Lecture Notes for Topic 4: Economic Growth

James R. Maloy, Department of Economics, Spring 2020

Readings: Froyen, Chapter 20 (8th Ed. Ch. 5) has a basic form of

the Solow model. These lecture notes present a simplified version

of a very detailed presentation of the Solow model from David

Romer’s Advanced Macroeconomics postgraduate textbook.

1 Introduction

The purpose of this section is to understand the determinants of long-run

economic growth. Economic growth is the change in output (GDP) over

time. Long-run growth theory is concerned with what is typically called

trend or potential GDP, not short-run business cycles which are simply

fluctuations around this trend. Long-run growth is due to changes in supply

factors which affect the production abilities of societies (recall the vertical

long-run AS from the classical model–most growth theories are founded in

classical theory). The focus in long-run growth is usually per capita; it is

true that more population will allow more total GDP, but economic growth

theorists are typically more interested in the factors that affect output per

capita, which is known as labour productivity.

There are many important questions in growth theory, which typically

centre around two key ways of looking at growth: inter-temporally or cross-

sectionally. The first is to look at a particular society over a period of

time–why does a particular nation have more per capita output today than

in, say, 1872? The second approach asks why GDP (and GDP growth rates)

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differ amongst nations at the same time–why is the US relatively affluent

while India is relatively poor? And furthermore, will India be able to catch

up in the future? The basic way to start answering these types of questions is

determine the factors which affect economic growth. Many of the much older

theories of economic growth focused on variables such as savings and capital

formation as the driving factors. However, the Solow Model, developed

in the 1950s and the original model behind much modern growth theory,

actually argues that such factors are not the most important things.

2 Assumptions and Variables in the Solow Model

Let us begin by identifying some variables:

Y = output

K = capital stock

N = labor

A = effectiveness of labour. This is basically a variable that encom-

passes all factors that determine how effective labor is, such as knowledge,

technology, etc. Often this is just simplified to technology only.

AN = effective unit of labor

y = YAN : This is defined as output per unit of effective labor.

k = KAN : This is a similar measure of capital per unit of effective labor.

Note the difference between Y and y, and K and k. They are distinct

items; do not get them confused. Now that we have some basic ideas, let us

define an aggregate production function:

Y = F (K,AN) (1)

Output is a function of the capital stock and the amount of effective

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labour. More specifically, we can define a Cobb-Douglas production function

as:

Y = Kα(AN)1−α (2)

Note that this production function has constant returns to scale (the

exponents sum to 1). This greatly simplifies the model. However, the model

relies not on the aggregate production function Y , but on the intensive form

production function, y. Recalling our definition of y from above, dividing

the above production function by AN yields:

y = Y

AN = Kα(AN)1−α

AN =

Kα

ANα = (

K

AN )α = kα (3)

Therefore, the intensive form production function is:

y = kα (4)

This production function gives output per unit of effective labour. Graph-

ically, it looks like the familiar production function from topic 2, only note

that the axes are different; the vertical axis is y and the horizontal axis is k.

To make this a model of growth, we must introduce time into the model.

Specifically, all of our variables in the original production function (for total

output Y ) are functions of time (t):

Y = [K(t)]α[A(t)N(t)]1−α (5)

Therefore, output at time t depends on that period’s capital stock, labor

force, and the level of technology and other factors that determine labour

effectiveness. In order to finish building the model, Solow makes some as-

sumptions about the growth rates of these items. The growth rate is calcu-

lated by taking a time derivative, i.e. a derivative of each item with respect

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to time. A time derivative is symbolised by placing a dot over the variable.

Much of this math involved in this requires some knowledge of differential

equations, which if you have not taken Calc II you will not know how to do,

so don’t get bogged down in worrying about where these equations come

from as it really isn’t necessary to understand the model.

1. Assumption 1: The labor force N grows at a constant, exogenously

given rate n.

2. Assumption 2: The level of effectiveness of labor A grows at a constant,

exogenously given rate g.

3. Assumption 3: Investment in each period is some fraction of output.

We can assume that investment is equal to the fraction of output that

is saved, e.g. all savings are channelled to investment (like in the

classical loanable funds theory with a balanced government budget).

This can be expressed as I(t) = sY (t). Therefore, the amount invested

each period is determined by the savings rate s, where 0 ≤ s ≤ 1. The

savings rate s is also assumed to be constant and given exogenously.

4. Assumption 4: Finally, we must indicate how the (total) capital stock

K changes over time, i.e. the value of K̇. It is assumed that the rate

of change of the capital stock is a function of the level of investment

(from assumption three) and the depreciation of the existing capital

stock. This can be given by K̇(t) = sY (t) − δK(t)

Therefore, the rate of change of the capital stock over time is the

level of investment minus the amount of depreciation, δ. Note that

0 ≤ δ ≤ 1. It is also taken exogenously.

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3 The Solow Equation and the Solow Diagram

We are now ready to formulate the “Solow equation”. The key to the Solow

model lies with the time derivative of k (capital per unit of effective labor).

Specifically, this is given by

k̇(t) = ˙

( K(t)

A(t)N(t) ) (6)

This is solved by differentiating and plugging in values for the variables.

The mechanics are given by Romer but rather than working it out, which

makes no one happy except for the one person in this course who actually

enjoys the chain rule, I will save us a headache and simply assert that this

will yield the key Solow equation. Using the Cobb-Douglas intensive form

production function from equation 4 above, this is given by:

k̇(t) = skα − (n+ g + δ)k (7)

This equation has two components. The first component indicates that

savings increases the rate of change of capital per unit of effective labor over

time. Recalling assumption three above, this is because savings is channelled

into investment in new capital goods. The second component (which is

subtracted) indicates that the growth of labor n, labour effectiveness g, and

depreciation δ (see assumptions 1, 2 and 4 respectively) all decrease the rate

of change of capital per unit of effective labour. Showing each of these parts

of the equation separately on a graph (the Solow diagram) will help this to

make sense.

The Solow diagram shows capital per unit of effective labor k on the

horizontal axis and investment per unit of effective labor on the vertical

axis (which is a fraction of output per unit of effective labor y, as noted

5

above in assumption 3). Recalling our production function, we note that

the first term in the Solow equation is just a fraction s of the production

function. Therefore, it is sloped just like the production function. The

second component is a linear function, since we assumed that n, g and δ

are constant. Note that the two lines intersect. We shall call the value of k

where the curves intersect k∗.

It is now necessary to explain these curves in more detail. The curved

line, skα, is the actual level of investment. Since we assumed that all savings

are chanelled into investment, the fraction s of output that is saved must

be the level of investment by definition. The straight line, given by (n +

g+ δ)k is what is called break-even investment. It is the level of investment

needed simply to maintain k at its current level. What does that mean?

Well, suppose initially that there are 5 units of capital K and 5 units of

effective labour AN . Therefore, k is equal to 1. Now suppose that you have

population growth n. Specifically, suppose that now AN = 6 due to this

increase in N . In order to maintain k = 1, you must invest in one more

unit of K to make K also equal to 6. This is why it is called break-even

investment. It is the amount of investment that is needed to maintain k at

its current level. Looking back at the Solow diagram, note that at values

of k below k∗, actual investment exceeds break even investment. Therefore,

the amount of capital per unit of effective labour is growing, e.g. k is getting

bigger. Looking back at the Solow equation, note that the first term is bigger

than the second term so the rate of change of k is positive. If k is greater

than k∗, then actual investment is below break even investment, and k is

getting smaller; from the Solow equation, the second term is larger then the

first term and the rate of change of k is negative. The model thus implies

that k converges to k∗, and k will be constant at that point. This is what

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is known as the steady state or balanced growth path. The actual value of

k∗ can be calculated by setting the Solow equation equal to zero (since k

is constant at the steady state so its rate of change is zero) and solving for

k. Note that just because the economy reaches a steady state in which k

becomes constant does not mean the economy as a whole is not growing;

it just means that, in the steady state, all of the variables are growing at

a constant rate. The implication is that regardless of its initial starting

position, the economy will move to a long-run steady state, with constant

growth. We will now turn to a description of the steady state, and also

discuss how changes in the values of savings and other variables will cause

the economy to converge to a new steady state.

7

It was mentioned above that all variables grow at a constant rate in the

steady state. Some simple calculations can show that these growth rates

are:

Variable Growth Rate

N n

A g

K n+ g

AN n+ g

Y n+ g

K/N g

Y/N g

k 0

y 0

The most interesting conclusion is that output Y is growing at a constant

rate n + g. This indicates that in the steady state, the growth rate of

output depends on the growth rate of population and labour effectiveness.

Furthermore, the growth rate of output per worker Y/N depends only on

the growth rate of labour effectiveness A. This may come as a surprise; it

is often thought that the level of savings determines the growth rate of the

economy. The Solow model indicates that savings do not affect the growth

rate of output in the steady state. What we will see is that saving levels do

effect the level of output, but not the rate of growth of output in the steady

state.

Changes in savings will only affect growth rates in the short run transi-

tion period. If output was growing at 5% before the increase in savings rates,

it will be growing at 5% once the new steady state is reached. This does not

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mean that savings does not affect the economy—quite the contrary. During

the transition period, variables will be affected. This temporary change has

permanent effects on the level of variables. To think of an example, suppose

that every year you get a 10% raise on the previous year’s pay, which we say

is $100 in 2015. Now suppose that in 2016 you get a special bonus that dou-

bles your income to $200. The next year, you go back to getting 10% raises,

but this is now 10% of a much bigger value than it would be otherwise—it

is 10% of $200 instead of $110! Therefore, although after the transition

period the growth rate is back to 5%, you are at a higher level of income

thanks to the bonus. A similar type of thing is occurring with savings in the

Solow model. The Solow model indicates that, once the new steady state is

achieved, output will be growing at the same rate as before, say 5%, but the

change in savings will affect the level of output, and an increase in savings

means that we are now taking 5% of a larger level of output than we would

have had if savings had not increased. A few simple graphs can capture the

essence of a change in the savings rate. Suppose initially the economy is in

the steady state, and at time t0 there is a permanent increase in the savings

rate. This can be shown as:

We know from the Solow diagram that capital per unit of effective labour,

k, increases to the new steady state. Once the new k∗ is attained, the rate

of change of k is again zero.

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The effects of the increase in savings on the natural logarithm of Y/N

can now be seen. Output per unit of labour climbs until the new steady

state is reached, when it becomes parallel to the original growth path, once

again growing at a rate of g.

4 Conclusions of the Solow Model

We have spent considerable time deriving and working with the Solow model,

but have yet to draw any fundamental conclusions. By going a bit deeper

than we have here, it can be shown that the Solow model indicates two

main sources of differences in output per worker Y/N over time (or across

nations). Differences in capital per worker K/N and differences in the ef-

fectiveness of labour A will affect the value of output per worker Y/N .

Furthermore, the model indicates that only growth in the effectiveness of

labour can cause a permanent change in growth rates. Before Solow, many

theorists had suggested that differences in capital stocks per worker were the

reason why some nations are rich and others poor; the Solow model indicates

that different capital stocks alone can not account for differences between

nations, or for that matter differences in a particular nation over time. The

key seems to rest with differences in the effectiveness of labour. However,

the Solow model assumes that the value of A and the growth rate of A are

exogenous and constant; therefore, it takes as given the very thing which

10

causes economic growth! Furthermore, Solow just treats A as a catch-all

phrase that incorporates everything except capital. A can include knowl-

edge, technology, human capital, property rights, attitudes towards work, or

anything that can determine labour effectiveness; often it is just simplified

to technology (Froyen does this) although Solow was not that specific. This

is a key weakness in the model; the model indicates that growth depends

on A but does not give any indication of what A is or how it is determined,

which has led to later theories that expand on this concept.

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