CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

CHAPTER SIX
MODELS OF MACROECONOMICS GROWTH
Chapter Objectives:
 At the end of this chapter, you will be able to:
 Recognize the existence of huge difference in countries economic performance and
societies living standards;
 Explain the growth implication of the Harrod-Domar Model;
 Appreciate the role of saving, investment, growth of labor force and technology in
economic growth:
 Distinguish the differences b/n the Solow model and the endogenous model and
analyze the respective growth implications;
 Demonstrate a high level of awareness about engines of economic growth;
 Analyze why societies in different countries experience differing levels of living
standards.
6.1. Introduction: Modeling Economic Growth

Basic Questions:
1. Why some countries are richer than other countries?
2. What exactly is the engine of growth? Why some countries do exhibit
sustained growth while others not?

To measure economic growth, economists use data on gross domestic product, which
measures the total income of everyone in the economy. Over the past century, most
countries in the world have enjoyed substantial economic growth. Real incomes have
risen from generation to generation, and these higher incomes have allowed people to
consume greater quantities of goods and services. Higher levels of consumption have led
to a higher standard of living.

However, growth rates are not same across countries and over time. There are 215
countries in the world, whose GDPs are growing at different rates. In any given year, we
can observe large differences in the standard of living among countries. Table 5.1 shows
income per person in 2014 of some world countries. The United States tops the list with
an income of $54, 630 per person. Ethiopia has an income per person of only $570—nearly
1 percent of the figure for the United States.

1
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

Table: 6.1: International Differences in the Standard of Living: 2014

Country Income per Person Country Income per Person
(in U.S. dollars) (in U.S. dollars)
United States 54, 630 India 1, 582

Canada 50, 230 Egypt 3, 365

Germany 47, 774 Nigeria 3, 203

Mexico 10, 326 Kenya 1, 358

Brazil 11,726 Ethiopia 570

 Basic Facts of Economic Growth

a. There is enormous variation in per capita income across economies.
 Countries at the top of the world income distribution are more than thirty times as
rich as those at the bottom.
 For instance, PCI (at PPP) in the US is more than 36 times higher than PCI in
Ethiopia in 2014.
b. Rates of economic growth vary substantially across countries.
 Eg: GDP growth rate was, on average, 10% in Ethiopia but 5.5% for Singapore over
the last decade.
c. Growth rates do vary over time.
d. A countries relative position in the world distribution of per capita incomes is not
immutable.
 Countries can move from being “poor” to being “rich” (Eg: East Asian countries) &
vice versa (Eg: Argentina).

But why do growth rates and performances differ ever so much? Every country has a
different starting point. The supply of labor, the stock of resources, physical capital and
human capital, the level of infrastructure, the climate conditions, political and
institutional environment, the volume of international trade, and the current GDP are
some of the main aspects influencing the growth of countries.

Economists try to model the relationships and the development of these and other
variables over time in trying to explain growth dynamics. They try to reveal how decisions
made by economic agents, the value of the variables in the past, present and future, affect
growth rates and achievements.

2
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

Our goal in this chapter is, therefore, to understand what causes these differences in
income over time and across countries by examining different growth models.

6.2. Models of Economic Growth

6.2.1. The Harrod-Domar Growth Model
The Harrod-Domar growth model is based on the work by these two authors in the
1930’s. They developed their models independently, but they arrive at the same
results. The model emphasizes the key role of investment in determining economic
growth. Their argument can be stated as: Every economy must save a certain
proportion of its national income, if only to replace worn-out or impaired capital
goods. However, in order to grow, new investments representing net additions to the
capital stock are necessary. If we assume that there is some direct economic
relationship between the size of the total capital stock, K, and total GDP, Y—it follows
that any net additions to the capital stock in the form of new investment will bring
about corresponding increases in the flow of national output, GDP.

Notice that this relationship, known in economics as the capital-output ratio. If we define
the capital-output ratio as and assume further that the national net savings ratio, s, is a
fixed proportion of national output and that total new investment is determined by the
level of total savings, we can construct a simple model of economic growth:

Model Assumptions

1. Net saving (S) is some proportion, s, of national income (Y) such that we have the
simple equation: = − − − − − − − − − − − − − −( )
2. Capital stock (K) is proportional to output (Y): i.e.,
= − − − − − − − − − − − − − −( )
3. We assume a closed economy (no trade or capital flows). Then savings S must be used for
investments I, and investments can only come from savings. Hence they must be equal:
= −−−−−−−−−−−−−−−( )
4. In year t, the stock of capital is . The year after, year t+ 1, the capital stock is .
Changes in the capital stock come from investments and the depreciation (wearing
out) of the capital stock. Depreciation occurs at a constant rate , so an amount of
capital disappears every year. Hence, the capital accumulation equation is:
= + −
3
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

∆ = − = − − − − − − − − − − −( )

Now, having the above assumptions, we can solve for the growth rate of capital and/or
output. Therefore, the growth rate of capital is defined as
− ∆
= =
−
= = −
And we know that = , and = , so substitution yields:

= −

= − − − − − − − − − − − − −( )
As we are ultimately interested in the rate of growth of GDP, we use that = .Then
we get that growth in GDP is
− + − + −
= = =

This indicates that GDP and capital grow at the same rate. Hence, growth in GDP is
= − − − − − − − − − − − − −( )
This equation states that, to grow, economies must save and invest a certain proportion
of their GDP. Countries with lower savings and hence investment will end up at lower
level of GDP. Thus, the more countries can save and invest, the faster they can grow.

 Evidence in support of the H-D Model

As we know Chinese growth has been impressive registering a growth rate of 10% per year
for over 30 years beginning in the early 1980s. Many international observers view that
high savings as successful experience supporting China’s rapid growth. The average rate of
the world saving is 21%, less than half of that in China, which is averaged 47.4% in the last
decade. In particular, china’s savings to GDP ratio was 30.5%, 34.7%, 40.9%, and 47.4% in
the 1970s, 1980s, 1990s, and 2000s, respectively. Researchers indicated that this high level
of saving over the years was one major source of capital accumulation and rapid long-term
growth in china.

4
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

6.2.2. The Neo-Classical model: The Solow Growth model

Robert Solow (1956), with the publication of his work “A Contribution to the Theory
of Economic Growth”, provides a good growth model. His 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.
The model is built around two equations: a production function and a
capital accumulation equation. As our first step in building the model, we
examine how the supply and demand for goods determine the accumulation of
capital. To do this, we hold the labor force and technology fixed. Later, we
relax these assumptions, first by introducing changes in the labor force and then
by introducing changes in technology.

6.2.2.1. Accumulation of Capital in the Solow Model

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.

I. The supply of goods and the production function.

The supply of goods in the Solow model is based on the following production
function: Which states output depends on the capital stock and the labor force.

= ( , ) −−−−−−−−−−−−−( )
The Solow growth model assumes that the production function has constant returns
to scale. Thus, recall that a production function has constant returns to scale if

= ( , ) for positive value, z

That is, if we multiply both capital and labor by z, we also multiply the amount of output
by z.
Production functions with constant returns to scale allow us to analyze all
quantities in the economy relative to the size of the labor force. Besides, Per
capita output is claimed to be the best measure of welfare and hence we are
interested in explaining output per worker. With this interest in mind, we can
rewrite the production function in equation [1] in terms of output per worker and
capital per worker.

5
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

Divide both sides of equation [1], to obtain: = ,

 This equation states that output per worker / is a function of capital per worker
/ . So, from now on, let’s denote all quantities in per worker terms in lower
case letters. Thus, = / is output per worker and = / is capital per worker.
Equation [1] can, thus, be rewritten as:

= ( ) − − − − − − − − − − − −( ) ,
Where ( ) = ( , ).

This production function is graphed below. With more capital per worker, firms produce
more output per worker. However, there are diminishing returns to capital per worker:
each additional unit of capital we give to a single worker increases the output of that
worker by less and less. This amount is the marginal product of capital .
Mathematically, we write: = ( + )− ( )

Note that in Figure 5-1, as the amount of capital increases, the production function
becomes flatter, indicating that the production function exhibits diminishing
marginal product of capital.
The production function
Figure: 6.1: The Production Function
shows how the amount of
Output per
capital per worker k
worker, y output, f(k)
determines the amount of
output per worker y = f (k).
 Implication: Countries with
1 small capital stock (k) are
more productive => grow
faster

Capital per worker, k

II. The Demand for Goods and the Consumption Function

Solow assumes that the economy is closed, so that saving equals investment, and the
only use of investment is to accumulate capital. The demand for goods in the Solow
model comes from consumption and investment. In other words, output per worker
is divided b/n consumption per worker and investment per worker :

6
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

= + −−−−−−−−−−−−( )
This equation is the national income accounts identity for the economy. It omits
government purchases (which for present purposes we can ignore) and because it
expresses , , and as quantities per worker.

The Solow model assumes that the consumption function takes the simple form:

= ( − ) −−−−−−−−−−( )
Where, s, the saving rate.
This consumption function states that consumption is proportional to income. Each year
a fraction (1-s) of income is consumed, and a fraction s is saved.
To see what this consumption function implies for investment, substitute ( − ) for
in the national income accounts identity:
= ( − ) +

Rearrange the terms to obtain

= −−−−−−−−−−−−− ( )
This equation states that investment, like consumption, is proportional to income.
Since investment equals saving, the rate of saving s is also the fraction of output
devoted to investment.

III. Growth in the Capital Stock and the Steady State

Having introduced the two main ingredients of the Solow model-the production
function and the consumption function-we can now examine how increases in the
capital stock over time result in economic growth.
Two forces cause the capital stock to change:
 Investment: The capital stock rises as firms buy new plants and equipment.
 Depreciation: The capital stock falls as some of the old capital wears out.

Recall that investment per worker = .

Let’s substitute the production function for y from [2], and then we can express
investment per worker as a function of the capital stock per worker:
= ( ), − − − − − − − − − − − − ( )

This equation relates the existing stock of capital k to the accumulation of new
capital i. Figure 5-2 shows this relationship. This figure illustrates how, for any value of ,
the amount of output is determined by the production function ( ), and the allocation of
that output b/n consumption and saving is determined by the saving rate .

7
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

Output per Output, f(k)

worker, y Output per
worker, y
Consumption
per worker, c Investment, ( )

Investment
per worker, i

Capital per worker, k

Figure: 5.2: Output, Consumption, and Investment
Note: The saving rate s determines the allocation of output b/n consumption and investment.
At any level of capital k, output is f(k), investment is sf(k), and consumption is f(k) – sf(k).

To incorporate depreciation into the model, we assume that a certain fraction  of

the capital stock wears out each year. We call  the depreciation rate. So, depreciation is
proportional to the capital stock.
Now, we can express the impact of investment and depreciation on the capital stock in
the following equation:
= −
. ., ∆ = − −−−−−−−−−−−( )
where ∆k is the change in the capital stock between one year and the next. Because
investment equals ( ), we can write this as:

∆ = ( ) − − − − − − − − − − − − − − −( )
Equation [8] is termed as the capital accumulation equation.
Figure 5-4 graphs the terms of this equation—investment and depreciation—for different
levels of the capital stock k.
 The higher the capital stock, the greater will be the amounts of
investment and output. Yet the higher the capital stock, the greater will
also be the amount of depreciation.

Figure 5.3 shows that there is a single capital stock at which the amount of
investment equals the amount of depreciation. If the economy has this capital stock,
the capital stock will not change over time because the two forces acting to change it –
investment and depreciation – just balance. That is, at this level of the capital stock,
 = . We call this the steady-state level of capital (long-run equilibrium of the
economy) and designate it as k*.

8
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

 Implication: Approaching the Steady State

 An economy at the steady state will stay there.
 An Economy which is not in the steady state will go there => convergence to the
constant level of output per worker over time. But an economy not at the steady
state will go there. That is, regardless of the level of capital with which the
economy begins, it ends up with the steady-state level of capital. In this sense, the
steady state represents the long-run equilibrium of the economy.
To see why an economy always ends up at the steady state, suppose that the economy
starts with less than the steady-state level of capital, such as level in Figure 5-3. In
this case, the level of investment exceeds the amount of depreciation. Over time, the
capital stock will rise and will continue to rise —along with output ( )—until it
approaches the steady state ∗ .
Similarly, suppose that the economy starts with more than the steady-state level of
capital, such as level . In this case, investment is less than depreciation: capital is
wearing out faster than it is being replaced. The capital stock will fall, again
approaching the steady-state level. Once the capital stock reaches the steady state,
investment equals depreciation, and there is no pressure for the capital stock to either
increase or decrease.

: 6. : , ,
The steady-state level of
capital k* is the level at
which investment equals
depreciation- indicating
that the amount of
capital will not change
over time.
 Below k*, investment
exceeds depreciation, so
the capital stock grows.
 Above k*, investment is
less than depreciation,
so the capital stock
shrinks.

9
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

Numerical Example: Approaching the steady state

Let’s use a numerical example to see how the Solow model works and how
the economy approaches the steady state. For this example, we assume that
the production function is: =
To derive the per worker production function ( ), divide both sides of the
Y K 1 / 2 L1 / 2
production function by L: =
L L
1/ 2
K
Substitute for / , and rearrange to obtain: y =  
L
/
Since = / , this becomes = = k
Output per worker is the square root of capital per worker.

To complete the example, we assume that 30% of output is saved (s = 0.3),

that 10% of the capital stock depreciates every year (  = 0.1), and that the
economy starts with 4 units of capital per worker (k = 4). We can now
examine what happens to this economy over time.
Recall that:  = ( )− 
Since  = in the steady state, we know that
= ( ∗) −  ∗
k* s
or, equivalently, =
f ( k *) 
Substituting the values above, we obtain
k* 0.3
= =3
k* 0.1
∗
Square both sides to find = .
 The steady-state capital stock is 9 units per worker.

How Saving Affects Growth

Evidences tell us that the international difference in economic performance is partly

due to differences in saving rates among countries.
Hence, to understand the effect of savings, let’s consider what happens to an economy
when the saving rate increases in the Solow model. Fig. 5.4 shows such a change. We
assume that the economy begins in a steady state with saving rate s1 and capital stock
∗
. The saving rate then increases from s1 to s2, causing an upward shift in the ( )
curve. At the initial saving rate s1 and the initial capital stock, ∗ , the amount of
investment just offset the amount of depreciation. The moment after the saving rate

10
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

rises, investment is higher, but the capital stock and depreciation are unchanged.
Therefore, investment exceeds depreciation. The capital stock will gradually rise until
the economy reaches the new steady state, ∗ , with a higher capital stock and a higher
level of output.
The Solow model shows that the saving rate is a key determinant of the steady-
state capital stock. If the saving rate is high, the economy will have a large capital
stock and a high level of output. If the saving rate is low, the economy will have a
small capital stock and a low level of output.
According to the Solow model, higher saving leads to faster growth, but only in the
short run. An increase in the rate of saving raises growth until the economy reaches
the new steady state. If the economy maintains a high saving rate, it will also maintain
a large capital stock and high level of output, but will not maintain a high rate of
3. ... raises
per capita
, &
income.
∗ = ( )

∗
=
∗ , ( )

∗ , ( )
∗
=
1. An increase
in the saving
rate raises
investment,
…

∗ ∗ Capital per worker, k

2….Causing the
capital stock to
grow towards a
new steady state.
Figure: 6.4: Saving, Capital Stock & Economic Growth
Note: An increase in the saving rate, s implies that the amount of investment for
any given capital stock is higher. It therefore shifts the saving function
upward. At the initial steady state ∗ , investment now exceeds depreciation.
The capital stock rises until the economy reaches a new steady state ∗ with
more capital and output.

11
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

The Golden Rule Level of Capital

As mentioned above, 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. Note that: when choosing a steady
state, the policymaker’s goal is to maximize the wellbeing of the society (where wellbeing
is measured by consumption level).
 The steady-state value of k that maximizes consumption is called the Golden
Rule Level of Capital and is denoted by k*gold.
To find steady-state consumption per worker, we begin with the national income
accounts identity: = +

And rearrange it as: = –

Since, we want to find steady-state consumption; we substitute steady-state values
for output and investment. From [2] Steady-state output per worker is ( ∗ ),
where ∗ is the steady-state capital stock per worker. Furthermore, since
investment is equal to depreciation, at the steady state. Substituting ( ∗ ), for
and ∗ for , we can write steady state consumption per worker as:
∗ ∗
= ( ∗) − −−−−−−−−−−−−( )
 This equation states that steady-state consumption is the difference between
steady-state output and steady-state depreciation.
 Equivalently, because, consumption per worker is the difference between
output per worker 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
 The slope of the production function is the marginal product of capital . The slope
∗
of the line is . Because these two slopes are equal at k*gold, a condition that
characterizes the golden rule level of capital is = .
 At the Golden Rule level of capital, the marginal product of capital equals the
depreciation rate.
o If − > 0, then increases in capital increase consumption, so k* must be
below the Golden Rule level.
o If − < 0, then increases in capital decrease consumption, so k* must be
above the Golden Rule level.

Figure 5.5 graphs steady-state output and steady-state depreciation as a function of the
steady-state capital stock. Steady-state consumption is the gap b/n output and
depreciation. This figure shows that there is one level of the capital stock—the
Golden Rule level, ∗ —that maximizes consumption.

12
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

Steady-state Steady state

output & deprn, k* The economy’s output is used for
depreciation consumption or investment.
 In the steady state, investment
equals depreciation. Therefore,
steady-state consumption is the
Steady-state difference b/n output f(k*) and
output, f(k*) depreciation ∗ .
 Steady-state consumption is
maximized at the Golden Rule
steady state. The Golden Rule
capital stock is denoted ∗ , &
∗ Steady-state capital the Golden Rule consumption
per worker, k* is denoted ∗ .
Below the Golden NB: At the Golden Rule
Rule steady Above the Golden level of capital, the
state, increases Rule steady state,
increases in steady- production function and
in steady-state
capital, raise state capital the k* line have the same
steady-state reduces steady- slope, and consumption is
consumption. state consumption. at its greatest level.

Figure: 6.5: Steady State Consumption

What is the golden rule level of savings?

Figure: 5.6:The Saving

Rate & the Golden Rule:
 There is only one saving
rate that produces the
Golden Rule level of capital
∗
. Any change in the
saving rate would shift the
( ) curve & would move
the economy to a steady
state with a lower level of
consumption.

Keep in mind that the economy does not automatically gravitate toward the Golden Rule
steady state. If we want any particular steady-state capital stock, such as the Golden Rule,
13
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

we need a particular saving rate to support it. The figure below shows the steady state if
the saving rate is set to produce the Golden Rule level of capital. If the saving rate is
higher than the one used in this figure, the steady-state capital stock will be too high. If
the saving rate is lower, the steady-state capital stock will be too low. In either case,
steady-state consumption will be lower than it is at the Golden Rule steady state.

Golden Rule Steady State: A Numerical Example

Consider the production function in our earlier example; = √ , and assume that 1o% of
the capital stock depreciates each year. Then, calculate the golden rule level of steady
state capital?

Solution: At the golden rule level, the slope of the production function in per capita
terms equals with the rate of depreciation. That is, =

Therefore, with =√ , = . Then, = .

√ √

∗
Solve for k and we obtain: = units

6.2.2.2. Population Growth in the Solow Model

The basic Solow model shows that capital accumulation, by itself, 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 that we observe in most parts of
the world, we must expand the Solow model to incorporate the other two sources of
economic growth —population growth and technological progress. In this section we add
population growth to the model.

The Steady State with Population Growth

How does population growth affect the steady state?

To answer this question, we must discuss how population growth, along with investment
and depreciation, influences the accumulation of capital per worker over time. As we
have noted before, investment raises the capital stock, and depreciation reduces it. But
now there is a third force acting to change the amount of capital per worker: the growth
in the number of workers causes capital per worker to fall.

14
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

We continue to let lowercase letters stand for quantities per worker. Thus, = / is
capital per worker, and = / is output per worker. Keep in mind, however, that the
number of workers is growing over time.

The change in the capital stock per worker (i.e., our capital accumulation equation) now
is: ∆ = ( )−( + ) Discrete time version

or ̇ = ( )−( + ) Continous time version

Proof:
We use the usual production function: = ( ) − − − − − − − − − (1)
The other key equation of the Solow model is an equation that describes how capital
accumulates. Thus, the aggregate capital accumulation equation is given by:
̇ = − −−−−−−−−−( )

Note that notationally, ̇ =

 According to this equation, the change in the capital stock, ̇ , is equal to the amount of
gross investment, sY, less the amount of depreciation that occurs during the production
process, .

NB: The term on the left-hand side of equation [2] is the continuous time version of
– , that is, the change in the capital stock per "period". We use the 'dot' notation
to denote a derivative with respect to time.

 To study the evolution of output per person in this economy, we rewrite the
capital accumulation equation in terms of capital per person.

Recall that = , Take the log & differentiate with respect to time:
⇒ = −
⇒ = −
̇ ̇ ̇
⇒ = − − − − − − − − − − −( )

NB: the derivative of logarithm of a variable with respect to time gives its growth
rate.
̇
From the capital accumulation equation of [2], we have = − . Substituting this
expression into [3],

15
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

̇ ̇
⇒ = − − , where is the growth in labour force,

̇
⇒ = − −
∴ ̇ = −( + ) −−−−−−−−−−−−−−−−−−−−−( )

 This equation shows how investment, depreciation, and population growth influence
the per-worker capital stock. Investment increases , whereas depreciation and
population growth decrease .
We can think of the term ( + ) as defining Break-Even Investment —the
amount of investment necessary to keep the capital stock per worker constant.
Break-even investment includes the depreciation of existing capital, which equals .
It also includes the amount of investment necessary to provide new workers with
capital. The amount of investment necessary for this purpose is, because there are
new workers, and because is the amount of capital for each worker. The equation
shows that population growth reduces the accumulation of capital per worker much
the way depreciation does. Depreciation reduces by wearing out the capital stock,
whereas population growth reduces k by spreading the capital stock more thinly
among a larger population of workers.
Per capita ∗ = ( )
income,
Investment, ( + )
& break-even
investment
( )

Capital per worker, k

∗

The steady state

)
Figure: 6.7: Population Growth in the Solow Model
Note: For the economy to be in a steady state, investment ( ) must offset the effects
of depreciation & population growth ( + ) . This is represented by the crossing
of the two curves.

16
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

An economy is in a steady state if capital per worker is unchanging. If is less than

∗
(a steady state level), investment is greater than break-even investment, so rises.
If is greater than ∗ , investment is less than break-even investment, so falls. In the
steady state, the positive effect of investment on the capital stock per worker exactly
balances the negative effects of depreciation and population growth. That is, at ∗ ,
∗
∆ = = ∗ + ∗.
The Effect of Population Growth in the Solow Model
 As the figure below presents, an increase in the rate of population growth from
to reduces the steady-state level of capital per worker from ∗ to ∗ . Thus, the
Solow model predicts that economies with higher rates of population growth will
have lower levels of capital per worker and therefore lower per capita incomes.
 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:
i. Some of it, ( ∗ ), replaces the depreciated capital,
ii. The rest, ( ∗ ), provides new workers with the steady state amount of

capital.

Per capita ∗
= ( )
income, 1. An increase
∗ ( + )
Investment, in the rate of
& break-even population
investment ( + ) growth …

( )

3…thus, reduces
the steady-state
per capita income.
∗ ∗ Capital per worker, k

2…reduces the
steady-state
capital stock.
Figure: 6.8: The Impact of Population Growth in the Solow Model

Because ∗ is lower, and because ∗ = ( ∗ ), the level of output per worker ∗ is also
lower. Thus, the Solow model predicts that countries with higher population
growth will have lower levels of GDP per person.

17
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

Finally, population growth affects our criterion for determining the Golden Rule
(consumption-maximizing) level of capital. To see how this criterion changes, note that
consumption per worker is; = − .

Because steady-state output is ( ∗ ), and steady-state investment is ( + ) ∗, we can

express steady-state consumption as
∗
= ( ∗ ) − ( + ) ∗.
∗
The level of that maximizes consumption is therefore, the one at which

= + ,

or equivalently, − =

 That is, in the Golden Rule steady state, the marginal product of capital net of
depreciation equals the rate of population growth.

Properties of the steady state

 An economy is in a steady state when per capita income and capital are constant.

Suppose the production function is given as:

= − − − − − − − − − − − − − −( )
And we have the steady-state condition that ̇ = 0. so, we can solve for the steady state
quantities of capital per worker and output per worker.
From our production function, we have
= − − − − − − − − − − − − − − − −( )
Therefore, by substitution we obtain:
̇ = –( + ) ,
and setting this equation equal to zero yields:

⇒ =( + )

⇒ =
( )

∗
⇒ = ----------Steady state capital per worker

Substituting this into the production function reveals the steady-state quantity of output per
worker, ∗ :
∗
⇒ = -----------Steady state output per worker

18
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

This equation reveals the Solow model's answer to the question "Why are we so poor and
they so rich?" countries that have high saving/investment rates will tend to be richer,
ceteris paribus. Such countries accumulate more capital per worker, and countries with
more capital per worker have more output per worker. Countries that have high
population growth rates, in contrast, will tend to be poorer, according to the Solow
model.
Economic Growth in the Solow Model with population Growth

What does economic growth look like in the steady state of this simple version of the
Solow model? The answer is that there is no per capita growth in this version of the
model! Output per worker is constant in the steady state. Output itself, Y, is growing, of
course, but only at the rate of population growth. This can be shown as:

⇒ = , take log both sides, and then differentiate with respect to time
⇒ = −
⇒ = −
Since the growth in per capita output is zero at the steady state
⇒ = − ,
̇ ̇
⇒ = ⇒ =

Thus, this model fails to predict a very important fact: that economies exhibit sustained
per capita income growth. In this model, economies may grow for a while, but not
forever. For example, an economy that begins with a stock of capital per worker below its
steady-state value will experience growth in and along the transitional path to the
steady state. Over time, however, growth slows down as the economy approaches its
steady state, and eventually growth stops altogether.
To see that growth slows down along the transition path, notice two things.
1st, from the capital accumulation equation:
̇ = − ( + ) , and from equation (2) above we have, = , thus:
̇
⇒ = −( + )
̇
⇒ = − −( + )

19
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

 Therefore, because is less than one, as rises, the growth rate of

gradually declines.

2nd, the growth rate of is proportional to the growth rate of , so that the same
statement holds true for output per worker- i.e.
= , ⇒ =
⇒ =
̇ ̇
⇒ =
 This indicates that when the growth rate of is zero at the steady state, output growth
would also be zero. Thus, the model doesn't explain growth at the steady state.

As a Conclusion
The Solow model developed so far shows how saving and population growth determine
the economy’s steady-state capital stock and its steady-state level of income per person.
As we have seen, it sheds light on why countries that save and invest a high fraction of
their output are richer than countries that save and invest a smaller fraction, and why
countries with high rates of population growth are poorer than countries with low rates of
population growth.

What the model cannot do, however, is explain the persistent growth in living standards
we observe in most countries. In the model we developed so far, when the economy
reaches its steady state, output per worker stops growing. To explain persistent growth,
we need to introduce technological progress into the model.

6.2.2.3. Technology and the Solow Model

To generate sustained growth in per capita income, we need to introduce technological

progress to the model. This is accomplished by adding a technology variable, A, to the
production function:

= ( , )= ( ) − − − − − − − − − − − −( )

Entered this way, the technology variable A is said to be "labour-augmenting" or

"Harrod-neutral". Technological progress occurs when A increases over time – a unit of
labour, for example, is more productive when the level of technology is higher.
An important assumption of the Solow model is that technological progress is
exogenous: in a common phrase, technology is like "manna from heaven", in that it
descends upon the economy automatically and regardless of whatever else is going on in
the economy. Instead of modelling carefully where technology comes from, we simply
20
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

recognize for that there is technological progress and make the assumption that is
̇
growing at a constant rate, = = where g is a parameter representing the growth
rate of technology.

 Note that we use lower case letters for per capita terms and lower case with a hat
for per effective worker levels.
Per capita levels:
( , )
= , = ⇒ = = ( ), = & =

Per effective worker levels:

( , )
= , = ⇒ = = ( ), ̂ = & =

The capital accumulation equation in this model is the same as before- i.e.
̇ = −
̇
= − −−−−−−−−( )
 Capital grows with investment ( = ), and a part of it is destroyed each period with
depreciation ( ).
To see the growth implication of this model, first rewrite the production function [1]
in terms of output per worker:

⇒ = ( , ) = ( )
( , )
⇒ = =

⇒ =
⇒ =
 Taking the logs and differentiating the above expression with respect to time:

= +( − )

= +( − )

̇ ̇ ̇
= +( − ) −−−−−−−−−−−−−−−( )

21
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

Notice from the capital accumulation equation [2] that the growth rate of K will be constant
if and only if Y/K is constant. Furthermore, if Y/K is constant, y/k is also constant
( . ., = = ) and most important, y and k will be growing at the same rate. A
situation in which capital, output, and consumption are growing at constant rates is
called a Balanced Growth Path.

Along the balanced growth path, = . Substituting this relationship into equation [3]:
= +( − )
Since = along the balanced growth path

= +( − )
( − )=( − )
= =
 That is, along the balanced growth path in the Solow model, output per worker and
capital per worker both grow at the rate of exogenous technological change, g.

Notice that in the simple model (i.e., the model without technology), there was no
technological progress, and therefore there was no long-run growth in output per worker
or capital per worker; = = .
 The Solow model with technology reveals that technological progress is the
source of sustained per capita growth.

The Solow Diagram with Technology

In the case where we introduce technology in the Solow model, the only important
difference is that the variable is no longer constant in the long run. So we have to write
our differential equation in terms of another variable. The new state variable will be
= . Notice that this is equivalent to and is obviously constant along the
balanced growth path because = = . The variable therefore represents the
ratio of capital per worker to technology (or capital per effective worker). We refer to this
as the "capital-technology" ratio (b/c the numerator is capital per worker).

Rewriting the production function, = ( ) in terms of per effective

worker, we obtain:

= =

= − − − − − − − − − − − − − − − −[ ]

Where, = = and = = .

22
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

Note: is called the "output-technology ratio" (output per effective labour).

Equation [4] implies that output per effective worker depends on capital per
effective worker.

Recall that: = . If we log linearize this, and differentiate with respect to time (to

get growth rate of ):

⇒ = −( + )

⇒ = − +

̇ ̇ ̇ ̇
⇒ = − − −−−−−−−−−−−−−−−−−( )
̇
From the aggregate capital accumulation equation [2], we know that = −

Substituting this into the above expression, we get:

̇
⇒ = − − −

̇
⇒ = − − −

̇
⇒ = − − −

∴ ̇ = −( + + )
Or, ̇ = ( )−( + + ) − − − − − − − − − − − − − − − − −[ ]
(Fundamental Differential Equation for Capital Stock Accumulation in the
Neoclassical Solow Model with Exogenous Technological Progress)
 As before, the change in the capital stock ̇ equals investment minus break-even
investment( + + ) .
 Now, however, because = / , Break-Even Investment (minimum amount of
investment required to keep capital per-effective worker constant) includes three terms:
to keep capital-per effective worker, constant, each period, new investment
amounted to is needed to replace depreciating capital, is needed to provide
capital for new workers, and is needed to provide capital for the new “effective
workers” created by technological progress.

23
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

Equation [6] would give the Solow diagram with technology as given below. There is one
level of , denoted ∗ , at which capital per effective worker and output per effective
worker are constant.
Investment, = ( )
break-even
investment ( + + )

The steady state

∗
Capital per effective worker,
Figure: 6.9: Technological Progress and the Solow Growth Model

 Note that: In the steady state, investment, exactly offsets the reductions in
attributable to depreciation, population growth, and technological progress.
If the economy begins with a Capital per effective worker that is below its steady-state
level, the Capital per effective worker will rise gradually over time because the amount
of investment being undertaken exceeds the amount needed to keep the Capital per
effective worker constant.
The introduction of technological progress modifies the criterion for the Golden Rule
(consumption-maximizing) level of capital. The Golden Rule level of capital is now
defined as the steady state that maximizes consumption per effective worker.

To see how this criterion changes, note that consumption per effective worker is;
= − ̂

Because steady-state output is ( ∗ ), and steady-state investment is ( + + ) ∗, we

can express steady-state consumption as

∗ ∗
= −( + + ) ∗.
∗
The level of that maximizes consumption is therefore, the one at which

= + + ,

or equivalently, − = +

24
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

 That is, in the Golden Rule steady state, the marginal product of capital net of
depreciation equals the rate of population growth plus the rate of technology growth.

Steady State Solutions and Growth rates /Effects of Technological Progress/

The steady-state capital and output-per effective worker is determined by

the production function and the steady-state condition – i.e. ̇ = . Setting ̇ = we can
solve for ∗ as follows:
⇒ = −( + + )
⇒ =( + + )
From equation [2], we have =
⇒ =( + + )

⇒ =
+ +

∴ ∗
= Steady State Capital per effective worker
+ +

To see what this expression implies about capital per worker, rewrite the equation as:

= =

=
∗
( )= ( )

−
∗( )= ( ) − − − − − − − − − − − − − − − −[7]
+ +
Since s, n,g and are all constants, capital per worker grows with growth in
technology.
Finding the growth rate (e.g., if = . , thus, = + )
, = + =
 Thus, the growth rate of capital per worker is the growth rate of labour
augmented technology (at a rate, )
∗
 Substituting the steady state in to the production function, = , we get the
Steady state output per-effective worker.

∴ ∗
= Steady State Output per effective worker
+ +

25
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

To see what this expression implies about output per worker, rewrite the equation as

= =
=
∗
( )= ( )
−
∗( )= ( ) − − − − − − − − − − − − − − − −[8]
+ +
 Since s, n, g, and are all constants, output per worker grows with growth
in technology.
o Finding the growth rate, we get: = + =
 Thus, the growth rate of output per worker is the growth rate of labour
augmented technology (at a rate, ).

An interesting result from equation [8] is that changes in the investment rate or the
population growth rate affect the long-run level of output per worker but do not affect the
long-run growth rate of output per worker.

Conclusion on the Effect of Technological progress on Economic Growth

With the addition of technological progress, the Solow model explains the sustained
increases in standards of living. That is, we have shown that technological progress can
lead to sustained growth in output per worker. By contrast, a high rate of saving leads to a
high rate of growth only until the steady state is reached. Once the economy is in steady
state, the rate of growth of output per worker depends only on the rate of technological
progress.
 According to the Solow model, only technological progress can explain
persistently rising living standards.

26
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

Summary: Steady-State Growth Rates in the Solow Model with Technological Progress

Variable Symbol Steady State level formula Steady State

Growth Rates
Capital per effective ∗
= ∗ =
∗
worker =
+ +
Output per effective ∗
= ∗
∗ =
worker =
+ +
Capital per worker ∗
= = .
∗ =
∗
( )= ( )
+ +
Output per worker ∗
= = . ∗
∗ =
( )= ( )
+ +
∗ 1
Aggregate Capital, = . . 1− = +
= () ( )
+ +
∗
Aggregate Output, = . . 1− = +
= () ( )
+ +

The Solow Model: Evaluation

How does the Solow model answer the key questions of growth? First, the Solow model
appeals to differences in investment rates and population growth rates and (perhaps) to
exogenous differences in technology to explain differences in per capita incomes. Why
some countries are so rich and others so poor? According to the Solow model, it is
because the rich countries invest more and have lower population growth rates, both of
which allow the rich countries to accumulate more capital per worker and thus increase
labour productivity. This hypothesis is supported by data across countries of the world.

Second, why do economies exhibit sustained growth in the Solow model? The answer is
technological progress. Without technological progress, per capita growth will eventually
cease as diminishing returns to capital set in. Technological progress, however, can offset
the tendency for the marginal product of capital to fall, and in the long run, countries
exhibit per capita growth rate of technological progress.

How, then, does the Solow model account for differences in growth rates across
countries? At this juncture, the Solow model appeals to the transitional dynamics. An
economy with a capital-technology ratio below its long-run level will grow rapidly until

27
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

the capital-technology ratio reaches its steady-state level. This reasoning may help
explain why some countries such as Japan and Germany, which had their capital stocks
wiped out by World War II, have grown more rapidly than the United States over the last
fifty years. Or it may explain why an economy that increases its investment rate will grow
rapidly as it makes the transition to a higher output-technology ratio. This explanation
may work well for countries such as South Korea, Singapore, and Taiwan which increased
their investment rates dramatically since 1950.

6.2.3. Endogenous Growth Theory

From the Solow growth model we saw that for an economy that exhibits CRS that simply
accumulating factors of production will not lead to permanently growing values of
capital, output, and consumption per capita. We also saw that while there was some
evidence supportive of the Solow growth model there was also quite a bit of evidence
which suggested that quite a bit still needed explaining, especially why countries kept on
growing and what was causing technological change to occur.

Endogenous Growth theory began with the thesis of Paul Romer at the University
of Chicago in 1986, and has flourished ever since. His main insight was that technological
change is caused by the deliberate actions of people and that it responds to economic
incentives as does any economic activity. He and others after him also explored what
impact this view of technological change would have for economic growth.

 Models that try to explain the source and progress of technological progress
often go by the label endogenous growth theory, because they reject the
Solow model’s assumption of exogenous technological change.
o Growth in this model is driven by technological change that arises from
intentional investment decisions on R&D by profit-maximizing agents.

Some Characteristics of Technology:

 It requires huge investment: E.g: the creation of Boeing 747 jet aircraft, cost was
about $1.2 billion.
 Free rider problem: ownership of R&D outcomes (technology) is difficult. There is
certain degree of non-exclusiveness problem. This in turn discourages reinvestment
on R&D.
 Capital Spillover Effects: Romer’s central insight was that the benefits of some
forms of capital not only accrued to the individual with the capital but spilled over to
others. Now, let’s look how human capital produces Spillovers on others as an

28
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

example. That is, why the accumulation of human capital may not only benefit the
individual accumulating it, but others around him or her:

i. Education:

Education increases the skills and intellectual capabilities of people and it is possible that
there are spillovers in knowledge generation between the more educated. Knowledge
generation is probably much greater in specialist research labs, than if each person
worked alone because they can discuss ideas with other skilled people in the area they are
working on. It may also be easier to acquire human capital when the people around you
have a lot of human capital themselves (i.e. they help a person learn much faster (that is
at a lower cost) than if the person had to learn by themselves).
ii. Learning-By-Doing

As firms produce more of a good, people in the work-force learn how to make it more
efficiently. People share their experiences (that is, their human capital) with each other,
helping them avoid errors and helping them find better ways of producing the goods.
Thus, the human capital acquired by a person not only is an input directly into producing
output, but also acts to increase the level of technology of the economy due to the
resulting capital spillovers.
Note that: Capital spillovers imply that the aggregate production function exhibits
Increasing Returns to Scale, contrary to the Solow model.
Implications for Government Policies
1. Government Actions May Affect Growth Rates
What happens if the government subsidizes R&D? This causes that the cost of
innovation to decreases, so that innovative activity increases. This is likely to lead to a
higher rate of technological progress, and, therefore, a higher growth rate in output
per worker.
2. Importance of Property Rights: For people and firms to have an incentive to invest
in R&D and create new techniques or products, they need to have some security over
the ownership of the capital they sacrifice to create.

29
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

A Simple Endogenous Growth Model: The AK Model

The model’s structure is the same as the Solow growth model with the exception of the
production function (I.e., The production function is modified so that = ). Thus, the
aggregate production function is now:
= −−−−−−−−−−−[ ]
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.
Notice that this production function does not exhibit the property of 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.
 Now let’s see how this production function relates to economic growth. As before,
we assume a fraction s of income is saved and invested. We therefore describe
capital accumulation with an equation similar to that we used previously:

̇ = − −−−−−−−−−[ ]
 This equation states that Capital grows with investment ( = ), and a part of it is
destroyed each period with depreciation ( ).

Combining equation [2] with the = production function, we obtain, after a bit of
manipulation,
̇ ̇
= = − −−−−−−−[ ]

̇
What determines the growth rate of output, ?

Diagram for the AK Model

30
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

 Notice from [3] that, as long as > , the economy’s income grows forever, even
without the assumption of exogenous technological progress.
Consider the above figure drawn for the AK model.
 The line reflects the amount of investment that has to occur just to replace
the depreciation of the existing capital stock.
 The curve is total investment as a function of the capital stock. Notice that
because Y is linear in K, this curve is actually a straight line, a key property of
the AK model. We assume that total investment is larger than total
depreciation, as drawn.
Consider an economy that starts at point . In this economy, because total
investment is larger than depreciation, the capital stock grows. Over time, this growth
continues: at every point to the right of , total investment is larger than
depreciation. Therefore, the capital stock is always growing, and growth in the
model never stops.
̇ ̇
 The simple algebra = = − reveals a key result of the AK

growth model: the growth rate of the economy is an increasing function of

the investment rate. Therefore, government policies that increase the
investment rate of an economy permanently will increase the growth rate of
the economy permanently.
This result can be interpreted in the context of the Solow model with < 1. Recall
that in this case, the line is a curve, and the steady state occurs when =
(since we have assumed n = 0). The parameter α measures the “curvature” of the
curve: if α is small, then the curvature is rapid, and intersects at a “low” value of
K*. On the other hand, the larger α is, the further away the steady-state value, K*, is
from K0. This implies that the transition to steady state is longer. The case of = is
the limiting case, in which the transition dynamics never end. In this way, the AK
model generates growth endogenously. That is, we need not assume that anything
in the model grows at some exogenous rate in order to generate per capita growth-
certainly not technology.

Thus, a simple change in the production function can alter dramatically the predictions
about economic growth.
 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.

31
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

But is it reasonable to abandon the assumption of diminishing returns to capital? The

answer depends on how we interpret the variable K in the production function,
= . If we take the traditional view that K includes only the economy’s stock of
plants and equipment, then it is natural to assume diminishing returns. Giving 10
computers to each worker does not make the worker 10 times as productive as he or
she is with one computer. Advocates of endogenous growth theory, however, argue that
the assumption of constant (rather than diminishing) returns to capital is more
palatable if is interpreted more broadly.
Perhaps the best case for the endogenous growth model is to view knowledge as
a type of capital. Clearly, knowledge is an important input into the economy’s
production—both its production of goods and services and its production of new
knowledge. Compared to other forms of capital, however, it is less natural to assume
that knowledge exhibits the property of diminishing returns. (Indeed, the increasing
pace of scientific and technological innovation over the past few centuries has led
some economists to argue that there are increasing returns to knowledge.)
 If we accept the view that knowledge is a type of capital, then this endogenous
growth model with its assumption of constant returns to capital becomes a
more plausible description of long-run economic growth.

Conclusion

Endogenous growth theories attempt to explain the rate of technological progress by

explaining the decisions that determine the creation of knowledge through research and
development. By contrast, the Solow model simply took this rate as exogenous. In the
Solow model, the saving rate affects 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, many endogenous growth models in essence
assume that there are constant (rather than diminishing) returns to capital, interpreted to
include knowledge. Hence, changes in the saving rate can lead to persistent growth.

Review Exercises
1. Explain the growth implication of the Harrod-Domar growth model.
2. In the Solow model, how does the saving rate affect the steady-state level of income?
How does it affect the steady-state rate of growth?
3. Why might an economic policymaker choose the Golden Rule level of capital?

32
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

4. In the Solow model, how does the rate of population growth affect the steady-state
level of income? How does it affect the steady-state rate of growth?
5. In the Solow model with technology, what determines the steady-state rate of growth
of income per worker?
6. How do you determine whether an economy has more or less capital than in the
Golden Rule steady state in the Solow model with technology?
7. How does endogenous growth theory explain persistent growth without the
assumption of exogenous technological progress? How does this differ from the Solow
model?

II. Workout Questions

1. Country A and country B both have the production function, = ( , ) = / / .

A. Does this production function have constant returns to scale? Explain.
B. What is the per-worker production function?
C. Assume that neither country experiences population growth nor technological
progress and that 5 percent of capital depreciates each year. Assume further that
country A saves 10 percent of output each year and country B saves 20 percent of
output each year. Using your answer from part (b) and the steady-state condition
that investment equals depreciation, find the steady-state level of capital per worker
for each country. Then find the steady-state levels of income per worker and
consumption per worker.
2. Consider an economy described by the production function = ( , ) = . .
A. What is the per-worker production function?
B. Assuming no population growth or technological progress, find the steady-state
capital stock per worker, output per worker, and consumption per worker as
functions of the saving rate and the depreciation rate.
C. Suppose the saving rate, s= 40 percent, and depreciation rate, δ= 5 percent.
i. Calculate the steady state level of capital per worker and output per worker.
ii. What are the golden rule levels of capital accumulation and consumption for
this economy?
iii. Explain whether the economy’s steady state capital is more or less than the
Golden Rule steady state?
3. An economy described by the Solow growth model has the following production
function: = .
A. Solve for the steady-state value of as a function of , , , and .
B. A developed country has a saving rate of 28 percent and a population growth rate
of 1 percent per year. A less-developed country has a saving rate of 10 percent

33
 CHAPTER SIX: MODELS OF MACROECONOMICS GROWTH

and a population growth rate of 4 percent per year. In both countries, = 0.02
and = 0.04. Find the steady-state value of for each country.
C. What policies might the less-developed country pursue to raise its level of
income?
4. Suppose the production function of two countries (developed and less developed) is given
/ /
as; = ( , )= . The capital stock grows as the usual Neo-classical capital
accumulation equation (i.e., = ̇ = − ), where δ is depreciation rate, s is saving rate,
and t is time.

A. Assuming that there is no technological progress, find the steady-state capital

stock per worker, output per worker as functions of the saving rate, population
growth rate and the depreciation rate (2.5 points).
B. A developed country has a saving rate (s) of 36 percent and a population growth
rate (n) of 1 percent per year. A less-developed country has a saving rate of 18
percent and a population growth rate of 4 percent per year. In both countries 5
percent of the capital stock depreciates each year.
 Find the steady-state value of capital stock and output per worker and the
corresponding steady state consumption per worker for each country (3
points).
C. In accordance with the Solow Growth Model, what are the implications of the
results in A and B above? Explain what policies might the less-developed country
pursue to raise its per capita income? (2 points).
5. Suppose the production function of a hypothetical economy is given by =
( ) ; where is output, is total labour force, is total capital stock, and is
labour-augmenting technology. The capital stock of this hypothetical economy grows

as the usual Neo-classical capital accumulation equation (i.e., = ̇ = − ),

where δ is depreciation rate, s is saving rate, and t is time.

A. Find the formula for the growth rate of output per worker, and capital per
worker. (3 points)
B. Suppose β= 2/3, s= 25 percent, δ= 3 percent, the growth rate of the labour force is
1.5 percent, labour-augmenting technology is growing at 3 percent annually.
 What is the growth rate of capital and output per worker for this economy?
(2 points)
C. What are the golden rule level of capital stock and consumption for this
economy? (2 points).

34