Chapter 4 Utility

Introduction Last chapter we talk about preference, describing the ordering of what a consumer prefers. For a more convenient mathematical treatment, we turn this ordering into a mathematical function.

Utility Functions A utility function U: R+nR maps each consumption bundle of n goods into a real number that satisfies the following conditions: x’ x” U(x’) > U(x”) x’ x” U(x’) < U(x”) x’ ~ x” U(x’) = U(x”). p p

Utility Functions Not all theoretically possible preferences have a utility function representation. Technically, a preference relation that is complete, transitive and continuous has a corresponding continuous utility function.

Utility Functions Utility is an ordinal (i.e. ordering) concept. The number assigned only matters about ranking, but the sizes of numerical differences are not meaningful. For example, if U(x) = 6 and U(y) = 2, then bundle x is strictly preferred to bundle y. But x is not preferred three times as much as is y.

An Example Consider only three bundles A, B, C. The following three are all valid utility functions of the preference.

Utility Functions There is no unique utility function representation of a preference relation. Suppose U(x1,x2) = x1x2 represents a preference relation. Consider the bundles (4,1), (2,3) and (2,2). U(2,3) = 6 > U(4,1) = U(2,2) = 4. That is, (2,3) (4,1) ~ (2,2). p

Utility Functions p Define V = U2. Then V(x1,x2) = x12x22 and V(2,3) = 36 > V(4,1) = V(2,2) = 16. So again, (2,3) (4,1) ~ (2,2). V preserves the same order as U and so represents the same preferences. p

Utility Functions p Define W = 2U + 10. Then W(x1,x2) = 2x1x2+10. So, W(2,3) = 22 > W(4,1) = W(2,2) = 18. Again, (2,3) (4,1) ~ (2,2). W preserves the same order as U and V and so represents the same preferences. p

Utility Functions f ~ f ~ If U is a utility function that represents a preference relation and f is a strictly increasing function, then V = f(U) is also a utility function representing . Clearly, V(x)>V(y) if and only if f(V(x)) > f(V(y)) by definition of increasing function. ~ f ~ f

Ordinal vs. Cardinal As you will see, for our analysis of consumer choices, an ordinal utility is enough. If the numerical differences are also meaningful, we call it cardinal. For example, money, weight, height are all cardinal.

Utility Functions & Indiff. Curves An indifference curve contains equally preferred bundles. Equal preference  same utility level. Therefore, all bundles on the same indifference curve must have the same utility level.

Utility Functions & Indiff. Curves

Utility Functions & Indiff. Curves

Goods, Bads and Neutrals A good is a commodity which increases your utility (gives a more preferred bundle) when you have more of it. A bad is a commodity which decreases your utility (gives a less preferred bundle) when you have more of it. A neutral is a commodity which does not change your utility (gives an equally preferred bundle) when you have more of it.

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

Some Other Utility Functions and Their Indifference Curves Consider V(x1,x2) = x1 + x2. What do the indifference curves look like? What relation does this function represent for these two goods?

Perfect Substitutes x2 x1 + x2 = 5 x1 + x2 = 9 x1 + x2 = 13 V(x1,x2) = x1 + x2. x1 5 9 13 These two goods are perfect substitutes for this consumer.

Some Other Utility Functions and Their Indifference Curves Consider W(x1,x2) = min{x1,x2}. What do the indifference curves look like? What relation does this function represent for these two goods?

Perfect Complements x2 45o W(x1,x2) = min{x1,x2} 8 min{x1,x2} = 8 5 3 min{x1,x2} = 3 x1 3 5 8 These two goods are perfect complements for this consumer.

Perfect Substitutes and Perfect Complements In general, a utility function for perfect substitutes can be expressed as u (x, y) = ax + by; And a utility function for perfect complements can be expressed as: u (x, y) = min{ ax , by } for constants a and b.

Some Other Utility Functions and Their Indifference Curves A utility function of the form U(x1,x2) = f(x1) + x2 is linear in just x2 and is called quasi-linear. For example, U(x1,x2) = 2x11/2 + x2.

Quasi-linear Indifference Curves x2 Each curve is a vertically shifted copy of the others. x1

Some Other Utility Functions and Their Indifference Curves Any utility function of the form U(x1,x2) = x1a x2b with a > 0 and b > 0 is called a Cobb-Douglas utility function. For example, U(x1,x2) = x11/2 x21/2, (a = b = 1/2), and V(x1,x2) = x1 x23 , (a = 1, b = 3).

Cobb-Douglas Indifference Curves x2 All curves are hyperbolic, asymptoting to, but never touching any axis. x1

Cobb-Douglas Utility Functions By a monotonic transformation V=ln(U): U( x, y) = xa y b implies V( x, y) = a ln (x) + b ln(y). Consider another transformation W=U1/(a+b) W( x, y)= xa/(a+b) y b/(a+b) = xc y 1-c so that the sum of the indices becomes 1.

Marginal Utilities Marginal means “incremental”. The marginal utility of commodity i is the rate-of-change of total utility as the quantity of commodity i consumed changes; i.e.

Marginal Utilities For example, if U(x1,x2) = x11/2 x22, then

Marginal Utilities and Marginal Rates of Substitution The general equation for an indifference curve is U(x1,x2) º k, a constant. Totally differentiating this identity gives

Marginal Utilities and Marginal Rates of Substitution It can be rearranged to

Marginal Utilities and Marginal Rates of Substitution can be furthermore rearranged to This is the MRS (slope of the indifference curve).

A Note In some texts, economists refer to the MRS by its absolute value – that is, as a positive number. However, we will still follow our convention. Therefore, the MRS is

MRS and MU Recall that MRS measures how many units of good 2 you’re willing to sacrifice for one more unit of good 1 to remain the original utility level. One unit of good 1 is worth MU1. One unit of good 2 is worth MU2. Number of units of good 2 you are willing to sacrifice for one unit of good 1 is thus MU1 / MU2.

An Example Suppose U(x1,x2) = x1x2. Then so

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

MRS for Quasi-linear Utility Functions A quasi-linear utility function is of the form U(x1,x2) = f(x1) + x2. Therefore,

MRS for Quasi-linear Utility Functions MRS = - f′(x1) depends only on x1 but not on x2. So the slopes of the indifference curves for a quasi-linear utility function are constant along any line for which x1 is constant. What does that make the indifference map for a quasi-linear utility function look like?

MRS for Quasi-linear Utility Functions x2 Each curve is a vertically shifted copy of the others. MRS = -f(x1’) MRS = -f(x1”) MRS is a constant along any line for which x1 is constant. x1’ x1” x1

Monotonic Transformations & Marginal Rates of Substitution Applying a monotonic (increasing) 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 & Marginal Rates of Substitution For U(x1,x2) = x1x2 , MRS = -x2/x1. Create V = U2; i.e. V(x1,x2) = x12x22. What is the MRS for V? which is the same as the MRS for U.

Monotonic Transformations & Marginal Rates of Substitution More generally, if V = f(U) where f is a strictly increasing function, then The MRS does not change with a monotonic transformation. Thus, the same preference with different utility functions still show the same MRS.