MACRO-ECONOMIC THEORY. A Mathematical Treatment

Size: px

Start display at page:

Download "MACRO-ECONOMIC THEORY. A Mathematical Treatment"

Clementine O’Brien’
5 years ago
Views:

1 MACRO-ECONOMIC THEORY A Mathematical Treatment

2 Allen Some Other ELBS Low-Priced Editions MATHEMATICAL ANALYSIS FOR ECONOMISTS Macmillan Allen MATHEMATICAL ECONOMICS Macmillan Bridger and FAMINE IN RETREAT? Dent de Soissons Dasgupta and COST-BENEFIT ANALYSIS Macmillan Pearce Giles MARKETING Macdonald & Evans Hague MANAGERIAL ECONOMICS Longman Hanson A DICTIONARY OF ECONOMICS AND COMMERCE Macdonald & Evans Hanson MONETARY THEORY AND PRACTICE Macdonald & Evans Hanson TEXTBOOK OF ECONOMICS Macdonald & Evans Hicks VALUE AND CAPITAL Oxford University Press Lipsey AN INTRODUCTION TO POSITIVE ECONOMICS Weidenfeld Meade THE THEORY OF INTERNATIONAL Oxford ECONOMIC POLICY, Vols. I and II University Press Marshall PRINCIPLES OF ECONOMICS Macmillan Paish and Culyer BENHAM'S ECONOMICS Pitman Prest PUBLIC FINANCE IN THEORY AND PRACTICE Weidenfeld Robinson THE ACCUMULATION OF CAPITAL Macmillan Stafford MATHEMATICS FOR ECONOMISTS Macdonald & Evans Stillwell, Lipsey and Clarke WORKBOOK TO ACCOMPANY THE FOURTH EDITION OF 'AN INTRODUCTION TO POSITIVE ECONOMICS' Weidenfeld Stone and Stone NATIONAL INCOME AND EXPENDITURE Bowes & Bowes Stonier and Hague A TEXTBOOK OF ECONOMIC THEORY Longman Thirlwall GROWTH AND DEVELOPMENT Macmillan Second Edition

3 MACRO-ECONOMIC THEORY A M athelnatical Treatnzent R.G.D.ALLEN [Jj[5J ~ Palgrave Macmillan

4 R. G. D. Allen 1967 All rights reserved. No part of this publication may be reproduced or transmitted, in any form or by any means, without permission. First edition 1967 Reprinted 1968, (with corrections) 1970, (with corrections) 1973 ELBS edition first published 1973 Reprinted 1975, 1979 Published by THE MACMILLAN PRESS LTD London and Basingstoke Associated companies in Delhi Dublin Hong Kong Johannesburg Lagos Melbourne New York Singapore and Tokyo ISBN ISBN (ebook) DOI / The paperback edition of this book is sold subject to the condition that it shall not, by way of trade or otherwise, be lent, re-sold, hired out, or otherwise circulated without the publishers' prior consent in any fonn of binding or cover other than that in which it is published and without a similar condition including this condition being imposed on the subsequent purchaser.

5 Contents Preface Me xi 1 Introduction 1.1 Macro-Economics Stocks and Flows: Time Dimensions Fonnulation of Models Continuous versus Period Analysis Necessary and Sufficient Conditions Variables of the Models Plan of Text THE TOOLS 2 Consumption and Saving Functions 2.1 The Circular Flow of Income The Consumption Function Lagged Consumption Functions The Product Market: Flow Conditions The Relation of Saving to Investment The Multiplier Particular Forms of the Saving Function The Technology of the Economy 3.1 Capital and Output-Capital Ratios Fixed Coefficients Linear Programming Fonn Production Functions Distribution of the Product Elasticity of Substitution between Factors The Cobb-Douglas Production Function The C.E.S. Production Function Investment Functions 4.1 Detenninants of Investment The Rate of Return: Discounting Varying Rates of Return Investment and the Rate of Interest D.C.F. versus Fixed Pay-off Period Investment and Income The Acceleration Principle Investment and Profits 73 75

6 vi CONTENTS 5 Economic Dynamics page 5.1 Statics and Dynamics A Simple Illustration Methods of Economic Dynamics Lags: Period Analysis Lags: Continuous Analysis Error-Adjustment Conditions Consumption and Investment Lags Closed-Loop Systems: Block Diagrams SHORT-PERIOD EQUILIBRIUM: KEYNES AND THE CLASSICS 6 Classical Macro-Economic Theory 6.1 The Short-Period Problem A Classical Model of Income and Employment A Particular Case Saving and Investment The Marginal Efficiency of Investment Diagrammatic Representation Keynesian Models 7.1 Income as the Central Variable Simple Keynesian Model: the Multiplier The Keynesian Theory of Income Determination Diagrammatic Representation The Labour Market The Complete Keynesian Model Unemployment Keynes and the Classics Aggregate Demand 8.1 Government Expenditure and Taxation Government: the Multiplier Government: the IS and LM Schedules Diagrammatic Representation Taxes Varying with Income Inflation A Two-Sector Model The Dynamic Multiplier 9.1 Keynesian Dynamics The Dynamic Multiplier: Period Analysis Diagrammatic Representation Variable Autonomous Investment Continuous Analysis Inventory Cycles Fixed versus Working Capital

7 CONTENTS vii LONG-PERIOD EQUILIBRIUM: GROWTH MODELS 10 Capital Accumulation 10.1 Growth 10.2 Fixed Cofficients: Continuous Analysis 10.3 Autonomous Expenditure and the Stationary State 10.4 Fixed Cofficients: Period Analysis 10.5 Stability: the Knife-Edge Problem 10.6 Stability: Continuous Analysis 10.7 Stability: Period Analysis 10.8 The Parameters and their Values 11 Simple Growth Models 11.1 Basic Harrod-Domar Model 11.2 Continuous Analysis: Fixed-Coefficients Version 11.3 Multiplier-Accelerator Version 11.4 Period Analysis 11.5 Variants of the Harrod-Domar Model 11.6 A Linear Programming Model Basic Neo-Classical Model 11.8 Basic Kaldor (Keynesian) Model 12 Two-Sector Growth Models 12.1 Price Equations 12.2 Quantity Equations 12.3 Basic Two-Sector Model 12.4 Case: Classical Saving Function 12.5 Case: Proportional Saving Function 12.6 Stability 13 Technical Progress 13.1 Various Forms of Technical Progress 13.2 Neutrality of Disembodied Technical Progress 13.3 Constant Returns 13.4 Case: Fixed Coefficients 13.5 Case: Cobb-Douglas Production Function 13.6 Non-Constant Returns 13.7 Technical Progress Functions 13.8 Embodied Technical Progress 14 N eo-classical Growth Models 14.1 The Neo-Classical Approach 14.2 A General Neo-Classical Model 14.3 Constant Returns 14.4 Diagrammatic Representation 14.5 Model with Differential Saving 14.6 Particular Production Functions 14.7 Stability 14.8 A Neo-Classical Two-Sector Model page

8 viii CONTENTS 15 Vintage Models page 15.1 Models with Embodied Technical Progress Vintage Capital with Factor Substitution The Aggregate Production Function Vintage Model: Factor Substitution The Capital Stock and its Valuation Vintage Model: Fixed Labour Requirements Obsolescence Vintage Model: Fixed Coefficients Kaldor (Keynesian) Models 16.1 Development of the Basic Model Kaldor's Non-Vintage Model Steady-State Solution Case: Classical Saving Function Kaldor's Vintage Model Steady-State Solution Valuation of the Capital Stock Summary MEDIUM-PERIOD DISEQUILIBRIUM: CYCLE MODELS 17 Multiplier-Accelerator Models 17.1 Disequilibrium Models Formulation: Continuous Analysis Parameters of Multiplier-Accelerator Models Case: Investment Lag Only Phillips' Model Period Analysis Samuelson-Hicks Model: Elementary Case Economic Regulation 18.1 Autonomous Expenditure Intrinsic and Forced Oscillations Stabilisation Policy Stabilisation in the Multiplier Model Stabilisation in the Multiplier-Accelerator Model Solution by Laplace Transforms Numerical Examples Trade-Cycle Theory 19.1 Oscillations and the Trade Cycle Some Theory of the Trade Cycle A Mixed Difference-Differential Model Solution of the Model A Non-Linear Accelerator Goodwin's Non-Linear Model

9 CONTENTS 20 Models of Cyclical Growth 20.1 Cycles and Growth 20.2 Phillips'Model: Product Market 20.3 Product and Money Markets 20.4 Solution: Steady-State Growth 20.5 Cyclical Growth 20.6 Bergstrom's Model 20.7 Differential Equation of the Model 20.8 Solution: Growth and Cycles : Solutions Index ix page

10 Preface T HIS book on macro-economic theory is confined to a treatment of deterministic models in contrast to the stochastic models of econometrics, and is largely limited to positive economics as opposed to the problems of planning and optimalisation. The simpler parts of the text are designed for use in undergraduate courses for economics specialists in their third year and the rest is aimed at first-year graduates. Apart from the extensive applications to international trade, most of the theory of positive macroeconomics needed by a mathematically-inclined student is included in one place or another in the development. In reaching decisions on what to include and on how to pursue the analysis, I have been greatly indebted to my colleagues at the London School of Economics, and in particular to Dr. S. A. Ozga for advice from his much greater experience of developing and teaching macro-economics. I offer no excuse for the assumption that the mathematical techniques employed need no separate exposition. In this day and age it is generally recognised that serious theoretical work in economics requires a substantial mathematical equipment. One of the advantages of sticking to positive macro-economics is that the mathematics needed are relatively simple. Anyone who has worked through a book on the calculus and related topics, such as my Mathematical Analysis for Economists (1938) or T. Yamane, Mathematicsfor Economists (1962), can handle the techniques used here. He may, however, need a little more on differential equations, which can be obtained from one or other of several good texts on the subject, and some specialised treatment of difference equations as presented in S. Goldberg, Introduction to Difference Equations (1958). Whatever may be thought about combining macro- and micro-analysis in teaching, and of mixing theory with applications, there is a case for an exposition, as attempted here, of macro-theory on uniform lines, establishing the common and solid basis of all macro-models of equilibrium and disequilibrium. Much has been written about the Keynesian short-period analysis and on models of long-period growth. More recently, the attention of mathematical economists and econometricians has turned to such optimalisation problems as those based on the Turnpike Theorem. The present text aims at a uniform treatment of equilibrium in the short-

11 xii PREFACE period and in the long-period, and it goes on to develop models of disequilibrium. In doing so, it stops a good deal short of the Turnpike. There is, in my view, a large area to be explored, lying between growth theories and optimalisation, and having to do with disequilibrium in the medium-run, as analysed in the last main section here. This area is worth intensive cultivation, not only by theorists but also in econometric terms. I hope that not all econometricians are committed to pushing along the Turnpike. I should add a few words of caution for those primarily concerned with economic policy. The models considered here, despite their formidable appearance at times, are many stages removed from policy applications. They attempt to explain how things work in precise terms. But just because of their precision, they are not easily used in any consideration of how the working of the economic system should be viewed in practice, when the strategic variables change from one situation to another. Moreover, in order to obtain precision, the models are drastically simplified to keep down the number of variables and parameters. The policy-maker may not find the strategic variables he wants in the short list tait8t for theoretical models. To quote Sir Eric Roll, in an address on the Department of Economic Affairs, we may be in danger of succumbing to macroeconomicosis ' in economic policy: it is almost as if we had completed a cycle; and having become used to the notion that Governmental economic management had to rely on the macro-economic analysis to which Keynes had introduced U8, we began to realise its limitations and the need to add to it a more selective approach'. London School of EconotTtia MflTch 1966 R. G. D. ALLEN

INFLATION, SAVING AND GROWTH IN DEVELOPING ECONOMIES

INFLATION, SAVING AND GROWTH IN DEVELOPING ECONOMIES By the same author Growth and Development with Special Reference to Developing Economies INFLATION, SAVING AND GROWTH IN DEVELOPING ECONOMIES A. P.