VIEWS: 3 PAGES: 39 POSTED ON: 10/8/2011 Public Domain
Utility UTILITY FUNCTIONS A preference relation that is complete, reflexive, transitive and continuous can be represented by a continuous utility function (as an alternative, or as a complement, to the indifference “map” of the previous lecture). Continuity means that small changes to a consumption bundle cause only small changes to the preference (utility) level. UTILITY FUNCTIONS A utility function U(x) represents a preference relation if and only if: p x0 x1 U(x0) > U(x1) x0 p x1 U(x0) < U(x1) x0 ~ x1 U(x0) = U(x1) UTILITY FUNCTIONS is an ordinal (i.e. ordering or Utility 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 Consider the bundles (4,1), (2,3) and (2,2). Suppose (2,3) (4,1) ~ (2,2). p 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 Anindifference 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 x2 (2,3) > (2,2) = (4,1) x2 U6 U4 x1 xx11 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. UTILITY FUNCTIONS and INDIFFERENCE CURVES x2x2 U6 U4 U2 xx1 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. UTILITY FUNCTIONS If (i) U is a utility function that represents a preference relation; and (ii) f is a strictly increasing function, then V = f(U) is also a utility function representing the original preference function. Example? V = 2.U 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). GOODS, BADS and NEUTRALS Utility Utility function Units of Units of water are water are goods bads x’ Water Around x’ units, a little extra water is a neutral. UTILITY FUNCTIONS Cobb-Douglas Utility Function U x1 , x2 x x (a 0, b 0) a b 1 2 Perfect Substitutes Utility Function U x1 , x2 ax1 bx2 Note: MRS = (-)a/b Perfect Complements Utility Function U x1 , x2 minx1, x2 Note: MRS = ? UTILITY Preferences can be represented by a utility function if the functional form has certain “nice” properties Example: Consider U(x1,x2)= x1.x2 1. u/x1>0 and u/x2>0 2. Along a particular indifference curve x1.x2 = constant x2=c/x1 As x1 x 2 i.e. downward sloping indifference curve UTILITY Example U(x1,x2)= x1.x2 =16 X1 X2 MRS 1 16 2 8 (-) 8 3 5.3 (-) 2.7 4 4 (-) 1.3 5 3.2 (-) 0.8 3. As X1 MRS (in absolute terms), i.e convex preferences COBB DOUGLAS UTILITY FUNCTION Any utility function of the form U(x1,x2) = x1a x2b with a > 0 and b > 0 is called a Cobb- Douglas utility function. Examples U(x1,x2) = x11/2 x21/2 (a = b = 1/2) V(x1,x2) = x1 x23 (a = 1, b = 3) COBB DOUBLAS INDIFFERENCE CURVES x2 All curves are “hyperbolic”, asymptoting to, but never touching any axis. x1 PERFECT SUBSITITUTES Instead of U(x1,x2) = x1x2 consider V(x1,x2) = x1 + x2. PERFECT SUBSITITUTES x2 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. PERFECT COMPLEMENTS Instead of U(x1,x2) = x1x2 or V(x1,x2) = x1 + x2, consider W(x1,x2) = min{x1,x2}. PERFECT COMPLEMENTS x2 45o 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/corners on a ray from the origin. MARGINAL UTILITY Marginal means “incremental”. The marginal utility of product i is the rate-of-change of total utility as the quantity of product i consumed changes by one unit; i.e. U MU i xi MARGINAL UTILITY U=(x1,x2) MU1=U/x1 U=MU1.x1 MU2=U/x2 U=MU2.x2 Along a particular indifference curve U = 0 = MU1(x1) + MU2(x2) x2/x1 {= MRS} = (-)MU1/MU2 MARGINAL UTILITY E.g. if U(x1,x2) = x11/2 x22 then U 1 1/ 2 2 MU 1 x1 x2 x1 2 MARGINAL UTILITY E.g. if U(x1,x2) = x11/2 x22 then U 1 1/ 2 2 MU1 x1 x2 x1 2 MARGINAL UTILITY E.g. if U(x1,x2) = x11/2 x22 then U 1/ 2 MU 2 2 x1 x2 x2 MARGINAL UTILITY E.g. if U(x1,x2) = x11/2 x22 then U 1/ 2 MU 2 2 x1 x2 x2 MARGINAL UTILITY So, if U(x1,x2) = x11/2 x22 then U 1 1 / 2 2 MU 1 x1 x 2 x1 2 U MU 2 2 x1 / 2 x 2 1 x2 MARGINAL UTLITIES AND MARGINAL RATE OF SUBISITUTION The general equation for an indifference curve is U(x1,x2) k, a constant Totally differentiating this identity gives U U dx1 dx2 0 x1 x2 MARGINAL UTLITIES AND MARGINAL RATE OF SUBISITUTION U U dx1 dx2 0 x1 x2 rearranging U U dx2 dx1 x2 x1 MARGINAL UTLITIES AND MARGINAL RATE OF SUBISITUTION U U dx2 dx1 x2 x1 rearranging d x2 U / x1 d x1 U / x2 This is the MRS. MU’s and MRS: An example Suppose U(x1,x2) = x1x2. Then U (1)( x2 ) x2 x1 U ( x1 )(1) x1 x2 d x2 U / x1 x2 so MRS d x1 U / x2 x1 MU’s and MRS: An example x2 x2 U(x1,x2) = x1x2; MRS x1 8 MRS(1,8) = - 8/1 = -8 6 MRS(6,6) = - 6/6 = -1. U = 36 U=8 1 6 x1 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? (Hopefully, nothing) MONOTONIC TRANSFORMATIONS AND MRS For U(x1,x2) = x1x2 the MRS = (-) x2/x1 Create V = U2; i.e. V(x1,x2) = x12x22 What is the MRS for V? V / x1 2 2 x1 x2 x2 MRS V / x2 2 2 x1 x2 x1 which is the same as the MRS for U. MONOTONIC TRANSFORMATIONS AND MRS More generally, if V = f(U) where f is a strictly increasing function, then V / x1 f (U ) U / x1 MRS V / x2 f '(U ) U / x2 U / x1 . U / x2 So MRS is unchanged by a positive monotonic transformation.