This is “Myerson-Satterthwaite Theorem”, section 18.2 from the book Beginning Economic Analysis (v. 1.0).
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The lemons problem is a situation where the buyers are relatively uninformed and care about the information held by sellers. Lemons problems are limited to situations where the buyer isn’t well-informed, and these problems can be mitigated by making information public. In many transactions, the buyer knows the quality of the product, so lemons concerns aren’t a significant issue. There can still be a market failure, however, if there are a limited number of buyers and sellers.
Consider the case of one buyer and one seller bargaining over the sale of a good. The buyer knows his own value v for the good, but not the seller’s cost. The seller knows her own cost c for the good, but not the buyer’s value. The buyer views the seller’s cost as uniformly distributed on the interval [0,1], and, similarly, the seller views the buyer’s value as uniformly distributed on [0,1].The remarkable fact proved by Roger Myerson and Mark Satterthwaite (“Efficient Mechanisms for Bilateral Trade,” Journal of Economic Theory 28 [1983]: 265–281) is that the distributions don’t matter; the failure of efficient trade is a fully general property. Philip Reny and Preston McAfee (“Correlated Information and Mechanism Design,” Econometrica 60, no. 2 [March 1992]: 395–421) show the nature of the distribution of information matters, and Preston McAfee (“Efficient Allocation with Continuous Quantities,” Journal of Economic Theory 53, no. 1 [February 1991]: 51–74.) showed that continuous quantities can overturn the Myerson-Satterthwaite theorem. Can efficient trade take place? Efficient trade requires that trade occurs whenever v > c, and the remarkable answer is that it is impossible to arrange efficient trade if the buyer and seller are to trade voluntarily. This is true even if a third party is used to help arrange trade, provided the third party isn’t able to subsidize the transaction.
The total gains from trade under efficiency are $\underset{0}{\overset{1}{\int}}{\displaystyle \underset{0}{\overset{v}{\int}}v-c}\text{\hspace{0.17em}}dc}\text{\hspace{0.17em}}dv={\displaystyle \underset{0}{\overset{1}{\int}}\frac{{v}^{2}}{2}}dv=\frac{1}{6}\text{.$
A means of arranging trade, known as a mechanismA means of arranging a trade.,A mechanism is a game for achieving an objective, in this case to arrange trades. asks the buyer and seller for their value and cost, respectively, and then orders trade if the value exceeds the cost and dictates a payment p by the buyer to the seller. Buyers need not make honest reports to the mechanism, however, and the mechanisms must be designed to induce the buyer and seller to report honestly to the mechanism so that efficient trades can be arranged.Inducing honesty is without loss of generality. Suppose that the buyer of type v reported the type z(v). Then we can add a stage to the mechanism in which the buyer reports a type, which is converted via the function z to a report, and then that report is given to the original mechanism. In the new mechanism, reporting v is tantamount to reporting z(v) to the original mechanism.
Consider a buyer who actually has value v but reports a value r. The buyer trades with the seller if the seller has a cost less than r, which occurs with probability r.
$$u(r,v)=vr-{E}_{c}p(r,c)$$The buyer gets the actual value v with probability r, and makes a payment that depends on the buyer’s report and the seller’s report. But we can take expectations over the seller’s report to eliminate it (from the buyer’s perspective), and this is denoted E_{c}p(r, c), which is just the expected payment given the report r. For the buyer to choose to be honest, u must be maximized at r = v for every v; otherwise, some buyers would lie and some trades would not be efficiently arranged. Thus, we can concludeWe maintain an earlier notation that the subscript refers to a partial derivative, so that if we have a function f, f_{1} is the partial derivative of f with respect to the first argument of f. ${\frac{d}{dv}u(v,v)={u}_{1}(v,v)+{u}_{2}(v,v)={u}_{2}(v,v)=r|}_{r=v}=v\text{.}$
The first equality is just the total derivative of u(v,v) because there are two terms: the second equality because u is maximized over the first argument r at r = v, and the first-order condition ensures u_{1} = 0. Finally, u_{2} is just r, and we are evaluating the derivative at the point r = v. A buyer who has a value v + Δ, but who reports v, trades with probability v and makes the payment E_{c}p(v, c). Such a buyer gets Δv more in utility than the buyer with value v. Thus, a Δ increase in value produces an increase in utility of at least Δv, showing that $u(v+\Delta ,v+\Delta )\ge u(v,v)+\Delta v$ and hence that $\frac{d}{dv}u(v,v)\ge v\text{.}$ A similar argument considering a buyer with value v who reports v + Δ shows that equality occurs.
The value u(v,v) is the gain accruing to a buyer with value v who reports having value v. Because the buyer with value 0 gets zero, the total gain accruing to the average buyer can be computed by integrating by parts ${{\displaystyle \underset{0}{\overset{1}{\int}}u(v,v)}dv=-(1-v)u(v,v)|}_{v=0}^{1}+{\displaystyle \underset{0}{\overset{1}{\int}}(1-v)\left(\frac{du}{dv}\right)}dv={\displaystyle \underset{0}{\overset{1}{\int}}(1-v)vdv}=\frac{1}{6}\text{.}$
In the integration by parts, dv = d – (1 – v) is used. The remarkable conclusion is that if the buyer is induced to truthfully reveal the buyer’s value, the buyer must obtain the entire gains from trade. This is actually a quite general proposition. If you offer the entire gains from trade to a party, that party is induced to maximize the gains from trade. Otherwise, he or she will want to distort away from maximizing the entire gains from trade, which will result in a failure of efficiency.
The logic with respect to the seller is analogous: the only way to get the seller to report her cost honestly is to offer her the entire gains from trade.
The Myerson-Satterthwaite theoremA theorem that shows private information about value may prevent efficient trade. shows that private information about value may prevent efficient trade. Thus, the gains from trade are insufficient to induce honesty by both parties. (Indeed, they are half the necessary amount.) Thus, any mechanism for arranging trades between the buyer and the seller must suffer some inefficiency. Generally this occurs because buyers act like they value the good less than they do, and sellers act like their costs are higher than they truly are.
It turns out that the worst-case scenario is a single buyer and a single seller. As markets get “thick,” the per capita losses converge to zero, and markets become efficient. Thus, informational problems of this kind are a small-numbers issue. However, many markets do in fact have small numbers of buyers or sellers. In such markets, it seems likely that informational problems will be an impediment to efficient trade.
Let h(r, c) be the gains of a seller who has cost c and reports r, h(r, c) = p(v, r) – (1 – r)c.
Noting that the highest cost seller (c = 1) never sells and thus obtains zero profits, show that honesty by the seller implies the expected value of h is 1/16.