Lectu re Notes in Economics and Mathematical Systems

Managing Editors: M. Beckmann and W. Krelle

276

Michael R. Baye Dan A. Black

Consumer Behavior, Cost of Living Measures, and the Income Tax

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo

Editorial Board

H.Albach M. Beckmann (Managing Editor) P. Dhrymes G. Fandel J. Green W. Hildenbrand W. Krelle (Managing Editor) H.P.Kunzi K.Ritter RSato U.Schittko P.Schonfeld RSelten

Managing Editors

Prof. Dr. M. Beckmann Brown University Providence, RI 02912, USA

Prof. Dr. W. Krelle Institut fOr Gesellschafts- und Wirtschaftswissenschaften der Universitat Bonn Adenauerallee 24-42, 0-5300 Bonn, FRG

Authors

Prof. Michael R Baye Department of Economics Texas A&M University College Station, TX 77843, USA

Prof. Dan A. Black Department of Economics University of Kentucky Lexington, KY 40506, USA

ISBN-13: 978-3-540-16797-6 e-ISBN-13: 978-3-642-46587-1 001: 10.1007/978-3-642-46587-1

Library of Congress Cataloging-in-Publication Data. Baye, Michael R., 1958- Consumer behavior, cost of living measures, and the income tax. (Lecture notes in economics and mathematical systems; 276) Bibliography: p. 1. Income tax. 2. Consumers. 3. Cost and standard of living. I. Black, DanA. II. Title. III. Series. HJ4629.B39 1986339.4'186-20315

ISBN-13: 978-3-540-16797-6 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting; reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich.

© Springer-Verlag Berlin Heidelberg 1986

2142/3140-543210

TO M'LISSA AND SUSAN

PREFACE

This material is based upon work supported by the National

Science Foundation under grant #SES-8410190. Any opinions, findings, and conclusions or recommendations expressed in this publication are

those of the authors and do not necessari~y reflect the views of the National Science Foundation. This support was crucial to the

completion of this project, and we are grateful for it. As is usually the case when doing academic research, we are

also indebted to a number of individuals. Robert Gillingham, John Greenlees, Jack Triplett, and Paul Harte-Chen freely gave of their

time to share their ideas concerning income-based cost of living indices. Seminar participants at the BLS, the University of Karlsruhe, and Tilburg University provided insightful comments on preliminary

portions of the manuscript. Bill Stober provided encouragement, and Desmond Lo and Albert Tsui read parts of the manuscript. We owe a special thanks to Bert Balk for providing detailed handwritten comments on a preliminary draft. Evelyn Buchanan and Audrey Abel did an

excellent job of typing and retyping numerous drafts of the manuscript. Finally, a very warm thanks to our wives, for enduring.

PREFACE

PART I.

PART II.

PART III.

PART IV.

PART V.

CONTENTS

Page

PRELIMINARIES ••••••••.••••••••••••• It • • • • • • • • • • • • • • • • • 1

Chapter One: Introduction ••••••••••••••••••••••••••• 2 Chapter Two: Mathematical Preliminaries ••••••••••••• 20

CONSUMER BEHAVIOR ••••••••••••••.••••••••••••••••••••• 28

Chapter Three: Consumer Behavior in tne Absence of an Income Tax ••.•••.........•..•.......... ; ....... 29

Chapter Four: Consumer Behavior in the Presence of an Income Tax................ ... . . . . . . . . . . . . . . . . .. 43

ECONOMIC INDICES IN THE ABSENCE OF TAXATION •••••••••• 57

Chapter Five: The Konus and Frisch Expenditure-Based Cost of Living Indices •••••••••••••••••••••••.•• 58

Chapter Six: Wage and Nonwage Income-Based Measures of the Cost of Living in the Absence of an Income Tax ............................... -...... 69

ECONOMIC INDICES IN THE PRESENCE OF TAXATION ••••••••• 76

Chapter Seven: A Simple Taxable Income-Based Cost of Living Index .•............ " ................ It. 77 Chapter Eight: The Full Income and Nonwage Income-Based Cost of Living Indices •••••••••.••••••••• 84

THE THEORY IN PRACTICE............................... 91 Chapter Nine: Topics in Applied Analysis •••••••••••• 92 Chapter Ten: Changes in Government Goods and the Tax Code ••••••..•...•••.•.•.••••.•••••••••••••••• 108

BIBLIOGRAPHY ••••••••••••••••••••••••••••••••••••••••••••••••••••• 11 2

PART I

PRELIMINARIES

CHAPTER ONE: INTRODUCTION

Income based measures of the cost of living have become increas­ingly important in recent years (cf. Cagan and Moore (1981), Gilling­ham and Greenlees (1983), Kay and Morris (1984), and Triplett (1983». The basic idea behind the concept of an income-based cost of living index is to compa're the levels of pretax income necessary to maintain some base standard of living under alternative price and tax regimes. This is in contrast to an expenditure-based index of the cost of living, which is designed to compare the expenditures (exclusive of taxes) necessary to maintain the base standard of liv­ing under 'alternative price regimes.

In order to illustrate the important distinction between these two concepts, consider a ceteris paribus doubling of all prices. In

this case the expenditures net of taxes necessary to maintain the base standard of living doubles, and hence an expenditure based mea­sure of the cost of living will also double. But if pretax income doubled and taxes are unindexed and progressive, one could not afford the base standard of living because the doubling of pretax income would lead to a higher marginal tax rate and a higher real tax bill. Consequently, a doubling of all prices will, ceteris paribus, more than double the income-based cost of living index.

Income-based measures of the cost of living are important for two primary reasons. First, as noted by Cagan and Moore (1981), they are the relevant index concept to be used in income escalation mea­sures because such cost of living adjustments are usually a percent­age of existing pretax income. Escalation measures based on an ex­

penditure-based cost of living index will lead to an underadjustment of income. The importance of an income-based cost of living index for purposes of escalating income has been mitigated somewhat in the United States, due to the ~ecent indexation of the tax codes. None­theless, many countries and indeed many states within the United States presently have un indexed tax codes, and the construction of an income-based cost of living index for purposes of income escalation remains important.

Secondly, the income-based concept of the cost of living is im­portant because it is the correct means of deflating a time series of data on pretax income. Because of the absence of such a measure of the cost of living, economists typically deflate pretax income (or

3

GNP) by using an expenditure-based measure of the cost of living, or

more frequently the CPI approximation to an expenditure-based index. Recently, Kay and Morris (1984) and Gillingham and Greenlees

(1983) have done pioneering empirical research on incorporating taxes

into a price index for Britain and the United States, respectively. While neither of these studies constructed an income-based cost of living index, which requires the estimation of consumer preferences, they do suggest that the distinction between income and expenditure­

based cost of living indices is empirically significant. For ex­ample, Gillingham and Greenlees found that between 1967 and 1981 the average annual rate of change in their TPI (an approximation to an income-based index) was 7.6%, whereas the average annual rate of

change in the CPI (an approximation to an expenditure-based index) was 6.8%.

In this monograph, we develop the theory of income-based cost of living indices. Our methodology differs in three respects from the

studies mentioned above. First, the existing studies are empirical

in nature, while our focus is on theoretical issues. Second, the ex­

isting studies incorporating taxes into a price index are not based

on consumer preferences, and hence do not allow for substitutions

among commodities as prices and tax codes change. As such, they are approximations to an income-based cost of living index. We develop

the theory of income-based cost of living indices from a model of maximizing behavior, and thus our analysis allows for substitutions

among goods as the incentives to consume the goods change over time. Third, and perhaps more importantly, existing approximations to

an income-based cost of living index do not account for the labor­leisure decisions of consumers. Our analysis of consumer behavior explicitly models the labor-leisure decision, and as such, our income­

based cost of living indices allow consumers to substitute between

goods and leisure as the ~ftertax wage changes. Because leisure

is, in effect, a tax deductible good, one might expect such substitu­

tions to be amplified as marginal tax rates increase. In order to accomplish our purpose, it is necessary for us to

develop the theory of consumer behavior in the presence of an income tax, and to provide an examination of traditional cost of living in­

dex theory. Because of the technical nature of our results, it is useful to provide a descriptive survey in this introduction of works on (1) consumer behavior in the presence of nonlinear budget sets,

4

and (2) cost of living indices. Our introduction concludes with an

overview of our monograph.

1. Consumer Behavior with Nonlinear Budget Sets

In order to construct cost of living indices that allow for sub­

stitutions among goods and leisure, it is necessary to specify the

behavior of the consumer. Unless the income tax system is propor­

tional, the budget constraint is nonlinear. In Figure 1.1, we depict

a simplified budget set of a consumer that faces a proportional in­

come tax, and in Figure 1.2, we depict the budget set of a consumer

that faces a progres~ive income tax that is piecewise linear. In

Figure 1.1, 6 represents the total endowment of time, P is the price

of consumption, and T represents the marginal tax rate. The solid

line represents the budget set with the proportional income tax and

the broken line represents the budget set of the consumer without

taxes. Note that the presence of a proportional income tax has two

effects. First, it reduces the income of the consumer. Second, the

tax on income reduces the price of leisure. Holding utility con­

stant, the presence of the proportional income tax will in-

crease the quantity of leisure consumed. The negative income effect,

however, (assuming that leisure is a normal good) reduces the quant­

ity of leisure consumed. Thus, the total effect of the proportional

income tax on labor supply is ambiguous.

As far as traditional theory is concerned, the presence of a

proportional income tax presents no major problems: If the price of

leisure and nonlabor income is weighted by one minus the marginal tax

rate to obtain the "effective price" of leisure, then the resulting

demand system satisfies all the classical properties. Mark Killings­

worth (1983), in his excellent review of the labor supply literature,

notes that several empirical studies assumed that the tax system was

proportional. Killingsworth calls these works "first generation

studies." Perhaps the most important of the first generation studies

is Abbott and Ashenfelter (1976). This study estimated consumer pre­

ferences over eight broad commodity groups, including leisure.

Abbott and Ashenfelter's preferences parameters have been used fre­

quently in other studies. For instance, Pencavel (1977, 1979) used

these estimates to construct his real wage estimates. Similarly,

Cole and Harte-Chen (1985) used the data and methodology in their

study.

Consumption Good

, " ......

"-' ........ /

"-, " ......

5

Slope =

, SI (I-T)w',

ope = ,

w P

P , ')

(I-r)y/P {

FIGURE 1.1

I I I y/ P

8 Leisure

BUDGET SET WITH A PROPORTIONAL TAX

Consumption Good

\ Y

C

SI (I-TC) w ope=--­

p

I I I I I I I I I I I

6

SI (I-TB)w ope =--­

p

1\ '---__ ,.. __ A ...... __ ._ ..... ) ~ Y Y- 0

B A

FIGURE 1.2 BUDGET SET WITH A PIECEWISE LINEAR TAX

Leisure

7

One weakness of this approach is apparent in Figure 1.2. When

leisure consumption is in region A, the tax rate is lower than when it is in region B. The marginal tax rate is even higher when leisure consumption is in region C. Thus the after tax wage may vary across consumers even when all consumers are paid an identical before tax wage rate. This creates a legion of theoretical and econometric problems. Killingsworth calls works that address these (and similar) issues that arise from progressive taxation "second generation studies."

The works of Hausman (1981, 1985) and Wales and Woodland (1979)

have made significant theoretical and empirical contributions in solving this problem; see Hausman (1985) for a good review of the most recent literature. Intuitively, the approach consists of solv­

ing for solutions associated with various segments of the budget con­straint. In Figure 1.2, there would be three potential "traditional" solutions associated with each of the three segments. The consumer then selects the solution that maximizes utility. This approach al­lows the recovery of preferences when the consumer faces a piecewise linear budget constraint.

2. Cost of Living Indices A. Expenditure-Based Indices in the

Absence of a Labor Supply Decision

To most people, price indices such as the Laspeyres (1871) and Paasche (1874) indices are measures of the cost of living. But price indices that are based on a particular market basket of goods do not allow for substitutions among commodities as relative prices change. As such, these price indices are noneconomic indices in the sense that they are not based on behavior. While such noneconomic indices may be useful, they are technically not a measure of the cost of living.

Functional or economic cost of living indices, terms originally coined by Frisch (1936) and Samuelson and Swamy (1974), respectively, have been a focus of price index research for over one half of a cen­tury. This approach explicitly recognizes that, since a consumer reacts to changes in relative prices by altering behavior, price in­dices based on "fixed weights" (such as the Laspeyres and Paasche) do

not accurately reflect changes in an individual's "true" cost of liv­ing. Thus functional price indices take into account the changes in consumer behavior that arise from changes in relative prices.

8

The expenditure function of the individual has been the primary ingredient in constructing functional price indices. For the most part tbe Konus index (1939) or so called "true" cost of living index is the most widely recognized functional price index. This index compares the minimum expenditures in two periods necessary to buy a given level of utility. Afriat (1977), Diewert (1981), Pollak (1971), Samuelson and Swamy (1974), and others have done important theoretical research on this index, and recently Braithwait (1980) has estimated the Konus index for a relatively large system of demand equations.

In Figure l.~, we illustrate the basic concept of the Konus in­dex. Point A represents the consumer's selection of the two goods in the initial period, which requires a nominal level of expenditure equal to yO. In the next period, prices increase reflecting some "inflation" in the economy. But there is also a change in relative

prices: the price of good one has increased more than the price of good two. The price increase causes the budget line, represented now by the dotted line CD, to shift back toward the origin, which would make the consumer worse off unless nominal expenditures increase. Keeping relative prices constant, we may increase nominal income, which shifts the budget line out until the new budget line is tang­ent to the old indifference curve, uO. If we let yl denote the level of expenditures necessary to purchase the new consumption bundle (point B), the Konus index value is simply given by (yl/yO).

Rather than holding an arbitrary market basket fixed, the Konus index fixes the consumer's level of well-being. This allows the con­sumer to substitute among commodities as relative prices change, and hence the Konus index is an economic index. Further, by holding the

consumer's level of utility fixed, we have a rigorous definition of real income: the consumer's real income is determined by the level of utility it will purchase.

Unfortunately, the reported value of the Konus index generally depends on the prescribed level of utility. This means that if pol­icy agents disagree as to the "proper" utility base, or if consumer incomes are dispersed, different values of the Konus index emerge~ there is no unique Konus index.

It turns out that the existence of homothetic prefer-ences is necessary and sufficient for the ·Konus index to be independ­ent of base utility. Samuelson and Swamy remark on this fact: "Empirical experience is abundant that the Santa Clause hypothesis of

Consumption Good

9

FIGURE l.3

THE COST OF LIVING QUESTION IN COMMODITY SPACE

Good 2

10

homotheticity in tastes ••• is quite unrealistic" (1974, p. 592).

Despite the above caveats and ambiguities the Konus index has

received widespread acceptance. Seldom examined in the literature is an alternative economic index proposed by Frisch (1932) and later examined by Theil (1975), Baye (1983), and Balk (1985): the Frisch cost of living index. The Frisch cost of living i~dex utilizes an individual's expenditure function but compares the ma~ginal costs of utility in two periods. The economic rationale behind the Frisch in­dex is two fold. First, since many economic decisions depend on mar­

ginal costs, it is argued that a cost of living index should depend on marginal costs. Second, the marginal cost of utility is the recipro­cal of the marginal utility of income. Thus the Frisch cost of liv­ing index implicitly compares the marginal utility of income in two

periods. Frisch argues that changes in the marginal utility of in­come provides a useful measure of welfare change since the marginal utility of income should decline as "real income" rises. This argu­ment is also presented in t1)e context of his "money flexibility" (the elasticity of the marginal utility of income).

B. Income-Based Cost of Living Indices in the Presence of a Labor Supply Decision

The cost of living issue becomes more complex when we con-sider the labor supply question (see Cleeton, 1982). In Fiqure 1. 4, we illustrate the cost of living problem with endogenous labor supply. The consumer initially selects bundle A along the budget line denoted RS. In order to purchase this consumption bundle the consumer needs full income FO = yO + wO Ii, where y represents nonlabor

income, w is the wage rate, and Ii is once again the total endowment of time. After a change in the price of the consumption good, the wage rate, and the level of nonlabor income, the budget line shifts

down to GH. If we follow the approach of the Konus index, we wish to

shift the budget line out until it is tangent to the indifference

° curve u. How should the budget line be shifted out?

If we increase full income until the budget line is tangent to the indifference curve at point B, we may use this level of full in­come, denoted Fl, to construct the full income index value, Fl/FO.

As we have not changed the relative prices of leisure and the con­sumption good, this is equivalent to providing the consumer with suf­

ficient nonlabor income to reach the indifference curve uO at point B. The new level of nonlabor income, ~ may be used to construct the nonlabor income index value, yl/yO (where we assume that f > 0).

Consumption Good

R

11

.............. A -...-...--~ , , -...... -

G

o q;. , ~ (~ , ~ _ ,.,Qi' -- ~ , ---- , - ....... ----

s

H

8 Leisure

FIGURE 1.4 COST OF LIVING QUESTIONS IN LEISURE-GOODS SPACE

12

Although under both index concepts the consumer selects the consump­tion bundle point B, the two index values are not, in general, equal.

Yet, another approach (Pencave1 1977, 1979) to the index ques­tion is to hold the level of non1abor income and the price of the consumption good fixed, and increase the wage rate until the budget line is tangent to the indifference curve. The increase in the wage rate causes the budget line to pivot about point H until it is tan-

gent to the indifference curve at point C. 1 w , may be used to construct the real wage

This wage rate, denoted index value, wI/wOo Each

of the index concepts provides a measure of the cost of livingJ each of the concepts is an economic indexJ but each of the index concepts ignores taxation. In this monograph we extend these index concepts

to include income taxation.

3. Overview of the Monograph In this monograph we provide a self-contained guide to the

theory of consumer behavior and the construction of measures of the cost of living. We consider the traditional framework that ignores taxes, and also extend the theories to allow for an income tax. We assume the reader is familiar with calculus and set theoretic con­

cepts. The monograph is divided into five main parts and ten chapters.

Part I provides an overview and presents some mathematical pre­liminaries. Part II presents the modern theory of consumer behavior, and.examines the theory of consumer behavior in the presence of an income tax. In Parts III and IV, these behavioral concepts are used to derive economic index numbers in the absence of taxation and in the presence of an income tax, respectively. Part V touches upon practical issues that arise in applying the theory derived in earlier parts. The following is a summary of the remaining chapterR of the monograph.

Chapter Two: In this chapter we provide a review of the major mathe­matical results that we use in our analysis. We have attempted to provide either sketches of proofs or references for readers who wish to find proofs of these propositions.

Chapter Three: This chapter provides a review of the neoc1a~sical model of consumer behavior with labor supply in the absence of

13

taxes. Our presentation, following the modern approach, makes heavy use of the duality between the expenditure minimization problem and the utility maximization problem of the consumer.

We define the indirect utility function, which conveniently sum­marizes the preferences of the consumer. We demonstrate that the in­direct utility function is nondecreasing in nonlabor income, nonin­creasing in the prices of commodities, nondecreasing in the wage rate, and is homogenous of degree zero in all prices and nonlabor in­come. Similarly, we define the (indirect) expenditure function and demonstrate the properties of the expenditure function. These pro­perties include: (a) the expenditure function is nondecreasing in all prices: (b) the expenditure function is linearly homogenous in all prices: (c) the expenditure function is concave in all prices: and, (d) the expenditure function is a nondecreasing function of the utility level. In contrast with most published derivations of the properties of these functions (such as Varian, 1984), however, we al­low the consumer to also choose the level of labor supply.

We also define the Marshallian and Hicksian demand functions. We demonstrate how the Marshallian and Hicksian demand functions may be derived from the indirect utility and expenditure functions

via Antonelli-Roy's Identity and Shephard's Lemma. We then proceed to derive the properties of the Hicksian and Marshallian demand func­tions. We prove that the Hicksian demand functions are homogenous of degree zero in all prices, that the share-weighted price elas­

ticities sum to zero, that the cross price effects are symmetric, and the own price effects are nonpositive •. We also prove that the Marshallian demand functions are homogeneous of degree zero in prices and nonlabor income, that they satisfy Cournot aggregation, and that they satisfy Engel aggregation. In addition, we derive two forms of

the well known Slutsky equation that are valid in the presence of a labor supply decision.

Chapter Four: In this chapter, we modify the neoclassical model to allow for the presence of an income tax. We follow the outline of Chapter Three as closely as possible to allow the reader to see how

the presence of an income tax alters the neoclassical results. Since

income taxes are a function of earned income, the income tax has the effect of distorting the price of leisure compared to other goods. This distortion alters many of the properties of neoclassical ana­

lysis.

14

If we assume that the income tax is progressive, the indirect utility function is still nondecreasing in nonlabor income, nonin­creasing in the prices of commodities, and nondecreasing in the wage rate. However, the indirect utility function is no longer homogenous of degree zero in prices and nonlabor income, as increases in income may push the consumer into a higher tax bracket.

The presence of an income tax leads to a distinction between the full income compensation function and the expenditure function. While the expenditure function provides the minimum level of expendi­tures on goods and leisure necessary to achieve a given level of utility, the full income compensation function provides the level of full income necessary to purchase goods and leisure, and to pay the consumer's tax bill necessary to achieve a given level of utility. In the absence of an income tax, the full income compensation function would be equivalent to the expenditure function. We show that the full income compensation function is nondecreasing in the prices of goods, nondecreasing in the wage rate, and nondecreasing in the level of utility. The function is not homogenous of degree one .in all prices, however, unless the tax function is proportional.

The derivative of the full income compensation function with re­spect to the price of a good weighted by one minus the marginal tax rate provides the Hicksian demand for that good. The derivative of

. the full income compensation function with respect to the wage rate directly provides the Hicksian demand function for leisure. This modification of Shephard's Lemma provides us with a means of recover­ing the Hicksian demand functions in a world with taxes. We demon­strate that the Hicksian demand functions are homogenous of degree. zero in all prices only when the tax system is proportional. While the cross price effects are symmetric among goods, it is no longer the case that the derivative of the Hicksian demand function for leisure with respect to the price of a good is equal to the derivative of that good's Hicksian demand function with respect to the wage rate. Thus, the neoclassical symmetry restriction fails in the presence of income taxation.

The indirect utility function may be used to recover the Marshallian demand functions, although Antonelli-Roy's Identity must , be altered slightly due to the presence of income taxation. Similar-ly, the Slutsky equation is altered slightly because of the presence of the income tax: when the derivative of a Marshallian demand func­tion is taken with respect .to the price of a good, the income effect

15

must be weighted by one over one minus the marginal tax rate. More­

over, the Cournot and Engel aggregation results are altered by the

income tax, and the demand functions are homogeneous of degree zero

only when the tax system is proportional. We also derive two tax

analogues of the Slutsky equation.

Chapter Five: In this chapter we review the properties of the Konus

and Frisch cost of living indices and present some new results that

indicate that the Frisch index may act as a bound for the Konus in­

dex. We assume, in this chapter, that there are no taxes on income.

The Konus index is the ratio of the expenditure function in two peri­

ods required to maintain a fixed standard of living. The index sat­

isfies several desirable properties, which we refer to as the Eich­

horn-Voeller axioms. We define the Laspeyres and Paasche price in­

dices and indicate that Laspeyres and Paasche indices may act as

bounds on properly selected Konus indices. Indeed, we demonstrate

that if preferences are homothetic, then the Laspeyres and Paasche

indices are an upper and lower bound for the Konus index, respective­

ly, and if preferences are Leqntief, then the Laspeyres, Paasche, and

Konus indices are equivalent.

In general, the Konus index depends upon the base period level

of utility. But if preferences are homothetic, we show that the

Konus index is independent of base period utility. Thus, if prefer­

ences are homothetic, identical persons with different incomes will

have the same measure of the cost of living.

We then introduce the Frisch cost of living index, which is de­

fined as the ratio of marginal expenditures necessary to purchase an

additional unit of utility in the two periods. The Frisch cost of

living index satisfies a modified version of the Eichhorn-Voeller

axioms. Like the Konus, the index is generally a function of the

base period level of utility. The Frisch cost of living index, how­

ever, is independent of base period utility if preferences are of the

Gorman polar form, which is less restrictive than the assump-

tion of homotheticity that is needed for the Konus index to be inde­

pendent of base period utility. As the Gorman polar form of prefer­

ences is the most general form of preferences to allow for exact

linear aggregation, this is an important result.

The Frisch cost of living index also provides a bound on the

Konus index. We prove that if the Konus index is increasing

in base period utility, then the Frisch cost of living index is an

16

upper bound for the Konus index. If the Konus index is a decreasing function of base period utility, then the Frisch cost of living index is a lower bound for the Konus index. Finally, if the Konus index is independent of the level of utility, which implies that preferences are homothetic, then the Frisch and Konus cost of living indices are

equivalent.

Chapter Six: The focus of this chapter is on income-based cost of living indices in the absence of an income tax. The concepts we consider allow the consumer to alter the labor supplied as relative prices change, but because of the assumed absence of an income tax, there are no tax induced distortions in the consumer's labor-leisure decision.

We begin by defining the nonlabor income function that provides the minimum level of nonlabor income necessary to achieve a given level of utility. The nonlabor income function is nondecreasing in the prices of goods, non increasing in the wage rate, nondecreasing in the level of utility, and homogeneous of degree one in. all prices. Similarly, we define the minimum wage function, which provides the minimum wage necessary for the consumer to achieve a given level of utility given the prices of goods and the level of nonlabor income. We demonstrate that the minimum wage function is nondecreasing in the prices of goods, nonincreasing in the level of nonlabor income, and homogenous of degree one in the prices of goods and nonlabor income.

The nonlabor income function may be used to define the nonlabor income index. The nonlabor index is the ratio of the current peri­od's nonlabor income function to the base period's nonlabor income function. This index, which satisfies the Eichhorn-Voeller axioms,

is a complete welfare ordering. It will allow us to determine whether the consumer is better or worse off under current prices or past prices, given a common level of nonlabor income. Similarly, we may use the minimum wage function to define the real wage index. The real wage index is the ratio of the minimum wage function in the cur­rent period to the minimum wage function in the base period. The real wage index also satisfies the Eichhorn-Voeller axioms and is a complete welfare ordering.

Chapter Seven: In this chapter we begin to consider the problem of

constructing a cost of living index when there is income taxation.

We begin by considering the case where labor supply is exoge-

17

nous. The individual then will seek to maximize his utility subject

to the budget constraint that his expenditures on goods must equal

his total income minus his tax bill.

We define the gross expenditure function as a function that

relates the minimum pretax income necessary to provide a given level

of utility when. the individual must pay taxes on his income.

We demonstrate that the gross expenditure function is homothetic in

the prices of goods, nondecreasing in the prices of goods, quasicon­

cave in prices, and nondecreasing in the level of utility. If

all prices in the economy are doubled, then the gross expenditure

function more than doubles if taxes are progressive, less than

doubles if taxes are regressive, and exactly doubles if taxes are

proport ional.

The gross expenditure function may be used to define the taxable

income-based cost of living index. The taxable income-based cost of

living index is the ratio of the current period's gross expenditure

function to the base period's gross expenditure function. This index

provides the factor by which base period taxable income must be ad­

justed in order to leave the consumer equally well off after a price

change. We demonstrate that if the tax system is progressive, then a

doubling of all prices will result in the taxable income-based cost

of living index more than doubling, which reflects the bracket creep

associated with progressive income taxation. Moreover, when the

Konus index is greater than one, the Konus index acts as a lower

bound for the taxable income-based cost of living index. Like the

Konus cost of living index, the taxable income index will in general

depend upon base period utility. But if preferences are homothetic

and the tax system is proportional, then the taxable income-based

cost of living index is independent of the level of base period util­

ity.

Chapter Eight: In this chapter we extend the index ideas introduced

in Chapter Seven by allowing for a labor supply decision. We begin by

defining the full income-based cost of living index as the ratio of

the full income compensation function in the current period to the

full income compensation function in the base period. In the absence

of taxation, this index would be equivalent to the Konus index. With

a progressive income tax, a doubling of all prices will more than

double the value of the index. This is because the compensation

has pushed the consumer into a higher tax bracket. Like the Konus

index, the full income index generally depends upon the base period

18

utility level. If the tax system is proportional and preferences are homothetic, then the full income cost of living index is independent of base period utility.

To compare the value of the full income index with more tradi­

tional measures of the cost of living, we also define the expenditure­based cost of living index in the presence of an income tax. The ex­penditure-based cost of living index is just the ratio- of expendi­tures on goods and leisure in the current period to the expenditures

on goods and leisure in the base period. As such the index is ana­

logous to the Konus index, except that we have incorporated the ef­fects of income taxation on the labor supply and consumption deci­

sions of the consumer. We demonstrate that the rate of change in the

full income cost of living index may be divided into two components.

The first component is the rate of change in the expenditure-based cost of living index weighted by the ratio of one minus the average tax rate to one minus the marginal tax rate. When the tax system is

progressive, this ratio will be greater than one, indicating that the rate of growth in the full income-based cost of living index will ex­ceed the rate of growth in the expenditure-based cost of living index when there is inflation. But the second component is the opposite of

the rate of change in the expenditures on leisure weighted by two terms: the first is the fraction of full income spent on leisure and

the second is ratio of the marginal tax rate to one minus the mar­ginal tax rate •. As the second term is negative in periods of price

increases, this indicates that the rate of change in the full income

cost of living index may be greater or less than the rate of change in the expenditure-based cost of living index.

We next define the nonwage income cost of living index as the

ratio of the nonlabor income function in the current period to the minimum nonlabor income function in the base period. We demon­

strate that the full income index is equal to the weighted sum of the nonwage income cost of living index and the ratio of the current wage

to the wage in the base period. This result demonstrates the close

relationship between the full income and the nonwage income cost of living index.

Finally, we extend the real wage index to incorporate the ef­

fects of income taxation. We demonstrate that with a progressive in­

come tax, a doubling of the prices of goods and nonlabor income will

more than double the value of the real wage index in the presence Of an income tax.

19

Chapter Nine: This chapter addresses the issue of recovering prefer­ences from consumption data in a way that allows one to incorpor-

ate demographic effects and taxation into an aggregate measure of the cost of living. As families of different sizes and age compositions

may have systematically different preferences, we provide a method of

accounting for demographic variations that is consistent with exact linear aggregation. Similarly, families with differing incomes will face differ~nt tax rates. We demonstrate the restrictions on prefer­

ences necessary to allow for exact linear aggregation in this situa­tion. Both of these results exploit a general theorem that is due to Muellbauer.

Second, we prove a general theorem for constructing complete

demand systems to use in applied work. The actual number of com­plete demand systems is relatively small, but we demonstrate that a linear combination of expenditure functions is also an expenditure function. The resulting expenditure functions may be used to gener­

ate a complete demand system, which can be used in estimation. The chapter concludes with an empirical example that illustrates these results, as well as the index concepts derived in earlier chapters.

Chapter Ten: The monograph concludes with a discussion of how, in principle, the theory may be expanded to allow for changes in tax codes over time (perhaps due to indexation). Also discussed is the issue of incorporating the benefits of taxation (government provided

goods and services) into the index concepts.

CHAPTER TWO: MATHEMATICAL PRELIMINARIES

We let Rn denote Euclidean n-space, and let R:+ denote the

positive orthant. The notation "XES" means "X is an element of set

S." A set is taken to be a collection of objects. For example, we

express the positive orthant in set notation as

Throughout this study, for vectors x, YERn , "x ~y" means "xi ~Yi for all i and Xj > Yj for some j." Similarly, for two sets S and V,

"S c V" means that "XES implies XEV;" that is, S is a subset of V.

The notation S c V means every x in the set S is also contained in V,

but there exists an element of V that is not contained in S; in other

words, S is a proper subset of V.

Definition 2.1: A set S c Rn is said to be a convex set if x, yES

and 6E [0, 1] implies 6x + (1-6 )YES.

In Figure 2.1, set A is convex, but set B is not convex.

Definition 2.2: A set S c Rn is said to be a closed set if

every sequence {x.} c S with 1 im {X.} + x implies XE S. 1 i +co 1

In Figure 2.2, set C is closed; set D is not closed.

Definition 2.3: A set S c Rn is said to be a bounded set if there

exists an r > 0 such that S.£ {XERn !IIX - xli < r}.

Definition 2.4: A set S c Rn is said to be a compact set if it is

closed and bounded.

Definition 2.5: A function F : S + R , S .£ R n, is said to be a con­

tinuous function at Xo if for all E > 0, there exists a OER++

such that

II x - Xo II < 0 impl ies ! f (x) - f (xO) I < E.

If this condition holds for all XOES, then F is said to be a

continuous function on S.

21

Set A

Set B

FIGURE 2.1

CONVEX AND NON CONVEX SETS

22

Set C

Set D

FIGURE 2.2 CLOSED AND OPEN SETS

23

Theorem 2.1 (Weierstrass): Let F be a continuous function from S +

R, and suppose S ~ Rn is nonempty and compact. Then F has a maximum and a minimum.

Proof: See Takayama, p. 29.

The Weierstrass Theorem is important in economics because of the

assumption of maximizing (or minimizing) behavior on the part of

economic agents. The Weierstrass Theorem ensures that, given certain regularity conditions, the maximization (minimization)

problems have a solution.

Definition 2.6: A function F : Rn + R is said to be homogeneous of

degree k if for all AE:R++, F(AX) = A kF(x).

If, for example, a function F(x) is homogeneous of degree one, then a doubling of the components of x will lead to the doubling

of the value of the function.

Theorem 2.2 (Euler): Suppose F : S + R is differentiable and homo­

geneous of degree k on S ~ Rn. Then

+ aF(x) x --ax;- 2 + • •

Moreover, the function gi(x) _

(r-l) •

Proof: See Eichhorn (1978, p. 71).

• • + rF(x) •

aF(x) is homogeneous of degree ax.

1

Definition 2.7: A function is said to be homothetic if it may be

written as an increasing function of a function that is homo­

geneous of degree k.

For example, the function y = x~xi is both homogenous (of degree a fl a+fl) and homothetic, whereas the function z = In[xl x2 ) is homothetic

but not homogeneous.

Definition 2.8: A function F : Rn + R is said to be separable if n.

there exist functions G : Rm + Rand gi : R 1 + R [i = 1, 2,

•••• m) and mutually exclusive and exhaustive subvectors VI' v 2 ,

v such that m

24

Definition 2.9: A function F

separable if

Rn + R is said to be strongly

Definition 2.10: A function F : S + R, S ~ Rn, is called a concave function if for all x, yES and aE [0, 1),

F ( a x + (1- a ) y) ~ a F ( x) + (1-a) F ( y ) •

A function G is a convex function if - G is concave.

In Figure 2.3, function F(') depicts a concave function. The

function G(') is not concave.

Definition 2.11: An nxn matrix A is said to be negative semidefinite

if and only if xAx' < 0 for all XE Rn •

Theorem 2.3: A twice continuously differentiable function F is concave if and only if the Hessian matrix

H

• • F lIn

• • F 2n

• F nn

_ 2 is negative semidefinite, where Fij = a F/aXiaX j •

Proof: See Henderson and Quandt, pp. 375-377.

Theorem 2.4: Suppose F : Rn + R is a twice continuously differenti­

able concave function. Then a2F/aX~ < 0 for all i = 1, 2, 1. -

• •• n.

Proof: By Theorem 2.3 the Hessian matrix, H, of the function F sat­isfies xHx' < 0 for all xERn. If we let e ERn be the vector i with a one in the ith component and zero's elsewhere, we get

eiHei Fii - a 2F/axf .: 0 as required. Q.E.D.

Definition 2.12: A function F : Rn + R is said to be a quasiconcave function if and only if {XERn IF(X) > y} is convex for all YER. A function G is said to be quasiconvex if -G is quasiconcave.

In Figure 2.4, the function F(') is quasiconcave (but not concave),

and the function G(') is quasiconvex (but not convex). One can show that a concave (convex) function is quasiconcave (quasiconvex), but a

quasiconcave (quasiconvex) function need not be concave (convex).

F

F(y)

F(8x+(I-8) y)

8 F ( x ) + (I - 8) F ( y )

F(x)

G

25

--F(z)

x 8x+(I-8)y y z

G (z)

z

FIGURE 2.3

CONCAVE AND NONCONCAVE FUNCTIONS

26

F

y

F ( x)

{t<IF(x)~y} x

G

G(x)

y

x

FIGURE 2.4

QUASICONCAVE AND QUASICONVEX FUNCTIONS

27

Theorem 2.5 (Young): Let F : Rn + R be twice continuously differ­entiable. Then a2F/aX.aX. = a 2F/aX.aX. for all i, j = 1, 2,

1 J J 1 • •• n.

Proof: See Olmsted, p. 263.

Theorem 2.6 (Implicit Function Theorem): For X€~, y€Rn, let

1 F (y: x) 0

~2(y: x) 0

Fn(y: x) = 0

be a system of n equations to be solved for the n unknowns (the components of y) in terms of the m remaining variables (the components of x). If the functions Fl, ••• Fn are continuously

. •• • ( ) Rn+m dl.fferentiable l.n a nel.ghborhood of a pOlnt xO' Yo € satisfying the above equations, and if the Jacobian determinant

aF l (IF 1 (IF 1

aYl aY2 aYn

IJI a(Fl, Fn) aF 2 (IF 2 (IF 2 -

~Yl :Yn a (Yl' aYn) :Y2 . . aFn aFn a Fn ay

n aYn aYn

is non-vanishing at (xO' YO), then there exist continuously differentiable implicit functions

Y = f 2 (x) .2

Yn fn(x)

that satisfy the system of equations in a neighborhood of

(xo' yO)·

Proof: See Olmsted, pp. 327-329.

PART II

CONSUMER BEHAVIOR

CHAPTER THREE: CONSUMER BEHAVIOR IN THE ABSENCE OF AN INCOME TAX

In this chapter we present the modern approach to the classical theory of consumer choice. The consumer's objective is to select the (n + 1) bundle of goods and leisure that maximizes his well being.

We let X = (Xo ' Xl' ••• Xn) denote the vector of commodities. The element Xo shall be interpreted as leisure, and is contained on the interval [0,6]. The scalar 6 represents the individual's total endowment of time. The Xk's [k = 1, 2, ••• n] are interpreted as commodities, and are elements of R+. Thus the consumption set of the

individual is taken to be n :: {X E [0, 61 x R~}.

Associated with the vector of commodities is an (n + 1) vector

of strictly positive prices, denoted P = (Po' PI' ••• Pn ). The element Po rep"resents an implicit price of leisure, for as the individual consumes leisure he forgoes the opportunity to work. Thus, the implicit cost of one unit of leisure is the wage rate that the individual would have received had he worked. The remaining Pk's [k = 1, 2, ••• n] are strictly positive scalars that represent the ex­plicit unit prices of commodities.

Given this structure, we now define several concepts of inter-

est.

Definition 3.1: The consumer's full income, F, is defined as

where y is the individual's exogenous nonlabor income.

Full income represents the total income the individual has to allo­cate among purchases of leisure and the other n commodities. Note that F is homogeneous of degree one in (y, PO).

Definition 3.2: The budget set, B(P, F), is defined as

B(P, F):: {XEn I p·x ~F}.

The budget set thus contains those elements of the consumption set that are affordable for given prices and full income. The budget set

may be viewed as the consumer's choice set.

proposition 3.1: The budget set:

(a) is compact:

30

(b) is homogeneous of degree zero. That is, satisfies

B(9P, 9F) :: B(P, F);I and

(c) is convex.

Proof: (a) Follows from Definition 2.4 and the fact that P and F

are strictly (b) B( 9P, 9F)

positive.

- {XEO I 9p· X S.9F}. - {XEO p. X < F}. - B(P, F).

* ** * (c) Suppose X , X E B(P, F). We must show 9X

* ** + (1-9)X **

E B(P, F) for 9E [0, 1]. Now X , X E B(P, F) implies

* ** * P • X < F and P • X < F. Hence, 9P • X < 9F and

** (1-9) P • X S. (1-9) F. Adding these expressions we

* ** obtain p. [9X + (1-9)X ] S.9F + (1-9) F = F.

* ** Hence, 9X + (1-9)X E B(P, F). Q.E.D.

While the budget set defines consumption bundles that are feas­ible, we have yet to characterize the preferences of the individual. We shall impose restrictions on the consumer's preferences that are stronger than necessary in order to simplify the analysis. While many of the results may be derived under weaker assumptions, there is

little gain in economic insight in doing so.

We shall assume that the consumer's preference ordering is rep­

resented by a twice continuously differentiable, regular strictly * ** * ** quasiconcave function U : 0 + R. Thus, if U(X ) > U(X ) for X , X

* ** E 0, the consumer prefers X to X

* Definition 3.3: The consumer's upper contour set, denoted C(u), is defined as

* C ( u) = {XE 0 I U (X) ~ u} •

Thus, the upper contour set defines those elements of the consumption set that yield a level of utility at least as good as u. Important­

ly, the upper contour set is invariant to monotonic transformations

of the utility function. That is, if ~ : R + R satisfies ~'(.) > 0, * then C(u) = {XEO I U(X) ~ u} contains the same elements as the set

31

* C(CP (u» = {ue:Q I cP (U(X» ~ cP (u)}.

In this sense, the utility function is an ordinal representation of

preferences. The quasiconcavity of the utility function implies (by

Definition 2.12) that the upper contour set is convex. The addition­

al assumption that the utility function is regular strictly quasicon­

cave rules out "flats" in the level sets and also rules out points of

satiation.

The objective of the consumer is to select the commodity bundle

that maximizes utility, subject to the constraint that the bundle be

affordable. Formally, the consumer attempts to

m~x {u ( X) I Xe: B (P, F)}.

Since U(·) is continuous and B(P, F) is nonempty and compact, Theorem

2.1 ensures that a solution to the consumer's problem exists. The

solution value depends on the level of prices and full income, and is

termed the indirect utility function because, at the optimum, utility

indirectly depends on prices and income.

Definition 3.4: The indirect utility function, denoted V(P, F), is

defined as

V(P, F) - M~X {U(X) I p. X < F}.

While the notation V(P, F) is convenient, one should keep in

mind that, using Definition 2.1,

V(P, F) " V(P, y + oPo)'

Differentiation of this identity yields

Proposition 3.2: The following identities hold:

(a) av av

I for i = 1, 2, n -- = ...

ap. ap. 1 1 F

(b) av av

I +

av 0 -- =

apo apo aF F

(c) av av -

aF ay

(d) av av P ao - aF 0

In Proposition 3.2, result (b) demonstrates two competing effects of

an increase in the wage rate (price of leisure). The first effect is

32

negative because for a given level of full income, an increase in the

price of leisure reduces the choice set. The second term offsets

this effect because an increase in the wage rate increases full in-

come.

Proposition 3.3: The indirect utility function, V(P, y + 6PO)'

(a) is nondecreasing in y; (b) is non increasing in Pk for all k = { 1 , 2, ... , n} ;

(c) is nondecreasing in PO; (d) is homogeneous of degree zero in (P, y) ;

Proof:

(a) Let yO > yl. Note that the budget set with nonlabor income

yO contains the budget set with nonlabor income yl Thus the

consumer may at least choose the same bundle with nonlabor . 0 . hId . bl d b Th t' 1 . t . Income y as WIt y an POSSI y 0 etter. us, u 1 1 Y IS nondecreasing in y.

(b) Let pO > pI and let P~ = P~. Note that the budget set under

prices pI contains the budget set under prices pO Thus, the

consumer may at least choose the same commodity bundle under prices pI and possibly do better. Thus, utility is nonin­

creasing in the prices of goods. o 1 (c) Let Po > PO and let all other prices remain unchanged. Note

that the budget set under the higher wage contains the budget set under the lower wage. Thus, the consumer may at least

choose the same bundle of commodities under P~ as under p~, and utility is nondecreasing in the wage rate.

(d) The function V(P, F) is clearly homogeneous of degree zero in (P, y) since B(P, F) is homogeneous of degree zero by

Proposition 2.1. Q.E.D.

Note that these properties are due to properties of the budget set and the fact that consumers are maximizers. Few restrictions are

needed on preferences. To summarize, the consumer (weakly) prefers more nonlabor income, lower prices of commodities and a higher wage rate, and the consumer does not suffer from a "money illusion."

An alternative way of viewing consumer behavior is to assume

that the consumer selects the consumption bundle that minimizes the expenditures nece"ssary to achieve a given level of uti! ity. Formal­ly, the consumer attempts to

33

* m~n {p • X I XeC(u)}.

The solution value defines the minimum expenditures on goods and leisure required to purchase utility level u, and is termed the (in­direct) expenditure function because, at the optimum, expenditures

depend indirectly on prices and the preselected level of utility.

Definition 3.5: The (indirect) expenditure function, denoted e(P, u), is defined as

e(P, u) = min {p • X I U(X) ~ u}. XeO

Proposition 3.4: The expenditure function is (a) nondecreasing in P; (b) homogeneous of degree one in P; (c) concave in P; (d) nondecreasing in u.

Proof:

(a) By way of contradiction, let one or more elements of P

increase, so that pI ~ pO, and let the expenditure function

decrease. Then pI • xl < pO • xO, where xi is the expenditure

minimizing vector associated with price vector pi. But this im­

plies that pO • xl < pO • xO, which is a contradiction as xO

does not minimize the cost of achieving u.

(b) Follows immediately from Definition 3.5.

(c) Let (P, X) and (P', X') be two cost minimizing price-com-

modity pairs associated with u. Define pII = ep + (l-e)p', and

let X" be the cost minimizing bundle associated with P". By

definition, P • X" > p • X and P' • X" > p' • X'. Thus, we have

[ep + (1-e )P'] • X" = e(pII, u) ~ ee(P, u) + (1-e) e(P', u),

which establishes the result.

* ** * * * ** (d) Note that u > u implies C(u ) c C(u ). Hence,

* e(P, u ) - min {p • X XeO

> min {p • X - XeO

** =e(P,u·).

* * XeC(u )}

* ** Xe C( u ) }

C. E. D.

34

Thus, increases in prices do not decrease the expenditure required to achieve a given level of utility: a doubling of all prices doubles

expenditures required to achieve a given level of utility. Moreover, because the consumer substitutes among goods, average prices result in higher expenditures than variable prices. In order to achieve a higher level of utility the consumer must spend more money.

One may also establish the following useful result about the ex­

penditure function.

Proposition 3.5: The expenditure function is factorable into func­tions w(P) and feu) if and only if preferences are homothetic, i.e., e(P, u) = f(u)w{P) if and only if U(X) = F(g(X», where for ),>0, g(>'X) = >'g(X) and F' (.) > O.

Proof: We shall first establish the "if" part of the Proposition. Suppose preferences are homothetic. Then

e(P, u) :: Min {p • X IV(X) > u} X

Min {p . X Ig(X) .?. F-I(u)} X

Min { -1 ( X ) Ig( X ) I} F (u)P' > X F- 1 -1 (u) F (u)

F-1(u) Min {p • z Ig(Z) > l} [where Z X 1 -z F-I(u) F-l(u) w(P)

- feu) w(P)

as required. To establish the "only if" part of the Proposition, we first note:

Fact 3.1 (cf. Diewert, 1982):

and quasiconcave function Definition 3.5, then U(X)

If U(X) is a continuous, nondecreasing, n

of XER++, and e(p,u) is defined as in

may be recovered (up to a monotonic transformation) via the formula

U(X) = Max {u I p·X .:. e(P,u) for every P> 0 }. u

35

Given Fact 3.1, suppose e(P,u} = f(u}1/i(P} for some monotonic function f (.) • Define u*;: f (u), so that

U* (X);: f (U(X})

Hence for A> 0,

U* (AX) ;: f (U(AX})

Max {u*/ p·X ~ u*1/i(P} for every P> 0 }. u*

Max {u* / p. (AX) > u*1/J (P) for every P> O} u*

Max {AU*/A / p·X > (U*/A) I/J (P) and P> O} U*/A

W*(X}.

That is, U*(AX} = AU*(X}. But U*(X) = f(U(X», so that we have -1

U(X} = f (U*(X}), where U* is homogeneous of degree one in X. But this is the definition of a homothetic function given in Definition 2.7, thus establishing the "only if" part of the Proposition. Q.E.D

Note that since utility is an ordinal relationship, Proposition 3.5 implies that if preferences are homothetic the expenditure function

may also be written as e(P,u} = uW(p). Thus, optimal expenditures per util [e(P,u)/u) depend only on prices.

The model of utility maximization gives rise to a solution vect­or for the optimum quantities of leisure and goods.

Definition 3.6: The vector valued function that solves the problem of utility maximization subject to the affordability constraint

is called the Marshallian demand vector, and is denoted m(P, F}.

Formally, m(P, F} = argmax {U(X) Ip • X ~ F}, where the vector valued function m(P, F) has component functions mk(p, F)

for k = 0, 1, ••• n.

While the notation m(P, F) is convenient, one should keep in

mind that, using Definition (3.1),

36

m(P, F} :; m(P,y + OPo ).

Differentiate this identity to obtain:

Proposition 3.6: The following identities hold:

(a) omk

-omk for i 1, 2, ... n

o P. o P. I F and k 0, 1, 2, ....

1

omk

a Po -(b)

(c) omk

oy

(d) omk

00

1

omk

I F oP O

+

(lm k for k

(IF

omk -- P for k of 0

omk for 0, 1, _0 k

of

0, 1, •.• n;

0, 1, ••• n.

n

... n

The model of expenditure minimization also gives rise to a solution vector for the optimum quantities of leisure and goods.

Definition 3.7: The vector valued function that solves the problem

of minimizing expenditures subject to the constraint that

utility be no less than some pre-selected level is called the Hicksian, or compensated, demand vector, and is denoted hip, u). Formally, h(P, u) :; argmin {p • X I U(X) ~ u}, where the vector valued function h(P, u) has component functions hk(P, F) for k = 0, 1, •.• , n.

The following identities, which are immediate, shall be fre­quently used in our analysis:

(3.1) V{P, F) - U{m(P, F) )

(3.2) e{P, u) - P . h(P, u)

(3.3) e(P, V(P, F» - F

(3.4) V{P, e(P, u » - u

(3.5) h(P, u) - m{P, e(P, u) )

(3. 6) m(P, F) - h(P, V{P, F»

The expenditure function provides a convenient way of summariz­ing the Hicksian demand funtions.

37

Proposition 3.7: The Hicksian demands for goods and leisure, denoted hk(P, u) [k = 0, 1, ••• , n], may be obtained by Shephard's Lemma:

h (P, u) = <le(P, u) k ClP k

Proof: Let h' be the cost minimizing way of buying u given price vector P'. It follows that

g(P, u) ;: e(P, u) - P • h' < 0

and that the function g(P, u) is maximized at pl. Taking the

derivative of g(P, u) and evaluating at P' we have

<lg(P, u) , <lPk pI

= u) ,

or

<le(P, U)I = h ' = h (P', u), <lP k p' k k

which establishes the result.

pI - h I

k 0,

Q.E.D.

Similarly, we may recover the Marshallian demand equations from the indirect utility function.

Proposition 3.8: The Marshallian demand functions for goods and leisure, denoted mk(p, F) [k = 0, 1, •••• n], may be obtained by

Roy's Identity:

m (P, F) = k <lV(P, F)

<IF

_ <lV(P, F) ,

<lPk . F

Proof: Let xO maXlmlze utility given (P , F). Thus, we have u = -----U(Xo) ;: V(P, e(P, u». Holding utility fixed at u, we may

differentiate with respect to the kth price to get

o ;: :;k , + <IV <Ie

F aF <I Pk •

Noting Proposition 3.7 and identity (3.6), we have

xO m(P,F) k k

<IV

I <I Pk F <IV <IF

which establishes the result. Q.E. D.

38

Thus, the indirect utility function and the expenditure function pro­vide enough information to recover the Marshallian and Hicksian de­

mand equations, res.pectively. We now characterize properties of Hicksian and Marshallian

demand functions.

Proposition 3.9: The Hicksian demand functions satisfy

(a)

(b)

(c)

(d)

Proof:

(a)

< 0 for k = 0, 1, ••• n;

n n

k=LOYJ'k = L uJ'YJ'k j=O

P,h, and a - ~F J;

j

0, 1, 2, ••• n;

0, where

heap, u) hIP, u) for all a > O.

The concavity of the expenditure function and Theorem 2

imply ~ < O. ap2

By Proposition 2.4, this implies ahk - < O. aPk -

k

2.4

This implies

(c) To establish the first equality, apply Theorem 2.2 to hj{P, u) to obtain

n L

k=O

and then divide the equation by hj(P, u). To establish the second equality, we note that the necessary condition for expenditure minimization requires

P j au Pk axk '

39

If we let u - U(h(P, u» and differentiate with respect to Pk we get

o

1 n Substituting the necessary conditions yields ~

Pk j=O

n so that upon elasticizing we get ~ a Y O.

j=O j jk

0,

(d) Recall that the expenditure function is homogeneous of de­

gree one in P, so that by Theorem 2.2 its partial derivatives are

homogeneous of degree zero in P. As the Hicksian demand func­

tions are the partial derivatives of the expenditure function, it

follows that the Hicksian demand functions are homogeneous of

degree zero in prices. Q.E.D.

Result (a) reveals that an increase in the price of a good or

leisure will not result in an increase in the expenditure minimizing

demand for the good. Thus, Hicksian demand curves slope downward (in

price-commodity space). Result (b) reveals the symmetry of Hicksian

price effects. It is natural to interpret goods i and j as comple­

ments (substitutes) when an increase in the price of good i results

in decrease (increase) in the Hicksian demand for good j. Result (b)

indicates that if good j is a complement (substitute) for good i,

then good i is a complement (substitute) for good j. The third pro­

perty implies that sum of the Hicksian price elasticities weighted by

their respective budget shares sums to zero, while the final property

establishes that the Hicksian demand functions depend upon relative

prices, not absolute prices. In this sense, there is no money illu­

sion.

Now consider the properties of the Marshallian demand functions.

Proposition 3.10: The Marshallian demand functions

(a) are homogeneous of degree zero in (P, y) ;

n amk F Pkm k (b) satisfy ~ 13 r = 1, where r - and 13 k - --;

j =0 j j k aF ffik F

40

amk P. n 1

(c) I: fI.E .• fI • , where E - -1 j=O ) )1 1 ki a P. F mk 1

Proof:

a) If we rewrite the definition of Marshallian demand func­

tions as

m(P, F) = argmax {U(X) : (p/F) • X < I},

and note that F is homogeneous of degree one in (P, y), then the

homogeneity of the Marshallian demand equations follow at once

from the definition.

(b) If we differentiate the budget equation with respect to F

we have

This

or

n amk I: P __ = 1.

k=O k aF

may be rewritten as

n Pkm k am k I:

k=O F aF

n I: fI r = 1,

k=O k k

F --mk

1,

which establishes the result.

(c) If we differentiate the budget equation with respect to Pk we have

n amj I: P - ~, j=O j aPk F

which may be rewritten as

n amj P k I: fI -- - fI .

j=O j a Pk F mj k

But this may be rewritten as

n I: fI E = - fI ,

j=O j jk k

which establishes the result. Q.E.D.

41

Property (al demonstrates that a doubling of all prices and nonlabor income does not change Marshallian demands. Thus, consumers are free from money illusion. The second property indicates that the budget share weighted sum of income elasticities is equal to one, while the final property indicates that the budget share weighted sum of cross

price elasticities is equal to the own budget share. Note that the first property rests on the homogeneity of degree zero of the budget

set in (P, y), whereas the second and third properties rely on the assumption of nonsatiation and the linearity of the budget con­straint. Of course, one could redefine the results of Proposition

3.10 in terms of nonlabor income by using Proposition 3.6. The following proposition summarizes the well-known Slutsky

Equation.

Proposition 3.11: The total effect of a price change may be decom­

posed into substitution and full income effects:

= m,

F J

Proof: Note that

hk ( P, u l = mk ( P , e ( P, u».

Differentiate this identity with respect to P j to obtain

= + F

Rearranging this and using Shephard's lemma yields

Clmk Clhk Clmk = - --m , ,

ClP, F ClPj ClF J J

which establishes the result. Q.E.D.

The first term of the Slutsky equation is the substitution ef­fect, it represents the change in consumption of the kth good given a

change in price for the jth good while holding utility constant. But price changes will, by Proposition 3.3, alter the consumer's util­

ity. The second component of the Slutsky equation captures this ef­

fect.

42

Note that by Proposition 3.6, we may obtain an alternative form of the Slutsky Equation.

Proposition 3.12: The total effect of a price change may be decom­posed into substitution and nonlabor income effects:

for k j

0, 1, 1, 2,

!l

!l

for k = 0, 1, ••• n.

Proof: Substitute the identities in Proposition 3.6 into the full income version of the Slutsky equation (Proposition 3.11) and

rearrange. Q.E.D.

CHAPTER FOUR: CONSUMER BEHAVIOR IN THE PRESENCE OF AN INCOME TAX

We now consider the implications of allowing taxes to be a func­tion of the income earned by individuals. If the government wishes to. redistribute income by its tax policy, government may decide to use a nonlinear tax, taxing wealthy individuals more than poorer in­dividuals. This leads to the question: how should government define how wealthy an individual is? Perhaps ideally government could base the income tax on the individual's full income, F. This would allow

government to base taxes on the market spending potential of each in­dividual, and to construct a tax bill that really redistributes util­ity.

In practice, most income taxes are based on the observed -income of the individual, y + PO(6 - XO). This has the effect of allowing the consumption of leisure to reduce the income tax bill. To make. this argument explicit, let the individual's tax bill, Z, be a given

by

where once again, F = Y + 6PO is full income. The function T(') will determine how the income tax bill will be distributed across income

classes. We now place some limitations on the tax function to simplify

our analysis. The first restriction that we shall make is that T(O)

< O. This assumption admits the negative income tax that has re-

ceived considerable treatment elsewhere, see Hausman (1981). Second, we shall assume that T(E) ~ E, where E = Y + PO(6-XO) is taxable in­come. Thus, government will not impose a tax bill greater than the individual's taxable income. Finally, we assume that the marginal tax rate is less than one, so that the individual will get to keep at least some fraction of any aoditional income earned.

Given these restrictions, we may now define the consumer's choice

set in the presence of an income tax.

Definition 4.1: In the presence of a tax function, T, the budget set, denoted B(P, F: T), is defined as

B(P, F: T) = (xeQ I P • X + T(F - POXO) ~ F}.

Thus, the consumer must allocate his full income among the purchase of goods, leisure, .and the payment of the tax bill.

44

Proposition 4.1: If T(·) is convex, then B(P, Fi T) is convex. ~ ~

Proof: Let X*,X £ B(P, Fi T). We claim X = 9X* + (1-9)X £ B(P, F: T)

for 9£ [0, 11. Now P • X* + T(F - POXO) ~ F and P • X + A ~

T(F - PoX o ) ~ F implies P • X + 9T(F - POXO) + (1-9) T(F - POX O)

< F. Let Z* = F - POXO and

9 T ( Z *) + (1-9) T ( Z) > ~

Hence, P • X + T(F -

Z - F - POXO. Convexity of T(·) im­

T(Z), where ~ = 9Z* + (1-9)Z = F -

POX o ) < F. That is, ie £ B(P, Fi T).

Q.E.D.

Proposition 4.2: If T(·) is convex and T(O) < 0, then B(9P, 9F: T) £ B(P, Fi T) for 9 > 1.

Proof: Suppose X*£ B(9P, 9F: T) and let 9 > 1. We must show X*£

B(P, F: T). Now X*£ B(9P, 9F: T) implies

which implies

P • X* + ~- T (9 F - 9 P X*) < F. 9 0 0 -

Hence it is sufficient to show

~ T(BF - BP X*) > T(F - P X*). B 00 - 00

By convexity of T(·),

~ T(BF - BP X*) + (1 - ~) T(O) BOO B

> T (~( B F - B P X*) + (l - ~)( 0) ) BOO B

T (~ (B F - B P X*». BOO .

As T ( 0) < 0, we ha ve

~ T(BF - BPOXO) ~ T(F - POXO). Q.E.D.

Note that under a proportional tax (i.e., when T(·) is linear), B(BP,

BFi T) = B(P, F: T): otherwise, a proportional change in all prices

and full income will generally shrink the budget set, and thus, lead

to behavioral effects. As we shall see, this shrinking of the budget

set shall alter a number of the neoclassical properties of consumer

behavior.

Having characterized the budget set in the presence of an income

tax, we are now ready to examine the implications of taxation on con-

45

sumer behavior. Before doing so, we shall make two additional as­

sumptions. First, we shall assume that the tax function, T(·), is indeed convex. For convenience, we also assume that T(·) is twice continuously differentiable: if the tax function were piecewise linear, the derivatives would exist almost everywhere.

The assumption that the tax function is convex implies that we restrict the tax system to be proportional or progressive. The term "progressive" is ambiguous, having at least two well accepted defini­

tions. One definition, which is a very common one used in the literature, requires that the average tax rate be an in-creasing function of income. If we let E denote income, this re­

quires that

T'(E) - T(E) > 0, E

or that the marginal tax rate exceeds the average tax rate. Note that if we let EO be a deduction given to the taxpayer, then the tax system

T( Z) ° < T < 1 and E > EO otherwise,

is a progressive tax system. The second definition, proposed by Blum and Kalven (1963), is more restrictive: it requires that the margin­

al tax rate increases with income. Hence, the above tax system is progressive under the first definition, but not the second defini­

tion. In our analysis we shall adopt the following simple notion of

"progressive."

Definition 4.2: A tax function T(·) is said to be progressive if

it is strictly convex.

We now examine the effects of the tax system on consumer behav­

ior. The objective of the consumer is to select the commodity bundle (i.e., leisure - goods pair) that maximizes utility subject to the

constraint that the bundle be affordable. Of course, the presence of an income tax affects what is affordable. Formally, the consumer at­

tempts to

mix {U(X) I X E B(P, F: T>}

Once again, the fact that B(P, F: T) is a nonempty compact and convex

set ensures that a solution to the consumer's problem exists. The

46

solution value now depends on the level of prices, full income, and the structure of the tax function.

Definition 4.3: The indirect utility function in the presence of an income tax, denoted V(P, F: T) is defined by

V(P, F: T) ;: max {U(X) X

P • X + T(F - P X ) < F}. o 0

One must bear in mind that F ;: Y + 6PO' so that

V(P, F: T) ;: V(P, y + 6PO: T).

As a consequence, the following proposition holds:

Proposition 4.3: The following identities hold in the presence of an income tax:

(a) av av

I for i 1, 2, n -- - ... aPi aP i F

(b) av av I + av 6 apO - aPo F aF

(c) av av -ay aF

(d) av = av P ar aF 0

Proposition 4.4: In the presence of a progressive income tax, the indirect utility function, V(P, 6PO + y: T),

Proof:

(a) is nondecreasing in y:

(b) is nonincreasing in Pk, for all k = {I, 2, ••• , n}: (c) is nondecreasing in PO: (d) is not homogeneous of degree zero in (P, y). In

particular,

veep, eF: T) ~ V(P, F: T) for e > 1.

(a) It is sufficient to show that increases in y do not shrink the choice set: ie, for yl > yO,

B(P, yO + .spo: T) £. B(P, yl + 6PO: T).

Now B(P, y + 6PO: T) = {XgO Ip • X + T(y + PO(o-XO» ~ y + opo}.

Since the marginal tax rate is less than unity, increases in y must expand the opportunity set.

Parts (b) and (c) may be similarly deduced.

47

(d) . By Proposition 4.2, B(6P, SF; T) 2. B(P, F;T) when S > 1.

Thus, proportionate increases in all prices and nonlabor

income shrink the budget set, thus resulting in lower utility. Q.E.D.

The presence of progressive income tax thus alters a fundamental

neoclassical result, namely, the consumer's indifference to the price-income level. Indeed, it is this result that has recently led to the indexation of tax codes -- a topic we shall address in Chapter Ten. However, let us point out an alternative manner in which the homogeneity of degree zero of the function may be maintained.

Proposition 4.5: If the income tax code is proportional, so that

T(F - POXO) = T[F - POXOl for some scalar T, 0 < T < 1, then

V(SP, SF; T) _ V(P, F; T) for all S > O.

Proof: As the discussion following proposition 4.2 reveals, B(SP, SF: T) = B(P, F; T) when the tax function is proportional. Hence, proportionate changes in all prices and nonlabor income

do not alter the choice set, nor the level of utility. Q.E.D.

A dual model of consumer behavior is to assume that the consumer selects the consumption bundle that minimizes the full in­come required to purchase a fixed utility level. Formally, this amounts to

mkn {F I X£B(P, F: T) and X£C*(u)}.

The solution thus defines the min~mum expenditures on leisure, goods, and taxes required to purchase utility level u, and is termed the

full income compensation function.

Definition 4.4: The full income compensation function is defined as

p(P, u; T) = min {F I P • X + T(F - P X ) < F and U(X) > u}. X 0 0 - -

Proposition 4.6: The full income compensation function is (a)

(b)

(c)

(d)

nondecreasing in Pk for k = 1, 2, •••• n:

nondecreasing in Po; nondecreasing in u; not homogeneous of degree one in P (unless T(') proportion-

a1.) In particular, p (SP, u: T) > Sp (P, u; T) for S > 1

when T is progressive.

Proof:

(a)

(b)

(c)

48

1 _ ° 1 ° Let Po = Po and P > P. Then

B(pl, F; T) ~ B(PO, F; T), so that

lI(pl, u; T) _ min {F XEB(pl, F; T) and U(X) > u} X

> min {F I XEB(PO, F; T) and U(X) > u} X -

;: II(PO, u: T).

1 ° 1 ° Let Po > Po and Pk = Pk[k = 1, 2, ••• nl. For fixed F,

B(pl, F: T) ~ B(PO, F: T),

so that

lI(pl, u: T) - min {F X

XEB(pl, F: T) and U(X) > u}

> mkn {F I XEB(pO, F: T) and U(X) > u}

;: II(PO, u: T).

Suppose ul > uO. Then c*(u l )

> min {F X

XE B( P,

;: II(P, uO; T).

° ~ C*(u ) so that

r': or) and XEc*(ul)}

(d) For S > 1, II ( S P, u; T) - mi n {F X

XEB(SP, F; T) and XEC*(U>}

XEB(SP, SF; T) and XEC*(U)} > min {F X

> min {SF I XEB(P, F; T) and XEC*(U>} - X

;: SII(P, u; T). Q.E.D.

Once again, the presence of a progressive income tax alters a funda­

mental neoclassical result: a doubling of all prices more than dou­

bles the full income necessary to obtain a given utility. This is

because the increase in the price level increases not only the re­

quired expenditures on goods and leisure, but in the presence of pro­

gressive income tax, it also increases the real tax bill.

Interestingly, since F ;: y + oPO' we may use the full income

compensation function to define a nonlabor income compensation func­

tion:

nefini.tion 4.5: The nonlabor (or nonwage) income compensation

function, denoted y(P, Ui T), is defined as

y(P, u; T) ;: II (P, Ui T) - 0 PO.

49

The next proposition follows immediately from Definition 4.5.

Proposition 4.7: The following identities hold in the presence of an

income tax:

(a) 3y(P, u: T) 3J.l(P, u: T) -

3P k 3P k

(b) ay(p, u: T) aJ.l(p, u; T) - o : -ap O aPO

(c) a y( P, u: T) aJ.l(p, u: T) -

au au

The properties of the nonlabor income compensation function are

summarized in the following proposition:

Proposition 4.8: The nonlabor income compensation function is

(a) nondecreasing in Pk for k = 1, 2, •.• n;

(b) nonincreasing in PO;

(c) nondecreasing in u;

(d) not homogeneous of degree one in P (unless T(·) is propor­

tional). In particular, if T is progressive,

y(9P, u; T) ~ 9y(P, u: T) .for 9 > 1.

Proof: (a), (c), and (d) follow from Propositions 4.6 and 4.7. (b)

may be proved directly along the lines of Proposition 4.6. Q.E.D

The models of utility maximization and full income minimization

in the presence of an income tax give rise to two alternative types

of solution vectors for the optimum quantities of leisure and goods.

Definition 4.6: The vector valued function that maximizes utility

subject to the affordability constraint in the

presence of an income tax is denoted M(P, F: T). Formally,

M(P, F; T) - argmax {U(X) I P . X + T(F - POXo) < F},

where the vector valued function ~("o , p. 'J') hil.S component

functions Mk(P, F; T} for k = 0, 1, ... n.

While the notation M(P, F; T} is convenient, just as in the last

chapter, full income is a function of the wage rate and nonlabor in-

come. Thus

M(P, F; T} - M(P, y + oPo; T).

50

Proposition 4.9: The following identities hold in the presence of an income tax:

(a) aMk aMk for i 1, 2, ... n -- -api aP i F and k 0, 1, 2, ... n

(b) ClMk aMk aMk aPO - apO

+ _15

F aF

(c) aM k ClM k

for k 0, 1, 2, - .. . n • ely elF

(d) elM k elMk

for k 0, 1, 2, aT - --P = ... n elF 0

Definition 4.7: The vector valued function that solves the problem of minimizing the full income required to purchase a fixed

level of utility in the presence of an income tax is denoted H(P, u; T). Formally,

H(P, u; T) = argmin {F I P • X + T(F - POXO) ~ F and U(X) > u},

(4.1)

(4.2)

(4.3)

(4.4) (4.5)

(4.6)

where the vector valued function H(P, u; T) has component

functions Hk(P, u; T) for k = 0, 1, ... n.

The following identities are immediate:

V(P, F; T) - U(M(P, F; T» lI(P, u; T) - P . H(P, u; T) + T[lI (P, u; T) - POHO(P, lI(P, V(P, F· , T) ; T) - F

V(P, 1I (P, u; T) ; T) - u H(P, u; T) - M(P, lI(P, u; T) T)

M(P, F; T) - H(P, V(P, F; T) T)

u; T) ]

The full incoTTl"! com.pAnf;!l.tion function nrovides a convenient way

of obtaining Hicksian demands in the presence of an income tax:

Proposition 4.10: In the presence of an income tax the Hicksian demands for goods and leisure, denoted Hk(P, u; T) [k = 0, 1,

••• n] may be obtained from the full income compensation

function as follows:

H (P, Ui T) = d}J(P, Ui T) [l-T ' (·)] for k k elPk

H (P, u; T) o

1, 2, ••• n

51

Proof: Let xO be the full income minimizing way of buying utility level u given prices, pO, and tax function, T. Let F(P, XO) solve the implicit function

Application of the implicit function theorem reveals

Cl F(P, XO)

ClP k

aF(p, XO) and

ClPO

for k 1, 2, ••• n

Now since ~(P, u; T) is the full income minimizing way of buying utility level u = U(XO), we have

g(P, u) = ~(P, u; T) - F(P, Xo) < 0.

° By construction, ~(P ,

a~ (P, u; T) a Pk

so that Cl~(P, u; a Po

T) xO

° a~ (P, u; T) xo

and k aPk [l-T' (. )] •

Thus,

0,

Q.E.D.

Similarly, in the presence of an income tax we may recover the Marshallian demand functions from the indirect utility function.

Proposition 4.11: In the presence of an income tax, the Marshallian demands for goods and leisure, denoted Mk(P, F; T) [k = 0, 1,

•.• n] may be obtained by the following formulas:

M (P, F; T)

° av(p, F;

ClV(P, F; a Po

av(p, F; ClF

T)

T) , F

T)

M (P, F; T) k av(p, F; T)

ClF

[1 - T'(')] for k 1, 2, ••• n.

52

Proof: For a given u we have V(P, jJ(P, u: T) ;T) - u, so that

av

I

+ av ajJ 0 for all k.

a Pk F aF a Pk

Now for k = 0 we have, using the previous theorem and the

identity relating Hicksian and Marshallian demands,

H = M = o 0

When k * 0, we have

ajJ Hk -,--.,.~-:--

aP k [1 - T ' (·)],

av

av aF

so that M k

a P k

av aF

[1 - T I (. )] for k 1, 2, ••• n.

Q.E.D.

Proposition 4.12: In the presence of an income tax, the total effect

of a, price change may be decomposed into substitution and full

income effects as follows:

aMk

IF

aHk aM - M k for k 0, 1,

apo ... , n

a Po 0 aF

aMk

IF =

aH k M. aM k for j 1, 2, ... , n J and k 0, 1, n •

ap. ap. aF ... ,

J J [1 - T' (. )]

Proof: Note that Hk(P, u; T) - Mk(P, jJ(P, u; T);T). Differentiating

this identity with respect to P. yields J

Using the tax analogue of Shephard's Lemma and the identity relating Marshallian and Hicksian Demands, we may re-write this as

F

53

and

M. J

[l - T'(.)] 3F

for j = 1, 2, ••• , nand k = 0, 1, ••• , n.

Q.E.D.

Note that, using Proposition 4.9, we may obtain an alternative form

of the Slutsky Equation.

Proposition 4.13: In the presence of an income tax, the total effect

of a price change may be decomposed into substitution and

nonlabor income effects

aMk aHk M. aM k J a P j ap. [1 - T' ( • ) ] ay

J

aM k aH k (M <5 )

aM k - -a Po apo 0 ay

for all k = 0, 1, 2, . . . , n and j 1, 2, ... , n •

The presence of an income tax alters fundamentally many of the

neoclassical properties of consumer behavior. We have already demon­

strated that modifications to Roy's identity and Shephard's Lemma are

required. More importantly, however, we demonstrated that in the

presence of an income tax, the indirect utility function will in gen­

eral not be homogeneous of degree zero and the full income compensa­

tion function is not homogeneous of degree one. Further implications

of an income tax for the Hicksian and Marshallian demands are sum­

marized in the following two propositions:

Proposition 4.14: In the presence of an income tax, the Hicksian

demands satisfy the following properties:

(a) H(6P, u: T) = H(P, u: T) for 6 > 0 if and only if the tax

function is homogeneous of degree one.

aH. aH. (b) ~ _J for all i, j 1, 2, ... n

a P. ap. J 1

aHa aH. --* _J for all j 1, 2, . . . n • ap. aPO

J

Proof: (a) Follows from the fact that B(P, F: T) is homogeneous of

degree zero in (P, F) if and only if the tax function is

homogeneous of degree one.

54

(b) using the tax analogue of Shephard's Lemma,

and

Hence,

3H. but _J

dP O

Thus,

However,

so that

[1

[1

aHo --* ap.

H j

J

aH. _J ap.

1

H (.) o

H (.) j

-T'(·)]

- T'(·)]

dH. J

ap o·

3\1

d Po

[l_T,(.)]d\l(.) forj dP.

1,2, ••. n. J

d 2\1 d\l d\l d HO + [- T"(·)] [-- - P -- - H ]

a P oa P j a P. d Po Od Po 0 J

aHo H.T"(·) Po aHo -- + J d P. a po·

J [1 - T'(·)]

a\l a\l [l-T'(·)] --- + [- T"(·) -]

ap. a P. a P .ap. 1 J

a 2\1

J 1

8. H . T" (. ) 1 J

which is symmetric in i, j 1,2, .•• n.

Hence, aH. _J 3P.

1

aH. 1

ap. J

for i, j 1,2, .•• n. Q.E.D.

Because the tax system treats leisure as deductible, the sym­

metry of cross price effects fails. In particular, because the tax

system distorts the price of leisure relative to the price of goods,

the symmetry fails between leisure and other goods. But when looking

only at nondeductible goods, symmetry is maintained. Thus, if good i

is a substitute (complement) for good j (i, j = 1, 2, ••. , n), then

good j is a substitute (complement) for good i. However, the fact

that leisure is a substitute (complement) for good j need not imply

good j is a substitute (complement) for leisure.

Proposition 4.15: In the presence of an income tax, the Marshallian

demands satisfy the following Properties:

(a) M(ep, eF: T) ;: M(P, F: T) for all e > 0 if and only if the

tax function is homogeneous of degree one.

(b)

(c)

(d)

Proof:

55

n Ef:lr +(I-T'(o)]f:lr

k=1 k k 0 0 (1 - T'(o)]

where

n

r j

aM j 1"

elF M. J

and 13. -J

Ef:le: +(I-T'(o)]f:le: k=1 k kj 0 OJ

elMk I P. - J = -- -. d P j F Mk

where e: kj

n

P.M. J J F

- 13 for all j '" 0, j

Ef:le: +(I-T'(o)]f:le: _[l-T'(o)]f:l. k=1 k kO 0 00 0

(a) Follows from the properties of B(P, F; T).

(b) From the budget constraint,

n E P M (P, F; T) + P M (P, F; T) + T[F - P M (P, F; T)] - F.

k=l k k 0 0 0 0

Differentiate this identity with respect to F to get

n aMk aMO EP -+[1-T'(o)]P [1-T'(o)],

k=1 k elF 0 a F

which imp 1 ies

n E 13 r + (l - T'(o» 13 r [1 - T'(o)].

k=l k k 0 0

(c) Differentiate the budget constraint identity with respect to

P. (j J

M + j

Thus,

n E

k=1

or

n

'" 0)

n E P

k=l

to obtain

elM k elMO --+ [1 - T' (0)] P

oP. k elP. 0 J J

+ [1 - T'(o)] --F

Ef:le: +(I-T'(o»f:le: =-13 k=l k kj 0 OJ j

o.

P. -P.M. J J J -- --

MO F

(d) Differentiate the budget constraint identity with respect to

Po to get

n E P

k=1 k oPo + [1 - T'(o)] P

F 0 oPO F

+[l-T'(o)]M o

00

56

Rearranging we have

n elMO r P

k=l k elPo + [l-T'C-)] P =-M [l-T'C·)].

o el Po 0 F

But this is just

or n

F

rile: +[I-T'C-)]IIe: =-[I-T'C·)]1I 0 • k=l k kO 0 00

Q.E_D.

The nonlinearity of the budget equation that results from income

taxation and the tax deductibility of leisure alters many of the re­sults that are derived from classical theory. In particular, the

Hicksian demands are not homogeneous of degree zero in money prices; the matrix of first derivatives of the Hicksian demand functions is not symmetric; and the Marshallian demand functions are not homogene­ous of degree zero in prices and nonlabor income.

PART III

ECONOMIC INDICES IN THE ABSENCE OF TAXATION

CHAPTER FIVE: THE KONUS ~ND FRISCH EXPENOITURE-BASEO COST OF LIVING INDICES

In order to solidify important concepts, we shall first present

some index concepts in the absence of an income tax. The key to con­

structing an expenditure-based cost of living index is the expendi­

ture function examined in Chapter Three, namely

e(P, u);: min {p. X I U(X) > u}. X -

This function defines the minimum expenditures on goods and leisure

necessary to purchase the standard of living corresponding to u.

The Konus (1939) expenditure-based cost of living index compares

the expenditures on goods necessary to maintain the living standard u

under price regimes pO and pl. When u corresponds to the utility in

the base period, the Konus expenditure-based cost of living index de­

fines the factor by which base period expenditures must be adjusted

after movement in prices in order to leave the consumer as well off

after the price change as before.

Definition 5.1: The Konus expenditure-based cost of living index is

defined as

e(pl, u) o e(P , u)

where pO is a vector of base period prices and pI is a vector of

current prices.

The Konus index is often called the "true" cost of living index be­

cause it compares the expenditures in two periods necessary to

achieve a given level of utility. As we shall demonstrate, however,

one must be careful in interpreting the Konus index as "the" true in­

dex. Various formulations of the cost of living question will produce

various "true" indices.

Let us first address the properties of the Konus index.

Proposition 5.1: The Konus index, K(PO, pI, u), satisfies the Eich­

horn-Voeller axioms, namely:

(a) Homogeneity: o 1 for hE: R++, K(P , hP , u)

o 1 hK(P , P , u).

59

(b) Monotonicity: K(PO, 01 p , u) > K(PO, pI, u) if :p1 > pI:

K(PO, pI, u) < K(PO, pI, u) if pO > pO.

(c) Dimensionality: For Ae:R++, K(APO , A pI, u) = K(PO, pI, u) •

(d) Identity: K(PO, pO, u) = 1.

Proof: (a) through (c) trivially obtain from Proposition 3.4 and (d) from the definition of the Konus cost of living index.

Q.E.D.

Thus, the Konus index doubles when all prices double: it does not de­

crease when one or more prices increase: it is invariant to the unit of account: and it equals one when prices do not change over time.

The Konus index depends on preferences: it allows for substitu­tions among goods and leisure as relative prices change. It is of

interest to compare the Konus index with two widely used indices that do not depend on preferences.

Definition 5.2: The Laspeyres expenditure-based price index is de-fined as

L(pO , pI, xO ) pI . xO

-pO ° ' . X

where pO and XO are vectors of base period prices and quantities and pI is the vector of current prices.

Definition 5.3: The Paasche expenditure based price index is defined

as

where pI and xl are vectors of current period prices and quantities and pO is the vector of base period prices.

The Laspeyres and Paasche indices are widely used as measures of the

cost of living. Strictly speaking, they are not cost of living in­

dices but rather price indices. Both possess the advantage of being relatively easy to compute. This simplicity is not without cost, as the next few propositions indicate.

60

Proof: Let xi denote the expenditure minimizing way of buying u O

given price vector pi. Then

1 uO) _ e(P , < ° uO} e(P ,

since xO is not necessarily the expenditure minimizing bundle

given price pl. Q.E;D.

Proposition 5.2 reveals the well-known result that the Laspeyres in-

dex is "biased upwards". This result may be illustrated graphically

as follows. i .

Let X denote the expenditure minimizing way of buying o i o 0 ° ° n 00 u given P • Thus e(P , u ) = P • X - r P X , and so on for

j=O j j other superscripts.

The traditional diagram illustrating Proposition 5.2 is present­

ed in Figure 5.1. The initial bundle xO minimizes the expenditure

necessary to achieve utility level uO, which is e(po, uO). Given a

price change from pO to pI, the new bundle xl minimizes the expendi­

ture necessary to achieve utility level uO, which is e(pl, uO). How­

ever, pi • xO passes through the initial bundle, and therefore . th b' . f ° 1 . h 19nores e su stltutlon rom X to X as prlces c ange.

The diagram in Figure 5.1 does more to show why pl. x O > pl. xl

than to provide a direct comparison of the Laspeyres and Konus in­

dices. In order to directly compare these indices graphically, con-

° ° ° ° 110 sider the case in which P = (PO' PI' ••• , Pn ) and P = (PO' P l , ... ,

pO) where pI > ° This is the n ' ° PO' case

sure increases. In this ° case, e(P ,

1 ° 1 1 lIn ° 1 e(P , u } = P • X = P X + r P X • ° ° j=l j j

in

° u )

which only the price of lei-

° ° n ° ° - P . X = r P X and j=O j j

Viewing p~, ••• , P~ and xO as given constants, define the func-

° n 00 ° 1 ° 1 ° ° tion f(P } = P X + r P X. Thus, L(P , P , X } = f(P }/P • X • ° ° ° j=l j j ° The graph of f(P O} is shown in Figure 5.2. For any value of PO'

f(PO} gives the value of the numerator in the Laspeyres price index.

n ° ° Note also that the intercept of the function is r P X • j=l j j

In Figure

though it holds

unlike f(P O} it

5.2 the expenditure function is also graphed. Al-

all prices constant

is nonlinear in Po ° (at P j ) except that of leisure,

since it allows the X.'s to opti­J

Xl I

X?

61

I I

___ L __ I I I I I

X~ X~ Xo

FIGURE 5.1

THE SUBSTITUTION BIAS IN COMMODITY SPACE

o

62

FIGURE 5.2

pi o

THE SUBSTITUTION BIAS IN PRICE-EXPENDITURE SPACE

63

mally adjust to changes in PO' Moreover, note that at Pg, f(Pg)

° ° e(P , u ).

We may now graphically compare the Konus and Laspeyres indices. The slope of the line labeled K(PO, pl, uO) gives the value of the

Konus price index. To see this, obtain from Pg the value e(pO, uO)

on the vertical axis and then use the 45° reference line to map this value onto the horizontal axis. Thus, the slope of the l'ine through OA is e(pl, uO)/e(pO, uO) = K(PO, pl, uO). Similarly, the slope of

the line labeled L(PO, pI, XO) gives the value of the Laspeyres price index, since the slope of the line through OB is pl. xO/pO. xO = L(PO, pl, XO). Figure 5.2 thus provides a mechanism with which to

directly compare the Konus and Laspeyres price indices. Moreover, unlike Figure 5.1, Figure 5.2 enables one to directly observe the role that the concavity of the expenditure function plays in causing the Laspeyres index to be "biased upwards". This analysis suggests:

Proposition 5.3: If the expenditure function is of the form

n e(P, u) = u r P a then

i=O i i'

K(PO, pl, uO) = ° 1 ° L(P , P , X ). Proof: Application of Shephard's Lemma reveals

h (P U O) =_ . , 0,1, ••• ,n, 1

so that Q. E.D.

Thus, the Laspeyres index is greater or equal to the Konus index, and

is equal to the Konus index if there are no substitution effects. Note that if prices move proportionately then the Konus and Laspeyres

indices also coincide. This result may be verified by noting that

Laspeyres index, like the Konus index, is homogeneous of degree one in current prices. In the presence of relative price changes and substitution effects, however, the Laspeyres index is an upper bound for the Konus index for base period utility.

We next demonstrate that the Paasche index provides a lower

bound for an appropriately defined Konus index.

64

Proof: Let xi be the expenditure minimizing way of buying u l given

price vector pi. Then

since xl is not necessarily the expenditure minimizer given pO. Q.E.D.

Propositions 5.2 and 5.4 together might lead one to believe that

the Konus index is always bounded between the Laspeyres and Paasche

indices. This is not always the case. To see this, note that

and

<

e(pl, u O)

o uo) e(P ,

lUI) e(P ,

o ul) e(P ,

<

but the two Konus indices depend, in general on two different utility

levels. Can the Konus index ever be independent of the level of

utility?

Proposition 5.5: K(PO, pI, u) is independent of u if and only if

preferences are homothetic.

Proof: ( ... ) 1 u l ) I u 2 ) e(P , e(P , 0 u l )

-0 e(P , e(P , u 2 )

I u ,

e(·) is factorable into functions of price alone and utility

alone, Le., e(P, u) ;: h(u)1/I(P). By Proposition 3.5, this is

both necessary and sufficient for homotheticity.

(+) Again using Proposition 3.5,

;: h(u)1/I(P), which in turn implies

homotheticity implies I

K(PO,pl,u);:~. 1/1 (pO)

Propositions 5.2, 5.4, and 5.5 thus imply:

Proposition 5.6: If preferences are homothetic, then

e(P, u)

Q.E.D.

010 0 I ° I I L(P , P , X ) ~ K(P , P , u) ~ n(P , P , X ) for all u.

65

When preferences are homothetic, the Konus index is bounded from above by the Laspeyres index and bounded from below by the Paasche

index. When preferences do not allow substitutions among commodities

and leisure, the upper and lower bounds converge to the Konus index value. Thus, we have:

Proposition 5.7: If the expenditure function is of the form

n 010 0 1 e(P, u) = u r P a , then L{P , P , X ) = K{P , P , u)

i=O i i

An alternative expenditure-based cost of living index that also depends on preferences is the Frisch index. The ,Frisch expenditure­based cost of living index compares the marginal expenditures requir­ed to buy an additional unit of utility under alternative price reg­imes.

Definition 5.4: The Frisch expenditure-based cost of living index is

defined as

ae(pl, u)/au o ae{P , u)/au

Note that the conditions required for the existence of the Frisch in­dex (namely differentiability) are stronger than those required for the existence of the Konus index.

Proposition 5.8: The Frisch cost of living index, 8(PO, pl, u), sat­isfies a modified generalization of the Eichhorn-Voeller axioms. specifically:

(a) Homogeneity: ( 0 'pl ,) (0 1 ) for AE R++, 8 P , 1\ U = A8 P , P , u • (b) Conditional Monotonicity: If each good is normal (ahi/au > 0) , then

8 (pO, '1 P , u)

8 (pO, pl, u)

(c) Dimensionality: for

(d) Identity: 8(PO, pO,

pl, u) if pl > pl:

pl, u) if pO > pO.

0 AER++,!:J.{AP, Apl, u) !:J. (pO, pl, u) •

u) = 1.

Proof: We begin by proving (a) and (c). By Proposition 3.4 we have

e(AP, u) = Ae{P, u). Differentiating this identity with respect to u obtains ae(AP, u)/au = Aae(P, u)/au, from which (a) and (c)

above easily follow. Consider now (b) above, and note that by

66

Shevhard's Lemma 3[ae(P, u)/au]/aPi = ahi(p, u)/au where hi(P, u) is the Hicksian demand function for good i, and hence, ah i ('P, u)/au is positive (negative) as good i is normal (inferior). Therefore, if all goods are normal, ae(P, u)/au is increasing in P, and hence (b) directly obtains. Q.E.Q.

Remark: Normality of all goods is a necessary and sufficient con-dition for (b) to hold for all price changes, but for some price changes normality may not be necessary for (b) to hold.

The next issue to be addressed in this section is the relative magnitude of the Konus and Frisch indices. In terms of policy ana­lysis, compensation based on an index that is slightly less than another index can result in expenditure differences of billions of dollars. The following proposition establishes the relative magni­

tudes of the two measures of price change.

Proposition 5.9: 6(PO, pl,u) ~ K(PO, pI, u) as aK(pO, pI, u)/au ~ 0.

Proof: Differentiating K(') with respect to u we get

aK(pO, pI, u)/au ~ ° as [e(pO, u)] [3e(pl, u)/3u] ~ [e(pl, u)] [3e(pO, u)/3u].

Noting that ae(P, u)/3u is positive for P€R~; and dividing both

sides by [ae(pO, u)/3u] [e(pO, u)] obtains the form of the pro-position. Q.E.D.

Thus, whether the value of the Frisch cost of living index is

greater or less than the Konus index depends on whether the Konus in­dex is increasing or decreasing in base utility.

On economic grounds, both the Konus and Frisch cost of living indices have some intuitive appeal and provide alternative ways of interpreting price changes. In light of the previous proposition it

is interesting to determine when the two indices provide equivalent measures of price changes. The answer is given in the following pro­position.

Proposition 5.10: K(PO, pI, u) = 6(PO, pI, u) for all pO, pl€R~~1 if

and only if preferences are homothetic.

67

Proof: (+) By Proposition 3.5, homotheticity implies e(P, u) _

h(u)~(P) in which case we easily find K(PO, pI, u) =

b(PO, pI, u) _ ~(pl) ~(pO)

(+) K(PO, pI, u) = 6(PO, pI, u) for all pO, plER~:l implies

ae(pO, u)/au _ ae(pl, u)/au for all pO, plERn+l. That is, for e(pO, u) e(pl, u) ++

PERn+l , de(P, u)/au = A(U) for some function A(U). Rewriting ++ e(P,u)

this as ae(P, u) = A(U)aU and integrating we get log'[e(p, u)] :: e(P, u)

log[h(u)] + log[~(P)], or e(P, u) = h(u)~(P). By Proposition

3.5, this is sufficient for homotheticity. Q.E.D.

Thus, homotheticity is necessary and sufficient for the Frisch and

Konus measures of price change to be equivalent for all price

changes. Homotheticity eliminates the issue of whether to use the

Frisch or Konus index. Moreover, under homotheticity, the Konus in­

dex (or equivalently the Frisch index) has two interpretations: the

ratio of total expenditures necessary to obtain a given level of

utility and the ratio of marginal expenditures necessary to do so.

Finally, note that if preferences are homothetic, the Frisch index is

unconditionally monotonic in prices.

We demonstrated that the Konus index value generally depends on

the base utility level. Similarly, the Frisch index also generally

depends on the specified level of base utility. The following pro­

position establishes that the necessary and sufficient conditions for

the Frisch cost of living index to be independent of base utility are

less restrictive than the conditions required for the Konus index to

be independent of base utility.

Proposition 5.11: 6(PO, pI, u) is independent of u if and only if the

expenditure function is of the form

e(P, u) = alP) + g(u)~(P).

Proof: Sufficiency obtains directly from the definition, so we pro­

ceed with necessity. Suppose

ae(pl, ul)/au

<le(pO, ul )/au

68

Cle(pl, u 2 )/ClU

Cle(pO, u 2 )/au

for all pO, pIER~:l and ul, U2ER. Then Cle(P, u)/au is factor­

able into functions.f(u) and 1jJ(P), Le., ae(P, u)/au:: f(u)1jJ(P)

for some feu) and 1jJ(P). Therefore, Cle(P, u) :: f(u)1jJ(P)Clu which,

upon integration, yields e(P, u) :: 1jJ(P)g(u) + a(P). Q.E.D.

CHAPTER SIX: WAGE AND NONWAGE INCOME -BASED l1EASURES OF THE COST OF LIVING IN THE ABSENCE OF AN INCOME TAX

In this section we examine income-based measures of the cost of

living in the absence of an income tax. Recall that, from the budget

constraint in the absence of an income tax,

P • x, so that

Definition 6.1: The non1abor income function, is defined as

yep, u ) ;: min X

{y I y = P • X - Poo and U(X) > u}.

Thus, given prices and a standard of living, the non1abor income

function defines the minimum amount of nonlabor income required to

buy that standard of living. Note that the non1abor income function

may take on positive or negative values.

In Proposition 6.1 we demonstrate that the higher the prices of

goods, the hiqher the non labor income required to buy a given standard

of living: the higher the wage rate, the lower the non1abor income

required to buy a given level of utility: and the greater the desired

standard of living, the greater the required non1abor income. Final­

ly, if all prices double, the minimum nonlabor income required to buy

a fixed standard of living also doubles.

Proposition 6.1: The non1abor income function is

(a) nondecreasing in Pk for k = 1, 2, ••• n:

(b) nonincreasing in PO:

(c) nondecreasing in u:

(d) homogeneous of degree one in P.

Proof: Note that yep, u) ;: e(P, u) - Poo.

Hence, properties (a), (c), and (d) follow directly from

Proposition 3.4. To establish (b), note that

° ;: h (P, u ) - ° < O. o

We now define a minimum wage function. First, note that

y + P (0 - X ) o 0

n I: P X •

k=l k k

Q.E.D.

70

Assuming that Xo < 0, we may rearrange this to obtain

n P = (L P X - y)/(o - X ). o k=l k k 0

Definition 6.2: The minimum wage function is defined as

w(P, y, u) n

_ min {p I P = ( L P X X 0 0 k=l k k

U(X) ~ u },

y)/(o - X } and o

where P (PI' P 2 , •.• , Pn ). Note that the minimum wage func­

tion is well defined only if nonlabor income is insuffi-

cient to achieve the utility level u.

Proposition 6.2: The minimum wage function

(a) is homogeneous of degree one in (P, y);

(b) is nondecreasing in P;

(c) is nonincreasing in y;

Proof: (a) n

w(9P, 9y, u) := min {( L 9P X - 9y)/(0 - Xo) I U(X) > u} X k=l k k

- 9w(P, y, u).

-0 (b) By way of contradiction, assume that p' > P and the value

of the minimum wage function fell. Given y, let X' be the o

bundle selected under prices p' and let X be the bundle select--0

ed under prices P. By assumption

n 0 0 0 n ( L P X - y)/(o - X ) > ( L P'X' - y)/(o - X'). k=l k k 0 k=l k k 0

But because the bundle xO is the minimizing bundle, we have

n 0 n 0 XO) ( L P X' - y)/(o - X') > ( L P xO - y)/(o -

k=l k k 0 - k=l k k o • This implies that

n 0 n ( L P X' - y)/(o - X') > ( L p'x' - y)/(o - X' ) • i=l k k 0 k=l k k 0

Thus,

n 0 L (P - P')X' > 0

k=l k k k ' _ _0

which is a contradiction as p' > P and X, is nonnegative.

(c) Works similar to (b). Q. E. D.

71

Thus, in the absence of an income tax, a doubling of the prices of

goods and non1abor income doubles the wage necessary to maintain a

give~ standard of living. Moreover, all else equal, higher prices

mean a higher wage rate is needed to maintain a fixed living stand­

ard. A higher nonwage income lowers the wage necessary to maintain

the fixed standard of living.

Both the minimum wage function and the minimum non1abor income

function can be used to construct alternatives to the expenditure

based indices.

Definition 6.3: The non1abor income-based cost of living index

is defined as

° ° where yIP , u ) * 0.

y(p1, uO)

y(PO,uO)

Thus, the non1abor income index provides the factor by which the base

period non1abor income must be adjusted after movement in prices so

as to leave the consumer equally well off in the base and current

° ° period. Note that the index is defined only if yIP , u ) is not zero.

Definition 6.4: The real wage index is defined as

WI ~P , ~1 ° 1 0)_ t:,y,y,u = -1

w(P , --0

w(P ,

1 y ,

° y ,

° u )

° u )

The real wage index provides the factor by which the wage rate must

be adjusted after price or income changes to maintain the base level

of util ity.

Both indices are complete welfare measures in the sense that we

can determine whether the consumer is better off or worse off given

the movement in relevant prices. Consider first the non1abor income

index. If the value of the index is less than one, the consumer is

strictly better off under current prices than under base prices.

Formally,

Proposition 6.3: The non1abor income index satisfies

72

(a) y ( • ) > 1 if and only if V[pO, ° ° o pOl yep , u ) +

° >

1 ° V[P , yep , uO ) 1 +oPol.

( b) y ( • ) < 1 if and only if V[p o , y(po, uO) + cpol

° <

Proof: We prove case (a): the proof of the other case is very similar.

If y(.) ~ 1, then

V[po , y(po, uO) + OP~l 1 1 ° 1 v[P , yep , u ) + oPOl and

1 ° ° ° yep , u ) ~ yep , u ).

As the indirect utility function is monotonic in full income, we

have

° ° ° ° 100 1 V[P , yep , u ) + oPol ~ V[P , yep , u ) + oPol.

Conversely, if

V[po, ° yep , uO ) + 6Pgl ?. V[pl, ° yep , u 0) + 6P~l and

V[po , ° yep , uO ) + OPgl = V(pl, 1 yep , uO) 1 + oPO)'

then

1 ° ° ° yep , u ) ~ yep , u )

because the indirect utility function is monotonic in y. Thus,

which complete the proof. Q.E.D.

The nonlabor income index defines the factor by which base period in­come must be multiplied so as to leave the consumer equally well off after a price change. Note that when there is economic growth, so

that the wage rate grows at a faster rate than the overall price level, the nonlabor income index will take on values less than one. The index may be zero, or even negative, if there is sufficient

growth.

Now consider the real wage index, W(·). The real wage index is also a complete welfare ordering in the sense that if the real wage

73

index is less than one, the consumer is better off under current prices and income than under base period prices and income. For con-

veniance, let us adopt the notation V(P, y + oPo) = V*(P, PO' y),

where once again, P = (PI' P2 , .•• Pn ).

ProQosition 6.4: The real wage index satisfies

-1 0 1 --0 0 0 (a) W(· ) > 1 if and only if V*(P , w

0' y ) < V*(P , w

0' y ).

(b) W( • ) 1 if and only if v*(pl, yl) ~ V* (pO, o ' < w , w , y ),

i -i i where w - w(p , y , u) for i = 1, 2.

Proof: We shall prove case (b); the other case may be proved in a simi­

lar manner. If W(·) ~ 1, then

-1 1 1 V*(P , w , y )

--0 0 0 0 1 V*(P , w y) and w > w

Because V*(·) is nondecreasing in the wage rate,

-1 0 1 --0 0 0 V*(P , w , y ) > V*(P , w , y ) .

Conversely, if

_1 0 1 --0 0 0 -1 1 V*(P , w , y ) > V*(P w , Y ) and V*(P , w , y

--0 0 0 V*(p , w , y ) ,

then

--0 0 0 -1 1 0 w(P , Y , u ) > w(P , Y , u )

by the monotinicity of V*(·). This implies that

-1 1 0 --0 0 0 w (P , Y , u ) /w (P , Y , u ) = W ( .) < 1,

which establishes the result.

1 )

Q. E.D.

The real wage index provides the factor by which the current wage

rate must be multiplied to leave the consumer equally well off after

a change in the price of commodities and the change in the consumer's

nonlabor income. The real wage index is well defined as long as the

minimum wage function is not zero. If the minimum wage function is zero, however, the real wage index may not be well defined. In par­ticular, if

74

-1 1 -0 ° ° v* ( p , 0, y ) > V* ( p , w , y ),

then the minimum wage function cannot be defined. Thus, if there is

a sufficiently large increase in nonlabor income, the consumer may be

able to surpass the base period level of utility without working. In

this case there does not exist a real wage index.

Not only are the minimum wage and nonlabor income indices com­

plete welfare orderings, they also satisfy the Eichhorn-Voeller

axioms, as the next two propositions demonstrate. Proposition 6.5: The nonlabor income index satisfies the following

properties: (a)

(b)

Homogeneity: For A > ° y(pO, ApI, u O) =: AY(PO, pI, u O)

Monotonicity: y(pO, pI, u O) ~ y(pO, pI, uO) if P~ pI and

° pI > pI; y(pO, pI, ° ° 1 ° . '0 ' 1 u ) ~ Y(P , P , u ) 1f Po = Po and

pO < pO. The inequalities are reversed if

only the wage increases.

(c) Dimensionality: For A > 0, Y<ApO, ApI, u O) =: y(pO, pI, uO);

(d) Identity: y(pO, pO, uO) =: 1.

Proof: Properties (a) and (c) follow from the homogeneity of degree one of the non labor income function, and property (b) from the

monotonicity of y(P, u ) in P. Property (d) is a consequence of

the definition of Y(·). Q.E.D.

Proposition 6.6: The real wage index satisfies the following· properties:

(a) -Homogeneity: For A > 0,

-0 -1 ° 1 -0-1 ° 1 W(P , AP , Y , Ay ) _ AW(P , P , Y , Y ).

(b) Monotonicity:

-0 ° 1 -0 ° 1 W(P , pI, Y , Y ) > W(P , p" , y , y ) if pI > Pili

-1 ° 1 -1 ° 1 W( pI, P , Y , Y ) ~ W(P", P , Y , Y ) if pI < pOI -0 --l ° -0 --l ° W(P , P , Y , y') ~ W(P , P , Y , y") if y' < y": -0 -1 1 -0 -1 1

W(P , P , Y I , Y ) ~ W(P , P , y", Y ) if y' < y". (c) Dimensionality: For A > 0,

-0 --1 ° 1 -0 _1 ° 1 W( AP , AP , Ay , Ay ) - W(P , P , Y , Y ) .

-0 -D ° ° (d) Identity: W(P , P , Y , Y ) - 1.

75

Proof: Properties (a) and (c) follow from the homogeneity of degree one of the minimum wage function, and property (b) from its monotonicity. Property (d) is a consequence of the definition

of WC·).

Thus, both indices are complete welfare measures and satisfy the Eichhorn-Voeller axioms. Both indices, therefore, appear to be use­ful, depending upon the question to be addressed. The real wage in­dex treats the level of nonwage income and the prices of commodities as fixed, while the non1abor income index treats the level of wages and the prices of commodities as given. The real wage index does suffer from the disadvantage of not being well defined when non1abor income grows sufficiently to achieve the base period of utility with­out working. This may be a problem when trying to construct a long time series of the real wage index.

PART IV

ECONOMIC INDICES IN THE PRESENCE OF TAXATION

~HAPTER SEVEN: A SIMPLE TAXABLE INCOME-BASED COST OF LIVING INDEX

As a means of introducing the concept of an income-based measure of the cost of living, it is useful to note that in the absence of

taxes, the expenditure function and the full income compensation

function correspond. That is,

e(P, u) = pIP, u: T)

where e(') in the expenditure function defined in Chapter Three, and p ( .) is the income-compe'nsa tion funct ion def i ned in Chapter Four.

Hence, in the absence of an income tax the Konus expenditure­based cost of living index may be equivalently written as

1 K(PO, pl, u) = p(P, u: T)

° p (P , u: T)

This formulation emphasizes that, in the absence of an income tax, the Konus expenditure-based cost of living index is also an income­

based cost of living index: i.e., it compares the amounts of full in­come required under alternative price regimes to buy a fixed level of utility. The distinction between an expenditure-based cost of living

index and an income-based cost of living index arises because of the existence of a tax code, summarized for simplicity by the function T that maps pretax income, Y, into taxes paid T(Y).

Before we define an income based measure of the cost of living in the presence of taxes and a labor supply decision, it should be

emphasized that there are several alternative concepts of the income

based index. For example, one could construct an index designed to measure the adjustment to pretax wage income necessary to maintain a

fixed standard of living, or alternatively, a measure of the adjust­ment to nonwage income necessary to maintain a fixed standard of

living. As noted by Triplett (1983), each of these indices is de­

signed to serve different functions; there is no single income-based cost of living index to serve all purposes.

In this chapter we address the construction of a taxable income­based cost of living index when the consumer's income is exogenous (as

is the case for Social Security recipients). This assumption will be relaxed in Chapter Eight, where we allow some income to be derived

from the labor market. Let P = (PI' P2 , ••• ,Pn) and X = (Xl 'x2 ' ••• ,Xn ) denote the relevant prices and goods, and assume utility depends solely

on X. Let e(F, u) be defined analogous to Definition 3.5.

78

Definition 7.1: A tax rule is a function that maps pretax income, denoted Y, into taxes paid, denoted T(Y). A tax rule T is said to be a simple income tax if: i) it is twice continuously differentiable: ii) Ya = r(y) = Y - T(Y) > 0 and T(O) = 0, where r(·) is the after tax income function and Ya is after tax in­come, iii) 0 < T'(Y) < 1, iv) Y is solely exogenous.

Definition 7.2: A simple income tax is said to be regressive, pro­portional, or progressive as T"(') < 0, T"(') = 0, and Tn(.) > 0, respectively.

The degree of progression is defined in terms of nominal pretax income, Y, which will be a crucial distinction below.

Proposition 7.1: Given a simple income tax, T: (a) the after· tax income function, r(Y), is one-to-one, (b) the pretax income function, G(Y ) = r- l , exists:

a (c)

(d)

the pretax income function satisfies G'(Y ) > 1 for all a Ya eR+, the pretax or G"(Y )

a

income function satisfies G"(Y »0, G"(Y ) = 0, a a < 0 as T(') is progressive, proportional, or re-

gressive, respectively: (e) G(O) = O.

Proof: (a) It is implied by the definition of r(Y) that r'(y) > O.

Hence, r(y) is strictly increasing on R+, which implies

that r(.) must be one-to-one. (b) As r(.) is strictly increasing on R+, the implicit function

theorem ensures that G(') exists and is strictly increas­ing.

(c) Note that G(Y ) = r- l implies G'(Y ) = [l/r'(y)]. By de~ a a

finition 7.1,0 < r'(Y) < 1, which implies G'(Y a» L (d) Since G"(Y) =_[l/r'(y)]2 r"(y) and r'(y) > 0, sgn[G"(Y)] a a

= -Sgn[f"(Y)] = Sgn[ T"(Y)]. (e) The pretax income necessary to provide an after tax income

of zero is zero. Q.E.D.

The pretax income function, G('), indicates the amount of pre­tax income required to provide a given level of after tax income. It is useful to define an expenditure function that determines the mini-

79

mum total expenditures required to achieve a given level of utility,

where some of the expenditures are on taxes.

Definition 7.3: The gross expenditure function, eG(p, u), is defined

min {G(P • X) I U(X) ~ u}. as e (P, u) G X

The following identities are immediate:

(7.1) eG(p, u) - G[e(P, u»);

(7.2) eG(p, u) - e(P, u) + T[eG(p, u») ;

(7.3) Ya - e(P, u) ;

(7.4) Y - eG(p, u) •

The properties of the gross expenditure function, summarized in

the following two propositions, will be useful in the characteriza­

tion of an income-based cost of living index.

Proposition 7.2: The gross expenditure function, eG(p, u):

(a) is homothetic in P;

(b) is nondecreasing in P

(c) is quasiconcave in P; (d) is nondecreasing in u;

(e) may be written as G(w(u)h(P» if and only if preferences

are homothetic.

Proof:

(a) By identity 7.1, eG(p, u) = G[e(P, u»). As e(P, u) is

homogeneous of degree one in P and G'(·) > 0, eG(p, u) is

homothetic in P. (b) Since eG(p, u) :::: G[e(p, u») with G' (.) > 0, and as e(P, u)

is nondecreasing in P, e G(·) is nond~creasing in P.

(c) Since G(·) is monotonic and e(P, u) is concave (and hence

quasiconcave) eG(p, u) is quasiconcave.

(d) G(·) is monotonic and e(p, u) is nondecreasing in u, and

hence eG(p, u) is nondecreasing in u.

(e) Follows immediately from Proposition 3.5 and identity 7.1.

ProQosition 7.3: For the scalar ). > 1 : (a) eG().p, u) > ).eG(p, u) if the tax rule is progressive;

(b) eG().p, u) ).eG(p, u) if the tax rule is proportional;

(c) eG().p, u) < ).eG(p, u) if the tax rule is regressive.

For 0 < ). < 1, the inequalitites are reversed.

80

Proof: (a) If the tax rule is progressive G(·) is strictly convex by

. i (-i ° 1 Proposition 7.1. That lS, for e - e P , u): SG(e)+(l-S)G(e» G(seO+(1-S)e1),0 < 6 < 1. Taking e l = ° yields 6eG(po, u) >

eG(6po, u). Define P :: 6po so that 6eG [(l/6)'P, u] > eG(p, u). Defining (1/6) :: A > 1, we have eG(AP, u) > AeG(p, u), as re­quired. (b) If the tax rule is proportional we may write T(Y) = aY,

where a is a constant, ° < a < 1. Hence by identity 7.2, - I-e (P, u) :: (--)e(P, u), which is homogeneous of degree 1 in P.

G I-a (c) If the tax rule is regressive, then eG(p, u) :: G[e(p, u»)

-0 is strictly concave in P by Proposition 7.1, or 6eG(p , u) + -1 -0 -1 -0 (1-6) eG(p , u) < eG(6P + (1-6)P , u). Again, define P

to be the zero vector, let A :: [1/(1-6)], and let P:: (1-6)pl

to get eG(AP, u) < A eG(P, u).

Q.E. D.

One may use these foundations to define a particular type of in­

come-based cost of living index.

Definition 7.4: The taxable income-based cost of living index is

-1 ° eG(p , u )

eG(po , uO)

Thus, the taxable income-based cost of living index defines the factor by which base period taxable income must be adjusted and given to the consumer in the current period in order to enable him to pur­chase the same level of utility after movement in prices. The pro­perties of the taxable income-based cost of living index are sum­marized below.

Proposition 7.5: The taxable income-based cost of living index sat­isfies the following properties:

(a) Bracket Creep (Fall): For A > 1,

-0 -1 {>} IT'O-l KG(P , AP , u) < AKG\P, P , u)

progressive as tax rules are { proportional }.

regressive

For ° < A < 1 the inequalities are reversed~

81

( b) -0 -

Monotonicity: KG(P, P, u) u) if15~15: ~

- -1 KG(P, P , u) u) if 15 ~ 15: (c) Dimension Sensitivity:

-0 -1 -0 -1 -0 -1 n KG(~P , ~P , u) = KG(P , P , u) for all P , P ER++ and ~ > ° if

and only if the tax function is proportional. -0 -0 (d) Identity: KG(P, P , u) = 1.

Proof: (a) Suppose the tax system is progressive and ~ > 1. Then

-1 ° 1 ° K (pO, ~pl, uO) _ G(~e(P, u » > ~G(e(p, u» G G(e(pO, uO» G(e(pO, uO»

-0 -1 ° - ~KG(P , P , u ),

where the inequality follows from Proposition 7.3. The other cases similarly obtain.

(b) Follows from Proposition 7.2 and themonotonicity of the gross expenditure function.

(c) Follows from the fact that eG(') is homogeneous of degree one if and only if the tax system is proportional.

(d) Follows from the definition of KG(')' Q.E.D.

Results (a) through (c) may be interpreted as follows. Property (a) states th~t when tax rules are progressive (regressive) a doubling of all prices more than doubles (less than doubles) the taxable income­based cost of living index because, given that more income is needed to purchase the bundle of goods at higher prices, relatively more (less) income is needed since the consumer's tax burden will have in­creased (decreased). Property (b) simply reveals that price in­creases lead to an increase in the taxable income-based cost of living index. The fact that KG is, in general, dimension sensitive reflects the fact that, given nonproportional tax rules, nominal val­ues matter. In fact, only when tax rules are proportional do the standard Eichhorn-Voeller axioms hold for all price changes.

Consider next the magnitude of the taxable income-based cost of living index relative to the Konus index.

82

proposition 7.5: K(pO , -1 u) 1, -0 -1 u) ~ K(PO, pl, u) For P , > KG(P , P , <

{ progressive

} . K(PO , -1 as tax rules are proportional For P , u) < 1, the regressive

inequalities are reversed.

Proof: We provide a proof for the case of progressive taxes when K(') > 1; the other cases easily obtain. By Proposition 7.3, eG(KP, u) > KeG(p, u) since K > 1 by assumption. By definition,

-1 -0 e(P , u) = K e(P , u) so that

K [pO pl u) G . ' ,

= G[Ke{p°, u)]

G[e(pO, u)] -0 G[e(P , u)] -0-1 > K = K( P , P , u)

G[e(p°, u)]' Q.E.D.

Thus, when tax rules are progressive, the Konus index under­states the taxable income based cost of living index during infla­tionary periods. In light of Propos~tion 5.2, this implies that the Laspeyres expenditure-based price index is not an upper bound on the taxable income-based cost of living index. However, in light of Pro­position 5.4, the Paasche index (when u = ul ) is a lower bound for the taxable income-based cost of living index.

Another implication of Proposition 7.5 is that the Konus and taxable income-based indices are equivalent measures of the cost of living when the tax system is proportional. In other words, in the presence of a proportional tax, the Konus index answers two ques­tions: i) "By what factor must base period expenditures be adjusted to compensate for a price change?" and ii) "By what factor must base period taxable income be adjusted to compensate for a price change?" Just as the Konus index depends upon the base level of utility, the taxable income-based index also depends, in general, upon the base

period utility level.

Proposition 7.6: The taxable income-based cost of living index is independent of base utility for all price changes if and only if preferences are homothetic and tax rules are proportional.

Proof: Sufficiency is obtained trivially since under proportional tax rules KG = K, and homotheticity implies the Konus index is independent of u. To prove necessity, suppose KG is independent

of u, i. e. ,

83

G(e(pl, u» suppose

G(e(pO, u»

and u*ER. Then G[e(~, u)] = .(u)h(~). By identity 7.2 and by

definition, eCP, u) := .(u)h(P) - T[.(u)h(P)] is homogeneous of

degree one in P. Thus, T(') and h(') must both be homogeneous of degree 1, implying tax rules are proportional and preferences are homothetic. Q. E. D.

Interestingly, Propositions 5.10, 7.5, and 7.6 reveal that if

the taxable income-based cost of living index is independent of base utility, then the taxable income-based cost of living index, the Konus index, and the Frisch index are all equivalent. Thus we may

state

Proposition 7.7: If preferences are homothetic and the tax system is proportional, then

-0 -1 -0 -1 _ -0-1 KG ( P , P , u) := K (P , P , u) = f::, (P , P , u).

Moreover, the indices are bounded from above by the Laspeyres

index and from below by the Paasche index.

CHAPTER EIGHT: THE FULL INCOME AND NONWAGE INCOME-BASED COST OF LIVING INDICES

In Chapter Seven we examined the construction of an income based cost of living index in the presence of an income tax, but where the consumer's income was solely exogenous. This chapter considers the concept of income based cost of living indices when the consumer's income depends upon an endogenously determined labor supply. Recall

that P = (PO' P) and X = (XO' x). Recall that total endowment of time is given by 6, and that the

budget constraint under taxes is given by

A more convenient expression may be obtained by using full income (F

- y + 6PO)' so that the constraint becomes

This equation emphasizes the tax deductible nature of the consumption of leisure. Moreover, differentiation of this equation with respect to Xe reveals that the effective price of leisure is [1 - T ' (' )]PO' where T ' (') denotes the effective marginal tax rate.

In this section we shall focus on an income-based index concept that is based on a very broad notion of income. This index, which we

call the "full" (pretax) income based cost of living index, treats the value of the endowment of leisure as one component of income. The other component is nonwage income.

The full income based cost of living index is based on the full income compensation function introduced in Chapter Four, namely

\I(P, u: T) :: min {F X

P • X < F - T(F - P X ) and U(X) ~ u}. o 0

Given prices and an arbitrarily selected utility level, this func­

tion defines the minimum level of full income necessary to purchase the given standard of living. Thus, the construction of \1(') allows for substitutions among goods and leisure as relative prices change.

Given \1('), it is natural to define the (pretax) full income based cost of living index as the function as follows.

Definition 8.1: In the presence of taxes, the full income based cost of living index is defined as

85

1 0 ,(pO, pI, uO: T) E ~(P , u : T)

o 0 ~ (P , u : T)

Thus, the full income-based cost of living index defines the ratio of

minimum pretax full incomes under alternative price and wage regimes necessary to purchase the given standard of living. In other words, ,(.) denotes the factor by which base period full income must be ad­justed after movement in prices so that, after the payment of the tax bill, the base standard of living may be maintained.

The following proposition is an immediate consequence of Propo­sition 4.6.

Proposition 8.1: In the presence of taxes, the full income-based cost of living index satisfies the following properties:

(a) Bracket Creep (Fa1l): For).. > 1,

010> 010 ,(P , )..p , u : T) < )",(P , P , u : T) as

{ progressive

} tax rules are proportional regressive

For 0 < ).. < 1 the inequalities are reversed.

(b) Monotonicity: , (pO, pI, 0 T) > ,(pO, -1 0 T) u : P , u : if pI >

-1 p :

,( pO, ;1, uO: T) > ,(pO, pI, 0 T) u : if pO ~ pO:

(c) Dimension sensitivity:

( 0 1 0 ) &(pO, pI, 0 ) f 1 , )..p , )..p , u : T = 'I' U : T or a 1 o 1 n+l p , p ER++ and).. > 0 if and only if the ,tax function

is proportional.

(d) o 0 Identity: ,(P, P , u: T) - 1.

Proposition 8.2: In the presence of taxes, the full income-based cost of living index is independent of base utility if and only if preferences are homothetic and the tax function is proportion­al.

Proof: Under a proportional tax, effective prices are independent of

behavior so that with homotheticity,

(l-t)~(P, u: T) E e(P*, u) E h(u)g(P*),

where p* = «l-T)PO' P) and T is the marginal tax rate. Hence,

86

To prove necessity, suppose ~(.) is independent of u, i.e., sup­pose

1 P (P , u, T) 1 p(P , u*: T) o

P (P , u, T) o p(P , u*: T)

Then pep, u: T) ~(u)g(P: T) for some ~(u). But using the tax analogue of Shephards Lemma, this implies Hep, u: T) = ~(u)f(P: T), so that by the budget constraint identity,

T[~(u)g(P: T) - ~(u)fO(P: T)] + ~(u) P • f(P: T) = ~(u)g(P: T).

This implies that T(·) must be homogeneous of degree one. But in turn, this implies virtual prices are independent of behavior

and

(l-T)P(P , u: T) = e(P*, u) = ~(u)g(p*),

which is sufficient for homotheticity. Q.E.D.

An alternative formula based on the full income-based cost of living index may be derived as follows. Recall that H(P, u: T) de­notes the compensated (Hicksian) demands that solve the full income ~inimization problem in the presence of an income tax. The market value of the goods and leisure necessary to buy the standard of living, u, is given by

C(P, u: T) = P • H(P, u: T).

Thus we may define the following market expenditure-based cost of living index in the presence of an income tax and a labor supply de­cision.

Definition 8.2: In the presence of taxes, the market valued expendi­ture-based cost of living index is defined by

o u : T) -

o u : T)

o u : T)

Thus, Ic(·} defines the factor by which base period market expendi­tures must adjust after movement in prices to maintain a fixed stand-ard of living. This formula is useful when measuring the rate of change in the full income-based cost of living index. To see this, note that, at the optimum,

87

(8.1) C(P, u; T) :: )J(P, u; T) - T[)J(P, u; T) - CO(P, u; T)],

where CO(P, u; T) :: POHO(P, u: T) denotes compensated expenditures on leisure. Assuming prices vary over time, differentiate equation (8.1) with respect to time to obtain

where dots over variables denote their time derivatives. Letting T :: T'(') denote the effective marginal tax rate, this may be rearranged

to yield

. )J

(l - T) C {_T __ } . o 1 - T

Dividing by full income, we have

(8.2) . !:I.. = )J

c C {t} {_I _} {~} _ {~} {_T _} {~} . C 1 - T II Co 1 - T )J

Let (C/II) :: (I-A), where A is the average tax rate (total taxes as a fraction of pretax full income) and let (CO/II) = a, where a is the· fraction of full income spent on the tax deductible good (leisure).

Noting that .wII :: ~/II and Wc :: C/C are simply the rates of change in

~ and Ic' respectively, and Wo :: CO/CO is the rate of change in com­pensated expenditures on leisure, equation 8.2 may be written more compactly as

(8.3) W II

= W {~} _ W {_T_} a. C l-T Ol-T

Equation (8.3), which we call the fundamental equation of the "full" income-based cost of living index, relates the rate of change­

in the full income-based cost of living index to the rate of change in an expenditure-based cost of living index and the rate of change in compensated expenditures on leisure, the tax deductible good. Thus, W represents the rate of change in full income necessary to

II maintain a constant standard of living as prices, wages, and tax

rates change. The term Wc is the rate of change in the market valued expenditure-based cost of living index (where some of the expendi­tures are on leisure), and Wo is the rate of change in compensated expenditures on leisure.

In the presence of taxes it is clear that the expenditure-based cost of living index will differ from the income-based cost of living index. When government does not tax the endowment of time (i.e., in

88

the absence of a head tax), the average tax rate appearing in equa­tion (8.3) will be less than the typically defined a~erage tax rate, which is the ratio of taxes paid to actual earned and unearned in­come. Hence, if the tax system is progressive in the sense that the typically defined measure of the average tax rate is less than the marginal tax rate, it will also be progressive in the sense that A < T. Consequently, in the presence of a progressive tax, the coeffi­cient of Wc in equation (8.3) will exceed unity, i.e., [(l-A)/(l-T)] > 1. This, taken alone, suggests that the rate of change in the in­come-based measure of the cost of living exceeds the rate of change

in the expenditure-based measure of the cost of living. Note, how­ever, that the second term in equation (8.3) offsets to some extent this effect. Specifically, because expenditures on leisure are tax deductible, the government absorbs a fraction of the increased cost of purchasing leisure. The importance of this component of the rate of change in the income-based cost of living index depends on the marginal tax rate as well as the fraction of full income allocated to leisure, the tax deductible good. In short, the inclusion of leisure into an income-based cost of living index will lead to a reduction in the rate of change in pretax income necessary to maintain a fixed standard of living, provided compensated expenditures on leisure in­crease over time.

As an alternative to the full income-based cost of living index, we_may define an index that is based on nonwage income.

Definition 8.6: In the presence of taxes, the nonwage income-based cost of living index is defined as

y(pl, uO: T) 0, ° y(P , u ; T)

where y(.) is the nonlabor income compensation function defined in Definition 4.5.

Thus, the nonwage income-based cost of living index defines the factor by which base period nonwage income must be adjusted in order to maintain a fixed standard of living after movement in prices.

The full income index and the nonwage income index are quite similar: each treats the movements in prices as exogenous. Which one should be used? The following proposition indicates that the choice between the two indices is really a question of how much "weight" to place on a wage index relative to a nonlabor income in­dex.

89

Proposition 8.3: The full income-based cost of living index may be

viewed as a weighted average of the nonwage income-based cost of living index and a relative wage index. specifically,

o 1 0 4>(P, P,u ;T) - 61 (.) + (1-6) Y

o 0 0 0 where, 6 = y(P,u ;T)/JdP,u ;T) denotes nonlabor income as a fraction of full income.

Proof: Using definition 4.5,

1 0 T) + opl 4>(pO, I 0

y(P , u ; 0 P , u ; T) -0 0 T) + 0 pO y(P , u ; 0

0 0 T) }

I 0 T) } {y( P , u ; {y( P , u ;

-0 II (pO. 0 T) y(pO T) u u ;

opO pI + { 0 {~}

0 0 II(P , U : T) pO

0

Q. E. D.

We may also define a minimum wage function in the presence of an in­come tax.

Definition 8.4: In the presence of an income tax, the minimum wage function is defined as

w(P, y, o

u :

o U(X) > u}, where P = (PI' P2 , ••• Pn ).

The minimum wage function defines the lowest possible wage that will allow the consumer to achieve a level of utility uOgiven the vector of commodity prices and the level of nonlabor income, recognizing

that the consumer must pay taxes on his income. The properties of the minimum wage function are summarized below.

Proposition 8.4: The minimum wage function in the presence of a

progressive income tax:

(a)

(b)

(c)

(d)

is nondecreasing

is nonincreasing

is nondecreasing

satisfies w( ap,

for a > 1.

90

in P;

in y;

in u;

ay, u; T) > aw{p, y, u; T) •

Proof: The proof is similar to the proof of Proposition 4.6.

Thus, in the presence of an income tax, higher prices increase

the wage compensation needed to maintain a fixed living standard. In

contrast, a higher level of nonlabor income lowers the wage compensa­

tion required to maintain a fixed standard of living. Finally, a

doubling of all prices and nonlabor income more than doubles the wage

required to maintain a fixed utility level, because of the increased

tax burden under a progressive income tax.

The minimum wage function may be used to construct a wage index

in the presence of an income tax.

Definition 8.5: The real wage index in the presence of an income tax

is defined as

-1 1 0 -0 -1 0 1 0 w{P , Y , u ; T)

W(P , P , Y , Y , u ; T) --0 0 0

w{p , y , u ; T)

Proposition 8.5: In the presence of a progressive income tax, the

real wage index satisfies the following properties: -1

a) nondecreasing in P ;

b) nonincreasing in yl;

c) Price Level Sensitivity: For a > 0,

-0 -1 1 0 -0 -1 0 1 0 W{9P , ap ,ay; ay u T) * W(P P, Y , Y , u ; T)

d) Bracket Creep: For 9 > 1: -0 -1 0 1

W(P , ap , y , ay , o -0 -1 0 1

u ; T) > aW(p , P , Y , Y , o

u ; T)

Proof: These properties are direct consequences of the properties of

the minimum wage function. Q.E.D.

PART V

THE THEORY IN PRACTICE

CHAPTER NINE: TOPICS IN JI.PPLIED ANALYSIS

In the previous chapters, "'Ie have explored how the presence of

income taxation may alter several of the neoclassical properties of

consumer demand functions and labor supply functions. As of yet,

however, we have not discussed how one may empirically estimate the

demand and supply functions when consumers face nonlinear budqet sets. While the literature is now quite extensive, the interested reader

may find the recent work of Hausman (1985) a most useful review of the labor supply literature.

1. The "Virtual" Approach

We have seen in previous cha.pters that the nonlinearity of the budget equation that results from income taxation and the tax deduct­ibility of leisure alters many of the results that are derived from

classical theory. In particular, the Hicksian demands are not homo­

geneousof degree zero in money prices; the matrix of first derivatives

of the Hicksian demand functions are not symmetric; and the Marshal­

lian demand functions are not homogeneous of degree zero in prices and nonlabor income.

One method of "avoiding" the complexities that arise due to non­linearities of the budget set is the so called "virtual" approach. This approach expresses behavioral functions (such as demand equa­

tions) as functions of effective, or virtual prices. This approach is outlined below.

Consider first the maximization problem

max U(X) s.t. P • X + T(F - POXO) $ F. X

The Lagrangean is given by

L = U(X) + A{ F - P • X - T[F - POXO]} ,

where A is the Lagrange multiplier. Assuming an interior solution,

the first-order conditions are given by

for k =1, 2, .... n;

~ - AP [1 - T' (.)] 0 ax 0 o F - P • X - T(F - POXO) o.

93

Let us d.fine the price vector p* 5 ([1 - T' (o)]Po' Pl , P2 000

Pn)o If we evaluate p* at the solution value for T' (0) and consider the problem

max U(X) Sot. p*. X ~ Z, X

where Z equals F - T(F - POXo) + T' (.)POXO at the optimum, then the solutions to both problems are equivalent (see Figure 9.1). The ad­vantage of considering the second problem is that the budget set is linear in P*, while the budget set of the first problem is nonlinear in P.

One can use this technique to define the virtual Marshallian demand functions,

M*(P*, Z) 5 argmax {U(X) I p* • X = Z} and the virtual indirect utility function,

V*(P*, Z) = max {U(X) I P* . X = Z}.

The virtual Marshallian demand functions and the virtual indirect utility function satisfy all the classical properties, except that one must use the virtual prices (P*) and virtual income, Z, rather than the actual ones.

Similarly, we may define the virtual Hicksian demands,

H*(P*, u) 5 argmin {p* X I U (X) ~ u},

and the virtual expenditure function

e*(P*, u) 5 min {p* • X I U(X) ~ u} .

The virtual Hicksian demand functions and the virtual expenditure function satisfy all of the neoclassical properties, except that one must use virt'ual prices (P*) instead of actual ones. It should be emphasized, however, that the virtual prices and virtual income are exogenous to the economist -- not the consumer.

2. Aggregation Across Households in the Presence of an Income Tax The "virtual" approach will allow economists to estimate the

preferences of individuals using micro data. Often, however, policy makers or economists would prefer a single index, or ~ay have only aggregate data available. In this section, we explore the restrictions necessary to allow us to estimate preferences using aggregate data in the presence of an income tax.

Consider household s, whose preferences are represented by the utility function US(XS), where XS = (xg, x~, x~, ... X~) is an (n+l)-vector of cO~IDodities, and xg denotes the household's consumption of'leisure. The household is endowed with 0 units of

Other Goods

Linearized Constraint

'/ ...... ......

Actual Constraint

94

FIGURE 9.1 THE LINEARIZED CONSTRAINT

8 Leisure

95

time to allocate between work and leisure. The wage rate is given by

PO' and the prices corresponding to XS are given by the (n+l)-vector,

P = (PO' Pl , P2 , '" Pn )·

It is assumed that household s attempts to maximize utility sub­

ject to the constraint that total expenditures cannot exceed total

after-tax income. Formally, the constraint is given by

(9.1)

where yS denotes the household's nonwage income, T(') is the tax

function (schedule), and (6 - X~) is the labor supplied by household

s.

In the United States, the nature of the progressive tax schedule gives rise to a piecewise-linear budget constraint consisting of

"kinks. " As such, it is not generally possible to fi.nd closed form

expressions for demand functions, and consumers may locate at the

kinks. We ignore these problems and utilize the linear segment of

the budget constraint that corresponds to the observed point in XS

space in order to derive explicit functional forms for demand funct­

ions (see Figure 9.1). In this case, the tax function for household

s maybe written as

T = TS + 'Is [(6 - X~) Po + y s - BS),

where BS is the smallest level of taxable income that puts household

s in the 'Is marginal tax bracket, TS is the tax paid on BS dollars

of taxable income, and 'Is is the observed marginal tax rate for

household s. As a consequence, the maximization of utility subject

to constraint (9.1) yields the same solution as the maximization of

utility subject to the linearized constraint

(9.2)

where z's is commonly referred to as "virtual" income.

Defining ws " (1 - ,s)Po to be the effective after-tax wage of

household s (the "virtual" wage), P (P l , P2 , ... , Pn ), and -s s s s X = (Xl' X2 ' , Xn ), equation (9.2) may be written compactly as

(9.3) P . XS + wsX~ .$ zs.

The maximization of utility subject to (9.3) yields the virtual

Marshallian demand functions

]':1~(w ,"P , zS) [i = 0, 1, ... n) 1. s

for leisure and goods, which were discussed above. For our purposes,

96

it is useful to use duality theory to describe household s's prefer­

ences through the virtual expenditure function defined by

(9.4) eS(ws ' ~, u ) ~ min {V • XS + wsX~ I US(Xs ) ~ us}. s X,

By construction, the compensated demands that solve equation (9.4)

will correspond to the solution to the utility maximization problem.

The form of the expenditure function in equation (9.4) allows

preferences, effective (aftertax) wages and (virtual) incomes to

vary across households. While this generality provides an element

of realism, more structure is necessary before one may estimate

preferences from aggregate data. Specifically, we must specify

functional forms for the e S (.) 's that permit consistent aggregation

across all S households in the economy when there is variation across

households in expenditure patterns, wages, and incomes. Importantly,

we have imposed enough structure so as to utilize the following im­

portant theorem:

Theorem 9.1 (Muellbauer): For exact linear aggregation under

optimizing behavior in the presence of wages and incomes

that vary across households, necessary and sufficient con­

ditions for household and aggregate expenditure functions

are that they have the following forms:

Household: s -

e (ws ' P u) =wBUA(P) +w B(P) +d (P) s s s s s

Aggregate: e ( rl, ~, u) = WSUA (P) + WB (P) + D (p)

where B is a constant, A(P) is homogeneous of degree 1 - S, B (P) is homogeneous of degree 0, d. s (P) is homogeneous of

degree one, and D(P) = (l/S) S~lds(P).

Proof: See Muellbauer, 1981.

Note that the conditions of the theorem allow for some variation in

preferences across households (through the ds{P) terms) and for

variation in aftertax wages across households. For example, one

may impose the above aggregation restrictions on the functional form

proposed by Blundell and Ray (1984) and choose:

d (P) s

n B, ,nIP,l. l.= l.

n ,LIS. l.= l.

s where r i = m s d (s s La. ·c ., an c l' c 2'

j=l ~) )

97

s •.. c m) denotes a vector of exo-

genous demographic characteristics (such as family size or age of

family members) specific to household s. The technique of allowing

preferences to depend on the r~'s in this fashion is known as linear ~

demographic translating (cf. Pollak and Wales, 1981).

Applying Shephard's Lemma and substituting the indirect utility

function in place of each us, we then average the Marshallian expend­

itures over households to obtain per household expenditures on leisure

(9.5) lS s n n ~~ '¥O := ('8)s~tsXo = vlyoO + (30[Z - j~l i~lYijPiPj - WYOO

£P. Ta .. c.) i=l ~ j=l ~) )

and per household expenditures on commodity k

(9.6)

where' Z

W

T

S P XS l: k k

s=l

- WyOO - £ P. ~ a .. C.) + Pk ~ ak.c., i=l ~j=l ~J J j=l J)

1 S s ('8) l:c).;

s=l

1 S s ('8) l: L •

s=l

Po (1-T); and

Thus, aggregate per capita expenditures on goods and leisure depend

solely on the averages of the variables that vary across households,

specifically the "average" marginal tax rate, average income, and the

average of each demographic variable. For example, if cj is the

number of family members in the household, this variable affects

average expenditures only through Cj ' the average family size in the

economy.

One may also make use of a "trick" developed by Abbott and

Ashenfelter (1976, p. 397), which eliminates the necessity of speci­

fying a value of 0 prior to estimation. Define LS := 0 - X~, Y~o -* * Yoo - 0, and Z := Z - oW (note that Z is independent of 0).

98

Substituting these expressions in equations (9.5) and (9.6) yields, after a bit of algebra,

* - '1'0

and

n m - ;!lP; .L la . . C.]

.L .L J= ~J )

* n m m - Wy 00 - i;lPi j;l aijc j ) + Pk j;lakjCj for k 1, 2, ... n.

* n * * * Noting that '1'0 + k~l 'I'k = Z , the share equations are So = 'I'o/Z and

* Sk = 'I'k!-Z' (for k = 1, 2, ... , n). Because the shares add to unity,

one may delete (for example) the leisure equation before performing

the estimation. A caveat should be mentioned, however. The 'I'k'S

* are observable in the aggregate, but '1'0 is unobservable. (Statisti-S s S s s

cal agencies collect s~l PiXi but not s~lPO(l-T )L ). Consequently,

* one may use as a proxy for the "unobservable" 'l'0 the observable

S s * (l-T) s~lPOL in the construction of Z , which is needed even when

the leisure equation is deleted. This problem was not encountered

by Abbott and Ashenfelter because they assumed a proportional tax rate.

The estimation of preferences yields estimates of the compensated demands that solve virtual expenditure minimization. For expositional

simplici ty, ignore differences across households, and let H (W, i? , u) and e(W, P, u) denote the value of these compensated demands and

virtual expenditure function, respectively. By construction,

o -0 0 0 -0 0 0 0 -0 0 C (PO' P , u ; T) = e (W , P , u ) + Po THO (W , P , u )

so that information about the virtual expenditure function may be used to obtain the base period value of the market valued expenditure function, which is needed to construct the index number concepts. Moreover, note that given (P~, pl) there exists an effective marginal

tax rate, say i, such that

99

-1 - 1 0 Let E(P , (l-T)PO' u ) denote the value of the right hand side of

this equation. Unfortunately, T is unobservable. Specifically,

given (P~, pI), we obs~rve Tl, which is the effective marginal tax

rate in period 1. During times of economic growth, Tl will be asso­

ciated with a higher level of utility, u l , than in the base period.

We might expect that the observed marginal rate, Tl, will exceed ~, which is needed to construct the full income-based cost of living

index. What is the direction of the bias resulting from our use of

the observable Tl rather than the unobservable T in our construction

of the indices?

Application of Shephard's Lemma reveals that

dE d'l'

This implies that

~ O.

t-t t -t tt 0 C(PO' P , u ; T) .s E(P , (l-T )PO' u ) for t 0, 2, 2, ••.•

Hence, while the value of C(.) is unobservable for t ~ 0, we may ob­

tain a preference dependent upper bound via the observable E(·),

which is based on data from the virtual expenditure function. More­

over, the size of the upward bias varies directly with the respon­

siveness of labor suppiy to changes in the aftertax wage rate. It

is important to note that this complication arises because of the

allowance for a progressive income tax.

3. Generating Functional Forms

In empirical applications of index numbers it is often useful

to obtain parameter estimates of the underlying preferences of eco­

nomic agents (cf. Braithwait, 1981). While there have been recent

developments in the nonparametric approach to the analysis of consumer

choice (cf. Varian, 1982 and Manser and McDonald, 1984), we shall dis­

cuss in this section the parametric approach to demand analysis.

There are four basic ways in which functional forms for complete

100

demand systems (vi~tua1 or neoclassical) may be specified. The first approach is the direct approach, where some functional

form of the direct utility function, U{X), is postulated. The Mar­

sha11ian demand functions, Mi(P, F) [i = 0, 1, ... nl are then ob­

tained by maximizing the chosen utility function subject to the

budget constraint. For example, if U{X) is taken to be Cobb-Douglas, so that

flO fl1 fln n U(X) = Xo Xl ... Xn ' i~ofli 1,

then the vector

M(P, F) - argmax {U(X) I P . X ~ F}

has components

Mi (P, F) for i = 0, 1, '" n.

Given 'observations on prices and expenditures, the parameters of the

utility function (the fli'S) may be easily estimated using standard

econometric techniques.

The second approach to specifying functional forms for complete

demand systems is to select functions Mi(P, F) [i = 0, 1, ..• nl that satisfy the four integrability conditions:

(9.7)

(9.8)

(9.9)

Adding Up:

Homogeneit::l:

S::lmmetr::l:

where K .. -~J

n i~OPiMi (P,

M (SP, SF)

Kij - K .. J~

F

F) = F;

- M(P, F) i

for all it j = 0, 1, ... aMi (P, F)

+ M j (p, F) aF

(9.10) Negative Semidefiniteness:

i) Z 'KZ < 0 for all Ze:Rn+l ;

n ii) i~OP i Kij o for all j 0, ., •.. ni and

n

iii) j~OPjKij = 0 for all i 0, ., '" n,

where K - [I" ... ionJ nO nn

n,

101

It is known (cf. Samuelson, 1950) that if these four conditions are

satisfied, then the Mi (·) 's comprise a complete demand system gener­ated by some utility function under constrained utility maximization.

Moreover, one can, in principle, "integrate back" to recover the underlying utility function that generated the demand functions. As

an example, consider the functions (which we already know are gener­ated by Cobb-Douglas preferences):

(9.11)

M2 (p, F)

It may easily be verified that these functions satisfy the four inte­

grability conditions. Now since, at the point of constrained utility

maximization, the marginal rate of sUbstitution equals the price

ratio, we have

=

so that using equations (9.11),

Writing this as a differential equation, namely,

+ 0,

and integrating yields,

But this is merely a monotonic transformation of

The remaining two methods of specifying functional forms of

preferences is to use duality to describe preferences through a valid

indirect u~i1ity function or expenditure function. Once these

functional forms are specified, demand equations may be obtained via

102

either Roy's identity or Shephard's Lemma, and these functional forms may again be estimated using standard techniques. Consider, for ex­ample, the "function

(9.12) 1.

Application of Shephard's Lemma reveals

(9.13) =

Noting that F = e(P, u), solving equation (9.12) for u, and substi­tuting this in equation (9.13), we optain the Marshallian demand functions

Alternatively, one may specify a valid functional form for an in­direct utility function, such as

F -V(P, F) _

n

k~O

and apply Roy's identity to get

O,l, ••. ,n.

There has been recent emphasis on specifying preferences via the specification of a valid expenditure function. Two recent and notable examples are the works of Deaton and Muellbauer (1980) and Blundell and Ray (1984). In practice, there are a limited number of published structural forms of valid expenditure functions. We shall demonstrate below how one may generate valid expenditure functions from known expenditure functions.

Definition 9.1: A function e:R~!l X R + R+ is said to be a valid neo­classical expenditure function if it satisfies the following properties:

(a) e(pl, u) ~ e(pO, u) if pl ~ pO;

(b) e(ep, u) = ee(P, u) for all e > 0; (c) e(P, u) is concave in P; (d) e(P, u) is nondecreasing in u.

103

Given this definition, we now state and prove the following

proposition:

Proposition 9.1: Let 1 e , 2 e , em denote m valid neoclassical ex-

penditure functions, and let ~k be positive scalars. Then the

function

e* ::

is a valid neoclassical expenditure function.

Proof: Ne must show that e*(P, u) satisfies the four properties

listed in Definition 9.1. Note that

Hence properties (a), (b), and (d) follow directly from the properties k of each e (P, u), and (c) follows from the fact that the sum of con-

cave functions is a concave function. Q.E.D.

Let us close this section with a couple of simple examples il­

lustrating Proposition 9.1. First, let 1 n

e (P, u) = ui~OPiai; a k > 0

and

2 e (p, u)

Then for 0 < ~ < 1, the function

is a valid neoclassical expenditure function.

Now consider

1 e (P I u) 1

and

Then the resulting expenditure function, setting a l = a 2 just

which is just a form used by Blundell and Ray (1984).

1, is

104

4. An Example

In this section we provide an example of how aggregate data may

be used, to estimate an index. The function that we shall use is

the one Blundell and Ray (1984) suggest. The form of the aggregate

"expenditures" on leisure, given the aggregation restrictions, is

- ljIo

n m - l'~lP1' .Ela, .C.] J= 1J J

and the form of the aggregate expendit.ure on good k is

¥ (P P )~ + S [z*- ¥ ¥ (P P)~ W j=lYkj k i k j=l i=lYij i j - YOO

n m n - i~lPi j~laijCj] + Pk j~lakjCj'

As this system contains n[(n+l)/2 + m + 1] + 1 parameters (where n

is the number of commodities), it is necessary to restrict the number

of commodities and demographic variables used. As a result, we used

leisure and three broad commodity groups (durables, nondurables, and

services) in the estimation. In addition, we used household size

and number of children per household as demographic variables. The

resulting system has 16 independent parameters. We used annual U.S.

data from 1947 to 1980, and obtained commodity group expenditures

and demographic variables from various u.s. government agencies.

The data on average marginal tax were obtained from Barroand Sahasakul

(1983). A more detailed description of the data is available in our

1986 pap'er.

The full information, maximum likelihood estimates of the share

equations are reported in Table 9.1. Note that (1) all of the Si's are

statistically significantly positive, indicating services, durables

and nondurables are all normal goods. Leisure is required to be a

normal good given our selection of functional form; (2) we performed

log likelihood ratio test and rejected both the Blundell and Ray form

without demographics and the linear expenditure system with demo­

graphics. This suggests that the inclusion cross price effects and

demographics variables significantly improve the explanatory power

of the model.

Commodity

Group

t3 i

Leisure 0.120 ( -)

Durab1es 0.157 (0.043)

Nondurables 0.237 (0.024)

Services 0.486 (0.081)

105

TABLE 9.1

Maximum Likelihood Parameter Estimates with Asymptotic Standard Errors

Nondemo raphic Parameters

YiO Yil Yi2 Yn

-2730.009 * * * (210 .100)

* 817.296 -144.269 552.685 (1013.723) (1013.723) (1611.026)

* -2524.630 1304.159 (4544.457) (1790.947)

* -7555.611 (10684.68)

Demographic Parameters

Family Size Children

ail a i2

761.806 -2268.528 (1274.176) (1403.242)

2896.550 -3682.970 (1665.260) (2086.808)

6127.100 -9846.641 (4653.296) (5556.855)

* These parameters are restricted to be zero by the aggregation restrictions.

Convergence Criteria: 0.01

Method of Iteration: Gauss

Number of Iterations: 18

Loglikelihood Value: 299.225

Initialized Parameter Values: Y22 = Y33

-2000.00

131 13 2 = 133 = 0.20

100.00

Y12 Y13 = Y23 = -50.00

0.00

106

In Table 9. 2 we provide estimates of the rates of change of the income- and expenditure-based cost of living indices for the period 1970 to 1980. The two are related by the formula, derived in Chap­ter 8,

(9.14) { l-A } TI _ { _T_ } eTI I-T C I-T 0

where A is the average tax rate, e is the fraction of total expendi­ture allocated to leisure, and TIO is the rate of change of expendi­tures on leisure.

A comparison of the first two columns of Table 9.2 demonstrates that the distinction between the expenditure- and income-based cost of living indices may be substantial. In 1980, for instance, the difference ,,,as over 1.1 percent. Thus, income would have had to in­crease by 9.521 percent in 1980 for expenditures to increase at 8.404. Moveover, the average difference between the two indices for the 11-year period was 0.645 percent annually.

This difference may be interpreted as the "inflation tax" because

both TIc and TI~ allow for substitution among commodities and leisure. Thus, any difference between the two indices is due strictly to taxa­tion. In the last columns of Table 9.2 we present the two components of the income-based index, corresponding to the first and second terms of the right side of equation (9.14). The term

- [T/l-T)]eTIo reflects the tax deductibility of leisure. The deducti­bility of leisure substantially lowers the "inflation tax," but in no year is this term sufficiently large so as to make the inflation tax negative.

Year

1970

1971

1972

1973

1974

1975

1976

1977

1978

1979

1980

107

TABLE 9.2

Empirical Comparison of Income and Expenditure-Based Cost of Living Indices, 1970 - 1980.

Annual Percentage Annual Percentage Composition of the Change in the Change in the Full Income-Based

Expenditure-Based Full Income-Based Cost of Living Index Cost-of-Living Index Cost-of-Living Index

(ITc) (IT}J) IT {I-A} C 1-, -IT {-'-}8 o 1-,

5.257 5.624 6.791 -1.666

4.677 4.994 6.031 -1.037

5.088 5.198 '6.573 -1.375

6.975 7.399 9.243 -1.844

7.711 8.688 10.411 -1. 723

5.718 6.694 7.816 -1.122

6.383 6.854 8.808 -1.954

6.704 7.383 9.343 -1.960

7.250 7.977 10.538 -2.560

8.673 9.697 12.357 -2.661

8.404 9.521 12.274 -2.753

CHAPTER TEN: CHANGES IN GOVERNMENT GOODS AND THE TAX CODE

Throughout this monograph we have focused on behavioral and index concepts that view government services and the tax system as fixed. In practice, the consumer may receive benefits from taxes (eg., public goods), and the government may from time to time change the tax code. This chapter provides a brief look at these issues, and shows how earlier results may be extended to deal with the issues. We consider first the issue of government goods, and then the issue

of tax code changes. Finally, some preliminary results concerning the indexation of tax codes are presented.

A. EXOGENOUS GOVERNMENT SERVI~ES

Let Q denote a vector of government provided goods and services that are exogenous the the atomistic consumer. If preferences are defined over private goods (X) and these government goods (Q), then

it is clear that the resulting behavioral equations (eg., demand and compensation functions) will also depend on the exogenous level of Q. If Q changes over time, the consumer's welfare may change.

It is easy to extend the index concepts developed in earlier chapters to incorporate changes in Q. For example, one may define an income compensation function, say V*(P, Q, u: T), analogous to Definition 4.4, and use the compensation function to construct an index that includes any benefits of taxation that arise due to changes in Q:

° u : T)

° u : T)

This index is similar to the one given in Definition 8.1, except that it explicitly recognizes that the level of government services may differ in the two periods, and thus alter the required change in full income needed to maintain a fixed standard of living. In this sense, one may view the indices developed in earlier chapters as conditional cost of living indices because they are conditional on a fixed level of government services.

B. EXOGENOUS TAX CHANGES

In practice, the government alters the tax code from time to time, and these effects may also be incorporated into the index concepts. Let TO and Tl denote the tax codes existing at two points in time. Then, for example, one may define a full income-based

109

cost of living index that explicitly accounts for the change in the

tax code: 1 ° Tl)

**(P ° , pl ° TO, Tl) jl(P , u :

I , u : - ° ° TO) jl (P , u :

Here, jl(') is defined as in Definition 4.1, and thus, the index defines the factor by which the consumer's base period full income must be adjusted after movement in prices and a change in the tax code in order to maintain a fixed standard of living. Depending

upon how much the tax code changed, there mayor may not be "bracket creep."

C. ENDOGENOUS TAX CHANGES

Several governments have recently indexed income tax codes to a

price index. With an indexed tax code, tax code changes are endogenous in the sense that they are determined by changes in prices (usually,

a simple price index). We present in this section a treatment of the effects of an indexed tax code.

Definition 10.1: A function n: R~~+2 + R++ is said to be a price

index if it satisfies the following four axioms:

(1. 5 I

Monotonicity:

n(pO, PI > n(pO, pI) if p > pI

n(pO, P) ~ n(pl , P) if pO > pI

Linear Homogeneity:

° ° nIP , AP) = h(P , PI, Ae:R++

Identity:

n(pO, pO) = I

Dimensionality:

n(APO, AP) = n(pO, PI, Ae:R++

Two examples of indices satisfying the above definition include:

n(pO, P) a . P ae:Rn+1 = pO ++ a .

and

° P. tl i

n {~} n 'II(P , P) n tl . e:R 1 I: tl . I

i=O p? 1 ++ i=O 1

1

110

Since pO is to be viewed as fixed, we shall write w(pO, P) - w(P) for notational convenience.

Definition 10.2: A tax function, T[F - POXO; w(P)] is said to be indexed to price index w(P) if T[B(F - POXO)7 w(BP») E BT[F -

POX07 w(P»).

A natural example of an indexed tax code is

T[F - POX07 w(P)] = w(P) ·f[(F - POXo)/w(P)],

where f[·] is some function. Note that T I [.], the marginal tax rate, is homogenous of degree zero in full income and prices when the tax system is indexed.

Definition10.3: The indexed budget set, denoted B(P, F7 w(P», is defined as B(P, F7 w(P» - {XeS'! I p. X + T[F - POX07 w(P») < F}.

Proposition10.~: If the tax system is indexed to price index w(P), then

B (P, F 7 w ( P» - B ( B P, B F 7 w ( B P» f or all S > O.

Proof: Suppose X*eB(P, F7 w(P». Then P • X* + T(F - POX07 w(P» ~

F so that, multiplying through by S and using Definition 10.3, we have

SP • X* + T(BF - SPOX07 w(BP» .$.. SF.

That is, X*eB( BP, SF7 w (BP». Similarly it is easy to show X*eB(SP, SF7 w(SP»

=> X*eB(P, F7 w(P». Q.E.D.

We summarize key results in the following proposition.

Proposition 10.2: Under an indexed tax system:

(a) Marshallian demands are homogeneous of degree zero in prices and full income 7

(b) The indirect utility function is homogeneous of degree zero in prices and full income7

(c) The Hicksian demands are homogeneous of degree zero in prices;

111

(d) The full income compensation function is homogeneous of

degree one in prices;

(e) All of the indices introduced in Chapters Five through Eight satisfy the homogeneity and dimensionality axioms.

Proof: All of the above properties are due to the zero degree homogeneity of the indexed budget set.

The upshot of the last part of Proposition 10.2, of course, is that the income based cost of living indices do not suffer from "bracket creep"

when the tax code is indexed.

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Multiple Criteria Decision Methods and Applications Selected Readings of the First International Summer School Acireale, Sicily, September 1983 Editors: G.Fandel, J.Spronk

1985. 56 figures, 35 tables. XIV, 402 pages. ISBN 3-540-15596-1

This book provides selected readings of the first international summer school on multiple criteria decision making, held in Acireale, Sicily, in September 1983. Its aim is ~o ~ive a . state-of-the-art survey of multiple cntena decI­sion methods, applications and software. It is addressed to interested students, academic researchers beginning ill fields such as computer science, operational research and management science and staff members ill government and industry illvolved in planning and decision makirIg. The first part of the book is devoted to the philosophy of multiple criteria decision making and to a survey of solution approaches for discrete problems. The second part is concerened with an evalua­tion of the usefulness of multiple criteria deci­sion makirIg in practice.

E.Schlicht

Isolation and Aggregation in Economics 1985. XI, 112 pages. ISBN 3-540-15254-7

Contents: The Setting of the Argum'ent. - On Isolation. - The Moving Equilibrium Method. - Econometric Implications. - The Nature of Macroeconomic Laws. - Epilogue: Economic Imagination. - References. - Author Index. -Subject Index.

M. Sattinger

Unemployment, Choice and Inequality 1985. 7 figures, 49 tables. XIV, 175 pages. ISBN 3-540-15544-9

This book examilles the earnmgs inequality generated when job search is used to assign workers to jobs. It explaills the differences ill earnmgs which are observed among otherwise identical workers and which are a substantial proportion of earnings inequality. Unlike some previous treatments, it distinguishes between choice and random outcomes as sourc;es of earnmgs differences. First a model is developed in which workers sear~h for jobs in a Markov process with two states, employment and unemployment. Firms at the same time search for workers and generate the wage offer distribution. This model is then used to study the costs of unem­ployment, the distribution ofunempl?yment and the distribution of wage rates. Usmg U.S. census data, costs of unemployment are found to exeed foregone wages. The distribution of accepted wages is shown to differ from the distribution of wage offers. Earnings illequality is then related to the distribution of un employ­ment, wage offers and reservation wages. With data from the U.S. census, estimates are found for the contributions of choice and random outcomes to earnings inequality. The book provides a systematic treatment of a source of inequality that has been neglected in the past, namely the earnings differences that arise for otherwise identical workers. It relates this illequality to the problem solved by job search, that of assigning worker to jobs.

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