Instructions:
Complete all questions. Show working for all parts. Answers do not have to be integers: assume all variables (goods, prices) are infinitely divisible. Submit answers in PDF format, with your name, ID, and email address included (if you would like feedback on your work).
1. Consider a perfectly competitive industry in which the inverse demand is given by p(y) = 2001 - 2y and each firm has the following cost function:
1/3 y3 + 18 for y > 0
c(y) =
0 for y = 0
(a) In the long-run equilibrium, what price will be charged for the product? How many firms will operate in this market?
(b) A tax of 6 dollars per unit sold is now imposed on each firm operating in this market. In the long-run equilibrium, what price will be charged for the product? What total quantity will be sold? How many firms will operate in this market?
(c) Suppose instead that a monopolist operates in this market. There is no tax im- posed on the monopolist, and he can produce the product at a constant marginal cost of c. What price will the monopolist charge (as a function of c)? Can this price be below the price you found in part a?
2. The following game can be used to model strategic interactions between two players who are competing over a contested resource of value V . Each player can choose to take an aggressive stance ("hawk") or a peaceful one ("dove").
- If both players play "dove", they split the resource, V , equally.
- If one player plays "hawk" while the other plays "dove", the player who played "hawk" gets the entire resource, V , to himself, while the player who played "dove" gets 0.
- If both players play "hawk", they split the resource, V , equally. But a fight results, and each player also incurs a cost, C.
- V > 0 and C > 0.
(a) What must be true about the relationship between C and V for Hawk, Hawk to be a Nash equilibrium?
(b) Suppose V = 2 and C = 1. Find all Nash equilibria.
3. Consider a market with the inverse demand p(y) = 7 - y. Two firms operate in this market as Cournot competitors. Both firms have the cost function c(y) = y + F. If both firms are making nonnegative profits in the Cournot equilibrium, the fixed cost F must be no greater than ___.
4. Firm A produces good A and its profit is denoted as πA. Firm B produces good B and its profit is denoted as πB . The two firms' profit functions are as follows:
πA = 9A - 2A2
πB = 6B - B2/50 - A2
Suppose the government recognises that Firm A's production imposes a negative ex- ternality on Firm B, and to try to remedy this, mandates that Firm B can charge Firm A a price of P for every unit of A that Firm A produces. What P would arise in equilibrium (i.e. what is the market clearing P )? What quantity of A would be produced? (Hint: Start by writing down both firms' profit maximisation problems with the compensation scheme in place.)