The Solow Growth
Model
Model Background
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
the accumulation of capital is a possible engine of
long-run economic growth
Building the Model: goods market supply
• We begin with a production function and assume constant
returns.
Y=F(K,L)
so… zY=F(zK,zL)
• By setting z=1/L we create a per worker function.
Y/L=F(K/L,1)
• So, output per worker is a function of capital per worker. We
write this as,
y=f(k)
Building the Model: goods market supply
• The slope of this function
is the marginal product of
capital per worker.
MPK = f(k+1)–f(k)
• It tells us the change in
output per worker that
results when we increase
the capital per worker by
one.
y
MPK =
change in y
change in k
y=f(k)
Change in y
Change in k
k
Building the Model:
goods market demand
• We begin with per worker consumption and investment.
(Government purchases and net exports are not included in the Solow
model). This gives us the following per worker national
income accounting identity.
y = c+I
• Given a savings rate (s) and a consumption rate (1–s) we can generate a consumption function.
c = (1–s)y
…which makes our identity,
y = (1–s)y + I …rearranging,
i = s*y
…so investment per worker
equals savings per worker.
Steady State Equilibrium
• The Solow model long run equilibrium occurs at the
point where both (y) and (k) are constant. These are
the endogenous variables in the model.
• The exogenous variable is (s).
Steady State Equilibrium
• By substituting f(k) for (y), the investment per worker function
(i = s*y) becomes a function of capital per worker
(i = s*f(k)).
• To augment the model we define a depreciation rate (d).
• To see the impact of investment and depreciation on capital
we develop the following (change in capital) formula,
dk = i – dk
…substituting for (i) gives us,
dk = s*f(k) – dk
The Solow Diagram graphs the production function and the capital
accumulation relation together, with Kt on the x-axis:
Investment,
Depreciation
At this point,
dKt = sYt, so
Capital, Kt
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, dK = 0
Investment, depreciation
Depreciation: d K
Investment: s Y
Net investment
K0
K*
Capital, K
Suppose the economy starts at K0:
•The red line is above the
green at K0:
Investment,
Depreciation
•Saving = investment is greater
than depreciation at K0
•So ∆Kt > 0 because
•Since ∆Kt > 0, Kt increases
from K0 to K1 > K0
K0
K1
Capital, Kt
Now imagine if we start at a K0 here:
Investment,
Depreciation
•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 K0
Capital, Kt
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
No matter where we start,
we’ll transition to K*!
At this value of K, dKt = sYt, so
K*
Capital, Kt
Changing the exogenous variable - savings
• We know that steady state is
Investment,
Depreciation
dk
at the point where s*f(k)=dk
s*f(k*)=dk*
s*f(k)
• What happens if we increase
s*f(k*)=dk*
savings?
• This would increase the
slope of our investment
function and cause the
function to shift up.
• This would lead to a higher
steady state level of capital.
• Similarly a lower savings
rate leads to a lower steady
state level of capital.
s*f(k)
k*
k**
k
We can see what happens to output, Y, and thus to growth if we
rescale the vertical axis:
Investment,
Depreciation, Income
• Saving = investment and
depreciation now appear
here
Y*
• 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*
K*
Capital, Kt
The Solow Diagram with Output
At any point, Consumption is the difference between Output and
Investment: C = Y – I
Investment, depreciation,
and output
Output: Y
Y*
Consumption
Depreciation: d K
Y0
Investment: s Y
K0
K*
Capital, K
Conclusion
• The Solow Growth model is a dynamic model that allows
us to see how our endogenous variables capital per worker
and output per worker are affected by the exogenous
variable savings. We also see how parameters such as
depreciation enter the model, and finally the effects that
initial capital allocations have on the time paths toward
equilibrium.
• In the next section we augment this model to include
changes in other exogenous variables; population and
technological growth.
Model Background
• As mentioned, the Solow growth model allows us a dynamic
view of how savings affects the economy over time. We also
learned about the steady state level of capital.
• Now, we assume policy makers can set the savings rate to
determine a steady state level of capital that maximizes
consumption per worker. This is known as the golden rule
level of capital (k*gold)
Building the Model:
• We begin by finding the steady state
consumption per worker.
From the national income accounts
identity,
y = c + i
we get
c=y–i
• We want steady state “c” so we
f(k*),dk*
substitute steady state values for both
output (f(k*)) and investment which
equals depreciation in steady state (dk*)
giving us
c*=f(k*) – dk*
•
Because, consumption per worker is the
difference between output and
investment per worker we want to choose
k* so that this distance is maximized.
•
This is the golden rule level of capital
k*gold
•
dk*
f(k*)
c*gold
k*
k*gold
A condition that characterizes the golden
Below k*gold,
rule level of capital is
increasing k*
MPK = d
increases c*
Above k*gold,
increasing k*
reduces c*
Building the Model:
• While the economy moves
toward a steady state it is
not necessarily the golden
rule steady state.
• Any increase or decrease
f(k*),dk*
dk*
in savings would shift the
sf(k) curve and would
result in a steady state with
a lower level of
consumption.
f(k*)
sgoldf(k*)
sgoldf(k*)
k*
k*gold
To reach the
golden rule
steady state…
The economy
needs the right
savings rate.
The Transition to the Golden Rule Steady State
• Suppose an economy starts
with more capital than in the
golden rule steady state.
•
This causes an immediate
increase in consumption
and an equal decrease in
investment.
Output, y
•
Over time, as the capital
stock falls, output,
Consumption, c
consumption, and investmentInvestment, i
fall.
•
The new steady state has a
higher level of consumption
than the initial steady state.
t0
At t0, the savings
rate is reduced.
Time
The Transition to the Golden Rule Steady State
• Suppose an economy starts
with less capital than in the
golden rule steady state.
•
•
•
This causes an immediate
decrease in consumption
and an equal increase in
investment.
Over time, as the capital
stock grows, output,
consumption, and
investment increase.
Output, y
Consumption, c
The new steady state has a
higher level of consumption
than the initial steady state.
Investment, i
t0
At t0, the savings
rate is increased.
Time
Conclusion
• In this section we used our knowledge that savings
affects the steady state and chose the savings rate to
maximize consumption per worker. This is known as
the golden rule level of capital (k*gold)
• In the next section we augment this model to include
changes in other exogenous variables; population and
technological growth.
Model Background
• As mentioned, the Solow growth model allows us a
dynamic view of how savings affects the economy
over time. We learned about the steady state level of
capital and how a golden rule steady state level of
capital can be achieved by setting the savings rate
to maximize consumption per worker. We now
augment the model to see the effects of population
growth and technological progress.
Steady State Equilibrium
• By expanding our model to include population growth our model
more closely resembles the sustained economic growth
observable in much of the real world.
• To see how population growth affects the steady state we need
to know how it affects the accumulation of capital per worker.
When we add population growth (n) to our model the change in
capital stock per worker becomes…
dk = i – (d+n)k
• As we can see population growth will have a negative effect on
capital stock accumulation. We can think of (d+n)k as breakeven investment or the amount of investment necessary to keep
capital stock per worker constant.
• Our analysis proceeds as in the previous presentations. To see
the impact of investment, depreciation, and population growth
on capital we use the (change in capital) formula from above,
dk = i – (d+n)k …substituting for (i) gives us,
dk = s*f(k) – (d+n)k
Steady State Equilibrium with
population growth
• At the point where both (k)
and (y) are constant it
must be the case that,
dk = s*f(k) – (d+n)k = 0
…or,
s*f(k) = (d+n)k
…this occurs at our
equilibrium point k*.
Like depreciation, population
growth is one reason why the
capital stock per worker shrinks.
Investment
Break-even
Investment
Break-even
investment (d+n)k
s*f(k)
s*f(k*)=(d+n)k*
Investment
k*
At k* break-even
investment equals
investment.
k
The impact of population growth
An increase
in “n”
• Suppose population growth
changes from n1 to n2.
• This shifts the line
Investment
Break-even
growthInvestment
(d+n2)k
representing population
and depreciation upward.
•
•
At the new steady state k2*
capital per worker and output
per worker are lower
The model predicts that
economies with higher rates of
population growth will have
lower levels of capital per
worker and lower levels of
income.
…reduces k*
(d+n1)k
s*f(k)
k 2*
k 1*
k
The efficiency of labour
• We rewrite our production function as…
Y=F(K,L*E)
where “E” is the efficiency of labour. “L*E” is a
measure of the number of effective workers. The
growth of labour efficiency is “g”.
• Our production function y=f(k) becomes output
per effective worker since…
y=Y/(L*E) and k=K/(L*E)
• With this augmentation “dk” is needed to replace
depreciating capital, “nk” is needed to provide
capital to new workers, and “gk” is needed to
provide capital for the new effective workers
created by technological progress.
Steady State Equilibrium with population
growth and technological progress
• At the point where both (k)
and (y) are constant it
must be the case that,
dk = s*f(k) – (d+n+g)k = 0
…or,
s*f(k) = (d+n)k
…this occurs at our
equilibrium point k*.
Like depreciation and population
growth, the labour augmenting
technological progress rate causes the
capital stock per worker to shrink.
Break-even
investment Investment
Break-even
Investment
(d+n+g)k
s*f(k)
s*f(k*)=(d+n)k*
At k* break-even
investment equals
investment.
Investment
k*
k
The impact of technological progress
• Suppose the worker
An increase
in “g”
efficiency growth rate
changes from g1 to g2.
• This shifts the line
representing population
growth, depreciation, and
worker efficiency growth
upward.
•
At the new steady state k2*
capital per worker and
output per worker are lower.
•
The model predicts that
economies with higher rates
of worker efficiency growth
will have lower levels of
capital per worker and lower
levels of income.
Investment
Break-even
Investment
(d+n+g2)k (d+n+g )k
1
s*f(k)
k 2*
…reduces k*
k 1*
k
Effects of technological progress on the golden rule
• With technological progress the golden rule level of capital is
defined as the steady state that maximizes consumption per
effective worker. Following our previous analysis steady state
consumption per worker is…
c* = f(k*) – (d + n + g)k*
• To maximize this…
MPK = d + n + g
or
MPK – d = n + g
• That is, at the Golden Rule level of capital, the net marginal
product of capital MPK – d, equals the rate of growth of total
output, n+g.
Steady State Growth Rates in the Solow Model with
Technological Progress
Variable
Symbol
Steady-State Growth
Rate
Capital per
effective worker
k=K/(E*L)
0
Output per
effective worker
y=Y/(E*L)=f(k) 0
Output per
worker
Y/L=y*E
g
Total output
Y=y(E*L)
n+g
Conclusion
• In this section we added changes in two exogenous
variables (population and technological growth) to the
Solow growth model. We saw that in steady state
output per effective worker remains constant, output
per worker depends only on technological growth, and
that Total output depends on population and
technological growth.
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.
The Model Summarized
Production
The production function:
- has constant returns to scale in capital and labor
- has an exponent of less than 1 (let’s say 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
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
Summary
1. The starting point for the Solow model is the
production model with constant returns to scale in
capital and labor with a diminishing marginal
product of capital
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.
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.
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.
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.
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.
CHAPTER 5 The Solow Growth Model