901 Notes: 8.doc John E. Walker Department of Economics Clemson University

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1 9 Notes: 8.doc Joh E. Walker Departmet of Ecoomcs Clemso versty THE COPLETE EPIRICAL IPLICATIONS OF THE THEORY OF CONSER BEHAVIOR The Law of Demad s largely a emprcal fact of ature. I practce we have a dea about how to emprcally estmate demad curves, ad we kow that emprcally, demad curves always have a egatve prce effect. The Law of Demad ca be derved graphcally ad ths smple exposto s suffcet gudace to desg a cotrolled expermet that tests the law both ts basc ad eve more sophstcated dmesos. I our cotrolled expermet, we frst exame the substtuto effect whch says that whe relatve prces chage oe drecto, cosumpto chages the other. We fd ths to be true. Also, we kow from the graphcal exposto of the theory that whle a postve prce effect s theoretcally possble, t ca oly occur whe there s a suffcetly large, egatve come effect to offset the pure substtuto effect. Hece, we are left wth the somewhat puzzlg questo of why we eed to derve the theory of cosumer behavor more detal? If emprcal demad curves obey the smple law of demad, what more ca we lear from a complcated theory. The aswer to ths query s the purpose of ths lecture. THE SET-P The theory of cosumer behavor s a set of predctos about observable pheomea. These predctos are derved from a model whch dvduals are assumed to maxmze utlty subect to lmted purchasg power. Sce utlty s uobservable, the challege of the theory s derve emprcally meagful statemets predctg cosumer behavor. To summarze the model we have a utlty fucto ( ) that raks all combatos of the set of goods. The cosumer's obectve fucto s max x ( x,..., x )..,, s t = Px = l q The frst ad suffcet secod order codtos (FOC & SSOC) of ths problem mply that cosumer behavor wll be predcted by equatos of the followg sort: = () x = x ( P,..., P, ) The problem wth eqt () s that the model does ot yeld ay drect predctos about cosumer behavor. That s, comparatve statc aalyss does ot detfy the sgs of the partal dervatves of the demad fuctos defed by eqt (). Ths problem s remeded by examg the dual of the maxmzato problem. Every optmzato problem has a dual or a mrror mage. I ths case maxmzato of utlty subect to a expedture costrat ca be looked upo as mmzato of expedtures subect to a utlty costrat. () m lq = Px.. (,..., ) x s t = x x = (3) The superscrpts refer to the parameter that forms the costrat the optmzato problem. Revsed: September 7, 3.T. aloey

2 9 Notes: 8.doc Joh E. Walker Departmet of Ecoomcs Clemso versty The FOC & SSOC mply soluto fuctos for the model the form of: x = x ( P,..., P, ) (4) THE THEORETICAL RESLTS It turs out that these fuctos have comparatve statc mplcatos. That s, there are several dervatves that ca be detfed. Frst we ca show that x / <. I words, ths says that cosumpto of a good creases whe ts prce falls f the cosumer s forced to move alog a utlty cotour. By tself ths s ot a emprcally meagful statemet because we caot measure utlty drectly or determe whether a cosumer s costraed to stay at a gve level of utlty. However, because the dervatve s sgable, we ca put the result to good use. I addto to the egatve sg of the ow prce effect, we ca show that the cross prce effects are symmetrcal the utlty-costat demad fuctos. That s, x / = x /. Ths result derves from Youg's Theorem. Sce the order of cross partal dfferetato does ot matter, the coeffcet matrx the comparatve statc aalyss s symmetrc. I the expedture mmzato problem, the vector of costats has oly oe term. Thus, the symmetry comes to play. Now we apply the Dualty Theorem. Dualty meas that every maxmzato problem ca be looked at as a mmzato problem. The dualty theorem says that the soluto to the maxmzato problem s detcal to the soluto of ts mmzato dual whe the costrat to the maxmzato problem s approprately defed. Cosder the optmzed value of expedtures the problem defed by eqt (3). Ths ca be wrtte as = Px + µ ( x,..., x = ( P,..., P, ) (5) = Eqt (5) s called the drect expedture fucto because t expresses the mmzed expedture level ecessary to acheve o as a fucto of the parameters of the problem. The dualty theorem says that by makg ( ) the value of the come costrat the maxmzato problem defed by eqt (), the the optmzed value of utlty that problem wll be equal to o. That s, = ( P,..., P, ) ( P,..., P, )= (6) whch makes tutve sese. But more mportatly ad more rgorously, the behavor fuctos for the x 's are solved smultaeously for the prmal ad the dual. Smply put, ths meas that the levels of the soluto values are the same. That s, x = x as ca be see the graph the two goods case. But t also meas that equatos () ad (4) are equal at that pot. x ( P,..., P, ( P,..., P, )) = x ( P,..., P, ) Equato () s observable but ot predctve. Equato (4) s predctve but ot observable. Equato (7) combes the best of both worlds. (7) Revsed: September 7, 3.T. aloey

3 9 Notes: 8.doc Joh E. Walker Departmet of Ecoomcs Clemso versty THE EPIRICAL APPLICATIONS Slutsky Equato Eqt (7) allows us to derve emprcally meagful statemets about cosumer behavor. Frst s the Slutsky equato. Dfferetate eqt. (7) wth respect to P. The result looks famlar: x + x = x < (8) The prce effect from the ordary demad curve plus the come effect tmes a scalar s equal to the slope of the compesated demad curve. The evelope theorem helps defe ths scalar. It says that / = x, ad by eqt (7) tself, x = x. Hece, x + = x x x < (9) Eqt (9) s emprcally meagful because every term o the left-had-sde of the equals sg s observable. Ths s the behavor of dvduals whe moey come ad prces are accouted for. These are partal dervatves. We observe a chage cosumpto wth respect to a prce chage holdg come costat. The we observe a chage cosumpto wth respect to a come chage holdg prces costat. Eqt (9) tells us how to add up these effects. Result.: The Slutsky equato show equato (9) says that the ow-prce effect plus the come effect weghted by the cosumpto level must be egatve. I elastcty terms the Slutsky aggregato of the prce ad come effects takes the form + S = < where s the percetage chage the cosumpto of the th good wth respect to a chage the th prce hold come costat, s come elastcty, S s the budget share of the th good, ad s the ow-prce elastcty alog the compesated demad curve. Result.: I elastcty form, the Slutsky equato says that the sum of the ow-prce elastcty plus come elastcty tmes budget share must be egatve. The emprcal ature of ths hypothess s revealed whe we apply the theorem to estmated values of the prce ad come effects. Commoly, the prce ad come effects are estmated usg a lear structure for the demad equato: x = a + bp + c + dp ultply each term equato (9) by P /x, ad the secod term o the left by /. Revsed: September 7, 3 3.T. aloey

4 9 Notes: 8.doc Joh E. Walker Departmet of Ecoomcs Clemso versty Lear regresso returs coeffcet values for x /, whch s labeled b above, ad for x /, whch s c. The Slutsky equato says that -b must be bgger tha c tmes the level of cosumpto. If t s ot, the demad curve s msestmated (or the theory s wrog). Furthermore, the Slutsky equato tells us a cosumpto rage over whch the estmated demad curve apples. Result.3: Lear demad curves are oly cosstet wth utlty maxmzg cosumer behavor the rage of x < -b/c. Specfcally ths meas that lear demad curves may ot be theoretcally cosstet for low prce ad hgh quatty combatos. These same kds of statemets ca be made wth regard to estmated elastctes. Elastctes ca be computed from lear demad curves or estmated drectly usg a log trasformato. Ether way Result. apples. Result.4: Demad curves are oly vald the rage where budget share of the th good s less tha - /. I practce ths meas that log-lear demad curves are theoretcally vald at super hgh prce levels whe demad s elastc ad at super low prce levels whe demad s elastc. Cross Prce Effects The cross prce effects for the utlty costat demad curves are symmetrc. That s, x = x The ature of ths result ca be see clearly the graph of the two-good world. As the prce rato betwee the two goods chages ad cosumpto sldes alog the dfferece curve, the chage the prce rato ca be vewed as ether a chage the prce of oe good or the other. Hece, the chage cosumpto of caused by a chage the prce of must equal the chage the cosumpto of wth respect to a chage the prce of. We ca emprcally explot ths symmetry result by lookg at the ordary demad curves. The Slutsky equato s the ow prce dfferetato of eqt (7). The cross prce dfferetato has a very smlar form. x + x = x () Substtutg based o the evelope theorem gves x + = x x x () Revsed: September 7, 3 4.T. aloey

5 9 Notes: 8.doc Joh E. Walker Departmet of Ecoomcs Clemso versty Based o the symmetry of the utlty-costat demad curves, we ca wrte: x + = x x x = x = x + x x () or elastcty form 3 S + = + (3) S Eqt (3) shows a terestg aomaly the theory. The observable substtutablty or complemetarty of x for x whe lookg at the demad for good ca reverse whe we exame the demad for good. Eve though the utlty costat cross effects are symmetrc, observed behavor s codtoed by the come effect that s assocated wth ay prce chage. So, for stace for the Chese, fsh s a complemet to meat, but meat s a substtute for fsh. That s, whe the prce of meat goes up, Chese people eat less fsh. However, whe the prce of fsh goes up, they eat more meat. Seems odd, eve for the Chese. Result.: The observed cross-prce effects betwee two goods foud ordary demad fuctos eed ot work the same drecto across demad fuctos. Not oly are the observed effects ot recprocal, as they are the utlty-costat demad relatos, they ca be of opposte sg. Good ca be a complemet to, whle good s a substtute for. However we ca ote the followg from eqt. (3): If < ad >, the >. Thus we have a emprcal statemet about come elastctes relatve to cross-prce effects. Result.: If the cross-prce elastcty of for s egatve whle for s postve, the good (the complemet to ) must have a hgher come elastcty. For the case of the Chese cosumpto of meat ad fsh, the theory says that because of the sgs of the gross cross prce elastctes, the come elastcty of fsh must be larger tha the come elastcty of meat. There are a couple of other thgs that we ca pot out lookg at ths form. Result.3: If the cross-prce effects betwee two goods are equal, the the come elastctes must be equal. Note that the statemet s about the lear prce effects compared to the come elastctes. 4 The result s derved from eqt.. The cross-prce terms subtract out. Cross multplyg the 3 ultply both sdes of eqt. () by P /x ad by P /x whch gets the cross-prce elastcty terms. ultply the come effect terms by / to get the come elastctes. Cross multply the left over x ad x terms ad dvde both sdes by. Ths gves a budget share multpler o both sdes whch ca be used to cacel the budget share term ext to the come elastctes. 4 Be careful here to dstgush betwee the terms called effects ad the elastcty expressos. The partal of x wth respect to p s ot the same thg as the elastcty of good wth respect to prce. Elastcty s a weghted effect. Revsed: September 7, 3 5.T. aloey

6 9 Notes: 8.doc Joh E. Walker Departmet of Ecoomcs Clemso versty cosumpto level terms ad dvdg both sdes by gves =. Thus, whether the goods are substtutes or complemets, the goods must have detcal come elastctes f they have recprocal cross-prce effects. Result.4: If the come elastctes are equal, the the cross-prce effects betwee two goods must be equal. Ths s also most easly see eqt. (). Remember to ote the dfferece betwee elastctes ad effects. Result.5: Iferor goods wll have relatvely large cross-prce elastctes real umbers. Ths says that feror goods wll be less complemetary f both goods are complemets ad more substtutable f the cross prce elastcty s postve. There must be more substtuto potetal to make up for the egatve come effect. If < eqt (3), the has to take up the slack whe good s pared wth substtute goods that have ormal come effects. Homogeety Because come-costat demad curves are homogeeous w.r.t. come ad prces, we ca employ Euler's Theorem to make aother emprcal statemet. We ca show that observable demad curves are homogeeous of degree zero by dervg the FOC ad SSOC at oe set of prces ad come ad comparg these the FOC ad SSOC derved at aother set that dffers by a factor of proporto k. If the FOC ad SSOC of the two problems are detcal, the the soluto equatos are homogeeous of degree zero moey ad prces. 5 Euler's theorem says that the sum of the partals tmes the levels of the varables equals the degree of homogeety. For demad curves ths meas or elastcty form L N O Q x P + = P x =,...,,... (4) + = =,, (5) The emprcal value of eqt (5) allows for a specfcato test of estmated demad curves. 6 Note that eqt (5) s a restrcto placed o a sgle demad curve. It apples to the demad for oe good wth respect to all of the argumets that demad fucto. Sce the sum of the elastctes s zero whe all relevat prce varables are cluded, the observed sum of estmated elastctes s a specfcato test. 5 You should be able to derve ths result. 6 Note that the result gve by eqt (5) s the same as the result that the sum of the prce elastctes for a compesated demad curve must sum to zero. Revsed: September 7, 3 6.T. aloey

7 9 Notes: 8.doc Joh E. Walker Departmet of Ecoomcs Clemso versty Result 3.: The sum of ow-prce, cross-prce, ad come elastctes wth a demad fucto must sum to zero. Emprcally ths meas that the more the sum of ow-prce, cross-prce, ad come elastctes dffers from zero, the greater s the bas of omtted varables. I order to explore ths result, let s retur to the rat expermet results. I that exercse we were able to estmate a prce effect ad a come effect. If we look at oe pot o the demad curve ad evaluate eqt. (4), we ca form a estmate of the cross prce effect. For stace, f we look at the demad curve related to a come of 6 presses, at the prce of 5 presses for root beer, the ow prce effect tmes prce s - ad the come effect tmes come s +3, hece the cross prce effect tmes the prce of Colls x must be - for the homogeety codto to hold. Also, ths mples that the tercept value of the lear demad form must be 6. The homogeety result appled to emprcal demad curves presets somewhat of a paradox. Notce that f we had evaluated the demad curve at the prce of presses for root beer, the weghted ow-prce effect would have bee -4. Sce the weghted come effect s stll +3, for the homogeety codto to hold at that pot, the weghted cross prce effect would have to be +. Sce the prce of Colls x has ot chaged, the mplcato s that the cross prce effect chaged. Ths s cosstet wth a lear specfcato of all effects. Paradoxcal or ot, ths represets a lmtato of the lear estmatg form. That s, usg a lear estmatg form for a sgle demad curve, the homogeety codto caot be mposed as a estmato costrat for every observato. Noetheless, the homogeety codto ca be tested for ad mposed as a costrat at the mea of the data o the depedet varables. Cosder a estmatg form of the sort: Q = α + β P + β P + γ If the true demad curve s Q (P, P, P 3, ) the our estmated α = α+ β 3 P 3 f estmated ubased fasho. However, we expect that t wll be based as wll the estmates of the other parameters because the prces are lkely to be correlated. Eve so, f [ βp+ βp+ γ] / Q=, the the estmated tercept term s a ubased estmate of the true α, whch s the md-pot of the cosumer s choces lear form. Result 3.: If two markets are related oly by the cross prce effects ther demads ad they are substtutes, the the markets wll have a well defed ot equlbrum. Ths result s the pot of the study questo below. It s most easly see where we assume there s oly oe other prce the demad curve ad that other good s a substtute. Eqt. (4) tells us that the weghted ow-prce effect, whch s aturally egatve, must be as large as the sum of the weghted cross prce effect ad come effect. A suffcet codto for the ot equlbrum s that the product of the ow-prce effects be bgger tha the product of the cross prce effects. Ths wll be the case based o eqt. (4) above. Eve though the prce effects eqt. (4) are weghted by the prces, the algebra works out. Revsed: September 7, 3 7.T. aloey

8 9 Notes: 8.doc Joh E. Walker Departmet of Ecoomcs Clemso versty Cosder a smple demad ad supply model two related markets: Q D = a I - b P + c P Q D = f I - g P + h P Q S = d + e P Q S = + P P s prce, Q s quatty ad I s come. The coeffcets are sged explctly. The equlbrum codto prevals: quatty demaded equals quatty suppled both markets. a) What s the effect of a supply shft market? That s, derve the comparatve statc results o prce both markets wth respect to a parametrc chage the supply codto market. b) What codtos must be mposed to get a determate soluto? c) Are these codtos cosstet wth the theory of cosumer behavor? Homogeety ca also be appled to the compesated or utlty-costat demad curves. tlty costat demad curves are homogeeous of degree zero prces. That s, f all prces crease by a costat factor, the slope of the budget costrat does ot chage ad the optmal cosumpto budle remas the same. Formally, we derve the result usg Euler's Theorem: x = P Because we kow that the ow prce effect, x postve umber. Hece,, s egatve, the rest of the terms must sum to a Result 3.3: Every good has at least oe et substtute. Budget Costrat 7 Oe last set of emprcal statemets ca be derved by substtutg the demad curves to the budget costrat ad dfferetatg. I elastcty form, dfferetatg wth respect to yelds ad wth respect to P yelds S = (6) = S = S (7) = Eqts (6) ad (7) are restrctos o the estmato of systems of demad fuctos. These are mplcatos about come ad prce elastctes across demads. 7 Slberberg (99)&()sectos.5-.6; Itrlgatar, Bodk, Hsao (996)-Chapter 7-sectos Layard ad Walters (978) secto 5.. Ncholso (989) Chapter 7; Ncholso () Chapter 4. Revsed: September 7, 3 8.T. aloey

9 9 Notes: 8.doc Joh E. Walker Departmet of Ecoomcs Clemso versty Eqt. (6) tells us what must happe f come goes up. If come goes up, the come elastctes detfy how cosumpto of each good chages. However, we kow that the budget costrat has to be satsfed overall. Equato (6) says, Result 4.: The sum of the budget share weghted come elastctes must sum to oe. Equato (7) speaks to the effect of a chage oe prce across demad fuctos. It s best see whe cosderg the two-good case. Let there be good ad everythg else, good. Rewrtg eqt. (7) gves: S = ( + ) (8) S Ths shows that the chage the cosumpto of everythg else whe the prce of good chages s a fucto of the prce elastcty of good. Ths dea ca be summarzed as follows: Result 4.: For elastc goods, substtutes domate complemets; for elastc goods, complemets domate substtutes. Cosder the case where good s prce elastc. Whe the prce of good creases total spedg o good falls. Ths extra moey must be spet elsewhere. I the two good world, t s spet o the composte commodty. Hece, the demad fucto for the composte commodty, whe the prce of good oe creases, cosumpto of the composte commodty creases. The stuato s reversed whe good s prce elastc. ore elaborately, let there be four goods, beer, we, peauts, ad everythg else. Now let beer be prce elastc, peauts be a complemet to beer, we a substtute, ad let the budget shares of beer, we, ad peauts all be the same. What ca we say about the relatve purchases of we ad peauts? Rewrtg eqt (7) we have S S S + + = ( + ) > where < (9) S S S for beer defed as good, peauts as, we as 3, ad the rest as 4. Let the effect of a chage the prce of beer be cofed to oly we ad peauts. That s, let e 4 be zero. To satsfy equato (9), we expedtures (the substtute) must react more tha peaut purchases (the complemet) as the prce of beer chages. (October 993) Revsed: September 7, 3 9.T. aloey

10 9 Notes: 8.doc Joh E. Walker Departmet of Ecoomcs Clemso versty GLOSSARY OF TERS AND RESLTS x Ow-prce effect--slope of ordary demad curve x Icome effect--shft ordary demad curve x x x x + = x x x Pure substtuto effect--slope of compesated demad curve Gross cross-prce effect--shft ordary demad curve Net cross-prce effect--shft compesated demad curve < Slutsky equato expressed lear effects + S = < Slutsky equato expressed elastctes x + = x x x = x = x + x x Cross-prce Slutsky equato S + = + Cross-prce Slutsky elastcty terms S + + = Homogeety of ordary demad fucto + = xp P x P Homogeety of compesated demad fucto S+ S = Budget costrat requremet for come elastctes across ordary demad curves ( + ) κ + κ = Budget costrat requremet for prce elastctes across ordary demad curves Revsed: September 7, 3.T. aloey

11 9 Notes: 8.doc Joh E. Walker Departmet of Ecoomcs Clemso versty BDGET SHARES The theory of cosumer behavor s ofte expressed terms of budget shares. From ths perspectve t s mportat to cosder how come ad prce chages affect budget shares. Budget share ca be expressed terms of the demad fucto: S = px( p, p, ) Frst, let's dffferetate wth respect to come: S p x px = Now cosder what t meas f the budget share remas costat whe come chages. S If = the p x px x = =, or other words, come elastcty s equal to x oe: x =. x Smlarly for prce elastcty: S p x x = +. p p S If p = the p x p x + =, or ow prce elastcty equals mus oe: p x = x p If come ad ow prce elastctes equal ad, by the homogeety codto, the cross prce elastcty must be zero. Revsed: September 7, 3.T. aloey

12 9 Notes: 8.doc Joh E. Walker Departmet of Ecoomcs Clemso versty A Graphcal Presetato of Result. 8 If the cross-prce elastcty of for s egatve whle for s postve, the good (the complemet to ) must have a hgher come elastcty. Cosder two goods as show below. Let the prce of good go dow whch causes the dvdual to chage the cosumpto choce from pot b to pot a. I so dog, good s revealed to be a complemet to good (.e., x /p <). Next cosder a prce decrease good oe. Ths causes the dvdual to chage the cosumpto choce from pot a to pot c. I so dog, good s revealed to be a substtute for good (.e., x /p >). Fally, otce that the frst ad thrd budget costrats are parallel. Hece, the movemet from pot b to pot c measures a come effect. Also otce that based o the algmet of the pots, the complemet (good ) has a come effect that s ecessarly larger tha the substtute (good ). 8 Thaks go to Hazhe L for ths presetato. Revsed: September 7, 3.T. aloey

13 Setup Slutsky Symmetry Homogeety Budget Costrat maxlx ( x,..., x ), =, q Law of Demad: Cross prce effects from Demad Curves are Across Demad Substtuto effect s eg. compesated demad homogeeous of degree Codtos st.. = Px Dowward slopg curves are equal. zero moey ad prces. = compesated demad. FOC & SSOC => x = x ( P,..., P, ) Ordary Demad Curves x + = x x x + S = < < Cross Slutsky x = x x + = x x x + x x Euler's Theorem Lx P O x P + = P =,...,,... N Q + = =,, Dfferetate the budget costrat at the optmal values wth respect to a prce or come. Dualty: For every maxmzato problem, there s a mrrormage mmzato problem. Ca't cosume too much or sped too much: - / Ordary cross prce effects are ot ecessarly symmetrc If gross cross prce effects are equal come elastctes must be equal. Elastctes ordary demad curve must sum to zero. Emprcal test Omtted varable bas = S = S = ( ) S x = x ( P,..., P, ) Compesated Demad If costrats are matched: Dualty Theorem x (.) = x (.) For: x = a + bp + c + dp x < -b/c must be true for ormal prce ad come effects. S + = + S Implcatos for relatve come elastctes Graph For elastc goods, substtutes domate complemets; for elastc goods, complemets domate substtutes. S = => x = x S = => p x = p x p

Lecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model

Lecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model Lecture 7. Cofdece Itervals ad Hypothess Tests the Smple CLR Model I lecture 6 we troduced the Classcal Lear Regresso (CLR) model that s the radom expermet of whch the data Y,,, K, are the outcomes. The