UTILITY Chapter 11 What is Utility A way

UTILITY

Chapter 11 What is Utility? • A way of representing preferences • Utility is not money (but it is a useful analogy) • Typical relationship between utility & money: 2

UTILITY FUNCTIONS • A preference relation that is complete, reflexive, transitive and continuous can be represented by a continuous utility function. • Continuity means that small changes to a consumption bundle cause only small changes to the preference (utility) level.

UTILITY FUNCTIONS • Utility is an ordinal (i. e. ordering or ranking) concept. • For example, if U(x) = 6 and U(y) = 2 then bundle x is strictly preferred to bundle y. However, x is not necessarily “three times better” than y.

UTILITY FUNCTIONS and INDIFFERENCE CURVES • Suppose I have two variables I want to consider. For example, cheapness of tuition and ranking of dept (higher represents better). • Consider the bundles (4, 1), (2, 3) and (2, 2). • Suppose (2, 3) > (4, 1) ~ (2, 2). • Assign to these bundles any numbers that preserve the preference ordering; e. g. U(2, 3) = 6 > U(4, 1) = U(2, 2) = 4. • Call these numbers utility levels.

UTILITY FUNCTIONS and INDIFFERENCE CURVES • An indifference curve contains equally preferred bundles. • Equal preference same utility level. • Therefore, all bundles on an indifference curve have the same utility level.

UTILITY FUNCTIONS and INDIFFERENCE CURVES • So the bundles (4, 1) and (2, 2) are on the indifference curve with utility level U º 4 • But the bundle (2, 3) is on the indifference curve with utility level U º 6

UTILITY FUNCTIONS and INDIFFERENCE CURVES Notice you have three things you want to represent (x 1, x 2 and the utility – can’t do in two dimensions) x 2 (2, 3) > (2, 2) = (4, 1) Utility curve=6 Helps us to predict intermediate values Uº 6 Utility curve=4 Uº 4 x 1 xx 1 1

UTILITY FUNCTIONS and INDIFFERENCE CURVES • Comparing more bundles will create a larger collection of all indifference curves and a better description of the consumer’s preferences. • Marginal rate of substitution (MRS) : How many units of x 2 are required to get a unit of x 1. It is marginal, because it depends on where you are currently. • It is actually the slope of the indifference curve.

UTILITY FUNCTIONS and INDIFFERENCE CURVES x 2 x 2 Uº 6 Uº 4 Uº 2 xx 1 1

UTILITY FUNCTIONS and INDIFFERENCE CURVES • Comparing all possible consumption bundles gives the complete collection of the consumer’s indifference curves, each with its assigned utility level. • This complete collection of indifference curves completely represents the consumer’s preferences.

UTILITY FUNCTIONS and INDIFFERENCE CURVES • The collection of all indifference curves for a given preference relation is an indifference map. • An indifference map is equivalent to a utility function; each is the other.

GOODS, BADS and NEUTRALS • A good is a commodity unit which increases utility (gives a more preferred bundle). • A bad is a commodity unit which decreases utility (gives a less preferred bundle). • A neutral is a commodity unit which does not change utility (gives an equally preferred bundle). • Often it changes depending on amount.

GOODS, BADS and NEUTRALS Utility Units of water are goods x’ Utility function Units of water are bads Water Around x’ units, a little extra water is a neutral.

Come up with a utility function for these bundles • Picking best job: starting salary & job satisfaction • Picking best home: cost & square footage • Picking best restaurant: Quantity of food & cost • Picking best treatment: Time Until Results & cost

Perfect Substitutes: Example • X 1 = pints; X 2 = half-pints; • U(X 1, X 2) = 2 X 1 + 1 X 2. • U(4, 0) = U(0, 8) = 8. • MRS = -(2/1), i. e. individual willing to give up two units of X 2 (half-pints) in order to receive one more unit of X 1 (pint). • In this case the MRS is constant – it doesn’t depend on the values of X 1 or X 2

COBB DOUGLAS UTILITY FUNCTION • Any utility function of the form U(x 1, x 2) = x 1 a x 2 b with a > 0 and b > 0 is called a Cobb-Douglas utility function. • Examples U(x 1, x 2) = x 11/2 x 21/2 (a = b = 1/2) V(x 1, x 2) = x 1 x 23 (a = 1, b = 3)

COBB DOUBLAS INDIFFERENCE CURVES x 2 All curves are “hyperbolic”, asymptotic to, but never touching any axis. x 1

PERFECT SUBSITITUTES x 2 x 1 + x 2 = 5 13 x 1 + x 2 = 9 9 x 1 + x 2 = 13 5 V(x 1, x 2) = x 1 + x 2 5 9 13 x 1 All are linear and parallel.

PERFECT COMPLEMENTS x 2 8 5 3 45 o W(x 1, x 2) = min{x 1, x 2} = 8 min{x 1, x 2} = 5 min{x 1, x 2} = 3 3 5 8 x 1 All are right-angled with vertices/corners on a ray from the origin.