ECON2001 Microeconomics Lecture Notes Term 1

1 ECON2001 Lecture NotesECON2001 MicroeconomicsLecture NotesTerm 1 Ian PrestonBudget constraintConsumers purchase goodsqfrom within a budget setBof affordable bun-dles. In the standard model, pricespare constant and total spending has toremain within budgetp q ywhereyis total budget. Maximum affordablequantity of any commodity isy/piand slope qi/ qj|B= pj/piis constantand independent of total practical applications budget constraints are frequently kinked or discon-tinuous as a consequence for example of taxation or non-linear pricing. If theprice of a good rises with the quantity purchased (say because of taxation abovea threshold) then the budget set is convex whereas if it falls (say because of abulk buying discount) then the budget set is not demandsThe consumer s chosen quantities written as a function ofyandpare theMarshallianoruncompensateddemandsq=f( y, p)Consider the effects of changes inyandpon demand for, say, theith good.

2 Total budgety the path traced out by demands asyincreases is called theincomeexpansion pathwhereas the graph offi(y, p) as a function ofyiscalled theEngel curve we can summarise dependence in the total budget elasticity i=yqi qi y= lnqi lny if demand for a good rises with total budget, i>0, then we say itis anormalgood and if it falls, i<0, we say it is aninferiorgood if budget share of a good,wi=piqi/y, rises with total budget, i>1,then we say it is aluxuryorincome elasticand if it falls, i<1, wesay it is anecessityorincome inelastic own pricepi the path traced out by demands aspiincreases is called theoffercurvewhereas the graph offi(y, p) as a function ofpiis called thedemand curve1 Ian Preston, 2007 ECON2001 Lecture Notes we can summarise dependence in the (uncompensated) own priceelasticity ii=piqi qi pi= lnqi lnpi if uncompensated demand for a good rises with own price, ii>0,then we say it is aGiffengood if budget share of a good rises with price, ii> 1, then we say it isprice inelasticand if it falls, ii< 1, we say it isprice elastic other pricepj, j6=i we can summarise dependence in the (uncompensated) cross priceelasticity ij=pjqi qi pj= lnqi lnpj if uncompensated demand for a good rises with the price of another, ij>0, then we can say it is an (uncompensated)substitutewhereasif it falls with the price of another, ij<0, then we can say it is an(uncompensated) complement .

3 These are not the best definitions ofcomplementarity and substitutability however since they may not besymmetric ieqicould be a substitute forqjwhileqjis a complementforqj. A better definition, guaranteed to be symmetric, is one basedon the concept of compensated demands to be introduced upWe know that demands must lie within the budget set:p f(y, p) consumer spending exhausts the total budget then this holds as an equality,p f(y, p) =y,which is known asadding upBy adding up not all goods can be inferior not all goods can be luxuries not all goods can be necessitiesAlso certain specifications are ruled out fordemand systems. It is not possi-ble, for example, for all goods to have constant income elasticities unless theseelasticities are all 1. Otherwisepiqi=Aiy i, say, and 1 = iAi iy i 1for allbudgetsywhich is impossible unless all i= 1.

4 This does not rule out constantelasticities for individual Preston, 2007 ECON2001 Lecture NotesThere are also restrictions on price effects - for example, if price of somegood goes up then purchases of some good must be reduced so no good can bea Giffen good unless it has strong the same factor does not affect preferences or thebudget constraint so choices should not be affected either, assuming thatyandpinfluence choice only through the budget demands should therefore behomogeneousof degree zero:f( y, p) =f(y, p)for any > the consumer has a preference relation%whereqA%qBmeans qAis at least as good asqB . For the purpose of modelling demand this canbe construed as an inclination to choose the bundleqAover the bundleqB. Formodelling welfare effects the interpretation needs to be strengthened to includea link to consumer weak preference relation suffices to define strict preference and indiffer-ence if we let %and-be equivalent to , %and be equivalent to.

5 We want the preference relation to provide a basis to consistently identify aset of most preferred elements in any possible budget set and for this we needassumptions. CompletenessEitherqA%qBorqB%qA. This ensures that choice ispossible in any budget set. TransitivityqA%qBandqB%qCimpliesqA%qC. This ensures thatthere are nocyclesin preferences within any budget these ensure that the preference relation is apreference curvesFor any bundleqAdefine the weakly preferred setR(qA) as all bundlesqBsuch thatqB%qA the indifferent setI(qA) as all bundlesqBsuch thatqB qA3 Ian Preston, 2007 ECON2001 Lecture NotesTo make these sets well behaved we make the technical assumption: ContinuityIfqA%qBandqB%qCthen there is a bundle indifferent toqBon any path joiningqAtoqC.

6 This rules out discontinuous jumps is violated by the example oflexicographicpreferences. Say thatthere are two goodsq1andq2and that the consumer prefers one bundle toanother if and only if it either has more ofq1or the same amount ofq1andmore ofq2. Such preferences do not satisfy continuity and indifferent sets aresingle further assumption rules out consumers ever being fully satisfied: NonsatiationGiven any bundle there is always some direction in whichchanging the bundle will make the consumer better this is true then indifferent sets have no thick regions to them and we canvisualise them as indifference functionsA utility functionu(q) is a representation of preferences such thatqA%qBif and only ifu(qA) u(qB). A utility function exists if preferences give acontinuous functions are not however unique since ifu(q) represents certain pref-erences then any increasing transformation (u(q)) also represents the samepreferences.

7 We say that utility functions of indifference curvesWe now consider assumptions which put some actual shape on indifferencecurves. For example: MonotonicityLarger bundles are preferred to smaller monotonicity, indifference curves must slope down. This slope isknown as the marginal rate of substitution (MRS).Monotonicity corresponds to increasingness of the utility functionu(q). Anindifference curve is defined byu(q) being constant and therefore the MRS isgiven byMRS =dq2dq1 u= u/ q1 u/ q2which is obviously negative if u/ q1, u/ q2>0. Convexity qA+ (1 )qB%qBifqA%qBand 1 0. This saysthat weakly preferred sets are convex or, equivalently, MRS is Preston, 2007 ECON2001 Lecture NotesConvexity can be interpreted as capturing taste for variety.

8 It says that aconsumer will always prefer to mix any two bundles between which they areindifferent. The corresponding property of the utility function is known are said to behomotheticifqA qBimplies that qA qBforany >0. Graphically this means that higher indifference curves are magnifiedversions of lower ones from the origin. This is a strong restriction that wouldrarely be made in practice but it is useful to consider as a reference case. It is nota restriction on the shape of any one indifference curve but on the relationshipbetween indifference curves within an indifference preferences are homothetic then marginal rates of substitution are constantalong rays through the origin. This is only true for homothetic preferences andthis is usually an easy way to check whether given preferences are there exists a homogeneous utility representationu(q) whereu( q) = u(q) then preferences can be seen to be homothetic.

9 Since increasing transfor-mations preserve the properties of preferences, then any utility function whichis an increasing function of a homogeneous utility function also represents ho-mothetic is another strong restriction based on a similar idea. Prefer-ences are quasilinear if implies qA1+ qB1+ .In other words adding the same amount to one particular good preserves indif-ference. This means that higher indifference curves are parallel translations oflower ones. In this case, marginal rates of substitution are constant along linesparallel to there exists a utility representationu(q) such thatu(q1, q2, ..) =q1+F(q2, q3, ..), say, then preferences are quasilinear. This is also true of anyutility functions which are increasing transformations of functions with Examples Perfect substitutesu(q1, q2) =aq1+bq2: The MRS is a/band isconstant.

10 Indifference curves are parallel straight lines. These are theonly preferences which are homotheticandquasilinear. Perfect complementsu(q1, q2) = min[aq1, bq2]: Indifference curves areL-shaped with the kinks lying on a ray through the origin of preferences are homothetic but not Preston, 2007 ECON2001 Lecture Notes Cobb-Douglas:u(q1, q2) =alnq1+blnq2: Preferences are homothetic,indifference curves are smooth and MRSaq2/bq2is diminishingRevealed preferenceSuppose the consumer choosesqAat pricespAwhenqBwas cheaper:pA qA> pA say thatqAis (directly) revealed preferred toqB. TheWeak Axiom ofRevealed Preference (WARP)says that the consumer would never then chooseqBat pricespBwhenqAwas affordable:pB qA pB is an implication of consumer Axiom of Revealed Preference (SARP)says that there should beno cycles in revealed preference eg we should never findqArevealed preferred toqB,qBrevealed preferred toqCandqCrevealed preferred toqA(or any longercycle).