Question 1. The Solow Model of Economic Growth 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

QUESTION

Question 1. The Solow Model of Economic Growth
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?

ANSWER

Equations:
(1) Yt = Kt^0.4(AtNt)^0.6: This equation represents the production function in the Solow growth model. It states that the level of output (Yt) at time t is determined by the capital stock (Kt) raised to the power of 0.4, multiplied by the level of technology (At) and the labor force (Nt), both raised to the power of 0.6.

(2) St = sYt: This equation represents the saving function, where the level of savings (St) is a fraction (s) of the output (Yt). The parameter s represents the saving rate, which is the proportion of output saved.

(3) Kt+1 = (1 – δ)Kt + It: This equation represents the capital accumulation equation. It states that the capital stock at time t+1 (Kt+1) is equal to the sum of the current capital stock (Kt) multiplied by the depreciation rate (δ) plus investment (It). Investment represents the portion of output that is allocated to increasing the capital stock.

(4) Nt+1/Nt = 1 + gN, gN = 0.01: This equation represents the population growth rate equation. It states that the ratio of the population at time t+1 (Nt+1) to the population at time t (Nt) is equal to 1 plus the population growth rate (gN). In this case, the population growth rate is 0.01, indicating a 1% annual population growth.

(5) At+1/At = 1 + gA, gA = 0.02: This equation represents the technological change equation. It states that the ratio of the level of technology at time t+1 (At+1) to the level of technology at time t (At) is equal to 1 plus the technological growth rate (gA). In this case, the technological growth rate is 0.02, indicating a 2% annual technological progress.

b) The goods market equilibrium condition for the model is that aggregate output (Yt) equals aggregate consumption (Ct) plus aggregate investment (It). Mathematically, Yt = Ct + It.

c) Combining the goods market equilibrium condition with equations (1) through (3), we can 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. By substituting Yt from equation (1) into the goods market equilibrium condition and rearranging, we get:

It = sYt = sKt^0.4(AtNt)^0.6

From equation (3), we know Kt+1 = (1 – δ)Kt + It. Substituting the expression for It, we have:

Kt+1 = (1 – δ)Kt + sKt^0.4(AtNt)^0.6

This equation describes the change in the capital stock between dates t and t+1 based on the level of investment and the initial capital stock.

d) To express the equation derived in (c) in terms of transformed variables, we divide each variable by AtNt:

Kt+1 / (AtNt) = (1 – δ)(Kt / (AtNt)) + s(Kt / (AtNt))^0.4

The transformed equation describes the change in the capital stock per effective worker between dates t and t+1. It is determined by the ratio of investment per effective worker to the initial capital per effective

worker, and the saving rate per effective worker.

e) In a long-run, steady-state equilibrium of this economy, the variables are not changing over time. The capital stock per effective worker, the output per effective worker, and other relevant variables reach a constant level. The steady state occurs when the capital stock per effective worker does not change.

In the diagram, the horizontal axis represents the capital stock per effective worker (K / (AtN)), and the vertical axis represents output per effective worker (Y / (AtN)). The diagram shows a positively sloped line representing the production function (equation 1), and a flat line representing the savings line (sY / (AtN)). The steady-state equilibrium is where the two lines intersect. At this point, the capital stock per effective worker remains constant, and the economy operates at its potential output level.

The condition on the equation derived in (d) that would measure or capture this steady state is that the change in the capital stock per effective worker between dates t and t+1 is zero. In other words, Kt+1 / (AtNt) – Kt / (AtNt) = 0, indicating that the capital stock per effective worker remains constant.

f) In the steady state equilibrium:
– The growth rate of aggregate output is zero since output per effective worker is constant.
– The growth rate of the aggregate capital stock is zero since the capital stock per effective worker is constant.
– The growth rate of aggregate investment is zero since investment is equal to depreciation.
– The growth rate of aggregate savings is zero since savings are equal to investment.
– The growth rate of output per worker and capital per worker is zero since these variables are constant.
– The growth rate of output per effective labor unit is zero since it is the same as output per worker.
– The growth rate of capital per effective labor unit is zero since it is the same as capital per worker.