Chapter Four Utility

Utility Functions   A utility function U(x) represents a preference relation if and only if: x’ x” U(x’) > U(x”) x’ x” U(x’) < U(x”) x’  x” U(x’) = U(x”). ~   

Utility Functions u Utility is an ordinal concept; i.e. it is concerned only with order. u E.g. suppose U(x) = 6 and U(y) = 2. Then the bundle x is strictly preferred to the bundle y. It is not true that x is preferred three times as much as is y.

Utility Functions & Indiff. Curves u Consider the bundles (4,1), (2,3) and (2,2).  Suppose that (2,3) (4,1)  (2,2). u Let us assign to these bundles any numbers that preserve this preference ordering; e.g. U(2,3) = 6 and U(4,1) = U(2,2) = 4. u We will call these numbers utility levels. 

Utility Functions & Indiff. Curves u An indifference curve contains only equally preferred bundles. u Bundles are equally preferred if and only if they have the same utility level. u Therefore, all bundles in an indifference curve have the same utility level.

Utility Functions & Indiff. Curves  So the bundles (4,1) and (2,2) both lie in the indifference curve with the utility level U  while the bundle (2,3) lies in the indifference curve with the higher utility level U  6. u On an indifference map diagram, this preference information looks as follows:

Utility Functions & Indiff. Curves U  6 U  4 (2,3) (2,2)  (4,1) x1x1 x2x2 

Utility Functions & Indiff. Curves U  6 U  4 U  2 x1x1 x2x2

Utility Functions   U(x 1,x 2 ) = x 1 x 2 (2,3) (4,1)  (2,2). u Define W = 2U 

Utility Functions   U(x 1,x 2 ) = x 1 x 2 (2,3) (4,1)  (2,2). u Define W = 2U   Then W(x 1,x 2 ) = 2x 1 x so W(2,3) = 22 and W(4,1) = W(2,2) = 18. Again, (2,3) (4,1)  (2,2). u W preserves the same order as U and V and so represents the same preferences.  

Goods, Bads and Neutrals u A good is a commodity unit which increases utility (gives a more preferred bundle). u A bad is a commodity unit which decreases utility (gives a less preferred bundle). u A neutral is a commodity unit which does not change utility (gives an equally preferred bundle).

Some Utility Functions and Their Indifference Curves u Instead of U(x 1,x 2 ) = x 1 x 2 consider V(x 1,x 2 ) = x 1 + x 2. What do the indifference curves for this “perfect substitution” utility function look like?

Perfect Substitution Indifference Curves 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.

Other Utility Functions and Their Indifference Curves u Instead of U(x 1,x 2 ) = x 1 x 2 or V(x 1,x 2 ) = x 1 + x 2, consider W(x 1,x 2 ) = min{x 1,x 2 }. What do the indifference curves for this “perfect complementarity” utility function look like?

Perfect Complementarity Indifference Curves 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 on a ray from the origin. W(x 1,x 2 ) = min{x 1,x 2 }

Some Other Utility Functions and Their Indifference Curves u A utility function of the form U(x 1,x 2 ) = f(x 1 ) + x 2 is linear in just x 2 and is called quasi- linear. u E.g. U(x 1,x 2 ) = 2x 1 1/2 + x 2.

Quasi-linear Indifference Curves x2x2 x1x1 Each curve is a vertically shifted copy of the others.

Some Other Utility Functions and Their Indifference Curves u 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. u E.g. 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-Douglas Indifference Curves x2x2 x1x1 All curves are hyperbolic, asymptoting to, but never touching any axis.

Marginal Utilities u Marginal means “incremental”. u 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 u E.g. if U(x 1,x 2 ) = x 1 1/2 x 2 2 then

Marginal Utilities u E.g. if U(x 1,x 2 ) = x 1 1/2 x 2 2 then

Marginal Utilities u So, if U(x 1,x 2 ) = x 1 1/2 x 2 2 then

Marginal Utilities and Marginal Rates-of-Substitution  The general equation for an indifference curve is U(x 1,x 2 )  k, a constant. Totally differentiating this identity gives

Marginal Utilities and Marginal Rates-of-Substitution rearranged is

Marginal Utilities and Marginal Rates-of-Substitution rearranged is And This is the MRS.

Marg. Utilities & Marg. Rates-of- Substitution; An example u Suppose U(x 1,x 2 ) = x 1 x 2. Then so

Marg. Utilities & Marg. Rates-of- Substitution; An example MRS(1,8) = - 8/1 = -8 MRS(6,6) = - 6/6 = -1. x1x1 x2x U = 8 U = 36 U(x 1,x 2 ) = x 1 x 2 ;

Marg. Rates-of-Substitution for Quasi-linear Utility Functions u A quasi-linear utility function is of the form U(x 1,x 2 ) = f(x 1 ) + x 2. so

Marg. Rates-of-Substitution for Quasi-linear Utility Functions  MRS = - f  (x 1 ) does not depend upon x 2 so the slope of indifference curves for a quasi-linear utility function is constant along any line for which x 1 is constant. What does that make the indifference map for a quasi- linear utility function look like?

Marg. Rates-of-Substitution for Quasi-linear Utility Functions x2x2 x1x1 Each curve is a vertically shifted copy of the others. MRS is a constant along any line for which x 1 is constant. MRS = - f(x 1 ’) MRS = -f(x 1 ”) x1’x1’ x1”x1”

Monotonic Transformations & Marginal Rates-of-Substitution u Applying a monotonic transformation to a utility function representing a preference relation simply creates another utility function representing the same preference relation. u What happens to marginal rates-of- substitution when a monotonic transformation is applied?

Monotonic Transformations & Marginal Rates-of-Substitution u For U(x 1,x 2 ) = x 1 x 2 the MRS = - x 2 /x 1. u Create V = U 2 ; i.e. V(x 1,x 2 ) = x 1 2 x 2 2. What is the MRS for V? which is the same as the MRS for U.