Economic Growth Theory: Capital, Knowledge, and Economic Stuctures [1 ed.] 0815388683, 9780815388685

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Economic Growth Theory    WEI-BIN ZHANG

Economic Growth Theory Capital, Knowledge, and Economic Structures

WEI-BIN ZHANG

ISBN 978-0-815-38868-5

www.routledge.com  an informa business

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10/28/2017 6:25:29 PM

ECONOMIC GROWTH THEORY

Economic Growth Theory

Capital, Knowledge, and Economic Stuctures

WEI-BIN ZHANG

Ritsumeikan Asia Pacific University, Japan

First published 2005 by Ashgate Publishing Reissued 2018 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN 711 Third Avenue, New York, NY 10017, USA Routledge is an imprint of the Taylor & Francis Group, an informa business © Wei-Bin Zhang 2005 Wei-Bin Zhang has asserted his right under the Copyright, Designs and Patents Act, 1988, to be identified as the author o f this work.

All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. A Library of Congress record exists under LC control number: 2005927747 Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Publisher’s Note The publisher has gone to great lengths to ensure the quality of this reprint but points out that some imperfections in the original copies may be apparent. Disclaimer The publisher has made every effort to trace copyright holders and welcomes correspondence from those they have been unable to contact. ISBN 13: 978-0-815-38868-5 (hbk) ISBN 13: 978-1-351-15944-9 (ebk)

Contents

List o f Figures List o f Tables Preface A cknowledgements 1

2

3

Economic Growth and Growth Theory 1.1 Economic Growth 1.2 Economic Growth Theory 1.3 The Structure o f the Book The One-Sector Growth (OSG) Model 2.1 Behavior o f Producers 2.2 Behavior o f Consumers 2.3 On the Generalized Keynesian Consumption Function 2.4 Dynamics in Capital-Labor Ratio 2.5 Existence o f a Stable Steady State 2.6 Productivity Improvement, Population Growth and the APS 2.7 The Dynamics with the Cobb-Douglas Production Function 2.8 The OSG M odel with the CES Production Function 2.9 The OSG M odel with the Leontief Production Function 2.10 The OSG Model with General Utility Functions 2.11 The OSG Model with Preference Dynamics 2.12 The OSG Model in Discrete Time 2.13 On the OSG Model Appendix A.2.1 Proving Theorem 2.4.1 A .2.2 The General OSG Model in Discrete Time Traditional Growth Theories and the OSG Model 3.1 The Harrod-Domar Model 3.2 The Solow Model and the OSG Model 3.3 The Life Cycle Hypothesis and the Generalized Keynesian Consumption Function in the OSG Model 3.4 The Permanent Income Hypothesis and the Generalized Keynesian Consumption Function 3.5 The Ramsey Growth Model 3.6 The Ramsey M odel and the OSG M odel 3.7 Poverty Traps Generated in the Solow M odel 3.8 On the Utility Function in the OSG Model

ix xiii xv xix 1 1

4

10 17 18

22 27 32 34 36 41 46 50 52 59 62

66 67 67

68 73 74 76 81 83

86 91 93 97

VI

Economic Growth Theory Appendix A.3.1 The Golden Rule of Capital Accumulation in the Solow Model

100 100

Some Extensions of the OSG Model 4.1 Exogenous Technological Change 4.2 Endogenous Time and the OSG Model 4.3 Simulating Evolution of Time Value and Distribution 4.4 Public Goods and Returns to Scale 4.5 Home Production in the OSG Model 4.6 Environment and Growth 4.7 Population and Capital Accumulation 4.8 Money and Economic Growth Appendix A.4.1 The General OSG model with Endogenous Labor A.4.2 Proving Lemma 4.6.1

103 103 108 114 118 128 131 137 143 149 149 155

5 Income Distribution and Growth 5.1 Growth with Income Transfers 5.2 Does Inequality Accelerate Growth? 5.3 Properties of the Dynamic System 5.4 The Distribution Policy and the Equilibrium 5.5 Human Capital and the Distribution Policy 5.6 Dynamics of the Loren Curve and the Kuznets Curve 5.7 Endogenous Time in the Two-Group Model 5.8 The OSG Model with Multiple Consumers 5.9 Simulating the Three-Group OSG Model 5.10 The OSG Model with Sexual Division of Labor 5.11 On Modeling Group Differences Appendix A.5.1 Stability Conditions in Section 5.3

157 160 167 170 173 177 182 186 191 193 195 204 205 205

6 Structural Changes and Development 6.1 The Uzawa Two-Sector Model 6.2 Refitting the Uzawa Model 6.3 Behavior of the Two-Sector Model 6.4 A Generalization of the Ricardian Growth Model 6.5 Discussion on Growth Rates 6.6 Simulating Economic Structural Change 6.7 The Propensity to Hold Wealth and Economic Structure 6.8 Engel’s Law and Structural Change 6.9 A Two-Capital Growth Model 6.10 Economic Evolution with Multiple Sectors Appendix A.6.1 Proving Proposition 6.4.1

209 211 214 217

4

222

230 231 235 239 240 244 244 244

C on ten ts

vii

7 Growth, Unemployment and Welfare 7.1 The Welfare Economy with Unemployment 7.2 Dynamics in Capital-Worker Ratios 7.3 Equilibrium of the Economy 7.4 The Unemployment Policy 7.5 The Efficiency Wage Theory 7.6 Applying the Efficiency Wage Theory to the OSG Model 7.7 Other Causes of Unemployment Appendix A.7.1 A General OSG Model with the Efficiency Wage Theory

249 251 255 258 261 263 268 273 276 277

8 Learning by Doing, Education and Learning by Leisure 8.1 The AK-OSG Model with Learning by Doing 8.2 The AK-Ramsey Model 8.3 The Lucas Growth Model with Education 8.4 Growth with Multiple Ways of Learning 8.5 Growth Rates and the Education Policy 8.6 Lower Propensity to Save and Higher Growth Rates 8.7 More Than ‘All Work and No Play Make Jack a Dull Boy’ 8.8 The Dynamic Properties of the Model 8.9 Leisure and the Equilibrium Economic Structure 8.10 Growth, Human Capital and Unemployment 8.11 Human Capital and Economic Development Appendix A.8.1 Endogenous Education in the OSG Model

281 283 286 288 294 301 304 306 309 310 315 322 323 323

A.8.2 The Dynamics in Terms of K ( t ) and H ( t )

327

A.8.3 Proving Proposition 8.8.1 A.8.4 Growth with Human Capital and Unemployment

328 329

9 Research and Learning by Doing 9.1 Growth with Learning by Doing and Research 9.2 Growth Rates and the Research Policy 9.3 Intermezzo for Multiple Equilibria and Poverty Traps 9.4 Equilibrium of the Growth Model with Knowledge 9.5 The Policy on Scientists’ Payment 9.6 The Job Amenities and Economic Structure 9.7 Knowledge Accumulation and Economic Structure 9.8 Economic Growth with Knowledge and Multiple Groups 9.9 On Growth with Endogenous Knowledge

335 337 346 352 358 364 367 369 372 375

10 New Growth Theories and Monopolistic Competition 10.1 Development with Monopolistic Competition 10.2 Product Variety and Growth 10.3 Variety of Consumer Goods and Growth 10.4 The Schumpeterian Creative Destruction 10.5 Growth with Improvements in Quality of Products

379 381 387 392 396 401

viii

Economic Growth Theory 10.6 Learning by Doing and Research 10.7 The New Growth Theory and Linearized Knowledge Growth

408 416

11 Complexity of Economic Growth 11.1 Preference Change with the OSG Model 11.2 Economic Chaos as Conclusion

421 421 429

Bibliography Index

435 453

List of Figures

2

1.1 1.2 1.3

An Illustration of Cross-Country Economic Growth The Dynamics of a Malthusian Economy The Malthusian Economy with Technological Change

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11

Intensive Form of the Aggregate Production Function Indifference Curves and the Propensity to Hold Wealth The Optimal Choice at Time t Optimal Saving with Different Propensities to Own Wealth The Impact of Changes in the Interest Rate The APS is Dependent on A and K / Y The Generalized Keynesian Consumption Function and Wealth Evolution of Capital-Labor Ratio in the OSG Model The Dynamics of Growth Rate in the OSG Model Impact of Productivity Improvement The Impact of Changes in A, n and t k

21

2.12

The Dynamics of the OSG Model with a = 0.3

43

2.13 2.14

The Rate of Saving in Terms of the Current Income The Dynamics of k{t), f ( t ) , c(t) and sit) when a = 0.45

43 43

7 7

25 27 27 28 31 32 35 36 38 39

2.15

The Equilibrium Values of c, s, y and y as A Changes

45

2.16

The Parameter, a, and the Equilibrium Values of c and y

45

2.17 2.18

The Equilibrium Value of c is Related to A and a The CES Production Function with a = b = 1/2,

46 48

2.19

Sustained Growth with 0 < o < 1 and AAball

2.20 2.21

The Per Capita Leontief Production Function The Growth Rate of kit) when AA < £k + n

51 52

2.22

The Growth Rate of k (t ) when AA = £k + n

53

2.23

The Growth Rate of kit ) when AA > £k + n

53

2.24 2.25 2.26

Preference Changes that Keep the APS Constant The Capital-Labor Ratio and Propensity to Own Wealth The Dynamic Interdependence Between k(t) and / ( / )

61 61 61

2.27 2.28

Continued but Slow Growth The Dynamics in the OSG Model

66

3.1 3.2 3.3

Evolution of Capital-Labor Ratio in the Solow Model Lifetime Consumption and Saving The Dynamics of the Ramsey Model

78 82 89

+n

49

62

Economic Growth Theory

X

3.4

Two Equilibrium Points with S* > 0

96

3.5

A Single Poverty Trap with 0 and x2 > 0, 0 < X < 1, and for any c , if F(xx) > c and F(x2) > c , then:

F[Xxx + (1 - X)x2] > c ,

2 The description of behavior of producers and production sectors follows the traditional approach (e.g., Burmeister and Dobell, 1970, Zhang, 1990, Azariadis, 1993).

Economic Growth Theory

20

with equality iff xY = x 2. A concave function appears like a hill. Consider, for example, setting an eggshell on end and cutting it in half. The top half would constitute a concave function. For the production function F(K, N) we define the homogeneity of degree n for capital and labor inputs as follows: F(XK, XN) = XnF ( K , N ) where X is an arbitrary non-negative number. When n = 1, we say that the production function has constant returns to scale. It is linearly homogeneous or homogeneous of degree one. It simply means that if all inputs are changed in a given proportion, then output also changes in the same proportion. The same definition applies to any number of inputs. When we assume output to be related to infrastructures, knowledge, land, in addition to labor and capital, we have the same concept of homogeneity. When n > ( 0.

The One-Sector Growth (OSG) Model

21

(iv) The Euler Theorem holds KFk + NF n = F. We depict intensive form f ( k ) of the aggregate production function in Figure 2.1. As we move out to the right along the production function, output per worker increases as the capital/labor ratio k(t) rises. The shape of f ( k ) in the figure reflects the assumption that there are diminishing returns to increases in k(t). The increment to output per worker declines as capital per worker rises. The slope of the production function becomes flatter from left to right. This means that although more capital always leads to more output, it does so at a decreasing rate.

Figure 2.1 Intensive Form of the Aggregate Production Function We assume (identically numerous) one production sector. Its goal of economic production is to maximize its current profit: n{t) = p(t)F(t) - r(t) K(t) - w(t)N(t) where p(t) is the price of product, r(t) is the rate of interest, and w{t) is the wage rate. We assume that the output good serves as a medium of exchange and is taken as numeraire. We thus set p{t) = 1 and measure both wages and rental flows in units of the output good. The rate of interest and wage rate are determined by markets. Hence, for any individual firm r and w are given at each point of time. The production sector chooses the two variables K and N to maximize its profit. Maximizing n with regards to K and N as decision variables yields: r = FK = f ( k ) , w = FN = f ( k ) - kf'(k). We assume that factor markets work quickly enough so that our system always displays competitive equilibrium in factor markets. Thus we always have w(t) = FN

Economic Growth Theory

22

and r(t) = FK. These equations tell that the production factors are paid according to their marginal product. Since we assumed that the production function is homogenous of degree one, we have KFk + NF n = F , or: rK + wN = F.

(2.1.3)

This result means that the total revenue is used up to pay all factors of the production. We thus conclude that if the production function is homogenous of degree one, the ‘adding-up requirement’ is satisfied.

2.2 Behavior of Consumers Consumption and saving are important to growth. With regard to growth, the division of society’s resources between current consumption and various types of savings - in physical capital, human capital and research and development - is central to improvement of living standard in the long term. Consumers’ decisions on goods and service consumptions, time distribution or labor supply are subject to a wide and complex range of influences. Some of these influences are general in their effect while others represent the idiosyncrasies of particular households. In economics, the characteristics of an individual consumer are described by preference, utility function, or demand functions. Consumers make decisions on choice of consumption levels of services and commodities as well as on how much to save (in terms of material and educational terms). There is no single purpose for people to make saving. Wealth may be accumulated for different reasons such as the capitalist spirit, old age consumption, providing education for children, power and social status. In order to provide proper description of endogenous saving, we should know how individuals perceive the future. Different from the optimal growth theory in which utility defined over future consumption streams is used, we do not explicitly specify how consumers depreciate future utility resulted from consuming goods and services. In our approach, preferences for future consumption are reflected in the consumer’s current preference structure over current consumption and saving. Consumers obtain income: Y = rK + wN = F

(2.2.1)

from the interest payment rK and the wage payment wN. We call Y the current income in the sense that it comes from consumers’ daily toils (payment for human capital) and consumers’ current earnings from ownership of physical capital. The current income is equal to the total output as we neglect any taxes at this initial stage. The sum of money that consumers are using for consuming, saving, or transferring are not necessarily equal to the temporary income because consumers can sell wealth to pay, for instance, the current consumption if the temporary income is not sufficient for buying food and touring the country. Retired people may live not only on the interest

The One-Sector Growth (OSG) Model

23

payment but also have to spend some of their wealth. The total value of wealth that consumers can sell to purchase goods and to save is equal to K(t). Here, we assume that selling and buying wealth can be conducted instantaneously without any transaction cost. The gross disposable income is equal to: r

= Y + K.

(2.2.2)

The gross disposable income is used for saving and consumption and for paying the depreciation of the wealth. We assume that consumers pay the depreciation of capital goods which they own. The total amount is equal to SkK(t) where Sk (0 < Sk < 1) is the depreciation rate of physical capital. At each point of time, consumers would distribute the total available budget among saving S(t), consumption of goods C(t), and payment for depreciation SkK(t). The budget constraint is given by: C (0 + SkK(t) + S(t) = r (0 = Y(t) + K(t).

(2.2.3)

Since the consumer has to pay the depreciation 6kK ( t \ we call: Y(t) + K(t) - SkK(t) the disposable income, which equals the net income minus the depreciation loss. We denote the DPI by: Y{t) = Y(t) + K(t) - SkK( t) = rK(t) + w(t)N(t) + SK(t)

(2.2.4)

where S = 1 - Sk. In our model, at each point of time, consumers have two variables to decide. A consumer decides how much to consume and to save. Equation (2.2.3) means that consumption and saving exhaust the consumers’ disposable personal income, i.e.: C (0 + S(t) = Y(t).

(2.2.5)

The slope of the budget line is equal to - 1, i.e. dSI dC = - 1. This means that if the consumers are going to increase their consumption of goods C(t), they have to reduce the saving that exactly equals C(t). We assume that utility level U(t) that the consumers obtain is dependent on the consumption level C(t) of commodity and the net saving S(t). We use the CobbDouglas utility function to describe consumers’ preferences:

24

Economic Growth Theory

(2.2.6)

U(t) = C t ( t ) S \ t ) , £ A > 0 ,

in which t, and X are respectively the propensities to consume goods and to own wealth.3 It is convenient to describe preferences graphically by using a construction of indifferent curves. The difference curve through a bundle (C(t),S(t)) consists of all bundles of the variables that leave the consumer indifferent to the given bundle. The marginal rate o f substitution (MRS) measures the slope of the indifference curve at a given bundle of variables. It can be interpreted as the rate at which a consumer is just willing to substitute a small amount of saving for consumption goods. Taking the total differential of the utility function, keeping utility constant, yields: r)TI r)TI dU(C, S) = — dC + — dS = 0, dC dS

(2.2.7)

where we omit time index. By equations (2.2.6) and (2.2.7), we obtain: MRS

dC

dC

dS

=

XC

(2 .2 .8)

The algebraic sign of the MRS is negative: if you save more you have to consume less in order to keep the same level of utility. The preferences represented by equation (2.2.6) is described in Figure 2.2. In Figure 2.2a, we have illustrated the indifference curves for t, - 0.5 and X = 0.5. In Figure 2.2b, we have illustrated the indifference curves for t, = 0.2 and X = 0.8. Different values of the parameters t, and X lead to different shapes of the indifference curves. The greater the propensity to own wealth is, the more patient the consumer is. The consumer described in Figure 2.2a is less patient than the consumer in Figure 2.2b. A monotonic transformation of the utility function will represent exactly the same preferences. Here, by monotonic transformation we mean a way of transforming one set of numbers into another set of numbers in a way that preserves the order of the numbers. Mathematically, if F(U) is a monotonic transformation, then:

3 This utility function was initially proposed by Zhang in the early 1990s. After having analyzed almost all the main dynamic economic models of different schools in the literature, I found out that the traditional approaches to consumer behavior is the obstacle to producing an analytically tractable framework for a general economic theory of capital, population, and knowledge with structures and heterogeneous households over time and space. My approach was proposed to surmount the obstacle. After having applied this alternative approach to consumer behavior in the last decade, I have demonstrated that it proves to be effective in analyzing various economic issues which are difficult to analyze by the traditional approaches. Keynes (1936) observed: ‘the difficulty lies, not in the new ideas, but in escaping from the old ones, which ramify ... into every comer of our minds.’

The One-Sector Growth (OSG) Model t/(C „S ,) > t/(C 2,S 2) if and only if F [f/(C „5 1)]>

f

25

[[/(C2,5 2)].

The monotonic transformation represents the preference in the same way as the original utility function U(C, S ) .

Figure 2.2 Indifference Curves and the Propensity to Hold Wealth Applying the natural log: F(U) = In U, which is a monotonic transformation, to the original Cobb-Douglas utility function, we have: U(C, S) = £ ln (c ) + A In(5). The indifference curves for this utility will look just like the ones for the Cobb-Douglas function. Applying the monotonic transformation: F{U) = U' Ki +x\ to equation (2.2.6) yields: U(C, S) = C ^ S \ i + A = l, 0 < £ , i < l , where

(2.2.9)

Economic Growth Theory

26

1 =

■f-j, ^+X

A=

X

Z +X'

This means that we can always take a monotonic transformation of the Cobb-Douglas utility function that makes the exponents sum to 1. We can always require t, + A - 1 in equation (2.2.6) without losing the generality. The households determine C(t) and S(t). In the remainder of this chapter, assume t, + A - 1. At the optimal choice (C(t),S(t)), the indifference curve is tangent to the budget line. Since the slope of the budget line is equal to - 1 and the slope of the indifference curve at the point is equal to - 0, 0 < b < 1,

(2.3.2)

where a and b are constant, C(t) is real consumption at time t , and Y(t) is real disposable income (which is the same as the current income in our model), which equals GNP minus taxes. It can be seen that if we swap the real disposable income in the Keynesian model with the disposable personal income in our model, we see that the Keynesian consumption function with a - 0 is identical to the consumption derived from our rational choice assumption. Since Y in the Keynesian consumption function is the current consumption in our model, we cannot see further relations between the two approaches without further exposition. The parameter b is the marginal propensity to consume, which measures the increase in consumption in association with per unit increase in disposable income. The intercept a measures consumption at a zero level of disposable income. Because of the intercept, the Keynesian consumption is not a proportional relationship between consumption and income. The ratio of consumption to income is termed the average propensity to consume (APC), i.e.: APC = C IY = b + ~ . Y The average propensity to consume declines as income increases. The average propensity to consume is greater than the marginal propensity to consume, by the amount of a/ Y. The ratio of saving to income is termed the average propensity to save (APS), i.e.: APS = —

Y

- = 1- b - - . Y

(2.3.3)

We have APC + APS = 1. As Y increases, the average propensity to save rises. The Keynesian consumption function is based on the assumption that consumption reacts only to actual levels of income. As shown in the next chapter, the permanent income hypothesis would relax the Keynesian assumption. We now show that the Keynesian assumption is related to but different from the implications of our rational choice theory with consumption and saving. One of the implications of the Keynesian assumption is that the intercept a is independent of any change in wealth and other factors. Consider two persons, A and B, who started to work 5 years ago with the same conditions and the same preference. Their preferences were invariant during the period. Person A has accumulated a large amount of wealth during this period; but person B accumulates little. At the end of the period, they both lose the job. If the interest rate is almost zero, both persons A and B have no income. According to the Keynesian consumption, at this moment when they

Economic Growth Theory

30

lose the job or are retired, persons A and B should enjoy the same level of consumption (which is given by the intercept a ). Nevertheless, intuitively this is invalid as person A is a rich man while person B is poor. It is reasonable to see that person A consumes more than person B. Our theory will solve this problem. We connect the two theories by treating a in the Keynesian model as a wealthrelated variable. If we assume that the intercept a is dependent on wealth and marginal propensity to propensity b is related to the propensity to consume in the following way: a = %SK(t\ b = 4,

(2.3.4)

then the Keynesian consumption function is identical to the consumption function in the OSG model. We may call our consumption function as a generalized Keynesian consumption function. We can define the APC and APS, denoted by c{t) and s ( t \ respectively, for the OSG model in the same way as in the Keynesian consumption function. In the OSG model:

m

W

= A P s , s + e‘ K - K =

Y

l ¥

+

x

a

Y

:

-

x

=

Y

( 2 .3 .5 )

where we use £ + X =1. It should be noted that according to the definition of the APS: s(t) = APS = S ( 0 + W ) ~ K{t) = K{t) + W Y(t) Y{t)

)

The APC in the OSG model rise as wealth increases or as current income declines; The APS in the OSG model rise as wealth falls or as current income rises. It should be noted that APC + APS = 1. It is possible for the APS to be negative in the OSG model. As shown late, the propensities to consume and to own wealth can actually become endogenous variables. We illustrate a relationship among the propensity to own wealth, the wealth-income ratio, and the average propensity to save in Figure 2.6 with Sk = 0.05. The ranges of values of the propensity to hold wealth and the wealthincome ratio are respectively 0.4 < A < 0.8 and 1 < K I Y < 6. It is direct to show: - 3.02 < s < 0.61. The highest saving rate for the given ranges of X and K / Y is 61 percent out of the current income. The highest dissaving rate for the given ranges of X and K / Y is 302 percent out of the current income (which may occur in special situations, for instance, when the current income is low). When both X and K / Y are low, the APS is low. If K / Y is high, the APS may be high even for low values of X.

The One-Sector Growth (OSG) Model

31

I f a country has a low w ealth-incom e ratio and low propensity to hold wealth, then it tends to use up not only its current income but also use up its wealth.

Figure 2.6 The APS is D ependent on A and K / Y Em pirical studies have been done to estimate the relationship between consum ption and current income. Contrary to K eynes’ claims, no consistent and stable relationship has been identified. Instead, it has been dem onstrated that consum ption is not only related to current income but also ‘perm anent incom e’ (which is reflected by w ealth in our approach). Our m odel is incongruous w ith the im plications o f the Keynesian consum ption function. It seems that our m odel is m ore consistent with historical data, for instance, for the United States. It is argued that ‘although there has been continued growth in real GNP in the United States and other industrialized countries, there has been no tendency for the A PC to decline and the APS to rise. The shares o f consum ption and saving in income have been relatively constant for over a century, as becam e apparent w hen estimates o f GNP and output shares extending back into the nineteenth century becam e available in the early post-W orld W ar II period.’ 6 According to the OSG model:

C _ s - SK _ AY - SK _ yl C

“

~ %

f

'j

1

UJ

(2.3.6)

The O SG model predicts that the shares o f consum ption and saving in income are constant only w hen the ratio Y / K o f current income and assets is invariant as time passes. Fortunately, this is confirm ed for the U.S. data. As indicated in Froyen, the ratio o f wealth to current income is approxim ately 4.75, estim ated for the U.S. economy; K / Y = 4. 7 5 .7 Substituting this relation into equation (2.3.6) yields:

c s 6 7

A B

'D

(S} £ / 0.2105 +

Source: Froyen (1999: 280). See Froyen (1999: 285).

32

Economic Growth Theory

which is constant. For a fixed level o f wealth, Figure 2.7 describes the relation between consumption and current income in the OSG model. We see that if the wealth is constant during a study period, the generalized Keynesian consumption function is the same as the Keynesian consumption function. However, in a growth model, wealth is accumulated so that the consumption line moves upwards in Figure 2.7, which is confirmed for the U.S. economy. In Figure 2.7, below the current income level Y, consumption exceeds the current income; hence part o f wealth will be consumed (here, we neglect depreciation). In this range the APC is greater than 1. Above the income level 7 , the APC is less than 1. As the wealth increases, the value o f the ‘parameter’

a will raise; the generalized

Keynesian curve shifts upward. The intersection o f the 45° -lin e and the generalized Keynesian consumption curve shifts toward the left in the horizontal direction.

Figure 2.7 The Generalized Keynesian Consumption Function and Wealth

2.4 Dynamics in Capital-Labor Ratio It appears reasonable to consider population as independent o f economic conditions, as a first approximation. Here, we assume that the population dynamics is exogenously determined in the following way:

N( f )=nN(t ) i.e., g N = n,

(2.4.1)

where n is a constant and gx (= x / x) is the growth rate o f variable x . We use the symbol gx to stand for the growth rate in the rest o f the book if without special explanation. Here, we neglect variations in labor productive potential due to, for

The One-Sector Growth (OSG) Model

33

instance, technical progress, education, learning through working, and learning through leisure. The assumption of a constant population growth rate has been widely used in the literature of economic growth. This rules out economic factors that may affect birth and death rates. The change in the households’ wealth is equal to the net saving minus the wealth sold at time t, i.e.: k ( t ) = s ( t ) ~ K(t).

(2.4.2)

The above equations along with equations (2.1.1), (2.1.2), (2.1.10), and (2.4.1) determine all the variables, K(t), C(t), S(t), N(t), F(t), r(t), w(t),U(t), in the system. We call this dynamic system (with proper initial conditions) the one-sector growth (OSG) model. The OSG model deals with the economic questions, such as markets for factor services, the reconciliation of independently determined saving and investment desires, production conditions, and determination of population growth and labor force participation rates. We now rewrite the dynamics in terms of per capita. From equation (2.2.10) and Y(t) = F{t): S(t) = XY(t) = A{F(t) + SK(t)).

(2.4.3)

Inserting the above equation into equation (2.4.2) yields: K(t) = XF{t) + X8K{t) - K(t) = XF(t) - £kK(t), where we use: t, + X =1, 0 if k is non-negative. It can be seen that once the capital per capita k(t) is determined, all the variables in the system, such as K, F, Y, C, w, r and U can be calculated accordingly. We now examine some properties of equation (2.4.6).

34

Economic Growth Theory

2.5 Existence of a Stable Steady State This section shows that the economy will eventually arrives at a steady state/equilibrium - a situation in which output per worker y(t) (= Y(t)/ N ( t ) \ consumption per worker c{t) (= C{t)l N ( t ) \ and capital stock per worker k(t) don’t change over time. The dynamics is depicted in Figure 2.8. Since equation (2.4.6) contains two terms, Af(k) and + ri)k, we plot them as two separate curves. The ( 4 + n)k term, a linear function of k(t), will show up in the figure as a straight line, with a zero vertical intercept and a slope equal to {£,k + n). The Xf(k) term plots as a curve that increases at a decreasing rate. Based upon these two curves, the value of the change rate dk! dt for each value of k(t) can be measured by the vertical distance between the two curves. The two curves intersect when the capital-labor ratio arrives at steady state. As shown by the figure, the capital-labor ratio will always reach a positive steady state. For instance, if k(t) starts at some value less than k \ Af(k) is greater than

+ n)k. When this extra amount Xf(k) - ( 4 + n)k is converted into capital,

the capital-labor ratio will rise. As indicated by the arrows, k(t) increases toward k \ We can similarly explain falling of k(t) when it starts at some value greater than k \ The conditions imposed on f ( k ) imply that the right-hand side of equation (2.4.6) is positive for small k(t). The right-hand side first increases and then decreases, and eventually becomes and remains negative. These imply that there exists a unique k* > 0 such that k \ t ) > 0 when k'(t) < k \

k'(t) = 0 when k'(t) = k \ and

k'(t) < 0 when k'(t) > k*. Thus k* is a globally asymptotically stable equilibrium for k(t). Starting from any initial level of capital per capita, k(t) moves monotonically to a predetermined capital intensity defined by the intersectional point of: # (* ) = & + « > •

(2-5-1)

This condition leads to the zero change rate of capital intensity: k(t) = 0. If the economy starts at any level lower than k' (apart from the trivial possibility at k = 0 ), it experiences positive growth until it reaches at k " . When the economy has reached the stationary capital intensity, capital per capita will remain the same as time passes, but the stock of capital K(t) remains growing infinitely at the same predetermined rate as the labor force n. The sustainable growth rate o f the model is exogenously given by n. This can be confirmed by: K(t) = kN0em, F(t) = f ( k ) N 0e", C(t) = c(k)N (>e"'.

(2.5.2)

The One-Sector Growth (OSG) Model

35

We now formally describe the properties o f the dynamic system. The proof o f the following theorem is provided in Appendix A.2.1.

Theorem 2.5.1 (The existence o f a unique equilibrium and the condition for stability) If 6 and X satisfy:

0 < ^ - p < / ' (0),

(2.5.3)

then there exists a unique positive value k* such that: # ( * * ) = (& + n )k \ The equilibrium point k* is asymptotically stable in the region k > 0.

Figure 2.8 Evolution of Capital-Labor Ratio in the OSG Model We have examined the dynamic properties o f the OSG model. It has a long run steady state at which growth rate o f per capita consumption is zero. W e now examine ‘transitional dynamics’ - a time-dependent process o f how per capita income converges toward its long-run steady state. Dividing equation (2.4.6) by k(t), we obtain growth rate gk(t) o f per capita capital as:

36

Economic Growth Theory „

tt\

8ki }

=

MM k(t)

¥ ( m )

k(t)

(& + 4

Here, g t (r) stands for growth rate of per capita capital at time t. The above equation says that the growth rate of per capita capital equals the difference between two terms, A f Ik and t,k + n, which we plot against k in Figure 2.9. The first curve is a downward-sloping curve and the second term is a horizontal line. The vertical distance between the curve and the line equals the growth rate of per capita capital. As shown before, there is a unique equilibrium. The figure shows that to the left of the steady state, the curve lives above the line. Hence, the growth rate is positive and k increases over time. As k rises, the growth rate declines. Finally, k achieves at k ’ as the growth rate becomes zero. An analogous argument demonstrates that if the system starts from the right of the steady state, the growth rate is negative. As k declines, the growth rate rises and finally becomes zero.

Figure 2.9 The Dynamics of Growth Rate in the OSG Model 2.6 Productivity Improvement, Population Growth and the APS In the OSG model, the capital-labor ratio will reach the steady state. Once we determine k \ we directly solve all the variables. As the steady state is given by:

# (* ’ ) = fe + i.e.: /( * * ) =

1+ n

-Sf=U\

A=

1+ n

X

- s,

(2.6.1)

The One-Sector Growth (OSG) Model

37

where we use £ + X =1 and b = 1 - bk, and k * is dependent on values o f A, bk, n, and the technology represented in f ( k) . We now examine impact o f changes in these exogenous conditions on the equilibrium. First, consider an improvement in productivity. This is reflected by upward shift in the per-worker production function because an improvement in productivity means that at any prevailing capital-labor ratio, each worker can produce more output. Figure 2.10 shows a shift from the original production function f { k \ to a new production function f ^ k ) with productivity improved. As shown in the figure, the effect o f this improvement increases the level o f the steady state. The initial steady state, as discussed in Figure 2.10, corresponds to ( k\ f (k*)). As the line Xk is not affected and the production function is shifted upward at the prevailing k(t) by the productivity improvement, the intersection o f the two curves determines a new equilibrium (& *,/(£*)) which is higher than the original steady state. To further illustrate the mechanism o f growth, we note that the APS (‘the saving rate’ in the Solow model) is given by:

m

(2 .6.2)

s=A -£S—

At the prevailing capital-labor ratio, an improvement in productivity w ill raise the APS, i.e.:

s.(k) =

X- & - T -

/m

>

A-

m > s(k).

The productivity improvement raises output per worker and the saving rate out o f the current income (= output in the OSG model), saving per worker also rises at the prevailing capital-labor ratio. The original equilibrium will be disturbed. The capitallabor ratio will rise until to a new equilibrium at which the two curves intersect. Overall, a productivity improvement raises steady state in two ways. First, it increases the output for any given level o f k(t). Second, it causes the long-run capita-labor ratio to rise. We showed that an improvement in productivity raises the steady state capital-labor ratio and per-worker output. We now examine effects o f the change on the other variables. First, we note that the main variables can be expressed in terms o f per capita:

c = ^ = t ( f + Sk), s = ^ = A(f + Sk), N N r = f'(k), w = f{k) - kf'(k), s = X - & - £ f(k)

(2.6.3)

38

Economic Growth Theory

Figure 2.10 The Impact of Productivity Improvement Through the traditional comparative statics analysis, we can directly determine the effects of a productivity improvement upon the consumption, the saving out of the DPI, the rate of interest, the wage rate, and the APS. As the equilibrium values of k and f ( k ) rise, c and s increase. Technological progress betters long-term living standard. We now examine effects of change in A, n, and 6k on the steady state. From equation (2.6.3), an increase in A or a fall in n and Sk reduces A. A reduction in A shifts the linear line Ak passing through the origin downward, resulting in a new equilibrium with higher values of (fc,*, f(k*)) than the original equilibrium

(*’,/(**».

We explain the relationship between population growth and a country’s output, consumption, and capital per worker. The OSG model’s answer to this question is shown in Figure 2.11. The effects of a slow-down of population growth increases the level of the steady state. The initial steady state corresponds to ( k\f (k* )). As the line

Ak is shifted downward and the production function remains the same at any prevailing k(t ) by the slow-down of population growth, the intersection of the two curves determines a new equilibrium {k\,f{k\)) which is higher than the original steady state. A reduction in population growth means that workers are entering the labor market less rapidly than before. As per worker is equipped with more capital, output per worker tends to increase as n falls. As shown in Figure 2.11, as n falls the steady state rises. By equation (2.6.3), we conclude that:

dc dn

e/ V

Q\ dk 'dn

.

ds dn

V

Q\dk ' dn

_

dr dn

r„dk dn

The One-Sector Growth (OSG) Model ds_ dw = - ¥ " ^ - > 0, dn dn dn

f 2^

j

eh

f 2 dn

< 0.

39 (2.6.4)

As n falls, consumption per capita and wage rate increase; rate of interest rate and the APS fall. The OSG model asserts that the policy should aim to reduce population in order to raise living standards. An extension of the OSG model to include technology as an endogenous variable may give counterarguments to this conclusion. If the dynamic system exhibits increasing returns to scale, policies to increase population growth will raise living standards because more people make more contributions to knowledge accumulation; an improvement in productivity would increase living standards. Another important determinant of long-run living standards in the OSG model is the propensity to save X. As illustrated in Figure 2.11, an increase in X shifts the steady state to a higher level. From equation (2.6.3), we see that when X rises, s and w increase, but r falls. With £ = 1 - X, we rewrite the APS as:

Figure 2.11 The Impact of Changes in X, n and Sk

f

s = 1+ V

Sk )

m)

As X rises, the APS rises at the original level of capital-labor ratio. The increase in the saving rate raises saving at every level of the capital-labor ratio. Graphically, the linear line shifts downward. The new steady-state capital-labor ratio corresponds to the intersection of the new linear line and the per-worker production function. The higher propensity to save has increased the new steady-state capital-labor ratio. At the new

Economic Growth Theory

40

steady state, we cannot directly judge whether the APS rises or falls as an increase in the capital-labor ratio reduces the APS. To check how the two opposite forces - one increasing the APS and the other reducing the APS - balance at the new steady state, we take derivatives of the above equation to obtain:

ds dX

~

k f

w%8 dk f 2 dX

— = 1 + 8 --------- ^ ------ = 1 +

8k +»

in which we apply the equilibrium condition, £ + X = 1, and: w dk V 'd X

1+ n

I T

9

w =f- k f\

We see that the impact on the APS is ambiguous if we don’t provide additional requirements on the parameters and functional forms. The change in consumption per capita is given by: - = - f - S k + Z ( f' + S ) ^ - . dX J cix The impact on consumption is ambiguous. We have been concerned with impact of exogenous changes. We have not yet explained why and how these changes occur. We are left with little understanding of why the world’s standard of living stagnated until the industrial revolution that occurred around the year 1800; why it has grown rapidly from then until today. We are also left with little understanding of why in some countries like the United Kingdom and Sweden the standard of living has grown much more slowly than that in countries like Singapore and Korea in the 1980s. It has become clear that although we have succeeded in constructing an analytically consistent framework of economic growth, the theory is not adequate for explaining well-observed growth phenomena around the world. In the remainder of this book, step by step, we introduce other important determinants and mechanisms to the OSG model. If we neglect the importance of technology differences, the OSG model provides little insight into persistent income differences between countries. For instance, real income per capita in a rich country like the United States is many times higher than in a poor country like China. The OSG model states that if two economies have small technological gaps, there are only two reasons for long-term differences in per capita income - propensity to save and population growth rate. However, even very large differences in the propensity to save or rate of population growth cause only small variations in per capita income. To take one example, increasing the propensity to save by 20 percent and reducing the rate of population growth by half would boost per capita income not enough to explain differences per capita income among countries by magnitudes on the order of 1.0 to 10.0. In other words, large variations observed in the world cannot be explained without considering

The One-Sector Growth (OSG) Model

41

differences in other determinants of economic growth. To illustrate the point, consider the case that two economies have an identical Cobb-Douglas production function Y = K ^ N ™ . We have Y I N = {KI M y . Accounting for the tenfold difference in per capita income observed in the real world requires a difference of a factor of 10,000 in capital per capita. This huge difference in capital per capita is not observed among economies. Another prediction from the OSG model is that there should be inter-country as well as interregional convergence. By the law of diminishing returns to inputs, poor capital-scarce countries should exhibit higher rates of return to capital. This can be seen in Figure 2.8. Since growth rate is determined by the gap between the two curves in Figure 2.8, the OSG model concludes that nations that had been poor should have enjoyed fast growth rates than nations that were rich before. If propensities to save and population growth rates are the same across countries, per capita incomes in poor countries should grow faster, and eventually living standards in all countries must be converge. According to the OSG model, if disparities between rich and poor countries persist over time, it is must be because things, such as propensities to save, and population growth rates are not equal. However, in the world at large convergence has not been observed. It seems that income and wealth gaps tend to enlarge among individuals and regions in many parts of the world. The above discussions mean that countries may have different technologies. Another explanation for the failure of the OSG model is that it may have neglected other determinants of economic growth in the formation of the model. We will see that these other determinants include technological change, human capital improvement, cultural attitudes toward work, education, institutional structure and institutional performance, interregional and international trade, infrastructure, resources, and environment. 2.7 The Dynamics with the Cobb-Douglas Production Function This section solves the OSG model when the production function is taken on the CobbDouglas production function: F(t) = A K a ( t )Nfi(t), a, p > 0, a + p = 1,

(2.7.1)

where A is a number measuring overall productivity, and a and /5 are parameters. The parameter A is often referred to total factor productivity or simply productivity. For any values of capital and labor, an increase in productivity x% implies an x% increase in output. In the above analysis, the production function is assumed to be invariant over time. Changes in A can be caused by improvement in production technology, institutional transformations or any change in the economy that allows capital and labor to be utilized differently in terms of efficiency. If the state of technology is allowed to improve, the production function has to be dully modified. As shown in Appendix A.2.1, under perfect competition the parameter a corresponds to the share of income received by owners of capital, whereas labor

Economic Growth Theory

42

receives a share of income equal to /5. The actual shares of income received by capital and labor can be used to estimate values of a. The total income paid to capital is rK (= (xF) and paid to labor is wN (= fiF). The factor shares, r K /w N = a / /5, are constant for all levels of output. We summarize the OSG model in per capita terms under equation (2.7.1): f = Ak°, r - aAk'^, w = pAka, c = g(Aka + Sk),

s = 4 # “ +&),

s=X-$8— , A

k = AAka - (gt + n)k.

(2.7.2) (2.7.3)

We now simulate this model, specifying the parameters as a = 0.3, n = 0.015, X = 0.55, and Sk =0.015. The population grows at annual growth rate of 1.5 percent and capital depreciates at rate of 1.5 percent. The propensity to own wealth is 0.55, which may be unreasonably low for a rich economy. We will discuss possible change of X later on. We don’t consider any technological change here and specify A = 1. The parameter a is set at 0.3. With initial conditions of £(0) = 0.7, we run the dynamics of the economy for 25 years. Figure 2.12a describes the dynamics of per-capita capital and per-capita income. The per-capita capital and per-capita income exhibit similar pattern of growth - in initial stages they grow very rapidly. The growth rates of these two variables are demonstrated in Figure 2.12c. As the per-capita consumption and saving are positively proportionally related to the per-capita capital and per-capita income, they grow in the same pattern k{t) and y(t), as demonstrated in Figure 2.12b. The wage rate grows not rapidly even during the initial stage of fast economic growth and it becomes stationary after a few years. Similarly but in the opposite direction, the rate of interest declines in the initial years but becomes stationary soon. Usual economic data about saving rate are defined in terms of the current income. The saving illustrated Figure 2.12b is not the saving out of the current income but from the disposable income. Changes in the rate of saving in terms of the current income over the 25 -year period are illustrated Figure 2.13. Its initial value is high, about 37 percent out of the current income; then it declines first rapidly and then becomes stationary. Its equilibrium value is still 30 percent. As we omit technological change, economic growth is not sustainable in the sense that it becomes stationary after a few years rapid growth. We can illustrate impact of changes in any parameter. For instance, Figure 2.14 depicts the dynamics of the per-capita wealth, income, consumption and saving when a = 0.45. We observe that the increase in a from 0.3 to 0.45 leads to increases in the equilibrium values of these variables. We can analytically solve all the variables. Equation (2.7.3) is a Bernoulli equation in the variable k(t). Inserting z{t) = k p{t) into equation (2.7.3) yields

The One-Sector Growth (OSG) Model

Figure 2.12 The Dynamics of the OSG Model with a = 0.3

Figure 2.13 The Rate of Saving in Terms of the Current Income

Figure 2.14 The Dynamics of k(t ), f(t), c(t) and s(t) when a = 0.45

43

44

Economic Growth Theory 2

+ (1 - « )( 4 + n)z = (1 - a)AA

(2.7.4)

which is a standard first-order linear differential equation. The solution is given by: z(/) = 2(0)

aA

+»)

&+»J

aA

(2.7.5)

&+«

Substituting z{t) = k fi(t) back to equation (2.7.5), we obtain: r k \ t ) = k p(0) -

aA & +«

\

e~P(Zk +n)< +

aA

(2.7.6)

where k(0) is the initial value of the capital-labor ratio k(t). This solution is what determines the time path of k(t). Once we know k ( t\ all the other variables are explicitly determined at any point of time. As t —> + the exponential expression will approach zero. Consequently, letting t -> + °o yields the unique steady-state capital ratio:

k =

U

y//? (2.7.7)

4 +«

The capital-labor ratio will approach a constant as its equilibrium value. This steady state, as shown in the preceding section, varies directly with the propensity to save A, the technology A, and inversely with the propensity to consume £, the population growth rate n, and the capital depreciation rate Sk. We mentioned that a rise in the propensity to own wealth may either increase or reduce consumption. We now simulate the model to demonstrate how the equilibrium values of y and c vary as the propensity to own wealth changes. We specify the parameters as follows: a = 0.3, n = 0.015, 6k = 0.03, and A = 0.8. Using: y = A k a, c = %{Aka + 8 k \ s = x{Aka + S k \ we depict how c, s, y and y

y = c + s,

vary as A changes for A e [0.01, 0.99]. The

consumption per capita increases as A rises until A reaches 0.5628; after A = 0.5628, the consumption per capita falls as A rises. The simulation shows that

45

The One-Sector Growth (OSG) Model

from the long-run perspective, it is desirable to have a ‘proper’ propensity to own wealth. The saving per capita, the current income, and the disposable income rise as A rises. A national economy may definitely become rich in this model by increasing the propensity to own wealth. If an economy ‘over-saves’, its income rises but consumption falls. By the way, we may also examine how the equilibrium values of the variables change as other parameters vary. Figure 2.16 shows how the equilibrium values of the consumption per capita and the current income per capita change as a varies for 0.1 < a < 0.9. In the two plots, we specify the parameters values as n -

0.015,

6k =

0.03, and

A =

0.8. In Figure 2.16a, when we take

that the equilibrium values fall as

Figure 2.15 The Equilibrium Values of c,

With value of

n c

= 0.015,

Sk -

is related to

a

X

and

a

a

A

A

= 0.60,

rises.

s, y

and

y

as

A

Changes

, and the Equilibrium Values of

0.03, and

0.53, we see

rises. In Figure 2.16b, when we take

a

we see that the equilibrium values increase as

Figure 2.16 The Parameter,

A =

c

and

y

= 0.8, Figure 2.17 shows how the equilibrium

(with the plot ratios (0.5, 0.5, l)).

Economic Growth Theory

46

Figure 2.17 The Equilibrium V alu e o f c is R elated to A and a

2.8 The O SG M odel w ith the C ES P roduction F unction W e exam ined behavior o f the dynam ic system w hen the production function is the C obb-D ouglas form. W e now study behavior o f the OSG m odel w hen the production function is specified in other tw o - CES and L e o n tie f- forms. It is known that the elasticity o f substitution for the Cobb-D ouglas function is everywhere equal to unity. To avoid this strict property, economists sometim e use another form o f production function which, while still implies a constant elasticity o f substitution (CES), can yield a constant elasticity o f substitution other than 1. The C ES production function is: F it) = M b K i D f + (1 - 4 ( 1 w here 0
- °° The formation implies that the shares o f K ( t) and N ( t) in total product approach respectively b and

The One-Sector Growth (OSG) Model 1 - b as o

47

- oo. It can shown that the elasticity of substitution between capital and

labor is l / ( l - o ) As o -> -

the production function approaches a fixed-

proportions technology: F{t) = min [bK(t\ (l - fc)W(0l where the elasticity of substitution is zero. As o —» 0 ,, the production function approaches the Cobb-Douglas form: F(t) = (constant) K(t )aN ( t y ~ a, where the elasticity of substitution is unity. For o = 1, the production function is linear: F{t) = A{a(bK{tj)+ (1 - a)[(l - b)N(t)]}, where the elasticity of substitution is infinite. Dividing the two sides of equation (2.8.1) by N(t) yields: (2 .8.2)

f ( k ) = A(ab°ka + B ) la, where fi = ( l - c/)(l - b f . The marginal and average products of capital are: f ( k ) = Aab°{ab° +

A * ) = A(ab° + B k ' 0) ' 0. k

Both f ' ( k ) and f ( k ) / k are positive and diminishing in k(t) for all values of o. Figure 2.18 depicts the behavior of the CES production function with: a- b-

o e [-2,1],

k e [0.01,150j.

As the rest of the OSG model is the same as before, using these expressions and equations (2.6.3), we can represent all the variables in terms of k(t). Here, we are interested in the dynamics of the growth rate of per-capita capital. Substituting equation (2.8.2) into equation (2.4.6) yields: k(t) = M ( p b ak(t)a +

b

) ' ° - (gk + n)k{t).

The growth rate of per capita capital is:

48

E c o n o m ic G ro w th T heory

1

F igure 2.18 The CES P roduction F unction w ith a = b = 1/2

(2.8.3)

g k(t) = AA(aba + B k (t)~ a ) ' a - (gk + n).

The above equation says that the growth rate o f per capita capital equals the difference between two terms, AA(aba + B k ~ ° y a and

+ n. I f we graph versus k , the first

curve is a dow nw ard-sloping curve and the second term is a horizontal line. The vertical distance betw een the curve and the line equals the grow th rate o f per capita capital. First consider the case 0 < o < 1. As the elasticity o f substitution betw een capital and labor is equal to l / ( l - o ) ,

the elasticity is high for 0 < o < 1. It is

straightforward to show:

lim f \ k ) = lim ^ £ -»°° k -* > °

h

-

=Aba', a > 0 , lim /'(& ) = lim ^->o

=«>.

As k goes to infinity, the marginal and average products approach a positive rather than zero. This hints that the OSG m odel with the CES production function may generate a non-zero steady-state growth. Figure 2.19 shows the dynam ics o f the growth rate when: A A baira >

+ n.

The One-Sector Growth (OSG) Model

49

The condition guarantees that the curve X f Ik always lies above the £k + n line. As shown in the figure, the growth rate is always positive. The growth rate will approach to a positive constant: g k{k) = XAbaVa - {§k + n) as k ->< .

It should be noted that if XAbalra
0 and dUIds > 0 for each (c, 5) satisfying the constraint set in (2.10.2). Suppose that (c \ s ' ) maximizes U on the constraint set. Then, there is a scalar X > 0 such that: dU dc

(C* , 5*) 0 and s * > 0, then: dU / * *\ dU / * *\ t * — {c , s )= A , — (c ,5 )= A . dc d^ The budget constraint is binding, c + s = y. Conversely, suppose that U is a C 1 function which satisfies the monotonicity assumption and that (c*, / ) > 0 and the first order conditions. If U is C 2 and if:

0

pi - 1 1

1

1

u cac

u t - 2U - U

I/_

-U

>0

U

then (c*, s ' ) is a strict local solution to the utility maximization problem. If U is quasiconcave and Vf/(c, 5) for all (c, s ) ? ( c \ s*) then (c*, j *) is a global solution to the problem. The proof of this proposition and other general properties of the problem can be found in advanced textbooks on microeconomics.10

10 See Mas-Colell, et al. (1995).

The One-Sector Growth (OSG) Model

55

We now specify some properties of U to obtain explicit conclusions. We require

U a C 2 function, and satisfy Uc > 0, Us > 0 for any (c, s)> 0. Construct the Lagrangian:

L(c,s,A) = U(c,s) + A(y - c - s). The first-order condition for maximization is:

Uc = Us = X , y - c - s = 0.

(2.10.3)

The bordered Hessian for the problem is: 0

1

\H\ = 1 u t 1 Usc

1

ucs = 2 U u

cs

- Ucc - U,.

ss

The second-order condition tells that given a stationary value of the first-order condition, a positive |//j is sufficient to establish it as a relative maximum of U. It is known that the bordered Hessian is identical with the endogenous-variable Jacobian. Hence, if |//| is not equal to zero, we can directly apply the implicit function theorem to the problem. That is, the first-order condition has a solution as C 1 functions of the disposable income y . Taking the derivatives of equations (2.10.3) with respect to y yields:

V - * U dy

*=£/

dy

dy

dy

, dc ds 1= — + — .

dy

dy

We solve these functions:

ds _ Usc~ Ucc dy 2 Ua - U a - U j

dc _ Ux - Uss dy 2Ucs - Uss - Ucc'

We see that 0 < ds/ dy 0 .

(2.10.7)

When k is approaching zero, y (= / {k) + dk) is also approaching zero, and hence s(j)) is coming near zero. As 0 < s' (y) < 1 and f ' ( k )

as k -» 0:

lim^> ^•(oy(o) ^ ) >1

*■*0 (l + n)k

(l + n)

( 2. 10.8)

When k is approaching positive infinity, y is coming to positive infinity. As 0 < s'(y) < 1 and f ' { k ) -> 0 as k -> + °°, we have:

k

lim

s'(+ -)(/'(+ ~) + S) „ (1 + »)

(l + n)k

Taking derivatives of equation (2.10.7) with respect to k yields:

d/ dk < 0 for k > 0. To prove this, we use ds / dy 0. The equation, O (k) = 0 for k > 0 has a unique solution because of equations (2.10.8), (2.10.9), and dldk< 0. We now demonstrate that the unique stationary state is stable. For the steady state to be stable, the following conditions must prevail: (l + n )k\ dk

k =k"

= s'(y)(/'(£) + £)-(! + «)< o.

(2.10. 10)

From the equilibrium condition and inequalities (2.10.9), we have:

m t ) + * ) = s ' ( f m t ) + * ) fc}, t~t0

where G is some given function and H(s) is a function describing the ‘weighted average’ impact of income on the consumption propensity. The above equation implies that a household’s propensity to consume is dependent on the household’s level K(t) of capital stocks, current income Y ( t \ consumption structure C(t), and current preference structure