1. 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: ljlau@stanford.edu; WebPages: http://www.stanford.edu/~ljlau

2. 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: ljlau@stanford.edu; WebPages: http://www.stanford.edu/~ljlau

3. Lawrence J. Lau, Stanford University3 One-Sector Closed Economy Models u Optimizing Models of Growth u Specification of an objective function u Consumption versus savings u Choice of technique u Allocation of investment u Descriptive Models of Growth

4. Lawrence J. Lau, Stanford University4 Assumptions u Consumers u Representative consumer with time preference u Infinite lifetime u Existing level of output per capita exceeds the subsistence level of consumption per capita (otherwise no capacity for savings) u Production u One-sector aggregate production function as a function of capital stock and labor u Investment u Distribution u Exogenously determined rate of growth of population u CAPITAL ACCUMULATION IS THE LINK BETWEEN THE PRESENT AND THE FUTURE

5. Lawrence J. Lau, Stanford University5 The Harrod-Domar Model u Production function with fixed coefficients (no substitution possibilities) u Y = min { a K K, a L L}

6. Lawrence J. Lau, Stanford University6 One-Sector Model with Neoclassical Production Function u Production function with smooth substitution possibilities u Cobb-Douglas production function u Constant-Elasticity-of-Substitution (C.E.S.) production function u Special case of elasticity of substitution greater than unity

7. Lawrence J. Lau, Stanford University7 The Neoclassical Model of Growth (Solow) u Production Function u One-sector aggregate production function as a function of capital stock and labor Y = F(K, L) u Consumers u Representative consumer with time preference u Infinite lifetime u Existing level of output per capita exceeds the subsistence level of consumption per capita u No income-leisure choice u Consumption and savings behavior C = (1-s) Y; S = sY; where s is the savings rate, assumed to be constant

8. Lawrence J. Lau, Stanford University8 The Neoclassical Model of Growth (Solow) u Producers u Competitive maximization of profits u Investment behavior I = S u Equilibrium in output markets C + I = Y u Equilibrium in factor markets u Full employment of capital and labor u Population growth L = L 0 e nt u Capital accumulation u The link between present and future

9. Lawrence J. Lau, Stanford University9 The Neoclassical Model of Growth (Solow) Capital Accumulation

10. Lawrence J. Lau, Stanford University10 The Assumption of Constant Returns to Scale u Let y  Y/L; k  K/L. Under constant returns to scale, F( K, L) = F(K, L) u Let =1/L, then, F(K/L, L/L) = F(K, L)/L, or u 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. Lawrence J. Lau, Stanford University11 Differential Equation of k

12. Lawrence J. Lau, Stanford University12 Differential Equation of k: The Equation of Motion

13. Lawrence J. Lau, Stanford University13 Differential Equation of k

14. Lawrence J. Lau, Stanford University14 Existence of Steady-State Growth

15. Lawrence J. Lau, Stanford University15 Existence of Steady-State Growth

16. Lawrence J. Lau, Stanford University16 The Inada (1964) Conditions u The marginal productivity of capital approaches infinity as capital approaches zero, holding labor constant u The marginal productivity of capital approaches zero as capital approaches infinity, holding labor constant u The Inada conditions are sufficient, but not necessary, for the existence of a steady state u It is possible to replace the second Inada condition by the following (at the cost of possible non-existence of a steady-state) u The marginal productivity of capital approaches a constant as capital approaches infinity, holding labor constant

17. Lawrence J. Lau, Stanford University17 The Role of Strict Monotonicity and Strict Concavity of the Production Function u Strict monotonicity of F(K, L) implies strict monotonicity of f(k) u Strict concavity of F(K, L) implies strict concavity of f(k) u Twice continuous differentiability of F(K, L) implies twice continuous differentiability of f(k) u Essentially of K and L implies that f(0) = 0 u The Inada conditions imply that f’(k) approaches infinity as k approaches zero and f’(k) approaches zero as k approaches infinity u 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. Lawrence J. Lau, Stanford University18 The Role of Strict Monotonicity and Strict Concavity of the Production Function u 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 u 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 u The function is strictly concave in k so that its slope is always declining u 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 u 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. Lawrence J. Lau, Stanford University19 Comparative Statics of the Steady State u Comparative statics with respect to s u The effect on the steady-state rate of growth--none u The effect on the steady-state level of k--positive u Hence the effect on the steady-state level of y—positive u Comparative statics with respect to n u The effect on the steady-state rate of growth—positive u The effect on the steady-state level of k—negative u Hence the effect on the steady-state level of y--negative u Comparative statics with respect to  u The effect on the steady-state rate of growth--none u The effect on the steady-state level of k--negative u Hence the effect on the steady-state level of y—negative

20. Lawrence J. Lau, Stanford University20 The Case of Purely Labor-Augmenting (Harrod-Neutral) Technical Progress u Production Function u 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 u Consumers u C = (1-s) Y; S = sY; where s is the savings rate, assumed to be constant

21. Lawrence J. Lau, Stanford University21 The Neoclassical Model of Growth (Solow) u Producers I = S u Equilibrium in output markets C + I = Y u Equilibrium in factor markets u Full employment of capital and labor u Population growth L = L 0 e nt u Capital accumulation

22. Lawrence J. Lau, Stanford University22 The Neoclassical Model of Growth (Solow) Capital Accumulation

23. Lawrence J. Lau, Stanford University23 The Assumption of Constant Returns to Scale u 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, u F(K/Le  t, Le  t /Le  t ) = F(K, Le  t )/Le  t, or u 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. Lawrence J. Lau, Stanford University24 Differential Equation of k

25. Lawrence J. Lau, Stanford University25 Differential Equation of k

26. Lawrence J. Lau, Stanford University26 Existence of Steady-State Growth

27. Lawrence J. Lau, Stanford University27 Steady State in the Case of Purely Labor- Augmenting Technical Progress u Since K/Le  t =K/L 0 e (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 u 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  u The capital/“augmented” labor ratio is constant, but the actual capital/labor ratio grows at the rate 

28. Lawrence J. Lau, Stanford University28 The Case of a Non-Constant Savings Rate u Let s  g(y) with g’(y)  0 u g’(y) approaches zero for y  some y* u Consider the function sf(k)-(  +n)k = g(f(k))f(k)-(  +n)k = f*(k)-(  +n)k u 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 u 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 u 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. Lawrence J. Lau, Stanford University29 Existence of Steady-State Growth

30. Lawrence J. Lau, Stanford University30 The Case of a Non-Constant Savings Rate u Let s  g(r/p, y), where r/p is the rate of return on capital u g(.) is assumed to be continuously differentiable and weakly monotonically increasing with respect to r/p and y u r/p = f’(k) under the assumption of competitive profit maximization u 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 u If for some k 1, f*(k 1 ) -(  +n)k 1  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 k 2, f*(k 2 ) -(  +n)k 2  0, then a steady state exists and is stable. u 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. Lawrence J. Lau, Stanford University31 Alternative Sets of Sufficient Conditions u Conditions on f*(k) u There exists k 1 and k 2, k 1  k 2,such that f*(k 1 ) -(  +n)k 1  0; f*(k 2 ) -(  +n)k 2  0 u Conditions on f*’(k) u lim f*’(k) as k approaches zero is strictly greater than (  + n) u lim f*’(k) as k approaches plus infinity is strictly less than (  + n)

32. Lawrence J. Lau, Stanford University32 The Independence of the Steady-State Rate of Growth from the Savings Rate u R. M. Solow (1956) u The importance of Inada’s second condition--the marginal product of capital approaches zero as the quantity of capital (relative to labor) approaches infinity u 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. Lawrence J. Lau, Stanford University33 Two-Gap Models u How to overcome short and medium-term constraints on economic development and growth? u How to jump-start a stagnant economy? u Two-gap models are not intended for long-run or steady-state analysis u Open economy versus closed economy u Constraints on savings u Net imports can augment domestic savings and enable higher domestic investment in an economy with low real GNP and/or low savings u Constraints on imports: u Foreign exchange revenue (exports, foreign investment, loans, foreign aid)

34. Lawrence J. Lau, Stanford University34 A Simple Two-Gap Model u Production Function u One-sector aggregate production function as a function of capital stock and labor Y = F(K, L) u Consumers u 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 u Producers u Investment behavior (X and M are perfect substitutes in this one-good model) I = S + M -X

35. Lawrence J. Lau, Stanford University35 A Simple Two-Gap Model u Equilibrium in output markets C + I + X - M= Y u Equilibrium in factor markets u Full employment of capital and labor u Population growth L = L 0 e nt u Capital accumulation u The link between present and future

36. Lawrence J. Lau, Stanford University36 A Simple Two-Gap Model: Capital Accumulation

37. Lawrence J. Lau, Stanford University37 A Simple Two-Gap Model: The Savings and Foreign Exchange Gaps u The savings gap--nonnegativity of net investment (or net investment per unit labor) I -  K = G(Y) + M -X -  K  0 (nK) u 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 u The foreign-exchange gap M  X + FC, where FC = Foreign aid, foreign investment and foreign loans u Increasing FC allows M to increase, other things being equal, thereby relieving both constraints u 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. Lawrence J. Lau, Stanford University38 Extensions of the Two-Gap Model u Imports can affect an economy more directly and more significantly--exports and imports are not really perfect substitutes: u Output may depend on both domestic capital stock and imported inputs (capital or intermediate goods) u Fixed investment may depend on imported capital and intermediate inputs

39. Lawrence J. Lau, Stanford University39 Alternative Specifications of Two-Gap Models u Production Function u 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) u A heterogeneous capital stock model--the aggregate production function as a function of domestic and imported capital stocks and labor Y = F(K D, K M, L) u Drawback: two capital accumulation equations will be needed u Investment function u (Fixed) investment is constrained by both the availability of financial savings and actual physical imports (of capital equipment)

40. Lawrence J. Lau, Stanford University40 Implications on Export Orientation u 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 u In order to increase gross imports, exports must be increased (in order to increase net imports, exports can be decreased) u Moreover, the ability to export makes an economy much more attractive to foreign investors and lenders because it facilitates potential repatriation

41. Lawrence J. Lau, Stanford University41 Refinements of One-Sector Models u Heterogeneous capital goods u Human capital u Wage-productivity relations u Endogenous population growth u Overlapping generations u Endogenous technical progress u Non-purely labor-augmenting technical progress and the existence of a steady state u Two- and multi-sector models