Week 6: Consumer Theory Part 1 (Jehle and Reny, Chapter...

Week 6: Consumer Theory Part 1 (Jehle and Reny,Chapter 1)

Tsun-Feng Chiang*

*School of Economics, Henan University, Kaifeng, China

November 2, 2014

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Consumer Theory Primitive Notions

1.1 Primitive Notions

Consumer theory is the foundation of economics, and a study ofconsumer choice. There are four building blocks in any model ofconsumer choice. They are the consumption set, the feasible set, thepreference relation, and the behavioral assumption.

A consumption set or choice set represents the set of allalternatives, or complete consumption plans, that the consumer canconceive, whether some of them will be achievable or not. Let eachcommodity be measured in some infinitely divisible units. Let xi ∈ Rrepresent the number of units of good i . We assume that onlynonnegative units of each good are meaningful. We let x = (x1, · · · , xn)be a vector containing different quantities of each of the n commoditiesand call x a consumption bundle or consumption plan. Aconsumption bundle is x ∈ X is thus represented by a point x ∈ Rn

+. Tosimplify things, there are basic requirements of a consumption set X :

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Assumption 1.1 Minimal Properties of the Consumption Set, X

∅ 6= X ⊂ Rn+.

X is closed.X is convex.0 ∈ X

When the consumption bundle x is conceivable and achievable giventhe economic realities the consumer faces, then we say x is in afeasible set, denoted by B. It is clear B is a subset of X that remainsafter we have accounted for any constraints on the consumer’s accessto commodities due to the practical, institutional, or economic realitiesof the world.A preference relation typically specifies the limits, if any, on theconsumer’s ability to perceive in situations involving choice, the form ofconsistency or inconsistency in the consumer’s choices, andinformation about the consumer’s tastes for the different objects ofchoice.

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Finally, the model is closed by specifying some behavioralassumption which supposes that the consumer seeks to identify andselect an available alternative that is most preferred in the light of hispersonal tastes.

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Consumer Theory Preference and Utility

1.2 Preference and Utility

The section explains how to derive utilty by preference relations.Consumer preferences are characterized axiomatically. Formally, werepresent the consumer’s preferences by a binary relation, %, definedon the consumption set, X . If x1 % x2, we say that "x1 is at least asgood as x2," for this consumer. The following two axioms set forthbasic criteria with which those binary comparisons must conform.

AXION 1: Completeness. For all x1 and x2 in X , either x1 % x2 orx2 % x1.

Axiom 1 formalizes the notion that the consumer has the ability todiscriminate and the necessary knowledge to evaluate alternatives. Itsays the consumer can examine any two distinct consumption plans x1

and x2 and decide whether x1 is at least as good as x2 or x2 is at leastas good as x1.

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AXION 2: Transitivity. For any three elements x1, x2 and x3 in X , ifx1 % x2 and x2 % x3, then x1 % x3

Axiom 2 gives a very particular form to the requirement that theconsumer’s choices be consistent. These two axioms together implythat the consumer can completely rank finite number of elements in theconsumption set, X , from best to worst. We summarize the view thatpreferences enable the consumer to construct such a ranking bysaying that those preferences can be represented by a preferencerelation.

Definition 1.1 Preference RelationThe binary relation % on the consumption set X is called a preferencerelation if it satisfies Axiom 1 and 2.

There are two additional relations that we will use in our discussion ofconsumer preferences. Each is determined by the preference relation,%, and they formalize the notions of strict preference andindifference.

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Definition 1.2 Strict Preference RelationThe binary relation � on the consumption set X is defined as follows:

x1 � x2 if and only if x1 % x2 and x2 � x1.

The relation � is called the strict preference relation induced by %, orsimply the strict preference relation when % is clear. The phrasex1 � x2 is read "x1 is strictly preferred to x2."

Definition 1.3 Indifference RelationThe binary relation ∼ on the consumption set X is defined as follows:

x1 ∼ x2 if and only if x1 % x2 and x2 % x1.

The relation ∼ is called the indifference relation induced by %, orsimply the indifference relation when % is clear. The phrase x1 ∼ x2 isread "x1 is indifferent to x2."

Using these two supplementary relations, we can establish something7 / 28


very concrete about the consumer’s ranking of any two alternatives.For any pair x1 and x2, exactly one of three mutually exclusivepossibilities exists: x1 � x2, or x2 � x1, or x1 ∼ x2.

Now we use the preference relation to define some related sets. Thesesets focus on the single alternative in the consumption set andexamine the ranking of all other alternatives relative to it.

Definition 1.4 Sets in X Derived from the Preference Relation

Let x0 be any point in the consumption set, X . Relative to any suchpoint, we define the following subsets of X :

1. % (x0) ≡ {x|x ∈ X ,x % x0}, called the "at least as good as" set.2. - (x0) ≡ {x|x ∈ X ,x0 % x}, called the "no better than" set.3. � (x0) ≡ {x|x ∈ X ,x � x0}, called the "preferred to" set.4. ≺ (x0) ≡ {x|x ∈ X ,x0 � x}, called the "worse than" set.5. ∼ (x0) ≡ {x|x ∈ X ,x ∼ x0}, called the "indifference" set.

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A hypothetical set of preferences satisfying Axiom 1 and 2 is sketchedin Figure 1.1 (see the next slide) for X = R2

+. We pick any pointx0 = (x0

1 , x02 ) representing a consumption plan consisting of a certain

amount x01 of commodity, together with a certain amount x0

2 ofcommodity 2. From Axiom 1, it is assumed the consumer knows howto compare different points (or consumption bundles) with x0. FromAxiom 2, it is assumed that he knows how to rank his preferences ofthese points. Given Definition 1.4, Axioms 1 and 2 tell us that theconsumer must place every point in X into one of three mutuallyexclusive categories relative to x0: � (x0), ≺ (x0), or ∼ (x0)

The preference in Figure 1.1 may seem rather odd. First we see thethe gap, or the open area. This irregularity can be ruled out byimposing an additional requirement on preferences:

AXION 3: Continuity. For all x ∈ Rn+, the "at least as good as" set,

% (x), and the "no better than" set, - (x), are closed in x ∈ Rn+.

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Figure 1.1 (Left): Hypothetical preferences satisfying Axioms 1 and 2;

Figure 1.2 (right): Hypothetical preferences satisfying Axioms 1, 2 and 3

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Because � (x0) and ≺ (x0) are closed, so is ∼ (x0) because the latteris the intersection of the former two. Consequently, Axiom 3 rules outthe open area in the difference set in Figure 1.1. Thus we have Figure1.2 (see the last slide).

Another irregularty is the thick region for the set ∼ (x0) in Figure 1.2.We use the following weak axiom to deal with it:AXION 4′: Local Nonsatiation. For all x0 ∈ Rn

+, and for all ε > 0, thereexists some x ∈ Bε(x0) ∩ Rn

+ such that x � x0.Axiom 4′ says that within any vicinity of a given point x0, no matter howsmall that vicinity is, there will always be at least one other point x thatthe consumer prefers to x0. It rules out the possibility of having "zonesof indifference," such as that surrounding x1 in Figure 1.2. Thepreferences depicted in Figure 1.3 (see the next slide) do satisfyAxiom 4′ as well as Axioms 1 to 3. After applying Axiom 4′ to the casesof � (x0) and ≺ (x0), the thick regions are all eliminated as shown inFigure 1.4 (see the next slide).

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Figure 1.3 (Left): Hypothetical preferences satisfying Axioms 1, 2, 3 and 4′;

Figure 1.4 (right): Hypothetical preferences satisfying Axioms 1, 2, 3, and 4′

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AXION 4: Strict Monotonicity. For all x0,x1 ∈ Rn+, if x0 ≥ x1 then

x0 % x1, while if x0 � x1, then x0 � x1.

Axiom 4 says that if one bundle contains at least as much of everycommodity as another bundle, then the one is at least as good as theother. Moreover, it is strictly better if it contains strictly more of everygood. Axiom 4 eliminates the possibility that the indifference sets in R2

+

"bend upward," or contain positively sloped segments. It also requiresthat the "preferred to" sets be "above" the indifference sets and that"the worse than" sets be "below" them. For example, in Figure 1.4,points like x2 located in southwest quadrant of x0 should be worse tox0 because x0 � x2; and points like x1 located in southwest quadrantof x0 should be preferred to x0 because x1 � x0. Therefore, a set ofpreferences satisfying Axioms 1, 2, 3, and 4 is given in Figure 1.5 (seethe next slide).

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The indifference set in Figure 1.5 has both regions convex andconcave to the origin. To have the familiar shape of preference, weneed one final axiom on tastes. We will state two different versions ofthe axiom and then consider their meaning and purpose.

Figure 1.5: Hypothetical preferences satisfying Axioms 1, 2, 3 and 4

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AXION 5′: Convexity. If x1 % x0, then tx1 + (1− t)x0%x0 for allt ∈ [0,1] .A slightly stronger version of this is the following:AXION 5: Strict Convexity. If x1 6= x0 and x1 % x0, thentx1 + (1− t)x0�x0 for all t ∈ (0,1) .

Now we pick x1 and x2 on the indifference set ∼ (x0) in Figure 1.5.and find a point xt which is a convex combination of x1 and x2. Axiom5′ says xt should be either in ∼ (x0) or � (x0). Axiom 5 says xt shouldbe in � (x0) only. Both Axioms 5′ and 5 rule out the cases where theindifference set concave to the origin. Thus we can obtain anindifference set depicted in Figure 1.6. (see the next slide). (Whatwould the graph look like as long as Axiom 5′ is imposed but Axiom 5isn’t?) The thrust of Axiom 5′ or Axiom 5 is to forbid the consumer frompreferring extremes in consumption. Axiom 5′ requires that anyrelatively balanced bundle as xt be no worse than either of the twoextremes between which the consumer is indifferent.

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Axiom 5 goes a bit further and requires that the consumer strictlyprefer any such relatively balanced consumption bundle to both of theextreme s between which she is indifferent.

Figure 1.6: Hypothetical preferences satisfying Axioms 1, 2, 3, 4, and 5′ or 5

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Another way to describe the implications of convexity for consumer’stastes focuses attention on the "curvature" of the indifference setsthemselves. When X = R2

+, the (absolute value of the) slope of anindifference curve is called the marginal rate of substitution, or MRS. The slope measures, at any point, the rate at which the consumer iswilling to give up x2 in exchange for x1, such that he remains indifferentafter the exchange.Axiom 5′ requires that the MRS to be either constant or decreasing,where the latter one means the consumer wants to have more x1 tocompensate a given amount of loss in x2 because he has relativelymuch x1 and little x2. Axiom 5 goes a bit further and requires that theMRS to be strictly diminishing. The indifference curve in Figure 1.6display this property, sometimes called the principle of diminishingmarginal rate of substitution in consumption.

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The Utility Function

In modern theory, a utility function is simply a convenient device forsummarizing the information contained in the consumer’s preferencerelation. When we would like to employ calculus methods, it is easierto work with a utility function. A utility function is defined formally asfollows.

Definition 1.5 A Utility Function Representing the Preference Relation%

A real-valued function u : Rn+ → R is called a utility function

representing the preference relation %, if for all x0,x1 ∈ Rn+,

u(x0) ≥ u(x1)⇐⇒ x0 % x1.

Thus a utility function represents a consumer’s preference relation if itassigns higher number to preferred bundles.

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Having a continuous real-valued utility function can simplify theanalysis of many problems. The preference relations used in the lastsection can guarantee the existence of a continuous real-valued utilityfunction.

Theorem 1.1 Existence of a Real-Valued Function Representing thePreference RelationIf the binary relation % is complete, transitive, continuous, and strictlymonotonic, there exists a continuous real-valued function, u : Rn

+ → R,which represents %.

With Theorem 1.1, we can represent preferences in terms of acontinuous utility function. But this utility representation is neverunique. If some function u represents a consumer’s preferences, thenso too will the function v = u + 5, or the function v = u3, because eachof these functions ranks bundles the same way u does. For preferencerepresentation, the rankings of consumption bundles are moremeaningful than the value of utility associated with these consumption

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bundles. If we transform the function u to another function v and theorder of every consumption bundle is preserved, we call it monotonictransform. The following theorem tells us an utility function stillrepresents the original preference relations after the monotonictransforms.

Theorem 1.2 Invariance of the Utility Function to Positive MonotonicTransformsLet % be a preference relation on Rn

+ and suppose u(x) is an utilityfunction that represents it. Then v(x) also represents % if and only ifv(x) = f (u(x)) for every x, where f : R→ R is strictly increasing on theset of values taken on by u.

Any additional axioms we impose on preferences will be reflected asadditional structure on the utility function representing them. Thefollowing theorems are some of them,

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Theorem 1.3 Properties of Preferences and Utility Functions

Let % be represented by u : Rn+ → R. Then:

u(x) is strictly increasing if and only if % is strictly monotonic.u(x) is quasiconcave if and only if % is convex.u(x) is strictly quasiconcave if and only if % is strictly convex.

We had known there exist continuous real-valued utility functions byaxioms imposed on preferences. However, to use calculus tools for theutility functions, they should be differentiable. This implies a more strictrestriction should be imposed on preferences. To our convenience, weare content to simply assume that the utiltiy function is differentiablewhenever necessary.Once the utility function is differentiable, the first-order partialderivative of u(x) with respect to xi is called the marginal utility ofgood i . For the case of two goods, we can derive the MRS in terms ofthe two goods’ marginal utilities. To see this, consider any bundle

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x1 = (x11 , x

12 ). We can write x1

2 as a function of x11 , that is, x1

2 = f (x11 ).

Therefore, as x11 varies, the bundle (x1

1 , x12 ) = (x1

1 , f (x11 )) traces out the

difference curve through x1. Consequently, for all x11

u(x11 , f (x

11 )) = constant (eq.1)

Now the MRS of good one for good two at the bundle x1 = (x11 , x

12 ),

denoted MRS12(x11 , x

12 ), is the absolute value of the slope of the

indifference curve through (x11 , x

12 ). That is,

MRS12(x11 , x

12 ) ≡ |f ′(x1

1 )|= −f ′(x11 ) (eq.2)

, because f ′ < 0. Take the derivative of (eq.1) with respect to x11 ,

∂u(x11 , x

12 )

∂x11

+∂u(x1

1 , x12 )

∂x12

f ′(x11 ) = 0 (eq.3)

(eq.2) and (eq.3) together give

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MRS12(x1) =∂u(x1)/∂x1

1

∂u(x1)/∂x12

Be more general, if we have x ∈ Rn+, we can define the MRS of good i

for good j as the ratio of their marginal utilities,

MRSij(x) =∂u(x)/∂xi

∂u(x)/∂xj

Again this formula tells us the rate at which xi can be substituted for xjwith no change in the consumer’s utility. When u(x) is differentiableand preferences are strictly monotonic, the marginal utility of everygood is virtually always strictly positive. That is, ∂u(x)/∂xi > 0 for alli = 1, · · · ,n. When preferences are strictly convex, the MRS betweentwo goods is always strictly diminishing along any level surface of theutility function.

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1.3 The Consumer’s Problem

A consumer have viewed the consumption set X = Rn+. Her inclination

and attitudes toward consumption bundles x are described by thepreference relation %. The consumer’s circumstances limit thealternatives she is actually able to achieve, and we collect these alltogether into a feasible set B ⊂ Rn

+. The consumer’s problem is to findthe most preferred alternatives in the fesible set according to herpreference relation. Formally, the consumer seeks

x∗ ∈ B such that x∗ % x for all x ∈ B (Pr .1)

To make further progress, we make the following assumptions that willbe maintained unless stated otherwise.

Assumption 1.2 Consumer PreferencesThe consumer’s preference relation % is complete, transitive,continuous, strictly monotonic, and strictly convex on Rn

+. Therefore,by Theorem 1.1 and 1.3 it can be represented by a real-valued utilityfunction, u, that is continuous, strictly increasing, and strictlyquasiconcave on Rn

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Assumption 1.2 (Continued)function, u, that is continuous, strictly increasing, and strictlyquasiconcave on Rn

+.

In the two-good case, preferences like these can be represented by anindifference map whose level sets are nonintersecting, strictly convexaway from the origin, and increasing northeasterly, as depicted inFigure 1.8.(see the next slide).

Next we assume the consumer is in the market economy. By amarket economy, we mean an economic system in which transactionsbetween agents are mediated by markets. The transactions are sonumerous that no individual agent has power to affect the prices. Tothe consumer, a price pi for each commodity i is fixed to him. Wesuppose that prices are strictly positive, so pi > 0, i = 1, · · · ,n, orp� 0 by vector notation.

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Figure 1.8: Indifference map for preferences satisfying Assumption 1.2

The consumer is endowed with a fixed money income y ≥ 0. Thepurchase of xi units of commodity i at price pi per unit requires anexpenditure of pixi dollars, the requirement that expenditure notexceed income can be stated as

∑ni=1 pixi ≤ y , or more compactly, as

p · x ≤ y .

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A feasible set B is the set of consumption bundle afford to theconsumer given his income y ,

B = {x|x ∈ Rn+,p · x ≤ y}.

B is also called budget set. In the two-good case, B consists of allbundles lying inside or on the boundaries of the shared region inFigure 1.9.

Figure 1.9: Budget set, B = {x|x ∈ Rn+,p · x ≤ y}, in the case of two

commodities.

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By Assumption 1.2 and the assumptions on the feasible set, totalexpenditure must not exceed income. We can recast the consumer’sproblem in (Pr. 1) as

maxx∈Rn+u(x) s.t . p · x ≤ y . (Pr .2)

Note that if x∗ solves this problem, then u(x∗) ≥ u(x) for all x ∈ B,which means that x∗ % x for all x ∈ B. That is, solution to (Pr .2) areindeed solutions to (Pr .1). The converse is also true.

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