Transcript growth rate of output per head

Macroeconomic Theory
Chapter 7
Exogenous Growth Theory
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
 Long run economic growth determines how living standards
change. This chapter introduces exogenous growth theory,
beginning with the Solow-Swan model. The role of capital,
savings, population growth and technological progress in
determining the equilibrium levels of output per worker in the
economy is also elaborated.
 There are two key concepts in growth theory, the steady state or
balanced growth, and the transitional dynamics (what happens
when steady state is disturbed).
 The growth model in the long run is characterized as one of
steady state growth or as a balanced growth path in which output
and employment grow at constant proportional rates and in
which net savings (and investment) is a constant share of output,
and the growth rate of output per-capita is zero.
 A key characteristic of the Solow-Swan model is that technological
progress is an exogenous force.
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
 Key concepts in the discussion of income disparities are those of
absolute and conditional convergence.
 A much weaker notion of diversion is labeled conditional
divergence; countries would not be expected to converge to the
same living standards unless they are similar in important
respects, such as having the same saving rate. This means that
poor countries will not catch up and achieve the living standards
of the rich countries unless they are able to change the
determinants of their steady state.
 Although the Solow-Swan model provides the foundation for
growth theory, there has been much work on developing models
in which the growth of living standards is endogenous, rather
than the result of exogenous growth.
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
Fig 13.1
1– high rates of growth
2– high rates of growth
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
 The Solow-Swan model
 Some concepts:
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Capital is productive, but it does not need to be productive all
the time (it can be withdrawn), inventories of intermediate and
finished products are also part of the capital stock.
Agents face a trade off between consuming some resources
today or transforming them into capital, i.e., produce
tomorrow.
Capital earns a return, so it can be rented at a real rate R.
Capital depreciates, i.e., it becomes less productive as time
passes. It is assumed that capital depreciates at a constant
range δ, the real interest rate r =R – δ. Since δ is constant R
and r move together.
Capital is a rival good, i.e., the usage of one unit of capital by
an individual necessarily means that no one else can use it.
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
 The production function
 The simple production function takes the form Y=f(K,L), where
the production function is smooth and the MPK (FK) and MPL
(FL) are positive and diminishing. The production function relates
flows of output to flows of capital and labor services. The
assumption of diminishing returns to capital plays an important
role in defining the difference between models of exogenous and
endogenous growth.
 We shall assume constant returns to scale, i.e., both inputs and
outputs are increased by the same factor, F(θK, θL)= θF(K,L),
where θ is a positive constant. We can write the production
function in the intensive form (per worker terms, y=Y/L, k=K/L,)
as:
Y=Y/L=1/LF(K,L)=F(K/L,L/L)=F(k,1)=F(k)
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
 The constant returns to scale implies that average and marginal
returns depend only on the factor ratio k, K/L. it follows then
from the CRS assumption that APK, and MPK are decreasing
functions of K/L (APL and MPL are increasing functions of K/L).
We have;
APK=f(k)k and APL=f(k)
MPK=f’(k) and MPL=f(k-f’(k)k.
 In practice it is always convenient to use the Cobb-Douglas
production function given by;
Y=KαL1-α
Y/L=Kα/L L/Lα
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
 How labor and capital inputs change over time
 We shall assume that labor force grows at a constant growth n. if
we work in continuous time the growth rate is:
n=L./L=(dl/dt)/L (growth rate of labor input: exogenous)
 Which implies that the labor force grows exponentially and for
any initial level L0, the level of labor force is Lt=L0exp(nt)
 By contrast, the growth of the capital stock depends on economic
factors. In a closed economy (no borrowing) S=I, and a constant
saving growth rate s,
I=sF(K,L)
 Where I is gross investment. To see how capital stock changes
(dk/dl or K.) we need to deduct depreciation from gross
investment, i.e.
K.=I-δK
 Where δ is the rate of capital appreciation.
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
 Where δ is the rate of capital depreciation. By incorporating the
condition that I=S
K.=I-δK=sF(K,L)-δK.
(13.6)
 By dividing through by K we have
gK=K./K=sY/K-δ=sAPK-δ. (growth rate of capital)
 Which says that the growth of capital stock depends on the APK
and is therefore a declining function of the capital labor ratio (k).
 Steady state or balanced growth
 The capital stock is growing at the same rat of the labor force n
gK=n
 The capital labor ratio will be constant. This is called the steady
state capital-labor ratio k*. The steady state requires
sY/K-δ=n
(growth of capital=growth of labor)
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
 Which implies
ν*=K*/Y*=k*/y*=s/(n+δ) (steady state capital-output ratio: Domar’s
formula)
 Domar’s formula provides an explicit expression for the steadystate capital output ratio, which does not depend on the particular
form of the production function. The results of the Solow-Swan
growth model can be summarized so far as:
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In steady state growth, output and capital grow at the same rate as the
exogenously given growth rate of the labor force. There is no growth in
output per capita in the steady state
The capital output ratio in the steady state is higher, the higher is the saving
rate and the lower are the labor force growth rate and depreciation.
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
 There is a second way of characterizing the steady-state growth
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path. By dividing through both side of eq 13.6 by L we obtain;
K./L=sf(k)-δk.
13.7
In order to rewrite our eq in terms of k only, we note that;
k./k=K./K-L./L
Multiplying each side by K/L and simplifying gives
k.=K./L-kn
So K./L=k.+kn. Substituting this expression into 13.7 and
rearranging we obtain the fundamental Solow Equation of Motion
which describes how capital per worker varies over time;
k.=sf(k)-(δ+n)k.
13.8
The first term of this equation shows the extent to which
investment is adding to the capital stock per worker. The second
term show the amount of capital needed to offset depreciation
(δk) and to equip additions of to L at existing k, (nk).
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
 Note that if there is no saving (s=0) then k.=-(δ+n), that is k
would be falling under the pressures of:
1.
2.
An increasing population, n>0
Capital depreciation, δ>0.
 If sf(k)>(δ+n)k, capital per worker increases because investment
per head is greater than the reduction in capital per head due to
an increasing population and depreciation. If sf(k)<(δ+n)k,
capital per worker decreases because investment per head is
smaller than the reduction in capital per head due to an
increasing population or depreciation.
 Fig 13.3 plot the two parts of the RHS of eq 13.8. since s is a
fraction, the shape of the sf(k) curve is given by the properties of
the production function. As δ and n are constant the second term
is a line from the origin with a slope of δ+n. the point of
intersection of the two curves is given by the level of capital per
worker (k*) where sf(k*).
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
Fig 13.3
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
 At this point k.=0 and y.=0, i.e. the level of capital per worker and
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output are constant.
This defines a steady state in the Solow-Swan model: at k*, both
k and L grow at the same constant rate n. in fig 13.3 the steady
state level of output per head (y*) and consumption per head (c*)
are shown. The steady state level of output per head is given by:
y*=f(k*)
(steady-state output per head)
While the steady state level of consumption per head is also
constant and is a fraction of the steady state income
c*=f(k*)-sf(k*)=(1-s)f(k*) (steady state consumption per head)
What is the growth rate of output gY=Y./Y? according to Domar
formula, output grows at the same rate as does capital and labor.
Differentiating Y=F(K,L), with respect to time;
Y.=FKK.+FLL.
where FK=dF(K,L)/dK and FL=dF(K,L)/dL.
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
 Divide both sides by Y to obtain the growth rate of output on LHS
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gY=FKK./Y+FLL./L
Multiply each of the RHS in terms of growth rates,
gY=(FKK/Y)gK+(FLL/Y)gL
Which with a constant rate of labor, n, can be written as:
gY=σKgK+σLn (growth rate of output)
σK=FKK/Y, σL= FLL/Y
The growth rate of output is a weighted average of the growth
rates of L and K.
The alternative Solow diagram (fig13.4), is useful for analyzing
the gY.
In the model so far, the steady state is characterized by constant
level of output per worker y*, if population and capital growth
are the same, the capital per worker remains constant at k* and
output per worker y*.
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
Fig 13.4
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
 The intuition for this result is clear: diminishing returns to K
ensure that extra unit of K per worker produce less and less; at
the same time depreciation does not diminish. Eventually all
savings must will be deployed to replace the existing capital, i.e.,
when sf(k*) cuts the (δ+n)k line in fig 13.3, the point at which the
economy will be in eq with k.=0.
 Transitional dynamics: what happens out of equilibrium?
 In fig 13.3 when capital labor ratio is below the steady state,
capital intensity will increase and vice versa. The economy will
adjust until k=k*. It is useful to present the model in a slightly
different way. Consider gK=k./k, the growth rate of capital per
worker. Divide eq 13.8 by k to obtain
gk=sf(k)/k-(δ+n)
=s.APK-(δ+n)
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
 The two components of the RHS are plotted in fig 13.5. in the
steady state when k=k*, sf(k*)/k*=δ+n, thus the growth rate of
capital per worker is given by the vertical distance between the
tow curves. Given the growth rate of L, n, capital stock per
worker also grows. The economy moves from A to B. similarly, for
high k, capital stock per worker will decline.
 Following the same procedure we used for working out the growth
rate of Y, we have
gy=(MPK/APK)gk=σKgK
(growth rate of output per head)
 Where with competitive markets, σK is capital’s share of output
and is positive and less than one. In the steady state, the equation
still holds because gy=gk=0
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
Fig 13.5
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
 Policy experiment 1: a rise in the saving rate
 At time t1, there is an exogenous rise in the saving rate, fig 13.6
traces the impact of this shock and show how k and y adjust as
the economy moves to a new steady state. We only observe a
positive growth rates in per worker variables if the economy
experiences a parameter shift. Once the system returns to its
steady state we are back on the balanced growth path with zero
growth of output per capita.
 Along with the rise in capital per worker and productivity, there is
a rise in the capital output ratio. This means that the faster capital
stock growth generated by the higher savings rate fades away as
depreciation eventually swallows up all the additional investment
effort.
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
Fig 13.6
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
 Policy experiment 2: a rise in the population growth rate.
 Population growth suddenly and permanently goes up due to an
exogenous change in fertility. Fig 13.7 shows the impact of this
shock on the system, it traces how k and y adjust to the new
steady state. At k** capital per worker and output per worker is
lower than at t1, capital intensity declines.
 What happens to the growth rate of output in this economy? To
see this we use alternative Solow diagram in fig 13.8.
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
Fig 13.7
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
Fig 13.8
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
 These experiments leave an open question of what generates the
type of sustained growth documented in chapter 1. the model can
account for this by either
 The response of very long adjustment periods as the economy
transits from one steady state to another
 The presence of repeated shocks, e.g., a steadily rising savings
share or a steadily falling population growth rate.
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University
 Solow growth accounting: measuring the impact of technology
Macroeconomic Theory
Prof. M. El-Sakka
CBA. Kuwait University