Solow Growth in Economics

1.  In this chapter, we learn:
 how capital accumulates over time, helping us understand economic growth.
 the role of the diminishing marginal product of capital in explaining differences in
growth rates across countries.
 the principle of transition dynamics: the farther below its steady state a country is, the
faster the country will grow.
 the limitations of capital accumulation, and how it leaves a significant part of
economic growth unexplained.
 The Solow growth model is the starting point to determine why growth differs across
similar countries
 it builds on the Cobb-Douglas production model by adding a theory of capital
accumulation
 developed in the mid-1950s by Robert Solow of MIT, it is the basis for the Nobel Prize
he received in 1987
 capital stock is “endogenized”: converted from an exogenous to an endogenous
variable.
 the accumulation of capital is a possible engine of long-run economic growth

2.  Start with the Cobb-Douglas production model and add an equation describing the
accumulation of capital over time.
Yt = F(Kt ,Lt ) = A Kt
1/3Lt
2/3
Production
 The production function:
 has constant returns to scale in capital and labor
 has an exponent of one-third on capital  decreasing returns to capital
 Variables are time subscripted—they may potentially change over time
 Output can be used for either consumption (Ct) or investment (It): Yt = Ct + It
Capital Accumulation
 the capital stock next year equals the sum of the capital started with this year plus the
amount of investment undertaken this year minus depreciation
 Depreciation is the amount of capital that wears out each period ~ 10 percent/year

3. Labor
 the amount of labor in the economy is given exogenously at a constant level, L
Investment
 the amount of investment in the economy is equal to a constant investment rate,
s, times total output, Y
It = s Yt
 Total output is used for either consumption or investment
 therefore, consumption equals output times the quantity one minus the investment
rate
Ct = (1 - s) Yt

4. The Model Summarized

5. Prices and the Real Interest Rate
 If we add equations for the wage and rental price, the MPL and the MPK would pin
them down:
w = MPL, r = MPK
 the real interest rate, r, (measured in constant dollars, not in nominal dollars)
= the amount a person can earn by saving one unit of output for a year
= the amount a person must pay to borrow one unit of output for a year measured
in constant dollars, not in nominal dollars
 saving is the difference between income and consumption: St = Yt - Ct = It
» Saving equals Investment
 In the long-run when output is at full-employment potential output, a unit of
saving is a unit of investment, which becomes a unit of capital:
 therefore the return on saving must equal the rental price of capital
 the rental price of capital is the real interest rate which is equal to the
marginal product of capital
r = MPK

6.  To solve the model, write the endogenous variables as functions of the parameters of
the model and graphically show what the solution looks like and solve the model in
the long run.
 combine the investment allocation equation with the capital accumulation
equation
 net investment is investment minus depreciation
 substitute the supply of labor, L, into the production function:
 These two equations, the capital accumulation relation and the production
function, are all we need to solve the Solow model
(change in capital) (net investment)
Solving the Solow Model

7. Investment,
Depreciation
Capital, Kt
At this point,
dKt = sYt, so
The Solow Diagram graphs the production function and the capital
accumulation relation together, with Kt on the x-axis:

8. The Solow Diagram:
When investment is greater than depreciation, the capital stock increases
The capital stock rises until investment equals depreciation:
At this steady state point, ΔK = 0
Depreciation: d K
Investment: s Y
Investment, depreciation
Capital, K
K*
Net investment
K0

9. Suppose the economy starts at K0:
Investment,
Depreciation
Capital, Kt
K0
•The red line is above the
green at K0:
•Saving = investment is greater
than depreciation at K0
•So ∆Kt > 0 because
•Since ∆Kt > 0, Kt increases
from K0 to K1 > K0
K1

10. Now imagine if we start at a K0 here:
Investment,
Depreciation
Capital, Kt
K0
•At K0, the green line is above the
red line
•Saving = investment is now less
than depreciation
•So ∆Kt < 0 because
•Then since ∆Kt < 0,
Kt decreases from K0 to K1 < K0
K1

11. We call this the process of transition dynamics:
Transitioning from any Kt toward the economy’s
steady-state K*, where ∆Kt = 0 and growth ceases
Investment,
Depreciation
Capital, Kt
At this value of K,
dKt = sYt, so
K*
No matter where
we start, we’ll
transition to K*!

12. We can see what happens to output, Y, and thus to growth if we
rescale the vertical axis:
Investment,
Depreciation, Income
Capital, Kt
K*
• Saving = investment and
depreciation now appear
here
• Now output can be
graphed in the space
above in the graph
• We still have transition
dynamics toward K*
• So we also have
dynamics toward a
steady-state level of
income, Y*
Y*

13. The Solow Diagram with Output
At any point, Consumption is the difference between Output and
Investment: C = Y – I
Investment, depreciation,
and output
Capital, K
Y0
K0
Y*
K*
Consumption
Depreciation: d K
Investment: s Y
Output: Y

14.  in the steady state, investment equals depreciation. If we evaluate this equation at the
steady-state level of capital, we can solve mathematically for it
 the steady-state level of capital is positively related with the investment rate, the
size of the workforce and the productivity of the economy
 the steady-state level of capital is negatively related to the depreciation rate
Solving Mathematically for the Steady State
 In the steady state:

15.  Once we know K*, then we can find Y* using the production function:
•This solution also tells us about per capita income in the steady state, y*, and per capita
consumption as well, c*
c* = y* - sy* = (1 – s) y*
 notice that the exponent on the
productivity parameter is greater
than in the production function
 higher productivity
parameter raises output as in
the production model.
 higher productivity also
implies the economy
accumulates additional
capital.
 the level of the capital stock
itself depends on
productivity

16. The Capital-Output Ratio
 the capital to output ratio is given by the ratio of the investment
rate to the depreciation rate:
 while investment rates vary across countries, it is assumed that the
depreciation rate is relatively constant
Looking at Data through the Lens of the Solow Model

17. Empirically, countries with higher investment
rates have higher capital to output ratios:

18. Differences in Y/L
 the Solow model gives more weight to TFP in explaining per capita output
than the production model does
 Just like we did before with the simple model of production, we can use this
formula to understand why some countries are so much richer
 take the ratio of y* for a rich country to y* for a poor country, and assume
the depreciation rate is the same across countries:
 Now we find that the factor of 45 that separates rich and poor country’s
income per capita is decomposable into:
 A 103/2 = 18-fold difference in this productivity ratio term
 A (30/5)1/2 = 61/2 = 2.5-fold difference in this investment rate ratio
 In the Solow Model, productivity accounts for 18/20 = 90% of differences!
45 = 18 x 2.5

19.  the economy will settle in a steady state because the investment curve, sY, has
diminishing returns
 however, the rate at which production and investment rise is smaller as the
capital stock is larger
 a constant fraction of the capital stock depreciates every period, which implies
depreciation is not diminishing as capital increases
 In summary, as capital increases, diminishing returns implies that production and
investment increase by less and less, but depreciation increases by the same amount
d.
 Eventually, net investment is zero and the economy rests in steady state.
 There are diminishing returns to capital: less Yt per additional Kt
 That means new investment is also diminishing: less sYt = It
 But depreciation is NOT diminishing; it’s a constant share of Kt
Understanding the Steady State

20.  there is no long-run economic growth in the Solow
model
 in the steady state: output, capital, output per person,
and consumption per person are all constant and
growth stops
both constant
Economic Growth in the Solow Model

21.  empirically, economies appear to continue to
grow over time
thus capital accumulation is not the engine of
long-run economic growth
 saving and investment are beneficial in the short-
run, but diminishing returns to capital do not
sustain long-run growth
 in other words, after we reach the steady state,
there is no long-run growth in Yt (unless Lt or A
increases)

22. Investment, depreciation
Capital, K
New investment
exceeds depreciation
Depreciation: d K
K**
K*
Old investment: s Y
An Increase in the Investment Rate

23.  the economy is now below its new steady state and
the capital stock and output will increase over time by
transition dynamics
 the long run, steady-state capital and steady-state
output are higher
 What happens to output in response to this increase in
the investment rate?
the rise in investment leads capital to accumulate over
time
this higher capital causes output to rise as well
output increases from its initial steady-state level Y*
to the new steady state Y**

24. The Behavior of Output
Following an Increase in s
Investment, depreciation, and output
Capital, K
Y*
K*
Y**
K**
Depreciation: d K
Output: Y
New
investment:
s ‘Y
Old
investment:
s Y
(a) The Solow diagram with output.

25. The Behavior of Output
Following an Increase in s (cont.)
(b) Output over time.
Y*
Y**
Output, Y
(ratio scale)
Time, t
2000 2020 2040 2060 2080 2100

26. A Rise in the Depreciation Rate
 the depreciation rate is exogenously shocked to a higher
rate
 the depreciation curve rotates upward and the investment
curve remains unchanged
 the new steady state is located to the left: this means that
depreciation exceeds investment
 the capital stock declines by transition dynamics until it
reaches the new steady state
 note that output declines rapidly at first but less rapidly as it
converges to the new steady state

27. A Rise in the Depreciation Rate
Investment, depreciation
Capital, K
Depreciation
exceeds
investment
K** K*
New
depreciation:
d ‘K
Old
depreciation:
d K
Investment: s Y

28.  What happens to output in response to this increase
in the depreciation rate?
the decline in capital reduces output
output declines rapidly at first, and then gradually
settles down at its new, lower steady-state level Y**

29. The Behavior of Output
Following an Increase in d
Investment, depreciation,
and output
Capital, K
New depreciation: d‘K Output: Y
Investment:
s Y
Old depreciation: d K
Y**
K**
Y*
K*
(a) The Solow diagram with output.

30. The Behavior of Output
Following an Increase in d (cont.)
Output, Y
(ratio scale)
Time, t
2000 2020 2040 2060 2080 2100
Y**
Y*
(b) Output over time.

31. The Principle of Transition Dynamics
 when the depreciation rate and the investment rate
were shocked, output was plotted over time on a ratio
scale
ratio scale allows us to see that output changes more
rapidly the further we are from the steady state
as the steady state is approached, growth shrinks to
zero

32. CHAPTER 5 The Solow Growth Model
 the principle of transition dynamics says that the
farther below its steady state an economy is, in
percentage terms, the faster the economy will grow
 similarly, the farther above its steady state, in
percentage terms, the slower the economy will grow
 this principle allows us to understand why economies
may grow at different rates at the same time

33. Understanding Differences in Growth
Rates
 empirically, OECD countries that were relatively poor in
1960 grew quickly while countries that were relatively rich
grew slower
 if the OECD countries have the same steady state, then the
principle of transition dynamics predicts this
 looking at the world as whole, on average, rich and poor
countries grow at the same rate
 two implications: (1) most countries have already reached
their steady states; and (2) countries are poor not because
of a bad shock, but because they have parameters that yield
a lower steady state

36. Strengths and Weaknesses of the Solow Model
 The strengths of the Solow model are:
1. It provides a theory that determines how rich a country is
in the long run.
2. The principle of transition dynamics allows for an
understanding of differences in growth rates across
countries.
 The weaknesses of the Solow model are:
1. It focuses on investment and capital, while the much more
important factor of TFP is still unexplained.
2. It does not explain why different countries have different
investment and productivity rates.
3. The model does not provide a theory of sustained long-run
economic growth.

37. Summary
1. The starting point for the Solow model is the
production model of Chapter 4. To that framework,
the Solow model adds a theory of capital
accumulation. That is, it makes the capital stock an
endogenous variable.
2. The capital stock is the sum of past investments.
The capital stock today consists of machines and
buildings that were bought over the last several
decades.

38. 3. The goal of the Solow model is to deepen our
understanding of economic growth, but in this it’s
only partially successful. The fact that capital runs
into diminishing returns means that the model does
not lead to sustained economic growth. As the
economy accumulates more capital, depreciation
rises one-for-one, but output and therefore
investment rise less than one-for- one because of the
diminishing marginal product of capital. Eventually,
the new investment is only just sufficient to offset
depreciation, and the capital stock ceases to grow.
Output stops growing as well, and the economy
settles down to a steady state.

39. 4. There are two major accomplishments of the Solow
model. First, it provides a successful theory of the
determination of capital, by predicting that the
capital-output ratio is equal to the investment-
depreciation ratio. Countries with high investment
rates should thus have high capital-output ratios,
and this prediction holds up well in the data.

40. 5. The second major accomplishment of the Solow
model is the principle of transition dynamics, which
states that the farther below its steady state an
economy is, the faster it will grow. While the model
cannot explain long-run growth, the principle of
transition dynamics provides a nice theory of
differences in growth rates across countries.
Increases in the investment rate or total factor
productivity can increase a country’s steady-state
position and therefore increase growth, at least for a
number of years. These changes can be analyzed
with the help of the Solow diagram.

41. CHAPTER 5 The Solow Growth Model
6. In general, most poor countries have low TFP levels
and low investment rates, the two key determinants
of steady-state incomes. If a country maintained
good fundamentals but was poor because it had
received a bad shock, we would see it grow rapidly,
according to the principle of transition dynamics.

42. Investment in South Korea
and the Philippines, 1950-2000
Investment rate (percent)
Year
U.S.
South Korea
Philippines

43. The Solow Diagram
Investment, depreciation,
and output
Capital, K
Y0
K0
Y*
K*
Output: Y
Depreciation: d K
Investment: s Y

44. Output Over Time, 2000-2100
Output, Y
(ratio scale)
Time, t
2000 2020 2040 2060 2080 2100
Y*
Y0