Suppose that a firm produces output using two factors of production, whose market prices are w1 = 1 and w2 = 4, respectively. Let x] and 12 denote the amounts used of the two factors and f ($1, X2) = 2x2 2 the firm's production function. The market is competitive. (a) Write down the firm's total costs. (b) Recall the following cost minimization condition: MP(x1, 12) W1 MP2(21, x2) W2 Using the condition, derive the relationship between the optimal levels of I, and $2. (It would be a form of $1 = kx2 with some constant number k.) (c) Plug x1 = kx2 that you found in (b) into the production function y = f(X1, X2) = $1 12t find the relationship between the optimal level of 12 and the given output level y. (It would be a form of 12 = my with some constant number m.) (d) Find the relationship between the optimal level of 21 and the given output level y by plugging T2 = my that you found in (c) into X1 = kx2 that you found in (b). (e) Plug both x1 = my and $2 = ky from (c) and (d) into the firm's cost function in (a) to find the minimized cost function c(y). (f) If done correctly, you would have found c(y) = 4y. From this cost function, derive fixed costs (F), variable costs (Co(y)), average variable costs (AVC(y)), average fixed costs( AFC(y)), average costs(AC(y)), and marginal cost(MC(y))