The true production function in the city of Gotham is Y = AKN1-a where K, N...

 

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The true production function in the city of Gotham is Y = AKN1-a where K, N are inputs of capital and hours worked by production workers. Unfortunately, Gotham is plagued by crime, which forces the representative firm in the city to hire y security guards per production worker. Security guards earn the same wage as production workers. The representative firm in Gotham must solve the problem max AK N-a K.N "-wN(1+y)-rK 1. Solve for the profit maximizing choice of the K/N ratio as a function of w,r., o. A. Does a higher y raise or lower K/N for a given set of values for the remaining quantities? Provide intuition. 2. Suppose there is a fixed amount of capital K = 1 and a fixed amount of total labor I that can be allocated either to production or to security work. (a) Solve for the wage rate that equates labor demand and labor supply in equilibrium. To do this, i. Write down the firm's FOC for production labor given that it solves the problem above. ii. Rearrange the FOC to obtain production labor demand by the firm as a function of the wage. What is total labor demand (including security guards) associated with this level of production labor demand? iii. Equate total labor demand with total labor supply, and solve for the wage rate. iv. Your answer should be a function of A.Z.a.7. (b) What is the impact of more extreme crimes (which would require a higher ) on the wages of an average production worker? Provide Intuition. 3. An economist sitting in Boston, Massachusetts tries to estimate TFP A in Gotham. She observes total output Y, total capital K = 1 and total labor I, but mistakenly assumes that all labor is production labor. She tries to estimate TFP using the production function and plugs in the total labor supply I instead of N. What is the economist's estimate of TFP? How does it compare to true TFP? Provide Intuition. 4. Suppose you are told that crime reduces TFP directly by demotivating workers and reducing their effective labor supply. The evidence you are shown is a negative correlation between TFP estimated by the method above and crime rates across countries. Is this evidence sufficient to convince you of the proposed explanation? The true production function in the city of Gotham is Y = AKN1-a where K, N are inputs of capital and hours worked by production workers. Unfortunately, Gotham is plagued by crime, which forces the representative firm in the city to hire y security guards per production worker. Security guards earn the same wage as production workers. The representative firm in Gotham must solve the problem max AK N-a K.N "-wN(1+y)-rK 1. Solve for the profit maximizing choice of the K/N ratio as a function of w,r., o. A. Does a higher y raise or lower K/N for a given set of values for the remaining quantities? Provide intuition. 2. Suppose there is a fixed amount of capital K = 1 and a fixed amount of total labor I that can be allocated either to production or to security work. (a) Solve for the wage rate that equates labor demand and labor supply in equilibrium. To do this, i. Write down the firm's FOC for production labor given that it solves the problem above. ii. Rearrange the FOC to obtain production labor demand by the firm as a function of the wage. What is total labor demand (including security guards) associated with this level of production labor demand? iii. Equate total labor demand with total labor supply, and solve for the wage rate. iv. Your answer should be a function of A.Z.a.7. (b) What is the impact of more extreme crimes (which would require a higher ) on the wages of an average production worker? Provide Intuition. 3. An economist sitting in Boston, Massachusetts tries to estimate TFP A in Gotham. She observes total output Y, total capital K = 1 and total labor I, but mistakenly assumes that all labor is production labor. She tries to estimate TFP using the production function and plugs in the total labor supply I instead of N. What is the economist's estimate of TFP? How does it compare to true TFP? Provide Intuition. 4. Suppose you are told that crime reduces TFP directly by demotivating workers and reducing their effective labor supply. The evidence you are shown is a negative correlation between TFP estimated by the method above and crime rates across countries. Is this evidence sufficient to convince you of the proposed explanation?

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