# ps5 - Economics 202 Principles of Macroeconomics Name...

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Economics 202 Name: _______________ Principles of Macroeconomics Professor Melick Problem Set #5 Due Monday February 18, 2008 This problem set is meant to provide you with a solid understanding of the Solow growth model as presented in Chapter 6 of Abel and Bernanke (2005). The first part of this problem set develops the equations that characterize steady state equilibrium while the second part deepens your understanding of the model through an Excel example. The last section sheds some light on the optimal or “Golden Rule” steady state equilibrium. I. Theory - Simple Solow model We start in the simplest possible setting – there is no government and the economy is closed. In this case, the two main building blocks of the Solow model are the equilibrium condition d d d d d d t t t t t t t t S I or Y C I or Y C I = - = = + and the capital accumulation identity K K I d K t t t t + + - 1 The first equation tells us that the economy is in equilibrium in period t if desired savings is equal to desired investment. The capital accumulation identity simply states that next year’s capital stock must be equal to this year’s capital stock plus any investment over this year minus depreciation over the year. As discussed in class, the equilibrium solution for the model must be done in per worker terms, so we define the following variables t t t t t t Y K y k N N = = 1. If we assume that the labor force ( ) t N grows at rate n and that savings ( ) t S is just a constant fraction s of income ( ) t Y , use the space below to show that t t I S = implies that ( ) ( ) t t t k d k n y s - + + = + 1 1 1 .

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The economy reaches a steady state when capital per worker and output per worker are constant, that is when y y y t t = = + 1 and k k k t t = = + 1 . We denote steady state equilibrium values as * y and * k . 2. Use the space below and your answer to question 1 to show that the steady-state equilibrium condition in the Solow model is written as ( ) * * k d n y s + = To make any numerical, as opposed to graphical, headway we must assume a functional form for the production function. As you might imagine, we turn to our old friend the Cobb-Douglas function α α - = 1 t t t N K A Y . 3. In the space below, put the production function into per worker terms, showing that it can be written as ( ) α t t k A y = . 4. In the space below, use the per worker production function to substitute for per worker output in ( ) * * k d n y s + = to show that the equilibrium, steady state, capital per worker can be written as α - + = 1 1 * d n s A k .
5. Given the expression for steady state capital per worker, use the space below to show that steady state output per worker can be written as α α - + = 1 * d n s A A y . II. Spreadsheet Example - Simple Solow model We can gain more intuition about the Solow model by constructing a spreadsheet version of the model and demonstrating most of the points made in Chapter 6. First we will need to supply values for the parameters of the model ( s n d A , , , , α ). NOTE: Throughout, we will use data for 2006 , since

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