Post on 20-Dec-2015
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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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