Econ 154a Problem Set 5 Solution

Econ 154a Problem Set 5 Solution - Problem Set 5 -...

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Problem Set 5 - Solutions Problem 1 Solow Model with per worker production function : y = Af ( k ) . given by: y = Ak α , if k < 1 y = λ Ak α if k > =1 s=0.1 ; A=1 ; n=0.05 ; d=0.1 ; α =0 . 3; λ =2 . 1. Graph Solow Model - Problem 1 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Investment (n+d)k Figure 1: The second curve, ( n + d ) k gives the amount of investment needed to keep the capital per worker constan for the following reasons: f rst, n is the population growth. So, in each period the number of workers increases at rate n and for the ratio K/N be kept constant it’s necessary that capital also increases at least at rate n also. Second, there is depreciation in this economy. This means that in each period a fraction of the current capital is lost, disappears. Therefore, this lost capital has to be replaced in order to keep the level of capital per worker constant. 1

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2. First, let’s identify the points in which the two curves plotted in the graph cross, i.e., the steady-states. Note that this occurs when: sAf ( k ss )=( n + d ) k ss Substituting the production function, we have that: sA ( k ss ) α =( n + d ) k ss if k < 1 s λ A ( k ss ) α n + d ) k ss if k > =1 Solving for k ss for each case, ( k ss ) α 1 = ( n + d ) sA if k < 1 ( k ss ) α 1 = ( n + d ) λ sA if k > =1 Finally, k ss ( n + d ) sA ) 1 α 1 if k < 1 k ss ( n + d ) λ sA ) 1 α 1 if k > =1 Substituting the values for the parameters, given above, we get that: k ss sA ( n + d ) ) 1 1 α if k < 1 k ss s λ A ( n + d ) ) 1 1 α if k > =1 k ss 0 . 1 0 . 15 ) 1 1 0 . 3 =0 . 56 if k < 1 k ss 0 . 2 0 . 15 ) 1 1 0 . 3 =1 . 51 if k > =1 The dynamics of k can be described as follows. Whenever the curve of investment, i.e., sAf ( k ), is higher than ( n + d ) k the level of k is increasing, since there is more investment than necessary to keep the k constant. Then, whenever the curve of investment is lower than ( n + d ) k we know that k is decreasing. As just shown above, there are two values for which the two curves cross, i.e., there are two steady-states in this economy. One at lower levels of k and the other at hogher levels. Following the graph, we see that for values of k< 1and 0 . 56 , k is increasing. When k . 56 the capital per worker is constant characterizing the steady-state. For values of k lower than 1 but higher than 0.56, the capital per worker is decreasing. Using the same reasoning, we get that for values of k higher than 1 but lower than 1.51 the capital per worker increases. When k . 51 , capital per worker is constant and consists of a steady-state. And f nally, for values of k higher than 1.51 capital per worker is decreasing.
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This note was uploaded on 07/18/2008 for the course ECON 154 taught by Professor Bjoernbruegemann during the Fall '07 term at Yale.

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Econ 154a Problem Set 5 Solution - Problem Set 5 -...

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