What is the solution to this problem?

Two firms, V E G (V for short) and Y U M (Y for short) make Vegemite. Suppose firm V has production function q V = K 1/4L 1/4 , while firm Y has production function q Y = 2K 1/4L 1/4 . Both firms operate in perfectly competitive Vegemite and input markets, so that they are price takers in both output and input markets. Denote the price of Vegemite by p, the price of labor (wage rate) by w and the price of capital (rental rate) by r .

(a) What are the cost functions for the two firms?

(b) In what sense is one firm always more efficient than another? Which one?

(c) Suppose the wage rate is w = 1 and the price of capital is r = 9. What is the profit maximizing choice of output for the two firms when p = 12?

(d) Suppose Vegemite is only made by firms V and Y . Maintaining the assumption that the firms behave competitively (i.e., take the Vegemite price p as given), what is the industry supply curve? Use the input prices from part 1c.

(e) Suppose firm V acquires firm Y , and so has access to both its own production function as well the production function of firm Y . What is its new cost function? Will firm V only produce using the most efficient production function? Assuming firm V still takes the price of p = 12 as given, how much will it produce?