A monopolist sells a product whose quality is unknown to consumers before they buy. It is common knowledge however that the product’s quality, s, is drawn from a uniform distribution on the unit interval. There is continuum of consumers with a total mass of 1. Each consumer is interested in buying at most 1 unit and gets a utility of s - p if he buys and 0 otherwise. The monopoly can send its product to inspection before consumers buy. The cost of inspection is c. The inspection perfectly reveals s to consumers with probability α. With probability 1- α, it reveals nothing in which case consumers cannot tell whether the monopolist has sent its product to inspection or did not.

1. Write out the monopoly’s profit if it sends its product to inspection and if it does not.

2. Prove that the monopoly will send its product to inspection if and only if s is above some threshold ŝ.

3. Compute the expected quality of the monopoly’s product if consumer do not see the inspection results (you should compute a weighted

average of two conditional expected values: one for cases in which s < ŝ and one for s > ŝ; the weights depend on the probability that s < ŝ and the probability that s > ŝ and that the inspection reveals nothing).

4. How does the expected quality that you computed in (3) vary with α? Explain the intuition.

5. Given your answer to (3), compute the threshold ŝ.

6. How does s vary with c and with? Explain the intuition.