FAMILY BARGAINING: A STACKELBERG APPROACH Joaquín Andaluz, Miriam Marcén and JoséAlbertoMolina University of Zaragoza

SUMMARY 1. INTRODUCTION 2. FRAMEWORK 3. ANALYSIS AND RESULTS 4. CONCLUSIONS

1. INTRODUCTION A. The microeconomic analysis of family decisions: . Unitary models: . The consensus model: Samuelson 1954 . The altruistic model: Becker 1974 . Family bargaining models: . The income received by husband and wife have separate impacts. . Is essential the definition of the threat point.

B. Definition of the status quo: -Divorce (Manser and Brown 1980, McElroy and Horney 1981). . -Non-cooperative solution (Lundberg and Pollak 1993, Chen and Wolley 2001) C. Non-cooperative family bargaining models: -The repeated interaction between agents over time can tacitly lead to the cooperative solution (folk theorem).

D. Objective: -To analyse the dynamic aspects of the non-cooperative family bargaining models assuming that the spouses take their decisions in a sequential way (Stackelberg equilibrium). E. Question: -How the individual discount factor influences on the decisions as well as on the gains of well-being derived from cooperation?

2. FRAMEWORK Where: d = discount factor xj = private consumption of j 2.1. Objective functions: Where: d = discount factor xj = private consumption of j Q = q1 + q2 = Family good, with qj = percentage of hours that agent j devotes to the provision of the family good.

2.2. Preferences Where:

3.ANALYSIS 3.1 THE ONE SHOT GAME - The non-cooperative equilibrium is sequential: Spouse 1: Leader, Spouse 2: Follower - Non-cooperative equilibrium: Levels of consumption and the provision of the family good:

Levels of utility

The repeated of the game gives rise to multiple equilibria. 3.2. COOPERATION The repeated of the game gives rise to multiple equilibria. The repeated interaction between agents can tacitly lead to Pareto efficient solutions (folk theorem). Both spouses can implicitly create a punishment strategy: Trigger strategy.

Conditions of sustainability:

Nash bargaining solution:

Proposition 1 The contribution to the family good of the spouse who decides in the second place (follower) is increasing with respect to the discount factor: The discount factor influences in an ambiguous way the contribution to the family good of the spouse who decides in the first place (leader):

Proposition 2 For both spouses, an increase in the discount factor can increase or reduce the level of utility in the bargaining solution:

4. CONCLUSIONS 1. The higher the discount factor of the follower, the greater is the set of sustainable agreements. 2. The contribution to the family good of the follower is increasing with respect to the discount factor. 3. The discount factor influences in an ambiguous way the contribution of the leader and the surplus derived from the bargaining.