Chapter 3Axioms of Rational ChoiceSlide 3Slide 4UtilitySlide 6Slide 7Slide 8Economic GoodsIndifference CurvesMarginal Rate of SubstitutionSlide 12Indifference Curve MapTransitivityConvexitySlide 16Utility and the MRSSlide 18Marginal UtilitySlide 20Deriving the MRSDiminishing Marginal Utility and the MRSMarginal Utility and the MRSExamples of Utility FunctionsSlide 25Slide 26Slide 27Slide 28CES Utility FunctionSlide 30Slide 31Slide 32Slide 33Homothetic PreferencesNonhomothetic PreferencesMonotonic TransformationImportant Points to Note:Slide 38Slide 39Chapter 3PREFERENCES AND UTILITYCopyright ©2002 by South-Western, a division of Thomson Learning. All rights reserved.MICROECONOMIC THEORYBASIC PRINCIPLES AND EXTENSIONSEIGHTH EDITIONWALTER NICHOLSONAxioms of Rational Choice•Completeness–If A and B are any two situations, an individual can always specify exactly one of these possibilities:•A is preferred to B•B is preferred to A•A and B are equally attractiveAxioms of Rational Choice•Transitivity–If A is preferred to B, and B is preferred to C, then A is preferred to C–Assumes that the individual’s choices are internally consistentAxioms of Rational Choice•Continuity–If A is preferred to B, then situations “close to” A must also be preferred to B–Used to analyze individuals’ responses to relatively small changes in income and pricesUtility•Given these assumptions, it is possible to show that people are able to rank in order all possible situations from least desirable to most•Economists call this ranking utility–If A is preferred to B, then the utility assigned to A exceeds the utility assigned to BU(A) > U(B)Utility•Utility rankings are ordinal in nature–They record the relative desirability of commodity bundles•Because utility measures are nonunique, it makes no sense to consider how much more utility is gained from A than from B•It is also impossible to compare utilities between peopleUtility•Utility is affected by the consumption of physical commodities, psychological attitudes, peer group pressures, personal experiences, and the general cultural environment•Economists generally devote attention to quantifiable options while holding constant the other things that affect utility–ceteris paribus assumptionUtility•Assume that an individual must choose among consumption goods X1, X2,…, Xn•The individual’s rankings can be shown by a utility function of the form:utility = U(X1, X2,…, Xn)•Keep in mind that everything is being held constant except X1, X2,…, XnEconomic Goods•In the utility function, the X’s are assumed to be “goods”–more is preferred to lessQuantity of XQuantity of YX*Y*Preferred to X*, Y*WorsethanX*, Y*??Indifference Curves•An indifference curve shows a set of consumption bundles among which the individual is indifferentQuantity of XQuantity of YX1Y1Y2X2U1Combinations (X1, Y1) and (X2, Y2)provide the same level of utilityMarginal Rate of Substitution•The negative of the slope of the indifference curve at any point is called the marginal rate of substitution (MRS)Quantity of XQuantity of YX1Y1Y2X2U11UUdXdYMRSMarginal Rate of Substitution•MRS changes as X and Y change–reflects the individual’s willingness to trade Y for XQuantity of XQuantity of YX1Y1Y2X2U1At (X1, Y1), the indifference curve is steeper.The person would be willing to give up moreY to gain additional units of XAt (X2, Y2), the indifference curveis flatter. The person would bewilling to give up less Y to gainadditional units of XIndifference Curve Map•Each point must have an indifference curve through itQuantity of XQuantity of YU1U2U3U1 < U2 < U3Increasing utilityTransitivity•Can two of an individual’s indifference curves intersect?Quantity of XQuantity of YU1U2ABCThe individual is indifferent between A and C.The individual is indifferent between B and C.Transitivity suggests that the individualshould be indifferent between A and BBut B is preferred to Abecause B contains moreX and Y than AConvexity•A set of points is convex if any two points can be joined by a straight line that is contained completely within the setQuantity of XQuantity of YU1The assumption of a diminishing MRS isequivalent to the assumption that allcombinations of X and Y which are preferred to X* and Y* form a convex setX*Y*Convexity•If the indifference curve is convex, then the combination (X1 + X2)/2, (Y1 + Y2)/2 will be preferred to either (X1,Y1) or (X2,Y2)Quantity of XQuantity of YU1X2Y1Y2X1This implies that “well-balanced” bundles are preferredto bundles that are heavily weighted toward onecommodity(X1 + X2)/2(Y1 + Y2)/2Utility and the MRS•Suppose an individual’s preferences for hamburgers (Y) and soft drinks (X) can be represented byYX  10 utility •Solving for Y, we getY = 100/X• Solving for MRS = -dY/dX:MRS = -dY/dX = 100/X2Utility and the MRSMRS = -dY/dX = 100/X2•Note that as X rises, MRS falls–When X = 5, MRS = 4–When X = 20, MRS = 0.25Marginal Utility•Suppose that an individual has a utility function of the formutility = U(X1, X2,…, Xn)•We can define the marginal utility of good X1 bymarginal utility of X1 = MUX1 = U/X1 •The marginal utility is the extra utility obtained from slightly more X1 (all else constant)Marginal Utility•The total differential of U isnndXXUdXXUdXXUdU ...2211nXXXdXMUdXMUdXMUdUn ...2121•The extra utility obtainable from slightly more X1, X2,…, Xn is the sum of the additional utility provided by each of these incrementsDeriving the MRS•Suppose we change X and Y but keep utility constant (dU = 0)dU = 0 = MUXdX + MUYdY•Rearranging, we get:YUXUMUMUdXdYYX// constantU•MRS is the ratio of the marginal utility of X to the marginal utility of YDiminishing Marginal Utility and the MRS•Intuitively, it seems that the assumption of decreasing marginal utility is related to the concept of a diminishing MRS–Diminishing MRS requires that the utility function be quasi-concave•This is independent of how utility is measured–Diminishing marginal utility depends on how utility is measured•Thus, these two concepts are differentMarginal Utility and the MRS•Again, we will use the utility function5050 .. utility YXYX •The marginal utility of a soft drink ismarginal utility = MUX = U/X = 0.5X-0.5Y0.5•The marginal utility of a hamburger ismarginal utility = MUY = U/Y