Exercises (60 points) 21. (10 points) Consider a Buyer (B) and a seller (S) in a garage sale market in which has a lawn mower for sale. Price is not posted on the article itself but is revealed by S at either a high price PH or low price PL where the marginal cost of selling the machine is a > 0, where P > PL - c. B valuen the machine at v> 0 and offers either the high price pil or the low price PL (for simplicity suppose there are only those two prices) where Sand B reveal simultaneously the price to trade Utility for B is defined to be U-P if trade occurs 0 if trade does not occur U { for j =HL while profit for S is defined to be SP-c - { if trade occurs if trade does not occur -- for y = H, L. Trade occurs only if the price offered by B is greater or equal than the price demanded by S. If price offered by B is strictly greater than that demanded by then the transaction occurs at the midpoint between the two offers i.e. Po It price offered by B is lower than that demanded by then the transaction does not occur. The normal form representation of the strategic interaction is as follows 0.- PH PE , - Pro PL -PL 0 a. (3 points) Consider the case of simultaneous choices under complete information in which it is common knowledge for B (row player) and S (column player) the information in the payoff matrix. Find the pure strategy Nash equilibrium when v > PH Determine the joint surplus (buyer plus eller surpluses) at equilibrium Determine if equilibrium outcomes are Pareto efficient. b. (3 points) Consider now that S chooses either PH or p in a first stage and then B in the second stage observes the posted price by S and chooses either to match the offer (offers the same price) or not (offers a different price). Assume complete information and v > P# Draw the extensive form game with payoffs in the terminal nodes. Determine then the subgame Nash equilibrium by backward induction (4 points) Determine the normal form representation of the dynamic game of part Find the Nash equilibria of this game and verify if there are NE that are not subgame perfect C Exercises (60 points) 21. (10 points) Consider a Buyer (B) and a seller (S) in a garage sale market in which has a lawn mower for sale. Price is not posted on the article itself but is revealed by S at either a high price PH or low price PL where the marginal cost of selling the machine is a > 0, where P > PL - c. B valuen the machine at v> 0 and offers either the high price pil or the low price PL (for simplicity suppose there are only those two prices) where Sand B reveal simultaneously the price to trade Utility for B is defined to be U-P if trade occurs 0 if trade does not occur U { for j =HL while profit for S is defined to be SP-c - { if trade occurs if trade does not occur -- for y = H, L. Trade occurs only if the price offered by B is greater or equal than the price demanded by S. If price offered by B is strictly greater than that demanded by then the transaction occurs at the midpoint between the two offers i.e. Po It price offered by B is lower than that demanded by then the transaction does not occur. The normal form representation of the strategic interaction is as follows 0.- PH PE , - Pro PL -PL 0 a. (3 points) Consider the case of simultaneous choices under complete information in which it is common knowledge for B (row player) and S (column player) the information in the payoff matrix. Find the pure strategy Nash equilibrium when v > PH Determine the joint surplus (buyer plus eller surpluses) at equilibrium Determine if equilibrium outcomes are Pareto efficient. b. (3 points) Consider now that S chooses either PH or p in a first stage and then B in the second stage observes the posted price by S and chooses either to match the offer (offers the same price) or not (offers a different price). Assume complete information and v > P# Draw the extensive form game with payoffs in the terminal nodes. Determine then the subgame Nash equilibrium by backward induction (4 points) Determine the normal form representation of the dynamic game of part Find the Nash equilibria of this game and verify if there are NE that are not subgame perfect C