Intra-Household Decision Making under Incomplete Information: Modeling Income-Hiding between Spouses

Carolina Castilla Department of Economics Colgate University 13 Oak Drive, Hamilton, NY 13346 Email: [email protected]

Abstract: I use a collective bargaining model with non-cooperative threat points within the marriage to show that, when household resources are not perfectly observed by both spouses, there is a threshold change in bargaining power needed to induce revelation. In deciding to reveal or hide a monetary transfer, the wife faces a trade-off between increasing her own discretionary spending and increasing her bargaining power within the household. In equilibrium, the wife chooses to hide the transfer if revealing it does not increase her bargaining power enough to offset the loss in discretionary expenditure. As bargaining power is a function of the relative threat points, when household allocations default to separate spheres, income hiding is less likely to occur when the wife free-rides, relative to the case where she provides the public good.

Key words: incomplete information, intra-household bargaining, income hiding. JEL Classification: D13, D82, J12.

Intra-Household Decision Making under Incomplete Information: Modeling Income-Hiding between Spouses

1.

Introduction

Empirical studies on household bargaining have documented non-cooperative behavior (Chen, (2005); Ashaf, (2009)) and inefficient allocations within households (Udry, (1996)). It is often argued that, because families involve long-term, repeated inter- actions and caring, households will realize there are opportunities for Pareto improvement and therefore cooperation will evolve over time (Browning and Chiappori, (1998)). However, these opportunities may diminish if information asymmetries exist. Spouses, even those living under the same roof, can face information asymmetries as well and these can result in hiding (Ashraf, (2009)). I illustrate this with a simple two-stage game. In the first stage, the wife receives a monetary transfer that is unobserved by her husband, and she must decide whether to reveal the transfer or to hide it. In the second stage, spouses bargain over the allocation of resources between private and public good consumption. In deciding to reveal or hide the transfer, the wife faces a trade-off between increasing her own discretionary spending and increasing her bargaining power within the household. If she hides the transfer, the wife may spend the entire amount without influence from her spouse. However, as household public goods are observable by both spouses, if the wife is to successfully hide her additional income, she can spend it only on private consumption, which is unobservable. Conversely, if she reveals the transfer, the wife can increase her influence over intra-household allocation decisions, but her income will effectively be taxed via the bargaining process.

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2.

Household Bargaining under Incomplete Information

To derive the equilibrium allocations that result when spouses bargain over public and private consumption jointly, I draw from the Browning and Chiappori (1998) collective bargaining model. Asymmetric information is introduced by allowing everything but the wife’s monetary transfer to be common knowledge, such that the decision to hide can be thought of as the step prior to choosing to pool income every time new resources are available to each household member. This transfer is independent of household members’ labor market decisions, such as gifts, bonuses, or government transfers. For simplicity, the transfer and the wife’s private consumption choices are assumed to be observable with probability zero. However, her husband can perfectly infer the increase in resources through the household good allocation, which is perfectly observed. There are two family members, the wife (f) and the husband (m). Both family members have preferences over consumption of one private (or personal) good, denoted household public good,

, and one

. The household resource allocation decision is made in two stages. In

the first stage the wife receives a monetary transfer (T) that is not observed by her husband. The wife must decide whether to reveal the transfer or to spend at her discretion. In the second stage, household members bargain over allocations conditional on observable resources. I allow household members to bargain over all allocations, and assume they can negotiate binding agreements with zero transaction costs. Both family members face the same price for private goods which is normalized to 1 (one can think about the private good as being money for discretionary expenditure), and p is the price

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for the household good. Preferences over own consumption are represented by an egotistic utility function, Ui, which is separable in

and : for i = f,m

Where

. The functions .

, implying

and

and

satisfy

(1)

,

,

,

, and

are normal goods. In (1) I assume that the family

members have the same functional form for simplicity. The household public good is assumed to be non-rival in utility. For instance, child outcomes provide utility to both members of the household, while clothing provides utility only to the person consuming it. The equilibrium is solved by backwards induction. The objective function of the household is the bargaining power weighted sum of each member’s utility: (2) Where

is the bargaining power of spouse f and

bargaining power of spouse m, and

if the wife reveals, and

is the if she hides. Thus T only

influences bargaining power when it is revealed. This is the weight given to each spouse’s utility in the household welfare function when bargaining, and it is partially determined by each spouse’s outside options through income, as well as by resources originally brought into the marriage and distribution factors (z). The bargaining weight is used as a generic way to incorporate the existence of a threat point. Consistent with both non-cooperative equilibria within marriage and divorce threat points, the wife’s bargaining power is assumed to be increasing in the transfer.

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In the second stage, the collective household’s problem is to maximize (2) subject to the aggregate budget constraint

. This is solved assuming that both

spouses are better off cooperating than under the threat points1. (3) Solving the system of first order conditions with respect to

and

, yields the demand for the

household good and the demand for private consumption. The optimal demands respond to changes in aggregate income (i.e. income pooling feature) and to changes in individual income through the resulting changes in bargaining power.

(4) In the first stage, the wife decides whether to reveal the transfer or to hide if from her husband. She faces the following tradeoff: if she hides, she can get increase her discretionary expenditure relative to the case when she reveals. If she reveals, she can increase her bargaining power but both her private and household good consumption will be effectively taxed by bargaining power. Proposition 1: Given bargaining power

,

and T, there exists a strictly positive threshold change in

such that for any

income-hiding is the Subgame Perfect Nash

Equilibrium iff

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This is not a strong assumption given that spouses are bargaining over all allocations, such that the public good provision will be efficient (at least when all information is revealed). To successfully hide the transfer, the wife must act as if it had not occurred.

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Corollary 1: Given

,

and T, as

is strictly negative, whereas when

approaches zero, the threshold level of bargaining power tends to 1 it is positive.

In Proposition 1, the wife compares the change in utility per unit change in T when she reveals and when she hides. In equilibrium, there exists a strictly positive threshold change in bargaining power needed to induce revelation. Corollary 1 indicates that the threshold level of bargaining power is increasing in initial bargaining power, implying that the threshold is more difficult to overcome as initial bargaining power increases. The result is intuitive because when the wife’s bargaining power is high, the household good allocation is going to be close to what she prefers, thus on the margin the benefit per unit of revelation is not as high.

3.

Incentives to Hide Income in the presence of Non-Cooperative Threat Points within Marriage

Lundberg and Pollak (1993) suggest that households may revert to a non-cooperative outcome within marriage when transaction costs associated with cooperation are high, or divorce is unfeasible. In this section, I consider the case where spouses threaten to regain control over their own resources while staying in the marriage if an agreement is not reached. This is modeled through a voluntary contributions game where each spouse decides separately how to allocate her resources between the household good and private consumption, taking the other’s contribution as given. The optimization problem of spouse i is to maximize the objective function (3) subject to her own budget constraint taking j’s household good purchases,

, where

, and

, as given. The optimization problem is given by: for i,j = f,m 5

(4)

,

Solving the system of reaction functions that result from the Kuhn-Tucker conditions simultaneously yields the Nash equilibrium. I focus on the corner solutions, as these would be possible threat points to the aforementioned bargaining game. Bargaining power is a function, among other things, of the relative utilities attainable in the threat points when the transfer is revealed, such that

Proposition 2: Given

. It is assumed that

,

, bargaining power increases more when the wife free-rides

relative to the case when her husband free rides

When the husband free-rides

and

.

, the wife’s utility increases as a result of the transfer

because both public good and her private consumption increase. The husband’s utility only changes through the increase in her provision of the public good. Thus, the wife’s utility increases more relative to her spouse’s. Her bargaining power will only increase if implying the ratio of initial utilities needs to be small. In the case where the wife free-rides , her transfer only affects her own utility because her husband’s allocations are independent of her resources. Thus, the wife’s bargaining power increases when she reveals the transfer. The threat of free-riding gives the wife more bargaining power per unit of transfer and as a result she is less likely to hide relative to the case where she provides the public good (as long as her initial utility is large enough relative to her husband’s).

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4.

Conclusions

I illustrate the incentives to hide income when household resources are not perfectly observed by both spouses through a simple model. In equilibrium, there exists a strictly positive threshold of change in bargaining power needed to induce revelation, which is consistent with the notion of bargaining power being a function of distribution factors unaffected by the presence of additional resources. When the threat point is a non-cooperative outcome within marriage and household allocations default to separate spheres, the change in bargaining power is larger when the wife free-rides relative to the case where her husband free-rides. Because revelation depends upon the responsiveness of bargaining power to the transfer, income hiding is more likely if the wife were to provide the public good if cooperation fails. As efficiency is attained when household bargaining is cooperative and all resources are revealed, allocations are more likely to be efficient when the wife’s threat is to free-ride. Future research should consider the implications of spouses interacting in a repeated game where reputation can have a significant influence on the possibility to hide, and retaliation from the uninformed spouse is allowed.

The author would like to thank Joyce Chen for her helpful advice and suggestions, as well as Benjamin Anderson, Michael Sinkey, and Gregory Howard for their comments.

Appendix: Proofs Proof of Proposition 1: Spouse f hides the transfer from m if and only if

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Where Simplifying the above expression yields

where M

.

,

Proof of Proposition 2: Define

such that

Case (I): If

. Assume and

the change in utility per unit change

in the transfer for each spouse is Case (II): If

.

, such that

, the change in each spouse’s utility is given by such

because

.

given the assumption of

that

.

REFERENCES Ashraf, Nava. 2009. “Spousal Control and Intra-household Decision Making: An Experimental Study in the Philippines.” American Economic Review, 99(4): 1245–1277.

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Browning, Martin, and Pierre-André Chiappori. 1998. “Efficient Intra-Household Allocations: A General Characterization and Empirical Tests.” Econometrica, 66(6): 1241–78. Chen, Joyce. 2005. “Migration and Imperfect Monitoring: Implications for Intra-Household Allocation.” American Economic Review Papers & Proceedings, 96(2): 227 - 231. Chen, Ziqui, and Wooley, Frances. 2001. “A Cournot-Nash Model of Family Decision Making.” The Economic Journal, 111(474): 722-748. Lundberg, Shelly and Robert Pollak. 1993. “Separate Spheres Bargaining and the Marriage Market.” The Journal of Political Economy. 101(6): 988-1010. Udry, Christopher. 1996. “Gender, Agricultural Production, and the Theory of the Household.” The Journal of Political Economy, 104(5): 1010-1046.

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