Writing Goods Market Equilibrium Condition for given Solow Model

Question1

Consider the following Solow growth model with technological change and population

growth:

𝑌𝑡 = 𝐾𝑡

0.4(𝐴𝑡𝑁𝑡)0.6 (1)

𝑆𝑡 = 𝑠𝑌𝑡, 0 < 𝑠 < 1 (2)

𝐾𝑡+1 = (1 − 𝛿)𝐾𝑡 + 𝐼𝑡 (3)

𝑁𝑡+1

𝑁𝑡

= 1 + 𝑔𝑁, 𝑔𝑁 = 0.01 (4)

𝐴𝑡+1

𝐴𝑡

= 1 + 𝑔𝐴, 𝑔𝐴 = 0.02. (5)

a) Explain in words what each of these equations means or describes.

b) Write down the goods market equilibrium condition for the model.

c) Combine the goods market equilibrium condition with equations (1) through (3) to find

an equation that describes the change in the capital stock between dates t and t+1 in terms

of the levels of inputs to production at date t. Explain in words what determines this change

over time; whether it is positive or negative or zero.

d) Now take each variable in the model and divide it by 𝐴𝑡𝑁𝑡. Use these transformed

variables to re-express the equation you derived in c) as an equation that describes the

change in the capital stock per effective worker between dates t and t+1. Explain in words

what determines this change over time; whether it is positive or negative or zero.

e) Define and describe in words a long-run, steady state equilibrium of this economy.

Depict a long-run, steady state equilibrium in a diagram and label the diagram carefully.

What condition on the equation that you derived in d) would measure or capture this

steady state?

f) In the steady state equilibrium, what will be the numerical values of the growth rates of

aggregate output, the aggregate capital stock, aggregate investment, and aggregate savings?

What will be the numerical values of the growth rate of output per worker, and capital per

worker? What will be the numerical values of the growth rate of output per effective labor

unit and capital per effective labor unit?

g) What would be the qualitative impact of an increase in s for the steady state level of

capital per effective worker and output per effective worker? Show this in a diagram. What

would be the qualitative impact of an increase in s for the steady state growth rates of

output, capital, savings and investment?

h) Now compute the effects for steady state capital and output per effective worker of an

increase in s from 0.1 to 0.2 when the depreciation rate is 0.1. How do these effects differ

quantitatively from the example that we studied in class in which 𝑌𝑡 = 𝐾𝑡

0.5(𝐴𝑡𝑁𝑡)0.5 when

we increased s from 0.1 to 0.2?

i) Compute the value of capital per effective worker over time, for at least 10 years,

resulting from an increase in s from 0.1 to 0.2, starting from an initial steady state at date 0

to a new steady state. Roughly how long do you think it would take to get “very close” to

the new steady state?

j) What is the numerical value of the savings rate, s, that maximizes steady state

consumption per effective worker – the Golden Rule savings rate?

k) What is the quantitative impact for the steady state levels of capital and output per

effective worker of an increase in the growth rate of technological progress from 0.2 to 0.3?

What is the quantitative impact for the steady state growth rates of capital and output per

effective worker of an increase in the growth rate of technological progress from 0.2 to 0.3?

What is the quantitative impact for the steady state growth rate of capital and output per

worker of an increase in the rate of technological progress from 0.2 to 0.3?

l) How could a government raise the rate of technological progress?

 

Question2

Now suppose that the production function is given by

𝑌𝑡 = 𝐾𝑡

0.2(𝐴𝑡𝑁𝑡)0.8. (1’)

How does this affect your answers to Question 1 parts h), i), j), and k)? Does the different

production function change any of your other answers to Question 1?

 

Question3

Assume the Cobb-Douglas production function (1) determines aggregate real GDP.

a) Derive a “growth rate” version of (1) which shows how the growth rate of real GDP

is determined by the growth rates of technology, employment, and capital.

b) Now collect at least ten years of annual data on:

Real GDP (https://fred.stlouisfed.org/series/GDPCA allows download into excel

file)

Employment (https://fred.stlouisfed.org/series/B4701C0A222NBEA

allows download into excel file)

Capital (https://fred.stlouisfed.org/series/RKNANPUSA666NRUG allows download

into excel file).

c) Compute the annual growth rate of each of the data series you downloaded or

otherwise collected in b). Compute the implied annual growth rate of technology at

each date. Now compute the sample average annual growth rates of all four

d) Using the equation you derived in a), and your results in c), what fraction of real

GDP annual average growth over your sample period is accounted for by i) capital

growth, ii) employment growth, and iii) technological change.

 

Answer

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