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  But the bundle (2,3) is on the indifference curve with utility level U  6

UTILITY FUNCTIONS and INDIFFERENCE CURVES Notice you have three things you want to represent (x1, x2 and the utility – can’t do in two dimensions) U  6 U  4 (2,3) > (2,2) = (4,1) xx1 1 x2x2 x1x1 x2x2 Utility curve=6 Utility curve=4 Helps us to predict intermediate values

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 x2 are required to get a unit of x1. 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 U  6 U  4 U  2 xx11xx11 x2x2x2x2

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

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 ) = 2X 1 + 1X 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 1 1/2 x 2 1/2 (a = b = 1/2) V(x 1,x 2 ) = x 1 x 2 3 (a = 1, b = 3)

COBB DOUBLAS INDIFFERENCE CURVES x2x2 x1x1 All curves are “hyperbolic”, asymptotic to, but never touching any axis.

PERFECT SUBSITITUTES x1x1 x2x2 x 1 + x 2 = 5 x 1 + x 2 = 9 x 1 + x 2 = 13 All are linear and parallel. V(x 1,x 2 ) = x 1 + x 2

PERFECT COMPLEMENTS x2x2 x1x1 45 o min{x 1,x 2 } = min{x 1,x 2 } = 5 min{x 1,x 2 } = 3 All are right-angled with vertices/corners on a ray from the origin. W(x 1,x 2 ) = min{x 1,x 2 }