Chapter 3. Preferences and Utility

Transcription

1 Chapter 3 Preferences and Utilit Microeconomics stdies how individals make choices; different individals make different choices n important factor in making choices is individal s tastes or preferences How an individal forms his tastes is not an economic isse de gstibsnon est disptandm (there is no acconting for tastes), economists nevertheless assme that preferences are rational in the sense that the possess the following properties: o Completeness: If and are an two choices, the individal can alwas make a choice between the two That is, one of the following three possibilities mst be specified: is preferred to, is preferred to, and and are eqal o Transitivit: If is preferred to and is preferred to C, then it mst be is preferred to C o Monotonicit: If contains more of each commodit (good) than, then mst be preferred to o Continit: If is preferred to, then sitations sitabl close to mst also be preferred to How reasonable are these assmptions (aioms)? o The completeness assmption reqires that an individal can alwas tell apart different choices and make the decision Since it takes work and serios reflection to find ot one s own preferences, this aiom sggests that individals make onl meditated choices o Transitivit is at the heart of the concept of rationalit Withot it, mch of economic theor wold not srvive Kahneman and Tversk (984) provide a nmber of interesting eamples where the transitivit aiom is violated The framing problem, change-of-tastes, and nonrational decisions o Monotonicit seems to be the least demanding assmption as long as we are dealing with goods not bads This aiom is also Ch3 Ch3 call nonsatiation assmption hman beings desires can never be satisfied or there is no bliss point o Continit seems to be jst a technical reqirement; it is in fact ver important in establishing the eistence of tilit fnctions This assmption states that an individal s preferences cannot ehibit jmps classical eample of discontinos preference is the leicographic preference relation Consider two choices and consisting of two commodities and : = (, ) and = (, ) is preferred to if and onl if either > or = bt > C C The tilit eistence theorem: if a preference relation is complete, transitive and continos, then the preference relation can be represented b a tilit fnction () in the sense that is preferred to ( ) > ( ) Ch3 3 Ch3 4

2 The theorem is proved throgh the following graph [α(), α()] C 45 α() = () D In general, an individal s tilit fnction is a fnction of everthing he consmes That is tilit = (,, z,,all other things) In this corse, however, we limit the discssions to the case of two commodities ( and ) We are invoking the sal certeris paribs assmption Eamples of tilit fnctions o Perfect sbstittes (linear): (, ) = α + with α, > o Perfect complements (fied proportion): (, ) = min{ α, } o Cobb-Doglas tilit: (, ) α = o CES (constant elasticit of sbstittion) tilit: δ δ δ (, ) = ( α + ) with δ The properties of tilit fnction (,) o It is assmed to be differentiable to an desired degree; note this is a stronger assmption than continit o It is single valed Ch3 5 Ch3 6 (, ) (, ) o The marginal tilities and are both positive, reflecting the assmption of monotonicit o The marginal tilities, however, are diminishing, ie, (, ) (, ) < and < This propert is referred to as the law of diminishing marginal tilit o Graphicall, tilit fnction is increasing and concave in each commodit consmed Note that we are not saing that (,) is concave in (,) final assmption on (,): diminishing marginal rate of sbstittion o For an given tilit level, eqation (, ) = contains all bndles of (,) that ield the same tilit level In other words, all soltions to (, ) = are indifferent to the individal since the all generate the same amont of tilit o Graphicall, (, ) = depicts what is called the indifference crve It is negativel sloping for two goods, reflecting the aiom of monotonicit: to consme more of a commodit and maintain the level of tilit, some of the other commodit mst be given p The direction of the preference is northeast: the frther awa the indifference crve is from the origin, the higher tilit it represents Two distinct indifference crves never cross; this is a reslt of transitivit and monotonicit o We frther assme that the indifference crve is conve, ie, it bends towards the origin That also amonts to assme that the tilit fnction is qasi-concave (for or prpose, this assmption is sfficient to ensre optimal decision making) Ch3 7 Ch3 8

3 o The slope of the indifference crve reflects the term of echange between and to maintain the constanc of tilit: C d Slope = < d = = o The marginal rate of sbstittion (MRS) is defined as the absolte vale of the slope of the indifference crve, ie, d MRS = Like the echange rate, it is defined as a positive d = term o MRS indicates the qantit of to be given p for each additional nit of consmed, holding tilit constant o working formla for MRS: taking total differentiation of (, ) =, we have d = d + d =, and ths, d (, ) MRS = = d (, ) = = o The meaning of the assmption of diminishing MRS alanced consmption is better: a balanced consmption between two etreme consmptions improves tilit: Ch3 9 Ch3 consmption C is on a higher indifference crve than both and s the qantit of consmed increases along the indifference crve, less and less qantit of needs to be given p in order to acqire one more nit of Utilit fnctions, indifference crves and MRS: some eamples o Perfect sbstittes: (, ) = α + with α, > MRS = α = ; the indifference crve is a straight line o Perfect complements: (, ) = min{ α, } Its potential marginal tilities are = α and = and the can be realized onl if two goods are consmed in the proportion of = Otherwise the actal marginal tilit is α smaller or zero The indifference crve is L-shape with the verte at = MRS at the verte is (on the vertical α portion of the indifference crve, MRS = and on the horizontal portion MRS = ) This means the two goods cannot be sbstitted o Cobb-Doglas tilit: (, ) α = α α α MRS = = = ; the indifference crve is a α hperbola crve o CES (constant elasticit of sbstittion) tilit: MRS δ α = = δ δ δ (, ) = ( α + ) ; the indifference crve is also a hperbola crve Note that the CES fnction incldes all Ch3 Ch3

4 previos fnctions as special cases: δ corresponds perfect complements; δ = corresponds perfect sbstittes; and δ corresponds the Cobb-Doglas tilit fnction o Perfect sbstittes and perfect complements are two etreme cases and the are rare in realit In a sense, most tilit fnctions sch as the Cobb-Doglas are bonded between the two etremes (, ) min{ α, } = δ δ δ (, ) = ( α + ) (, ) α = (, ) = α + The law of diminishing marginal tilit vs the law of diminishing MRS o The are two independent assmptions To see this, take derivative of MRS wrt, Ch3 3 Ch3 4 d( / ) ( + d / d) ( + d / d) = = d d [ + ( / )] [ + ( / )] = + = 3 The law of diminishing MRS reqires <, bt the law of d diminishing marginal tilit onl allows < and < and so < is not garanteed t if additionall, >, then d < will be achieved The meaning of > and is it a d reasonable assmption? Of corse, the law of diminishing MRS does not ield diminishing marginal tilit o Note that the condition < amonts to reqiring d + < In mathematics, a two-variable fnction satisfing this latter condition is referred to as qasiconcave We will see in the net chapter that this qasi-concavit plas a critical role in optimal consmption decisions o Each tilit fnction can onl represent one preference relation; each preference relation, however, can be represented b different tilit fnctions In particlar, (, ) and v (, ) = h [ (, )] represent the same preference relation since v h'[ (, )] MRSv = = = = MRS In this sense, a tilit v h'[ (, )] fnction is niqe onl p to a monotonic transformation Ch3 5 Ch3 6

5 o Finall, a fnction f (, ) is homogenos of degree n if for an a>, faa (, ) = af n (, ); a fnction g (, ) is homothetic if it can be written as g (, ) = hf [ (, )] for some homogenos fnction f (, ) ll tilit fnctions we have encontered so far are homothetic and, in fact, the are homogenos of degree zero If a tilit fnction (, ) is homothetic, then its MRS is a fnction of the ratio between and consmed; it does not depends on the absolte qantities Please tr to prove this reslt if o like to have a little etra eercise Eamples of non-homothetic tilit fnctions: (, ) = + ln α and (, ) = ( ) ( ) with, > Eercises Using Ecel to graph a tpical indifference crve for the following tilit fnctions and determine whether the have conve indifference crves a (, ) = + ln b (, ) = c (, ) = d (, ) = ma{ a, a} + min{, } where < a < One wa to show conveit of indifference crves is to show that, for an two points (, ) and (, ) on an indifference crve with the tilit level, the tilit + + associated with the midpoint (, ) is at least as great as Use this approach to discss the conveit of the indifference crves for the following three fnctions a (, ) = min{, } b (, ) = ma{, } c (, ) = + Ch3 7 Ch3 8 3 Draw indifference crves for the following people or phenomenon (here the assmptions that we make ma not alwas tre; also some commodities ma be bads) a Linda sas: I likes cat bt don t care dogs that mch b nn sas: I enjo beer and pretzels, bt after beers, an additional beer makes me sick c Popcorn is addictive the more o eat, the more o want d Jeff, a 5-ear-old bo, hates spinach bt loves cand e John sas: I get no satisfaction from once of vermoth or 3 onces of gin, bt once of vermoth and 3 onces of gin reall trn me on 4 Derive marginal tilities and MRS for the following tilit fnctions and also check whether the laws of diminishing marginal tilities and diminishing MRS are satisfied a (, ) = + α b (, ) = ( ) ( ) with, > c (, ) = ( α δ + δ ) δ d (, ) = + ln Ch3 9