Short Run and Long Run Cost Function We consider the same production function is X=F(K,L)=3K^1/3L^1/3. Define the wage cost as w and the rental rate of capital as r. The price of the product is defined as p. 1. Consider a short run case where the amount of capital is fixed at K = K ¯ . In this case, the short run firm cost is wL + rK ¯. The firm produces output x using K ¯ and L. Calculate the demand for labor input L for output x in the short run. 2. The cost function explains the relationship between total cost C and the firm output x. Solve for the short run cost function (where K ¯ is fixed) by substituting L into the firm cost expression. 1 3. From the short run cost function solved in the previous question, we consider the short run firm profit maximization problem: given market price of production p and factor prices {w, r}, the firm maximizes its profit by choosing the output of production x. Solving this short-run profit maximization problem, and derive the short run supply curve. 4. In the long run, the usage of capital can be changed as well. Thus, the long run firm cost is wL + rK. For a firm producing output x with K and L, calculate the new cost-minimizing usage of inputs K∗ and L∗. 5. Solve for the long run cost function (where K is flexible) by substituting K and L in question 4 into the long-run cost expression. 6. Solve the firm profit maximization problem in the long run and derive the long run supply curve.