We introduced the topic of differentiated products in a simple 2-firm Bertrand price setting model in which each firm’s demand increases with the price of the other firm’s output. The specific context we investigated was that of imperfect substitutes.
A: Assume throughout that demand for each firm’s good is positive at p =MC even if the other firm sets its price to 0. Suppose further that firms face constant MC and no fixed costs.
(a) Suppose that instead of substitutes, the goods produced by the two firms are complements— i.e. suppose that an increase in firm j ’s price causes a decrease rather than an increase in the demand for firm i ’s good. How would Graph 26.3 change assuming both firms end up producing in equilibrium?
(b) What would the in-between case look like in this graph — i.e. what would the best response functions look like if the price of firm j ’s product had no influence on the demand for firm is product?
(c) Suppose our three cases—the case of substitutes (covered in the text), of complements (covered in (a)) and of the in-between case (covered in (b)) —share the following feature in common: When p j = 0, it is a best response for firm i to set p = p >MC. How does p relate to what we would have called the monopoly price in Chapter 23?
(e) In which of the three cases might it be that there is no equilibrium in which both firms produce?
B: Consider identical firms 1 and 2, and suppose that the demand for firm is output is give by xi (pi, pj) = A−αpi −βpj. Assume marginal cost is a constant c and there are no fixed costs.
(a) What ranges of values correspond to goods xi and xj being substitutes, complements and in between goods as defined in part A of the exercise.
(b) Derive the best response functions. What are the intercepts and slopes?
(c) Are the slopes of the best response functions positive or negative? What does your answer depend on?
(d) What is the equilibrium price in terms of A, α, β and c. Confirm your answer to A(d).
(e) Under what conditions will only one firm produce when the two goods are relatively complementary?