A mass of consumers is uniformly distributed along the interval [0, 1]. Two firms, A and B, are located at points 0 and 1 respectively. We denote by p; the price of firm i € A, B. A consumer located at point x € [0, 1] obtains utility U₁(x) = u - PA - tx² if he consumes from firm A, and UB(x)=u-PB-t(1-x)² if he consumes from firm B. In the following, we assume that the gross utility u is sufficiently high, so that the market will be covered and all consumers will get positive utility in equilibrium. Both firms have a cost function equal to T;(q) = (1+X)qi, where you should substitute X for the last number of your student ID number.
a- Assume firms set their prices simultaneously. Solve for the Nash equilibrium prices, and compute the equilibrium profits.
b- Now, assume a Stackelberg timing, where firm A is the leader. Explain briefly why we should not use the Nash equilibrium concept to solve this game, and solve for the equilibrium prices and profits.
c- Compare the results obtained in parts (b) and (c) and explain the intuition for such difference. Are the equilibria efficient?
