Preferences and Utility Chapter 3 Preferences and Utility

Consumer Behavior ● Theory of consumer behavior Description of how consumers allocate incomes among different goods and services to maximize their well-being. Consumer behavior is best understood in these distinct steps: Consumer Preferences Utility Budget Constraints Consumer Choices

Premises of Consumer Behavior Individual preferences determine the amount of pleasure people derive from the goods and services they consume. Consumers face constraints or limits on their choices. Consumers maximize their well-being or pleasure from consumption, subject to the constraints they face. Copyright © 2012 Pearson Addison-Wesley. All rights reserved.

ALTERNATIVE MARKET BASKETS Consumer Preferences Market Baskets ● Market basket (or bundle) List with specific quantities of one or more goods. ALTERNATIVE MARKET BASKETS A 20 30 B 10 50 D 40 E G H MARKET BASKET UNITS OF FOOD UNITS OF CLOTHING To explain the theory of consumer behavior, we will ask whether consumers prefer one market basket to another.

Axioms of Consumer’s Behavior 1) Completeness 2) Transitivity 3) Continuity 4) More is Better.

Axioms of Consumer’s Behavior 1) Completeness Preferences are assumed to be complete. In other words, consumers can compare and rank all possible baskets. if A and B are any two market baskets, an individual can always specify exactly one of these possibilities: A is preferred to B B is preferred to A A and B are equally attractive Note that: These preferences ignore costs conditions.

Axioms of Rational Choice 2) Transitivity Preferences are transitive. Transitivity means that if a consumer prefers basket A to basket B and basket B to basket C, then the consumer also prefers A to C

Axioms of Rational Choice 3) Continuity if A is preferred to B, then situations suitably “close to” A must also be preferred to B

Axioms of Rational Choice 4) More Is Better - all else being the same, more of a commodity is better than less of it (always wanting more is known as nonsatiation). Marginal utilities is positive for all commodities. Good - a commodity for which more is preferred to less, at least at some levels of consumption. Goods are assumed to be desirable—i.e., to be good. Bad - something for which less is preferred to more, such as pollution.

Utility Given these assumptions, it is possible to show that people are able to rank all possible situations from least desirable to most. Economists call this ranking utility if A is preferred to B, then U(A) > U(B)

Utility: Nonuniqueness of utility measures Utility rankings are ordinal in nature they record the relative desirability of commodity bundles it makes no sense to consider how much more utility is gained from A than from B It is also impossible to compare utilities between people

Utility: The ceteris paribus assumption Utility is affected by the consumption of physical commodities psychological attitudes peer group pressures personal experiences the general cultural environment Economists generally devote attention to quantifiable options while holding constant the other things that affect utility ceteris paribus assumption

Utility from consumption of goods Assume that an individual must choose among consumption goods x1, x2,…, xn We can show his rankings using a utility function of the form: utility = U(x1, x2,…, xn; other things) Often “other things” are held constant utility = U(x1, x2,…, xn) We can assume the individual is considering two goods, x and y utility = U(x,y)

Economic Goods In the utility function, the variables are assumed to be “goods” more is preferred to less Quantity of y Preferred to x*, y* ? y* Worse than x*, y* Quantity of x x*

Economic Goods using market baskets Because more of each good is preferred to less, we can compare market baskets in the shaded areas. Basket A is clearly preferred to basket G, while E is clearly preferred to A. However, A cannot be compared with B, D, or H without additional information.

AN INDIFFERENCE CURVE The indifference curve U1 that passes through market basket A shows all baskets that give the consumer the same level of satisfaction as does market basket A; these include baskets B and D. Our consumer prefers basket E, which lies above U1, to A, but prefers A to H or G, which lie below U1.

Indifference Curves An indifference curve shows a set of consumption bundles among which the individual is indifferent Quantity of y Combinations (x1, y1) and (x2, y2) provide the same level of utility y1 y2 U1 Quantity of x x1 x2

Properties of Indifference Maps Bundles on indifference curves farther from the origin are preferred to those on indifference curves closer to the origin. There is an indifference curve through every possible bundle. Indifference curves cannot cross. Indifference curves slope downward. QUESTION: Can you tell which of the assumptions of consumer preferences determine each of the 4 properties? 18

Marginal Rate of Substitution The slope of the indifference curve at any point is called the marginal rate of substitution (MRS). Its value will have a negative sign. Quantity of y y1 y2 U1 Quantity of x x1 x2

Marginal Rate of Substitution MRS changes as x and y change reflects the individual’s willingness to trade y for x Quantity of y At (x1, y1), the indifference curve is steeper. The person would be willing to give up more y to gain additional units of x At (x2, y2), the indifference curve is flatter. The person would be willing to give up less y to gain additional units of x y1 y2 U1 Quantity of x x1 x2

Marginal Rate of Substitution The Marginal Rate of Substitution Example: U = Ax2+By2; MUx=2Ax; MUy=2By (where: A and B positive) MRSx,y = MUx/MUy = 2Ax/2By = Ax/By Marginal utilities are positive (for positive x and y) Marginal utility of x increases in x; Marginal utility of y increases in y

Indifference Curve Map Each point must have an indifference curve through it Quantity of y U1 U2 U3 Increasing utility U1 < U2 < U3 Quantity of x

Transitivity Can two of an individual’s indifference curves intersect? Quantity of y The individual is indifferent between A and C. The individual is indifferent between B and C. Transitivity suggests that the individual should be indifferent between A and B But B is preferred to A because B contains more x and y than A C B U2 A U1 Quantity of x

Impossible Indifference Curves

Impossible Indifference Curves Transitivity Lisa is indifferent between e and a, and also between e and b… so by transitivity she should also be indifferent between a and b… but this is impossible, since b must be preferred to a given it has more of both goods. itos per semester r , Bur B e b I 1 a I Z , Pizzas per semester

Impossible Indifference Curves (cont.) Lisa is indifferent between b and a since both points are in the same indifference curve… But this contradicts the “more is better” assumption. Why? As, b has more of both and hence it should be preferred over a. itos per semester b r , Bur B a I Z , Pizzas per semester

Impossible Indifference Curves (cont.) Consumer is indifferent between b and a since both points are in the same indifference curve… But this contradicts the “more is better” assumption since b has more of both and hence it should be preferred over a. itos per semester r b , Bur B a I Z , Pizzas per semester

Types of Ranking: Example Students take an exam. After the exam, the students are ranked according to their performance. 1) An ordinal ranking lists the students in order of their performance (i.e., Harry did best, Joe did second best, Betty did third best, and so on). 2) A cardinal ranking gives the mark of the exam, based on an absolute marking standard (i.e., Harry got 80, Joe got 75, Betty got 74 and so on). 3) Alternatively, if the exam were graded on a curve, the marks would be an ordinal ranking.

THE MARGINAL RATE OF SUBSTITUTION The magnitude of the slope of an indifference curve measures the consumer’s marginal rate of substitution (MRS) between two goods. In this figure, the MRS between clothing (C) and food (F) falls from 6 (between A and B) to 4 (between B and D) to 2 (between D and E) to 1 (between E and G).

The Marginal Rate of Substitution ● Marginal rate of substitution (MRS) Maximum amount of a good that a consumer is willing to give up in order to obtain one additional unit of another good. CONVEXITY The decline in the MRS reflects the assumption regarding consumer preferences: a diminishing marginal rate of substitution. When the MRS diminishes along an indifference curve, the curve is convex.

MARGINAL UTLITIES AND MARGINAL RATE OF SUBISITUTION The general equation for an indifference curve is U(x1,x2) º k, a constant Totally differentiating this identity gives rearranging This is the MRS.

MU’s and MRS: An example Suppose U(x1,x2) = x1x2. Then so

MU’s and MRS: An example U(x1,x2) = x1x2; x2 8 MRS(1,8) = -(8/1) = -8 MRS(6,6) = -(6/6) = -1. 6 U = 36 U = 8 x1 1 6

UTILITY Example U(x1,x2)= x1.x2 =16 1 16 2 8 (-) 8 3 5.3 (-) 2.7 X1 X2 MRS 1 16 2 8 (-) 8 3 5.3 (-) 2.7 4 4 (-) 1.3 5 3.2 (-) 0.8 As X1  MRS  (in absolute terms), i.e convex preferences

MONOTONIC TRANSFORMATIONS AND MRS Applying a monotonic transformation to a utility function representing a preference relation simply creates another utility function representing the same preference relation. What happens to marginal rates-of-substitution when a monotonic transformation is applied?

MONOTONIC TRANSFORMATIONS AND MRS For U(x1,x2) = x1x2 the MRS = (-) x2/x1 Create V = U2; i.e. = x12x22 What is the MRS for V? which is the same as the MRS for U.

Special Functional Forms Cobb – Douglas Utility Perfect Substitutes Perfect Complements CES Utility: includes the other three as special cases Quasi linear

Special Functional Forms 1. Cobb-Douglas: U = Axy where:  +  =, > or < than 1; A, , positive constants MUX = Ax-1y Axy-1   MUY = MRSx,y = (y)/(x) “Standard” case Note: marginal rate of substitute only depends on the ratio of X and y not on the total amounts of X and y. So indifference curves for different levels of Utility look identical to each other no matter how far away from the origin they are.

Special Functional Forms y Example: Cobb-Douglas All curves are “hyperbolic”, asymptoting to, but never touching any axis. Preference Direction IC2 IC1 x

Special Functional Forms Perfect Substitutes: U = Ax + By Where: A, B positive constants   MUx = A MUy = B MRSx,y = A/B 1 unit of x is equal to B/A units of y everywhere (constant MRS).

Perfect Substitution Indifference Curves x2 The indifference curves will be linear. The MRS will be constant along the indifference curve. x1 + x2 = 5 13 x1 + x2 = 9 9 x1 + x2 = 13 5 V(x1,x2) = x1 + x2. 5 9 13 x1 All are linear and parallel.

Special Functional Forms Perfect Complements: U = A min(x,y)   where: A is a positive constant. MUx = 0 or A MUy = 0 or A MRSx,y is 0 or infinite or undefined (corner)

Perfect Complementarity Indifference Curves : Fixed Proportion of Consuming Goods W(x1,x2) = min{x1,x2} 8 min{x1,x2} = 8 5 min{x1,x2} = 5 3 min{x1,x2} = 3 3 5 8 x1 All are right-angled with vertices on a ray from the origin. Only by choosing more of the two goods together can utility be increased.

Imperfect Substitutes The standard-shaped, convex indifference curve in panel lies between these two extreme examples. Convex indifference curves show that a consumer views two goods as imperfect substitutes. B I Z

Special Functional Forms Quasi-Linear Preferences: U = v(x) + Ay Where: A is a positive constant. U(x1,x2) = f(x1) + x2 is linear in just x2 and is called quasi-linear. MUx = v’(x) = V(x)/x, where  small MUy = A E.g. U(x1,x2) = 2x11/2 + x2. "The only thing that determines your personal trade-off between x and y is how much x you already have." *can be used to "add up" utilities across individuals*

• • Special Functional Forms Example: Quasi-linear Preferences y Example: Quasi-linear Preferences (consumption of beverages) Each curve is a vertically shifted copy of the others. IC2 IC’s have same slopes on any vertical line IC1 • • x

3.2 Curvature of Indifference Curves Imperfect Substitutes Between extreme examples of perfect substitutes and perfect complements are standard-shaped, convex indifference curves. Cobb-Douglas utility function (e.g. ) indifference curves never hit the axes. Quasilinear utility function (e.g. ) indifference curves hit one of the axes.

DOUBT next slides

Examples of Utility Functions CES Utility (Constant elasticity of substitution) utility = U(x,y) = x/ + y/ when   1,   0 and utility = U(x,y) = ln x + ln y when  = 0 Perfect substitutes   = 1 Cobb-Douglas   = 0 Perfect complements   = - Doubt all three

Examples of Utility Functions For the CES utility function, the elasticity of substitution () is equal to 1/(1 - ) Perfect substitutes   =  Fixed proportions (perfect complements)   = 0