www.elsevier.com / locate / econbase
Two further empirical implications of
Auspitz–Lieben–Edgeworth–Pareto complementarity
*
Christian E. Weber
Department of Economics and Finance, Albers School of Business and Economics, Seattle University, Seattle, WA98122, USA
Received 6 April 1999; accepted 21 October 1999
Abstract
I show that the assumptions of strong concavity and Auspitz–Lieben–Edgeworth–Pareto complementarity imply that for any good either all other goods must be gross substitutes or all other goods must be gross complements and that all goods must be compensated substitutes. 2000 Elsevier Science S.A. All rights reserved.
Keywords: ALEP complementarity; Gross cross price effects; Compensated cross price effects
JEL classification: D11; C61
1. Introduction
Let x be an n-vector of commodities, x5[x , x , . . . , x ], and assume that a household’s
1 2 n
preferences over different xs can be represented by a twice differentiable utility function, u5u(x).
Economists have long understood that if u(x) exhibits diminishing marginal utility and additive separability, so that u (x),0 and u (x)50 for i not equal to j, then all goods will be normal and thus
ii ij
have downward sloping demand curves. Pareto (1892) showed that these assumptions imply that demand curves slope downward. Later, Slutsky (1915), Hicks and Allen (1934), Schultz (1935), and Liebhafsky (1969) showed that they also imply that all goods are normal.
Two other implications also follow from the joint assumptions of diminishing marginal utility and additive separability: firstly, either all goods must be gross substitutes for good j in the sense that an increase in the price of good j increases the income constant, or uncompensated demands for all other
*Tel.:11-206-296-5725; fax: 11-206-296-2486. E-mail address: cweber@seattleu.edu (C.E. Weber)
goods, or all goods must be gross complements for good j in the sense that an increase in the price of
1
good j decreases the uncompensated demands for all other goods. Secondly, all goods must be compensated substitutes. Pareto (1892, 1909) appears to have been the first to state the proposition on gross cross price effects. It is implicit in Slutsky and was stated explicitly by Hicks (1956). Similarly, while one can prove the proposition on compensated cross price effects using results in Slutsky
2
(1915), the first explicit statements of this result appear to be due to Hicks and Allen (1934). More recently, Chipman (1977) has shown that the Slutsky–Hicks–Allen result on normality still holds under less restrictive conditions than additive separability. He shows that if the utility function is strongly concave, so that its Hessian matrix, denoted U(x), is negative definite, then all goods will be normal and thus have downward sloping demand curves if all goods are weak Auspitz and Lieben, (1989)–Edgeworth (1925)–Pareto (1892) (ALEP) complements, so that u (x)$0. Since a negative
ij
definite matrix has negative elements on the diagonal (i.e. u (x),0), this generalization permits u(x)
ii ij
to be non-zero if it is positive and not too large.
I show here that both the Pareto–Hicks result on gross cross price effects and the Slutsky–Hicks– Allen result on compensated substitutability also hold for Chipman’s more general, strongly concave
3
utility functions with weak ALEP complementarity between all goods. The motivation for Chipman’s paper, on which the present note builds, was to show that the twin assumptions of strong concavity and ALEP complementarity have definite empirical implications. The purpose of this note is to develop those empirical implications further.
1
Defining gross substitutability and complementarity by the impact of a change in the price of good j on the demand for all goods rather than by the impacts of changes in the prices of other goods on the demand for good j is important since, unlike compensated cross price effects, gross cross price effects are generally not symmetric and thus may have different signs, so that the definition of two goods as gross complements or gross substitutes may depend on which price is assumed to change.
2
It was later restated by Wilson (1939), Intriligator (1971), and Silberberg (1972). Weber (1999) discusses the relationship between the results which Slutsky (1915) derived for additively separable utility functions and those of later writers. Additive separability has continued to interest economists. For example both Pollak and Wales (1981) and Lewbel (1986) have explored the connection between additive separability and the theoretical plausibility of demand functions which incorporate the Prais and Houthakker (1955) procedure for accounting for demographic variables in demand functions. Blackorby et al. (1978) discuss applications of general functional separability, of which additive separability is a special case, to the econometric analysis of systems of demand equations; aggregation across commodities, prices, and economic agents; and numerous other theoretical and empirical problems in economics.
3
2. ALEP complementarity and cross price effects
2.1. Gross cross price effects
I begin by examining the signs of gross cross price effects.
Let p5[ p , p , . . . , p ] and I denote, respectively, the vector of prices which the household faces
1 2 n
and household income. LetV denote the set of price-income pairs with p .0, i[h1, 2, . . . nj and
I i
n
I.0. The household’s Marshallian or uncompensated demand function will be denoted h:V →R .
I 1
The range of h will be denoted X5hx: x5h( p, I ) for some ( p, I )[VIj. We have:
Theorem 1. Assume that the household’s preferences generate a single-valued, differentiable demand
function, h( p, I ), defined on V and satisfying the budget identity: I
household’s preferences over X can be represented by a realvalued, twice differentiable function u:
0 1
Proof. Denote the indirect utility function by v( p, I )5u(h( p, I )) and assume that it is twice
differentiable. Then under the conditions stated in the theorem, the first order conditions for maximizing u(x) subject to (1) are given by
0
so that the envelope theorem implies that v( p, I ) is the Lagrange multiplier from the constrained I
utility maximization problem.
Differentiating each of the n different versions of (2) with respect to p yields:j
n ≠
2
Rewriting (3) in matrix form for j given yields:
U [h( p, I )]h? 5[d v 1v p], or: parentheses on the right hand side is an n vector. As Chipman (1977) notes, McKenzie (1960) has
21
shown that assumptions (i) and (ii) imply that the inverse matrix, U [h( p, I )] , has all non-positive entries with at least one negative entry in each row. Thus, since each element of the vector which
21
post-multiplies U [h( p, I )] has the same sign with the possible exception of the row for which j5k,
every element of h? must have the same sign except possibly the one for which j5k. That is, either j
all goods are gross substitutes for good j or all goods are gross complements for good j, which was to be shown.
2.2. Compensated substitutability
Next, I consider compensated cross price effects. Again, let p5[ p , p , . . . , p ] be the vector of
1 2 n
prices which the household faces, but now assume that the household chooses x to minimize its
0
money expenditure, E5p?x, subject to the constant utility constraint, u(x)5u . LetV denote the set u
0
of price-utility pairs such that p .0, i[h1, 2, . . . nj and u is finite. The household’s Hicksian or i
n
compensated demand function will be denoted s: V →R . The range of s will be denoted X5hx: u 1
0 0
x5s( p, u ) for some ( p, u )[Vuj. In this case, we have:
Theorem 2. Let the household’s preferences be such as to generate a single-valued, differentiable 0
compensated demand function, s( p, u ), defined on V and satisfying: u
0 0 0
u(s( p, u ))5u for all ( p, u )[Vu. (6)
0 0 0 n
Let V be a neighborhood whose image X 5s(V ) is in the interior of R . If the household’s
u u 1
0 0 1
preferences over X can be represented by a real valued, twice differentiable function u: X →R , 0
differentiable. Then under the conditions stated in the theorem, the first order conditions for minimizing expenditure subject to (6) are given by:
0 0 0 0 0
e ( p, u )u (s( p, u ))u k 5pk ( p, u )[Vu, k[h1, 2, . . . nj, (7)
0
so that the envelope theorem implies that e ( p, u ) is the Lagrange multiplier form of the constrainedu0 expenditure minimization problem. Differentiating (7) with respect to p yields:j
n ≠
where d is again the Kronecker delta, and: jk
The last equality in (9) follows from Young’s theorem and the envelope theorem.
Recall that assumptions (i) and (ii) imply that all goods must be normal:≠h ( p, I ) /≠I.0 for all j
0 0
j[h1, 2, . . . nj. But, as Silberg (1972) has pointed out, ≠s ( p, u ) /≠u and ≠h ( p, I ) /≠I have the
j j
same sign as long as the marginal utility of income,v( p, I ), is positive. Thus, assumptions (i) and (ii) I
0 0
also imply ≠s ( p, u ) /≠u .0. j
Substituting (9) into (8) and rewriting (8) in matrix form for a given j yields:
0
function, so that the expression in parentheses on the right hand side is a column n vector. Recall that
0 21
assumptions (i) and (ii) imply that U [s( p, u )] has all non-positive entries with at least one negative entry in each row. Thus, since every element of the vector which post-multiplies U [s( p,
0 21
u )] is negative except the one for which j5k, every element of s must be positive except the one ij
for which j5k. That is, all goods must be compensated substitutes, which was the result to be shown.
4
3. Conclusion
Additively separable utility and diminishing marginal utility imply that: (1) all goods must be normal and hence have downward sloping demand curves; (2) all goods must be compensated substitutes; (3) for price j, all gross cross price effects, ≠h ( p, I ) /≠p must have the same sign for
i j
5
i[h1, 2, . . . nj, i not equal to j. The combined implications of Chipman (1977) and this paper are that each of these results also holds for a more general class of utility functions which includes additively separable utility functions with diminishing marginal utility as a special case.
Acknowledgements
I would like to thank John Chipman, Eugene Silberberg, and the editor for their very helpful comments on several earlier versions of this paper. Their suggestions improved the exposition considerably. It goes without saying that any errors which may remain are entirely my own responsibility.
References
Auspitz, R., Lieben, R., 1889. Untersuchungen uber die theorie des preises, Duncker and Humblot, Leipzig.
Blackorby, C., Primont, D., Russell, R.R., 1978. Duality, Separability, and Functional Structure: Theory and Economic Applications, North-Holland, New York.
Chipman, J.S., 1977. An empirical implication of Auspitz–Lieben–Edgeworth–Pareto complementarity. Journal of Economic Theory 14, 228–231.
Edgeworth, F.Y., 1925. The pure theory of monopoly. In: Papers Relating to Political Economy, Vol. 1, MacMillan, London, pp. 111–142.
Hicks, J.R., 1956. A revision of demand theory, Oxford University Press, Oxford.
Hicks, J.R., Allen, R.G.D., 1934. A reconsideration of the theory of value. Economica N.S. 1, 52–76, and 196–219. Houthakker, H.S., 1960. Additive preferences. Econometrica 28, 244–257.
Intriligator, M.D., 1971. Mathematical Optimization and Economic Theory, Prentice-Hall, Englewood Cliffs, NJ. Lewbel, A., 1986. Additive separability and equivalent scales. Econometrica 54, 219–222.
Liebhafsky, H.H., 1969. New thoughts about inferior goods. American Economic Review 59, 931–934.
McKenzie, L., 1960. Matrices with dominant diagonals and economic theory. In: Arrow, K.J., Karlin, S., Suppes, P. (Eds.), Mathematical Methods in the Social Sciences, 1959, Stanford University Press, Stanford, CA, pp. 47–62.
Pareto, V., 1892. Considerazioni sui principii fondamentali dell’economia politica pura, Part 3. Giornale degli Economisti 5, 119–157.
Pareto, V., 1909 (1971), Manual of political economy, translated by A.S. Schwier (Augustus M. Kelley, New York). Pollak, R.A., Wales, T.J., 1981. Demographic variables in demand analysis. Econometrica 49, 244–256.
Prais, S.J., Houthakker, H.S., 1955. The Analysis of Family Budgets, Cambridge University Press, Cambridge. Samuelson, P.A., 1947. Foundations of Economic Analysis, Harvard University Press, Cambridge, MA. Schultz, H., 1935. Interrelations of demand, price, and income. Journal of Political Economy 43, 433–481.
5
Silberberg, E., 1972. Separability and complementarity. American Economic Review 62, 196–197.
Slutsky, E.E., 1915. On the theory of the budget of the consumer. Giornale degli Economisti 51, 1–26, Reprinted in: G.J. Stigler and K.E. Boulding (Eds.), A.E.A. readings in price theory. Richard D. Irwin, Chicago, pp. 27–56.
