Growth Theory Notes
Professor Mark Pingle
These notes describe how to construct a standard growth theory model of the Solow-Swan
type. They also illustrate how one can use the model to derive theoretical conclusions about how
the economy works. The model is dynamic, meaning time plays an essential role. In particular, the
Solow-Swan model captures the impact of capital accumulation as saving flows into investment
over time. As Domar noted, when he constructed a precursor to the Solow-Swan model, investment
has a “dual character.” It provides “demand” in the present as investment goods are purchased, and
it provides additional “supply” in the future as the investment goods accumulate as capital and
thereby enhance production. The model can be used to predict how changes labor, capital, and
technology effect production and consumption over time, and predict how the economy’s real
interest rate level and real wage level will be effected.
Constructing a growth theory model of the Solow-Swan type involves (1) specifying a
production process, and (2) specifying a capital market equilibrium condition. The usual approach
involves first presenting the production process and capital market in the aggregate, or for the
economy as a whole. Then, the model is converted to a per worker, or per capita, basis, because it
makes the model easier to analyze. These notes proceed by first construction a model of
production, and then constructing a model of the capital market.
We extend the most basic model, so we can examine how the capital accumulation process
affects the real wage level, real interest rate level, and average consumption level. We can predict
the real wage and real interest rate levels by imposing the assumption that producers maximize
profit as they choose the levels of labor and capital to employ. We can track consumption because
it is the residual of income, after saving. The real wage, real interest rate, and consumption
variables are “auxiliary variables,” labeled as such because they are not core to the model, but rather
are predicted from equations that are added onto the core model.
After the production and capital market models have been constructed, and after the
auxiliary equations are presented for the real wage, real interest rate, and consumption, these notes
proceed by summarizing the model, both in the aggregate form and in the per capita form.
Constructing the production and capital market portions of the growth model separately should help
the student see how one actually constructs a model. Formally summarizing the models should also
be beneficial to the student in that it allows one to see, in an overarching way, what exogenous
forces drive the model versus what endogenous economic outcomes are determined by the model.
Once the model has been summarized, the model analyzed. The model reduces to a single
variable, nonlinear differential equation. So, the analysis involves using the mathematical tools
available for examining such a system. In particular, we can examine the model’s steady state, and
we can perform a comparative static analysis to see how changes in the exogenous variables of the
model impact the endogenous economic outcomes. We can determine what restriction must hold
in order for the model’s steady state to be stable. The Inada conditions are discussed, because they
are usually the conditions imposed to ensure that the steady state is stable.
One interesting characteristic of the Solow-Swan model is that capital can over accumulate,
so that the economy is inefficient. We discuss this dynamic inefficiency by examining what is
referred to as the Golden Rule, which is a steady state where consumption per capita is maximized.
This sets the state for studying optimal growth theory, where the savings rate is not exogenous to
the model, but rather is set by consumers within the model in a way that maximizes their well being.
1
We can characterize the behavior of the economy as it approaches the steady state. This
involves learning how to apply a Taylor series expansion. In economic terms, the analysis allows
one to identify what will make an economy converge more quickly toward its steady state. We
show how this is done.
We conclude these notes by examining how well the Solow-Swan model explains the
observed growth of real world economies. A primary prediction of the model is that the per person
output level of two economies should converge to the same level, if the two economies have the
same characteristics. Do we tend to see this “conditional convergence” in the real world?
We begin with production.
1. Production
The level of output Y depends upon the level of capital K and the level of “effective labor” AL :
P1
Y = F ( K , AL )
Production exhibits constant returns to scale
P2
λY = F (λK , λAL )
Production restrictions
P3
∂Y
>0
∂K
∂Y
>0
∂[ AL ]
also written FK > 0
More capital leads to more output
also written FAL > 0
More labor, or more
labor effectiveness, leads to more output
∂  ∂Y 
< 0 also written FKK < 0
∂K  ∂K 
∂  ∂Y 
< 0 also written FAL, AL < 0
∂[ AL ]  ∂[ AL ]
Diminishing returns to capital
Diminishing returns to effective labor
Converting the aggregate form to per capita form
1
Y
K
K
Y
, so P2 becomes
= F(
,1) . Then, define k ≡
and y ≡
, so we can write
AL
AL
AL
AL
AL
the prior equation as y = F (k ,1) . Then, defining a new function f ( k ) = F (k ,1) , we can
write y = f (k ) . That is, if production exhibits constant returns to scale, then the production function
given as P1 implies
Set λ =
P4
y = f (k )
To complete converting the production function to per capita form, we must determine what
the restrictions P3 imply for the per capita production function P4. Rewriting P4 using the
2
aggregate variables, we have
Y
K
= f( ).
AL
AL
Now, differentiate, allowing Y and K to change,
1
1
dY
dY = f '
dK . Rearranging, we have
= f ' . Because we held variables other
AL
AL
dK
than Y and K constant, this is a partial derivate result. So, we can more appropriately write
∂Y
∂Y
= f ' . Using this last result, we can see that the restriction
> 0 in P3 implies
∂K
∂K
obtaining
P5
f '> 0 .
To find the implication of the restriction FKK < 0 , we need to differentiate FK = f ' (
K
)
AL
1
. Since AL > 0 by assumption, the restriction
AL
< 0 in the aggregate model, implies in the per capita the restriction
with respect to K . Doing so, we obtain FKK = f ' '
FKK
P6
f ''< 0 .
To summarize, production in the per capita model is characterized by the production
function P4 and the restrictions P5 and P6. For reasons that will become clear, it is also common to
impose the “Inada conditions” on the production function, in addition to conditions P5 and P6. The
Inada conditions can be written
P7
f ( 0) = 0 ,
lim f ' ( k ) = +∞ ,
k →0
lim f ' ( k ) = 0
k → +∞
Together the restrictions P5-P7 indication the production function is such that the per capita
production level y depends upon the per capita capital level k as shown in the figure below.
Figure 1: The shape of the per capita production function, assuming the Inada conditions
y
k
2 Capital Market
It is assumed people save a constant fraction s of their income Y so that the saving level S can be
written:
3
S = sY
CM1
Capital accumulates as investment I occurs. However, capital also depreciates. It is assumed
capital depreciates at the constant rate δ , so with capital level K , the amount of depreciation at
time t is δK . When convenient, we can recognize that the level of capital depends upon time by
addition the time argument. Doing so, we would write the level of capital as K (t ) , rather than just
writing K . The instantaneous change in capital is given by dK / dt = K ' (t ) . It usually is an extra
notational burden to include the time argument, so it is common to drop the time argument and
write the change in capital as just K ' . (In many presentations, dK / dt is written as K& . We will
use K ' for dK / dt here because the apostrophe emphasizes that this change is a derivative.)
Because the change in capital is equal to the level of investment, less the amount of depreciation,
we can write
K ' = I − δK .
CM2
With no government or foreign sector in this model, the capital market clears when saving equals
investment. The standard approach is to assume the capital market clears, which implies
I =S.
CM3
Substituting CM1 and CM2 into CM3, the capital market can be reduced to the single equation
CM4 K '+δK = sY .
Converting the aggregate form to per capita form
Using the definitions k ≡
CM5 K '
K
Y
and y ≡
, we can divide condition CM4 by AL and rewrite it as
AL
AL
1
+ δk = sy .
AL
To complete the conversion to per capita form, we must find a way to replace the term K '
CM5. This can be accomplished using the definition k ≡
1
in
AL
K
. First, take the natural log of both
AL
K
to obtain ln(k ) ≡ ln( K ) − ln( A) − ln( L) . Then, differentiate with respect to time,
AL
k ' K ' A' L'
obtaining ≡
− − . Then, introduce the assumptions
k K A L
sides of k ≡
CM6
A'
=g
A
and
CM7
L'
= n,
L
4
which indicate we are assuming that the effectiveness of labor (level of technology) grows at rate
g , while the labor level (or population) grows at rate n . Using these assumptions, we can
k' K'
K'
k
1
write ≡
− g − n . Next, multiply by k to obtain k ' ≡
k − gk − nk . Recognizing
=
,
k
K
K
K AL
1
K'
we can write k ' ≡
from CM5, condition
− gk − nk . Using this last condition to eliminate K '
AL
AL
CM5 becomes
CM8
k '+ gk + nk + δk = sy .
Equation CM8 is the capital market equilibrium condition for the model, in per capita form.
3. Auxilliary Conditions
Assuming the output Y is sold at the price level P , the revenue generated from the sale of
output in the economy (i.e., Gross Domestic Product) is PY . Assuming the wage per effective
labor unit is W , the “wage bill”, or total cost of labor, is WAL . Assuming the payment for renting
one unit of capital is R , the “capital bill,” or total cost of capital, is RK . Thinking of there being
one large producer, it then follows that the profit Π earned from production can be written
AC1 Π = PY − WAL − RK .
Substituting the production function P1 for the output level, condition AC1 becomes
AC2 Π = PF ( K , AL ) − WAL − RK .
To approximate the assumption that markets are competitive, we think of firms as being able to
freely choose the labor and capital levels, while taking the prices as given. For profit to be
maximized with respect to the input choice, the derivative of the profit function AC2, with respect
to the variable, must equal zero. The partial derivatives of the profit function with respect to labor
and capital are
AC3
∂Π
= PFK − R
∂K
and
∂Π
= PFAL − W .
∂[AL ]
Setting these partial derivatives equal to zero, we obtain the necessary conditions for profit
maximization, which are
AC4 FK =
R
,
P
FAL =
W
.
P
The conditions AC4 indicate that the payment to each factor of production, in real terms, must equal
the factor’s marginal product.
From the work leading up to condition P5, we know FK = f ' ( k ) . Defining the real interest rate
R
level as r = , we can rewrite the capital condition from AC4 as
P
5
AC5 r = f ' (k ) .
We also want to write the labor condition from P4 in a per capita form. To do so, we need a per
capita representation of the partial derivative FAL . To obtain such a representation, note that
Y
K
K
= f ( ) , which further implies Y = [ AL ] f ( ) . Differentiating this last
y = f (k ) implies
AL
AL
AL
condition
with
respect
to
the
quantity
AL ,
we
find


∂
[Y ] = ∂ [AL] f ( K ) = f (k ) + [AL] f ' (k ) − K2  = f (k ) − f ' (k )k . Defining the
FAL =
∂[AL ]
∂[ AL] 
AL 
 [ AL] 
real wage level as w =
W
, the result FAL = f (k ) − f ' (k )k implies we can rewrite the labor
P
condition from AC4 as
AC6 w = f ( k ) − f ' ( k ) k .
Together, conditions AC5 and AC6 relate the real interest and real wage levels to the per capita
capital stock level, telling us what the real interest and real wage levels must be if producers
maximize their profits in competitive markets.
To obtain an auxiliary equation for the consumption level, note that, if there is no government to tax
away income and no foreign sector where income can be transferred, then the total real income
generated from the sale of output must either be spent on consumption or saved; i.e.,
AC7 Y = C + S .
Using condition CM1 to eliminate the saving level, we obtain Y = C + sY . Defining per capita
consumption as c = C /[ AL ] , by dividing AC7 through by AL and rearranging terms, we find
AC8 c = [1 − s ]y .
The equations AC5, AC6, and AC8 are the auxiliary equations that we will add to equations P4 and
CM6 to construct or model in per capita form.
4. Summary of Models
Summary of growth model in aggregate form:
A1
Y = F ( K , AL) ,
A2
A3
A4
A5
A6
A7
A8
A9
S = sY ,
K ' = I − δK
I =S
r = FK ( K , AL )
w = FAL ( K , AL)
Y =C+S
A' = gA
L' = nL ,
λY = F (λK , λAL ) , FK > 0 , FAL > 0 , FKK < 0 , FAL, AL < 0
0 < s <1
0 <δ <1
g≥0
n≥0
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Endogenous (9): Y , S , K ' , I , r , w , C , A' , L'
Exogenous (4): s , δ , g , n
Predetermined (3): K , L , A
Initial Conditions (3): K 0 , L0 , A0
When the economy begins at time t = 0 , the initial levels for capital, labor, and technology are
given as K 0 , L0 , and A0 . Similarly, at any particular time t > 0 , the initial levels for capital,
labor, and technology are predetermined (which means already determined) as K , L , and A . The
changes in capital, labor, and technology are exogenously determined by the model at each point in
time.
The following “neoclassical” story explains how the model’s nine endogenous variables are
determined. The change in technology A' is assumed to grow at a constant rate, as given by
equation A8. (That is, this model does not explain why technology changes.) The changes in
capital K ' and labor L' are determined by equations A5 and A6 as producers choose the optimal
changes in labor and capital in response to newly observed levels for the real interest rate r and real
wage w . With labor and capital levels predetermined, the production function A1 determines the
level of output Y . This output is sold and generates real income. From this income equation A2
determines the savings level S . The investment level is determined by equation A3, being equal to
the additional capital K ' demanded by produces less depreciation. The real interest rate r is
assumed to adjust to clear the capital market (i.e., equate investment demand with the level of
saving), meaning the real interest rate level is determined by equation A4. The economy’s labor
supply is assumed to exogenously increase at rate g , as given by equation A9. Because the change
in labor supply given by equation A9 must equal the change in labor demand obtained from
equation A6, one can think of equation A9 as determining the real wage level w , the real wage
level that clears the labor market. Equation A7 determines the level of consumption C as being
the amount of real income that is not saved.
This growth model in aggregate form is difficult to analyze because it is a nonlinear differential
equation system with three state variables. The constant returns to scale assumption is significant
analytically because it allows this three state variable system to be reduced to the single state
variable per capita form system. (The per capita form is also referred to as the intensive form.)
Summary of growth model in per capita form, or intensive form:
PC1
PC2
PC3
PC4
PC5
y = f (k )
f '> 0 ,
k ' = sy − gk − nk − δk ,
r = f ' (k )
w = f (k ) − f ' (k )k
c = [1 − s ]y
f ''< 0 ,
f (0) = 0 , lim f ' ( k ) = +∞ ,
k →0
0 < s <1, 0 < δ <1,
lim f ' ( k ) = 0
k → +∞
g ≥0, n≥0
Endogenous (9): y , k ' , r , w , c
Exogenous (4): s , δ , g , n
Predetermined (3): k
Initial Conditions (3): k 0
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The growth model in per capita form has the same exogenous variables as the model in aggregate
form. These variables---the savings rate s , the depreciation rate δ , the rate of technological
improvement g , and the population growth rate n --- are the driving force of the model. The
primary explanation offered by the model for different long run observed economic outcomes (i.e.,
different values for the endogenous variables of the model) is different values for the exogenous
variables of the model.
Because the initial levels K 0 , L0 , and A0 are given in the aggregate model, we can use these to
obtain the initial per capita capital stock level k 0 for the per capita model. The level of the per
capita capital stock k is predetermined at each point in time.
Having told the classical story for how the economy works for the aggregate model, here it will
be explained how the endogenous variables for the per capita model are determined from the
model’s five equations. With k predetermined, and exogenous variables given, equation PC1
determines the per capita output level y ; equation PC3 determines the real interest rate level r , and
equation PC4 determines the real wage level w . Know the per capita output level y from equation
PC1, we can obtain the per capita consumption level from equation PC5. Knowing the levels y and
k , equation PC2 determines the change in per capita capital stock k ' .
5. Reduced form for the Per Capita Model
Using equation PC1 to eliminate the y from equation PC2, we obtain the following model.
RF1 k ' = sf (k ) − gk − nk − δk
Endogenous (9): k '
Exogenous (4): s , δ , g , n
Predetermined (3): k
Initial Conditions (3): k 0
The only endogenous variable in this model is the change in per capita capital stock k ' . Equation
RF1 is a nonlinear differential equation with the single state variable k . Using standard techniques
for analyzing a differential equation of this type, we can learn how the path for k will evolve over
time, and we can learn how changes in the exogenous variables will affect that path. Once we
know the path followed by the variable k , we can use the following “auxiliary” equations to
determine the paths of y , r , w , and c .
RF2
y = f (k )
RF3 r = f ' ( k )
RF4 w = f ( k ) − f ' ( k ) k
RF5 c = [1 − s ]y
6. Analyzing the Steady State
8
Finding the Steady States
The steady state for the state variable k is the state where the variable is not changing, or where
k ' = 0 . Setting k ' = 0 in RF1, we know
SS1 sf ( k ) = [g + n + δ ]k .
There are five variables in equation SS1, the model’s four exogenous variables and the state
variable. Equation SS1 determines the steady state value(s) of the state variable as a function of the
exogenous variables. One value of k for which equation SS1 holds is k = 0 . This steady state is
not so interesting in that this result is simply telling us that if no capital initially exists, then non will
accumulate over time, (which is because capital is assumed to be essential in the production
process). Let k denote any steady state that may exist such that k is positive. Using the Inada
conditions, we can deduce that such a positive solution to equation SS1 exists, and we can conclude
it is unique. This is the steady state that will be of interest to us.
To understand why the Inada conditions ensure a unique steady state, consider Figure 2 below. The
right side of equation SS1 is linear in k , with a zero intercept, as shown in the figure. The
assumption f (0) = 0 implies the left side of equation SS1 also has a zero intercept. The
conditions f ' > 0 and f ' ' < 0 ensure that the production function increases at a decreasing rate as k
increases, as shown in the figure. If it were true that f ' (0) < g + n + δ , then the only steady state
would be associated with k = 0 , because the two curves sf ( k ) and [g + n + δ ]k would only
intersect at the point (0,0) in Figure 2. The Inada condition lim f ' ( k ) = +∞ ensures that the sf ( k )
k →0
curve initial rises above the [g + n + δ ]k curve. The Inada condition lim f ' ( k ) = 0 ensures that the
k → +∞
sf ( k ) curve eventually intersects the [g + n + δ ]k curve. Because f ' > 0 and f ' ' < 0 for all k , we
know that this intersection at a capital level k > 0 only occurs once, which implies k is unique, as
shown in Figure 2.
Figure 2: The steady state for the Solow-Swan growth model
y
[g + n + δ ]k
f (k )
sf (k )
k
k
Stability of the Steady State
The Inada conditions also ensure that the steady tate associated with k = k is stable.
A steady state is “locally stable” if the state variable moves toward the steady state value for the
variable when it is in the “neighborhood” of the steady state. Define the function Ψ ( k ) so
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k ' = Ψ (k ) . Notice that, if Ψ ' ( k ) < 0 , then the steady state k is locally stable. This is because
Ψ ' ( k ) < 0 implies k ' > 0 when k < k and k ' < 0 when k > k .
For our model given by SS1, Ψ ( k ) = sf ( k ) − [g + n + δ ]k . So, Ψ ' ( k ) = sf ' ( k ) − [g + n + δ ] .
Because Ψ ' ( k ) = sf ' ( k ) − [g + n + δ ] , our steady state k is locally stable if sf ' (k ) − [g + n + δ ] < 0 .
Notice, in Figure 2, that the slope of the sf ( k ) curve, at the steady state k = k , is less than the
slope of the [g + n + δ ]k curve. That is, sf ' (k ) < [g + n + δ ]k . So, we find Ψ ' ( k ) < 0 , meaning our
steady state k is locally stable.
Examining Figure 2, we can also see that the steady state k = 0 is locally unstable. In the
neighborhood of k = 0 , with k positive, notice in Figure 2 that the slope of the sf ( k ) curve is
greater than the slope of the [g + n + δ ]k curve. This implies sf ' (k ) − [g + n + δ ] > 0 , which implies
Ψ ' ( k ) > 0 . If Ψ ' ( k ) > 0 in the neighborhood of k = 0 , then k is increasing in that neighborhood,
meaning it increasing away from the steady state value of zero.
Figure 3 is a phase diagram. The traditional phase diagram for a single state variable differential
equation is constructed by plotting the state variable along the horizontal axis and plotting the time
derivative of the state variable along the vertical axis. So, for our model k is plotted horizontally
and dk / dt = k ' is plotted vertically. The function Ψ ( k ) shows how k ' depends upon k . The
Inada conditions tell us that the function Ψ ( k ) is positive and diverging to infinity as k approaches
zero, and they tell us Ψ ( k ) decreases as k increases, eventually becoming negative, as shown in
Figure 3. The steady state value for k is where Ψ ( k ) equals zero. The arrows along the horizontal
axis are presented to emphasize k is increasing when k < k but decreasing when k > k .
Figure 3: Phase diagram for the Solow-Swan growth model
Ψ (k )
k
k
Ψ (k )
To summarize, we know that the steady state k = 0 is locally unstable and the steady state k = k is
locally stable. The Inada conditions indicate that the path taken over time by the state variable k is
as shown in Figure 3. If the initial capital stock level k 0 is between zero and k , then k increases
over time and approaches k . Alternatively, if k 0 is greater than k , then k decreases over time and
approaches k . Figure 4 shows these two potential time paths for k by plotting k as it depends
upon time.
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Figure 4: Two possible time paths for the state variable k in the Solow-Swan growth model
y
k0 > k
k
k0 < k
t
Comparative Static Analysis
Comparative static analysis allows us to learn how the steady state value for the per capita capital
variable depends upon the exogenous variables. We can then take this knowledge and use it to
learn how other economic outcomes are affected by the exogenous variables.
Suppose we want to know how the steady state capital stock value k is affected by the savings rate
s . Equation SS1 determines the value of k as a function of the variable s and the other exogenous
variables. We can find out how s affects k by differentiating SS1, allowing both s and k to
change. We the differential concept rather than the derivative concept. Doing so, we obtain
sf ' dk + fds = [g + n + δ ]dk . The differential dk is the change in k that must occur in response to
ds , which is the change in s , in order for equation SS1 to still hold true. Collecting the dk terms,
we have fds = [[g + n + δ ] − sf ']dk , and solving for dk we have dk = f /[[g + n + δ ] − sf ']ds
Dividing through by ds , which is really saying we are taking the derivative of SS1 with respect to
s , we find
SS2
dk
f
=
ds [g + n + δ ] − sf '
Because we have held all exogenous variables fixed, except for s , the “comparative static
multiplier” dk / ds in SS2 is actually a partial derivative result, so we might more properly write it
as ∂k / ∂s . This multiplier result shows the marginal effect of s on k .
We typically are interested in the qualitative and quantitative characteristics of a comparative
static multiplier. The qualitative question is whether the sign of the multiplier is positive or
negative. Because the Inada conditions ensure the steady state is stable, we know the denominator
of SS2 is positive. Because f = f ( k ) for all positive k , we know the numerator is positive. With
numerator and denominator positive, we know the multiplier dk / ds is positive. That is, our
qualitative result is that an increase in the savings rate will increase the per capita capital stock.
The quantitative question involves examining what factors will make the multiplier larger in
magnitude (i.e., further away from zero) or smaller in magnitude (i.e., closer to zero). A large
magnitude multiplier indicates that the endogenous variable is sensitive to a change in the
exogenous variable, while a small magnitude indicates insensitivity. Examining the multiplier
dk / ds , we see, for example, k is more responsive to a change in s when the depreciation rate δ is
11
smaller. This is intuitively sensible in that capital should accumulate more readily in response to
more saving when it does not wear out as fast.
To obtain a comparative static result for each exogenous variable, we can differentiate SS1 while
allowing all variables to change. Doing so, we obtain sf ' dk + fds = [g + n + δ ]dk + kdg + kdn + kdδ .
Collecting dk terms and solving for dk , we find
SS3
dk =
f
1
1
1
ds −
dg −
dn −
dδ .
[g + n + δ ] − sf ' [g + n + δ ] − sf '
[g + n + δ ] − sf '
[g + n + δ ] − sf '
This total differential, algebraically rearranged in this manner, allows one to readily determine the
marginal effect of each exogenous variable on the endogenous variable. Notice that by setting
dg = 0 , dn = 0 , and dδ = 0 in SS3, we readily obtain the comparative static multiplier SS2 for the
savings rate. With the formulation SS3, the coefficient on the exogenous differential is the
comparative static multiplier for the particular exogenous variable. Taking another example, the
coefficient on dn is 1 /[[g + n + δ ] − sf '] . This implies dk / dn = −1 /[[g + n + δ ] − sf '] , obtained
mathematically from SS3 by setting ds = 0 , dg = 0 , and dδ = 0 , which means in this case we are
allowing the population growth rate to change but are holding fixed the savings rate, the rate of
technical change, and the rate of depreciation. It is evident that the multipliers dk / dg and dk / dδ
also equal − 1 / [[g + n + δ ] − sf '] . From this we learn that g , n , and δ each have the same
marginal impact on per capital stock level. An increase in any of these rates will decrease the per
capita capital stock (since [g + n + δ ] − sf ' > 0) ), and the size of [g + n + δ ] − sf ' determines the
magnitude of this change.
Once the comparative static multipliers for k are obtained, we can use the auxiliary equations to
obtain comparative static results for the auxiliary exogenous variables. For example, if we totally
differentiate RF3, we obtain dr = f ' ' dk . Because f ' ' < 0 , we know r and k move in opposite
directions when there is an exogenous shock that changes the steady state value of k . Rather than
taking the differential of RF3, we can take the derivative with respect to saving, and obtain
dr / ds = f ' ' dk / ds . Using the multiplier result dk / ds in SS2, we have
SS4
dr
f
= f ''
< 0.
ds
[g + n + δ ] − sf '
This multiplier result tells us that the real interest rate level is sensitive to the level of saving, with
the real interest rate tending to decline when the savings rate increases. As an example of looking
at sensitivity, the presence of the second derivative f ' ' tells us that, a change in the savings rate
will not impact the real interest rate level much if there are only slight diminishing returns to capital
(i.e., f ' ' near zero).
One important comparative static result obtainable from the Solow-Swan growth model is the
“factor price frontier.” When steady state level of k changes because of an exogenous shock, both
the real interest rate level r and the real wage w change. One way to examine how r and w
change relative to one another is by comparing dw / ds to the dr / ds result in SS4. By
differentiating RF4 and using the result SS2, you should be able to show
12
SS5
dw
f
= − f '' k
>0.
ds
[g + n + δ ] − sf '
Examining SS4 and SS5, we see that an increase in the savings rate increases the real wage level
but reduces the real interest rate level. Or, one can say a higher savings rate helps workers but hurts
capitalists who earn interest. Differentiating RF3 and RF4, we find dr = f ' ' dk and dw = − f ' ' kdk .
Using these last two conditions,1 it follows that
SS6 dw = − kdr .
Condition SS6 describes the factor price frontier. Because k is positive, condition SS6 tells us
that the real wage and real interest rate do move in opposite directions in response to any exogenous
shock. Further, we know that the relative magnitude of the impact depends upon the size of the per
capita capital stock. When the per capita capital stock is low, for example near zero, a shock that
greatly changes the interest rate level will not much affect the real wage level. In contrast, when the
per capita capital stock is very large, then a shock that only slightly affects the real interest rate will
tend to affect wages very much.
We can use the diagram in Figure 4 to examine the comparative statics effect of a change in one
of the parameters. The example shown in Figure 5 is an increase in the savings rate s . Consistent
with the comparative static result obtained above dk / ds = f /[[g + n + δ ] − sf '] > 0 , we see in Figure
5 that an increase in the parameter s shifts the per capita savings curve up and increases the steady
state capital stock. Because we know the steady state is stable, the increase in the savings rate
implies the economy would now be converging to the per capita capital stock level k 2 rather than
k1 . We can see f ( k 2 ) > f ( k1 ) in Figure 5, indicating the increase in the savings rate increases the
steady state average standard of living in the economy. We can see f ' ( k 2 ) < f ( k1 ) in Figure 5,
indicating the increase in the savings rate decreases the economy’s steady state real interest rate
level. (Can you see a geometric way to show that the steady state real wage increases, which it
does?)
Figure 5: The steady state for the Solow-Swan growth model
y
[g + n + δ ]k
f (k )
s 2 f (k )
s1 f (k )
k1 k 2
1
k
One can also use conditions SS3 and SS4 to derive SS6.
13
Growth Rates of the Aggregates in the Steady State
In the steady state, the state variable k is constant. We can examine the implication of k being
constant by using the natural log and some differentiation. Taking the natural log of both sides of
k = K /[ AL ] , we have ln(k ) = ln( K ) − ln( A) − ln( L) . Differentiating with respect to time, we then
have k ' / k = K ' / K − A' / A − L' / L . Because k ' = 0 in the steady state, this last condition reduces to
K ' / K = A' / A + L' / L . Since we assume A' / A = g and L' / L = n , we find that the growth rate of the
aggregate capital stock in the steady state is K ' / K = g + n .
From the auxiliary equations RF2 and RF5, we know that, if k is constant, then y and c are
also constant. Because y = Y /[ AL ] and c = C / [AL ] , we can use the same process used in the last
paragraph to show Y ' / Y = g + n and C ' / C = g + n . Thus, in the steady state, capital, output, and
consumption each grow at the same rate. By assumption the rate of growth in population (or the
labor force or employment) is L' / L = n , which is lower than the growth rates for capital, output,
and consumption as long as there is improving technical change ( g > 0 ). These steady state
modeling outcomes are consistent with the observed growth patterns of developed nations. Output
capital, and consumption tend to grow at about the same rate, and this rate is higher than the labor
growth rate. The Solow-Swan model is well regarded, in part, because of its ability to replicate the
growth paths followed by these important macroeconomic variables.
Per Capita Levels versus Per Capita Growth Rates
The quantity y = Y /[ AL ] is output per effective labor unit. Rearranging this equation, we find
output per person is Y / L = Ay . Similarly, consumption per person is C / L = Ac . Since the per
capital values y and c are constant in the steady state, the growth in per person output and per
person consumption must come from the growth in the labor effectiveness variable A . In
particular, taking the natural log of Y / L = Ay , we have ln(Y / L) = ln( A) + ln( y ) . Differentiating
with respect to time then yields [Y / L ]' / [Y / L ] = A' / A , which implies [Y / L ]' /[Y / L ] = g . Similarly,
one can show the growth rate of per person consumption is [C / L ]' /[C / L ] = g . So, we find that
the only factor affecting the steady state growth rate of per person output and consumption is the
rate of technical change g . The other exogenous variables s , n , and δ affect the levels of Y / L
and C / L because they affect the levels of y and c . However, changes in s , n , and δ do not
affect the long run growth rates of per person output and consumption because y and c only change
discretely at the points in time when s , n , or δ change.
To understand why this important, consider two economies. Assume the two are in the stable
steady state, possess the same initial capital stock, have identical production functions, and have
identical values for the exogenous variables up to a certain point in time. Then at a time t = t1 ,
assume economy 2 experiences an increase in the savings rate so that s2 > s1 for all time t > t1 .
Assume no other exogenous changes occur. What will the paths for output per person look like for
the two economies?
Figure 6 shows the time paths for per person output Y / L for the two economies. Up to time
t = t1 the two economies follow the same steady state path. At time t = t1 , the savings rate of
economy 2 increases. This implies there is a new higher steady state path for economy 2 as shown
in the figure. Economy 2 would not jump to the new steady state value, but rather would converge
14
toward it over time, as shown in the figure. At any point in time after t = t1 , the Y / L level for
economy 2 is higher than that for economy 1. However, the growth rate of Y / L is equal to g along
the new steady state growth path, as it is along the original path. Thus, while economy 2 will
consistently have a higher per person output level, the growth rate of Y / L for economy 2 will
eventually again approach the growth rate for economy 1. (It is evident and can be carefully shown
that the growth rate for economy 2 will be higher than that for economy 1 as economy 2 transitions
from the original steady state path to the new steady state path.) As shown in Figure 5, economy 1
will still achieve the same Y / L level as economy 1, it will just take longer. Thus, an increase in
the savings rate can be thought of as a way to jump the economy forward along its growth path.
Economy 2 will reach the output per person level [Y / L ]1 at time t 2 , while economy 1 will reach
that level at time t 3 .
Figure 6: Effect of an increase in the savings rate s on the path of per person output Y / L
y
Steady state path for s2
Steady state path for s1
[Y / L]1
[Y / L]0
t
t1
t 2 t3
Summary of what we can learn from Steady State Analysis
From equation SS3, we can determine how a change in any one of the exogenous variables will
affect the steady state per capita capital stock. Given those effects, we can then determine how the
auxiliary variables of the model will be affected. The auxiliary equations RF2-RF5 indicate the
steady state per capita capital stock k is directly related to the per capita output level y , per capita
consumption level c , real wage level w , but inversely related to the real interest rate level r . Given
the assumptions on the production function, equation SS3 indicates the steady state capital stock
level:
• increases when the savings rate s increases;
• decreases when the rate of technical change g increases;
• decreases when the rate of depreciation δ increases;
• decreases when the rate of population growth n increases.
The steady state real wage level and per capita output and consumption levels change in the same
direction, while the real interest rate level changes in the opposite direction.
The results in the previous paragraph may make it seem like technological improvement, or an
increase in the rate of technical change g , is not good. The opposite is true, and the possible
confusion lies in the fact that the per capita levels k , y , and c in the model are per effective unit of
labor. As shown above, growth rate of output and consumption per person depend only upon the
rate of technical improvement g . So, while the other exogenous variables are important for
determining the levels of output and consumption per person, the growth rates of these variables
only depend upon the rate of technical change.
15
In the steady state, the aggregate output, capital, and consumption levels grow faster than the
rate of population growth, by an amount equal to the rate of technical change. The real interest rate
remains steady. The real wage per unit of effective labor remains steady, but the real wage per
person grows at the rate of technical change.
These results give us some reasons for why one economy may outperform another. Differences
in the rate of technological improvement is the primary answer to the question of why one economy
grows faster than another, and for why real wages in one economy grow faster than in another. All
other things equal, an economy with a higher savings rate, lower depreciation rate, or lower
population growth rate would have a higher standard of living, measured in terms of either the level
of output per person or level of consumption per person. Because output per person or
consumption per person would typically be considered a reasonable measure of long term economic
health, the Solow Swan model points long term economy policy toward actions that will:
• enhance the rate of techological improvement (and this is most important);
• increase the savings rate;
• reduce the rate of depreciation; and
• reduce the rate of population growth;
7. The Golden Rule Consumption Level and Dynamic Inefficiency
Because c = [1 − s ]y and y = f ( k ) , we know c = [1 − s ] f ( k ) = f ( k ) − sf (k ) . In the steady state,
equation SS1 indications sf (k ) = [g + n + δ ]k . Using this and the previous equation, we know the
steady state per capita consumption level can be written
GR1 c = f ( k ) − [g + n + δ ]k
Equation GR1 is intuitive. Per capita consumption is equal to per capita output less per capita
investment. Investment is the output that must be sacrificed in the present, and not consumed, in
order to build additional production capacity in the future. The question that interests us is, “What
savings level s provides the highest possible steady state per capital consumption level. Because
we know from our analysis above that a higher savings rate produces a higher per capita capital
stock, equation GR1 indicates that there is a tradeoff to increasing the savings rate: A higher output
level, which increases consumption, verses a higher investment level, which decreases
consumption. The maximum steady state consumption level is called the Golden Rule level of
consumption.
To find the condition associated with the Golden Rule, we need to differentiate condition GR1
with respect to the savings level, recognizing that the per capita capital stock level depends upon the
savings level. Doing so, we obtain
GR2
dc
dk
dk
= f ' ( k ) − [g + n + δ ] .
ds
ds
ds
Setting this derivative equal to zero, we find f ' ( k ) = g + n + δ must hold as long as dk / ds > 0 ,
which we would expect give our result SS2 above. Because we know r = f ' ( k ) from condition
RF3, we can eliminate f ' ( k ) from the equation f ' ( k ) = g + n + δ to obtain
GR3 r = g + n + δ .
16
Equation GR3 is a “necessary condition,” or a condition that must hold in order for an economy
in a steady state to be experiencing the Golden Rule consumption level, or maximizing its per capita
consumption level over time. The condition has an empirical implication. If we think the economy
in question is in a steady state, or close to one, then the real interest rate level should be higher than
the rate of population growth, if the economy is to be following the Golden Rule. For example, if
the depreciation rate is 5%, the rate of technical improvement 2%, and the rate of population growth
1%, then the economy’s real interest rate would have to 8% for the economy to be following the
Golden Rule.
This analysis indicates that an economy can be dynamically inefficient, meaning it can have a
savings rate that is less efficient over time than it could be. In particular, if the savings rate is above
the Golden Rule rate, then the steady state capital stock would be above the golden rule capital
stock. Assuming capital can be disposed of freely, such an economy must be inefficient, because
by disposing of capital to reduce the per capita capital stock level, the economy could go to the
golden rule capital stock and increase consumption per capita for each point in time out into the
future. To stay at the Golden Rule level of consumption, the savings rate would have to be reduced
to the Golden Rule rate. That is, dynamic inefficiency is associated with excessive saving. Notice
that, because more capital is associated with a lower real interest rate, in order for the steady state
capital stock to be above the golden rule, r < g + n + δ must hold in when there is dynamic
inefficiency. Thus, a possible sign of this dynamic inefficiency would be a low real interest rate
level.
When r > g + n + δ , the steady state capital stock must be below the Golden Rule. There is no
inefficiency in this case. This is because the per capita capital stock needs to increase to get to the
golden rule, and the only way to do that immediately would be to make the existing consumers
sacrifice. Thus, this change is not “Pareto improving.” A Pareto improving change improves the
well being of some without hurting anyone else. The reason there is dynamic inefficiency when
r < g + n + δ is that a Pareto improving action exists. However, no such action exists when
r > g + n + δ . So, we are left with the conclusion that a steady state is dynamically efficient if
and only if
GR4 r ≥ g + n + δ .
8. Using a Taylor Series to Approximate the Rate of Growth as the Economy Approaches the
Steady State
We know from our analysis above that a stable steady state exists for the Solow-Swan economy.
This implies that when the per capita capital stock variable C is not in the steady state, it will be
approaching it. We know from our analysis above what the growth rates for various variables will
be when the economy is in the steady state. However, if the economy is not in the steady state, the
growth rate for any given variable will be different than in the steady state. If economies in the real
world are approaching steady states, it would be good to understand how their growth rates might
change as the steady state is approached. The method for doing this is to determine the behavior of
the state variable, here k , and then draw conclusions about the behavior of the other variables of
interest.
17
A Taylor series expansion is used to obtain the desired result. In general, Taylor’s Theorem tells
us that, under some conditions that regularly tend to hold, we can approximate the value G ( k ) to
any desired degree of accuracy by a polynomial expanded around a specific value k . In particular,
TS1 G ( k ) = G (k ) +
[
]
[
G ' (k )
G ' ' (k )
k −k +
k −k
1!
2!
]
2
+ ...
The approximation occurs when the higher order terms are dropped. Quite often, the approach is to
drop terms except for the linear term, so that the approximation is
[
]
TS2 G ( k ) ≈ G ( k ) + G ' ( k ) k − k ,
since 1! = 1 .
The function of k we want to approximate is G ( k ) = k ' / k . We know from equation FR1 that
k ' = sf (k ) − [g + n + δ ]k , so we want an approximation of k ' / k = sf (k ) / k − [g + n + δ ] , or
TS3 G ( k ) = sf ( k ) / k − [g + n + δ ].
Using TS2, the approximation is
[
]
 sf ' (k ) sf (k ) 
TS4 G (k ) ≈ G (k ) + 
− 2  k −k .
k 
 k
In the steady state, sf ( k ) = [g + n + δ ]k so G ( k ) = 0 , and we can rewrite TS4 as
 sf ' (k ) g + n + δ 
TS5 G (k ) ≈ 
−
 k − k , or
k
 k

[
]
replacing G ( k ) with k ' / k , we have
TS6
k '  sf ' (k ) − [g + n + δ ]
≈
 k −k .
k 
k

[
]
[
]
Because k is a constant, we know that the coefficient on k − k in TS6 is constant as
k changes. The Inada conditions ensure sf ' (k ) < g + n + δ , so we know this coefficient is
negative. Thus, consistent with our work above, we know the growth rate k ' / k is negative when k
is above the steady state level k , and positive when k < k . However, the approximation TS6 tells
us more, Because the coefficient sf ' (k ) − g + n + δ / k is constant we know the growth rate k ' / k
decreases to zero as k approaches its steady state value k . In other words, the growth rate of the
per capita capital stock is proportional to its distance from the steady state.
[
[
]
]
The size of the coefficient sf ' (k ) − g + n + δ / k determines the speed of the convergence to the
steady state, and it is possible to relate the size of this coefficient to the share of output being paid to
18
capital. Notice that we can take the production relationship y = f ( k ) and write it as
y = f ' ( k )k + f ( k ) − f ' ( k ) k . Dividing though by y , we obtain 1 = [ f ' (k )k ]/ y + [ f (k ) − f ' (k )k ]/ y .
Using RF3 and RF4, we then have 1 = [rk ]/ y + [w]/ y . The first term on the right side of this last
equation is the share of output paid to capital, and the second term is the share paid to labor.
Denoting the capital and labor shares as α K and α L , it follows that α K = [rk ] = [ f ' (k )k ]/ f (k ) , and
α L = [w]/ y = [ f ( k ) − f ' ( k )k ]/ f (k ) , and so α K + α L = 1 .
[
]
Returning to the coefficient sf ' ( k ) − g + n + δ / k , we can replace the variable s with using the
[
][
]
steady state condition SS1, Because s = [g + n + δ ]k / f ( k ) , we can rewrite the coefficient as
[g + n + δ ][[ f ' (k )k ]/ f (k ) − 1]/ k .
Noticing [ f ' (k )k ] / f ( k ) is capital’s share of output and defining
α K = [ f ' (k )k ] / f (k ) , to be the steady state share, we can rewrite the coefficient as
[g + n + δ ][α K − 1]/ k . Noting that α K − 1 = −α L must hold, we can rewrite the growth rate
approximation TS6 with the new coefficient as
TS7
k'
α [g + n + δ ]
≈ − L
 k −k .
k
k


[
]
We see in condition TS7 that an economy will converge faster when the share of output
allocated to labor is higher, (or equivalently when the share allocated to capital is lower). The rate
of convergence will also be higher for economies with higher rates of technical change, higher rates
of population growth and higher rates of depreciation.
9. Theory and Observations: How well does the Solow-Swan Model Explain the Facts
In 1963, economist Nicholas Kaldor presented the following list of facts that roughly
characterize the pattern of growth exhibited by developed economies for decades leading up to
1963:
The Solow-Swan model has become important, in part, because of its ability to reproduce these
growth patterns.
As noted above, when an economy is not in a steady state, it will tend to converge toward it.
This convergence property of the model suggests that real world economies might converge. The
term “convergence” or “ β convergence” is used to describe the expectation that a less developed
economy will grow faster than a developed economy, so that the gap between the per person output
levels for the two economies diminishes. The evidence for convergence, in general, is weak. Over
very long periods of time, country differences in per person incomes have actually widened. Over
the past few decades, they have remained roughly stagnant.
To be fair to the Solow-Swan model, however, we would not expect convergence for economies
with different exogenous parameters. For example, we know that a higher rate of technical change
(higher level for the parameter g in our model) is the primary factor associate with a higher rate of
per person output growth. We would not expect two economies to converge if one has a higher rate
of technical change. If one of the other parameters of the model differs, but the rate of technical
change is the same, we would expect per person output growth rates to converge, but the per person
levels may vary. “Conditional convergence” or “conditional β convergence” is used to describe the
19
idea that we would expect economies to converge that have the same underlying parameters, but not
if the underlying parameters vary.
Conditional convergence has significant explanatory power. For example, there is evidence that
the rates of growth among the difference states in the United States have converged over time.
Among the nations of Europe and North America, there is evidence that, over a 100 year period,
those developed nations with lower per person output levels grew faster, though this evidence is
more controversial.
From the Solow-Swan model, we get the important conclusion that changes in either the
saving rate or population growth rate cannot explain persistent growth. If technology is not
improving, per person output will increase when either the savings rate increases or the rate of
population growth decreases. However, this increase in per person output is a one time increase.
The Solow-Swan model tells us that only a steady improvement in technology can explain the
persistent growth in per person output that we observe in many western economies.
The Solow-Swan model does not tell how a nation can increase the rate of technological
improvement. That is what the “new growth theory” or “endogenous growth theory” seeks to
accomplish.
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