Consumer Theory - Utility Representation & Ordinary Demand

Transcription

1 Cosumer Theory - tlty Represetato & Ordary Demad tlty Represetato tlty Represetato, () - use fucto to represet prefereces so we ca use optmzato for proofs; fucto cotas formato about prefereces but fucto tself ("utls") mea othg Formally - fucto () represets prefereces R f () (y) R y Ordal - rags matter; orderg based o fucto must be the same as wth prefereces Not Cardal - magtudes (values) do't matter; e.g., () (y) meas othg other tha () > (y); also () - (y) meas othg other tha > or < for rag fte Number of Fuctos - f we fd oe fucto that wors there wll be a fte umber of fuctos that wor because of trasformatos tlty Trasformato - doma s R (e.g., (,,..., )), rage s a umber; trasformato s aother fucto V() F(()) that does somethg to the utlty umber (ot ) Eamples - V() a() b (lear); V() [()] ; V() [()] ; V() l () creasg Trasformato - eample of type of trasformato that s vald for utlty represetatos because t preserves the rags; two deftos (use ether): ) () > (y) F(()) > F((y)) ) dfd > (but eed to mae sure F(()) ests) Eamples - lear trasformato vald f a > ; other three trasformato lsted above vald oly f () > Vald ropertes - oly propertes that ca be obtaed from utlty represetato are those that hold for all utlty represetatos (.e., varat or ordal); these are propertes that rely o the preferece relato R (.e., rags); also called ordal property V F Frst Dervatve - >... loo at V() F(()): > for all creasg trasformatos V() (dfd > ) ay property based o frst dervatve s vald Secod Dervatve - >... loo at V() F(()): V F( ) F ( ) F( ) ; ow the secod term s postve for all creasg trasformatos, but do't ow aythg about the frst term; could be > or < ay property based o secod dervatve s ot vald Rato of Trasformatos - ths s vald; utlty trasformato does't chage rato of frst dervatves (.e., slopes of dfferece curves) F V V F Cocavty - f s cocave, that meas level curves are creasg at a decreasg rate, but we oly care that they are creasg; rate does't matter; ca have a trasformato that eeps level curves creasg, but has them at a creasg rate (.e., cove); ca 3; V, ; V ; V of

2 chec hessas; f H s egatve defte, do't have to have H V be egatve defte cocavty s ot vald Quascocavty - cocavty shows the relatoshp betwee level cures (.e., how to label them), but quascocavty shows the shape of the level curves (.e., set of pots above the level curve s cove); so QC oly depeds o prefereces so propertes based o QC are vald (proved ths the log way HW) () (y) ( ( - )y) (y).e., R y ( ( - )y) R y dfferece Curves dfferece Map - set of dfferece curves dfferece Curve - C() {y: y }; useful to llustrate argumets ad results Defto sg refereces - C() boudary of R () Defto sg tlty Represetato - C() {: () costat} ropertes - to be equvalet to prefereces wth the fve propertes must have: a C goes through every budle ad Cs are o-tersectg, dowward slopg, cotuous, "th" les that are cove to the org roof (of o-tersectg): Assume two Cs tersect as show pcture y ad z By trastvty of prefereces y z z But by mootocty of prefereces y z because y has more of both goods C caot tersect y Ordary Demads Budget Set, B(,) - specfc class of alteratve set A; B(,) {: ad } amout spet o good ; come;... ropertes - Cove - B(,) ad y B(,) ( - )y B(,) Closed - boudares are cluded because of ad defto Bouded - f < ad > (or demad < supply at ) Demad Fucto, o (,) - o (,) C(B(,),R); t's a fucto f prefereces are strctly cove (.e., C(B(,),R) s uque budle), otherwse t's called a demad correspodece because there ca be multple budles () for each prce Observable - we care about demad because t ca be observed (ule prefereces) ropertes - 5 propertes mae demad fucto equvalet to "stadard cosumer". Complete - o (,) defed for all > ad < < Assumg s fed (for ow) ad depedet of. "Sort of" Cotuous - f prefereces (R) are strctly cove, the o (,) s cotuous Small chage budget le from w strctly cove pref. does't chage optmal soluto much D(,) s cotuous D D of

3 3. Addg p roperty - f prefereces (R) are strctly cove, the o (,) mplcato - demads ot urelated; f you ow frst - demads, the last demad s gve by solvg formula above for o (,): o (, ) o o (, ) (, ) Secod mplcato - lmts o demad: o (,) 4. Homogeeous of Degree rces ad come - proportoal chages prces ad come have o effect o demad; o (t,t) o (,) roof: B(,) ; B(t,t) t t t t B(,) B(t,t) 5. Cove - f prefereces (R) are cove, the o (,) s cove set; f R s strctly cove, the o (,) s sgle budle Other ropertes - there are others (e.g., dowward slopg); come from comparatve statcs (hard way) or through dualty optmzato Comparatve Statcs - start wth choce set based o prefereces: C(B(,),R) whch ca be wrtte as optmzato of utlty represetato: Ma () s.t., ad Lagraga - L () - ( - ) K-T Codtos - gore frst order codtos for goods wth zero value; also ow > because mootocty of prefereces says soluto o boudary L L ( ),,,..., ad Ecoomc terpretatos - L MB MC - margal utlty (beeft) C Slope Budget Le Slope - Margal Rate of Substtuto - - tmes slope of dfferece curve; tae total dfferetal of (, ) ad solve for d d : d d d d Slope of Budget Le -... ; tae total dfferetal wth respect to ad : to fd d d (slope of budget le wth respect to ad ): d d d d margal cost s $ lost ($ut) s value of $ lost (utls$) Lst K-T Codtos - equatos, uows (wll come bac to these for comparatve statcs wrt ) () - () of

4 4 of Comparatve Statc for come - Totally dfferetate K-T codtos wth respect to (whch goes to ad ) Note: The last le says t's mpossble for all goods to be feror se frst order codto ad substtute for last term frst equatos ad all terms ( ) st equato Wrte wth matrces: ) ( Frst matr s a bordered hessa! Solve for usg Cramer's Rule: BH Techcally requremet for quascocave s (-) BH, but part s problem here so we usually assume suffcet codto (.e., (-) BH > ), ths case ad (-) (-) sg of s determed by sg of umerator:

5 ( ) Varable Case - sg of s same as sg of Could try to terpret ths by usg substtutescomplmets (propertes of ) ad dmshg margal utlty ( ), but secod dervatve s ot varat property so these coclusos would't hold for all utlty trasformatos we ca oly say that overall sg of ( - ) s ordal (varat to utlty trasformato) ad the same as sg of Drectly - feror Good - quatty demaded decles as come creases; < come Cosumpto Curve (CC) - shows how cosumpto chages wth come; f t slopes bac towards the vertcal as, s feror that rage; f t slopes dow towards the horzotal as, good s feror that rage; Note: ths also meas ca't have all goods be feror. CC feror feror maes budget les shft up (slope does't chage) dfferece Curve for feror Good - the Cs pctured here are cotuous, dowward slopg "th" les that do't tersect ad are cove to org; there are smlar Cs that go through every budle there est prefereces that lead to feror goods Real World - arrowly defed goods lead to feror goods (e.g., relatvely less epesve cuts of meat; relatvely cheap compact cars); more broadly defed goods wll ot be feror (e.g., food, trasportato); does't have to be feror everywhere (e.g., cheap car could go from ormal to feror ad bac to ormal) Share of come - share of come spet o good : s Necessary Good - amout spet as percet of come decles as come rses; ds d < ; techcally ths would clude feror goods, but usually do t clude them (.e., also requre > ) Luury Good - amout spet as percet of come creases as come creases; ds d > Homothetc Good - amout spet as percet of come stays costat whe come chages; has utary come elastcty Chage Share - tae dervatve of s ( s fucto of so use cha rule) ormal (d ca feror ormal 45 o CC 5 of Stay o 45 o le s homothetc below meas s luury ( ecessty); above meas s ecessty ( luury)

6 6 of d ds Multply frst term by ( )( ) ( ), d ds ε ε, > maes ds d > (luury) ad ε, < maes ds d < (ecessty) ds d ε, feror Good < < < Necessty > < (,) Homothetc > Luury > > > Lmted come - s so sum of s 's equals costat, f some s creases (.e., luury good wth ds d > ), the some other s must decrease (.e., ecessary or feror good);.e., f there s a luury good there has to be a ecessary or feror good Comparatve Statc for rces - Result - most mportat result of cosumer theory; prce chage has both come ad substtuto effects Ta Cut Eample - substtuto effect people wor more; come effect people wor less; do t ow whch effect s domat ad t's dffcult to measure emprcally because tme horzo (come effects may tae loger to observe, but by the ta codes chages aga) Totally dfferetate K-T codtos (lsted earler) wth respect to (geeralzes to ay ); Note: sce comparatve statcs loos at optmal soluto, eters to fuctos for optmal decso varables (.e., (,) ad (,)) (, 3,..., ) se frst order codto ad substtute for last term frst equatos ad all terms ( ) st equato; the frst eq move the to the rght had sde: (, 3,..., ) come elastcty product rule product rule

7 7 of Multply last eq by ad put frst term o other sde: Wrte wth matrces: ) ( Frst matr s a bordered hessa; t wll always be a BH for costraed optmzato... the trc s fgurg out what the other terms loo le Solve for usg Cramer's Rule: BH Epad by cofactors usg the frst colum: BH BH ) ( ) ( The frst term s the ow substtuto effect (S ) ad the secod s smply tmes : Slutsy Equato: S geeral: S Ow Substtuto Effect (S ) - frst term from above; tmes prcpal mor of BH dvded by determat of BH; prcpal mor has rowcol removed so t has opposte sg as BH ad S < (because we assume () s quascocave) Substtuto Effect come Effect

8 c More formally - S geeral: S.e., effect of prce o good whle holdg utlty costat Ha Compesato - order to eep utlty costat wth a chagg prce, we eed a crease come to offset the hgher prce; do't ow how much t s uless we ow full dfferece map Cross Substtuto Effect (S ) - effect of prce o good ; smlar to S above oly t's a dfferet row ad col used to epad the dervatve so the term the umerator s ot a prcpal mor S s ucerta The sg of depeds o ; for a ormal good ( > ), t s clear that < (.e.,... stadard law of demad); for a feror good, however, < so the effect of prce s determate; usually the substtuto effect domates so the law of demad holds, but there s the possblty that t wo't Gffe Good - feror good whch come effect domates so > (.e., volates law of demad); amed after frst perso to "observe" ad ; stuato was relad durg potato fame; Brtsh law essetally prohbted mports; ths lmted substtutes so S would be small; for potatoes relad would be large; potatoes are a feror good so Gffe's story s plausble, but "Gffe dd't say t ad t dd't happe, but t's plausble"; Gffe goods are hard to observe ad are more a rrtat to theorsts by mag results ambguous Other roblems - there are other cases wth upwards lopg demad curves (.e., > ); usually they are thgs whch demad s based o perceved qualty through prce so utlty s't ust (), but (,) whch we 8 of c chose to gore our aalyss because we assumed ad were depedet; eample: damods Slutsy Equato Graphcally - whe, the budget le gets steeper (assumg o the horzotal as) come Effect - the ew budget le s below the old oe so the orgal budle s ow feasble; essetally the cosumer s ow "poorer" Substtuto Effect - used to have ; sce, that o loger holds ad cosumers wll substtute away from good to reestablsh equlbrum Coceptual Vew - pcture creasg come to offset hgher prce (e.g., socal securty cost of lvg allowace); o the graph, that meas move the ew budget le up utl t's taget to the orgal dfferece curve; ths would be pot C; the move from pot C to pot B s the come effect; the move from pot A to pot C s the substtuto effect Quatty Matters - tal amout of good beg cosumed matters to dvdual because the more he cosumes, the more he'll be mpacted by a chage prce Not Measurable - ths coceptual vew s't operatoal because we'd eed to ow the cosumer's full dfferece map; all we ow s a specfc pot, but there could be lots of dfferece curves through that pot (each havg a dfferet come ad substtuto effect) Coceptual budget le wll hgher prce (stepper slope), but come off sets so cosumer s o same dfferece curve B C ' A erso who cosumes more of wll requre greater compesato to mae up for '

9 Substtutes - goods that could be used place of each other Gross - defed by cludg the come effect; > (.e., ) Net - does't clude come effect; also called Ha substtutes; S > Complmets - goods used together; defed as gross ad et ust le substtutes ecept ad S < Formal Deftos - formal defto of substtutes s debated; wat somethg that's ordal, symmetrc ( sub for sub for ), ad ubased (goods are equally lely to be substtutes or complmets) Classcal Substtutes - based o ; ot good because t's based o secod dervatve whch s ot varat (.e., ot cosstet for prefereces) Ha Substtutes - defed usg S ; ths s ordal ad symmetrc (S S... taes lots of algebra to prove); based because f there are ust two goods, the best you ca get s S (depedet) eve f the goods are perfect complmets (ca ever get S < ) Gross Substtutes - chec for symmetry: does? use Slutsy equato? S S S cacels S ; cross the ad ad multply both sdes by to get?? ε, ε, gross substtute defto s symmetrc oly f the goods have the same come elastctes (.e., homothetc prefereces); t's possble that come elastctes are so dfferet that & have dfferet sgs dfferece Curves - close to straght les are substtutes ad close to rght agles are complmets Nothg's erfect - o defto really wors so pc oe based o the cotet (e.g., whchever wors best whe tryg to "say somethg terestg" comparatve statcs) Ha Terms (S ) - ot drectly observable; do't ow how much to compesate, but ca compute them drectly by rewrtg Slutsy eq: S ; s drectly observed, the other terms are computed Slutsy Matr - matr of all the Ha Terms; puts restrcto o demad; () dagoal elemets ; () symmetrc; (3) egatve sem-defte S S S S Checg Demad ropertes - loo at fve propertes lsted earler ad the chec the Slutsy matr... but ths s hard stuff erfect complmets, S 9 of

10 Elastcty (ε y, ) - elastcty of y wth respect to ; percetage chage y dvded by percetage chage Dvde By What? - y y - y... do we dvde by y or y? some say to dvde by the average (arc elastcty) ot Elastcty - y ; related to slope, but ot the same % y y dy d dy ε y, % y y y d Why se t? - depedet of uts (does't chage whe you chage uts le slope does); sometmes comes up comparatve statcs of