Economics 216: The Macroeconomics of Development

1
Economics 216The 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 ljlau_at_stanford.edu WebPages
  http//www.stanford.edu/ljlau

2
Lecture 6Models of Economic DevelopmentOne-Sect
or 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 ljlau_at_stanford.edu WebPages
  http//www.stanford.edu/ljlau

3
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

4
Assumptions

• Consumers
• 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)
• Production
• 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

5
The Harrod-Domar Model

• Production function with fixed coefficients (no
  substitution possibilities)
• Y min aKK, aLL

6
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

7
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

8
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

9
The Neoclassical Model of Growth (Solow)Capital
Accumulation
10
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 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

11
Differential Equation of k
12
Differential Equation of kThe Equation of
Motion
13
Differential Equation of k
14
Existence of Steady-State Growth
15
Existence of Steady-State Growth
16
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

17
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) approaches a
  constant

18
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

19
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
  ypositive
• Comparative statics with respect to n
• The effect on the steady-state rate of
  growthpositive
• The effect on the steady-state level of
  knegative
• 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
  ynegative

20
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

21
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

22
The Neoclassical Model of Growth (Solow)Capital
Accumulation
23
The Assumption of Constant Returns to Scale

• Let 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,
• F(K/Le?t, Le?t/Le?t) F(K, Le?t)/Le?t, or
• y 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

24
Differential Equation of k
25
Differential Equation of k
26
Existence of Steady-State Growth
27
Steady State in the Case of Purely
Labor-Augmenting Technical Progress

• Since 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 ?

28
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

29
Existence of Steady-State Growth
30
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

31
Alternative Sets of Sufficient Conditions

• Conditions on f(k)
• There exists k1 and k2, k1 ? k2,such
  that f(k1) -(?n)k1 ?0 f(k2) -(?n)k2 ?0
• 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)

32
The Independence of the Steady-State Rate of
Growth from the Savings Rate

• R. M. Solow (1956)
• The importance of Inadas 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))

33
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)

34
A Simple Two-Gap Model

• Production Function
• One-sector aggregate production function as a
  function of capital stock and labor Y
  F(K, L)
• Consumers
• Consumption and savings behavior (CSY) 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

35
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

36
A Simple Two-Gap ModelCapital Accumulation
37
A Simple Two-Gap ModelThe 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)

38
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

39
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)

40
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

41
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