Problem Set for Macroeconomic Theory I at University of Southern California, Assignments of Introduction to Macroeconomics

Download Problem Set for Macroeconomic Theory I at University of Southern California and more Assignments Introduction to Macroeconomics in PDF only on Docsity! UNIVERSITY OF SOUTHERN CALIFORNIA Department of Economics ECON 505: Macroeconomic Theory I Fall 2008 Problem Set 2 Due: Wednesday, 09/17/2008 You are encouraged to work in groups on the assignment; however, you must turn in an individual solution. Where applicable, be sure to show your work and briefly explain it in words. Question 1 You can always do the following exercise using EXCEL; however, it will be a very good exercise to be able to do the basics in MATLAB. Use data for output, labor and capital for the U.S for the period 1947-2006.1 Start with a production function that tells us what output tY will be at some particular time t as a function of the economy’s stock of capital tK , its labor force tL , and the economy’s total factor productivity tA . The Cobb- Douglas form of the production function is:   1tttt LKAY Assume that the capital share (α) is 1/3. α) is 1/3. ) is 1/3. a) Generate the series for productivity. b) Generate the following tables for the selected time periods. Please show your calculations. Table 1: Growth Accounting in the United States Average growth rate per year (%) 1947- 1957 1957- 1967 1967- 1977 1977- 1987 1987- 1997 1997- 2006 1947- 2006 Labor share * Labor growth ? ? ? ? ? ? ? Capital share * Capital growth ? ? ? ? ? ? ? Total Input Growth ? ? ? ? ? ? ? Productivity Growth ? ? ? ? ? ? ? Output Growth ? ? ? ? ? ? ? Table 2: Relative Contributions to Growth (%) 1 Finding the data on the BEA and BLS websites is part of the exercise here, so be sure to note your data sources at some point in your answer. If you have trouble finding the data or questions about which data to use, please email Nate. 1 1947- 1957 1957- 1967 1967- 1977 1977- 1987 1987- 1997 1997- 2006 1947-2006 Capital ? ? ? ? ? ? ? Labor ? ? ? ? ? ? ? Productivity ? ? ? ? ? ? ? c) Interpret your results in a single page. Question 2 Consider the production function BLAKY  , where A and B are positive constants. a) Is this production function neoclassical? Which of the neoclassical conditions does it satisfy and which ones do it not? b) Write output per person as a function of capital per person. What is the marginal product of k ? What is the average product of k ? In what follows, we assume that population grows at the constant rate n and that capital depreciates at the constant rate . c) Write down the fundamental equation of the Solow-Swan model. Question 3 Consider an economy with a population growth rate n, with constant returns to scale in production, and in which individuals save a constant fraction s of their income. Assume there is no depreciation of capital. Suppose that the (α) is 1/3. continuous time) aggregate production function is     1, LKLKF a) Show that the differential equation describing the behavior of the capital stock per capita is given by 2 where 2k , denotes the input of capital, 2l , the input of labour. The function f is not necessarily Cobb-Douglas; but it is homogeneous of degree one and must satisfy the additional assumption, which is given by Assumption. For any positive finite 2k and 2l and for any positive finite number x there exists  1,0 such that  22 , lkfx   . 1. Interpret the assumption. 2. Prove the following proposition Proposition. If the population is constant, if all technical progress is ruled out and if the Assumption is satisfied then survival in Solow’s sense is possible if and only if   . Question 6 Consider a two-sector version of the neoclassical growth model, where one sector produces the consumption good ct with technology  ctctctt nkfzc , and the other sector produces the investment good itwith technology  itititt nkfzi , Assume f is CRS, strictly concave and increasing in both argument. Note that f is the same in both sectors. The superscripts c and i denote the consumption sector and the investment sector, respectively. 5 Aggregate capital is it c tt kkk  , the sum of the capital used in the consumption and investment sectors, and it accumulates according to the law of motion ttt ikk  )1(α) is 1/3. 1  . Aggregate labor is i t c tt nnn  , the sum of the labor supplied to the consumption and investment sectors. The representative household has period utility )1,(α) is 1/3. tt ncu  , with time separable preferences and discount factor β. All markets are competitive and both labor and capital are free to move across sectors. Let ctp be the price of the consumption good at time t and let itp be the price of the investment good at time t. Show that this economy aggregates into a one sector growth model, i.e. the production structure can be summarized by an aggregate production technology  tt nkfz , ^ . In equilibrium, what is the interpretation of the ratio        i t c t p p ? 6