Solow Growth Model

Solowgiin ediin zasgiin usultiin onol

1. The Solow Growth Model
TheSolow Growth Modelis a model ofcapital accumulationin a pure production economy: there are no
prices because we are strictly interested in output = real income. Everyone works all the time, so there is no
labor/leisure choice. In fact, there is no choice at all: the consumer always saves a fixed portion of income,
always works, and owns the firm so collects all “wage” income and profit in the form of all output. We will
not need to model the “consumer”.
We assume all peoplework all the time, and we assume they save, hence invest, a fixed portion of their
income. There is no government, hence no taxation nor subsidies; this is aclosed economy, so there is no
trade. Since there are no prices there is no need for money: thereare no financial markets, etc.
This model, then, is a model that captures the pure impact savings = investment has on the long run
standard of living = per capita income. Since we allow for population growth, this model may be called the
Blue Lagoon Model(i.e. as opposed to Robinson Crusoe, two people can reproduce).
Ingredients: Consumers and Firms. All consumers own the firms, so consumers receive all output, and
therefore all profit and rent.
Aggregates: Output = Real Income =Ytin periodt.
CapitalStock =K
t;
Population Size =Nt= Labor Supply (since everyone works all the time).
Consumption =Ct;
Savings =St;
Investment =I.t.
Per Capita: Output = Real Income =y
t=Y
t/N
tin periodt.
Capital Stock =kt=Kt/Nt;
Consumption =ct=Ct/Nt;
Savings =st=St/Nt;
Investment =i.t=It/Nt;
THE POPULATION = THE LABOR SUPPLY
The population grows at a constant raten:
tt
t
tt
t
NnN
N
NN
Nn )1(%
1
1
1






THE FIRM
The firm produces according to a Cobb-Douglas production function:













tt
t
t
t
t
t
t
t
t
t
tt
tt
t
t
tt
t
t
ttt
Aky
Ak
N
K
A
N
K
A
N
N
N
K
A
NN
NK
Ay
N
NK
A
N
Y
NAKY




























:CapitaPer
:Aggregate
1
1
1
1
1
1

2. THE CONSUMER : CONSUMPTION AND SAVINGS
The consumer saves the fractionsof income:
ttttttt SCYYsCsYS  hence)1(and
Notice this reflects the fact that there is no government (no taxes), and no imports/exports (no trade).
EQUILIBRIUM GROWTH
1. POPULATION
If the population size starts atN0, then
021
)1()1)(1()1( NnNnnNnN
t
ttt



We need only know the initialN
0and the growth ratento find the population size in any periodt.
2. SAVINGS AND INVESTMENT
Since there is no government (no taxes), there are no imports/exports (no trade), consumers receive
everything from the firms, and there are no financial markets, savings is simply investment (the only place
consumers can put their “money”, which is actually output, is simply back into the firm):
tt
SI
3. CAPITAL ACCUMULATION
Aggregate capital grows according to the followinglaw of motion:
ttt
IKdK 

)1(
1
Next periods capital stock is this periods discounted for depreciation (d= depreciation rate), plus whatever
was invested.
Use the above production function and savings = investment identity to deduceper capita capital
accumulation evolves according to:






1
1
)1(
)1(
)1(
)1(
ttt
tt
tt
ttt
NsAKKd
sYKd
SKd
IKdK

3. 




ttt
t
t
t
t
tt
t
t
t
tt
tt
t
t
t
tt
t
t
t
t
sAkkdkn
N
K
sA
N
K
d
N
K
N
N
NN
NK
sA
N
K
d
N
NK
sA
N
K
d
N
K









)1()1(
)1(
)1()1(
1
1
11
1
11
1
hence

ttt k
n
sA
k
n
d
k











11
1
1
THE STEADY STATE
The per capita capital stock growths, but at a decreasing rate: see below. Eventually growth converges to
zero. In this long-runsteady state
*
1
kkk
tt


We can explicitly solve fork
*
as follows:
 



dn
sA
k
k
n
sA
n
dn
k
n
sA
n
d
n
n
ksA
n
d
k
n
sA
k
n
d
k




































1
*
1
*
1
*
1
****
1
1111
1
1
1
1
1
1
11
1
hence
)1/(1
*
StateSteady









dn
sA
k
The steady state level of real income, consumption, savings and investment can all be deduced fromk
*
:
)1/(
**
)1/(
***
)1/(
**
)1()1(
)(































dn
sA
Asysc
dn
sA
sAsysi
dn
sA
AkAy
EXAMPLE #1
Consider three economies which differ in their savings rate and/or population growth rate:

4. 175.07.30.02..3
175.07.25.08..2
175.07.25.02..1
00
00
00



AKNdsn
AKNdsn
AKNdsn



Economy #2 has a highpopulation growth rate; economy #3 has a high savings rate. Per capita capital
accumulation and the steady state capital stocks are
123.46
07.02.
13.
.3
7.716
07.08.
125.
.2
54.59
07.02.
125.
.1
)75.1/(1
*
)75.1/(1
*
)75.1/(1
*






























k
k
k
SolowGrowth Model
per capita capital k(t) evolution
An inc re a s e in the
s a vings ra te
inc re a s e s s te a dy
s ta te k. A highe r
po pula tio n gro wth
ra te de c re a s e s
s te a dy s ta te k.
0
10
20
30
40
50
60
70
80
90
100
110
120
130
0 50 10 0 150
Time t
k
=
K
/
N
s = .25, n = .02
s = .25, n = .08
s = .30, n = .02
1. CAPITAL GROWTH TOWARD THE STEADY STATE
Use the definition of growth
t
tt
t
k
kk
k




1
1
%
and the capital accumulation formula

5. 
ttt
sAkk
n
d
k 









1
1
1
to deduce


































































1
1
11
1
1
1
1
1111
11
11
1
1
1
1
1
1
1
1
1
1
111
1
%
t
t
tt
t
t
ttt
t
tt
t
kn
sA
n
dn
k
n
sA
n
nd
k
n
sA
n
n
n
d
k
n
sA
n
d
k
n
sA
n
d
k
k
n
sA
kk
n
d
k
kk
k
Positive Growth
We need to verify that growth is positive as long as the capital stockkis less than the long-run steady state
level:
tt
tt
t
t
k
dn
sA
k
dn
sA
dn
k
sA
n
dn
kn
sA
kn
sA
n
dn
k






























)1/(1
1
11
1
1
1
1
1
1
when
0
1
11
%positiveisGrowth




Growth Declines askIncreases
Also, the growth equation shows



















1
1
1
11
%
t
t
kn
sA
n
dn
k
In other words, askincreases, the growth rate declines. That mathematically verifies the above plots:kis
clearly increasing, buta decreasing rate.
EXAMPLE #2
The growth plots for Example #1 are:

6. Solow Growth Model
per capita capital growth: %k
0%
5%
10%
15%
20%
0 50 10 0 150
Time t
g
=
%

k
s = .25, n = .02
s = .25, n = .08
s = .30, n = .02
2. COMPARATIVE STATICS
We can alter the underlying economies in order to inspect what Solow’s model predicts about differing
countries.
2.1 Highern
The first example clearly shows the for two countries that are identical in every way, except population
growth, the higher growth rate leads to slower capital growth and a lower long run level of per capita
capital.
2.2 Highers
The first example clearly shows the for two countries that are identical in every way, except the savings
rate, the higher savings rate generates a higher capital growth rate and a higher long run level of per capita
capital.
2.3 Identical Economies, with Difference Initialk
0
Consider two economies, identical in every way except their initial capital stock per capita:
415.07.4.03..
15.07.4.03..
00
00


KANdsnB
AKNdsnA


Thus, Economy B starts with four times the level of capital per capita as Economy A.
Notice

7. 
  
AAAAABBB
AAA
kk
n
sA
k
n
d
k
n
sA
k
n
d
k
n
sA
k
n
d
k
k
n
sA
k
n
d
k
10000001
001
11
1
4
11
1
4
11
1
11
1













































because 4

> 1 for any 0 << 1. And so on.
Economy B will (almost) always have more capital. Why “almost”? Because their steady-states are
identical. Thesteady state does not depend on initial conditions, and only on technology, saving and
population growth parameters.
Both have
16
1.
4.
07.03.
14.
2)5.1/(1)1/(1
*























dn
sA
k
Solow Growth Model
per capita capital k(t) evolution:
Two Identical Economies:
one starts with 4x the level of k
0
5
10
15
20
1 51 10 151
Time t
k

=
K
/
N
s = .5, n = .03, k0 = 1
s = .5, n = .03, k0 = 4
Thus, even economies that are otherwise identical, one can nevertake-overthe other.
3. THE GOLDEN RULE OF SAVINGS
Although a higher savings rate generates more per capita capital with whichto generate more output, a
constant increase inscannot always lead to a higher level of per capita consumption. Clearly a very high
savings rate implies people are consuming very little of that very large amount of output that is being
produced.
Imagine very hard working people, but very hungry, planting a corn crop, and harvesting the corn. Instead
of eating a lot they plant 80% of the kernels. The result is an enormous crop the next season. They again eat
very little, invest 80% of the output toward production, and so on. The total level of output per capital sky-
rockets, but is welfare truly higher? If we measure welfare by consumption clearly it is not.

8. The steady state level of consumption is

)1/(
)1/(*
)1/(
***
)1(
)1(
)1()1(StateSteady
























dn
A
ssc
dn
sA
s
ksysc
Thus, the long-run level ofper capita consumption is a function of the savings rates:
)1/(
)1/(*
)1()(












dn
A
sssc
EXAMPLE #3
Consider Economies #1 and #2 in Example #1. We can plot
*
c(s) as a function ofs:
Steady State Per Capita Consumption c(s)
as a Function of the Savings Rate s
The Go ld en Rule
d o es no t d ep end
o n n. The Rule s =
alp ha = .75
g enerat es t he
hig hes t
p o s s ib les t ead y
s t at e c. A s o ciet y
wit h a hig h
p o p ulat io n
0
2 0
4 0
6 0
8 0
10 0
12 0
14 0
16 0
0 %10 %2 0 %3 0 %4 0 %50 %6 0 %70 %8 0 %9 0 %10 0 %
Savings Rate s
c
(
s
)
s = .2 5, n = .0 2
s = .2 5, n = .0 8
There is a clear tug-of-war: a highersmeansless consumption, but allows for higher long-run capital
accumulation and therefore higher steady state income. The optimal savings rate is theGolden Rule: the
savings rate the optimizes steady state per capita consumption:
0)1(
1
1:
0)1(
1
:
0)(:FOC
)1(max)(max
1
)1/(
1)1/(
)1/(
)1/(
*
s
)1/(
)1/(
10
*
10







































ss
dn
A
ss
dn
A
s
sc
dn
A
sssc
ss











9. 









s
sss
ss
ss
:RULEGOLDEN
)1(:
)1()1(:
)1(
1
:
Compare the result to the plots above. For two otherwise identical economies with the same= .75, the
Golden Rule iss= .75, and indeed the steady state per capita consumption is optimizes in both cases
identically ats= .75.
EXAMPLE #4
Although the Solow Model does not itself build in any sense of a “business cycle”, or suggest anything like
a shock (all plots clearly show the model predicts perfectly uniform growth!), we ourselves can introduce a
shock at any time.
Consider a scenario where an economy experiences two negative shocks, one small and one large, and a
medium positive shock. The small negative shock (att= 50) is causes by news that a sector (e.g. the energy
sector) has been substantially over invested, profits arepredicted to be negative, and therefore investment
capital is withdrawn:kfalls by 20%. The large negative shock (att= 100) is due to a massive storm:kfalls
by 50%. The positive shock (att= 150) is due to a capital infusion by the International Monetary Fund:k
instantly increases by 15%.
S ol ow Growth Mode l
pe r capi ta capi tal k (t) e vol u ti on :
Ne gati ve s h ock at t = 50: k drops by 20%.
Ne gati ve sh ock at t = 100: k drops by 50%.
Posti i ve sh ock at t = 150: k i n cre ase s by 15%
k(t)
Ne g at ive S ho c ks
r e d uc e t he c a p it a l
s t o c k, b ut g r o w t h is
hig her a t lo we r k, a nd
t he s t e a d y s t a t e k is
a lw a ys t he s a me .
y(t)
0
10
20
30
40
50
60
1 5 1 10 1 15 1
Time t
k
=
K
/N
k(t ) y(t )
The Solow Model is a very simple model in the final analysis: at whatever point or state the economy is in,
growth immediately occurs (fast ifkis small, slow ifkis large), while the standard of livingyslowly
approaches the long-run steady state.
SOLOW MODEL AND REALITY
The Solow Model predicts growth is always positive, but slowly declines to zero. Economies with a high
population growth rate can never take-over. In fact, otherwise identical economies where one simply starts
with a smaller level of per capita capital, will never take-over to the other economy.

10. Nevertheless, the Solow Model does correctly predict that higher population growth rates, and lower
savings = investment rates are associated with lower growth levels, and lower standards of living.