Transcript K - Sapienza

POLITICAL ECONOMY OF GROWTH
SECS-P01, CFU 9
Finance and Development
academic year 2016-17
7. THE SOLOW MODEL
Roberto Pasca di Magliano
Fondazione Roma Sapienza-Cooperazione Internazionale
[email protected]
Model Background
• The Solow growth model is the starting point to
determine why growth differs across similar countries
– it builds on the Cobb-Douglas production model by
adding a theory of capital accumulation
– developed in the mid-1950s by Robert Solow of MIT,
it is the basis for the Nobel Prize he received in 1987
– the accumulation of capital is a possible engine of
long-run economic growth
Building the Model: goods market supply
• We begin with a production function and assume constant returns.
•
Y=F(K,L)
so… zY=F(zK,zL)
• By setting z=1/L it is possible to create a per worker function.
•
Y/L=F(K/L,1)
• So, output per worker is a function of capital per worker.
•
y=f(k)
Building the Model: goods market supply
• The slope of this function is
•
the marginal product of
capital per worker.
MPK = f(k+1)–f(k)
It tells us the change in
output per worker that
results when we increase
the capital per worker by
one.
y
change
in
y
MPK

change
in
k
y=f(k)
Change in y
Change in k
k
Building the Model:
goods market demand
• Begining with per worker consumption and investment
(Government purchases and net exports are not included in the Solow
model), the following per worker national income accounting
identity can be obtained:
•
y = c+I
• Given a savings rate (s) and a consumption rate
(1–s) a consumption function can generated:
c = (1–s)y
…which is the identity. Then
y = (1–s)y + I …rearranging,
i = s*y
…so investment per worker
equals savings per worker.
Steady State Equilibrium
• The Solow model long run equilibrium occurs at the point
•
•
where both (y) and (k) are constant.
The endogenous variables in the model are y and k.
The exogenous variable is (s).
Steady State Equilibrium
In order to reach the stady state equilibrium:
•By substituting f(k) for (y), the investment per worker function (i =
s*y) becomes a function of capital per worker (i = s*f(k)).
•By adding a depreciation rate (d).
•The impact of investment and depreciation on capital can be
developed to evaluate the need of capital change:
•
dk = i – dk
•
…substituting for (i)
dk = s*f(k) – dk
The Solow Diagram equilibrium
production function, capital accumulation (Kt on the x-axis)
Investment,
Depreciation
At this point,
dKt = sYt, so
Capital, Kt
The Solow Diagram
When investment is greater than depreciation, the capital stock increase until investment
equals depreciation. At this steady state point, dK = 0
Investment, depreciation
Depreciation: d K
Investment: s Y
Net investment
K0
K*
Capital, K
Suppose the economy starts at K0:
•The red line is above the
Investment,
Depreciation
green at K0:
•Saving = investment is
greater than depreciation at K0
•So ∆Kt > 0 because
•Since ∆Kt > 0, Kt increases
from K0 to K1 > K0
Capital, Kt
K0
K1
Now imagine if we start at a K0 here:
Investment,
Depreciation
•At K0, the green line is above the
red line
•Saving = investment is now less
than depreciation
•So ∆Kt < 0 because
•Then since ∆Kt < 0,
Kt decreases from K0 to K1 < K0
Capital, Kt
K 1 K0
We call this the process of transition dynamics:
Transitioning from any Kt toward the economy’s
steady-state K*, where ∆Kt = 0 and growth ceases
Investment,
Depreciation
No matter where we start, we’ll transition to K*!
At this value of K,
dKt = sYt, so
Capital, Kt
K*
Changing the exogenous variable - savings
• We know that steady state is
Investment,
Depreciation
dk
at the point where s*f(k)=dk s*f(k*)=dk*
• What happens if we
s*f(k)
s*f(k)
s*f(k*)=dk*
increase savings?
• This would increase the
•
slope of our investment
function and cause the
function to shift up.
This would lead to a higher
steady state level of capital.
• Similarly a lower savings
rate leads to a lower steady
state level of capital.
k*
k**
k
We can see what happens to output, Y, and thus to growth if we
rescale the vertical axis:
• Saving = investment and
Investment,
Depreciation, Income
depreciation now appear
here
• Now output can be
Y*
graphed in the space
above in the graph
• We still have transition
dynamics toward K*
• So we also have
dynamics toward a
steady-state level of
income, Y*
K*
Capital, Kt
The Solow Diagram with Output
At any point, Consumption is the difference between Output and
Investment: C = Y – I
Investment, depreciation,
and output
Output: Y
Y*
Consumption
Depreciation: d K
Y0
Investment: s Y
K0
K*
Capital, K
Conclusion
• The Solow Growth model is a dynamic model that
allows us to see how our endogenous variables
capital per worker and output per worker are
affected by the exogenous variable savings.
• We also see how parameters such as
depreciation enter the model, and finally the
effects that initial capital allocations have on the
time paths toward equilibrium.
• In other section the dynamic model is improved in
order to include changes in other exogenous
variables; population and technological growth.