Screening with endogenous preferences

Transcription

1 Screenng wth endogenous preferences Lz Chen MIT Casey Rothschld Wellesley College July 2015 Abstract A general framework s developed for studyng screenng n many-agent dscrete type envronments wheren each agent s preferences depend endogenously on the allocatons receved by the other agents. Applcatons nclude optmal ncome taxaton, performance contractng wth across-worker externaltes, and nsurance wth endogenous rsks. The soluton to the prncpal s problem s analyzed by decomposng t nto an nner problem wth fxed preferences and an outer problem whch determnes preferences. Because the outer problem s typcally dscontnuous at ponts where the preferences of two or more types endogenously concde, the prncpal frequently fnds t optmal to select allocatons whch nvolve two or more types wth endogenously concdng preferences, even though such allocatons are ex-ante hghly unusual. Assumng that types are strctly ordered by ther sngle-crossng preferences s, therefore, not nnocuous n endogenous preference envronments. The authors thank Eduardo Azevedo, Domnk Sachs, Floran Scheuer, Glen Weyl and partcpants n the Cologne Taxaton Theory Conference for helpful comments on earler drafts. Any remanng errors are our own. Casey thanks the Radclffe Insttute for Advanced Study at Harvard Unversty for generous support.

2 1 Introducton Prncpal-agent contractng models of screenng developed by Rothschld and Stgltz 1976, Mrrlees 1971, Mussa and Rosen 1978, and others have been appled to a wde varety of economc settngs wth nformatonal asymmetres. Examples nclude nsurance contractng, the desgn of redstrbutve ncome taxes, and compensaton va performance contracts. Ths paper develops a technque for extendng prncpal-agent screenng models to allow for across-agent nteractons specfcally, to allow for the possblty that each agent s preferences over contracts may depend on the contracts receved by other agents. 1 We dscuss three man applcatons. Frst, dfferent ncome earners may be employed n complementary or otherwse nteractng actvtes, as n Stgltz 1982, so that the earnngs of workers n one actvty can ndrectly affect the wages of workers n another sector. Second, frms typcally hre and offer performance contracts to multple workers who nteract n teams and effort by one ndvdual wthn that team may affect the measured performance of the other ndvduals on that team. Thrd, an nsured s exposure to rsk may depend on other nsureds coverage. For example, one ndvdual s coverage level may affect her accdent avodance effort, whch n turn may affect another ndvdual s accdent rsk and hence preferences over coverage levels. Smlarly, exposure to fnancal rsk from a not-at-fault accdent wth another drver may depend on the coverage level of that other drver. We ncorporate agents wth nterdependent preferences nto screenng problems by allowng each agent s preferences to depend, endogenously, on other agents contracts through a possbly mult-dmensonal functon X of all agents contracts. We make standard sngle crossng assumptons about preferences that facltate the applcaton of standard screenng technques at any gven value x of the functon X. We then solve the prncpal s problem by decomposng t nto two problems: an nner problem for a fxed x but whch requres the allocatons to be consstent wth X = x and an outer problem for determnng the optmal value of x and hence ndvduals endogenous preferences. We combne frst order necessary condtons for these two problems to provde a general characterzaton of 1 Our paper can alternatvely be thought of as extendng Segal 1999 who, lke us, studes a prncpal-agent problem wth multple agents and across-agent externaltes to screenng contexts wth an unnformed prncpal. 1

3 the optmal dstortons of ndvdual allocatons away from frst best. Two dstnct types of optma are possble: separatng optma, and partal- poolng optma. A separatng optmum occurs at a value of x for whch the types are strctly ordered by ther sngle crossng preferences. A poolng optmum occur at a value of x for whch the types are only weakly ordered because two or more types have endogenously concdent preferences. A central result of our paper s to establsh that poolng optma are surprsngly common: despte the fact that the set of x-values whch nvolve preference-poolng s genercally sparse, t wll frequently be optmal to choose such an x. For example, consder an optmal tax context, where an ndvdual s market wage w s a suffcent statstc for her preferences over ncome-consumpton bundles y, c. Our results ndcate that when ndvduals wages depend endogenously on the earnngs or efforts of other workers, t wll frequently be optmal for a Paretan socal planner to choose an ncome tax code that endogenously leads two or more ndvduals to have dentcal wages. The ntuton for ths poolng s common result les n a falure of lowerhemcontnuty n x of the set of ncentve compatble allocatons at values of x wth pooled types. Fgure 1 llustrates the basc ntuton by depctng the topology of ncentve compatblty between two types 1 and 2 n a standard optmal tax framework where contracts can be descrbed by the par y, c of pre-tax ncome y and after-tax ncome c and where ndvduals preferences over contracts are gven by uc, y/w, where w {w 1, w 2 }. The fgure depcts three dstnct cases. In case a, w 2 > w 1, so type 2 has the hgher wage and hence the shallower ndfference curve under standard assumptons. In case b w 2 = w 1. In case c, and w 2 < w 1. In each case and for any fxed type-1 allocaton y, c 1, ncentve compatblty requres that type 2 s allocaton y, c 2 les on or below the type-1 ndfference IC 1 through y, c 1 and on or above the type-2 ndfference curve IC 2 through y, c 1. When w 2 > w 1, y, c 2 must therefore le n the shaded wedge n Panel a above and to the rght of y, c 1. When w 2 < w 1, y, c 2 must le n the shaded wedge n Panel c below and to the left of y, c 1. When w 1 = w 2, IC 1 and IC 2 concde, and ncentve compatblty requres that y, c 2 les on these concdent ndfference curves. Now suppose that the wages of the two types vary smoothly wth some scalar parameter x, and that, for some x, w 1 x = w 2 x, w 2 x > w 1 x for x < x, 2

4 ,,,,,,, a w 2 > w 1 b w 2 = w 1 c w 2 < w 1 Fgure 1: Dscontnuous ncentve constrants as a force for a poolng optma and w 2 x < w 1 x for x > x. As x x from below, the ncentve compatble wedge n Panel a of Fgure 1 collapses to the rght-hand half of the concdent curve IC 1 = IC 2 n Panel b. Smlarly, as x x from above, the ncentve compatble wedge n Panel c collapses to the left-hand half of the same curve. Fnally, suppose that for some as-yet-unspecfed objectve functon for the prncpal there s a unque optmal allocaton y, c 1, y, c 2 gven x, and further that ths unque allocaton has y, c 2 y, c 1, as depcted n Panel b of Fgure 1. Per the topology of Panel c, there s no ncentve compatble allocaton near y, c 1, y, c 2 for any x = x + ε for any ε > 0, however small. On the other hand, per Panel a s topology, we typcally can expect to fnd feasble allocatons close to y, c 1, y, c 2 for x = x ε for suffcently small ε. Ths suggests that the value Tx of the prncpal s optmzed-gven-x objectve wll be left-contnuous and wll exhbt a downward jump dscontnuty to the rght of x. Fgure 2 plots an objectve functon for a smple optmal-tax model we develop n greater detal n Secton 5.3 below. It exhbts exactly ths sort of downward jump dscontnuty. The computatonal example plotted n Fgure 2 s a smple three-type extenson of Stgltz s 1982 model of optmal redstrbutve taxaton wth two varetes of labor whch are complements n a constant-returns-to-scale producton functon. As n Stgltz, there s a low-skll type type 0 and a hgh-skll type type 2 whose labor supples are complementary to each other. In addton, there s a thrd type type 1 who has hgh-skll and whose effectve labor effort s a perfect substtute for type 0 s labor effort and thus complements type 2 s. Labor markets are compettve, so each worker s pad a wage equal to her margnal product. Constant- 3

5 Tx x Fgure 2: An example wth a jump-poolng optmum returns-to-scale therefore mples that the rato x of effectve effort of type 2 to the combned effectve efforts of types 0 and 1 s a suffcent statstc for each types wage. As descrbed below, the parameters of ths computatonal example are set up so that, at x = x = 1, the wages of types 1 and 2 concde. Exactly as n Panel b of Fgure 1, the Rawlsan optmum at x = 1 for ths parameterzed model nvolves y, c 2 y, c 1. Intutvely, ths s because type 2 s labor s a complement to the worst-off type 0 s labor, whle type 1 s labor s a substtute for t. As such, dstortng type 2 s labor supply up and dstortng type 1 s labor supply down both ndrectly redstrbute to type 0 by rasng her wage. Snce x < x mples w 2 > w 1 and x > x mples w 2 < w 1, the argument followng Fgure 1 apples: the optmum at x can be approxmated for x < x but not for x > x, and the value functon exhbts the downward jump-dscontnuty at x = 1 that s readly apparent n Fgure 2. Also apparent s that the global optmum occurs at x = x = 1, where two types have ted wages, and that the optmalty of ths pooled-wage optmum s robust to small changes n the underlyng model. The optmalty of ted wages s not a knfe-edge result. Proposton 2 below formalzes and generalzes the precedng ntuton: jumpdscontnutes n the value functon occur at ted-wage values x of the parameter 4

6 x precsely when t s optmal to separate the ted types at x. 2 When such separaton s optmal, these jump-dscontnutes wll frequently yeld robust local optma, and global optma are thus lkely to nvolve ted wages even f ted wages mght appear rare, ex-ante. A number of other papers, ncludng Weymark 1987, Brett and Weymark 2008, 2011, and Smula 2010, analyze the comparatve statcs of optmal taxaton wth respect to the dstrbuton of wages n a dscrete type framework. These comparatve statcs are qualtatvely smlar to our analyss of the outer problem, snce both analyses consder how the optmal tax vares as the wage dstrbuton changes. Our problem dffers fundamentally from thers precsely because our outer-problem wage varaton s an endogenous choce for the prncpal. wage poolng s common result arses because t s frequently optmal for the prncpal to choose a poolng allocaton. Absent ths actve choce, abstractng from wage poolng by assumng fully order wages as the earler papers do s reasonable. Moreover, and relatedly, even f wages of two dstnct are ted n the exogenously-varyng wage settngs of these earler papers, then t wll typcally be optmal to bunch the ted types at a common allocaton rather than separatng them as n Panel b of Fgure 1. 3 Indeed, n the three-type optmal tax example underlyng Fgure 2, t s precsely the endogenety of preferences the fact the the contracts gven to the two dstnct types wth ted wages types dfferentally affect the preferences of the thrd type that ncents the prncpal to assgn dfferent contracts to the two ted-wage types. In exogenous-wage envronments, there s no such ncentve, so the ted types can typcally be treated as a sngle more common type. 4 Our We vew our poolng s common result as havng at least two mportant m- 2 Guesnere and Seade 1982 show that related dscontnutes qute generally arse when the prncpal ndrectly optmzes over the contract menu rather than drectly optmzng over ncentve compatble allocatons. The dscontnutes they dentfy arse because ndvdual types wll jump to dstnct, far-away contracts n response to local contractng changes, leadng to a correspondng jump n the prncpal s objectve. In the present context we nstead consder drect ncentve-compatble allocatons, but a smlar effect arses at ponts where wages are endogenously ted: a small change n x dscontnuously changes the set of ncentve compatble allocatons, whch forces a jump n the drectly-assgned contracts and hence n the prncpal s objectve. 3 Vz Blackorby et al 2007, Lemma 4 and Weymark 1986, Secton 5, for the basc argument. 4 For comparatve statcs over x n exogenous settngs, t s potentally mportant to note, however, that the falure of lower hemcontnuty n the ncentve constrants can cause a knkdscontnuty n the value functon over x. See Secton 5.4 for a related dscusson. 5

7 plcatons for theoretcal and appled work on optmal taxaton and elsewhere. Frst, t calls for cauton n assumng strctly-ordered wages n the dscrete-type optmal tax envronments wth endogenous wages that are ncreasngly mportant n appled work vz Ales et al Strctly-ordered wages can be justfed ex-ante by suffcently strong assumptons about wages for example, Ales et al. make 2015 an absolute-advantage assumpton whch exogenously fxes the wage order of types whle stll allowng changes n the absolute and relatve magntude of ndvdual types wages but not by appeals to genercty. Second, t s mportant for ratonalzng the connecton between dscrete type models and ther contnuum lmts. For example, terms related to the overlappng wages of dstnct types feature promnently n Rothschld and Scheuer s 2013, 2014a, 2014b work on optmal ncome taxaton wth multdmensonal skll heterogenety. Wage overlap s generc n ther contnuum-of-types model; absent our result, t mght seem hghly unusual, and hence puzzlng, wth dscrete types. Ths paper proceeds as follows. Secton 2 descrbes our basc analytcal framework. Secton 3 llustrates how the framework can be appled n varous economc applcatons, namely: optmal ncome tax settngs wth endogenous wages, optmal contractng n a producton-n-teams multtask settng, and nsurance markets wth ex-ante moral-hazard and accdent preventon costs whch are endogenous to the effort preventon actvtes of other agents. Secton 4 contans our man analyss. We defne the nner and outer problems, characterze the frst order condtons for both, and then combne them to provde a characterzaton of the optmal dstortons for the dfferent types for separatng optma where there are no types wth dentcal preferences. We then dscuss poolng optma where one or more types endogenously have the same preferences. Proposton 2 presents our central analytcal result about jump-dscontnutes n the value functon at pooled wages. Secton 5 provdes several explctly worked examples, ncludng ones whch feature separatng optma and ones whch feature poolng optma. We then relate our work to Rothschld and Scheuer 2014b n Secton 6 before concludng. 6

8 2 Setup Consder an economy wth N types of agent = 1,..., N, each wth probablty mass f. Type agents have preferences uc, y, w x. 1 The notaton n 1 s talored to a Mrrleesan 1971 ncome tax settng, where c s consumpton, y s ncome, and w s a wage, whch depends endogenously on some x. The value of x s a functon X Y : R N R K wth K N, whch maps the N-vector Y of total ncomes wth components f E[y ] f ȳ of the varous types of agents nto a parameter vector whch s suffcent for determnng {w } N =1. Wthn the ncome-tax settng and more generally, w can be nterpreted as any measure that affects the margnal rate of substtuton between c and y e.g., a dsutlty of labor parameter. We dscuss addtonal applcatons n Secton 3. We assume that uc, y, w has u c > 0, u y < 0, u w > 0, and satsfes the sngle crossng property: w u/ y < 0. 2 u/ c We also assume that w x s contnuously dfferentable n x. The prncpal s goal s to maxmze surplus g Y, C, where c E[c ] s the average consumpton of -types, g s a producton functon, assumed ncreasng n each component of Y and decreasng n each component of C, and C s the vector wth components f E[c ] f c. 5 The prncpal must provde a mnmum utlty ū X Y to type- agents. Note that we allow ū to depend endogenously on X Y. The prncpal can observe only the consumptons c and ncomes y of each agent, but not that agent s type. We adopt the standard mechansm desgn approach of treatng the prncpal s problem as one of drectly assgnng ncentve compatble allocatons y, c to all agents. We start wth a formal defnton of allocaton. Defnton 1. Let N = {0, 1,..., N} be the set of all types. A type-profle s a correspon- 5 The case wth g Y, C = ȳ c f s of partcular nterest, and we focus on t n much of our analyss. 7

9 dence p : I 2 R2 mappng each type nto a measurable set of y, c pars. An allocaton s a collecton Π N p of type profles p and probablty measure M on p for each type. An allocaton s smple f p s a sngleton for all. Incentve compatblty of an allocaton wth type profles Π N p then requres that: u c, y, w X Y u ĉ, ŷ, w X Y, y, c p, ŷ, ĉ N p. 3 The prncpal must offer each ndvdual some mnmum utlty level ū : u c, y, w X Y ū X Y y, c p. 4 The followng lemma allows us to restrct attenton to smple allocatons, whch wll do from ths pont forward. Lemma 1. If u s quasconcave then, for any allocaton Π N p and Π N M satsfyng 3 and 4, there exsts a smple allocaton y, c, N satsfyng 3, 4 and g Y, C g Y, C, where Y s the vector wth components f E M [ỹ ] and Y s the vector wth components f y. Proof. Let ũ denote the utlty of all types n the ntal allocaton, whch, by 3, s well-defned. Take y E M [ỹ ]. Take c such that uc, y, wx Y = ũ X Y. By constructon, Y = Y, so the new allocaton leaves wages and reservaton utltes unchanged. So 4 remans satsfed. By quasconcavty of u, c E M [ c ], so the prncpal s objectve s at least weakly mproved. Fnally, ncentve compatblty of the new allocaton follows from sngle crossng and the fact that [nf{ỹ c s.t. ỹ, c p}, sup{ỹ c s.t. ỹ, c p}]. ỹ We can thus wrte the prncpal s problem as: max y, c g Y, C 5 subject to u c, y, w X Y ū X Y 6 and u c, y, w X Y u c j, y j, w X Y, j 7 8

10 Before characterzng the soluton to ths problem we dscuss some applcatons that ft n ths general framework. 3 Applcatons Ths general framework can be appled to a broad range of economc problems. We dscuss three here: an optmal tax applcaton, a multtask-n-teams applcaton, and an nsurance market applcaton. 3.1 Optmal Taxes wth Multdmensonal Skll Heterogenety Stgltz 1982 consders a smple optmal tax model wth two types whose labor efforts are complementary. We frst construct a smple example usng our notaton whch s consstent wth Stgltz s framework; we return to ths example n secton 5.1. Suppose there are two equal-mass types = 1, 2, wth wages gven by w 1 x = [ ] [ x 1 2 and w 2 x = x ] 2, wth x = X y = 1 y2 2 4 y 1. Both types have utlty functon uc, y, w = c 1 y 2 2 w and a mnmum utlty of ū = 0. The 1 prncpal s objectve s to maxmze =1,2 2 y c. It s straghtforward to show see Secton 5.1 for detals that the optmum wll occur at an x for whch w 1 x < w 2 x. Ths and ū 1 = ū 2 = 0 mples that the only bndng ncentve constrant s the one constranng type 2 not to mtate type 1. It also mples that only type 1 s ndvdual ratonalty constrant s bndng. By strong dualty, the prncpal s problem s equvalent to that of maxmzng type 1 s utlty subject to downward ncentve compatblty 7 and a resource constrant =1,2 1 2 y c R for some R. The prncpal s problem s thus equvalent to an optmal tax problem for a Rawlsan socal planner. In fact, we show n secton 5.1, that ths optmal tax problem s a specal case of Stgltz 1982 model wth [ ] producton functon Y = 3 E E 1 2 2, where E = y /w measures agents effort. Rothschld and Scheuer 2013 extend Stgltz 1982 to a general two-sector economy wth a contnuum of types wth preferences uc, y/w who endogenously choose to work n one of the two sectors. As n Stgltz aggregated sectoral efforts E θ and E ϕ are complements n a constant returns to scale producton 9

11 technology YE = YE θ, E ϕ, and workers are assumed to earn a wage equal to ther margnal product. Each ndvdual s characterzed by a skll vector θ, ϕ drawn from a contnuous dstrbuton, and each chooses to work n the sector whch provdes the hghest wage; ndvduals thus acheves the wage w ρ max{θ Y/ E θ, ϕ Y/ E ϕ } where ρ = E θ /E ϕ. The dscrete analog of ther model s readly subsumed nto our framework the only complcaton beng that one needs to translate between wrtng wages n terms of ncomes y versus efforts E. See appendx C for detals. Rothschld and Scheuer 2014b generalze ths model to have an arbtrary number of sectors, arbtrary output functons YE 1,..., E K = Y E, and arbtrary return functons r k E 6. Ths generalzed model can be smlarly adapted to our framework, as we dscuss n Secton Multtask n Teams In ths secton, we adapt to a team settng the framework of Baker 2002, who studes an optmal ncentve contractng problem for an agent who can pursue multple dfferent types of work but can only be compensated based on an mperfect undmensonal performance measure. 7 We show that ths adaptaton s a specal case of our framework. Suppose there s a contnuum of N types of worker = 1,, N. Each can smultaneously pursue K dstnct tasks k = 1,, K. Condtonal on some parameter vector β, wth elements βk > 0, total frm output depends lnearly on the total effort devoted to each task, va Y = k β k f e k. The frm cares about total output net of total compensaton Y f c. The frm wshes to ncent effort through performance contracts, but cannot drectly observe ndvdual effort or types. The frm can only observe the un-dmensonal performance measure y = β k e k = β e. The frm s problem s to choose a non-lnear compensaton scheme cy, or, equvalently, a menu of y, c pars. Workers preferences depend separably on ther compensaton utlty vc and on a type-specfc cost of effort functon 1 α m e 1,..., e K ; x, where m s assumed to be homogenous of degree α henceforth HODα and strctly quasconvex n e and may 6 Wth the property that k r k EE k = Y E. 7 See also Hölmstrom and Mlgrom

12 also depend on some parameter vector x. Snce e s unobservable, an ndvdual of type who wshes to acheve a gven level of her total performance measure y wll chooses the e that mnmzes m e; x subject to β e = y. Snce m s HODα, the functon m y; x, β mn e m e; x subject to β e = y s proportonal to y α. Hence, w x, β y α /m y; x, β 1 α, s well defned. We can thus wrte a worker s utlty as 1 α uc, y, w x, β = vc y/w α x, β. 8 The frm s goal s to maxmze f y c subject to ncentve compatblty and some partcpaton constrants of the form uc, y, w x, β ū. For a fxed x and an exogenous β, ths s a standard screenng problem. We can endogenze preferences by allowng β and m to depend endogenously, va some aggregator x = ˇX e, on the efforts of others. The drect dependence of m on the efforts of others captures the possblty that the efforts of others may affect, postvely or negatvely, the ease wth whch a gven ndvdual can accomplsh any gven task. The ndrect effect, va β, may capture producton complementartes. For example, f no one s puttng n effort e 2 at makng a gven product user-frendly, then engneerng efforts e 1 may be wasted. In formally: β 1 may be ncreasng n aggregated efforts e 2. 8 For any gven y, fxng any gven x pns down β x and hence the optmal effort vectors e y, β x, x of all types. These effort vectors n turn determne ˇX e 1,, e N. We can then defne X y as the fxed pont of ths mappng x { e } N =1 ˇX. Insofar as ths fxed pont mappng s well defned and we gve an explct example n secton 5.4 where t s, ths problem s a specal case of our general framework. 8 More generally, endogenzng β allows for qute general non-lnear producton functon. 11

13 3.3 Insurance Provson wth Endogenous Avodance Cost For our thrd applcaton, consder a smple monopoly nsurance market wth exante moral-hazard n the sprt of Bond and Crocker Each ndvdual k = 1,, N faces a potental loss of sze D out of an ntal wealth W. The probablty p of experencng the loss depends on the ndvdual s costly self-protecton effort. Wthout loss of generalty, parameterze effort negatvely by p. Insurance contracts offer ndemntes n the loss state n exchange for statendependent premums. We can descrbe such contracts n terms of the consumptons c L and c NL nduced n the loss and no loss states. Indvdual preferences over such contracts are gven by the ndrect utlty functon: Ûc L, c NL, θ = max p [0,1] puc L + 1 puc NL hp, θ, 9 where u s a standard utlty functon wth u x > 0 and u x < 0 and hp, θ a type-specfc cost-of-preventon functon wth h/ p < 0 and 2 h/ p 2 > 0, and 2 h/ θ p < 0. A contract c L, c NL can equvalently be unquely descrbed by = uc NL uc L and V NL = uc NL. The unque soluton p, θ to 9 s ndependent of V NL n ths formulaton. It s nonncreasng n, snce a bgger utlty gap ncents greater effort and hence lower rsk, and t s nondecreasng n θ and strctly decreasng n and strctly ncreasng n θ except at the p = 0 and p = 1 corners. In, V NL space, preferences are U, V NL, θ = V NL p, θ hp, θ, θ. U s ncreasng n V NL and decreasng n snce p + p,θ > 0. By the envelope theorem, the margnal rate of substtuton between V NL and s gven by p, θ at any pont. Hence, sngle crossng s satsfed except for types who are at the same corner. Fnally, the partcpaton constrants are: U, V NL, θ ū k ÛW D, W, θ k, 10 and the rsk-neutral monopolst-prncpal s expected proft from the sale of a contract menu k, V k NL to a set of types θ k, k = 1,, N wth probablty masses f k 12

14 s: N [ ] [ ] f k W p k, θ k D + u 1 VNL k k 1 p k, θ k u 1 VNL k. k=1 11 All of the precedng s ndependent of whether the θ k are endogenous or exogenous. We mght reasonably expect, however, that greater accdent preventon by others reduces the cost of achevng any gven rsk level through one s own costly effort. A smple way to model ths s to take θ k to be some functon θ k p, where p s the vector of rsks hence efforts of the N types. Snce, for any gven θ, the optmal p s a monotonc functon of, we can typcally equvalently wrte nstead θ k.e., types preferences depend endogenously on the vector of allocatons to the ndvduals n the economy. See Secton 5.2 for an explctly worked example. Ths model s agan a specal case of our general model, wth θ w, y, and V NL c. 4 Analyss The only non-standard element of the prncpal s problem s the endogenety of the preference parameters w. Condtonal on the value of X Y = x, the w are determned, and the problem s essentally standard. Ths motvates a decomposton nto an outer problem for the optmal x and an nner problem whch takes x as fxed and mposes the addtonal consstency constrant X Y = x. For smplcty, we focus n the man text on the smpler case wth exogenous ū and a lnear objectve functon f y c. Appendx D has a general dervaton; the results are entrely analogous. 13

15 The nner problem n ths smpler case s: T x max y, c f y c subject to 12 u c, y, w x ū 13 u c, y, w x u c j, y j, w x, j, and 14 X Y = x, 15 where 15 s the consstency constrant: snce x s fxed for the nner problem, the prncpal must choose an allocaton y, c consstent wth ths x. The outer problem s then to choose x to maxmze T x. We defne Γ x as the set of all y, c satsfyng 13, 14, and 15. It wll be useful to defne Z x as the set of all allocatons consstent wth 15: Z x { y, c X Y = x} Solvng the nner problem Snce fxng x fxes w x for all, we temporarly suppress the argument x for notatonal convenence n ths subsecton Prelmnary results Under the followng mld regularty condton, whch we assume to hold hereafter, exstence of a soluton to the nner problem s straghtforward. Assumpton 1. The functon u s strctly quasconcave, and for any ū and any fnte w, there exsts a fnte y, c such that MRSy, c u yc,y u c c,y = 1. An mmedate mplcaton of ths assumpton s that y c along any ndfference curve as y ±. Ths ensures that we can restrct attenton to a compact subset of the set Γ x of feasble allocatons gven x and yelds the followng lemma. Lemma 2. For any x, a soluton to the nner problem exsts. Proof. See Appendx A.1. 14

16 The followng lemma collects some smple mplcatons of ncentve compatblty and sngle-crossng for the soluton to the nner problem Lemma 3. In any soluton to the nner problem, the followng propertes hold 1. For any two types and j: c c j y y j ; 2. For any two types and j: w > w j c c j ; 3. For any three types, j and k wth w w j w k : uc, y, w uc j, y j, w and uc j, y j, w j uc k, y k, w j together mply uc, y, w uc k, y k, w ; 4. For any three types, j and k wth w w j w k : uc k, y k, w k uc j, y j, w k and uc j, y j, w j uc, y, w j together mply uc k, y k, w k uc, y, w k. Proof. See Appendx A.2. Lemma 3 establshes the standard result that ncentve compatble allocatons must be monotoncally ncreasng n w, and that local ncentve constrants are suffcent. Importantly, note that when w = w j, the lemma permts c > c j, c j > c, or c = c j The nner problem: frst order condtons By Lemma 3, we can restrct ncentve constrants to the set of local ncentve constrants. Wthout loss of generalty, order the types so that wages are nondecreasng n. Defne τ 1 MRS. Ths s nterpretable as the mplct tax wedge at s allocaton.e., as the dstorton away from the frst best, at whch MRS = 1. Denote by ξ, η,j j = ± 1, and µ the respectve multplers on the ndvdual ratonalty constrants 13, the ncentve constrants 14, and the consstency constrants 15. As above, we wrte Y = f1 y 1,..., f K y N. Fnally, denote by MRS j the margnal rate of substtuton u y c, y, w j /u c c, y, w j for a j type gven s allocaton. The followng lemma uses the frst order condtons from the Lagrangan formulaton of the nner problem to characterze the optmal tax wedges τ for any gven x. It s most easly expressed n terms of f τ whch s the vector wth components f τ. 15

17 Lemma 4. For any gven x, the optmal tax wedge satsfes: f τ = X µ + D, 17 where D s the N-vector wth elements D = η j, u c c, y, w j j { 1,+1} N j MRS MRS 18 and X s the N K matrx wth, k th element x k y. Proof. See appendx A.3. Equaton 17 reveals two sources of dstortons: the term D and the term X µ. The former s an entrely standard dstorton due to ncentve effects. The latter arses from the endogenety of preferences, as captured by consstency constrants. We wll next use the outer problem to characterze X µ and show that t can be nterpreted as a modfed Pgouvan correctve term. 4.2 Characterzng the outer problem Towards solvng the outer problem of fndng the optmal x.e., the x that maxmzes T x from the nner problem t s useful to defne X = { x = j wth w x = w j x},.e., the set of all x wth ted types. We make three techncal assumptons. The frst ensures that ted wages are unusual from an ex-ante perspectve: Assumpton 2. = 0 for all = j and all x X wth w x = w j x. w x w j x Under Assumpton 2, the wage rato w x /w j x s not locally dentcal to unty. In partcular, for any x X, there s some smooth curve passng through x along whch the ted wages are pulled apart. The parttons X and R K \ X naturally dvde optma nto two classes: separatng optma, whch occur at some x R K \ X and are characterzed by N dstnct wages; and poolng optma, whch occur at some x X and are characterzed by fewer than N dstnct wages. 16

18 Our second techncal assumpton s suffcent for establshng dfferentablty of T x at ponts x / X. Assumpton 3. At any x / X and y, c satsfyng the KKT condtons, the correspondng vector of Lagrange multplers ξ, µ, η s unque. It s well known that the unqueness of KKT multplers s equvalent to the Strct Mangasaran-Fromowtz Constrant Qualfcaton SMFCQ condton see for example Kyparss By Theorem 7 n Morand et al. 2015, SMFCQ together wth approprate smoothness assumptons on u and w and compactness of a restrcton of Γ x, guarantees that T x s drectonally dfferentable, and the Envelope Theorem apples. We explot ths n computng the drectonal dervatve of T x n the followng secton. The thrd techncal assumpton wll be useful n ensurng that the constrant set Γ x s reasonably well-behaved at x X. The followng defnton s necessary: Defnton 2 Perverseness. The pont x s sad to be non-perverse f there s a neghborhood V of x such that for each x V there s a soluton y, c to the nner problem and a sequence y n y such that y n, c satsfes 15 for all n and y n = y n j for all n and all = j. Otherwse, x s sad to be perverse. Assumpton 4. All x X are non-perverse. Note that Γ x s qute generally upper-hemcontnuous n x / X. Lower hemcontnuty can fal for three reasons. Frst, t wll typcally fal at x X, snce, as we dscuss n the ntroducton and n Proposton 2 below, the set R x of allocatons satsfyng 13 and 14 s dscontnuous at such ponts. Second, t can fal at x / X where Z x s dscontnuous. Fnally, t can potentally fal at perverse x / X, even when Z x s contnuous. At non-perverse x / X where Z x s contnuous, Γ x s contnuous, and, by Berge s Maxmum Theorem, T x s contnuous. 9 9 To see why assumpton 4 s useful, note that at non-perverse x, the optmum can always be approxmated by non-bunchng allocatons whch gve dstnct types dfferent allocatons; moreover, snce 15 does not depend on c, the optmum can be approxmated by non-bunchng allocatons n the nteror of R x. Hence, Γ x = Cl [Int R x Z x] at non-perverse x. Theorem n Border 1989 and the fact that closure preserves lower hemcontnuty then mples Γ x s lower hemcontnous. 17

19 4.2.1 Decomposng Tx n the outer problem We now compute Tx usng the Envelope Theorem and the Lagrangan formulaton. Specfcally, the proof of the followng lemma dfferentates the nnerproblem Lagrangan wth respect to x holdng fxed c and z uc, y, w x for all. Suppose a small change n x leads to a change dw n the type s wage. Then y y wll change by w dw = u wc,y,w x z,c u y c,y,w x dw. We denote u wc,y,w x u y c,y,w x by u w uy. It s also useful to defne an analogous term for the j type mtatng : j u wc,y,w j x u y c,y,w j x. u w u y Fnally, defne τ p as the N-vector wth elements τ p j f u w u y k X k Y j w X k. 19 These elements are nterpretable as Pgouvan correctve taxes on agents of the X varous types j. To see ths, note that the term k w k Y measures the amount by j X k whch the wage of types would change n response to a small ncrease n Y j, and u w uy measures the amount by whch y would change n response to a wage change, holdng constant s utlty and consumpton. Hence τ p j s a measure of how much the total output s ndrectly affected by wage changes nduced by a p margnal ncrease n earnngs by a non-atomc type- ndvdual. We defne f τ as the vector wth components f τ p by analogy wth f τ defned n secton Lemma 5. When T x s dfferentable, ts dervatve satsfes: where I = N =1 j { 1,+1} {1,,N} X T x = X P µ f τ p + X I, 20 η j, u y c, y, w j u w u w j x y and where P = C X I K, wth C s the K N matrx wth elements C j = 18 j u w u y w x, 21 u w w uy x j

20 and I K s the K K dentty matrx, Proof. See Appendx B Combnng the nner and outer problems We wll now combne the frst order condtons from lemmas 4 and 5 for the nner and outer problem to characterze the optmal dstortons at separatng optma. To that end, let Q = I N X C be an N N matrx. We assume Q s nvertble. 10 Proposton 1. If T x s dfferentable at a separatng optmum, then f τ = D + Q 1 f τ p Q 1 X I 22 Proof. Follows drectly from lemmas 4 and 5 and the necessary condton T = 0. Equaton 22 can be understood as decomposng the optmal dstorton for each type nto three components. The frst, D, s the standard dstorton that arses from ncentve compatblty. The other two components are a result of the endogenety of preferences. One s a Pgouvan correcton f τ p. It s modfed by the term Q 1, whch can be understood as a general equlbrum correcton. Recall that τ p s nterpretable as the negatve of the externalty caused by a small ncrease n Y = f y, computed holdng each type s utlty and consumpton constant. Specfcally, an ncrease n Y changes X Y, and therefore the preferences of all ndvduals of all types j. When preferences change, holdng utlty and consumpton constant requres changes n y j. These changes n turn change X Y, whch change preferences, whch change y, and so forth. The term Q translates between the partal equlbrum effects of the change n y and the general equlbrum effect after all these rounds of changes have taken place. The fnal component whch s modfed by Q 1 for the same reason s a Stgltz effect. It arses because a change n y dfferentally affects the preferences of dfferent types; accordngly, t ndrectly relaxes or tghtens ncentve constrants. Equvalently, one can thnk of the Stgltz term XI as a term capturng 10 If Q s sngular, then the fxed pont mappng nduced by the consstency constrants, holdng c and z constant as we change x, s not well-defned. 19

21 the ndrect redstrbuton whch occurs as a result of the change n preferences nduced by a change n y. Ths s a partcularly transparent nterpretaton when the preference parameter w s a wage. 4.4 Poolng optma The precedng analyss apples only to separatng optma,.e., when t s optmal to choose an x / X. Away from X, the value functon T x s typcally wellbehaved, snce the ncentve and ndvdual ratonalty constrants are contnuous n x at such ponts. Per Fgure 1, the ncentve constrants are n general not lower hemcontnuous at x X. We frst show that ths can mply jump dscontnutes n T x for x X, and we characterze precse condtons under whch such jump dscontnutes wll occur. The followng defntons wll be useful n ths characterzaton. Defnton 3. An x X at whch w x = w j x for some, j = s sad to be unusual f there exst two dstnct solutons y, c and y, c such that c > c j and c < c j. Otherwse, x s sad to be usual. Defnton 4. An x X at whch w x = w j x for some, j = s sad to be degenerate f there exsts a soluton y, c such that c = c j. Otherwse, x s sad to be nondegenerate. Fgure 1 and the assocated dscusson n the ntroducton, can be used to provde ntuton for these defntons n the one-dmensonal x case. 11 Graphcally, an x X would be unusual f n addton to the optmum depcted n Panel b wth y, c 2 lyng above and to the rght of y, c 1, there were a second optmum wth y, c 2 lyng below and to the left of y, c 1. An x X would be degenerate f the optmum at x nvolved bunchng,.e., had y, c 2 concdng wth y, c 1. In ether case, an optmum allocaton at x can be approached as x x both from above and below, so we expect the value functon Tx to be contnuous at x. At usual, nondegenerate x, the lmt allocaton as x x ether from the rght or the left wll approach a strctly sub-optmal allocaton, and the value functon Tx wll have a jump dscontnuty at x. 11 Appendx E dscusses usualness and provdes smple prmal condtons whch mply t. 20

22 The followng proposton formalzes ths ntuton for a general-dmensonal X. For smplcty, the proposton s stated only for those x wth exactly two ted types. 12 Proposton 2. If x X s usual and nondegenerate, then T x has an one-sded jump dscontnuty at x. Specfcally, T x s contnuous at x vewed as a pont n half space { x R N w x w j x} or { x R N w x w j x}, but not both. Conversely, f x X s unusual or degenerate, then T x s contnuous at x. Proof. See appendx B.2 Even when x X s unusual or degenerate, and, per Proposton 2, T x s contnuous, T may stll be poorly behaved. In partcular, T can fal t be dfferentable at such ponts because the lnear ndependence constrant qualfcaton condton fals as two ncentve constrants concde. So, even f T s contnuous, t can stll have knk-dscontnutes at x X. These jump- and knk-dscontnutes n T x can be nterpreted as forces pushng towards poolng optma. Indeed, n the one-dmensonal case, x X wll be a robust local maxmum of T whenever T jumps up at x and T x < 0 to the rght of x. Symmetrcally, x X wll be a robust local maxmum of T whenever T jumps down at x and T x > 0 to the left of x. Fnally, x X wll be a robust local maxmum f T s contnuous at x and T x < 0 to the rght of x and T x > 0 to the left. Suppose, for example, that x X s a local maxmum of Tx because T jumps down at x and T x > 0 to the left of x. Let lm x x T x C. Then nstead of the optmzaton condton for the optmal tax τ from equaton 22, we wll have f τ = D + Q 1 f τ p Q 1 X I + Q 1 C. 23 In other words, we can thnk of the jump-dscontnuty at the optmal poolng allocaton as addng an extra dstorton. Smlarly, we can thnk of knk dscontnutes as addng an extra dstorton R. 12 Ths s completely nnocuous f X s one-dmensonal snce t holds genercally n ths case, but t s a substantve restrcton wth a multdmensonal X. Wth consderable notatonal clutter, the proposton s readly generalzed to the many-ted-type case, when the same logc pushes for addtonal dscontnutes and addtonal poolng. 21

23 We now provde a number of examples whch llustrate both separatng and poolng optma and the varous components of the dstortons. 5 Illustratve Examples In ths secton, we return to the three applcatons dscussed n Secton 3. The frst two examples optmal taxes a la Stgltz 1982 and nsurance provson feature separatng optma. We then present two addtonal examples wth poolng optma: a multtask-n-teams applcaton and a second optmal tax example n the sprt of Rothschld and Scheuer 2014b. 5.1 Relaton to Stgltz 1982 As dscussed n Secton 3.1, Stgltz 1982 s essentally a two-type specal case of our framework. A key dfference s that Stgltz gnores the possblty of ted wages by tactly assumng that one type s more productve than the other whch s problematc n lght of endogenous productvty. In ths secton, we show formally that the model dscussed n Secton 3.1 s ndeed a specal case of Stgltz We then solve for the optmum and show that wages can n prncple be ted, but, n practce they wll not be. To recap: There are two equal-mass types = 1, 2, wth wages gven by w 1 x = [ ] [ x 1 2 and w 2 x = x ] 2, and x = X y = 4 1 y2 2 y 1. Both types are assumed to have uc, y, w = c 1 y 2 2 w and a mnmum utlty of ū = 0. The prncpal s objectve functon s 2 =1 1 2 y c. Note that w 1 /w 2 = 2 1 x, and defne E = 1 y 2 w. The consstency constrant 15, namely y 2 /y 1 = 2 x mples y 2 /y 1 = 4w 1 /w 2 and x = 1 4 y2 y 1 2 y2 /w = 2 y 1 /w 1 E 2 E 1. 22

24 The total ncome earned n the economy s: Y = 1 2 y 1 + y 2 = w 1 E 1 + w 2 E 2 [ ] = E 2 3 E 1 E [ ] 1 E [ 1 E 2 = 3 E E ] 2 3 E 1 2 2, 24 whereby w = Y E. In other words, the problem can equvalently be formulated as a Stgltz economy wth producton functon 24. Wages are ted at ˆx = 4. For x > ˆx resp. < w 1 x > w 2 x resp. <, and ncentve compatblty requres y 1 y 2, resp.. The consstency constrant, re-wrtten as y 2 = 2x 1 2 y1, 25 mples y 2 4y 1 > y 1 for any x ˆx = 4. Any value x > ˆx s therefore nfeasble for the prncpal, snce 25 would requre y 2 > y 1, whch s ncompatble wth ncentve compatblty. Moreover, snce, per 25, y 2 > y 1 at ˆx, Proposton 2 mples that Tx s left-contnuous at ˆx. For x < ˆx, standard arguments show that only the downward ncentve compatblty and type-1 mnmum utlty constrants bnd. The prncpal s nner problem sans the upward ncentve constrant s a concave optmzaton problem when re-wrtten n terms of the utltes v and ncomes y of the two types. The unque soluton has y 1 x = x 1 w 1 x 2 + x 4x, y 2 x = 4x x w 2 x 2 w 1 x 2 w 2 x 2 and Lagrange multplers η 21 = 1 2 and ψ 1 = 1 respectvely assocated wth the bndng ncentve constrant and the bndng mnmum utlty constrant n the nner problem. Assumpton 3 s therefore satsfed, so T x s well defned. Straghtforward calculatons usng the Lagrangan formulaton of the nner problem, the envelope theorem to compute T x holdng c and y /w constant, and the frst 23

25 Tx Tx optmum x Fgure 3: Value functon for a Stlgltz 1982-lke separatng optmum example order condtons for the nner problem mply T x = y2 1 dw 1 w1 3 dx + 1 y 2 2 dw 2 2 w2 3 dx 1 4 y 2 w y 2 x. 26 Hence, lm x ˆx=4 T x = y 1 dw 1 w1 3 dx < y 1 dw 1 w1 3 dx y2 dw2 /dx 2 y 1 dw 1 /dx x x 1 < 0. + y 2 4x 1 y 2 w 2 2 The optmum thus occurs for some x < ˆx wth x 1 4 for consstency. It s safe, per Stgltz 1982 to assume that w 1 < w 2 when computng the optmum. Indeed, Tx s smooth throughout the doman of consstent x s. It s depcted n Fgure Insurance wth Endogenous Avodance Costs Consder next the nsurance example from secton 3.3, wth the followng addtonal assumptons: The rsk avodance dsutlty functon s gven by hp, θ = θ/p. The parameter θ depends endogenously on some x va θ k = θ k x = α k x. 24

26 The value of x s endogenously gven by a type-weghted average of the rsk avodance efforts: x = f p, θ. Indvdual optmzaton for any gven V NL uc NL and uc NL uc L mples p, θ = θ/ 1 2 and UV NL,, θ = max p {V NL p hp, θ} = V NL 2 θ. Hence, x = f θ / 1 2 = x f α / 1 2, and we can equvalently wrte x = X = f α / In other words, ths example s nested wthn our general framework, wth θ takng the role of w, and V NL, playng the role of c, y. 13 Note that θ θ j = α α j, so wage overlap of dstnct types s mpossble here: preferences are endogenous, but the orderng of preferences concdes wth the orderng of the exogenous α parameters An Optmal Tax Model wth a Jump-Poolng Optmum The precedng examples nvolved separatng optma. We now further develop the smple three-type tax model wth a poolng optmum dscussed n the ntroducton. Suppose the economy conssts of three equal measure of workers wth uc, y, w = y2 1/α c y/w 3. Let x = X y = y 1 +y 0 for some α < 0. Let wages be gven by [ 1 α [ 1 α [ 1 α w 0 x = x ] α α, w 1 x = x ] α α, and w 2 x = x ] α α. Work n the dual formulaton wth a socal planner wth a Rawlsan socal welfare functon and a balanced budget requrement. Snce utlty s quaslnear, t s straghtforward to show that ths s equvalent to a prmal formulaton wth a lnear objectve and some common reservaton utlty ū for all three types. Fgure 2 n the ntroducton plots the value functon Tx for ths problem for α = 0.1. The fgure clearly ndcates that the optmum occurs at a jump- 13 Strctly speakng, ths nestng only works f we exogenously assume that all ndvduals of the same type receve the same contract rather than rely on Lemma 1. Ths s because the average wthn a type of c, y s not a suffcent statstc for the frms profts. 14 Ales et al dscuss a generalzed dscrete type Stgltz 1982 model wth the same property: wages are endogenous, but because of an absolute advantage assumpton, the orderng of types s exogenously fxed and wage overlap s mpossble. In an appendx, they also treat a contnuum-of-types case wth wage overlap. 25

27 dscontnuty at the pont x = 1, where w 2 x = w 1 x. 15 The mechancal ntuton for ths jump-dscontnuty s straghtforward, and mrrors the ntuton from Fgure 1 n the ntroducton. To wt: at x = 1, y 2 = y 1 + y 0. It turns out to be optmal to have y 0 > 0; hence, y 2 > y 1 at x = 1. Snce w 2 s decreasng and w 1 ncreasng n x, y 2 y 1 for any x > 1. The optmum at x = 1 thus cannot be approxmated by any feasble sequence of allocatons as x 1 from above so the value functon necessarly jumps down at x = 1. The underlyng ntuton s also reasonably straghtforward. It s easest to see n an equvalent Stgltz 1982-lke formulaton of the same problem, namely: a problem wth the producton functon YE θ, E φ = [ 1 2 Eα φ Eα θ ]1/α, where E θ = θ e θ and E φ = φ e φ are the aggregate efforts n two sectors, and where there are three equal measure worker types wth sklls θ, φ 0 = 0, 1, θ, φ 1 = 0, 2, θ, φ 2 = 2, 0. We omt the straghtforward calculatons, akn to those n secton 5.1, establshng equvalence. In ths alternatve formulaton, t s clear from the producton functon that type 2 s effort ncreases the wages of the other two types whle reducng the wages of other type 2s and vce-versa for types 0 and 1. Intutvely, ths provdes an extra ncentve to up-dstort 2s effort and earnngs and down-dstort the effort and earnngs of type 0 and type 1, n order to ndrectly redstrbute towards the worstoff type 0s. At x and to the left of t, ths ncentve manfests n y 2 > y 1. To the rght of x, y 2 y 1 by ncentve compatblty, and the objectve Tx drops dscretely. Towards unpackng ths ntuton and relatng t to the formalsm n the precedng secton, Fgure 4 plots the optmal tax dstortons on types 1 and 2 from the nner problem, as a functon of x, just to the left of the optmum x. The dashed lnes show the total tax dstorton on the two types. Per the precedng ntuton, type 1 faces a postve tax so her earnngs are dstorted down whle type 2 faces 15 The fgure omts socal welfare for x > x, where x s the x at whch w 2 = w 0. It can be verfed, per the followng argument, that such solutons yeld strctly lower socal welfare. For x > x, y 1 y 0 y 2 and c 1 c 0 c 2 by ncentve compatblty. Hence, c 2 < Y/3, where Y s total output. The consstency constrant for α = 0.1 and y 2 > 0 then requres x = y 1 + y 0 /y 2 10 > 2 10 hence the omsson from Fgure 2. Type 2 s utlty, c 2 e2 3, s no greater than Y/3 e3 2 = 3Ke 2 e2 3, where Kx = x α 1/α as s computed from an equvalent Stgltz 1982-lke formulaton of the same problem see followng text. Usng the fact that K x < 0 and x > 2 10, we can bound type 2 s utlty for x > x by max e2 Kxe 2 e2 3 = 2Kx1.5 2K =.0026, whch s clearly below the optmum n Fgure 2. 26

28 Tax/Dstorton Type 1: Informaton Dstorton Type 1: "Stgltz" Dstorton Type 1: Total Dstorton Type 2: Informaton Dstorton Type 2: "Stgltz Dstorton" Type 2: Total Dstorton Fgure 4: Tax dstorton components near x = 1 x a negatve tax and has up-dstorted earnngs. The fgure also decomposes ths total dstorton nto the components per equatons 22 and 23 dvded through, componentwse, by f. Snce ndvduals are are pad ther margnal product of effort n ths example, the Pgouvan component τ p s dentcally zero, and t s omtted from the plot. Recall that the component D, graphed as the sold lnes n Fgure 4, s a standard nformaton dstorton. It corresponds to the dstorton that would appear n a model wthout endogenous preferences. The nformaton dstorton for type 2 s dentcally 0. Snce type 2 s the hghest wage type over the plotted range, ths corresponds to the standard no dstorton at the top result. Type 1, who s the the ntermedate wage type, faces a postve nformaton dstorton, as s to be expected wth a progressve socal welfare functon. Ths dstorton converges to 0 as x 1 and type 1 s wage converges to type 2 s. The Stgltz dstorton correspondng to the I term n equaton 22 s plotted as the dashed lnes. It s large n magntude for both types, but wth opposte sgns. Intutvely, dstortng type 2 towards hgher effort compresses the wage dstrbuton and benefcally redstrbutes towards the worst-off type 0. Smlarly, dstortng type 1 towards lower effort benefcally compresses the wage dstrbuton. It s clear from the fgure that the total dstorton for both types s smaller n magntude than the sum of the three component dstortons from equaton