Sufficient Statistics Revisited

Transcription

1 Suffcent Statstcs Revsted Henrk J. Kleven Prnceton Unversty and NBER August 2018 Abstract Ths paper revews and generalzes the suffcent statstcs approach to polcy evaluaton. The dea of the approach s that the welfare effect of polcy changes can be expressed n terms estmable reduced-form elastctes, allowng for polcy evaluaton wthout estmatng the structural prmtves of fully specfed models. The approach s based on three assumptons: that polcy changes are small, that there are no non-government dstortons, and that a set of hgh-level restrctons on the envronment and preferences can be used to reduce the number of elastctes to be estmated. We generalze the approach n all three dmensons. It s possble to present transparent suffcent statstcs formulas under very general condtons that allow for large reforms and for a wde range of non-government dstortons. However, the estmaton requrements ncrease consderably. In the case of dscrete polcy reform, we have to estmate both elastctes and elastcty changes due to the reform. If such elastcty changes cannot be estmated, then t s necessary to assume so-elastc preferences (or another parametrc form) makng the approach fully structural. In the case of non-government dstortons, the suffcent statstcs nclude elastctes and adjusted tax wedges that combne observable government dstortons wth estmable non-government externaltes and nternaltes. I thank Felx Berbrauer, Claus Krener, Magne Mogstad, Emmanuel Saez, and Owen Zdar for comments. Contact nformaton for the author: kleven@prnceton.edu.

2 1 Introducton The last decade of research on polcy evaluaton and welfare analyss has seen an exploson n the use of a term: suffcent statstcs. The dea of the suffcent statstcs approach s that the welfare effects of polcy changes can be expressed n terms of estmable elastctes, allowng for polcy evaluaton wthout makng parametrc assumptons or estmatng the structural prmtves of fully specfed models. Chetty (2009) coned the phrase, argung that the approach combnes the best of reduced-form and structural approaches: credble dentfcaton of causal effects and the ablty to make welfare predctons. To llustrate the rsng popularty of ths termnology, Fgure 1 shows the fracton of NBER workng papers n publc economcs that refer to the suffcent statstcs approach. The graph s based on the study of language trends n publc economcs by Kleven (2018). The suffcent statstcs termnology was rarely used untl the late 2000s, but takes off after the publcaton of Raj Chetty s paper. It has been on a steep upward trend snce then. Whle the termnology s new, the ntellectual orgns of the suffcent statstcs approach are extremely old. Economsts have expressed optmal tax polcy and deadweght loss n terms of demand and supply elastctes snce the early days of normatve publc fnance theory. 1 modern developments have been pvotal for the recent nfluence of the approach. One development s the dentfcaton revoluton n emprcal work over the last two or three decades. Ths work has allowed for clear and credble dentfcaton of the reduced-form effects of publc polcy usng quas-expermental research desgns. The other development s a set of theoretcal contrbutons on optmal polcy and welfare measurement that have clarfed the general prncples under whch welfare can be wrtten as a functon of reduced-form elastctes. The fundamental nsght s that, because of envelope condtons from household and frm optmzaton, the welfare effect of small polcy changes can be expressed as a fscal externalty the mpact of changed behavor on the government budget and s therefore governed by behavoral elastctes nteracted wth 1 Ramsey (1927) and Corlett & Hague ( ) dscussed the role of elastctes for optmal tax polcy. Harberger (1964) popularzed the measurement of deadweght loss usng elastcty-based approxmatons (Harberger trangles), but he was not the frst to expose the basc deas. Hotellng (1938) provdes an analyss of Harberger trangles before Harberger, labellng t the classcal argument and credtng Duput (1844) wth the underlyng deas. Hnes (1999) provdes a knowledgeable revew tracng the ntellectual hstory of Harberger trangles. The emprcal estmaton of the demand and supply elastctes relevant for deadweght loss calculatons orgnates prmarly wth the large body of work by Martn Feldsten from the late 1960s onwards. Two 1

3 observable tax-transfer rates. 2 Ths paper revsts the foundatons of the suffcent statstcs approach, clarfes ts advantages and lmtatons, and provdes a number of generalzatons. As t stands, the suffcent statstcs approach reles on three key assumptons. The frst assumpton s that the polcy change beng analyzed s small, whch n prncple means nfntessmal or at least close enough to nfntessmal for frst-order approxmatons to be precse. The second assumpton s that government polcy s the only thng that stands between the actual equlbrum and the frst-best equlbrum. In other words, there are no non-government externaltes or nternaltes that would be affected by behavoral responses to polcy reforms. Fnally, there s a thrd set of assumptons on the decson envronment (e.g., dynamcs, uncertanty, and polcy nstruments) and on preferences (e.g., separablty assumptons and quas-lnearty). These assumptons are not theoretcally necessary, but they are mportant for emprcal mplementaton as they determne the exact set of suffcent statstcs for that partcular settng. 3 Ths paper generalzes the suffcent statstcs approach n all three dmensons. We cast the analyss n the language of taxaton, but the framework can capture non-tax polces as well. To begn wth, keepng the assumptons of small reforms and no non-government externaltes, we present a suffcent statstcs formula that s very general n terms of the envronment and preferences. Ths formula holds for any type of tax system and tax reform, and t allows for dynamcs and general equlbrum effects. However, the problem wth ths general approach s that the parameter space s very large: t ncludes the compensated own- and cross-prce elastctes as well as the ncome elastctes of every good at each pont n tme. A suffcent statstcs approach based on ths many parameters s nfeasble. We therefore smplfy the parameter space by mposng more structure on tax polcy and preferences. An advantage of startng from a general formulaton s that we can see very clearly how dfferent combnatons of assumptons generate the smple suffcent statstcs formulas used n practce. Ths ncludes the Harberger-style formulas expressed n terms of a sngle suffcent statstc summarzed by Chetty (2009). 4 2 The fscal externalty property underles most of the normatve publc fnance lterature, but ts exact role for welfare measurement and the emprcal mplcatons have crystallzed more recently. The property s crucal for the suffcency of the elastcty of taxable ncome for welfare (Feldsten 1999). See Saez (2004) and Kleven & Krener (2005) for clear expostons of the general prncple. 3 Ths last set of assumptons make the suffcent statstc language slghtly odd, because the elastctes are only suffcent condtonal on the hgh-level structural assumptons beng made. 4 Startng from a general formula also allows us to compare suffcent statstcs approaches to structural approaches. The latter can be vewed as an alternatve way of smplfyng the parameters space, namely by makng parametrc assumptons that reduces the hgh-dmensonal elastcty space to a few structural prmtves. 2

4 We then relax the other key assumptons of the approach, allowng for large reforms and for non-government dstortons. It s possble to provde transparent and ntutve suffcent statstcs formulas for those more general cases, but the estmaton requrements ncrease consderably. We hghlght two man results. Frst, t s possble to provde a trapezod approxmaton of the welfare effect of large reforms, whch depends on the same elastcty (or elastctes) as the smallreform formula as well as the change n the elastcty (or elastctes) created by the reform. The suffcent statstcs are therefore a set of elastcty levels and elastcty changes. As we dscuss, gven the dffcultes of reachng a consensus on elastcty levels, t may be unrealstc to hope for a consensus on elastcty changes. If so, we have to mpose more structure. The smplest soluton s to assume quas-lnear, so-elastc preferences, but ths s of course a fully parametrc approach. Exstng suffcent statstcs approaches are mplctly based on such preferences, because they analyze dscrete polcy reforms and do not account for elastcty changes. 5 Second, t s possble to provde suffcent statstcs results under a very general formulaton of non-government externaltes by re-defnng the tax wedges to nclude any unnternalzed utlty effect of behavoral changes. Our formulaton allows for atmospherc externaltes, relatve consumpton and relatve labor supply externaltes, nternaltes due to psychologcal aspects, and many other effects. 6 Crucally, the redefned tax wedges are not drectly observable, but have to be estmated and n fact may be harder to estmate than the behavoral elastctes. For example, whle there may be some degree of consensus on the earnngs elastcty at the top of the dstrbuton, there may be less consensus on the degree to whch these earnngs responses represent socally productve effort as opposed to, say, rent-seekng or rat race effects. In any case, our formulas show that the set of suffcent statstcs for polcy evaluaton nclude behavoral elastctes along wth externalty-adjusted tax wedges on each margn of response. In the suffcent statstcs sprt, welfare evaluaton does not requre a fully specfed model of each dfferent market mperfecton. It s suffcent to estmate reduced-form gaps between prvate and socal prces n conjuncton wth the behavoral elastctes that we normally estmate. The paper s organzed as follows. Secton 2 lays out the framework, secton 3 characterzes the welfare effect of small reforms absent any non-government externaltes, secton 4 generalzes the analyss to large reforms, secton 5 generalzes the analyss to allow for non-government 5 Assumng so-elastc preferences, the standard suffcent statstcs formula stll needs to be modfed when reforms are large. Ths s because the formula has to account for the changng tax wedge over the dscrete reform path. 6 As a specfc example, our framework allows for the type of wage barganng externaltes modelled by Pketty et al. (2014). 3

5 dstortons, and secton 6 concludes. 2 Model There s a contnuum of ndvduals ndexed by. There s a dscrete set of goods ndexed by j = 0,..., J. The set of goods may nclude consumpton and labor supply at dfferent ponts n tme as well as dfferent types of consumpton and labor supply at a pont n tme. If the settng s dynamc, goods may also nclude wealth at dfferent ponts n tme or bequests. Utlty s gven by u ( x 0,..., x ) J = u ( x ). (1) The budget constrant s gven by J x j + T ( x 0,..., x ) J = y, (2) j=0 where pre-tax prces are normalzed to one, or rather where we nterpret x as a vector of pre-tax expendtures and pre-tax earnngs (rather than consumpton and labor supply quanttes). Prces are then emboded n utlty (as a functon of expendtures/earnngs) and any heterogenety n prces/wage rates are accounted for n u (.). 7 The tax functon T (.) embodes all taxes and transfers, t may be nonlnear, and t may feature non-separabltes between dfferent arguments. For example, f good j denotes consumpton at tme t (such that T / x j ncludes captal taxes) and good j + 1 denotes labor earnngs at tme t (so that T / x j+1 ncludes labor taxes), then T (.) s not separable n j and j + 1 under an ncome tax (.e., 2 T / x j x j+1 = 0). We wll assume, however, that T (.) s pecewse lnear, so that margnal tax rates are constant wthn brackets. Denotng margnal tax rates by T / x j τ j, we can rewrte the budget as follows J ( ) 1 + τ j x j = Y, (3) j=0 where Y y + J j=0 τ j x j T ( x 0,..., x J ) s vrtual ncome. 7 Ths parsmonous specfcaton mples that we avod carryng prce notaton n the dervatons. Whle ths mples that we are takng pre-tax prces as gven, there s n fact no loss of generalty here. Gven the assumpton of perfect competton, tax-nduced prce changes (ncdence) affect only the dstrbuton of real ncomes, they do not affect effcency. Because the suffcent statstcs approach deals wth the measurement of effcency, the results we provde are vald under any arbtrary ncdence. 4

6 Each household maxmzes utlty (1) subject to the budget constrant (3). Denotng the Lagrange multpler of ths optmzaton program by λ, the frst-order condton for x j s gven by u λ ( 1 + τj) = 0, (4) x j where we have used that Y / x j = 0.8 The uncompensated demand and supply functons mpled by equaton (4) can be wrtten as x j = ( x j 1 + τ 0,..., 1 + τj, Y ). Indrect utlty may then be defned as follows v ( 1 + τ0,..., 1 + τj, Y ) = u ( x ( τ 0,..., 1 + τj, Y ),..., x ( J 1 + τ 0,..., 1 + τj, Y )). (5) The dervatves of ndrect utlty have well-known propertes that wll become useful later. The margnal utlty of ncome Y and of tax prces 1 + τ k (takng Y as gven), respectvely, are equal to v Y = λ, v (1 + τ k ) = λ x k. (6) In dervng these results, we use the budget constrant as well as the frst-order condtons. The second dervatve s, of course, Roy s dentty. Fnally, we also have the Slutsky decomposton,.e. x j (1 + τ k ) = where tlde denotes compensated demand/supply. x j x j (1 + τk ) x k Y, (7) 3 Welfare Effect of Small Reforms To study the effects of reform, we specfy tax polcy as a functon of a treatment parameter θ,.e. we wrte tax lablty as T ( x 0,..., x J, θ) and margnal tax rates as τj (θ) j. Changes n treatment θ may capture any arbtrary set of changes n τ0,..., τ J and n T (.), wthn the class of pecewse 8 Equaton (4) s the optmalty condton for agents locatng wthn brackets,.e. condtonal on not bunchng at a knk pont between brackets. We gnore bunchng at knk ponts throughout the analyss, because such local responses (mass ponts) have no frst-order mpact on aggregate welfare. That s, whle bunchng s wdely studed as an emprcal approach to uncover behavoral elastctes (see e.g., Saez 2010; Kleven 2016), bunchng mass s not mportant for welfare n and of tself. 5

7 lnear polces. 9 Ths secton focuses on small reforms (dθ 0), whle the next secton generalzes the analyss to large reforms. We start by calculatng the money-metrc effect on utlty of such polcy reforms,.e. dv /dθ λ. We have dv dθ = J j=0 Usng equaton (6), ths may be rewrtten to v ( ) dτ j 1 + τj dθ + v dy Y dθ. (8) dv /dθ λ = J j=0 x dτj j dθ + dy dθ. (9) From the defnton of vrtual ncome,.e. Y (θ) y + J j=0 τ j (θ) x j T ( x 0,..., x J, θ), we have dy J dθ = dτj j=0 dθ x j T θ. (10) Insertng (10) nto (9), we obtan dv /dθ λ = T θ. (11) Hence, the utlty effect of any arbtrary, small reform equals the mechancal revenue effect. Ths central result follows from envelope condtons (as emboded n equaton (6)) and the assumpton of no other externaltes than those operatng through the government budget. To move from ndvdual welfare to socal welfare, we specfy a socal welfare objectve W (θ) = ω v (θ) d + T (θ) d, (12) where ω s a Pareto weght on ndvdual, and s the margnal value of government revenue (or the average socal margnal utlty of ncome n the populaton). Dfferentatng W (θ) and usng equaton (11), we obtan [ dt = dθ ] T g d = θ ( dt dθ T ) + ( 1 g ) T d, (13) θ θ }{{}}{{} effcency equty 9 The same flexble specfcaton of polcy reform was used by Kleven & Krener (2002, 2006), Essa et al. (2006, 2008), and Hendren (2016). 6

8 where g ω λ denotes the socal welfare weght on ndvdual. The socal welfare weghts are equal to 1 on average, and ther varaton n the populaton summarzes the government s preferences for equty. 10 Equaton (13) splts the total welfare effect nto an effcency effect (frst term on the rght-hand sde) and an equty effect (second term on the rght-hand sde), the latter beng governed by g. Absent equty concerns (g = 1 ), the second term equals zero. The suffcent statstcs approach s about the measurement of effcency, not equty. It s therefore useful to hghlght the result on effcency as a proposton: Proposton 1 (Fscal Externalty). The effect of any small tax reform on economc effcency equals = g =1 [ dt dθ T ] d, (14) θ namely the dfference between the total and mechancal revenue effects, whch corresponds to the behavoral revenue effect ( the fscal externalty ). The effcency effect of any small reform of the tax-transfer system (wthn the class of pecewse lnear systems) equals the fscal externalty from behavoral responses to the reform. 11 The ntuton for ths result s smple. Because agents are optmzng and there are no non-tax externaltes, behavoral responses to small reforms have no frst-order effects on utlty. The only frst-order effect comes from an externalty that operates through the government budget: When agents adjust behavor to avod hgher taxes, they create tax revenue leaks that mpose a fscal externalty on the rest of the populaton (the potental transfers they can receve are now lower). We have cast the analyss n the language of taxaton, but the underlyng envelope theorem logc extends to any form of publc polcy gven the same general assumptons. The logc of Proposton 1 underscores much of the normatve publc fnance lterature, ncludng optmal taxaton (e.g. Damond & Mrrlees 1971; Damond 1998; Saez 2001), deadweght loss measurement (e.g. Harberger 1964; Brownng 1987; Feldsten 1999; Goulder & Wllams 2003; Kleven & Krener 2005; Essa et al. 2006, 2008), the margnal cost of publc funds (e.g. Brownng 1976; Slemrod & Ytzhak 1996, 2001; Kleven & Krener 2006), socal nsurance (e.g. Baly 1978; Chetty 2006; Kolsrud et al. 2017), and the suffcent statstcs approach (Chetty 2009). Whle the theoretcal 10 Throughout the paper, we cast the analyss n the language of taxes/transfers and equty, but everythng can be restated n the language of socal nsurance and consumpton smoothng. In that case, W (θ) would represent expected utlty over dfferent states of the world, and the weghts g capture the benefts from consumpton smoothng across good and bad states. 11 See also Kleven & Krener (2005, 2006) and Essa et al. (2006, 2008) for detaled analyses of ths pont. 7

9 property was always there behnd the curtans, the crystallzaton of the deep and general prncple ncludng ts mplcatons for emprcal work has emerged more clearly over the last couple of decades. Leavng asde any potental concerns about the underlyng assumptons, should we conclude the theoretcal analyss here and focus on the fscal externalty as the target for emprcal work? Hendren (2016) argues that we should, re-castng the fscal externalty n dfferent language. Specfcally, usng the tax functon T ( x 0,..., x J, θ), we can rewrte equaton (14) as = g =1 J τjx j j=0 ( d log x j dθ ) d, (15) where d log(x j) dθ s labelled the polcy elastcty by Hendren (2016). He argues that such polcy elastctes should be the object of nterest for appled welfare analyss. To be precse, gven we do not estmate ndvdual-level elastctes, the object of nterest for emprcal studes would be ε P j = τ j x d log(x j) j dθ d,.e. the revenue-weghted average polcy elastcty for each good j. Two remarks on such an approach are worth makng, a pedantc one and a substantve one. Frst, the pedantc remark: Remark 1 (Elastctes are Irrelevant for Assessng Actual Reform). When measurng the welfare effect of an actual polcy reform, t s unnecessary to estmate elastctes. Consder the polcy treatments θ 0, θ 1 where dθ = θ 1 θ 0 0, and assume random assgnment to treatments. The welfare effect can be estmated as E { T ( x 0 (θ 1 ),..., x J (θ 1 ), θ 1 ) θ1 } E { T ( x 0 (θ 0 ),..., x J (θ 0 ), θ 1 ) θ1 } = E { T ( x 0 (θ 1 ),..., x J (θ 1 ), θ 1 ) θ1 } E { T ( x 0 (θ 0 ),..., x J (θ 0 ), θ 1 ) θ0 }. (16) The assumpton of random assgnment s not mportant for the conceptual pont (absent randomzaton one would use a dfferent estmator). Whatever the estmaton approach, f the sole objectve s to measure the welfare effect of an actual reform experment, t s unnecessary to estmate elastctes or suffcent statstcs. The outcome of nterest s drectly measurable. However, evaluatng actual reforms s almost never the sole objectve of polcy debate or academc dscourse. Assessng polcy reform s nstead about comparng dfferent counterfactual scenaros, whch leads to the second and substantve remark: 8

10 { } dτ Remark 2 (Polcy Elastctes). Consder polcy reforms R = 0 dθ,..., dτ J dθ, T θ. Because polcy I elastctes are functons of R, they can be used only to measure the welfare effect of an actually mplemented reform R (compared to the counterfactual of no reform). They cannot be used to assess the welfare effect of any other counterfactual reform that could be mplemented. Under the polcy elastcty approach to emprcal research, economsts would know only the aggregate welfare effect of hstorcal reforms. Wth ths nformaton alone, we would not be able to provde any advce on future reforms, unless they exactly replcate or reverse hstorcal reforms. The basc lmtaton of polcy elastctes s that they are externally nvald by constructon. To assess polcy desgn, we have to express the fscal externalty n terms of prce and ncome elastctes that are externally vald. To derve a suffcent statstcs formula based on (potentally) externally vald elastctes, we go back to the fscal externalty expresson n equaton (14). Usng the tax functon T ( x 0,..., x J, θ), ths can be rewrtten as = g =1 J τj j=0 [ J k=0 ] x j dτk (1 + τk ) dθ + x j dy Y d. (17) dθ Moreover, usng dy dθ = J dτk k=0 may state the followng: dθ x k T θ, the Slutsky decomposton, and rearrangng terms, we Proposton 2 (Suffcent Statstcs for Small Reforms). The effect of any small tax reform on economc effcency can be wrtten as = g =1 J j=0 [ J k=0 τjx jε dτk /dθ jk 1 + τk τjx jηj ] T / θ Y d, (18) where ε jk x j Y Y x j x j 1+τk (1+τk) x j s the Hcksan prce elastcty of good j wrt. the prce on good k, and η j s the ncome elastcty of good j. Therefore, condtonal on a set of observables (tax parameters, { } expendture and earnngs levels), the suffcent statstcs for evaluatng reform are ε jk, η j j,k,. Ths formula s completely general gven the assumptons of small reforms and no non-polcy mperfectons. It s a general equlbrum result as t accounts for dstortons n all markets and allows for any possble cross-effects between markets. We take prces as gven n the dervaton, but ncorporatng prce changes (tax ncdence) would not change the formula. Under the assumpton 9

11 of perfect competton, general equlbrum prce changes redstrbute across agents, but do not mpact economc effcency. There s a problem, however. Because the goods vector n general ncludes dfferent types of consumpton and labor supply over tme (.e., j captures both goods type and tme), the parameter space s n general very large. The suffcent statstcs approach s therefore nfeasble wthout more structure. To make progress, we have to reduce the dmensonalty of the problem by ether () restrctng the tax polcy space or () restrctng behavoral responses. Most suffcent statstcs approaches do both, although ths s often left mplct. By startng from a general formulaton, we are able to see exactly how dfferent sets of assumptons lead to smple suffcent statstcs expressons. We consder ths n the next secton. 3.1 Many Roads Lead To Harberger To begn wth, consder the baselne model presented n Chetty (2009). Three assumptons are made there: () utlty s quas-lnear, () only one good s taxed, and () the tax s lnear. The frst assumpton mples η j to = 0 j,. Assumng that good 0 s the taxed good, equaton (18) smplfes = ε 0 g =1 τ τ 0 dτ 0 dθ, (19) where ε 0 [ x 0 ε ] 00 d s the demand-weghted average Hcksan elastcty n the populaton. Ths s a Harberger-style formula for the margnal deadweght loss of taxaton n whch ε 0 s the suffcent statstc for welfare analyss. 12 The sense n whch ε 0 s suffcent s of course condtonal on the underlyng assumptons. But t s possble to consder alternatve restrctons on tax polcy space or preferences that gve rse to a smlar formula. Keepng the assumpton of quas-lnearty, we can relax the assumpton that only one good s taxed. Assume that goods 0,..., J 0 are taxed at rate τ 0, whle goods J 0 + 1,..., J are taxed at rate τ 1. We may then normalze τ 1 to zero (and adjust τ 0 accordngly) wthout loss of generalty. In ths case we obtan the Harberger formula (19) once more, except that we have to redefne the elastcty as ε 0 [ ] J 0 j=0 J 0 k=0 x j ε jk d. Ths s a demand-weghted elastcty across goods 0,..., J 0 (nstead of only good 0) wth respect to the tax rate τ 0 on all those goods (nstead of only on good 0). Agan, a sngle elastcty s suffcent for welfare analyss, but the elastcty s dfferent 12 Unlke Harberger-trangle approxmatons, equaton (19) s an exact formula that holds for any tax rate τ 0 and any functonal form for preferences (gven quas-lnearty). Tradtonal Harberger-trangle formulas assume ether that taxes are small or that demand functons are lnear. 10

12 than before. We have descrbed two cases: one where only good 0 s taxed, and one where goods 0,..., J 0 are taxed at a unform rate whle goods J 0 + 1,..., J are taxed at another unform rate. To see the practcal dfference between these two cases, consder the taxaton of labor ncome. The frst case may be nterpreted as a statc model n whch good 0 s labor supply and goods 1,..., J are dfferent consumpton goods. The second case may be nterpreted as a dynamc model n whch goods 0,..., J 0 are labor supples n dfferent perods (taxed at a constant rate over tme), whle goods J 0 + 1,..., J are consumpton n dfferent perods (taxed at a constant rate over tme). 13 These two models gve rse to the same Harberger formula by renterpretng the statc earnngs-weghted labor supply elastcty to a lfetme earnngs-weghted labor supply elastcty wth respect to a permanent tax. Alternatvely, the second case may be nterpreted as capturng mult-dmensonal labor supply choces (hours worked, effort, occupaton, tranng, etc.), whch jontly determne labor earnngs taxed at rate τ 0. In ths case, the suffcent statstc ε 0 [ ] J 0 j=0 J 0 k=0 x j ε jk d s the elastcty of total labor ncome governed by all the underlyng margns of behavor, or the elastcty of taxable ncome n the language of Feldsten (1995, 1999). If we allow for dynamcs, ths s the lfetme elastcty of taxable ncome wth respect to a permanent ncome tax. Ths result generalzes Feldsten s analyss. 14 There are other ways of obtanng the smple suffcent statstcs formula n (19). If we assume both quas-lnearty (η j = 0) and no cross-effects (ε jk = 0 for j = k), then the result s obtaned when only one tax s changng (and ths tax s lnear). In other words, by addng the assumpton of no cross-effects, we can relax the assumpton that only one good s taxed to an assumpton that only one tax s changng n the reform. By makng a stronger assumpton on the tax system, t s possble to obtan a smple Harbergerstyle formula wthout quas-lnear preferences. Specfcally, under a lnear proportonal tax system (no lump-sum taxes or transfers), we have T = J k=0 τ k (θ) x k and therefore T θ = J k=0 dτ k dθ x k. Insertng ths nto (18) and usng (7), we obtan a welfare formula that depends only on prce elastcty terms, but where those prce elastctes are Marshallan rather than Hcksan. Assum- 13 The assumpton of unform consumpton taxaton rules out captal taxes. 14 The reason why we can express effcency n terms of a sngle suffcent statstc even n the presence of cross-effects (such as shftng responses) s that all margns of behavor are taxed ether at the rate τ 0 or not at all. Wth cross-effects (shftng) between bases that are taxed at dfferent non-zero rates, we need to estmate more parameters. Equaton (18) provdes the general formula. Saez (2004), Saez et al. (2012), and Pketty et al. (2014) consder an example wth two bases taxed at dfferent (non-zero) rates and shftng between the two. 11

13 ng that goods 0,..., J 0 are taxed at rate τ 0 whle the rest of the goods are untaxed, we obtan the Harberger-style formula (19) wth ε 0 beng a Marshallan elastcty. The fundamental challenge we face s that a general suffcent statstcs approach (one based on equaton (18)) would rely on too many parameters to be feasble. Actual suffcent statstcs approaches smplfy the parameter space often to just one parameter by makng hgh-level assumptons about preferences and tax polcy. The alternatve s to smplfy the parameter space by assumng an explct parametrc form for u (.),.e. to take a structural approach. It s useful to contrast the suffcent statstcs and structural approaches n a specfc example. Consder the case where the goods vector ncludes labor supples and consumpton n dfferent perods,.e. u = u (l 0,..., l T ; c 0,..., c T ). Assumng that l t and c t are taxed at constant rates over tme, we can express the welfare effect of tax reform as equaton (19) where the suffcent statstc s the lfetme earnngs-weghted labor supply elastcty. The assumpton of a constant tax rate on consumpton rules out captal taxes. Alternatvely, we may assume that u (.) has a tractable form. We mght consder, say, a nested CES functon u (f (c 0, l 0 ),..., f (c T, l T )) wth three parameters: an ntratemporal elastcty of substtuton between consumpton and labor σ 1, an ntertemporal elastcty of substtuton σ 2, and a dscount factor δ. These three parameters (along wth prces) control all the suffcent statstcs n equaton (18). We no longer have to make restrctons on tax polcy space, but can be more general n ths dmenson. Ths ncludes allowng for captal taxes. Besdes the dfferences n assumptons, a crucal dfference between these two approaches les n the data requrements. The suffcent statstcs approach calls for the estmaton of a lfetme earnngs-weghted labor supply elastcty. It s challengng to fnd the tax varaton and data allowng for the estmaton of ths long-run parameter. By makng parametrc assumptons, the structural approach allows for the estmaton of welfare effects usng shorter-run varaton n the data. As long as the tax varaton s rch enough to separately dentfy the structural prmtves (σ 1, σ 2, δ), the model can generate the full lfecycle effects necessary to calculate welfare. Of course, these calculatons are meanngful only f the parametrc assumptons are correct. The precedng dscusson hghlghts that the key trade-off when choosng methodology s between data requrements and parametrc assumptons. How to strke ths trade-off depends on the queston and settng. It s worth mentonng that, n practce, a common dfference between the suffcent statstcs and structural approaches s that the former s based on exogenous (e.g. quas-expermental) varaton, whle the latter s based on observatonal and potentally endogenous varaton. However, ths dvde s largely ndependent of the deeper conceptual trade-offs 12

14 dscussed here; t reflects cultural dfferences across research strands. The structural approach could (and should) target quas-expermental moments Tax-Base Changng Reforms A key feature of many tax reforms s that they change both tax rates and tax bases. The framework s suffcently general to analyze tax base changes. For example, base broadenng corresponds to ntroducng a postve tax rate on some good k that was ntally taxed at rate zero,.e. dτ k dθ > 0 where τ k = 0 ntally. Typcally, the newly ntroduced good s taxed at the same rate as a set of other goods already n the tax base,.e. dτ k dθ = τ k where k s a good already n the tax base. If the ntal tax base s taxed at a large rate, τ k >> 0, then a base broadenng reform s necessarly a large reform. Therefore, the small-reform assumpton made thus far s more tenuous when consderng tax base changng reforms, and so the generalzaton to large reforms n the next secton s mportant. Nevertheless, t s possble to acheve a number of key nsghts on base broadenng wthn the small-reform framework as we now show. To smplfy, we focus on stuatons where preferences are quas-lnear and taxes are lnear. We start by consderng a case where only good 0 s taxed ntally, and where the reform changes the tax rate on good 0 and brngs good 1 nto the tax base. In ths case, the general welfare formula (18) smplfes to where ε 0 = ε 0 g =1 τ τ 0 dτ 0 dθ + ε 01 τ τ 1 dτ 1 dθ, (20) [ x 0 ε ] 00 d and ε01 [ x 0 ε ] 01 d. If the ntal good n the tax base and the newly ntroduced good are substtutes (complements), then ε 0 and ε 01 have the opposte (same) sgns. Accordng to conventonal wsdom n publc fnance, t s better to collect a gven amount of revenue by taxng broad base at a low rate than by taxng a narrow base at a hgh rate. The result n equaton (20) formalzes the condton under whch ths folk theorem s true. To see ths, consder a reform that lowers the tax rate on the exstng base ( dτ 0 dθ the base ( dτ 1 dθ < 0) and at the same tme broadens > 0). The own-prce elastcty ε 0 s negatve, whle the cross-prce elastcty ε 01 s postve (negatve) f the two goods are substtutes (complements). In the case of substtutablty, both terms of equaton (20) are postve and the base-broadenng reform necessarly ncreases effcency. In the case of complementarty, we have offsettng effects on effcency. 16 Therefore, 15 For example, Jakobsen et al. (2018) develops a structural, quas-expermental approach for studyng the long-run effects of wealth taxes, an area where a suffcent statstcs approach s not feasble n practce. 16 In ths case, the net effect on effcency depends, besdes the strength of complementarty ε 01, on the magntude 13

15 we are able to re-state the conventonal wsdom n a precse and ntutve fashon: t s always effcent to broaden the tax base and lowerng the tax rate f the exstng and new elements of the tax base are substtutes, but not necessarly f they are complements. Ths s related to the classc logc of Corlett & Hague ( ). 17 To see ths pont more starkly, assume for smplcty that goods 0 and 1 are the only goods n the economy. In ths case, homogenety of degree zero of compensated demands mples ε 01 = ε 0 (usng Euler s Theorem). Hence, equaton (20) can be rewrtten as = ε 0 g =1 { τ0 dτ τ 0 dθ τ 0 dτ } 1. (21) 1 + τ 1 dθ We arrve once more at a welfare formula wrtten n just one suffcent statstc, ε 0, but the tax rate term (n curly brackets) wth whch we multply that elastcty s very dfferent. Because both the elastcty and the tax rate term are negatve, the base broadenng reform always ncreases effcency. Another way of lookng at the mportance of tax bases s to compare the welfare effect of tax rate ncreases under a narrow base and under a broad base, respectvely. If only good 0 s taxed (narrow base), then the welfare effect s gven by the Harberger formula (19). If both goods 0 and 1 are taxed at rate τ (broad base), then the welfare effect (18) can be wrtten as = [ ε ε 01 + ε 1 ] g =1 τ 1 + τ dτ dθ, (22) where we use that x 0 ε 01 = x 1 ε 10 due to Slutsky symmetry and the fact that the two goods are taxed at the same rate τ. Comparng the narrow-base formula (19) to the broad-base formula (22), we see that there are two key dfferences. Frst, the tax rate s lower under the broad base for a gven revenue requrement (condtonal on beng below the Laffer pont). Ths makes the effcency cost of tax ncreases smaller under the broad base. Second, the elastcty term s dfferent under the broad base. The broad-base elastcty ε ε 01 + ε 1 s smaller or larger than the narrow-base elastcty ε 0 dependof tax rate changes dτ0 dθ, dτ1 dθ. If the reform s revenue neutral, the latter depends on the budgets shares of the dfferent goods. 17 Whle our example was phrased n terms of goods (demands), the result apples equally to ncomes (supples). For example, f the goods are two dfferent labor ncome components l 0 and l 1, we have x 0 = l 0 and τ 0 = τ l0 and smlarly for good 1. The own-wage elastcty ε 0 s negatve (because good zero s mnus labor supply) and the cross-wage elastcty ε 01 s postve f the two labor components are substtutes. Accountng for these sgns, a smlar reasonng apples to base broadenng n ncome taxaton. 14

16 ng on the extra components ncluded n the base. The broad-base elastcty may be smaller (n absolute value) when substtutes are ncluded n the base (n whch case ε 01 has the opposte sgn of ε 0, ε 1 ). 18 On the other hand, the broad-base elastcty s larger when complements are ncluded. Agan, ths connects the effects of base broadenng wth classc Corlett-Hague reasonng. 3.3 An Optmal Tax Trck Techncally, t s just a small step from the tax reform formulas presented above to optmal taxaton. A necessary condton for an optmal tax system s that there exsts no small reform that can ncrease welfare. 19 Therefore, we must have dw dθ = 0 for any θ. From equaton (13), ths mples ( = g 1 ) T d, (23) g =1 θ where the left-hand sde s the effcency effect and the rght-hand sde s the equty effect. For example, n the specal case underlyng the Harberger-style result (19), we can rewrte ths to ( τ 0 g 1 ) x 0 = d, (24) 1 + τ 0 ε 0 where the numerator corresponds to the covarance between socal welfare weghts g and demand for the taxed good x 0.20 If there are several goods taxed at rate τ 0, then x 0 s replaced by the total demand for the taxed goods. Ths formula s a classc nverse elastcty rule tradng off effcency losses (denomnator) aganst equty gans (numerator). Results such as (24) make t temptng to say and many papers do that ε 0 s a suffcent statstc for optmal taxaton. However, ths s a trck, because the fact that we are consderng the optmal tax system changes the game. The condton holds at the optmum, and therefore ε 0 s the elastcty at the optmal pont. A pror ths can be any pont. Therefore, whle t s formally correct that ε 0 at the optmal pont s a suffcent statstc, knowng ths statstc requres global knowledge of demand functons. Ths mples a fully structural approach rather than a suffcent 18 For example, f goods 0 and 1 are the only two goods n the economy, they are necessarly substtutes. In ths case, we have ε 01 = ε 10 = ε 1 (usng Slutsky symmetry, dentcal tax rates, and homogenety of degree zero of compensated demands). Ths gves a broad-base elastcty equal to ε 0 ε 1, whch s smaller n absolute value than the narrow-base elastcty. 19 See Saez (2001) and Pketty & Saez (2013) for detaled expostons of the tax reform approach to dervng optmal tax rates. 20 To see ths, recall that the socal welfare weghts average to 1 n the populaton (E [ ] [ numerator corresponds to E g x 0 E g ] [ ] E x 0. [ g ] = 1), mplyng that the 15

17 statstc approach. 21 Many optmal tax papers assume so-elastc preferences, whch s a specfc parametrc form. Despte ths lmtaton, t remans useful to express optmal tax rules n terms of elastctes, because of the ntuton ths provdes and because so-elastc preferences may be a natural benchmark. 4 Welfare Effect of Large Reforms The suffcent statstc approach s exact only for nfntessmal reforms, but real-world reforms are never nfntessmal. If they were, we would not be very nterested n them. How small do reforms have to be for the suffcent statstc approach to be nformatve? Can we formulate a suffcent statstc approach for larger reforms? These are mportant questons, because the alternatve to assessng large reforms a fully structural approach s less transparent and reles on potentally strong parametrc assumptons. When reforms are large, the exstng suffcent statstc approach corresponds to a frst-order Taylor approxmaton of socal welfare. Therefore, a natural way to mprove the welfare analyss of large reforms would be to consder hgher-order Taylor approxmatons. Whle we consder such Taylor approxmatons later, we start by developng another approach to the welfare analyss of large reforms. As before, we specfy the hgh-dmensonal polcy space n terms of a treatment parameter θ,.e. we wrte the tax functon as T = T ( x 0,..., x J, θ) and the margnal tax rates as τj (θ) for all, j. Specfcally, by defnng the margnal tax rates as τj + θ τ j where τ j s the reform-nduced tax rate change, the pre-reform polcy corresponds to θ 0 = 0 and the post-reform polcy corresponds to θ 1 = 1. The tax rate changes τj may be large. We start from the observaton that the dscrete welfare change between θ 0 = 0 and θ 1 = 1 can be wrtten as the ntegral of the margnal welfare changes between those two ponts,.e. W = W (1) W (0) = 1 0 dw dθ dθ. (25) Because we have prevously characterzed the margnal welfare changes under very general con- 21 Motvated by these challenges, Kleven (2004) develops a dfferent approach to optmal taxaton n whch the optmal polcy can be expressed n terms of potentally observable proxes for the (hard-to-estmate) prce elastctes. Based on Gary Becker s theory of the allocaton of tme, these proxes are factor shares n household producton/consumpton actvtes. One mght be tempted to call ths a suffcent statstcs for the suffcent statstcs approach. At the same tme, t s a more structural approach as t reles on the Becker household producton model. 16

18 dtons (n Propostons 1-2), we are able to provde an (almost) exact formula for the welfare effect of large reforms n terms of elastctes. Focusng on the pure effcency effect, we have Proposton 3 (Suffcent Statstcs for Large Reforms: Almost Exact). The effcency effect of a dscrete reform from regme θ 0 = 0 to regme θ 1 = 1 can be wrtten as W 0 = g = dθ, (26) g =1 where 0 = (0) s the margnal value of government revenue at the ntal polcy θ 0 = 0, and where 0 g =1 J j=0 [ J k=0 ( τ j + θ τ j ( τ j + θ τj ) x j (θ) ε jk (θ) ) x j (θ) η j (θ) T / θ Y (θ) τ k 1 + τk + θ τ k ] d. (27) The elastctes ε jk, η j and demands x j are measured at regme θ (0, 1). Equaton (27) s an approxmaton only because t assumes 0 (θ) between θ 0 = 0 and θ 1 = 1. Ths proposton provdes an (almost) exact formula for the welfare effect of large reforms as a functon of elastctes. The formula s too general to be useful for polcy evaluaton, but t does hghlght the dmensons n whch the standard suffcent statstcs approach may get t wrong. The standard approach measures the fscal externalty based on the ntal tax wedge nteracted wth the ntal elastcty, whle the exact formula depends on the path of wedges and elastctes between θ 0 = 0 and θ 1 = 1. Therefore, the potental sources of error come from changng wedges and changng elastctes over the reform path. To obtan addtonal nsght, we smplfy the analyss n two dmensons. Frst, nstead of consderng the exact ntegral of margnal welfare effects n equaton (26), we consder a trapezod approxmaton of ths ntegral. We have W 1 { dw (0) + 2 dθ } dw (1). (28) dθ Compared to the standard suffcent statstcs approach (whch assumes that the margnal welfare effect s constant over the reform path), the trapezod approxmaton n (28) allows to change, but n a lnear fashon. The error made by ths approxmaton depends on the degree of convexty or concavty of the margnal welfare effect. 17

19 Second, we smplfy the analyss by mposng more structure on preferences and tax polcy. Consder the specal case underlyng the standard suffcent statstcs formula (19) for small reforms. Ths specal case assumes quas-lnear utlty and a sngle tax rate τ 0 on taxed goods. In ths case, we have W 0 1 g =1 2 { ε 0 (0) Ths allows us to state the followng: τ τ 0 τ 0 + ε 0 (1) } τ 0 + τ 0 τ 0. (29) 1 + τ 0 + τ 0 Proposton 4 (Suffcent Statstcs for Large Reforms: Trapezod). Assume quas-lnear utlty and a sngle tax rate τ 0 on taxed goods. In ths case, the trapezod approxmaton of the effcency effect of large reforms can be wrtten as W 0 g =1 τ 0 ε 0 τ τ { [ τ0 ε τ 0 ] + ε 0 [ ]} τ 0 τ0 + ε 0 τ 0, (30) 1 + τ τ 0 where ε 0 = ε 0 (0) s the elastcty at the ntal polcy and ε 0 = ε 0 (1) ε 0 (0) s the elastcty change due to the polcy. The frst term on the rght-hand sde s the standard small-reform formula (19), whle the second term s the large-reform adjustment. Condtonal on the observable tax parameters, the suffcent statstcs for welfare are the elastcty level ε 0 and the elastcty change ε 0. The welfare effect can be wrtten as the standard frst-order effect plus an adjustment term. The adjustment term reflects the two errors made by the frst-order approach: the error comng from the change n the tax wedge over the reform path and the error comng from the potental change n the elastcty. The correcton for these errors has three elements: the change n the tax wedge nteracted wth the ntal elastcty, the change n the elastcty nteracted wth the ntal tax wedge, and the change n the elastcty nteracted wth the change n the wedge. It seems reasonable to assume that, for most reforms, the last element (the two changes nteracted) s small. 22 Proposton 4 shows that two parameters provde suffcent statstcs for evaluatng large reforms: the elastcty level ε 0 and the elastcty change ε 0. All other parameters n the formula are 22 Proposton 4 clarfes the condtons under whch the standard frst-order approach to evaluatng large reforms s precse. Equaton (30) shows that, all else beng equal, the frst-order approach wll be relatvely precse when the τ ntal elastcty ε 0 s small and when the ntal wedge 0 1+τ 0 s small. In envronments wth small elastctes and small wedges, we may be comfortable usng the frst-order approach to assess even very large reforms. Conversely, n settngs where ntal wedges are large and agents tend to be very elastc, the frst-order approach may be very mprecse. 18

20 drectly observed. Whle the result s conceptually smple, the emprcal challenge s that estmatng elastcty changes due to polcy reforms s not an easy task. Gven the dffcultes of reachng a consensus on elastcty levels n many settngs, how can we hope to reach a consensus elastcty changes? Gven the current state of emprcal knowledge, we have to mpose more structure on the problem n order to assess the welfare mpact of large reforms. Ths mples makng a parametrc assumpton that restrcts ε 0. In fully parametrc approaches that starts from a specfc functonal form for utlty, there s a set of prmtves that determne both ε 0 and ε 0. The smplest approach s to assume so-elastc utlty, n whch case we have ε 0 = 0. Under ths assumpton, we obtan Proposton 5 (Suffcent Statstcs for Large Reforms: Iso-Elastc). Assume quas-lnear, so-elastc utlty and a sngle tax rate τ 0 on taxed goods. In ths case, the trapezod approxmaton of the effcency effect of large reforms can be wrtten as W ε 0 g =1 { τ0 + 1 [ ]} 1 + τ 0 2 τ0 τ 0, (31) 1 + τ 0 a standard Harberger-style formula usng a modfed tax wedge, τ 0 1+τ [ τ0 1+τ 0 ]. Ths characterzaton retans the smple Harberger-style structure, but usng an adjusted wedge that accounts for the dscreteness of the reform. How bg are the quanttatve mplcatons of ths generalzaton? As an example, consder a commodty tax of 10% that s ncreased to 30%. In ths case, the ntal wedge equals τ 0 1+τ 0 = 0.09 whle the adjusted wedge equals τ 0 1+τ [ τ0 1+τ 0 ] = The adjusted wedge s almost 80% larger, and so the estmated welfare cost wll be almost 80% larger too. As another example, consder a labor ncome tax of 50% that s reduced to 30%. When consderng the taxaton of supples, a postve tax rate corresponds to a negatve value of τ 0. Denotng the labor ncome tax by τ l, we have τ 0 = τ l. In ths case, the ntal wedge s gven by τ l 1 τ l = 1, whle the adjusted wedge s gven by τ l 1 τ l [ τl 1 τ l ] = The adjusted wedge s about 30% smaller, and so the estmated welfare gan wll be 30% smaller. To conclude, the standard frst-order approach understates the welfare costs of tax ncreases and overstates the welfare gan of tax reductons. As demonstrated by the numercal examples, the errors can be qute large for large reforms The numercal examples provded here are not unrealstc. For example, the labor ncome tax cut corresponds roughly to the tax changes for hgh-ncome earners n the Tax Reform Act of 1986 n the US. When analyzng such reforms, f we want precse welfare calculatons, we have to modfy the suffcent statstc formula to account for the changng wedge along the reform path. 19

21 As an alternatve to these trapezod approxmatons, we may consder hgher-order Taylor approxmatons. Specfcally, we now develop a second-order Taylor approxmaton, showng that ths gves smlar results as the trapezod approxmaton. Around the ntal polcy θ 0 = 0, the second-order Taylor approxmaton of socal welfare s gven by W (θ) W (0) + dw (0) θ + 1 d 2 W (0) dθ 2 dθ 2 θ 2. (32) Focusng on effcency (g = 1 ), the welfare effect of reform, W = W (1) W (0), s approxmately equal to W 0 g =1 dw (0) /dθ g =1 2 d 2 W (0) /dθ 2 0 g =1. (33) To compare the Taylor and trapezod approaches, let us agan consder the specal case underlyng the standard suffcent statstcs formula (19). In ths case, we have and Therefore, we have d 2 W (0) /dθ 2 0 dw (0) /dθ 0 g =1 = d ε 0 dθ = ε 0 g =1 τ τ 0 τ 0, (34) τ τ 0 τ 0 + ε 0 τ 0 (1 + τ 0 ) 2 τ 0. (35) Proposton 6 (Suffcent Statstcs for Large Reforms: 2nd-Order Taylor). Assume quas-lnear utlty and a sngle tax rate τ 0 on taxed goods. In ths case, the second-order Taylor approxmaton of the effcency effect of large reforms can be wrtten as where ε 0 d ε 0 dθ W 0 ε 0 g =1 τ τ 0 τ { ε 0 s the elastcty change due to the reform. τ 0 (1 + τ 0 ) 2 + ε 0 τ τ 0 } τ 0, (36) As n Proposton 4, the welfare effect can be wrtten as the standard frst-order effect plus an adjustment term capturng the dscreteness of the reform. The adjustment term under the secondorder Taylor approxmaton s slghtly dfferent than under the trapezod approxmaton, because the Taylor expanson starts from the ntal polcy θ 0 = 0 nstead of combnng the ntal and the new polces. Rather than usng the exact change n the wedge [ τ0 1+τ 0 ], equaton (36) uses the approxmaton τ 0 (1+τ 0 ) 2. Moreover, the Taylor approxmaton does not nclude the changes n the 20

22 elastcty and wedge nteracted. These dfferences may make the Taylor expanson less precse. We have consdered welfare formulas for large reforms usng trapezod and second-order Taylor approxmatons. The two approaches yeld roughly smlar results. Ths s natural gven that the Taylor approxmaton assumes that welfare s a second-order polynomum, whle the trapezod approxmaton assumes the dervatve of welfare s lnear. It would be possble to ncrease the precson of these approxmatons by consderng hgher-order Taylor expansons or by applyng the trapezod rule to a fner parttonng of the ntegraton nterval. However, ths would be emprcally pontless as the requred moments would be too dffcult to estmate. If we are concerned by hgher-order effects, t s more natural to go fully structural. In fact, the relatvely smple approaches developed here may already be beyond our emprcal reach. There are dfferent vews one could take on the approaches developed n ths secton. One vew s that, when consderng large reforms (whch we almost always do), the analyss has clarfed the suffcent statstcs that need to be estmated. The elastcty s not enough, we need reform-nduced elastcty changes as well. Another vew s that elastcty changes are mpossble to estmate persuasvely, so we have to assume that compensated elastctes are constant over large ranges. In ths case, we have seen that the suffcent statstcs approach corresponds to the case of so-elastc and quas-lnear utlty, whch s a partcular parametrc form. In ths sense, the suffcent statstcs approach s a structural approach. 5 Welfare Effect wth Non-Government Dstortons The suffcent statstcs lterature assumes that government polcy s the only source of economc neffcency. That s, the mposton of taxes or transfers s the only reason for a wedge between prvate and socal ncentves, and therefore the welfare effect can be summarzed by the fscal externalty. The assumpton that the fscal externalty s the only externalty s often unrealstc and we wll now relax ths. It s possble to provde elegant suffcent statstcs results n the presence of non-government externaltes or nternaltes, but the estmaton requrements ncrease consderably. We specfy utlty as follows u ( x 0,..., x J; E 0,..., E J ), (37) 21