A monopoly operates for two periods and produces a homogenous good whose quality is either high or low (the monopoly cannot choose the quality of the good). In the first period, the quality of the good is unobserved by consumers and their demand is q₁ = s₁ - p₁, where s1 is the perceived quality of the good and p₁ is the price in period 1. In the second period, the quality of the good becomes common knowledge and the demand for the good is q₂ = 4 - p₂ if the quality is high and q₂ = 2 - p₂ if the quality is low, where p2 is the price in the second period. The per unit cost of production is 1 in the first period, and 1 - ɣq₁ in the second period, where ɣ is a positive constant that reflects a learning-by-doing effect: the more the firm produces in period 1, the lower is its per unit cost in period 2. Assume that ɣ = 1=4 if the monopoly produces a high quality product and ɣ = 1=2 if the monopoly produces a low quality product. For simplicity, assume that there is no discounting.

1. Solve the monopoly’s problem in period 2 and compute the monopoly’s profit at the optimum, taking q1 as given (recall that q₁ determines the per-unit cost of production in period 2).

2. Write out the sum of the monopoly’s profits in periods 1 and 2 as a function of p₁, given the monopoly’s type, assuming that consumers believe that (i) s₁ = 4, and (ii) s₁ = 2.

3. Now suppose that in period 1 the monopoly chooses a price, p₁, and a level of uninformative advertising, A. Solve for the strategy of a low type monopoly in a separating equilibrium.

4. Let A(p₁) define, for each period 1 price p₁, the minimal amount of advertising  required by a high quality monopoly in order to deter a low quality monopoly from mimicking it. Given your answers to parts (2) and (3), compute A (p₁) and show it in a figure. Moreover, compute the prices at which A (p1) crosses the horizontal axis. Explain the meaning of these crossing points.

5. Solve for the price that a high quality monopoly will charge in a Pareto undominated separating equilibrium (one where a high quality monopoly advertises just enough to induce separation, or more precisely, one where consumers believe that the monopoly must be of a high quality if they observe a pair (p₁,A) which is a weakly dominated strategy for a low quality monopoly) and compute the amount of advertising that it will choose.

6. Compare your answer in part (5) to the optimal strategy of a high quality  monopoly in the full information case (the case where the quality is common knowledge even in period 1). Does the monopoly under price or overprice in equilibrium, relative to the full information case? Explain why the price distortion could serve as a signal for quality in this particular case.