CHAPTER 3
THE THEORY OF THE CONSUMER
We will define a consumer as an individual or group of individuals who each possess income and values and who purchase goods and services, and who generally perform labor for this income. The fundamental problem of the consumer is one of determining the quantity of goods and services to purchase by the given prices for goods and services and the income of the consumer. The consumer will be assumed to make this choice such that the value, or satisfaction or utility, derived from consuming the goods and services is the greatest possible. We will assume, in this chapter, a single consumer, and we will also assume that the consumer is “rational” by being aware of alternative goods and services that may be purchased and capable of evaluating their worth in terms of their value and cost.
To do this we will postulate the existence of a utility function that measures the satisfaction or value derived from alternative goods and services. We will examine some concepts of utility maximization and determine demand functions that measure the quantity of goods and services a consumer will purchase by their price. Then we will examine various properties of this demand function. This will enable us to combine the theory of the firm and the theory of the household and determine various supply–demand or market equilibrium conditions, which is the subject of Chapter 4.
3.2 ECONOMIC UTILITY THEORY AND ITS AXIOMS
Suppose that we wish to purchase a quantity of meat or tofu for preparing a meal for a fixed number of people. It seems almost intuitive that prudent consumers would reduce the quantity of meat or tofu purchased as the price of meat or tofu rises and substitute an increased quantity of other items of lesser cost. Alternatively, suppose that we have the choice of spending n dollars for a trip to Europe or on a new automobile. We decide on the trip to Europe. This is because, in this specific instance, a trip to Europe has greater utility to us as a consumer than does an automobile. We assume that the consumer is able to make choices “as if” there existed a utility function that could be used to determine them.
We would like to determine consumer demand for various goods and services and we adopt certain, generally reasonable, axioms of the theory of rational choice to accomplish this. First, we define the consumer’s vector or bundle of goods and services, which we will call the commodity bundle
of n available commodities, or goods and services, for the consumer to purchase. We will not worry about fractional numbers of commodities, such as one and one-half automobiles, but we must assume that only nonnegative commodities are purchased, such that xi ≥ 0 for all I, or x ≥ 0. The commodity space is the set of all possible commodity bundles C = {x|x ≥ 0}, and this commodity space is a closed convex set in the nonnegative part of the n-dimensional Euclidean space.
The “numbers” that we assign to various commodity bundles are irrelevant as long as any bundle xi that is preferred over any other bundle xj is assigned a higher utility number, such that we have U(xi) > U(xj) if and only if (iff or IFF) xi > xj. In the foregoing expression, we require only ordinal utility functions. For many purposes, however, we need to trade off commodities. Then we will typically need to know whether the differences between utility levels are comparable. Thus, as a specific comparison, we will be able to determine whether U(xi) − U(xj) = U(xk) − U(xi), and this will require a cardinal utility function. An ordinal utility function will provide information only about preference orderings among commodity bundles. A cardinal utility function will provide information about differences between utility functions in terms of their value; for example, we would be able to say that one values the difference between a 10-day vacation in Miami and one in Atlanta as equivalent to the difference between a 10-day vacation in Miami and one in Las Vegas through the use of a cardinal utility function.
The requirement that the relative magnitudes of the differences between utilities of individual commodities have explicit meaning is essential in dealing with normative or prescriptive issues that involve economic systems. It requires interpersonal comparisons of values, something that is not easily accomplished. However, this is needed if we are to decompose a utility function into various attributes and aggregate attribute scores to obtain the overall utility.
A utility function U(x) is effectively ordinal if it can be replaced by any monotonically increasing transformation of itself and still preserve all the properties of the utility function. A cardinal utility function U(x) can be replaced only by an affine function of itself and the properties of the utility function as an indicator of preference are preserved. Although many of our developments in utility theory in this chapter apply to ordinal utilities, we find it convenient to view utilities as if they were cardinal utilities. Our initial discussions here apply to ordinal utilities and, since cardinal utilities are certainly ordinal, they apply also to cardinal utilities. We state four axioms applicable to ordinal or cardinal utilities. The first two are the most important and they are as follows:
Axiom 1. The Preference and Indifference Axiom
The choice between two commodity bundles depends on the consumer’s basic primitive notions of preference or taste. We use the symbol > to denote is preferred to, the symbol ∼ to denote is indifferent to, and the symbol ≥ to denote is preferred to or indifferent to. Thus, when we write x1 > x2 we mean that commodity bundle 1 is preferred to commodity bundle 2; when we write x1 ∼ x2 we mean that we are indifferent between commodity bundles 1 and 2; and when we write x1 ≥ x2 we mean that we prefer or are indifferent to commodity bundles x1 and x2.
This axiom assumes that preferences can be expressed, that every pair of commodities gives rise to a choice. Also, we assume that the weak preference relation ≥ is complete so that there are no gaps in commodity space. Finally, we assume that weak preference relations are continuous. This continuity condition ensures that there will be no rapid jumps in preference for very small changes in bundle composition.
Axiom 2. The Transitivity Axiom
We assume that preferences and indifferences are transitive. This requires the following:
a. If we prefer bundle x1 to bundle x2 and bundle x2 to bundle x3, then we must prefer bundle x1 to bundle x3; that is, if x1 > x2 and x2 > x3, then, by transitivity, x1 > x3.
b. If we are indifferent between bundles x1 and x2 and between bundles x2 and x3, then we must be indifferent between bundles x1 and x3; that is, if x1 ∼ x2 and x2 ∼ x3, then x1 ∼ x3.
c. If x1 is preferred to or indifferent to x2 and x2 is preferred or indifferent to x3, then x1 is either preferred or indifferent to x3; that is, if x1 ≥ x2 and x2 ≥ x3, then x1 ≥ x3.
Axioms 1 and 2 represent rationality suppositions for a consumer.1 If these are not assumed, various maladies result, the most common of which is that the consumer may be converted into a money pump.
Example 3.1:
To illustrate a consequence of intransitivity, suppose that we can find a person who will say, “I prefer beer to milk, I prefer milk to water, and I prefer water to beer.” If we can get the consumer to agree to set a price differential converting the preference to an indifference such that we have the cardinal relations beer = milk+25¢, milk = water+15¢, and water = beer+10¢, then we can convert the consumer into a money pump. We start by giving the consumer a glass of water; then we offer to trade the consumer a glass of milk for the glass of water and 15¢. The consumer should agree, since they are indifferent between the two choices. Now we offer to trade the consumer a glass of beer for the milk if they will give us 25¢. Again, the consumer is indifferent between these, and so a trade is made. Finally we offer to trade the consumer a glass of water in exchange for the beer and l0¢ and the consumer agrees that this is reasonable, since indifference exists. In this process the consumer starts with a glass of water and winds up with a glass of water. We make 50¢ and can continue this pumping operation if we so desire. The consumer is happy to accommodate this, assuming that they truly have this preference intransitivity!.
The two axioms of transitive preference and transitive indifference and the fact that we have assumed that preference relations are complete and continuous allow us to define a utility function, a real continuous function defined in the commodity space C. The primary properties of an ordinal utility function are: U(x1) > U(x2) iff x1 > x2, U(x1) = U(x2) iff x1 ∼ x2, and U(x1) ≥ U(x2) iff x1 ≥ x2. This utility function is defined with respect to various attributes of the bundle purchased. Satisfaction values depend on this time interval and the attributes of the commodity bundle.
Example 3.2:
Considerable care must be exercised in combining preferences based on ordinal utilities. Even though individual preferences may well not be intransitive, there does not generally exist any reasonable basis on which to combine ordinal preferences for aspects of various alternative commodity bundles; intransitive preferences may therefore easily result.
To illustrate this, suppose that we wish to establish a preference ordering among three automobiles: A, B, and C. We determine that cost, performance, and style are the attributes of importance to us. Suppose that the preference ordering with respect to cost is such that A is preferred to B, which is preferred to C; that is, A > cB > cC. Suppose further that B is preferred to C, which is preferred to A with respect to performance, or B > pC > pA. Finally, suppose that C is preferred to A, which is preferred to B with respect to style, or C > sA > sB. It might not seem unreasonable to prefer car i over car j if it is better on a majority of the attributes of importance. So we prefer car A to B since it is better on two of the three attributes. Similarly, we prefer car B to C since it is better on two of the three attributes. A common method of judgment consists of making pairwise comparisons and discarding the inferior alternative from further consideration. Usually transitivity is assumed and the failure to examine potentially disconfirming comparisons may lead us to select alternatives that we believe are “best” but which may be in fact inferior. Here, for example, we have determined that B > C; so we discard C from further comparison. Then we could determine that A > B and infer transitivity such that we believe that A > C. But this is not correct, since C is preferred to A on two out of the three attribute scores! The problem is that we are using deficient judgment heuristics2 and that these are not recognized.
The use of cardinal utilities, which allows expressions of preference weights across attributes and the expression of alternative preferences within attributes, is potentially capable of resolving difficulties associated with the poor judgment heuristics that are often associated with attempts to combine ordinal preferences, often by making binary preference comparisons across two alternatives and rejecting the one judged inferior. Here, for example, we may determine the following utility scores of the alternative commodity bundles on the three attributes:
These scores are fully consistent with the ordinal preferences given earlier. If we can assume a linear aggregation rule U = wcUc+wpUp+wsUs with which to obtain a single scalar cardinal utility function for the three cars, then we will have for the utilities of the three automobiles: U(A) = wc+0.5ws, U(B) = 0.5wc+wp, and U(C) = 0.5wp+ws. For convenience we assume that the weights are in the interval 0 to 1 and that they sum to 1, wc+wp+ws = 1, such that the foregoing utilities become U(A) = 0.5+0.5wc−0.5wp, U(B) = 0.5wc+wp, and U(C) = 1.0−wc−0.5wp. With these utilities, we can obtain any preference ordering that we wish by appropriate selection of the weights. However, all preferences will be transitive. If we are to obtain, for example, A > B > C, this requires U(A) > U(B) > U(C), or wp<1/3 and wc+wp > 2/3. These two inequalities also infer that wc > 1/3 and ws<1/3.
As we have noted, utility functions are not unique. For example, if an ordinal utility function for a given bundle is U1(x1), then any monotonically increasing function of the utility U1 is also a valid utility function. An affine, or linear, transformation of cardinal utility functions U1(x) = a+bU(x), for b > 0, is often used to scale utilities to the range 0–1 or 0–100. Cardinal utilities are unique only up to affine transformations. In many cases the anchor on the cardinal utility scale is such that the constant term must be zero and we can only have a linear transformation U1(x) = bU(x). We are now in a position to state the final two formal axioms of economic utility theory.
Axiom 3. The Nonsatiation Axiom
If bundle x1 contains at least as much of any constituent commodity as bundle x2, then x1 must be preferred or indifferent to x2. Thus, the utility of x1 must be greater than the utility of x2. So x1 > x2 implies x1 > x2 and x1 > x2 implies U(x1) > U(x2). Also, x1 ≥ x2 implies x1 > x2 and x1 ≥ x2 implies U(x1) ≥ U(x2).
Axiom 4. The Strict Convexity Axiom
The axiom of continuity ensures that there will be at least one commodity bundle with preference equivalent to x3 that can be made up of a combination of bundles x1 and x2, where x1 > x3 > x2. This axiom3 can also be stated in a slightly different form as x1 > ax1+(1−a)x2 > x2 for all 0<a<l if x1 > x2.
3.3 PROPERTIES OF UTILITY FUNCTIONS
Economic utility functions have a number of important properties. In this section we will investigate some of them, based on the assumption that we are dealing with cardinal utility functions. We assume that utility functions are twice differentiable with respect to the independent variable x, the commodity bundle. In Chapter 2 the symbol x is used to represent the inputs to the production process, which will generally consist of commodities (goods) and factors (services). Here we use it to represent the commodity bundle, which generally could also consist of goods and services. The first derivative of the utility function with respect to x is known as the marginal utility MU(x):
From the nonsatiation axiom, we see that marginal utilities are always positive, or MU(x) > 0. At each and every point in the commodity space C, the marginal utility must increase for changes in any single component of the commodity bundle, in that
We examine the strictly convex indifference curve defined by
Along this indifference curve the derivative or gradient with respect to the commodity bundle must be zero such that we have the requirement that 
. More importantly, the differential must be zero on a surface of constant utility, and so 
. We can rewrite this expression, since we recognize the definition of the marginal utility in the expression, as MUT(x) dx = 0 or
This simply says that the sum of the changes in utility produced by a unit change in xi times the change dxi will be zero. Thus for the two-dimensional case, if we decrease x1 such that dx1 is negative, then dx2 must be positive as the marginal utilities are always positive and Equation 3.4 will hold. The marginal rate of commodity substitution MRCS is an important quantity that can be obtained from this relationship. For the two-commodity case we have, from Equation 3.4,
and for the general case, where we keep all commodity bundle components constant except for xi and xj and trade an increase in consumption of xi for a decrease in consumption of xj (or vice versa), we have
We will obtain some important implications of these properties of utility functions in Section 3.4. A number of special utility functions can be examined. The case of strictly complementary commodities exists when for two commodities, use of x1 requires the use of ax1 units of x2 for a fixed a and excess purchases of x2 do not increase utility. The utility function for strictly complementary commodities is U(x1, x2) = f[min(ax1, x2)]. If the utility function is additive in individual utilities, we have
and we say that the commodities are utility independent. A more general form of utility independence is represented by the multiplicative form
This expression reduces to the additive form of Equation 3.7 when K = 0, and to the multiplicative form
when K = ∞, ki = 0, kiK = 1.
3.4 THE FUNDAMENTAL PROBLEM OF THE CONSUMER
In general the consumer will have a given budget or income I to spend on a commodity bundle of goods and services x. The fundamental problem for the consumer is to pick a particular commodity bundle that maximizes satisfaction, value, or utility for the consumer. We note that a realistic consumer problem would be more difficult than this, as there would be questions of savings, taxes, and the likes involved. Extensions to include these broader aspects of economic behavior will be considered later; here our objective is to obtain a reasonably simple, although incomplete, picture of the economic behavior of the consumer.
We assume that a constant known price vector
of prices per unit item for the items in the commodity bundle has been established. Both the price vector p and the budget constraint I are nonnegative. The consumer’s utility function has been established and so the fundamental problem of the consumer is to maximize consumer utility
subject to the budget constraint
We assume a positive budget and positive prices and require that the consumer not accept any negative goods, and so we have the constraint
We recognize this as a simple problem in nonlinear optimization for which the Kuhn–Tucker conditions of Table 2.4 are directly applicable. We form the Lagrangian by adjoining the inequality constraint of Equation 3.11 to the cost function of Equation 3.10 and obtain
Use of the Kuhn–Tucker conditions obtained in Chapter 2,
or direct use of Table 2.4 leads to the necessary (and sufficient) conditions for optimality:
where the variables x and λ are evaluated at the optimum values 
and 
.
We can draw some very important general conclusions from this result. We note that we must have x ≥ 0 from Equation 3.21 and our initial constraint on the commodity bundle. For x = 0 we see that Equation 3.18 is satisfied and so we must require, from Equation 3.17, that
Here, we need the partial derivative signs to define marginal utility, since utility is an implicit function of I. When x > 0, we see that we must have, from Equation 3.18,
This relation will generally be the applicable relation rather than Equation 3.23, since the consumer will normally purchase a nonzero bundle. The important conclusion to be gained from this is that the ratio of marginal utility to price is the same constant for all components in the commodity bundle vector such that, from Equation 3.24,
It is not reasonable that all the marginal utilities be zero (unless all the prices are infinite), and so we see that the Lagrange multiplier will not, in general, be zero. From Equation 3.22 we see that λ must then be positive. Thus we must have, from Equation 3.20,
which simply states that the consumer will spend their entire budget or income I. This is a most reasonable conclusion, in that it is, under the mildest of conditions, guaranteed by the nonsatiation axiom. There is no penalty, in our problem description thus far, for spending the entire budget. We can increase the quantity of the bundle x by spending more. The nonsatiation axiom insists that this will necessarily increase utility. Of course, we could introduce savings as a term in the utility function and we would then devote a portion of the budget to savings.
Example 3.3:
The results of this optimization effort have particular significance for the case of a two-dimensional bundle. From Equations 3.25 and 3.26 we have
and
If we consider the isoquants, or indifference curves of constant utility where U(x) = constant, we can obtain the slope of these lines, as in Equation 3.6,
By substituting the values of the marginal utilities from Equations 3.27 and 3.28 into Equation 3.30, we obtain, after canceling the Lagrange multiplier,
or
We see that the slope of the utility indifference curve at the point where it touches the budget line, Equation 3.29, is −p1/p2. This is to be expected. Figure 3.1 illustrates the tangency solution for this classic optimization problem of the consumer.
Figure 3.1. Two-Dimensional Graphic Solution for the Problem of the Consumer.
An interesting and useful interpretation can be given to the Lagrange multiplier λ. We combine Equation 3.25, which can be written as
and the partial derivative with respect to I of Equation 3.26,
so as to eliminate pi. We then obtain the expression
This very important result indicates that the Lagrange multiplier for this problem is the marginal utility of income MUI,
where we have dropped the optimality symbol for the sake of simplicity.
The nonlinear algebraic equations 3.17 to 3.22 will not, in general, be easy to solve analytically, and there are a number of nonlinear programming algorithms we can use to obtain numerical solutions. The result of this is that we obtain a solution for the optimum commodity bundle as a function of price and income,
and this is called a demand curve.4 It is a simple matter to show that the demand equation is homogeneous in price and income such that the optimum commodity bundle purchased remains the same, regardless of the same proportionate change in price and income; we have for all positive a,
Thus we see that the demand for a commodity will depend on relative prices and real income. If we let a in Equation 3.35 equal the price of the ith component in the commodity bundle, we obtain
Alternately, we can set a = I−1 to obtain for Equation 3.35
This is a sometimes used normalized expression for a consumer with unit budget.
Example 3.4:
As our second example of consumer optimization, let us suppose that the consumer’s utility function is linear in the commodity vector and given by
We also assume that we have a budget constraint given by
The nonsatiation axiom ensures that the consumer will spend the entire budget. We are able to solve this linear programming problem by simple intuitive means. We should determine the component of the commodity bundle that has the maximum marginal utility–cost ratio and spend the entire budget on this item. Thus we find
Also we find the i that yields the largest ratio. We will denote this amount as 
. Thus, we purchase an amount given by
The resulting maximum utility is given by
The utility function we have assumed here is not strictly concave and does not satisfy the principle of diminishing marginal utility. Thus we do not obtain for this example the result that the ratio of marginal utility to price is constant for all consumer bundle components.
Example 3.5:
As a fairly complete illustrative example that has an analytical solution, let us consider a quadratic consumer utility function
in which a > 0, the B matrix is symmetric negative definite, and
The consumer’s budget constraint is
We have indicated earlier in this chapter that the nonsatiation principle guarantees that the consumer spends the entire budget. Thus we need only consider the equality constraint in Equation 3.40. But we must be very careful to ensure that all the conditions on which our previous results are established are valid. One of these conditions is that the marginal utility always be greater than zero, or in equation form that
Unfortunately, this will not always be true for sufficiently large x since B is negative definite. Thus we cannot necessarily assume that the nonsatiation axiom holds for U(x) of Equation 3.38, and it then will not be always true that all of the budget I is spent. When the budget constraint does not apply, we have λ = 0 in Equation 3.22. Then Equation 3.18 becomes, since x ≠ 0,
and we see that this is just the result for unconstrained maximization of Equation 3.38. This unconstrained maximization using Equation 3.38 yields
as the optimum commodity bundle. The total unconstrained purchases using this 
are the unconstrained purchases pTx = −pTB−1a. If this is not greater than the budget, unconstrained purchases are going to be less than or equal to I, and then of Equation 3.42 is indeed the optimum solution.
There is likely going to be a modeling problem inherent in this example if these results hold. If the budget I is sufficiently large such that we do not need to use the entire budget, we are in the utility satiation region. A utility function of this sort is unrealistic. Thus, if our assumed utility function is to be a reasonable model for the range of commodity bundles that provide feasible solutions, we must restrict the budget to a value less than that spent on the unconstrained purchases. So we will assume that 
, or from Equation 3.43,
for this example. We will then, under this restriction, use the entire budget I.
The Kuhn–Tucker conditions are directly applicable here, but we will use a somewhat simpler approach, since we assume a budget equality constraint, to obtain the commodity bundle that maximizes utility. We adjoin the equality constraint of Equation 3.40 to the cost function of Equation 3.38 using a Lagrange multiplier and obtain
The necessary conditions of optimality are, at 
and 
,
From this, we obtain the relationship that, under optimality conditions, the ratio of the marginal utility for the ith bundle component to the ith bundle component price is a constant for all i. For the specific problem considered here, we obtain for Equation 3.46
Thus, we have
We must adjust such that the equality constraint of Equation 3.47
is satisfied. Premultiplying Equation 3.48 by pT results in
Thus, we have on solving for ,
as the value of the Lagrange multiplier. The optimum commodity bundle is obtained by substituting Equation 3.50 into Equation 3.48 to obtain
For this particular example the commodity bundle increases linearly with respect to increases in budget or income I. It is interesting to note that the optimum commodity bundle is not a linear function of price, however. We note from Equation 3.51 that it is possible to obtain negative components of in the foregoing for sufficiently small I. These are not reasonable values, as they violate Equation 3.21. We do not allow the consumer to sell back some items to get money to pay for other items. We shall not explore problem solution for this case in any depth. Generally, it is sufficient to set these computed negative elements of x equal to zero, delete the corresponding coefficients for the zero x elements, and use the Kuhn–Tucker conditions to find a solution to the problem.
Example 3.6:
As a special case of Example 3.5, let us consider the simplest case where the commodity bundle is a one vector. For this scalar case the results simplify considerably and we have
where the last equation follows from the general optimal result of Equation 3.51. We note that x for this example is just the value that satisfies the equality budget constraint. However, when MU(x) = a+bx becomes zero (or less), then we should use the inequality constraint budget and actually obtain a larger utility by using less than the full budget. Of course, this is because of the (now) unrealistic utility function. Thus we have, from Equation 3.44, that when I >− pa/b, we should use the unconstrained optimum 
. Figures 3.2 and 3.3 illustrate salient features of this example for the particular case where a = l and b = −0.1.
Figure 3.2. Utility Function for Example 3.5.
Figure 3.3. Optimal 
versus I/p for Example 3.5.
Example 3.7:
We may also obtain some interesting graphical results for the two-commodity case. For the case where there are two commodities, we may rewrite Equation 3.51 as
where
We consider the specific case where
such that
The unconstrained optimum solution is
Figure 3.4 shows contours of constant utility, the utility indifference curves; also shown is a budget line for I = $100, p1 = $1, p2 = $5. The optimum can be determined either from Equations 3.52 to 3.54 or by careful construction of the curves in Fig. 3.4 as
Figure 3.4. Budget Lines and Utility Indifference Curves for Example 3.6.
and we see that both are less than the components of . The amount of money required to purchase is pTx = $133.33 and the consumer has only $100.
It is of great interest to determine the behavior of the consumer if the prices p1 and p2 change. If, for example, p1 = 0, we obtain from Equations 3.52 through 3.54
and we see that the full consumer budget is now being spent on commodity x2 and so we use whatever value of x1 results in unconstrained maximization of U(x). Since b11 is negative, the consumer always purchases a positive number of the free commodities x1. As will generally be the case, the size of the commodity bundle decreases with increasing price. Problems involving consumer sensitivity to price changes are very important and we will turn our attention to this in Section 3.5.
Example 3.8:
In this example, we examine the sufficiency requirements for consumer maximization of a utility function expressed as a general quadratic of the form
where the budget constraint is given by
and where B is symmetric and possesses an inverse. To proceed, we define the Lagrangian, which is a function of the variables x and λ:
We replace x and λ with 
and 
and assume that δx and δλ are sufficiently small such that the three-term Taylor series
is a valid approximation. For the case where δx and δλ are completely arbitrary, the necessary conditions for an extremum (either maximum or minimum) of L are that
For L(x, λ), or U(x) subject to the constraint, to be maximum, we require that the quadratic expression
be negative, or at least nonpositive, for arbitrary δx and δλ with x ≥ 0. This is equivalent to the requirement that ∂2L/∂x2 be negative definite subject to the inequality constraint of Equation 3.58. We could use this requirement as the sufficiency requirement associated with the use of Table 2.2.
Equations 3.61 and 3.62 yield, assuming that the full budget is spent,
and
from which we obtain, as in Example 3.5, the optimum commodity bundle
Here we assume that the solution to Equation 3.66 yields ≤ 0. This result clearly shows us that B must have an inverse, but it does not require that B be either positive or negative definite matrix; in fact, it can be indefinite.
Either the full budget I or less than this amount will be spent. For the case where less than the full budget is spent, there is really no need for the constraint of Equation 3.58. The necessary conditions for optimality are obtained directly from Equation 3.64 as the unconstrained optimum
and the sufficiency conditions become
and we see that we must require B to be negative definite.
For the case where the full budget is spent, the necessary conditions Equations 3.64 and 3.65 prevail. From Equation 3.57 we see that if B is negative definite, then surely 
is negative definite and
Thus we see that negative definiteness of B is sufficient to establish a maximum of U(x) under full budget utilization conditions, regardless of the budget path.
Other conditions can also be obtained from the general requirement that
be negative definite. One of these is that the first n−1 trailing principal minors of this matrix alternate in sign, with the sign of the first principal minor being (−l)n. Thus if U(x) = x1x2 such that
we need to examine the first trailing principal minor of the matrix
which is just the determinant of the entire matrix. This is 2p1p2. The sign of this is positive and so we see that the B matrix does not necessarily have to be negative definite. The indefinite matrix used here satisfies the sufficiency condition quite nicely.
We are really just determining the conditions under which 
along the budget income line. For U = x1x2 we need to examine
In general, this expression is not definite in sign. However, since I = pTx, we have 0 = pTδx or p1δx1+p2δx2 = 0 for the example we are considering. Thus δx2 = −p1δx1/p2 and
This expression is surely negative for all positive prices, and so we see that U(x) = 2x1x2 is a valid utility function for utility maximization. This utility function has a convex utility indifference curve, as we may easily verify.
Example 3.9:
We now consider a logarithmic utility function of the form
which is strictly quasi-convex for all positive ai (and xi). We wish to maximize this utility function subject to, because of the applicable nonsatiation axiom, the equality budget constraint given by
We adjoin the equality constraint of Equation 3.68 to the cost function of Equation 3.67 and write the Lagrangian as
We obtain necessary (and sufficient) conditions for an optimum from ∂J/∂x = 0 and ∂J/∂λ = 0 as
Multiplying both sides of Equation 3.69 by 
and summing over all i from i = 1 to n yields
Thus, we have, on substituting the value of 
obtained in Equation 3.71 into Equation 3.69,
The ai are assumed to be fixed by the consumer’s utility function. Here, the amount of a commodity that a consumer purchases will vary inversely with the price of the commodity. A very interesting property of this result is that the budget allocation to each commodity is
In many ways this is a very attractive utility function, because this result makes calibration of the utility function a relatively simple matter once we invoke its form. For example, suppose that we observe purchase data, averaged perhaps over a year, for a consumer, in the table shown next:
| Commodity number | Number of commodities purchased | Total price paid for commodities |
| 1 | 250 | $1000 |
| 2 | 300 | 450 |
| 3 | 1000 | 300 |
| 4 | 650 | 250 |
| 5 | 30 | 500 |
| Total | 2230 | $2500 |
We have, on invoking the utility function of Equation 3.67 and using Equation 3.73, the results 
and 
. Because of the nonuniqueness of utility functions we are free to pick at any convenient value and so we choose = 2500. Thus for this consumer that we have modeled, we obtain the vectors
Product substitutability is a general function of economic life. If the ai in this example are truly constants, then the utility function chosen is somewhat unrealistic in that the consumer always denotes a fixed percentage of their total budget to the purchase of each commodity. We can make the ai a function of the price of the commodities in the bundle and obtain product substitutability. It would not seem unreasonable to let 
, where pin is the fixed nominal or normal price of commodity i, pi the actual price of item i, and bi a constant. It is also not at all unreasonable that consumer utility for a commodity depend on the price of the commodity. This is the substance, in effect, of the theory of revealed preference, from which a consumer’s utility indifference curve can be obtained. According to this theory, the fact that a consumer purchases a bundle x1 rather than x2 only indicates a consumer preference for x1 over x2 if the total price paid for x1 is greater than that for x2. In other words, where we use > for revealed preference, we have x1 > x2 iff 
for all p1.
There are two axioms of revealed preference: a weak axiom that states that revealed preferences are asymmetric, in that if x1 > x2, it is not possible for x2 > x1, and a strong axiom that states the transitivity of revealed preferences, in that if x1 > x2 and x2 > x3, then it must be true that x1> x3.
A number of extensions of the basic theory of the consumer are possible. For example, consumers not only desire to purchase commodities but also desire leisure. Thus we might postulate a utility function for a commodity bundle x and leisure time l and denote it by U(x, l). Income and leisure are both desirable and the substitution of income for leisure to maintain constant utility is given by the expression
We may alternately write this as
We may maximize consumer utility in much the same way as before. If we let t represent the total amount of available time, W the time devoted to work, and w the wage rate, then we see that we wish to maximize
subject to the constraints that the total time t is fixed, and given by the sum or work time W and leisure time l:
Also, the total income for commodity purchase is determined by the amount of time worked and is given by
We can eliminate Equation 3.76 by substituting W from this equation into Equation 3.77 to obtain
We have thus obtained the result that maximization of Equation 3.75 subject to the constraint of Equation 3.78 is the problem of the consumer with leisure. To obtain a solution, we adjoin Equation 3.78 to Equation 3.75 and obtain
The necessary conditions for optimality
result. The first two of these equations may be combined to eliminate the Lagrange multiplier λ such that we have
This is equivalent to the expression
Thus, we see that the rate of substitution of budget income for leisure is the wage rate. This is a potentially useful result, although obtained under simplified assumptions, in that it tells us something about how much leisure a consumer would like. This is the sort of result possible from the theory of the consumer and we will examine issues like this further in Chapter 5.
We may modify the basic problem of the consumer in any number of ways that are basically similar to our modification to include the utility for leisure. Some of these are suggested for exploration in the problems at the end of this chapter. Others will be considered in Chapter 5 and the subsequent chapters.
Example 3.10:
For our next example in this section we consider the very general translog (transcendental logarithmic) utility function as defined by the log quadratic function
where a is a scalar, b a vector, and C a matrix. All three are assumed constant. The vector z is comprised of the logarithm of all commodities and leisure:
We formulate precisely the same optimization of consumer utility problem that we have just discussed. Here, Equations 3.80 to 3.82 are the necessary conditions for consumer optimality. We obtain from Equation 3.85, for i = 1, 2,..., n,
where Ci is the ith row of the matrix C. Using Equations 3.80 to 3.82 results in
We multiply both sides of Equation 3.89 by xi and sum over i from 1 to n. Use of Equation 3.91 then gives us
Substituting for λ in Equation 3.89 with this result leads us to the relation
Substitution of the value of λ obtained in Equation 3.92 into Equation 3.90 results in
Because of the linear way in which the parameters b and C enter into Equations 3.93 and 3.94, we can use linear regression techniques, such as those discussed in statistics, forecasting, or system identification texts, to identify the b and C parameters. In so doing we will have identified the parameters in the utility function of Equation 3.85.
Equations 3.93 and 3.94 are the central results of this example. From these equations we obtain the optimum commodity demand bundle x and the optimum amounts of leisure or work. An explicit solution of Equations 3.93 and 3.94 is not possible unless C = 0. In this special case we obtain from Equations 3.93 and 3.94
as the optimum consumer commodity demand bundle and leisure supply.
In this section, we have examined a relatively large number of examples. It would be instructive to consider some particular numerical cases. This is the purpose of some of the problems at the end of this chapter.
3.5 SENSITIVITY AND SUBSTITUTION EFFECTS
Often optimum economic consumer operating conditions have been determined and one or more parameters, most often income and prices, change. If these changes are small, we can make a linear perturbation of the original problem equations and then make a relatively simple determination of changes in the optimum consumer bundle as a function of the changed parameters. This linear perturbation about operating conditions is often termed a sensitivity analysis. To accomplish this sensitivity analysis we recall some of our results from Section 3.4 concerning the theory of the consumer.
The fundamental problem of the consumer is to maximize utility
subject to the budget constraint
where we assume that prices are nonnegative and that the full budget I will be spent. The Kuhn–Tucker optimality conditions for this problem are, where we recall that the optimum commodity bundle could and perhaps should be written as 
to give explicit indication to the fact that the optimum bundle is a function of budget and prices, given by
A brief introduction to sensitivity-based analysis will provide useful results that we will use on the consumer demand problem. We obtain sensitivity analysis relations for an equation
by letting the values x and z equal nominal values plus a perturbation,
and then expanding the resulting function in a Taylor series about Δx and Δz and then dropping terms higher than first order in Δx and Δz to yield5 the result
The nominal solution (xn, zn) should satisfy Equation 3.101, and so we have
We have a linear relationship between Δx and Δz given, from the foregoing, by
assuming, of course, that the matrix inverse in this equation exists.
We will now apply these results to the necessary conditions for consumer optimality of Equations 3.99 and 3.100. We let
We next note that, to terms of first order,
Here we assume that the optimum consumer bundle is that which results when the budget income and prices are nominal. Inserting Equations 3.106 through 3.110 into Equations 3.99 and 3.100 results in the expressions
We subtract the identities of Equations 3.99 and 3.100 when they are evaluated at the nominal budget and prices In and p, respectively, and at the resulting from the foregoing two equations. This yields, after neglecting the Δλ Δp and ΔpT Δx products of second-order terms, the expression
On dropping the superscripts for convenience, we have the expressions
If the utility function obeys the restriction we imposed earlier, the 
expression is negative definite. All the components of −p are negative since no price is negative. Thus the inverse of the A matrix exists. We have, as we may easily verify,6
where
We see from Equation 3.113 that we can write the relation between the commodity bundle change and changes in the budget and prices as
We can show7 that the scalar coefficient α is, from Equation 3.33,
Thus, we see that this coefficient is the rate of decrease of the marginal utility of income. Also, we can easily obtain a set of partial differential equations equivalent to Equation 3.117. They are
An interesting interpretation can be obtained from Equation 3.117 if we consider the effect of a compensated change in price where the income is adjusted to keep the consumer utility a constant. If U(x) is a constant, then ΔU(x) = 0 and
From Equation 3.99 we see that the foregoing becomes, at the consumer optimal commodity bundle,
From Equation 3.100 we obtain the required change in income
which becomes, because of Equation 3.122,
This states the required relation between the budget income and prices to keep utility a constant. It states a compensated income change for preservation of constant utility. This is a very important relation in that it demonstrates a key property of the demand functions obtained from consumer optimization for maximum utility; demand functions are homogeneous of degree zero in budget I and prices p. Thus we see that if all prices and consumer income are multiplied by the same positive constant a, then the optimum consumer bundle is unchanged and not a function of a.
We denote the solution for the optimum commodity bundle change under constant utility conditions as (Δx)const. Substituting Equation 3.124 into Equation 3.117, we see that this is given by
The (Δx)const obtained from this relationship is often called the substitution effect of a compensated change in price on demand. The other terms in (Δx) Equation 3.117 are called (1) the income effect (Δx)income, which represents the effect of a change in income on demand, and (2) the change in demand quantity due to a change in income or budget from that required to purchase . We have
and
Or, in words, total effect equals substitution effect plus income effect.
We can obtain a partial differential equation corresponding to Equation 3.125 as
such that Equation 3.120 becomes, using the foregoing and Equation 3.119,
From Equation 3.128 we see that the matrix (∂x/∂p)const is symmetric and negative semidefinite. By equating the ijth coefficient of (∂x/∂p)const in Equation 3.129 to the jith coefficient of (∂x/∂p)const, we obtain the interesting result8 that, for any components in the commodity bundle,
This equation is known to economists as the Slutsky equation and can be used to account for the changes in the demand for one component in the commodity bundle due to changes in prices for another commodity bundle component.
Postmultiplying either Equation 3.125 or 3.128 by p, we easily see that, because of the definition of α in Equation 3.116,
This has a very interesting interpretation, since all prices p are positive. All elements of any row of (∂x/∂p)const cannot have the same sign. The elements on the main diagonal must be negative.9 Thus at least one of the off-diagonal elements must be a positive quantity. We say that commodities are substitutes if the off-diagonal element is positive, or
and that commodities are complementary if the off-diagonal element is negative, or
If commodities are substitutes, and utility is constant, an increase of price of one commodity will lead to an increased demand for the other commodity. All commodities must have at least one substitute, otherwise it will not be possible to keep the utility function invariant. For complementary commodities, an increase in price of one commodity will lead to a decreased demand for the other commodity. For example, coffee and tea are likely substitute commodities, whereas guns and bullets are likely complementary commodities.
The concept of substitute and complementary commodities is applicable only when utility is held invariant by the price–budget income relationship of Equation 3.124. When price and budget income are not so constrained, we say that a commodity is normal if the corresponding main diagonal component of ∂x/∂p is negative:
It is called a Giffen commodity if it is positive:
It is rare that a commodity is Giffen. A commodity is termed superior if
A commodity is called inferior if
For example, meat would likely be considered by many to be a superior commodity, and potatoes an inferior commodity.
Elasticities play an important role in the theory of the household, as they did in the theory of the firm. We define the (cross) elasticity of the demand for commodity i with respect to the price of commodity j as
The elasticity of a given commodity with respect to its own price will usually be negative (again, the Giffen commodity is rare). Such an elasticity is generally called the price elasticity of demand and is defined by
The (price) elasticity of demand will depend on the shape of the demand curve and also on the equilibrium position on the demand curve. Let us examine three relevant examples here. More are present in the set of problems at the end of the chapter.
Example 3.11:
For the particular case of a linear demand curve xi = a−bpi, we easily see that the price elasticity of demand is
and we see that the price elasticity of demand varies from 0 to −∞ as the demand and price vary from a and 0 to 0 and a/b. For the case where b = 0, such that the demand is constant and independent of price, we have 
. This zero elasticity is called a completely inelastic demand. The other extreme is where the price is extremely insensitive to demand and a constant price results. For this case a = ∞ and b = ∞ in such a way that a/b = pi. Here the price elasticity of demand is −∞ and we say that the demand is completely elastic. To avoid obtaining negative elasticities for normal commodities, many authors define elasticities as the negative of the ratio of the percentage of change in quantity to the change in price. Figure 3.5 illustrates typical demand curves and commonly used names for the price elasticity of demand.
Figure 3.5. Price Elasticities for Various Demand Curves.
Also, we can write for the elasticity of the demand for commodity i with respect to budget income
Either by using the necessary conditions for consumer optimality of Equations 3.99 and 3.100 or by postmultiplying Equation 3.129 by p and using Equation 3.131, we obtain
In component form this is
or
and in terms of the price and income elasticities we have
Thus we see that the sum of the income elasticity for commodity j and the summed demand cross-elasticity for commodity j with respect to the price of commodities i (for all i) is zero for all j.
Other interesting and useful results can also be obtained here. Premultiplying Equation 3.119 by pT we see that
which says that
or
Thus the sum of total expenditures for each commodity times the income commodity sensitivity is the total budget income. This is called the Engel aggregation condition by economists.
If we premultiply Equation 3.129 by pT and use the results in Equations 3.131 and 3.144, we obtain
This is often known as the Cournot aggregation condition. In scalar form this says that
or
So we see that the demanded commodity xj is just the negative sum of the commodity price-weighted change of demanded quantities with respect to price.
Let us now suppose that budget income I is constant but that there is a change Δpi in the price of the ith commodity. We might wish to obtain the ratio of the change in expenditure for commodity i to the original expenditure for commodity i. We have
Thus, we see that if 
, we will spend less for commodity i if the price of commodity i rises. If ε >− 1, a price rise for commodity i will cause an increase in expenditure for commodity i. Thus we see that if we have a highly elastic demand such that the magnitude of the price elasticity is large, 
, we are dealing with either commodities for which substitutes can easily be found or optional commodities that we can easily do without. If the magnitude of the price elasticity is small, 
, we say that we have a low demand or inelastic demand and we are dealing with a commodity that is either quite necessary or a very small part of the overall commodity bundle.
Sensitivity concepts are especially important in economic systems analysis and assessment. It is rare that we will have precise parameter information in realistic real-world situations. With sensitivity analysis we can determine which parameters are most influential in determining system behavior such that we can allocate parameter measurement resources appropriately. To accomplish sensitivity analysis on other than very simple problems requires the use of a digital computer for detailed computations, as the associated analysis efforts can become quite laborious, as we easily sense by the mathematics in this section.
Example 3.12:
Suppose that the utility function for n goods is given by
and that the consumer income is I. We easily obtain the optimum commodity bundles from the Lagrangian expression
as
This equation represents the demand equations for this example. Each commodity is normal and superior. The commodities are independent in that the demand function for one good does not depend on the price of another. Thus no good is a “substitute” or “complement” for another good if we obtain this with respect to changes in price for a fixed income or without regard for utility. We may remove this potential confusion over the meaning of “substitutability” by use of the terms normal and superior.
The elasticities are
Thus, we see that the Engel aggregation condition of Equation 3.145 is satisfied. Equation 3.129 becomes
From application of the results of Equation 3.132 here, we see that the commodities are substitutes. Again we must emphasize that this concept of substitutability is applicable if and only if we keep utility invariant by appropriate adjustment of the price–income relationship.
Example 3.13:
Consider now the utility function
with the budget equation
From the Lagrangian, which is given here by the expression
we obtain the necessary conditions for utility maximization and the resulting optimal purchases as
We can easily obtain the results
From Equation 3.129 we obtain
In this example, we see that the commodities are normal, superior, and substitutes for one another.
In this chapter we have examined some elementary concepts from the microeconomic theory of the consumer. We introduced the concept of a utility function with which to express the satisfaction or value the consumer obtains from alternative goods and services. A number of axioms of utility theory were postulated to model a rational consumer. We examined the necessary conditions for a consumer to maximize utility subject to a constraint of a fixed budget income and with given prices for goods and services. The results of this optimization are important in that we obtain the consumer demand function for goods and services in terms of their price and the consumer budget or income for goods and services.
Finally, we established a number of important ancillary concepts using the necessary conditions for optimum consumer behavior. We obtained some sensitivity results concerning optimum consumer demand functions, as well as some linearized properties of the demand function. In Chapter 4 we will combine the results of this chapter concerning the consumer together with those of Chapter 2 concerning the firm and establish some important microeconomic supply–demand equilibrium properties.
We have not, in this chapter, considered problems of consumer demand where the outcomes involved with purchasing decisions are associated with risk and uncertainty. This is a very interesting and pertinent topic, but to go into it would greatly increase the size and scope of this book. Nor have we considered problems that involve preferences over time. Chapter 6 is devoted to a brief discussion of this topic where we examine cost–benefit and cost–effectiveness analyses and assessments.
PROBLEMS
1. Show pictorially the reasons for the following:
a. The indifference relation ∼ is symmetric and transitive.
b. The strict preference relation > or < is symmetric and transitive.
c. U(x1) = U(x2) if and only if x1 ∼ x2.
d. U(x1) > U(x2) if and only if x1 > x2.
2. For the marginal rate of substitution defined by Equation 3.8,
Show that the utility function is additive only if
and that the demand for a commodity depends only on the price of the commodity and the total budget available for commodities.
3. Do any of the following utility functions have convex indifference curves? What is the marginal utility of each utility function?
4. Show that the utility functions
can both describe the same preferences, in that one is a monotonic transformation of the other.
5. What is the optimum commodity bundle for a consumer with the following utility functions and budgets?
6. A consumer purchases the commodity bundle x1 = 5, x2 = 10 at prices p1 = 3, p2 = 1. The same consumer also purchases the bundle x1 = 8, x2 = 36 at prices p1 = 10, p2 = 6. Is the consumer behavior consistent with the revealed preference concept?
7. Demonstrate that the necessary conditions of optimum consumer behavior are unchanged by any monotonically increasing transformation of the utility function.
8. For a two-commodity bundle, can both bundle components be superior or inferior goods?
9. Demonstrate for the case of additive independent utilities that
a. there are no complementary commodities,
b. there are no inferior commodities, and
c. there are no Giffen commodities.
10. We have defined substitute and complementary commodities in this chapter. Discuss the use of the complementary utility of commodities i and j
and the substitute utility of commodities i and j
as a replacement for our earlier definitions. Consider a simple quadratic utility function to illustrate your conclusions.
11. Contrast our definition of substitute and complementary commodities i and j to the alternate definitions ∂xi/∂pj > 0 and ∂xi/∂pj<0. Consider a simple quadratic utility function to illustrate your result.
12. For the case where consumer utility is U(x) = 0.5xTAx where A is positive semidefinite and where the purchase cost for a consumer bundle x is given by b = xTp with p as the price vector for the commodity bundle, find the
a. optimum consumer bundle to maximize U(x) where b ≤ bmax and
b. optimum consumer bundle to minimize the purchase cost, b = xTp, subject to the constraint that U(x) ≥ Umin. Show that these two problems are mathematical equivalents.
13. How will the revenue to the firm producing a commodity change as a function of price changes and the price elasticity? Equation 3.147 is an appropriate vehicle for your discussion.
14. What are the optimum commodity bundles for each of the utility functions of Problem 3 where there is a budget constraint that will limit consumer spending?
15. The following are the results of monotonically increasing transformations:
Suppose that the price vector is pT = [0.2 0.6]. What are the optimum commodity bundles for each utility function if we assume an income I = 20? Can these utility functions be interpreted as ordinal utility functions or must they be interpreted as cardinal utility functions? Do your conclusions change if we have the utility given by 
?
16. For each of the utility functions of Problem 15, is commodity 1 superior or inferior? Are commodities 1 and 2 complements or substitutes?
17. Suppose that a consumer’s utility function is
Determine the optimum commodity bundle as a function of the consumer income I and the price vector p. Are any of these commodities inferior, superior, complements, or substitutes? Be sure to examine the conditions for optimality. Is there anything especially significant about I ≤ 2p1?
18. Suppose that consumer utility is a function of the income of the consumer and the consumer’s desire for leisure. Suppose further that the consumer has a constant marginal utility of income and a decreasing marginal utility of leisure and that the utility function is separable in leisure and commodities.
a. Show that a straight vertical or horizontal line will cross the utility indifference curves for income and leisure at points of constant slope.
b. Show that the effect of an income tax will result in less work and more leisure for the consumer.
c. Illustrate these results for the case where U(I, l) = aI+bl−cl2 with a, b, and c constant and where I = income and l = leisure.
19. A consumer has a utility function for products and leisure U(x, l) and desires to maximize this utility by choice of the commodity bundle x and leisure l. This must be accomplished subject to the constraint pTx = ℐ+w
, that the total dollar amount of products purchased must be equal to the nonwage income ℐ plus the consumer’s wages w times the number of hours worked . The time available, 24 h per day, must be the sum of the hours worked per day and the hours of leisure per day l: 24 = +l. Prices, wages, and nonwage income are fixed.
a. What are the necessary conditions for utility maximization?
b. What is the optimum number of work hours for the consumer as a function of p, w, and ℐ if U(x, l) = x+36l−l2?
20. A pure exchange economy is one in which there is no production and H households consume and exchange N commodities by bartering a number of items of a given commodity for a number of items of another commodity. Each household has an initial endowment of commodities 
and seeks to increase its utility by exchanging a portion of this initial endowment for other commodities. Commodity prices p are fixed and so the consumer total initial worth is given by 
. The worth of the commodities purchased and consumed must be equal to the initial worth of the consumer given by 
where qh may include some commodities in the initial endowment. Each household desires to maximize utility as given by 
, subject to the total worth constraint. Equilibrium requirements for the H households are 
.
a. What are the necessary conditions for utility maximization for each consumer?
b. What are the equilibrium conditions for two households and two commodities where
c. What do these results become when you change the above two matrices to the identity matrix?
d. Determine (i) the optimum exchange price ratio p1/p2 and (ii) the optimum final commodity distributions q1 and q2 for the various conditions assumed here.
21. In times of scarcity, consumers must live not only within their individual budget or income constraints but also within constraints imposed by rationing. Develop the necessary conditions for maximization of the utility function U(x) subject to a budget-rationing constraint pTx ≤ I and a point-rationing constraint rTx ≤ R. How will the rationing of some commodities affect the consumption of those not rationed?
22. Demonstrate the validity of Equation 3.118. You may find the derivation leading to Equation 3.33 helpful in doing this.
23. Develop sensitivity and substitution effects for the theory of the firm that parallel our results in Section 3.5 concerning the theory of the household.
24. The effect of an ad valorem tax of 100ν%, such that the total tax collected is νpTx, can be considered by modifying the consumer budget constraint to I > (1+ν)pTx. The effect of an income tax IT is to reduce the consumer disposable income such that the budget constraint is given by the expression I−IT = pTx. Investigate the effects of these two taxes on consumer utility. You may find it helpful to assume a quadratic utility U(x) = 0.5xTAx. Be certain to indicate for the case of specified tax revenue whether consumer utility is greater with an income tax or an ad valorem tax. Can you explain your conclusion by substitution and income effects? Demonstrate your results graphically.
25. Suppose that two commodities x1 and x2 are perfect substitutes and that the price p1 of x1 is higher than the price p2 of x2. What will be the relative ratio of the two commodities in a consumer commodity bundle?
26. Will it necessarily be true that consumer utility will decrease with an increase in the price of a commodity?
27. For the quadratic utility function U(x) = aTx+0.5xTBx and the constraint xTp ≤ I
a. What is the marginal rate of technical substitution?
b. What is the demand price elasticity?
c. What is the demand price cross-elasticity?
d. What is the condition(s) under which a commodity could be (i) normal, (ii) Giffen, (iii) superior, or (iv) inferior?
Assume that there are only components in the commodity bundle and obtain explicit results for this case.
28. Repeat Problem 27 for the case of a translog utility function such as that used in Example 3.10.
29. A person has the utility function given by the expression U(x, l) = x0.5+2l0.5 where x is the quantity of goods produced and l is leisure. The person’s income depends on the wage rate and time devoted to work. There is a maximum amount of time T available and so the income earned is given by I = w(T−l). The price of the goods is p. Determine the labor supply and commodity demand functions. Suppose that there is an income tax rate t such that the actual income available for purchases is I = (1−t)w(T−l). How does this affect the labor supply and commodity demand functions?
30. The utility for food x1 and clothing x2 is given by U(x1, x2) = x1x2. The price of clothing is 1 and income is 250. What is the effect of a drop in the price of food from 2 to 1.5? How have consumers benefited? Illustrate your results graphically.
31. Is there an increase in the price of clothing that would leave consumer utility unchanged in Problem 30? Make these computations using both the fundamental theory of the consumer and the sensitivity results presented in this chapter.
32. The utility function of a consumer is given by U(x) = x1x2+x3x4. Prices are fixed and the consumer has a fixed income I. What is the optimum commodity bundle? Can there be any inferior goods? Can there be any complementary goods? Can there be any substitute goods?
33. Outline a numerical procedure to obtain the sensitivity-based results of Section 3.5 for cases where the consumer optimization problem cannot be explicitly solved. Use these results to find substitution effects for the case where U(x1, x2, x3) = x10.5+x2x3, I = 100, p1 = p2 = p3 = 1.
34. A person’s utility function is given by 
where 
. The person is income constrained and prices are known constants.
a. If pi increases, what are the possible behavior patterns for xi?
b. If pi increases, what are the possible behavior patterns for xj?
c. Determine possible “demand curves” for cases a and b for the two-dimensional case.
d. Is it possible to have any inferior commodities in this example? Any Giffen commodities?
e. Repeat parts a to d for the two-dimensional case where the utility function is multiplicative and of the form U(x1, x2) = k1U1(x1)+k2U2(x2)+k1k2KU1(x1)U2(x2).
BIBLIOGRAPHY AND REFERENCES
The texts referenced in Chapter 2 are basic references for this chapter, as well as for Chapter 2. Interesting discussions of the utility and preference concepts may be found in the following dated but seminal references.
Debreu G. Theory of value. New York: Wiley; 1959.
Edwards W. The theory of decision making. Psychol Bull 1954;5:380–417.
Edwards W. Behavioral decision theory. Annu Rev Psychol 1961;12:473–498.
Samuelson PA. Consumption theory in terms of revealed preferences. Econometrica 1948;15:243–253.
1We should remark that these are normative assumptions or assumptions for a prescriptive theory of consumer behavior. In a descriptive sense, consumers are often unintentionally nontransitive or intransitive. There is a wealth of literature on this subject and, sadly, we will be unable to cover it here in any detail due to lack of space. We will devote some more attention to this subject in Chapters 5 and 10.
2Discussion on this topic in detail would carry us far away from our principal objectives. The reader is referred to Sage’s ‘Behavioral and organizational’ considerations in the design of information systems and processes for planning and decision support. IEEE Trans Syst Man Cybern 1981;SMC-11(9):640–678, for a somewhat dated discussion of these issues, as well as references to much related literature. Also see Kahneman D, Tversky A, editors. Choices, values and frames. Cambridge: Cambridge University Press; 2000. We examine these issues further in Chapter 5.
3The two statements of the axiom are equivalent as we easily see by letting x3 = ax1+(1−a)x2 such that the ith component of x3 contains ax1,i+(l−a)x2,i units of commodity i.
4It is clear that the resulting demand curve is a useful prescriptive or normative result in that it describes how a consumer should behave. Often these results are essentially correct for descriptive behavior. However, in many cases, people do not behave as they “should”.
5To simplify the notation, we define
We use a similar definition for the derivative with respect to z.
6We simply calculate A−1A = I, the identity matrix, and use this relation.
7See Problem 22.
8See Problem 30. This result may also be obtained by using the necessary conditions, given by Equations 3.99 and 3.100, for combining optimum consumer behavior.
9The matrix 
is negative semidefinite. Also, an element on the main diagonal of this matrix is its own substitution effect.