Question

Acme is a monopolist who faces inverse market demand function P (Q, y) = 100 ? 2Q + y, where y is the quality level of Acme’s product. Acme has cost function function C(Q) = 20Q.
  
1. For now, let Acme’s quality level be predetermined at y = 80.
   (1a) Determine Acme’s optimal output level and profits.
   (1b) Determine aggregate surplus.
   (1c) Determine Acme’s output level such that aggregate surplus is maximized.What are Acme’s profits in this case?

2. Suppose quality is costly. Specifically, assume that Acme must pay innovation cost I(y)= (1/4)(y^2). Thus, Acme’s total profits are x(Q,y)=P(Q,y)Q?C(Q)?I(y). Assuming Acme is allowed to act like a monopolist, we will work out Acme’s optimal quality choice, y?.
   (2a) First step is to calculate Acme’s optimal quality conditional variable profits, vx(y) = P (Q?, y) Q? ? C(Q?), where Q? is the monopolist’s output level given            inverse demand function P (Q, y) and cost function C(Q).

2ai. Determine Q? as a function of y.   

2aii. Determine vx(y) by inserting Q? into the variable profits expression.

   (2b) Profits can now be boiled down to a simple quality choice, x(y) = vx(y)?I(y). The optimal choice of y is the one that maximizes x(y). This is the point where the slope of the profit function is exactly equal to zero. Hence, the optimal quality choice satisfies, x?(y?) = 0. This implies, (d(vx(y?))/dy) ? (d(I(y?))/dy) = 0 ? vx?(y?) = I?(y?)
       2bi. Determine the derivative of vx(y).
       2bii. Determine the derivative of I(y) .
       2biii. Solve for y?.
   (2c) Determine aggregate surplus (remember to subtract innovation cost).

3. Suppose, for any given quality level y, Acme is forced to deliver an output level that maximizes aggregate surplus for the given quality level y.
   (3a) What is Acme’s optimal innovation choice, y?, in this case?
   (3b) Determine aggregate surplus for this case.

4. Is it preferable for a social planner to grant Acme a patent for its innovation and thereby allow it to act like a monopolist?

Answer 1