1. The document provides an overview of the Solow growth model, which shows how capital accumulation, labor force growth, and technological advances interact in an economy and affect total output. 2. It examines how the model treats the accumulation of capital over time and how savings, depreciation, population growth, and technological progress influence the long-run capital stock and output. 3. The model predicts that economies with higher savings rates or population growth rates will reach different steady-state levels of capital and output per worker.

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A PowerPoint™Tutorial
to Accompany macroeconomics, 5th ed.
N. Gregory Mankiw
®
Economic Growth

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The Solow Growth Model is designed to show how
growth in the capital stock, growth in the labor force,
and advances in technology interact in an economy,
and how they affect a nation’s total output of
goods and services.
Let’s now examine how the
model treats the accumulation
of capital.

4. 4

5. 5
The production function represents the
transformation of inputs (labor (L), capital (K),
production technology) into outputs (final goods
and services for a certain time period).
The algebraic representation is:
Y = F ( K , L )
The Production FunctionThe Production Function
IncomeIncome isis some function ofsome function of our given inputsour given inputs
Let’s analyze the supply and demand for goods, and
see how much output is produced at any given time
and how this output is allocated among alternative uses.
Key Assumption: The Production Function has constant returns to scale.
z zz

6. 6
This assumption lets us analyze all quantities relative to the size of
the labor force. Set z = 1/L.
Y/ L = F ( K / L , 1 )
OutputOutput
Per workerPer worker
isis some function ofsome function of the amount ofthe amount of
capital per workercapital per worker
Constant returns to scale imply that the size of the economy as
measured by the number of workers does not affect the relationship
between output per worker and capital per worker. So, from now on,
let’s denote all quantities in per worker terms in lower case letters.
Here is our production function: , where f(k)=F(k,1).y = f ( k )

7. 7
MPK = f (k + 1) – f (k)
yy
kk
f(k)
The production function shows
how the amount of capital per
worker k determines the amount
of output per worker y=f(k).
The slope of the production function
is the marginal product of capital:
if k increases by 1 unit, y increases
by MPK units.
1
MPK

8. 8
consumptionconsumption
per workerper worker
dependsdepends
onon savingssavings
raterate
(between 0 and 1)(between 0 and 1)
OutputOutput
per workerper worker
consumptionconsumption
per workerper worker investmentinvestment
per workerper worker
y = c + iy = c + i1)
c = (1-c = (1-ss)y)yc = (1-c = (1-ss)y)y2)
y = (1-y = (1-ss)y + i)y + iy = (1-y = (1-ss)y + i)y + i3)
4)
i =i = ssyyi =i = ssyy Investment = savings. The rate of saving s
is the fraction of output devoted to investment.

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Here are two forces that influence the capital stock:
• Investment: expenditure on plant and equipment.
• Depreciation: wearing out of old capital; causes capital stock to fall.
Recall investment per worker i = s y.
Let’s substitute the production function for y, we can express investment
per worker as a function of the capital stock per worker:
i = s f(k)
This equation relates the existing stock of capital k to the accumulation
of new capital i.

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Investment, s f(k)
Output, f (k)
c (per worker)
i (per worker)
y (per worker)
The saving rate s determines the allocation of output between
consumption and investment. For any level of k, output is f(k),
investment is s f(k), and consumption is f(k) – sf(k).
yy
kk

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Impact of investment and depreciation on the capital stock: ∆k = i –δk
Change in
Capital Stock
Investment Depreciation
Remember investment equals
savings so, it can be written:
∆k = s f(k)– δk
δk
kk
δk
Depreciation is therefore proportional
to the capital stock.

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Investment
and Depreciation
Capital
per worker, k
i* = δk*
k*k1 k2
At k*, investment equals depreciation and
capital will not change over time. Below k*,
investment
exceeds
depreciation,
so the capital
stock grows.
Below k*,
investment
exceeds
depreciation,
so the capital
stock grows.
Investment, s f(k)
Depreciation, δ k
Above k*, depreciation
exceeds investment, so the
capital stock shrinks.
Above k*, depreciation
exceeds investment, so the
capital stock shrinks.

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Investment
and
Depreciation
Capital
per worker, k
i* = δk*
k1* k2*
Depreciation, δ k
Investment, s1f(k)
Investment, s2f(k)
The Solow Model shows that if the saving rate is high, the economy
will have a large capital stock and high level of output. If the saving
rate is low, the economy will have a small capital stock and a
low level of output.
An increase in
the saving rate
causes the capital
stock to grow to
a new steady state.
An increase in
the saving rate
causes the capital
stock to grow to
a new steady state.

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The steady-state value of k that maximizes consumption is called
the Golden Rule Level of Capital. To find the steady-state consumption
per worker, we begin with the national income accounts identity:
and rearrange it as:
c = y - i.
This equation holds that consumption is output minus investment.
Because we want to find steady-state consumption, we substitute
steady-state values for output and investment. Steady-state output
per worker is f (k*) where k* is the steady-state capital stock per
worker. Furthermore, because the capital stock is not changing in the
steady state, investment is equal to depreciation δk*. Substituting f (k*)
for y and δ k* for i, we can write steady-state consumption per worker as
c*= f (k*) - δ k*.
y - c + i

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c*= f (k*) - δ k*.
According to this equation, steady-state consumption is what’s left
of steady-state output after paying for steady-state depreciation. It
further shows that an increase in steady-state capital has two opposing
effects on steady-state consumption. On the one hand, more capital
means more output. On the other hand, more capital also means that more
output must be used to replace capital that is wearing out.
The economy’s output is used for
consumption or investment. In the steady
state, investment equals depreciation.
Therefore, steady-state consumption is the
difference between output f (k*) and
depreciation δ k*. Steady-state consumption
is maximized at the Golden Rule steady
state. The Golden Rule capital stock is
denoted k*gold, and the Golden Rule
consumption is c*gold.
δk
kk
δk
Output, f(k)
c *gold
k*gold

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Let’s now derive a simple condition that characterizes the Golden Rule
level of capital. Recall that the slope of the production function is the
marginal product of capital MPK. The slope of the δk* line is δ.
Because these two slopes are equal at k*gold, the Golden Rule can
be described by the equation: MPK = δ.
At the Golden Rule level of capital, the marginal product of capital
equals the depreciation rate.
Keep in mind that the economy does not automatically gravitate toward
the Golden Rule steady state. If we want a particular steady-state capital
stock, such as the Golden Rule, we need a particular saving rate to
support it.

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The basic Solow model shows that capital accumulation, alone,
cannot explain sustained economic growth: high rates of saving
lead to high growth temporarily, but the economy eventually
approaches a steady state in which capital and output are constant.
To explain the sustained economic growth, we must expand the
Solow model to incorporate the other two sources of economic
growth.
So, let’s add population growth to the model. We’ll assume that the
population and labor force grow at a constant rate n.

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Like depreciation, population growth is one reason why the capital
stock per worker shrinks. If n is the rate of population growth and δ
is the rate of depreciation, then (δ + n)k is break-even
investment, which is the amount necessary
to keep constant the capital stock
per worker k.
Investment,
break-even
investment
Capital
per worker, k
k*
Break-even
investment, (δ + n)k
Investment, s f(k)
For the economy to be in a steady state
investment s f(k) must offset the effects of
depreciation and population growth (δ + n)k. This
is shown by the intersection of the two curves. An
increase in the saving rate causes the capital stock
to grow to a new steady state.
For the economy to be in a steady state
investment s f(k) must offset the effects of
depreciation and population growth (δ + n)k. This
is shown by the intersection of the two curves. An
increase in the saving rate causes the capital stock
to grow to a new steady state.

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Investment,
break-even
investment
Capital
per worker, k
k*1
Investment, s f(k)
(δ + n1)k
An increase in the rate of population growth shifts the line
representing population growth and depreciation upward. The new
steady state has a lower level of capital per worker than the
initial steady state. Thus, the Solow model
predicts that economies with higher rates
of population growth will have lower
levels of capital per worker and
therefore lower incomes.
k*2
(δ + n2)k
An increase in the rate
of population growth
from n1 to n2 reduces the
steady-state capital stock
from k*1 to k*2.
An increase in the rate
of population growth
from n1 to n2 reduces the
steady-state capital stock
from k*1 to k*2.

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The change in the capital stock per worker is: ∆k = i –(δ+n)kThe change in the capital stock per worker is: ∆k = i –(δ+n)k
Now, let’s substitute sf(k) for i: ∆k = sf(k) – (δ+n)k
This equation shows how new investment, depreciation, and
population growth influence the per-worker capital stock. New
investment increases k, whereas depreciation and population growth
decrease k. When we did not include the “n” variable in our simple
version– we were assuming a special case in which the population
growth was 0.
Now, let’s substitute sf(k) for i: ∆k = sf(k) – (δ+n)k
This equation shows how new investment, depreciation, and
population growth influence the per-worker capital stock. New
investment increases k, whereas depreciation and population growth
decrease k. When we did not include the “n” variable in our simple
version– we were assuming a special case in which the population
growth was 0.

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In the steady-state, the positive effect of investment on the capital per
worker just balances the negative effects of depreciation and
population growth. Once the economy is in the steady state,
investment has two purposes:
1) Some of it, (δk*), replaces the depreciated capital,
2) The rest, (nk*), provides new workers with the steady state amount
of capital.
Capital
per worker, k
k*k*'
The Steady State
Investment,s f(k)
Break-even Investment,(δ + n) k
Break-even investment,(δ + n') k
An increase in the rate
of growth of population
will lower the level of
output per worker.
sf(k)

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• In the long run, an economy’s saving determines the size
of k and thus y.
• The higher the rate of saving, the higher the stock of capital
and the higher the level of y.
• An increase in the rate of saving causes a period of rapid growth,
but eventually that growth slows as the new steady state is
reached.
Conclusion: although a high saving rate yields a high
steady-state level of output, saving by itself cannot generate
persistent economic growth.
Conclusion: although a high saving rate yields a high
steady-state level of output, saving by itself cannot generate
persistent economic growth.

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The Production Function is now written as:
Y = F (K, L × E)
The term L × E measures the number of effective workers.
This takes into account the number of workers L and the efficiency
of each worker E. Increases in E are like increases in L.

25. 25Capital
per worker, k
k*
The Steady State
Investment,
sf(k)
(δ + n + g)k
Technological progress causes E to grow at the rate g, and L grows
at rate n so the number of effective workers L × E is growing at rate
n + g.
Now, the change in the capital stock per worker is:
∆k = i –(δ+n +g)k, where i is equal to s f(k)
Technological progress causes E to grow at the rate g, and L grows
at rate n so the number of effective workers L × E is growing at rate
n + g.
Now, the change in the capital stock per worker is:
∆k = i –(δ+n +g)k, where i is equal to s f(k)
Note: k = K/LE and y=Y/(L × Ε).
So, y=f(k) is now different.
Also, when the g term is added,
gk is needed to provided capital
to new “effective workers”
created by technological progress.
Note: k = K/LE and y=Y/(L × Ε).
So, y=f(k) is now different.
Also, when the g term is added,
gk is needed to provided capital
to new “effective workers”
created by technological progress.
sf(k)

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Labor-augmenting technological progress at rate g affects the Solow
growth model in much the same way as did population growth at rate
n. Now that k is defined as the amount of capital per effective worker,
increases in the number of effective workers because of technological
progress tend to decrease k. In the steady state, investment sf(k)
exactly offsets the reductions in k because of depreciation, population
growth, and technological progress.

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Capital per effective worker is constant in the steady state. y = f(k)
output per effective worker is also constant. But the efficiency of
each actual worker is growing at rate g. So, output per worker,
(Y/L = y × E) also grows at rate g. Total output Y = y × (E × L)
grows at rate n + g.
Capital per effective worker is constant in the steady state. y = f(k)
output per effective worker is also constant. But the efficiency of
each actual worker is growing at rate g. So, output per worker,
(Y/L = y × E) also grows at rate g. Total output Y = y × (E × L)
grows at rate n + g.

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Steady-state consumption is maximized if
MPK = δ + n + g,
rearranging, MPK - δ = n + g.
That is, at the Golden Rule level of capital, the net marginal
product of capital, MPK - δ, equals the rate of growth of total
output, n + g. Because actual economies experience both
population growth and technological progress, we must use this
criterion to evaluate whether they have more or less capital than at
the Golden Rule steady state.
The introduction of technological progress also modifies the
criterion for the Golden Rule. The Golden Rule level of capital is
now defined as the steady state that maximizes consumption per
effective worker. So, we can show that steady-state consumption
per effective worker is:
c*= f (k*) - (δ + n + g) k*c*= f (k*) - (δ + n + g) k*

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An important prediction of the neoclassical model is this:
Among countries that have the same steady state,
the convergence hypothesis should hold:
poor countries should grow faster on
average than rich countries.

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?
The Endogenous Growth Theory rejects
Solow’s basic assumption of exogenous technological change.

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Start with a simple production function: Y = AK, where Y is output,
K is the capital stock, and A is a constant measuring the amount of
output produced for each unit of capital (noticing this production
function does not have diminishing returns to capital). One extra unit
of capital produces A extra units of output regardless of how much
capital there is. This absence of diminishing returns to capital is
the key difference between this endogenous growth model and the
Solow model.
Let’s describe capital accumulation with an equation similar to those
we’ve been using: ∆K = sY - δK. This equation states that the change
in the capital stock (∆K) equals investment (sY) minus depreciation
(δK). We combine this equation with the production function, do
some rearranging, and we get: ∆Y/Y = ∆K/K = sA - δ

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∆Y/Y = ∆K/K = sA - δ
This equation shows what determines the growth rate of output ∆Y/Y.
Notice that as long as sA > δ, the economy’s income grows forever,
even without the assumption of exogenous technological progress.
In the Solow model, saving leads to growth temporarily, but diminishing
returns to capital eventually force the economy to approach a steady
state in which growth depends only on exogenous technological progress.
By contrast, in this endogenous growth model, saving and investment can
lead to persistent growth.

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Solow growth model
Steady state
Golden rule level of capital
Efficiency of labor
Labor-augmenting technological progress
Endogenous growth theory