Question

In: Economics

Posted Price by a Buyer: In this problem, everything is as in problem 3, except that the buyer is the one making an offer to the seller.

Posted Price by a Buyer: In this problem, everything is as in problem 3, except that the buyer is the one making an offer to the seller. To be clear, the Bayesian game now proceeds as follows: first, the

 buyer announces a price at which he will purchase the object from the seller and, second, the seller either accepts or rejects this offer. If the seller accepts the buyer's offer, the buyer pays the announced price to the seller and the seller transfers the object to the buyer. Otherwise, if the seller rejects the buyer's offer, no cash transfer or trade occurs, and both players get zero utility. As before, suppose that the seller's and buyer's types, t, and ts, respectively, are independently drawn from the uniform distribution on [0, 1]. Also, suppose that the buyer commits to his take-it-or-leave-it offer; that is, he does not make a new offer if the seller rejects his initial offer.

 a. What is the seller's type-t, best response? In other words, at what prices would the seller accept the buyer's offer and at what prices would the seller reject the buyer's offer. Assume that the seller acoepts the buyer's offer if the seller is indifferent between accepting and rejecting the buyer's offer.

 b. Suppose that the buyer offers to buy the object at price p.

 i. What is the buyer's utility assuming the seller acoepts his offer?

 ii. What is the probability that the seller accepts the buyer's offer?

 iii. What is the buyer's expected utility when offering to buy the object at price p?

 c. i. Based on your answer to part b.iii., what is the optimal offer price p*?

 ii. What is the buyer's expected utility when offering to buy the object at price p*?

 d. Based on your answer to part c.i., you should notice that certain types of sellers are completely "priced out", i.e. they will never accept the buyer's offer, regardless of the buyer's type. Identify this set of seller types.



Solutions

Expert Solution

The given game is a type of one-period ultimatum game. In this type of game one player gives an offer another player and then the second player has the choice to either accept or reject the offer.

The given game has two players: player one = ts, and player 2 = tb

The total pie shared =1 and distribution among two players is uniform between [0,1]

Now, ts makes a “Take-it-or-leave-it” offer to tb, let’s say (s)he chooses to offer 1-s to tb.

tb can now either accept the offer, in which case the payoffs for both the players are (s,1-s), or tb can reject ts’ offer in which case the payoffs for the players are (0,0).

Below is the depiction of the described one-shot sequential game:

where P-1 is ts and P-2 is tb

Now, for any 1-s>0, tb will accept the offer as the payoff on accepting would be higher than that on rejecting, and at 1-s=0, tb will be indifferent towards accepting or rejecting the offer.

Hence, P-1 will offer the smallest fraction of the pie possible to P-2, let’s say it is e which tends to 0. Thus, the Sub-game Perfect Equilibrium of the game will be (1,0).

Therefore, in ultimatum games, the first-mover holds maximum bargaining power and has an infinite advantage over the second-mover.

The best response for ts = offer 1 i.e. sell at 1

So payoff to seller = 1

So payoff to buyer = 0


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