### Chapters

#### 2. Consumer Theory

Behind every supply and demand curve is an army of producers and consumers making their own decisions. For consumers, their decisions are driven, quite simply, by what they want! All consumers make decisions to maximize their utility.

**PREFERENCES**

Preference is the ordering of alternatives based on a consumer’s relative utility, a process which results in an optimal "choice" (whether real or theoretical). The character of the individual preferences is determined purely by taste factors, independent of considerations of prices, income, or availability of goods.

Life is like a shopping center. The consumer enters it and sees lots of goods, in various quantities, that she might buy. A consumption bundle, or a bundle for short, is a combination of quantities of the various goods (and services) that are available. For instance, a consumption bundle might be 2 apples, 1 banana, 0 cookies, and 5 diet sodas. We would write this as (2, 1, 0, 5). Of course the consumer prefers some consumption bundles to others; that is, she has tastes or preferences regarding those bundles.

We will make various assumptions about a consumer’s feelings about alternative consumption bundles. We will assume that when given a choice between two alternative bundles, the consumer can make a comparison. (This assumption is called completeness.) We will assume that when looking at three alternatives, the consumer is rational in the sense that, if she says she likes the first better than the second and the second better than the third, she will also say that she likes the first better than the third. (This is part of what is called transitivity.) We will examine other basic assumptions that economists usually make about a consumer’s preferences: one says that the consumer prefers more of each good to less (called monotonicity), and another says that a consumer’s indifference curves (or sets of equally-desirable consumption bundles) have a certain plausible curvature (called convexity). We will describe and discuss the consumer’s rate of tradeoff of one good against another (called her marginal rate of substitution)

**UTILITY**

A utility function is a numerical representation of how a consumer feels about alternative consumption bundles: if she likes the first bundle better than the second, then the utility function assigns a higher number to the first than to the second, and if she likes them equally well, then the utility function assigns the same number to both. We will analyze utility functions and describe marginal utility, which, loosely speaking, is the extra utility provided by one additional unit of a good. We will derive the relationship between the marginal utilities of two goods and the marginal rate of substitution of one of the goods for the other.

**CONSUMERS PREFERENCES AND UTILITY**

The Nature of Consumer Preferences

How consumers make choices is an important question. To answer this, we first need to know the nature of consumer preferences. Suppose there are several market baskets (or commodity bundles), say A, B, C, etc, each of which consists of some combination of goods. A consumer should have some preferences over these baskets. We make the following assumptions about preferences:

(1) Completeness. For any two baskets, the consumer should be able to compare them. In other words, if A and B are any two baskets, the consumer should be able to say whether she likes A better, or B better, or she is indifferent between the two (she equally likes them).

(2) Transitivity. If a consumer prefers A to B, and prefers B to C, then she should prefer A to C.

(3) Nonsatiation. The consumer always prefers more to less of a commodity.

**Utility and Utility Functions**

The level of satisfaction a consumer receives from a certain choice is often referred to as the utility of that choice. When a consumer prefers A to B, we say she gets higher utility from A than from B. For each consumption bundle, we can assign a numerical value to it which preserves the preference ordering, and we call this correspondence between preference and real values the utility function.

Utility function is defined such that for any baskets A and B: U(A) > U(B) if the consumer prefers A to B; U(A) = U(B) if the consumer is indifferent between A and B.

There are two types of preference rankings: ordinal rankings and cardinal rankings. Ordinal rankings give us information about the order in which a consumer ranks baskets, while cardinal rankings give us information about the intensity of a consumer’s preference. Utility is an ordinal ranking of commodity bundles. Thus the utility function for a certain preference system is not unique

**Analysis for A Single Good**

If a consumer consumes only one good, the utility function can be denoted as U(y), where y is the amount of the good consumed. For instance:

U(y) = 10vy.

Marginal utility is the change in utility caused by an incremental change in the consumption of a good.

MUY = DU/DY.

Example: When U(y) = 10vy , we have MUY = 5/vy.

The Principle of diminishing marginal utility: As consumption of a good increases, the marginal utility of that good will eventually decrease after some point.

Analysis of multiple goods:

**Indifference Curves**

A useful way to describe a consumer's preference is indifference curves. An indifference curve is the set of points representing the market baskets among which the consumer is indifferent. Alternatively, an indifference curve is the set of points representing the market baskets that give the consumer the same level of satisfaction.

Example. The derivation of indifference curves for a consumer.

Properties of indifference curves:

(1) All indifference curves are downward-sloping.

(2) The further an indifference curve is away from the origin, the higher level it represents of a consumer's satisfaction.

(3) Two indifference curves cannot cross each other.

These properties are derived from our assumptions about consumer's preferences.

Example. The shape of indifference curves. What happens if the assumption of nonsatiation is violated?

Marginal Rate of Substitution

Example. Utility function with two goods: U(X,Y) = XY. Another one: U(X,Y) = 2XY.

The consumer is usually willing to substitute amounts of one good for another. If you have 5 units of apple and 12 units of orange, you might be equally happy to have 6 apples and 10 oranges instead. In other words, you are willing to give up 2 units of orange in order to have one more unit of apple. In this particular case, we say your marginal rate of substitution (of apple for orange) is 2. In general, we define:

__Marginal rate of substitution (of X for Y)__ is the number of units of Y a consumer is willing to give up to get one more X.

MRS can be measured by the negative slope of the indifference curve. MRSXY = ‑DY/DX.

In general, MRS varies at different points of an indifferent curve. Two ways MRS can be calculated.

(1) If you know the functional form of an indifference curve, say XY=10, then you can calculate MRS at each point of the indifference curve by obtaining the slope of the indifference curve at each point.

Example. What is MRSXY if the indifference is given by XY=10?

(2) If you do not know the functional form of the indifference curve, but you do know some points on it, then your calculation can be based on these points. In the apple and orange example above, when you have 5 units of apple and 12 units of orange, your MRS of apple for orange is ‑(10‑12)/(6‑5) = 2.

MRS is usually diminishing. As more apples (X) are consumed, the consumer becomes less

willing to reduce the consumption of another good (Y) for even more apples (X).

When the consumption of one good changes alone, utility changes too. Marginal utility is the change in utility divided by an incremental change in consumption of one good alone.

MUX = DU/DX; MUY = DU/DY.

Example: If U(2,3) = 5, U(2.5,3) = 8, what is MUx at X=2?

Example: If U(X,Y) = XY, then MUX = Y, and MUY = X.

Example: If U(X,Y) = Ö(XY), then MUX = (1/2)×Ö(Y/X), and MUY = (1/2)×Ö(X/Y)

There is an important relationship between marginal rate of substitution and the ratio of MUX over MUY: MRSXY = ‑DY/DX = MUX/MUY

That is, MRSXY is equal to MUX divided by MUY. Intuition: the larger MUX compared with MUY, the more Y you are will to give up to increase the consumption of X.

Example: If MUX = 6, MUY = 3, what is the MRSXY?

Example: If U(X,Y) = XY, what is MRSXY? If U(X,Y) = Ö(XY), what is MRSXY? What do the results imply about the indifference curves for these two utility functions?

**Representation of Preferences by a Utility Function**

A consumer's preferences can be represented by a utility function if they satisfy properties P.1 through P.4, and one additional property called *continuity*. Continuity is probably the least intuitive property of preferences, yet it is not implausible.

### P.5 The "Continuity" Property

Preferences are continuous if the set of all choices that are at least as good as a choice x' and the set of all choices that are no better than x' are both closed sets. In the notation of sets, this is written as {x : x x'} and {x : x' x} are both closed.

One definition of a closed set is that any sequence of points in the set that converges, converges to a point of the set. In this context, that means that for a sequence of points {xn} with n = 1, 2, 3, ..., if x xn for every xn and if xn converges to some consumption point x', then x x'.

Figure 1 shows an example of this. In the figure, if x xn for every n, and if xn converges to x', then continuity implies that x x'.

Figure 1: Sequence of points xn that converge to x'.

**Representation Theorem**

**Representation Theorem**

If a consumer has a preference relation that is complete, reflexive, transitive, strongly monotonic, and continuous, then these preferences can be represented by a continuous utility function u(x) such that u(x) > u(x') if and only if x x'.

Proof : Let e = (1, 1, ..., 1). For each x, define u(x) by u(x) e x. Then u(x) is a utility function for the preferences if

1. such a function u(x) exists,

2. the function u(x) is unique, and

3. u(x) > u(x') if and only if x x'.

Let B = {a: a e x}. If x = (x1, x2), let y = (max{x1, x2}, max{x1, x2}). Then strong monotonicity implies that y x, so B is not empty. Let W = {a: x a e}. Then 0 is an element of W, so W is not empty. By completeness, B W = {a : a 0}. Both B and W are closed sets, from property P.5 (continuity), so B W is not empty. Therefore, there is some a such that a e x. By strong monotonicity, if a' > a, then a' e x, and if a' < a, then x a' e, so a is unique. Let u(x) = a. So u(x) exists and it is unique.

Next, we want to show that u(x) represents the preferences . Suppose that x and y are two consumption levels and u(x) = ax where a x e x. Let u(y) = a y where a y e y. If a x > a y then by monotonicty a x e a y e. By transitivity, x a x e a y e y. Finally, if x y, then a x e a y e so that a x > a y.

**Examples of Consistent Choices**

For the choice example with two commodities, when prices change, the choice of a consumer typically changes as well. The example of inconsistent choices demonstrates that not all patterns of choice are consistent with the preference properties P.1 - P.4.

Figure 2: Regions where choices are consistent or inconsistent.

In the example of inconsistent choices, both choices lie along the portion of the budget that is shown in red in figure 2. In the analysis that demonstrates that the pattern of choices xa and xb is inconsistent with the properties of preferences, the argument depends on the fact that xa is chosen when xb is available, and xb is chosen when xa is available. This can only happen when the choices lie in the red part of the budget lines in figure 7. (Strong monotonicity implies that each choice lies on the budget line rather than in the interior of the budget set, so that if the two choices were on the red portion of the budget set, there would be one on each of the two segments.) If either of the two choices lies on the blue part of the budget lines, then there is a utility function that is consistent with the choices.

There are three different ways that at least one of the choices can be on the blue portion of the budget line. Examples of utility functions that are consistent with choices are shown for each of these three cases in the figures below.

Figure 3: Utility function that rationalizes choices xa and xb.

Figure 3 shows an example of two consumption choices that are consistent with a utility function. In this example, xa is now on the blue portion of its budget set.

Figure 4: Utility function that rationalizes choices xa and xb.

Figure 4 shows another example of two consumption choices that are consistent with a utility function. In this example, xa is on the blue portion of its budget set, and xb is also on the blue portion of its budget set.

The last case is similar to the case in figure 3, but and xa is on the red part of its budget set and and xb is on the blue portion of its budget set. This case is really the same as the one in figure 8 though, with the labels and axes for xa and xb interchanged.

These examples are intended to convey the idea that utility functions can be used to generate many patterns of choices, though not all choices can come from maximization of a utility function. The next section, which describes some of the basic results from "revealed preference" theory, approaches in a more formal way this question of when observations can be rationalized by a utility function.

**Revealed Preferences (WARP, SARP, and GARP)**

Revealed preference theory provides a direct test of the utility model of preferences. A very simple application of the concepts of revealed preference was used to demonstrate that the two choices made in the example of inconsistent choices could not be produced by a consumer who chooses the consumption levels to maximize a utility function.

That example though is the simplest case of choices that are inconsistent with a utility function. The property that is violated in that example is called the Weak Axiom of Revealed Preference (WARP). Before providing the definition of WARP, some background and notation is helpful.

We say that a utility function u(x) *rationalizes *the choice x if at the given prices, the chosen level of consumption has utility that is at least as high as any other available choice. Stated formally, if prices are given by the vector pi = (p1i, p2i, ..., pni), and if the chosen consumption levels of goods 1, 2, ..., n are xi = (x1i, x2i, ..., xni), then u(xi) u(x) for any consumption level x such that pi xi pi x. When xi is chosen and pi xi pi x so that x is an available choice, we say that xi is *directly revealed preferred *to x. This is written xi R0 x.

### Weak Axiom of Revealed Preference (WARP)

WARP If xi R0 xj and xi is not equal to xj, then it is not the case that xj R0 xi.

In the example of inconsistent choices, we had both xa R0 xb and xb R0 xa. The analysis of the inconsistency in that example shows directly why these two choices are inconsistent with the existence of a utility function.

**Strong Axiom of Revealed Preference (SARP)**

**Strong Axiom of Revealed Preference (SARP)**

WARP is one implication of choices that are consistent with a utility function, but there are other implications as well. Another implaication, called the Strong Axiom of Revealed Preference (SARP) is closely related to WARP.

If x 1 R0 x 2, x 2 R0 x 3, and so on until x n - 1 R0 x n, then we say that x 1 is *revealed preferred *to x n. This is written as x 1 R x n.

The primary difference between SARP and WARP is that SARP not only rules out two choices that are both directly revealed preferred to one another, but it also rules out chains of choices that ultimately lead to two choices that are each revealed preferred to one another.

SARP If xi R xj and xi is not equal to xj, then it is not possible to have xj R xi.

In the example below, there are three different prices, and three choices. For each pair of prices and choices, there is no violation of WARP, but there is a cycle that violates SARP.

Example: The first price and associated consumption choices are p1 = (2, 3, 3) and x1 = (3, 1, 7). The second price and choices are p2 = (3, 2, 3) and x2 = (7, 3, 1), and the third pair are p3 = (3, 3, 2) and x3 = (1, 7, 3).

None of these price and consumption pairs violate WARP. This can be verified directly by observing that for the price p1, x2 is available but x3 is not available. Since x1 is not available at the price p2, there is no violation of WARP that involves x1.

When the price is p2, the consumption level x3 is affordable, but when the price is p3, x2 is not available, so there is no violation of WARP that involves x2.

The problem with this set of choices is with the cycle x1 R0 x2, x2 R0 x3, and x3 R0 x1. The combination of the first two of these show that x1 R x3, and from the third, x3 R x1, so SARP is violated but WARP is not.

This example leads to a problem similar to the example of inconsistent choices, and the demonstration that this example is inconsistent with the existence of a utility function that represents choices is similar.

**Generalized Axiom of Revealed Preference (GARP)**

**Generalized Axiom of Revealed Preference (GARP)**

SARP is very close to a characterization of the implications of utility maximization. If there is always a single consumption level that maximizes utility, then SARP does characterize all of the implications of utility maximization. The Generalized Axiom of Revealed Preference (GARP) covers the case when, for some price level, there may be more than one level of consumption that maximizes utility.

In order to state GARP, one more term is useful. If pi xi > pi x and xi is the chosen consumption level, then xi is *strictly directly revealed preferred *to x.

GARP If xi is revealed preferred to xj, then xi cannot be strictly directly revealed preferred to xi.

**The GARP Exercise**

**The GARP Exercise**

The objective of the GARP exercise is to provide students with a price for each of several commodities, and a budget. The student is then asked to select a level of consumption for each of the commodities. This choice problem is repeated several times with different prices. The resulting choices can then be analyzed to determine whether the choices are consistent with the existence of a utility function, that is, to determine whether there is any violation of GARP. If there is a violation of GARP, the analysis reports the number of violations.

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