Distributional Equity and the Optimal Structure of Public Prices Martin S. Feldstein The American Economic Review, Vol. 62, No. 1/2. (Mar. - May, 1972), pp. 32-36. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28197203%2F05%2962%3A1%2F2%3C32%3ADEATOS%3E2.0.CO%3B2-V The American Economic Review is currently published by American Economic Association.

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Distributional Equity and the Optimal

Structure of Public Prices

Several recent papers (\Tilliam Baumol a n d I)n\.itl Bradford, A l h l ~Lerner, a ;\vinash Ilixit) have restated the rules originally derived 11y Frank Kamsey and 11.Boiteux for optimal pricing by a public enterprise that protluces several gootls and t h a t must satisfy n budget constraint. Seither the liamsey-Boiteux papers nor the more recent tliscussions deal directly with the important tlistributional aspects of puljlic pricing. Ziamsey actually developecl his study as a derivation of the optimal excise taxes to be levied on a single inctivitlual. Boiteuv specified his analysis to include an optimal lump sum income retlistrihution as well as the optimal pricing of pul~licly produced goods. T h e more recent papers have also limited their analysis to the derivation of prices that achieve Paretian eficiency. In practice, optimal lump sun1 redistribution is impossible ancl the distributional aspect oi public pricing is an important policy consideration. T h e Ramsey-Boiteux rule is therefore inadequate. T h e current paper eltends these earlier results on public enterprise pricing by explicitly incorporating tlistrihutional ecjuity.I I. T h e Distributional Characteristic and Optimal Pricing

It is suficient t o consider a public prise that produces tJvo gootls2 and . S(pl, p,, y) them a t prices pl and p ~ Let *

I'rofessor of econoniics. IIarvard Yniversitv. This cluestion is also considered by flerhert hlohring and I'eter 1)iamond and James 3Iirrlees. llohring's etli~ation (30). 1). 703, ant1 1)inmontl ant1 ;\Iirrleesl equation (65), 11. 266, are equivalent nlternzttive statements of the first-order optimality conclition of equation (4) helow. 2 , . l lie problem of peak hour or p c ; ~ kseason pricing is of course a special case of pricing different gootls.

be the traditional consumer surplus of a household with income y that can purchase the goods a t prices pl and p4. Let the distrihution of household incomes he represented by the relatiite density function f(j1) ; i.e., if there are S households in the population being served, the number in a small interval around y,, is ,Tf(y,,)d~~. Finally, let the marginal social utility of a dollar t o a householcl with income y 11e denotecl 11'(j1).3 I t will be assumed that the marginal utility of income t o a household is unafiectetl by the prices cliargetl by the public enterprise. This approximation seems justified for any practical application. ' The appropriate welfare makinland is the weighted sum of the household consumer surpluses, weighting by the marginal social utility of income t o that household :' (1)

IT7

=

x

S

y . ~ ( p lp?. , y)zlf(y)i(~)dy

0

It' must be maximized subject to the con3 The term "marginal social utility" is used here tn denote the derivative of the socinl \\-e1f;tre function n i t h respect to the income of the householtl. Since there arc i ~ n l yit finite numher of households, this tleriviiti\-e is \\.ell defined; the continuous density function,i(,vl ancl the associated integrals defined helon. shoultl IIC regardetl as al11)roximations. Constant i l ' ( ~ ifor each household ~jermitsdoiric the analysis in terms of consumer surplus n-ithout sl~ecifying any particular Hicksian (letinition. I t ztlso makes it reasi,n,Irle to assume that the arcument of the tiiar~innl social utility function is money Lcorne, i.c., that ;('(!I is not affected 1)y changes in pl ant1 p?. Since only the first-order conditions are relevant. the an;~l\.siscan I)e developed without the use of consumer surl)lus b) umrk ing with the indirect utilit~-function; this al)pro;tch is follon.ed in m y forthcoming article on the pricing of pul~licintermediate goods. " This assurlles that for the other gootls to \\-hich the rilarginal derilarld is transferred there is no producer H arl)erger, surpkisor excise tax revenue; see Di\it, .I,('. anti Lerner.

F E L D S T E I N : PUBLIC PRICES straint that revenue minus proctuction cost be equal to a specified amount (B). If q2(p1,pa, y ) is the quantity of good i purchnsetl by a household with income y,6 the total quantity sold of good i is:

Letting C(Q1, Qz) be the total production cost, the constrained maximant1 is the Lagrangian expression

T o derive the first-order conditions for a maximum we make use of the result t h a t dS(pl, pz, y) apt = - q.(y), the quantity of good i purchaseti I)y a household with inconle F. The basic first-order conditions can be written :

33

T h e tlistributional characteristic of good i is defined by the ratio

The ratio R, is a weighted average of the marginal social utilities, each household's marginal social utility weighted by that household's consumption of gootl i. The conventional welfare assumption that u f ( y ) declines as y increases implies that the value of R, will be greater for a necessity than for a luxury. T h e higher the income elasticity of ctemantl for a good, the lower the value of R,. The next section illustrates an approach to making R , an operational measure. First, however, we derive a numher of results that do not depend on any specific parametric representation of R,. Equations ( i a ) antl (411) can be expressed in terms of the R,'s and simplified by denoting the price elasticity :IS

and by employing the Slutsky relati011 E,,= E , , ~ , Q , / ~ , Q , .The ' first-order conditions are then :

where m , = aCIdQ,, the marginal cost of good i. X convenient concept for introducing consitlerations of distributional equity in the analysis of optimrtl prices antl taxes is the distvibz~tiotzal characteristic of a good. "his im1)licitly a s u m e s that all other prices remain constant.

T h e use of the Slutsky compensated tiemarit1 relation e,,=q,plQj,'ptQ,. ignores the income effect. If this is taken into account, the quantity al.S1(R2-A) -arSi(Rl-A), where a , is the income elasticity of demand and S , is the share of income spent o n gootl i, must he added to the numerator and subtracted from the tlenominator of the right-hand side of equation (7). Since .Ti and S? can he expected to he quite small and the weighted difference even smaller, this correctioi~for the income e8ect is likely to be of no practic;rl significance.

These may be solvetl explicitly to yieltl the rclative "profit" or "tax" rates:

I n the special case in which the distributional characteristics are irrelevant, i.e., R1=K2, equation ('7) yields the hasic Iiamsey rule :

tive profits" is the protluct of 1) an efficiency factor (the Ramsey ratio of price elasticities), antl 2) n tlistril~utionalequity ~ ~ the relative factor. Since E ~ ~ , ' isE positive, tax rates or profit rates vary with the correspontling R,'s. T h e higher the value of K,, i.e., the more that the consunlption of the good is concentraled in low income families, the lower should 1)e the relative price of t h a t good. Ecluation (9) (and more generally etjuation (8)) provitles a precise statement of how the liamsey-tloiteus efficiency prices should be modifietl to reflect this principle of tlistril~utional equity. The derivative of the optimal tax ratio of equation (9) with respect to X

If, moreover, the cross price elasticities are zero el?= 0), we ol~tainthe familiar rule t h a t the tax rates should be inversely proportional t o the own price elasticities. T h e definition of R1 in ecluation (5) sho-izs how unlil~elyit is that the distributional characteristics will be irrelevant. The ratios R1 and R ? will IIC equal only if shows t h a t a n increase (decrease) in X 1) the marginal social utility of income is raises (lowers) the relative price of gootl 1 the same for a11 householtls, or 2) the relaif R1 ertceetls K ? ; i.e., if the consumption of tive quantities purchasetl of the two gootls 1 is more concentratetl in low income gootl is the same for all householtls, or 3) some families than the consumption of gootl 2 . extremely iniprobahle balancing of tlificrSince X is the shatlow price of the 1)utlget ences in quantities ,tnd social utilities constraint, an increase in the recjuiretl occurs. In general, therefore, the relc1' t'~ v e A. Therefore as the recluirecl surplus raises optimal prices will reflect R1, R?, antl A. surplus increases (or as the subsitly tleSince X is the shadow price of the hutlget creases), the price of the good with the constrairlt (a% d B = -A), the relative oplower income elasticity rises relative to t h e timal prices will depend on the size of t h e with the higher income price of the good deficit or surplus that the enterprise is e l a s t i ~ i t y . ~ ~ o w income e r iamilies conrecluired t o ha\-e. This is in contrast to the t r i l ~ u t e an increasecl share of total revenue Iianisey rule (equation (8)) in which the a larger surplus is recjuiretl. as relative prices do not change with variations in the hutlget constraint. 11. An Operational Measure o f the In the special case of zero cross-elasDistributional Characteristic ticity of clemand ( t 1 2 = 0 ) , i t is easy t o This section suggests how a n explicit provitle an intuitive interpretation of the operational expression for the tlistriburole of the tlistril~utional characteristics tional characteristic can he derived by antl the huclget constraint. Ecjuation ( 7 ) adopting reasonable parametric forms for now implies: the three functional relations in terms of which R, is defined: 'This ratio of optimal "tax rates" or '(rela-

This change in relative prices never imljlies that the e less expensive good l~ecornesthe more e ~ l ~ e n s i vunlei. the lo\\-cr X,becomes the higher I<,.

Ii, for a n y value of a , a n d causes t h e R , t o be m o r e sensitive t o t h e tiistrihutional characteristics. If t h e approximations suggested in t h i s section a r e considered satisfactory for a particular problem, equation (1 7 ) provitles a simple m e t h o d of calculating each R , in t e r m s of available income tlistrihution parameters (7 anti a,!), a n easily estimable income elasticity of tlemancl ( a , ) ancl a n intuitively n a t u r a l representation of t h e normative tlistrihutional jutigement (?I). T h e optimal prices can t h e n he calculatetl b y solving ecluations (6a), (6b), ancl t h e l ~ u t i g e tconstraint of ecjuation (3).ln 111. Conclusion

T h e brief discussion in this p a p e r ha. inclicatetl ho\v considerations of tiistrihutional e q u i t y can h e inclutied explicitly anti operationally in t h e derivation of t h e o p t i m a l prices for public eilterprises o r regulated industries. -1 n u m b e r of questions a r e left unanswered. F o r example, how shoultl goods be priced t h a t a r e soltl t o industrial firms r a t h e r t h a n t o households? IYhat pricing rules a r e a p p r o p r i a t e if price tiiscrimination ancl m u l t i p a r t tariffs a r e LAnti how should t h e optimal pricing rule be modifietl if t h e clemantl is shifted t o goods t h a t a r e tasecl or producetl h y puhlic enterprises? T h e In T h e rele\.ant lirice elasticities of erjuation (6) cannot remain constant if the I~asicS i u t s k . relation is to I I satisfied. ~ Equatioll (61 is correct for the elasticity values that prevail a t the optitnun1 point. 111practice, constant values may IIC an adequate nl)prosimation within the relevant range. If not. some more general demand structure must he considered; see, e.g., h.1'. Barten. See 1,'eldstein for the theory of the optimal two-part tariif and a specific alil~lication.

"

g r o w t h of t h e puhlic sector anti t h e increasing concern w i t h distributional e q u i t y m a k e i t i m p o r t a n t t o develop a m o r e complete t h e o r y of puhlic pricing t h a t incorporates considerations of clistriljutional equity. REFERENCE:, A. P. Barten, ('on.umer Demailtl l'unctio1-1. Under Contlitions of 4lmo.t .\dditi\e Preferences." E ~ O I Z ( tIiIi (~(Ir (,Jan 1963. 3-3,1 38. \Y'. J. Baumol and D. F. Bradford, bgOptin~al Departure. from RIarginal C0.t I'ricing." 1ii1(i. 1G011 R ( i ' . , J ~ n e1970. 60,265-83 h3. Boiteux, '.Sur la ge+tion des llonopole. I'ublic+ aytreint* a l'equilibre hudgetaire." Eco~~ollrc.triccr, Jan. 195h. -34,2 2 30. P. A. Diamond and J. A. Mirrlees, '%Optimal I1 Tax 'I'axation ant1 E ' ~ ~ l ~ 1)roduction: lic Rules," .-liirc3r.E c o r ~ .Kczl., June 1971, 6 1 , 2(~1-78. A. I(. Dixit, '.On the Optimum Structure of

Ei-oil. K C T I June ., Commodity Taxes." .4?71c>r. 1970. 611, 2 9 5 3 0 1 . hl. S. Feldstein, %'Equityand EHiciency in l'ub-

lic Sector I'ricing: The Optimal TITO-Part Tariff," ()uart. J. E C O I I .forthcoming. , , '%TheI1ricing of Public Intermediate Goods." J. P z ~ b l Econ., . forthcoming. A. C. Harberger, ,'Taxation. Resource .\llocation ancl LVelfare," in J . Due, etl.. T h c Rolr, of Dirczct ( I I I ~ I/~dir(,i-tT(I.c.('silz tlz(' F(~d(,rcrlI ~ ( ' ~ ' c ISITI S~ ~( C ' ' IPrinceton II, 1961. J . R. Hicks, T'trlllc. o11d C'(zpit(11,Zd etl.. S e w York 1916. A. P. Lerner, "On Optimal Taxes with an U,'ntasable Sector." .lr11c3r.Ecoiz. Kczl., June - -

--

1970, 6 0 , 2 3 1 9 3 . H. hlohring, '&Thel'eak Loatl Problem with Increasing Returns and Pricing Constraints." .-1711t.r.Ecolz. I\'t.sl., Sept. 1970, 60,6 9 3 705. F. Rarnsey, *'.\ Contribution to the Theory of Taxation," Ecoir. J., l l a r . 1927, 37, 47-61,

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You have printed the following article: Distributional Equity and the Optimal Structure of Public Prices Martin S. Feldstein The American Economic Review, Vol. 62, No. 1/2. (Mar. - May, 1972), pp. 32-36. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28197203%2F05%2962%3A1%2F2%3C32%3ADEATOS%3E2.0.CO%3B2-V

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[Footnotes] 1

The Peak Load Problem with Increasing Returns and Pricing Constraints Herbert Mohring The American Economic Review, Vol. 60, No. 4. (Sep., 1970), pp. 693-705. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28197009%2960%3A4%3C693%3ATPLPWI%3E2.0.CO%3B2-9 1

Optimal Taxation and Public Production II: Tax Rules Peter A. Diamond; James A. Mirrlees The American Economic Review, Vol. 61, No. 3, Part 1. (Jun., 1971), pp. 261-278. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28197106%2961%3A3%3C261%3AOTAPPI%3E2.0.CO%3B2-4

References Consumer Demand Functions under Conditions of Almost Additive Preferences A. P. Barten Econometrica, Vol. 32, No. 1/2. (Jan. - Apr., 1964), pp. 1-38. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28196401%2F04%2932%3A1%2F2%3C1%3ACDFUCO%3E2.0.CO%3B2-K

NOTE: The reference numbering from the original has been maintained in this citation list.

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Optimal Departures From Marginal Cost Pricing William J. Baumol; David F. Bradford The American Economic Review, Vol. 60, No. 3. (Jun., 1970), pp. 265-283. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28197006%2960%3A3%3C265%3AODFMCP%3E2.0.CO%3B2-B

Sur la gestion des Monopoles Publics astreints a l'equilibre budgetaire M. Boiteux Econometrica, Vol. 24, No. 1. (Jan., 1956), pp. 22-40. Stable URL: http://links.jstor.org/sici?sici=0012-9682%28195601%2924%3A1%3C22%3ASLGDMP%3E2.0.CO%3B2-C

Optimal Taxation and Public Production II: Tax Rules Peter A. Diamond; James A. Mirrlees The American Economic Review, Vol. 61, No. 3, Part 1. (Jun., 1971), pp. 261-278. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28197106%2961%3A3%3C261%3AOTAPPI%3E2.0.CO%3B2-4

On the Optimum Structure of Commodity Taxes Avinash K. Dixit The American Economic Review, Vol. 60, No. 3. (Jun., 1970), pp. 295-301. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28197006%2960%3A3%3C295%3AOTOSOC%3E2.0.CO%3B2-S

Optimal Taxes with an Untaxable Sector Abba P. Lerner The American Economic Review, Vol. 60, No. 3. (Jun., 1970), pp. 284-294. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28197006%2960%3A3%3C284%3AOTWAUS%3E2.0.CO%3B2-O

The Peak Load Problem with Increasing Returns and Pricing Constraints Herbert Mohring The American Economic Review, Vol. 60, No. 4. (Sep., 1970), pp. 693-705. Stable URL: http://links.jstor.org/sici?sici=0002-8282%28197009%2960%3A4%3C693%3ATPLPWI%3E2.0.CO%3B2-9

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A Contribution to the Theory of Taxation F. P. Ramsey The Economic Journal, Vol. 37, No. 145. (Mar., 1927), pp. 47-61. Stable URL: http://links.jstor.org/sici?sici=0013-0133%28192703%2937%3A145%3C47%3AACTTTO%3E2.0.CO%3B2-K

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