# problem set 3 answers - Economics 100B UCSC Abhijit Sen...

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Economics 100B Abhijit Sen Gupta U C S C S u m m e r 2 0 0 6 Answer Key to Problem Set 3 1) Problem 4, Chapter 10 of Blanchard a) 81 49 9 7 63 YK N == = × = b) So capital doubles to 162 and labor doubles to 98 ' 2 2 162 98 9 2 7 2 63 2 126 2 N Y = × = × = = c) Since by doubling the labor and capital we obtained twice the original output the production function exhibits constant returns to scale. (Try when you reduce capital and labor by half) d) N = Divide both sides by N 1 2 N NN N N = ⇒= ⎛⎞ ⎜⎟ ⎝⎠ e) Let K/N = 4. Then Y/N is given as () 1 2 1 2 42 Y N = = When K/N increases to 8, Y/N is given as 1 2 1 2 8 2.828 Y N = = f) The relationship between output per worker and capital per worker does not exhibit constant returns to scale. As we double the capital per worker from 4 to 8, output per worker rises but does not double.

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g) The answer is not the same because in part (c) we doubled both capital and labor and therefore the output doubled. In part (f) we have doubled only capital. Thus while part (c) tells us that the production function exhibits constant returns to scale, part (f) tells us that the production function exhibits diminishing returns to capital. h) Yes the graph looks exactly like figure 10.5. As we increase capital keeping labor constant thereby increasing capital per worker, output and hence output per worker increases but by a smaller proportion. 2) Problem 6, Chapter 11 of Blanchard The production function is given by 12 33 YKN = a) The production function does exhibit constant returns to scale. Lets increase (decrease) both inputs by a constant x. Then the new output is given as () ' ' ' ' Yx K x N K x N KN Y = ⇒= Thus the output increases (decreases) by the same amount. b) To see if there is decreasing returns or not we need to look at the second derivative 22 52 2 2 1 0 3 21 2 0 9 Y K Y K δ −− = => =− × =− < Since the first derivative is greater than zero it means that as we increase capital holding labor constant output increases. However, since the second derivative is negative it means that an increase in capital increases output but at a decreasing rate.
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## This note was uploaded on 06/26/2008 for the course ECON 100B taught by Professor Yisun during the Summer '07 term at UCSC.

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problem set 3 answers - Economics 100B UCSC Abhijit Sen...

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