401exam2answers-10 - ECON 401 Autumn 2010 SECOND...

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ECON 401 Hartman Autumn 2010 SECOND EXAMINATION with answers 1. Consider a Solow-Swan model with labor augmenting technical change. The savings rate, s , is fixed. The production function is 1/2 1/2 8( ) YKA L where Y is output, K is the capital stock, L is the labor input, and A is the level of the labor augmenting technical change parameter. Capital accumulates according to K I K   where I is gross investment and is the depreciation rate. The population grows at the constant proportional rate n , and each person provides one unit of labor services so that / LL n . Labor augmenting technical change occurs at the rate g so that / AA g . There is no government or international trade, and therefore we must have YCI  . a. Let / ( ) kKA L denote capital per unit of effective labor, and let / kd t . Derive the equation that gives k in terms of k and the parameters of the model. (To receive full credit, you must show the steps required to derive the equation.) Answer: Note first that 1/2 1/2 1/2 88 YK A L y k AL AL AL     . Since saving must equal investment in this model, we must have Y C sY K K which, after dividing through by AL, becomes sY K K AL AL AL or 1/2 8 K sk k AL  . Now, 2 () KAL LA AL K K L A K K kn g k AL AL L A AL AL   so that K g k AL   , and therefore 1/2 ) k n gg k so that 1/2 ) ks k ng k  . b. Suppose that 0.03 n , 0.12  , 0.05 g , and 0.25 s . What is the steady state value of /( ) L ? What is output per unit of effective labor in the steady state? What is consumption per unit of effective labor in the steady state? Answer: Let 0 k and substitute for n, g, , and s to see that the steady state value of k satisfies 1/2 20 . 2 0 kk . It follows that the steady state value of k is *2 2 (2 / 0.2) (10) 100 k . Output per unit of effective labor is ** 1 / 2 1 / 2 8( ) 8(100) 80 yk and consumption per unit of effective labor is * 1 / 2 1 / 2 (1 ) (.75)8( ) (.75)8(100) 60 cs y k  . c. Does output per person grow in the long run? If so, at what rate? Answer: Yes, Y/L does grow in the long run. It grows at the same rate as A, which is 0.05 or 5% per period. 2. Suppose the production function is 1/2 1/2 8 L where Y is aggregate output, K is the capital stock, and L is the input of labor. Capital evolves according to K I K   where I denotes gross investment and 0.08 . The population grows at the constant proportional rate of 2% per period, and each person provides one unit of labor services so that / 0.02 LL . There is no government or international trade, and therefore we must have . Denote aggregate consumption by C , and consumption per person by / cCL . There is no technical change.
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a. It is convenient to work with capital per person, / kKL
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401exam2answers-10 - ECON 401 Autumn 2010 SECOND...

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