Rational Behavior?The Indifference CurveProperties of Indifference CurvesProperties of Indifference CurvesMarginal Rate of SubstitutionMarginal Rate of SubstitutionUtilityUtilityUtility (Cont’d)Utility FunctionsProperties of Utility FunctionsMRS, AGAINDiminishing MRSSummaryThe Budget & Consumer ChoiceConsumer’s BudgetU.S. HOUSEHOLD EXPENDITURES, 2001 Yearly after tax income: $42,362 Yearly total expenditures: $40,900 Source: U.S. Census BurOPPORTUNITY SET & BUDGET LINE BUDGET LINECONSUMER’S BUDGET AN EXAMPLE An Example (Cont’d)An Example (Cont’d)CHANGES IN PRICES AND INCOME INCOME CHANGE INCOME CHANGEPRICE CHANGE PRICE CHANGE MORE COMPLICATED BUDGETS MORE COMPLICATED BUDGETSMORE COMPLICATED BUDGETS MORE COMPLICATED BUDGETSPREFERENCES AND UTILITYFundamental Problem of Micro-Economics: satisfying unlimited wants with scarce resourcesWhat a consumer wants (preferences)=> What a consumer actually consumes (choice)What a consumer can afford (budget)The Approach: perfectly-rational consumers pursuing their self interestRATIONAL BEHAVIORTHREE KEY ASSUMPTIONSConsumer can make a decision (“completeness”)Either a preferred to b, or b preferred to a, or indifferent between the two:Consumer is consistent (“transitivity”)If a preferred to b, and b preferred to c, then a preferred to c:Consumer prefers more to less (“monotonicity”)If a has more of all goods than b, then a is preferred to b:or or ~ab ba ab;;if and , then ab bc ac;; ; if and , then ab abXXYY ab>>;;Rational Behavior?• Limitation of homo economicus– Transitivity experiments– Biases caused by “anchoring,” status quo, regret, “halo” effects• Interdependencies– One consumer’s preferences depend on how another ranks bundles, or on their consumption (bandwagon, snob effects)• Habit formation– A consumer’s preferences depend on how much of the good they consumed in the pastThe Indifference Curve• All bundles among which a consumer is indifferent– “Indifference map” is all of a consumer’s indifference curves– All bundles (Xa, Ya) such that:(Xa, Ya) ~ (X0, Y0)Properties of Indifference Curves1. Every bundle is on some indifference curve2. Two indifference curves never cross3. An indifference curve is not “thick”4. Indifference curve slopes downward4. a bundle that has more of all goods is on a higher indifference curve (“no satiation”)Properties of Indifference CurvesMarginal Rate of Substitution• Question– How much more of a good (e.g., Y) would a consumer require to compensate them for loss of a unit of another good (e.g., X)• Measurement– MRS measures willingness to make this substitution:XYMRS dY dX=−Marginal Rate of SubstitutionUtility• The holy grail for 19thcentury economists– Measure a person’s happiness in “utils”– Make comparison of different bundles, and between consumers• Modern notion of utility– Indicates relative (ordinal rankings, not strength of preferences– Each indifference curve assigned a different number, with higher indifference curves getting higher numbersUtilityUtility (Cont’d)• Utility function– Assigns a number to each bundle that represents a consumer’s preferences:– Utility number can take any value, even negative– Along an indifference curve:u(X,Y)=u0(a constant)if and only if ( ) ( )ab ua ub>;Utility Functions• Examples of utility functions– Perfect substitutes: u(X,Y) = 2X+3Y– Perfect compliments: u(X,Y) = minimum {X,Y}– Smooth, symmetric: u(X,Y) = XYProperties of Utility Functions• Invariant to an increasing transformation:v(X,Y) = f (u(X,Y))Where f is an increasing function: df/dx > 0Ex: a positive, linear transformation:A + B*u(X,Y) where B > 0• Increasing in any good (holding all others fixed): Marginal utility: MUx= du/dX = du(X,Y)/dX > 0MRS, AGAINExpressing MRS mathematically• recall definition of an indifference curve:u(X, Y) = u0(a constant) • totally differentiating, we get:dUX/ dX ∗dX + dUY/ dY∗dY = du0= 0• rearranging: MRSXY= - dY / dX= (dUX/ dX) / (dUY/ dY)= MUX/ MUYMRS, AGAINDiminishing MRS• a consumer needs less of a second good (e.g., Y) to compensate for giving up a unit of some good (e.g., X), the more of that good she has to begin with• mathematically, MRSXYdecreases with increases in X (along a given indifference curve)• indifference curve is “convex to origin” = prefer “mixtures”Examples• u(X,Y) = 2X + 3Y => MRS = MUX/MUY = 2/3• u(X,Y) = XY => MRS = MUX/MUY= Y/XSummary1. Preferences of a “rational consumer” assumed to satisfy 3 conditions: completeness, transitivity, monotonicity.2. Indifference curves summarize a consumer’s preferences, and indifference maps have certain properties if they satisfy the 3 conditions3. Utility functions represent a consumer’s preferences by assigning numbers to bundles to indicate rank, but they are unique only up to an increasing transformation.4. Marginal rate of substitution measures a consumer’s willingness to trade off between two goods, expressed as the ratio of the marginal utilities of the two goods.The Budget & Consumer ChoiceMajor Issues• Characterize the consumer’s “opportunity set” and her “budget line”• Examine how budget line changes when income and prices change• Write down and solve the consumer’s choice problem as utility maximization to budget constraintConsumer’s Budget• Money income– I is money income available to buy goods– Could include loans, credit cards, money value of assets (even “knowledge”)• Nominal prices•pX= price of X = music albums (CDs, downloads)•pY= price of Y = movies (tickets, rental, PPV)• Note that 1/price is the number of units that can be bought with $1• Expenditures•pXX + pYY = expenditure on entertainment• Money can also be saved. All I does not have to be expendedU.S. HOUSEHOLD EXPENDITURES, 2001Yearly after tax income: $42,362Yearly total expenditures: $40,900Source: U.S. Census BureauFood $5,904Housing $12,248Transportation $8,672Health care $2,239Entertainment $1,958OPPORTUNITY SET & BUDGET LINE• The “opportunity set”– all bundles that are affordable (given income and prices) pXX+ pYY<I– Both income and prices both assumed known• The “budget line”– bundles on the “frontier” of the opportunity set:PXX + pYY= I– intercepts equal maximum of a good that can be purchased – e.g., if all money spent on music (none on movies), then can buy I /
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