Lecture # 05

Consumer Preferences and the Concept of Utility (cont.)

Lecturer: Martin Paredes

Indifference Curves (end)The Marginal Rate of SubstitutionThe Utility FunctionMarginal Utility Some Special Functional Forms

Definition:An Indifference Curve is the set of all baskets for which the consumer is indifferent

Definition: An Indifference Map illustrates the set of indifference curves for a particular consumer

Completeness Each basket lies on only one indifference curve

Monotonicity Indifference curves have negative slopeIndifference curves are not thick

xyA

xyPreferred to AA

xyPreferred to AALess preferred

IC1xyPreferred to AALess preferred

IC1xyAB

3. Transitivity Indifference curves do not cross

4. Averages preferred to extremes Indifference curves are bowed toward the origin (convex to the origin).

xyAIC1Suppose a consumer is indifferent between A and C

Suppose that B preferred to A.

BC

xyABIC1IC2CIt cannot be the case that an IC contains both B and C Why? because, by definition of IC the consumer is:Indifferent between A & CIndifferent between B & C Hence he should be indifferent between A & B (by transitivity).=> Contradiction.

xyABIC1

xyAB(.5A, .5B)IC1

IC2xyAB(.5A, .5B)IC1

There are several ways to define the Marginal Rate of Substitution

Definition 1:It is the maximum rate at which the consumer would be willing to substitute a little more of good x for a little less of good y in order to leave the consumer just indifferent between consuming the old basket or the new basket

Definition 2: It is the negative of the slope of the indifference curve:MRSx,y = dy (for a constant level of dx preference)

An indifference curve exhibits a diminishing marginal rate of substitution:

The more of good x you have, the more you are willing to give up to get a little of good y.

The indifference curves Get flatter as we move out along the horizontal axisGet steeper as we move up along the vertical axis.

Example: The Diminishing Marginal Rate of Substitution

Definition:The utility function measures the level of satisfaction that a consumer receives from any basket of goods.

The utility function assigns a number to each basketMore preferred baskets get a higher number than less preferred baskets.

Utility is an ordinal conceptThe precise magnitude of the number that the function assigns has no significance.

Ordinal ranking gives information about the order in which a consumer ranks basketsE.g. a consumer may prefer A to B, but we cannot know how much more she likes A to BCardinal ranking gives information about the intensity of a consumers preferences.We can measure the strength of a consumers preference for A over B.

Example: Consider the result of an exam

An ordinal ranking lists the students in order of their performance E.g., Harry did best, Sean did second best, Betty did third best, and so on. A cardinal ranking gives the marks of the exam, based on an absolute marking standard E.g. Harry got 90, Sean got 85, Betty got 80, and so on.

Implications of an ordinal utility function:Difference in magnitudes of utility have no interpretation per seUtility is not comparable across individualsAny transformation of a utility function that preserves the original ranking of bundles is an equally good representation of preferences.eg. U = xyU = xy + 2U = 2xyall represent the same preferences.

10 = xyxy20525Example: Utility and a single indifference curve

Example: Utility and a single indifference curve 10 = xy20 = xyxyPreference direction20525

Definition:The marginal utility of good x is the additional utility that the consumer gets from consuming a little more of xMUx = dU dx It is is the slope of the utility function with respect to x.It assumes that the consumption of all other goods in consumers basket remain constant.

Definition:The principle of diminishing marginal utility states that the marginal utility of a good falls as consumption of that good increases.

Note: A positive marginal utility implies monotonicity.

Example: Relative Income and Life Satisfaction (within nations)Relative Income Percent > SatisfiedLowest quartile70Second quartile78Third quartile82Highest quartile85Source: Hirshleifer, Jack and D. Hirshleifer, Price Theory and Applications. Sixth Edition. Prentice Hall: Upper Saddle River, New Jersey. 1998.

We can express the MRS for any basket as a ratio of the marginal utilities of the goods in that basket

Suppose the consumer changes the level of consumption of x and y. Using differentials:dU = MUx . dx + MUy . dyAlong a particular indifference curve, dU = 0, so:0 = MUx . dx + MUy . dy

Solving for dy/dx:dy = _ MUxdx MUyBy definition, MRSx,y is the negative of the slope of the indifference curve:MRSx,y = MUx MUy

Diminishing marginal utility implies the indifference curves are convex to the origin (implies averages preferred to extremes)

Example:U= (xy)0.5MUx=y0.5/2x0.5MUy=x0.5/2y0.5

Marginal utility is positive for both goods:=> Monotonicity satisfiedDiminishing marginal utility for both goods=> Averages preferred to extremesMarginal rate of substitution:MRSx,y = MUx = y MUy xIndifference curves do not intersect the axes

Example: Graphing Indifference Curves IC1xy

IC1IC2xyPreference directionExample: Graphing Indifference Curves

Cobb-Douglas (Standard case)U = Axy where: + = 1; A, , positive constants

Properties:MUx = Ax-1y MUy = Axy-1MRSx,y = y x

Example: Cobb-DouglasIC1xy

IC1IC2xyPreference directionExample: Cobb-Douglas

2. Perfect Substitutes:U = Ax + By where: A,B are positive constants

Properties:MUx = AMUy = BMRSx,y = A(constant MRS) B

Example: Perfect Substitutes (butter and margarine) x0yIC1

x0yIC1IC2Example: Perfect Substitutes (butter and margarine)

x0yIC1IC2IC3Slope = -A/BExample: Perfect Substitutes (butter and margarine)

3. Perfect Complements:U = min {Ax,By} where: A,B are positive constants

Properties:MUx = A or 0MUy = B or 0MRSx,y = 0 or or undefined

Example: Perfect Complements (nuts and bolts)

x0yIC1

Example: Perfect Complements (nuts and bolts)

x0yIC1IC2

4. Quasi-Linear Utility Functions:U = v(x) + Aywhere: A is a positive constant, and v(0) = 0

Properties:MUx = v(x) MUy = AMRSx,y = v(x) (constant for any x)A

xy0IC1Example: Quasi-linear Preferences (consumption of beverages)

Example: Quasi-linear Preferences (consumption of beverages)ICs have same slopes on anyvertical linexy0IC2IC1

Characterization of consumer preferences without any restrictions imposed by budget

Minimal assumptions on preferences to get interesting conclusions on demandseem to be satisfied for most people. (ordinal utility function)