Suppose that firm i has the Cobb-Douglas production function Yi = AiL? i (1) where Yi is output, Li is Labor input, Ai > 0 is total factor productivity and ? ? (0, 1) is the output elasticity of labor. Every firm receives the same non-random price P > 0 for each unit Yi and pays a wage of Wi = 1 for each unit of labor Li. a) Suppose that each firm knows Ai, ?, P , and the cost $1 of each unit of labor, and that it chooses a positive value of Li to maximize profits. What is the firm's optimal choice of Li? b) Using 1, show that we can write log(Yi) = ?0 + ?1log(Li) + Ui where Ui is an unobservable variable with E(Ui) = 0 and ?0 is an unknown parameter. c) Suppose that we observe an i.i.d. sample (Yi, Li) for a cross-section of firms i = 1, , N . Assume that each firm is behaving optimally according to the rule derived in part a). Let ?? be the slope coefficient of the regression in a regression of Log(Yi) on log(Li) and a constant. Find the probability limit of ?? d) Is ?? generally consistent for ?? If not, is its probability limit larger or smaller than ?, or is it ambiguous given the provided information? Discuss the intuition behind these findings. e) Assume now that the price Wi that firm i pays for each unit of labor Li varies across firms and is independent of the total factor productivity Ai. Suppose that we observe an i.i.d sample (Yi, Li, Wi) for a cross-section of firms i = 1, , N , and suppose, as in part a), that each firm chooses Li to maximize its profit (knowing P , Ai, ? and Wi). Find a consistent estimator of ? and justify its consistency

(D V (D \\_/ Suppose that we observe an i.i.d. sample (Yg, Li) for a cross-section of rms 2' = 1, - - - ,N. Assume that each rm is behaving optimally according to the rule derived in part a). Let B be the slope coefcient of the regression in a regression of Log(Y;-) on log(L,;) and a constant. Find the probability limit of Is [:3 generally consistent for [3? If not, is its probability limit larger or smaller than [3, or is it ambiguous given the provided information? Discuss the intuition behind these ndings. Assume now that the price Wi that rm i pays for each unit of labor L1- Varies across rms and is independent of the total factor productivity Ai. Suppose that we observe an i.i.d sample (YhLi, W%) for a crosssection of rms 2' = 1, - - - ,N, and suppose, as in part a), that each rm chooses L; to maximize its prot (knowing P, Ai, B and W1). Find a consistent estimator of 5 and justify its consistency. Suppose that firm i has the Cobb-Douglas production function Yi = AiL; (1) where Y; is output, Li is Labor input, A; > 0 is total factor productivity and B E (0, 1) is the output elasticity of labor. Every firm receives the same non-random price P > 0 for each unit Y; and pays a wage of Wi = 1 for each unit of labor Li. a) Suppose that each firm knows Ai, B, P, and the cost $1 of each unit of labor, and that it chooses a positive value of Li to maximize profits. What is the firm's optimal choice of Li? b) Using 1, show that we can write log(Yi) = Bo + Bilog(Li) + Ui where U is an unobservable variable with E(Ui) = 0 and Bo is an unknown parameter