Harrington Chapter 13 Exercise 8. There are n 3 doctors who have created a partnership. In each period, each doctor decides how hard to work.

4. Harrington Chapter 13 Exercise 8.There are n ≥ 3 doctors who have created a partnership. In each period, each doctor decides how hard to work. Let eti denote the effort chosen by doctor i in period t, and assume that eti can take 1 of 10 levels: 1, 2, ..., 10. The partnership’s profit is higher when the doctors work harder. More specifically, total profit for the partnership equals twice the amount of total effort:Profit=2×(et1 +et2 +...+etn)A doctors payoff is an equal share of the profits, less the personal cost of effort,which is assumed to equal the amount of effort; thus,Doctor i’s payoff = (1/n) × 2 × (et1 + et2 + ... + etn) − etiThis stage game is infinitely repeated, where each doctors payoff is the present value of the payoff stream and doctor is discount factor is δi.a. Assume that the history of the game is common knowledge. That is, in period t, the past choices of effort for all doctors over periods 1, ..., t − 1 are observed. Derive a subgame perfect Nash equilibrium in which each player chooses an effort e∗ > 1. (Take e∗ > 1 as a constant.)b. Assume that the history of the game is not common knowledge. In period t, only the total effort (eτ1 +eτ2 +...+eτn), in period τ is observed by all players (for all τ ≤ t−1). (By the assumption of perfect recall, a player knows his own past effort, but you can ignore that information.) Find a subgame perfect Nash equilibrium in which each player chooses an effort e∗ > 1. (Take e∗ > 1 as a constant.)

4. Harrington Chapter 13 Exercise 8.There are n ≥ 3 doctors who have created a partnership. In each period, eachdoctor decides how hard to work. Let eti denote the effort chosen by doctor iin...