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Homework 2 GDP Solow Growth Model October 24, 2023 Due date: November 6th 2023 Your homework must be submitted electronically using the course bruinlearn page. Submissions must be in PDF format. In your answers, you should also describe the steps/reasoning leading to your final solutions. Question I – Solow Model without Population or Technology Growth Consider the Solow growth model with no population growth and no technology growth, i.e., n = x = 0. Output is created by a Cobb-Douglas production function combining Labor, L t , and capital, K t , such that output Y t is given by Y t = A t K α t L 1 - α t . Recall that, without population growth, L t = L 0 and assume that L 0 = 1. Furthermore, recall that, without technology growth, A t = A 0 and assume that A 0 =1. The law of motion for capital is given by K t +1 =(1 - δ ) K t + sA t K α t L 1 - α t . (1) Assume that the savings rate is s =0 . 2, the depreciation rate is δ =0 . 1, and that the capital share is α =0 . 3. 1. Use equation (1) to solve for the steady state level of capital, K ss , and show that K ss = h s δ i 1 1 - α . What is the steady state level of capital? (Replace the numbers in the expression) 2. Suppose that K 0 =1. Does capital grow or fall over time? What is the maximum level of capital this economy will ever have? 3. Suppose that K 0 =5. Does capital grow or fall over time? What is the maximum level of capital this economy will ever have? 4. Suppose that this economy was at its steady state level of capital K ss when it is hit by a natural disaster that destroys 10 percent of the capital stock. This catastrophe means that the economy must start anew with K 0 = 0 . 9 × K ss . Use equation (1) to compute K t for the next 10 years. How long does it take to recover at least half of the lost capital stock, i.e., in which year is K t ≥ 0 . 95 × K ss . 5. Suppose that the government conducts a program of incentivizing investment and is able to take s = 0 . 21, i.e., the government increases the savings rate by 1 percentage point. Use the same K 0 as in question 4, and compute the new path K t for the next 10 years. Now, how long does it take to recover at least half of the lost capital stock, i.e., in which year is K t ≥ 0 . 95 × K ss . Is convergence faster or slower with this new savings rate? Explain why. 1

Question II – Solow Model with Population or Technology Growth Consider the Solow growth model with population growth and technology growth. Suppose that x = 0 . 1 and that n =0 . 02, i.e., A t =(1+0 . 1) A t - 1 , and L t =(1+0 . 02) L t - 1 . Output is created by a Cobb-Douglas production function combining Labor, L t , and capital, K t , such that output Y t is given by Y t = A t K α t L 1 - α t . (2) Recall that e ffi cient units of labor are given by ˜ L t = A 1 1 - α t L t . Assume that δ =0 . 1, s =0 . 2, and α =0 . 3. 1. What is the growth rate of ˜ L t , i.e., find ˆ x ≡ ˜ L t ˜ L t - 1 - 1 . 2. Recall that capital per-e ffi cient worker ˜ k t = K t / ˜ L t varies over time according to ˜ k t +1 = (1 - δ ) ˜ k t + s ˜ k α t 1+ˆ x . Use this equation to show that the steady state level of capital per e ffi cient worker is given by ˜ k ss =  s δ +ˆ x bracketrightbigg 1 1 - α . Replace the numbers for s , δ , ˆ x and α to find the value of capital per e ffi cient worker in steady state. 3. Use the production function (2) to show that output per e ffi cient worker, ˜ y t = Y t / ˜ L t , is given by ˜ y t = ˜ k α t . What is output per-e ffi cient worker if capital per-e ffi cient worker is at its steady-state value? 4. Ifcapitalper-e ffi cientworkerisatitssteadystatevalue,doesoutputpere ffi cientworkergrowovertime? Recall that output per worker is given by y t = Y t /L t . Use our definition of output per-e ffi cient worker to replace Y t and write y t = ˜ y t ˜ L t L t . What is the growth rate of output per worker g y = y t y t - 1 - 1 if capital per-e ffi cient worker is at its steady-state value? 5. Suppose that the savings rate in this economy increases to s = 0 . 3. What happens to steady-state level of capital per-e ffi cient worker? Is it higher, lower, the same? 6. Continue assuming that the savings rate in this economy increases to s = 0 . 3. What happens to steady-state level of output per-e ffi cient worker? Is it higher, lower, the same? 7. Recompute the growth rate of output per worker g y with the higher savings rate s = 0 . 3. Is it higher, lower, the same? Use your results to comment on how does the savings rate a ff ect the growth rate of output per capita in this model. 2