Virtual willingness to pay

By the revelation principle (Section 3.2), it is sufficient to consider an incentive compatible direct mechanism. In other words, regardless of the mechanism constructed by the intermediary, given the equilibrium of the mechanism, we can construct an equivalent incentive compatible direct mechanism, where the buyer and the supplier report their respective valuations to the intermediary, and the intermediary determines if the trade is to take place. If so, she determines the buyer s payment and the suppliers revenue. Otherwise, the players take their outside options in a direct matching market. Let T(/3,p, w) represents the direct revelation mechanism, where /3(s, v) is the probability that the trade will take place, p(s, v) is the expected payment to be made by the buyer to the intermediary (the asked price), and w(s, v) is the expected payment from the intermediary to the supplier (the bid price), where s and v are the valuations given by the supplier and buyer, respectively. As mentioned above, the intermediary is aware of the buyer and the supplier s outside options as random variables characterized by distributions G and F, respectively. Based on this information the intermediary establishes the buyer s virtual willingness to pay follows ... 

Step 2. The intermediary is subject to the same information asymmetry as the players. Based on the players reports, and the probability distributions G and F characterizing the asymmetric information, the intermediary constructs a virtual willingness to pay the buyer a virtual op-... 

Based on the supply chain intermediation framework outlined above, the intermediary devises a direct mechanism T P p,w) to maximize her profit (3.35), subject to incentive compatibility and individual rationality. First, the intermediary must determine P s, v) which specifies whether the trade is to take place given the reported s, v and her knowledge of their virtual opportunity costs and virtual willingness to pay. Given the simple form of the profit function (3.35) it is straightforward to find a profit maximizing P subject to individual rationality as follows ... 

In essence, to maximize her own profit the intermediary must restrict the trade to profitable situations, as indicated by the difference between the buyer s virtual willingness to pay and the supplier s virtual opportunity cost. For instance, suppose s and v are both uniformly distributed on the unit interval. Then, based on the above mechanism the trade takes place if and only if 5(1 ) = 2v — 1 >2s = 5(5). Or equivalently, v — s >., i.e., the trade takes place iff the buyer s valuation exceeds the supplier s valuation by... 

Thus, the intermediary s profit is determined by the difference between the buyers virtual willingness to pay and the supplier s opportunity cost. If the trade is to take place, there must be at least one buyer i such that " i vi) > = s). The intermediary could ensure that this is the case by stating a reserve price Us(v) > s. It can be shown that if i(ui) is a monotone strictly increasing function of Vi, for every ieM, one possible solution to the above auction maximization problem is as follows ... 

Figure 3.3. Interpretation of the Bidder i s Virtual Willingness to Pay (Bulow and Roberts (1989)...

In reality, virtually no public policy is predicated on Mr. Stockman s brand of cost-benefit analysis, let alone on "willingness to pay." Social security, defense spending, welfare, the Vietnam War, the space program — whatever their merits, any of the major policy decisions of the last few decades illustrate that there are things we do collectively as a nation simply because we believe they should be done, even though no one of us would do them individually. We have preferred to let our human values guide our public policy, even in cases where that has strained our pocketbooks. 

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