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A difficulty in the search for Giffen behavior.

Introduction

Recently, Gilley and Karels [1991] have analyzed Giffen behavior for a household whose choices of meat, M, and potatoes (spuds), S, are restricted by both a money income constraint and a minimum calorie constraint. Assuming that both constraints are binding, they show that M and S are chosen independently of the utility function since the two binding linear constraints in two unknowns uniquely determine both M and S. Gilley and Karels [1991] also state a condition on the prices and calorie contents of the two goods such that S is inferior and Giffen. Hence, they suggest that empirical searches for Giffen goods focus on cases where two goods are chosen subject to two binding constraints.

After examining the two-good case, Gilley and Karels [1991] introduce a third good, X, into their analysis. With this addition to the model, the budget and calorie constraints become:

[P.sub.M] M + [P.sub.S] S + [P.sub.X] X [less than or equal to] R, (1)

[C.sub.M] M + [C.sub.S] S + [C.sub.X] X [greater than or equal to] N, (2)

where [P.sub.I] and [C.sub.I] are, respectively, the money price and calorie content of food L R is the finite income available to the household, and N is the household's finite minimum calorie requirement. Assuming that both constraints are binding at the optimum, the weak inequalities in (1) and (2) are replaced by equations. Following this modification, the authors state two conditions on the prices and calorie contents of the three foods which they claim jointly imply that S is still Giffen. These are:

[P.sub.S] / [P.sub.M] [less than] [C.sub.S] / [C.sub.M] and [P.sub.S] / [P.sub.X] [less than] [C.sub.S] / [C.sub.X]. (3)

Before examining whether these two conditions necessarily imply Giffen behavior for good S, it may be helpful to rearrange them in the following equivalent form:

[C.sub.M] / [P.sub.M] [less than] [C.sub.S] / [P.sub.S] and [C.sub.X] / [P.sub.X] [less than] [C.sub.S] / [P.sub.S]. (4)

Condition (4) implies that good S provides more calories per dollar spent than either good M or good X. For example, one might imagine X to be green vegetables although Gilley and Karels [1991] do not offer this interpretation. Thus, Gilley and Karels [1991] claim, in essence, that the food which provides the most calories per dollar spent must be a Giffen good. In their words [p. 186], "the good most efficient in providing subsistence exhibits Giffen characteristics." They also claim [p. 188] that "(t)he argument is virtually independent of consumer preference orderings." Similar claims are also made on pages 183-4.

The purpose of this paper is to show that this conclusion must be qualified. Adding a third good restores a degree of freedom to the utility maximization problem which is lost in the two-good, two-constraint case.(1) The problem is that, by themselves, constraints (1) and (2) cannot determine the unique {M, S, X} triple chosen by the household even when both hold as equalities. Instead, these equations jointly define a line segment (that is, infinitely many combinations of M, S, and X) in the non-negative orthant of M-S-X space (i.e., for M, S, X [greater than or equal to] 0). Although Gilley and Karels [1991] discuss carefully how this line segment moves when prices and income change, they do not discuss how the household chooses to consume at exactly one point on it or how the point chosen changes when prices and income change.

A natural assumption - which Gilley and Karels [1991] do not pursue - would be that in the three-good, two-constraint case, the household chooses which bundle to purchase by maximizing a quasi-concave utility function defined over the three goods subject to the two relevant constraints. In this case, traditional comparative statics methods again yield interesting results - which they do not if there are the same number of both goods and binding constraints - and the Slutsky equation with a negative own substitution effect reappears. Because of the negative substitution effect, which Gilley and Karels [1991] do not discuss, S may or may not be Giffen in the three-good model if there are fewer than three independent constraints on possible choices of the goods.(2) The point of this paper is to pursue this approach and determine the extent to which the results of Gilley and Karels still hold in the three-good case.

Minimum Subsistence and a Third Good

Implications of the Second Constraint

Throughout their paper, Gilley and Karels [1991] work with inequalities such as (3) to develop conditions for a good to be inferior or Giffen. However, some interesting results can be obtained without any a priori restrictions on prices, calorie contents, or the utility function. These emerge solely from the imposition of the second constraint on the household's choices. Thus, assume that the parameters of the calorie constraint - [C.sub.M], [C.sub.S], [C.sub.X], and N - are all constant and totally differentiate the calorie constraint with respect to I, where I is one of the parameters of the budget constraint, I [element of] {[P.sub.M], [P.sub.S], [P.sub.X] R}. This yields:

[C.sub.M] [Delta]M / [Delta]I + [C.sub.S] [Delta]S / [Delta]I + [C.sub.X] [Delta]X / [Delta]I = 0. (5)

Since the [C.sub.i] variables are all non-negative,(3) (5) implies that at least one good must be inferior. At least one of [Delta]M/[Delta]R, [Delta]S/[Delta]R, and [Delta]X/[Delta]R must be negative regardless of the nature of the utility function. Furthermore, each good with a downward-sloping demand curve must have at least one gross substitute and each Giffen good - if there are any - must have at least one gross complement. Note, for example, that [Delta]X/[Delta][P.sub.x] [greater than] 0 implies either [Delta]M/[Delta][P.sub.x] [less than] 0 or [Delta]S/[Delta][P.sub.x] [less than] 0 or both, indicating that one or both of M and S must be a gross complement for X.

Since Gilley and Karels [1991] are interested specifically in Giffen behavior, they do not stress inferiority per se. Nonetheless, it is an interesting and important implication of their model that the mere existence of the second constraint implies that at least one good must necessarily be inferior regardless of the utility function.

Utility Maximization

Gilley and Karels [1991] do not introduce a utility function into their analysis and the implication of (5) that at least one good must be inferior when the household faces a second binding constraint is completely independent of the nature of the preference ordering. Thus, it is of some interest to ask what additional restrictions on behavior follow from utility maximizing behavior and whether the results claimed by Gilley and Karels [1991] for the three-good, two-constraint model necessarily follow from utility maximizing behavior. This section develops the comparative statics implications of utility maximizing behavior in the three-good, two-constraint case. To do the comparative statics analysis, assume that the household utility function is:

U = U(M, S, X), (6)

where U() has positive first derivatives and is strictly quasi-concave. The money income constraint is (1) and the minimum calorie constraint is (2). It is assumed that both constraints are binding at the optimum.

The first- order conditions for maximizing (6) subject to the constraints (1) and (2) are the constraints and:

[U.sub.S] = [[Lambda].sub.1] [C.sub.S] + [[Lambda].sub.2] [P.sub.S], (7)

[U.sub.M] = [[Lambda].sub.1] [C.sub.M] + [[Lambda].sub.2] [P.sub.M], (8)

[U.sub.X] = [[Lambda].sub.2] [P.sub.X] + [[Lambda].sub.2] [P.sub.X], (9)

where [U.sub.1] is the derivative of U() with respect to I and [[Lambda].sub.1] and [[Lambda].sub.2] are Lagrangian multipliers for the calories and income constraints, respectively. Define [U.sub.IJ] to be the cross partial derivative of the utility function with respect to goods I and J - the derivative of the marginal utility of good I with respect to good J. Then the second- order condition for a maximum requires that the bordered Hessian determinant:

[Mathematical Expression Omitted]

be negative. For a proof and discussion of the second- order conditions in the multiple constraint case, see Simon and Blume [1994].

To determine conditions under which S is inferior and Giffen, totally differentiate (1), (2), and (7)-(9) and rearrange the results in the matrix equation:

[Mathematical Expression Omitted]. (10)

To determine the impact of a change in income, R, or the price of potatoes, [P.sub.s], on the quantity of potatoes demanded, one must solve for the comparative statics partial derivatives, [Delta]S/[Delta]R and [Delta]S/[Delta][P.sub.s]. To do so, apply Cramer's rule to (10). After simplifying somewhat, this yields:

[Mathematical Expression Omitted]. (11)

[Mathematical Expression Omitted], (12)

where [Delta][S.sup.c]/[Delta][P.sub.s] in the second line of (12) is the compensated own price effect. This effect is negative by virtue of the second- order condition.

Even assuming diminishing marginal utility for goods M and X ([U.sub.MM], [U.sub.XX] [less than] 0), the sign of the expression in (11) is unknown without further restrictions on the signs of the three cross partial derivatives of the utility function, [U.sub.MS], [U.sub.MX], and [U.sub.SX]. Thus, it is not possible to determine whether S is inferior based solely on assumptions imposed on the [P.sub.i] and [C.sub.i] variables. This establishes the following.

Proposition 1

Without further restrictions on the utility function, S need not be inferior when a third good is added to the two-good, two-constraint model.

Perhaps more importantly, (12) shows that adding a third good to the analysis restores the Slutsky equation as fundamental for understanding the effects of price changes. The Slutsky equation does not appear in the analysis of Gilley and Karels [1991]. The sign of the first term in (12) - the income effect - is unknown without further restrictions but the second term - the substitution effect - is negative by the second- order condition on the bordered Hessian determinant. This proves the following.

Proposition 2

Introducing a third good into the model restores a negative substitution effect to the impact of a change in the price of S on the quantity of S demanded. If S is inferior, the slope of the demand curve may take either sign. It need not be positive. If S is normal, the demand curve for S necessarily slopes downward.

However, (11) and (12) do yield two further interesting results which Gilley and Karels [1991] do not pursue. First, using (11), one can develop a sufficient condition for S to be inferior. Assume that U(M, S, X), is additive: [U.sub.MS] = [U.sub.MX] = [U.sub.SX] = 0.(4) Assume also that M and X are subject to diminishing marginal utility: [U.sub.MM], [U.sub.XX] [less than] 0. Under these assumptions, (11) becomes:

[Mathematical Expression Omitted]. (13)

[C.sub.M] and [C.sub.X] are positive, [U.sub.MM] and [U.sub.MM] are assumed negative, and the denominator is negative by the second- order condition. Thus, (13) shows that S is inferior if, but not only if, the utility function is additive and:

([C.sub.M][P.sub.S] - [C.sub.S][P.sub.M]) [less than] 0, and ([C.sub.X][P.sub.S] - [C.sub.S][P.sub.X]) [less than] 0.(14)

After rearranging slightly, the conditions in (14) become the same as those in (3) above, which Gilley and Karels [1991] claim are sufficient by themselves to yield Giffen behavior. Thus, as long as one is willing to impose the strong assumptions that the utility function is additive and that meat and good X are subject to diminishing marginal utility in addition to (3) - or equivalently, (14) - then Gilley and Karels [1991] are correct in asserting that S is necessarily inferior in the three-good model,(5) although it still need not be Giffen.

This result is unexpected. In the single-constraint case, the combination of additive utility and diminishing marginal utility for all goods implies that all goods are normal.(6) In the three-good, two-constraint case; however, these conditions lead to the opposite conclusion. Combined with condition (14), additivity and diminishing marginal utility for M and X imply that S must be inferior - not normal.

Gilley and Karels [1991] show that in the two-good case, the first inequality in (14) implies that S is inferior. Adding a third good, X, to the model does not alter this result if the utility function is additive and M and X have diminishing marginal utility. In addition, Gilley and Karels [1991] also claim that the conditions in (14) yield Giffen behavior for S in the three-good case. However, the negative substitution effect in (12) means that while these conditions - along with additivity and diminishing marginal utility - do imply that S is inferior, they do not imply that S must be Giffen.

Finally, refer again to (12). The whole point of this equation is to show that the Slutsky equation with a negative substitution effect reappears when a third good is added to the two-good, two-constraint model. In order for S to be Giffen, it must be inferior and the negative substitution effect must be sufficiently small. Interestingly, (12) suggests the following condition under which the substitution effect, [Delta][S.sup.c] / [Delta][P.sub.s], is approximately zero:

[Mathematical Expression Omitted]. (15)

If condition (15) holds, then the own substitution effect of a change in [P.sub.S] is small so that if S is inferior, Giffen behavior for S becomes more likely. If this condition holds, then meat and good X provide very nearly the same number of calories per dollar spent. Intuitively, this makes meat and good X very similar in terms of their objective characteristics, price, and calorie content. That is, they are very nearly the same good. Thus, if condition (15) holds, one is very nearly back in the two-good, two-constraint case where S is both inferior and Giffen as long as the first inequality in (14) holds.

Conclusion

No household spends its entire income on meat and potatoes, beer and whisky, public and private transportation, or cake and bread.(7) Adding a third good to the two-good, two-constraint model analyzed by Gilley and Karels [1991] implies that traditional comparative statics analysis - including the Slutsky equation with a negative own substitution effect - is required to study own price effects. Thus, good S - which must be Giffen in the two-good model - might not be Giffen if a third good is added to the model, although the presence of the second binding constraint implies that at least one good must still be inferior. Furthermore, S continues to be inferior if it provides more calories per dollar than either of the other two goods and the utility function is additive. Otherwise, the sign ambiguities of the one-constraint case apply as long as there are more goods than constraints.

Where should economists search empirically for inferior goods and Giffen behavior? Gilley and Karels [1991] suggest cases where two or three goods are chosen subject to two constraints. They are correct in the two-good case and in the search for inferior goods. However, in the three-good, two-constraint case, a negative substitution affect appears in the comparative statics analysis of the demand curve so that an inferior good need not be Giffen.

Seattle University. The author thanks Tim Sorenson and Ed Tower for helpful comments on an earlier draft of this paper.

Footnotes

1. As noted above, in this case, the two linear constraints determine the unique quantities demanded of the two goods independently of the utility function.

2. If the household were choosing three goods subject to three binding constraints, then the constraints would again suffice to determine the quantities of each good chosen if all three constraints were binding.

3. The [C.sub.i] variables are assumed to be non-negative rather than positive since [C.sub.x] = 0 indicates that X is a composite commodity that represents all nonfood items purchased by the household.

4. An additive utility function may be written: U(M, S, X) = [u.sub.1](M) + [u.sub.2](S) + [u.sub.3](X).

5. This condition is sufficient but not necessary since S may be inferior even when the utility function is not additive.

6. This proposition has long been known [Slutsky, 1915]. Graduate texts often include a proof as an exercise [Gravelle and Rees, 1981; Intriligator, 1971; Jehle, 1991; Katzner, 1988; Nicholson, 1989; Russell and Wilkinson, 1979; Silberberg, 1990; Varian, 1984].

7. These are just a few of the examples commonly used to motivate Giffen behavior in the presence of a second constraint on the household's choices [Spiegel, 1994].

References

Gilley, Otis; Karels, Gordon. "In Search of Giffen Behavior," Economic Inquiry, 29, 1, January 1991, pp. 182-9.

Gravelle, Hugh; Rees, Ray. Microeconomics, New York, NY: Longman Group, Ltd., 1981.

Intriligator, Michael D. Mathematical Optimization and Economic Theory, Englewood Cliffs, NJ: Prentice Hall, 1971.

Jehle, Geoffrey A. Advanced Microeconomic Theory, Englewood Cliffs, NJ: Prentice Hall, 1991.

Katzner, Donald W. Walrasian Microeconomics: An Introduction to the Economic Theory of Market Behavior, Reading, MA: Addison-Wesley Publishing Co., 1988.

Nicholson, Walter. Microeconomic Theory: Basic Principles and Extensions, 4th ed., Orlando, FL: Dryden Press, 1989.

Russell, R. Robert; Wilkinson, Maurice. Microeconomics: A Synthesis of Modern and Neoclassical Theory, New York, NY: John Wiley and Sons, 1979.

Silberberg, Eugene E. The Structure of Economics: A Mathematical Analysis, 2nd ed., New York, NY: McGraw Hill Inc., 1990.

Simon, Carl; Blume, Lawrence. Mathematics for Economists, New York, NY: W. W. Norton and Co., 1994.

Slutsky, Eugen E. "On the Theory of the Budget of the Consumer," Giornale degli Economisti, 51, July 1915, pp. 1-26 (reprinted in Stigler, George J.; Boulding, Kenneth E., eds. A.E.A. Readings in Price Theory, Chicago, IL: Richard D. Irwin, Inc., 1952, pp. 27-56).

Spiegel, Uriel. "The Case of a 'Giffen Good'," Journal of Economic Education, 25, 2, Spring 1994, pp. 137-48.

Varian, Hal R. Microeconomic Analysis, 2nd ed., New York, NY: W. W. Norton and Co., 1984.

Christian E. Weber Seattle University. The author thanks Tim Sorenson and Ed Tower for helpful comments on an earlier draft of this paper.
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Author:Weber, Christian E.
Publication:Atlantic Economic Journal
Date:Sep 1, 1997
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