901 Notes: 11.doc Department of Economics Clemson University

Transcription

1 901 Notes: 11.doc Department of Economcs Clemson nversty Consumer's Surplus 1 The dea of consumer's surplus s to attempt to measure n money terms the value of consumpton of a good from the nformaton contaned n the demand curve. The noton s to take the area under the demand curve and attrbute to ths the welfare mplcaton that ths s how much the good s worth to the consumer n total. The dea seems to make sense when we thnk about utlty constant demand curves of the sort (.). Let the prce of the th good go from hgh, 1, to low, 0. The area under the utlty constant demand curve s: z 1 - d 1 0 Recall that from the envelope theorem the dervatve of the mnmzed ependture level (M ) w.r.t. prce n the utlty constant problem s the consumpton level of the th good. 2 From ths we can wrte: z z - d = - M (, ) Hence, the defnte ntegral s well defned as: d 2 z 1 - d = M 1 = M ( 0) M ( 1) The ntegral to the left of the utlty constant demand curve between two prces can be nterpreted as the amount of money that a lower prce 0 s worth to the consumer startng from the hgher prce, 1. That s, how much money wll the low prce reduce the consumer s ependtures assumng that the consumer stays on the same ndfference curve. Operatonally, however, ths s not obvously useful. The problem s that we cannot observe utlty and, hence, cannot hold t constant. On the other hand, observable demand curves are based on holdng money ncome constant. nfortunately these demand curves do not have the property of defnte ntegraton found n the utlty constant demand curves (see append). That s, we cannot dentfy the area under the money constant demand curve as meanng anythng. 3 Of course, f the ncome effect for good s zero, then the ordnary demand curve les on top of the compensated demand curve and the area under one s the same as the other. Even 1 See Slberberg (1990) p ; p , secton 11.5; Slberg (2000) p ; p ; Layard and Walters (1978) p ; Varan (1992) Chapter 10; Ncholson (1998) pp ; Ncholson (2000) p Stars are used to desgnate the functons epressng optmzed values n terms of the parameters of the problem. Where there s some ambguty regardng the arguments n these optmzed functons, as s the case n terms of the demand curves, then more specfc notaton s used. The arguments of these functons are not generally eplctly noted. 3 The problem from an old comprehensve eamnaton nvolvng corn and rats hghlghts ths pont. A formal dervaton s found n the append to ths lecture. Revsed: October 3,

2 901 Notes: 11.doc Department of Economcs Clemson nversty so, we stll appeal to the ntegraton of the compensated demand curve, as gven n eqt. (3) above, for the nterpretaton of what ths are means. Whle the area under the ordnary demand curve does not mean anythng, we are not completely at a loss. From the Slutsky equaton, we know that the slope of utlty constant demand curves can be computed from the money constant ones. The amount that a consumer wll pay for a lower prce s computed along the money constant demand curve by determnng the shft n the money constant demand curve that results from the lump sum payment to get the lower prce. Consder Fgure 1. Shown there s a prce change n good from p 0 to p 1. The consumer starts at pont a and moves to b. ont c s the consumer s choce gven a compensatng adjustment n ncome that would keep the consumer on the orgnal utlty contour. In ths sense we can call that ncome shft, measured n y unts n the vertcal plane (because the prce of y stays constant), the amount that the consumer would have to be pad to wllngly accept the hgher prce p 1 startng from the prce p 0. The vertcal dstance between c and b s ths y unt measure. Ths compensatng ncome adjustment s also measured as the area under the compensated demand curve. The compensated demand curve s pctured n the second frame as the lne segment ac. The area under t s really the area to the left because we draw demand n a Marshallan way. The ordnary demand curve s gven by the lne segment ab. An alternatve measurement s the compensated demand curve shown n the second frame of Fgure 1 as bd. Ths s the compensated demand curve bult along the utlty contour assocated wth the consumer s second choce, the choce condtoned on the new prce p 1. Ths compensated demand curve answers the queston, how much would the consumer be wllng to pay to keep the orgnal prce p 0 nstead of sufferng the hgher prce p 1? The answer s the area under the compensated demand curve bd between the two prces. Ths area s also measured n the top frame n y unts as the vertcal dstance between the ponts a and d. Though t s not precsely seen n the top frame, the area under the compensated demand curve bd s less than the area under the compensated demand curve ac. In general, consumers must be compensated more to wllngly accept a prce ncrease than they wll pay to avod such a fate. Ths asymmetry of consumer behavor whch s a commonplace observaton s not an anomaly but rather a result of consumer theory. 4 Notce that the area under the ordnary demand curve, ab n the lower panel, measures nothng n ths dscusson. However, t s an appromate average of the areas under the two dfferent compensated demand curves. Moreover, the slope of the ordnary demand curve along wth ts ncome effect gves us a measure of the slope of the compensated demand curves va the Slutsky equaton. On ths bass, I wll defne Consumer Surplus as the area under the compensated demand curve. Ths s not a unque defnton for ths term. Some people defne consumer surplus dfferently. Moreover, many people call the area under the compensated demand curve somethng else as well. The student s smply alerted to ths nconsstently n the jargon of 4 For nstance, t has often been noted n epermental surveys that the payment consumers requre to wllngly partcpate n a drug test that has a 1:100 chance of kllng them s larger than the amount that consumers wll pay to acqure a drug that wll completely cure them when they have a 1:100 chance of dyng. Rather than beng evdence of rratonal behavor, we have shown that ths behavor s completely predctable based on standard consumer theory. Revsed: October 3,

3 901 Notes: 11.doc Department of Economcs Clemson nversty economcs and s counseled to smply state one's own defnton when speakng about ths problem. I have adopted the area under the compensated demand curve as consumer surplus because t s logcally consstent and arguably observable from nformaton contaned n the ordnary demand curve. Moreover, t has mplcatons about the behavor of ndvduals. Consumer Surplus and rce Indces The queston of consumer surplus s closely lnked to the noton of prce ndces. There are two standard defntons of prce ndees. One s called the Laspeyres nde. It s defned as p p 1 0, 0 0 that s, the effect of the prce change s defned n terms of the orgnal consumpton bundle desgnated by 0, where p 0 s the orgnal prce vector and p 1 s the new set of prces. The other prce nde s aache. 5 Here the prce change s defned n terms of the new consumpton bundle. Fgure 2 uses our two good graph to dentfy the Laspeyres prce nde. Let the prce of go up so that the consumpton choce shfts from a to b. The effect of the prce change as measured by the Laspeyres prce nde s shown n the lower panel as p 1 -p 0 tmes a. Ths s the large rectangle etendng outsde of the demand curves. Ths ncome loss s also shown n the upper panel as t-r. The mtgatng effect of the consumer's reacton n choosng less s t-s so that the true loss due to the hgher prce of s s-r. Of course, ths s the area under the compensated demand curve ac n the lower panel. Notce that calculatng the loss by the area under the ordnary demand curve understates the loss n consumer welfare. Compensatng and Equvalent Varatons 6 A consstent defnton of the terms compensatng and equvalent varatons can be found usng the area under the compensated demand curve. Let s start wth the defntons used by Layard & Walters on p. 151: The compensatng varaton (CV) s the amount of money we can take away from an ndvdual after an economc change, whle leavng hm as well off as he was before t. The equvalent varaton (EV) s the amount of money we would need to gve an ndvdual f an economc change dd not happen, to make hm as well off as f t dd. The dfference between CV and EV can be thought of n terms of the clam on the state of nature that s ascrbed to the consumer. For CV, the consumer has a clam to the pror state of nature; n EV, the latter. For the prce change shown n Fgure 1, f the ntal poston s a, then CV defnes 5 Donald McClosky clamed that Laspeyres s pronounced 'la sparce' whch rhymes wth 'scarce'. aasche rhymes wth squash. 6 Layard and Walters (1978) p ; Varan (1992) secton Revsed: October 3,

4 901 Notes: 11.doc Department of Economcs Clemson nversty the consumer s welfare change from the hgher ndfference curve, whle EV defnes the welfare change n reference to the lower ndfference curve. Note that both CV and EV can be postve or negatve, but they are both of the same sgn. Ther sgn s the same as the drecton of the welfare change. Agan from Fgure 1, a prce change from a to b s a welfare declne. Both CV and EV are negatve. CV s gven by the area under the compensated demand curve shown n the lower panel as the lne segment ac. It s negatve. EV s the area under the compensated demand curve bd. CV s a bgger negatve number than EV, but ths means that n real number space EV>CV. If the prce change were reversed so that prces fell from pont b to pont a, then welfare would be ncreasng. Both CV and EV would be postve. CV would be defned n terms of the lower utlty level; EV n terms of the hgher. CV would then be the area under the compensated demand curve bd whle EV would be the area under ac. EV>CV. The rule s: For normal goods and measured n both postve and negatve numbers, EV s always bgger than CV. Consumer Surplus and All-or-Nothng Demand 7 It s clear that the precse applcaton of consumer surplus to a partcular problem depends crtcally on the precse queston that s beng asked. To add to ths basc confuson, the defnton of consumer surplus vares from one wrter to the net. The conventon adopted here s to defne consumer surplus as the area under a compensated demand curve. Ths s the defnton that Slberberg uses as well as Layard & Walters. However, Layard & Walters use the term a lttle more restrctvely. For them, consumer surplus s the prce-quantty rectangle under an all-or-nothng demand curve. All-or-nothng demand answers the queston, What s the mamum amount that a consumer would pay for a partcular quantty of? Or, what s the most that a seller can possbly charge for a partcular block or bundle of? Say the consumer would pay R 1 for 1. Ths represents a pont on the all-or-nothng demand wth 1 on the horzontal as and {R 1 / 1 } on the vertcal as. The ntercept along the vertcal as s called the ecluson prce (a concept that we wll dscuss more n a moment). Layard & Walters measure the all-or-nothng demand n the followng way. I have reproduced the Layard & Walters graph (ther fgure 5-11, p. 149) n Fgure 3; t s not drawn eactly as they do as I wll eplan n a moment. By ther defnton, the all-or-nothng payment that the consumer s wllng to pay for 0 unts of s gven by the dstance ab. The argument goes lke ths: If the consumer were allowed to buy at a prce assocated wth the slope of the lne ma, call ths p, the consumer would choose 0 unts. The consumer would be on the ndfference curve at m f prced out of the market. (The ecluson prce s a prce hgh enough so that the slope of the budget constrant makes m the optmal choce for the consumer.) The vertcal dstance between the two ndfference curves, ab, s the y measure of the value of ths consumpton choce. That s, ab represents the mamum amount that a consumer would be wllng to pay n addton to how much s already beng spent n order to have the good, p 0. Ther argument seems to make sense, but careful nspecton of Fgure 3 shows that t s not true n ths pcture. In Fgure 3 the consumer would be wllng to pay cm n order to have the 7 Layard and Walters (1978) p Revsed: October 3,

5 901 Notes: 11.doc Department of Economcs Clemson nversty rght to buy the good at the prce p. Ths s the area under the compensated demand curve startng at the ecluson prce down to prce, p. Snce cm > ab, to say that ab s the mamum the consumer would pay for the rght to buy at the prce, p, s to say that s a bad from 1 to 0, whch t s not. So the argument appears flawed. It turns out that n ther fgure, s depcted as havng an ncome elastcty of zero. Hence, 1 and 0 are the same pont. Moreover, n ther mathematcal dervaton, they do defne the all-of-nothng demand as the area under the compensated demand curve, so everythng s ok. As a pont of nterest, Layard & Walters defne compensatng and equvalent varatons as the dstances mc and em, whch are consumer surplus measured as the ntegrals under the compensated demand curves for the two ndfference curves relatve to the ecluson prces. Layard & Walters clam that the dstance mc s bounded, that s, t cannot be nfntely large, whereas, em can be. Implctly, they are assumng that pont m has been observed and pont e has not. Obvously f pont e ests, the equvalent varaton s bounded. If there s no ecluson prce for the ndfference curve upon whch pont a les, then the equvalent varaton s unbounded. The queston hnges on the value of the ecluson prce: If the ndfference curve s asymptotc to the vertcal as, the ecluson prce s nfnte and the value of both the compensatng and equvalent varatons are nfntely large. Summary of Consumer Theory When you came nto h.d. prce theory, one of the thngs that you were supposed to know s the prncples level, tet book treatment of the ncome and substtuton decomposton of a prce change. Ths analyss s portrayed n commodty space. In detal you were supposed to understand all the possble wggles and waggles of ths analyss. Second, you were supposed to know how to deal wth demand curves n what mght be called practcal terms. For nstance, f you were gven a lnear demand curve, you were supposed to be able to derve the margnal revenue functon, compute elastcty, and dentfy the area under the demand curve n terms of margnal revenue and elastcty. In ths class we concern ourselves wth the precse mappng between these two sets of analyss. A part of the ncome-substtuton analyss that s only shown n some tests s the quantty mappng of the prce and substtuton effects to prce and consumpton space. That s, f you portray a prce and a substtuton effect n commodty space for the good on the horzontal as and then draw the related prce and consumpton ponts below, you are now supposed to know eactly how these curves are related. They are related through the Slutsky equaton. Ths relaton s not trval. Its precson s requred n, for nstance, the computaton of consumer's surplus as dscussed above. MTM 10/96 Revsed: October 3,

6 901 Notes: 11.doc Department of Economcs Clemson nversty An Append on Integratng Ordnary Demand Curves The area under the money constant demand curve s: z 1 - M d 4 0 From the envelope theorem the dervatve of the mamzed utlty level w.r.t. prce n the money constant problem s the consumpton level of the th good tmes the optmzed value of the lagrangan multpler n that problem, (, M). So here we can wrte: λ z z - d = M λ (, M ) The ntegral on the rght-hand sde of eqt. (5) s well defned only f we can factor l out of the ntegral z 1 0 z d = 0 d 1 (, M ) 1 (, M ) 1 0 d = (, M ) λ λ λ Ths factorng s only possble f λ s not a functon of, that s, only f λ s a constant. If t s then eqt. (6) says that the ntegral under the money constant demand curve s the money equvalent of the change n utlty receved as the prce of the good changes. We have not talked much about λ, but we know from the envelope theorem that t s the margnal change n utlty when the money constrant changes, that s, λ s the margnal utlty of money. The queston, then, s, What does t mean to say that the margnal utlty of money s a constant? In the utlty mamzaton problem, λ s a functon of prces and money. In fact t s a homogeneous functon of degree -1 n these varables. If all prces and money ncrease by a factor of t, λ falls by ths factor. From Euler's Theorem we can wrte: n = 1 λ + λ M M = - L NM 1 0 λ 7 If the margnal utlty of money s constant w.r.t. all prces, eqt. (7) says that the elastcty of w.r.t. M s -1. λ M = M λ O Q 5 6 λ Gong back to the partal of mamzed utlty w.r.t. the th prce and w.r.t. M, we have = - λ M and M = - λ 9 Revsed: October 3,

7 901 Notes: 11.doc Department of Economcs Clemson nversty The second cross partals are: L NM M = - M + M 2 2 M M λ λ = O Q M = - λ = whch are equal by Young's Theorem. Convertng eqt. (10) to elastcty form gves: Substtutng from eqt. (8) leaves us wth: L NM λ M - + M λ M M M M M M M M O Q = = 1 12 Ths result says that f all ncome elastctes are equal to 1, then the margnal utlty of money s a constant. In other words, n order to ntegrate money constant demand curves and make any sense out of them, all goods must respond unformly to an ncrease n money. Ths result s true for some homothetc utlty functons. 8 But, even n ths case the noton of the area under the money constant demand curve s not partcularly meanngful. Whle t s a consstent dollar measure of utlty for some utlty functons, t does not answer any real world questons. 8 For some, but not all. The standard Cobb-Douglas form utlty functon s homothetc. However the margnal utlty of money s a functon of prces. Even though all ncome elastctes are untary, the ntegral of the Marshallan demand curve has no economc nterpretaton. Revsed: October 3,

8 901 Notes: 11.doc Department of Economcs Clemson nversty Fgure 1 Revsed: October 3,

9 901 Notes: 11.doc Department of Economcs Clemson nversty Fgure 2 Revsed: October 3,

10 901 Notes: 11.doc Department of Economcs Clemson nversty Fgure 3 Revsed: October 3,

11 901 Notes: 11.doc Department of Economcs Clemson nversty Some notes on the calculaton of consumer surplus: The approach adopted by most people n addressng the problems on p. 17 of the problem lst follows the method gven by Layard & Walters, bottom p That s, we calculate the slope or elastcty of the compensated demand curve and then etrapolate for the gven prce change usng a lnear appromaton. I want to work through an eample and then dscuss the alternatve of ntegratng to fnd the compensated demand curve and consumer surplus. Consder the case of a prce ncrease of 20%. Let the ordnary own-prce elastcty be -1.12, the ncome elastcty be 1.34, and the consumpton share be.13. (Ths s the eample of the prce ncrease n fsh under the assumpton that fsh demand n the nted States s the same as meat demand n Chna.) Assume that ncome s $31,000. nder these assumptons, compensated ownprce elastcty s Consumpton n dollars before the prce change s $4030. A 20 percent prce ncrease would cause ependtures to go up by $806. However, consumers reduce ther purchases. Gven the compensated prce elastcty of -.95, a 20 percent prce ncrease would mply a 19 percent quantty reducton. Hence, the compensated ependture ncrease after the prce ncrease would be $653, (.e., 81 percent of $806). The deadweght loss n consumer surplus, assumng a lnear appromaton, s ½ ( ), or $ The total compensatng varaton, or consumer surplus at stake n the proposed prce ncrease s $ [If you draw a smple pcture, I thnk that ths wll be planly obvous.] Now, consder the alternatve of ntegratng to fnd the compensated demand curve. Snce, ε 11 = we can wrte dp = ε 11 dp = (.) Note that the compensated own-prce elastcty s negatve, so we must appeal to a constant of ntegraton to get postve consumpton at the ntal pont. Intalzng at 1 gves = k Aln where k s the constant of ntegraton and A s the product of consumpton tmes the compensated prce elastcty. Snce prce s ntalzed to 1, the log of ntal prce s zero, and k s Revsed: October 3,

12 901 Notes: 11.doc Department of Economcs Clemson nversty amount of consumpton at the ntal pont,.e., The value of A s ε 11 (=-.95) tmes 4030 or sng ths demand curve, we can estmate the quantty reducton due to a prce ncrease by settng to 1.2. Consumpton along the compensated demand curve s 3335, down from 4030 at the ntal level. sng a lnear appromaton between these two ponts, the loss n consumer surplus due to the ncrease n prce s estmated to be $737. We can be a lttle more precse by ntegratng. The ntegral of over the proposed prce change s: dp = ( k Aln ) [ ( ln )] 1 1 dp= k A 1 where agan k s the ntal consumpton level, 4030, and A s Evaluatng the ntegral gves $734. The ntegraton approach gves a slghtly hgher level of consumer surplus than my basc lnear appromaton, but all three estmates are close to each other. MTM 10/14/2003 More on estmatng consumer surplus: Consder the utlty functon = = 1, n. Ths gves demand functons of the form: 1 M = n The ndrect utlty functon s M = n n By dualty, 1 1 ( ) M = n n n From whch we get Revsed: October 3,

13 901 Notes: 11.doc Department of Economcs Clemson nversty 1 1 n 1 M + 1 n n = j = j (.) For smplcty, ntalze all prces ecept to one, and suppress the subscrpts. The defnte ntegral of compensated demand s b a b + 1 n n n dp = dp = n n a 1 b a Alternatvely, we can ntegrate the Slutsky equaton from the ordnary demand. For our demand functon the Slutsky equaton looks lke M 1 1 = + = + 2 M n n Substtute the demand epresson for n the last term on the rght and ntegrate to get the compensated demand functon: 1 M M dp = dp 2 2 n n whch gves an estmate of compensated demand 1 M M ˆ = k 2 + n n where k s the constant of ntegraton. The constant of ntegraton s ntalzed to make equal the observed level of consumpton. The defnte ntegral of our estmate of compensated demand looks lke ths: b a M M ˆ dp = ln( ) + k 2 n n b a Let s compare consumer surplus calculated from the eact compensated demand curve wth that from the Slutsky estmate. Let M be $100,000, be $5,000, n be 20, and be 1. (Ths s somethng lke demand for cars. I suspect that people wth $100K ncome spend around $5K per year on cars.) Now ask how much consumers would have to be compensated to suffer a doublng of the prce from $5K to $10K. Revsed: October 3,

14 901 Notes: 11.doc Department of Economcs Clemson nversty From the eact compensated demand curve, the value s $3526. From the Slutsky estmate, the value s $2542. These are not very close because of nonlnearty and the sze of the prce change. As the prce change converges to zero, the two methods gve the same answer. Reconsder the utlty functon. Ths gves demand functons of the form: = 1, n = 1 M = = S n M That s, each good takes up 1/n th of the budget. The ndrect utlty functon s M = n n and by dualty, the ependture functon s: 1 1 ( ) M = n n n Let the prces of all goods ecept good equal one and drop the subscrpt. Then we can wrte: Substtutng for, we have: S b S S b ( ) ( ) [ S CS = M = b a ] a S = S S a S ( ) 1 S M S / a S S S b CS = [ b a ] = M 1 S S a A problem: Clemson nversty offers the followng room and board optons to students. The average room charge s $2960 per academc year. The most epensve housng s $4230 per year. The least epensve s $2000. The basc meal plan allows students to eat 15 meals per week for $996. (There are a total of 21 meals per week served n the dnng halls. The average student eats 15 meals out of 21.) The unversty estmates that personal epenses amount to $1666 per year. Revsed: October 3,

15 901 Notes: 11.doc Department of Economcs Clemson nversty Assume that ordnary prce and ncome elastctes are (-1) and (+1) respectvely for all goods, and that scholarshps or parents pay for tuton and educatonal epenses so that a student s budget s allocated to housng, meals, and personal tems. a. How much wll a student lvng n the most epensve unversty housng spend on meals? How much wll a student lvng n the least epensve unversty housng spend on personal tems? (10 ponts) b. The unversty offers a meal plan that covers all 21 meals per week (100% meal plan) for $1072. Is ths a good deal for the average student? Eplan. (10 ponts) c. What s the mamum prce that the unversty could charge the average student for the 100% meal plan? (15 ponts) d. If parents pay housng costs, so a student s budget s only meals and personal tems, what effect does ths have on the value of the 100% meal plan? (10 ponts) See <ConsumerSurplusCalcs.ls>. Revsed: October 3,