- competitive economy with n consumption goods, and a single form of labor which is only input

Size: px

Start display at page:

Download "- competitive economy with n consumption goods, and a single form of labor which is only input"

Dina Matthews
6 years ago
Views:

1 APPLIED WELFARE ECONOMICS AND POLICY ANALYSIS Commodity Taxatio Basic problem i commodity taxatio: if a social welfare fuctio is assumed, whas the choice of commodity tax rates that will maximize social welfare subject to a reveue costrait? Ramsey (1927) developed first solutio to problem - result overlooed util rediscovered by Samuelso i 1951; moder theory developed by Diamod ad Mirlees (1971) The Ramsey Rule (i) Basic Assumptios: - competitive ecoomy with cosumptio goods, ad a sigle form of labor which is oly iput - there is a sigle household, whose prefereces are represeted as a idirect utility fuctio (there could be a populatio of idetical households)

2 - give costat returs to scale, competitio esures that pre-tax price of good p i : p i c i w, i 1,..., (1) c i is a techical coefficiet describig amout of labor required to produce oe uit of good i, ad w is wage rate - labor, which is utaxed, is chose as the umeraire, ad w is fixed at a costat value, implyig a set of fixed pre-tax prices for cosumptio goods - cosumer prices q i, are pre-tax prices plus taxes: q i p i, i 1,..., (2) - the reveue costrais: R (3) where is cosumptio of good i; to esure marets clear, R is used to purchase labor which the produces othig thas traded

3 - prefereces of the sigle household give by: U V (q 1,...,q,w,I ) (4) (for the prices ot to be distorted by the taxes, i i worth of labor must be wasted) household cosumes goods, ad supplies labor used i productio ad by the govermet; because of the competitive assumptio ad costat returs, there is o proficome, ad, so lump-sum icome I is zero (ii) Derivatio of Ramsey Rule - optimal tax problem is give by maximizatio proble m: max {t1,...,t } V (q 1,...q,w,I ) s.t. R (5) The Lagragea is: V ( q 1,...,q,w,I ) [ R ] (6) - the first-order coditio for choice of tax rate o good is:

4 t V t where [ x V V, q t q q ] 0 t (7) Re-arragig (7): V [ x t q ] (8) Such a coditio has to hold for all goods, i.e., utility cost of raisig tax reveue should be i same proportio to margial reveue raised by the tax Usig Roy s idetity: V q V I x x (9) I is lump sum icome, ad is margial utility of icome, evaluated at I=0 - substitutig (9) ito (8):

5 x [ x q ] (10) - defiig the Slutsy equatio: q S i x I (11) where S i = i /q, the compesated demad for good i, ad the chage i ucompesated demad with respect to icome I is evaluated at I=0 - usig (11), ad re-arragig (10): or [S i x I ] [ S i [ ] x ] x x I (12) - factorig out x : S i [ 1 I ] x (13)

6 - symmetry of the Slutsy substitutio matrix implies S i = S i, so re-writig: S i x, where [ 1 I ] (14) - (14) is the Ramsey rule for a system of optimal commodity taxes, (14) havig to hold for all goods, = 1,..., - multiplyig both sides of (14) by t, ad summig over : 1 t S i R (15) - as the Slutsy matris egative semi-defiite, left-had side of (15) is egative, so has the same sig as R

7 - give defiitio of S i, S i is a approximatio of total chage i compesated demad for good due itroductio of tax system from positio of iitially o taxes The Ramsey rule ca the be writte: d S i x, 1,..., (16) - optimal tax system should be such that compesated demad for each good is reduced i same proportio relative to pre-tax positio d is Mirlees (1976) idex of discouragemet, which, by Ramsey rule is the same for all goods (iii) Implicatios - Ramsey rule implies that as proportioal reductio i compesated demad should be same for all goods, the goods that are price ielastic will be more highly taxed

8 - this would imply high taxes for goods such as food ad housig, implyig low-icome households would pay a larger proportio of their icomes i taxes - result follows from the sigle-household assumptio, i.e., maximizatio is oly cocered with efficiecy, ot equity - as lump-sum taxes are ruled out, commodity taxes resul substitutio effects that distort optimal choices, however, Ramsey rule esures losses are miimized Iverse Elasticities Rule (i) This rule was the oe that domiated the textboos for may years util the rediscovery of Ramsey s rule - essetially, this rule assumes o cross-price effects betwee taxed goods, so taig (8), ad usig Roy s idetity: x [ x q ] (17)

9 - o cross-price effects implies /q = 0, i, so (17) becomes: x [ x t x q ] (18) - re-arragig (18), ad dividig by q = p +t : t [ ] [ x q ] (19) p t q x - this ca be re-writte as: t p t [ ] 1 d (20) (ii) This rule implies proportioal rates of tax should be iversely related to the price elasticity of demad of good d - this is a extreme iterpretatio of the Ramsey rule, ecessities beig highly taxed May ouseholds Extedig sigle-household ecoomy to may, o-idetical households, itroduces equity

10 (i) Follow a variat of the Dia mod-mirlees (1971) ecoo my which has the same productio techology as before - ecoomy cosists of households, each havig a idirect utility fuctio: U h V h (q 1,...,q,w,I h ) (21) U h vary amog households, otherwise reduces to sigle-household problem - reveue costrait beco mes: R i (22) - social welfare fuctio is a Bergso swf, defied over the idirect utilities: W W ( V 1 (.),...,V (.) ) (23) - the problem is ow to fid set of commodity taxes that maximizes the swf subject to (22)

11 - first-order coditio for a optimal tax o good is: W V V h t h [ - usig Roy s idetity, (24) ca be re-writte as: where h =W/V h. h, made up of the effect of a chage i h s utility o social welfare ad h s margial utility of icome h - Diamod ad Mirlees deote this the social margial utiltiy of icome for h - usig Slutsy equatio agai, ad re-arragig (25): h S h i 1 [ h 1 [ i ] 0 (24) q i ] (25) q i t ] i I h (26) * see the appedix

12 - left-had side of (25) is approximately the proportioal chage i aggr egated compesated demad for good (geeralizes left-had side of (16)) - right-ha d terms show reductio i demad for should be smaller whe: (a) demad for is cocetrated amog households with high h - this reflects a cocer for equity, i.e.,reduces rates of tax levied o ecessities that low icome households cosume (b) dema d is cocetrated amog those whose tax pay mets chage a lot as icome chages - reflects a efficiecy cocer, i.e., if ot, taxes would have to be higher to meet reveue costrait, icre asig distortios

13 (26) ca be re-writte i a maer closer to the Ramsey rule: S h i [ x h [ i I h ] ] (27) where: x is the mea level of cosumptio of across households - defie: b h h i I h - b h is et social margial utility of icome - et as it measures both gai i social welfare h from a icrease i h s icome ad icrease i tax paymets by h from a icrease i icome, i.e., ivolves both efficiecy ad equity effects

14 - (27) ca be re-arraged to give: S h i [ 1 b h x ] (28) * see the appedix - reductio i aggregate compesated demad for th commodity due to commodity taxes should be iversely related to correlatio betwee b h ad, i.e., give b h reflects equity cocers, goods with high b h s should be taxed less Reductio to Ramsey Rule - Tax rule i (28) will collapse to the Ramsey rule whe all households are give same social valuatio, or because tax system caot discrimiate betwee households - Suppose b h = b all h=1,...,, right-had side of (28) becomes -[1-b], ad tax rule describes a efficiet tax system oly, i.e. proportioal reductio i demad will be the same for all goods - occurs if all households are valued equally ad

15 x is same for all implies o good is cosu med disproportioately by rich ad po or househo lds, i.e., idetical Egel curves - o way of subsidizig households with high b h s without subsidizig those with low b h s - com modity taxatio problem is oe of sigal extractio, i.e., state is attemptig to ifer household prefereces ad edo wmet from purchases i order to tax each household accordig to their circumstaces - if o sigal ca be extracted, system focuses o effic iecy oly Productio Efficiecy - from coditios for Pareto efficiecy, ow productio efficiecy occurs whe ecoomy is o its productio frotier, ad there is equality of firms margial rates of techical substitutio

16 - Dia mod ad M irlees (1971) proved the productio efficiecy lem ma : if ecoo my is com petitive, equilibrium with optimal com modity taxatio is o the frotier of the pro ductio set - implies itermediate goods should ot be taxed, which cotrasts with the predictios of Lipsey ad Lacaster (1956) which suggest distortios iduced by commo dity taxes sho uld be offset by a similar distortio i iput prices - focus o a sigle household, 2 good ecoo my, where oe good is a cosumptio good, the other a iput such as labor (see Figure 1) - vertical axis depicts output of cosu mptio good, horizo tal axis iput use - give costat returs to scale, sha ded area is ecoo my s productio set, where frotier is mo ved left by amout a to reflect the tax reveue required i uits of the iput - household supplies iput, ad cosu mes output, so its budget costrais upward-slopig, ad, i absece of lump-sum taxes, it goes through origi

17 - budget costrait correspods to optimal posttax set of prices q - as supplyig labor gives disutility, idifferece curves are do w ward-slopig - output/i put combiatios household is willig to trade to at these prices is give by the price cosu mptio locus or offer curve - optimal tax equilibrium is highest poit o the offer curve thas i productio set, at poit e, o idifferece curve I 0 - I 1 is both preferred ad feasible, but ca oly be achieved by the use of a lump-sum tax, which is ruled out by assumptio - why is the optimum o frotier? (a) if at f, household utility ca be icreased by reducig use of labor, eepig output costat, which is feasible (b) if at g, output ca be icreased without usig more labor, which is feasible

18 - suppose labor is used to produce a itermediate good, ad this is the a iputo producig cosumptio good - there is a direct li betwee uits of labor ad itermediate good, allowig prefereces ad budget costrait to be defied directly o labor ad cosumptio good - productio efficiecy implies itermediates should ot be taxed, as it would violate Pareto efficiecy i productio, firms margial rates of techical substitutio ot beig equal Some Numerical Results - tax rules as defied do ot have precise implicatios, ad, hece, o practical policy applicatios - limited amout of research doe o developig methodology for applyig the rules - Ray (1986), Murty ad Ray (1987) ad Sriivasa (1989) calculated optimal commodity taxes for may-household ecoomy, usig Idia data - 80% tax reveue raised by idirect taxatio

19 - to calculate tax rates, first have to specify a swf, procedure has bee to use a additive social welfare fuctio, ad the defie a social utility of icome fuctio for each household: - deote aggregate expediture of h as µ h, which is equal to icome, social utility of icome beig: U h K [ µ h ] 1v 1 v v 1 or K log µ h v 1 (29) where = costat relative iequality aversio - with additive social welfare fuctio, W/V h =1, so h, margial social utility of icome is: h U h µ h K [ µ h ] v (30) - households i sample raed by expediture, lowest expediture (icome) beig firs raig - settig 1 =1 for lowest expediture household: h µ1 µ h v (31) * see the appedix

20 - (31) implies that as µ h icreases relative to µ 1, h declies at rate v - as higher values of v reduce h for h>1, v reflects cocer for equity - method fixes h s exogeously, ad ca be determied by observed expediture levels i data set, ad cocer for equity ca be varied parametrically - give h s, optimal taxes are determied from (22) ad (25); requires specificatio of demad system ad estimatio of system from data set - estimated demad fuctios are the substituted ito (22) ad (25), allowig solutio for taxes - Murty ad Ray (1987) estimated a demad system based o the idirect utility fuctio: V h (.) µ h 1 1 i w 2 h p w o h (32) LES system, where = subsistece requiremets, umerator is discretioary allocatio, ad deomiator is weighted geometric mea of prices

21 - (32) permits evaluatio of separability assumptios, i.e., if 2 ad are zero, there is separability betwee goods ad leisure i utility, ad o-separability betwee goods if tax rates calculated o basis of estimated values of i, i, ad 1 give i Ray (1986), 0 was set at zero, ad a value of 2 assumed - defiig as the wage as a proportio of expediture, which is imposed o the aalysis, results from Murty ad Ray are show i the table for v=2 ad 2 = Table 1: Optimal Tax Rates Item = 0.05 = 0.1 Cereals Mil ad mil products Edible oils Meat, fish, eggs Sugar ad tea Other food Clothig Fuel ad light Other o-food

22 - redistributio occurs through products that are su bsidized - Atiso ad Stiglitz (1972) calculated optimal tax rates satis fyig the Ra msey rule i a sigle household ecoomy; usig tw o types of demad system, results show ed food ad ret bear highest tax rates, w hile durable goods bear the low est

EC426 Class 5, Question 3: Is there a case for eliminating commodity taxation? Bianca Mulaney November 3, 2016

EC426 Class 5, Question 3: Is there a case for eliminating commodity taxation? Bianca Mulaney November 3, 2016 EC426 Class 5, Questio 3: Is there a case for elimiatig commodity taxatio? Biaca Mulaey November 3, 2016 Aswer: YES Why? Atkiso & Stiglitz: differetial commodity taxatio is ot optimal i the presece of