Consumer Theory. 1 Consumption set. 2 Preferences and utility. These notes essentially correspond to chapter 1 of Jehle and Reny.

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1 Consumer Theor These notes essentall correspond to chapter of Jehle and Ren. Consumpton set The consumpton set, denoted X, s the set of all possble combnatons of goods and servces that a consumer could consume. Some of these combnatons ma seem mpractcal for man consumers, but we allow the possblt that a consumer could have a combnaton of 500 Ferrars and 400 achts. Assume there s a xed number of goods, n, and that n s nte. Consumers ma onl consume nonnegatve amounts of these goods, and we let x R + be the amount consumed of good. Note that ths mples that the consumpton of an partcular good s n ntel dvsble. The n-vector x conssts of an amount of each of the n goods and s called the consumpton bundle. Note that x X and tpcall X R n +. Thus the consumpton set s usuall the nonnegatve n-dmensonal space of real numbers. Standard assumptons made about the consumpton set are:.? 6 X R n +. X s closed 3. X s convex 4. The n-vector of zeros, 0 X. The feasble set, B (soon we wll call t the budget set), s a subset of the consumpton set so B X. The feasble set represents the subset of alternatves whch the consumer can possbl consume gven hs or her current economc stuaton. Generall consumers wll be restrcted b the amount of wealth (or ncome or mone) whch the have at ther dsposal. Preferences and utlt The basc buldng bloc of consumer theor s a bnar relaton on the consumpton set X. The partcular bnar relaton s the preference relaton %, whch we call "at least as good as". We can use ths preference relaton to compare an two bundles x ; x X. If we have x % x we sa "x s at least as good as x ". We wll mae some mnmal restrctons about the preference relaton %. In general, our goal wll be to mae the most mnmal assumptons possble. We mae two assumptons about our preference relaton %:. Completeness: For an x 6 x n X, ether x % x or x % x or both.. Transtvt: For an three elements x, x, and x 3 n X, f x % x and x % x 3, then x % x 3. Completeness means that the consumer can mae choces or ran all the possble bundles n the consumpton set. Transtvt mposes some mnmal sense of consstenc on those choces. The boo lsts a thrd assumpton, re exvt. However, f the preference relaton % s complete and transtve, then we can also show that t s re exve. Re exvt smpl means that an element of the consumpton set s at least as good as tself, or x % x for all x X. These ver basc assumptons comprse the condtons of ratonalt n economc models. The noton of ratonalt n economcs s one that s often msunderstood all we assume for a "ratonal" economc agent s that preferences are complete and transtve (and re exve).

2 x x A thc ndfference set. x Whether or not the bul of socet consders a choce to be a good one (sa a brght orange tuxedo at a formal event), f an ndvdual agent possesses complete and transtve preferences then that consumer s consdered to be ratonal. Now that we have establshed the preference relaton % we can de ne () the strct preference relaton and () the nd erence relaton. De nton The strct preference relaton,, on the consumpton set X s de ned as x x f and onl f x % x but not x % x. De nton The nd erence relaton,, on the consumpton set X s de ned as x x f and onl f x % x and x % x. Note that nether nor s complete, both are transtve, and onl s re exve. Once we have and we can see that ether x x, x x, or x x. Thus, the consumer s able to ran bundles of goods. However, these assumptons of completeness, transtvt, and re exvt onl mpose some mnmum order on the ranng of bundles. We wll mpose a lttle more structure on our consumer s preferences. Assumpton: Contnut. For all x R n +, the "at least as good as" set, % (x), and the "no better than set" - (x) are closed n R n +. Contnut s prmarl a mathematcal assumpton, but the ntutve reason behnd mposng t s so that sudden preference reversals do not happen. Assumpton: Local Nonsataton. For all x 0 R n +, and for all " > 0, there exsts some x B " x 0 \R n + such that x x 0. When local nonsataton s assumed, ths means that there s some bundle close to a spec c bundle whch wll be preferred to that spec c bundle. There s nothng n local nonsataton that spec es the drecton of the preferred bundle. What local nonsataton does s rule out "thc" preferences.

3 Assumpton: Strct monotonct. For all x 0 ; x R n +, f x 0 x then x 0 % x, whle f x 0 >> x then x 0 x. In a prncples or ntermedate mcroeconomcs class ths s what we would call the "more s better" assumpton. Note that when an ndvdual compares bundles of goods, f the ndvdual has bundle x 0 wth more of at least one good (and the same level of all other goods) than s n x, then ths ndvdual deems x 0 at least as good as x. And f x 0 has more of all goods than x, then the ndvdual strctl prefers x 0 to x. Assumpton: Convext. If x % x 0, then tx + ( t) x 0 % x 0 for all t [0; ]. Assumpton: Strct convext. If x 6 x 0 and x % x 0, then tx + ( t) x 0 x 0 for all t (0; ). Ether of these assumptons rules out concave to the orgn preferences. The ntuton behnd these convext assumptons s that consumers (generall) prefer balanced consumpton bundles to unbalanced consumpton bundles. Thus, snce these convex combnatons of consumpton bundles provde a more balanced consumpton plan, the consumer would prefer them. Alternatvel, thn about an partcular nd erence set n R +. The slope of an nd erence curve s called the margnal rate of substtuton. If we have strct monotonct and ether form of convext then ths means that the margnal rate of substtuton should not ncrease as we move from bundles along the same nd erence whch have a lot of good to those whch have relatvel less of good. Thus, when the consumer has a lttle of good he should be wllng to gve up more of good to get an extra unt of good than when he has a lot of good. As a summar, the assumptons of completeness, transtvt, and re exvt are the bass for the ratonal consumer. The assumpton of contnut s prmarl a mathematcal one to mae the problem slghtl more tractable. The remanng assumptons represent assumptons about a consumer s tastes.. Utlt The utlt functon s a nce wa to summarze preferences, partcularl f one wants to use calculus methods to solve problems (as we wll want to do). We can establsh results that show that wth a mnmal amount of structure that there wll be a utlt functon whch represents our preference relaton %. De nton 3 A real-valued functon u : R n +! R s called a utlt functon representng preference relaton %, f for all x 0 ; x R n +, u x 0 u x () x 0 % x. The queston s whch assumptons that we made about our preference relaton wll be needed to establsh that a utlt functon whch represents % exsts? There s a theorem whch states that all we need s completeness, transtvt, re exvt, and contnut. Note that monotonct, convext (of an tpe), and local nonsataton are NOT needed to guarantee the exstence of a utlt functon whch represents %. Theorem 4 If the bnar relaton % s copmlete, re exve, transtve, and contnuous then there exsts a contnuous real-valued functon, u : R n +! R, whch represents %. We wll not go through the proof of ths theorem but we wll use the result. Note that n the boo the provde the proof of a slghtl less general result n Theorem 3. as the assume strct monotonct. Agan, note that strct monotonct s NOT requred to ensure the exstence of a utlt functon whch represents %. When specfng utlt functons economsts are prmarl concerned wth preservng ordnal relatonshps, not cardnal ones. Thus, two utlt functons whch preserve the order of preferences over bundles wll be vewed the same UNLESS the cardnalt of the utlt functon s mportant for a partcular applcaton. So f there are two utlt functons, u (x ; x ) x + x and v (x ; x ) x + x + 5 t should be clear that the resultng utlt level from the same (x ; x ) bundle s hgher n v () than n u (). However, snce the order of preferences over bundles s preserved between the two utlt functons, the are generall vewed the same b economsts. Now consder the same functon u (x ; x ) x + x and another functon g (x ; x ) x x. If we loo at the table for three d erent bundles of x and x we see that: Note that ths text refers to the margnal rate of substtuton as a postve number even though the slope wll be nonpostve for convex preferences. 3

4 x x u (x ; x ) g (x ; x ) Snce u (8; 0) > u (; ) but g (; ) > g (8; 0), we can see that these utlt functons do not preserve the order of preferences for the bundles so that the are not vewed as the same b economsts. Gven ths noton of ordnalt of utlt functons, we have the followng theorem. Theorem 5 Let % be a preference relaton on R n + and suppose u (x) s a utlt functon that represents t. Then v (x) also represents % f and onl f v (x) f (u (x)) for ever x, where f : R! R s strctl ncreasng on the set of values taen on b u. We have been developng a model of the consumer based upon the preference relaton % and some assumptons (hopefull realstc) about the preference relaton. We would le to represent our preference relaton % wth a utlt functon (so that we can use calculus to solve the problem). Based on a ratonal preference relaton %, we ensure that the utlt functon has certan propertes when we mpose monotonct and convext on our preference relaton %. Theorem 6 Let % be represented b u : R n +! R. Then:. u (x) s strctl ncreasng f and onl f % s strctl monotonc. u (x) s quasconcave f and onl f % s convex 3. u (x) s strctl quasconcave f and onl f % s strctl convex In order to facltate ndng a soluton to the consumer s problem we mpose d erentablt of the consumer s utlt functon u (). Le contnut, d erentablt s a mathematcal assumpton. When u () s d erentable, we can nd the rst-order partal dervatves. The rst-order partal dervatve of u (x) wth respect s called the margnal utlt of good. We can now de ne the margnal rate of substtuton (MRS) between two goods as the rato of the margnal utltes of the two goods. So the margnal rate of substtuton of good for good s: MRS (x) () What the MRS tells us s the rate at whch we can substtute one good for the other, eepng utlt constant. 3 Consumer s problem The consumer s general problem s to choose x X such that x % x for all x X. However, when X R n + ths smpl means that the consumer chooses an n nte amount of all goods. Thus, we restrct the consumpton set to a feasble set B X R n +. The consumer s problem then s to choose x B such that x % x for all x B. Snce t s easer to wor wth utlt functons than preference relatons, we mae the followng assumptons about our preference relaton %. Assume the preference relaton % s complete, re exve, transtve, contnuous, strctl monotonc, and strctl convex on R n +. Ths means that % can be represented b a real-valued utlt functon that s contnuous, strctl ncreasng, and strctl quasconcave on R n Maret econom In a maret econom the consumer wll face a prce vector p, where there s one prce for each of the n goods. We assume that the prce vector p s strctl postve, or p >> 0 so that each p > 0. Also, the prce vector s xed and exogenous to the consumer s decsons therefore, an ndvdual consumer has no mpact on the prce of ANY good. The consumer also has a xed amount of mone > 0. Ths s an endowment (for Ths s ust an assumpton that can be changed. 4

5 x /p B /p x now), meanng that the consumer smpl receves ths sum of mone. The consumer CANNOT spend more than ths partcular amount of ncome. The combnaton of postve prces, nte ncome, and the assumpton that the consumer cannot spend more than hs ncome restrct the consumpton set, X, to the feasble set B. Thus, the consumer s budget constrant s gven b: Wth the budget constrant we can now create the budget set B, where: nx p x () B xx R n +; px (3) When there are goods, the budget set B s: Because of our assumptons about % and ts relatonshp to u (), we can formulate the consumer s problem as: max xr n + u (x) subect to px (4) Thus, our consumer s problem s an nequalt constraned maxmzaton problem. The soluton to the problem, x, s the x such that u (x ) u (x) for all x B. Gven the relatonshp between % and u (), ths means that x % x for all x B and that x solves our orgnal consumer s problem wth the preference relaton. Note that the partcular soluton x wll depend upon the parameters of the problem, or the prces and ncome that the consumer faces. Thus we wll wrte x as x (p; ), wth x (p; ) representng the partcular quantt of good. A few general results. We "now" that the optmal bundle x (p; ) wll le on the budget constrant, or where px. Ths s because preferences are strctl monotonc. If the consumer chooses a bundle of goods on the nteror of the budget set (not along the budget constrant), then that consumer wll alwas be able to nd another bundle that s preferred to the chosen bundle (because there s some feasble bundle wth more of both goods). Also, because > 0 and x 6 0, we now that the consumer consumes a postve amount of at least one good. Snce % s assumed to be strctl convex, the soluton x (p; ) wll be unque, 5

6 so that x (p; ) s a demand functon that spec es the amount of each good a consumer wll choose gven prce vector p and ncome level. We call these x (p; ) the Marshallan demand functons. To nd them, smpl solve the nequalt constraned maxmzaton problem b settng up the Lagrangan, d erentatng, and solvng for each of the x. L (x; ) u (x) + [ px] (5) Assumng that x >> 0, we now there s a 0 such that (x ; ) (x) p 0 ; :::; px 0 (7) [ px ] 0 (8) Snce we are assumng % s strctl monotone we have px 0, whch leaves us wth n + equatons and n + unnown. Whle t s possble that ru (x ) 0 t s unlel that ths s so we assume ru (x ) 6 0. So we wll > 0 for at least one ; :::; n. Snce p > 0, we have that > 0 because from: For an two goods we can rewrte ths (x p (x ) > 0 (x (x @u(x p () Recall ) s the margnal rate of substtuton between goods and. Thus, at the optmum, the MRS between goods and wll be equal to the slope of the budget constrant. Ths s smpl the mathematcal result that one would see n an ntermedate mcroeconomcs class. The gure llustrates that the consumer optmum s where the nd erence curve s tangent to the budget constrant (pont E), whle also showng wh other ponts cannot be optmal. If the consumer were at pont G, there are man bundles that are strctl preferred to G (ncludng pont E). Whle pont F s on the budget constrant t s not optmal as t s nd erent to pont G (snce t les on the same nd erence curve) and we have alread seen that G s not optmal. 3.. Example Consder the utlt functon u (x ; x ) x x. The prces of good s p > 0 and the prce of good s p > 0. The consumer has ncome > 0. The consumer s problem s: max x x x s.t. p x + p x (3) 0;x 0 We can form the Lagrangan, d erentate wth respect to x, x, and, and nd the soluton as: D erentatng we have: L (x ; x ; ) x x + [ p x p x ] (4) x x p 0 x x p 0 p x p x 0 (7) [ p x p x ] 0 (8) 6

7 E G I F I Good B 7

8 Agan, the budget constrant wll hold wth equalt so: Smplfng we have: x x p 0 x x p 0 p x p x 0 x x p x x x x p x p x p x xp p Now, substtutng nto the budget constrant we have: p x p xp p 0 p x p x 0 p x + p x x p + p x + x To nd x we smpl plug x bac nto x xp p : p (+)p x xp p x x (9) (0) () p (+)p p () (+)p Thus, f we have done the calculus and algebra correctl, we have: x (p; ) (3) ( + ) p We can chec that at the optmum we have: x (p; ) MRS p or MU p ( + ) p (4) MU p (5) MU x x (6) MU x x (7) Substtutng x and x and dvdng b the respectve prces we have: MU p MU p (8) x x x x p p (9) x x p p (30) (+)p p ( + ) p p 8 (+)p p (3) ( + ) p p (3)

9 Techncall we should chec to mae sure that the nteror soluton IS the optmal soluton, and that there s not a better soluton at a corner (when ether x 0 and x p or x p and x 0). However, n ths problem u (x ; x ) x x, so f ether x or x equals 0 then the utlt functon s unde ned. So the consumer should bu at least some small postve amount of each good wth ths utlt functon. Usng,, p 5, p 0, and 50, the followng pcture s a two-dmensonal representaton of the consumer s problem: x x 3.. Second example Now suppose that u (x ; x ) x + x, wth prces p > 0 and p > 0 respectvel, and > 0. start b settng up the Lagrangan and followng our steps: We can L (x ; x ; ) x + x + [ p x p x ] (33) D erentatng we p 0 p 0 (35) p x p x 0 (36) [ p x p x ] 0 (37) Whle the budget constrant wll stll hold wth equalt, combnng the rst two equatons we get: p p (38) Snce ths condton does not depend on x or x t wll onl be true for certan parameters. If the parameters are such that p p, then an combnaton of x and x such that the budget constrant holds wth equalt wll be a soluton to the problem (n ths case we have a Marshallan demand correspondence, not a Marshallan demand functon). However, f p 6 p, then the consumer would le to spend all of hs ncome on ether x or x. Thus we can chec the "corners" to see whch gves hgher utlt. The next secton dscusses checng corner solutons more generall. 9

10 3..3 A more general example Whle we ma not have a guarantee of an nteror soluton, we stll want to restrct x 0 and x 0. So, our consumer s problem s stll to maxmze utlt subect to hs budget constrant, but now we have the addtonal constrants that x 0 and x 0. Wrtng ths out for a two good problem we have: max u (x ; x ) s.t. p x + p x, x 0, x 0. x ;x We have alread seen ths general example for an nequalt constraned optmzaton problem. For our spec c problem, we need all of the nequalt constrants as constrants. Snce x 0 and x 0 are alread wrtten n ths manner, that ust leaves rewrtng the budget constrant as p x p x 0. Now we can form the Lagrangan: L (x ; x ; ; ; 3 ) u (x ; x ) + [ p x p x ] + [x ] + 3 [x ] We wll now have a full set of Kuhn-Tucer condtons for both our choce varables and our @x p + 0; x 0; p + 3 0; x 0; p x p x 0; 0; p x p x 0 x 0; 0; x 3 x 0; 3 0; 3 x 0 Note that n ths case we have complementar slacness condtons for the choce varables because we are uncertan as to whether or not the constrants are bndng. Techncall we would have 3 cases to chec, one for each possble combnaton of x, x,,, and 3 beng ether strctl postve or zero. However, we now that the budget constrant wll bnd, and we now that ether x 0 or x 0 so we reall onl have to chec f x 0 or x 0 (wth more goods, for example three goods, we would stll have to chec whether x x 0 and x 3 > 0, x x 3 0 and x > 0, or x x 3 0 and x > 0). In general, the process I would use to nd the optmal value would be to set up the Lagrangan functon and assume an nteror soluton (or argue that the soluton must be nteror) and then chec the potental corner solutons. If ou cannot nd a unque optmal nteror soluton (whch would be the case f wth our lnear utlt functon example we had p p ), then I would suggest checng the varous "corners". Contnung wth the lnear functon example, f u p ; 0 > u 0; p, then the consumer would choose to consume onl x. Ths would be true f: > (39) p p To mae t easer to see that ths s optmal, assume. consume onl the good that s less expensve. Then the consumer would smpl choose to 4 Addtonal formulatons of the consumer s problem We wll loo at two addtonal formulatons of the consumer s problem. In the rst we create the consumer s ndrect utlt functon. The ndrect utlt functon possesses a few useful propertes that we can tae advantage of. In the second we formulate the consumer s problem as an expendture mnmzaton problem. In ths problem, the consumer s goal s to set a target level of utlt and then nd the bundle that mnmzes expendture. 4. Indrect utlt functon When we set up the consumer s utlt functon we have the consumer maxmzng u (x) b choosng a bundle of goods. For an set of prces p and ncome the consumer chooses x (p; ) that maxmzes u (x). The value of the utlt functon at x (p; ) s the maxmum utlt for a consumer gven prces p and ncome. 0

11 We can de ne a functon that relates the maxmum value of utlt to the d erent prce vectors and ncome levels a consumer ma face. De ne a real-valued functon v : R n+ ++! R as: v (p; ) max xr n + u (x) s.t. px (40) The functon v (p; ) s called the ndrect utlt functon because the consumer s not drectl maxmzng v but ndrectl maxmzng v b maxmzng u. If u (x) s contnuous and strctl quasconcave, then there s a unque soluton to ths optmzaton problem, and that s the consumer s demand functon x (p; ). There s a relatonshp between v (p; ) and u (x): v (p; ) u (x (p; )) for some prce vector p and ncome level. There are a number of propertes the ndrect utlt functon possesses and the are summarzed n the theorem below: Theorem 7 If u (x) s contnuous and strctl ncreasng on R n +, then v (p; ) s:. Contnuous on R n ++ R +. Homogeneous of degree zero n (p; ) 3. Strctl ncreasng n 4. Decreasng n p 5. Quasconvex n (p; ) 6. Ro s Identt: If v (p; ) s d erentable at p 0 ; 0 p 0 ; 6 0, then: x p 0 ; p0 (p 0 ; 0 ; ; :::; n The rst ve ponts are smpl restrctons on v (p; ) gven restrctons on u (x). The sxth pont greatl smpl es ndng the consumer s Marshallan demand functon f the ndrect utlt functon s nown. Whle we wll not prove these results (proofs and setches of proofs are n the text), we wll wor through an example usng the followng ndrect utlt functon: v (p; ) 3 3 p p p3 3 wth > 0 and Ths ndrect utlt functon can be found b solvng the consumer s maxmzaton problem and substtutng the Marshallan demands nto the utlt functon. Consder: where > 0 and so that: v (p; ) u (x ; x ; x 3 ) x x x3 3 (4) We now that: x (p; ) p (4) 3 3 (43) p p p 3 v (p; ) p p p3 3 v (p; ) 3 3 p p p3 3 (44) (45)

12 To show that v (p; ) s homogeneous of degree zero, we have: v (tp; t) v (tp; t) 3 3 t (tp ) (tp ) (tp 3 ) t t ++3 p p p3 3 v (tp; t) 3 3 p p p3 3 v (tp; t) v (p; ) To show that v (p; ) s ncreasng n we smpl nd the partal dervatve wth respect to and show that ths dervatve s strctl (p; 3 3 p p p3 3 Ths partal dervatve s strctl greater than 0 snce p > 0 and > 0. To show that v (p; ) s decreasng n p, we can nd the partal dervatve wth respect to an prce we rewrte as: v (p; (p; To provde an example of Ro s dentt we have: p + > 0 p p p p p < 0 Tang the (p; p p + p p x (p; ) x (p; ) p + x (p; ) p p + p p p p p p p p Note that these demand functons are smlar to those that we found when worng through the standard maxmzaton problem when u (x ; x ) x x. Recall that the demand functons n that example were x (p; ) (+)p and x (p; ) (+)p. If we mpose + (as we have done n the ndrect utlt functon example), we have the same form for the demand functons. So the ndrect utlt functon for the Cobb-Douglas utlt functon (wth 3 goods) s v (p; ) 4. Expendture functon p p p 3 3 Wth the standard utlt maxmzaton problem we assume that the consumer has a xed budget constrant and then determnes what the maxmum level of utlt can be acheved gven p and. However, we can.

13 formulate a smlar problem where the consumer xes a target level of utlt and then chooses the ncome level whch mnmzes expendture to attan that target level of utlt. In essence, the nd erence curve s xed and the consumer s shftng the budget constrant bac and forth trng to nd the lowest possble cost to acheve hs target utlt level. Formall we can de ne the expendture functon as: e (p; u) mn xr n + px subect to u (x) u (46) The soluton to the expendture mnmzaton problem s nown as the Hcsan demand functon, of x h (p; u). If we nd x h (p; u), then we now that e (p; u) p x h (p; u) because ths s the bundle whch mnmzes expendture for utlt level u and prces p. What these demand functons tell us s how purchases change when we hold utlt constant and there s a prce change of one good. Let s use u (x) x x x3 3 wth to nd the Hcsan demands. Set up the Lagrangan: L (x ; x ; x 3 ; ) p x + p x + p 3 x 3 + [u x x x3 3 ] (47) D erentatng: We nd p p 3 p 3 3 x x x3 x x3 3 x x 3 0 (48) 3 0 (49) 0 (50) u x x x3 3 0 (5) p x x x3 3 p x x x3 3 (5) x p x p (53) x x p p (54) We also have: p 3 3 x 3 3 x x p x x x3 3 (55) x 3 p 3x p 3 (56) x 3 3x p p 3 (57) 3

14 Pluggng nto the utlt constrant we have: u x p p x 3 3 x p 0 (58) p 3 3 x p 3 x x p u (59) p p 3 3 p 3 p u (60) p p 3 3 p p3 u x (6) p 3 p x ++3 u p p3 3 3 p 3 3 p3 u p p p x (6) x (63) u p p p 3 3 x (64) u p p p 3 3 u p p3 3 p 3 3 x (65) x (66) To clarf, we are ndng the Hcsan demands when solvng the expendture mnmzaton problem, so We can then nd that: x h (p; u) u p p3 3 p 3 3 x h (p; u) u p p3 3 p 3 3 x h 3 (p; u) u p p 3p As we dd wth the ndrect utlt functon, we have a set of results wth the expendture functon: Theorem 8 If u () s contnuous and strctl ncreasng, then e (p; u) s:. Zero when u taes on the lowest level of utlt n U. Contnuous on ts doman R n ++ U 3. For all p >> 0, strctl ncreasng and unbounded above n u. 4. Increasng n p 5. Homogeneous of degree n p. 6. Concave n p 7. If u () s also strcl quasconcave, we have Shephard s lemma: e (p; u) s d erentable n p at p 0 ; u 0 wth p 0 >> 0, p 0 ; u (67) (68) x h p 0 ; u ; :::; n (70) 4

15 Agan, we wll tae these wthout proof although the text has proofs. Expendture s zero when utlt s at ts lowest possble level there s no need to spend an mone to acheve that level. As u ncreases, expendture must ncrease (holdng prces constant). Also, expendture s unbounded as utlt ncreases. If prces (or one prce) ncrease, then expendture does not decrease. There are also homogenet, concavt, and contnut results. Fnall, we can derve the Hcsan demand functons drectl from the expendture functon. Recall that the expendture functon s essentall px, or p x + p x + ::: + p n x n. Thus we can smpl d erentate the expendture functon to nd the Hcsan demands. As an example, consder the followng expendture functon: e (p; u) up p p where The expendture functon s zero when u 0 (from a Cobb-Douglas ths s the lowest level of utlt we can have). It s strctl ncreasng n u and unbounded above n u. It s ncreasng n an prce. For homogenet of degree n prces we have: For Shephard s lemma, we have: 4.. Relatng v (p; ) and e (p; u) e (tp; u) u (tp ) (tp ) (tp 3 ) p e (tp; u) ut p t p t up e (tp; u) t++3 p p (7) (7) (73) (74) e (tp; u) te (p; u) (p; u) u p x h (p; u) (76) There s a relatonshp between v (p; ) and e (p; u). If we x p and and let u v (p; ). As per the de nton of v (p; ), u s the maxmum utlt that can be attaned when prces are p and ncome s. Also, f the consumer wshes to acheve utlt level u at prces p, then the consumer wll be able to acheve that level of utlt wth expendture. But the expendture mnmzaton functon tells us the LEAST amount of expendture needed to acheve utlt level u at prces p. Thus, we would need: e (p; u) (77) e (p; v (p; )) (78) We can perform a smlar analss f we x p and u. We now that at prces p and utlt u, we wll need: v (p; ) u (79) v (p; e (p; u)) u (80) Now we have a theorem explanng the relatonshp between e (p; u) and v (p; ). Theorem 9 Let v (p; ) and e (p; u) be the ndrect utlt functon and expendture functon for some consumer whose utlt functon s strctl ncreasng and contnuous. Then for all p >> 0, 0, and u U:. e (p; v (p; )). v (p; e (p; u)) u 5

16 Agan, we forgo the proof and use an example. We wll use So: Also: e (p; v (p; )) v (p; ) 3 3 p p p3 3 e (p; u) up p p p p p p p p3 3 (8) (8) (83) e (p; v (p; )) (84) v (p; e (p; u)) up 3 p p p p p (85) v (p; e (p; u)) u (86) Theorem 0 Gven that u () s strctl ncreasng, contnuous, and strctl quasconcave, the followng relatons hold between the Marshallan and Hcsan demand functons when p >> 0, 0, and u U.. x (p; ) x h (p; v (p; )). x h (p; u) x (p; e (p; u)) Ths rst relatonshp states that the Marshallan demand for prces p and ncome are dentcal to the Hcsan demand at prces p when utlt s v (p; ). Alternatvel, the Hcsan demands at p and u are equal to the Marshallan demands when prces are p and expendture s gven b e (p; u). As an example: x h (p; u) u p x h (p; v (p; )) p p p p p p (87) (88) x h (p; v (p; )) p p (89) x h (p; v (p; )) p (90) x h (p; v (p; )) x (p; ) (9) Also: x (p; ) (9) p up p p x (p; e (p; u)) (93) p x (p; e (p; u)) u p p (94) x (p; e (p; u)) x h (p; u) (95) 6

17 5 Propertes of consumer demand In prncples and ntermedate mcroeconomcs ou tpcall stud demand functons rst and then (perhaps) utlt or consumer theor. Here we have started wth the utlt functon and used the utlt functon to derve demand functons. Rght now we are concerned wth ndvdual demand functons. When ou studed them at the undergraduate level, we smpl stated thngs le "The Law of Demand states that there s an nverse relatonshp between the prce of a good and ts quantt demanded". You were usuall ased to tae that on fath, and ntutvel t maes sense holdng everthng else constant, f ou rase the prce of a good ts quantt demanded should fall. Now, we wll see that the demand functons used n the undergraduate classes were a drect result of the assumptons that we have been mang about our preference relaton %, our utlt functon u (), and our feasble set. Two of the most basc concepts are relatve prces and real ncome. Economsts are more concerned wth relatve prces rather than actual prces, as consumers care about the quantt of mone onl n terms of the amount of goods and servces a partcular amount of mone can bu (people have lttle utlt for the actual good "mone", other than that t serves as a medum of exchange b whch the can purchase goods). Thus, we can dscuss prces n terms of relatve prces namel, we can x the prce of one good (call t the numerare) and then denomnate all other goods n that numerare. Economsts also dscuss purchasng power n terms of real ncome. If one ndvdual has $0,000 and the other has $00,000 then we tend to thn that the person wth $00,000 s better o than the one wth $0,000. Ths s true f th prces are the same, but f the ndvdual wth $0,000 faces a prce vector that s 00 of the prce vector that the ndvdual wth $00,000 (so that the person wth $00,000 faces prces that are 00x hgher), then the person wth $0,000 wll be better o than the person wth $00,000 because the person wth $0,000 can purchase more goods. If consumer preferences are complete, transtve, re exve, strctl monotonc, and strctl convex, then Marshallan demand functons are homogeneous of degree zero (whch essentall means that f ou ncrease all prces and wealth b the same proporton the consumer s Marshallan demand does not change) and budget balancedness holds (the budget constrant holds wth equalt). 5. Income and substtuton e ects Whle ndng the soluton to the UMP or the EMP s an mportant step, man economsts focus on what happens when somethng changes n the economc sstem. We wll begn b dscussng prce changes n Hcsan demand, as Hcsan demand sats es the law of demand (prce ncreases, quantt demanded decreases) whle Walrasan demand ma or ma not. However, Hcsan demand s a functon of an unobservable varable, utlt. Walrasan demand, however, s a functon of the observable varables (or at least varables that we mght be able to observe) prce and wealth (or ncome). We have that the own-prce dervatves of Hcsan demand are nonpostve because Hcsan demand follows the compensated law of demand. Ths h (p; u) Recall that wth a Hcsan demand change we are determnng how much quantt demanded falls when prce ncreases b eepng the consumer on the same nd erence curve (or at the same utlt level).gven that our nd erence curves are downward slopng, t s necessarl the case that Hcsan demand decreases (f we have a d erentable utlt functon and are at an nteror soluton) as the gure above shows or remans at zero (f we have a utlt functon that s nond erentable and are at an nteror soluton we sta at the same pont, thn of perfect complements or f we are at a corner soluton). We can also show ths mathematcall as we (p; u) x h e (p; (p; @p Ths s because the expendture functon s a concave functon. 7

18 Good A x A x A x B x B Good B 8

19 Now consder the cross-prce dervatve of Hcsan demand x h (p; u) wth respect to the prce of good, p. 0 then goods and are complements or complementar goods, because as the prce of good ncreases the Hcsan demand for good decreases. Thus we are consumng less of good and less of good when p 0 then goods and are substtutes because as the prce of good ncreases the Hcsan demand for good ncreases. Note that f the cross-prce dervatve s equal to zero then the goods could be class ed as ether substtutes or complements. However, consder what t means f the cross-prce dervatve trul s zero a change n p has no e ect on the Hcsan demand for good. Thus the two goods could be class ed as ndependent. We now that there must be at least one good whch whch has a nonpostve substtuton e ect for an spec c good n the econom. To see ths, consder the -good case. If the prce of good ncreases, then the consumpton of good wll decrease (unless the consumer s at a corner soluton) because Hcsan demand follows the compensated law of demand. Now, f the consumer s to reman at the same utlt level, and he s consumng less of good, then he must consume more of good. (p;u) 5.. Decomposng Hcsan demand changes The purpose of usng Hcsan demand s because Hcsan demand follows the compensated law of demand. But, we cannot observe Hcsan demand because one of ts arguments s unobservable (utlt level). We can explot the relatonshp between Hcsan demand and Walrasan demand to obtan nformaton on prce e ects. 3 Proposton (The Sluts Equaton) Suppose that u () s a contnuous utlt functon representng locall nonsatated preference relaton % de ned on X R N +. Then for all (p; ) and u v (p; ) we h (p; (p; ) (p; ) x (p; ) for @ Or, rewrtng n terms of the cross-prce e ect of the Walrasan (p; (p; u) (p; ) x (p; ) for @ Proof. We now that x h (p; u) x (p; e (p; u)) at the optmal soluton to the consumer s problem. We can d erentate wth respect to p and evaluate at p and u. Statement + x h. Chan rule for d erentaton (p; u). Earler result on relaton of e (p; u) to x h (p; u) 3. x h (p; u) x (p; e (p; u)) x (p; ) 3. Earler result on relaton of x h (p; u) to x (p; ) 4. e (p; u) 4. Earler result on relaton of e (p; u) 5. x (p; ) 5. Substtuton For the Walrasan demand, the change n quantt of good wth respect to a change n the prce of good s nown as the Total E ect of the change n prce of good. The total e ect s decomposed h the Substtuton E ect and the Income (or Wealth) x (p; ). The Substtuton E ect s the change n quantt demanded of good due to the fact that good s now relatvel more (less) expensve f the prce of another good (sa good ) ncreases (decreases) when the prces of all other goods sta the same. Thus, f the prce of a good ncreases, we would expect that a consumer would purchase more of a second good because s now a relatvel less expensve substtute (unless of course the goods are complements). The Income E ect s the change n quantt demanded of good due to the fact that 3 For a recent reference on usng the Sluts equaton n emprcal wor, see Fsher, Shvel, and Buccola (005). Actvt Choce, Labor Allocaton, and Forest Use n Malaw. Land Economcs, Vol. 8:4 9

20 x Budget constrant when p ncreases x h Hcsan compensaton budget constrant x Intal budget constrant x SE ( ) TE (+) x IE ( ) Fgure : Decomposng the e ect of a prce change on a G en good. the consumer has control over how he spends hs wealth. There need be no actual change n for there to be an ncome e ect, but f the prce of good ncreases, then the consumer ma not ust decde to reduce consumpton of good at the rate of the prce ncrease. For example, f p doubles, the consumer ma eep consumpton of good the same and smpl reduce consumpton of good to ts prevous level, but s not requred to act n ths manner. The consumer ma cut consumpton b more (or less) than and adust consumpton of good accordngl. The consumer ma even ncrease the amount of good when a prce ncrease occurs ths s the case of a G en good, and t occurs because the Income E ect overwhelms the Substtuton E ect. Consder the Sluts equaton for a change n the own-prce of a (p; (p; u) (p; ) x h We now that f p ncreases that the Substtuton E ect wll be negatve. However, there s no such restrcton on the Total E as t ma be postve or negatve (recall the case of G goods). It wll be postve f the Income E ect s more negatve than the Substtuton E ect (remember, ths s an OWN-prce equaton, so the Hcsan demand must decrease when ts own prce ncreases). In ths case, we have a G en > 0. Thus, t s an usuall large negatve ncome e ect that s drvng the G en good result. Tpcall one would thn that ncome e ects would be postve (we now that x (p; w) 0, so ). Ths s ust the dervatve of the Walrasan demand functon wth respect to wealth, and usuall f wealth ncreases consumers consume more of a good (hence the reason we call these goods normal goods ). However, f a good s a G en good then t must have a wealth e ect negatve enough to overwhelm the negatve substtuton e ect. Thus an good that s a 0

21 G en good must be an nferor good. However, ths does not mean that all nferor goods are G en goods f the dervatve of the Walrasan demand s negatve (so that the good s nferor), t s possble that the wealth e ect s LESS negatve than the substtuton e ect. In ths case, whle the good s nferor, ts total e ect wll stll be negatve. Fgure shows the e ect of a prce change of a G en good decomposed nto ts total, substtuton, and ncome e ects. The ntal budget constrant s n blac and the optmal bundle s represented b x. The new budget constrant after an ncrease n the prce of good x s gven n red and ts optmal consumpton bundle s represented b x. The blue budget constrant s the budget constrant that returns the consumer to hs orgnal utlt after the prce change and the optmal bundle s represented b x h. Now, the total e ect s smpl the change n good x when ts prce changes, so we compare the quantt of x consumed under the ntal budget constrant wth the quantt consumed under the budget constrant when p ncreases. Note that there s an INCREASE n consumpton of x when p ncreases thus we have a G en good (the exact "equaton" to nd ths s quantt of x consumed at bundle x mnus quantt of x consumed at bundle x ). To nd the ncome e ect, compare the quantt of x consumed under the new budget constrant wth the quantt of x consumed under the budget constrant wth the new relatve prces that returns the consumer to hs ntal utlt level (the Hcsan compensaton budget constrant as t s labeled). Agan, to nd ths tae the quantt of x consumed at x h and subtract the quantt of x consumed at x. The substtuton e ect s smpl the change n consumpton of x at x to consumpton of x at x h (tae the amount of x consumed at x h and subtract the amount of x consumed at x ). Note that snce ths s an own-prce e ect on Hcsan demand t must be negatve. We can do the exact same analss for good x when the prce of good x ncreases. For good x, ts total e ect s negatve, whle ts substtuton e ect s postve (onl two goods so the must be substtutes) but ts ncome e ect s MORE postve than ts substtuton e ect, leadng to the negatve total e ect. 4 Now, there are a few addtonal results that rel on the Hessan matrx of the expendture functon e (p; u). If we tae the dervatve of e (p; u) once wth respect to p we wll obtan a row vector of length N, where N s the number of goods (we wll have one dervatve for each of the N goods). Recall that our Hcsan demand wthout a subscrpt, x h (p; u) s reall a vector of Hcsan demands, one for each good, or x h (p; u) x h (p; u) x h (p; u) for the two-good world. Alternatvel, we could So the Hcsan demand functon s nothng more than the gradent of the expendture functon n pure math terms. Note that the vectors are the h (p; u). The Hessan matrx s smpl an N N matrx of second partal dervatves. For our two-good world, we would have: Now, a e (p; (p; u) @p # h Proposton Suppose that u () s a contnuous utlt functon representng locall nonsatated preference relaton % on the consumpton set X R L +. Suppose also that x h (; u) s contnuousl d erentable at (p; u) and denote ts N N dervatve matrx b (p; u). Then. (p; u) D pe (p; u). (p; u) s a negatve semde nte matrx. 3. (p; u) s a smmetrc matrx. 4. (p; u) p 0: We have alread dscussed the rst result. The second and thrd results have to do wth the fact that snce e (p; u) s a twce contnuousl d erentable concave functon, t has a smmetrc and negatve semde nte Hessan matrx. The fourth result follows from Euler s formula snce x h (p; u) s homogeneous of degree zero n prces. Homogenet of degree zero mples that prce dervatves of Hcsan demand for 4 Obtanng the correct sgn for these e ects ma be a lttle confusng. The e s to tae the amount of the good at the NEW bundle, and subtract the amount of the good at the orgnal bundle. # :

22 an good, when weghted b these prces, sum to zero. Euler s formula states f x h (p; u) s homogeneous of degree zero n prces and wealth, then: h # h (p;u) h # (p; u) p p 0 p h p p As for the terms smmetrc and negatve semde nte matrx, a smmetrc matrx s smpl a matrx that equals ts transpose (to transpose a matrx smpl tae the rst column of the orgnal matrx and mae that the rst row of the transpose, then tae the second column of the matrx and mae that the second row of the transpose, etc.). So, for our two-good world, f (p; u) s our matrx and T (p; u) s ts transpose, Or h (p; u) T (p; h # h Notce that the two o dagonal elements are swtched. Thus, snce (p; u) s or usng the expendture functon e(p;u) use to d erentate wth rst the result wll be the same. Thus, t does not matter whch prce ou For a refresher on the de nton of negatve semde nteness, tae a loo at the mathematcal appendx. That (p; u) s negatve semde nte ensures us that the own-prce dervatves of the Hcsan demand functon are less than or equal to zero (note that the own-prce dervatves of the Hcsan demand functon are the elements along the dagonal of (p; u)). 5. Elastctes One nal concept that we wll dscuss whch s commonl used n economcs s elastct. Elastct s a untless measure, and t tells us how responsveness quantt changes are to changes n prces or ncome (or, more generall, a "dollar" measure). B de nton, elastctes are determned as the rato of the percentage change of one varable to another. There are three common elastctes n consumer theor: own-prce elastct of demand, cross-prce elastct, and ncome elastct. 5.. Prce elastctes The general formula for a prce elastct s: (p; x (p; ) If we have then we have an own-prce elastct of demand. In 0 as most goods are NOT G en goods. As such, we tpcall tae the absolute value of own-prce elastct of demand (snce t s negatve). If 0 " < we sa that demand s nelastc. Ths smpl means that there s not as large a percentage change n quantt as there s n prce. If " then we sa that demand s elastc, or that the percentage change n quantt s larger than that of prce. If we have 6 then we have a cross-prce elastct of demand how responsve s the quantt of good to a change n the prce of good. Note that cross-prce elastct of demand can be postve or negatve. If " > 0 then we have substtute goods, as a prce ncrease n good wll cause more of good to be consumed (consumers substtute awa from good towards good ). If " < 0 then we have complements or complementar goods, as the consumer s now bung less of good n response to a prce ncrease n good. If " 0, then the two goods are ndependent. 5.. Income elastct The general formula for ncome elastct (p; x (p; )

23 Agan, ncome elastct ma be postve or negatve. If 0 then we have a normal good as purchases of good ncrease (or sta the same) as ncome ncreases. If < 0 then we have an nferor good as consumers are purchasng less of good despte more ncome. 3