A buyer has value v, for a potential acquisition and believes the seller's reservation price has the cumulative probability distribution F(v). The buyer chooses P to maximize its expected profit: = (P)Pr(P accepted) (P)F(P).

Find the buyer's marginal profit and set it equal to zero. Show that the buyer's optimal price satisfies P v-F(P)/f(p), where f(v) dF(v)/dv is the associated density function. Note that the buyer shades down its value in making its optimal bid.