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(30 points) Suppose that QED Inc.'s production function for widgets is given by F(L, K) = [1/2K1/2, where L is the quantity of labor, K is the quantity of capital, PL is the price of labor, and Px is the price of capital. a) In the short run, suppose that capital is fixed at 400 units; i.e., KsR = 400. Derive QED's total product of labor, TPL, marginal product of labor, MPL, and average product of labor, APL, functions. Suppose that the price of labor is $25 per unit (PL = $25) and the price of capital is $36 (Px = $36). In addition, suppose that QED faces the widget demand function given by the linear equation P(Q) = 425 - Q. Derive QED's (short run) total revenue and marginal revenue functions, TR(Q) and MR(Q). Derive the marginal revenue product of labor function, MRPL, and the marginal factor cost of labor function, MFCL. [Note: your MRP, function should be written as a function of L, not Q.]zCalculate the SR profit maximizing quantity of labor LsR and the resulting SR output level, QSR. Calculate QED's SR total cost at QsR, SRTC(Qs), price, PSR, and profits at QsR; II(QsR). Calculate the price elasticity of demand at the point (QSR, PSR). b) Using the cost minimization approach and given the output constraint Q = F(L,K) = [ 1/2K1/2 and the input prices PL and Px (no longer set equal to the above initial values), construct the Lagrangian function and derive the firm's input demand functions for labor, L = g(Q, PL, Px), and capital. K = h(Q, PL PK). Derive QED's long run total cost function, LRTC(Q,PL,PK). Calculate the value of the Lagrangian multiplier, 2, and give an economic interpretation of 2. c) In the long run, suppose that the prices of labor and capital remain at $25 and $36, (PL = $25 and and PK = $36). Calculate QED's long run profit maximizing output level, QIR*, and verify that QIR* is a maximum and not a minimum. Calculate the optimal input bundle, (LR*, KUR*), for producing the quantity QIR* . Calculate QED's long run profits at QIR*, II(QIR"), and the price that QED will charge in the long run, PLR*