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Problem 3 (25 pts) Firm F uses two inputs with quantities 3:1 and 3:2 and prices 1.01 and 102 to produce output y with price p per unit, Via the production function y = f($1, $2) = $1 ($02 + 4502 (332)2 (a) Assume that both inputs are variable and write down the rm's long-run prot maxi- mization problem. Let p = 2 from now on. (b) Solve the rm's problem in part (a) for the prot-maximizing input demands as func- tions of 101 and 1U2. Show that the quantity demanded of each input is decreasing in the input's price. (0) Show that pM P1 = 101 where M P1 is the marginal product of input 1. Explain the economics meaning of this condition. ((1) Suppose now that input 2 is xed at 3:2 = 1 and the input prices are W1 = 1 and 102 = 2. Write down and solve the rm's short-run prot maximization problem. For these input prices, are the rm's long-run prots (part b) larger or smaller than its short-run prots? Explain why