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SOLOW GROWTH MODEL
The theory of economic growth is studied by solow growth model In this model
growth is studied with savings population growth and technological progress , capital
etc
THE ACCUMULATION OF CAPITAL
DD and SS for goods determine the accumulation of capital . Labor and technological
are assumed to be fixed . By studying DD and SS, we can see how much output got
produced and how was it distributed.
DD and SS for goods- It determines how much output is produced and how this
output is distributed
SS of goods – Y=FCK1L SS is a function of output based on K and fixed labour
Solow model assumes constant returns, to the scale .So we divide the whole function
by L
Y/L=F(K/L ,1)
(since 1 is a constant can be ignored )
So we have constant returns we have fixed ratio between K/L and B/L
Therefore y=F(k)
y=Y/L and k=K/L
Therefore, slope =MPk=f(k+1) -f(k)
output workers
f(K)=y
Slope = 𝑀𝑃𝑘
Capital /workers
As the amount of capital increases the production function becomes flatter so
production function exhibits diminishing returns to capital ie when k(k/l) is low we have
high productivity but as k increases the productivity decreases and production
increases slowly.
DEMAND FOR GOODS-2 sector model HH and Firms and no govt.
Therefore Y = C+I
( we divide it by L)
Implies y=c+i and c=(1-s)y
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Implies y=(1-s) y gives i =sy
Savings as a fraction of output=Investment
Growth in k stock and steady state
K stock determines a country’s output but it can change over time and these changes
lead to economic growth
2 forces that influence k stock :
a) Depreciation
b) In
In refers to the expenditure on new plant and equipment depreciation is the
wearing out of old k and it causes k stock to fall.
CnFn
i=sy
Production fn Y=f(k)
Implies i=s f(k)
So, accumulation of new k is based on stock of old k
Investment
F(k) output
S f(k) savings/investment
Depreciation -lets say k stock depreciation by a constant amount 8 years .so, if a
capital lasts 20 years then it depreciates by 5% year
depri./workers
∂/ workers
dk=i-∂k ( Change in Capital)
dk=s f (k) -∂.k
k/workers
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steady state is where the economy is stable and has a tendency to reach there
sf(k)
k1
k*
k2
k/workers
At K1 , In >Depreciation and therefore capital increases and reaches K*
At K2 , In <Depreciation and therefore capital decreases and reaches K*
For egY=K ½ L ½
Y/L =k ½ L ½ /L
S =30%
implies y= √𝑘
S=10%
and k=4
Then each year economy’s Y=√𝑘 =2 and depreciation =10%
At k=9 the economy is in the steady state
INCREASE IN SAVINGS RATE (S Increase )
The economy even though, it is in steady state that has an increase in S then
automatically s f(K) increases and therefore I increases and the economy reaches a
new level of k, as the depreciation of new k is also higher.
k1*
k2*
k/worker
THE GOLDEN RULE LEVEL OF CAPITAL: please refer to the video.
At each savings rate(s) we have different k and therefore by changing S we get
various steady statets and the govt. chooses that one which maximizes Cn. This cant
happen automatically
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The economy states at K* and policymakers increases and therefore k also increases
to k*+1 then MPk = f(k*+1) – f(k*) and then s also increases .Thus, MPk -S>0 implies
increase in capital increases Cn.
Numerically y=√𝑘
K*/√𝑘 = S/0.1 *=10%
K* =100S2, by hit and trial with’s’values we can find k* gold.
OR
MPk =∂y/∂k =1/2√𝑘 =0.1 =S
Implies 10=2√𝑘
K=25
S=0.5 or 50%
TRANSITION TO GOLDEN STATE {steady state w/o golden Cn}
The economy states at another level .
A) WITH TOO MUCH CAPITAL- Policy makers should be in as reducing at T0
they will able to achieve a lower S
T0=no longer and steady state
Y
Cn
Cn immediately increases
In
In immediately decreases.
T0
Since at steady , i=Depreciation and now s decreases and I decreases therefore i<
depreciation implies will no longer a steady state .Gradually this will make k fall and
therefore y will also fall ,Cn will also fall and I will also fall.They will keep on falling
till they reach a new steady state .
So, now after T0 Cn is higher than before and also at new steady state Cn is higher
than at older steady state.
T0
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So when initially Cn decreases implies current generation is worse off and later
when Cn increase future generation is benefited.
POPULATION GROWTH
Till now we assumed labor to be constant. But now we remove assuming that
and we let labor grow at a constant rate n
SO now since ‘n’ is growth rate of population to keep k=K/L constant k has to
increase by the same rate L does and depreciation does.
Therefore, Dk=i-(s+n)k
(s+n) k is the amount of I necessary to keep overall k constant
Therefore, ∆K= sf(k) –(s-n)k= 0 at steady state
Y
(n+∂)K
Sf(k)
K*
K
EFFECTS OF POPULATION ( GROWTH)
1. It tells us about the sustained economic growth (since population is
growing at n, total Y is also increasing and total K is also increasing at n
y=Y/L is constant and k =K/L is also constant
2. Implies disparity b/w nations
When population increases from n1 to n2 then
output
K2*
3. C=y-I
k1*
K
(golden rule )
C*=f(k*) –(s+n)k*
Or MPk =s+n
Or MPk-n=S
Numerically n=0.1 S=0.1 s=?
Y=√𝑘
Then MPk =1/2√𝑘 =0.1+0.1
Or 2√𝑘 =5
K=(2.5)2 =6.25
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Efficiency of labor-Now the labor is also becoming more and more
efficient each year OE represents the society’s knowledge about
production method it keeps on improving each year.
Y = FCK1LxE
LxE measures the effective number of workers the increasing in efficiency
is same as increase in number of workers Lets say , efficiency of labor
grows at some constant rate g. This is called labor augmenting
technological progress labor force grows@ n and efficiency increases @g
therefore the increase in no. of effective workers is by n+g
BALANCED GROWTH
Technological progress causes the values of many variables to rise
together in steady state.This is called balanced growth
CONVERGENCE
Economies around the world have diff purchasing power or standard of
living.But according to solow growth model countries may start off poor
but later on will catch up with richer economies .This property is called
convergence. Solow model explains that if 2 economies have same S and n ,
g and same production function then they may stark at different k but
soon they will converge.
But if S is diff in 2 economies then they don’t converge .So, international
diff in income/person can be attributed to1. Difference in factor of production function such as quantity of physical
and human capital
2. Difference in the efficiency with which they use these factor
So it can be because of diff in production function or because of less
capital accumulation .And studies have found that these are +vely co
related because of following reasons :
a. An efficient economy may encourage capital accumulation
b. Capital accumulation may induce greater efficiency
c. They both are driven by 3rd common factor’
POLICIES TO PROMOTE GROWTH
If MPk >n+g+s then any increase in S implies i> depreciation and therefore increase in
k and increase in economic growth and therefore a new steady state with higher Cn
Suppose k=2.5 y : Sk=0.1y
MPk=0.3y/k
Then S=Sk/k = 0.1y/2.5y =0.04
And MPk = MPk . k/K =0.3y/2.5 y=0.12
MPk=n+g+s
012=0.04+n+g implies n+g=0.08
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Because n+g=0.08 is greater than whats there in the economy therefore S should
increase
Beyond the solow model: Endogeneous Growth model
Production function Y=A.k
Since A is a constant there are constant there are constant returns to scale if k is
multiplied by any no. it can be checked.One extra unit of capital ‘k’ produces a unit of
output regardless of no. of units Solow model had assumed diminishing returns but
this model has constant returns.
∆k=sY-sK
∆k=SAK=Sk
∆k/k=sA -S=∆y/Y
So long sA > S , economy grows forever , even w/o technological progress.
In solow growth model , diminishing returns set in and therefore steady state is
reached but constant returns makes no steady state so, depending upon production
function assumption we can take about the economic growth.But if the traditional
view is believed then the economy will have diminishing returns and if a broad view is
taken then the constant returns seems appropriate.
2 Sector model
Y=FCk1(1-u)LE production function in manufacturing
∆E=g(u).E
production function in research
∆k=sY-SK (capital accumulation)
U=total labor force in university (research)
1-u=total labor force in manufacturing
E=efficiency
G=growth in M
Economy exhibits constant returns to scale If we double k and E then we will get
double Y therefore this model can also have constant or persistent growth w/o n and
g
SIMILARITIES WITH SOLOW
If n is held constant then E increases with constant g(u)same as solow and rest of
the model also assembles solow.
There are 2 important variables in this model .S and u they affect the level of income
but only u affects the steady state. This explains how growth happens in the economy
.
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