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Unformatted text preview: Economics 1011b Problem Set 3 Professor Aleh Tsyvinski The problem set is due next Tuesday, March 13, by 5 pm in your TF’s mailbox in Littauer. Late problem sets will not be accepted. You can work in groups (discuss the solutions, etc). However, you must write the solutions by yourself. Please write down all derivations/explanations. For clarifications (not answers) please contact Leon Berkelmans ([email protected]). Exercise 1. Solow Growth Model Suppose that the aggregate production function of the economy is Y ( t ) = F ( K,L ) and follows constant returns to scale, and we are in a continuous time world. Suppose that the population of the economy grows at rate n and people save a constant share of income, s that then gets invested. There is no technical change. Capital depreciates at rate δ . Let lower case letters denote per capita variables, and dots above a variable indicate time derivatives. 1. Show that marginal product of capital is just a function of the capital to labor ratio, which we will denote k . Solution: Y = F ( K,L ) = LF ( K L , 1) Therefore ∂Y ∂K = L 1 L F 1 ( K L , 1) = F 1 ( k, 1) where the above line was done by the chain rule and F 1 indicates the derivative of F with respect to its first argument. 2. Denote the function from part 1 r(k). Suppose that lim k → r ( k ) = ∞ , lim k →∞ r ( k ) = 0 and r ( k ) < 0. Write the expression for the amount0....
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