The wedding anniversary of a husband and wife is fast approaching, and each is deciding how much to spend. Let gH denote the amount that the husband spends on his wife gW and the amount the wife spends on her husband. Assume that they have agreed that the most each can spend is 500. A players’ strategy set is then the interval [0,500]. A spouse enjoys giving a bigger gift, but doesn’t like spending money. With that in mind, the husband’s payoff function is specified to be
The payoff function can be understood as follows: The benefit from exchanging gifts is captured by the term Since “men are boys with bigger toys,” this benefit increases with the size of the wife’s gift:
The “warm glow” the husband gets from giving his wife a gift is reflected in the term which increases with the size of his gift:
Alas, where there are benefits, there are costs. The personal cost to the husband from buying a gift of size gH is represented by the term or in his payoff function. Thus, we subtract this cost from the benefit, and we have the husband’s payoff function as described. The wife’s payoff function has the same general form, though with slightly different numbers:
These payoff functions are hill shaped.
a. Derive each spouse’s best-reply function and plot it.
b. Derive a Nash equilibrium.
c. Now suppose the husband’s payoff function is of the same form as the wife’s payoff function: