# Growth theory Swan-Solow model (exogenous growth )

Document Sample

```					Growth theory
Swan-Solow model (exogenous growth)

•   Production function: Y = AF ( K , AL) where A=technology that is ‘labour-augmenting’ or ‘Harrod-
neutral’, K=capital stock, L=labour force
•   The production function has constant returns to scale overall, but diminishing returns to scale from each
factor individually

•   Technology is exogenous – determined outside the model – ‘like manna from heaven’ – and grows at a
constant rate g
•   There is only a single good being produced, and therefore no place for trade

Y
•   In terms of income per capita: y =     = F ( K , A) (assumes population = labour force)
L
•   In Cobb-Douglas form: y = k α A1−α
•   Therefore growth in per capita income is proportional to the change in the capital stock of the economy

•   A situation in which capital, output, consumption and population are growing at constant rates is called a
balanced growth path
•   Along a balanced growth path, output per worker and capital per worker both grow at the rate of
exogenous technological change, g

•                                                      ɶ K =k
The ratio of capital (per worker) to technology is k =
AL A

•   The change in the capital-technology ratio is determined by gross investment less the amount of capital
needed to replace depreciation and the amount of capital needed for new workers (capital widening), and
the capital needed for new ‘effective workers’ because of technological growth:
ɺ
ɶ                    ɶ
•    k = sy − (d + n + g )k
ɶ
•   Gross investment per worker = sy as savings=investment and all investment is used to accumulate capital

The Solow diagram:

•   A steady state equilibrium of capital per worker is reached where k = k *

•   If the economy is out of equilibrium it is on the transition path to the steady-state
•   If sy > (d + n)k then the amount of investment exceeds the amount needed to keep capital-technology
ratio constant, so capital deepening occurs until k = k *
•   If sy < (d + n)k then the amount of investment provided is less than the amount needed to keep the
capital-technology ratio constant, and so the capital-technology ratio falls until k = k *
•                                                             ɶ
When the economy reaches the point where sy = (d + n + g )k the economy is in steady state and grows
ɶ
along a balanced growth path at rate g

•   The steady-state value of output per-worker, y* is then determined by the production function

•   The difference between the steady-state investment per worker sy* and the steady-state output per
worker y* is given by the steady-state consumption per-worker

ɺ
ɶ                   ɶ
• k = sy − (d + n + g )k
ɶ
ɺ
• k = sk α − (d + n + g )k
ɶ                     ɶ
1

•
ɺ
ɶ         ɶ 
Setting k = 0 =0, k * = 
s    1−α

n+d +g 
a
  s    1−α
•   Substituting into the production function, y *(t ) = A(t )         , explicitly noting the dependence
n+d +g 
of y and A on time

Therefore the steady-state level is higher if:-
• Technological progress is higher
• Savings/investment rates are higher
• The rate of depreciation is lower
• Population growth is lower

•   Empirically it does tend to be the case that countries with high investment rates are generally richer than
those with low investment rates and countries with higher population growth rates are generally poorer
The effects of an increase in the savings rate :

•   The increase in the savings rate raises the growth rate temporarily as the economy transits to the new
ɶ

•   When the output-technology ratio reaches its new steady state growth returns to its long-run level of g

•   Thus, policy changes targeted at savings/investment can have no long-run growth effects, only level
effects

Convergence

•   Without technological progress, there can be no sustained growth in the Solow model
•   Growth slows down as the economy approaches its steady state and eventually stops altogether

•   ɺ
k = sy − (d + n)k
ɺ
k
•     = sk α −1 − (d + n)
k
ɺ
k
•   Because α <1, as k rises,     gradually declines
k

Absolute convergence:
If countries have access to identical production technologies, identical savings rates and identical population
growth, then their economies will converge to the same steady-state capital-technology ratio

•   In getting to this level, countries that are further away from the steady-state will exhibit faster growth than
•   Once at the steady-state all countries growth rates will be equal, and proportional to g, which will be
identical everywhere because of technology transfer

Evidence for this hypothesis comes from post-war Japan, which had seen massive capital destruction, but
retained similar levels of technology, population growth and savings to other developed countries, and grew
rapidly, seeing output per capita converge with other major economies

Conditional convergence:
If countries have access to identical production technologies, and identical population growth, then their
economies will converge to the same rate of growth at their own respective steady-states

•   Conditional convergence takes into account differences in savings rates

Evidence for this hypothesis comes from the fact that there has been little absolute convergence worldwide,
but countries with similar population growth such as India and Nigeria have converged to similar growth
rates at their own steady states
Y=AK model (endogenous growth)
This is a highly simplified growth model that endogenizes the source of growth, and thus makes for useful
comparison with the Solow model

Y      K
•   The production function is:     =A
L      L
•   Here, K is a broader measure of capital, exhibiting constant returns to scale rather than the diminishing
returns that characterised the Solow model
•   This can be justified by perhaps including human capital, innovation and R&D, all of which may have
increasing returns to scale

∂y
•   The marginal product of capital is given by      =A
∂k

•                                              ɺ
Capital accumulation over time is given by k = sAk − dk

ɺ
k
•   Therefore     = sA − δ
k

Hence, all that is needed for an economy to ensure perpetual growth is for sA > δ

Therefore the rate of growth increases if:
• The savings rate increases
• The level of technology increases
• The rate of capital depreciation decreases

•   Here there is no steady-state to converge to as before
•   There will only be convergence in growth rates if sA is the same everywhere

•   The feature of conditional convergence can be retained in the model with constant returns to capital in
the long run, as demonstrated by Jones and Manuelli (1992)
Past questions
TT 2004-5
Is it true that the Neoclassical (Solow) growth model predicts a decreasing growth rate of output while the
Endogenous growth tradition postulates a constant growth rate of output?

TT2003-4
Would you expect to see convergence in levels of income per capita across countries over time?

LV 2003-4
“Poor countries should grow faster than rich ones.”
“Savings rate does not affect long term economic growth rate, hence can be ignored by policy makers.”
Are these statements correct in light of economic theory?

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 42 posted: 1/4/2010 language: English pages: 6