Macroeconomics
Macroeconomics
4th Edition
ISBN: 9780393602487
Author: Jones, Charles I.
Publisher: W. W. Norton & Company
Question
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Chapter 6, Problem 9E

a)

To determine

Economic interpretation for the equation.

a)

Expert Solution
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Explanation of Solution

If the output elasticity with respect to idea is ½, it implies that the percentage point increase of idea will lead to a 0.5 percentage increase in the output level. Also, the discovery of new ideas will lead to a growth rate in idea to diminish. It can be derived as follows:

Yt=At(12)LytYtAt=12At(12)LytYtAt=14At(32)Lyt<0

b)

To determine

Growth rate of knowledge in the economy.

b)

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Explanation of Solution

The growth rate of knowledge implies the growth rate of idea in the economy. Then, derive the growth rate of idea as follows:

 At+1=z¯AtLatΔAt+1=z¯Atl¯L¯ΔAt+1At=z¯l¯L¯Growth rate of idea(ga)=z¯l¯L¯

c)

To determine

Growth rate of output per person in the economy.

c)

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Explanation of Solution

Derive the growth rate of output per person in the economy as follows:

Lyt=Lat=L¯Lyt=l¯L¯=L¯Lyt=(1l¯)=L¯Yt=At12LytYt=At12(1l¯)L¯YtL¯=At12(1l¯)

Where, Yt=YtL¯

Yt=At12(1l¯)

Take logs and rewrite the equation as follows:

lnYt=12lnAt+ln(1l¯)

Suppose ln(1l¯) is a constant term, with respect to the time, differentiate it as follows:

ΔYtYt=12ΔAtAt+0gy=12gagy=12z¯l¯L¯

Therefore, the growth rate of output per person is calculated as follows:

gy=12ga=z¯l¯L¯2

d)

To determine

Output per person at each point in time.

d)

Expert Solution
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Explanation of Solution

Suppose,

ga=z¯l¯L¯

The stock of knowledge can be equated as follows:

At=A0(1+ga)t

The production function per worker can be derived as follows:

Yt=At12(1l¯)Yt=[A0(1+ga)t]12(1l¯)Yt=A012(1+ga)t2(1l¯)

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