6.5ExercisesExercise 6.1 (Technological Progress and Long-Run Growth). Consider aSolow economy with population growth and technological progress. Theevolution of the capital stock per efficiency unit of labor, denoted k1, is givenby the law of motion(1+n)(1+ g)kt+1 = (1 – 8)kt +of(kt).Capital per efficiency unit of labor is defined as k, = Kt/(LEt), where Ktdenotes the stock of physical capital, L, denotes population, and E is a tech-nological factor. Population grows at the rate n and the technological factorgrows at the rate g. The subscript t denotes time, measured in years. Theparameters d E (0, 1) and o > 0 denote, respectively, the depreciation rateof capital and the savings rate. The function f(k;) represents the produc-tion technology. Specifically, let Y, denote output and yt = Yt/(LEt) denoteoutput per efficiency unit of labor. Then yt = f(kt). Assume thatf(kt) = /k.1. Find the steady-state stock of capital per efficiency unit of labor, de-noted k*, as a function of the parameters n, g, 8, and o.2. Suppose that population grows at 2 percent per year and that thetechnological factor grows at 1.5 percent per year. Assume further Intermediate Macroeconomics, Chapter 6 d169that the depreciation rate is 10 percent and that the saving rate is 25percent. Find the steady-state level of output per efficiency unit oflabor, denoted y*. For the remaining questions of this exercise, assumethat yt = y*.3. How many years does it take for output to double?4. How many years does it take for output per capita to double?5. How many years does it take for the stock of capital to double?

Note: Since the question holds more than four subparts we are restricted to answer only first three…

6.5 Exercises Exercise 6.1 (Technological Progress and Long-Run Growth). Consider a Solow economy with population growth and technological progress. e evolution of the capital stock per efficiency unit of labor, denoted kt, is given by the law of motion (1+n)(1+g)kt+1 = (1 – 8)kt +of(kt). Capital per efficiency unit of labor is defined as k, = K/(LE:), where Kt denotes the stock of physical capital, L, denotes population, and Et is a tech- nological factor. Population grows at the rate n and the technological factor grows at the rate g. The subscript t denotes time, measured in years. The parameters o E (0, 1) and o > 0 denote, respectively, the depreciation rate ivil of capital and the savings rate. The function f(k,) represents the produc- tion technology. Specifically, let Y, denote output and y = Yt/(LEt) denote output per efficiency unit of labor. Then yt = f(kt). Assume that f(k) = Vk. 1. Find the steady-state stock of capital per efficiency unit of labor, de- noted k*, as a function of the parameters n, g, 8, and o. 2. Suppose that population grows at 2 percent per year and that the technological factor grows at 1.5 percent per year. Assume further

6.5 Exercises Exercise 6.1 (Technological Progress and Long-Run Growth). Consider a Solow economy with population growth and technological progress. e evolution of the capital stock per efficiency unit of labor, denoted kt, is given by the law of motion (1+n)(1+g)kt+1 = (1 – 8)kt +of(kt). Capital per efficiency unit of labor is defined as k, = K/(LE:), where Kt denotes the stock of physical capital, L, denotes population, and Et is a tech- nological factor. Population grows at the rate n and the technological factor grows at the rate g. The subscript t denotes time, measured in years. The parameters o E (0, 1) and o > 0 denote, respectively, the depreciation rate ivil of capital and the savings rate. The function f(k,) represents the produc- tion technology. Specifically, let Y, denote output and y = Yt/(LEt) denote output per efficiency unit of labor. Then yt = f(kt). Assume that f(k) = Vk. 1. Find the steady-state stock of capital per efficiency unit of labor, de- noted k*, as a function of the parameters n, g, 8, and o. 2. Suppose that population grows at 2 percent per year and that the technological factor grows at 1.5 percent per year. Assume further

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Chapter1: Making Economics Decisions

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