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UISKOFIIOHS Question 1: Expanding the Span of Control Consider the undistorted model with a continuum of firms i 6 [0,1] where Y,- = A}\"'L\" and the labor supply (and the number of firms) is endogenously determined by free entry into entrepreneurship. The parameter 0 represents the "span of controlll of the manager of the rm (how many workers a manager can manage effectively). Suppose you created a new piece of software that helped entrepreneurs manager their workers, causing an infinitesimal increase in a'. 3. Calculate the impact on the number of firms in the economy. Would the number of firms increase or decrease? Hint: infinitesimai' increase = partiai derivative in. What is the intuition for your answer to part a? Question 2: Span~ofcontro| with Capital Suppose we add capital to the model: 1'; = xii\"(K'ILI'f'T' for each firm i E [0,1]. Assume firms hire workers at wage w and rent capital at price R. Assume Ag ~ U[O,1] a. Derive the profit of rm i as a function of managerial ability, wages, and rental prices 1 . b. Suppose total capital and total labor are fixed (no free entry into entrepreneurship): )0 Kg d: = K and J: Lidi = L. Derive an expression for Y =f01 1'} (if as a function of K and L. That is, derive the aggregate production function. c. Suppose total capital and labor are NOT fixed, but instead that there is free entry into entrepreneurship. If they are a worker, they supply 1 infinitesimal unit of capital to the entrepreneurs in return for rental price R, and 1 infinitesimal unit of labor in return for the wage w. Write down the three equations you would have to solve to find the number of workers, the equilibrium wage, and the equilibrium rental price. (You do NOT have to solve the equations.)