Transcript Lecture6 - Stanford University

Economics 216:
The Macroeconomics of Development
Lawrence J. Lau, Ph. D., D. Soc. Sc. (hon.)
Kwoh-Ting Li Professor of Economic Development
Department of Economics
Stanford University
Stanford, CA 94305-6072, U.S.A.
Spring 2000-2001
Email: [email protected]; WebPages: http://www.stanford.edu/~ljlau
Lecture 6
Models of Economic Development:
One-Sector Models
Lawrence J. Lau, Ph. D., D. Soc. Sc. (hon.)
Kwoh-Ting Li Professor of Economic Development
Department of Economics
Stanford University
Stanford, CA 94305-6072, U.S.A.
Spring 2000-2001
Email: [email protected]; WebPages: http://www.stanford.edu/~ljlau
One-Sector Closed Economy Models
 Optimizing
Models of Growth
 Specification
of an objective function
 Consumption versus savings
 Choice of technique
 Allocation of investment
 Descriptive
Models of Growth
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Assumptions

Consumers




Production





Representative consumer with time preference
Infinite lifetime
Existing level of output per capita exceeds the subsistence level of
consumption per capita (otherwise no capacity for savings)
One-sector aggregate production function as a function of capital stock and
labor
Investment
Distribution
Exogenously determined rate of growth of population
CAPITAL ACCUMULATION IS THE LINK BETWEEN THE
PRESENT AND THE FUTURE
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The Harrod-Domar Model
 Production
function with fixed coefficients (no substitution
possibilities)
 Y = min {aKK, aLL}
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One-Sector Model with Neoclassical
Production Function
 Production
function with smooth substitution possibilities
 Cobb-Douglas
production function
 Constant-Elasticity-of-Substitution (C.E.S.) production function
 Special
case of elasticity of substitution greater than unity
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The Neoclassical Model of Growth (Solow)

Production Function


One-sector aggregate production function as a function of capital stock and
labor
Y = F(K, L)
Consumers





Representative consumer with time preference
Infinite lifetime
Existing level of output per capita exceeds the subsistence level of
consumption per capita
No income-leisure choice
Consumption and savings behavior
C = (1-s) Y;
S = sY; where s is the savings rate, assumed to be constant
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The Neoclassical Model of Growth (Solow)
 Producers
 Competitive
maximization of profits
 Investment behavior
I=S
 Equilibrium
in output markets
C+I=Y
 Equilibrium in factor markets
 Full
employment of capital and labor
 Population
growth
L = L0ent
 Capital accumulation
 The
link between present and future
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The Neoclassical Model of Growth (Solow)
Capital Accumulation
dK

K
 I  K, where is the rate of depreciation
dt
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The Assumption of Constant Returns to Scale
 Let
y  Y/L; k  K/L. Under constant returns to scale,
F(K, L) =  F(K, L)
 Let
=1/L, then,
F(K/L, L/L) = F(K, L)/L, or
= Y/L = F(k, 1)  f(k), the “intensive” form of the
production function, expressing output per unit labor as a
function of capital per unit labor
y
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Differential Equation of
k
k  d K  1  dK  K dL
dt L L dt L2 dt
dK
 I  K  S  K  sY  K  syL - kL;
dt
dL d
nt
nt
 L 0 e  nL 0 e  nL; so that
dt dt
k  sy  k  nk  sf(k)  (  n)k
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Differential Equation of
The Equation of Motion
k:
k  sf(k)  (  n)k
Given an initial condition on k, say k(0)  k 0 ,
the entire future time path of k is determined.
But if k is determined, so is y. Moreover, given
k 0 , y 0 is also known and vice versa. Thus y(t)
can be expressed as a function of y 0 , the initial
value of y.
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Differential Equation of
k
k  sf(k)  (  n)k  0
implies that K/L is a constant over time, so that
capital grows at the same rate as labor, which is
assumed to be the same as the exogenously given
rate of population growth of n.
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Existence of Steady-State Growth
The Case of the Constant Savings Rate
2
1
k
0
k*
-1
-2
dk/dt=sf(k) - (d+n)k
-(d+n)k
sf(k)
s*f(k)
dk/dt=s*f(k)-(d+n)k
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Existence of Steady-State Growth
The Case of the Constant Savings Rate
2
1
dk/dt=sf(k) - (d+n)k
-(d+n)k
sf(k)
-(d+n)*k
dk/dt=sf(k)-(d+n)*k
k
0
k*
-1
-2
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The Inada (1964) Conditions
 The
marginal productivity of capital approaches infinity as
capital approaches zero, holding labor constant
 The marginal productivity of capital approaches zero as
capital approaches infinity, holding labor constant
 The Inada conditions are sufficient, but not necessary, for
the existence of a steady state
 It is possible to replace the second Inada condition by the
following (at the cost of possible non-existence of a
steady-state)
 The
marginal productivity of capital approaches a constant as
capital approaches infinity, holding labor constant
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The Role of Strict Monotonicity and Strict
Concavity of the Production Function
 Strict
monotonicity of F(K, L) implies strict monotonicity
of f(k)
 Strict concavity of F(K, L) implies strict concavity of f(k)
 Twice continuous differentiability of F(K, L) implies twice
continuous differentiability of f(k)
 Essentially of K and L implies that f(0) = 0
 The Inada conditions imply that f’(k) approaches infinity
as k approaches zero and f’(k) approaches zero as k
approaches infinity
 f’(k) is therefore a continuously differentiable, positive and
strictly decreasing function of k, taking values within the
range infinity and zero--for
sufficiently large k, f(k)
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approaches a constant
The Role of Strict Monotonicity and Strict
Concavity of the Production Function





The function sf(k)-( + n)k considered as a function of k is
monotonically increasing for small positive values of k because of
the Inada condition
The function sf(k)-( + n)k considered as a function of k is
monotonically decreasing for sufficiently large positive values of k
again because of the Inada condition
The function is strictly concave in k so that its slope is always
declining
For sufficiently small values of k, the function is positive; for
sufficiently large values of k, the function is dominated by -( + n)k
and is hence negative
Given strict concavity, which implies continuity, the function must
be equal to zero for some k*, and only for that k*--there is a unique
value of k = k* for which sf(k)-( + n)k = 0
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Comparative Statics of the Steady State
 Comparative
statics with respect to s
 The effect on the steady-state rate of growth--none
 The effect on the steady-state level of k--positive
 Hence the effect on the steady-state level of y—positive
 Comparative
statics with respect to n
 The effect on the steady-state rate of growth—positive
 The effect on the steady-state level of k—negative
 Hence the effect on the steady-state level of y--negative
 Comparative
statics with respect to 
 The effect on the steady-state rate of growth--none
 The effect on the steady-state level of k--negative
 Hence the effect on the steady-state level of y—negative
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The Case of Purely Labor-Augmenting
(Harrod-Neutral) Technical Progress
 Production
Function
 One-sector
aggregate production function as a function of capital
stock and labor
Y = F(K, Let), where  is the exogenously given rate of purely
labor-augmenting technical progress
 Consumers

C = (1-s) Y;
S = sY; where s is the savings rate, assumed to be constant
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The Neoclassical Model of Growth (Solow)
 Producers
I=S
 Equilibrium
in output markets
C+I=Y
 Equilibrium in factor markets
 Full
employment of capital and labor
 Population
growth
L = L0ent
 Capital accumulation
Lawrence J. Lau, Stanford University
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The Neoclassical Model of Growth (Solow)
Capital Accumulation
dK

K
 I  K, where is the rate of depreciation
dt
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The Assumption of Constant Returns to Scale
y  Y/Let; k  K/Let, respectively output per unit
“augmented labor” and capital per unit “augmented labor”.
Under constant returns to scale,
 Let
 F(K/Let,
Let/Let) = F(K, Let)/Let, or
= Y/Let = F(k, 1)  f(k), the “intensive” form of the
production function, expressing output per unit
“augmented labor” as a function of capital per unit
“augmented labor”
y
Lawrence J. Lau, Stanford University
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Differential Equation of
k
d K
1 dK
K
 t dL
t
k
 t 
 2 2 t (e
  Le )
t
dt Le
Le
dt L e
dt
dK
t
 t dL
 syLe -  kLe ;
 nL; so that
dt
dt
k  sy   k  (n+ )k  sf(k)  (  n+ )k
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Differential Equation of
k
k  sf(k)  (  n   )k  0
implies that K/Le t , output per unit " augmented labor" ,
is a constant over time, so that capital grows at the same
rate as augmented labor, which grows at the sum of the
rate of population growth, n, and the rate of labor
augmentation,  .
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Existence of Steady-State Growth
The Case of Purely Labor-Augmenting Technical Progress
2
sf(k)
-(d+n+c)k
1
sf(k)-(d+n+c)k
k
0
-1
-2
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Steady State in the Case of Purely LaborAugmenting Technical Progress
K/Let =K/L0e(n+)t is equal to a constant in steady
state, K must also be growing at the same rate of (n+) as
“augmented labor”. By constant returns to scale, the rate
of growth of real output is also (n+), independent of the
value of s
 The rate of growth of real output per unit “augmented
labor” is therefore 0, but the rate of growth of real output
per unit (actual, unaugmented) labor is 
 The capital/“augmented” labor ratio is constant, but the
actual capital/labor ratio grows at the rate 
 Since
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The Case of a Non-Constant Savings Rate






Let s  g(y) with g’(y)  0
g’(y) approaches zero for y  some y*
Consider the function sf(k)-(+n)k = g(f(k))f(k)-(+n)k
= f*(k)-(+n)k
For sufficiently large k (and therefore y), g’(y) = 0, the behavior of
f*(k)-(+n)k is therefore similar to that of sf(k)-(+n)k with s a
constant
For sufficiently small k (and therefore y), if g’(y) approaches a
constant as y approaches zero, then the behavior of f*(k)-(+n)k is
again similar to that of sf(k)-(+n)k with s a constant
f*(k)-(+n)k is therefore positive for small k and negative for large k
and therefore must be equal to 0 for some k*
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Existence of Steady-State Growth
The Case of a Non-Constant Savings Rate
2
1
k
0
k*
-1
dk/dt=s(f(k))f(k) - (d+n)k
-(d+n)k
s(f(k))f(k)
-2
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The Case of a Non-Constant Savings Rate






Let s  g(r/p, y), where r/p is the rate of return on capital
g(.) is assumed to be continuously differentiable and weakly
monotonically increasing with respect to r/p and y
r/p = f’(k) under the assumption of competitive profit maximization
Consider the function sf(k)-(+n)k = g(f’(k), f(k))f(k)-(+n)k =
f*(k)-(+n)k; its behavior determines whether a steady state exists
If for some k1, f*(k1) -(+n)k1 0, that is, the savings in the economy
exceed the depreciation and the dilution (due to the growth of the
labor force) of capital; and for some k2, f*(k2) -(+n)k2 0, then a
steady state exists and is stable.
Condition II is generally satisfied because g(.) is bounded by, say, 0.5
from above and 0 from below, and f(k) is strictly concave, f*(k)(+n)k is therefore eventually negative for large k
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Alternative Sets of Sufficient Conditions
 Conditions
on f*(k)
exists k1 and k2, k1  k2,such that
f*(k1) -(+n)k1 0; f*(k2) -(+n)k2 0
 There
 Conditions
on f*’(k)
 lim
f*’(k) as k approaches zero is strictly greater than (+n)
 lim f*’(k) as k approaches plus infinity is strictly less than (+n)
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The Independence of the Steady-State Rate of
Growth from the Savings Rate
 R.
M. Solow (1956)
 The importance of Inada’s second condition--the marginal
product of capital approaches zero as the quantity of
capital (relative to labor) approaches infinity
 If the marginal product of capital has a lower bound, then
the steady-state rate of growth may depend on the savings
rate (Rebelo (1991))
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Two-Gap Models





How to overcome short and medium-term constraints on economic
development and growth?
How to jump-start a stagnant economy?
Two-gap models are not intended for long-run or steady-state
analysis
Open economy versus closed economy
Constraints on savings


Net imports can augment domestic savings and enable higher domestic
investment in an economy with low real GNP and/or low savings
Constraints on imports:
 Foreign exchange revenue (exports, foreign investment, loans,
foreign aid)
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A Simple Two-Gap Model

Production Function


Consumers


One-sector aggregate production function as a function of capital stock and
labor
Y = F(K, L)
Consumption and savings behavior (C+S=Y)
C = (1-s) Y;
S = sY; where s is the savings rate, not necessarily constant, more generally,
one can write
S = G(Y), where G(.) is a non-decreasing function of Y
Producers

Investment behavior (X and M are perfect substitutes in this one-good model)
I = S + M -X
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A Simple Two-Gap Model
 Equilibrium
in output markets
C + I + X - M= Y
 Equilibrium in factor markets
 Full
employment of capital and labor
 Population
growth
L = L0ent
 Capital accumulation
 The
link between present and future
Lawrence J. Lau, Stanford University
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A Simple Two-Gap Model:
Capital Accumulation
  dK  I  K, where  is the rate of depreciation
K
dt
 G(Y)  M  X  K
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A Simple Two-Gap Model:
The Savings and Foreign Exchange Gaps




The savings gap--nonnegativity of net investment (or net investment
per unit labor)
I - K = G(Y) + M -X - K  0 (nK)
 The net investment required to increase K and Y sufficiently so
that domestic savings can become a sustaining source of domestic
investment and capital accumulation
The foreign-exchange gap
M  X + FC, where FC = Foreign aid, foreign investment and
foreign loans
Increasing FC allows M to increase, other things being equal,
thereby relieving both constraints
Increasing X also helps, provided M is also increased at the same
time (that is why even export-oriented developing countries run
trade deficits in their early phases of development)
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Extensions of the Two-Gap Model
 Imports
can affect an economy more directly and more
significantly--exports and imports are not really perfect
substitutes:
 Output
may depend on both domestic capital stock and imported
inputs (capital or intermediate goods)
 Fixed investment may depend on imported capital and
intermediate inputs
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Alternative Specifications of Two-Gap Models

Production Function


One-sector aggregate production function as a function of capital stock, labor,
and the quantity of imports (of intermediate inputs)
Y = F(K, L, M)
A heterogeneous capital stock model--the aggregate production function as a
function of domestic and imported capital stocks and labor
Y = F(KD, KM, L)


Drawback: two capital accumulation equations will be needed
Investment function

(Fixed) investment is constrained by both the availability of financial savings
and actual physical imports (of capital equipment)
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Implications on Export Orientation



These alternative specifications incorporate the recognition that it is
not only net imports, but also gross imports, that matter. In other
words, exports and imports are not perfect substitutes
In order to increase gross imports, exports must be increased (in
order to increase net imports, exports can be decreased)
Moreover, the ability to export makes an economy much more
attractive to foreign investors and lenders because it facilitates
potential repatriation
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Refinements of One-Sector Models
 Heterogeneous
capital goods
 Human
capital
 Wage-productivity relations
 Endogenous population growth
 Overlapping generations
 Endogenous technical progress
 Non-purely labor-augmenting technical progress and the
existence of a steady state
 Two- and multi-sector models
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