Utility Maximization

In this chapter we begin our examination of consumer behavior. In the theory of a competitive firm, the supply and demand functions were derived from a model of profit-maximizing behavior and a specification of the underlying technological constraints. In the theory of the consumer we will derive demand functions by considering a model of utility-maximizing behavior coupled with a description of underlying economic constraints.

7.1 Consumer preferences

We consider a consumer faced with possible consumption bundles in some set X, his consumption set. In this book we usually assume that X is the nonnegative orthant in Rk, but more specific consumption sets may be used. For example, we might only include bundles that would give the consumer at least a subsistence existence. We will always assume that X is a closed and convex set.

The consumer is assumed to have preferences on the consumption bundles in X. When we write x >z y, we mean "the consumer thinks that the bundle x is at least as good as the bundle y." We want the preferences to order the set of bundles. Therefore, we need to assume that they satisfy certain standard properties.

COMPLETE. For all x and y in X, either x X y or y h x or both. REFLEXIVE. For all x in X, x ^ x.

TRANSITIVE. For all x, y, and z in X, if x y and y > 2,, then x z.

The first assumption just says that any two bundles can be compared, the second is trivial, and the third is necessary for any discussion of preference maximization-, if preferences were not transitive, there might be sets of bundles which had no best elements.

Given an ordering ^ describing "weak preference," we can define an ordering of strict preference simply by defining x >- y to mean not y >: x. We read x >- y as "x is strictly preferred to y." Similarly, we define a notion of indifference by x ~ y if and only if x X y and y y x.

We often wish to make other assumptions on consumers' preferences; for example.

CONTINUITY. For all y in X, the sets {x : x t. y} and {x : x ^ y} are closed sets. It follows that {x : x >- y} and {x : x -< y} are open sets.

This assumption is necessary to rule out certain discontinuous behavior; it says that if (xl) is a sequence of consumption bundles that are all at least as good as a bundle y, and if this sequence converges to some bundle x*, then x* is at least as good as y.

The most important consequence of continuity is this: if y is strictly preferred to z and if x is a bundle that is close enough to y, then x must be strictly preferred to z. This is just a restatement of the assumption that the set of strictly preferred bundles is an open set. For a brief discussion of open and closed sets, see Chapter 26, page 478.

In economic analysis it is often convenient to summarize a consumer's behavior by means of a utility function; that is, a function u : X —> R such that x >- y if and only if u(x) > u(y). It can be shown that if the preference ordering is complete, reflexive, transitive, and continuous, then it can be represented by a continuous utility function. We will prove a weaker version of this assertion below. A utility function is often a very convenient way to describe preferences, but it should not be given any psychological interpretation. The only relevant feature of a utility function is its ordinal character. If u(x) represents some preferences >; and / : R —> R is a monotonic function, then f(u(x)) will represent exactly the same preferences since f(u(x)) > f(u(y)) if and only if u(x) > u(y).

There are other assumptions on preferences that are often useful; for example:

STRONG MONOTONICITY. If x > y and x ^ y, then x X y.

Weak monotonicity says that "at least as much of everything is at least as good." If the consumer can costlessly dispose of unwanted goods, this assumption is trivial. Strong monotonicity says that at least as much of every good, and strictly more of some good, is strictly better. This is simply assuming that goods are good.

If one of the goods is a "bad," like garbage, or pollution, then strong monotonicity will not be satisfied. But in these cases, redefining the good to be the absence of garbage, or the absence of pollution, will often result in preferences over the re-defined good that satisfies the strong monotonicity postulate.

Another assumption that is weaker than either kind of monotonicity is the following:

LOCAL NONSATIATION. Given any x in X and any e > 0, then there is some bundle y in X with |x — y| < e such that y X x.1

Local nonsatiation says that one can always do a little bit better, even if one is restricted to only small changes in the consumption bundle. You should verify that strong monotonicity implies local nonsatiation but not vice versa. Local nonsatiation rules out "thick" indifference curves.

Here are two more assumptions that are often used to guarantee nice behavior of consumer demand functions:

CONVEXITY. Given x, y, and z in X such that x X z and y X z, then it follows that tx + (1 — t)y X ® for all 0 < t < 1.

STRICT CONVEXITY. Given x + y and z in X, if* X z and y X z, then tx + (1 - t)y X z for allO <t < 1.

Given a preference ordering, we often display it graphically. The set of all consumption bundles that are indifferent to each other is called an indifference curve. One can think of indifference curves as being level sets of the utility function; they are analogous to the isoquants used in production theory. The set of all bundles on or above an indifference curve, {x in X : x X y}, is called an upper contour set. This is analogous to the input requirement set used in production theory.

Convexity implies that an agent prefers averages to extremes, but, other than that, it has little economic content. Convex preferences may have indifference curves that exhibit "flat spots," while strictly convex preferences

1 The notation |x — y| means the Euclidean distance between x and y.

have indifference curves that are strictly rotund. Convexity is a generalization of the neoclassical assumption of "diminishing marginal rates of substitution."

EXAMPLE: The existence of a utility function

Existence of a utility function. Suppose preferences are complete, reflexive, transitive, continuous, and strongly monotonie. Then there exists a continuous utility function u : R+ —► R which represents those preferences.

Proof. Let e be the vector in R+ consisting of all ones. Then given any vector x let u(x) be that number such that x ~ u(x)e. We have to show that such a number exists and is unique.

Let B = {i in R : te h *} and W = {t in R : x >: ie}. Then strong monotonicity implies B is nonempty; W is certainly nonempty since it contains 0. Continuity implies both sets are closed. Since the real line is connected, there is some tx such that txe ~ x. We have to show that this utility function actually represents the underlying preferences. Let u(x) = tx where txe ~ x u(y) = ty where tye ~ y.

Then if tx < ty, strong monotonicity shows that txe -< tye, and transitivity shows that x ~ txe -< tye ~ y.

Similarly, if x X y, then txe X tye so that tx must be greater than ty.

The proof that u(x) is a continuous function is somewhat technical and is omitted. I

EXAMPLE: The marginal rate of substitution

Let u(xi,... ,Xk) be a utility function. Suppose that we increase the amount of good i; how does the consumer have to change his consumption of good j in order to keep utility constant?

Following the construction in Chapter 1, page 11, we let dxi and dxj be the changes in Xi and Xj. By assumption, the change in utility must be zero, so

OXi OXj

This expression is known as the marginal rate of substitution between goods i and j.

The marginal rate of substitution does not depend on the utility function chosen to represent the underlying preferences. To prove this, let v(u) be a monotonic transformation of utility. The marginal rate of substitution for this utility function is dxj dxi v'{u)

du(x)

du(x)

dxi

dxi

du(x)

du(x)

dxj

dxj

7.2 Consumer behavior

Now that we have a convenient way to represent preferences we can begin to investigate consumer behavior. Our basic hypothesis is that a rational consumer will always choose a most preferred bundle from the set of affordable alternatives.

In the basic problem of preference maximization, the set of affordable alternatives is just the set of all bundles that satisfy the consumer's budget constraint. Let m be the fixed amount of money available to a consumer, and let p = (pi, • • ■ ,pjt) be the vector of prices of goods, 1, • • •, k. The set of affordable bundles, the budget set of the consumer, is given by

B = {x in X : px < m.} The problem of preference maximization can then be written as:

Let us note a few basic features of this problem. The first issue is whether there will exist a solution to this problem. According to Chapter 27, page 506, we need to verify that the objective function is continuous and that the constraint set is closed and bounded. The utility function is continuous by assumption, and the constraint set is certainly closed. If Pi > 0 for i = 1,..., k and m > 0, it is not difficult to show that the constraint set will be bounded. If some price is zero, the consumer might want an infinite amount of the corresponding good. We will generally ignore such boundary problems.

The second issue we examine concerns the representation of preferences. Here we can observe that the maximizing choice x* will be independent of the choice of utility function used to represent the preferences. This is because the optimal x* must have the property that x* y x for any x in B, so any utility function that represents the preferences >; must pick out x* as a constrained maximum.

Third, if we multiply all prices and income by some positive constant, we will not change the budget set, and thus we cannot change the set of optimal choices. That is, if x* has the property that x* >z x for all x such that px < m, then x* ^ y for all y such that ipy < trn. Roughly speaking, the optimal choice set is "homogeneous of degree zero" in prices and income.

By making a few regularity assumptions on preferences, we can say more about the consumer's maximizing behavior. For example, suppose that preferences satisfy local nonsatiation; can we ever get an x* where px* < m? Suppose that we could; then, since x* costs strictly less than to, every bundle in X close enough to x* also costs less than to and is therefore feasible. But, according to the local nonsatiation hypothesis, there must be some bundle x which is close to x* and which is preferred to x*. But this means that x* could not maximize preferences on the budget set B.

Therefore, under the local nonsatiation assumption, a utility-maximizing bundle x* must meet the budget constraint with equality. This allows us to restate the consumer's problem as w(p, m) = max u(x)

The function v(p, m) that gives us the maximum utility achievable at given prices and income is called the indirect utility function. The value of x that solves this problem is the consumer's demanded bundle: it expresses how much of each good the consumer desires at a given level of prices and income. We assume that there is a unique demanded bundle at each budget; this is for purposes of convenience and is not essential to the analysis.

The function that relates p and m to the demanded bundle is called the consumer's demand function. We denote the demand function by x(p, to). As in the case of the firm, we need to make a few assumptions to make sure that this demand function is well-defined. In particular, we will want to assume that there is a unique bundle that maximizes utility. We will see later on that strict convexity of preferences will ensure this behavior.

Just as in the case of the firm, the consumer's demand function is homogeneous of degree 0 in (p. to). As we have seen above, multiplying all prices and income by some positive number does not change the budget set at all and thus cannot change the answer to the utility maximization problem.

As in the case of production we can characterize optimizing behavior by calculus, as long as the utility function is differentiable. The Lagrangian for the utility maximization problem can be written as

C. = u(x) - A(px - m), where A is the Lagrange multiplier. Differentiating the Lagrangian with respect to xz gives us the first-order conditions

In order to interpret these conditions we can divide the ith first-order condition by the jth first-order condition to eliminate the Lagrange multiplier. This gives us du(x')

The fraction on the left is the marginal rate of substitution between good i and j, and the fraction on the right might be called the economic rate of substitution between goods t and j. Maximization implies that these two rates of substitution be equal. Suppose they were not; for example, suppose du(x*) dxl = 1 , 2 = p, du(x*) 1^1 p' t -^-J

Then, if the consumer gives up one unit of good i and purchases one unit of good j, he or she will remain on the same indifference curve and have an extra dollar to spend. Hence, total utility can be increased, contradicting maximization.

Figure 7.1 illustrates the argument geometrically. The budget line of the consumer is given by {x : piXi +p$X2 = to}. This can also be written as the graph of an implicit function: x2 = Tn/p2 - (p\/p2)xi. Hence, the budget line has slope —pi/p2 and vertical intercept m/p2. The consumer wants to find the point on this budget line that achieves highest utility. This must clearly satisfy the tangency condition that the slope of the indifference curve equals the slope of the budget line. Translating this into algebra gives the above condition.

Preference maximization. The optimal consumption bundle Figure will be at a point where an indifference curve is tangent to the 7.1

budget constraint.

Finally, we can state the condition using vector terminology. Let x* be an optimal choice, and let dx be a perturbation of x* that satisfies the budget constraint. Hence, we must have p(x* ± dx) = m.

Since px = m, this equation implies that pdx = 0, which in turn implies that dx must be orthogonal to p.

For any such perturbation dx, utility cannot change, or else x* would not be optimal. Hence, we also have

which says that Du(x') is also orthogonal to dx. Since this is true for all perturbations for which pdx = 0, we must have D«(x*) proportional to p, just as we found in the first-order conditions.

The second-order conditions for utility maximization can be found by applying the results of Chapter 27, page 494. The second derivative of the Lagrangian with respect to goods i and j is d2u{x)/dxldxJ. Hence, the second-order condition can be written as h(D2u(x*)h < 0 for all h such that ph = 0. (7.1)

This condition requires that the Hessian matrix of the utility function is negative semidefinite for all vectors h orthogonal to the price vector. This is essentially equivalent to the requirement that tt(x) be locally quasiconcave. Geometrically, the condition means that the upper contour set must lie above the budget hyperplane at the optimal x*.

As usual the second-order condition can also be expressed as a condition involving the bordered Hessian. Examining Chapter 27, page 500, we see that this formulation says that (7.1) can be satisfied as a strict inequality if and only if the naturally ordered principal minors of the bordered Hessian alternate in sign. Hence,

0

-Pi

~P2

~Pl

«11

«12

~P2

«21

«22

0

~Pl

~P2

~P3

-Pi

"11

«12

«13

~P2

«21

«22

«23

~P3

«31

«32

7.3 Indirect utility

Recall the indirect utility function defined earlier. This function, v(p,m), gives maximum utility as a function of p and m.

Properties of the indirect utility function.

(1) u(p, m) is nonincreasing in p; that is, if p' > p, w(p',m) < v(p,m). Similarly, u(p, m) is nondecreasing in m.

(3) v(p,m) is quasiconvex in p; that is, {p : v(p,m) < k) is a convex set for all k.

(4) v(p, to) is continuous at all p 0, to > 0. Proof.

(1) Let B = {x : px < to} and B' = {x : p'x < m} for p' > p. Then B' is contained in B. Hence, the maximum of u(x) over B is at least as big as the maximum of u(x) over B'. The argument for to is similar.

(2) If prices and income are both multiplied by a positive number, the budget set doesn't change at all. Thus, v(tp,tm) = i;(p, rri) for t > 0.

INDIRECT UTILITY 103

(3) Suppose p and p' are such that i;(p,m) < k, v(p',m) < k. Let p" = ip+ (1 — i)p'. We want to show that v(p", m) < k. Define the budget sets:

B = {x : px < m} B' = {x : p'x < to} B" = {x : p"x < to}

We will show that any x in B" must be in either B or B'\ that is, that B U B' D B". Assume not; then x is such that fpx + (1 - t)p'x < to but px > to and p'x > to. These two inequalities can be written as ipx > tm (1 - f)p'x > (1 - t)m.

Summing, we find that ipx+ (1 - i)p'x > m which contradicts our original assumption. Now note that v(p",m) = max u(x) such that x is in B"

since BUB' D B"

< k since v(p, to) < fc and v(p', to) < k.

(4) This follows from the theorem of the maximum in Chapter 27, page 506.

In Figure 7.2 we have depicted a typical set of "price indifference curves." These Eire just the level sets of the indirect utility function. By property (1) of the above theorem utility is nondecreasing as we move towards the origin, and by property (3) the lower contour sets are convex. Note that the lower contour sets lie to the northeast of the price indifference curves since indirect utility declines with higher prices.

We note that if preferences satisfy the local nonsatiation assumption, then u(p, m) will be strictly increasing in to. In Figure 7.3 we have drawn the relationship between v(p, to) and to for constant prices. Since u(p, to) is strictly increasing in to, we can invert the function and solve for to as a function of the level of utility; that is, given any level of utility, u, we can read off of Figure 7.3 the minimal amount of income necessary to achieve utility u at prices p. The function that relates income and utility in this way—the inverse of the indirect utility function—is known as the expenditure function and is denoted by e(p, u).

price 1

Figure Price indifference curves. The indifference curve is all those

7.2 prices such that v(p,m) = k, for some constant k. The lower contour set consists of ail prices such that v(p, m) < k.

Figure Utility as a function of income. As income increases indi-

7.3 rect utility must increase.

An equivalent definition of the expenditure function is given by the following problem:

The expenditure function gives the minimum cost of achieving a fixed level of utility.

The expenditure function is completely analogous to the cost function we considered in studying firm behavior. It therefore has all the properties we derived in Chapter 5, page 71. These properties are repeated here for convenience.

Properties of the expenditure function.

(5) If h(p, u) is the expenditure-minimizing bundle necessary to achieve suming the derivative exists and that pi > 0.

Proof. These axe exactly the same properties that the cost function exhibits. See in Chapter 5, page 71 for the arguments. I

The function h(p, u) is called the Hicksian demand function. The Hicksian demand function is analogous to the conditional factor demand functions examined earlier. The Hicksian demand function tells us what consumption bundle achieves a target level of utility and minimizes total expenditure.

A Hicksian demand function is sometimes called a compensated demand function. This terminology comes from viewing the demand function as being constructed by varying prices and income so as to keep the consumer at a fixed level of utility. Thus, the income changes axe arranged to "compensate" for the price changes.

Hicksian demand functions axe not directly observable since they depend on utility, which is not directly observable. Demand functions expressed as a function of prices and income axe observable; when we want to emphasize the difference between the Hicksian demand function and the usual demand function, we will refer to the latter as the Marshallian demand function, x(p, m). The Marshallian demand function is just the ordinary market demand function we have been discussing all along.

7.4 Some important identities

There Eire some important identities that tie together the expenditure function, the indirect utility function, the Marshallian demand function, and the Hicksian demand function.

Let us consider the utility maximization problem

Let x* be the solution to this problem and let u* = u(x*). Consider the expenditure minimization problem e(p, u*) = min px such that u(x) > u*.

(5) If h(p, u) is the expenditure-minimizing bundle necessary to achieve for i = 1,..., k as-

suming the derivative exists and that pi > 0.

An inspection of Figure 7.4 should convince you that in nonperverse cases the answers to these two problems should be the same x*. (A more rigorous argument is given in the appendix to this chapter.) This simple observation leads to four important identities:

(1) e(p, v(p,m)) = m. The minimum expenditure necessary to reach utility u(p, m) is m.

(2) ?;(p, e(p, u)) = u. The maximum utility from income e(p, u) is u.

(3) Xj(p, m) = hi(p,v(p,m)). The Marshallian demand at income m is the same as the Hicksian demand at utility v(p,m).

(4) hi(p,u) = ^¿(p, e(p,u)). The Hicksian demand at utility u is the same as the Marshallian demand at income e(p, u).

This last identity is perhaps the most important since it ties together the "observable" Marshallian demand function with the "unobservable" Hicksian demand function. Identity (4) shows that the Hicksian demand function—the solution to the expenditure minimization problem—is equal to the Marshallian demand function at an appropriate level of income— namely, the minimum income necessary at the given prices to achieve the desired level of utility. Thus, any demanded bundle can be expressed either as the solution to the utility maximization problem or the expenditure minimization problem. In the appendix to this chapter we give the exact conditions under which this equivalence holds. For now, we simply explore the consequences of this duality.

It is this link that gives rise to the term "compensated demand function." The Hicksian demand function is simply the Marshallian demand functions for the various goods if the consumer's income is "compensated" so as to achieve some target level of utility.

A nice application of one of these identities is given in the next proposition:

Roy's identity. If x(p, ra) is the Marshallian demand function, then dv(p, m)

Xi{p'm) = ~d^m) f°r i = dm provided, of course, that the right-hand side is well defined and that Pi > 0 and m > 0.

Proof. Suppose that x* yields a maximal utility of u* at (p*,m*). We know from our identities that x(p*,m*) = h(p*,u*). (7.2)

Maximize utility and minimize expenditure. Normally, Figure a consumption bundle that maximizes utility will also minimize 7.4

expenditure and vice versa.

From another one of the fundamental identities, we also know that u* = v(p, e(p, u*)).

This identity says that no matter what prices are, if you give the consumer the minimal income to get utility u* at those prices, then the maximal utility he can get is u*.

Since this is an identity we can differentiate it with respect to pi to get

Rearranging, and combining this with identity (7.2), we have

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