MACROECONOMICS+ACHIEVE 1-TERM AC (LL)
MACROECONOMICS+ACHIEVE 1-TERM AC (LL)
10th Edition
ISBN: 9781319467203
Author: Mankiw
Publisher: MAC HIGHER
Question
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Chapter 9, Problem 5PA

(a)

To determine

The per worker production function.

(a)

Expert Solution
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Explanation of Solution

The production function is given using Equation (1) as follows:

F(K,L)=AKαL1α (1)

 According to the Solow growth model with technological progress, the output per worker is y and the capital per worker is k.

 According to the Solow growth model, the output per worker can be calculated using Equation (3) as follows:

Output per worker=Total outputNumber of workers (2)

The output per worker can be calculated by substituting the respective values in Equation (2) as follows:

Output per worker=F(K,L)L=AKαL1αL=A(KL)αy=Akα

Thus, the output per worker is given as Akα.

(b)

To determine

The ratio of steady state income in Country R to Country P.

(b)

Expert Solution
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Explanation of Solution

In the steady state, the condition for the steady state value of income is given as follows:

sy=(δ+n+g)k (3)

Substitute the value of y in Equation (3) to obtain the steady state value of capital per worker as follows:

sAkα=(δ+n+g)kkkα=sAδ+n+gk1α=sAδ+n+gk=(sAδ+n+g)11α

Thus, the steady state value of capital per worker is k=(sAδ+n+g)11α.

The steady state value of output per worker can be calculated as follows:

y=A(sAδ+n+g)α1α=A11α(sδ+n+g)α1α

Thus, the steady state value of capital per worker is y=A11α(sδ+n+g)α1α.

Given that the saving rate in Country R is SR=0.32 and in Country P is SP=0.1. The rate of population growth in Country R is nR=0.01 and in Country P is nP=0.03. The rate of technological progress, g=0.02 and the rate of depreciation is 0.05.

The ratio of steady-state income per worker in Country R to the Country P can be calculated as follows:

yRyP=((SRδ+nR+g)(SPδ+nP+g))α1α=((0.320.05+0.01+0.02)(0.10.05+0.03+0.02))α1α=4α1α

Thus, the ratio of steady-state income per worker in Country R to the Country P is as follows:

4α1α.

(c)

To determine

Comparison of income in the two countries.

(c)

Expert Solution
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Explanation of Solution

 We know that the ratio of steady state income in both countries is given as follows:

 yRyP=4α1α (4)

 When the value of α=13, the ratio can be obtained as follows:

 yRyP=413(113)=412=2

 Thus, the income per worker in Country R is two times higher than the income per worker is Country P.

(d)

To determine

The difference in income per worker in both countries.

(d)

Expert Solution
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Explanation of Solution

 It is given that the steady state income in Country R is 16 times greater than the income of Country P.

 yRyP=4α1α16=4α1α(4)2=4α1α2=α1α22α=α2=3αα=23

 Thus, the value of the capital’s share of income should be 2/3. This could be due to the reason that capital includes both human capital and physical capital. There is also a possibility that the factor productivity of both countries would be different largely due to the difference in the values of other parameters which is assumed to be constant in the given situation. This is the reason for the difference in the income gap in the two countries.

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The country of Swan is very wealthy, with a high level of per capita income and capital. The country of Solow is quite poor, with low levels of per capita income and capital. Both countries have the same production function, Y = Af(K, L), and both countries are described by the Solow–Swan growth model. The two countries will converge to the same level of per capita income only if: a. Swan and Solow are able to trade goods and services b. Swan donates a sufficient amount of its income to Solow c. the population growth rate and depreciation rate are the same d. the saving rate, the population growth rate and the depreciation rate are the same
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