4. The growth rates of capital and output. Given the production function: Assume

Question

4. The growth rates of capital and output. Given the production function: Assume that Nis constant and equal to 1. if z rates of z and x. x", then gPa g, where g, and g, are the growth a. Given the growth approximation here, derive the the growth rate of capital. relation between the growth rate of output and b. Suppose we want to achieve output growth equal to 2% per year, what is the required rate of growth of capital? c. In (b), what happens to the ratio of capital to output overtime? d. Is it possible to sustain output growth of 2% forever in this economy? Why or why not? 5. Consider the following statement and explain why you agree or disagree with this statement? "The Solow model shows that the saving rate does not affect the growth rate in the long run, so we should stop worrying about the low U.S. saving rate. Increasing the saving rate wouldn't have any important effects on the economy." a. b. The right to exclude saving from income when paying income taxes leads to a higher output per person. Higher labor productivity allows firms to produce more goods with the same number of workers and thus to sell the goods at the same or even lower prices. That's why increases in labor productivity can permanently reduce the rate of unemployment without causing inflation. c.

Explanation / Answer

a) It is given that z = xa implying that growth of z is a times the growth rate of x.

Functionally,

gz = agx ................... eq i

We have been given the production function as follows:

Y = (KN)0.5 where N is constant and equal to one. Hence N0.5 = 1

Thus,

Y = (KN)0.5 =(K)0.5 .................. eq ii

The above is relationship between output and capital.

Now from eq i (growth approximation) and eq ii ( relationship between output and capital)

gY = 0.5gk

It implies that the growth rate of output (Y) is 0.5 times the growth rate in capital (k).

b) The realtionship between growth rate of output and growth rate of capital, that we derive in part a, is as follows -

gY = 0.5gK

Now to have 2% growth in output (Y) , put gY equals to 2%

gY = 0.5gK

2%= 0.5gK

gK = 2%/0.5 = 4%

Hence to have 2% growth in output it is required to have 4% growth in capital.

c) Overtime with increase in output by 2% the capital will be required to increase by 4% given that capital does not depreciate then the ratio of output to capital will decrease overtime.

d) No, it is not possible to have 2% growth rate forever with this production function because in absence of technological progress and assumption of depreciation of capital there is no way to have perpetual output growth in the long run.

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