, Wi, where
Wi ' Ui%sUj ' [u(x i)%v(x h)]%s[u(x j)%v(x h)]; i, j ' m, f .
(3)

We assume that s 0 [0, 1], which means that no one cares about her partner more than herself. A crucial point about formulation (3) is that the partners' preferences are not paternalistic. Each cares about the other's utility, not her partner's consumption choices per se. Woolley (1988) calls preferences of this form caring preferences, a term which has been adopted by Chiappori (1992) and others. Caring preferences involve cardinalization and interpersonal comparability. Yet this assumption is appropriate, given our focus on income transfers. We do in fact observe transfers of income within households (Woolley and Marshall, 1994), and it seems reasonable to assume that people, when making such transfers, are able to compare the utility they receive from consumption with that their spouse receives. The focus on income transfers also justifies the assumption of non-paternalism. When the higher income partner is non-paternalistic, he will be indifferent between making income and in-kind transfers to the lower income partner. Although some authors (for example Becker, 1996: 237) have
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suggested the amount of caring between spouses is influenced by factors such as the ease of divorce, in this paper we treat caring as exogenous, and analyze the effect of varying levels of caring on household decision making. In what follows we will first analyse a situation where no transfer is allowed between the family members, i.e., p is assumed to be 0. We will derive the Nash equilibrium in a game where the two family members choose their expenditure on private and household goods (xi and xih) noncooperatively and will call it the pre-transfer equilibrium. We then allow transfer between family members. We first consider voluntary transfers, then go on to examine the Nash bargaining solution with regard to the amount of transfer, using the pre-transfer equilibrium as the threat point. We examine how the amount transferred and consequently each spouses consumption is affected by bargaining power, caring between spouses, and each spouses income level.

2. Pre-transfer Resource Allocation In this section we characterize the allocation of resources within the household prior to any income transfers, that is, constraining transfers between the partners to zero (p =0). Because one spouses expenditure on household goods enters into the other's welfare function (see equations (2) and (3)), each partner's spending on the household good will depend on how much she expects the other to spend. We resolve the interdependence between the partners' decisions using the Cournot-Nash solution concept. Each spouse maximizes his or her own well-being, taking the other's behaviour as given.

The optimization problem of spouse i is then to maximize objective function (3) subject to the budget constraint (1) (with p being set to 0) and taking partner j's household goods purchases, xjh, as given. We can solve (1) for xi and substitute the resulting expression into (3). The optimization problem of spouse i can then be written as:
max Ii&x i h Ij&x j h h h W '[u( ) % v(x i %x j )] % s[u( )%v(x i h%x j h)] x i h$0 i p p
i j

(4)

A non-negativity constraint is imposed on xih in (4) because a corner solution is possible. The KuhnTucker condition to this problem is:
MWi Mx i h
h MW 1 ) Ii&x i u( ) # 0; x i h $ 0; x i h( i ) ' 0. pi pi Mx h i

'

(1%s)v )(x i h%x j h) &

(5)

Solving the two Kuhn-Tucker conditions for m and f simultaneously yields the Nash equilibrium in this case. Two distinct types of equilibria are possible, as shown in Figure 1. Equilibrium A in Figure 1 is an "interior solution", that is, one in which both spouses purchase household goods. Equilibrium B is a "corner solution" in which the better off spouse makes all the household purchases, and the less well-off spouse uses her income to finance her private expenditures only. Spouse is reaction curve (I= m, f) in Figure 1 is derived from setting MWi/Mxih = 0 in (5). It can be shown that both reaction curves are negatively sloped and with xfh on the vertical axis spouse ms reaction curve is steeper than that of spouse f. Furthermore, for any given xmh on spouse fs reaction curve,

Mx f h MIf

'

u ))(x f) p f2(1%s)v ))(x h)%u ))(x f)

> 0;

(6)

in other words, a fall in spouse fs income leads to a downward shift in her reaction function. Hence, we would expect that the equilibrium moves from an interior equilibrium (A) towards a corner solution (B) as spouse fs income decreases. We can, in fact, prove that this will always be the case, as stated in Proposition 1: Proposition 1.3 Given Im, there exists Ia in the interval (0, Im) such that the pre-transfer Nash equilibrium is a corner solution (xfh = 0) if If # Ia, but an interior solution ( xfh > 0) if If > Ia. The income level Ia is that at which spouse f is just indifferent between contributing and not contributing to the household public good. We will refer to Ia as the contribution threshold income level. The properties of a pre-transfer equilibrium are different for corner and interior solutions, as demonstrated in the next proposition: Proposition 2. In a pre-transfer Nash equilibrium, i) At a corner solution (If # Ia): a) An increase in If raises xf but has no effect on xm, xmh and xh; b) An increase in Im raises xm, xmh, and xh but has no effect on xf; c) An increase in s raises xmh and xh, reduces xm, but has no effect on xf. ii) At an interior solution (If > Ia):
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a) An increase in Ii (I = f or m) raises xi, xj, xih and xh, but reduces xjh. b) An increase in s raises xmh, xfh, and xh, but reduces xm and xf. Proposition 2.i.a and 2.i.b follow from the observation that, in the pre-transfer corner solution, spouse f spends all of her income on xf, while spouse m spends his income on xm and xmh. Since all goods are normal, an increase in either spouses income leads to an increase in consumption of those goods which he or she purchases. Proposition 2.i.c. follows from the fact as s increases, spouses care more for each other, therefore derive relatively more welfare from household public goods. The intuition behind Proposition 2.ii is similar to that behind Proposition 2.i. However Proposition 2.ii.a differs from 2.i.a in that it describes the free-riding which occurs in a Cournot-Nash equilibrium; as one spouses spending on household public goods increases, the other decreases. 3. Voluntary Income Transfers In this section we remain within the non-cooperative, Cournot-Nash, framework, but allow one spouse to voluntarily transfer income to the other. With a transfer p from m to f, the disposable incomes of f and m become If + p and Im - p, respectively. Hence , with Ii replaced by If + p (for I=f) or Im - p (for I=m), condition (5) continues to govern spouse is choice of xih. Intuitively, we expect that if spouse fs income is above the contribution threshold, If > Ia, so that an interior solution prevails in the pre-transfer equilibrium, a small income transfer from m to f will have no effect on the consumption of this family. Spouse f simply increases her spending on household goods by the amount of the transfer, and m decreases his spending by the same amount. Indeed, it can be shown that in the case If > Ia, (Mxfh/Mp) = 1 and (Mxmh/Mp) = -1; as a result, Proposition 3. A small transfer from m to f has no effect on xf, xm, and xh if If > Ia.
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Proposition 3 is an application to the household of Warrs (1983) and Bergstrom, Blume and Varians (1987) finding that income redistribution is irrelevant in the presence of privately provided public goods. Our finding differs from that of Konrad and Lommerud (1995: 594), who establish non-neutrality in the case where spouses differ in their productivity in public good production. In our model, if we allowed the spouses to face different prices for household goods, we would also find non-neutrality. However we differ from Konrad and Lommerud (1995) in that, in our model, non-neutrality does not establish a case for government redistribution. Rather, at an interior solution, the spouse which faced the higher price for household goods would voluntarily transfer income to the spouse facing the lower household goods price. Since transfers are irrelevant when If>Ia, we will focus our analysis of transfer on the case If # Ia. Let Wi(p) (I = f, m) denote spouse is welfare after m makes a transfer p to f. Then Wi(0) represents is welfare in the pre-transfer equilibrium. Since spouse f is the recipient of the transfer p, it is clear that with a small transfer Wf(p) > Wf(0). Furthermore, Wm(p) > Wm(0) is also possible for a small transfer because m cares about the utility level of spouse f. Indeed, if the income level of f is sufficiently low, a small transfer from m to f will raise the utilities of both spouses. Formally: Proposition 4. For any given s in the open interval (0, 1), there exists Ib in the interval (0, Ia) such that a small transfer raises ms welfare (i.e., MWm(p)/Mp*p=0 > 0) if If < Ib but a small transfer reduces ms welfare (i.e., MWm(p)/Mp*p=0 < 0) if If 0 ( Ib , Ia). However, Ib = 0 if s = 0, and Ib = Ia if s = 1. We term Ib the transfer threshold income level, as it is the level of spouse fs income at which spouse m is just indifferent between making and not making an transfer to her. The parameter s determines the
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length of the interval ( Ib , Ia). As implied by Proposition 4, the interval does not exist when s = 1 but the interval expands to ( 0, Ia) if s = 0. When s = 1, m cares as much about the utility of f as that of himself. In this case, m will always want to make a transfer to f if a corner-solution would prevail in the absence of the transfer. When s = 0, m is not concerned about fs utility at all. It is not surprising that in this case m would never want to make any transfer to f. In the voluntary transfer equilibrium, spouse m chooses an amount of transfer that maximizes his own welfare, i.e., he sets MWm(p)/Mp = 0. Using equations (1), (3), and (5) we can write this firstorder condition as:
1 ) If%p h ) su ( ) ' (1%s)v )(x m pf pf

(7)

Equation (7), along with ms optimization condition with regard to xmh,

1 ) Im & p & x m ' u( ) pm pm
h

h ) (1%s)v )(x m

(8)

form a system of equations that determines the voluntary transfer from spouse m to spouse f, denoted pv, and also the household expenditure patterns. Proposition 5 describes the properties of the transfer: Proposition 5. The household will be at a voluntary transfer equilibrium, in which spouse m makes a transfer to spouse f (pv > 0), as long as If < Ib. The amount of transfer decreases as spouse fs income rises, but increases as spouse ms income rises.
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The voluntary transfer equilibrium is the equilibrium described in Beckers (1974) famous Rotten Kid Theorem. Formally: Proposition 6. (Rotten kid theorem re-stated). If If<Ib, then xf, xm and xh in the voluntary transfer equilibrium maximize Wm, given the aggregate household budget constraint. As a corollary, we can observe that the level of household goods provided will be efficient. Since household expenditures maximize the welfare of the higher income spouse, and all goods are normal, we find: Proposition 7. Suppose spouse fs income is such that If < Ib. As the income of spouse f, If, rises, both total expenditure on the household good and expenditure on private goods of the two persons rise. The voluntary transfer equilibrium is interesting because it describes precisely the circumstances under which the households behaviour can be represented by the preferences of a single individual. However, it is vulnerable to the criticisms levied at Beckers (1974) qualitatively similar approach. The voluntary transfer equilibrium will be an appropriate characterization of intra-household relationships if the higher income spouse has all the bargaining power, and is able to impose the transfer that maximizes his or her welfare. It will also hold if the higher income spouse is a player in an asymmetric bargaining game in which he can offer the others all-or-nothing choices (Pollak, 1985: 599). In many instances, however, the lower income spouse does have alternative. She does not have to accept the transfer offered by her spouse. She can refuse, and bargain over the size of the transfer offered. In the next

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section of the paper, we describe a generalized bargaining framework, which allows for a range of bargaining outcomes, including the voluntary transfer outcome as a special case. 4. Nash bargained income transfers In this section we model the case where the amount of transfer is determined within the household through some implicit or explicit bargaining process. We begin by setting up a generalized Nash bargaining framework. We then describe conditions under which positive Nash bargained transfers take place, and the effect of the transfer on each spouses contribution to the household public good. We find that, when income transfers are determined through intra-household bargaining, the resulting resource allocation is, in general, not Pareto efficient, and the Rotten Kid theorem results, recovered in section 3, no longer hold. We model the households decision on the amount of transfer as the solution to the following generalized Nash bargaining problem:
max p [Wf(p) & Wf(0)]a [Wm(p) & Wm(0)]1&a .
(9)

In the above problem, (Wf(0), Wm(0)) represents the disagreement point; no transfer would be made if they cannot reach an agreement on the value of p. The parameter a 0 [0, 1] measures the relative bargaining power of spouse f. Spouse f has all the bargaining power in the family if a =1. In the other extreme, spouse m has all the bargaining power if a = 0. The parameter a is determined partly by law, institutional practices, and cultural norms. For example, legal restrictions on the right of women to control property lower womens bargaining power, as do institutional practices such as the refusal of banks (historically) to give loans to married women
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without a husbands co-signature, and cultural norms, for example, women should love, honour, and obey their husbands.4 Individual men and women may also have idiosyncratic sources of bargaining power such as education, family ties, and physical health. In marriage, as in any other institution, each agent may be able to use strategies such as selective efficiency, agenda setting, and exploitation of informational asymmetries to increase his or her bargaining power. In our analysis we do not analyze the various sources of intra-household bargaining power. Instead, we undertake the less ambitious task of describing intra-household resource allocations for various exogenously determined a. Having set up the bargaining framework, we can now describe the circumstances under which a positive transfer will take place. Let pn denote the Nash bargaining solution to (9). Proposition 4 implies that if spouse fs income is between the transfer threshold and the contribution threshold, If 0 ( Ib , Ia), any positive transfer will make spouse m worse off. Nash bargaining cannot result in an outcome which is worse for either spouse than their disagreement point, therefore we have: Corollary 1. If If 0 ( Ib , Ia), the Nash bargaining solution is pn = 0.

It is useful to draw out the difference between the interior equilibrium, in which both spouses are contributing to the household good (If>Ia), and the corner solution (Ib<If<Ia). In the interior equilibrium, spouse ms welfare is not affected by a small transfer because the entire transfer will be used by f to purchase the household good. At the corner solution, an income transfer from m to f would have real effects on consumption. But it would be spent on fs private good, which, in ms calculation, does not generate as much as utility as keeping the money to himself. As a result, spouse m does not want to make any transfer to f. In neither the interior equilibrium nor the corner solution does changing the

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models assumptions from a voluntary transfer to a generalized bargaining framework change the models predictions. The Nash bargaining solution involves a positive transfer if fs income is below the transfer threshold, that is If < Ib . We begin by describing the effect of transfers on individual contributions to the household public good, and then use our results to describe the effect of various parameters on the size of intra-household transfers. Before any transfers are made, spouse m is the sole contributor to the household good. A transfer from m to f raises fs but reduces ms disposable income, therefore changing the households consumption bundle. Depending on the size of the transfer, there are three possible scenarios. First, spouse m may remain the sole contributor to the household good, which we will term a pro-male outcome. Second, both spouses may contribute to the household good, which we will term a egalitarian outcome. Finally, it is possible for spouse f to become the sole contributor to the household good, which we call a pro-female outcome. We use the terms pro-male and pro-female advisedly. When spouse m (f) is the sole contributor to the household good, his (her) welfare is strictly greater than with the other two types of outcomes. This can be seen intuitively by observing that, if spouse m is not contributing to the household good, it is because he is constrained with respect to spending on his private good. He only contributes to the household good when he has achieved his preferred level of private goods expenditure. To characterize the circumstances under which each of these outcomes arise, we define two critical values of p: pN and pO. Let pN be the largest transfer with which spouse f does not contribute to the household good (xfh = 0). Any higher p will create an egalitarian outcome, with both spouses contributing to the household good. Using the first order conditions to the individual welfare
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maximization problems (5), we can see that spouse f will begin to contribute to the household public good when the transfer, pN, equalizes the marginal utility she receives from expenditures on private goods and expenditures on the household good.
h ) ) 1 ) If%p 1 ) Im&p &x m h u( ) ' u( ) ' (1%s)v )(x m ) pf pf pm pm

(10)

We define pO (> pN) as the transfer level at which spouse m ceases contributing to the household good. Using first order conditions for individual welfare maximization (5) once more, pO is defined by

h )) )) 1 ) If%p &x f 1 ) Im&p u( ) ' u( ) ' (1%s)v )(x f h) pf pf pm pm

(11)

For p < pN, spouse m is the sole purchaser of the household good for the family. As p is increased to the interval (pN, pO), both spouses contribute to the household good. Within this interval, any change in p has no real effect. An increased transfer to f will be used to increase fs purchase of the household good so as to keep the combined household expenditure constant, as in Proposition 3. As p is further raised to above pO, spouse f becomes the sole purchaser of the public good for the family. Although we can define a transfer large enough to reverse the profile of household goods contributions, could such a transfer ever be the outcome of intra-household bargaining? We now characterize the solution to the Nash bargaining game, and describe how the outcome changes with the key parameters in our model. From an inspection the bargaining problem (9), it is obvious that
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bargaining power, a, influences the size of the transfer. On the other hand, the amount that m is willing to give to f also depends on how much he cares about her utility level (i.e., the value of s), and each spouses level of income. We will start with a brief comment on the special, dictatorial, case a = 0, and then turn to the general case a 0 [0, 1]. The case a = 0 is worth special attention for several reasons. First, it is relatively easy to analyse. Second, it has important empirical relevance to societies where the property and legal rights of women are restricted, as has been the case historically in Western and other societies, for example, with the doctrine of coverture. Third, when a = 0, the generalized Nash bargaining problem in (9) is equivalent to the voluntary transfer equilibrium, where spouse m chooses a transfer to maximize his welfare. The voluntary transfer equilibrium is a useful benchmark case. Comparing the conditions under which a voluntary transfer takes place (7) with the conditions under which spouse f will begin to contribute to the household good (10), we conclude that the dictatorial transfer will be less than pN. Spouse m will continue to be the sole contributor to the household good. This is true regardless of the degree of sympathy, s, between the spouses. However, as s approaches one, the three outcomes converge. When s=1, the household maximizes Wf+Wm, and the quantity of household goods, male private goods and female private goods purchased is independent of the incomes of the spouses. Next, we consider the general case where a may take on any value in the [0,1] interval. The first-order condition to the Nash bargaining problem (9) is:
MWf(p) Mp MWm(p) Mp

a[Wm(p) & Wm(0)]

'& (1 & a)[Wf(p) & Wf(0)]

(12)

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The general case is more complicated to analyse because, as spouse f gains more bargaining power, in equilibrium she may be able to extract a transfer equal to pN, and create an egalitarian outcome, or even a transfer greater than pO, creating a pro-female outcome. Whether f is able to extract such a large transfer depends on, in addition to her bargaining power, the amount of caring between spouses. If the caring between spouses is below a some critical threshold, which we denote sN, ms welfare may drop to the level of the disagreement point before the amount of transfer reaches pN. In this case, the equilibrium outcome will always be pro-male. Formally: Proposition 8. For any given If < Ib, there exists sN0 (0, 1) such that Wm(pN) > Wm(0) if s > sN but Wm(pN) < Wm(0) if s < sN.

The significance of Proposition 8 is twofold. First, it implies that there is some critical level of caring above which an egalitarian outcome, or possibly a pro-female outcome, can be reached through intrahousehold bargaining. Second, it shows that caring worsens spouse ms bargaining position, all else being equal, by making the no-transfer disagreement point less attractive. When caring is below the critical threshold, s < sN, the amount of transfer will never exceed pN, and the outcome will remain pro-male. Although this case is similar to the dictatorial case with a = 0 in that both are favorable to spouse m, we cannot generalize the effects of exogenous change in fs income If on consumption from the dictatorial to the general case.

19

Proposition 9. Consider the case If < Ib and s < sN. The Nash bargaining solution pn is less than pN. A small increase in fs income will reduce pn, raise ms expenditure on both household and private good, but have ambiguous effect on fs private expenditure. An increase in spouse fs income decreases the transfer she receives and, because funds are diverted from the transfer into other forms of spending, ms household and private goods expenditures rise. Indeed, the transfer f receives may fall by more than the amount her income increases, which is why the impact of income increases on fs private expenditures is ambiguous. Transfers would be reduced more than dollar for dollar if the increase in spending on household goods resulted in an increase in her utility sufficient to compensate for lower private expenditures.5 The following proposition considers the same pro-male outcome, and demonstrates that, when m has only limited caring for f, her ability to improve her position through gains in bargaining power is restricted. Proposition 10. Consider the case If < Ib and s < sN. There exists a critical a a such that if a > a a, the amount of transfer will make spouse m just indifferent between the equilibrium with the transfer and the equilibrium in the absence of transfer (i.e., Wm(pn) = Wm(0)), and any increase in the relative bargaining power of f has no real effect. Over the range in which bargaining power matters, however, an improvement in fs bargaining position will have real effects on household demands. Proposition 11 On the other hand, if a < a a, a Nash-bargained transfer will increase ms wellbeing (i.e., Wm(pn) > Wm(0)). Assuming that uNNN(@) > 0 and vNNN(@) < 0, a small
20

increase in the bargaining power of f will raise the amount of transfer and fs expenditure on her private good, but reduce ms expenditure on the household good and his private good. Proposition 11 is significant in that it shows the rotten kid theorem result restated in Proposition 6, namely that household consumption patterns reflect the preferences of the higher income spouse, with the corollary that an efficient level of household goods is provided, do not hold for a>0. Instead, an increase in spouse fs bargaining power, moves the household towards a more egalitarian outcome, at the expense of efficiency in household public goods production. Next we turn to the case where s > sN. Because Wm(pN) > Wm(0), it is possible, then, for spouse f to receive a transfer larger than pO and become the sole purchaser of the household good. This will happen if fs bargaining power is large enough. Proposition 12 Consider the case If < Ib and s > sN. There exists a critical a b 0 (0, 1) such that pn is less than pN if a < a b, but pn exceeds pO if and a > a b.

When s > sN and a < a b, the equilibrium is pro-male. The comparative static results in Proposition 9 apply to this case as well. When a=a b the transfer will be between pNand pO. The exact size of the transfer is indeterminate, as all transfers in this range result in the same, egalitarian, solution. The egalitarian outcome is a knife-edge solution, which will only prevail under very special circumstances. The next proposition deals with the pro-female outcome, which occurs when a > a b.6

21

Proposition 13.

Consider the case If < Ib, s > sN and a > a b. Assume that vNNN< 0. An increase in fs bargaining power raises the amount of transfer and her expenditure on both public and private goods but reduces ms expenditure on his private good.

Proposition 13 shows how, in a pro-female equilibrium, an increase in fs bargaining power moves the household closer to the outcome that would prevail with a female dictator. Household good purchases and spending on fs private good increase because the households purchases come closer to reflecting a single persons preferences those of the female household head. To summarize the results in this section, we have shown that the amount of transfer between spouses depends on their relative income levels (If versus Im), their relative bargaining power (a), and the degree to which they care about each other (s). When the spouses have relatively equal income levels (Im $If > Ia), there is no need for a transfer because it has no real effect. In the other extreme, when they have very unequal incomes (If < Ib), transfers will occur and are increasing (or nondecreasing) in a and s.

5. Comparison with the Unitary Model of family In this section we contrast the results from our model to those arising when the household is assumed to maximize a single welfare function subject to a single budget constraint - the "unitary" model of the household. In the unitary model, the household maximizes a household welfare function W=W(xm,xf,xh) subject to a single budget constraint
p fx f%p mx m%x h ' If%Im
(13)

22

Optimization yields demand functions of the form xi = xi(pf, pm, Im+If). A key property of this form of demand functions is that they depend on total household full income, not the division of (full) income between spouses. In our model, however, division of income between spouses does matter for a wide range of parameter values. It matters, first, in that the changes in the intra-household distribution of income alters the demand for the household good, and for each spouses private good, holding total household income constant. For example, in terms of spending on the household good, Propositions 2, 7 and 9 imply that, holding Im fixed, xh will vary with If as illustrated in Figure 2. As a corollary, governments can influence household demands by targeting transfers. In the unitary model, it should not make any difference which family member receives child benefits, pension income, or any other government transfer. By way of contrast, in our model, if the household is in a no-transfer equilibrium, If 0 (Ib, Ia), Proposition 2 and Corollary 1 imply that a government transfer made to spouse m and a government transfer to spouse f have completely different effects on the familys spending pattern. A government transfer to f raises her private expenditure but has no effects on the total household expenditure and ms private expenditure. A transfer to m, on the other hand, has no effect on fs private expenditure but raises the total household expenditure and ms private expenditure. To conduct a rigorous comparison of our model with the unitary model, we consider a small change in the division of a given level of family income. Suppose that fs income is increased by g dollars at the same time as ms income is decreased by g dollars. The unitary model suggests that this change in the division of family income should have no effects on the familys spending pattern. In our

23

model the effects of such a redistribution of income will, as hinted above, depend on fs income If (relative to Im), as well as parameters s and a. Proposition 14. a) At an interior equilibrium, where If $Ia, the change in the division of family income will have no effect on household expenditure (xh); b) In a no-transfer equilibrium, where If 0 (Ib, Ia), the change in the division of family income will reduce household expenditure (xh); c) At a positive transfer equilibrium, where If < Ib, the change in the division of family income will i) have no effect on household expenditure (xh) if there is a female dictator (a = 1 and s > sN), a male dictatorial household head (a = 0), or if an egalitarian equilibrium results from household bargaining (a = a b and s > sN). ii) in general change household expenditure (xh) for all other values of s and a. In particular, the change raises household expenditure if a > a a and s < sN.

Figure 3 illustrates the results in Proposition 14 in terms of relative income ratio If/Im. We see that household expenditure curve is downward sloping if relative income ratio is between Ib/Im and Ia/Im, but remains flat if relative income ratio is between Ia/Im and 1. If relative income ratio is below Ib/Im, the slope of the household expenditure curve depends on the values of a and s. The curve is flat if one of the conditions in part c-i) of Proposition 14 is satisfied; otherwise, the curve is in general not
24

flat. Part c-ii) of Proposition 14 gives one set of conditions under which the curve is upward-sloping. But for some other parameter values we cannot rule out the possibility of a downward-sloping curve. Proposition 14 states a change in the division of income will affect the household expenditure unless the conditions in a) or c-i) are satisfied. Part a) represents a situation where the incomes of the two spouses are relatively equal and both spouses contribute household expenditure even in the absence of a transfer p. A change in the division of income will merely change the amount contributed by each spouse but not the combined expenditure. The last set of conditions in part c-i), a = a b and s > sN, represents a similar situation. In this case the amount of transfer pn is such that both spouses contribute to household expenditure; a change in the division of income will merely bring an offsetting change in the amount of transfer. The first two sets of conditions in part c-i) of Proposition 14 describe a situation where either f or m has all the bargaining power in the family. In this case the spouse with bargaining power effectively has complete control over the total family income (Im + If), and he (she) can achieve his (her) most-desired spending pattern through appropriate choice of transfer p. He (she) will respond to a change in the division of income by making an offsetting change in transfer.7 If, on the other hand, both spouses have some bargaining power, the amount of transfer is determined by the balance of bargaining power and the position of the threat point matters in the Nash bargaining problem. A change in the division of income moves the threat point and hence has, in general, real effects on the spending pattern of the family.

6. Testing the Model

25

In this section we describe the predictions of the model for family spending pattern as a function of relative earnings (If/Im), and compare available empirical evidence to the predictions of the model. We then discuss alternative methods of testing the model. The first thing we should note is that there are two ways to generate variations in relative income If/Im and the two methods have different implications for the relationship between demand and relative income. The first method is to keep the income of one spouse, say Im, constant and vary the income of the other spouse If. The second method is to keep the total income of the family constant and look at different divisions of the same total income between two spouses. In terms of empirical testing, the first method involves taking a sample of families in which the income levels of the higherearning spouse are roughly the same but other spouse has different income levels, while the second method is an analysis of families with the same total income but different divisions of the total incomes. Figures 2 and 3 show that these two methods imply qualitatively different relationships between xh and If/Im. As a result, it would not be appropriate to take a sample of families with different personal incomes and different total family income and try to infer a statistical relationship between xh on If/Im. Another point we should note is that parameter a, which measures the relative bargaining power of two spouses, plays a role in determining the relationship between demands and relative income. Figure 3 shows that when relative income level is below Ib/Im, xh can be increasing in, decreasing in, or independent of If/Im, depending on the value of a. This suggests that empirical tests should take into consideration social, economic, and cultural factors that may influence bargaining power within families.

26

Recent studies have found shifts in household expenditure patterns as spouses relative income changes consistent with the predictions of the model. Phipps and Burton (1993), using the Canadian family expenditure survey, find that relative share of husband and wife in family income shapes commodity demands for seven out of 12 commodities studied. Browning, Bourguignon, Chiappori and Lechene (1994), again using Canadian family expenditure survey data, find that as the wife's share of income increases, so too does her share of expenditures, just as Figure 3 predicts. Lundberg, Pollak and Wales (1997) find that the replacement of the U.K. child tax allowance received by the father with child benefits paid to the mother resulted in a significant increase in expenditure on children's clothing. An alternative method of testing the model is to examine the flow of financial resources within household. The model makes a number of predictions about household financial management patterns. First, when one partner has a sufficiently low income, so the family is at a positive transfers solution, the model predicts that she will receive an income transfer from the other spouse, but will not have full access to all household resources. Second, at a no-transfers corner solution, we would expect the lower income spouse's own earnings to be reserved for her use - we would not expect the higher income spouse to have access to these moneys. Third, when the partners are at an interior solution, they may adopt any one of a variety of household financial management systems. Since expenditures on household goods effectively pool their resources, we would expect to see a number of these households contributing their income to a common pool, adopting a "pooling" or "shared management" budgeting system. However the partners may also choose to manage their incomes independently, each making their own chosen contribution to household public goods out of their own earnings.

27

Until recently most studies of family financial management have been small, sociological case studies. The findings of these studies have, however, been broadly consistent with the hypotheses of the model. For example, Pahl (1989) has found that the independent management system is most common in dual-earner couples, who would be expected to be indifferent between shared and other forms of management. Allowance or whole wage systems, which involve a transfer from one spouse to the other, appear in single-earner couples, where one partner is more likely to have a low enough income that transfers of income occur voluntarily, as the theory predicts. However, as more data on household financial management becomes available, economists have now begun to use this to examine transfers, for example, Woolley and Marshall (1994) and Dobbelsteen and Kooreman (1997). Our model provides a rigorous theoretical foundation for such empirical work. 7. Conclusions The model developed in this paper is, in one sense, a straightforward extension of the theory of rational choice. All that we have done is recognize that within households there are often two or more rational decision makers. Yet this small change has radical implications. It can be seen that the distribution of income between household members matters, and will generally affect commodity demands. The model developed in this paper provides a theoretical explanation for recent empirical work, which has found expenditures on certain items, such as food, clothing, or children's clothes, tend to increase as female incomes rise. It also develops a framework within which it is possible to analyse the targeting of benefits to men and women, to mothers or to fathers.

28

Endnotes
1.As one referee points out, male and female incomes matter in the unitary model to the extent that an increase in female incomes resulting from increased hours of work may be associated with a substitution of market goods for home produced goods. Our focus is not on this type of income change but rather on income changes resulting from changes in government targeting of benefits or from changes in employment income of full-time, full year workers which might arise from, say, a move from individual to joint taxation (or vice versa). 2.Although the model applies equally well to partners in same-sex relationships. 3. The proofs of all propositions are in Appendix. 4.Sen 1990 and Agarwal 1994 contain excellent discussions of the determinants of intra-household

bargaining power
5.We do not model labour supply in this paper, however the disincentive effects created by such transfer

changes are certainly merit further exploration

6. As in the case s < sN, it is possible to have Wm(pn) = Wm(0) if a is large. Proposition 13 deals with only the case where Wm(pn) > Wm(0). 7. If spouse f has full bargaining power, however, her choice over transfer is constrained by that ms welfare after the transfer cannot fall below Wm(0). When this constraint is binding (see Proposition 10), she will not be able to completely offset the effect of a change in the division of income through her choice of transfer. That is why in part c-i) of Proposition 14 an additional condition, s > sN, is attached to the case a = 1.

29

Bibliography Agarwal, Bina 1994 A Field of One's Own: Gender and land rights in South Asia, Cambridge: Cambridge University Press. Apps, Patricia. 1981. A Theory of Inequality and Taxation Cambridge: Cambridge University Press. Becker, Gary S. 1974. "A Theory of Social Interactions", Journal of Political Economy, 82, 1063-1093. Becker, Gary S. 1996. Accounting for Tastes Cambridge and London: Harvard University Press. Bergstrom, T. 1989. "A Fresh Look at the Rotten Kid Theorem and Other Household Mysteries" Journal of Political Economy, 97, 1138-59. Bergstrom, T. 1996 Economics in a Family Way Journal of Economic Literature 34(4): 1903-1934. Bergstrom, T. 1997 A Survey of Theories of the Family in Handbook of Population and Family Economics Amsterdam: North-Holland. Bergstrom, T. L. Blume and H. Varian 1987. "On the Private Provision of Public Goods" Journal of Public Economics, 29, 25-49. Bragstad, T. 1989. "On the significance of standards for the division of work in the household," mimeo, University of Oslo. Browning, M. F. Bourguignon, P-A Chiappori and V. Lechene. 1994. "Incomes and Outcomes: A Structural Model of Intra-Household Allocation" Journal of Political Economy, 102, 1067-1093. Chiappori, P-A. 1988. "Nash-Bargained Household Decisions: A Comment" International Economic Review, 29, 791-796. Chiappori, P-A. 1992. "Collective Labor Supply and Welfare" Journal of Political Economy, 100, 437-467. Del Boca, D and C. J. Flinn. 1994. "Expenditure Decisions of Divorced Mothers and Income Composition" Journal of Human Resources 29(3): 742-761.

30

Dobbelsteen, Simone and Peter Kooreman (1997) Financial Management, Bargaining and Efficiency within the Household; An Empirical Analysis De Economist 145(3): 345-366. Kanbur, R. and L. Haddad. 1994. Are Better Off Households More Unequal or Less Unequal Oxford Economic Papers 46, 445-458. Hoddinott, John and Lawrence Haddad. 1992. "Does Female Income Share Influence Household Expenditure Patterns? mimeo, Trinity College, University of Oxford. Konrad, K.A. and K.E. Lommerud. 1995. "Family Policy with Non-cooperative families" Scandinavian Journal of Economics, 97, 581-601. Konrad, K.A. and K.E. Lommerud. 1996. The Bargaining Family Revisited Centre for Economic Policy Research Discussion Paper No. 1312. Leuthold, Jane. 1968. "An empirical study of formula income transfers and the work decisions of the poor," Journal of Human Resources, 3, 312-23. Lundberg, S. and R.A. Pollak. 1993. "Separate Spheres Bargaining and the Marriage Market" Journal of Political Economy, 101, 988-1010. Lundberg, S. and R.A. Pollak. 1996. Bargaining and Distribution in Marriage Journal of Economic Perspectives 10(4), 139-158. Lundberg, Shelly, Robert Pollak and Terence Wales. 1997. Do Husband and Wives Pool their Resources? Evidence from the U.K.Child Benefit Journal of Human Resources, 32(3), 463-480. Manser, M. and M. Brown. 1980. "Marriage and household decision-making," International Economic Review, 21, 31-44. McElroy, M. J. and M. B. Horney. 1981. "Nash bargained household decision-making," International Economic Review, 22, 333-349. McElroy, Marjorie B. 1990. "The Empirical Content of Nash-Bargained Household Behavior" The Journal of Human Resources, 25, 559-583. Ott, Notburga. 1995. Fertility and Division of Work in the Family: A game theoretic model of household decisions in E. Kuiper and J. Sap (ed.) Out of the Margin: Feminist Perspectives on Economics London and New York: Routledge. Pahl, Jan 1989. Money and Marriage Basingstoke, Hampshire: MacMillan Education.
31

Parker, Hermoine and Holly Sutherland. 1991. Child Tax Allowances? A comparison of child benefit, child tax reliefs, and basic incomes as instruments of family policy STICERD Occasional Paper 16, STICERD, London School of Economics, London. Phipps, Shelley A and Peter S. Burton. 1993. "What's Mine is Yours: The Influence of Male and Female Incomes on Patterns of Household Expenditure" Department of Economics, Dalhousie University. Pollak, R. 1985. "A Transactions Cost Approach to Families and Households" Journal of Economic Literature 23(2): 581-608. Sen, Amartya K. 1990. Gender and Cooperative Conflicts in Irene Tinker (ed.) Persistent Inequalities: Women and World Development, New York: Oxford University Press, 123-149. Thomas, D. 1990. "Intra-household Resource Allocation: An Inferential Approach" Journal of Human Resources, 25: 635-663. Ulph, David, 1988. "A general noncooperative Nash model of household behaviour," mimeo, University of Bristol. Warr, Peter. 1983. The Private Provision of Public Good is Independent of the distribution of Income Economics Letters, 13: 207-211. Woolley, F.R. 1988. Woolley, F. R. and J. Marshall. 1994. "Measuring Inequality Within the Household" Review of Income and Wealth 40(4): 415-431. Woolley, Frances. 1996. "A Strategic Model of Household Labour Supplies" mimeo, Carleton University.

32

Appendix Proof of Proposition 1. Equation (5) implies that a corner solution (xfh = 0) prevails as long as
h (1%s)v )(x m )<

1 ) If u( ) pf pf

(A1)

for some positive xmh. For If close to 0, (A1) holds because uN(0) = 4. On the other hand, if If = Im, (1/pf)uN(If/pf) < (1/pm)uN((Im-xmh)/pm) = (1+s)vN(xmh). Furthermore, uN(If/pf) decreases as If rises. Therefore, there exists a unique Ia in the interval (0, Im). Proof of Proposition 2. If If # Ia, xf = If/pf, which is an increasing function of If but is independent of Im and s. The value of xm = xmh is solved from spouse ms optimization condition:
h )' (1%s)v )(x m h 1 ) Im&x m u( ) pm pm

(A2)

The results in the proposition are obtained by conducting comparative statics on the above condition. If If > Ia, the equilibrium is determined by the following equations:
h %x f h) ' (1%s)v )(x m h I &x h 1 ) Im&x m 1 u( ) ' u )( f f ). pm pm pf pf

(A3)

The results in the proposition are obtained by conducting comparative statics on this equation system. In particular,
Mx i h MIi ' u ))(x i)[p j2(1%s)v ))%u ))(x j)] J1p ip j >0
(A4)

Mx i h MIj

'

u ))(x j)p i(1%s)v )) J1p j

<0

(A5)

where i, j = m, f, and
33

J1 '

u ))(x m) pm

p f(1%s)v )) %

u ))(x f) pf

[p m(1%s)v )) %

u ))(x m) pm

] > 0.

(A6)

Proof of Proposition 3. Using equations (A4) - (A5), we can verify that for a small p, Mxfh/Mp = Mxfh/MIf - Mxfh/MIm = 1, and Mxmh/Mp = Mxmh/MIf - Mxmh/MIm =-1. Proof of Proposition 4. Differentiate Wm(p) with respect to p:
MWm(p) Mp ' su )(x f) pf & u )(x m) pm
(A7)

Suppose p = 0. For If close to 0, uN(xf) approaches infinity. Hence given that s > 0, MWm(0)/Mp > 0 for If close to 0. On the other hand, at If = Ia, uN(xf)/pf = uN(xm)/pm, in which case MWm(0)/Mp < 0 as long as s < 1. Furthermore,
M2Wm(0) MIf Mp ' su ))(x f) p f2 <0,
(A8)

which implies that MWm(0)/Mp is decreasing in If for If in the interval (0, Ia). Therefore, there exists Ib such that MWm(0)/Mp > 0 if If < Ib but MWm(0)/Mp < 0 if If 0 (Ib, Ia). In the case s = 0, MWm(0)/Mp < 0 for all levels of If < Ia, in which case we can set Ib = 0. In the case s = 1, MWm(0)/Mp = 0 at If = Ia, in which case Ib = Ia.

Proof of Proposition 5. The first part of the proposition follows from the definition of Ib. Comparative statics on the system of equations (7) and (8) yields:
)) u ))(x m) Mpv su (x f) ' [p m(1%s)v )) % ] <0; MIf J2p f pm

(A9)

u ))(x m)p f(1%s)v )) Mpv '& >0; MIm J2p m

(A10)

where
34

J2 '&

su ))(x f) pf

[p m(1%s)v )) %

u ))(x m) pm

]&

u ))(x m) pm

p f(1%s)v )) <0.

(A11)

Proof of Proposition 6. The aggregate household budget constraint is pmxm + pfxf + xh = Im + If. It is straightforward to show that the first-order conditions to the problem of maximizing Wm subject to this household budget constraint are the same as equations (7) and (8). Proof of Proposition 7. Again, comparative statics on equation system (7) - (8) reveals that
h u ))(x m)su ))(x f) Mx h Mx m ' '& >0; MIf MIf J2p mp f

(A12)

Mx m MIf

'

h n su ))(x f)(1%s)v )) 1 M(Im&p &x m ) '& >0; MIf pm J2p f

(A13)

Mx f MIf

'

n u ))(x m)(1%s)v )) 1 M(If%p ) '& >0. J2p m p f MIf

(A14)

Proof of Proposition 8. It is obvious that if s = 0, Wm(0) > Wm(p) for any positive p including pN. Equations (10) and (A7) imply that if s = 1, Wm(p) achieves maximum at p = pN. Hence, Wm(0) < Wm(pN) if s = 1. Furthermore, for s 0 (0, 1),
M[Wm(p))&Wm(0)] Ms ' MWm(p)) Mp) Mp Ms % [u(x f(p))) % v(x h(p))) & u(x f(0)) & v(x h(0))] >0
(A15)

because u(xf(pN)) + v(xh(pN)) > u(xf(0)) + v(xh(0)), MWm(pN)/Mp < 0, and MpN/Ms < 0 (from comparative statics on (10)). Therefore, there exists a unique sN.

35

Proof of Proposition 9. Since pn has to satisfy Wm(pn) $ Wm(0), Proposition 8 implies that pn < pN. If a > a a (see Proposition 10 for the definition of a a), equilibrium is determined by (8) and the condition Wm(pn) = Wm(0). Let
J3 ' [p m(1%s)v )) % u ))(x m) MWm(pn) ] >0 pm Mp
(A16)

be the Jacobian associated with this system of equations. (Equation (12) implies that MWm(pn)/Mp < 0). Comparative statics reveals:
u ))(x m) u )(If/p f)&u )((If%pn)/p f) Mpn s ' [p m(1%s)v )) % ] <0 MIf J3 pm pf
h u ))(x m) [u )(If/p f)&u )((If%pn)/p f)] Mx h Mx m ' '& >0 MIf MIf J3p m pf

(A17)

(A18)

Mx m

) ) n (1%s)v )) [u (If/p f)&u ((If%p )/p f)] '& >0 J3 MIf pf

(A19)

Mx f

u ))(x m) su )(If/p f) u )(x m) > 1 )) ' [p (1%s)v % ][ & ] 0 MIf J3p f m pm pf pm <

(A20)

If a < a a, equilibrium is determined by the equations (8) and (12). It can be shown that the Jacobian of this system can be written as
J4 ' B[p m(1%s)v )) % u ))(x m) pm ]& Au ))(x m) pm ,
(A21)

where

36

A / a [Wm(pn)&Wm(0)]

[p m(1%s)v )&su )(x m)][u )))(x m)v ))%p mu ))(x m)v )))](1%s)
2 [p m (1%s)v ))%u ))(x m)]2 (1&s 2)u ))(x m)v ))

2 pm (1%s)v )) % u ))(x m) su )(x m) su )(x f) u )(x m) u ))(x m) ][ & ]%[Wf(pn)&Wf(0)] } %(1&a ){[(1%s)v )& pm pf pm p2
m

& a [Wm(pn)&Wm(0)]

(A22)

B/a (

su )(x f) u )(x m) u )(x f) su )(x m) u ))(x m)(p m(1%s)v )&su )(x m)) & )[ & & ] pf pm pf pm p [p 2(1%s)v ))%u ))(x )]
m m m

% % 2 2 (1%s)v ))%u ))(x m) (1%s)v ))%u ))(x m)]2 p f2 pm [p m u )(x f) su )(x m) su )(x f) u )(x m) su ))(x f) u ))(x m) %(1&a )[( % )] & )( & )%(Wf(pn)&Wf(0))( pf pm pf pm p2 p2 %a [Wm(pn)&Wm(0)][
f m

u ))(x f)

s(1%s)u ))(x m)v ))

u )))(x m)(1%s)v ))[p m(1%s)v )&su )(x m)]

] (A23)

It can be shown, after tedious and lengthy derivations, that J4 is positive as long as uNNN > 0 and vNNN < 0. Comparative statics on this system reveals that
u ))(x m) Mpn 1 ' C[p m(1%s)v ))% ] <0, MIf J4 pm
(A24)

Mx f MIf

'

Cu ))(x m) > 1 [J4%Cp m(1%s)v ))% ] 0, p fJ4 pm <

(A25)

37

Mx m MIf

'&

C(1%s)v )) >0, J4

(A26)

h Cu ))(x m) Mx h Mx m ' '& >0, MIf MIf J4p m

(A27)

where
C / (1&a ) [u )(x f)&u )(x f(0))] su )(x f) u )(x m) s(Wf(pn)&Wf(0)) [ & ][ &1] pf pf pm Wm(pn)&Wm(0) u ))(x f) [s(1&a )(Wf(pn)&Wf(0))%a (Wm(pn)&Wm(0))] & p f2 >0

(A28)

Proof of Proposition 10. It can be shown that

MWf(p) Mp '
2 )) v (1%s)v )(1&s 2)p m 1 ) If%p u( )&(1%s)v )% 2 pf pf (1%s)v ))%u ))(x m) pm

(A29)

which is positive at p = pN (see (10)). This implies that, if a = 1, spouse f would like to set p > pN. However, since Wm(pN) < Wm(0), the maximum p she can obtain is the one that solves Wm(p) = Wm(0). On the other hand, we know that Wm(pn) > Wm(0) if a = 0. Therefore, there exists a a such that Wm(pn) = Wm(0) if a > a a and Wm(pn) > Wm(0) if a < a a. Proof of Proposition 11. Comparative statics on (8) and (12) shows that if a < a a,

38

u ))(x m) MW (pn) MW (pn) Mpn 1 ' [p m(1%s)v ))% ][(Wf(pn)&Wf(0)) m &(Wm(pn)&Wm(0)) f ] >0 Ma J4 pm Mp Mp
h Mx m )) MW (pn) MW (pn) 1 u (x m) [(Wf(pn)&Wf(0)) m &(Wm(pn)&Wm(0)) f ] <0 J4 p m Mp Mp

(A30)

Ma

'

(A31)

Mx m Ma

'&

MW (pn) MW (pn) 1 (1%s)v ))[(Wf(pn)&Wf(0)) m &(Wm(pn)&Wm(0)) f ] <0 J4 Mp Mp

(A32)

and Mxf/Ma = (1/pf)(Mpn/Ma) > 0. Proof of Proposition 12. If s > sN, Wm(pO) > Wm(0). It is possible to have pn > pO, in which case f becomes the sole purchaser of the household good. Equation (11) implies that at pn = pO,
MWf(p))) Mp
h )) I &p)) 1 ) If%p &x f 1 )& su )( m ) >0. ' u( pf pf pm pm

(A33)

If a = 1, f will set pn > pO. We know that pn < pN if a = 0. Therefore, a b exists. Proof of Proposition 13. The equilibrium in this case is determined by equations (12) and
h n 1 ) If%p &x f u( ) ' (1%s)v )(x f h). pf pf

(A34)

The Jacobian of this system is:

u ))(x f) pf B )u ))(x f) pf

J5 ' A )[p f(1%s)v )) %

]%

(A35)

where AN and BN are messy expressions similar to A and B in (A22) and (A23). It can be shown that J5 > 0 if vNNN < 0. Comparative statics on this system reveals:
39

)) MW (pn) MW (pn) Mpn 1 u (x f) ' [ %p f(1%s)v ))][(Wf(pn)&Wf(0)) m &(Wm(pn)&Wm(0)) f ] >0 Ma J5 p f Mp Mp

(A36)

h )) MW (pn) MW (pn) Mx h Mx f 1 u (x f) ' ' [(Wf(pn)&Wf(0)) m &(Wm(pn)&Wm(0)) f ] >0 Ma Ma J5 p f Mp Mp

(A37)

Mx f Ma

'

MW (pn) MW (pn) (1%s)v )) [(Wf(pn)&Wf(0)) m &(Wm(pn)&Wm(0)) f ] >0 J5 Mp Mp

(A38)

and Mxm/Ma = - (1/pm)(Mpn/Ma) < 0. Proof of Proposition 14. Part a) follows from Proposition 3 and part b) follows from Proposition 2 and Corollary 1. Part c-i) is true because it can be shown that Mxh/Mg = 0 for the three cases identified here. In the case a > a a and s < sN,
)) ) ) Mx h u (x m) u (x m(0)) su (x f(0)) ' [ & ] >0. Mg J3p m pm pf

(A39)

In the case a < a a and s < sN (or in the case a < a b and s > sN), we can show that
h )) MW (pn) su )(x f(0)) Wf(pn)&Wf(0) 1 Mx h Mx m 1 u (x m) ' ' (1&a ) m { [ & ] Mg Mg J4 p m Mp pf Wm(pn)&Wm(0) s u )((x m(0)) Wf(pn)&Wf(0) (1&s 2)p mu )(x m(0))v ))(x h(0)) } [ &1]& & 2 pm Wm(pn)&Wm(0) (1%s)v ))(x h(0))%u ))(x m(0)) pm

(A40)

which in general is not equal to 0. We obtain a similar expression for Mxh/Mg in the case a > a b and s > sN.

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