A monopoly faces a continuum of consumers who are distributed uniformly on the unit interval (i.e., the "numberöf consumers in each given interval is the same). The total mass of consumers is 1. Each consumer is interested in buying at most one unit. Consumers differ in the way they perceive the monopoly’s output. Assuming for simplicity that the monopoly is located at point 0, the utility of a consumer who is located at some point x between 0 and 1 if he buys from the monopoly is r - p - tx², where p is the monopoly’s price, r > 0, and t > 0 is the transportation cost per-unit of distance. If a consumer does not buy, his utility is 0. The monopoly’s per-unit cost of production is c.

1. Given p, find the location (i.e., äddress") of the consumer who is just indifferent between buying and not buying. Show this consumer on a graph.

2. How high can p be such that the market will be still covered (i.e., every consumer will buy)?

3. Write the monopoly profit as a function of p (hint: distinguish the case where the market is covered from the case where the market is uncovered).

4. Show that if t < (r - c) =3, then at the profit maximum, the monopoly will choose a price that ensures that the market is covered.

5. Suppose that the monopoly incurs a fixed cost F whenever it opens a plant. Compute the monopoly’s maximal profit given the assumption that the market is fully covered by the monopolist and that the monopolist has a single plant at point 0.

6. Now suppose that the monopoly opens a second plant at point 1. The utility for the consumer who buys at this plant is r - p₂ – t (1- x)², where p₂ is the price of the product at the second plant. Compute the monopoly’s maximal profit in this case under the assumption that the market is covered (note that now the monopoly bears a fixed cost of 2F since it operates two plants).

7. Based on your answers in (5) and (6), compute the range of F for which the monopoly will operate two plants, one plant, or no plants (i.e., will exit the market altogether).

8. Explain the intuition for your result in (7). In particular, explain why the monopoly might benefit from opening a second plant at point 1.