Economerca Supplemenary Maeral SUPPLEMENT TO SOCIAL DISCOUNTING AND INTERGENERATIONAL PARETO (Economerca, Vol. 86, No. 5, Sepember 2018, 1537 1567) TANGREN FENG Deparmen of Economcs, Unversy of Mchgan SHAOWEI KE Deparmen of Economcs, Unversy of Mchgan THIS SUPPLEMENT consss of four pars: () he robusness of fndngs n he man paper wh respec o several man assumpons, () an alernave nerpreaon of nergeneraonal Pareo and s mplcaon on quas-hyperbolc dscounng, () a resul wh forward and backward ndvdual exponenal dscounng, and (v) a dscusson of he choce doman of he man paper. The supplemen uses defnons and noaons from he man paper. S1. DISCUSSION OF THE MAIN ASSUMPTIONS Our man fndngs are bul upon hree ses of assumpons: () he assumpons abou ndvdual preferences, () nergeneraonal Pareo and srong non-dcaorshp, and () he assumpons abou he planner s preference. In he frs, we have assumed ha a paren s dscoun funcon and nsananeous uly funcon are nhered by hs offsprng. Ths assumpon may or may no be realsc. I s helpful o undersand how our resuls depend on. In he second, nergeneraonal Pareo only has be when all ndvduals from he curren and fuure generaons agree. I s useful o undersand o wha exen nergeneraonal Pareo can be srenghened. In he hrd, we have requred ha he planner have an EDU funcon. Ths assumpon mposes resrcons on how he planner can aggregae ndvdual preferences. We examne wha resuls sll hold f we drop hs assumpon. We frs sae a more general verson of Lemma 1, whch also follows from Harsany (1955) and Fshburn (1984) drecly. LEMMA S1 Harsany (1955): Suppose each generaon- ndvdual s uly funcon akes he followng form: U (p) = δ (τ )u (p τ τ) and he planner s uly funcon n perod akes he followng form: U (p) = δ (τ )u (p τ τ) Tangren Feng: angren@umch.edu Shaowe Ke: shaowek@umch.edu 2018 The Economerc Socey hps://do.org/10.3982/ecta15011
2 T. FENG AND S. KE n whch δ ( ) and δ ( ) are dscoun funcons, and u ( τ) and u ( τ) are normalzed nsananeous uly funcons. The planner s preference ( ) T s nergeneraonally Pareo f and only f, n each perod T, here exss a fne sequence of nonnegave numbers (ω ( s)) N s such ha U = N =1 ω ( s)u s We sck o he assumpons abou ndvduals and he planner s uly funcons n he man paper, unless saed oherwse. S1.1. Inherng Dscoun Funcons and Insananeous Uly Funcons From Parens One mananed assumpon abou ndvdual preferences s ha each generaon- ndvdual s dscoun funcon δ and nsananeous uly funcon u are ndependen of. We show n hs subsecon ha hs assumpon can be removed whou changng our man fndngs. We analyze wo cases below. In he frs case, for any N and fne, suppose generaon- ndvdual s dscoun funcon s δ and nsananeous uly funcon s u ; ha s, we sll assume ha ndvdual nsananeous uly funcons do no depend on me. Fxng each generaon- ndvdual s dscounng uly funcon for any N and any naural number, our resul may requre us o vary he me horzon T. The resul below shows ha we can esablsh a posve resul ha s smlar o Theorem 2. THEOREM S1: Suppose each generaon- ndvdual s dscounng uly funcon has an nsananeous uly funcon u and a dscoun funcon δ such ha (A2) and (A3) hold and (u ) N s lnearly ndependen. Le he planner s nsananeous uly funcon u be an arbrary src convex combnaon of (u ) N. Then, 1. for each δ>max δ, he planner s nergeneraonally Pareo and srongly nondcaoral; 2. for each δ such ha for some,, δ<δ, here exss some T > 0 such ha f T T, he planner s no nergeneraonally Pareo. We wll prove hs heorem as a specal case of Theorem S2 below. Theorem S1 shows ha socal dscounng should sll be more paen han he mos paen ndvdual s longrun dscounng when ndvdual dscoun funcons may change across generaons. Snce generaon- ndvdual s dscoun funcon s now δ raher han δ, he cuoff for he socal dscoun facor becomes max δ. The second par of he heorem can be undersood as follows. Suppose he socal dscoun facor δ s below some generaon- ndvdual s long-run dscoun facor. Then, as we ncrease T, hs planner wll evenually volae nergeneraonal Pareo. One may wonder why we sll assume ha generaon- ndvdual s nsananeous uly funcon does no depend on. Le us assume ha generaon- ndvdual s nsananeous uly funcon s u. The example below shows ha hs assumpon wll lead o a rval negave resul ha has nohng o do wh dscounng. EXAMPLE S1: Suppose N = 1. Le generaon-1 ndvdual s nsananeous uly funcon be u 1, whch s lnearly ndependen of generaon-2 ndvdual s nsananeous uly funcon u 2. Snce he planner has an EDU funcon, her nsananeous uly funcon should never change. In he frs perod, he planner s nsananeous uly funcon
SOCIAL DISCOUNTING AND INTERGENERATIONAL PARETO 3 for perod-1 consumpon can only be u 1, because only he generaon-1 ndvdual cares abou perod-1 consumpon. The planner s nsananeous uly funcon for perod-2 consumpon, however, mus depend on boh u 1 and u 2 due o srong non-dcaorshp, whch means ha he planner s nsananeous uly funcon for perod-2 consumpon mus dffer from u 1. Therefore, s mpossble o requre ha he planner be nergeneraonally Pareo and srongly non-dcaoral. As can be seen n he example above, seems nevable ha he planner s nsananeous uly funcon should depend on me; ha s, he planner s nsananeous uly funcon for perod-τ consumpon should depend on τ. Indeed, one way o resore he posve resul s o allow he planner s nsananeous uly funcon o be u( τ). However, here s anoher way o resore he posve resul, whch s he second case we wan o analyze. For any N and fne, suppose generaon- ndvdual s dscoun funcon s δ and nsananeous uly funcon for perod-τ consumpon s u ( τ); ha s, f he planner s nsananeous uly funcon for perod-τ consumpon has o depend on τ, le us make he same assumpon for ndvduals. Noe ha ndvdual nsananeous uly funcons now depend on me, bu n a manner dfferen from Example S1. The planner s dscoun funcon s agan exponenal. These assumpons are parcularly suable n our seng. Recall ha each ndvdual only lves for one perod, and he cares abou fuure consumpon based on alrusm. Imagne ha u ( τ) s generaon-τ ndvdual s acual consumpon uly ha s, he uly ha generaon-τ ndvdual derves by consumng raher han from alrusm. Now, generaon- ndvdual s uly funcon s U (p) = δ (τ )u (p τ τ) whch means ha when he generaon- ndvdual alruscally cares abou generaonτ ndvdual s consumpon, he values he consumpon n exacly he same way ha generaon-τ ndvdual wll value for hmself. THEOREM S2: Suppose each generaon- ndvdual s dscounng uly funcon has nsananeous uly funcons (u ( τ)) τ and a dscoun funcon δ such ha (A2) and (A3) hold and (u ( τ)) N s lnearly ndependen for each τ T. Suppose, for some posve (λ ) N such ha λ N = 1, he planner s u( τ)= λ N u ( τ)for any τ T. Then, 1. for each δ>max δ, he planner s nergeneraonally Pareo and srongly nondcaoral; 2. for each δ such ha for some,, δ<δ, here exss some T > 0 such ha f T T, he planner s no nergeneraonally Pareo. PROOF: Par I Ths par s smlar o Par I of Theorem S1. Frs, we prove a lemma for one-ndvdual aggregaon. LEMMA S2: Assume ha N ={}. Suppose each generaon- ndvdual s dscounng uly funcon has nsananeous uly funcons u ( τ) and a dscoun funcon δ such ha (A2) and (A3) hold. Le he planner s nsananeous uly funcon be u ( τ) for any τ T. For any δ>max δ, he planner s nergeneraonally Pareo and srongly nondcaoral.
4 T. FENG AND S. KE PROOF: We wan o show ha for any δ>max δ, here exss a fne sequence of posve numbers (ω ( s)) T s such ha U (p) = δ τ u(p τ τ)= ω ( s)u s (p) for each T.Gvenanyδ>max δ, we can consruc (ω ( s)) T s accordng o he followng formula recursvely: 1 f s = s 1 ω ( s) = [ δ δ τ (s 1 τ) δ τ (s τ) ] (S1) ω ( τ) f s> Noe ha by assumng δ>max δ, ω ( s) > 0foranys and T. Then, U (p) = = ω ( s)u s (p) = ω ( s) δ s (τ s)u(p τ τ) τ=s ( τ ) δ s (τ s)ω ( s) u(p τ τ) We wan o prove ha U (p) = T τ δτ u(p τ τ). Clearly, for τ =, δ s(τ s)ω ( s) = ω ( ) = 1 = δ 0.Suppose,forsomeτ, wehaveprovenha τ δ s(τ s)ω ( s) = δ τ. We wan o prove ha for τ + 1, τ+1 δ s (τ + 1 s)ω ( s) = δ τ +1 (S2) To prove (S2), we only need o noce ha accordng o (S1), τ+1 δ s (τ + 1 s)ω ( s) = ω ( τ + 1) + τ δ s (τ + 1 s)ω ( s) τ [ = δδ s (τ s) δ s (τ + 1 s) ] ω ( s) + τ = δ δ s (τ s)ω ( s) = δ τ +1 τ δ s (τ + 1 s)ω ( s) By nducon, we know ha τ δ s(τ s)ω ( s) = δ τ for all τ, and hence U (p) = T δτ u (p τ τ). Q.E.D.
SOCIAL DISCOUNTING AND INTERGENERATIONAL PARETO 5 Nex, for any socal dscoun facor δ>max max δ, we can fnd (ω (s)) T N s such ha ω (s)u s (p) = δ τ u (p τ τ) for each N. Then, we know ha U (p) = = N N λ ω ( s)u s (p) = δ τ λ u (p τ ) =1 =1 δ τ N λ u (p τ τ)= δ τ u(p τ τ) =1 Par II We prove by conradcon. Suppose here exss an nergeneraonally Pareo planner wh socal dscoun facor δ<δ for some = and =. By nergeneraonal Pareo, for each T, here exss a fne sequence of nonnegave numbers (ω ( s)) N s such ha he followng equaly holds: δ τ u(p τ τ)= N τ ω ( s)δ s (τ s)u (p τ τ) =1 (S3) for any τ. Whenτ =, he above equaon reduces o u(p τ τ)= N ω τ ( τ)u (p τ τ) =1 (S4) for any τ T. Snce u( τ)= N λ u ( τ) for any τ T and (u ( τ)) N s lnearly ndependen, ω ( ) = λ > 0, for any and. Mulply δ τ o boh sdes of equaon (S4) and combne wh equaon (S3). We oban N ω τ ( τ)δ τ u (p τ τ)= =1 N τ ω ( s)δ s (τ s)u (p τ τ) =1 Snce (u ( τ)) N =1 s lnearly ndependen, he above equaon s equvalen o ω τ ( τ)δ τ u (p τ τ)= τ ω ( s)δ s (τ s)u (p τ τ) for any N, T,andτ. Le = and =, and rearrange he above equaons. We have δ τ = τ ω ( s ) δ s(τ s) ω τ ( τ )
6 T. FENG AND S. KE ω ( ) ( δ ) τ τ + ω ( s ) δ s(τ s) = λ δ ( ) τ λ +1 ω τ ( τ ) = δ ( τ ) (S5) for any τ> τ. However, we also know ha δ<lm τ δ (τ), here exss T such ha for any τ T, δ< τ δ (τ), whch conradcs (S5). Q.E.D. When we assume u( τ)= N λ u ( τ),wehaveassumedhaλ s do no depend on τ. In he socal choce leraure, some economss have argued ha wh normalzed ndvdual uly funcons, equal ularan weghs should be used (see Karn (1998), Dhllon and Merens (1999), and Segal (2000)). To some exen, hs s conssen wh our assumpon ha λ s do no change over me, alhough n our case, λ s may no be 1/N. In general, one may wan λ s o depend on me. In ha case, he fac ha he planner s dscoun funcon s exponenal wll mpose resrcons on how λ s may change over me. S1.2. Srenghenng Inergeneraonal Pareo The premse of nergeneraonal Pareo requres ha he curren generaon and fuure generaons reach a consensus. A naural way o srenghen nergeneraonal Pareo may be o requre ha he planner prefer one consumpon sequence over anoher f more han a ceran fracon of curren- and fuure-generaon ndvduals agree. 1 However, n hs case, how he planner aggregaes ndvdual preferences may dffer somewha from ularan aggregaon. Therefore, we srenghen nergeneraonal Pareo n he followng smple way whou devang from sandard ularansm. Le I N T be an arbrary subse of ndvduals across generaons. Le us weaken he premse of nergeneraonal Pareo by requrng ha he planner prefer a consumpon sequence p o q whenever ndvduals n I agree. Inergeneraonal Pareo and he srongly non-dcaoral propery are adaped as follows. DEFINITION S1: The planner s preference ( ) T s I-nergeneraonally Pareo f for any consumpon sequences p q (X) T,neachperod T, p s q for all ( s) I wh s mples p q,andp s q for all ( s) I wh s mples p q. DEFINITION S2: We say ha he planner s I-srongly non-dcaoral f for each T, U (p) = f ( U1 (p) U 1 T (p) U 2 (p) U 2 T (p) U N T (p) ) for some funcon f ha s (srcly) ncreasng n U s for any ( s) I. I s sraghforward o show ha under I-nergeneraonal Pareo, he planner s uly funcon can be wren as a weghed sum of he uly funcons of ndvduals n I. Below,we show ha under some assumpon abou I, posve resuls can sll be esablshed afer srenghenng nergeneraonal Pareo. 1 Ths srenghenng can ceranly be appled o curren-generaon Pareo as well.
SOCIAL DISCOUNTING AND INTERGENERATIONAL PARETO 7 The followng example shows why we need an addonal assumpon. Suppose N = 2 and ndvdual nsananeous uly funcons, u 1 and u 2, are lnearly ndependen. Assume ha I ={(2 1) (1 2)}; ha s, he planner wll gve generaon-1 ndvdual 1 and generaon-2 ndvdual 2 zero weghs. Then, he somewha rval negave resul, as n Example S1, appears agan. To see hs, noe ha n perod 1, he planner s nsananeous uly funcon for perod-1 consumpon mus be equal o u 2, because only generaon- 1 ndvduals care abou perod-1 consumpon and generaon-1 ndvdual 1 has been gnored. We have assumed ha he planner has an EDU funcon, n whch her nsananeous uly funcon never changes. Now, frs, n perod 1, he planner s nsananeous uly funcon for perod-2 consumpon s a src convex combnaon of u 1 and u 2, whch mus dffer from u 1 ; second, n perod 2, followng he same logc, he planner s nsananeous uly funcon for perod-2 consumpon mus be equal o u 1,whchsagan dfferen from u 1. Therefore, s hopeless o derve any posve resul. The heorem below mposes a smple assumpon o avod he example above, whch urns ou o be srong enough for us o esablsh a posve resul. For each T,le I := { N : ( ) I} be he se of generaon- ndvduals who may no be gnored by he planner, and le I := T I. THEOREM S3: Suppose I N T, and each generaon- ndvdual s dscounng uly funcon has an nsananeous uly funcon u {u θ } Θ θ=1 for some lnearly ndependen Θ- uple of nsananeous uly funcons (u θ ) Θ, and has a dscoun funcon δ θ=1 such ha (A2) and (A3) hold. Assume ha co({u } I ) = co({u θ } Θ θ=1 ) for any T. Le he planner s nsananeous uly funcon u be a src convex combnaon of (u ) I. Then, 1. for each δ>max I δ, he planner s I-nergeneraonally Pareo and I-srongly nondcaoral; 2. for each δ<mn I δ, here exss some T > 0 such ha f T T, he planner s no I-nergeneraonally Pareo. PROOF: Par I Wh an abuse of noaon, le Θ := {1 Θ}. For each θ Θ, lei θ := { N : u = u θ }, whch s he se of s whose nsananeous uly funcon s u θ. For each θ Θ and T,leI θ := { I θ : ( ) I} be he se of generaon- ndvduals who may no be gnored by he planner and whose nsananeous uly funcon s u θ.le I θ := T Iθ. We prove hs par n four seps. Frs, we aggregae ndvduals n each I θ no a new famly θ. Each generaon- famly θ has nsananeous uly funcon u θ ( ) and he followng dscoun funcon: δ θ (τ) = 1 δ I θ (τ); ha s, f a generaon- ndvdual may no be gnored by he planner, hs dscoun funcon δ ( ) eners famly θ s generaon- dscoun funcon δ θ ( ) wh a wegh equal o ha of oher generaon- ndvdual(s) n I θ. Noe ha generaon- famles dscoun funcons may change as changes. Nex, we prove a lemma on one-famly aggregaon ha s smlar o Lemma S2. LEMMA S3: Assume Θ ={θ}. Suppose each generaon- famly θ s dscounng uly funcon has an nsananeous uly funcon u θ ( ) and a dscoun funcon δ θ ( ). Le he planner s nsananeous uly funcon be u θ ( ). For any δ>max I θ δ, he planner s nergeneraonally Pareo and srongly non-dcaoral. I θ
8 T. FENG AND S. KE PROOF: We wan o show ha for any δ>max I θ δ, here exss a fne sequence of posve numbers (ω θ(s)) T s such ha U (p) = δ τ u θ (p τ ) = ω θ (s)u θ (p) s for each T.Gvenanyδ>max I θ δ, we can consruc (ωθ(s)) T s accordng o he followng formula recursvely: 1 f s = ω θ (s) = s 1 [ δ δ θ (s 1 τ) τ δθ(s τ)] τ ω θ (τ) f s> (S6) δ Noe ha f δ>max max θ (τ+1) τ, hen ω θ δ θ (τ) (s) > 0foranys and T.Wealsoknow ha δ θ (τ + 1) δ θ (τ) = δ (τ + 1) I θ δ (τ) = I θ δ (τ) δ (τ + 1) δ (τ) δ (τ) I θ δ (τ) max I θ δ (τ) δ (τ + 1) δ (τ) I θ max I θ δ (τ + 1) δ (τ) I θ max δ max I θ δ I θ I θ δ Therefore, max max θ (τ+1) τ max δ θ (τ) I θ δ. Hence, by assumng δ>max I θ δ, ωθ(s) > 0 for any s and T. The res of he proof s he same as n Lemma S2. Q.E.D. Thus, for any socal dscoun facor δ>max θ Θ max I θ δ, we can fnd (ωθ(s)) T θ Θ s such ha ω θ (s)u θ (p) = s δ τ u θ (p τ ) for each θ Θ. Consder any posve numbers (λ θ ) Θ such ha Θ θ=1 θ=1 λθ = 1. Togeher wh he weghs (ω θ(s)) T θ Θ s we have found above, he planner s uly funcon becomes U (p) = λ θ ω θ (s)u θ (p) = s λ θ δ τ u θ (p τ ) θ Θ θ Θ = δ (S7) τ λ θ u θ (p τ ) = δ τ u(p τ ) θ Θ n whch u(p τ ) = θ Θ λθ u θ (p τ ) can be any src convex combnaon of (u θ ) θ Θ. Las, we back ou he weghs (ω ( s)) T N s and show ha he planner has an EDU funcon, s I-nergeneraonally Pareo, and s I-srongly non-dcaoral under hese
SOCIAL DISCOUNTING AND INTERGENERATIONAL PARETO 9 weghs. We consruc (ω ( s)) T N s accordng o he followng formula: Then, 0 f ( s) / I ω ( s) = 1 λ θ ωθ(s) > 0 f( s) I I θ s N ω ( s)u s (p) = =1 = = = = N ω ( s) δ (τ s)u (p τ ) =1 θ Θ Is θ τ=s 1 λ θ I θ θ Θ s ωθ(s) δ (τ s)u (p τ ) τ=s Is θ τ=s λ θ ω θ (s) 1 I θ s δ (τ s)u (p τ ) λ θ ω θ (s) δ θ (τ s s)uθ (p τ ) θ Θ τ=s λ θ ω θ (s)u θ (p) = U s (p) = θ Θ δ τ u(p τ ) The frs equaly follows from he defnon of U s. The second equaly follows he consrucon of (ω ( s)) T N s. The fourh equaly follows he consrucon of δ θ s ( ). The ffh equaly follows from he defnon of U θ s. The las wo equales follow equaon (S7). Par II We prove by conradcon. Suppose here exss an I-nergeneraonally Pareo planner wh socal dscoun facor δ<mn I δ.byi-nergeneraonally Pareo, here exss a fne sequence of nonnegave weghs (ω ( s)) T N s such ha he followng equaly holds: δ τ u(p τ ) = τ I s ω ( s)δ (τ s)u (p τ ) (S8) for each T and τ. Combnng equaon (S8) wh he normalzaon assumpon, δ τ = τ ω ( s)δ (τ s) ω ( s)δ (τ s) I s I (S9) for each T and τ. We assume ha arg mn I δ ={ }. The proof can be easly exended o he case wh mulple mnma. The followng wo clams mus hold: 1. I; ha s, here exss T such ha I. 2. There exss T 1 such ha for any τ max{t 1 }, δ (τ ) δ (τ ) for any I.
10 T. FENG AND S. KE Consder he perod- planner. Le = n equaon (S9), and suppose τ max{t 1 }.Wehave δ τ = τ ( ω ( s)δ (τ s) ω ) ( δ ) τ I s I I ω ( ) δ ( τ ) δ ( τ ) However, we know ha δ<δ. Then, here exss T 2 such ha for any τ T 2, δ< τ δ. Therefore, f T max{t 1 T 2 }, here mus be a conradcon. Q.E.D. Noe ha we assume co({u } I ) = co({u θ } Θ θ=1 ) for any T. Ths s because we wan co({u } I ) o reman consan across o rule ou he example we dscuss before he heorem, and we wan o assume ha here s no redundan ype. The heorem seems dfferen from our prevous resuls ha have only one cuoff for he socal dscoun facor, bu n fac has a one-cuoff verson ha s smlar o our prevous posve resuls. However, he expresson of he cuoff wll become raher complcaed. 2 The curren verson s easer o undersand, and clearly shows ha f he socal dscoun facor s hgher han he hghes long-run dscoun facor among ndvduals who are no gnored n some generaon, hen we know ha he planner s nergeneraonally Pareo and srongly non-dcaoral. Agan, hs s no he only way o esablsh posve resuls. If he planner s nsananeous uly funcon s allowed o vary n a general way by akng he form of u ( τ), hen he addonal assumpon we need can be weaker. S1.3. Ularansm and Long-Run Socal Dscounng The man queson of hs paper s, f a planner has an EDU funcon, under wha condons s she nergeneraonally Pareo/ularan and srongly non-dcaoral? The fac ha an nergeneraonally Pareo/ularan planner has an EDU funcon ceranly mposes resrcons on how he planner may aggregae ndvdual preferences. On he one hand, economss ofen assume ha a planner has an EDU funcon, and here are many reasons o beleve ha hs s normavely appealng. Therefore, undersandng he answer o our man queson s mporan. On he oher hand, here are oher ways o examne he planner s aggregaon problem. For example, somemes economss may beleve ha he planner s uly funcon should be equal o he smple average of ndvduals dscounng uly funcons. However, because s unlkely ha he planner s dscoun funcon s exponenal n hs case, a choce abou wha o assume for he planner mus be made. A naural queson arses: If we now wan o allow he planner o aggregae ndvdual preferences n a flexble way n oher words, we only requre ha he planner be nergeneraonally Pareo/ularan and srongly non-dcaoral and do no requre ha her uly funcon be an EDU funcon wha nsgh from our man fndngs remans rue? The followng resul shows ha under hs dfferen requremen, he planner s dscoun 2 The cuoff for he socal dscoun facor n he one-cuoff verson should ake he maxmum across ypes and perods, and hen for each ype n each perod, ake he mnmal ndvdual long-run dscoun facor across all ndvduals who have he desred ype and are no gnored n ha perod.
SOCIAL DISCOUNTING AND INTERGENERATIONAL PARETO 11 facor should sll be hgher han he mos paen ndvdual s long-run dscoun facor. The resul assumes ha T =+. Some noaons and defnons for he case wh T =+ can be found n Secon A.8. THEOREM S4: Suppose T =+, eachgeneraon- ndvdual s dscounng uly funcon has an nsananeous uly funcon u and a dscoun funcon δ such ha (A2) and (A3) hold, and he planner s uly funcon n any perod T s U = δ (τ )u (p τ τ) for some dscoun funcon δ and (normalzed) nsananeous uly funcon (u ( τ)) τ such ha δ δ = lm (τ+1) τ δ = lm τ (τ) τ δ (τ) exss. If he planner s nergeneraonally ularan and srongly non-dcaoral, δ max δ. PROOF: SnceU = N =1 ω ( s)u s, we know ha δ (τ )u (p τ τ)= N τ ω ( s)δ (τ s)u (p τ ) =1 (S10) for any T and τ. Lep τ = x n equaon (S10). We have δ (τ ) = N τ ω ( s)δ (τ s) ( ω ( )δ (τ ) ω ) δ (τ ) =1 (S11) n whch := arg max δ.leτ n (S11) go o nfny. We have δ max δ. Thus, f he planner s nergeneraonally Pareo and srongly non-dcaoral, her longrun dscoun facor should agan be hgher han he mos paen ndvdual s long-run dscoun facor. Q.E.D. S2. AN ALTERNATIVE INTERPRETATION OF INTERGENERATIONAL PARETO AND A RESULT WITH QUASI-HYPERBOLIC DISCOUNTING We say ha he generaon- ndvdual has a quas-hyperbolc dscounng uly (QHDU) funcon f hs dscoun funcon sasfes { 1 fτ = 0 δ (τ) = β δ τ f τ {1 T 1} for some β (0 1] and δ > 0. In he leraure of me nconssency, economss somemes gnore he β parameer and use an EDU funcon wh a dscoun facor δ as he welfare creron of ndvdual who has a QHDU funcon. The nuon s ha because β s he cause of me nconssency, β should no ener he welfare creron. We show how our analyss provdes some foundaon for hs pracce. 3 Consder Corollary 1. If we nerpre he generaon-( + 1) ndvdual n our model as he fuure self of he generaon- ndvdual, Corollary 1 provdes some foundaon for he use of hs welfare creron. Assume ha ndvdual s he only ndvdual (N = 1) and has a quas-hyperbolc dscoun funcon. Accordng o Corollary 1, we mmedaely know ha any EDU funcon wh a dscoun facor ha s (srcly) greaer han δ s a 3 Recen papers by Drugeon and Wgnolle (2017) and Galper and Srulovc (2017) nroduce resuls smlar o he one we presen below.
12 T. FENG AND S. KE welfare creron ha s conssen wh nergeneraonal Pareo; ha s, f ndvdual n every perod agrees ha one consumpon sequence s beer han anoher, he welfare creron says ha he uly of he former s greaer han he laer. The followng resul s sronger han Corollary 1. I shows ha δ s ndeed he smalles dscoun facor such ha he correspondng EDU funcon s conssen wh nergeneraonal Pareo. PROPOSITION S1: Suppose each generaon- ndvdual has a QHDU funcon wh an nsananeous uly funcon u, β (0 1), and δ (0 1). Then, 1. for each δ mn δ, he planner s nergeneraonally Pareo and srongly nondcaoral; 2. for each δ<mn δ, here exss some T > 0 such ha f T T, he planner s no nergeneraonally Pareo. PROOF: The second par follows from Theorem 1. We only prove he frs par. LEMMA S4: Assume ha N ={}. Suppose ndvdual has a QHDU funcon wh parameers β (0 1), δ (0 1), and u. Then, here exss a cuoff δ(t) for each T such ha he planner s nergeneraonally Pareo and srongly non-dcaoral f and only f δ>δ(t). In addon, δ(t) s (srcly) ncreasng wh a lm δ. PROOF: The planner s nergeneraonally Pareo and srongly non-dcaoral f and only f here exss a fne sequence of posve weghs (ω ( s)) T s such ha he followng equaon holds: τ 1 ω ( τ)u(p τ ) + ω (s)β δ τ s u(p τ ) = δ τ u(p τ ) (S12) for any T and τ. We can solve (ω ( s)) T s from (S12) as follows: 1 f m = 0 ω ( + m) = δ m β m (1 β ) h δ h 1 β δm h f 1 m T h=1 (S13) Noe ha ω ( ) = 1 > 0, and he planner s nergeneraonally Pareo and srongly nondcaoral f and only f (ω ( + m)) T 1 m T s posve. We can rewre he second equaon of (S13) asω ( + m) = F m (δ β δ ),nwhchf s a degree-m polynomal of a sngle ndeermnae δ wh parameers β, δ.defne S(β δ T):= { δ R + : F m (δ β δ )>0forany1 m T 1 } Therefore, he planner s preference s nergeneraonally Pareo and srongly nondcaoral f and only f δ S(β δ T). We wan o show ha S(β δ T) s an nerval ha (srcly) shrnks o [δ + ) as T ncreases. Frs, we prove ha here exss a unque roo/cuoff x m (0 δ ] for F m (δ β δ ) such ha F m (x m β δ ) = 0, F m (δ β δ )<0forδ<x m,andf m (δ β δ )>0forδ>x m. We know ha F m (0 β δ ) = (1 β ) m 1 δ m < 0, F m (δ β δ ) = (1 β ) m δ m > 0, and F m s connuous. Therefore, he exsence of x m s guaraneed by Bolzano s heorem.
SOCIAL DISCOUNTING AND INTERGENERATIONAL PARETO 13 Also noe ha he funcon G m (δ β δ ) := δ m F m (δ β δ ) has he same roo as F m (δ β δ ),andg m (δ β δ ) s (srcly) ncreasng n δ because dg m (δ) dδ = β 1 β m k=1 k (1 β δ ) k δ k+1 By Rolle s heorem, here canno be more han one roo. Hence, he unqueness s proved. Second, we prove ha he cuoff sequence (x m ) m s (srcly) ncreasng and converges o δ. Nong ha G m+1 (δ β δ ) G m (δ β δ ) = β [ ] m+1 (1 β )δ < 0 1 β δ we have G m+1 (x m β δ ) G m (x m β δ )<0. By he defnon of (x m ) m, G m (x m β δ ) = G m+1 (x m+1 β δ ) = 0. Therefore, G m+1 (x m β δ )<G m (x m β δ ) = G m+1 (x m+1 β δ ). We also know ha G m (δ β δ ) s (srcly) ncreasng. Hence, x m+1 >x m. Now ha (x m ) m s bounded and (srcly) ncreasng, he convergence follows from he monoone convergence heorem. The only remanng par s o prove ha he lm of he cuoff sequence s δ. Suppose lm m x m = x. Then, x m <xfor all m>1. Snce G m (δ β δ ) s (srcly) ncreasng, we have > 0 G m (x m β δ )<G m (x β δ ) 0 < 1 β m (1 β ) h δ h 1 β x h h=1 β m (1 β ) h δ h 1 β x h < 1 h=1 m [ ] h (1 β )δ < 1 β x h=1 for any m>1. Gven ha (1 β )δ > 0, we mus have (1 β )δ < 1; oherwse, m x x m ncreases. Now, le m n (S14) go o nfny. We have + [ ] h (1 β )δ 1 β β [ (1 β )δ h=1 x (S14) ] h dverges as h=1 x (1 β )δ x δ x β 1 1 (1 β )δ x 1 β β (S15) In addon, snce x m <δ for all m, wehavex δ. Therefore, x = δ. Q.E.D. Lemma S4 saes ha for any fne T,neachperod, he planner can aggregae each ndvdual from he h generaon o he T h generaon so ha he aggregaed uly funcon s an EDU funcon wh a dscoun facor ha s slghly below δ. Then, we can
14 T. FENG AND S. KE apply he f par of Proposon 2 for N exponenal dscounng ndvduals, and oban a socal dscoun facor δ mn δ. Q.E.D. When T =+, we can assume n Proposon 4 ha ndvduals have QHDU funcons and oban a smlar resul. S3. THE CASE WITH BACKWARD DISCOUNTING The resul we nroduce below shows ha f ndvduals exponenally forward and backward dscoun consumpon, our man resuls connue o hold. Before proceedng, should be noed ha backward dscounng has no revealed-preference foundaon. Whenever we observe an ndvdual choosng, he pas s sunk; here are no choces (ye) ha allow he ndvdual o aler he pas. Therefore, we do no know how ndvduals hnk abou he pas from acual choce daa. However, economss have consdered he possbly ha ndvduals backward dscoun (see Sroz (1955), Capln and Leahy (2004), andray, Vellod, and Wang (2017)). Below, we analyze our aggregaon problem wh exponenal dscounng ndvduals who backward dscoun. Insead of assumng ha U (p) does no depend on pas consumpon, we assume ha he generaon- ndvdual dscouns boh pas and fuure by he same dscounng facor δ. DEFINITION S3: The generaon- ndvdual has an exponenal forward and backward dscounng uly funcon f hs uly funcon has he followng form: U (p) = τ=1 δ τ u (p τ ) (S16) n whch he dscoun facor δ (0 1), andu s he ndvdual s nsananeous uly funcon. Noe ha he negave resul, obvously, would connue o hold f we had assumed ha each generaon- ndvdual s uly funcon was U (p) = τ=1 δ τ u (p τ ) In ha case, he ndvdual s offsprng has exacly he same preference as he ndvdual. Ths s problemac, however, because he generaon-2 ndvdual wll value perod-1 consumpon even more han hs own perod-2 consumpon. The resul below demonsraes ha he assumpon ha he planner has an EDU funcon and nergeneraonal Pareo are compable when ndvduals exponenally forward and backward dscoun consumpon. The ypcal negave resul n he leraure only consders he planner s aggregaon problem n perod 1. The followng resul also focuses on he perod-1 aggregaon problem o hghlgh he dfference. PROPOSITION S2: Suppose each generaon- ndvdual has an exponenal forward and backward dscounng uly funcon wh dscoun facor δ and nsananeous uly funcon u such ha δ := max δ < 1. Le he planner s nsananeous uly funcon u be an arbrary src convex combnaon of (u ) N. Then, for each δ ( δ δ 1 ), he planner n perod 1 s nergeneraonally Pareo and srongly non-dcaoral.
SOCIAL DISCOUNTING AND INTERGENERATIONAL PARETO 15 PROOF: To prove he proposon, we consder he one-ndvdual case frs. LEMMA S5: Assume ha N ={}. Suppose each generaon- ndvdual has an exponenal forward and backward dscounng uly funcon wh dscoun facor δ (0 1) and nsananeous uly funcon u. Then, for each δ (δ δ 1 ), he planner n perod 1 s nergeneraonally Pareo and srongly non-dcaoral. PROOF: We wan o show ha for any δ (δ δ 1 ), here exss a fne sequence of posve weghs ω = (ω( 1) ω( 2) ω( T )) such ha he followng equaon holds: U 1 (p) = δ τ 1 u(p τ ) = ω( s)u s (p) (S17) Pluggng n U 1 (p) and U s (p), equaon(s17) becomes τ=1 s=1 δ τ 1 u(p τ ) = τ=1 ω( s) s=1 τ=1 δ τ s u(p τ ) = τ=1 s=1 ω( s)δ s τ u(p τ ); (S18) ha s, for each τ 1, δ τ 1 = s=1 Nex, we can rewre equaon (S19) as follows: ω( s)δ s τ (S19) A ω = δ (S20) n whch δ = (1 δ δ 2 δ T 1 ) and A = 1 δ δ 2 δ T 1 δ 1 δ δ T 2 δ T 1 δ T 2 δ T 3 1 Noe ha A s nverble. In parcular, 1 δ 0 0 δ 1 + δ 2 δ 0 0 δ 1 + δ 2 δ A 1 = 1 0 2 1 δ 0 δ 1 + δ 2 δ 0 0 δ 1 + δ 2 δ 0 0 δ 1
16 T. FENG AND S. KE We have ω = A 1 δ. If we can show ha ω 0, he lemma s proved. Showng ha ω 0 s equvalen o showng ha ω( 1) = 1 δ δ>0, ω( s) = δ s 2 [ δ +(1+δ 2)δ δ δ 2 ] > 0for2 s T 1, and ω( T) = δ δ T 2 + δ T 1 > 0, whch can be verfed because δ (δ δ 1 ). Q.E.D. Lemma S5 shows ha we can aggregae each ndvdual from he h generaon o he T h generaon no an EDU funcon wh any dscoun facor δ whn (δ δ 1 ). Now we can prove Proposon S2. For any socal dscoun facor δ ( δ δ 1 ), we can fnd (ω( s)) N s 1 such ha ω( s)u s (p) = s=1 δ τ 1 u (p τ ) for each N. Consder any posve numbers (λ ) N such ha N λ = 1. Togeher wh he weghs (ω( s)) N s 1 we have found above, he planner s uly funcon becomes τ=1 U 1 (p) = = N N λ ω 1 ( s)u s (p) = δ τ 1 λ u (p τ ) =1 s=1 =1 τ=1 δ τ 1 N λ u (p τ ) = δ τ 1 u(p τ ) =1 n whch u(p τ ) = N λ u (p τ ) s an arbrary src convex combnaon of (u ) N. Q.E.D. S4. RISK RESOLUTION The man model s choce doman s (X) T ; ha s, n each perod, here s a loery/probably measure over X. In many dynamc economc models wh uncerany, uncerany resolves over me. Below, we dscuss wha may change f we le uncerany resolve over me, mananng our assumpons abou ndvduals and he planner s uly funcons. For smplcy, assume ha T = 2andN = 1. In perod 2, he choce objec s agan a loery over X. Somemes, wll be called a perod-2 loery. To dsngush beween choce objecs n he man paper and n hs secon, here we call X oucomes and perod-1 choce objecs dynamc loeres. A dynamc loery s a loery over X (X). For example, wh probably 1/2, a dynamc loery p 1 yelds a perod-1 oucome x X and a perod-2 loery q 2 (X); wh probably 1/2, p 1 yelds a perod-1 oucome x and a perod-2 loery r 2 (X). Now, he se of dynamc loeres s (X (X)), raher han (X) 2. 4 However, (X) 2 can be vewed as a subse of (X (X)) ha consss of all dynamc loeres whose perod-2 loeres are ndependen of (he realzaon of) perod-1 oucomes. The followng smple example shows n wha sense, n perod 1, he planner s aggregaon problem under (X (X)) s he same as under (X) 2. Connue our example of p 1, q 2, r 2 above. Le q 2 be a loery ha yelds y y X wh equal probably. Le r 2 be 4 For any merc space Y,le (Y ) denoe he se of Borel probably measures on Y.Weendow (X) wh he Prohorov merc and X (X) wh produc opology.
SOCIAL DISCOUNTING AND INTERGENERATIONAL PARETO 17 a degenerae loery ha yelds z X. Frs, consder he generaon-1 ndvdual. A naural way o exend our perod-1 ndvdual uly funcon on (X) 2 o he new doman (X (X)) s as follows: V 1 ( p 1 ) = 1 ( v(x 1) + δv(q2 2) ) + 1 ( v x 2 2( 1 ) + δv(r 2 2) ) = 1 ( ( 1 v(x 1) + δ 2 2 v(y 2) + 1 2 v( y 2 ) )) + 1 ( v x 2( 1 ) + δv(z 2) ) n whch δ s he ndvdual dscoun facor and v( τ)s he perod-τ ndvdual nsananeous uly funcon. Noe ha he above equaon can be rewren as ( 1 V 1 ( p 1 ) = 2 v(x 1) + 1 2 v( x 1 ) ) ( 1 + δ 4 v(y 2) + 1 4 v( y 2 ) + 1 ) v(z 2) ; 2 ha s, he uly of p 1 (X (X)) s equal o he followng dynamc loery: In perod 1, he ndvdual consumes a 50 50 loery beween x and x, and n perod 2, he consumes a loery ha yelds y wh probably 1/4, y wh probably 1/4, and z wh probably 1/2. I s no dffcul o see he logc behnd hs observaon. In general, gven any p 1 (X (X)), we compue he margnal probably dsrbuon of perod-1 oucomes and call p 1 (X), and compue he margnal probably dsrbuon of perod-2 oucomes and call p 2 (X). Then, (p 1 p 2 ) s a dynamc loery whose perod-2 loeres are ndependen of perod-1 oucomes. I mus be he case ha V 1 ( p 1 ) = V 1 ((p 1 p 2 )), because V 1 s a me-addvely separable expeced uly funcon. Second, consder he generaon-2 ndvdual. Because we are examnng he perod-1 planner s problem, whch means he dynamc loery s rsk has no resolved, how does he planner evaluae he second generaon s uly of p 1? Arguably, V 2 ( p 1 ) = 1 ( 1 2 2 v(y 2) + 1 2 v( y 2 ) ) + 1 v(z 2) (S21) 2 seems o be a reasonable evaluaon wh probably 1/2, he second generaon s uly wll be 1 2 v(y 2) + 1 2 v(y 2), and wh probably 1/2, he second generaon s uly wll be v(z 2).Now,agan, ( V 2 ( p 1 ) = V 2 (p1 p 2 ) ) = 1 4 v(y 2) + 1 4 v( y 2 ) + 1 v(z 2) 2 Therefore, p 1 and (p 1 p 2 ) are equvalen for he planner n perod 1. The planner s perod-1 aggregaon problem under (X (X)) s he same as under (X) 2 here s a bjecon beween me-addvely separable expeced uly funcons defned on he doman wh and whou correlaon. As long as he perod-1 planner uses he same ularan weghs o aggregae ndvdual uly funcons, he planner s preference wll be he same n boh cases. Move on o perod 2 and connue our prevous example of p 1 and (p 1 p 2 ). Wh eher (X (X)) or (X) 2, he second generaon s uly funcon s defned on (X), because ndvduals do no care abou pas consumpon. Therefore, here s agan a (rval) bjecon beween generaon-2 ndvdual uly funcons defned on he doman wh
18 T. FENG AND S. KE and whou correlaon. The planner s perod-2 preference wll be dencal n boh cases as long as she uses he same ularan weghs for ndvduals. The analyss above can be exended o he case wh more perods and more ndvduals. In hs sense, focusng on consumpon sequences (X) T whou modelng how uncerany resolves over me s whou loss of generaly. However, should be noed ha wh p 1, he perod-2 loery s eher q 2 or r 2.Wh (p 1 p 2 ), no maer wha he frs generaon consumes, he perod-2 loery s p 2.Therefore, here wll be some ex pos dfference beween p 1 and (p 1 p 2 ) abou whch generaon consumes wha. However, hs dfference should no affec he perod-2 planner s aggregaon problem. Anoher ssue o be noed s ha n eher he case wh correlaon or he case whou, we only sudy wha he planner s objecve should be f she aggregaes ndvduals preferences. Ths exercse does no requre us o consder, for example, feasbly consrans. If he planner s problem s o maxmze some objecve under ceran consrans, correlaon may be mporan n he feasbly consrans. For example, f here s a echnologcal advancemen n he frs perod, we can ancpae a larger feasble se of consumpon n he fuure. Ths requres correlaon n he consrans. REFERENCES CAPLIN, A., AND J. LEAHY (2004): The Socal Dscoun Rae, Journal of Polcal Economy, 112 (6), 1257 1268. [14] DHILLON, A.,AND J.-F. MERTENS (1999): Relave Ularansm, Economerca, 67 (3), 471 498. [6] DRUGEON, J.-P., AND B. WIGNIOLLE (2017): On Tme-Conssen Collecve Choce Wh Heerogeneous Quas-Hyperbolc Dscounng, Workng Paper, Pars School of Economcs. [11] FISHBURN, P. (1984): On Harsany s Ularan Cardnal Welfare Theorem, Theory and Decson, 17(1), 21 28. [1] GALPERTI, S., AND B. STRULOVICI (2017): A Theory of Inergeneraonal Alrusm, Economerca, 85(4), 1175 1218. [11] HARSANYI, J. (1955): Cardnal Welfare, Indvdualsc Ehcs, and Inerpersonal Comparsons of Uly, Journal of Polcal Economy, 63 (4), 309 321. [1] KARNI, E. (1998): Imparaly: Defnon and Represenaon, Economerca, 66 (6), 1405 1415. [6] RAY, D., N. VELLODI, AND R. WANG (2017): Backward Dscounng, Workng Paper, New York Unversy. [14] SEGAL, U. (2000): Le s Agree Tha All Dcaorshps Are Equally Bad, Journal of Polcal Economy, 108 (3), 569 589. [6] STROTZ, R. (1955): Myopa and Inconssency n Dynamc Uly Maxmzaon, Revew of Economc Sudes, 23 (3), 165 180. [14] Co-edor Izhak Glboa handled hs manuscrp. Manuscrp receved 13 January, 2017; fnal verson acceped 17 May, 2018; avalable onlne 25 May, 2018.