An Economic Interpretation of LINEAR PROGRAMMING

Transcription

1 An Economic Interpretation of LINEAR PROGRAMMING

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3 An Economic Interpretation of LINEAR PROGRAMMING Quirino Paris A Revised and Updated Edition

4 AN ECONOMIC INTERPRETATION OF LINEAR PROGRAMMING Copyright Quirino Paris 2016 Softcover reprint of the hardcover 1st edition This book was originally published by Iowa State University Press in 1991, and a revised edition was published in All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No portion of this publication may be reproduced, copied or transmitted save with written permission. In accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, Saffron House, 6-10 Kirby Street, London EC1N 8TS. Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages. First published 2016 by PALGRAVE MACMILLAN The author has asserted their right to be identified as the author of this work in accordance with the Copyright, Designs and Patents Act Palgrave Macmillan in the UK is an imprint of Macmillan Publishers Limited, registered in England, company number , of Houndmills, Basingstoke, Hampshire, RG21 6XS. Palgrave Macmillan in the US is a division of Nature America, Inc., One New York Plaza, Suite 4500, New York, NY Palgrave Macmillan is the global academic imprint of the above companies and has companies and representatives throughout the world. ISBN: E-PDF ISBN: DOI: / Distribution in the UK, Europe and the rest of the world is by Palgrave Macmillan, a division of Macmillan Publishers Limited, registered in England, company number , of Houndmills, Basingstoke, Hampshire RG21 6XS. Library of Congress Cataloging-in-Publication Data Paris, Quirino. An economic interpretation of linear programming / Quirino Paris. pages cm Includes bibliographical references and index. 1. Programming (Mathematics) 2. Linear programming. I. Title. HB143.P dc A catalogue record for the book is available from the British Library.

5 To the fond memory of Gerald W. Dean Lucio De Angelis and Vincenzo Cosentino

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7 Contents List of Illustrations xiii Preface xvii Acknowledgements xix 1 Introduction to Linear Programming Economic Equilibrium Opportunity Cost What Is Linear Programming? An Example of Price-Taking Behavior Why Another Linear Programming Book? Linear Programming Problems A Very Important Pattern A Farmer s Problem A Second Encounter with Duality Revenue Max in LP: The Primal Problem Free Goods Cost Min in LP: The Dual Problem The Transposition Principle of LP Graphical Solution of LP Problems Slack Variables and Their Meaning A General Linear Program Three Crucial Assumptions Exercises Primal and Dual LP Problems The Structure of Duality Dual Pairs of LP Problems with Equality Constraints Dual Problems with Equality Constraints and Free Variables An Alternative Way to the Relation Between Equations and Free Variables Total Revenue Equals Total Cost Exercises Setting Up LP Problems Preliminaries Problem 1: A Small Family Farm Problem 2: The Expanded Family Farm Problem 3: A Grape Farm vii

8 viii CONTENTS 3.5 Problem 4: The Expanded Grape Farm Problem 5: A Sequential Production Problem 6: A Horticultural Farm Problem 7: A Detailed Example The Investment Problem Example of Investment Problem The Diet Problem Example of Diet Problem Gasoline Blending Exercises The Transportation and Transshipment Problems Introduction The Dual of the Transportation Problem A Numerical Transportation Problem The Transshipment Problem Multiple Optimal Solutions Economic Interpretation of the Transportation Problem Exercises Spaces, Cones, Bases and Extreme Points Introduction Two Ways of Looking at a System of Constraints A Primer on Vectors and Bases Addition of Two Vectors - Parallelogram Subtraction of Two Vectors - Parallelogram Multiplication of a Vector by a Scalar Addition of Vectors as Solution to Systems of Linear Equations - A Parallelogram Extreme Points Basis Feasible Bases and Feasible Solutions Orthogonality Between Two Vectors A More General Look at Orthogonality The Dual Definition of a Line or a Plane Exercises Solving Systems of Equations Introduction The Meaning of Solving a System of Linear Equations The Elimination Method The Pivot Method The Transformation Matrix Choice of a Pivot Another Numerical Example The Inverse of a Basis

9 CONTENTS ix 6.6 More on the Inverse of a Basis More on the Meaning of Solving a System of Equations - Economic Meaning A General View of the Pivot Method Exercises Primal Simplex Algorithm: The Price-Taking Firm Introduction to LP Algorithms The Primal Simplex Algorithm Ideas Underlying the Primal Simplex Method The Marginal Rate of Technical Transformation The Opportunity Cost of a Commodity Recapitulation Termination of the Primal Simplex Algorithm Dual Variables and the Dual Interpretation of Opportunity Costs Summary of the Primal Simplex Algorithm Another Try at Marginal Cost Intermediate Tableau Final Tableau General Structure of a Price Taker Problem and Linear Programming Exercises The Dual Simplex Algorithm Introduction The Dual Simplex Algorithm Termination of Dual Simplex Algorithm Application of Dual Simplex Algorithm Multiple Optimal Solutions Exercises Linear Programming and the Lagrangean Function The Lagrangean Method The Complementary Slackness Conditions Complementary Slackness at Work Exercises The Artificial Variable Algorithm Introduction Ideas of the Artificial Variable Algorithm Termination of the Artificial Variable Algorithm Another Application of the Artificial Variable Algorithm Exercises

10 x CONTENTS 11 The Artificial Constraint Algorithm Ideas Underlying the Artificial Constraint Algorithm Termination of the Artificial Constraint Algorithm Exercises The Diet Problem Revisited The Simplex Solution of Stigler s Problem The Optimal Diet from 77 Food Categories A Multiperiod Least-Cost Diet Problem Parametric Programming: Input Demand Functions Introduction Derivation of Input Demand Functions Connection with Linear Programming The Derived Demand for Input # Step 1: Increase in b 1 : The Rybczynski Theorem Step 1: Decrease in b Step 2: A Further Increase in b Step 2: A Further Decrease in b The Derived Demand for Input # Step 1: Increase and Decrease in b Exercises Parametric Programming: Output Supply Functions Derivation of Output Supply Functions The Supply Function for Output # First Increment of c 2,Δc First Decrement of c 2,( Δc 2 ) Second Increment of c 2,Δc Second Decrement of c 2,( Δc 2 ) The Stolper-Samuelson Theorem The Supply Function of Output # First Increase of Output Price c Second Increase of Output Price c First Decrease of Output Price c Second Decrease of Output Price c The Supply Function of a Commodity Not in the Optimal Basic Production Plan Symmetry in Parametric Programming Exercises Dealing with Multiple Optimal Solutions Introduction Choosing among Multiple Optimal Solutions Dual Multiple Optimal Solutions Problems with Multiple Primal and Dual Optimal Solutions

11 CONTENTS xi 15.5 Modeling to Generate Alternatives Solid Waste Management A Growing Problem A LP Model of Waste Management The Dual Problem Solution of the Waste Management Problem Parametric Analysis of Solid Waste Management The Choice of Techniques in a Farm Production Model Introduction A Numerical Farm Production Model The Derived Demands for Land and Labor Cattle Ranch Management A Static Model of Ranch Management A Multistage Model of Ranch Management The Measurement of Technical and Economic Efficiency The Notion of Relative Efficiency The Farrell Method Linear Programming and Efficiency Indexes Technical and Economic Efficiency in a Group of Small Family Farms Efficiency in Multiproduct Firms An Example of Efficiency in Multiproduct Farms Efficiency and Returns to Scale Decentralized Economic Planning When Is Planning Necessary? Decentralized Economic Planning and LP The Western Scheme of Economic Planning The Russian Scheme of Economic Planning A Comparison of Planning Schemes Theorems of Linear Programming Standard Linear Problems Canonical Linear Problems General Linear Problems Equivalence of Standard and Canonical Linear Programming Problems Existence of Solutions to Systems of Linear Equations Existence of Solution to Systems of Linear Inequalities Farkas Theorem - version Farkas Theorem - version Farkas Theorem - version

12 xii CONTENTS Other Important Existence Theorems Linear Programming The Nature of Solution Sets in LP Degeneracy Lines, Planes, Hyperplanes, Halfspaces Geometric View of Linear Inequalities Convexity Separating Hyperplane Convex Cones Positive Polar Cone Negative Polar Cone (Dual Cone) Bibliography Glossary Index

13 Illustrations Figures 1.1 The average and marginal cost functions with a nonlinear technology The average and marginal cost functions with a linear technology The primal optimal solution of a LP problem The dual optimal solution of a LP problem A line as the intersection of two closed half spaces The flow of information in a transportation problem The flow of information in a transshipment model The output space The input space Vector addition Subtraction of two vectors and multiplication by scalar Solution of system III by addition of two vectors The point p = (a, b) expressed with the Cartesian basis The point p = (a, b) expressed with a new basis The flow of information in linear programming Orthogonality of two vectors: inner product equals zero The unit circle, sines and cosines The law of cosines The geometrical relation between two vectors The length of a vector Pythagorean theorem The dual definition of a line The flow chart for the primal simplex method (a) Output space (b) Input space The initial feasible basis and basic feasible solution Representation of the second feasible basis and basic feasible solution Examples of marginal rates of technical transformation Feasible basis and basic feasible solution corresponding to extreme point C (a) Primal output space (b) Primal input space Dual price space xiii

14 xiv Illustrations 13.1 The inverse demand function of the ith input The direct demand function for the ith input The revenue function after the first optimal tableau The inverse demand function for input # The increase of b 1 in the input space The Rybczynski theorem illustrated The decrease of b 1 in the input space The revenue function after step The derived demand for input #1 after step The revenue function after step The derived demand for input #1 after step The revenue function with respect to input # The inverse demand function for input # A traditional representation of the output supply function The revenue function for output # The supply function for output # The output space (x 2, x 3 ) Partial revenue function and supply function for corn Complete revenue function for corn Complete supply function for corn The Stolper-Samuelson theorem illustrated The revenue function for c The supply function for x The symmetry of parametric programming Parametric analysis of land, type A soil Parametric analysis of land, type B soil Parametric analysis of labor Technical and economic efficiency Technical efficiency from a LP interpretation of the Farrell method Relations among economic efficiency, overall efficiency, and cost/saleable production Relations among technical efficiency, overall efficiency, and cost/saleable production The Western scheme of decentralized economic planning The feasible region of the jth sector The Russian scheme of decentralized economic planning A simplified graphic illustration of Farkas theorem Graphic representation of degeneracy in a three-dimensional space Geometric representation of solution of linear inequalities Illustration that 0 λ 1 for the line segment (x 1, x 2 ) Graphical representation of simplexes 3 and Geometric representation of separating hyperplane Separating hyperplane through a boundary point Convex set as intersection of its supporting half-spaces

15 Illustrations xv 21.9 Example of a finite cone in vector language Addition and intersection of finite cones Positive polar cone Negative polar cone Tables 1.1 Daily Allowance of Nutrients for a Moderately Active Man (weighing 154 pounds) Minimum Cost Annual Diet, August Nutritive Values of Common Foods per Dollar of Expenditure, August Various Notions of LP Problems in Geometrical, Analytical and Economic Languages The Investment Projects Prices and Nutritional Content of Seven Foods Transportation Unit Costs (dollars per ton) Objectives of LP Simplex Algorithms The Primal Simplex Algorithm The Simplex Algorithms The Artificial Algorithms The Optimal Primal and Stigler s Solutions of the Diet Problem The Dual Solution of the Diet Problem (cents) Nutritive Values of Common Foods per Dollar of Expenditure, August The Primal Specification of a Multiperiod Diet Problem The Dual Specification of a Multiperiod Diet Problem Multiple Optimal Solutions Diets for Various Percent Increases over the Minimum Cost Based on the 15 Foods of Table 1.3 (lb/day) Diets for Various Percent Increases over the Minimum Cost Based on the 77 Foods of Table 1.3 and Table 12.3 (lb/day) Solution and Range Analysis of the Waste Management Problem (units of hundred thousand tons) Parametric Analysis of the Objective Function Coefficients The Structure of a General Farm Production Model Initial Tableau of the Farm Production Model Primal Solution and Range Analysis of the Objective Function Dual Solution and Range Analysis of the Right-Hand Side Seasons, Range Forage, and Cow Status Cattle Requirements for Dry Matter and Crude Protein Initial Tableau of the Cattle Ranch Management Problem Primal Solution of the Cattle Ranch Management Problem Dual Variables of the Cattle Ranch Management Problem

16 xvi Illustrations 18.6 Parametric Analysis of Improved Range, Capital and Cows Assumed Growth of Seasonal Forage in a Fair Year (pounds of dry matter per acre) Partial Initial Tableau for a Multistage Model of the Cattle Ranch Management Problem Information for the Application of the Farrell Method (single output, normalized inputs) Efficiency Indexes for a Group of Small Family Farms (single output) Shadow Prices of Technical Efficiency Inputs and Saleable Production (natural units) Product Information for the Farrell Method Efficiency Indexes for a Sample of Multiproduct Family Farms (without product normalization) Returns to Scale in a Group of Small Family Farms

17 Preface Linear programming has brought to economists the meaning of duality. To be sure, economists have dealt for a long time with production and cost, quantities and prices, but, prior to 1950, the dual relations between technology and profits were known only to a handful of them. Duality, as it turns out, possesses a meaning that transcends linear programming and its economic interpretation. While searching for the origin of the word, I stumbled upon a book by Boole, who in 1859 wrote: There exists in partial differential equations a remarkable duality, in virtue of which each equation stands connected with some other equation of the same order by relations of a perfectly reciprocal character. It is not certain whether Boole was the first to introduce the word duality. The notion, if not the word, however, dates back to 1750 when Euler and Legendre were inventing methods for solving differential equations. Mathematicians, therefore, have been acquainted with duality for at least two centuries. As Boole s words anticipate, the striking feature of duality is made apparent by symmetrical relations that bind the various components of the problem. Symmetry has always fascinated artists and philosophers. Why should it fascinate mathematicians and scientists? Above all, why should it fascinate economists? The answer is one and the same for everybody: symmetry greatly reduces the set of theoretical specifications that may fit the description of reality. With a smaller number of such specifications, the search and the verification of the most plausible one seem closer at hand. And given the complexity of the reality they study, economists ought to accept symmetry as the principal guideline for their theoretical specifications, on a par with other scientists. Duality is a gateway to symmetry. We will attempt to illustrate this general statement throughout this book devoted to linear programming, one of the simplest forms of symmetry. xvii

18 xviii Preface This book began as a set of lecture notes developed over several years for an undergraduate class on linear programming at the University of California at Davis, attended mainly by students of economics and agricultural economics. It requires only a minimal knowledge of linear algebra and calculus. It emphasizes a geometric interpretation of the subject and makes extensive use of diagrammatic illustrations. To fully understand the geometry of the primal and dual spaces, the concept of a vector is necessary. All the definitions and algebraic operations required for a serious comprehension of the material are introduced in the main body of the text, avoiding the use of appendices. Duality, symmetry, and economic interpretation are the main focus of the book. Without the direct appeal to duality and symmetry, for example, it would have been exceedingly more difficult to present the dual simplex algorithm of chapter 8 and the derivation of supply functions in chapter 14. Students reaction to these chapters has been consistently characterized by disbelief followed by surprise and enthusiasm for having acquired the ability to grasp a notion of symmetry and its advantages. Personally, I consider them the high point of my linear programming course and book. The detailed treatment of the simplex algorithms, therefore, is presented to students not merely as a set of recipes for obtaining numerical solutions but, rather, for two more important reasons: because they provide the discipline to study and understand the symmetric relations embodied in linear programming and because every single step and component of the algorithms can be given a clear economic meaning, a feature not easily associated with other procedures. Parametric programming is given a prominent role as a method for deriving economic relations of demand and supply. Unusual attention was paid to the neglected topic of multiple optimal and near optimal solutions. This aspect greatly enhances the ability of linear programming to interpret the structure and the solutions of empirical problems. The most important (and challenging) economic topic, however, is the notion of opportunity cost evaluated in an environment of multiple inputs and multiple outputs. The last five chapters and the diet problem attempt to illustrate the wide applicability of linear programming, which is bounded, above all, by the user s imagination. This treatment of linear programming is centered around the economic behavior of a price taker and the four notions of demand (D), supply (S), marginal cost (MC) and marginal revenue (MR) which constitute the core of any economic subject dealing with the production side of an economy. The presentation of the material goes from a very elementary beginning to a certain level of sophistication. It relies consistently on the process of repetition following Aristotle s assertion that We are what we repeatedly do. Excellence then is not an act, but a habit.

19 Acknowledgements I wish to acknowledge my debt of gratitude to Michael R. Caputo for many inspiring discussions leading to new scientific horizons. Many undergraduate and graduate students contributed substantially to the structure of this book with their incisive questions regarding the meaning of traditional notions and their intuitive significance. Special thanks go to Qinxue Yao for her careful reading of the entire manuscript. xix