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Chapter 3

Preferences

Preferences (or tastes) refer to a consumer’s ability to compare or rank one commodity bundle over another. Because the idea of ranking one thing over another captures a relationship between two things, preferences are mod- eled using a mathematical concept called binary relations.1 For example, in a gathering of family members, ‘is a child of’ relates any person with her or his mother or father, should they be present in the gathering. Here ‘is a child of’ is a binary relation that relates some family members. In the realm of numbers, ‘is greater than’ (the symbol ‘>’) relates any two different num- bers; in the realm of a social network such as Facebook, ‘is a friend of’ relates some pairs of individuals on the network but not others. In consumer the- ory, we will work in the realm of the commodity space, X, comparing pairs of consumption bundles.

3.1 Binary Relations

Different commodity bundles will be represented by letters A, B, C, etc. We will denote a binary relation by ! to stand for ‘is at least as good as’ when comparing two bundles. So A ! B means ‘A is at least as good as B’ when a consumer likes bundle A as much as bundle B. From this primitive relation, we derive two other relations:

(a) ≻ (read as ‘is better than’ or ‘is preferred to’), and

(b) ∼ (read as ‘is indifferent to’). 1In the context of tastes, such a binary relation is sometimes called a preference relation.

Banerjee, Samiran. Intermediate Microeconomics : A Tool-Building Approach, Routledge, 2014. ProQuest Ebook Central, http://ebookcentral.proquest.com/lib/du/detail.action?docID=1783888. Created from du on 2020-03-20 08:09:25.

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36 Chapter 3

When we write A ≻ B meaning that the consumer prefers A over B, it is a shorthand for writing ‘A ! B and not B ! A’. Similarly, C ∼ D is a shorthand for ‘C ! D and D ! C’, meaning that the consumer is indifferent between C and D.2 In other words, the relations of strict preference and indifference between commodity bundles are derived from the primitive idea of weak preference, !. Henceforth, we will use the symbol ≻ to denote strict prefer- ence, ∼ for indifference, and ! for weak preference (which could be either indifference or strict preference).

3.2 Properties of Binary Relations

There are five properties we would like the “at-least-as-good-as” relation ! on a commodity space X to satisfy in order to capture a typical consumer’s preferences.

3.2.1 Regular preferences

We begin with three basic properties that we expect preferences to satisfy. Any binary relation satisfying the three properties P1–P3 below will be called regular.

P1 A binary relation ! is reflexive: for any commodity bundle A in X, it must be the case that A ! A.

The property of reflexivity is something of a “sanity” requirement: for any “sane” person, it seems reasonable to require that any bundle must be at least as good as itself. This is an innocuous assumption that does not restrict a person’s preferences much.

P2 A binary relation ! is total: for any two different bundles A and B in X, it must be the case that one of the following is true:

(1) A ! B, or (2) B ! A, or (3) both A ! B and B ! A.

2The symbol ≻ is called the asymmetric part of !, while ∼ is called the symmetric part of !.

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Preferences 37

A total preference relation requires that any two different bundles in X can be compared. If only (1) above were to be true (so A ! B but not the other way around), we say that A ≻ B. If only (2) held, the reverse would be true: B ≻ A. If (3) held, then we would say that A ∼ B.

Therefore, to assume that P2 holds is to claim that a consumer will be able to rank any two different bundles — say, a safari trip to Madagascar versus a skiing trip in the Alps — in one of these three ways, even when she has had no prior experience with either or has no basis for ranking them. This of course may be too much to expect in reality! If a person is unable to make such a ranking, we say her preference relation is partial: she may be able to compare some bundles and rank them, but she cannot rank all pairs of bundles.3

P3 A binary relation ! is transitive: for any three different bun- dles A, B and C in X, whenever A ! B and B ! C, it must be the case that A ! C is true.

Transitivity of a preference relation lies at the heart of our intuitive notion of rationality: we expect a “rational” consumer’s preferences to satisfy P3 which requires that if one bundle is at least as good as a second, and this second bundle is at least as good as a third, then the first must be at least as good as the third. So if a person likes a serving of icecream over a popsicle, and a popsicle over a glass of Kool-Aid, then transitive preferences imply that she must like the icecream over the Kool-Aid.4

Note that property P1 compares a bundle to itself, P2 compares two dif- ferent bundles, and P3 compares three or more bundles in a pairwise fashion.

3.2.2 Monotonicity

The fourth property that we expect preferences to satisfy is fairly intuitive and captures the idea that “goods are good”: more of a desirable commodity cannot make a consumer worse off.

3A binary relation is said to be complete if it is both reflexive and total. Some economists prefer to replace P1 and P2 with completeness instead.

4Whether people’s preferences in reality are transitive or not is an empirical issue. While it may certainly be true that both assumptions P2 and P3 are unrealistic, it is standard practice to maintain these assumptions at this level — presenting consumer theory without these assumptions is possible, but beyond the scope of an intermediate-level class.

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38 Chapter 3

x 1

x 2

4 7

Figure 3.1 Monotonicity

P4 A binary relation ! is monotonic: for any two bundles A and B in X, if A contains at least as much of each good as bundle B and more of at least one good, then weak monotonicity implies A ! B, whereas strict monotonicity implies A ≻ B.

Weak monotonicity is the idea that having more of at least one good in bundle A as compared to B should not make the consumer worse off: either the consumer is indifferent between A and B, or she prefers A over B. On the other hand, strict monotonicity means that having more of at least one good makes the consumer strictly better off, i.e., she strictly prefers the bundle A over B.

In Figure 3.1, given bundle A = (4, 3), monotonic preferences imply that any bundle to the northeast of A lying in the shaded blue area (including any bundle exactly north of A such as B = (4, 5), as well as to its east such as C = (7, 3)) leaves the consumer as well off (in the case of weak monotonic- ity) or better off (in the case of strict monotonicity). So (weak) monotonicity of preferences captures the idea that more is never worse and could actually be better. Once again, this may or may not be true in real life. For exam- ple, if good 1 is icecream and good 2 is chocolate, eating too much of either good might make a consumer sick (i.e., worse off), so more is not necessarily better.5

5Also note that when one or both commodities is a ‘bad’ instead of a good — garbage, for instance — more is not better.

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Preferences 39

Monotonicity of preferences helps to determine the direction in which preferences are increasing and also the direction in which they are decreas- ing. For instance, in Figure 3.1, any point to the northeast of A is better (or at least not worse), while every point to the southwest of A in the pink shaded area is worse (or at least not better). So with strictly monotonic preferences, a bundle that is indifferent to A must lie either in the northwest quadrant or in the southeast quadrant from A.

3.2.3 Convexity

This assumption on preferences is not obvious and is somewhat technical in nature. We introduce it at this point for the sake of completeness. It will be discussed in detail in section 3.6.3.

P5 A binary relation ! is convex: for any two bundles A and B in X where B ! A, if C is any bundle on the line segment joining bundles A and B, then C ! A.

This requirement states that if B is a bundle that is at least as good as A, then any weighted average of the commodities in bundles A and B cannot be worse than A.

3.3 Utility Representation of Preferences

Preferences as binary relations are somewhat abstract and hard to visualize. One way to make them concrete is to focus on preferences that can be repre- sented by a utility function. A utility function u attaches a number to each commodity bundle so that A ! B means that the number or ‘utility’ attached to bundle A is at least as large as the number attached to bundle B: A ! B implies u(A) ≥ u(B), and vice versa.6

If there is a utility function that represents a consumer’s preferences, then whenever this consumer ranks P ≻ Q, it must be that u(P) > u(Q) and vice versa; similarly, whenever R ∼ S, it must be that u(R) = u(S) and vice versa. In other words, a utility function maintains the same ranking between any two bundles as that given by the underlying preference relation: ≥ for !, > for ≻, and = for ∼.

6You may, for now, think of the number associated with a bundle as the level of satisfaction or “utility” from that bundle measured in some mythical units called ‘utils’. But as you will see in section 3.5, this interpretation is neither necessary, nor is it our preferred interpretation.

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40 Chapter 3

(12, 0) (16, 0)

(0, 12)

(0, 16)

u = 16-u = 12-

0 x 1

x 2

Figure 3.2 Indifference curves

To visualize what preferences look like when they can be represented by a utility function, consider the utility function u(x1, x2) = x1 + x2. Here good 1 is white eggs while good 2 is brown eggs and, at the risk of having a messy kitchen, we will assume as usual that they are both divisible. Since the utility of this consumer is the sum of the two goods, what the consumer cares about is not the color of the eggs but the total number of eggs. Suppose this consumer is baking a cake and needs a dozen eggs, i.e., her utility is fixed at ū = 12. Then the combinations of white and brown eggs that yield utility 12 constitute an indifference curve given implicitly by the equation x1 + x2 = 12 and drawn in Figure 3.2 as the line from (0, 12) to (12, 0).

An indifference curve joins all combinations of goods that give the con- sumer the same utility level. In this instance, if the consumer needs a dozen eggs to bake a cake, she does not care whether they are white or brown so long as they add up to 12. So a white egg here is a perfect 1 : 1 substitute for one brown egg, as can be inferred from the slope of the indifference curve of –1.

Note that the indifference curve that yields ū = 16 lies to the right of the first one. In fact, we can draw infinitely many indifference curves (all parallel to each other in this instance and having a slope of −1) for different levels of ū. Indifference curves that yield a higher utility lie to the northeast of the original indifference curve; similarly, those that yield a lower utility must lie to the southwest. So we attach arrows to the indifference curves to show the direction in which utility is increasing. no two indifference curves can cross

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Preferences 41

because then the bundle where the two intersect would simultaneously yield two different utility levels, which is impossible.

The family of all possible indifference curves taken together constitute the consumer’s preferences. Specifically, the family of these linear indiffer- ence curves are an instance of linear preferences that we will cover in more detail in section 3.4.1.

3.3.1 Marginal rate of substitution

We refer to the magnitude (i.e., the absolute value) of the slope of an indif- ference curve at a point as the marginal rate of substitution of good 1 for good 2 (abbreviated as MRS12 or just MRS) at that point. The marginal rate of substitution shows how much x2 a consumer is willing to give up in ex- change for one unit of x1 so as to remain on the same indifference curve. In the case of the utility function above, one brown egg can always be replaced by one white egg without affecting the utility level, so the marginal rate of substitution is a constant 1 at any point along any indifference curve.

In general, the MRS is given by the ratio of the marginal utilities:

MRS = MU1 MU2

, (3.1)

where MU1 = ∂u/∂x1 is the marginal utility of good 1 at a specific point on the indifference curve, and MU2 = ∂u/∂x2 is that for good 2 at the same point. The marginal utility of good 1 shows the additional satisfaction from increasing the consumption of good 1, keeping the consumption of good 2 fixed.7

7Equation (3.1) can be derived using some calculus. First, fix the indifference curve of interest by setting u(x1, x2) = ū, and then take its total differential (see section A.5.2 in the Mathematical appendix):

dū = ∂u ∂x1

dx1 + ∂u ∂x2

dx2.

Since we are moving along an indifference curve where the utility level remains the same, the change in the utility level dū = 0. Substituting zero on the left hand side and rearranging we get

− dx2 dx1

= ∂u/∂x1 ∂u/∂x2

where the left hand side is the negative of the slope of the indifference curve and the right hand side is the ratio of the marginal utilities in equation (3.1).

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3.4 Types of Preferences

We introduce four types of utility functions that are commonly used in eco- nomics and which show up in many applications.

3.4.1 Linear preferences

Linear preferences can be represented by the utility function

u(x1, x2) = ax1 + bx2 (3.2)

where a and b are arbitrary constants and give rise to linear indifference curves. When a and b are positive, indicating positive marginal utilities, both goods are desirable and the preferences are captured by negatively sloping linear indifference curves as shown in the left panel of Figure 3.3. Fixing the utility level at ū, a typical indifference curve has a vertical intercept of ū/b, a horizontal intercept of ū/a, and a slope of −a/b. The arrows pointing to the northeast show the direction in which utility is increasing. The marginal rate of substitution at any point on any indifference curve is a constant a/b, showing that a units of x2 need to be substituted by b units of x1 in order to remain on the same indifference curve. Preferences like these are therefore called perfect substitutes.

If both a and b are negative, both commodities are ‘bads’ and undesirable (such as garbage and nuclear waste), so utility increases in the southwest di-

0 x 1

x 2

u/b- u/b-

u/a-u/a-

– a/b

0 x 1

x 2

– a/b

bliss point

Figure 3.3 Linear preferences with goods and bads

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Preferences 43

rection, as shown by the reversed preference arrows in the right panel of Figure 3.3. The marginal rate of substitution is still a/b, so the bads are still perfect substitutes for each other. If these bads cannot be consumed in neg- ative amounts, then the origin (0, 0) yields the highest utility possible. A commodity bundle that yields a maximum utility is called a bliss point or a point of satiation.

If a > 0 but b = 0 so u(x1, x2) = ax1, then this consumer does not care about the quantity of good 2 at all. Here good 2 is called a neutral good and the indifference curves are vertical as shown in the left panel of Figure 3.4. Alternatively, if a = 0 but b > 0, then u(x1, x2) = bx2 and good 1 is now the neutral good with the indifference curves being horizontal as shown in the right panel of Figure 3.4. Once again arrows indicate the direction in which utility increases.

0 x 1

x 2

0 x 1

x 2

Figure 3.4 Linear preferences with neutrals

3.4.2 Leontief preferences

Leontief preferences refer to tastes when goods are perfect complements in consumption, i.e., they are consumed in fixed proportions, as in the case of four tires to each car, or a cup of milk to each bowl of cereal.

Suppose good 1 is cups of coffee and good 2 teaspoons of sugar. Nguyen, who is quite inflexible in how she likes her coffee, has Leontief preferences over coffee and sugar given by the utility function

u(x1, x2) = min{x1, 0.5x2}. (3.3)

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coffee

sugar

1 5

u = 1- 2

Figure 3.5 A Leontief indifference curve

If Nguyen wants to attain a utility level of ū = 1, she can reach this utility level from the combination (1, 2), which can be figured out by substitutingb· these values into equation (3.3) and taking the minimum of the two num- bers. The same utility can also be attained by the bundle (1, 5) or (1, 10): for 1 cup of coffee, any additional teaspoons of sugar beyond 2 are wasted and do not add to Nguyen’s utility. Plugging (5, 2) into the utility function also yields a utility level of 1 because the 2 teaspoons of sugar are only ade- quate for 1 cup of coffee. The remaining 4 cups do not provide her with any satisfaction because they do not have adequate amounts of sugar to go with them. Therefore, all of these points — (1, 2), (1, 5), (1, 10), and (5, 2) — lie on the same indifference curve, shown in Figure 3.5.

The shape of Nguyen’s indifference curve is the result of her preference to combine two teaspoons of sugar for every cup of coffee — no substitution between teaspoons of sugar and cups of coffee are possible, so the notion of marginal rate of substitution is meaningless in the case of perfect comple- ments. As a check of your understanding, draw another indifference curveb· for the utility level ū = 3 in Figure 3.5.

A general functional form that represents such preferences is given by the utility

u(x1, x2) = min{ax1, bx2}, (3.4)

where a and b are positive constants. Figure 3.6 shows two generic indiffer- ence curves for such a utility function. Note that the kinks of this family of indifference curves lie on a ray through the origin with slope a/b, signifying

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Preferences 45

0 x 1

x 2

a/b

Figure 3.6 Leontief preferences

that the two goods are consumed in a proportion of b units of good 1 to a units of good 2.

There is a simple algorithm for drawing indifference curves for any ‘min’- type utility function. To illustrate, suppose u(x1, x2) = min{2x1, 3x2} and we wish to draw the indifference curve for ū = 6. Follow these four steps:

1. Write 6 = min{2x1, 3x2}.

2. Solve each piece separately: from 6 = 2x1, obtain x1 = 3, and from 6 = 3x2, obtain x2 = 2.

3. Plot each piece, x1 = 3 and x2 = 2, as shown in Figure 3.7.

4. Take the ‘outer envelope’ of the lines, erasing the pieces to the south- west of (3, 2), shown with the serrated lines. The line segments that remain constitute the indifference curve for ū = 6.

You can verify that this is indeed the desired indifference curve by plug- ¶b ging in the (x1, x2) coordinates from the line segments that have not been erased to check that they yield ū = 6. Likewise, verify that any coordinate which was crossed out, say (0, 2), does not yield the desired utility level.

While the steps above are written for a specific Leontief utility function, they are easily generalized for any‘min’-type utility function.

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46 Chapter 3

u = 6-

x 1

x 2 x 1 = 3

x 2 = 2

Figure 3.7 Drawing Leontief indifference curves

3.4.3 Quasilinear preferences

Suppose a utility function is of the form

u(x1, x2) = f (x1) + x2 (3.5)

where the function f (x1) is strictly concave8 in x1 and linear in x2, hence the name quasilinear preferences.

In Figure 3.8, quasilinear indifference curves are drawn for u(x1, x2) =√ x1 + x2 which show the possibility of substitution between x1 and x2. The

marginal rate of substitution is MRS = MU1/MU2 = 1/(2 √

x1) which doesb· not depend on the level of x2. For any given level of x1, say x1 = 4, the MRS = 1/4 regardless of the value of x2 as shown by the slopes at A, B, C, and D. This means that all the indifference curves are ‘vertically parallel’: each indifference curve is essentially identical except vertically displaced. Analogously, a quasilinear utility of the form u(x1, x2) = x1 + g(x2) with g′′ < 0 has ‘horizontally parallel’ indifference curves. Thus the indifference curves for quasilinear preferences are either vertically or horizontally paral- lel.

In the case of preferences with vertically parallel indifference curves given by equation (3.5), f ′(x1) decreases as x1 increases when f (x1) is strictly con- cave. Since MRS = f ′(x1) here, as we move from left to right along an indif- ference curve and the quantity of x1 increases, there is diminishing marginal

8If the second derivative f ′′(x) < 0 for all x > 0, this guarantees that the function f is strictly concave in x. See section A.3 in the Mathematical appendix.

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Preferences 47

x 1

x 2

0 4 16

u = 20-

u = 12-

u = 8-

u = 4- A

Figure 3.8 Quasilinear preferences

rates of substitution. A similar logic also holds in the case of horizontally parallel indifference curves when u(x1, x2) = x1 + g(x2). Verify that the ¶b marginal rate of substitution now depends only on x2 and is also decreasing because g(x2) is strictly concave and the quantity of x2 decreases as we move from left to right along an indifference curve.

3.4.4 Cobb-Douglas preferences

Another type of preferences that allow for substitution possibilities are Cobb- Douglas preferences which are represented by the utility function of the form

u(x1, x2) = Axa1 x b 2, (3.6)

where a, b and A are positive constants. In Figure 3.9, we draw some of the indifference curves for the utility function u(x1, x2) = x1 x2. Note that the indifference curve for ū = 0 is the L-shaped indifference curve that coincides with the horizontal and vertical axes and has a kink at the origin. All indif- ference curves for positive levels of utility are smooth and allow continuous substitution possibilities between the two goods.

In general, MU1 = Aaxa−11 x b 2 and MU2 = Abx

a 1 x

b−1 2 , so the marginal rate ¶b

of substitution is MRS =

ax2 bx1

. (3.7)

As one moves along an indifference curve, say, from A to A′ in Figure 3.9, the MRS decreases since x2 decreases in the numerator while x1 increases in the

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48 Chapter 3

x 1

x 2

u = 121-

u = 64- u = 25-

u = 0-

Figure 3.9 Cobb-Douglas preferences

denominator. Therefore, there is diminishing marginal rate of substitution for both quasilinear and Cobb-Douglas utilities.

3.5 The Notion of Utility

Given that a consumer’s preferences can be represented by a utility function, how is one to interpret the notion of the associated ‘utility level’ of each con- sumption bundle? It turns out that the level of utility itself has no special significance so long as the preference ranking between any pairs of bundles is retained.

This more subtle but important idea can be illustrated by the following example. Let u represent a consumer’s preferences. We construct a new utility function v, where v = 2u. Suppose there are two bundles A and B such that A ! B. Then, by definition of the utility function u, it must be that u(A) ≥ u(B), which in turn implies that 2u(A) ≥ 2u(B), or v(A) ≥ v(B). Thus v retains the same ranking over A and B as u, even though the utility level under the function v is twice that of the level under u. Because both u and v represent the same preferences, both u and v will generate the same collection of indifference curves. The only difference between them is the label attached to each bundle: if the level of utility of a particular bundle is 10 under the utility u, then under v it will be 20 (twice as much) instead. But this difference is irrelevant from the viewpoint of preferences since u and v retain the same ranking between any two bundles. Therefore, the notion of

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Preferences 49

utility in economics is ordinal, not cardinal, where ordinality refers to the ranking(such as first, second, third, etc.) while cardinality refers to the level (such as 1, 2, 3, etc.).

The function v = 2u is of course not the only one that preserves the rank- ing of the utility function, u. In general, any function v = f (u) where the slope dv/du = f ′ > 0 when u > 0 will also be a utility representation of the same preferences. Such a function is called a positive monotonic transfor- mation of the utility u. Verify that the following functions are all examples of ¶b positive monotonic transformations of u and thus represent the same prefer- ences as u:

(a) v = ur for r > 0;

(b) v = ln u;

(d) v = eu.

Therefore, given the utility function u(x1, x2) = x1 + x2, the set of indif- ference curves which constitute the underlying preferences will not change if we transform this utility function to v(x1, x2) = (x1 + x2)2 or v(x1, x2) =√

x1 + x2; all of them generate the same family of linear indifference curves with slope −1.

Similarly, given a Cobb-Douglas utility u(x1, x2) = xa1 x b 2, the logarith-

mic transformation v(x1, x2) = a ln x1 + b ln x2 will generate the same set of indifference curves as the original utility function. Yet another common transformation of the Cobb-Douglas utility is v = ur where r = 1/(a + b), i.e.,

v(x1, x2) = x a

a+ b 1 x

b a+ b 2 = x

α 1 x

1−α 2

where α = a/(a + b). Therefore, the utility u(x1, x2) = x21 x2 generates the same indifference curves as the transformed utility v(x1, x2) = x

2/3 1 x

1/3 2 ; they

are two different ways of representing the same preferences. An important point to note is that given a utility function, the marginal

rate of substitution at any point on an indifference curve remains unchanged under positive monotonic transformations of the utility function. To see this, consider a utility function u(x1, x2) and fix a commodity bundle (x̄1, x̄2). At (x̄1, x̄2), the marginal rate of substitution from (3.1) can be written as

MRSu(x̄1, x̄2) = ∂u(x̄1, x̄2)

∂x1

! ∂u(x̄1, x̄2) ∂x2

, (3.8)

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50 Chapter 3

where the u-superscript on the lefthand side is a reminder that this is the MRS under the original utility u.

Now suppose v = f (u) is a positive monotonic transformation of u. The marginal rate of substitution at the same bundle under the new utility v is then given by

MRSv(x̄1, x̄2) = ∂v(x̄1, x̄2)

∂x1

! ∂v(x̄1, x̄2) ∂x2

. (3.9)

But using the Chain Rule, we can writeb·

∂v(x̄1, x̄2) ∂x1

= f ′(u) ∂u(x̄1, x̄2)

∂x1 ,

and similarly for ∂v(x̄1, x̄2)/∂x2. Therefore, equation (3.9) can be rewritten as

MRSv(x̄1, x̄2) = f ′(u)∂u(x̄1, x̄2)/∂x1 f ′(u)∂u(x̄1, x̄2)/∂x2

= MRSu(x̄1, x̄2) (3.10)

because the positive f ′(u) term in the numerator and denominator cancel out.

3.6 Utility, Preferences and Properties

When preferences can be represented by a utility function, which of the prop- erties P1–P5 of preferences are satisfied in general? What specific properties do the four types of utility functions introduced in section 3.4 have? These questions are explored in this section.

3.6.1 Regularity of preferences

If a consumer’s preferences can be represented by a utility function, then properties P1–P3 hold automatically, i.e., the preferences must be regular (reflexive, total and transitive). This is always true for any utility function, not just the ones introduced in section 3.4.

To see this, note that for any bundle A, given a utility function u, it is trivially true that u(A) = u(A), which implies that A ! A and so the prefer- ences must be reflexive.

For any two different bundles A and B, three cases are possible: either u(A) > u(B), or u(B) > u(A), or u(A) = u(B). The first case implies that A ≻ B; in the second case, B ≻ A; and in the last case, A ∼ B. This establishes that the preferences are total.

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Finally, for any three bundles A, B, and C, if A ! B and B ! C is true, then by definition u(A) ≥ u(B) and u(B) ≥ u(C). Concatenating these two inequalities, it follows that u(A) ≥ u(C), which implies that A ! C as transitivity requires.

3.6.2 Monotonicity of preferences

Monotonicity guarantees that more of one or both goods cannot make a con- sumer worse off. Strict monotonicity guarantees that more of one or both goods definitely make a consumer better off; it also rules out thick indiffer- ence curves. Thick indifference curves may arise, for example, from cogni- tive limitations since they imply that there is a range of bundles over which the consumer cannot discern a difference, for instance between a cup of cof- fee with 2 teaspoons of sugar on the one hand, and a cup of coffee with 2.5 teaspoons of sugar. Therefore she is indifferent between any bundle with one cup of coffee and amounts of sugar ranging from 2 to 2.5 teaspoons. Strict monotonicity rules out such preferences from consideration.

For the four types of preferences introduced in section 3.4, linear prefer- ences with positive values for a and b are strictly monotonic — verify that ¶b more of either good leaves the consumer on a higher indifference curve. However, if one of the goods is a neutral (as in Figure 3.4), the preferences are only guaranteed to be weakly monotonic: more of just one good is no longer sufficient to leave the consumer on a higher indifference curve.

Since Leontief preferences allow for the good to be neutral along a hor- izontal or vertical segment of an indifference curve, it follows that these preferences too are weakly monotonic, not strictly monotonic. Quasilinear preferences are strictly monotonic. Cobb-Douglas preferences, on the other hand, are strictly monotonic so long as positive amounts of both goods are consumed. For example, along the horizontal axis good 2 is not consumed, so the Cobb-Douglas utility is always zero, even if more of good 1 is con- sumed (see Figure 3.9). A similar logic holds along the vertical axis.

3.6.3 Convexity of preferences

Recall that a set is said to be convex if the line segment joining anytwo points in that set lies within the set.9 If this line lies strictly in the interior of the set,

9See section A.7.1 in the Mathematical appendix.

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A B'

x 1

x 2

(A)

x 1

x 2

(A)

Figure 3.10 Convex and strictly convex preferences

we say that the set is strictly convex; if some part of the line overlaps with the boundary, we say the set is convex.

Now suppose a consumer with Leontief preferences picks any bundle A. In Figure 3.10, draw the indifference curve that goes through A and then shade the set of bundles that are at least as good as A. The shaded set !(A) is called the weakly-better-than set of A, i.e., the set of bundles that are at least as good as A.10 Convex preferences then require that the set !(A) be a convex set. Verify that !(A) is a convex set by picking any point like Bb· where B ∼ A, or B′ where B′ ≻ A. This has to be true no matter where the initial A point happens to be.

All four types of preferences introduced in section 3.4 are convex. Linear and Leontief utilities represent convex preferences since the weakly-better- than sets have linear segments as shown in the left panel of Figure 3.10 in the case of Leontief preferences.

Quasilinear and Cobb-Douglas utilities, however, represent strictly con- vex preferences: any bundle that lies in the line segment between some point A and some other point B (so long as B ! A) must leave the consumer on a higher indifference curve. In the right panel of Figure 3.10, this is shown when preferences are Cobb-Douglas. The bundle B ∼ A, and any point like C that lies on the line segment joining them, must lie on a higher indifference curve, so C ≻ A. Similarly, with bundle B′ ≻ A, a point like C′ that lies on the line segment joining them is also preferred to A.

10The weakly-better-than set is also known as the weak upper contour set.

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There are two justifications that economists give for requiring preferences to be (strictly) convex. The first is that consumers prefer ‘combinations’ to ‘extremes’. In the right panel of Figure 3.10, suppose A is one ‘extreme’ bun- dle, consisting of 2 units of food and 6 of water, while point B = (6, 2) is another ‘extreme’ with lots of food and little water. Then strictly convex preferences guarantee that a consumer will prefer point C = (4, 4), the aver- age of the A and B bundles, over either of the extreme bundles.

The second justification is that consumers have diminishing marginal rates of substitution. In Figure 3.11, as a consumer incrementally increases her consumption of good 1 by single units in moving from A to B to C to D to E along one indifference curve, she gives up less and less amounts of good 2 (as shown by the magenta dotted heights a, b, c, and d, where a > b > c > d). Thus strictly convex preferences embody diminishing marginal rates of sub- stitution.

But why is it reasonable for a consumer to have diminishing marginal rates of substitution? The intuitive idea is that people tend to place a lower value on goods that they have in relative abundance. So at A, because she has relatively more of good 2, she is willing to give up the amount a to ac- quire one unit of good 1. But at B, while she still has relatively more of good 2, it is not as relatively abundant as before. So in moving to C, she is only willing to give up b < a units of good 2.

A B

C D

x 1

x 2

a b

c d

Figure 3.11 Diminishing marginal rate of substitution

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3.7 Special Topics◦

In this section, we present some optional material which is useful to have as a source of reference.

3.7.1 Preferences without utility representations

We began with our discussion of preferences as binary relations that satisfy P1–P5 and then went on to introduce utility functions that represent those binary relations in order to better visualize preferences as families of indif- ference curves. But can all preference relations that satisfy properties P1–P5 be represented by some utility function? The answer is no. While many preferences have utility representations, the notion of a preference relation is more general and there are many that cannot be so represented. One impor- tant class of such preferences is lexicographic preferences.

Lexicographic binary relations have a built-in hierarchy that dictates what a person cares about first, what she cares about second, and so on.11 Parents who feed their children first before they feed themselves exhibit such a hier- archy where the needs of their children come first and then their own.

To illustrate, suppose a consumer’s commodity space consists of bundles of food ( f ) and units of shelter (s) where she cares about food first and then shelter. Let !L (read as ‘is lexicographically at least as good as’) denote her lexicographic binary relation. Then given the bundles ( f1, s1) and ( f2, s2), ( f2, s2) !L ( f1, s1) means either

(a) f2 ≥ f1 regardless of the values of s1 and s2, or

(b) f2 = f1 and s2 ≥ s1.

This definition says that in comparing two bundles, this consumer first looks at the amount of food. A bundle with more food is always preferred. If two bundles have the same amount of food, then the one which provides more shelter is preferred.

In Figure 3.12, to determine whether B is lexicographically better than A, check the amount of food first. Bundle B has 6 units of food as opposed to A’s 4, therefore B !L A. Now compare bundle C to A. Both have the same

11The psychologist Abraham Maslow’s hierarchy of human needs is such an example. Ac- cording to him, the primary concern of humans is meeting physiological needs (food, water, sleep etc.), followed by security needs (shelter, employment, health, etc.), social needs (family, friendship, etc.), the need for esteem, and the need for self-actualization.

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Preferences 55

3 2 1

4 6

(A)L C

D B

Figure 3.12 Lexicographic preferences

amount of food, so check the units of shelter. Since C has 6 units of shelter as opposed to A’s 3 units, C !L A.

As an exercise, determine this consumer’s ranking between A and D, B ¶b and D, and B and C. Verify that the only bundle that is indifferent to A is A itself; any other point is either preferred to A, or A is preferred to it. Therefore there are no indifference curves here, just indifference points.

It is instructive to derive the weakly-better-than set of A for this lexico- graphic preference, shaded in Figure 3.12. The weakly-better-than set !L(A) includes the point A and all points with values of shelter greater than 3, and all bundles with more than 4 units of food.

3.7.2 Two more properties of preferences

We introduce two more properties of preferences that are somewhat tech- nical: continuity and homotheticity. While all preferences encountered in subsequent chapters will be continuous, not all will be homothetic.

Continuity

In section 3.6.3, we introduced the the weakly-better-than set. Analogously, we define a weakly-worse-than set of A, denoted by "(A), to consist of all bundles that A is at least as good as. For the preferences in the right panel of Figure 3.10 which shows the weakly-better-than set of A, we now show the weakly-worse-than set of A in Figure 3.13.

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0 x 1

x 2

(A)

Figure 3.13 The "(A) set

Before defining what we mean by continuous preferences, we need an- other definition: a set is said to be closed if it contains all of its boundary points.12

P6 A binary relation ! is continuous: for every bundle A in X, the sets !(A) and "(A) sets are closed.

Thus continuous preferences have closed weakly-worse-than and weakly- better-than sets. Continuity of preferences captures the notion that small changes in satisfaction can only result from small changes in consumption bundles.

Check that all the preferences that can be represented by utility functions considered in section 3.4 are continuous. However, the lexicographic prefer-b· ence from section 3.7.1 is not. To see this, consider the !L (A) set drawn in Figure 3.12. Here the point D = (4, 1) for instance is a boundary point of !L(A) but is not contained in !L(A). Hence, !L(A) is not a closed set and therefore the lexicographic preference is not continuous.

Homotheticity

This is a property of indifference curves, i.e., the preferences must be repre- sentable by a utility function.

12This is not meant to be a precise mathematical definition, but to convey the intuition behind the definition.

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P7 A binary relation ! is homothetic: for every bundle (x̄1, x̄2) in X, the MRS(x̄1, x̄2) = MRS(tx̄1, tx̄2) for all t > 0.

x 1

x 2

tx 2 -

tx 1 -

x 2 -

x 1 -

Figure 3.14 Homothetic preferences

Preferences are homothetic if the marginal rates of substitution along any ray through the origin are the same. This property is illustrated graphically in Figure 3.14 for Cobb-Douglas preferences. To derive this algebraically, recall from (3.7) that at the point (x̄1, x̄2),

MRS(x̄1, x̄2) = ax̄2 bx̄1

But at the point (tx̄1, tx̄2) along the ray from the origin through (x̄1, x̄2),

MRS(tx̄1, tx̄2) = atx̄2 btx̄1

= ax̄2 bx̄1

= MRS(x̄1, x̄2),

since t > 0. Linear preferences are also homothetic (the marginal rates of substitution

are all the same at any point on any indifference curve, so they must be the same along any ray through the origin in particular) but quasilinear prefer- ences are not.

Exercises

3.1. Richard is an old-fashioned Englishman who likes his tea with milk and sugar. Though he prefers more sugar to less, he cannot always

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distinguish between two cups unless the difference in the amount of sugar is less than a third (< 1/3) or more than one teaspoon (> 1). If there are two cups of tea where the difference is between a third and one teaspoon of sugar, he is indifferent (∼) between them.

(a) Is Richard’s strict preference relation (≻) reflexive, total and tran- sitive? Explain why or why not for each property.

(b) Is Richard’s indifference relation (∼) reflexive, total and transi- tive? Explain why or why not for each property.

(c) Is Richard’s weak preference relation (!) reflexive, total and tran- sitive? Explain why or why not for each property.

3.2. In Sichuan province in China, 1 black cat catches 5 mice, while 1 white cat catches 10 mice. Assuming that cats are not divisible and that a consumer only cares about how many mice are caught, draw one in- difference curve between black cats and white cats for catching 40 mice and one for 50 mice.

3.3. If the ‘parent’ binary relation ! is transitive, what does it imply about its ‘offspring’, ≻ and ∼? Prove the following results.

(a) If P ≻ Q and Q ≻ R, then P ≻ R. (b) If P ∼ Q and Q ∼ R, then P ∼ R. (c) If P ≻ Q and Q ∼ R, then P ≻ R. (d) If P ∼ Q and Q ≻ R, then P ≻ R.

(Hint: Recall that A ≻ B means both A ! B and not B ! A, while A ∼ B means both A ! B and B ! A. So in part (a), for example, to establish P ≻ R, you need to show that P ! R and not R ! P.)

3.4. Three friends, Anton, Bertil and Cecil, wish to go out for lunch to- gether. Their choices are between a pizzeria (P), a sandwich place (S), or a Chinese restaurant (C). Each person’s preference ranking over these three alternatives is regular and is given in the table below.

Anton Bertil Cecil C P S S S P P C C

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For example, Anton strictly prefers the Chinese restaurant over the sandwich place, the sandwich place over the pizzeria, and also the Chinese restaurant over the pizzeria (because his preferences are tran- sitive). From these individual rankings, we want to construct an over- all social rankingbased on pairwise comparison and majority rule. For instance, since two out of three (namely, Anton and Cecil) strictly pre- fer S over P, we say that this ‘society’ of three friends strictly prefers S over P. Derive the social strict preference ranking over the alternatives C, S, and P. Is this ranking transitive? Explain!

3.5. Professor Economicus wants a grader for her class this semester. She chooses from students who have taken her class before and looks for three qualities in a grader: speed, accuracy, and sense of humor. If student A is better than student B in two of these three qualities, she will strictly prefer A to B. She is trying to rank three students based on their qualities given in the following table.

Speed Accuracy (%) Humor Evgenievich fast 80 average

Freiherr average 95 funny Gustav slow 90 hilarious

Derive Professor Economicus’ preference ranking over these three stu- dents. Is ranking transitive? Explain why or why not.

3.6. For each of the following utility functions, draw indifference curves for different utility levels as indicated. Use arrows to show the direction in which utility is increasing. For parts (c)–(e), also draw the line(s) from the origin along which the kinks lie.

(a) u(x1, x2) = (x1 + 2x2)2 for ū = 4, 9

(b) u(x1, x2) = x2 − x1 for ū = 3, 4 (c) u(x1, x2) = min{2x1, x1 + x2} for ū = 4, 6 (d) u(x1, x2) = min{2x1 + x2, x1 + 2x2} for ū = 3, 6, 9 (e) u(x1, x2) = min{x1, x22} for ū = 1, 4, 9

(f) u(x1, x2) =

" x+ y if y< 4 4 + x if y≥ 4

for ū = 6, 8

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3.7. For each of the following utility functions defined over goods x and y, calculate the marginal rate of substitution for positive levels of both goods.

(a) u(x, y) = (x+ 2y)2

(b) u(x, y) = 2 ln x+ 3 ln y

3.8. Do any of the following functions qualify as a positive monotonic trans- formation? Explain why or why not.

(a) f (u) = −10 + 2u (b) f (u) = 10u − u2

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