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Consumer Theory Based on the Marginal Rate of Substitution Function.

BARBARA BROWN [*]

This paper presents an alternative structure of demand theory based on a marginal rate of substitution (MRS) function. The theory's new results include: 1) criteria are derived for goods to be normal/inferior, "ordinary"/Giffen, and substitutes/complements, for the n-goods case; 2) the total effect of a price change is decomposed into MRS and relative price (RP) effects, corresponding respectively to income and substitution effects for an own-price change but not for a cross-price change; 3) the RP effect of a cross-price increase is always positive; and 4) a good is a complement if and only if the MRS effect is negative and its absolute value is larger than the RP effect. Pedagogically, the new approach makes it possible to teach demand theory speedily and effectively because the MRS is a relatively concrete entity, the theory and its results are transparent, and the results of standard utility-based theory are derived far more easily. (JEL DOO)

Introduction

The structure of demand theory is that axioms of preference give conditions for the existence of a utility function; constrained maximization of utility gives quantities demanded of all goods; comparative-static analysis of the conditions for utility maximization gives the changes in demand resulting from budget parameter changes. An alternative structure of demand theory, to be presented in this paper, is based on a marginal rate of substitution (MRS) function: The axioms of preference and the constraint give conditions for the existence of a derived MRS function whose use gives rise to an equilibrium condition determining demands for all goods and then comparative-static changes in demand.

My precursor in basing demand theory on an MRS function was R. G. D. Allen, whose part of "A Reconsideration of the Theory of Value" [Hicks and Allen, 1934a, 1934b] is a formal mathematical presentation of demand theory based on an MRS function. Allen [1938] held the MRS to be the fundamental entity, with indifference curves and utility functions derivable only in the special case of integrability:

"In general... we cannot integrate the set of indifference planes into a complete set of indifference surfaces, and we cannot assume that any utility function exists. The assumption of a scale of preferences for small changes of purchases does not imply that a complete scale of preferences exists. The consumer can discriminate between small changes from his established purchases but need not be able to discriminate between widely different sets of purchases" [pp. 440-1].

Wold [1943] made it clear, however, that utility-based and MRS-based theory are equivalent to each other. Transitive preference must be assumed in MRS theory for the choice bundle to be a global maximum, transitivity implies integrability, and integrability implies the existence of an ordinal utility function. With that understanding, Allen's MRS-based demand theory was consigned to oblivion.

The analysis presented here differs from Allen's, centering on the consumer's MRS for a good when he has already chosen optimal amounts of all other goods, through sequential application to other goods of the equilibrium condition that MRS be equal to relative price (RP). Ultimately, there is a single equilibrium condition equation that gets totally differentiated to very easily give the comparative-static derivative (CSD) equations, which are simple in form and give rise to easy analysis, thus differing from Allen's difficult comparative statics.

Why present a new demand theory when there is already a perfectly good one? The theory given here is not intended to supplant existing theory or even to serve as an alternative. Rather, the two versions of theory should be taken as complements, one or the other more convenient for a particular purpose, the new theory giving new results which can be used together with the familiar results from existing theory.

The new results include:

1) Criteria are derived for goods to be normal/inferior, "ordinary"/Giffen, and substitutes/complements for the n-goods case. Up to now, there were only Johnson's [1913] and Vandermeulen's [1972] two-goods case criteria. The new criteria should prove useful in relating subjective natures of kinds of goods to conditions for demand increase or decrease, and in coming up with more satisfactory definitions of substitutes and complements.

2) The total effect of a price change can be decomposed into MRS and RP effects. These two effects respectively correspond to the income and substitution effects for an own-price change but not for a cross-price change.

3) The RP effect of a cross-price increase is always positive.

4) A good is a complement if and only if the MRS effect is negative and its absolute value is larger than the RP effect.

MRS-based theory, as compared with utility-based theory, has a number of virtues. It is relatively concrete: the MRS is a more concrete entity than is an indeterminate ranking number. The CSD equations, which are equivalent to CSD equations in utility-based theory, are much more directly and easily derived, as are most of the results. At the level of intuitive meaning, the theory and its results are transparent. These are pedagogical virtues, making it possible to effectively teach demand theory (using the two-goods case, as is almost always done) in much less time. Also, the transparency of MRS-based theory is likely to make demand theory as a whole more appealing to many economists.

The second section of this paper deals with the consumer's problem, the preference-order basis for the theory, and the basic (that is, foundational) MRS function. The third section arrives at the derived MRS function and uses it to establish the equilibrium condition. The fourth, fifth, and sixth sections cover the income, own-price, and crossprice CSDs and their implications. The last section is the concluding remarks.

Consumer's Problem, Preferences, and Basic MRS Function

This section sets out the axiomatic assumptions of preference and defines the basic MRS as a function of all n goods. Introduction of one axiom, that when [x.sub.1] increases while linked to its most-preferred amounts of other goods, its MRS decreases, is deferred to the third section, where the context helps make its content clear. The only departure from the standard analysis, in this section, is that the MRS is the primitive of the analysis, that is, there is no utility function and the MRS is not defined as a ratio of marginal utilities.

The consumer's problem is to choose a bundle of goods, consisting of amounts of goods [x.sub.1],..., [x.sub.n] to buy. His goal is to buy the one bundle among those available to him that he prefers over every other bundle. He has a given income, y, and there are prices, [p.sub.i], i = 1,..., n, of the goods. So the consumer's problem is to choose, from among all bundles such that:

[[[sigma].sup.n].sub.i=1][p.sub.i][x.sub.i] [less than or equal to] y,

the one bundle that he prefers over every other bundle.

The consumer has preferences: For any two bundles A and B, he either prefers A over B or prefers B over A or is indifferent between A and B. The preference relation is assumed to be transitive and complete, and to be monotonic, in the sense that if two bundles are identical except one bundle contains more of one good, the consumer prefers the bundle that contains more of the good.

The assumptions about the preference relation imply that for any bundle A, there exists another bundle B, with identical amounts of goods [x.sub.i], i = 3 ,...,n and one more unit of [x.sub.1] than in A, such that the consumer is indifferent between A and B. So for every bundle, there is an MRS.

Definition 1

The MRS for good [x.sub.j] in terms of good [x.sub.k] is the amount of [x.sub.k] whose loss would leave the consumer indifferent if he were acquiring one more unit of [x.sub.j], amounts of other goods held constant.

The paper focuses on the consumer's demand for one good, called [x.sub.1]. Since any good can be designated [x.sub.1], all conclusions are still perfectly general. Any other good is chosen to serve as the numeraire and is called [x.sub.2]. The consumer's MRS for [x.sub.1] is in terms of [x.sub.2]; it is the number of units of [x.sub.2] whose loss would leave the consumer indifferent if he were acquiring one more unit of [x.sub.1], amounts of other goods constant.

When the consumer is indifferent between bundles A and B and there is one more unit of [x.sub.1] in B than in A, both bundles containing identical amounts of goods [x.sub.i], i = 3,..., n, B must contain less of [x.sub.2] than is in A because of the monotonicity of the preference relation. So all MRSs are positive.

The assumptions about the preference relation imply the existence, for every bundle, of an MRS for [x.sub.1] in terms of [x.sub.2], and therefore the existence of an MRS function:

MRS = m([x.sub.1],...., [x.sub.n]).

This function is called the basic MRS function to distinguish it from what will be called the derived MRS function, to be introduced, whose arguments will include optimal amounts of goods. Reference to the MRS without specification of the goods involved is always reference to the MRS for [x.sub.1] in terms of [x.sub.2]. The basic MRS function is assumed continuous and differentiable.

The Derived MRS Function and Consumer Equilibrium

This section derives the condition for consumer equilibrium. The condition is derived through sequential use of the definitions of MRS and RP.

Theorem 1: The consumer will spend all available income.

Proof: The proof of this theorem, given in the Appendix, is exactly the same as in utility-based theory.

Definition 2

The RP of good [x.sub.j] in terms of good [x.sub.k] is the amount of [x.sub.k] the consumer would be prevented from buying, if he were to buy one more unit of [x.sub.j], if he were spending his entire income, amounts purchased of all other goods held constant. Alternatively and equivalently, the RP of [x.sub.j]in terms of [x.sub.k] is the additional amount of [x.sub.k] he would be able to buy if he were to buy one less unit of [x.sub.j], while spending his whole income, amounts purchased of all other goods held constant. The definition implies:

[RP.sub.jk] = [p.sub.j]/ [p.sub.k],

where [RP.sub.jk] denotes the RP of good [x.sub.j] in terms of good [x.sub.k].

Theorem 2: Given income and prices, for each amount of [x.sub.1] there are most-preferred amounts of the other goods [x.sub.2],..., [x.sub.n].

Proof: Hold [x.sub.1],...,[x.sub.n-1] fixed at [x.sub.1],...., [x.sub.n-1], where the bar over the variable denotes quantity is fixed. Define [x.sub.n] as the optimal quantity of [x.sub.n] for relevant amounts of all lower-numbered goods. By Theorem 1, all available income will be spent. Then:

[x.sub.n] = y - [sigma] [p.sub.i][x.sub.i]/[p.sub.n], I [not equal to] n.

Now let [x.sub.n-1] vary while still holding [x.sub.1], ..., [x.sub.n-2] fixed; [x.sub.n] will vary with [x.sub.n-1], so:

[MRS.sub.n-1,n] = m ([x.sub.1],...,[x.sub.n-2], [x.sub.n-1], y - [[[sigma].sup.n-1].sub.i=1] [p.sub.i][x.sub.i]/[p.sub.n]).

To show that the bundle with the amount of [x.sub.n-1] for which [MRS.sub.n-1,n] = [RP.sub.n-1,n] is the most-preferred available bundle, assert the axiom that [MRS.sub.n-1,n] decreases as [x.sub.n-1] increases. Then for amounts of [x.sub.n-1] smaller (larger) than the one for which [MRS.sub.n-1,n] = [RP.sub.n-1,n], [MRS.sub.n-1,n] [greater than] [RP.sub.n-1,n] ([MRS.sub.n-1,n] [less than] [RP.sub.n-1,n]). By the definitions of MRS and RP and the assumption of monotonicity, for an amount of [x.sub.n-1] for which [MRS.sub.n-1,n] [greater than] [RP.sub.n-1,n] ([MRS.sub.n-1,n] [less than] [RP.sub.n-1,n]), the consumer prefers a bundle containing more (less) [x.sub.n-1]. The bundle for which [MRS.sub.n-1,n] = [RP.sub.n-1,n] is therefore preferred over the bundles with both a little less and a tittle more [x.sub.n-1], and since the bundle with a little less (more) [x.sub.n-1] is preferred over bundles with even less (more) [x.sub.n-1], by the transitivity of the preference relation the bundle for which [MRS.sub.n-1,n] = [RP.sub.n-1,n] is preferred over all available bundles.

The next step is to prove that for [x.sub.1],..., [x.sub.n-3], there is a most-preferred amount of [x.sub.n-2], when each amount of [x.sub.n-2] would be consumed with its most-preferred amounts of [x.sub.n-1] and [x.sub.n]. There are most-preferred amounts of [x.sub.n-1] and [x.sub.n] for some amount of [x.sub.n-2], to be called [x.sub.n-2][degrees]. Let [x.sub.n-2] vary while holding [x.sub.n] constant at [x.sub.n][degrees]. [x.sub.n-1] will vary with [x.sub.n-2]. For [x.sub.n-2][degrees], if:

[MRS.sub.n-2,n-1] ([x.sub.1], ..., [x.sub.n-3], [x.sub.n-2], [x.sub.n-1], [x.sub.n]) [greater than] [RP.sub.n-2,n-1],

the consumer will prefer a bundle with more [x.sub.n-2] over this bundle. However, for the bundle with more of [x.sub.n-2], the consumer prefers over it, the bundle with this amount of [x.sub.n-2] and the most-preferred amounts of [x.sub.n-1] and [x.sub.n] for this amount of [x.sub.n-2]. Therefore, by the transitivity of the preference ordering, when [MRS.sub.n-2,n-1] [greater than] [RP.sub.n-2,n-1] for an amount of [x.sub.n-2], the consumer prefers over it a bundle with more of [x.sub.n-2], when amounts of [x.sub.n-2] are linked with their most-preferred amounts of [x.sub.n-1] and [x.sub.n].

For [x.sub.1],...,[x.sub.i-1], assume that when amounts of [x.sub.i] are linked with their most-preferred amounts of [x.sub.i+1], ..., [x.sub.n], that [x.sub.i]'s [MRS.sub.ij] decreases as [x.sub.i] increases.

By assumption of decreasing MRS, this larger amount of [x.sub.n-2] will have a smaller MRS than [x.sub.n-2][degrees]. If its MRS is still larger than RP, the consumer prefers over it a larger amount of [x.sub.n-2], which by transitivity is preferred over [x.sub.n-2][degrees] and all smaller amounts of [x.sub.n-2]. By the same line of reasoning, among amounts of [x.sub.n-2] with [MRS.sub.n-2,n-1] [less than] [RP.sub.n-2,n-1], the consumer prefers the smallest amount of [x.sub.n-2]. The amount of [x.sub.n-2] for which

[MRS.sub.n-2,n-1] = [RP.sub.n-2,n-1] is larger (smaller) than amounts in the near vicinity for which [MRS.sub.n-2,n-1] [greater than] [RP.sub.n-2,n-1] ([MRS.sub.n-2,n-1] [less than] [RP.sub.n-2,n-1]) so is preferred over such amounts of [x.sub.n-2], so by transitivity is preferred over all other amounts of [x.sub.n-2].

The proof above that there is a most-preferred amount of [x.sub.n-2] applies also to the choice of [x.sub.n-3], giving the conclusion that for [x.sub.1],...,[x.sub.n-4] fixed and with each amount of [x.sub.n-3] linked with its most-preferred amounts of [x.sub.n-2],...,[x.sub.n], the consumer prefers the amount of [x.sub.n-3] for which [MRS.sub.n-3,n-2] = [RP.sub.n-3,n-2] over all other amounts of [x.sub.n-3], and similarly, sequentially, for [x.sub.n--4],...,[x.sub.2].

The key axiom used in the proof of this theorem is: For [x.sub.1],..., [x.sub.i-1], when amounts of [x.sub.i] are linked with their most-preferred amounts of [x.sub.i+1],...,[x.sub.n], [x.sub.i]'S [MRS.sub.ij] decreases as [x.sub.i] increases. The assumption that the MRS decreases with [x.sub.i] corresponds to the diminishing MRS assumption of standard utility-based analysis in that both assumptions are necessary for interior solutions. These are nevertheless different assumptions: The translation into two-goods utility analysis of the present assumption is that indifference curves intersecting a budget line get flatter at the intersection points reading from left to right, whereas the standard assumption is a flattening of a single indifference curve reading from left to right.

Theorem 3: The MRS for [x.sub.i] can be written:

MRS = m([x.sub.1],[x.sub.2],...,[x.sub.n]).

Proof: Substitute [x.sub.2],...,[x.sub.n] into the equation for the basic MRS.

Theorem 4: The optimal amount of [x.sub.i], [x.sub.i], for fixed amounts of goods can be written as a function of the amounts of the fixed goods and of the parameters.

Proof: From Theorem 2:

[x.sub.n] = f([x.sub.1],...,[x.sub.n-1],y,[p.sub.1],...,[p.sub.n]).

The equilibrium condition for [x.sub.n-1] in Theorem 2 implies:

[x.sub.n-1] = f([x.sub.1],...,[x.sub.n-2],y,[p.sub.1]...,[p.sub.n]).

Substitution of this last equation for [x.sub.n-1] into the equation for [x.sub.n] gives:

[x.sub.n] = f([x.sub.1],...,[x.sub.n - 2],y,[p.sub.1],...,[p.sub.n]).

Similarly, solving for [x.sub.n-2] eliminates [x.sub.n-2] as an argument in its equation, and then substitution of the equation for [x.sub.n-2] into the equations for [x.sub.n-1] and [x.sub.n] eliminates [x.sub.n-2] as an argument in those equations as well. The technique can be repeated until each [x.sub.i] is a function of amounts of the fixed goods and the parameters.

Theorem 5: The MRS for [x.sub.1] can be written:

MRS = f([x.sub.1], [x.sub.2] ([x.sub.1], y, [p.sub.1],...,[p.sub.n]),..., [x.sub.n] ([x.sub.1],y,[p.sub.1],...,[p.sub.n])).

Proof: Take each amount of [x.sub.1] as a fixed amount of that good and find [x.sub.2],..., [x.sub.n] for that amount of [x.sub.1]. From Theorem 4, each [x.sub.i] is then a function of [x.sub.1] and the parameters.

The MRS function as derived in Theorem 5 will be called the derived MRS function, to distinguish it from the basic MRS function. The derived function gives the MRS for [x.sub.1] in terms of [x.sub.2], as a function of [x.sub.1] and the optimal amounts of each of the other goods for each amount of [x.sub.1], optimal amount of each of these goods expressed as a function of [x.sub.1] and the parameters. From this point on, the MRS function discussed is always the derived MRS function, unless otherwise noted.

Theorem 6: The quantity demanded of [x.sub.1] will be the quantity for which:

MRS = f([x.sub.1], [x.sub.2] ([x.sub.1],y,[p.sub.1],...,[p.sub.n]),...,[x.sub.n] ([x.sub.1],y,[p.sub.1],...,[p.sub.n])) = RP.

Proof: The method of proof used in Theorem 2 can be applied here.

Theorem 7: The demand functions for all the goods are written:

[[x.sup.*].sub.i] = f(y, [p.sub.1],...,[p.sub.n], i = 1,..., n.

Proof: By Theorem 6, [[x.sup.*].sub.i] (the quantity demanded of [x.sub.1]) will be the quantity for which [MRS.sub.12] ([x.sub.1],y,[p.sub.1],...,[p.sub.n] = [RP.sub.12] ([p.sub.1],[p.sub.2]). Solving this equation for [x.sub.1] gives:

[[x.sup.*].sub.i] = f(y,[p.sub.1],...,[p.sub.n].

Theorem 4 implied:

[x.sub.i] = f([x.sub.1], y,[p.sub.1],...,[p.sub.n], i = 2,...,n.

Substituting the demand function for [x.sub.1] into the equations for [x.sub.i] to get their demand functions gives:

[[x.sup.*].sub.i] = f(y,[p.sub.1],...,[p.sub.n]. i = 2,...,n.

The present analysis differs from utility-based demand theory in having its consumer equilibrium condition in the form of a single equation:

MRS([x.sub.1], [x.sub.2]([x.sub.1],y,[p.sub.1],...,[p.sub.n]),...,[x.sub.n]([x.sub.1 ],y,[p.sub.1],...,[p.sub.n])) = RP ([p.sub.1],[p.sub.2]) ,

where RP = [p.sub.1]/[p.sub.2]. The single-equation formulation makes it very easy to get the comparative-static results, given in the next sections.

The Income CSD

Theorem 8 gives the effect of an increase in income on demand for [x.sub.1]:

Theorem 8: The income CSD is:

[partial][[x.sup.*].sub.1]/[partial]y = -[[[sigma].sup.n].sub.i=2] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial]y/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.1]

Proof: Totally differentiate the equilibrium equation and set d[p.sub.i] = 0, i = 1,... , n. [partial][[x.sup.*].sub.1]/[partial]y is written in place of d[x.sub.1]/dy to emphasize that a CSD is the partial derivative of the demand function for [x.sub.1].

The standard determinant-form equations for [partial][[x.sup.*].sub.1]/[partial]y, [partial][[x.sup.*].sub.1]/[partial][p.sub.1], and [partial][[x.sup.*].sub.1]/[partial][p.sub.2], derived by Slutsky [1915] and presented by Hicks in the Mathematical Appendix to Value and Capital [1939], have been shown equivalent to the CSD equations given here (see Brown [1998]).

As for the meaning of the equation for [partial][[x.sup.*].sub.1]/[partial]y: It is as if the consumer considered what his situation would be like if, with his new, higher income, he were to buy his original amount of [x.sub.1] together with the new most-preferred amounts of the other goods for that amount of [x.sub.1]. The change in income changes the most-preferred amounts of all the other goods linked to the original-equilibrium amount of [x.sub.1], and the changes in the quantities of the [x.sub.1], i [not equal to] 1, change [x.sub.1]'s MRS and thereby create a difference between [x.sub.1] 's MRS and RP. Meanwhile, an increase in [x.sub.1] (keeping each amount of [x.sub.1] linked to its most-preferred amounts of the other goods), parameters constant at their original equilibrium values, would change [x.sub.1]'s MRS and therefore the difference between MRS and RP. So dividing the difference between MRS and RP that would be created by the income increase, if [x.sub.1] were held constant, by the change in that diff erence from increasing [x.sub.1] gives the change in [x.sub.1] that will restore equilibrium.

The income CSD equation is used to derive the condition for a good to be normal/inferior. A good is defined as normal (inferior) if [partial][[x.sup.*].sub.1]/[partial]y [greater than] 0 ([partial][[x.sup.*].sub.1]/[partial]y [less than] 0).

Theorem 9:

[partial][[x.sup.*].sub.1]/[partial]y [greater than]/[less than] 0 iff [[[sigma].sup.n].sub.i=2] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial]y [greater than]/[less than] 0.

Proof: The equation for [partial][[x.sup.*].sub.i]/[partial]y implies:

[partial][[x.sup.*].sub.i]/[partial]y [greater than]/[less than] 0 iff -[[[sigma].sup.n].sub.i=2] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial]y/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.i] [greater than]/[less than] 0.

To get an interior solution it was assumed that the denominator above is negative, which implies that the condition for [x.sub.1] to be normal/inferior is as stated in the theorem.

The condition states that a good will be normal (inferior) if, when income increases and quantity of the good itself is held constant, its MRS in terms of any one other good increases (decreases). This result was derived by Johnson [1913, p. 114] for the two-goods case. It is a new result for the n-goods case.

The Own-Price CSD

Theorem 10: The own-price CSD is:

[partial][[x.sup.*].sub.1]/[partial][p.sub.1] = [partial]RP/[partial][p.sub.1] - [[[sigma].sup.n].sub.i=2] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][p.sub.1]/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.1].

Proof: Totally differentiate the equilibrium condition equation, set dy = d[p.sub.i] = 0, i [not equal to] 1, and the conclusion follows directly.

As with the income CSD, a change in [p.sub.1] creates a disequilibrium, a difference between MRS and RP, of a specific magnitude for [x.sub.1] held constant. Increasing [x.sub.1] would bring about a reduction in the difference between MRS and RP. Dividing the "disequilibrium" by the "reduction in disequilibrium" from increasing [x.sub.1] gives the change in demand for [x.sub.1] with [p.sub.1], this being the change in [x.sub.1] necessary to restore equilibrium.

The own-price CSD equation decomposes the change in demand into two parts which will be called the MRS effect and the RP effect of the price change:

MRS effect = - [[[sigma].sup.n].sub.i=2] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][p.sub.1]/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.i] and RP effect = [partial]RP/[partial][p.sub.i]/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.1].

For this own-price change, these two parts are each identical in magnitude to the income effect and the substitution effect:

Theorem 11: The RP and MRS effects of an own-price change are, respectively, the substitution and income effects.

Proof: Proof of this theorem is in the Appendix. An alternative proof [Brown, 1998] consists in manipulating the two determinants in Slutsky's equation stating the decomposition into income and substitution effects, the manipulations showing that the income effect determinant equals:

- [[[sigma].sup.n].sub.i=2] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][p.sub.1]/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.1],

and that the substitution effect determinant equals:

[partial]RP/[partial][p.sub.1]/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.1].

The reason for giving new names to these two effects is that, while the two decompositions are identical for the own-price CSD, they are different for the cross-price CSD, as will be shown later.

MRS-based demand theory gives a very easy proof that the own-price RP/substitution effect is negative:

Theorem 12: The RP/substitution effect of an own-price increase is negative.

Proof: By definition and Theorem 11:

RP/substitution effect = [partial]RP/[partial][p.sub.1]/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.1].

To get an interior solution, it was assumed that the denominator above is negative. RP = [p.sub.1]/[p.sub.2], so:

[partial]RP/[partial][p.sub.1] = [partial]([p.sub.1]/[p.sub.2])/[partial][p.sub.1] = 1/[p.sub.2] [greater than] 0.

Hence, the RP/substitution effect is negative.

The analysis yields a simple explanation as to why the substitution effect is negative. The increase in the price of a good increases its RP and, thus, amount of the good held constant, creates a situation in which MRS [less than] RP. It was established in the discussion of consumer equilibrium that if, for a good, MRS [less than] RP, the consumer will prefer over it a bundle with less of that good.

The criterion for a good to be Giffen/ordinary is given in Theorem 13.

Theorem 13:

[partial][[x.sup.*].sub.1]/[partial][p.sub.1] [greater than]/[less than] 0 iff [[[sigma].sup.n].sub.i=2] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][p.sub.1] [greater than]/[less than] [partial]RP/[partial][p.sub.1].

Proof:

[partial][[x.sup.*].sub.1]/[partial][p.sub.1] [greater than]/[less than] 0 iff [partial]RP/[partial][p.sub.1] - [[[sigma].sup.n].sub.i=2] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][p.sub.1]/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.1] [greater than]/[less than] 0.

The conclusion follows directly from the negativity of the denominator, which has been assumed all along.

Thus, a good will be Giffen (ordinary) if, for [x.sub.1] held constant, the change in [x.sub.1]'s MRS is larger (smaller) than the change in [x.sub.1]'s RP. In the equilibrium before [p.sub.1] goes up, [x.sub.1]'s MRS and RP are equal. If the change in the MRS for [x.sub.1] held constant is bigger (smaller) than the increase in the RP, the consumer would find, if he contemplated buying a bundle with the original amount of [x.sub.1] after the price increase, that [x.sub.1]'s MRS would be larger (smaller) than its RP. He therefore would prefer a bundle containing more (less) of [x.sub.1] over this one.

Theorem 14: For a good to be a Giffen good, the income effect must be larger in absolute value than the substitution effect, and the good must be inferior.

Proof: From Theorem 13:

[partial][[x.sup.*].sub.1]/[partial][p.sub.1] [greater than] 0 iff - [[[sigma].sup.n].sub.i=2] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][p.sub.i]/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.1] [greater than] - [partial]RP/[partial][p.sub.1]/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i][partial][x.sub.1],

which says that for a good to be Giffen, the MRS/income effect must be larger than the RP/substitution effect in absolute value. From the proof of Theorem 11, the MRS/income effect is equal to - [x.sub.1][partial][[x.sup.*].sub.1]/[partial]y and, just seen, must be positive, so [partial][[x.sup.*].sub.1]/[partial]y must be negative.

The Cross-Price CSD

Theorem 15: The cross-price CSD is:

[partial][[x.sup.*].sub.1]/[partial][p.sub.2] = [partial]RP/[partial][p.sub.2] - [[[sigma].sup.n].sub.i=2] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][p.sub.2]/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.i].

Proof: Totally differentiate the equilibrium condition equation and set dy = d[p.sub.i] = 0, i [not equal to] 2.

Theorem 16: Decomposition of the cross-price CSD into MRS and RP effects does not correspond to Slutsky's s decomposition into income and substitution effects.

Proof: See Brown [1998] for proof of this theorem.

New results are given in Theorems 17, 18, and 19.

Theorem 17: The RP effect of a cross-price increase is always positive.

Proof:

Cross-price RP effect = [partial]RP/[partial][p.sub.2]/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.i]

The denominator above is negative by assumption. Also:

[partial]RP/[partial][p.sub.2] = [partial] ([p.sub.1]/[p.sub.2])/[partial][p.sub.2] = [p.sub.1]/[[p.sup.2].sub.2] [less than] 0.

Therefore the cross-price RP effect is positive.

Theorem 18: The cross-price RP effect equals -RP times the own-price RP effect.

Proof:

Own-price RP effect = [partial]RP/[partial][p.sub.1]/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.1] = [partial] ([p.sub.1]/[p.sub.2])/[partial][p.sub.1]/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.1] = 1/[p.sub.2]/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.1],

and

Cross-price RP effect = [partial]RP/[partial][p.sub.2]/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.1] = [partial] ([p.sub.1]/[p.sub.2])/[partial][p.sub.2]/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.1] = - [p.sub.1]/[[p.sup.2].sub.2]/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.1],

and therefore:

Cross-price RP effect = - [p.sub.1]/[p.sub.2] (own-price RP effect).

The theory also yields a criterion for a good to be a substitute/complement.

Theorem 19: A good will be a substitute (complement) if, with a cross-price increase, the change in [x.sub.1]'s MRS is large (smaller) than the change in [x.sub.1] 's RP, for [x.sub.1] held constant.

Proof: The cross-price CSD is:

[partial][[x.sup.*].sub.1]/[partial][p.sub.2] = [partial]RP/[partial][p.sub.2] - [[[sigma].sup.n].sub.i=2] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][p.sub.2]/[[[sigma].sup.n].sub.i=1][parti al]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.1].

Since the denominator by assumption is negative:

[partial][[x.sup.*].sub.1]/[partial][p.sub.2] [greater than]/[less than] 0 iff [[[sigma].sup.n].sub.i=2] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][p.sub.2] [greater than]/[less than] [partial]RP/[partial][p.sub.2].

Theorem 20: A good will be a complement if and only if the MRS effect is negative and if its absolute value is larger than the RP effect.

Proof: Theorem 15 implies:

[partial][[x.sup.*].sub.1]/[partial][p.sub.2] [less than] 0 iff - [[[sigma].sup.n].sub.i=2] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][p.sub.2]/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.1] [less than] - [partial]RP/[partial][p.sub.2]/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.1].

By Theorem 17, the RP effect is positive, so the further implication is that the MRS effect must be negative. Multiplication of the right-hand side terms in the inequality above by -1 reverses the inequality sign and thus gives the result that the absolute value of the MRS effect must be larger than the RP effect.

The one standard result of utility-based demand theory not yielded up by this MRS-based theory is cross-price substitution effect symmetry: the result that the substitution effect in [partial][[x.sup.*].sub.1]/[partial][p.sub.2] is equal to the substitution effect in [partial][[x.sup.*].sub.2]/[partial][p.sub.1]. The result follows from the assumption of equality, in utility-based theory, of [[partial].sup.2]U/[partial][x.sub.1][partial][x.sub.2], and [[partial].sup.2]U/[partial][x.sub.2][partial][x.sub.1], an equality that has no equivalent in MRS-based theory. Allen [1936] wrote, "[Slutsky's] plea for statistical evidence on this point is one that we can support most strongly." Should the weight of empirical evidence in this regard undermine the hold of utility-based demand theory, MRS-based demand theory would survive intact.

Concluding Remarks

In this paper I have presented a new version of demand theory, one based on the idea that for any bundle the consumer can state his MRS for one good in terms of another good. I have shown that consumer equilibrium can be thought of as achieved sequentially and that the consumer chooses the amount of each good for which its MRS equals its RP. Full consumer equilibrium is reached when, using this criterion, the consumer chooses his most-preferred amount of the last chosen good, given that he has already chosen, for each amount of this good, the most-preferred amounts of all other goods.

Perhaps the most striking aspect of this new approach to demand theory is that one can teach its two-goods version very quickly and easily, while conveying insight into the meaning of the critical CSD equations and therefore into the underlying behavior; try it!

Besides providing almost all the results given by standard demand theory, MRS-based analysis yields a number of new results:

1) New equations for the demand CSDs are of the form:

[partial][[x.sup.*].sub.1]/[partial]A = [partial](RP - MRS)/[partial]A/[partial]MRS/[partial][x.sub.1],

where A is any parameter.

2) For changes in own-price and cross-price, the change in demand decomposes into RP and MRS effects.

3) There are n-goods case criteria for a good to be normal/inferior, ordinary/Giffen, and substitute/complement.

4) The own-price RP effect is always negative.

5) The cross-price RP effect is always positive.

6) The cross-price RP effect on a good equals -RP times the own-price RP effect on that good.

7) A good will be a complement if and only if its cross-price MRS effect is negative and the absolute magnitude of that MRS effect is larger than its cross-price RP effect.

I hope that the normal/inferior and substitute/complement criteria will be used to come to an understanding of the subjective natures of these kinds of goods. More generally, perhaps the ideas herein will bring about some progress in attending to the unfinished business of demand theory.

(*.) Pace University--U.S.A. An earlier version of this paper was incorrectly printed in the May 1999 issue of IAER.

References

Allen, R. G. D. Mathematical Analysis for Economists, New York, NY: St. Martin's, 1938.

___. "Professor Slutsky's Theory of Consumers' Choice, "Review of Economic Studies, III, February 1936, PP. 120-9.

Brown, B. J. "Changes in Demand: The MRS and RP Effects," Pace University Economics Working Paper, 1998.

Hicks, J. R. Value and Capital, London, United Kingdom: Oxford University Press, 1939.

Hicks, J. R.; Allen, R. G. D. "A Reconsideration of the Theory of Value," (Part I), Economica, N.S., I, 1, February 1934a, pp. 52-76.

___. "A Reconsideration of the Theory of Value," (Part II), Economica, N. S., I, 2, May 1934b, pp. 196-219.

Johnson, W. E. "The Pure Theory of Utility Curves," The Economic Journal, 23, December 1913, PP. 483-513, reprinted in W. J. Baumol; S. M. Goldfeld, eds., Precursors in Mathematical Economics: An Anthology, London, United Kingdom: London School of Economics and Political Science, 1968.

Slutsky, E. "Sulla teoria del bilancio del consummatore," Giornale degli Economisti e rivista di Statistica, 51, 1915, pp. 1-26, translated in G. Stigler; K. Boulding, eds., Readings in Price Theory, Homewood, IL: Irwin, 1952.

Vandermeulen, D. C. "Upward Sloping Demand Curves Without the Giffen Paradox," American Economic Review, 62, June 1972, pp. 453-8.

Wold, H. "A Synthesis of Pure Demand Analysis, I," Scandinavisk Aktuarietidskrift, 26, 1943, pp. 85-118.

APPENDIX

Proof of Theorem 1: If the consumer (assumed to have a one-period life and no bequest motive) were buying a bundle such that some income remained unspent, he could instead be buying the identical bundle but with more of some one good. By the monotonicity of the preference relation, he prefers the bundle with more of the one good over the bundle with unspent income. So for every bundle with unspent income, there exists another available bundle that the consumer prefers over the bundle with unspent income. The consumer will therefore never demand a bundle that does not exhaust his income.

Proof of Theorem 11: In Slutsky' s decomposition, the own-price income effect equals -[x.sub.1] [partial][[x.sup.*].sub.1]/[partial]y and the substitution effect is the residual. So the following demonstration that the own-price MRS effect equals -[x.sub.1] [partial][[x.sup.*].sub.1]/[partial]y constitutes proof that the two decompositions are identical.

The proof involves deriving a useful expression for [partial][[x.sup.*].sub.1]/[partial]y. Let:

S = y-[p.sub.1][x.sub.1]/[p.sub.n],

so that:

[x.sub.n] = s - [[[sigma].sup.n-1].sub.i=2] [p.sub.i][x.sub.i]/[p.sub.n].

Then for [x.sub.n-1]'s MRS:

[MRS.sub.n-1,n] = m([x.sub.1],...,[x.sub.n-2],[x.sub.n-1],[x.sub.n](s,[p.sub.2],...,[p. sub.n],[x.sub.2],...,[x.sub.n-2],[x.sub.n-1])).

Solving for [x.sub.n-1] gives:

[x.sub.n-1] = f([x.sub.1],...,[x.sub.n-2],s,[p.sub.2],...,[p.sub.n],[RP.sub.n-1,n]) ,

and substituting the solution for [x.sub.n-1] into the equation for [x.sub.n] gives:

[x.sub.n] = f(s,[p.sub.2],...,[p.sub.n],[x.sub.1],...,[x.sub.n - 2],[RP.sub.n - 1,n).

Now, solve for [x.sub.n-2] and substitute the solution into the equations for [x.sub.n-1] and [x.sub.n]:

[MRS.sub.n-2,n-1] = m([x.sub.1],...,[x.sub.n-3],[x.sub.n-2],

[x.sub.n-1]([x.sub.1],...,[x.sub.n-3],[x.sub.n-2],s,[p.sub.2],...,[p. sub.n],[RP.sub.n-1,n]),

[x.sub.n]([x.sub.2],...,[x.sub.n-3],[x.sub.n-2],s,[p.sub.2],...,[p.su b.n],[RP.sub.n-1,n])),

[x.sub.n-2] = f ([x.sub.1],...,[x.sub.n-3],s,[p.sub.2],...,[p.sub.n],[RP.sub.n-1,n],[ RP.sub.n-2,n-1]),

[x.sub.n-1] = f ([x.sub.1],...,[x.sub.n-3],s,[p.sub.2],...,[p.sub.n], [RP.sub.n-1,n],[RP.sub.n-2,n-1],

and

[x.sub.n] = f (s,[p.sub.2],...,[p.sub.n],[x.sub.1],...,[x.sub.n-3],[RP.sub.n-1,n],[ RP.sub.n-2,n-1]).

Solving sequentially in this manner for the [x.sub.i], i = 2, ..., n, gives:

[x.sub.i] = f (s, [p.sub.2],...,[p.sub.n],[x.sub.1],[RP.sub.2,3],...,[RP.sub.n-1,n]).

So [x.sub.1]'s equilibrium condition is:

MRS ([x.sub.1],[x.sub.2] ([x.sub.1],s,[p.sub.2],...,[p.sub.n],[RP.sub.2,3],...,[RP.sub.n-1,n]) ,

...,[x.sub.n] ([x.sub.1],s,[p.sub.2],...,[p.sub.n],[RP.sub.2,3],...,[RP.sub.n-1,n]) ) = RP.

Totally differentiating the equilibrium condition and then setting appropriate parameters equal to 0 gives:

[partial][[x.sup.*].sub.1]/[partial]y = - [[[sigma].sup.n].sub.i=2] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial]s[partial]s/[partial]y/[[[sigma].sup.n].s ub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.1],

and

[partial][[x.sup.*].sub.1]/[partial][p.sub.1] = [partial]RP/[partial][p.sub.1]/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.1] - [[[sigma].sup.n].sub.i=2] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial]s [partial]s/[partial][p.sub.1]/[[[sigma].sup.n].sub.i=1] [partial]MRS/[partial][x.sub.i] [partial][x.sub.i]/[partial][x.sub.1].

By inspection, the first term on the right-hand side in the equation above is the RP effect. The implication is that the second term on the right-hand side is the MRS effect. Since [partial]s/[partial]y = 1/[p.sub.n] and [partial]s/[partial][p.sub.1] = -[x.sub.1]/[p.sub.n], we conclude that the own-price MRS effect equals -[x.sub.1][partial][[x.sup.*].sub.1]/[partial]y and is therefore equal to the own-price income effect.
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Comment:Consumer Theory Based on the Marginal Rate of Substitution Function.
Author:BROWN, BARBARA
Publication:International Advances in Economic Research
Geographic Code:1USA
Date:May 1, 2000
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