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Intermediate Macroeconomics: Homework 5*Do LeeNovember 4, 2020Due Wednesday, November 25th. Please upload your answers to NYU Classes under the Assignmentstab, or send your answers to Do Lee [email protected] 1.Numerical Problem 7, Chapter 6, page 240 of the 8th edition of the textbook, page 238 of the 9th editionof the textbook, page 245 in the 10th edition of the textbook.NOTE: for Numerical Problem 7, part a, assume the tax rate, t, is 0.Both population and the work force grow at the rate of n = 1% per year in a closed economy. Consumptionis C = 0.5(1 − t)Y , where t is the tax rate on income and Y is total output. The per-worker productionfunction is y = 8√k, where y is output per worker and k is the capital–labor ratio. The depreciation rateof capital is d = 9% per year. Suppose for now that there are no government purchases and the tax rateon income is t = 0.a. Find expressions for national saving per worker and the steady-state level of investment per workeras functions of the capital–labor ratio, k. In the steady state, what are the values of the capital–laborratio, output per worker, consumption per worker, and investment per worker?b. Suppose that the government purchases goods each year and pays for these purchases using taxeson income. The government runs a balanced budget in each period and the tax rate on income ist = 0.5. Repeat Part (a) and compare your results.Question 2.Analytical problems 6, Chapter 6, page 241 of the 8th edition of the textbook, pages 239 of the 9th editionof the textbook, page 246 in the 10th edition of the textbook.Suppose that total capital and labor both increase by the same percentage amount, so that the amountof capital per worker, k, doesn’t change. Writing the production function in per-worker terms, y = f(k),requires that this increase in capital and labor must not change the amount of output produced per*Econ-UG 12-001, Intermediate Macroeconomics, Fall 2020, Jess Benhabib.1worker, y. Use the growth accounting equation to show that equal percentage increases in capital andlabor will leave output per worker unaffected only if aK + aN = 1.NOTE: aK and aN are the capital and labor exponents of the Cobb-Douglas production function andrepresents the factor shares. See equation (6.2) of the textbook (8ed, 9ed) for more.Question 3.Analytical problems 7, Chapter 6, page 241 of the 8th edition of the textbook, pages 239 of the 9th editionof the textbook, page 246 in the 10th edition of the textbook.An economy has a per-capita production function y = Akah1−a, where A and a are fixed parameters, yis per-worker output, k is the capital–labor ratio, and h is human capital per worker, a measure of theskills and training of the average worker. The production function implies that, for a given capital–laborratio, increases in average human capital raise output per worker.The economy’s saving rate is s, and all saving is used to create physical capital, which depreciates atrate d. Workers acquire skills on the job by working with capital; the more capital with which theyhave to work, the more skills they acquire. We capture this idea by assuming that human capital perworker is always proportional to the amount of physical capital per worker, or h = Bk, where B is a fixedparameter.Find the long-run growth rates of physical capital, human capital, and output in this economy.2
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