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The Solow Growth Model- No Technical Change

I. Introduction We begin our formal study of modern macroeconomics with the Solow Growth model

named after Robert Solow who introduced it in 1956, and was awarded the Nobel Prize

for this work in 1987. The Solow Growth model is a good starting point for two reasons.

First, it is very easy to solve on account that it assumes that consumers save a fixed

fraction of their income. Indeed, as we shall see this makes the model mechanical. The

assumption of a fixed savings rate means that households are not choosing savings

optimally, and thus not necessarily maximizing utility. This simplification means the

model is not truly modern in the sense of being fully founded on microeconomics. There

are, however, profit maximizing firms and there is a general equilibrium. Later in these

chapters we will endogenize the savings of households. The version of Solow’s model

where savings are chosen optimally by utility maximizing households is called the

Neoclassical Growth model. A solution to this model with technological change and

population growth was completed by David Cass in 1965.

A second reason to start out with this model is that it turns out to be a good

measuring device for the purpose of testing theory and evaluating policy. It is a good

measuring device because it matches the long run-growth experience of the United States

since the start of the 20th

century. This is not surprising since Robert Solow (1956)

developed the model to capture the long run performance of the US economy.

Before turning to the Solow model it is instructive to review the idea of a model,

and to set clear the difference between endogenous variables, exogenous variables, and

parameters.

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A model is an abstraction of reality. No model, therefore, is true. All models will have

some variables that are endogenous and others that are exogenous. Additionally,

mathematical models will contain parameters.

An Endogenous Variable- is a variable whose value is determined within the model itself.

In macroeconomic models, among the endogenous variables are output, consumption,

wages, investment and interest rates.

An Exogenous Variable – is a variable whose value is assumed to be determined outside

the model. From the standpoint of the model, its value is taken as a given. Many

variables related to economic policy, such as the tax rate, government expenditures, and

growth rate of the money supply are examples of exogenous variables. Note that in

conducting policy analysis, the researcher will consider different values for these

endogenous variables.

In addition, the model contains parameters. These are like exogenous variables in that

their values are taken as given. Parameters are distinct from exogenous variables in that

they tend to represent things that are given by nature such as consumer preferences or

production technologies.

II. Model Structure We begin by analyzing Solow’s version of the growth model, where the savings rate of

the economy is treated parametrically. When the savings rate is treated parametrically,

the model is trivial to solve.

Almost every model is made up of people, usually referred to as households.

Additionally a model contains firms, and sometimes government. Households typically

supply inputs to firms, and buy final goods and service. Firms rent or buy inputs supplied

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by household to produce goods and services. Households and firms determine the supply

and demand for goods and inputs. Government policy also affects these supply and

demands either directly by having the government demand or supply some good, or

indirectly through altering the choices of households and firms.

People/Households

Initially, there are N0 people alive in our model world. We use Nt to denote the number

of people in the economy at date t. Typically, we assume that people have a utility

function which they try to maximize subject to a budget constraint. In the Solow growth

model, this has all been assumed away. The assumption is that people prefer more

consumption to less and save a constant fraction s of their income.

Demographics: For the Solow model, we assume that population growth is exogenously

determined with a constant rate of increase equal to 0n . More specifically,

tt NnN )1(1 (1)

Endowments: Each person in the economy is endowed with one unit of time each period

which he or she can use to work. Additionally, people are endowed with some capital

initially. In a narrow sense, capital is defined as the stock of machines and structures in

the economy. In a more general sense, capital is something of value which is used to

produce a good and service and is not completely used up in the process. The aggregate

endowment of capital in the economy is denoted by K0.

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Production Function: The economy produces a single final good using labor and

capital. The production function is given by

1])1[( NKAY t

tt (2)

The letter A is a parameter that reflects the efficiency at which a country uses its

resources to produce output. We call this parameter Total Factor Productivity (TFP).

The parameter γ is the rate of exogenous technological change. We shall first assume

that its value is zero, so that there is no technological change. After doing this and

gaining some intuition for the model’s mechanics, we shall assume a positive rate of

technological change.

Properties of Production Function

The results for the Solow model follow directly from the properties of the production

function. Figure 2.1 plots total output that an economy can produce as we vary its total

capital stock, Kt, but holding its population constant. There are two features of the figure

that are apparent. The first is that curve is upward sloping, namely output increases as

the capital input increases. This corresponds to the slope or first derivative of the

function being positive.

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The slope corresponds to the marginal (physical) product of the input. The

marginal physical product of a factor production is the increase in output associated with

an increase in the factor. Mathematically, it is the derivative of output with respect to

capital (or labor).

Figure 2 plots the marginal physical product of capital. The marginal product is

measured in terms of the output. For this production function, the marginal product is a

decreasing function that approaches zero in the limit. As we shall see, decreasing

marginal product that goes to zero is the critical property of the model that drives all of

the results.

Figure 2.1: Aggregate Production Function

Y

Kt

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The decreasing marginal product is a direct consequence of a second feature of the

production function, namely that it bows outward. This latter property is what is known

as called concavity. To say that a curve is concave means that the slope of the function

declines as we increase the independent variable, in this case K. Concavity corresponds

to a second derivative that is negative.

Concavity and the Law of Diminishing Returns

It turns out that concavity has a very important economic meaning. In particular,

concavity implies there are diminishing returns to the factor of production being

considered.

K

Figure 2.2: Marginal Product of Capital

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The Law of Diminishing Returns states that as one factor of production is increased,

holding all other factors and technology fixed, the increases in output associated with

increasing the factor (eventually) become smaller.

The law of Diminishing returns can also be restated in terms of the marginal product of

an input. Alternatively, the law of diminishing returns states that the marginal physical

product of an input (eventually) decreases.

Other Properties of Production Functions

Another property of production functions relates to the returns to scale. A production

function can either be characterized by Constant, Increasing, or Decreasing Returns to

Scale. To determine if a production functions has constant, increasing or decreasing

returns to scale, we change all of the inputs by the same fraction and determine by how

much output changes. If we double all the inputs and output exactly doubles, then the

production function is characterized by constant returns to scale; if we double all the

inputs and output increases by more than a factor 2, then the production function is

characterized by increasing returns to scale; finally, if we double all the inputs and output

increases by less than a factor 2, then the production function is characterized by

decreasing returns to scale.

Notice that the law of diminishing returns deals with the case of increasing only one

input, holding all other fixed while constant returns deals with the case of increasing all

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inputs by the same proportion. Thus it is separate for the returns to scale property of the

production function.

Capital Stock – The capital stock evolves according to the following equation:

ttt sYKK )1(1 (3)

In the above equation, δ is the depreciation rate parameter. Namely, δ represents the

fraction of machines and structures that wear out between periods. The letter, s, is the

savings rate. It is fixed and hence viewed as a parameter. The households always save a

fixed fraction of the economy’s output. (As we shall see, output in this economy is

exactly equal to household income.) The other fraction, (1-s), is of course consumed.

Summary of Solow Growth Model: Aggregate Variables The Solow Growth Model is completely described by the following four equations.

i. ttt YscN )1(

ii. 1])1[( NKAY t

tt

iii. ttt sYKK )1(1

iv. tt NnN )1(1

Solow Model – the Per Capita Variable Representation

For many purposes, it is more interesting and relevant to study the behavior of per capita

variables. This is because a change in the per capita variable is more informative of a

change in peoples’ welfare than the change in the aggregate. For example, we could have

an increase in total output with no change in per capita output on account of population

growth.

It is not difficult to transform the Solow model into its per capita representation.

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We begin by defining yY/N, kK/N. These are the per capita output and capital. In the

convention taken in this book, we will denote aggregate variables by upper case letters

and per capita by lower case letters.

To begin the transformation, we divide the aggregate production function given

by equation (ii) by Nt so as to obtain the following per capita production function:

t

t

t kAy )1()1( . (4)

This production function has the same shape as the aggregate production function

displayed in Figure 2.1.

The law of motion for the per capita capital stock equation is:

ttt sykkn )1()1( 1 (5)

This is obtained by dividing both sides of the equation for the aggregate capital stock

given by Equation (iii) by Nt . This yields

tt

t

t sykN

K )1(1 (6)

The only trick in getting from (6) to (5) is to express the left hand side of (6) in terms of

kt+1. To do this we use the population growth function which can be rewritten as

.)1( 1

1

tt NnN Substituting for Nt into (6) yields

tt

t

t sykN

Kn

)1()1(1

1 (7)

As Kt+1/Nt+1 is just kt+1, we now arrive at equation (5).

III. General Equilibrium

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Modern macroeconomics is based on general equilibrium analysis. That analysis is based

on the notion that all markets must clear simultaneously in the equilibrium. This is the

analysis we will use in this course. There are really three markets in the model: the labor

market, the capital rental market, and the goods market.

There is no money in the growth model. As such there are no nominal prices to be

determined. The prices in this economy are real prices. Namely, for each good that is

traded, its price is expressed in terms of another good in the economy. In what follows,

we will measure the prices of each good in the economy in terms of the final good, Y.

The payment to labor, wt, therefore, is the quantity of the final good paid to a unit of

labor. The payment to capital, rt, is the quantity of the final good paid to a unit of capital

rented by a firm from households. The price of the final good is trivially one.

In the Solow model, it is trivial to find the market clearing quantities of labor and

capital. This is because the supply of labor and the supply of capital are perfectly,

inelastic, i.e. vertical. As such, demand is irrelevant for determining equilibrium

quantities; labor demand and capital services demand only pin down the equilibrium

price.

Derivation of Labor Demand and Capital Services Demand.

Profit Maximization of Firms

Profits of the firm are defined as sales less wage payments less capital service payments.

More specifically,

Profits: ttttt

t

t KrNwNKA 1])1[( (8)

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Standard microeconomic theory states that profits are maximized at the point where the

marginal product of the input equals its marginal cost. The marginal product of labor or

capital is just the derivative of the production function with respect to that variable. The

marginal cost is the wage rate in the case of labor and the rental rate in the case of capital.

Thus, the profit maximizing conditions are:

Labor Demand: t

ttt

t

tN

YNKAw )1()1()1( )1( (9)

Capital Demand t

ttt

t

tK

YNKAr 11)1()1( (10)

Steady State or Balanced Growth Path Equilibrium

Often we will begin by characterizing a particular type of equilibrium which is referred to

as a steady state equilibrium or balanced growth equilibrium in the case that variables

grow. The steady state equilibrium is the easiest one to characterize as it requires that

variables either never change or if they change, change by a constant percentage every

period.

For the economy to be at the steady state equilibrium or its balanced growth path,

it must be the case that the economy starts off with the “right” initial conditions. To

illustrate, the concept of a steady state equilibrium and balanced growth equilibrium

consider the following simple demographic model.

Simple Demographic model

1. People live two periods and are either young or old.

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2. Young people each have n children.

3. Old people die at the end of the period; young people all survive into old age.

Example 1. Steady State Equilibrium.

Here we assume that each young person has one child and that initially there are 50

young people and 50 old people alive. This gives rise to the population dynamics shown

in the below table.

Steady State Example

t=0 t=1 t=2

Young 50 50 50

Old 50 50 50

Example 1 is clearly a steady-state equilibrium- the population is constant in each period

at 100. Moreover, there are always 50 young people and 50 old people in any period.

Example 2. No Steady State Equilibrium, but Convergence

Here we assume that each young person has one child and that initially there are 50

young people and 25 old people alive. This gives rise to the population dynamics shown

in the below table.

Non-Steady State Example

t=0 t=1 t=2

Young 50 50 50

Old 25 50 50

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This is not a steady state equilibrium as the total population and the distribution of young

and old agents is not the same in all periods starting with the first t=0. Although the

economy is not in its steady state in period 1, it reaches it in the second period. Hence

the economy converges to its steady state equilibrium.

Example 3. Balanced Growth Equilibrium

Here we assume that each young person has two children and that initially there are 100

young people and 50 old people alive. This gives rise to the population dynamics shown

in the below table.

Balanced Growth Example

t=0 t=1 t=2

Young 100 200 400

Old 50 100 200

This is an example of a balanced growth path. What we have is that the total population

as well the number of young people and old people all double each period.

Reinforcement of Concepts: If you understand these concepts, can you

say what the equilibrium would look like in the above example if we

started with an equal number of young and old people? Does this world

converge to the steady state where population doubles? What if we

allowed people to live 3 periods where each person has one child in the

middle period?

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IV. Solving the Model

We first solve the model when there is no exogenous technological change, so the γ=0.

We do this to gain some intuition for why it is important to allow for exogenous

technological change in the model. We first show that the economy has a steady state,

but not a balanced growth path equilibrium when TFP does not change. In showing this,

we are implicitly answering the question whether it is possible to have sustained

increases in per capita output in the long-run.

IV.a Steady State or Balanced Growth Path?

We can easily show with some algebra that no sustained growth is possible. Start

with the equation that gives the law of motion for the per capita capital stock,

ttt sykkn )1()1( 1 (11)

and divide both sides by kt. This yields

t

t

t

t

k

ys

k

kn )1()1( 1 . (12)

Now in a steady-state or balanced growth path, kt+1/kt=1+gk where gk ≥ 0. We now invoke

this condition and so we substitute our for kt+1/kt in the above equation. This yields

t

tk

k

ysgn )1()1)(1( (13)

In the above equation we purposely have moved (1-δ) to the left-hand side of the

inequality. By doing this, we ensure that there are no time subscripts associated with the

left-hand side of the equation, which means it is constant through the time. The right

hand side, therefore, must be a constant. The only way for this to be a constant is if y and

k grow at the same rate, g.

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We now use the per capita production function with result that

)1(// 11 gkkyy tttt to solve for long-run growth rate, g. Recall, that he per capita

production function is tt kAy . This relation holds for all periods, including t+1 so that

11 tt kAy . Taking the ratio of the date t+1 to date t per capita output yields

t

t

t

t

t

t

k

k

Ak

Ak

y

y 111 . (14)

As the growth rate of output and capital per person are the same, it follows that

)1(1 gg . Since 0 <θ < 1, the only value for the growth rate that satisfies this

equation is g =0. Thus, we have a steady-state equilibrium, rather than a balanced growth

path.

What is the intuition for this result that in the long-run it is not possible to have

increases in per capita output? In this version of the Solow model, there only way output

per capita can be increased is to give each person more capital. Recall, however, that the

marginal product of capital is decreasing in the stock of capital and goes to zero as the

capital stock approaches infinity. With such a low marginal product (i.e. increase in per

capita output), the increase in output associated with adding another unit of capital is just

too small to generate much of an increase in output. In fact, even if we were to save one

hundred percent of our output, the increase in output associated with a one unit

investment eventually would be so small, it would be insufficient to cover the

depreciation on the existing capital stock. Thus, there is a steady-state equilibrium with

zero growth of per capita output, per capita capital and per capita consumption.

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The key to this result, therefore, is the diminishing product of capital that goes to

zero in the limit. In the next chapter, we will see that one branch of growth models

breaks this diminishing returns so as to obtain sustained increases in output.

IV.b Steady State Solution and Comparative Statics

We have shown that the growth rate of all variables is zero if we happen to start out with

the “right” initial capital stock, ssk0 . Obviously, we would like to know what the value for

the steady state per capita capital stock is, and how it is affected by the parameters of the

model. For instance, if an economy has a high savings rate will it have a high steady state

capital stock and living standard? Comparisons of steady states (or balanced growth

paths) for alternative values of the exogenous variables or parameters is referred to as

Comparative Statics. We now examine the model steady states for alternative savings

rates, population growth rates and depreciation rates. lytically.

Graphic Solution

It is possible to characterize the steady state per capita capital stock graphically. We do

this by using the equation for the per capita capital stock and the result that kt+1=kt=kss

.

This is

sykkn )1()1( . (15)

Next, we us the production function and substitute out for y. This is

sAkkkn )1()1( . (16)

Rearranging terms yields,

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sAkkn )( . (17)

The steady state capital stock must satisfy this equation. The left hand side of this

equation represents the amount of savings that must be done to keep the per capita capital

stock fixed at some given level k. To keep the per capita capital stock at k, we must

replace the amount that wears out, δk and in addition give each new-born of which there

is number n, the k units of capital. The right hand side is the actual savings done in the

economy. The steady state capital stock is, thus, the capital stock that for which actual

savings equals the amount needed to keep the per capita capital stock fixed.

The solution can be shown graphically by plotting the actual savings curve and

the needed savings curve. The needed savings curve is just a straight line through the

origin with slope (n+δ). The actual savings curve mimics the shape of the per capita

production function. Because the savings rate s is less than one, the actual savings curve

lies everywhere below the per capita production function. The steady state capital stock

is shown graphically as the intersection of the two curves.

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Steady-State Comparisons

As the position of these two curves depends on n, s, A and δ, it follows that the steady

state capital stock will depend on their values. Comparisons of the steady state

associated with changes in the model parameters are referred to as comparative statics.

Graphically, a higher savings rate, s, or TFP, A, will have the effect of shifting out the

actual savings curve. With a higher savings curve, the steady state capital stock will be

higher. This is shown in the below figure:

k

Figure 2.3: Determination of Steady State Capital Stock

kss

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If the population growth rate or the depreciation rate were higher, then the steady state

capital stock per person would be lower. This is shown in the next figure, Figure 2.5,

where the effect of a higher n or δ is to increase the slope of the needed savings curve.

Figure 2.4: Increase in s or A

k

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Algebraic Derivation of Steady State Solution

Alternatively, we can solve out explicitly for the capital stock for which actual savings

equals the needed amount,

sAkkn )( . (18)

Dividing both sides by k and using the law of exponents yields

1)( sAkn . (19)

Dividing both sides by kθ-1

and then raising both sides to the power 1/(1-θ) allows us to

solve explicitly for kss

. This is

)1/(1

n

sAk . (20)

Figure 2.5: Increase in n or δ

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As the solution for kss

above shows, a higher savings rate, a higher TFP, a lower

population growth rate, or a lower depreciation rate all have the effect of increasing the

steady state per capita capital stock.

IV.b Transitional Dynamics:

A natural question to ask is what would the equilibrium look like if the economy did not

start out with per capita stock of capital given by (20)? Would it converge to this steady

state capital stock?

The answer in the Solow model is “yes”. The reason for this relates to the

diminishing returns property of the production function with respect to the capital input.

If you take two countries that are identical in every way except for their initial capital

stocks, the one with the lower initial capital stock has a higher marginal product of

capital. Thus, even though they save the same fraction of output, the one with the lower

initial capital stock experiences a larger increase in its output. As a result, this country

catches up with the other. Graphically, if we assume that country 1 is on the steady state

starting at time 0, but country 2 is not, over time the path of per capita output of the two

countries looks as follows:

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We can show convergence will occur either graphically or analytically. For the

graphical exposition, we can show convergence by using the needed savings and actual

savings curves that we used to graphically solve for the steady state capital stock (Figure

2.3). If the economy starts out at k0 < kss

, then actual savings exceeds the needed savings

to keep the capital stock per person fixed at k0. It follows that tomorrow’s capital stock,

k1, must be larger than k0. At k1, it is also the case that actual savings exceeds the needed

savings to keep the capital stock fixed at k1, so again it follows that the capital stock must

increase. As long as the capital stock is below the steady state level, it is the case that

actual savings is greater than the needed savings, so the per capita capital stock must

increase. As a result, the capital stock must converge to the steady state level.

Country 2

Country 1 kt

0

Figure 2.6: Convergence to the Steady State

time

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The reverse holds in the case in which k0 > kss

. We still have convergence, but from

above.

Analytically, we simply use the law of motion for the capital stock per capita

given by Equation (11) where we have substituted in for the per capita production

function. This is

ttt sAkkkn )1()1( 1 (21)

Our first step is to divide both sides by (1+n). This yields

ttt k

n

sAk

nk

11

11 (22)

Equation (21) expresses kt+1 as a function g(kt) given by the right hand side of the

expression, i.e.,

k

Figure 2.7: Convergence to Steady State

k0 k1 k2 kss

Needed for k0

Actual at k0

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ttt k

n

sAk

nkg

11

1)( . (23)

The first derivate of g(kt) is

111

1)('

tt k

n

sA

nkg . Given that θ>1, it follows that g

has a positive slope.

The second derivative is equal to 2)1(1

)(''

tt kn

sAkg . The second derivate is

negative given our assumption that 0<θ<1. A negative second derivative is the property

of a concave curve, i.e., given the curve increases, it bends inward, or more rigorosly, that

if you draw a line between any two points on the curve, the point on the line will lie

below the point on the curve. Furthermore, we have g(0) = 0. Thus the function g is a

strictly increasing, strictly concave function that originates from the origin.

Now a steady state is a capital stock where kt+1=kt. Graphically, points where the

value on the x-axis equals the value on the y-axis is just a line from the origin with a

slope equal to 1, or a 450 degree line out of the origin. The function g(kt) and the 45

0 line

are shown graphically in Figure 2.7. Due to the concavity, the function g(k) cuts the 450

from above which implies that the economy converges to the steady state starting with

any initial capital stock. You can see this by starting with some arbitrary k0, and then

finding k1 from the g(k0) function. Now use the 450 line to show k1 on the horizontal axis,

and then find k2. If you keep on doing this, you will see the economy comes into kss

. We

call this convergence property “Global stability”.

Reinforcement: If you understand this, try changing the shape of the

curve so that it cuts the 450 line from below. How many steady states

are there? Is it globally stable or instable?

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Conclusion

The important conclusion from this analysis is that absent technological changes,

sustained increases in living standards are not possible in the Solow Growth model.

Temporary growth is possible, as an economy transitions from a low steady state to a

higher one. The key feature of the Solow model that delivers this result is the

diminishing marginal product of capital that approaches zero in the limit. In the next

chapter, we will see how allowing for exogenous increases in technology changes this

conclusion.

References:

Robert Solow. 1956.

kt

g(kt)

kt+1

Figure 2.7: Transitional Dynamics

kss

450

k0

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