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An axiomatic approach to compensated demand
JOURNAL
OF ECONOMIC
THEORY
An Axiomatic
52, 111-122 (1990)
Approach
to Corn
SUSANNEFUCHS-&LEER Institut Kaiserstraj’e
fir Wirtschaftstheorie 12, O-7500 Karlsruhe
und Qperations Research, I, Federal Republic of Germany
Received October 18, 1988; revised November 9, 1989
Starting with hypotheses on functions which can be interpreted as income compensation functions, a theory of consumer behaviour is developed. If people act rationally they will behave according to these hypotheses. From these axioms many propositions, well-known from traditional neoclassical consumer theory, can be deduced, In particular, Shephard’s lemma can be proved within this theory. Journal of Economic Literature Classification Number: 022. 0 1990 Academic Press, Inc.
I. 1biTRoDucT10~
Consumer behaviour model analysis can be started in several different approaches. Traditional neoclassical consumer theory uses the approach of utility functions from which the demand functions can be derived. Another approach is due to L. McKenzie who starts with the preference relation of the individual [9]. Other well-known models, which are integrab~lity theory and the theory of revealed preference, assume hypotheses on demand functions from which the existence of a utility function can deduced. In all of the above mentioned models income compensation functions, based on a preference relation 3, are defined by NP,
x)=minIpy/
Y?=x>,
supposing that this minimum exists always. Under appropriate conditions these income compensation functions can be used as money-metric anility functions, representing consumer’s preferences. This was pointed out by Samuelson [ll], Varian [13], Weymark [14], Chipman and Moore Cl], Fuchs-Seliger [4, 5] and others. Recently S. Honkapohja introduced a model of consumer behaviour in which he started with functions that can be interpreted as income compensation functions [7, pp. 553-5551. In this paper the idea of ~o~ka~o~ja will be further developed by presenting a system of hypotheses on f~~~tio~s which can also be interpreted as income compensation functions. ne of 111 0022-0.531/90 83.W 642:52/M
Copyright Q 1990 by Academic Press, Inc All rights of reproduction in any form reserved
112
SUSANNEFUCHS-SELIGER
the hypotheses, which require rational behaviour, is very near to the Weak Axiom of Revealed Preference. Based on this system of axioms a relation will be defined which can be plausibly interpreted as the consumer’s preference relation. It will be shown that this relation is complete, transitive, convex, and continuous. Finally, it will also be demonstrated that Shephard’s lemma can be deduced, and thus one of the most important laws in demand theory follows from these axioms.
II. HYPOTHESESON INCOME COMPENSATIONFUNCTIONS We will base our investigation on the following system of axioms: The map M: rW: + x X -+ E4, XC Iw:, Xf 0, l b = M(p, x), satisfies the following properties: (CI) (CII)
vxex:
[Vp’pE[W:+:px~M(p,x)]; Vx, ye:X: [x#y and pxzM(p,
y), Vp~E[W:+ailp’~5!~+:
P’Y5 WP’, x)1; (CIII)
(i)
Vx, y E X: [3p” E IR; + : M(p’, x) = M(p”, y) - Vp E LRf++ :
MP, xl = M(P> Y)X (ii)
Vx, yEX MP, xl ’ WP, VII.
[3p”E
rWy+:
M(PO, x) > MPO, Y) * VP E q
+:
Remark. We can interpret M(p, x) as an income compensation function. Thus, M(p, x) is equal to the money income that an individual needs in order to achieve, in the price situation p, a commodity bundle which he likes at least as much as x. This interpretation is the reason we have assumed that X is a subset of the LQT. However, more generally we could have assumed that X is a subset of [w”. Then all the results of Sections III and IV would remain, and for Section V we would only have to assume additionally that X is bounded from below. If we consider the function M(p, x) = 2 dm for (p, x) E iw: + x I%: +, then we can easily see that this function satisfies (CI)-(CIII). For interpretation: (CI) states that in every price situation p the minimum income needed to achieve a commodity bundle equivalent to x is less or equal to the cost of x in that price situation p. (CII) states that if in every price situation p the individual’s minimum expenditure for a commodity bundle equivalent to y is not higher than for x, then there exists a price situation p’ where y is not more expensive than a commodity bundle equivalent to x. The interpretation of (CIII) is obvious: According to rational behaviour, in the price situation p1 we would not spend more ‘XER:,:
OXi>O, visn, x=(x I,..., X”), xs[w”,: ox;~O.
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COMPENSATED DEMAND
money for a commodity bundle equivalent to y than for a ~ornmodi~~ bundle equivalent to x, if at an earlier time we have spent more money for a commodity bundle equivalent to x than for a commodity bundle equivalent to y in the price situation p”. In the next section it will then be demonstrated that based on the above hypotheses a model of consumer behaviour can be established.
III. A PREFERENCERELATION GENERATEDBY M(g,x)
Assuming the hypotheses (CI)-(CIII) it will be shown now that we can Will define a relation which describes the preferences of the individual. see that this preference relation deduced from the consumer’s ome compensation function possesses nice properties like corn transitivity. The following lemma will help to show these properties. LEMMA 1. Condition (CIII)
implies
Proof Since M(p, x) and M(p, y) are defined for all x, y E X and PER”++, we have 3p: [M(p, x) 2 M(p, yf or M(g, y) 2 M(p, x)]. From this together with (CIII) immediately follows (Vp(Plq+:
WP, ~12 WP, Y))
or
(VpJpE[w”,+:
If in all price situations the commodity bundle x is at least as expensive as a commodity bundle which the individual estimates as y then according to rational behaviour the individual will prefer x to y or be indifferent. T suggests the following definition of a preference relation according to S. Honkapohja [7, p. 5541:
vx, yex:
xRy -\dp~R~+:px~AI(p,
We will now show that the income compensation represents the relation R.
y).
function
THEOREM 2. Under the conditions (CI)-(CIII) the function QnYPOEW++, represents consumer’s preferences; in other words, \J’x, Y E x,
XRY -a M(p’, xl 2 WP’,
~1.
114
SUSANNE FUCHS-SELIGER
Prooj Consider x, y E X, x # y and xRy. Since by the definition we have ‘dp: px 2 M(p, y), this together with (CI) and (CII) yields
of R
3p’: M(p’, y) 5 M(P’, x).
Using (CIII) we obtain Vp: M(p, x) 2 M(p, y) and thus, especially, MPO, x) 2 MPO, Y).
Conversely, assume M(p”, x) 2 M(p’, y). Then by (CIII), Vp E L’k?:+ , M(p, x) 2 M(p, y). Using (CI), from this result xRy immediately follows. Based on R, we can define the relations P (better than) and I (indifferent to) by vx, ye&x-I
xPy + xRy and 1 (yRx) xIy +xRy
and yRx.
We will now deduce nice properties of R. As a corollary to Theorem 2 we obtain COROLLARY.
0) (ii)
Under the assumption of (CI)-(CIII)
xpy 0 WP, x) > Wp, XIY -M(P,
3.
y), VP E RT +
x) = WP, Y), VPE RT +.
The proof immediately 2, and condition (CIII). THEOREM
we have, Vx, y E X,
follows from the definition of P and 4 Theorem
The conditions (Cl)-(CIII)
imply:
(01) R is complete and transitive; (j?)
if X is convex, then R is a convex relation;
(y)
R is continuous from above;
(6)
R is strictly monotonic, i.e., x 2 y A x # y * xPy.
Proof for (a). In view of Lemma 1, for every x, y E X we have (Vp: M(p, x)zM(p, y)) or (VP: M(p, y)>=M(p, x)), and in view of (CI), pz 2 M(p, z), Vz E X. Thus, the completeness of R immediately follows by the definition of R.
In order to show transitivity, consider x, y, z E X such that x # y, z and y # z, and xRy and yRz. From this we obtain VP: PX 2 M(p, Y)
and
PY 2 M( P, z).
(1)
COMPENSATED
DEMAND
115
Then in view of (CII), ( 1) implies 3p’ c rW: + : p’z 5 M(p’, y). This together with (CI) gives M(p’, z)<=M(p’, y). Thus, in view of (C~I~~~ VP: M(p, z) 5 M(p, y). This together with (1) implies VP: px 2 M(p, z). Thus XRZ. If x, y, z are not pairwise unequal, then transitivity also follows by the application of Lemma 1. Proo~~or (fl). In order to show that R is convex, assume xRy and x f y for any x, y E X. Hence we have
Using (CI) and (2) we obtain, for any AE ]O, l[, tips Ryb + :
Hence, (Ax+ (I-A)y)Ry.
Proof for (y). In order to show that R is upper semicontinuo consider any sequence (xk) c X such that xkRy and xk -+ x0 E X. Then for every PE Et:+, pxkz M(p, y). Since xk 3 x0 we get pxO 2 M(p >y) > Vp . Thus x”Ry. Proof for (6). Consider x, y such that x 2 y A x # y, Using (CI) we immediately obtain
Thus, xRy. Suppose yRx. Then py 2 M(p, x), ‘dp E W+ + . By (CII) there exists p’ such that p’x _I M(p’, y), contradicting (*). This proves (S). Remark. If X is a nonempty connected subset of o;S:, then, as in traditional consumer theory, we can conclude that there exists an upper semicontinuous ordinal utility function representing the relation follows, since R is a complete, transitive, and upper semi~o~tin~o~§ relation on R.
Using the above results, we will now start to compare our axiomatic foundation of a theory of consumer behaviour, based on income compensation functions, with the approach used by S. Honkapohja. His hy~otbe~~~ are the following [7, Proposition 7, p. 5531. The function m = (PYx) mapping KY++ xX into R has the properties: (Ml )
It is homogeneous of first degree and concave in p,
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SUSANNEFUCHS-SELIGER
(M2) It is continuous in (p, x). (M3) For all p~Iw:+ and XEX there exists JJGX such that px &y = M(p, x) and p’y 2 M(p’, x) for all p’ > 0. (M4) For all X, VEX: px2 M(p, y) for all p E R: + implies M(p, x) 2 M(p, y) for all p E rW: + . (M5) For all x, y EX either M(p, x) 2 M(p, y) for all p E rW: + or M(p, x) 5 M(p, y) for all p E lF2: + . (In Section II it was already mentioned that also in my approach X can be a subset of R”.) We can see from Lemma 1 that conditions (CIj(CIII) imply (M5). Using the definition of R, instead of (M4) one can write xRy * M(p, x) 2 M(p, y) for all p E rW: + . Thus in view of Theorem 2, conditions (CI)-(CIII) also imply (M4). Condition (Ml) is a well-known property of income compensation functions; thus in my model, the function M(p, x) should also obey it. In the next section we will see that conditions (CI)-(CIII) and the additional hypothesis (CV) also imply (Ml) and (M3 ). Now we are looking for a condition which guarantees that M(p, x) can be identified with the McKenzie income compensation function. This will be done in the next section.
IV.
INTRODUCTION OF A MCKENZIE-TYPE INCOME COMPENSATIONFUNCTION
According to traditional neoclassical consumer theory we can construct a McKenzie-type income compensation function on the basis of the relation R by m(p, xl = jff
(py I YRx),
vpEIW;+,
XEX.
We can immediately recognize that conditions (CIj(CIII), which imply that R is complete and transitive, also yield the following result. LEMMA 4.
The conditions (CI j(CII1) m(p, x) 2 M(P, x),
Prooj: inf,,,(pyl
If
impZy VXEX, vpdpkq,.
yRx, we have py 2 M(p, yRx} ZM(p, x), VxeX.
x),
Vp E R”,,.
Thus,
117
COMPENSATED DEMAND
In order to establish equality between y~(p, x) and M(p, x) we add the following axiom: VP E R”, + > Vx E X: 3z E X: zRx and pz = M(p, x).
(CIV)
The interpretation of (CIV) is obvious. From the conditions (CT)-(CIV) we can conclude that the function A&, x) coincides with the McKenzie-type income compensation function. THEOREM
5.
Conditions (CI)-(CIV)
imply
WP, x) = m(p, x),
V(p,X)ER”,+XX.
Proof. Since by (CIV) for any (p, x) E IL!: + x X there exists z E X such that zRx and pz = M(p, x), we have pz 2_m(p, x) and M(p, x) 2_ m(p, x), This together with Lemma 4 proves our theorem. Under the above conditions, from the equality between m(p, x) asa M(p, x) we can conclude that M(p, x) is concave with respect to p. THEOREM
6.
Under the hypotheses (CI)-(CIV)
we have
(a) M(p, x) is concave with respect to p, and thus continuous wit respect to p, (p)
1~M(p,x)=M(A*p,x),
v/l>o.
Prooffor (a). Concavity of M(p, x) with respect to p foIIows in the same way as in traditional neoclassical demand theory (see for instance Diewert 12, p. 53981). Moreover, since M(p, 2) is concave for any fixed jc and is defined on an open set, continuity of M(p, x) with respect to p follows from a well-known mathematical theorem (see Takayama Cl;?, p. 6581). Proof for (fi).
Since m(p, x) = M(p, x), the proof immediately follows.
Remark. Continuing the comparison between (Ml k(M5) an (CI)-(CIV) we see that Theorem 6 corresponds to (Ml). conditions (CI) and (CIV) imply condition (M3) of S. Honkapohja: By (CIV) we have that for all p > 0 and x E X there exists y E X such t yRx and py = M(p, x). Thus by (CI), px zp,y = M(p, x). Since yRx mexx that p’y 2 M(p’, x) for all p’ E W+ + , this together with the previous result gives (M3). Thus we have seen that conditions (CI)-(CTV) imply (Ml), (M3), ( and (M5). However, it seems that the hypotheses (CI)-(CIV) are transparent and describe the individual’s observable behaviour in a kess technical way than (Ml t(M5) do. Moreover, in Theorem 2 we could also show that M(p, x) represents the relation R. This property can be deduce
118
SUSANNE FUCHS-SELIGER
from (Ml)-(M5) tion 1, p. 2223).
under further conditions
(see for instance [14, Proposi-
If we additionally require that M(p, X) be continuous with respect to x, then we also get continuity of M(p, X) in (p, x), which corresponds to (M2). The proof of this property is an immediate consequence of the following lemma. LEMMA 7. Let X be the function f: Xx Y -+ continuous with respect with respect to y, then f
an interval of [w” and Y be an interval of W” and let [w, t = f (x, y) be continuous with respect to x and to y. IA additionally, f is increasing (decreasing)2 is jointly continuous with respect to (x, y).
The proof of Lemma 7 can be easily done (see [6] ). THEOREM
(CI)-(CIV) (CV)
8. Let X be an interval in R”, . Then under the conditions and M(p, x) is continuous with respect to x,
it follows that M(p, x) is continuous in (p, x), V(p, x) E iw: + x X. Proof. By Theorem 5, M(p, x) = m(~, x). Hence by the definition of m(p, x), M(p, x) is increasing with respect to p. This together with the continuity of M(p, x) with respect to p and with respect to x, in view of Lemma 7, implies that M(p, x) is continuous in (p, x). From Theorem 8 the continuity THEOREM
9.
of the relation R also follows.
Under the hypotheses of Theorem 8 the relation R is
continuous. ProoJ In view of Theorem 3 (y) it suffices to show that R is lower semicontinuous. Therefore, consider y E X and a sequence (xk) -c X such that yRxk, f/k, and xk + x0 E X. Then for every p E rW; + we have py 2 M(p, xk), Vk. Since M is continuous with respect to x, we obtain py 2 M(p, x0), vplF[W;+. The following condition implies strict convexity of the relation R: (CVI) The function M(p, x) is strictly quasiconcave with respect to x. Using this condition,
we obtain
THEOREM 10. Under the hypotheses of Theorem 3 (/?) and (CVI) relation R is strictly convex. 2 z 2 z’ means zi 2 zi Vi s n. z > z’ means respect to Y means y1 2.9 ==-fb, $1 Bfb,
zi ;s zi Vi s n. f(x, y) is increasing v’) (f(x, Y’) 5f(x, u’)).
(decreasing)
the with
119
COMPENSATED DEMAND
Proof. In view of Theorem 3 (j?) we know that R is convex. Assuming xRy, x # y, and checking the proof of Theorem 3 (p) once again, we see that by strict quasiconcavity of M(pT x) with respect to x, instead of (3) we obtain, VJ, E 10, 1 [: wiWn++.
M(p,ax+(i-a)Y)>M(p,~)
Hence (2x+(1-1)y)Py.
V. COMPENSATED DEMAND F-UNCTIONS Using the relation R one can define a compensated spondence in the usual way by g(p,x)=argmin{py/yRx},
demand corre-
V(p,x)ERBn,+ xx.
If X is a closed subset of !RT, then in view of Theorem 3, under the conditions (CI)-(CIII) we have that for all (p, x) E R”, i- xX, g(p, x) is well-defined. is strictly convex, Since under the conditions (CI)-(CIII) and (CVI) we immediately obtain: THEOREM 11. If Xc KY+ is closed, convex, and if(U)-(CIH) hold, then g(p, x) is single-valued. THEOREM
(@I)-(CIV)
and (CV1)
12. Let Xc KY+ be closed and convex and let the conditions hold, then the equality p . y = M(p, x), Vy ~g(p, x) is fuljXed.
Prsof. By the definition of g(p, x) we have p. y = m(p, x) Vy ~g(p, x)? and then, by Theorem 5, p . y = M(p, x). THEOREM 13. Under the conditions of Theorem 12, g is komoge~e~~s of degree zero in p, i.e., y Eg( tp, x) o y Eg(p, x), ‘dt > 0.
Prroof:
The proof of Theorem 13 is standard.
THEOREM 14. If (CV) and (CVI) are added to the assurn~~~o~s of Theorem 12, then g(p, x) is a continuous function.
Proof. Since under the assumptions of the theorem, R is ia continuous, strictly convex, and strictly increasing preference ordering, the continuity of g in (p, x) follows as in Nicaido [lo, p. 2993.
120
SUSANNE FUCHS-SELIGER
Remark. More generally, assuming conditions (Cl)-(CV), in view of Theorem 3 all the assumptions of Proposition 4 in [7, p. 5481 are fultilled, and thus, g(p, x) is an upper semicontinuous correspondence.
As an important consequence from the hypotheses (CI)-(CV) Shephard’s lemma. This means
15. Let XE rW; be closed and convex, and assume conditions and let g(p, x) be single-valued. Then
THEOREM
(CI)-(CV)
8M(P, x) aPj where gi is the jth component ofg(p, Proof:
we obtain
= gj (P, x)3 x) = (g,(p, x), .... g,(p, x)).
Consider any two price vectors p and p + dp. Then we obtain
AMP
+ dp, x) = Wp + 4, x) - M(p, x) = (P + AP) .g(p + 4, x) -p .g(p, x) =(P+~p)~(g(p+~p,x)-g(p,x))+dp~g(p,x).
(1)
By Theorem
12 and the definition of g we have that p .g(p + dp, x) We now can proceed as in Hurwicz and Uzawa [IS, pp. 121-1221 using that g(p, x) is continuous with respect to p and M(p, x) is concave with respect to p.
p .g(p, x) 10.
Remark. Since gj(p, x) is continuous with respect to p the above result implies that the partial derivative aM(p, x)/apj is also continuous with respect to y.
The following section deals with income compensation are linearly homogeneous in x.
functions which
VI. THE CASE OF HOMOGENEITY We will now investigate the case when the income compensation function is homogeneous of degree one in x. Therefore we assume: (CVII) THEOREM
(CI)-(CIII) v/l>o.
VA > 0: M(p, Lx) = RM(p, x), V(p, x) E lq + XX 16. Let X be a cone in rW:, and let M(p, x) satisfy and (CVII). Then the relation R is linear, i.e., xRyo ;IxR,Iy,
COMPENSATED
121
DEMAND
xRy ~vpEEIWn++:px~:(p,
y)
‘da > 0:
-QPE@“++,
VA> 0: p . ix 2 M(p, Ay)
0 IxRlly,
apx
2 .aM(p,
y)
--VPE&+,
VI?> 0.
17. Let X be a closed convex co~le in R”, f and as.mme and (CVII). Then g(p, x) is homogeneous of degree one in x, i.e., g(p, ax) = 2”. g(p, x), v’i, > 0. THEOREM
(@I)-(CIII) Proof
g(p, 1*x) = arg min (py ] yRdx) = arg min
p
= arg min (Apy’ I Ay’RAx} = arg min { Apy’ / y/Rx], =/Zargmin
by Theorem 16
{py’Iy’Rx)=II~g(p,x)=(3,yjy~g(p,x)).
VII.
SUMMARY
The preceding results have shown that the hypotheses (CI)-(CV) establish a reasonable model of consumer behaviour based on income compensation functions. We have seen that from the axioms of the model propositions can be deduced, which can also be demonstrated in traditional neoclassical consumer theory where utility functions are given However, in this model based on income compensation functions the problematic notion of utility can be avoided.
REFERENCES 1. J. S. CHIPMAN AND 3. C. MOORE, Compensating variation, consumer’s surp!us Amer. Econ. Rev. 70 (1980), 933-949. 2. E. DIEAVERT, Duality approaches to microeconomic theory, in “Handbook of
and welfare,
Mathematical Economics” (K. .I. Arrow and M. D. Intriligator, Eds.), Vol. II, pp. 53.5-591, NorthHolland, Amsterdam, 1982. 3. W. FENCHEL, “Convex Cones, Sets, and Functions,” lecture notes, Princeton University, Princeton, NJ., 1953. 4. S. FUCHS-SELIGER, On the role of income compensation functions as money-metric utility functions. in “Measurement in Economics” (W. Eichhorn, Ed.), Physica Verlag. Wiirzburg, 1987.
122
SUSANNE
FUCHS-SELIGER
5. S. FUCHS-SELIGER,Money-metric utility functions in the theory Mud SW. Sci. 18 (1989) 199-210. 6. S. FUCHS-SELIGERAND M. KRTSCHA, “An Alternative Approach Economics,” Discussion Paper No. 320, University of Karlsruhe, 7. S. HONKAPOHJA, On continuity of compensated demand, Inr.
of revealed preference, to Joint Continuity in 1987. Econ. Rev. 28 (1987),
545-557.
8. L. HURWICZ AND H. UZAWA, On integrabihty of demand functions, in “Preferences, Utility, and Demand” (J. S. Chipman et al., Eds.), pp. 114-148, Harcourt Brace Jovanovich, New York, 1971. 9. L. W. MCKENZIE, Demand theory without a utility index, Rev. Econ. Stud. 24 (1957), 185-189.
10. H. NICAIDO, “Convex Structures and Economic Theory,” Academic Press, New York/ London, 1968. 11. P. A. SAMUELSON,Complementarity-An essay on the 40th anniversary of the HicksAllen revolution in demand theory, J. Econ. Lit. 12 (1974) 1255-1289. 12. A. TAKAYAMA, “Mathematical Economics,” Dryden, Hinsdale, IL, 1974. 13. H. R. VARIAN, Non-parametric tests of consumer behaviour, Rev. Econ. Stud. 50 (1983), 99-100. 14. J. A. WEYMARK, Money-metric utility functions, Int. &on. Reo. 26 (1985) 219-232.
OF ECONOMIC
THEORY
An Axiomatic
52, 111-122 (1990)
Approach
to Corn
SUSANNEFUCHS-&LEER Institut Kaiserstraj’e
fir Wirtschaftstheorie 12, O-7500 Karlsruhe
und Qperations Research, I, Federal Republic of Germany
Received October 18, 1988; revised November 9, 1989
Starting with hypotheses on functions which can be interpreted as income compensation functions, a theory of consumer behaviour is developed. If people act rationally they will behave according to these hypotheses. From these axioms many propositions, well-known from traditional neoclassical consumer theory, can be deduced, In particular, Shephard’s lemma can be proved within this theory. Journal of Economic Literature Classification Number: 022. 0 1990 Academic Press, Inc.
I. 1biTRoDucT10~
Consumer behaviour model analysis can be started in several different approaches. Traditional neoclassical consumer theory uses the approach of utility functions from which the demand functions can be derived. Another approach is due to L. McKenzie who starts with the preference relation of the individual [9]. Other well-known models, which are integrab~lity theory and the theory of revealed preference, assume hypotheses on demand functions from which the existence of a utility function can deduced. In all of the above mentioned models income compensation functions, based on a preference relation 3, are defined by NP,
x)=minIpy/
Y?=x>,
supposing that this minimum exists always. Under appropriate conditions these income compensation functions can be used as money-metric anility functions, representing consumer’s preferences. This was pointed out by Samuelson [ll], Varian [13], Weymark [14], Chipman and Moore Cl], Fuchs-Seliger [4, 5] and others. Recently S. Honkapohja introduced a model of consumer behaviour in which he started with functions that can be interpreted as income compensation functions [7, pp. 553-5551. In this paper the idea of ~o~ka~o~ja will be further developed by presenting a system of hypotheses on f~~~tio~s which can also be interpreted as income compensation functions. ne of 111 0022-0.531/90 83.W 642:52/M
Copyright Q 1990 by Academic Press, Inc All rights of reproduction in any form reserved
112
SUSANNEFUCHS-SELIGER
the hypotheses, which require rational behaviour, is very near to the Weak Axiom of Revealed Preference. Based on this system of axioms a relation will be defined which can be plausibly interpreted as the consumer’s preference relation. It will be shown that this relation is complete, transitive, convex, and continuous. Finally, it will also be demonstrated that Shephard’s lemma can be deduced, and thus one of the most important laws in demand theory follows from these axioms.
II. HYPOTHESESON INCOME COMPENSATIONFUNCTIONS We will base our investigation on the following system of axioms: The map M: rW: + x X -+ E4, XC Iw:, Xf 0, l b = M(p, x), satisfies the following properties: (CI) (CII)
vxex:
[Vp’pE[W:+:px~M(p,x)]; Vx, ye:X: [x#y and pxzM(p,
y), Vp~E[W:+ailp’~5!~+:
P’Y5 WP’, x)1; (CIII)
(i)
Vx, y E X: [3p” E IR; + : M(p’, x) = M(p”, y) - Vp E LRf++ :
MP, xl = M(P> Y)X (ii)
Vx, yEX MP, xl ’ WP, VII.
[3p”E
rWy+:
M(PO, x) > MPO, Y) * VP E q
+:
Remark. We can interpret M(p, x) as an income compensation function. Thus, M(p, x) is equal to the money income that an individual needs in order to achieve, in the price situation p, a commodity bundle which he likes at least as much as x. This interpretation is the reason we have assumed that X is a subset of the LQT. However, more generally we could have assumed that X is a subset of [w”. Then all the results of Sections III and IV would remain, and for Section V we would only have to assume additionally that X is bounded from below. If we consider the function M(p, x) = 2 dm for (p, x) E iw: + x I%: +, then we can easily see that this function satisfies (CI)-(CIII). For interpretation: (CI) states that in every price situation p the minimum income needed to achieve a commodity bundle equivalent to x is less or equal to the cost of x in that price situation p. (CII) states that if in every price situation p the individual’s minimum expenditure for a commodity bundle equivalent to y is not higher than for x, then there exists a price situation p’ where y is not more expensive than a commodity bundle equivalent to x. The interpretation of (CIII) is obvious: According to rational behaviour, in the price situation p1 we would not spend more ‘XER:,:
OXi>O, visn, x=(x I,..., X”), xs[w”,: ox;~O.
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COMPENSATED DEMAND
money for a commodity bundle equivalent to y than for a ~ornmodi~~ bundle equivalent to x, if at an earlier time we have spent more money for a commodity bundle equivalent to x than for a commodity bundle equivalent to y in the price situation p”. In the next section it will then be demonstrated that based on the above hypotheses a model of consumer behaviour can be established.
III. A PREFERENCERELATION GENERATEDBY M(g,x)
Assuming the hypotheses (CI)-(CIII) it will be shown now that we can Will define a relation which describes the preferences of the individual. see that this preference relation deduced from the consumer’s ome compensation function possesses nice properties like corn transitivity. The following lemma will help to show these properties. LEMMA 1. Condition (CIII)
implies
Proof Since M(p, x) and M(p, y) are defined for all x, y E X and PER”++, we have 3p: [M(p, x) 2 M(p, yf or M(g, y) 2 M(p, x)]. From this together with (CIII) immediately follows (Vp(Plq+:
WP, ~12 WP, Y))
or
(VpJpE[w”,+:
If in all price situations the commodity bundle x is at least as expensive as a commodity bundle which the individual estimates as y then according to rational behaviour the individual will prefer x to y or be indifferent. T suggests the following definition of a preference relation according to S. Honkapohja [7, p. 5541:
vx, yex:
xRy -\dp~R~+:px~AI(p,
We will now show that the income compensation represents the relation R.
y).
function
THEOREM 2. Under the conditions (CI)-(CIII) the function QnYPOEW++, represents consumer’s preferences; in other words, \J’x, Y E x,
XRY -a M(p’, xl 2 WP’,
~1.
114
SUSANNE FUCHS-SELIGER
Prooj Consider x, y E X, x # y and xRy. Since by the definition we have ‘dp: px 2 M(p, y), this together with (CI) and (CII) yields
of R
3p’: M(p’, y) 5 M(P’, x).
Using (CIII) we obtain Vp: M(p, x) 2 M(p, y) and thus, especially, MPO, x) 2 MPO, Y).
Conversely, assume M(p”, x) 2 M(p’, y). Then by (CIII), Vp E L’k?:+ , M(p, x) 2 M(p, y). Using (CI), from this result xRy immediately follows. Based on R, we can define the relations P (better than) and I (indifferent to) by vx, ye&x-I
xPy + xRy and 1 (yRx) xIy +xRy
and yRx.
We will now deduce nice properties of R. As a corollary to Theorem 2 we obtain COROLLARY.
0) (ii)
Under the assumption of (CI)-(CIII)
xpy 0 WP, x) > Wp, XIY -M(P,
3.
y), VP E RT +
x) = WP, Y), VPE RT +.
The proof immediately 2, and condition (CIII). THEOREM
we have, Vx, y E X,
follows from the definition of P and 4 Theorem
The conditions (Cl)-(CIII)
imply:
(01) R is complete and transitive; (j?)
if X is convex, then R is a convex relation;
(y)
R is continuous from above;
(6)
R is strictly monotonic, i.e., x 2 y A x # y * xPy.
Proof for (a). In view of Lemma 1, for every x, y E X we have (Vp: M(p, x)zM(p, y)) or (VP: M(p, y)>=M(p, x)), and in view of (CI), pz 2 M(p, z), Vz E X. Thus, the completeness of R immediately follows by the definition of R.
In order to show transitivity, consider x, y, z E X such that x # y, z and y # z, and xRy and yRz. From this we obtain VP: PX 2 M(p, Y)
and
PY 2 M( P, z).
(1)
COMPENSATED
DEMAND
115
Then in view of (CII), ( 1) implies 3p’ c rW: + : p’z 5 M(p’, y). This together with (CI) gives M(p’, z)<=M(p’, y). Thus, in view of (C~I~~~ VP: M(p, z) 5 M(p, y). This together with (1) implies VP: px 2 M(p, z). Thus XRZ. If x, y, z are not pairwise unequal, then transitivity also follows by the application of Lemma 1. Proo~~or (fl). In order to show that R is convex, assume xRy and x f y for any x, y E X. Hence we have
Using (CI) and (2) we obtain, for any AE ]O, l[, tips Ryb + :
Hence, (Ax+ (I-A)y)Ry.
Proof for (y). In order to show that R is upper semicontinuo consider any sequence (xk) c X such that xkRy and xk -+ x0 E X. Then for every PE Et:+, pxkz M(p, y). Since xk 3 x0 we get pxO 2 M(p >y) > Vp . Thus x”Ry. Proof for (6). Consider x, y such that x 2 y A x # y, Using (CI) we immediately obtain
Thus, xRy. Suppose yRx. Then py 2 M(p, x), ‘dp E W+ + . By (CII) there exists p’ such that p’x _I M(p’, y), contradicting (*). This proves (S). Remark. If X is a nonempty connected subset of o;S:, then, as in traditional consumer theory, we can conclude that there exists an upper semicontinuous ordinal utility function representing the relation follows, since R is a complete, transitive, and upper semi~o~tin~o~§ relation on R.
Using the above results, we will now start to compare our axiomatic foundation of a theory of consumer behaviour, based on income compensation functions, with the approach used by S. Honkapohja. His hy~otbe~~~ are the following [7, Proposition 7, p. 5531. The function m = (PYx) mapping KY++ xX into R has the properties: (Ml )
It is homogeneous of first degree and concave in p,
116
SUSANNEFUCHS-SELIGER
(M2) It is continuous in (p, x). (M3) For all p~Iw:+ and XEX there exists JJGX such that px &y = M(p, x) and p’y 2 M(p’, x) for all p’ > 0. (M4) For all X, VEX: px2 M(p, y) for all p E R: + implies M(p, x) 2 M(p, y) for all p E rW: + . (M5) For all x, y EX either M(p, x) 2 M(p, y) for all p E rW: + or M(p, x) 5 M(p, y) for all p E lF2: + . (In Section II it was already mentioned that also in my approach X can be a subset of R”.) We can see from Lemma 1 that conditions (CIj(CIII) imply (M5). Using the definition of R, instead of (M4) one can write xRy * M(p, x) 2 M(p, y) for all p E rW: + . Thus in view of Theorem 2, conditions (CI)-(CIII) also imply (M4). Condition (Ml) is a well-known property of income compensation functions; thus in my model, the function M(p, x) should also obey it. In the next section we will see that conditions (CI)-(CIII) and the additional hypothesis (CV) also imply (Ml) and (M3 ). Now we are looking for a condition which guarantees that M(p, x) can be identified with the McKenzie income compensation function. This will be done in the next section.
IV.
INTRODUCTION OF A MCKENZIE-TYPE INCOME COMPENSATIONFUNCTION
According to traditional neoclassical consumer theory we can construct a McKenzie-type income compensation function on the basis of the relation R by m(p, xl = jff
(py I YRx),
vpEIW;+,
XEX.
We can immediately recognize that conditions (CIj(CIII), which imply that R is complete and transitive, also yield the following result. LEMMA 4.
The conditions (CI j(CII1) m(p, x) 2 M(P, x),
Prooj: inf,,,(pyl
If
impZy VXEX, vpdpkq,.
yRx, we have py 2 M(p, yRx} ZM(p, x), VxeX.
x),
Vp E R”,,.
Thus,
117
COMPENSATED DEMAND
In order to establish equality between y~(p, x) and M(p, x) we add the following axiom: VP E R”, + > Vx E X: 3z E X: zRx and pz = M(p, x).
(CIV)
The interpretation of (CIV) is obvious. From the conditions (CT)-(CIV) we can conclude that the function A&, x) coincides with the McKenzie-type income compensation function. THEOREM
5.
Conditions (CI)-(CIV)
imply
WP, x) = m(p, x),
V(p,X)ER”,+XX.
Proof. Since by (CIV) for any (p, x) E IL!: + x X there exists z E X such that zRx and pz = M(p, x), we have pz 2_m(p, x) and M(p, x) 2_ m(p, x), This together with Lemma 4 proves our theorem. Under the above conditions, from the equality between m(p, x) asa M(p, x) we can conclude that M(p, x) is concave with respect to p. THEOREM
6.
Under the hypotheses (CI)-(CIV)
we have
(a) M(p, x) is concave with respect to p, and thus continuous wit respect to p, (p)
1~M(p,x)=M(A*p,x),
v/l>o.
Prooffor (a). Concavity of M(p, x) with respect to p foIIows in the same way as in traditional neoclassical demand theory (see for instance Diewert 12, p. 53981). Moreover, since M(p, 2) is concave for any fixed jc and is defined on an open set, continuity of M(p, x) with respect to p follows from a well-known mathematical theorem (see Takayama Cl;?, p. 6581). Proof for (fi).
Since m(p, x) = M(p, x), the proof immediately follows.
Remark. Continuing the comparison between (Ml k(M5) an (CI)-(CIV) we see that Theorem 6 corresponds to (Ml). conditions (CI) and (CIV) imply condition (M3) of S. Honkapohja: By (CIV) we have that for all p > 0 and x E X there exists y E X such t yRx and py = M(p, x). Thus by (CI), px zp,y = M(p, x). Since yRx mexx that p’y 2 M(p’, x) for all p’ E W+ + , this together with the previous result gives (M3). Thus we have seen that conditions (CI)-(CTV) imply (Ml), (M3), ( and (M5). However, it seems that the hypotheses (CI)-(CIV) are transparent and describe the individual’s observable behaviour in a kess technical way than (Ml t(M5) do. Moreover, in Theorem 2 we could also show that M(p, x) represents the relation R. This property can be deduce
118
SUSANNE FUCHS-SELIGER
from (Ml)-(M5) tion 1, p. 2223).
under further conditions
(see for instance [14, Proposi-
If we additionally require that M(p, X) be continuous with respect to x, then we also get continuity of M(p, X) in (p, x), which corresponds to (M2). The proof of this property is an immediate consequence of the following lemma. LEMMA 7. Let X be the function f: Xx Y -+ continuous with respect with respect to y, then f
an interval of [w” and Y be an interval of W” and let [w, t = f (x, y) be continuous with respect to x and to y. IA additionally, f is increasing (decreasing)2 is jointly continuous with respect to (x, y).
The proof of Lemma 7 can be easily done (see [6] ). THEOREM
(CI)-(CIV) (CV)
8. Let X be an interval in R”, . Then under the conditions and M(p, x) is continuous with respect to x,
it follows that M(p, x) is continuous in (p, x), V(p, x) E iw: + x X. Proof. By Theorem 5, M(p, x) = m(~, x). Hence by the definition of m(p, x), M(p, x) is increasing with respect to p. This together with the continuity of M(p, x) with respect to p and with respect to x, in view of Lemma 7, implies that M(p, x) is continuous in (p, x). From Theorem 8 the continuity THEOREM
9.
of the relation R also follows.
Under the hypotheses of Theorem 8 the relation R is
continuous. ProoJ In view of Theorem 3 (y) it suffices to show that R is lower semicontinuous. Therefore, consider y E X and a sequence (xk) -c X such that yRxk, f/k, and xk + x0 E X. Then for every p E rW; + we have py 2 M(p, xk), Vk. Since M is continuous with respect to x, we obtain py 2 M(p, x0), vplF[W;+. The following condition implies strict convexity of the relation R: (CVI) The function M(p, x) is strictly quasiconcave with respect to x. Using this condition,
we obtain
THEOREM 10. Under the hypotheses of Theorem 3 (/?) and (CVI) relation R is strictly convex. 2 z 2 z’ means zi 2 zi Vi s n. z > z’ means respect to Y means y1 2.9 ==-fb, $1 Bfb,
zi ;s zi Vi s n. f(x, y) is increasing v’) (f(x, Y’) 5f(x, u’)).
(decreasing)
the with
119
COMPENSATED DEMAND
Proof. In view of Theorem 3 (j?) we know that R is convex. Assuming xRy, x # y, and checking the proof of Theorem 3 (p) once again, we see that by strict quasiconcavity of M(pT x) with respect to x, instead of (3) we obtain, VJ, E 10, 1 [: wiWn++.
M(p,ax+(i-a)Y)>M(p,~)
Hence (2x+(1-1)y)Py.
V. COMPENSATED DEMAND F-UNCTIONS Using the relation R one can define a compensated spondence in the usual way by g(p,x)=argmin{py/yRx},
demand corre-
V(p,x)ERBn,+ xx.
If X is a closed subset of !RT, then in view of Theorem 3, under the conditions (CI)-(CIII) we have that for all (p, x) E R”, i- xX, g(p, x) is well-defined. is strictly convex, Since under the conditions (CI)-(CIII) and (CVI) we immediately obtain: THEOREM 11. If Xc KY+ is closed, convex, and if(U)-(CIH) hold, then g(p, x) is single-valued. THEOREM
(@I)-(CIV)
and (CV1)
12. Let Xc KY+ be closed and convex and let the conditions hold, then the equality p . y = M(p, x), Vy ~g(p, x) is fuljXed.
Prsof. By the definition of g(p, x) we have p. y = m(p, x) Vy ~g(p, x)? and then, by Theorem 5, p . y = M(p, x). THEOREM 13. Under the conditions of Theorem 12, g is komoge~e~~s of degree zero in p, i.e., y Eg( tp, x) o y Eg(p, x), ‘dt > 0.
Prroof:
The proof of Theorem 13 is standard.
THEOREM 14. If (CV) and (CVI) are added to the assurn~~~o~s of Theorem 12, then g(p, x) is a continuous function.
Proof. Since under the assumptions of the theorem, R is ia continuous, strictly convex, and strictly increasing preference ordering, the continuity of g in (p, x) follows as in Nicaido [lo, p. 2993.
120
SUSANNE FUCHS-SELIGER
Remark. More generally, assuming conditions (Cl)-(CV), in view of Theorem 3 all the assumptions of Proposition 4 in [7, p. 5481 are fultilled, and thus, g(p, x) is an upper semicontinuous correspondence.
As an important consequence from the hypotheses (CI)-(CV) Shephard’s lemma. This means
15. Let XE rW; be closed and convex, and assume conditions and let g(p, x) be single-valued. Then
THEOREM
(CI)-(CV)
8M(P, x) aPj where gi is the jth component ofg(p, Proof:
we obtain
= gj (P, x)3 x) = (g,(p, x), .... g,(p, x)).
Consider any two price vectors p and p + dp. Then we obtain
AMP
+ dp, x) = Wp + 4, x) - M(p, x) = (P + AP) .g(p + 4, x) -p .g(p, x) =(P+~p)~(g(p+~p,x)-g(p,x))+dp~g(p,x).
(1)
By Theorem
12 and the definition of g we have that p .g(p + dp, x) We now can proceed as in Hurwicz and Uzawa [IS, pp. 121-1221 using that g(p, x) is continuous with respect to p and M(p, x) is concave with respect to p.
p .g(p, x) 10.
Remark. Since gj(p, x) is continuous with respect to p the above result implies that the partial derivative aM(p, x)/apj is also continuous with respect to y.
The following section deals with income compensation are linearly homogeneous in x.
functions which
VI. THE CASE OF HOMOGENEITY We will now investigate the case when the income compensation function is homogeneous of degree one in x. Therefore we assume: (CVII) THEOREM
(CI)-(CIII) v/l>o.
VA > 0: M(p, Lx) = RM(p, x), V(p, x) E lq + XX 16. Let X be a cone in rW:, and let M(p, x) satisfy and (CVII). Then the relation R is linear, i.e., xRyo ;IxR,Iy,
COMPENSATED
121
DEMAND
xRy ~vpEEIWn++:px~:(p,
y)
‘da > 0:
-QPE@“++,
VA> 0: p . ix 2 M(p, Ay)
0 IxRlly,
apx
2 .aM(p,
y)
--VPE&+,
VI?> 0.
17. Let X be a closed convex co~le in R”, f and as.mme and (CVII). Then g(p, x) is homogeneous of degree one in x, i.e., g(p, ax) = 2”. g(p, x), v’i, > 0. THEOREM
(@I)-(CIII) Proof
g(p, 1*x) = arg min (py ] yRdx) = arg min
p
= arg min (Apy’ I Ay’RAx} = arg min { Apy’ / y/Rx], =/Zargmin
by Theorem 16
{py’Iy’Rx)=II~g(p,x)=(3,yjy~g(p,x)).
VII.
SUMMARY
The preceding results have shown that the hypotheses (CI)-(CV) establish a reasonable model of consumer behaviour based on income compensation functions. We have seen that from the axioms of the model propositions can be deduced, which can also be demonstrated in traditional neoclassical consumer theory where utility functions are given However, in this model based on income compensation functions the problematic notion of utility can be avoided.
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and welfare,
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FUCHS-SELIGER
5. S. FUCHS-SELIGER,Money-metric utility functions in the theory Mud SW. Sci. 18 (1989) 199-210. 6. S. FUCHS-SELIGERAND M. KRTSCHA, “An Alternative Approach Economics,” Discussion Paper No. 320, University of Karlsruhe, 7. S. HONKAPOHJA, On continuity of compensated demand, Inr.
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545-557.
8. L. HURWICZ AND H. UZAWA, On integrabihty of demand functions, in “Preferences, Utility, and Demand” (J. S. Chipman et al., Eds.), pp. 114-148, Harcourt Brace Jovanovich, New York, 1971. 9. L. W. MCKENZIE, Demand theory without a utility index, Rev. Econ. Stud. 24 (1957), 185-189.
10. H. NICAIDO, “Convex Structures and Economic Theory,” Academic Press, New York/ London, 1968. 11. P. A. SAMUELSON,Complementarity-An essay on the 40th anniversary of the HicksAllen revolution in demand theory, J. Econ. Lit. 12 (1974) 1255-1289. 12. A. TAKAYAMA, “Mathematical Economics,” Dryden, Hinsdale, IL, 1974. 13. H. R. VARIAN, Non-parametric tests of consumer behaviour, Rev. Econ. Stud. 50 (1983), 99-100. 14. J. A. WEYMARK, Money-metric utility functions, Int. &on. Reo. 26 (1985) 219-232.