An extended form of quasilinear utility functions

Journal Pre-proof An extended form of quasilinear utility functions Toshihiro Matsuda

PII: DOI: Reference:

S0165-1765(19)30450-1 https://doi.org/10.1016/j.econlet.2019.108893 ECOLET 108893

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Economics Letters

Received date : 20 October 2019 Revised date : 3 December 2019 Accepted date : 7 December 2019 Please cite this article as: T. Matsuda, An extended form of quasilinear utility functions. Economics Letters (2019), doi: https://doi.org/10.1016/j.econlet.2019.108893. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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An Extended Form of Quasilinear Utility Functions

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Toshihiro Matsuda∗ Department of Economics, Otemon Gakuin University, 2-1-15 Nishiai, Ibaraki, Osaka, Japan

Abstract

We show an extended form of quasilinear utility functions, which is characterized by the Gorman polar form and the property that a consumer surplus, a

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compensating variation, and an equivalent variation coincide for the changes of prices and income under a certain condition.

Keywords: Quasilinear utility, Gorman polar form, Extension

1. Introduction

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JEL classification: D11

Although the indirect utility functions that generate the Gorman polar form are fully characterized by Gorman (1961), the corresponding direct utility functions which can be expressed in a closed form are limited to several cases, such as homothetic utility, quasilinear utility, Stone-Geary utility, and the general-

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5

ized CES utility.1 We give another subclass of these direct utility functions, which can be regarded as an extended form of quasilinear utility functions for the following reasons.

First, a utility in this subclass is the sum of two functions: one is possi10

bly nonlinear and the other is linear in all of the consumed goods. Hence, it includes a quasilinear utility as a special case, which is linear in one of the con-

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sumed goods. Secondly, for a consumer with this utility, as is the case with a quasilinear utility, the consumer surplus, the compensating variation, and the ∗ E-mail 1 See

address: [email protected] Pollak and Wales (1992) for details of these functions.

Preprint submitted to Journal of LATEX Templates

December 3, 2019

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equivalent variation coincide when measuring welfare changes along any path 15

of income and prices satisfying the condition which will be defined in Section

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3. Furthermore, we can completely characterize this subclass of utilities by the converse of above statements. If a utility takes the Gorman polar form and if three kinds of measures on welfare changes coincide along any path of income and prices satisfying the condition to be defined later, then its direct utility 20

function has an extended form of quasilinear.

2. Functional Form of Utility and Demand

We define an extended quasilinear utility function by

n ∑ ki k2 k3 kn xi + f ( x1 − x2 , x1 − x3 , . . . , x1 − xn ), k k k k1 1 1 1 i=1

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U (x) =

(1)

where x = (x1 , . . . , xn ) is a vector of goods consumed, ki is an arbitrary constant (k1 ̸= 0 ) and f is an arbitrary function of n − 1 variables without its Hessian 25

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determinant being identically zero. It is assumed that U (x) is locally wellbehaved for maximization purposes. If ki = 0 for all i ≥ 2, the function (1) coincides with the quasilinear utility.

If ki ̸= 0 for all i, we can show that the functional form of (1) is invariant for any permutations of variables. Since any permutation can be ex-

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pressed as a product of transpositions, it is sufficient to show the invariance for a transposition (x1 , xj ) with j > 1. Using the equality (ki /k1 )x1 − xi = (ki /kj )xj − xi − (ki /k1 )((k1 /kj )xj − x1 ), we have U (x) =

n ∑ ki k1 ki kn xi + F ( xj − x1 , . . . , xj − xi , . . . , xj − xn ), k kj kj kj i=1 1

where F is an arbitrary function that is derived from f . We now prove that the extended quasilinear utility has the Gorman polar

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form. Let us denote the price vector by p = (p1 , . . . , pn ) , and the income of a

30

consumer by I. For simplicity, we denote the term (ki /k1 )x1 − xi by Xi . A set of function Xi = gi (Y ) denotes the inverse mapping of a set of Yi = ∂Xi f (X) for i ≥ 2.

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Proposition 1. The demand function corresponding to the utility (1) can be expressed as xi (p, I) = ai (p) + bi (p)I, where n (∑

ks p s

s=1

ai (p) =

n (∑

ks ps

s=1

with Yi =

n (∑

ks ps

s=1

n )−1 (∑ s=2

n )−1 ∑ s=1

n )−1 ∑

k1 ps gs (Y ),

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a1 (p) =

s=2

ki ps gs (Y ) −

n ∑

ks ps gi (Y )

s=1

)

for i ≥ 2

(ks /k1 )(ki ps − ks pi ), and bi (p) = ki

Proof. The first order conditions for utility maximization yield

n (∑ s=1

ks ps

)−1

.

n ( ∑ ) −1 p−1 1+ (ks /k1 )∂Xs f = p−1 1 2 ((k2 /k1 )−∂X2 f ) = pj ((kj /k1 )−∂Xj f ) for j ≥ 3.

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s=2

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This system of equations can be written in a matrix format as      p1 k2 /k1 − p2 p1 + p2 k2 /k1 p2 k3 /k1 p2 k4 /k1 · · · p2 kn /k1 ∂X2 f           p3 −p2 0 ··· 0   ∂X3 f   p3 k2 /k1 − p2 k3 /k1             p4 0 −p2 ··· 0   ∂X4 f  =  p4 k2 /k1 − p2 k4 /k1  .       .. .. .. .. ..     ..   ..    .   . . . . . .      ∂Xn f pn k2 /k1 − p2 kn /k1 pn 0 0 ··· −p2 Adding (kj+1 /k1 ) times the jth equation to the first equation for all j ≥ 2 in

the matrix gives

n ∑ ) (k2 ps − ks p2 )ks /k1 2 . ps ks /k1 ∂X2 f =

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n (∑ s=1

s=1

Substituting this equality into the jth equation respectively for j ≥ 2, we have ∂Xi f =

n (∑ s=1

ks p s

n )−1 ∑ s=1

(ki ps − ks pi )ks /k1 .

(2)

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Differentiating (2) with respect to I, we obtain n ∑ s=2

(∂Xi Xs f )(∂I Xs ) = 0 for i ≥ 2.

Since the determinant of a matrix (∂Xi Xs f ) ̸= 0 by the definition of the utility (1), we have

∂I Xi = 0 for i ≥ 2. 3

(3)

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Combining the budget constraint p1 x1 +· · ·+pn xn = I with xi = (ki /k1 )x1 −Xi , n ∑

ps Xs + I =

s=2

n ∑

(ks ps /k1 )x1 (p, I).

s=1

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we can deduce

Differentiation with respect to I, together with (3), yields ∂I x1 (p, I) = k1

n (∑

ks p s

s=1

)−1

.

Substitution of this equality into (3), respectively for each i, gives ∂I xi (p, I) = ki

n (∑

ks p s

s=1

)−1

= bi (p),

(4)

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which implies that the demand functions are linear in expenditure. From (4), we have ai (p) = (ki /k1 )a1 (p) − Xi for i ≥ 2. Substituting these equalities into p1 a1 (p) + · · · + pn an (p) = 0, we obtain

ai (p) =

n (∑

(

s=1 n ∑

ks ps

n )−1 ∑

k1 ps Xs ,

s=2 n ∑

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a1 (p) =

s=1

ks ps

)−1 (

s=2

ki ps Xs −

n ∑ s=1

ks ps Xi

)

(5) for i ≥ 2.

The equalities (4) and (5), combined with (2), give the expression of ai (p) and

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bi (p) as stated in Proposition 1.

It follows immediately from Proposition 1 that the indirect utility V (p, I) of (1) is linear in I, which implies the Gorman polar form. When the number of goods, n, is two, x(p, I) can be expressed as ( ) x1 (p, I) = (k1 p1 + k2 p2 )−1 k1 p2 f ′−1 ((k2 p1 − k1 p2 )/(k1 p1 + k2 p2 )) + k1 I , ( ) x2 (p, I) = (k1 p1 + k2 p2 )−1 −k1 p1 f ′−1 ((k2 p1 − k1 p2 )/(k1 p1 + k2 p2 )) + k2 I . 3. Properties on Welfare Change

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Let us denote the connected open set in price-income space by Ω, where the

three measures (CS, CV, and EV) on welfare impact caused by changes of price and income will be compared in this and the following sections. We denote by 4

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γ the smooth curve in Ω between an initial price-income vector (pA , IA ) and a

(6)

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final one (pB , IB ). A consumer surplus (CS) is defined by a line integral ∫ CS ≡ − [x(p, I)dp − dI]. γ

Using a money metric indirect utility function µ(p; q, I), which gives the minimum expenditure at prices p for a consumer to be as well off as he/she would be facing prices q with income I, we can write a compensating variation (CV) as µ(pB ; pB , IB )−µ(pB ; pA , IA ), and an equivalent variation (EV) as µ(pA ; pB , IB )− 40

µ(pA ; pA , IA ).

In the following propositions, we consider a welfare change on the curve satisfying the condition

dp1 dpi dpn + · · · + ki + · · · + kn = 0, dt dt dt

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k1

(7)

where (p(t), I(t)) denotes a bijective parametrization of γ from (pA , IA ) = (p(tA ), I(tA )) to (pB , IB ) = (p(tB ), I(tB )).

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Proposition 2. Assume that a consumer has the utility function (1). If the curve γ satisfies the condition (7), then CS, CV, and EV coincide. In particular, 45

if pi and I are free to vary for all i with ki = 0, and pj is fixed for all j with kj ̸= 0, then the three kinds of measures coincide.

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The particular case described in Proposition 2 is an extension of the wellknown property of quasilinear utility. Proposition 2 can be proved by using integral forms of CV and EV, together with the result of Proposition 1. Proof. An integral form of CV can be written as ∫ pB − grad µ(p; pA , IA )dp + IB − IA , pA

which equals

−

∫

pB

pA

x(p, µ(p; pA , IA ))dp + IB − IA .

(8)

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Let the curve (p(t), I(t)) satisfy (7) on [tA , tB ]. Using x(p, I) = a(p) + b(p)I and bi (p) = (k1 )−1 ki b1 (p) in (8), we have ∫ tB ∫ tB n dp b1 (p(t)) ∑ dps CV = − a(p) dt − [µ(p(t); pA , IA ) ] dt + IB − IA . ks dt k1 dt tA tA s=1 5

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(9)

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By substituting (7) into the above equality, CV can be simplified to ∫ tB dp − a(p) dt + IB − IA . dt tA

Similarly, we can show that EV equals (9). According to the definition (6), CS can be written as ∫ tB ∫ tB n ∑ dp dps I(t)[ bs (p(t)) dt] + IB − IA , − a(p) dt − dt dt tA tA s=1 50

which also equals (9) through a similar simplification.

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4. Characterization

We will show that the extended quasilinear utility is characterized by the converse of Proposition 2. To prove this, we make the additional assumption that the indirect utility function is twice continuously differentiable in Ω. Proposition 3. Assume that the indirect utility takes the Gorman polar form,

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55

and it is twice continuously differentiable in Ω. If two of three measures (CS, CV, and EV) coincide along any curve in Ω satisfying (7), then its direct utility is an extended quasilinear function with the form (1). We write the Gorman polar form as V (p, I) = (I − α(p))β(p)−1 , where α(p) and β(p) are homogeneous of degree one. According to Gorman (1961),

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60

the corresponding demands can be written as xi (p, I) = ai (p) + bi (p)I such that ai (p) = ∂i α(p) − α(p)∂i β(p)/β(p) and bi (p) = ∂i β(p)/β(p). The proof of Proposition 3 is divided into a sequence of three lemmas. In Lemma 1, from the assumption, we derive a system of partial differential equations, which 65

determines β(p). Lemma 2 shows its solution, from which we can deduce an explicit function of the demand. In Lemma 3, it is proved that the corresponding

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direct utility is equivalent to the function (1).

Lemma 1. From the assumption of Proposition 3, it follows that ∂i β(p) = (ki /k1 )∂1 β(p) for i ≥ 2 in Ω, which is equivalent to bi (p) = (ki /k1 )b1 (p).

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Proof. Suppose first that CS = CV on any curve in Ω satisfying (7). Consider a smooth curve γ in the neighborhood of (pA , IA ), which is parametrized by

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(p(t), I(t)) in [tA , tB ] with (p(tA ), I(tA )) = (pA , IA ). We denote by γm a part of the curve γ(t) on [tA , tm ] with tm < tB . Since CS = CV also on γm , subtracting CS from CV gives ∫

n ∑ dps [I(t) − µ(p(t); pA , IA )][ bs (p(t)) ]dt = 0. dt tA s=1 tm

(10)

Differentiating (10) with respect to tm , we have

n ∑ dps ] = 0. [I(tm ) − µ(p(tm ); pA , IA )][ bs (p(tm )) dt s=1

k1

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We now construct a curve γ such that

dp1 dpi dpj + ki = 0, = 0 for j ̸= 1, i, dt dt dt

(11)

(12)

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and I(t) = µ(p(t); pA , IA ) + ε(t − tA ) with ε > 0, where ε is sufficiently small so that the curve γ could be included in Ω. Since 1 I(tm ) − µ(p(tm ); pA , IA ) > 0 for tA < tm ≤ tB , (11) implies b1 (p(tm )) dp dt +

i bi (p(tm )) dp dt = 0. Letting tm → tA , by the continuity of b(p, I), we have

dpi 1 b1 (p(tA )) dp dt + bi (p(tA )) dt = 0. Hence, using (12), together with the smooth-

75

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ness of γ, we obtain bi (pA ) = (ki /k1 )b1 (pA ). Since we can choose any point in Ω as (pA , IA ), this gives bi (p) = (ki /k1 )b1 (p) in Ω. Similar arguments can be applied to the case CS = EV .

Next we consider the case CV = EV . Since µ(p; q, I) = α(p) + (I − α(q))β(p)/β(q), subtracting EV from CV gives (IB − α(pB ))(1 − β(pA )/β(pB )) + (IA − α(pA ))(1 − β(pB )/β(pA )) = 0.

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Differentiation with respect to IB yields β(pA ) = β(pB ). Substituting p1A = ∑n ∑n k1−1 ( s=1 ks psB − s=2 ks psA ), which holds by (7), into β(pA ) = β(pB ) gives β(k1−1 (

n ∑ s=1

ks psB −

n ∑

ks psA ), p2A , . . . , pnA ) = β(p1B , . . . , pnB ).

s=2

7

(13)

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Differentiating (13) with respect to piB , we have n n ∑ ∑ (ki /k1 )∂1 β(k1−1 ( ks psB − ks psA ), p2A , . . . , pnA ) = ∂i β(p1B , . . . , pnB ). s=2

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s=1

(14)

Letting pB → pA in (14), by the continuity of ∂i β(p), we have ∂i β(pA ) = (ki /k1 )∂1 β(pA ). Since we can choose any point in Ω as (pA , IA ), this gives ∂i β(p) = (ki /k1 )∂1 β(p) in Ω. 80

Lemma 2. If ∂i β(p) = (ki /k1 )∂1 β(p), the demand function can be written as (∑n )−1 xi (p, I) = ai (p) + ki I. s=1 ks ps

Proof. The system of partial differential equations ∂i β(p) = (ki /k1 )∂1 β(p) for

re-

85

i ≥ 2 is Jacobian complete. Therefore, the adjoint system of total differential ∑n equations, which is completely integrable, is dp1 = i=2 (−ki /k1 )dpi . It has ∑n the integral s=1 ks ps = constant. Hence, the general solution of this system is ∑n β(p) = H( s=1 ks ps ), where H is an arbitrary function. From the homogeneity

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of β(p) with degree one, H must be a homogeneous linear function. We thus (∑n )−1 get bi (p) = ki . s=1 ks ps

Lemma 3. If the demand function is xi (p, I) = ai (p) + ki 90

(∑n

s=1

ks ps

)−1

I, the

corresponding direct utility is an extended quasilinear function with the form

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(1).

It is not obvious, for an arbitrarily given a(p), that there exists some function f (X) which satisfies the equalities in Proposition 1. To prove this, we first consider a system of partial differential equations such that ∂i f (X) must 95

satisfy for i ≥ 2. Next, we examine the integrability conditions of these partial differential equations, which ensure the existence of f (X).

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Proof. Since gi (Y ) = (ki /k1 )a1 (p) − ai (p), we have gi (Y ) = (ki /k1 )∂1 α(p) − ∂i α(p) for i ≥ 2.

(15)

By the definition of gi (Y ), the inverse mapping of Xi = gi (Y ) yields a system of partial differential equations for ∂i f (X). Since 8

∂(∂2 f,...,∂n f ) ∂(X2 ,...,Xn )

∂(g2 ,...,gn ) −1 = ( ∂(Y ) 2 ,...,Yn )

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from the inverse function theorem, together with the fact that a matrix is symmetric if its inverse matrix is symmetric, the integrability condition ∂i ∂j f (X) =

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∂j ∂i f (X) is equivalent to ∂j gi (Y ) = ∂i gj (Y ) for any i ̸= j. For the calculation of the partial derivative of gi (Y ), it is necessary to rewrite the right side of (15) as a function of Y . First, using the homogeneity of α(p) with degree one, we can write

α(p) = p1 L(p2 /p1 , . . . , pi /p1 , . . . , pn /p1 ),

(16)

where L is an arbitrary function. Substituting (16) and its partial derivative into (15), we have n ∑ s=2

(ps /p1 )(∂s L)] − ∂i L,

(17)

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gi (Y ) = (ki /k1 )[L −

which can be regarded as a function of pi /p1 . Next, solving the equation (2) for pi /p1 , we obtain

pi /p1 = (−k1 Yi + ki )(k1 +

n ∑

ks Ys )−1 .

(18)

s=2

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To shorten notation, we write zi for pi /p1 . From (17) and (18), we have ∂gi /∂zt = −(ki /k1 ) ∂zt /∂Yj = −kj zt (k1 +

s=2

∂j gi (Y ) = [(ki kj /k1 )

s=2

zs (∂t ∂s L) − ∂t ∂i L,

ks Ys )−1 for t ̸= j, ∂zj /∂Yj = (−kj zj − k1 )(k1 +

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Hence, we obtain

n ∑

n ∑

n ∑ n ∑

zt zs (∂t ∂s L) + kj

t=2 s=2

+ ki

n ∑

n ∑

n ∑ s=2

zs (∂s ∂i L)

s=2

zs (∂j ∂s L) + k1 (∂j ∂i L)][k1 +

n ∑

ks Ys ]−1 .

s=2

s=2

Since ∂i ∂j L = ∂j ∂i L by the twice continuously differentiability of α(p), ∂j gi (Y )

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is invariant for a transposition (i, j). This implies ∂j gi (Y ) = ∂i gj (Y ). Acknowledgements

100

This research did not receive any specific grant from funding agencies in the

public, commercial, or not-for-profit sectors. 9

ks Ys )−1 .

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References

53-56. 105

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Gorman, W.M., 1961. On a Class of Preference Fields. Metroeconomica. 13,

Pollak, R.A. and Wales, T.J., 1992. Demand System Specification and Estima-

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tion. Oxford University Press, NewYork.

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Highlights ・ An extended form of quasilinear utility functions with a symmetry on the variables

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・New subclass of utilities with Gorman polar form

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・ Coincidence of consumer surplus, compensating variation, and equivalent variation