Some Notes on Consumer Theory

Some Notes on Consumer Theory. Introducton In ths lecture we eamne the theory of dualty n the contet of consumer theory and ts use n the measurement of the benefts of rce and other changes. Dualty s not a toc that s often not covered n undergraduate mcroeconomcs. Yet t s a owerful tool n the analyss of consumer behavour. It enables us to derve theoretcal results more easly. Much f not most aled demand analyss nowadays uses dualty. As ndcated above t s also used n the measurement of benefts n cost-beneft analyss. Fnally much of the materal we cover on dualty n the contet of the consumer has relevance to the frm. The frst art of the lecture outlnes the bascs of dualty theory. The second shows how t s used n the measurement of rce changes.. Dualty n consumer theory.. The rmal roblem and Marshallan demands In undergraduate mcroeconomcs consumers are tycally vewed as choosng a consumton bundle so as to mamse utlty u subect to ther budget constrant =M. Thus the roblem s Choose so as to mamse u subect to =M. To solve ths we set u the Lagrangan functon L = u + [ M ] λ dfferentate t wth resect to each of the and λ thereby obtanng the famlar condtons 3 = u u whch requre that the margnal rate of substtuton between good and good the sloe of the ndfference curve equals the rce rato of the these two goods the sloe of the budget lne. We thus derve an otmal consumton bundle whose value deends on the rce vector and the avalable ncome M. We can therefore talk of a demand functon for good of the form 4 D M = whch s known as the Marshallan demand functon. Ths gves us a start n the ma shown n Fg. Dualty Ma u s.t.. = m Mn. s.t. u = u

solve solve Marshallan demands = D m substtute ute Indrect utlty fn u m Roy s ID substtute substtute ute nverson Hcksan demands Shehard s Lemma = H u substtute ute Eendture functon mu The dervaton n the two-good case of the Marshallan demand functon s shown n Fg. The consumer starts on budget lne B at ont A reachng ndfference curve I. The rce of then falls from to gvng a new budget lne B and a new otmum at B on ndfference curve I. In the lower art of Fg onts a and b corresond to onts A and B n the uer art of Fg. The Marshallan demand for good curve asses through onts a and b and s labelled D. Changes n thus result n movements along D. Fg o B B B 3 A a C B I o I c o h b D Recall the locaton of D deends on the rces of the other goods n ths case ust and ncome M. For eamle f ncome rses and s a normal good the new otmum wll be on I somewhere to the left of B but to the rght of A. There wll be a new ont n the lower art of Fg corresondng to D labelled d that les above D. Through ont d there

3 wll be Marshallan demand curve for corresondng to the new hgher ncome level. Note though that f ncome were to say double and both rces were to double the demand for and wouldn t change. We say that the Marshallan demands are homogeneous of degree zero n rces and ncome together. More formally for any scalar θ> t s the case that D θθm=d M... The dual roblem and Hcksan demands The roblem above s known as the rmal roblem. An alternatve aroach known as the dual of the roblem above s to vew consumers as choosng a consumton bundle so as to mnmse the eendture requred to attan a secfc level of utlty u. Thus the dual roblem to s 5 Choose so as to mnmse M= subect to u=u. Ths s shown n the ma n Fg. The relevant Lagrangan for the dual roblem s 6 L + [ u u] = µ. Ths gves condtons 7 = u u whch are dentcal to those obtaned from the rmal roblem. Ths s llustrated n Fg 3. In the orgnal roblem the consumer moves along the budget lne untl the hghest attanable ndfference curve s reached. In the dual roblem she moves along the ndfference curve untl the lowest so-eendture lne s reached. Recall that lower so-eendture lnes are assocated wth lower eendture levels. Solvng the dual roblem wll roduce an otmal consumton bundle whose value wll deend on the rce vector as n the orgnal roblem but also on the target level of utlty chosen u. If as our target level of utlty we choose the level of utlty attaned n the orgnal roblem then t s clear from Fg 3 that the consumton bundle chosen n the dual roblem must be the same as that chosen n the orgnal roblem.e.. But of course the arguments of the demand functon are now and u rather than and M. Ths gves a demand functon assocated wth the dual roblem of the form 8 H u = whch s known as the Hcksan demand. The route from the dual roblem to the Hcksan demand functons s shown n the ma n Fg.

4 Fg 3 I M M M Sometmes Hcksan demand functons are also known as comensated demand functons because the consumer s utlty s held constant when rces change and hence the consumer s comensated for them. Ths s shown n Fg. We reduce as before but nstead of holdng M constant as we dd when dervng the Marshallan demand functon we hold utlty constant.e. we kee the ndvdual on ndfference curve I. Ths gves us budget lne B 3 and roduces the otmal bundle shown as C. Ths les to the rght of A gven convety of I but to the left of B assumng s normal. The move from A to C s of course the substtuton effect. We can trace out the Hcksan demand functon n the lower art of Fg. Pont c corresonds to ont C n the uer art of the dagram. We know that onts a and c le on the Hcksan demand curve corresondng to the utlty level assocated wth I. Ths Hcksan demand curve s labelled h. The sloe of the Hcksan demand curve H s the equal to the substtuton effect. We ll consder later the roertes of the Hcksan demand functon..3. The eendture functon Substtutng the Hcksan demand functons nto the obectve functon of the dual roblem gves 9 H u m u = =.

5 The functon mu s known as the eendture functon or cost functon. It shows the mnmum eendture requred to acheve a gven level of utlty condtonal on a artcular rce vector. Ths role of the eendture functon s shown n the ma n Fg. There are varous mortant roertes of the eendture functon we need to know. The frst s the followng: P: mu s homogeneous of degree one n rces. Ths means that f all rces double eendture doubles. Ths result s easly shown. F u at u' and at ' and denote the corresondng otmal bundle by '. Then the corresondng eendture functon s m'=m'u'. Pck a new rce vector k' where k>. Relatve rces don't change so f we hold u constant at u' t must be the case that ' s stll the otmal bundle. The new level of eendture s therefore equal to mk'u=k''=km'. Proerty P tells what haens when all rces change by the same roorton and we hold utlty constant. The net roerty tells us what haens when ust one rce changes and we hold utlty constant. The roerty s: P. The eendture functon s concave n rces. Ths s shown n Fg 4. If mu s strctly concave as shown the roerty means that f we double eendture less than doubles. Convety of ndfference curves guarantees strct concavty but convety sn t requred for non-strct convety. Fg 4 M u M u M u M 3 u K M u + K M u M u 3 3 = k + k We can rove concavty as follows. Suose we have two rce vectors and and suose that when utlty s fed at u the otmal bundles corresondng to these

6 vectors are and see Fg 5. We have the corresondng eendtures m u and m u. Net take a lnear combnaton.e. a weghted average of the rce vectors 3 = k + -k and let 3 be the otmal bundle when rces are 3 and utlty s u see Fg 5. Concavty means that the cost of reachng utlty level u at rce vector 3 must not be less than the weghted average of m u and m u see Fg 4. Thus concavty means m 3 u km u k m u +. By the aom of cost-mnmsaton ths must ndeed be true. The LHS of the nequalty n s equal to: [ ] m u = = k + k = k + k 3 3 3 3 3 3 Consder the terms 3 and 3. These ndcate the cost of bundle 3 at rces and resectvely. We know that bundles and 3 are all on the same ndfference curve corresondng to utlty level u. We also know that mnmses the cost of reachng u when rces are so we can be sure that the cost of buyng the bundle 3 when rces are cannot be less than the cost of buyng at the same rce vector cf. Fg 5. In other words 3 = m u. 3 Indeed f ndfference curves are conve as n Fg 5 5 wll be a strct equalty. By the same reasonng we can wrte: 4 3 = m u. Combnng 3 and 4 wth makes t clear that must be true.

7 Fg 5 = m u 3 m u 3 m u 3 3 = m 3 u = m u I The thrd roerty we need to know s Shehard's Lemma. Ths says that the artal dervatves of the eendture functon wth resect to rces are the Hcksan demand functons. Ths allows us to work back n Fg from the eendture functon to the costmnmsng demands that underle t see Fg. Shehard's Lemma stated formally s: P3. Shehard's Lemma: m = = H u. To rove ths consder an arbtrary rce vector a utlty level u and the corresondng vector of otmal choces =H u. Assocated wth ths s the cost functon m u. If we grah ths cost functon we get Fg 6. Suose we vary but kee all other rces unchanged and we kee the bundle unchanged at. Then by varyng we trace out the straght lne: = n 5 Z = H u + H u whch s the lne n Fg 7. Obvously at we have = n 6 Z = H u + H u = m u so that at m u and Z are tangental so that the sloe of the cost functon n Fg 7 s equal to the sloe of Z whch from eqn 5 s clearly ust. Ths roves the result and hels see the ntuton behnd Shehard's Lemma. Suose you buy unts of a week. The rce of then rses by. As a frst aromaton your eendture would have to

8 rse by to allow you mantan the same utlty. Fg 5 shows why ths s true only as a frst aromaton t s only true for nfntesmally small rce changes snce when the rce of rses you wll substtute away from see Fg 7. Fg 6 Fg 7 u H X here sloe = = Ζ u M + = Ζ = z u H u H u m I u M Z z + = = u m = u m =

9 Proertes P and P3 together have an mortant mlcaton: Hcksan demands are homogeneous of degree zero n rces. Thus doublng rces wth utlty held constant leaves the demand for good unchanged. That ths must be the case ought to be clear from Fg 3..4. The ndrect utlty functon and Roy s dentty We're ready now to ntroduce another concet the ndrect utlty functon. Ths s derved not from the dual roblem but from the orgnal roblem. To get the ndrect utlty functon we smly substtute the Marshallan demands = D M nto the utlty functon u to get 7 u... n u D M... Dn M u M = =. The functon u M shows mamsed utlty as a functon of the ultmate determnants of utlty rces and ncome. The dervaton of the ndrect utlty functon from the Marshallan demands s shown n Fg. The ndrect utlty functon s related to the eendture functon. Snce mu=m we can rearrange or "nvert" mu to gve u as a functon of and M. Ths wll gve us u=u M. The converse also ales we can nvert the ndrect utlty functon u= u M to get the eendture functon.e. M=mu. The lnk between the eendture functon and the ndrect utlty functon s also shown n Fg. There are other lnks between the varous functons n Fg worth mentonng. We mght wsh to generate Marshallan demands from the eendture functon. Ths we can do by substtutng the ndrect utlty functon nto the Hcksan demands: = h u = h u M = D M. 8 The whole thng can also be done n reverse startng wth the Marshallan demands we can substtute the eendture functon nto the Marshallan demands to get the Hcksan demands: 9 = D M = D m u = h u. These lnks are also shown n Fg. One last lnk s useful. We can work back from the ndrect utlty functon to the Marshallan demands. Snce the eendture functon s the nverse of the ndrect utlty functon we can wrte u m u u. It can be shown mathematcally. P4 tells us that the Hcksan demand for good s the dervatve of the cost functon wth resect to the rce of good. P tells us that the cost functon s homogeneous of degree one. It s the case that the dervatve of a functon that s homogeneous of degree n s tself homogeneous of degree n-. So t follows that Hcksan demands are homogenous of degree zero.

Dfferentatng eqn wth resect to allowng m to vary so as to hold u constant gves u u m + =. By Shehard's Lemma we have m H u H u D M = = =. But t s also true that.e. the amount demanded n equlbrum s the same rresectve of whether t s the Marshallan or Hcksan demand functon we're usng. Thus from we get: u = = u 3 Roy's dentty: H u D M The mortance of Roy s dentty s that t allows us to work back from the ndrect utlty functon to the Marshallan demands. Ths comletes the lnks between the varous functons n Fg..5. The Slutsky equaton The Slutsky equaton whch tells us nter ala that the effect of a rce change can be decomosed nto a substtuton effect and an ncome effect. The equaton s easly derved usng the results we have obtaned so far. Snce the budget constrant bnds M s equal to total eendture. Hence we can wrte 4 H u D m u = = Dfferentatng ths wth resect to allowng eendture to change so as to kee utlty constant we get 5 H D D m = + =...n The fnal term s equal to H u= by Sheherd s lemma. Substtutng ths n 5 and rearrangng gves us: 6 D H D = =...n whch s Slutsky s equaton. Consder frst the case where =.e. we are consderng the effect of a change n on the demand for. In ths case eqn 6 smly says that the sloe of the Marshallan demand functon s the sum of the substtuton effect H and the ncome effect D M. What can we say about the sgns of these terms? The frst we can sgn

unambguously. Recall Shehard s lemma above. Dfferentatng ths wth resect to gves: 7 m u H = whch gven that the eendture functon s concave means that H. Thus the substtuton effect cannot be ostve. The sgn of the second term deends on whether good s normal D > n whch case the Marshallan demand curve s defntely downward-slong or nferor D < n whch case t could be downward-slong uward-slong or flat deendng on the relatve szes of the two terms on the RHS of 6. The Slutsky equaton can be eressed n elastcty form. Kee = and multly 6 through by / and the second term on the RHS by M/M to get: 8 ε = σ s η where ε s the Marshallan demand elastcty σ s the Hcksan or comensated elastcty s = /M s the share of good n total eendture.e. ts budget share and η s the ncome elastcty of demand. Thus the ga between the Marshallan and Hcksan demand elastctes wll be smaller the smaller s the ncome elastcty of demand and the smaller s the good s budget share.