The Solow Growth Model (Part Three)

1
The Solow Growth Model (Part Three)

• The augmented model that includes population
  growth and technological progress.

2
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.

3
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?k i (dn)k
• As we can see population growth will have a
  negative effect on capital stock accumulation.
  We can think of (dn)k as break-even 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,?k i (dn)k substituting for (i)
  gives us,?k sf(k) (dn)k

4
Steady State Equilibrium with population growth
Like depreciation, population growth is one
reason why the capital stock per worker shrinks.

• At the point where both (k) and (y) are constant
  it must be the case that,?k sf(k) (dn)k
  0 or,sf(k) (dn)kthis occurs at our
  equilibrium point k.

InvestmentBreak-even Investment
sf(k)(dn)k
k
k
At k break-even investment equals investment.
5
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.

InvestmentBreak-even Investment
(dn2)k
(dn1)k

• 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.

sf(k)
k
k2
k1
reduces k
6
The efficiency of labour

• We rewrite our production function
  asYF(K,LE)where E is the efficiency of
  labour. LE is a measure of the number of
  effective workers. The growth of labour
  efficiency is g.
• Our production function yf(k) becomes output per
  effective worker sinceyY/(LE) and kK/(LE)
• With this augmentation dk 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.

7
Steady State Equilibrium with population growth
and technological progress
Like depreciation and population growth, the
labour augmenting technological progress rate
causes the capital stock per worker to shrink.

• At the point where both (k) and (y) are constant
  it must be the case that,?k sf(k) (dng)k
  0 or,sf(k) (dn)kthis occurs at our
  equilibrium point k.

Break-even investment (dng)k
InvestmentBreak-even Investment
sf(k)Investment
sf(k)(dn)k
At k break-even investment equals investment.
k
k
8
The impact of technological progress

• Suppose the worker efficiency growth rate changes
  from g1 to g2.
• This shifts the line representing population
  growth, depreciation, and worker efficiency
  growth upward.

An increase in g
InvestmentBreak-even Investment
(dng2)k
(dng1)k
sf(k)

• At the new steady state k2 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.

k
k2
k1
reduces k
9
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 isc f(k) (d n
  g)k
• To maximize thisMPK d n gorMPK d n
  g
• That is, at the Golden Rule level of capital, the
  net marginal product of capital MPK d, equals
  the rate of growth of total output, ng.

10
Steady State Growth Rates in the Solow Model with
Technological Progress
Variable Symbol Steady-State Growth Rate
Capital per effective worker kK/(EL) 0
Output per effective worker yY/(EL)f(k) 0
Output per worker Y/LyE g
Total output Yy(EL) ng
11
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.