1. 1 Bargaining & Markets u As before: Buyers and Sellers, δtp,δtp, δ t (1-p). u Matching: Seller meets a buyer with probability α. A buyer meets a seller with probability β. u Bargaining: Long term bargainng with breakdown of negotiations. Strategic Bargaining Steady State Market Strategic Strategic Bargaining Steady State Market

2. 2 Bargaining & Markets Strategic Bargaining Steady State Market 0 1/2 B/SS/B 0 αβ (1-α)(1-β) β(1-α) α(1- β) S,B have new partners B - newly matched S -unmatched S - newly matched B -unmatched S,B continue bargaining One period in the life of a matched pair

3. 3 Bargaining & Markets u V S, (V B ) The expected utility of an unmatched seller (buyer) u W S, (W B ) The Expected utility of a matched seller (buyer) Strategic Bargaining Steady State Market

4. 4 Bargaining & Markets Breakdown of negotiations occurs with probability Strategic Bargaining Steady State Market The seller’s expected payoff in the case of breakdown:

5. 5 Bargaining & Markets Strategic Bargaining Steady State Market The bargaining between a buyer and a seller is a sequential game with breakdown: 0 1/2 B/SS/B q 0 1-q q 0 AA A a period

6. 6 Bargaining & Markets Strategic Bargaining Steady State Market We seek an equilibrium in which all sellers use the same strategy, and all buyers use the same strategy. We seek an equilibrium in semi-stationary strategies: Strategies that may depend on the history of the bargaining within a match but are independent of who the partner is.

7. 7 Bargaining & Markets Strategic Bargaining Steady State Market 0 1/2 B/SS/B q 0 1-q q 0 AA

8. 8 Bargaining & Markets Strategic Bargaining Steady State Market 0 1/2 B/SS/B q 0 1-q q 0 AA B

9. 9 Bargaining & Markets Strategic Bargaining Steady State Market 0 1/2 B/SS/B q 0 1-q q 0 AA Alternative calculation B

10. 10 Bargaining & Markets Strategic Bargaining Steady State Market

11. 11 Bargaining & Markets Strategic Bargaining Steady State Market The solution: It can be shown that when all others use these strategies then this strategy is the best a player can do. A seller always demands x* and accepts y* or more. A buyer always offers y* and accepts 1-x* or more

12. 12 Bargaining & Markets Strategic Bargaining Steady State Market

13. 13 Bargaining & Markets u Buyers and Sellers, p, p, (1-p). δ=1 u Matching: Each seller meets a buyer. A buyer meets a seller with probability S/B. Matching is independent across periods u Bargaining: Rejection dissolves the match. Strategic Bargaining One time entry Strategic Strategic Bargaining One time entry

14. 14 Bargaining & Markets u Information: Agents know the time and they recognize all other agents present. They have no memory of past events. (imperfect Recall) Strategic Bargaining One time entry

15. 15 Bargaining & Markets There exists an equilibrium in which every agent proposes the price 1. A buyer accepts any price and a seller accepts price which is at least 1. Strategic Bargaining One time entry prove !!!

16. 16 Bargaining & Markets There exists an equilibrium in which every agent proposes the price 1. A buyer accepts any price and a seller accepts price which is at least 1. Strategic Bargaining One time entry prove !!! Although this is not the only equilibrium, all other equilibria lead to the objects being sold at p = 1.

17. 17 Bargaining & Markets Proof by induction on |S| Strategic Bargaining One time entry

18. 18 Bargaining & Markets Strategic Bargaining One time entry Why ???

19. 19 Bargaining & Markets Strategic Bargaining One time entry

20. 20 Bargaining & Markets Strategic Bargaining One time entry

21. 21 Bargaining & Markets Strategic Bargaining One time entry

22. 22 Bargaining & Markets Strategic Bargaining One time entry In any equilibrium, the sellers obtain the price 1, as in a competitive equilibrium

23. 23 Bargaining & Markets Strategic Bargaining One time entry