Example 22 Infinitely repeated version of Example 21
Let U(x, y) = xy, & = 2, and T = 24. We derived C& = (12, 24) with e& = 12 in the one-shot case (Example 2.1). At the partnership equilibrium, eP = 8 with CP = (16, 16). Note that U(C&) = 288 and U(CP) = 256. One Nash equilibrium for the infinitely repeated partnership has each partner supply 12 hours of effort in the first period and every subsequent period as long as the other partner supplied 12 hours in each previous period, but will supply 8 hours of effort in period t and every subsequent period if the other partner did not supply 12 hours of effort in period t — 1.
Suppose that partner 1 deviates from e = 12 in period t. What's the highest one-period utility that a partner can achieve when the other partner supplies 12 hours of effort? The answer is obtained by maximizing U(x, y) when y = 1/2 x 2 x (24 — x + 12). That is, we maximize x x (24 — x + 12) = 36x — x2. Using the formula for maximizing a quadratic (Section 1 of Chapter 2) or calculus yields x = 36/2 = 18. Then partner 1 will supply 6 hours of effort. Each individual's income will be1/ x 2(6 + 12) = 18 and partner 1's period t utility will be U(18, 18) = 324. (One can also solve for x by setting the MRS, which is y/x, equal to partner 1's opportunity cost of leisure, which is 1/2 x 2, and then using the budget constraint y = 24 — x + 12.)
By deviating in period t, partner 1 gets an increase in utility of at most 324 _ 288 = 36. But he is then punished by partner 2 in period t + 1 and every subsequent period. Partner 2 supplies 8 hours of effort in period t + 1 and beyond. We already know what partner 1's best one-shot response is because we have a unique Nash equilibrium of the one-shot game when each supplies 8 hours of effort. Therefore, each will supply 8 hours of effort from period t + 1 on. Hence the deviating partner will get a utility of at most 256 in each of those periods. Had he not deviated, utility would have been 288 each period. Therefore, by deviating partner 1 gets a utility bonus of at most 36 in period t but suffers a utility penalty of at least 288 _ 256 = 32 in every period after the tth. Discounting to period t, we find that deviating will not be profitable if
and this simplifies to 36 < 325/(1 _ 5). Therefore, deviating cannot benefit either player if 36 _ 365 < 325,or5 > 36/68 = 0.53. If the discount factor is 0.53 or higher then the trigger strategy specified in the first paragraph is a Nash equilibrium.
The efficient outcome can be sustained if the partnership lasts many periods and the partners are not too impatient. However, it is just one of the Nash equilibria in the infinitely repeated partnership. There are many other equilibria. At the other extreme, if both announce their intentions to supply in each period of the repeated game the amount of effort that emerges in the one-shot Nash equilibrium whatever the other does, then we have a Nash equilibrium of the repeated game. (This is true whatever the discount factor.)
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