Atkinson-Stiglitz and Ramsey reconciled: Pareto e cient taxation and pricing under a break-even constraint

Transcription

1 Abtract The Ramey tax problem examne the degn o lnear commodty taxe to collect a gven tax revenue Th approach ha been erouly challenged by Atknon and Stgltz (976) who how that (under ome condton) an optmal ncome tax make commodty taxe redundant Accordngly, the Ramey approach to taxaton condered a dated In the meantme, the Ramey ettng ha had a econd le a model o regulatory prcng Boteux (956) tude lnear prcng o a regulated mult-product monopoly that ha to cover ome xed cot through markup on the d erent product (equvalent to taxe) Whle the cope o regulaton ha declned, Ramey-Boteux prcng contnue to play an mportant role Th paper examne the optmal tax and regulatory prcng approache to Ramey prcng can be reconcled We ncorporate the two obectve o revenue rang or nancng the government expendture and a regulated rm xed cot nto a ngle ramework The rt maor leon o our tudy that the extence o a break-even contrant not only requre taxaton o good produced by the publc/regulated rm but alo the taxaton o other good Next, we conder two pecal cae or whch the Ramey model yeld mple and well-known reult; namely, the cae o ndependent Hckan and Marhallan demand curve In both the Ramey oluton mple the o-called nvere elatcty rule In the eparable Hckan demand cae, the prvate good (not ncluded n the breakeven contrant) contnue to go untaxed a n the Atknon and Stgltz ettng In the cae where Marhallan demand are ndependent, the e ect o the break-even contrant pll over to the other good who no longer go untaxed We contnue to get nvere elatcty rule; however, ther tructure d er rom the tradtonal expreon In partcular, there no covarance (or mlar) term that capture redtrbutve conderaton Intead, they contan tax revenue term that meaure the ocal value o the extra tax revenue generated rom demand varaton that ollow the (compenatng) adutment n dpoable ncome Fnally, we tudy the mot celebrated general reult obtaned n the Ramey model; namely, the (un)equal proportonal reducton n compenated demand property We nd that the redtrbutve conderaton are once agan replaced by tax revenue term

2 Atknon-Stgltz and Ramey reconcled: areto e cent taxaton and prcng under a break-even contrant Helmuth Cremer Touloue School o Economc (Unverty o Touloue and Inttut unvertare de France) Touloue, France Frouz Gahvar Department o Economc Unverty o Illno at Urbana-Champagn Urbana, IL 68, USA May, reved September relmnary and ncomplete leae do not quote or cculate wthout the author permon

3 Introducton The Ramey tax problem one o the cornertone o optmal tax theory The queton t examne that o degnng lnear (proportonal) commodty taxe to collect a gven tax revenue Labor ncome not taxed at all or ubect to a lnear tax rt analy o th problem due to Ramey (97) but mot o the ormal reult were derved n the 97 tartng wth the emnal paper by Damond and Mrrlee (97) Th wa ollowed by lterally hundred o paper Baumol and Bradord (97) and Sandmo (976) provde an nteretng htory o th ubect The man pont that th lterature make that, except n very pecal cae, commodty taxe hould not be unorm and that they hould be et to balance e cency and equty conderaton The e cency apect appear mot obvouly n the nvere elatcty rule derved when demand uncton are eparable (Hckan or Marhallan) Thee e cency drven tax rule whch are regreve becaue good wth low prce elatcte are oten necete conumed proportonately more by poorer houehold Th redtrbutve ba mtgated by the equty term n the tax rule (oten covarance term) Thee tend to ncreae the tax rate on good that are conumed proportonately more by rcher ndvdual Th approach to tax degn ha been erouly challenged by Atknon and Stgltz (976) who how that under ome condton an optmal ncome tax u cent to mplement any ncentve compatble areto-e cent allocaton In other word, commodty taxe are not needed (or hould be unorm), and the Ramey reult are merely an artact o the retrcton that ncome taxe mut be lnear (an ad hoc and ncontent aumpton gven the aumed normaton tructure) The Atknon and Stgltz (AS) reult ha ar reachng mplcaton or the degn o optmal tax ytem In partcular, In the orgnal Ramey problem, ndvdual are alke and there no ncome tax Wth heterogeneou ndvdual, one alo allow or a unorm lump-um tax or rebate (and pobly a lnear tax on labor ncome) The

4 whenever t apple, t mple that n-knd traner are not ueul and prce hould not be ued or redtrbuton (even n a econd bet ettng) It by now well undertood though that the AS reult alo ha t own lmtaton In partcular, t may not hold under uncertanty (Cremer and Gahvar, 995) or under mult-dmenonal heterogenety (Cremer, Gahvar and Ladoux, 998, and Cremer, eteau and Rochet, ) and redtrbuton through prce may then once agan be econd-bet optmal (Cremer and Gahvar, 998 and ) Stll, thee lmtaton notwthtandng, t ar to ay that the Ramey approach to taxaton condered a dated and no longer tate o the art It doe contnue to occupy a promnent place n all advanced textbook, but t taught manly a an ntroducton to tax degn and not becaue o t practcal relevance In the meantme, the Ramey ettng ha had a more or le ndependent econd or ome mght argue rt le a model o regulatory prcng In h poneerng paper, Boteux (956) tude lnear prcng o a regulated mult-product monopoly that ha to cover ome xed cot (or ntance the nratructure cot o the network) Th to be acheved through markup on the monopoly d erent product (equvalent to taxe) Formally, th problem equvalent to a Ramey tax model wth the xed cot playng the role o the government tax revenue requrement Whle the cope o regulaton ha declned over the lat decade, Ramey-Boteux (RB) prcng contnue to play an mportant role n the ector tll ubect to ome orm o regulaton promnent example the potal ector n the US where Ramey-Boteux prcng reman an mportant benchmark n regulatory hearng; ee Crew and Klendorer () A a matter o act, not only ha RB prcng kept t poton but t ha even ound new applcaton n ettng o regulatory reorm and market lberalzaton A For ntance, whle the orgnal Boteux model concern a monopoly, Ware and Wnter (985) how that generalzed RB rule preval n mperectly compettve market Furthermore, We ollow the termnology ued n the regulaton lterature but, n realty, th a qua- xed cot relevant alo n the long-run (a non convexty n the producton et)

5 La ont and Trole (99) argue that the network acce an ncumbent operator ha to provde to t uptream compettor hould be prced on the ba o RB logc Another nteretng reult, hown by Vogelang and Fnnger (979), that Ramey prce can be decentralzed through an teratve procedure baed on a global prce cap; ee alo La ont and Trole (, page 67) Fnally, the dea that prce ought to be ued or redtrbutve purpoe the ratonal or a great deal o regulatory polce ncludng ocal tar and more generally unveral ervce requrement; ee Cremer et al () or a dcuon o the theoretcal oundaton o thee polce and ther practcal mplementaton 3 To um up, optmal tax and regulatory prcng lterature appear to have dverged n the way they vew the mplcaton o the AS reult Th paper examne the two approache can be reconcled To th end, we ncorporate the two obectve o revenue rang or nancng the government expendture and a regulated rm xed cot nto a ngle ramework Whle the two decon are ormally equvalent when each addreed eparately, a comprehenve analy requre a un ed ramework Our ettng that o AS More precely, we conder a mxed taxaton ettng à la Chrtanen (984) that combne nonlnear ncome taxaton wth lnear prcng (taxaton) o conumpton good We aume that a ubet o the good are produced by a publc or regulated rm that ha to cover a xed cot through markup on the d erent commodte t ell 4 Th contrant gve re to a break-even contrant on the part o the rm Th come on top o the overall government budget contrant The queton th break-even contrant revve any o the Ramey prcng or taxaton eature (or good produced by the publc/regulated rm a well a or other good) 3 Whle t true that regulator and epecally competton authorte are oten reluctant to accept Ramey prcng argument, th not becaue o the AS reult Ther obecton more o legal and procedural nature In partcular, Ramey prce are oten vewed a dcrmnatory and ubect to normatonal problem when the operator better normed about demand condton than the regulator 4 Whether th rm publc, a n Boteux world, or prvate but regulated doe not matter Ether way, one mplctly aume that the rm revenue mut alo cover ome ar rate o return on captal 3

6 We tart by ormulatng the general problem a government ace when determnng ncentve compatble areto-e cent allocaton n a dcrete economy wheren ndvdual d er n ther productvty 5 Indvdual type and labor upple are not publcly obervable, but pre-tax ncome are Th rule out type-pec c lump-um taxe but allow or nonlnear ncome taxaton Indvdual conumpton level o good, whether ubect to regulaton or not, are not obervable o that nonlnear taxaton and/or prcng o good are not poble However, anonymou tranacton o good makng lnear commodty taxaton eable Th normaton tructure by now tandard n the taxaton lterature 6 The key eature o our etup that ome o the good are produced by a publc/regulated rm ubect to a break-even contrant Whle we optmze over all tax ntrument ncludng ncome, we are not concerned wth the properte o the ncome tax chedule Our am to tudy the commodty taxaton and prcng rule that emerge or good that are produced by publc/regulated rm a well a thoe produced ubect to no regulaton We derve thee rule or general preerence but concentrate on to the cae o weakly-eparable preerence between labor upply and good that underle the AS reult In th way, we demontrate the rt maor leon o our tudy; namely, that the extence o a break-even contrant not only requre taxaton o good produced by the publc/regulated rm but alo the taxaton o other good In other word, taxaton o prvately-produced good are generally needed to o et the dtorton created by the publc/regulated rm departure rom margnal cot prcng We then llutrate and elaborate on thee leon by tudyng a mple ramework wth one publcly-provded and two prvately-provded good Next, we recat and derve the Ramey rule or ettng wheren taxe (markup) are ued to nance both a revenue requrement and a xed cot (rather than only one o the two a n the tradtonal model) Whle th yeld predctable reult and not o much nteret n tel, t a ueul reerence pont It alo encapulate and hghlght 5 Our reult wll not change a contnuou dtrbuton o type are condered 6 Th amount to addng a nonlnear ncome tax to the Ramey tax/prcng problem 4

7 the e ect o the break-even contrant per e Then, returnng to our ettng, to gan a better ntuton nto the nature o the tax/prcng rule, we conder two pecal cae or whch the Ramey model yeld mple and well-known reult; namely, the cae o ndependent Hckan and Marhallan demand curve In both the Ramey oluton mple the o-called nvere elatcty rule In the eparable Hckan demand cae, we nd that the prvate good (not ncluded n the break-even contrant) contnue to go untaxed a n the Atknon and Stgltz ettng On the other hand, the prcng rule ued by the publc rm are purely e cencydrven Ramey rule Good are taxed nverely to ther compenated demand elatcty regardle o ther dtrbutonal mplcaton Redtrbuton taken care o by the ncome tax allowng the publc rm prce to be aduted or revenue rang (a n the Ramey model wth dentcal ndvdual) Reult become le predctable n the cae where Marhallan demand are ndependent Then the e ect o the break-even contrant no longer con ned to the good ubect to th contrant Intead, t pll over to the other good who no longer go untaxed We contnue to get nvere elatcty rule a n the Ramey model; however, ther tructure d er rom the tradtonal expreon one get n the Ramey model wthout a break-even contrant On the one hand, they are more complcated than the pure e cency rule On the other hand, there no covarance (or mlar) term that capture redtrbutve conderaton Intead, they contan tax revenue term that meaure the ocal value o the extra tax revenue generated rom demand varaton that ollow the (compenatng) adutment n dpoable ncome Thee term lead to predcton that are mlar to thoe comng rom redtrbutve term n the many-houehold Ramey model wthout a break-even contrant; namely, that good wth hgher demand elatcte hould be taxed more heavly (ee below) Fnally, we tudy what arguably the mot celebrated general reult obtaned n the Ramey model; namely, the (un)equal proportonal reducton n compenated de- 5

8 mand property We how that, n contrat to the ngle-houehold Ramey model, the reducton d er acro good Th n and o tel not partcularly urprng gven the preence o heterogeneou houehold More nteretngly, when compared to the many-houehold Ramey cae wthout the break-even contrant, we nd that the redtrbutve conderaton are once agan replaced by tax revenue term The model There are N type o ndvdual, ndexed ; : : : ; N, who d er n ther wage, w, but have dentcal preerence All good are produced at a contant margnal cot whch we normalze to one However, ome o the good are produced by a publc or regulated rm whch ncur a xed cot The rm contraned to break even by markng up t margnal cot 7 Denote the good that prvate ector produce by x (x ; x ; : : : ; x n ) and good that the publc rm produce by y (y ; y ; : : : ; y m ) Let p (p ; p ; : : : ; p n ) denote the conumer prce o x and q (q ; q ; : : : ; q n ) the conumer prce o y: Fnally, denote the commodty tax rate on x by t (t ; t ; : : : ; t n ) and the publc rm commodty-tax-cum-mark-up by ( ; ; : : : ; m ): We have p + t ( ; ; : : : ; n) and q + ( ; ; : : : ; m) Indvdual conumpton level are not publcly obervable but anonymou tranacton can be oberved Conequently, commodty taxe mut be proportonal and publc ector prce are lnear For the remanng varable, the normaton tructure the one typcally condered n mxed taxaton model; ee eg, Chrtanen (984) and Cremer and Gahvar (997) In partcular, an ndvdual type, w, and labor nput, L, are not publcly obervable; h beore-tax ncome, I w L, on the other hand, Conequently, type-pec c lump-um taxaton ruled out but non-lnear taxaton o ncome eable 7 Alternatvely one can thnk o a prvately owned regulated rm whoe prce are et to cover cot plu a ar rate o return on captal 6

9 To characterze the (contraned) areto-e cent allocaton we derve an optmal revelaton mechanm For our purpoe, a mechanm cont o a et o type-pec c beore-tax ncome, I, aggregate expendture on prvate good, c, a vector o conumer prce (ame or everyone) or x and or y, p and q To proceed urther, t neceary to conder the optmzaton problem o an ndvdual or a gven mechanm (p; q; c; I) Formally, gven any vector (p; q; c; I), an ndvdual o type maxmze utlty u u x; y; Iw ubect to the budget contrant p x + q y c The reultng demand uncton or x and y are denoted by x x p; q; c; Iw and y y p; q; c; Iw Subttutng n the utlty uncton yeld the ndrect utlty uncton v p; q; c; Iw u x p; q; c; Iw ; y p; q; c; Iw ; Iw : Thu, a -type ndvdual who agned c ; I wll have demand uncton and an ndrect utlty uncton gven by x x (p; q; c ; I w ); () y y p; q; c ; I w ; () v v p; q; c ; I w : (3) Smlarly, the demand uncton and the ndrect utlty uncton or a -type who clam to be o type k; the o-called mmcker, gven by x k x (p; q; c k ; I k w ); (4) y k y p; q; c k ; I k w ; (5) v k v p; q; c k ; I k w : (6) 7

10 areto-e cent (contraned) allocaton Denote the government external revenue requrement by R and the xed cot o publc rm by F: Contraned areto-e cent allocaton are decrbed, ndrectly, a ollow 8 Maxmze H v p; q; c ; I w (7) wth repect to p ; p 3 ; : : : ; p n ; q; c and I where are contant wth the normalzaton H The maxmzaton ubect to the reource contrant H "(I c ) + the break-even contrant m # (p )x + (q )y R + F; (8) " H m # (q )y F; (9) and the el-electon contrant v v k ; ; k ; ; : : : ; H: () Denote the Lagrangan expreon by L, and the Lagrangan multpler aocated wth the reource contrant (8) by ; the publc rm break-even contrant (9) by, and wth the el-electon contrant () by k We have 9 8 L < v + "(I m # 9 c ) + (p )x : + (q )y R F ; 8 " < m # 9 + (q : )y F ; + k (v v k ): () 8 Indrectly becaue the optmzaton over a mx o quantte and prce Then, gven the commodty prce, utlty maxmzng ndvdual would chooe the quantte themelve 9 Recall that p o that n (p )x (p )x k6 8

11 The rt-order condton o th problem wth repect to I ; c, or ; k ; ; : : : ; H; and p ; q ; or ; 3; : : : ; n; and ; ; : : : ; m; characterze the areto-e cent allocaton contraned both by the publc rm break-even contrant, the reource contrant, the el-electon contrant, a well a the lnearty o commodty tax rate They are derved n the Appendx One eature o th etup o note Wth x and y beng homogeneou o degree zero n p; q; and c; conumer prce can determned only up to a proportonalty actor Conequently, we do not optmze over p and et t value at p In the abence o the break-even contrant, th normalzaton wthout any lo o generalty Wth a bndng break-even contrant, t doe nvolve a retrcton, namely that we rule out an acro the board unorm ncreae n all (conumer) prce to enure that the publc rm revenue cover xed cot Whle th a theoretcal poblty n th ettng, t doe not appear to be a realtc coure o acton A a matter o act uch a polcy would be n contradcton wth the very dea o mpong a break-even contrant n the rt place 3 Atknon and Stgltz theorem and optmal commodty taxe In the tandard mxed taxaton model wthout the break-even contrant, aumng preerence are weakly eparable n good and labor upply, the Atknon and Stgltz (976) theorem on the redundancy o commodty taxe hold The partcular eature o eparablty that drve the AS reult the property that a -type who pretend to be In a ettng where tax polcy retrcted by normatonal conderaton only, the break-even contrant could be undone by a mple lump-um traner (rom the government to the operator) The RB approach rule out uch a traner n an ad hoc way Th re ect ome conderaton whch are not explctly addreed n the model For ntance, or poltcal economy reaon t requred that natural ga uer (and not the taxpayer) are reponble to nance the tranportaton network o ppelne and pumpng taton 9

12 o type k wll have the ame demand a type k: That, x k x k x (p; q; c k ); y k y k y p; q; c k : Th are becaue wth eparablty preerence take the ollowng orm u u x; y ; Iw : Under th crcumtance, the (condtonal) demand uncton or x and y pec ed n equaton () () and (4) (5) wll be ndependent o Iw o that x x y y p; q; c and p; q; c : Moreover, the ndrect utlty uncton too wll be weakly eparable n p; q; c and Iw and wrtten a v p; q; c ; Iw : The above property alo ha ar reachng mplcaton or optmal commodty taxe; both thoe produced by the publc rm a well a prvately To derve thee, ntroduce the compenated veron o demand uncton () () Spec cally, denote the compenated demand or a good by a tlde over the correpondng varable Let denote the (n + m ) (n + m ) matrx derved rom the Slutky matrx, aggregated over all ndvdual, by deletng t rt row @~x @ ~y @q ~y m : () C A We prove n the Appendx that n th cae optmal commodty taxe aty the ollowng equaton, t t n + + m B A y : (3) C A ym

13 Oberve that F, e there no xed cot, the break-even contrant rrelevant Under th crcumtance, and the rght-hand de o (3) cont o a vector o n +m zero One then obtan t and (margnal cot prcng) or all ; : : : ; n and ; : : : ; m, and return to the Atknon and Stgltz reult that commodty taxe are redundant Wth F >, the break-even contrant necearly volated under margnal cot prcng o that > In th cae, the rt (n ) lne on the o (3) contnue to be zero, but the other lne d er rom zero It then ollow that the oluton no longer mple all t and all are zero Nor wll t be the cae that the t are necearly zero Th latter pont, that the extence o a break-even pont requre not only the taxaton o good produced by the publc rm but alo the taxaton o prvately-provded good, conttute the rt maor leon o our tudy To ee the reaon or t, we reort to a mple pecal cae wth lmted number o good 3 Two prvately-produced good, one publc Under th mple tructure, and wth t ; t and are ound rom equaton (3) to be t +! y y A : It mmedately ollow rom the above + : Wth > to cover the xed < ; t ha the ame > ; one et t > : Th ncreae p and, wth t, ~y : On the < ; one et t < : Th lower p and, a a reult, ncreae ~y : Ether way then, one et t to ncreae ~y :

14 The break-even condton ncreae the prce o y above t margnal cot o that t conumpton le than optmal One attempt to revere th through t : A a general leon, taxaton o prvately-produced good are necetated to o et the dtorton created by havng to depart rom margnal cot prcng on the part o the publc rm The econd mportant leon we are eekng to anwer the extent to whch the prcng rule de ned by (3) reemble the tradtonal Ramey rule Tradtonally, however, the Ramey prcng rule are derved or ether a un ed government budget contrant (n the publc nance lterature), or or a publc rm (n the regulaton lterature), but not or the two together a we have done here Addng on a break-even contrant to the government budget contrant n the Ramey problem, however, doe not change the tructure o Ramey taxe or Ramey prcng rule Th eay to etablh For completene, and to etablh a benchmark or comparon, we do th n the next ecton 4 Benchmark: the many-conumer Ramey problem An ndvdual o type now maxmze h utlty u u x; y; L ubect to the budget contrant p x + q y w ( ) L + b; where the lnear ncome tax rate and b the unorm lump-um rebate The reultng demand uncton or x and y; and the upply uncton or L; are denoted by x x p; q; w ( ) ; b ; y y p; q; w ( ) ; b ; and L L p; q; w ( ) ; b : Subttutng n the utlty uncton yeld the ndrect utlty uncton v v p; q; w ( ) ; b u x p; q; w ( ) ; b ; y p; q; w ( ) ; b ; L p; q; w ( ) ; b :

15 Let denote the proporton o ndvdual o type n the economy and conder a utltaran ocal welare uncton o the orm H W v ; where W () ncreang, twce d erentable, and concave The optmzaton problem cont o maxmzng W v wth repect to p; q; b and ubect to reource and break-even contrant (8) (9) Oberve that wth x and y beng homogeneou o degree zero n p; q; w ( ) and b; conumer prce can determned only up to a proportonalty actor It or th reaon that we do not optmze over, ettng t equal to zero ; t @ ~y @p ~y @q ~y m : C A Manpulatng the rt-order condton o the government optmzaton problem, gven 3

16 n the Appendx, we prove there, t t n + + m where we alo prove that C B H x H x n H + y H + y m H : ; (4) C A Oberve that whle the tructure o taxe on prvate good and the good provded by the publc rm are not dentcal, the ame Ramey tax/prcng rule apply to both under th ettng Th the benchmark wth whch one hould compare and evaluate our reult under nonlnear ncome taxe and weak eparablty preented n equaton (3) The mot trkng d erence between the et o reult the lack o any dtrbutonal conderaton n (3) a compared to (4) wheren dtrbuton concern enter through : Th conttute the econd mportant leon o our tudy: The tax/prcng rule or both type o good, thoe that are produced prvately and thoe that re provded through the publc rm, are not a ected by redtrbuton concern To gan a better ntuton nto the nature o the tax/prcng rule, we next conder the two well-known pecal cae or whch the Ramey model yeld mple reult; namely, the cae o ndependent Hckan and Marhallan demand curve In both o thee cae, the Ramey oluton mple the o called nvere elatcty rule Followng thee cae, we examne the mot celebrated general reult o the Ramey model; namely the (un)equal proportonal reducton n compenated demand property The many-conumer Ramey problem condered here whle allowng or a unorm lump-um rebate, b: The lterature conder th problem or both cae when b and not preent The ame tax/prcng rule are derved n both cae The only d erence that when b preent, the optmzaton over b reult n : Th reult doe not hold when b not preent 4

17 5 areto e cent veru Ramey prcng rule 5 Zero cro-prce compenated elatcte Aume that Hckan demand are ndependent o that the compenated demand o any produced good doe not depend on the prce o the other produced good In th cae, the reduced Slutky matrx (where the lne and column pertanng to leure deleted) dagonal o that equaton (3) mpl e to t t n + C A + B y ~y : C A Conequently, or all ; ; : : : ; n and ; ; : : : ; m; t ; + q y + ~y e" ; where e" the abolute value o the -type own-prce elatcty o compenated demand or y : Or, or all ; ; : : : ; n and ; ; : : : ; m; t and + whch an nvere elatcty rule y ( + ) ye" ; (5) The above reult to be compared wth the cae o ndependent Hckan demand under lnear ncome taxe There, the correpondng reult are, or all ; ; : : : ; n 5

18 and ; ; : : : ; m; t + t + x x e + y ( + ) y e" Cov x ; x e ; (6) Cov y; y ( + ) ye" ; (7) where e the abolute value o the -type own-prce elatcty o compenated demand or x and Cov(:; :) denote covarance Oberve that n th tradtonal model, both type o good, are ubect to the nvere elatcty rule aduted or redtrbuton bene t (through the covarance term) However, wth nonlnear ncome taxe and weak-eparablty, all covarance term dappear Th mple that there wll be no tax on prvate good whle the good provded by the publc orm are ubect to the nvere elatcty rule that re ect pure e cency conderaton; ee equaton (5) To um, we nd that n th pecal cae, the prvate good (not ncluded n the break-even contrant) contnue to go untaxed a n the AS ettng On the other hand, the prcng rule ued by the publc rm are purely e cency-drven Ramey rule Good are taxed nverely to ther compenated demand elatcty regardle o ther dtrbutonal mplcaton Redtrbuton taken care o by the ncome tax allowng the publc rm prce to be aduted or revenue rang (a n the Ramey model wth dentcal ndvdual) It wll become clear below that the apparent mplcty o th rule to ome extent mleadng It obcure ome e ect whch are preent but happen to cancel out n th pecal cae We hall return to th ue n the next ubecton Thee dervaton are ound by ettng the cro-prce dervatve n equaton (A) (A) equal to zero and rearrangng the term The covarance nterpretaton ollow becaue o the reult that : 6

19 5 Zero cro-prce elatcte We now turn to the cae where Marhallan demand uncton are ndependent o that the demand or any gven good doe not depend on the prce o other (produced) good 3 To mply the prcng rule that obtan n th cae, t mpler to tart rom the ntermedate expreon (A) (A) gven n the Appendx rather than rom (3) Rearrangng thee expreon, makng ue o the weak-eparablty aumpton, and ettng all the cro-prce dervatve equal to zero, we obtan or all ; ; : : : ; n and ; ; : : : ; m; + x e t + + m y 4 e t @c ; y : (9) Beore mplyng thee expreon any urther, t normatve to delve nto ther nterpretaton Recall that we are conderng a compenated varaton n the tax rate uch that dc x dt or a varaton n t and dc y dq or a varaton n In other word, ndvdual dpoable ncome are aduted to keep utlty level contant or all ndvdual Wth utlty level unchanged, the mpact o the varaton on ocal welare entrely depend on the extra tax revenue (or pro t) t generate The let-hand de o (8) (9) meaure the ocal value o th extra tax revenue (or a varaton n t or n repectvely) Obvouly, when the tax ytem optmzed, th ocal value mut be equal to zero (otherwe welare could be ncreaed by changng the tax rate) To undertand th nterpretaton, aume one change c ater t or change 3 Much o the regulaton and IO lterature ue qua lnear preerence In that cae there are no ncome e ect and the dtncton between Hckan and Marhallan demand become rrelevant 7

20 Start wth a varaton n t Wth the tax revenue beng gven by t e x e + y A ; e and the cro-prce dervatve beng equal to zero, the change n t produce an extra tax revenue o x + Our compenaton rule requre x o th to be rebated to ndvdual4 The net change n revenue, mnu compenaton, equal : Th the rt expreon on the let-hand de o (8) At the ame tme, the x compenaton lead to an addtonal tax revenue o t e m e x A + x e x t e x e + e m y A : To convert thee tax revenue change nto ocal welare (meaured n unt o general revenue), one mut multply tax revenue varaton emanatng rom y-good by a actor o ( + ): Th becaue the revenue rom y-good enter both the global government budget contrant a well a the break-even contrant Th reult n the econd expreon on the let-hand de o (8) The let-hand de o (9) can be decompoed n a mlar way, except or one extra complcaton; namely the addtonal term () y In th exerce, y repreent the value o the reund to ndvdual When collected a a tax, th amount ha a ocal value o ( + ) y On the other hand, the reund cot only y (t come rom the general budget and ha no mpact on the break even contrant) 4 To ee th, oberve that c change accordng to dc x dt o that aggregate compenaton change by dc x dt : 8

21 Fnally, to eae the comparon wth tradtonal Ramey expreon, we can rewrte (8) (9) a nvere elatcty rule Introducng 3 H A x e t + + ; e 3 H B y e t y ; e where A and B meaure the ocal value o the extra tax revenue due to reund, wth B alo ncludng () y : We have, or all ; ; : : : ; n and ; ; : : : ; m; t + t + A x ; () B ( + ) y" ; () where and and " denote the abolute value o the -type own-prce elatcty o Marhallan demand or x and y : Expreon () () have a number o nteretng mplcaton, partcularly when compared to ther tradtonal counterpart Frt, the e ect o the break-even contrant no longer con ned to the good whch enter th contrant Intead, t pll over to the other good whch no longer go untaxed; compare wth the reult obtaned n Subecton 5 Second, we get nvere elatcty rule a n the Ramey model; albet wthout redtrbutve term Th become clear below The numerator o both expreon contan the tax revenue term A and B Recall that thee expreon meaure the ocal value o the extra tax revenue generated rom the demand varaton that ollow the (compenatng) adutment n dpoable ncome One may wonder why thee term where abent n Subecton 5 The key to undertandng th property that when Hckan demand are ndependent, the prcecum-ncome varaton we conder have by de nton no mpact on the demand o any o the other good And the e ect on the good under conderaton already capture n 9

22 the (compenated) elatcty term To um up, Subecton 5 ha gven mple reult, not becaue the d erent e ect were abent but becaue they happen to cancel out exactly under the condered aumpton Fnally, t nteretng to note that the counterpart o () () n the tradtonal Ramey model, under lnear ncome taxe and ndependent Marhallan demand, are or all ; ; : : : ; n and ; ; : : : ; m; t + t + A Cov x ; ; () x Thee relatonhp are derved n the Appendx B Cov y; ( + ) y" : (3) Comparon o thee relatonhp wth () () reveal that n our etup we have the exact expreon or optmal tax rate except or the covarance term whch vanh 53 roportonal reducton n compenated demand When there are cro-prce e ect, the Ramey model no longer gve reult that can be preented a mply a the nvere elatcty rule One o the popular way to preent the tax rule n term o proportonal reducton n compenated demand Th lead to the celebrated equal proportonal reducton reult n the pure e cency cae, whle the reducton depend on redtrbutve conderaton n the many houehold cae Wth eparable preerence, one can rearrange equaton (A) (A3) n the Appendx a ollow t e e e A A A A y :

23 whch can be wrtten a n e t e n e e + ~y e + @ x ; (4) y : (5) The let-hand de o (4) and (5) repreent the proportonal reducton n compenated demand o x and y repectvely Unlke n the ngle houehold Ramey cae t d er acro good A a matter o act, t proportonal to the compenated mpact o the condered good tax rate on the break even contrant Conequently we nd that lke n Subecton 5 redtrbutve conderaton are replaced by tax revenue or more precely revenue o the regulated rm Thee are to be compared wth, or all ; ; : : : ; n and ; ; : : : ; m; 5 n e t e n e @p e + e x x m y Cov x ; ; Cov y; : One gan mmedately oberve that the two et o ormula are dentcal except or the covarance term whch have nhed n equaton (4) and (5) 6 Concluon 5 Th ollow by rearrangng equaton (A) (A) n the Appendx, where we have ubttuted Cov x ; or x and Cov y ; or y :

24 Appendx A Frt-order characterzaton o the(contraned) areto-e cent allocaton: Rearrangng the term n (), and droppng the contant R and F; one may ueully rewrte the Lagrangan expreon a + k A v + # "(I c ) + (p )x k6 + ( + ) " m # (q )y k v k : k6 (A) The rt-order condton o th problem are, or ; k ; ; : : : ; H;

25 @L + k A v I " + + k6 " m # + ( + ) + " k A vc + + k6 " m # + ( + ) @ + k A + k6 # k v k I ; (A) # k v k c ; (A3) " (p e e e ( + ) m 4 (q @ + k A v + " # (p e e k6 e 3 + ( + ) m 4 (q + y # k v k ; ; 3; : : : ; n; (A4) k6 k v k ; ; ; : : : ; m; (A5) where the tax on x xed at zero and a ubcrpt on v denote a partal dervatve Equaton (A) (A5) characterze the areto-e cent allocaton contraned both by the publc rm break-even contrant, the reource contrant, the el-electon contrant, a well a the lnearty o commodty tax rate Optmal commodty taxe: Multply equaton (A3) by x, um over and add the reultng equaton to (A4) Smlarly, multply (A3) by y, um over and add the reultng equaton to (A5) Smplyng reult n the ollowng ytem o equaton or ; 3; : : : ; n and ; ; : : : ; m; 3

26 @L + k A v + x v c + k6 + ( + ) m 4 (q + k A v + yv c + k6 + ( + ) m 4 (q ) " e(p e )! 3 5 (p e ) @c k6! 3 e + e k v e + e k6 k v k!# + x vk c ;! + yv c k + (A6) y : (A7) Wth v + x v c rom Roy dentty, the let-hand de o (A6) how the mpact on the Lagrangan expreon L o a varaton n p when the dpoable ncome o ndvdual aduted accordng to dc x dt ; (A8) to keep ther utlty level contant Wth v + y v c, the let-hand de o (A7) how the ame compenated e ect or a varaton n q where dc y d : (A9) Thee compenated dervatve, (@L@p ) v v and (@L@q ) v v vanh at the optmal oluton 4

27 Make ue o Roy dentty to et, v k v k v + x v c ; + x k vc k ; v + y v c ; + y k v k c : Subttute thee value n equaton (A6) (A7), et p t and q, and dvde by : Upon changng the order o ummaton and urther mpl caton one arrve at, or all ; 3; : : : ; n; and ; ; : : : ; m; t e e + k x e t e k6 4 k y k6! x k vc k ; e + e m 4 y k vc k + Next, ung the Slutky equaton, y @q x @c ; ; whle makng ue o the ymmetry o the Slutky matrx, one can urther mply 5

28 (A) (A) to A e e e A e + @q A k x k6 A k y k6 x k vc k ; y k vc k (A) y; (A3) whch hold or all ; 3; : : : ; n; and ; ; : : : ; m: Fnally, ue the de nton o n () to wrte out equaton (A) (A3) n matrx notaton t k6 k x x k vc k t n + k6 k x n x k n vc k k6 k y y k vc k C A B m k6 k ym ym k vc k C A y ; C A ym (A4) Collectng the term nvolvng and premultplyng through by yeld: t k6 k x x k vc k t n + k6 k x n x k n v k c k6 k y y k v k c B A B m k6 C A k ym ym k vc k y ym Now, a hown n the text, weak-eparablty o preerence n labor upply and good mple that x k x k and y k y k or all ; ; : : : ; n and ; ; : : : ; m: 6 : C A (A5)

29 Th property reduce that the rt matrx on the rght-hand de o (A5) to zero o that (A5) mpl e to (3) a gven n the text Dervaton o (4), the Ramey tax rule: The Lagrangan expreon aocated wth maxmzng W v wth repect to p; q; b and ubect to contrant (8) (9) repreented by 8 L " W v < m # + (p )x : + (q )y b 8 " < m # 9 + (q )y F : ; : The rt-order condton o th problem are, R " # + (p ( + ) " m # (q + " # + (p e e e 3 ( + ) m 4 (q 5 ; ; ; : : : ; + " # (p e e 3 ( + ) m 4y + (q 5 ; ; ; : : : ; m: Ung the de nton o ; ; and n the text, the rt-order condton wth repect to b; equaton (A6), mpl ed to 7

30 : Next, ung Roy dentty, the rt-order condton (A7) (A8) are mpl ed to, or all ; ; : : : ; n; and ; ; : : : ; m;! 4 x + x e t e! 4 e + t e ; A5 : Ung the Slutky equaton and the ymmetry o the Slutky matrx, equaton (A9) (A) can alo be wrtten a 4 e 4 e t e @ex A A e + @q 3 A5 3 A5 x x ; + y (A) y: (A) Fnally, rewrtng equaton (A) (A) n matrx notaton, ntroducng and pre-multplyng by yeld equaton (4) Dervaton o () (3): Set the cro-prce dervatve n equaton (A9) (A) equal to zero and rearrange the term to get t + t + x x + y ( + ) y" : ; 8

31 Add and ubtract to to rewrte the above expreon a t + t + x x x y + () y y ( + ) y" : Rewrte x and y a covarance Then recall rom the de- nton o ; that we have n e t e@b + ( + : Subttute n above and make ue o de nton A and B : ; 9

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