Changes

* 1. Buyer makes investment, costing <math>x^2</math>. * 2. Buyer observes <math>v</math>* 3. Seller makes take-it-or-leave-it (TOILI) offer <i>without</i> seeing <math>v</math>. * 4. Buyer accepts or rejects.

Utility functions: The problem does not make reference to utility functions. I will assume that the buyer's utility is <math>x+v-x^2-P</math> where <math>P</math> refers to the price of the widget -- if the buyer chooses to buy. Otherwise his utility is zero. As for the seller: I will assume his utility is simply <math>P</math> (the price of the widget) if it is sold, and otherwise is zero. Note that both agents are risk neutral in this setup.

(a)Socially optimal level will be where marginal cost equals marginal benefits, or where social welfare is maximized. Marginal costs of investment are <math>2x</math>. Marginal benefits are <math>1</math>. These are equal where <math>x=1/2</math>.

(b)This solution requires backwards induction starting with step 4 above. * First, note that there is a cutoff price at which the buyer will accept or not. * Next, note that seller will correctly infer this (in expectation) and make a corresponding offer that will leave the buyer indifferent between accepting and rejecting the offer. * Lastly, note that buyer will correctly anticipate seller's step 3 behavior and make corresponding investment decision.