4. Consider the following extension of the model that we have used ...

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4. Consider the following extension of the model that we have used to discuss adverse selection with two types in the lecture with . € [0,1]: The cost function d(x) of the seller is differentiable with c(0) = 0, Cx (x) > 0 and cox(x) > 0. There are two types of consumers with types A1 < A2. Fraction fi > 0 of the consumers have type Ai. Willingness to pay u(x,Gi) is differentiable in . and satisfies u(0, 0i) = 0, Ux (x, 0) > 0, and Uxx (x,0i) < 0. Further, the single crossing property Ux (x,A1) < ux (x, A2) holds. If the consumers do not buy from the seller under consideration, they can obtain quality xo € [0,1] from some other source at price to, where ulxo, > to > c(x0) holds. (In the lecture we have restricted attention to the case in which the outside option is given by (xo,t to) = (0,0), SO that consumers who do not buy from the seller under consideration receive the utility u(x0, 0i) - to = 0.) Suppose throughout the following that the efficient qualities x1 and x2 satisfy 0 < x1 < x2 < 1. (a) Provide an economic interpretation of the inequalities u (xo, 0i) > to > c(xo) assumed above. (b) Formulate the participation and incentive constraints for the seller's profit maximization problem under hidden information. (c) Suppose xo = 1 holds, SO that the outside option for consumers is to obtain the highest possible quality. Determine which of the participation and incentive constraints will be binding in a solution to the seller's problem and which can be ignored. What does this tell you about the relationship between the qualities (1,2) solving the seller's reduced problem and the first best qualities (xi,x2)? (d) Suppose x1 < xo < x2 holds. Find the solution to the seller's problem under hidden information. How does this solution differ from the solution under complete information?

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