The Neoclassical Growth Models
Presented By :- Sanjukta Kar
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Introduction
I will discuss the Solow Swan model which points out the effects of saving, technological advance
and population expansion.
Next I go over the Ramsey, Cass and Koopmans's closed economy model with intertemporally
utility maximizing infinitely lived generations.
Lastly, I discuss a version of the overlapping generations model.
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The Solow Swan Model of Fixed Savings
Solow and Swan (1956) incorporated a supply side to the Keynesian aggregate demand framework.
Lets look at the modelThe new element which is of interest is technological progress. Technological shocks are classified as
Hicks Neutral with production function as—
where A is the exogenous productivity parameter,
Harrod Neutral with production function as—
where E is the labor augmenting technological shift parameter and EL is the supply of efficiency units of labor.
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The demand side--
The economy is closed and there is no government. Private savings S is a fixed fraction s of current
income Y. So we can write1.
We have a closed economy. So domestic savings= domestic investment. The capital accumulation
equation is given by –
2.
Where delta is the rate of depreciation.
The level of Harrod Neutral productivity E grows at the rate
3a.
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The population grows at the rate
(3b)
Our next step is to normalize all variables by the efficiency labor supply
“EL” . The capital to efficiency labor ratio is
(4)
We divide the capital accumulation equation by efficiency labor unit to get--
(5)
(6)
where (1+z) =(1+n) (1+g)
Re-writing equation 5 as
(7)
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where
function.
is the intensive form of the production
Equation (7) is the main equation of the Solow Model. The first term inside
brackets is the gross savings per efficiency worker in period t. The second
term inside brackets shows k decreases due to depreciation and the growth
in efficiency unit of labor. We divide the term in brackets by 1/(1+z) as any
effect on net investment in period t will be felt by efficiency workers in
period (1+t).
Now the Solow Swan model is represented diagrammatically--
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Comments - We have a concave production function and a concave
savings curve as savings is proportional to output. The distance between
the two curves is the consumption per efficiency unit of labor. The 45
degree line gives the savings needed to maintain any level of capital in
efficiency unit.
Dynamics
When
the savings are inadequate to maintain the
capital stock. So the capital per efficiency unit is falling. We have
. The opposite happens when capital stock is low. Overtime the
economy converges to the steady state where we have equality.
Eventually we get the long run equilibrium at which we have
1)
is constant
2) As EL grows at the gross rate
K and Y also
grow at the same rate.
3) Steady state per capita output Y/L grows at the rate of technological
advance
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Implications
Effect of a decline in savings rate
When savings fall capital efficiency labor ratio falls. Per capita income falls.
The growth of these variables fall temporarily. Eventually the economy
adjusts to its new lower levels and achieves its former level of growth.
Effect of a rise in the population growth rate
Reduces the steady state output per capita as
line rotates
upward as capital per efficiency unit falls.
Economic logic -- Solow model has the assumption of fixed savings rate.
So a higher population diminishes the capital resources.
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The Ramsey- Cass - Koopmans Growth Model with Infinitely Lived
Representative Dynasty
Let’s think of a world with identical and infinitely lived dynasties. Each
dynasty have a size L and they grow at an exogenous rate of (1+n) so that
we have
We have egalitarian dynasty planners who weigh the consumption of each
generation in proportion to its size. At each date t the representative
household maximizes
(8)
where ct is the consumption of the representative member alive at time t.
Condition for bounded dynasty utility is
As with Solow the labor augmenting technological progress is given by
There is no depreciation.
The budget constraint is
(9)
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Dividing by Lt we get
(10)
Here we are dealing with per worker production function.
Maximizing equation 8 with respect to equation 10 we get the first order Euler equation-
(11)
With isoelastic production function
(12)
we have the following
Euler equation
(13)
Dividing both sides of 10 and 13 by EL we get
(14)
(15)
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Steady State values
We get the steady state value of capital per efficiency labor when
consumption is not changing as given by the vertical line.
(16)
We get the steady state value of consumption per efficiency unit of labor
when capital is not changing as given by the hump shaped curve.
(17)
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Dynamics of the Model
To the right of the steady stock of capital the capital is above its steady state value, so
the marginal product of capital is below the steady state value of marginal product.
So consumption is falling. Consumption rises to the left of the vertical line.
If equation 14 is less than zero then consumption is above its steady state value. We
have a zone of rising consumption and falling capital.
For any initial level of capital per efficiency unit there is an unique consumption thus
enabling the economy to move along the saddle and converge to the steady state.
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Implications
Effect of a decline of the patience factor beta
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People consumes more and savings decline. `From equation 16 we find the
steady
state value of capital per efficiency unit falls as people become more
impatient i.e. they have a lower beta. This situation is analogous to the fall
in savings rate in the Solow Model.
A lower beta leads to an upward jump in consumption to the new saddle
path. But we find the capital stock has fallen as well.
So lower capital stock leads to a decline in output per efficiency unit and
consumption per efficiency unit and in the new long run position the new
consumption is lower than the initial steady state value.
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Effect of a rise in the population growth rate
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The
unchanged.
schedule shifts down but the
schedule is
At the new steady state we have
Unchanged capital stock per efficiency unit.
Unchanged output per efficiency unit
A decrease in consumption in efficiency units.
Difference with Solow - The path of output is doesn't fall as opposed to the
Solow Model.
with a fixed savings as we have forward looking planners who take care of
the utility of the future generation in making today’s consumption choice.
When they expect a higher population in the future they cut back on
consumption today to provide the same path of capital and output for their
progeny.
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An Overlapping Generations Growth Model
Assume individuals maximize utility according to the utility function
(18)
Total population = labor supply and grows at the rate n.
Each period agents earn income from wages and from renting out capital.
The budget constraint is
(19)
v is defined as family vintage.
Maximizing utility with respect to budget constraint we get the Euler
equation
(20)
Using
and
per capita terms.
=
we get the budget equation in
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Using
and
=
we get the budget equation in
per capita terms.
(21)
By using
we get the closed economy overlapping
generations model as
(22)
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Dynamics
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Like before a rise in population shifts the
curve downwards.
schedule moves upward.
We have a reduction in capital labor ratio like the Solow model.
As the present generation does not care about unborn future generations
the steady state capital stock will be inefficiently high.
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Comments
Questions?
Suggestions?
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