Macro - HW 3 answers

Macro - HW 3 answers - Answers to Homework #3 We have seen...

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Answers to Homework #3 We have seen that when the population is growing, the equation of motion of capital becomes: ( 29 ( 29 k n k f s k + - = δ , where s is the saving rate, n is the rate of growth of population, and is the depreciation rate. Now suppose that the aggregate production function is given by: ( 29 3 / 2 3 / 1 , L K L K F Y = = , and let y denote output per worker, and k denote capital per worker. a. (10 pts.) Using algebra (show your steps ) and the aggregate production function, show that the production function per worker in this case is given by: ( 29 3 / 1 k k f y = = Ans: To get the production function per worker we need to divide the aggregate production function with the number of workers. Doing so yields: = = = = = - - 3 / 1 3 / 1 3 / 1 3 / 1 3 / 1 1 3 / 2 3 / 1 3 / 2 3 / 1 L K y L K y L K y L K y L L K L Y 3 / 1 k y = b) (10 pts.) Using algebra (show your steps ) and the equation of motion of capital, show that the level of capital per worker at the steady-state in this case is given by: 2 / 3 * + = n s k Ans: We know that at the steady state capital per worker is constant so that 0 = k . This is the steady-state condition . According to the equation of motion of capital, the change in capital is the difference between actual investment and break-even investment, so that: ( 29 ( 29 k n k f s k + - = Using the equation of motion and the steady-state condition we get:
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( 29 ( 29 ( 29 + = + = + - = 3 / 1 3 / 1 3 / 1 * * * * * * 0 k k n s k n k s k n k s δ ( 29 ( 29 = + + = + = - 3 / 2 3 / 2 3 / 1 1 * * * k n s k n s k n s ( 29 2 / 3 2 / 3 3 / 2 2 / 3 * * + = = + n s k k n s . c. (10 pts.)
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Macro - HW 3 answers - Answers to Homework #3 We have seen...

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