In this exercise, you will explore the equilibrium of a market in which we know the cost function of the firms, as well as the demand function of consumers. You will first analyze the short-run equilibrium, and then the long-run equilibrium. Each firm in the industry has the same technology with cost function c(y)=k^2 +2y +y^2 where y is the quantity produced and k is some positive fixed cost. You should think of this cost function as coming out of a standard cost minimization problem with respect to the inputs.
(a) Derive the average cost c(y)/y, the average variable cost and the marginal cost c'(y) for y > 0. Graph all these cost curves in an appropriately labeled diagram.
(b) Draw the supply function in the graph, and write down the equation that represents it, y(p).
(c) Derive the aggregate supply function Y^S(p) by adding up the supply y(p) over the J firms that are in the market. Write down the expression for Y^S(p)
(d) Consider now the demand side of the market. For simplicity, assume a linear demand function: Y ^D(p) = A-Bp, where A and B are positive numbers, and Y^D represents the total quantity demanded in the industry. Assume that this comes from aggregation of the individual demand functions derived from maximization. Find the short-run equilibrium price p by equating Y^D and Y^S. You can assume that A and B are such that we are on the increasing part of the supply function, i.e. quantity produced is strictly positive. Find the short-run equilibrium industry production Y^*=Y^D(p^*) = Y^S(p^*).
(e) Under what conditions for A,B,J and k the firms will indeed produce a positive quantity of output? We maintain this assumption for the next items.
(f) In the short-run, what happens to Y^* and p^* as the fixed cost k increases? What is the intuition?
(g) Consider now the long-run equilibrium, in which firms are allowed to enter the market. Solve for the number of firms J^* that will enter into the market.
(h) How does the number of firms J^* depend on k, A and B?
(i) What is the equation of the long-run supply curve? Is it horizontal, increasing or decreasing?