Notes on Excess Burden (EB) most efficient lowest deadweight loss excess burden non-distorting tax system benchmark

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1 Notes on Exess Buren (EB) Our gol is to lulte the exess uren of tx system. This will llow us to juge one tx system ginst nother. All txes use inome effets simply euse they tke money wy tht oul hve een spent y the txpyer. One exmple of tx tht only uses inome effets is he tx, tx where everyone pys the sme mount regrless of their irumstnes. 1 One suh tx ws introue y Mrgret Thther in the erly 1990s in Gret Britin to reple the property tx in effet t the time. The he tx ws so unpopulr, tht her government fell n new eletions h to e hel. On the other hn, some txes use sustitution effets t the mrgin, in ition to the inome effets. Txpyers ten to sustitute wy from txe goos towr untxe goos, or from goos txe t high rte towr goos txe t low rte. As they o so, they espe some of the tx y this hnge in their eonomi ehvior. 2 It is this sustitution ehvior tht uses the hrmful exess uren effet, or the so-lle eweight loss, of tx. We wnt to mesure this sustitution ehvior n then ompre it ross ifferent txes. The most effiient tx system is the one with the lowest eweight loss. In orer to mke suh omprison, we will nee monetry mesure of exess uren. This les to lultion of the so-lle exess uren. To lulte this we must first lulte the equivlent vrition. There re other mesures tht re relte to the equivlent vrition. They re the ompensting vrition n onsumer's surplus nlysis, the esiest to pply eing onsumer's surplus nlysis. Uner ertin restritive onitions, onsumer's surplus is very lose to the theoretilly orret mesure, the equivlent vrition. We wnt to grphilly illustrte how to lulte the exess uren of istorting tx system versus non-istorting or lump sum tx system. We lmost lwys use non-istorting tx system s our enhmrk for omprison purposes. So, for exmple, if we re ompring two istorting tx systems, A n B, n we wnt to know whih is more effiient, we ompre oth to the non-istorting enhmrk system. Another enhmrk tht is use is the urrent istorting tx system. In tht se we re intereste in reforming the urrent tx system n look for hnges in the urrent tx system tht reue the exess uren. The most effiient tx reform is the one tht reues the exess uren of the system y the most. Tx reform nlysis, however, is more vne n we will not pursue it in this ourse. Auguste Dupuit first oneive of the ie of onsumer's surplus in Lter, Alfre Mrshll, the gret 19th entury eonomist, populrize the use of onsumer's surplus n thought it useful mesure of the impt of poliy on welfre. It eme quite populr in poliy isussions roun the turn of the entury n into the 1930s. Hrol Hotelling n Sir John Hiks first erive the theoretilly pproprite onitions for onsumer's surplus, the equivlent vrition, n the ompensting vrition. Pul Smuelson, in his isserttion, Fountions of Eonomi Anlysis, ritiize the onept of onsumer's surplus sine it ws not theoretilly rigorous n omine oth inome n sustitution effets in generl. Lter, nlysts suh s Hrerger, Chipmn n 1 Legen hs it tht uring the Frenh Revolution one h to py fee (or tx?) to relim the he of someone who h h it ut off. Hene the terminology, "he tx." I never got roun to reserhing this to fin ittion tht supporte it n, so, think of it s legen or myth of the Revolution. 2 Mny sttes rise the tx on eer in the erly 1990s in n ttempt to reue teen rinking. The en result ws rop in eer sles ouple with n inrese in the use of mrijun.

2 Moore, Dimon n Mfen, Diewert, Husmn, n Ky put the vrious onepts on theoretilly rigorous sis using vne tools of miroeonomis. We will present simple grphil nlysis in wht follows. There re severl steps in lulting the EB for n iniviul txpyer. Suppose there re two goos, n else, n we impose tx on. A tx on rises the onsumer prie of movie n will use the uget line to swivel in from to B', s epite elow euse reltive pries hnge when the tx is impose. First step: A = B 0 B' U To see this note tht the uget onstrint efore the tx is w = pm + qe, where w is lor inome, p is the prie per movie, M is the quntity of, q is the prie of else, n E is "goo" lle else. The two interepts for this eqution re the orere pirs (w/p, 0) n (0, w/q). (When E = 0, we hve w = pm, or M = w/p s the horizontl interept, n similrly for the other interept.) The uget onstrint fter the tx is w = (p + t)m + qe, where t is the tx rte on. The two interepts re (w/(p+t), 0) n (0, w/q). Only the horizontl interept is ffete y the tx. So the uget line swivels in from to B'. Intuitively, the tx reues inome n uses the uget "tringle" given y A0 to shrink to B0B'. For simpliity, ssume q = 1. The onsumer will move from point to point in the igrm s the tx on is impose. Sine she is still onsuming some every month, the government will ollet tx revenue from her in the mount tm, where M is the tx se. Wht re we to ompre the istorting tx on to? Every tx uses n inome effet sine the txpyer hs less inome fter pying the tx. However, some txes lso use sustitution effet n hene n exess uren. A non-istorting tx only uses n inome effet. So if we ompre the istorting tx to the non-istorting tx, we n isolte the sustitution effet n mesure the exess uren of the istorting

3 tx on. A non-istorting tx oes not lter reltive pries ut only uses n inome effet. Therefore, it only uses the uget line to shift k in prllel wy. This is epite s the shift from A to CC' elow. If the non-istorting tx h een impose the onsumer woul hve hosen point. Seon step: w = A = B w - T = C U C' In the following igrm we hve omine the previous two for omprison purposes. Sine oth uget lines BB' n CC' go through the sme point, point, they rise the sme revenue t tht point. Thir step: T w = A = B w - T = C U U B' D' C' To prove this, note tht the onsumption unle (M, E) is the sme t point for the two uget lines n hene the two txes. Given this, onsier the following. The txpyer's uget uner the istorting tx is

4 w = (p +t) M + qe. The uget uner the lump sum tx is w - T = pm + qe, where T is the lump sum tx. Sutrt the seon eqution from the first, w - (w - T ) = (p + t) M + qe - (pm + qe), t point, or, T = t. M. So the two tx systems rise the sme tx revenue t point. Point is lso on the non-istorting tx uget line n so rises the sme revenue s the istorting tx t point. In ft, every point on the CC' uget line rises the sme mount of revenue. Also notie tht s we move own the BB' uget line M inreses n so the mount of tx revenue rise uner the tx on inreses. Finlly, the istne etween points A n C is the mount of the tx revenue ollete. However, there is one ifferene. Notie tht point, where the iniviul pys the lump sum tx, is on higher inifferene urve thn point, where the iniviul pys the istorting tx, even though oth txes rise the sme mount of tx revenue. The ifferene in utility, U - U, is the loss in welfre ue to the istortion use y the tx on, reltive to the non-istorting tx. We wnt to lulte monetry mesure of the ifferene in utility, U - U. First, we must lulte the equivlent vrition. To lulte the equivlent vrition we strt t point n sk how muh inome we must tke wy to hieve the sme utility t point ut t the originl pries. This gives us point n uget line DD'. Thus, the ugets BB' n DD' re equivlent in tht they hieve the sme level of utility so U = U. The equivlent vrition (EV) of the istorting tx system is the vertil istne etween A n D. Fourth step: EV A = B D B' D' U U Next, we must sutrt off the tx revenue ollete from the EV. This is euse the tx revenue ollete hs positive vlue to soiety. Rell tht the lump sum tx system ollets the sme mount of revenue s the istorting tx system. This is given y

5 point C on the vertil xis. Thus, the istne etween A n C is the tx revenue ollete. Finlly, the exess uren is efine s EB = EV - T Therefore, the exess uren is the istne etween points C n D on the vertil xis. It is ollr mesure of the ifferene in utility etween the istorting n non-istorting tx systems. (Also not tht n re on the sme inifferene urve so utility is the sme, U = U.) Fifth Step: A = B EV T EB C D B' D' U U C' Next, onsier the se where preferenes re 'L-shpe' so tht there re no sustitution possiilities. The tx on moves the txpyer to point s efore. However, the lump sum tx tht rises the sme mount of revenue lso moves the txpyer to point, where =. Now points n from the other igrms oinie. There is no exess uren in this se. Why? Beuse there is no wy the onsumer n A = B C B' D' C'

6 sustitute wy from n espe prt of the tx. Note tht oth txes use the sme inome effets. This is euse they rise the sme mount of revenue. This is suggestive tht the exess uren of tx epens on sustitution possiilities t the mrgin. Without suh possiilities, there is no eweight loss. The most effiient tx system is the one with the smllest eweight loss for given mount of tx revenue, ll other things onstnt. We n lulte the exess uren for every tx system n then hoose the system for whih it is the smllest, yet rises enough revenue to meet the government's nee. This woul e the most effiient tx system. The ie is to lulte the exess uren for eh tx system for n iniviul txpyer. This gives us ollr mount tht is equivlent to the loss in utility from the istortions of the tx system for the iniviul txpyer. The one with the lowest exess uren is the most effiient system for tht txpyer. Aing up ross ll txpyers n hoosing the system with the lowest totl exess uren woul presumly give us the most effiient tx system. There re severl prolems with this nlysis, however. First, we require miro t on eh txpyer. Unfortuntely, we will proly not hve this sort of t for everyone. We will proly only hve mrket t tht ggregtes ross ll people uying the txe goos. Impliitly, when suh t is use, it is ssume tht we n up ross ifferent iniviuls to lulte the equivlent vrition of tx. Seon, even if we h t on iniviuls, n we relly the exess uren for one txpyer to tht of nother? Doing so involves vlue jugment euse it presumes the two iniviuls re rnke in the sme wy y soiety. However, this kin of rnking nnot e one ojetively. To simply the numers together ssumes tht the people re to e trete in the sme wy. In ft, soiety might feel ifferently out the two people. For exmple, if one is rih n the other poor, mny woul suggest weighting the poor person's eweight loss greter thn the rih person's loss when ing them up. So equity will lwys e n issue. Inee, mny goos tht hve low exess uren re lso neessities n thus hve very low-inome elstiity. This mens tht poor people uy more of these goos thn rih people s prt of their uget, e.g., hmurger. Effiieny woul hve us impose high txes on suh goos euse the exess uren is low. However, imposing high txes on the poor might e viewe s inequitle. This is known s the equity - effiieny treoff. Finlly, we n relte the previous igrms to the stnr eweight loss or exess uren tringle. The following emn igrm mthes up with the fifth step. Essentilly, the onsumer prie rises y the tx rte t. D w is the norml oservle emn urve tht hols inome w onstnt n llows the prie to hnge. Notie tht s prie inreses n we move up this emn urve, the onsumer is worse off in the sense tht they re moving to lower inifferene urves. So utility is hnging long the usul emn urve. As prie rises, utility flls, n s prie flls, utility inreses. On the other hn, D u is the utility onstnt emn urve tht hols the utility level onstnt inste. Rell from the fifth step tht points n re on the sme inifferene urve. However, the slope is steeper for thn, refleting the higher prie of m ue to the tx. These points orrespon to the points n in the igrm elow. The strting point is point gin. At point we impose the tx on n the uget line swivels in s epite in the previous igrms n the prie rises in the emn igrm elow. We then oserve the onsumer moving from point to point in

7 ll the igrms. We wish to lulte the exess uren using the emn urve elow. If we use the usul emn urve tht hols inome onstnt, we woul estimte the exess uren of the tx s the tringle e. However, this woul not mth up with step 5 n the EB lulte there, whih is orret sine it ompres points n. Inste, we shoul use the utility onstnt emn urve tht goes through points n. The orret lultion of the exess uren is tringle e. The onsumer's surplus overestimtes the exess uren of the tx in this se. Tringle e oinies with our previous lultion of the EV n EB in step 5, i.e., istne CD is ollr mesure of the EB of the istorting tx n this oinies with the tringle e elow. p t Revenue e EB D w D u M Rell tht the exess uren of tx or susiy is use y rtifiilly hnging reltive pries. Whenever there is hnge in reltive pries two effets re rete, sustitution effet n n inome effet. When prie rises, people nturlly sustitute wy from the goo towr other ommoities s muh s possile. This is where the istortion omes from. An, when prie rises, there is less rel inome for uying else n the onsumer's utility flls. When the prie of movie p rises, m flls euse of the sustitution effet n m flls euse of the inome effet if m is norml goo. Wht if m is n inferior goo? The sustitution effet still works in the iretion of reuing m. However, m inreses euse of the inome effet! If this ltter effet is lrger in mgnitue thn the sustitution effet, we woul oserve m inresing s its prie inrese n emn woul pper to slope upwr! This is iretly pplile to the se of goos in inelsti emn. Consier the se of n inelsti emn urve, sy. The prie rises y the tx rte on n oserve emn oesn't hnge t ll mking the oserve emn urve perfetly inelsti. As the prie rises, we move from to in the two igrms elow. Does this men there is no exess uren? No. Wht it mens is the sustitution effet n the inome effet re perfetly offsetting one nother n the goo is n inferior goo. As prie inreses, onsumers sustitute wy from the goo n m flls. However, sine it is n inferior goo n rel inome hs fllen, they uy more of it euse of the inome effet. The two effets nel one nother so oserve emn oes not respon. The true exess uren is the gp in the she lines on the left n this oinies with the tringle on the right. The orret wy to lulte the exess

8 uren of tx is to lulte the tringle uner the utility onstnt emn urve, not the oserve emn urve. E p EB t Revenue D w D u (To see tht this is n inferior goo, notie tht if we were t in the left igrm n gve the onsumer more inome she woul move to n then to point s inome ontinue to inrese. As inome inreses, she is onsuming less M n more E.) Severl remrks re in orer. If istorting txes re impose on other goos s well, we n up the EB for eh tx for the iniviul onsumer to get totl exess uren for the entire tx system for the iniviul onsumer. Why? Beuse they re ll mesure in ollrs. Seon, it is n open question s to whether we n ompre the EB for one onsumer with tht of nother. If ll iniviuls were ientil we oul. In tht se it mkes sense to up the EB ross ll txpyers to get totl EB. If ll iniviuls only iffer in their inome we still oul up ross txpyers uner ertin onitions hving to o with solving the so-lle ggregtion prolem. This is eyon the sope of this lss, however. Finlly, the interprettion of our ollr mesure of the EB is the following. Compre two tx systems, one tht istorts eonomi eision-mking t the mrgin n one tht oes not. The EB is the mount of inome we woul hve to give the onsumer so she is s well off onfronting the istorting tx system s she woul e if she pi the non-istorting tx inste. To lulte this, ompute the exess uren tringle uner the utility onstnt emn urve. An importnt pplition of this is to lor supply. Oserve lor supply funtions ten to e very prie inelsti. This is euse of offsetting inome n sustitution effets. When the wge rises, there is sustitution towr work n wy from leisure n onsumption. An rel inome rises when the wge inreses. If leisure is norml goo, leisure will inrese n work will fll. So the two effets work in o opposite iretions on lor supply. Rell our moel of lor supply. Utility epens on leisure n onsumption, U(L, C) n the uget onstrint is given y A + 24w = wl + C, where the prie of onsumption is equl to one, A is sset inome, n essentilly the wge w is the opportunity ost of leisure. The time onstrint is 24 = H + L where H is hours worke. This informtion is grphe elow. Note tht the oserve supply of lor urve tht hols inome onstnt S I is perfetly inelsti t point. The emn for lor is ssume to e perfetly elsti t the going wge w. Hit the mrket with tx n the M M

9 wge flls to w - t. The uget line swivels own through point ut lor supply ppers to e unffete y the rop in the net wge. Con A + 24w EB A U U 24 H A lump sum tx tht genertes the sme mount of revenue is given y the she line through points n. The gp etween U n U is mesure of the eweight loss or exess uren of the istorting lor inome tx reltive to the non-istorting tx. A ollr mesure of this is given y the istne etween the two she prllel lines on the vertil xis on the left, enote EB. Notie tht leisure is norml goo. If we strt t point n give the onsumer more inome, she will move from to to, n we woul oserve leisure inresing n lor supply flling. The following igrm mthes up with the previous one. The strting point is point. Impose the tx n emn shifts own ut we o not oserve the gent hnging her lor supply. The exess uren of the lor inome tx is given y the tringle. Notie tht the utility onstnt supply urve is upwr sloping. This reflets the sustitution effet. L

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