UTILITY MAXIMIZATION AND CHOICE

UTILITY MAXIMIZATION AND CHOICE

In this chapter we will examine the basic model of choice that economists use to explain indi-viduals’ behavior. That model assumes that individuals who are constrained by limited in-comes will behave as if they were using their purchasing power in such a way as to achieve thehighest utility possible. That is, individuals are assumed to behave as if they maximized util-ity subject to a budget constraint. Although the specific applications of this model are quite var-ied, as we will show, all of them are based on the same fundamental mathematical model, andall arrive at the same general conclusion: To maximize utility, individuals will choose bun-dles of commodities for which the rate of trade-off between any two goods (the MRS) is equalto the ratio of the goods’ market prices. Market prices convey information about opportunitycosts to individuals, and this information plays an important role in affecting the choices ac-tually made.

4C H A P T E R

Utility Maximization and Lightning Calculations

Before starting a formal study of the theory of choice, it may be appropriate to dis-pose of two complaints noneconomists often make about the approach we will take.First is the charge that no real person can make the kinds of “lightning calcula-tions” required for utility maximization. According to this complaint, when movingdown a supermarket aisle, people just grab what is available with no real pattern orpurpose to their actions. Economists are not persuaded by this complaint. Theydoubt that people behave randomly (everyone, after all, is bound by some sort ofbudget constraint), and they view the lightning calculation charge as misplaced. Re-call, again, Friedman’s pool player. He or she also cannot make the lightning cal-culations required to plan a shot according to the laws of physics, but those laws stillpredict the player’s behavior. So too, as we shall see, the utility-maximization modelpredicts many aspects of behavior even though no one carries around a computerwith his or her utility function programmed into it. To be precise, economists as-sume that people behave as if they made such calculations, so the complaint thatthe calculations cannot possibly be made is irrelevant.

Altruism and Selfishness

A second complaint against our model of choice is that it appears to be extremelyselfish—no one, according to this complaint, has such solely self-centered goals. Al-though economists are probably more ready to accept self-interest as a motivatingforce than are other, more Utopian thinkers (Adam Smith observed, “We are notready to suspect any person of being deficient in selfishness”1), this charge is also mis-placed. Nothing in the utility-maximization model prevents individuals from derivingsatisfaction from philanthropy or generally “doing good.” These activities also can beassumed to provide utility. Indeed, economists have used the utility-maximizationmodel extensively to study such issues as donating time and money to charity, leavingbequests to children, or even giving blood. One need not take a position on whethersuch activities are “selfish” or “selfless” since economists doubt people would under-take them if they were against their own best interests, broadly conceived.

An Initial Survey

The general results of our examination of utility maximization can be stated succinctly:

Utility maximization To maximize utility, given a fixed amount of income tospend, an individual will buy those quantities of goods that exhaust his or hertotal income and for which the psychic rate of trade-off between any two goods(the MRS) is equal to the rate at which the goods can be traded one for theother in the marketplace.

OPTIMIZATION PRINCIPLE

92 Par t I I Choice and Demand

1Adam Smith, The Theory of Moral Sentiments (1759; reprint, New Rochelle, NY: Arlington House, 1969),p. 446.

That spending all one’s income is required for utility maximization is obvious.Since extra goods provide extra utility (there is no satiation) and since there is noother use for income, to leave any unspent would be to fail to maximize utility.Throwing money away is not a utility-maximizing activity.

The condition specifying equality of trade-off rates requires a bit more explana-tion. Because the rate at which one good can be traded for another in the marketis given by the ratio of their prices, this result can be restated to say that the indi-vidual will equate the MRS (of X for Y) to the ratio of the price of X to the price of Y (PX/PY). This equating of a personal trade-off rate to a market determinedtrade-off rate is a result common to all individual utility-maximization problems(and to many other types of maximization problems). It will occur again and againthroughout this text.

A Numerical Illustration

To see the intuitive reasoning behind this result, assume that it were not true thatan individual had equated the MRS to the ratio of the prices of goods. Specifically,assume that the individual’s MRS is equal to 1, that he or she is willing to trade 1 unit of X for 1 unit of Y and remain equally well off. Assume also that the priceof X is $2 per unit and of Y is $1 per unit. It is easy to show in this case that the in-dividual can be made better off. Give up 1 unit of X and trade it in the market for 2 units of Y. Only 1 extra unit of Y was needed to keep the individual as happy asbefore the trade—the second unit of Y is a net addition to well-being. Therefore,the individual’s spending could not have been allocated optimally in the first place.A similar method of reasoning can be used whenever the MRS and the price ratioPX/PY differ. The condition for maximum utility must be the equality of these twomagnitudes.

The Two-Good Case: A Graphical Analysis

This discussion seems eminently reasonable, but it can hardly be called a proof.Rather, we must now show the result in a rigorous manner and, at the same time,illustrate several other important attributes of the maximization process. First wetake a graphic analysis. Then we take a more mathematical approach.

Budget Constraint

Assume that the individual has I dollars to allocate between good X and good Y. IfPX is the price of good X and PY is the price of good Y, then the individual is con-strained by

PXX � PYY � I. (4.1)

That is, no more than I can be spent on the two goods in question. This budget con-straint is shown graphically in Figure 4.1. The individual can afford to choose onlycombinations of X and Y in the shaded triangle of the figure. If all of I is spent ongood X, it will buy I/PX units of X. Similarly, if all is spent on Y, it will buy I/PY unitsof Y. The slope of the constraint is easily seen to be �PX/PY.

Chapter 4 Utility Maximization and Choice 93

First-Order Conditions for a Maximum

This budget constraint can be imposed on the individual’s indifference curve mapto show the utility-maximization process. Figure 4.2 illustrates this procedure. Theindividual would be irrational to choose a point such as A—he or she can get to ahigher utility level just by spending some of the unspent portion of income. The as-sumption of nonsatiation implies that a person should spend all of his or her in-come in order to receive maximum utility from it. Similarly, by reallocatingexpenditures, the individual can do better than point B. Point D is out of the ques-tion because income is not large enough to purchase D. It is clear that the positionof maximum utility is at point C, where the combination X*, Y* is chosen. This isthe only point on indifference curve U2 that can be bought with I dollars; no higherutility level can be bought. C is a point of tangency between the budget constraintand the indifference curve. Therefore at C,

slope of budget constraint � ��

PP

Y

X� � slope of indifference curve

� �ddXY� �U � constant

(4.2)

or

� � �U � constant� MRS (of X for Y ). (4.3)

dY�dX

PX�PY


The Individual’s Budget Constraint for Two Goods

Those combinations of X and Y that the individual can afford are shown in the shaded triangle. If, as we usually assume,the individual prefers more rather than less of every good, the outer boundary of this triangle is the relevant constraintwhere all of the available funds are spent either on X or on Y. The slope of this straight-line boundary is given by �PX/PY.

Quantity of X0

Quantityof Y

I � PXX � PYY

IPX

IPY

FIGURE 4.1

Our intuitive result is proved—for a utility maximum, all income should be spentand the MRS should equal the ratio of the prices of the goods. It is obvious fromthe diagram that if this condition is not fulfilled, the individual could be made bet-ter off by reallocating expenditures.

Second-Order Conditions for a Maximum

The tangency rule is only a necessary condition for a maximum. To see that it is nota sufficient condition, consider the indifference curve map shown in Figure 4.3.Here a point of tangency (C ) is inferior to a point of nontangency (B). Indeed, thetrue maximum is at another point of tangency (A). The failure of the tangency con-dition to produce an unambiguous maximum can be attributed to the shape of theindifference curves in Figure 4.3. If the indifference curves are shaped like those inFigure 4.2, no such problem can arise. But we have already shown that “normally”shaped indifference curves result from the assumption of a diminishing MRS.Therefore, if the MRS is assumed to be diminishing, the condition of tangency isboth a necessary and sufficient condition for a maximum.2 Without this assumptionone would have to be careful in applying the tangency rule.


A Graphical Demonstration of Utility Maximization

Point C represents the highest utility level that can be reached by the individual, given the budget constraint. The combi-nation X*, Y* is therefore the rational way for the individual to allocate purchasing power. Only for this combination ofgoods will two conditions hold: All available funds will be spent; and the individual’s psychic rate of trade-off (MRS) willbe equal to the rate at which the goods can be traded in the market (PX/PY).

Quantity of X0

Quantityof Y

I � PXX � PYY

Y*

X*

D

C

A

B

U1

U1

U2

U2

U3

U3

FIGURE 4.2

2In mathematical terms, because the assumption of a diminishing MRS is equivalent to assuming quasi-concavity, the necessary conditions for a maximum subject to a linear constraint are also sufficient, aswe showed in Chapter 2.

Corner Solutions

The utility-maximization problem illustrated in Figure 4.2 resulted in an “interior”maximum, in which positive amounts of both goods were consumed. In some situ-ations individuals’ preferences may be such that they can obtain maximum utilityby choosing to consume no amount of one of the goods. If someone does not likehamburgers very much, there is no reason to allocate any income to their purchase.This possibility is reflected in Figure 4.4. There utility is maximized at E, where X � X* andY � 0—any point on the budget constraint where positive amounts ofY are consumed yields a lower utility than does point E. Notice that at E the budgetconstraint is not precisely tangent to the indifference curve U2. Instead, at the op-timal point the budget constraint is flatter than U2, indicating that the rate at whichX can be traded for Y in the market is lower than the individual’s psychic trade-offrate (the MRS). At prevailing market prices the individual is more than willing totrade away Y to get extra X. Because it is impossible in this problem to consume neg-ative amounts of Y, however, the physical limit for this process is the X-axis, alongwhich purchases of Y are 0. Hence, as this discussion makes clear, it is necessary toamend the first-order conditions for a utility maximum a bit to allow for corner


Example of an Indifference Curve Map for Which the Tangency ConditionDoes Not Ensure a Maximum

If indifference curves do not obey the assumption of a diminishing MRS, not all points of tangency (points for whichMRS � PX/PY) may truly be points of maximum utility. In this example tangency point C is inferior to many other pointsthat can also be purchased with the available funds. In order that the necessary conditions for a maximum (that is, thetangency conditions) also be sufficient, one usually assumes that the MRS is diminishing; that is, the utility function isstrictly quasi-concave.

Quantity of X

Quantityof Y

I � PXX � PYYA

C

B

U1

U1

U2

U2

U3

U3

FIGURE 4.3

solutions of the type shown in Figure 4.4. Following our discussion of the generaln-good case, we will show how this can be accomplished.

The n-Good Case

The results derived graphically in the case of two goods carry over directly to thecase of n goods. Again it can be shown that for an interior utility maximum, theMRS between any two goods must equal the ratio of the prices of these goods. Tostudy this more general case, however, it is best to use some mathematics.

First-Order Conditions

With n goods, the individual’s objective is to maximize utility from these n goods:

utility � U(X1, X2, . . . , Xn), (4.4)

subject to the budget constraint:3

I � P1X1 � P2X 2 � . . . � PnXn (4.5)

or

I � P1X1 � P2X 2 � . . . � PnXn � 0. (4.6)


Corner Solution for Utility Maximization

With the preferences represented by this set of indifference curves, utility maximization occurs at E, where 0 amounts of goodY are consumed. The first-order conditions for a maximum must be modified somewhat to accommodate this possibility.

Quantity of X

Quantityof Y

U1 U2 U3

E

X*

FIGURE 4.4

3Again, the budget constraint has been written as an equality here because, given the assumption of non-satiation, it is clear that the individual will spend all available income.

Following the techniques developed in Chapter 2 for maximizing a function sub-ject to a constraint, we set up the Lagrangian expression

� � U(X1, X 2, . . . , Xn) � �(I � P1X1 � P2X 2 � . . . � PnXn). (4.7)

Setting the partial derivatives of � (with respect to X1, X2, . . . , Xn and �) equal to 0 yields n � 1 equations representing the necessary conditions for an interior maximum:

� � �P1 � 0 (4.8)

� � �P2 � 0

...

� � �Pn � 0

� I � P1X1 � P2X 2 � . . . � PnXn � 0.

These n � 1 equations can usually be solved for the optimal X1, X2, . . . , Xn and for� (see Example 4.1 to be convinced that such a solution is possible).

Equations 4.8 are necessary but not sufficient for a maximum. The second-orderconditions that ensure a maximum are relatively complex and must be stated in ma-trix terms (see the Appendix to Chapter 2). However, the assumption of strict quasi-concavity (a diminishing MRS in the two-good case) is sufficient to ensure that anypoint obeying Equations 4.8 is in fact a true maximum.

Implications of First-Order Conditions

The first-order conditions represented by Equations 4.8 can be rewritten in a vari-ety of interesting ways. For example, for any two goods, Xi and Xj, we have

� . (4.9)

But in Chapter 3 we showed that the ratio of the marginal utilities of two goods isequal to the marginal rate of substitution between them. Therefore, the conditionsfor an optimal allocation of income become

MRS (Xi for Xj) � . (4.10)

This is exactly the result derived earlier in this chapter; to maximize utility, the in-dividual should equate the psychic rate of trade-off to the market trade-off rate.

Interpreting the Lagrangian Multiplier

Another result can be derived by solving Equations 4.8 for �:

� � � � . . . � (4.11)�U/�Xn�

Pn

�U/�X2�

P2

�U/�X1�

P1

Pi�Pj

Pi�Pj

�U/�Xi��U/�Xj

��

�U��Xn

��Xn

�U��X2

��X2

�U��X1

��X1


or

� � � � . . . � .

This equation says that at the utility-maximizing point, each good purchased shouldyield the same marginal utility per dollar spent on that good. Each good thereforeshould have an identical (marginal) benefit to (marginal) cost ratio. If this were nottrue, one good would promise more “marginal enjoyment per dollar” than someother good, and funds would not be optimally allocated.

Although the reader is again warned against talking very confidently aboutmarginal utility, what Equation 4.11 says is that an extra dollar should yield thesame “additional utility” no matter which good it is spent on. The common valuefor this extra utility is given by the Lagrangian multiplier for the consumer’sbudget constraint (that is, by �). Consequently, � can be regarded as the mar-ginal utility of an extra dollar of consumption expenditure (the marginal utilityof “income”).

One final way to rewrite the necessary conditions for a maximum is

Pi � (4.12)

for every good i that is bought. This equation says that for every good that an in-dividual buys, the price of that good represents his or her evaluation of the util-ity of the last unit consumed. The price obviously represents how much theindividual is willing to pay for that last unit. In Chapter 5 (and elsewhere) we willmake considerable use of this result when discussing the value of a good to a consumer and the “consumer surplus” received by some purchasers when theyare able to buy a good for less than the maximum amount they would be will-ing to pay.

Corner Solutions

The first-order conditions of Equations 4.8 hold exactly only for interior maximafor which some positive amount of each good is purchased. When corner solutions(such as those illustrated in Figure 4.4) arise, the conditions have to be modifiedslightly.4 In this case, Equations 4.8 become

� ��Pi � 0 (i � 1 . . . n), (4.13)

and, if

� � �Pi � 0, (4.14)

then

Xi � 0. (4.15)

�U��Xi

��xi

�U��Xi

��Xi

MUx i��

MUx n�Pn

MUx 2�P2

MUx 1�P1


4Formally, these conditions are called the “Kuhn-Tucker” conditions for nonlinear programming. For amore complete explanation, see A. K. Dixit, Optimization in Economic Theory, 2nd ed. (New York: OxfordUniversity Press, 1990).

To interpret these conditions, we can rewrite Equation 4.14 as

Pi � � . (4.16)

Hence, the optimal conditions are as before, except that any good whose price (Pi)exceeds its marginal value to the consumer (MUx i

/�) will not be purchased (Xi � 0).Thus, the mathematical results conform to the commonsense idea that individualswill not purchase goods that they believe are not worth the money. Although cor-ner solutions do not provide a major focus for our analysis in this book, the readershould keep in mind the possibilities for such solutions arising and the economicinterpretation that can be attached to the optimal conditions in such cases.

EXAMPLE 4.1

Cobb-Douglas Demand Functions

As we showed in Chapter 3, the Cobb-Douglas utility function is given by

U(X, Y ) � X�Y �, (4.17)

where, for convenience,5 we assume � � � � 1. We can now solve for the utility-maximizing values of X and Y for any prices (PX, PY) and income (I ). Setting up theLagrangian expression

� � X�Y� � �(I � PXX � PYY ) (4.18)

yields the first-order conditions

� �X��1Y� � �PX �0 (4.19)

� �X�Y��1 � �PY � 0

� I � PXX � PYY � 0.

Taking the ratio of the first two terms shows that

� (4.20)

or

PYY � PXX � PXX, (4.21)

where the final equation follows because � � � � 1. Substitution of the first-ordercondition in Equation 4.21 into the budget constraint gives

I � PXX � PYY � PXX � PXX � PXX �1 � � � PXX; (4.22)1��

1 � ��

�

1 � ��

�

1 � ��

�

��

PX�PY

�Y��X

��

��Y

��X

MUx i��

��

�

XU

i

�

��


5Notice that the exponents in the Cobb-Douglas utility function can always be normalized to sum to onebecause U 1/(��) is a monotonic transformation.

solving for X yields

X* � ; (4.23)

and a similar set of manipulations would give

Y* � . (4.24)

These results show that an individual whose utility function is given by Equation4.17 will always choose to allocate � percent of his or her income to buying good X(that is, PXX/I � �) and � percent to buying good Y (PYY/I � �). Although this fea-ture of the Cobb-Douglas function often makes it very easy to work out simple prob-lems, it does suggest that the function has limits in its ability to explain actualconsumption behavior. Because the share of income devoted to particular goods of-ten changes significantly in response to changing economic conditions, a moregeneral functional form may provide insights not provided by the Cobb-Douglasfunction. We illustrate a few possibilities in Example 4.2.

Numerical Example. First, however, let’s look at a specific numerical example forthe Cobb-Douglas case. Suppose that X sells for $.25 and Y sells for $1.00 and thattotal income is $2.00. Succinctly then, assume that PX � .25, PY � 1, I � 2. Supposealso that � � � � 0.5 so that this individual splits his or her income equally betweenthese two goods. Now the demand Equations 4.23 and 4.24 imply

X* � �I/PX � .5I/PX � .5(2)/.25 � 4 (4.25)

Y* � �I/PY � .5I/PY � .5(2)/1 � 1

and, at these optimal choices,

Utility � X .5Y .5 � (4).5(1).5 � 2. (4.26)

Notice also that we can compute the value for the Lagrangian Multiplier associatedwith this income allocation by using Equation 4.19:

� � �X��1Y�/PX � .5(4)�.5(1).5/.25 � 1. (4.27)

This value implies that small changes in income yield about the same size changesin utility. For example, if income were to rise to I � 2.1 (with PX and PY unchanged),Equations 4.23 and 4.24 predict that X* � 4.2, Y * � 1.05 and the new level of util-ity would be

Utility � (4.2).5(1.05).5 � 2.10, (4.28)

which was predicted by the fact that � � 1.

QUERY: Would a change in PY affect the quantity of X demanded in Equation 4.23?Explain your answer mathematically. Also develop an intuitive explanation basedon the notion that the share of income devoted to good Y is a constant given bythe parameter of the utility function, �.

�I�PY

�I�PX


EXAMPLE 4.2

CES Demand

To illustrate cases in which budget shares are responsive to economic circum-stances, let’s look at two specific examples of the CES function. First, assume � � .5in the CES function. Then utility is given by

U(X, Y) � X .5 � Y.5. (4.29)

Setting up the Lagrangian expression

� � X .5 � Y.5 � �(I � PXX � PYY ) (4.30)

yields the following first order conditions for a maximum:

��/�X � .5X�.5 � �PX � 0 (4.31)

��/�Y � .5Y�.5 � �PY � 0

��/�� I � PXX � PYY � 0.

Division of the first two of these shows that

(Y/X ).5 � PX/PY . (4.32)

By substituting this into the budget constraint and using some algebraic manip-ulation, it is fairly easy to derive the demand functions associated with this utilityfunction:

X* � I/PX[1 � (PX/PY)] (4.33)

Y* � I/PY[1 � (PY/PX)]. (4.34)

Price Responsiveness. In these demand functions notice that the share of incomespent on, say, good X—that is, PXX/I � 1/[1 � (PX/PY)]—is not a constant, it de-pends on the price ratio PX/PY . The higher is the relative price of X, the smaller willbe the share of income spent on that good. In other words, the demand for X is soresponsive to its own price that a rise in the price reduces total spending on X. Thatthe demand for X is very price responsive can also be illustrated by comparing theexponent on PX in the demand function given by Equation 4.33 (�2) to that fromEquation 4.23 (�1). In Chapter 7 we will discuss this observation more fully whenwe examine the elasticity concept in detail.

A CES Function with Less Substitutability. Alternatively, let’s look at a demand func-tion with less substitutability6 than the Cobb-Douglas. If � � �1, the utility functionis given by

U(X, Y ) � �X�1 � Y�1, (4.35)

and it is easy to show that the first-order conditions for a maximum require

Y/X � (PX/PY).5. (4.36)


6One way to measure substitutability is by the elasticity of substitution, which for the CES function is given by � � 1/(1 � �). Here � � .5 implies � � 2, � � 0 (the Cobb-Douglas) implies � � 1, and � � �1 implies � � .5. See also the discussion of the CES function in connection with the theory of pro-duction in Chapter 11.

Again, substitution of this condition into the budget constraint, together with somealgebra, yields the demand functions

X* � I/PX[1 � (PY/PX).5] (4.37)

Y* � I/PY[1 � (PX/PY).5].

That these demand functions are less price responsive can be seen in two ways.First, now the share of income spent on good X—PXX I � 1/[1 � (PY/PX).5]—responds positively to increases in PX. As the price of X rises, this individual cutsback only modestly in good X, so total spending on that good rises. That the de-mand functions in Equations 4.37 are less price responsive than the Cobb-Douglasis also illustrated by the relatively small exponents of each good’s own price (�.5)in Equations 4.37. Overall then, the CES function allows us to illustrate a wide va-riety of possible relationships between two goods.7

QUERY: Do changes in income affect expenditure shares in any of the CES functionsdiscussed here? How is the behavior of expenditure shares related to the homo-thetic nature of this function?

Indirect Utility Function

Examples 4.1 and 4.2 illustrate the principle that it is often possible to manipulatethe first-order conditions for a constrained utility-maximization problem to solvefor the optimal values of X1, X2, . . . , Xn. These optimal values in general will dependon the prices of all the goods and on the individual’s income. That is,

X*1 � X1(P1, P2, . . . , Pn, I ) (4.38)

X*2 � X2(P1, P2, . . . , Pn, I )...

X*n � Xn(P1, P2, . . . , Pn, I ).

In later chapters we will analyze in more detail this set of demand functions, whichshow the dependence of the quantity of each Xi demanded on P1, P2, . . . , Pn and I.Here we use the optimal values of the Xs from Equations 4.38 to substitute in theoriginal utility function to yield

maximum utility � U(X*1, X*2, . . . , X*n) (4.39)

� U[X*1(P1, P2, . . . , Pn, I ),

X*2(P1, P2, . . . , Pn, I ),

. . . X*n(P1, P2, . . . , Pn, I )]

� V(P1, P2, . . . , Pn, I ).


7These relationships for the CES function are pursued in more detail in Problem 4.9 and in Ex-tension E4.3.

In words, because of the individual’s desire to maximize utility, given a budget con-straint, the optimal level of utility obtainable will depend indirectly on the prices ofthe goods being bought and on the individual’s income. This dependence is re-flected by the indirect utility function V. If either prices or income were to change,the level of utility that can be attained would also be affected. Sometimes, in bothconsumer theory and in many other contexts, it is possible to use this indirect ap-proach to study how changes in economic circumstances affect various kinds of out-comes, such as utility or (later in this book) firms’ costs.

EXAMPLE 4.3

Indirect Utility in the Cobb-Douglas

In the numerical illustration of Example 4.1 we found (Equations 4.25)

X* � (4.40)

Y* � .

Substituting these into the utility function gives

maximum utility � U(X*, Y*) � (X*).5(Y*).5 (4.41)

� � �.5

� �.5

� . (4.42)

With I � 2, PX � .25 and PY � 1, Equation 4.42 shows that maximum utility can beindirectly computed as

maximum utility � � 2, (4.43)

which is the same value we derived from the direct utility function. More generally,notice in Equation 4.42 that increases in income raise (indirect) utility, whereas in-creases in either of the prices cause utility to fall. By stating utility as a function ofsuch “outside forces” as prices and income, it is possible to study explicitly theseforces’ effects on well-being.

Lump Sum Principle. The indirect utility concept is very useful for studying the im-pact of taxes on an individual’s utility. For example, it is straightforward to illus-trate that “lump sum” principle that general income taxes reduce utility to asmaller extent than do single commodity taxes that yield the same revenue to thegovernment. In the present case, suppose the government were to adopt a $.50income tax. Equation 4.43 shows that this would reduce the individual’s indirectutility from 2.00 to 1.50. A tax on good X of $.25 would raise the same revenues,because Equation 4.25 shows that when PX rises from $.25 to $.50, purchases fall

2��2(.25).5(1).5

I�2P .5

XP .5Y

I�2PY

I�2PX

I�2PY

I�2PX


to 2. Hence, tax collections are $.50. With the sales tax, the individual’s indirectutility is now

maximum utility � � � 1.41, (4.44)

which falls short of utility under the income tax. The reason is that a sales tax altersindividuals’ choices in two ways—by reducing purchasing power and by changingrelative prices. The income tax has only the first effect and is therefore less harm-ful. Additional material related to the lump sum principle is discussed in Problems4.7 and 4.8.

QUERY: The indirect utility function in Equation 4.42 shows that a doubling of in-come and all prices leaves utility unchanged. Explain why that is a general propertyof all indirect utility functions.

Expenditure Minimization

In Chapter 2 we pointed out that many constrained maximum problems have as-sociated “dual” constrained minimum problems. For the case of utility maximiza-tion, the associated dual minimization problem concerns allocating income in sucha way as to achieve a given utility level with the minimal expenditure. This problemis clearly analogus to the primary utility-maximization problem, but the goals andconstraints of the problems have been reversed. Figure 4.5 illustrates this dual expenditure-minimization problem. There the individual must attain utility levelU2—this is now the constraint in the problem. Three possible expenditure amounts(E1, E2, and E3) are shown as three “budget constraint” lines in the figure. Expen-diture level E1 is clearly too small to achieve U2, hence it cannot solve the dual prob-lem. With expenditures given by E3, the individual can reach U2 (at either of the twopoints B or C), but this is not the minimal expenditure level required. Rather, E2 clearly provides just enough total expenditures to reach U2 (at point A), and this is in fact the solution to the dual problem. By comparing Figures 4.2 and 4.5,it is obvious that both the primary utility-maximization approach and the dual expenditure-minimization approach yield the same solution (X*, Y*)—they aresimply alternative ways of viewing the same process. Often the expenditure-minimization approach is more useful, however, because expenditures are directlyobservable, whereas utility is not.

A Mathematical Statement

More formally, the individual’s dual expenditure-minimization problem is tochoose X1, X2, . . . , Xn so as to minimize

total expenditures � E � P1X1 � P2X2 � . . . � PnXn, (4.45)

subject to the constraint

utility � U2 � U(X1, X2, . . . , Xn). (4.46)

2��2(.50).5(1).5

I�2P .5

X P .5Y


The optimal amounts of X1, X2, . . . , Xn chosen in this problem will depend onthe prices of the various goods (P1, P2, . . . , Pn) and on the required utility level U2.If any of the prices were to change or if the individual had a different utility “tar-get,” another commodity bundle would be optimal. This dependence can be sum-marized by an expenditure function.

Expenditure Function The individual’s expenditure function shows the mini-mal expenditures necessary to achieve a given utility level for a particular set ofprices. That is,

minimal expenditures � E(P1, P2, . . . , Pn, U ). (4.47)

This definition shows that the expenditure function and the indirect utility func-tion are inverse functions of one another (compare Equations 4.39 and 4.47). Bothdepend on market prices but involve different constraints (income or utility). Inthe next chapter we will see how this relationship is quite useful in allowing us toexamine the theory of how individuals respond to price changes.

DEFINITION


The Individual’s Dual Expenditure-Minimization Problem

The dual of the individual’s utility-maximization problem is to attain a given utility level (U2) with minimal expenditures.An expenditure level of E1 does not permit U2 to be reached, whereas E3 provides more spending power than is strictlynecessary. With expenditure E2 the individual can just reach U2 by consuming X* and Y*.

Quantity of X

Quantityof Y

C

A

B

X*

Y*

U2

E3

E2E1

FIGURE 4.5

EXAMPLE 4.4

Expenditure Function from the Cobb-Douglas

Returning yet again to the Cobb-Douglas utility function, the individual’s dualproblem is to minimize

E � PXX � PYY (4.48)

subject to

utility � U� � X .5Y .5, (4.49)

where U� is the utility target.The Lagrangian expression for this problem is

� � PXX � PYY � �(U� � X .5Y .5), (4.50)

and the first-order conditions for a minimum are

� PX � .5�X�.5Y .5 � 0 (4.51)

� PY � .5�X .5Y�.5 � 0

� U� � X .5Y .5 � 0.

These can again be solved by moving the terms in � to the right and dividing:

� � (4.52)

or

PXX � PYY, (4.53)

which is precisely the same first-order condition we had before (see Equation 4.21with � � � � .5). Now, however, we wish to solve for expenditures as a function ofPX, PY, and U—that is, we wish to eliminate X and Y from Equation 4.48. This willgive us the type of expenditure function we defined earlier in this section. Althoughthe algebra here isn’t difficult, it is important to keep this goal in mind because itis easy to become confused about whether you have found a solution. SubstitutingEquation 4.53 into the expenditure function yields

E � PXX* � PYY* � 2PXX* (4.54)

so

X* � (4.55)

and, similarly,

Y* � . (4.56)E

�2PY

E�2PX

X�Y

.5�X .5Y�.5

��.5�X�.5Y .5

PY�PX

��

��Y

��X


But, the utility target requires

U� � (X*).5(Y*).5, (4.57)

so

U� � � �.5

� �.5

� . (4.58)

Hence, we have the function

E � 2U�P .5XP .5

Y (4.59)

as the minimum expenditure necessary to reach U�. If, as before, U� � 2, PX � .25,and PY � 1, we have a required expenditure of

E � 2(2)(.25).5(1).5 � 2. (4.60)

Notice this was the original value for income with which we started this problem.We know that this income level is indeed just sufficient to attain a utility level of 2.Of course, as the expenditure function in Equation 4.59 shows, a higher utility tar-get would require greater expenditures. Similarly, an increase in PX or PY would alsorequire greater expenditures to attain a given utility target. Without such added ex-penditures, the utility target would have to be reduced—the individual would beworse off. Looked at in another way, the expenditure function shows how much ex-tra purchasing power this person would need to compensate for a rise in the priceof a good. In later chapters we will make some use of this property of the function.

QUERY: A doubling of PX and PY in Equation 4.59 will precisely double the expen-ditures needed to reach U�. Technically, this function is “homogeneous of degreeone” in the prices of the two goods (see footnote 1 in Chapter 5). Is this a propertyof all expenditure functions?

Summary

In this chapter we examined the basic economic model of utility maximization sub-ject to a budget constraint. Although we approached this problem in a variety ofways, all of these approaches lead to the same basic result:

• To reach a constrained maximum, an individual should spend all available in-come and should choose a commodity bundle such that the MRS between anytwo goods is equal to the ratio of those goods’ market prices. This basic tangencywill result in the individual equating the ratios of the marginal utility to marketprice for every good that is actually consumed. Such a result is common to mostconstrained optimization problems.

• The tangency conditions are only the first-order conditions for a constrainedmaximum, however. To ensure that these conditions are also sufficient, the in-dividual’s indifference curve map must exhibit a diminishing MRS. In formalterms, the utility function must be strictly quasi-concave.

• The tangency conditions must also be modified to allow for corner solutions inwhich the optimal level of consumption of some goods is zero. In this case, the

E�2P .5

XP .5Y

E�2PY

E�2PX


ratio of marginal utility to price for such a good will be below the common mar-ginal benefit–marginal cost ratio for goods actually bought.

• A consequence of the assumption of constrained utility maximization is that theindividual’s optimal choices will depend implicitly on the parameters of his orher budget constraint. That is, the choices observed will be implicit functions ofall prices and income. Utility will therefore also be an indirect function of theseparameters.

• The dual to the constrained utility-maximization problem is to minimize the ex-penditure required to reach a given utility target. Although this dual approachyields the same optimal solution as the primal constrained maximum problem,it also yields additional insight into the theory of choice. Specifically, this ap-proach leads to expenditure functions in which the spending required to reacha given utility target depends on the goods’ market prices.

Problems4.1Each day Paul, who is in third grade, eats lunch at school. He likes only Twinkies (T ) andOrange Slice (S), and these provide him a utility of

utility � U(T, S) � �TS�.

a. If Twinkies cost $.10 each and Slice costs $.25 per cup, how should Paul spend the $1 hismother gives him in order to maximize his utility?

b. If the school tries to discourage Twinkie consumption by raising the price to $.40, by howmuch will Paul’s mother have to increase his lunch allowance to provide him with the same level of utility he received in part (a)? How many Twinkies and cups of Slicewill he buy now (assuming that it is possible to purchase fractional amounts of both ofthese goods)?

4.2a. A young connoisseur has $300 to spend to build a small wine cellar. She enjoys two vin-

tages in particular: an expensive 1987 French Bordeaux (WF) at $20 per bottle and a lessexpensive 1993 California varietal wine (WC) priced at $4. How much of each wineshould she purchase if her utility is characterized by the following function?

U(WF , WC) � W 2F

/3W 1C

/3.

b. When she arrived at the wine store, our young oenologist discovered that the price of the 1987 French Bordeaux had fallen to $10 a bottle because of a decline in the value of the franc. If the price of the California wine remains stable at $4 per bottle, how much of each wine should our friend purchase to maximize utility under these alteredconditions?

4.3a. On a given evening J. P. enjoys the consumption of cigars (C ) and brandy (B) accord-

ing to the function

U(C, B) � 20C � C2 � 18B � 3B2.

How many cigars and glasses of brandy does he consume during an evening? (Cost is noobject to J. P.)

b. Lately, however, J. P. has been advised by his doctors that he should limit the sum ofbrandy and cigars consumed to 5. How many glasses of brandy and cigars will he con-sume under these circumstances?


4.4a. Mr. Odde Ball enjoys commodities X and Y according to the utility function

U(X, Y ) � �X2 � Y�2�.

Maximize Mr. Ball’s utility if PX � $3, PY � $4, and he has $50 to spend.Hint: It may be easier here to maximize U 2 rather than U. Why won’t this alter your results?

b. Graph Mr. Ball’s indifference curve and its point of tangency with his budget constraint.What does the graph say about Mr. Ball’s behavior? Have you found a true maximum?

4.5Mr. A derives utility from martinis (M ) in proportion to the number he drinks:

U(M) � M.

Mr. A is very particular about his martinis, however: He only enjoys them made in the exactproportion of two parts gin (G ) to one part vermouth (V ). Hence, we can rewrite Mr. A’sutility function as

U(M) � U(G, V ) � min � , V �.a. Graph Mr. A’s indifference curve in terms of G and V for various levels of utility. Show

that regardless of the prices of the two ingredients, Mr. A will never alter the way hemixes martinis.

b. Calculate the demand functions for G and V.c. Using the results from part (b), what is Mr. A’s indirect utility function?d. Calculate Mr. A’s expenditure function; for each level of utility, show spending as a func-

tion of PG and PV.Hint: Because this problem involves a fixed proportions utility function you cannot solvefor utility-maximizing decisions by using calculus.

4.6a. Suppose that a fast-food junkie derives utility from three goods: soft drinks (X ), ham-

burgers (Y ), and ice cream sundaes (Z ) according to the Cobb- Douglas utility function

U(X, Y, Z) � X .5 Y .5 (1 � Z).5.

Suppose also that the prices for these goods are given by PX � .25, PY � 1, and PZ � 2 andthat this consumer’s income is given by I � 2.a. Show that for Z � 0, maximization of utility results in the same optimal choices as in Ex-

ample 4.1. Show also that any choice that results in Z � 0 (even for a fractional Z) re-duces utility from this optimum.

b. How do you explain the fact that Z � 0 is optimal here? (Hint: Think about the ratioMUz/Pz.)

c. How high would this individual’s income have to be in order for any Z to be purchased?

4.7In Example 4.3 we used a specific indirect utility function to illustrate the lump sum princi-ple that an income tax reduces utility to a lesser extent than a sales tax that garners the samerevenue. Here you are asked to:a. Show this result graphically for a two-good case by showing the budget constraints that

must prevail under each tax. (Hint: First draw the sales tax case. Then show that thebudget constraint for an income tax that collects the same revenue must pass throughthe point chosen under the sales tax but will offer options preferable to the individual.)

b. Show that if an individual consumes the two goods in fixed proportions, the lump sumprinciple does not hold because both taxes reduce utility by the same amount.

c. Discuss whether the lump sum principle holds for the many-good case too.

G�2


4.8The lump sum principle discussed in Example 4.3 can be applied to transfers, too, but in thiscase it may be easier to use expenditure functions.a. Consider the expenditure function given by Equation 4.59 in Example 4.4. How much

would it cost the government (in terms of extra expenditures for this person) to raiseutility from 2.0 to 2.5 with unchanged prices? If the government wished to permit indi-viduals to attain the same utility target by subsidizing the cost of hamburgers, whatshould the hamburger subsidy be? How much will such a subsidy cost the government?

b. Explain intuitively and with a graph why the income transfer in part (a) proves to be alower cost way of raising utility than does the hamburger subsidy.

c. Is the lower cost of lump sum transfers a general result that applies to the many-goodcase as well?

4.9The general CES utility function is given by

U(X, Y ) � � .

a. Show that the first-order conditions for a constrained utility maximum with this functionrequire individuals to choose goods in the proportion

� � � .

b. Show that the result in part (a) implies that individuals will allocate their funds equallybetween X and Y for the Cobb-Douglas case (� � 0), as we have shown before in severalproblems.

c. How does the ratio PXX/PYY depend on the value of �? Explain your results intuitively.(For further details on this function, see Extension E4.3.)

4.10Suppose individuals require a certain level of food (X ) to remain alive. Let this amount begiven by X 0. Once X 0 is purchased, individuals obtain utility from food and other goods (Y )of the form

U(X, Y ) � (X � X 0)� Y�

where � � � � 1.a. Show that if I � PXX 0 the individual will maximize utility by spending �(I � PXX 0) � PXX 0

on good X and �(I � PXX 0) on good Y.b. How do the ratios PXX/I and PYY/I change as income increases in this problem? (See

also Extension E4.2.)

Suggested Readings

Barten, A. P., and Volker Böhm. “Consumer Theory.” In K. J. Arrow and M. D. Intriligator, eds., Hand-book of Mathematical Economics. Vol. II. Amsterdam: North-Holland, 1982.

Sections 10 and 11 have compact summaries of many of the concepts covered in this chapter.

Deaton, A., and J. Muelbauer. Economics and Consumer Behavior. Cambridge: Cambridge University Press,1980.

Section 2.5 provides a nice geometric treatment of duality concepts.

Dixit, A. K. Optimization in Economic Theory. Oxford: Oxford University Press, 1990.Chapter 2 provides several Lagrangian analyses focusing on the Cobb-Douglas utility function.

Hicks, J. R. Value and Capital. Oxford: Clarendon Press, 1946.Chapter II and the Mathematical Appendix provide some early suggestions of the importance of the expenditurefunction.

1��1PX

�PY

X�Y

Y �

��

X �

��


Mas-Colell, A., M. D. Whinston, and J. R. Green. Microeconomic Theory. Oxford: Oxford University Press, 1995.

Chapter 3 contains a thorough analysis of utility and expenditure functions.

Samuelson, Paul A. Foundations of Economic Analysis. Cambridge: Harvard University Press, 1947.Chapter V and Appendix A provide a succinct analysis of the first-order conditions for a utility maximum. Theappendix provides good coverage of second-order conditions.

Silberberg, E. The Structure of Economics: A Mathematical Analysis. 2nd ed. New York: McGraw-Hill, 1990.A useful, though fairly difficult, treatment of duality in consumer theory.

Theil, H. Theory and Measurement of Consumer Demand. Amsterdam: North-Holland, 1975.Good summary of basic theory of demand together with implications for empirical estimation.

Varian, H. R. Microeconomic Analysis. 3rd ed. New York: W. W. Norton, 1992.Sections 7.3–7.4 summarize the relationships between utility functions and expenditure functions.


Because data on budget shares are readily availablefrom studies of individuals’ consumption patterns,they can be used to shed light on underlying prefer-ences. Here we look at three specific utility functionsand show they have been used to study budget shares.Throughout our discussion, we will consider only thetwo-good (X and Y) case, though most results arereadily generalizable to many goods. Following cus-tomary notation, the share of income devoted to goodX(PXX/I) will be denoted by sX and sY � 1 � sX.

Before beginning, the connection betweenbudget shares and homothetic preferences should bementioned. In Chapter 3 we showed that for homo-thetic utility functions, the MRS depends only on theratio Y/X, not on the absolute levels of the goods. Be-cause utility maximization requires MRS � PX/PY forhomothetic functions, the price ratio will determinethe ratio Y/X. Hence, the budget shares themselveswill be determined solely by relative prices. If relativeprices do not change, budget shares will not changeeven when income fluctuates. Our examples of ho-mothetic functions (the Cobb-Douglas and the CES)illustrate this result, whereas the Linear ExpenditureSystem shows why nonhomothetic functions may bepreferable in some circumstances.

E4.1 Cobb-Douglas UtilityIf the utility function has the Cobb-Douglas form

U(X, Y ) � X�Y �, (i)

then Example 4.2 showed that the demand func-tions are

X � �I/PX (ii)

Y � �I/PY.Hence,

sX � PXX/I � � (iii)

sY � PYY/I � �

and the budget shares are constant for all possible relative prices. Although this feature of the Cobb-Douglas is one reason for its popularity in the study ofproduction (see Chapter 11), it does limit its suitabil-ity for the study of consumption. Budget shares inconsumption do not appear to be constant underchanging economic circumstances.

FoodEver since the pioneering studies of Ernst Engel inthe mid-nineteenth century, economists have been in-

terested in the share of income that consumers devoteto food purchases. Literally thousands of studies haveconfirmed that this share is indeed influenced by circumstances. Not only do food shares data exhibitEngel’s Law (�sX/�I � 0), but they also illuminatemany other aspects of consumer behavior. For exam-ple, Hayashi (1995) shows that the share of incomedevoted to foods favored by the elderly is significantlylarger in two-generation households in Japan than inone-generation households. Altruism appears to be asignificant feature of extended families in Japan.

Development economists sometimes make a dis-tinction between the share of income devoted to foodand the share of income devoted to nutrients. In principle, nutrients’ share of income might or mightnot follow Engel’s Law for the poorest people in de-veloped countries. If individuals choose increasinglynutrient-rich foods as their incomes rise, at the mar-gin nutrients’ share would exceed the share of food intotal income. On the other hand, if individuals opt fornutrient-poor foods as income rises, the situationwould be reversed. Behrman (1989) presents evidencethat an individual’s demand for an increasing varietyof food as income rises may interfere with the abilityof general economic improvement to raise the nutri-ent intake of poorest segments of the population.

E4.2 Linear Expenditure SystemA generalization of the Cobb-Douglas function thatincorporates the idea that certain minimal amountsof each good must be bought by the individual (X 0, Y0) is the utility function

U(X, Y ) � (X � X0)�(Y � Y0)� (iv)

for values of X � X0 and Y � Y0 and again � � � � 1.Demand functions can be derived from this util-

ity function in a way analogous to the Cobb-Douglascase by introducing the concept of supernumerary in-come (I*), which represents the amount of purchas-ing power remaining after purchasing the minimumbundle

I* � I � PXX0 � PYY0. (v)

Using this notation, then, the demand functions are

X � (PXX0 � �I*)/PX (vi)

Y � (PYY0 � �I*)/PY .

In this case, then, the individual spends a constantfraction of supernumerary income on each good once


EXTENSIONS

Utility Functions and Budget Shares

the minimum bundle has been purchased. Manipula-tion of Equation vi yields the share equations:

sX � � � (�PXX0 � �PYY0)/I (vii)

sY � � � (�PYY0 � �PXX0)/I,

which show that this demand system is not homo-thetic. Inspection of Equation vii shows the unsurpris-ing result that the budget share of a good is positivelyrelated to the minimal amount of that good neededand negatively related to the minimal amount of theother good required. Because the notion of necessarypurchases seems to accord well with real-world obser-vation, this linear expenditure system (LES), whichwas first developed by Stone (1954), is widely used inempirical studies.

Traditional PurchasesOne of the most interesting uses of the LES is to ex-amine how its notion of necessary purchases changeas conditions change. For example, Oczkowski andPhilip (1994) study how access to modern consumergoods may affect the share of income that individualsin transitional economies devote to traditional localitems. They show that villagers of Papua New Guineareduce such shares significantly as outside goods be-come increasingly accessible. Hence, such improve-ments as better roads for moving goods provide oneof the primary routes by which traditional culturalpractices are undermined.

E4.3 CES UtilityIn Chapter 3 we introduced the CES utility function

U(X, Y) � � (viii)

for � � 1, � � 0. The primary use of this function is toillustrate alternative substitution possibilities (as re-flected in the value of the parameter �). Budgetshares implied by this utility function provide a num-ber of such insights. Manipulation of the first-orderconditions for a constrained utility maximum with theCES function yield the share equations

sX � 1/[1 � PY/PX)K] (ix)

sY � 1/[1 � (PX/PY)K]

where K � �/(� � 1).The homothetic nature of the CES function is

shown by the fact that these share expressions dependonly on the price ratio, PX/PY. Behavior of the shares inresponse to changes in relative prices depends on thevalue of the parameter K. For the Cobb-Douglas case,� � 0 so K � 0 and sX � sY � 1⁄2, as we have found in sev-

eral examples. When � � 0, substitution possibilitiesare great and K � 0. In this case Equation ix shows thatsX and PX/PY move in opposite directions. If PX/PY rises,the individual substitutes Y for X to such an extent thatsX falls. Alternatively, if � � 0, substitution possibilitiesare limited, K � 0, and sX and PX/PY move in the samedirection. In this case an increase in PX/PY causes onlyminor substitution of Y for X, and sX actually rises be-cause of the relatively higher price of good X.

North American Free TradeCES demand functions are most often used in large-scale computer models of general equilibrium (seeChapter 16) that economists use to evaluate the im-pact of major economic changes. Because the CESmodel stresses that shares respond to changes in rela-tive prices, it is particularly appropriate for looking atinnovations such as changes in tax policy or in inter-national trade restrictions where changes in relativeprices are quite likely. One important recent area ofsuch research has been on the impact of the NorthAmerican Free Trade Agreement for Canada, Mexico,and the United States. In general, these models findthat all of the countries involved might be expected togain from the agreement, but that Mexico’s gains maybe the greatest because it is experiencing the greatestchange in relative prices. Kehoe and Kehoe (1995)present a number of computable equilibrium modelsthat economists have used in these examinations.8

ReferencesBehrman, Jere R. “Is Variety the Spice of Life? Implicationsfor Caloric Intake.” Review of Economics and Statistics (Novem-ber 1989): 666–672.

Green, H. A. Consumer Theory. London: The Macmillan Press,1976.

Hyashi, Fumio. “Is the Japanese Extended Family Altruisti-cally Linked? A Test Based on Engel Curves.” Journal of Polit-ical Economy (June 1995): 661–674.

Kehoe, Patrick J., and Timothy J. Kehoe. Modeling NorthAmerican Economic Integration. London: Klower AcademicPublishers, 1995.

Oczkowski, E., and N. E. Philip. “Household ExpenditurePatterns and Access to Consumer Goods in a TransitionalEconomy.” Journal of Economic Development (June 1994):165–183.

Stone, R. “Linear Expenditure Systems and Demand Analy-sis.” The Economic Journal (September 1954): 511–527.

Y �

��

X �

��


8The Research on the North American Free Trade Agree-ment is discussed in more detail in the Extensions to Chap-ter 16.

UTILITY MAXIMIZATION AND CHOICE

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