The Policy Elasticity

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1 The Polcy Elastcty Nathanel Hendren October, 2015 Abstract Ths paper llustrates how one can use causal effects of a polcy change to measure ts welfare mpact wthout decomposng them nto ncome and substtuton effects. Often, a sngle causal effect suffces: the mpact on government revenue. Because these responses vary wth the polcy n queston, I term them polcy elastctes, todstngushthemfromhcksanandmarshallan elastctes. The model also formally justfes a smple beneft-cost rato to measure the margnal value of publc funds correspondng to non-budget neutral polces. Usng exstng causal estmates, I apply the framework to fve polcy changes: top ncome tax rate, EITC generosty, food stamps, job tranng, and housng vouchers. 1 Introducton A large and growng lterature n economcs focuses on estmatng the causal effects of government polcy changes. Ths rse n expermental and quas-expermental methods have made sgnfcant advances n addressng the postve queston of what polcy changes do to behavor. But, translatng causal effects nto a normatve evaluaton of the polcy change runs nto an mmedate hurdle, expressed succnctly by Goolsbee (1999): The theory largely relates to compensated elastctes, whereas the natural experments provde nformaton prmarly on the uncompensated effects. Rarely do polcy changes hold everyone s utlty constant. Thus, the prevalng wsdom s that the causal effects of a polcy change are not the behavoral responses that are desred for a normatve analyss of that same polcy change. 1 Harvard Unversty and NBER (e-mal: nhendren@fas.harvard.edu). I would lke to thank Jeff Brown, Raj Chetty, Amy Fnkelsten, Don Fullerton, Peter Ganong, Adam Guren, Lous Kaplow, Wojcech Kopczuk, Erzo Luttmer, Emmanuel Saez, and semnar partcpants at the 2015 Tax Polcy and the Economy Conference, Chcago Booth School of Busness, Brown Unversty, The Unversty of Chcago, the Columba Tax Polcy Workshop, and the Mnneapols Federal Reserve for helpful comments. Fnancal support from the NBER Health and Agng Fellowshp, under the Natonal Insttute of Agng Grant Number T32-AG s gratefully acknowledged. Alex Olssen provded excellent research assstance. 1 For example, Feldsten (2012) crtqued the Mrrlees revew (Mrrlees et al. (2011)), an nfluental analyss of tax polcy nfluenced heavly by optmal tax theory and emprcal work, for usng causal nstead of compensated effects of tax polces: Whle decsons on the approprate sze of government must be left to the poltcal process, economsts can assst that decson by ndcatng the magntude of the total margnal cost of ncreased government spendng. That cost depends on the structure of taxes, the dstrbuton of ncome, and the compensated elastcty of 1

2 Ths paper clarfes how causal effects of polcy changes can be used drectly n welfare analyss of government polcy changes. Whle t s true that tradtonal welfare measures, such as the margnal excess burden (MEB) requre compensated behavoral responses, ths paper shows that nstead one can measure each ndvdual s wllngness to pay out of ther own ncome for a gven polcy change. The only behavoral response requred for calculatng ths measure of welfare s the causal mpact of the polcy a decomposton nto ncome effects, substtuton effects, or any other mechansm s not requred. Real-world polcy changes are often complex; therefore, these causal effects wll n general be nether a pure Hcksan nor Marshallan elastcty. Because these desred responses vary wth the polces n queston, I term them polcy elastctes. These are smply the dfference n behavor f the polcy s undertaken relatve to the counterfactual world n whch the polcy s not undertaken, precsely the textbook defnton of the causal effect of the polcy. What types of responses need to be estmated to understand the welfare mpact of a government polcy change? In the broad class of models n whch government taxaton s the only pre-exstng dstorton, a sngle causal effect s suffcent: the causal mpact of the behavoral response to the polcy on the government s budget. 2 The causal effect of the polcy on the government budget matters because of the envelope theorem, whch mples that behavoral responses to margnal polcy changes don t affect utlty drectly. However, to the extent to whch the prces faced by ndvduals do not reflect ther resource costs (e.g. f there are margnal tax rates on labor earnngs), behavoral responses mpose a resource cost on socety that has no mpact on the agent s utlty. If the government s the only dstorton between prvate prces and socal (resource) costs, the mpact of the behavoral response on the government s budget s the only behavoral response requred for welfare estmaton. 3 Of course, ths envelope theorem logc s not new t underscores almost all prevous lterature n emprcal welfare economcs ncludng Harberger (1964) s trangle and the suffcency of the taxable ncome elastcty n Feldsten (1999). 4 However, these papers often focus on calculatng MEB, whch reles on the compensated, not causal effect on the government budget. Here, I show that not only can one use the causal effects, but the resultng welfare framework has arguably more desrable features, such as the ablty to aggregate to measures of socal welfare usng the socal margnal utltes of ncome. 5 Acommonfeatureofmanygovernmentpolcychangessthattheyarenotbudgetneutral,at least n the short run. In such cases, one can compute a smple beneft/cost rato equal to the the tax base wth respect to a margnal change n tax rates. 2 To be precse, ths causal effect s suffcent for all components of the second dervatve of the utlty functon. See Footnote If the government s not the sole dstorton n the market, one needs to estmate the causal mpact on the other externaltes as well as ths fscal externalty. Ths ncludes not only tradtonal externaltes such as polluton, but also externaltes on one s self caused by mperfect optmzaton. Even n these more general models, the causal effects are suffcent for all behavoral responses; a decomposton nto ncome and substtuton effects s not requred. See Appendx C. 4 See also Kleven and Krener (2005) for a nce dscusson of the MEB approach and Essa et al. (2008); Essa and Hoynes (2011) for applcatons to the EITC. 5 In contrast, aggregaton of MEB measures to socal welfare requres knowledge of the ncome effects of the polcy change (Secton 2.7). 2

3 margnal socal welfare mpact of the polcy per unt of government revenue expended. 6 In practce, ths margnal value of publc funds (MVPF) has a very smple representaton. For polces whch affect taxes, transfers, or provde market goods, the beneft-cost rato s smply MVPF = 1 1+FE, where FE s the mpact of the behavoral response to the polcy on the government budget outlays per dollar of government expendture otherwse known as a fscal externalty. Polces that have postve effects on the government budget reduce the effectve cost of the polcy (FE < 0). For the provson of publc goods, one needs to adjust the formula for the rato of the wllngness to pay for the publc good or servce relatve to the benefcares ncome (as noted by Samuelson (1954)), yeldng a WTP slghtly more complcated formula: 1+FE, where WTP s the ndvduals wllngness to pay for the publcly-provded good out of ther own ncome. IllustratetheframeworktostudythewelfarempactofchangestofveU.S.polces: thetop margnal ncome tax rate, the generosty of the earned ncome tax credt (EITC), food stamps (SNAP), job tranng programs (JTPA), and housng vouchers (Secton 8). To do so, I use exstng causal effects to calculate the MVPF for these polces. For example, to study the mpact of rasng the top margnal ncome tax rate, I rely on the large lterature studyng the behavoral responses to such ncreases. Saez et al. (2012) and Gertz (2009) suggest md-range estmates that 25-50% of the mechancal revenue that s rased from ncreasng the top margnal ncome tax rate s lost due to the behavoral response to the polcy. Ths suggests a MVPF of taxng top earners of $1.33-$2. For the EITC generosty, there s a large lterature studyng the mpact of EITC expansons on labor earnngs (Hotz and Scholz (2003)). Exstng causal estmates suggest ncreasng EITC generosty leads to a cost that s ~14% above the mechancal cost due to behavoral responses. Ths suggests a MVPF of ncreasng EITC generosty of $0.88. One can use the MVPF to compare across polces usng Okun s leaky bucket experment (Okun (1975)): how much resources s socety wllng to lose to transfer from one person to another? 7 For example, consder the MVPFs from the EITC expansons and the top margnal ncome tax schedule. Combnng these suggests addtonal redstrbuton s desred f and only f one prefers $ n the hands of an EITC benefcary relatve to $1 n the hands of the rch (earnngs > $400K). From a postve perspectve, the exstng causal estmates of the behavoral responses to taxaton suggests the U.S. tax schedule mplctly values an addtonal $ to an EITC recpent as equvalent to $1 to someone subject to the top margnal ncome tax rate. 6 To the best of my knowledge, ths measure of the margnal value of publc funds (MVPF) was ntally proposed by Mayshar (1990) (see equaton 9 on page 267 of Mayshar (1990); also, see Slemrod and Ytzhak (1996, 2001) for smlar defntons). Here, I show that t has the unque feature that the only behavoral responses requred for ther measurement are the causal effects of the non-budget neutral polcy n queston. Ths contrasts wth more tradtonal defntons of the margnal cost of publc funds (MCPF) that seek to augment the standard wllngness to pay for the publc expendture (gven by a Samuelson condton) wth the dstortonary cost of rasng the requred tax revenue (Ballard and Fullerton (1992)). 7 In contrast, comparsons of MEBs across polces requres addng back n the ncome effects that were removed to form the MEB. Ths problematc feature of MEB was ntally shown by Damond and Mrrlees (1971). See also Auerbach and Hnes (2002) for a smple llustraton of ths on page 1370, equaton In ths sense, the aggregaton of welfare across people s easer wth the MVPF than the MEB. 3

4 Relaton to Prevous Lterature Ths paper s related to a large set of prevous lterature studyng the margnal welfare mpact of polcy changes and n partcular the defnton of the margnal value of publc funds adopted n Mayshar (1990), Slemrod and Ytzhak (1996, 2001) and Kleven and Krener (2006)). Relatve to ths lterature, the prmary contrbuton of ths paper s to llustrate why ths partcular defnton of the MVPF does not requre a decomposton of the causal effect of the polcy, and why other commonly used margnal welfare defntons rely on behavoral responses other than the causal effect, such as the compensated effect (as n the MEB analyss of Kleven and Krener (2005); Essa et al. (2008); Essa and Hoynes (2011)) or the causal effect of an augmented polcy that ncludes ahypothetcaltaxncreasetoclosethegovernmentbudgetconstrant(asnthetradtonalatknson- Stern-Stgltz-Dasgupta defnton of the MCPF n Stgltz and Dasgupta (1971); Atknson and Stern (1974); see also Ballard and Fullerton (1992)). 8 The paper s also related to the lterature on optmal taxaton. Whle orgnal optmal tax formulas generally sought to wrte optmal tax formulas usng underlyng structural prce and ncome elastctes (e.g. Damond and Mrrlees (1971) and Saez (2001)), more recent optmal tax lterature has noted that often one need not decompose behavoral responses nto underlyng structural components (e.g. Pketty and Saez (2013)), whch can ad both n the theoretcal expresson and emprcal mplementaton of optmal tax formulas. In ths sense, the paper bulds on the recent lterature on suffcent statstcs (Chetty (2009b)) and n partcular the suffcency of the taxable ncome elastcty (Feldsten (1999); Chetty (2009a)). It s well known that the taxable ncome elastcty s no longer suffcent n cases when there are responses to the polcy on multple tax bases wth dfferent margnal tax rates (e.g. captal and labor ncome (Saez et al. (2012)) or ntensve versus extensve margn responses (Kleven and Krener (2006))). However, the present analyss shows that the causal mpact of the behavoral response on the government budget (e.g. tax revenue) as opposed to the tax base (e.g. taxable ncome) remans suffcent even n cases where the behavoral response by ndvduals occurs on multple tax margns. Ths suggests focusng on the tax revenue mpacts, as opposed to taxable ncome, may be the most general emprcal approach for welfare analyss. The rest of ths paper proceeds as follows. Secton 2 presents the model, outlnes the welfare framework, and compares t to exstng lterature. Secton 3 draws upon exstng causal estmates to study the desrablty of changng the top margnal ncome tax rate, EITC generosty, food stamps, job tranng, and housng vouchers. Secton 4 concludes. The Onlne Appendx provdes some dervatons of the welfare formulas and also dscusses extensons of the model to non-margnal welfare analyss (Appendx B), externaltes (Appendx C), general equlbrum effects (Appendx D), and provdes further clarfcaton of the role of the Hcksan elastcty n prevous lterature (Appendx E). 2 Model Ths secton formalzes a general model of ndvdual behavor to llustrate the ablty to utlze causal effects, nstead of compensated effects, for normatve analyss of government polcy changes. The 8 In ths sense, t s related to Auerbach (1985) who noted the dfferent conceptual experments underlyng the MEB versus MCPF. 4

5 generalty of the model captures many realstc ssues faced n emprcal applcatons and also allows the model to nest many models n prevous lterature. Despte ts generalty, the model wll motvate a very smple beneft-cost rato n equaton (16) n Secton Ths beneft-cost rato s smply the ndvduals wllngness to pay for the polcy change out of ther own ncome, normalzed by the total government cost of the polcy nclusve of the mpact of behavoral responses on the government budget (.e. fscal externaltes) a term I call the margnal value of publc funds, MVPF = Indvduals WTP for Polcy Change Cost to Gov t = Beneft Cost When usng the MVPF, the behavoral responses requred for such a measure wll be causal, not compensated, effects. The model clarfes below how ths noton s dstnct from tradtonal measures of the margnal cost of publc funds (MCPF) and margnal excess burden (MEB), but algns wth adefntonproposedbymayshar(1990)andslemrodandytzhak(1996,2001). Ithenuseths formula to translate exstng causal effects for fve polces nto welfare statements about those polces n Secton Setup More formally, there exst a set of ndvduals n the populaton, ndexed by 2 I. They make two choces: they choose goods to consume, x = {x j } J X,andlaborsupplyactvtes,l = {l j } J L. 9 There also exsts a government that does three thngs: t provdes a range of publcly provded goods and servces to each ndvdual, G = {G j } J G,provdesmonetarytransferstoeachndvdual,T, n o and mposes lnear taxes 10 on goods, x = j x JX n o and labor supply actvtes, l = j l JL. Indvduals value ther goods, labor supply actvtes, and publcly provded goods and servces accordng to the utlty functon: whch s allowed to vary arbtrarly across people. 11 u (x, l, G ) (1) To smplfy the exposton, I assume a stylzed model of producton n whch one unt of any type of labor supply produces 1 unt of any type of good under perfect competton. ndvduals face a sngle lnear budget constrant 12 gven by (1 + x ) x apple Spendng on Goods 1 l l Earnngs + T {z} + y {z} Transfers Other Income Ths means that 9 For example, j can ndex tme so that l j s the labor supply of ndvdual n tme j. Or,l 1 could be labor suppled n wage work and l 2 could be labor suppled n the nformal (un-taxed) sector. 10 Because I focus on margnal polcy changes, the model can consder nonlnear tax settngs by nterpretng T as vrtual ncome and l j as the margnal tax on labor earnngs. 11 Note that these publcly provded goods could be market or non-market goods. For example, one can capture a settng where G s a market good by assumng the utlty functon has a form: u (x 1,x 2,G)=ũ (x 1,x 2 + G), sothatg and x 2 would be perfectly substtutable. 12 All vector multplcaton s the standard dot product (e.g. (1 + x ) x = P J X 1+ x j x j) (2) 5

6 The ndvduals expendture on goods, nclusve of the taxes x,mustbelessthantheearnngsfrom labor supply actvtes (plus taxes l ), transfers, T,andnon-laborncome,y. 13 Ths smplfed producton structure rules out many nterestng features that can easly be added to a more general model, ncludng mperfect competton (.e. producer surplus), producton externaltes (e.g. spllovers), and pecunary externaltes (n whch case real prces would not always be 1). 14 Iassumethemargnal cost to the government of producng publcly-provded goods s gven by c G = {c j } j for j =1,..,J G. 15 Each ndvdual takes taxes, transfers, non-labor ncome, and the provson of publcly-provded goods as gven and chooses goods and labor supply actvtes to maxmze utlty. Ths allows ndvduals to obtan a level of utlty V gven by V l, x,t, G,y = max u (x, l, G ) x,l s.t. (1 + x ) x apple 1 l l + T + y where V depends on taxes, transfers, ncome, and publcly provded goods. The choces of goods and labor supply actvtes generated by the agent s problem are denoted x m j x, l,t, G,y and l m j x, l,t, G,y. 16 Because the utlty functon s allowed to vary arbtrarly across people, t wll be helpful to normalze by the ndvdual s margnal utlty of whch s the Lagrange multpler from the type maxmzaton program. For measurng welfare, t wll also be helpful to defne the expendture functon, E u; l, x,t, G,ofndvdualto be the amount of ncome y requred for ndvdual to obtan utlty level u n a world wth taxes, transfers, and publcly provded good l, x,t, G. 17 The ndrect utlty functon provdes a measure of ndvdual s utlty; to move to socal welfare, IassumethereexstssomesetofParetoweghts,{ },foreachndvdual, sothatsocalwelfares gven by o W n l, x,t, G,y = Socal Welfare ˆ 2I V x, l,t, G,y d Weghted Sum of Indvdual Utltes Note that ths socal welfare functon n prncple depends on a very rch set of polcy choces by the government: t s an mplct functon of the taxes, transfers, and publcly provded goods to every 13 I allow (but do not requre) taxes and transfers to be ndvdual-specfc. Ths allows the model to nest the standard MEB experment that requres (potentally nfeasble) ndvdual-specfc lump-sum transfers to compare a polcy change to a frst-best allocaton. 14 See Appendx D for a dscusson of GE effects and Appendx C for a dscusson of externaltes. 15 Note ths nests the case of a pure publc good by assumng c G j = 1 and Gj s constant across. 16 N The superscrpt m refers to the fact that these are standard Marshallan demand functons. 17 Note that the standard dualty result mples: E V l, x,t, G,y ; l, x,t, G = y (3) 6

7 ndvdual n the economy. By allowng for an arbtrary set of Pareto weghts, one can allow for socal preferences for dfferent people n the economy (e.g. a preference for redstrbuton from rch to poor). These canoncal measures of ndvdual welfare, V,andsocalwelfare,W,provdeamappngfrom polcy choces of the government to measures of well beng. The remander of the paper wll seek to characterze the welfare mpact of small changes to government polcy. 2.2 Polcy Paths and Potental Outcomes Imagne the government makes a polcy change. To descrbe an arbtrary polcy change, I follow a setup that algns closely wth the canoncal defntons of causal effects as dfferences of potental outcomes (e.g. Angrst and Pschke (2008)). For the purposes of ths paper, and for comparson to earler lterature estmatng MEB, I consder small polcy changes. To be specfc, I defne a polcy path, P ( ). For any n a small regon near 0, 2 ( publcly provded goods to each ndvdual, P ( ) = n ˆ x ( ), ˆ l ( ), ˆT o ( ), Ĝ ( ) 2I where the ^ ndcates the polces are functons of., ), letp ( ) be the taxes, transfers, and (4) A polcy path can descrbe a polcy that ncreases/decreases taxes, ncreases spendng on a publc good, etc. In ths sense, ths path provdes a method for descrbng a wde array of polcy changes. I make two assumptons about how the polcy vares wth. Frst, I normalze the value of the polcy at =0to be the status quo: n ˆ x (0), ˆ l (0), ˆT o (0), Ĝ (0) n o2i = x, l,t, G 2I Second, I assume that the polcy path s contnuously dfferentable n (.e. exst and are contnuous n ). 18 dˆ x j, dˆ l j, d ˆT dĝj,and Intutvely, P ( ) traces out a smooth path of government polces, centered around the status quo. By usng ths path, one can easly consder polces that vary multple polcy parameters at the same tme. Gven a path P ( ), I consder the welfare mpact of followng the path, parameterzed by an ncrease n. Ths can be nterpreted as followng a polcy path or evaluatng a polcy drecton (e.g. ncreasng taxes on labor earnngs would be a path wth d labor l > 0, or ncreasng spendng on schools fnanced wth a cgarette tax would be a path wth d cgarettes x > 0 and dg schoolspendng > 0 at the same tme, etc.). 19 The normatve queston of nterest s should we follow the polcy path?. Before askng ths normatve queston, frst consder the postve queston of what the polcy change would do to behavor. 18 Ths does not requre that the behavoral response to the polcy be contnuously dfferentable. For notatonal convenence n the text, I wll assume the behavoral responses are contnuously dfferentable. However, n the emprcal applcaton to the study of the EITC expanson n Secton 3, I allow for extensve margn labor supply responses (whch s a key feature of the behavoral response to EITC expansons, and s known to be an mportant factor n MEB estmaton (Essa et al. (2008), Essa and Hoynes (2011))). 19 I have not specfed a scale/speed for the polcy path. In practce, one can normalze the speed of the polcy to one unt of a tax or one dollar of revenue rased, as llustrated n the applcaton n Secton 3. 7

8 Gven a polcy path, o I assume ndvduals choose goods and labor supply actvtes, ˆx ( ) ={ˆx j ( )} and ˆl ( ) =nˆlj ( ),thatmaxmzetherutltyunderpolcyp ( ).20 In the now-standard language of Angrst and Pschke (2008), ˆx ( ) and ˆl ( ) are the potental outcomes of ndvdual s choces of goods and labor supply actvtes f polcy world s undertaken. As moves away from 0, ˆx ( ) and ˆl ( ) trace out the causal effect of the polcy change on the ndvdual s behavor.. In addton to the ndvdual s behavor, the polcy wll also mpact the government budget. To keep track of these effects, let ˆt ( ) denote the net government resources drected towards type, ˆt ( ) = c G Ĝ ( ) + ˆT ( ) Net Resources Publc-Provded Goods Transfers ˆ x ( ) ˆx ( )+ˆ l ( )ˆl ( ) Tax Revenue where c G Ĝ ( ) s the government expendture on publcly provded goods to ndvdual, ˆT ( ) s the government transfers to type, andˆ x ( ) ˆx ( ) +ˆ l ( )ˆl ( ) s the tax revenue collected from ndvdual on goods and labor supply actvtes. Wth ths defnton of ˆt,thetotalmpactofapolcyonthegovernment sbudgetsgvenby dˆt 2I d. The analyss does not requre polces to be budget-neutral, but budget-neutralty of a polcy path could be mposed by assumng the polcy change has no aggregate mpact on the government budget: ˆ 2I dˆt d =0 8 (5) where dˆt s obtaned by takng the dervatve of equaton (5): dˆt dĝ = cg + d ˆT d h ˆ x ( ) ˆx ( )+ˆ l ( )ˆl ( ) The term c G dĝ s how much the polcy changes spendng on publcly provded goods; s how much the polcy ncreases drect transfers; and the last term s the mpact of the polcy on the net tax revenue from goods and labor supply actvtes. The mpact of the polcy on ndvdual behavor and on the government budget are related through the mechancal and behavoral mpact of the polcy on net tax revenue from goods and labor supply actvtes: d h ˆ x ( ) ˆx ( )+ˆ l ( )ˆl ( ) = dˆ x ˆx + dˆ l ˆl Mechancal Impact on Govt Revenue! + ˆ x dˆx + ˆ l dˆl 20 These can be calculated n theory by evaluatng the Marshallan demands at each : ˆx j ( ) = x m j ˆ x ( ), ˆ l ( ), ˆT ( ), Ĝ ( ) 8j =1..J X ˆlj ( ) = lj ˆ m x ( ), ˆ l ( ), ˆT ( ), Ĝ ( ) 8j =1..J L Behavoral Impact on Govt Revenue but n practce they can be estmated by lookng at the causal mpact of the polcy wthout knowledge of the underlyng mechansms. d ˆT (6) 8

9 The mechancal effect s the change n revenue holdng behavor constant. Ths would be the margnal budget mpact of the polcy f one dd not account for any behavoral responses. The behavoral mpact s the effect of the behavoral response to the polcy on the government s budget. 2.3 Defntons of Welfare Movng from postve to normatve analyss requres a defnton of welfare. A prmary am of the paper s to argue that one need not estmate the margnal excess burden (MEB) assocated wth a polcy change, and one can nstead focus on the margnal wllngness to pay for the polcy change. In Secton 2.6, I dscuss MEB; however I begn by defnng the ndvdual s wllngness to pay out of ther own ncome to follow the polcy path, as ths wll be the defnton of welfare that corresponds to the use of causal, not compensated, behavoral responses. Let ˆV ( ) denote the utlty obtaned by type under the polcy P ( ). The margnal mpact of the polcy on the utlty of ndvdual s gven by d ˆV =0. Normalzngbythemargnalutltyofncome, the ndvdual s own wllngness to pay (out of ther own ncome) for a margnal polcy change s gven by d ˆV =0. 21 Wth ths defnton of ndvdual welfare, aggregaton to socal welfare s straghtforward by summng across ndvduals usng ther Pareto weghts, dŵ =0 = 2I =0 d, where = s the socal margnal utlty of ncome of ndvdual (see Saez and Stantcheva (2013)). 22 Socal margnal utltes can be nterpreted n terms of Okun s classc bucket experment (Okun (1975)): Socety s ndfferent to transferrng 1 2 resources to ndvdual 2 as opposed to $1 to ndvdual 1. If 1 < 2, socety s wllng to lose resources n order to make a transfer from ndvdual 1 to ndvdual 2. In prncple, values of the socal margnal utlty of ncome are a subjectve matter. d ˆV However, Hendren (2014) provdes one potental alternatve that replaces these socal margnal utltes of ncome wth the nequalty deflator, whch measures the margnal prce of transferrng resources between 21 It s well-known that d ˆV =0 s equvalent to two other canoncal measures of welfare for margnal polcy changes. Frst, the equvalent varaton, EV ( ), ofpolcyp ( ) for type s the amount that the consumer would be ndfferent to acceptng n leu of the polcy change. EV ( ) solves V l, x,t, G,y + EV ( ) = ˆV ( ) Second, the compensatng varaton, CV ( ), ofpolcyp ( ) for type s the amount of money that must be compensated to the agent after the polcy change to brng her back to her ntal utlty level. CV ( ) solves V l ( ), x ( ),T ( ), G ( ),y CV ( ) = ˆV (0) It s straghtforward to verfy (e.g. Schlee (2013)) that: d ˆV =0 = d [EV] =0 = d [CV] =0 22 Note ths remans true even f the welfare weghts are not fxed and are functons of utlty levels, snce margnal polcy changes do not change the welfare weghts. For example, f W = G (V) d for a concave functon G, thenthe 2I socal margnal utlty of ncome would be = G 0 ˆV (0). 9

10 ndvduals usng modfcatons to the tax schedule. Under certan condtons, the use of such weghts corresponds to searchng for potental Pareto mprovements n the sprt of Kaldor (1939) and Hcks (1939), and does not requre a subjectve specfcaton of a socal welfare functon. For now, we magne the researcher has chosen a gven set of socal margnal utltes of ncome, ether subjectvely or usng the nequalty deflator of Hendren (2014). 2.4 Defnng Behavoral Responses: The Polcy Elastcty In prncple, the behavoral responses to the polcy can be drven by any number of underlyng reasons such as the mpact of changng prces, changng dsposable ncomes, or complementary effects from changes n the provson of publc goods and servces. A Marshallan elastcty holds dsposable ncome constant. A Hcksan elastcty holds utlty constant. A key beneft of these well-agreed upon elastcty defntons s that they allow the economst to state clearly what s beng held constant n the conceptual experment n queston. However, n practce polcy changes hold nether ncome nor utlty constant. All aspects of the polcy are varyng smultaneously whch could be nteractng to produce behavoral responses through apotentallycomplcatedunderlyngprocess. 23 To provde a vocabulary that allows one to make clear what s beng held constant n the conceptual experment n whch the polcy occurs, I defne the polcy response of x j and l j to be the local causal effect of the polcy on x j and l j. Smlarly, because emprcal estmaton often occurs n logs, I defne the polcy elastcty of x j and l j to be the local causal effect of the polcy on log (x j ) and log (l j ). Defnton 1. The polcy response of x j (or l j ) wth respect to polcy P ( ) s gven by dˆx j =0 (or dˆl j =0). The polcy elastcty of x j (or l j )sgvenbyˆ x j = dlog(ˆx j) =0 (or ˆ l j = dlog(ˆl j) =0 ) The polcy elastcty s smply the causal effect of the polcy n queston. Gven these defntons, 23 In prncple, the response to a gven polcy could be drven by a multtude of underlyng factors. To see ths, note that one can wrte these causal effects usng an ndvdual s Marshallan or Hcksan demand functons. For example, suppose x m j x, l,t, G,y s the Marshallan demand of ndvdual for good j. Then, dˆx j J = m j d x J x j + m j d x JGX x j j m j j Marshallan Expanson Smlarly, suppose x h j x, l,t, G,u s the Hcksan demand of ndvdual for good j (evaluated at utlty level u). Then, dˆx j = h j d x x j + h j d x JGX x j j h j j d Hcksan Expanson 10

11 the total mpact of the polcy change on government revenue n equaton (6) has three representatons:! d h ˆ x ( ) ˆx ( )+ˆ l ( )ˆl ( ) ˆ x dˆx + ˆ l dˆl Total - Mechancal Impact on Govt Revenue! = ˆ x dˆx + ˆ l dˆl Behavoral Impact on Govt Revenue (levels) = JX rjˆ x x j + j J L X j r l jˆ x j! Behavoral Impact on Govt Revenue (logs) where the weghts for the log responses, ˆr j x =ˆ j x ˆx j (or ˆr j x =ˆ jˆl l j x ), equal the government revenue on each good (or labor supply). (7) The compensated response In contrast to the polcy response, one can also defne the compensated response to the polcy, dˆxc j. To do so, one needs to subtract the ncome effects assocated wth the polcy change. dˆx c j = dˆx h j d } Income Effect = dˆx m j d Income Effect The compensated response to the polcy P s equal to the polcy response mnus the porton of the behavoral response that s due to changes n ncome or utlty. Ths ncome effect can be represented usng the Hcksan response to j d whch adjusts behavor for the change n the utlty nduced by the polcy, dv.alternatvely,thencomeeffectcanberepresentedusngthemarshallan response to d ˆV whch adjusts the polcy response for the change n the ncome-value of d ˆV the polcy change,. As llustrated n Equaton (8), the constructon of the compensated response to a polcy change s potentally much more dffcult than the polcy response. (8) Not only does one need to know the behavoral response to ncome )orutlty(@xh ), but to scale properly one also needs to know the utlty mpact of the polcy ( d ˆV d ˆV ) or the margnal wllngness to pay for the polcy change,.ass well-known, the compensated response to a gven polcy s the polcy response to an alternatve polcy, ˆP c,thatholdsndvduals utltesconstantvandvdual-specfclump-sumtransfers(auerbachand Hnes (2002)). The next secton llustrates that one can construct a comprehensve welfare framework for local evaluaton of government polcy changes that reles solely on the polcy elastctes of the polcy n queston, and does not requre knowledge of ether the compensated effect, or any of the underlyng marshallan or hcksan demand functons (condtonal on knowledge of the polcy response). 11

12 2.5 The Margnal Wllngness to Pay for the Polcy Change How much are ndvduals wllng to pay for a polcy change out of ther own ncome. Appendx A shows ths s gven by d ˆV =0 = = du dg dĝ + dt + dˆ x x + dˆ l l! du dg c G dˆt =0 + Net Resources! dĝ =0 + ˆ x dˆx + ˆ l dˆl Publc Spendng/ Mkt Falure Behavoral Impact on Govt Revenue (9) By the envelope theorem, the extent to whch ndvduals respond to the polcy change does not affect ther utlty drectly. Hence, the frst lne of Equaton 9 does not contan behavoral responses. However, behavoral responses are crucal when one attempts to account for the cost of the polcy, as shown n the second lne of Equaton 9. The frst term, dˆt,sstraghtforward: tsthechangennetgovernmentresourcesprovdedto ndvdual from the government, whch s the dfference between the change n spendng on publcly provded goods and transfers and the collecton of taxes on goods and labor supply actvtes. For budget neutral polces, recall that dˆt d =0;nthssense, dˆt captures the redstrbutve mpact of the polcy. These transfers ncrease socal welfare to the extent to whch those recevng the net transfer have hgher values of the socal margnal utlty of ncome than those who pay for the net transfer. The second term captures the value of any changes to publcly provded goods, dĝj =0. Ths s gven by the dfference between the wllngness to pay for the publcly provded goods and ther costs of producton, du dg c G dĝ =0 = P j c G j! dĝj =0. Ths component s well-known and popularzed n Samuelson (1954). One can nterpret ths number as the sze of the market neffcency beng addressed by the publcly provded goods. If the prvate market can effcently supply and allocate all goods, then agents would be able to pay c g to obtan a unt of a good that s j the publcly provded good, so that = c G j. If the prvate market does not provde such goods as effcently as the government (or vce-versa), then one needs to know the dfference between the costs and benefts of ts provson. The fnal term n Equaton 9 summarzes the mportance of behavoral responses. It s the mpact of the behavoral response to the polcy on the government s budget. It s a weghted sum of the polcy responses on behavor, dˆx j =0 and dˆl j =0, wth the weghts gven by the margnal tax rates. 24 The causal effect matters because of a fscal externalty. The envelope theorem guarantees that 24 Although ths causal effect s the mpact of a margnal change n the polcy, n practce causal effects are often measured usng dscrete changes n polces. Appendx B provdes ntutve condtons under whch the non-margnal causal effects (.e. ˆx j (1) ˆx j (0) nstead of dˆx j =0) can be used to measure the ndvdual s wllngness to pay for the polcy change. 12

13 behavoral responses do not affect utlty drectly; however, when prces do not reflect ther resource costs (as s the case wth taxaton), behavoral responses mpose a cost on those bearng the dfference between the prces faced by the ndvdual and ther resource costs. 25 Condtonal on calculatng ths fscal externalty, behavoral responses are not requred for measurng ndvdual s wllngness to pay for the polcy change. 26 Relaton to Feldsten (1999) If there s only one tax rate on aggregate taxable ncome and socal margnal utltes of ncome are the same for all types, then the aggregate taxable ncome elastcty s suffcent for capturng the behavoral responses requred for welfare analyss. 27 Ths nsght was recently popularzed n Feldsten (1999). I provde two clarfcatons to ths result. Frst, t s n general nether the Hcksan (compensated) nor the Marshallan (uncompensated) elastcty of taxable ncome that s desred for analyzng the welfare mpact of government polcy. Rather, t s the taxable ncome elastcty assocated wth the polcy n queston, whch depends on how the revenue s spent. Second, as s well known, the taxable ncome elastcty s not suffcent to the extent to whch ndvduals face multple tax rates. For example, f captal ncome s taxed at a dfferent rate than labor ncome, the elastcty of the sum of these two ncomes would not be suffcent (Saez et al. (2012)). If behavoral responses occur on both the partcpaton and ntensve margn, then the aggregate earnngs elastcty s not suffcent (Kleven and Krener (2006)). Moreover, one also needs to know the extent to whch polces affect consumpton of subsdzed goods or servces (e.g. enrollment n government programs such as SSDI or unemployment nsurance). Whle subsequent lterature tends to suggest a need for addng addtonal elastctes to the analyss 28, the present analyss shows that f one swtches the dependent varables n these analyses from the components of taxable ncome to aggregate tax revenue, suchadecompostonofthemechancsofthebehavoralresponsesnotrequred. 29 Of course, there are many reasons to be nterested n the mechansms drvng such a response; but calculatng the margnal welfare mpact of the polcy change n queston s not one of them. 25 As dscussed n Appendx C, f there are other externaltes one also requres an estmate of the mpact of the polcy on those externaltes as well. However, the causal effects reman the desred behavoral responses. 26 Adecompostonofcausaleffectsntoncomeandsubsttutoneffectsdoesnotgenerallyhelpmeasuretheszeof j whereas the sze of the market falure, et al. (1995)). 27 To see ths, note that f 1 = 2,then c G j. Income and prce effects depend on the Hessan (2nd dervatve) of the j c G j, depends on the frst dervatves of the utlty functon (Mas-Colell dx 1 1 dx 2 d =0 + 2 (x1 + x 2) =0 = 1 =0 28 For example, f there are both ntensve and extensve labor supply responses, one can compute both a partcpaton elastcty that s weghted by the average tax rates and an ntensve elastcty weghted by margnal tax rates (Kleven and Krener (2006)). If there are swtches between captal and labor ncome, one can compute the causal mpacts on each of these and weght by ther respectve tax rates. 29 Ths approach s taken by Chetty et al. (2013) who show the behavoral responses to the margnal ncentves nduced by the EITC lead to a 5% ncrease n government expendtures. 13

14 2.6 Relaton to MEB Snce Harberger (1964), the MEB framework s arguably the most common welfare framework for the evaluaton of the welfare mpact of changes to tax and transfer polces. 30 As dscussed n the ntroducton, calculatng welfare changes wth ths approach requres the compensated (Hcksan) elastcty. However, the compensated (Hcksan) elastcty s the causal effect of a polcy that holds utlty constant; under the present framework, t s therefore not suffcent for calculatng the welfare mpact of polces that actually change utltes. Ths secton clarfes the dfference between the margnal wllngness to pay measure of welfare whch reles on the polcy responses and the margnal excess burden that s known to rely on compensated effects. I begn by defnng the MEB n ths envronment. Let P ( ) be a polcy path. To compute the margnal excess burden to ndvdual from the polcy P ( ), letv =(v ) be a set of pre-specfed utltes. Most commonly, v s chosen to be the set of status quo utltes. Ths corresponds to the equvalent n varaton measure of MEB. 31 Now, defne the compensated polcy path, P v ( ), suchthatp v = ˆ x ( ), ˆ l ( ), ˆT o ( )+Ĉ( ; v), Ĝ ( ) where Ĉ ( ; u) s a compensaton provded to ndvdual such that V ˆ x ( ), ˆ l ( ), ˆT ( )+Ĉ( ; v), Ĝ ( ),y = v. Intutvely, P v ( ) s the same as the proposed polcy path, P ( ), wth the addton of ndvdual specfc lump-sum transfers, Ĉ ( ; v), thatholdagent s utlty constant at v. Now, let ˆt v denote the net government resources allocated to ndvdual under the compensated polcy P v ( ). FollowngthetextbookdefntonsofAuerbachandHnes(2002),theclassofmargnal excess burden measures are defned as MEB v = dˆt v =0 (10) Ths measures the amount of addtonal resources the government must gve to ndvdual n order to mantan ndvdual her utlty constant at v whle the polcy change s mplemented. 32 If the polcy 30 For example, Essa et al. (2008) and Essa and Hoynes (2011) apply the MEB framework to study the welfare mpact of recent expansons of the earned ncome tax credt n the US. 31 See Auerbach and Hnes (2002). Choosng v to be the utltes obtaned n the hypothetcal frst-best world wth no economc dstortons yelds the compensatng varaton measure of MEB. Of course, the dstncton between CV and EV measures of MEB depend on whether one s startng from the perspectve of the frst best or from the status quo. 32 An alternatve defnton of margnal excess burden s gven n the handbook chapter of Auerbach (1985) that preceded Auerbach and Hnes (2002). In ths chapter, the equvalent varaton MEB s defned as the margnal wllngness to pay for a hypothetcal polcy that s the same as the orgnal polcy but for whch the budget constrant s closed usng ndvdual-specfc lump-sum taxaton. To express ths defnton of MEB n the present framework, defne an augmented polcy path o P 85 = nˆ j l ( ), ˆ j x ( ) j j, ˆT ( ) ˆt ( ), Ĝ ( ) where ndvdual s forced to pay for net resources, ˆt ( ), provdedtoherbythepolcypath.gvenths,theequvalent varaton MEB from Auerbach (1985) s MEB 85 = P 1985 d ˆV =0 whch depends on compensated elastctes (snce the ndvdual must pay for the resource cost), but t s straghtforward to verfy that these are not fully compensated Hcksan elastctes snce the transfers are not guaranteed to hold utlty constant. 14

15 change s not desrable to ndvdual, she must be compensated to hold her utlty constant(so MEB s postve); conversely f the polcy change s good for ndvdual, the government must take away resources to hold her utlty constant (so MEB s negatve). If v s the status quo set of utltes (.e. the EV measure), then MEB s related to d ˆV =0 through the ncome effects that were removed to construct the MEB polcy experment. Let ˆx c j and ˆl c j denote the compensated choces of goods and labor supply actvtes under polcy path P v ( ). Then, the ncome effect component of the response to the polcy on x j s the dfference between the causal and dˆx c j compensated response: dˆx j =0 =0. Then, MEB s related to behavoral response to the compensaton on the government budget: d ˆV =0 through the mpact of the MEB v = d ˆV =0 dˆt {z } WTP Above Resource Cost INC (11) where INC = x dˆx dˆx c!! =0 =0 + l dˆl dˆl c =0 =0 Income Effects on Government Budget d ˆV =0 Also, f =0,thennomargnalcompensatonsprovdedtondvdual, sothat dˆx j =0 dˆx c j =0 =0and MEB v =0(and vce-versa). 33 Equaton (11) llustrates that MEB and the margnal wllngness to pay are related to each other through the ncome effects nduced by the lump-sum taxaton that hold ndvduals utlty constant n the MEB experment. In the specal case for whch there are no ncome effects, INC =0so that MEB s gven by the ndvdual s wllngness to pay above the resource costs for the polcy. More generally, calculaton of the MEB requres removng the porton of the causal effects that are due to the ncome effects assocated wth the polcy change. One reason MEB s a manstay n the welfare analyss toolkt s perhaps because t s a fundamental nput nto the optmal commodty taxaton analyss ntated by Ramsey (1927) and studed n detal n Damond and Mrrlees (1971). Ther results show that, n a model wth a representatve agent, the margnal excess burdens across commodtes are equated. Ths yelds the classc nverse elastcty rule for commodty taxaton: at the optmum, tax-weghted compensated prce dervatves for each commodty are equated. However, as shown n Appendx E, ths optmalty formula nvolves compensated responses because a necessary condton for taxes to be at an optmum s that small budget-neutral changes to taxes do not affect utlty. Hence, around the optmum, the polcy responses are compensated responses (.e. j dˆx =0 = dˆxc j =0 because utlty s not changng at the 33 If v s not the status quo utltes, no such relatonshp s guaranteed between MEB and d ˆV =0 j need to be computed n the alternatve world for whch V s the utlty level specfed n the MEB experment. 15

16 optmum) Aggregatng to Socal Welfare Heretofore, I have defned measures of the welfare mpact of a polcy change on an ndvdual, as measured by wllngness to pay (Equaton 9) and MEB (Equaton (10)). Here, I llustrate how these measures aggregate across ndvduals to measures of socal welfare. As noted n Secton 2.3, the margnal wllngness to pay for a polcy change can be aggregated to socal welfare usng the socal margnal utltes of ncome,. ˆ dw = d ˆV P =0 As shown n Equaton 9, the constructon of each term, d (12) d ˆV P =0,doesnotrequredecomposngbehavoral responses nto underlyng components such as ncome and substtuton effects. Of course, one must specfy socal margnal utltes of ncome,,nordertoprovdesuchaggregaton. Ass well-known, ratos of socal margnal utltes of ncome, j, measure one s wllngness to pay to transfer resources from ndvdual to ndvdual j and nest many forms of socal preferences (Saez and Stantcheva (2013)). As mentoned above, a potentally more objectve method of aggregaton of welfare mpacts across the ncome dstrbuton s to weght welfare by the margnal prce of transferrng resources across the ncome dstrbuton usng modfcatons to the ncome tax schedule, as suggested by Hendren (2014). Regardless of how one does the aggregaton, the margnal wllngness to pay measure of welfare, has the useful feature that t aggregates to socal welfare usng these socal margnal utltes of ncome,. In contrast, aggregaton of MEB across ndvduals s often more dffcult, even for budget neutral polces. Usng equaton (11), one can express the aggregate mpact ed by Damond and Mrrlees (1971), the aggregate mpact on socal welfare for a budget neutral polcy s gven by the sum of the net transfers to ndvduals, the margnal excess burden, and the ncome effects: dw = ˆ dˆt + MEB + INC d In other words, f one wshes to use the socal margnal utltes to aggregate to socal welfare, one needs to add back n the ncome effects that were removed from the polcy response n order to form the compensated response. The dffculty of aggregatng MEB across ndvduals was arguably well-documented by Damond and Mrrlees (1971). In the case of a representatve agent, the optmalty condton for polcy to be set optmally s dv =0, so that the ncome effects assocated wth the polcy change are zero at an 34 Moreover, away from an optmum, the causal effects from polces that change commodty taxes contnue to provde nformaton on the desrablty of changng commodty tax rates. In contrast, compensated elastctes defned not around the optmum wll not necessarly provde nformaton about the optmal commodty tax rate, as ths would requre an assumpton that the compensated elastctes are constant. 16 d ˆV P,

17 optmum (see Appendx E for further detals). But, wth heterogeneous ndvduals, the margnal welfare formulae no longer depend on compensated responses. 35 Ths s because small budget-neutral polcy changes does not hold the agents utltes constant at the optmum when there are heterogeneous agents. Some agents are better off; others are worse off. And, the aggregaton of MEB across ndvduals requres knowledge of the ncome effects assocated wth the polcy change (Damond and Mrrlees (1971)). 36 In ths sense, aggregaton of MEB across ndvduals s arguably more dffcult than aggregaton of margnal wllngness to pay, d ˆV P =0 of the ncome effects assocated wth the polcy change. 2.8 An MVPF for non-budget neutral polces, snce the latter does not requre knowledge If a polcy s budget neutral, then the aggregate mpact of the polcy on socal welfare (Equaton (12) and the components n Equaton 9) provdes a natural measure for the total welfare mpact of the polcy change. Such a measure depends on the polcy responses or polcy elastctes, and does not requre decomposton of behavoral responses nto ncome and substtuton effects, for example. In realty many polcy changes are not budget neutral, at least n the short run (.e. 2I dˆt P d 6= 0). For such polces, one needs to adjust the welfare framework to account for the total cost of the polcy. Indvduals wll generally be wllng to pay to obtan a transfer. In ths secton, I show that the framework provdes a natural justfcaton for computng a smple margnal value of publc funds (MVPF) as the welfare cost per dollar of government budget expended. I use the term MVPF to be dstnct from the tradtonal defnton of the MCPF whch generally does not rely on the causal effect of the polcy n queston, as dscussed n Secton 2.9. Normalzng socal welfare nto unts of ndvdual î s ncome, the MVPF s gven by: MVPFîP = î d ˆV P d =0 = dˆt P 2I d "Beneft" "Cost" (13) whch s the sum of the welfare mpact on each ˆV P =0 utltes of ncome,,andnormalzednuntsofdollarstondvdualî. 37 î,weghtedbythersocalmargnal There are many dfferent ways of constructng welfare measures for non-budget neutral polces (Fullerton (1991); Auerbach and Hnes (2002); Dahlby (2008)). Ths partcular defnton of the welfare mpact of non-budget neutral polces n equaton (13) was ntally proposed by Mayshar (1990) and also by Slemrod and Ytzhak (1996, 2001) and Kleven and Krener (2006). The MVPF has two key advantages. Frst, the needed behavoral responses depend solely on the causal, not compensated, effects of the non-budget neutral polces n queston. Ths contrasts wth 35 See Secton VII, page 268 of Damond and Mrrlees (1971). 36 See Auerbach and Hnes (2002) for a smple dervaton of ths on page 1370, equaton 3.24). 37 The î notaton denotes the unts of ncome used n the defnton; t s not the welfare mpact on type î. It s the welfare mpact on all types measured n unts of î s ncome. 17

18 both the MEB and also the tradtonal defnton of the margnal cost of publc funds, as dscussed below n Secton 2.9. Second, comparsons of MVPFs across polces correspond to comparsons of the socal welfare mpacts of polces. Gven any two non-budget neutral polces, P Tax and P Exp,letP( ) denote a polcy that ncreases spendng by $ on P Exp fnanced by a decrease n spendng by $ on P Tax. Then, the welfare mpact of ths combned polcy s gven by: dŵ P = î MVPFîP Exp MVPFîP Tax (14) so that polcy P Exp provdes a beneft of MVPFîP Exp per dollar of government revenue and a cost of MVPFîP Tax per dollar of government revenue. If MVPFîP Exp s greater (less) than MVPFîP Tax, then takng resources from the tax (expendture) polcy and usng t to fnance the expendture (tax) polcy wll mprove socal welfare. Identfyng heterogenety n the MVPF across dfferent polces s equvalent to dentfyng welfare-mprovng budget neutral polces. 2.9 Relaton to the MCPF As mentoned, the defnton of the MVPF s based on welfare measures of Mayshar (1990) and Slemrod and Ytzhak (1996, 2001); Kleven and Krener (2006). However, t dffers from the canoncal defnton of the margnal cost of publc funds. Followng the papers of Stgltz and Dasgupta (1971) and Atknson and Stern (1974), ths lterature seeks a number that can be used to adjust the standard Samuelson (1954) condton for the welfare cost of rasng the resources to fnance the publc expendture (Ballard and Fullerton (1992)). Ths defnton s the mpact of the behavoral response to the polcy on the government s budget of both ncreasng taxes and spendng resources on the publc good (.e. the fnal term n Equaton 9for a polcy that rases taxes and ncreases spendng on G). Suppose P Exp s non-budget neutral a polcy that ncreases spendng on a publc good G. The MCPF s ntended to provde an adjustment to the welfare measurement of the value of the publc spendng to account for the dstortonary cost of rasng tax revenue. For smplcty, consder a stylzed envronment wth one ndvdual and a lnear tax labor supply, l. Now, let P be a budget-neutral polcy that smultaneously ncreases spendng on G but that s fnanced by an ncrease n taxaton on labor supply. The aggregate welfare mpact s gven by d ˆV! c G dĝ + dˆl P {z } "MCPF" Samuelson Condton The frst term s the canoncal Samuelson condton for the provson of publc goods, whch measures the dfference between the wllngness to pay and the per-person cost for provdng the publc good. The second term s generally referred to as the margnal cost of publc funds (Fullerton (1991),Dahlby 18