1. Suppose we have a line of a length equal to one mile.
2. There are two firms, 1 and 2, one firm at either end of this line.
3. The two firms simultaneously set prices, p1and p2 , respectively.
4. The two firms have a constant marginal cost = c.
5. Each firm attempts to maximize its profit.
6. Customers are evenly distributed along the line, one at each point.
7. For simplicity, assume a total population of one (or you can think in terms of market shares if that helps).
8. Each potential customer buys exactly one unit, buying it either from firm 1 or from firm 2. So total demand is always exactly one.
9. Suppose we have a customer, Jane, at a position y on the line. She is a distance y from firm 1 and distance (1―y) from firm 2.
(Hint: y^2 is a transportation cost that increases rapidly with distance travelled y ; similar interpretation for (1―y)^2 )
10. Jane will:
a. Buy from firm 1 if: p1+ y^2 < p2+ (1―y)^2
b. Buy from firm 2 if: p1+ y^2 > p2+ (1―y)^2
c. Toss a fair coin if this is an exact equality.
Questions 2.1: Will either firm set its price below the marginal cost? Explain your answer.
Question 2.2: Suppose firm 2 sets a price p2. What should firm 1’s price be if it wants to capture the entire market?
Question 2.3: Suppose that prices p1 and p2 are close enough that the market is divided (not necessarily equally) between the two firms. Find Jane’s location if she’s exactly indifferent between buying from firm 1 and buying from firm 2. Use your answer (i.e., Jane’s location) to prove that firm 1’s demand is given by: 1(p1,p2)= (p2+―p1) / 2
Question 2.4: Now find firm 1’s BR. Recall that firm 1’s profit function can be written as: 1(p1,p2)=p11(p1,p2)―c1(p1,p2)
Question 2.5: Graph firm 1’s Best Response (BR) on the same plane as firm 2’s BR (you do not have to derive firm 2’s BR function, you can just invoke symmetry to deduce it).
Question 2.6: Find the NE algebraically.
Question 2.7: Explain how your NE corrects Bertrand’s conclusion. (Hint: set t=0 and see what happens.)