Reputational Incentives: A seller (player 1) and buyer (player 2) can engage in trade for two periods. The seller is good with probability 0.5. A good seller produces a high-quality product with probability π and a low-quality product with probability 1- π. The seller is strategic with probability 0.5.

A strategic seller chooses effort, e ∈ [0, 1], at a personal cost of c(e) = e² and produces a high-quality product with probability eπ and a low-quality product with probability 1- eπ. The buyer's payoff from a high-quality product is 1 and that from a low-quality product is 0. The game proceeds as follows: in each period t the seller makes an ultimatum price offer p_t to the buyer for his product. The buyer either accepts or rejects the offer.

If he rejects the offer then they move to the second period. If he accepts the offer then a strategic seller chooses e_t ≥ 0 and delivers the product. Neither the seller's type nor his choice of e_t (if he is strategic) is known to the buyer. After the product is delivered the buyer learns its quality.

a. If this game had only one period, what would be the effort level e and the price p in the unique perfect Bayesian equilibrium?

b. Imagine the game is repeated for two periods. Is there a perfect Bayesian equilibrium in which strategic sellers exert the level of effort e that you found in (a) in each period?

c. Find the unique perfect Bayesian equilibrium of the two-period game. Why is there a difference in the effort levels in each period?