Transcript The Solow Growth Model

The Solow Growth
Model (Part Three)
The augmented model that includes
population growth and technological
progress.
Model Background
• As mentioned in parts I and II, the Solow growth
model allows us a dynamic view of how savings
affects the economy over time. We learned about
the steady state level of capital and how a golden
rule steady state level of capital can be achieved by
setting the savings rate to maximize consumption
per worker. We now augment the model to see the
effects of population growth and technological
progress.
Steady State Equilibrium
• By expanding our model to include population growth our
model more closely resembles the sustained economic growth
observable in much of the real world.
• To see how population growth affects the steady state we need
to know how it affects the accumulation of capital per worker.
When we add population growth (n) to our model the change in
capital stock per worker becomes…
Δk = i – (δ+n)k
• As we can see population growth will have a negative effect on
capital stock accumulation. We can think of (δ+n)k as breakeven investment or the amount of investment necessary to keep
capital stock per worker constant.
• Our analysis proceeds as in the previous presentations. To see
the impact of investment, depreciation, and population growth
on capital we use the (change in capital) formula from above,
Δk = i – (δ+n)k …substituting for (i) gives us,
Δk = s*f(k) – (δ+n)k
Steady State Equilibrium with
population growth
• At the point where both
Like depreciation, population
growth is one reason why the
capital stock per worker shrinks.
(k) and (y) are constant it
Investment
must be the case that,
Break-even
Investment
Δk = s*f(k) – (δ+n)k = 0
…or,
s*f(k) = (δ+n)k
…this occurs at our
s*f(k*)=(δ+n)k*
equilibrium point k*.
Break-even
investment
(δ+n)k
s*f(k)
Investment
k*
At k* break-even
investment equals
investment.
k
The impact of population growth
An increase
in “n”
• Suppose population growth
changes from n1 to n2.
• This shifts the line
representing population
growth and depreciation
upward.
• At the new steady state k2*
capital per worker and output
per worker are lower
• The model predicts that
economies with higher rates
of population growth will have
lower levels of capital per
worker and lower levels of
income.
Investment
Break-even
Investment
…reduces k*
(δ+n2)k
(δ+n1)k
s*f(k)
k2*
k1*
k
The efficiency of labour
• We rewrite our production function as…
Y=F(K,L*E)
where “E” is the efficiency of labour. “L*E” is a
measure of the number of effective workers. The
growth of labour efficiency is “g”.
• Our production function y=f(k) becomes output
per effective worker since…
y=Y/(L*E) and k=K/(L*E)
• With this augmentation “δk” is needed to replace
depreciating capital, “nk” is needed to provide
capital to new workers, and “gk” is needed to
provide capital for the new effective workers
created by technological progress.
Steady State Equilibrium with population
growth and technological progress
• At the point where both
(k) and (y) are constant it
must be the case that,
Δk = s*f(k) – (δ+n+g)k = 0
…or,
s*f(k) = (δ+n)k
…this occurs at our
equilibrium point k*.
Like depreciation and population
growth, the labour augmenting
technological progress rate causes the
capital stock per worker to shrink.
Break-even
investment
Investment
Break-even
Investment
(δ+n+g)k
s*f(k)
s*f(k*)=(δ+n)k*
Investment
At k* break-even
investment equals
investment.
k*
k
The impact of technological progress
• Suppose the worker
An increase
in “g”
efficiency growth rate
changes from g1 to g2.
• This shifts the line
representing population
growth, depreciation, and
worker efficiency growth
upward.
Investment
Break-even
Investment
(δ+n+g2)k (δ+n+g )k
1
• At the new steady state k2*
s*f(k)
capital per worker and
output per worker are lower.
• The model predicts that
economies with higher rates
of worker efficiency growth
will have lower levels of
capital per worker and lower
levels of income.
k2*
…reduces k*
k1*
k
Effects of technological progress on the golden rule
• With technological progress the golden rule level of capital is
defined as the steady state that maximizes consumption per
effective worker. Following our previous analysis steady
state consumption per worker is…
c* = f(k*) – (δ + n + g)k*
• To maximize this…
MPK = δ + n + g
or
MPK – δ = n + g
• That is, at the Golden Rule level of capital, the net marginal
product of capital MPK – δ, equals the rate of growth of total
output, n+g.
Steady State Growth Rates in the Solow Model with
Technological Progress
Variable
Symbol
Steady-State Growth
Rate
0
Capital per
effective worker
k=K/(E*L)
Output per
effective worker
y=Y/(E*L)=f(k) 0
Output per
worker
Y/L=y*E
g
Total output
Y=y(E*L)
n+g
Conclusion
• In this section we added changes in two exogenous
variables (population and technological growth) to the
Solow growth model. We saw that in steady state
output per effective worker remains constant, output
per worker depends only on technological growth, and
that Total output depends on population and
technological growth.