3. A price-taking firm produces widgets (y), using capital (K) and labor (L). The per-unit price of capital is 2 and the per-unit price of labor is 4. The price of output is p. The firm's production function is given by, f(L, K) = L2Ki. (30 points) (a) Explain, in words, the difference between the short-run and long-run profit maximization problems. (5 points) (b) Suppose K is fixed in the short-run at K. Solve for the short-run factor demand for labor and the short-run supply function. (10 points) (c) Write down the first-order conditions for the firm's long-run profit maximization prob- lem. (5 points) (d) Solve for the firm's long-run factor demands and the long-run supply function. (10 points)