©2005, Southwestern Slides by Pamela L. Hall Western Washington University Consumer Preferences...

©2005, Southwestern

Slides by Pamela L. Hall

Western Washington University

Consumer Preferences

Chapter 2

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Introduction

Consumers are interested in consuming commodities that satisfy wants Food, shelter, and clothing

• Without these commodities there would probably be no happiness

Not all commodities bring happiness Bad water, leaky roofs, smelly clothes

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Introduction

Because consumers are faced with limited resources, not all their wants can be satisfied Must choose which wants to satisfy from limited

resources Consumer preferences are central to how

consumers behave to satisfy as many wants as possible

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Aim of Chapter To investigate how consumer preferences are employed by

consumers in making their individual commodity choices Allows us to relate individual consumer preferences to society’s

overall commodity choices No central authority is required for determining overall consumer

choices• Decentralized consumer choice underlies an efficient allocation of

resources

Will investigate properties associated with a number of utility functions Main objective—to derive demand functions based on consumer

preferences, prices, and income

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Aim of Chapter

Demand How much of a commodity consumers are willing and

able to purchase at a given price• May be willing and able to purchase 5 pounds of bananas at 59¢

per pound

• Can afford to purchase more bananas, but you are not interested in purchasing more

As price of a commodity declines Quantity demanded for commodity increases

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Law of Demand

While at supermarket, you notice a special on bananas for 39¢ per pound You are now willing to purchase 7 pounds

instead of 5 pounds Figure 2.1.2 illustrates law of demand

Inverse relationship between price and quantity demanded• Increase price, quantity demanded declines

• Decrease price, quantity demanded increases

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Figure 2.1 Law of Demand

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Introduction To determine consumer demand, large firms employ

applied economists to model demand for their products For example, American Express employs hundreds of

applied economists to model demand for their credit cards • Characteristics of consumers likely to default, commit fraud, or always

pay their minimum balances

Demand functions are based on assumption Consumers attempt to maximize their satisfaction by

consuming commodities• Out of an infinite combination of commodities, they choose a

particular set of commodities that maximize satisfaction Maximization is constrained by limited resources

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Commodity Bundles and Household Preferences

Foundation of consumer theory Model of consumer or household preferences

• Each household is a unit of society attempting to maximize its happiness based on its preferences for commodities

Commodity Particular good or service delivered at a specific time and at a

specific location

Commodities consumed at different times and locations should be viewed as distinct commodities However, in practice, economic models often involve some

aggregation over time and location (space) Assumption

• Commodities being aggregated are sufficiently similar

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Problem facing a household Deciding how much of each available commodity it should consume

Household’s objective Maximize satisfaction from commodities it consumes

• Given prices of all commodities and household’s limited resources Monetary constraint (income)

For developing models of consumer behavior, economists abstract by assuming a finite number of k commodities k could be restricted to just two commodities or be unrestricted

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Level of consumption may be zero for some commodities For example, consumers generally consume zero

levels of antique buses or trips into outer space A bundle comprising all the k commodities a

household may consume is called a commodity bundle, and is represented as

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Two-Commodity Assumption

Often assume a household is faced with a choice of only two commodities (x1 and x2) These commodities could be either

• Only two commodities a household consumes or

• Only two commodities it can vary

All other commodities are fixed in terms of some given quantity

Represent the commodity bundle as

Where all commodities to right of the bar | are considered fixed

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Two-Commodity Assumption

Two-commodity assumption can be generalized by assuming one of the commodities is a composite commodity—numeraire commodity—composed of all other commodities

In a graph of two commodities, x1 and x2, a commodity bundle is a point in commodity space Set of all possible commodity bundles

A household cannot consume a negative amount of a commodity Commodity space is represented by nonnegative quadrant

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Figure 2.2 Commodity space

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Preference Relation Objective of a household

To consume commodity bundle that yields highest satisfaction it can afford

A household’s choice of preferred commodity bundle depends on Commodity prices and household’s limited income Tastes and preferences of household

• Can be summarized by the preference relation, “is preferred to or indifferent to,” written as &

Indifference between x and y means household would be just as satisfied consuming x as it would be consuming y

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Figure 2.3 Gaps of indecision in the commodity space

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Preference Ordering Preference relation provides a method for modeling how a

household orders or ranks a set of bundles From most to least desirable Often done unconsciously But with large purchases it is also done consciously

• Such as buying an automobile

Without preference ordering a household cannot determine its preferred consumption bundle

Two assumptions—preference axioms—are required to order a set of bundles These assumptions are basic axioms in consumer theory

Axiom An assumption that is generally accepted as true

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Axiom 1: Completeness If x and y are any two commodity bundles, a household can

always specify exactly one of following

Commodity bundles within an area of household indecision cannot be ordered in terms of a household’s preferences

Completeness Axiom precludes areas of indecision Assuming household members completely understand contents of

each bundle and can always make up their minds• Households generally can make up their minds within their range of

common experience

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Axiom 2: Transitivity States that a household’s preferences for alternative

commodity bundles cannot be cyclical Example

• Partying Friday night is preferred to a Saturday football game

• Saturday football game is preferred to going to church on Sunday Going to church cannot be preferred to Friday partying

Necessary for any discussion of preference maximization Without this axiom, households cannot order commodity

bundles from most to least desirable Rationality means households can ordinally rank a set of

commodity bundles to maximize their satisfaction of wants given their limited resources

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Utility Functions Political theorist Jeremy Bentham introduced a ranking of commodity

bundles Represented by a utility function (U)

Utility Ability or power of a commodity or commodity bundle to satisfy wants when

a household consumes the commodity or bundle. Utility functions

Indicate how a household ranks commodity bundles Assigns numerical value to each level of satisfaction associated with each

commodity bundle Higher the preference ranking, larger the number assigned

Household then determines which bundle maximizes this utility function Given household’s limited income and fixed commodity prices

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Utility Functions

For graphical representation we assume that only two commodities can vary Hold all other commodities constant

Utility function is represented as U = U(x1, x2|x3, . . . , xk)

Suppressing fixed commodities x3 … xk

• U = (x1, x2)

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Characteristics of Utility Functions

Utility functions and indifference curves derived from them can take on a number of shapes Depending on particular assumptions concerning a

household’s preferences Classical shape of a utility function assumes a

commodity is desirable Greater amounts of the commodity increase utility

• Stated in Axiom 3 Nonsatiation

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Figure 2.4 Utility curve for the function U(x) = ln x

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Axiom 3: Nonsatiation More of a commodity is preferred to less Household can always do a little bit better by

consuming more of a commodity Such a commodity is termed a good (or desirable)

commodity As opposed to a bad (or undesirable) commodity

• Results in a decline in utility as more of the commodity is consumed

• More of the bad commodity is not preferred to less

Garbage by definition is a bad commodity One that yields negative utility

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Axiom 3: Nonsatiation Assuming utility function U(x) is differentiable

Nonsatiation Axiom requires that all first-order partial derivatives of utility function be positive

Increasing consumption of any commodity increases utility • Holding consumption of all other commodities constant

In other words, nobody is content with their current level of any commodity

For example, assuming a household ranks commodities by utility function U = U(x1, x2, . . . , xk) then

Represents extra utility obtained from consuming slightly more of xj

• Holding amounts consumed of all other commodities constant

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Axiom 3: Nonsatiation Value of marginal utility depends on point at which partial

derivative is to be evaluated How much x1, x2, …, xk household is currently consuming Only sign of MU is important

• Actual magnitude is meaningless since utility is an ordinal ranking

Figure 2.5 illustrates result of Nonsatiation Axiom for x1 and x2

Every point or bundle within positive quadrant represents a commodity bundle

States that given an initial commodity bundle x, every commodity bundle with more of at least one commodity will be preferred to x Shaded area in Figure 2.5 represents preferred set of bundles

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Figure 2.5 Nonsatiation, more is preferred to less

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Axioms Axioms 1 and 2 provide necessary assumptions for

household preference ordering Determine indifference sets of commodity bundles

• Within each set a household receives same level of satisfaction

A household is indifferent between any two bundles within an indifference set

• For example, a household may be willing to give up a six pack of beer for a pound of candy with no change in satisfaction

Axiom 3 provides direction of increasing utility given a change in a commodity bundle

These three axioms are all that is necessary for determining utility-maximizing bundle

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Table 2.1 Preference axioms

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Indifference Curves Indifference sets are a set of curves

Each indifference curve is a locus of points (commodity bundles) that yield same level of utility

Every point along an indifference curve represents a different combination of two commodities Each combination is equally satisfactory to a household

Each combination yields same level of total utility Commodity bundles can be represented by an

indifference space• Analogous to a relief map

Contour lines or curves represent equal levels of satisfaction or utility instead of equal elevation

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Indifference Curves Figure 2.6 shows an indifference space for a household consuming two

commodities, x1 and x2

Each of the indifference curves yield different levels of utility (U1,U2, and U3)

Commodity bundles x and y are on same indifference curve• Yield same level of utility even though they represent different combinations of

commodities

• More of commodity x2 and less of commodity x1 are consumed at x than y

Commodity bundle z is on a higher indifference curve and yields a higher level of total utility

According to Nonsatiation Axiom, increasing either or both of the commodities shifts household to higher and higher indifference curves Until household approaches global bliss

• Some households will never reach global bliss In this case, there will be an infinite number of indifference curves

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Figure 2.6 Indifference space, MRS(x2 for x1)

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Indifference Curves Between any two indifference curves are a

finite number of curves Household will not be able to distinguish

between two indifference sets that are very close to each other As sets diverge distinction will become apparent

Indifference curves cannot intersect Transitivity Axiom is violated if indifference

curves intersect

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Figure 2.7 Indifference curves cannot intersect

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Marginal Rate of Substitution (MRS)

The (generally) negative slope of an indifference curve implies If a household is forced to give up some x1, it must be compensated by an

additional amount of x2 to remain indifferent

A measure for this substitution is marginal rate of substitution (MRS) Defined as negative of indifference curve slope

For the two commodities x1 and x2 in Figure 2.6

Where U = constant (dU = 0) indicates that utility is being held constant as slope changes

• Represents a movement along an indifference curve

Marginal rate of substitution (x2 for x1) is defined as

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Figure 2.8 Indifference space, MRS(x1 for x2)

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How much a household is willing to pay in terms of x2 in order to consume more of x1 is measured by MRS(x1 for x2)

Results in flipping the axes MRS is directly related to a household’s marginal utilities for

each commodity Extra utility obtainable from consuming slightly more x1, x2, … , xk is

sum of additional utility provided by each of these increments

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Taking total differential of U = U(x1, x2, …, xk) gives

Concept of MRS changes level of only two commodities (say, x1 and x2), keeping household indifferent (dU = 0) Implies that all dx’s are equal to zero except dx1 and dx2

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Rearranging terms yields an application of Implicit Function Theorem Illustrates relationship of MRS to ratio of marginal utilities

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Strictly Convex Indifference Curves

Generally, most people prefer a combination of beer and munchies Vs. all beer and no munchies or all munchies

and no beer Strictly convex indifference curves indicate

this type of trade-off Based on Axiom 4

• Diminishing Marginal Rate of Substitution

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Axiom 4: Diminishing Marginal Rate of Substitution (Strict Convexity)

Diminishing MRS exists when value of MRS(x2 for x1) approaches zero as x1 increases

In Figure 2.9, as x1 increases, slope of indifference curve tends to zero Becomes less negative

Because MRS is the negative of the slope MRS decreases toward zero as x1 increases

For very low values of x1, a household is willing to give up a larger amount of x2 to get another unit of x1

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In Figure 2.9 household prefers average bundle z to two extreme bundles x and y (a > b)

Assumes indifference curves form a convex set of commodity bundles Yield at least same level of utility represented by an indifference

curve

As x1 increases, household is willing to give up less of x2 to obtain one more unit of x1

Similarly, as x2 increases, household is willing to give up less of x1 to obtain one more unit of x2

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Figure 2.9 Indifference curves with diminishing marginal rate of substitution

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A set of points is a convex set if Any two points within set can be joined by a straight line contained

completely within the set

Can determine an implication of Diminishing MRS Axiom Suppose a household is indifferent between following

bundles

Diminishing MRS Axiom states combination bundle will be preferred

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Strict convexity is equivalent to assumption of diminishing MRS Provided linear combinations of the commodities are

possible

Implication of diminishing MRS Well-balanced, diversified bundles of commodities are

preferred to bundles that are heavily weighted toward one commodity

• Household prefers averages to extremes

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Strictly Concave Indifference Curves

If a household’s preferences were represented by strictly concave preferences Household would prefer extremes to averages

Generally, households attempt to diversify Rules out these concave preferences Some rational choices imply concave preferences

• Examples Preferring either alcohol or driving to consuming both together Preferring grades of all As instead of some combination of As and

Bs

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Figure 2.10 Indifference curves with concave preferences

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Imperfect Substitutes Indifference curves represent an individual

household’s preferences for commodities Suppose a household may prefer consuming

relatively more of commodity x2 over commodity x1

For household to be willing to give up a small amount x2

• Would have to be given a relatively large quantity of x1

• Results in relatively flat indifference curves

Imperfect substitutes are characterized by the four preference axioms Most household preferences for most commodities fall into

this category

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Figure 2.11 Indifference curves where commodity x2 is relatively more preferred to x1

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Perfect Substitutes A utility function of form U = ax1 + bx2 represents preferences

associated with perfect substitutes a and b are positive parameters

For perfect substitutes

Slope of indifference curve does not change as one commodity is substituted for another

In general, perfect substitutes are associated with MRS = constant Violates Diminishing MRS Axiom

Indifference curves are parallel straight lines with a constant slope An example of perfect substitutes may be two different brands of cola

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Figure 2.12 Perfect substitutes

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Perfect Complements Perfect complements can be represented by

nondifferentiable utility function U = min(ax1, bx2) States that whichever value (ax1 or bx2) is smaller is the

level of utility Figure 2.13 illustrates perfect complements

If 1/a units of x1 and 1/b units of x2 are employed, U = 1 If we add more units of x1, say 2/a units, and hold x2

constant at 1/b• Utility remains constant at U = 1

As x2 increases, marginal utility of x2 is zero Violates Nonsatiation Axiom

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Figure 2.13 Perfect complements

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Perfect Complements For utility to increase, consumption of both

commodities must increase proportionally No possibility of substituting one commodity for

another Commodities must be consumed in same fixed

proportion MRS is therefore undefined where ax1 = bx2

Indifference curves have a kink where ax1 = bx2

• At this kink slope is undefined

Example of perfect-complement preferences• Right and left shoe where parameters a = b

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Bad Commodities

One a household does not like Cigarettes, for example

Figure 2.14 illustrates preferences for such a commodity along with another commodity, food

For household to be willing to consume additional cigarettes and maintain same level of utility Must be given more food along with increase in

cigarettes Bad commodities violate Nonsatiation Axiom

More is not preferred to less, given that more of the bad commodity reduces utility

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Figure 2.14 Bad commodities

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Bad Commodities

A utility function representing preferences for x2 as a bad commodity is U(x1, x2) = x1x2

-2

MRS(x2 for x1) = x2/(2x1) < 0 Indicating positively sloped indifference curves

Can determine concavity of these indifference curves by setting utility equal to a constant U = a, and solving for x2

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Bad Commodities

Evaluating the derivatives, we have

As x1 increases, MRS increases toward zero

• Becomes less negative

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Neutral Preferences Neutral commodity

One a household does not derive any utility from Does not affect level of utility A household only cares about the consumption of other commodities

Example—inert ingredients in many pharmaceutical products• A household would generally only be interested in active ingredients

U = U(active ingredients)

Figure 2.15 shows indifference curves are not affected by levels of a neutral commodity consumed Only an increase in consumption of active ingredient increases utility

Any increase in inert ingredient has no effect on satisfaction Axiom 3 is violated

Averages are not preferred to extremes Violates Axiom 4

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Figure 2.15 Neutral commodity

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Satiated Preferences

Figure 2.16 illustrate satiation point (bliss point) for two commodities All household’s wants for the two commodities are

satisfied

If more than two commodities exist, satiation point is only a local satiation point (local bliss) Prior to bliss point, all four axioms hold Once in global bliss, who cares which axioms hold?

• Are no longer limited resources, so allocation is no longer a problem

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Figure 2.16 Local satiation or local bliss point

©2005, Southwestern Slides by Pamela L. Hall Western Washington University Consumer Preferences...

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