Posted prices versus bargaining in markets_7

CHAPTER 9 The Role of the Trading Procedure 9.1 Introduction In this chapter we focus on the role of the trading procedure in determining the outcome of trade. The models of markets in the previous three chapters have in common the following three features. 1. The bargaining is always bilateral. All negotiations take place be-tween two agents. In particular, an agent is not allowed to make offers simultaneously to more than one other agent. 2. The termination of an unsuccessful match is exogenous. No agent has the option of deciding to stop the negotiations. 3. An agreement is restricted to be a price at which the good is ex-changed. Other agreements are not allowed: a pair of agents cannot agree that one of them will pay the other to leave the market, or that they will execute a trade only under certain conditions. The strategic approach has the advantage that it allows us to construct models in which we can explore the role of these three features. 173 174 Chapter 9. The Role of the Trading Procedure As in other parts of the book, we aim to exhibit only the main ideas in the field. To do so we study several models, in all of which we make the following assumptions. Goods A single indivisible good is traded for some quantity of a divisible good (“money”). Time Time is discrete and is indexed by the nonnegative integers. Economic Agents In period 0 a single seller, whom we refer to as S, and two buyers, whom we refer to as BH and BL, enter the market. The seller owns one unit of the indivisible good. The two buyers have reservation values for the good of vH and vL, respectively, where vH ≥ vL > 0. No more agents enter the market at any later date (cf. Model B in Chapter 6). All three agents have time preferences with a constant discount factor of 0 < δ < 1. An agreement on the price p in period t yields a payoff of δtp for the seller and of δt(v−p) for a buyer with reservation value v. If an agent does not trade then his payoff is zero. When uncertainty is involved we assume that the agents maximize their expected utilities. Information All agents have full information about the history of the mar-ket at all times: the seller always knows the buyer with whom she is matched, and every agent learns about, and remembers, all events that occur in the market, including the events in matches in which he does not take part. In a market containing only S and BH, the price at which the good is sold in the unique subgame perfect equilibrium of the bargaining game of alternating offers in which S makes the first offer is vH/(1+δ). We denote this price by pH. When bargaining with BH, the seller can threaten to trade with BL, so that it appears that the presence of BL enhances her bargaining position. However, the threat to trade with BL may not be credible, since the surplus available to S and BL is lower than that available to S and BH. Thus the extent to which the seller can profit from the existence of BL is not clear; it depends on the exact trading procedure. We start, in Section 9.2, with a model in which the three features men-tioned at the beginning of this section are retained. As in the previous three chapters we assume that the matching process is random and is given ex-ogenously. A buyer who rejects an offer runs the risk of losing the seller and having to wait to be matched anew. We show that if vH = vL then this fact improves the seller’s bargaining position: the price at which the good is sold exceeds p∗ . 9.2 Random Matching 175 Next, in Section 9.3, we study a model in which the seller can make an offer that is heard simultaneously by the two buyers. We find that if vH is not too large and δ is close to 1, then once again the presence of BL increases the equilibrium price above pH. In Section 9.4 we assume that in each period the seller can choose the buyer with whom to negotiate. The results in this case depend on the times at which the seller can switch to a new buyer. If she can switch only after she rejects an offer, then the equilibrium price is precisely pH: in this case a threat by S to abandon BH is not credible. If the seller can switch only after the buyer rejects an offer, then there are many subgame perfect equilibria. In some of these, the equilibrium price exceeds pH. Finally, in Section 9.5 we allow BH to make a payment to BL in ex-change for which BL leaves the market, and we allow the seller to make a payment to BL in exchange for which BL is committed to buying the good at the price vL in the event that S does not reach agreement with BH. The equilibrium payoffs in this model coincide with those predicted by the Shapley value; the equilibrium payoff of the seller exceeds p∗ . We see that the results we obtain are sensitive to the precise character-istics of the trading procedure. One general conclusion is that only when the procedure allows the seller to effectively commit to trade with BL in the event she does not reach agreement with BH does she obtain a price that exceeds pH. 9.2 Random Matching At the beginning of each period the seller is randomly matched with one of the two buyers, and one of the matched parties is selected randomly to make a proposal. Each random event occurs with probability 1/2, independent of all past events. The other party can either accept or reject the proposal. In the event of acceptance, the parties trade, and the game ends. In the event of rejection, the match dissolves, and the seller is (randomly) matched anew in the next period. Note that the game between the seller and the buyer with whom she is matched is similar to the model of alternating offers with breakdown that we studied in Section 4.2 (with a probability of breakdown of 1/2). The main difference is that the payoffs of the agents in the event of breakdown are determined endogenously rather than being fixed. 9.2.1 The Case vH = vL Without loss of generality we let vH = vL = 1. The game has a unique subgame perfect equilibrium, in which the good is sold to the first buyer to be matched at a price close to the competitive price of 1. 176 Chapter 9. The Role of the Trading Procedure Proposition 9.1 If vH = vL = 1 then the game has a unique subgame perfect equilibrium, in which the good is sold immediately at the price ps = (2−δ)2/(4−3δ) if the seller is selected to make the first offer, and at the price pb = δ(2 − δ)/(4 − 3δ) if the matched buyer is selected to make the first offer. These prices converge to 1 as δ converges to 1. Proof. Define Ms and ms to be the supremum and the infimum of the seller’s payoff over all subgame perfect equilibria of the game. Similarly, define Mb and mb to be the corresponding values for either of the buyers in the same game. Four equally probable events may occur at the beginning of each period. Denoting by i/j the event that i is selected to make an offer to j, these events are S/BH, BH/S, S/BL, and BL/S. Step 1. Ms ≥ (2(1−δmb)+2δMs)/4 and mb ≤ (1−δMs +δmb)/4. Proof. For every subgame perfect equilibrium that gives j a payoff of v we can construct a subgame perfect equilibrium for the subgame starting with the event i/j such that agreement is reached immediately, j’s payoff is δv and i’s payoff is 1−δv. The inequalities follow from the fact that there exists a subgame perfect equilibrium such that after each of the events S/BI the good is sold at a price arbitrarily close to 1−δmb, and after each of the events BI/S the good is sold at a price arbitrarily close to δMs. Step 2. mb = (1−δ)/(4−3δ) and Ms = (2−δ)/(4−3δ). Proof. The seller obtains no more than δMs when she has to respond, and no more than 1−δmb when she is the proposer. Hence Ms ≤ (2δMs+2(1− δmb))/4. Combined with Step 1 we obtain Ms = (2δMs +2(1−δmb))/4. Similarly, a buyer obtains at least 1−δMs when he is matched and is chosen to be the proposer, and at least δmb when he is matched and is chosen to respond. Therefore mb ≥ (1−δMs+δmb)/4, which, combined with Step 1, means that mb = (1−δMs +δmb)/4. The two equalities imply the result. Step 3. Mb ≤ 1−mb −ms. Proof. This follows from the fact that the most that a buyer gets in equilibrium does not exceed the surplus minus the sum of the minima of the two other agents’ payoffs. Step 4. Ms = ms = (2−δ)/(4−3δ) and Mb = mb = (1−δ)/(4−3δ). Proof. If the seller is the responder then she obtains at least δms, and if she is the proposer then she obtains at least 1−δMb.ByStep 3wehave1−δMb ≥ 1−δ(1−mb −ms), so that ms ≥ [2δms +2(1−δ(1−mb −ms))]/4, which implies that ms ≥ 1/2+δmb/[2(1−δ)] = 1/2+δ/[2(4−3δ)] = Ms. Finally, we have Mb ≤ 1−mb −ms = (1−δ)/(4−3δ) = mb. 9.2 Random Matching 177 By the same argument as in the proof of Theorem 3.4 it follows that there is a unique subgame perfect equilibrium in which the seller always proposes the price 1 − δMb = ps, and each buyer always offers the price δMs = pb. Note that the technique used in the proof of Step 1 is different from that used in the proofs of Steps 1 and 2 of Theorem 3.4. Given a collection of subgame perfect equilibria in the subgames starting in the second period we construct a subgame perfect equilibrium for the game starting in the first period. This line of argument is useful in other models that are similar to the one here. So far we have assumed that a match may be broken after any offer is rejected. If instead a match may be broken only after the seller rejects an offer, then the unique subgame perfect equilibrium coincides with that in the game in which the seller faces a single buyer (and the proposer is chosen randomly at the start of each period). The prices the agents propose thus converge to 1/2 as δ converges to 1. On the other hand, if a match may be broken only after a buyer rejects an offer, then there is a unique subgame perfect equilibrium, which coincides with the one given in Proposition 9.1. This leads us to a conclusion about how to model competitive forces. If we want to capture the pressure on the price caused by the presence of more than one buyer, we must include in the model the risk that a match may be broken after the buyer rejects an offer; it is not enough that there be this risk only after the seller rejects an offer. We now consider briefly the case in which the probability that a match terminates after an offer is rejected is one, rather than 1/2: that is, the case in which the seller is matched in alternate periods with BH and BL. Retaining the assumption that the proposer is selected randomly, the game has a unique subgame perfect equilibrium, in which the seller always pro-poses the price 1, and each buyer always proposes the price pb = δ/(2−δ). (The equation that determines pb is pb = δ(1/2 + pb/2).) A buyer accepts the price 1, since if he does not then the good will be sold to the other buyer. When a buyer is selected to make a proposal he is able to extract some surplus from the seller since she is uncertain whether she will be the proposer or the responder in the next match. If we assume that the matches and the selection of proposer are both deterministic, thenthe subgame perfect equilibrium depends onthe order in which the agents are matched and chosen to propose. If the order is S/BI, BI/S, S/BJ, BJ/S (for {I,J} = {L,H}), then the unique subgame perfect equilibrium is essentially the same as if there were only one buyer: the seller always proposes the price 1/(1 + δ), while each buyer always proposes δ/(1 + δ). If the order is BI/S, S/BI, BJ/S, S/BJ then in the unique ... - tailieumienphi.vn