Intermediate Microeconomics (22014)

I. ConsumerTheoryApplicationsIntermediate Microeconomics (22014)I. Consumer Theory ApplicationsInstructor: Marc Teignier-BaquéFirst Semester, 2011

I. ConsumerOutline Part I. Consumer Theory ApplicationsTheoryApplicationsTopic 0. ConsReviewTopic 1. Buyingand SellingTopic 2.IntertemporalChoiceTopic 3.Uncertainty1. Topic 0. Consumer Theory Review1.11.21.31.41.5Budget ConstraintsPreferencesUtility FunctionChoiceSlutsky Equation2. Topic 1. Buying and Selling3. Topic 2. Intertemporal Choice4. Topic 3. Choice under Uncertainty

I. ConsumerTheoryApplicationsTopic 0. ConsReviewBudgetConstraintsPreferencesUtility FunctionChoiceSlutsky EquationTopic 1. Buyingand SellingTopic 2.IntertemporalChoiceTopic 3.TOPIC 0. CONSUMER THEORY REVIEWBudget ConstraintsDe nitionsThe consumer'sbudget setis the set of all a ordablebundles,B (p1 , ., pn ; m) {(x1 , . . . , xn ) : x1 0, . . . , xn 0 and p1 x1 . . . pn xn m} .Thebudget constraintis the upper boundary of the budgetset.Uncertaintyx2m /p2Budget constraint: p1x1 p2x2 mx2 mp 1 x1p2p2Budget set: the collection of allaffordable bundles.BudgetdSetm /p1x1

I. ConsumerTheoryApplicationsTopic 0. ConsReviewBudgetConstraintsPreferencesUtility FunctionChoiceSlutsky EquationTopic 1. Buyingand SellingTopic 2.IntertemporalChoicePreferencesDe nitionsThe set of all bundles equally preferred to a bundle x' is theindi erence curvecontaining x'. The slope of theindi erence curve at x' is themarginal rate of substitution(MRS) at x', which is the rate at which the consumer is onlyjust willing to exchange commodity 2 for commodity 1':4x2MRS (x ) lim4x 0 4x1 0 1Topic 3.x’’x’ x” x”’z x2x’ Uncertaintyyx’’ x2z x1x”’yx1dx2dx1

I. ConsumerTheoryApplicationsTopic 0. ConsReviewBudgetConstraintsPreferencesUtility FunctionChoiceSlutsky EquationTopic 1. Buyingand SellingUtility FunctionDe nitionA Utility function U (x ) represents a preference relation ifand only ifx ' x U (x ') U (x )x ' x U (x ') U (x )x ' x U ( x ') U ( x )Topic 2.IntertemporalChoiceTopic 3.UncertaintyI The general equation for an indi erence curve isU (x1 , x2 ) k , where kis a constant. Totallydi erentiating this identity we obtain that the MRS isequal to the ratio of marginal utilities: U U Udxdx1 dx2 0 2 xU x1 x2dx1 x12

I. ConsumerTheoryApplicationsTopic 0. ConsReviewBudgetConstraintsPreferencesUtility FunctionChoiceSlutsky EquationTopic 1. Buyingand SellingTopic 2.IntertemporalChoiceTopic 3.UncertaintyChoiceDe nitionA decisionmaker chooses the most preferred a ordablebundle, which is called the consumer'sordinary demandorgross demand.I The slope of the indi erence curve at ordinary demand(x , x ) equals the slope of the budget constraint:12MRS (x1 , x2 ) p1 U / x1 p1 p2 U / x2 p2x2More preferredbundlesx2 *Affordablebundlesx1 *x1

I. ConsumerSlutsky EquationTheoryApplicationsTopic 0. Consprice change are always the sumpure substitution e ect and an income e ect .Changes to demand from aof aReviewBudgetConstraintsPreferencesUtility FunctionChoiceSlutsky EquationIPure substitution e ect :change in demand due onlyto the change in relative prices. What is the change inTopic 1. Buyingand Sellingdemand when the consumer's income is adjusted soTopic 2.that, at the new prices, she can only just buy theIntertemporalChoiceoriginal bundle? Topic 3.UncertaintyIIncome e ect:if, at the new prices, less income isneeded to buy the original bundle then real income isincreased; if more income is needed, then real income is decreased.

I. ConsumerSlutsky Equation graphicallyTheoryApplicationsx2Topic 0. ConsPure Substitution Effect OnlyReviewBudgetConstraintsPreferencesUtility FunctionChoiceSlutsky Equationx2 ’x2’’Topic 1. Buyingand Sellingx1 ’x1’’x1Topic 2.IntertemporalChoicex2Topic 3.UncertaintyAdding now the income effectx2 ’x2’’x1 ’x1’’x1

I. ConsumerTheoryApplicationsTopic 0. ConsReviewBudgetConstraintsPreferencesUtility FunctionChoiceSlutsky EquationSlutsky Equation formally(p1 , p2 ) be the initial price vector, m the level and (x1 , x2 ) the initial gross demand.I LetI De ne the Sltusky demand functionof goodiadjusted to give the consumer just enough to consumethe initial bundle(x1 , x2 ):Choicexis p10 , p20 , x1 , x2 xi p10 , p20 ; p10 x1 p20 x2 Topic 3.Uncertainty and SellingIntertemporalas the demandafter the price change when income isTopic 1. BuyingTopic 2.x1sincomeI Take the derivative with respect to xs1 p1 x1 p1 p1{zm}on both sides, x1 x1 x p1 m 1 x1s p1 {z} x1 xm 1} {zsubstitution e ect income e ect

I. ConsumerSlutsky Equation formallyTheoryApplicationsTopic 0. ConsReviewBudgetConstraintsPreferencesUtility FunctionChoiceSlutsky EquationTopic 1. Buyingand SellingTopic 2.Intertemporal x1sI The sign of the susbstitution e ect p1 is negative: thecange in demand due to the susbstitution e ect is theopposite to the change in pricep1 x1s ).(p1 x1s andI If the good is normal, the sign of the income e ect isalso negative: an increase in a price is like a decrease inChoiceincome, which leads to a decrease in demand; a price fallTopic 3.is like an income increase, which leads to an increase inUncertaintydemand. x s x1 x1 1 x1 p1 p1 m{z } {z}( )( ) {z}( )

I. ConsumerOutline Part I. Consumer Theory ApplicationsTheoryApplicationsTopic 0. ConsReviewTopic 1. Buyingand SellingEndowmentsNet DemandSlutsky EquationLabor SupplyTopic 2.IntertemporalChoiceTopic 3.Uncertainty1. Topic 0. Consumer Theory Review2. Topic 1. Buying and Selling2.12.22.32.4EndowmentsNet DemandSlutsky EquationLabor Supply3. Topic 2. Intertemporal Choice4. Topic 3. Choice under Uncertainty

I. ConsumerTOPIC 1. BUYING AND SELLINGTheoryApplicationsTopic 0. ConsReviewTopic 1. Buyingand SellingEndowmentsNet DemandSlutsky EquationLabor SupplyI So far, consumers' income taken as exogenous andindependent of prices. In reality, consumers' incomecoming from exchange by sellers and buyers.Topic 2.IntertemporalChoiceTopic 3.UncertaintyI How are incomes generated? How does the value ofincome depend upon commodity prices?I How can we put all this together to explain better howprice changes a ect demands?

I. ConsumerEndowmentsTheoryApplicationsI In this chapter, consumers get income from endowments.Topic 0. ConsReviewTopic 1. Buyingand SellingEndowmentsNet DemandSlutsky EquationLabor SupplyTopic 2.IntertemporalChoiceTopic 3.UncertaintyThis makes the budget set de nition change slightly.De nitionThe list of resource units with which a consumer starts is herendowment ,denoted byω (ω1 , ω2 ).De nitionsGiven p1 and p2 , the budget constraintwith an endowmentω (ω1 , ω2 )for a consumerisp1 x1 p2 x2 p1 ω1 p2 ω2 .and thebudget setis formally de ned as{(x1 , x2 ) x1 0, x2 0 and p1 x1 p2 x2 p1 ω1 p2 ω2 } .

I. ConsumerTheoryApplicationsEndowments and Budget SetsI Graphically, the endowment point is always on thebudget constraint.Topic 0. ConsReviewTopic 1. BuyingI Hence, price changes pivot the constraint around theendowment point.and SellingEndowmentsNet DemandSlutsky EquationLabor Supplyx2Topic 2.Intertemporalp 1 x 1 p 2 x 2 p 1 1 p 2 2ChoiceTopic 3.Budget Set before price changeUncertainty Budget Set after the price changep '1 x 1 p '2 x 2 p 1' 1 p '2 2 x1

I. ConsumerTheoryApplicationsTopic 0. ConsReviewTopic 1. Buyingand SellingEndowmentsNet DemandSlutsky EquationLabor SupplyNet demandDe nitionThe di erence between nal consumption and initialendowment of a given goodof good i .i , xi ωi , is called net demandI The sum of the values of net demands is zero:p1 x1 p2 x2 p1 ω1 p2 ω2 p1 (x1 ω1 ) p2 (x2 ω2 ) 0.Topic 2.IntertemporalChoiceTopic 3.x2x2UncertaintyAt prices (p1’,p2’) the consumersells units of good 2 to acquiremore of good 1 . The net demand ofgood 1 is, therefore, positive and thenet demanddd off goodd 2 isi negative.iAt prices (p1,p2) the consumer sellsunits of good 1 to acquire moreunits of good 2. The net demandof good 1 is, therefore,negative,and the net demand ofgoodd 2 isi positive.iix2 * x2 *p1 ( x1 1 ) p 2 ( x 2 2 ) 0p'1x1 p'2x 2 p1' 1 p'2 2 x1 *x1x1* x1

I. ConsumerTheoryApplicationsTopic 0. ConsReviewPrice o er curveDe nitionPrice-o er curvecontains all the utility-maximizing grossdemands for which the endowment is exchanged.Topic 1. Buyingand SellingEndowmentsNet DemandSlutsky EquationLabor Supplyx2Sell good 1, buy good 2Topic 2.IntertemporalChoiceTopic 3.Uncertainty Buy good 1, sell good 2 x1

I. ConsumerTheoryApplicationsSlutsky equation revisitedIn an endowment economy, the overall change in demanda pure substitutione ect , an (ordinary) income e ect , and an endowmentincome e ect.I Pure Substitution E ect : e ect of relative prices change.I Income E ect : e ect of original bundle cost change.I Endowment Income E ect: change in demand duecaused by a price change is the sum ofTopic 0. ConsReviewTopic 1. Buyingand SellingEndowmentsNet DemandSlutsky EquationLabor SupplyTopic 2.Intertemporalonly to the change in endowment value.ChoicePrice change from (p1’,p2’) to (p1”, p2’):Topic 3.x2UncertaintyPure substitution effectOrdinary income effectEndowment income effectx2 ’ 2x2 ”x1 ’ 1x1 ”x1

I. ConsumerTheoryApplicationsSlutsky Equation revisitedgood 1 andTopic 0. ConsReviewx1 (p1 , p2 ; m (p1 , p2 )) be the demand function ofm (p1 , p2 ) p1 ω1 p2 ω2 the money income.Then, the total derivative of x1 with respect to p1 isI LetITopic 1. Buyingdx1 x1 x1 ω .dp1 p1 m 1and SellingEndowmentsNet DemandSlutsky EquationLabor SupplyTopic 2.IntertemporalChoiceTopic 3.UncertaintyI Using that x1 p1dx1 dp1 x1s p xm x1 ,11 x1s p1 {z}we obtain x1 xm 1} {z x1 ω .m }1 {zsubstitution ord.-income end.-incomeI Rearranging,dx1 x1 x1 (ω1 x1 ).dp1 p1 m {z} {z}( )( )

I. ConsumerSlutsky equation revisitedTheoryApplicationsOverall change in demand of normal good (demand increasesTopic 0. ConsReviewTopic 1. Buyingand SellingEndowmentsNet DemandSlutsky EquationLabor SupplyTopic 2.IntertemporalChoiceTopic 3.with income) caused by own price change:I When income is exogenous, both the substitution and(ordinary) income e ects increase demand after anown-price fall; hence, a normal good's ordinary demandcurve slopes down (thus, Law of Downward-SlopingDemand always applies to normal goods when income isexogenous).UncertaintyI When income is given by initial endowments,endowment-income e ect decreases demand if consumersupplies that good (negative net demand); thus, if theendowment income e ect o sets the substitution andthe (ordinary) income e ects, the demand functioncould be upward-sloping!

I. ConsumerAn application: labor supplyTheoryApplicationsTopic 0. ConsReviewTopic 1. Buyingand SellingEndowmentsNet DemandSlutsky EquationLabor SupplyTopic 2.IntertemporalChoiceEnvironment description:I A worker is endowed withandRmeuros of nonlabor incomehours of time.I Consumption good's price ispc , and the wage rate is w .I Worker decides amount of consumption good, denotedbyC , and amount of leisure, denoted by R .Topic 3.UncertaintyBudget constraint:pc C m w R R mpc C wR {zwR} {z}Expenditures valueEndowment value

I. ConsumerLabor supply choiceTheoryApplicationsCTopic 0. ConsReviewTopic 1. Buyingand SellingEndowmentsNet DemandSlutsky EquationLabor Supply Budget constraint equation: wm w Rm w RpcC p cR p cC*Topic 2.Endowment pointIntertemporalChoiceTopic 3.Uncertaintym RR*leisuredemandedlaborsuppliedR

I. ConsumerTheoryApplicationsLabor supply curveE ect of a wage rate increase on amount labor supplied:I Substitution e ect: leisure relatively more expensiveTopic 0. Consdecrease leisure demanded / increase labor supplied.ReviewTopic 1. BuyingI (Ordinary) income e ect: cost original bundleand SellingEndowmentsNet DemandSlutsky EquationLabor Supplyincreases decrease leisure demanded / increase laborsupplied.I Endowment-income e ect: positive endowment incomeTopic 2.e ect because worker supplies laborIntertemporalChoiceTopic 3.Uncertainty demanded / increase labor supplied. Labor supply curve may bend backwards.decrease leisure