Redistributing the Gains from Trade through Nonlinear. Lumpsum Transfers


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1 Redistributing the Gins from Trde through Nonliner Lumpsum Trnsfers Ysukzu Ichino Fculty of Economics, Konn University April 21, 214 Abstrct I exmine lumpsum trnsfer rules to redistribute the gins from free trde under incomplete informtion, where the government does not know the chrcteristics of individuls. By considering trnsfer rules tht mke the mount of trnsfer dependent on trnsctions of individuls under utrky, I show tht there is no incentivecomptible nd budgetblncing trnsfer rule to redistribute the gins from free trde successfully, if the discount fctor is sufficiently lrge. Also, the pper exmines whether the mount of trnsfer should be dependent on trnsctions under utrky or trnsctions under free trde, nd shows tht the former is better when the discount fctor is smll or the government hs enough budget for redistribution. *Address: Okmoto, Higshindku Kobe , Jpn. Emil: 1
2 1 Introduction It is well known tht free trde cretes winners nd losers. However, with n pproprite lumpsum trnsfer, Preto improvement cn be chieved s country moves from utrky to free trde, s shown by Kemp (1962), Kemp nd Wn (1972), nd Grndmont nd McFdden (1972), mong others. 1 Essentilly, the ide of Pretoimproving lumpsum trnsfers is to let wht ech individul consumed under utrky ffordble under free trde, too. 2 Lter, Dixit nd Normn (1986) pointed out prcticl difficulties of implementing such lumpsum trnsfers. In implementing lumpsum trnsfer to mke everyone better off fter free trde, the government hs to figure out not only who re txed nd who re subsidized, but lso how much is txed or subsidized for ech individul. To do so, the government needs to collect informtion bout the chrcteristics of individuls. Moreover, when sked by the government, ech individul my not revel his chrcteristics tht the government is going to use to clculte the mount of trnsfer. As Dixit nd Normn (1986) write, individuls hve n incentive to mnipulte their behvior so s to misled the plnner bout these chrcteristics nd secure lrger net trnsfer. How do they mnipulte their behvior? Does tht mnipultion mtter? 3 Wong (1997) studied these questions in generl setting, nd rgued tht mnipultion does not mtter if the utrkic price does not chnge by such mnipultion. Recently, however, exmining the sme question in simpler setting of twogoods exchnge economy with qusiliner utility, Ichino (212) showed tht due to consumers mnipultion the utrkic price chnges to the directions unfvorble to those who lose by free trde. As result, lumpsum trnsfer rule tht llows everyone to consume under free trde wht they used to consume under utrky becomes ineffective in redistributing the gins from free trde. Moreover, in Ichino (212) it is shown tht ny trnsfer rules tht re linerly 1 These studies work on models of perfect competition. Under imperfect competition such s oligopoly, whether there re gins from free trde is less unmbiguous. See, for exmple, Dong nd Yun (21) nd Kemp (21). 2 Using such lumpsum trnsfers hs been one of the stndrd techniques to mke welfre comprison. For textbook tretment, see Wong (1995; pp ). 3 Che nd Fcchini (27) studied this kind of mnipultion in context of mrket liberliztion, while we study the mnipultion in the context of trde liberliztion. 2
3 dependent on utrkic consumption cnnot chieve Preto gins from trde. In this pper, we consider more generl clss of lumpsum trnsfer rules, including trnsfer rules nonlinerly dependent on utrkic consumption, nd explore if there exist ny trnsfer rules tht cn successfully redistribute the gins from free trde to everyone. For this purpose, we pply contrcttheory technique used by Feenstr nd Lewis (1991) (herefter, we refer it s FL (1991)). As explined bove, since the government does not hve complete informtion on individuls chrcteristics, it hs to sk the individuls to revel such informtion in order to clculte the mount of trnsfer. Then, under such circumstnce, individuls hve n incentive to mnipulte their behvior, or misreport their chrcteristics if doing so increses the mount of trnsfer to be received. To prevent mnipultion or misreporting, the government hs to design trnsfer rule tht induces truth telling. We therefore formulte welfremximiztion problem in which trnsfer rules should stisfy the incentive comptible constrint s well s the government budget constrint nd the everyoneismdebetteroff constrint. By solving the welfre mximiztion problem, we firstpointoutthtiftrnsferrules re to relte the mount of trnsfer to the mount of consumption under utrky, then the utrkic consumption should be distorted to prevent mnipultion or misreporting of the individuls. Then, we show tht there is no incentivecomptible nd budgetblncing trnsfer rules to chieve Preto gins from free trde if the discount fctor is sufficiently lrge. Intuitively, this is becuse the mount of subsidies given to losers of free trde hs to be lrge enough to compenste him not only for his loss from free trde but lso for the loss from distorted consumption under utrky. At the sme time, the mount of txes collected from the giners of free trde is limited becuse of distorted consumption. Also, we compre the outcome of our model with the outcome of FL (1991). As mentioned bove, in this pper, we consider trnsfer rules dependent on the consumption under utrky. On the other hnd, in FL (1991), trnsfer rules re dependent on the consumption under free trde. By compring the resulting welfre of our model nd FL(1991) s, we show tht the former gives higher welfre thn the ltter when the discount fctor is smll or the government hs enough budget for redistribution. 3
4 2 The Model We consider n exchnge economy with two goods, good x nd good y, where good y is the numerire. In this economy, there is continuum of individuls with the popultion normlized to one. The individuls re ssumed identicl, except the endowment of good x. Withω denoting the endowment of good x, the individuls in the economy re indexed by ω. For simplicity, ssume tht ω is uniformly distributed over the rnge of [, 1]. The utility function of n individul hving ω units of the endowment of good x is given by u(x + ω)+y +ȳ where x nd y denote excess demnd of good x nd y respectively, nd ȳ is the endowment of good y (which is the sme for everyone). To hve explicit solution, we ssume tht the subutility function u( ) is qudrtic: u (x + ω) =k + (x + ω) b 2 (x + ω)2. where k, >, b>. In this model, we hve two periods: in period, the economy is under utrky; in period 1, the economy is under free trde. The utility mximiztion problem of n individul is given by mx U (x, p, ω) =u(x + ω) px +ȳ x since the budget constrint px + y is lwys binding. The firstorder condition is u (x + ω) =p, from which the excess demnd function x (p, ω) =( p) /b ω is derived. Then, the utrkic equilibrium price p is determined from the mrketclering condition R 1 x (p, ω) dω =.SinceR 1 ωdω =1/2, solving this condition gives p =(2 b) /2. Let x (ω) x (p, ω) denote the excess demnd (or, we cll it trnsction ) of individul ω in the utrkic equilibrium, nd define the indirect utility of individul ω in the utrkic equilibrium U (ω) U (x (ω),p, ω). In period 1, when the economy llows free trde, the price of good x is equl to the 4
5 world price p (we ssume tht the economy is smll). Then, trnsction of individul ω in the freetrde equilibrium is given by x (ω) ( p ) /b ω, nd the indirect utility in the freetrde equilibrium by U (ω) U (x (ω),p, ω). We ssume tht p <p :theworld price of good x is smller thn the utrky price. 4 By this ssumption, x (ω) >x (ω). In ggregte, there re gins from free trde 5 : = = U (ω) dω U (ω) dω [U (x,p ) U (x,p )] dω + [U (x,p ) U (x,p )] dω +(p p ) [U (x,p ) U (x,p )] dω x dω > since the first term is positive (given p, the utilitymximizing trnsction is x )ndthe second term is zero becuse R 1 x dω =. However, the gins from trde do not fll eqully to everyone. Buyers of good x under utrky gin from free trde: U (ω) U (ω) =[U (x,p ) U (x,p )] + [U (x,p ) U (x,p )] =[U (x,p ) U (x,p )] + (p p ) x > since x >. On the other hnd, the sellers of good x under free trde lose from free trde: U (ω) U (ω) =[U (x,p ) U (x,p )] + [U (x,p ) U (x,p )] =(p p ) x +[U (x,p ) U (x,p )] <. provided tht x <. 4 For the prmeters of the utility function nd the world price, we impose the following restriction: (i) 2 >b. Thisistomketheutrkicpricepositive. (ii)p < (2 b) /2. This is from the ssumption of p <p. (iii) ( p ) /b < 1. This is to mke some individuls still sellers of good x under free trde. 5 Here, to simplify the nottion, we use expressions such s U (x,p ) to men U (x (ω),p, ω), by suppressing ω. Herefter, if there is likely no confusion, we write U (x, p) to denote U (x (ω),p,ω). 5
6 3 Lumpsum Trnsfers under incomplete informtion In order to redistribute the gins from trde, the government cn give the following lumpsum trnsfer z (ω) = (p p ) x (ω) to individul ω in period 1 (note tht z (ω) > is subsidy nd z (ω) < is tx). By this trnsfer, the budget constrint of individul ω in period 1 becomes p x + y (p p ) x (ω). It is esy to show tht this trnsfer mkes wht ech individul consumed under utrky ffordble under free trde s well: p x + y (p p ) x (ω) p x + y p x (ω) p x (ω) Therefore, everyone gets better off by free trde. Tht is, U (ω) +z (ω) >U (ω) for ll ω. In ddition, with this lumpsum trnsfer, the government budget is blnced since R 1 z (ω) dω = (p p ) R 1 x (ω) dω =. The discussion bove tht the free trde policy with the lumpsum trnsfer z (ω) mkes everyone better off, however, depends on n implicit ssumption tht the government cn figure out wht z (ω) s re. For the government to clculte z (ω), ithsto know ech individul s endowment. But this is not likely the cse. Without knowing the endowment of ech individul, the government is not ble to clculte x (ω), nd thus cnnot figure out z (ω). Then, the only wy for the government to find the mount of lumpsum trnsfer for ech individul is to observe his trnsction in period (under utrky). However, if ech individul knows tht the government is going to observe his trnsction in period to determine the mount of trnsfer for him, then he my wnt to chnge, or mnipulte his period trnsction in order to get lrger trnsfer in period 1. So, here is the question: Is it possible to chieve Preto gins from free trde through lumpsum trnsfers even when the government does not hve complete informtion on the chrcteristics of the individuls nd thus there cn be mnipultion of trnsction? To nswer the question formlly, let us now clrify the setting. As mentioned erlier, in the following nlysis we suppose tht the government does not know endowment of ech individul; the government just observes trnsction of ech individul in period (under utrky). In period, ll individuls know tht free trde will be llowed in period 6
7 1. They lso know tht lumpsum trnsfer z (x) will be given in period 1. Here, note tht the trnsfer depends on x, the observed trnsction of n individul in period. In this setting, the utility mximiztion problem of n individul should be n intertemporl one, nmely, mx x,x 1 U (x, p, ω)+δz (x)+δu x 1,p, ω (1) where x nd x 1 re respectively the period nd the period1 trnsction of good x, p is the previling price in period, nd δ is discount fctor δ 1 (we ssume everyone hs the sme discount fctor). By inspection of the mximiztion problem (1), it is obvious tht x 1 = x : the trnsction under free trde is the sme s the one we sw in Section 2. But the trnsction under utrky is now different from x becuse of the dependence of the lumpsum trnsfer on x. Thus, the utrkic equilibrium price, too, cn be different from p. We sy tht lumpsum trnsfer rule z (x) cn chieve Preto gins from free trde if the following inequlity holds for ll ω: mx x [U (x, p, ω)+δz (x)] + δu (ω) (1 + δ) U (ω). Ichino (212) exmined the cse of z (x) = (p p ) x, the lumpsum trnsfer rule to mke sure tht ll individuls cn consume under free trde wht they used to consume under utrky, nd found tht due to trnsction mnipultion Preto gins from trde cnnot be chieved by this trnsfer rule. Moreover, in Ichino (212) it is shown tht ny trnsfer rules tht re liner in x cnnot chieve Preto gins from trde. In this pper, we consider more generl clss of lumpsum trnsfers, including trnsfer rules nonlinerly dependent on x, nd explore if there re ny trnsfer rules tht cn successfully distribute the gins from free trde to everyone. For this purpose, we pply contrcttheory technique used by FL (1991), where the endowment of good x is tken s type of n individul not known to the government, nd the government is going to solve welfremximiztion problem to derive menu {x (ω),z(ω)}, i.e.,pirof trnsction nd trnsfer. 7
8 4 WelfreMximizing trnsfers 4.1 Welfre mximiztion problem Given menu {x (ω),z(ω)} for ω 1, if n individul whose endowment is ω picks prticulr pir {x (ω ),z(ω )}, his intertemporl utility is U x ω,p,ω + δz ω + δu (ω). By the reveltion principle, without loss of generlity we cn confine the set of pirs {x (ω),z(ω)} to be considered to those stisfying the following incentive comptibility (IC) constrints: U (x (ω),p,ω)+δz (ω) U x ω,p,ω + δz ω for ll ω, ω. (2) In ddition to the IC constrints, trnsctiontrnsfer menu {x (ω),z(ω)} hs to be designed so s to mke everyone better off by free trde. Tht is, the inequlity U (x (ω),p,ω)+δz (ω) U (ω)+δ (U (ω) U (ω)) for ll ω (3) hs to be stisfied. This is nlogous to the individul rtionlity (IR) constrint in stndrd contrcttheory model. So, in the pper we refer this s the IR constrint. Third, trnsfers should stisfy the government s budget (GB) constrint: z (ω) dω B (4) where B is some mount of budget the government cn use for trnsfer policy. Finlly, the mrketclering condition x (ω) dω = (5) determines the period equilibrium price p. 8
9 The welfre mximiztion problem of the government is set s follows: mx x(ω),z(ω) [U (x (ω),p,ω)+δz (ω)] dω + δb s.t. (2), (3), (4), nd (5), δz (ω) dω (6) where the objective function is the sum of the individul s utility nd the government surplus. Problem (6) is nturl extension of Ichino (212). However, here we do not solve problem (6). Insted, we solve somewht different problem defined below. mx x(ω),z(ω) [U (x (ω),p, ω)+δz (ω)] dω + δb [δz (ω) (p p ) x (ω)] dω (7) s.t. U (x (ω),p, ω)+δz (ω) U x ω,p, ω + δz ω for ll ω, ω, U (x (ω),p, ω)+δz (ω) U (ω)+δ (U (ω) U (ω)) for ll ω. δz (ω) dω (p p ) x (ω) dω δb. The welfre mximiztion problem (7) is different from problem (6) in how the period price is determined. In problem (6), the period price p is determined by the utrkic mrketclering condition R 1 x (ω) dω =. Therefore, the utrkic equilibrium price my be different from p, depending on the shpe of x (ω). On the other hnd, in problem (7), the period price is supported by the government t p, the utrkic equilibrium price under the complete informtion cse. To support the price t p, the government impose n import triff (if x (ω) > ) ornexportsubsidy(ifx (ω) < ) withthesizeof(p p ). Thtiswhywehveterm R 1 (p p ) x (ω) dω in the government s budget constrint. Simply put, in problem (6) utrky is mintined by prohibition of interntionl trde, while utrky is mintined by prohibitive triff (or export subsidy) in problem (7). Below, we present n outline of solving problem (7). Problem (6) is solved in similr fshion since it hs the sme structure s problem (7). Lter, we explin why we focus on problem (7) insted of problem (6). To mke expressions simpler, by busing nottion little bit, we define U (ω) 9
10 U (x (ω),p, ω)+δz (ω). As explined by FL (1991), the IC constrints of problem (7) is equivlent to (i) U (ω) =u (x (ω)+ω) nd (ii) x (ω) is nonincresing. By using condition (i) for the IC constrint, we cn construct the following constrined Hmiltonin by treting x (ω) nd z (ω) s control vribles nd U (ω) s stte vrible: H = [U (ω)+δb δz (ω)+(p p ) x (ω)] (8) +γ (ω) u (x (ω)+ω) +μ (ω)(u (ω) U (ω) δ (U (ω) U (ω))) +ρ (ω)(u (x (ω),p, ω)+δz (ω) U (ω)) +λ (δb δz (ω)+(p p ) x (ω)), where the second line is condition (i) of the IC constrint, nd gives the eqution of motion for U (ω) (we confirm tht condition (ii) of the IC constrint is stisfied fter hving derived solution). The third line is the IR constrint, the fourth line defines U (ω), ndthefifth line is the GB constrint. The mximum principle requires tht H x = (p p )+γ (ω) u + ρ (ω) u (x (ω)+ω) p + λ (p p )=, (9) H z = δ + ρ (ω) δ λδ =, (9b) H U = 1 μ (ω)+ρ (ω) =γ (ω), (9c) nd the trnsverslity conditions re γ () [U () U () δ (U () U ())] =, nd γ (1) [U (1) U (1) δ (U (1) U (1))] =. We let x (ω),z (ω) ª denote the solution, nd U (ω) utility level chieved by the solution. Here, we present n outline of deriving the solution. For the detil, see the ppendix. From (9b), ρ (ω) =1+λ is derived. Substituting this into (9c) gives γ (ω) =γ () 1
11 R ω μ (t) dt + λω. Then, pplying these expressions to (9), we cn derive eqution determining the trnsction schedule x (ω) u x (ω)+ω p = γ () R ω μ (t) dt + λω u, (1) 1+λ For the time being, ssume tht the IR constrint is binding only t ω =1(Lter, we explin it is ctully the cse when the government s budget B is lrge enough). Then, γ () = by the trnsverslity condition nd R ω μ (t) dt =since the IR constrint is not binding for ll ω < 1. Thus, (1) is simplified to u x (ω)+ω p = u mω where m = λ/ (1 + λ). Solvingforx, the following expression x (ω) =x n (ω) p b (m +1)ω (11) is derived. Here, we hve defined x n (ω) s solution when the IR constrint is not binding for ll ω < 1 (the superscript n for not binding ). Since the multiplier λ, x n (ω) is decresing in ω, stisfying condition (ii) of the IC constrint. Once x n (ω) is derived, from the IC constrint, nd from the IR constrint binding t ω =1,theperiod utility is determined s follows: U (ω) =U n (ω) U (1) + δ (U (1) U (1)) ω u (x n (ω)+t) dt. We cn interpret U (ω) s trget level of utility to be chieved, with trnsfer z (ω). On the other hnd, without trnsfer, the period utility of individul ω is U (x n (ω),p, ω). The trnsfer schedule is thus determined so s to fill the gp between the trget level nd the withouttrnsfer level: δz (ω) =δz n (ω) U n (ω) U (x n (ω),p, ω). (12) 11
12 Then, we cn clculte the GB constrint δz n (ω) dω (p p ) x n (ω) dω = δb, nd from this we cn determine m. This completes the solution. Now, we explin why we solved problem (7) insted of problem (6). Lemm 1 shows tht wht is chieved by solving problem (6) is chievble by problem (7). Lemm 1 Consider solution {ˆx (ω), ẑ (ω)} of problem (6) nd the welfre level chieved by the solution. There exists some z (ω) by which the sme welfre level nd the sme ˆx (ω) is implemented such tht {ˆx (ω),z(ω)} stisfies the constrints of problem (7). Proof. Consider solution {ˆx (ω), ẑ (ω)} of problem (6). In problem (6), the utility level of individul ω is equl to U (1) + δ (U (1) U (1)) R 1 ω u (ˆx (t)+t) dt. To chieve this trget level, the trnsfer schedule ẑ (ω) is clculted s follows: δẑ (ω) =U (1) + δ (U (1) U (1)) ω u (ˆx (t)+t) dt U (ˆx (ω),p,ω). With the sme ˆx (ω), the sme trget level of utility cn be chieved in problem (7) by the following trnsfer schedule δz (ω). δz (ω) =U (1) + δ (U (1) U (1)) ω u (ˆx (t)+t) dt U (ˆx (ω),p, ω). Since R 1 ˆx (ω) dω =by the mrket clering condition in problem (6), the ggregte trnsfers re the sme for ẑ (ω) nd z (ω), ndthetriff revenue in problem (7) is zero. Therefore, R 1 δẑ (ω) dω = R 1 δz (ω) dω R 1 (p p )ˆx (ω) dω. 4.2 Chrcteriztion of the solution Now we chrcterize the solution x (ω),z (ω) ª s follows, noticing tht the solution depends on the size of the government s budget B. When B is lrge nd the GB constrint is not binding, the solution is trivil. Since 12
13 m = λ/ (1 + λ) =if the GB constrint is not binding, from (11) we see tht x (ω) = ( p ) /b ω = x (ω). Thtis,thegovernmentcnchievethemostefficient outcome by letting everyone consume good x of the mount under the free trde. However, s the individuls consume x (ω)+ω when fcing p, their period utility without trnsfer is U (x (ω),p, ω), which is less thn U (ω). Since U (ω) <U (ω)+δ (U (ω) U (ω)) for lrge ω, the government hs to give positive trnsfer to those individuls to stisfy the IR constrint (see Figure 1). In prticulr, individuls with ω =1need to be compensted t lest by U (1) + δ (U (1) U (1)) U (x (1),p, 1) in order for the IR constrint to be stisfied. Suppose tht the government gives the individuls with ω =1exctly this mount of trnsfer so tht the IR constrint is just binding t ω =1. Then, the IC constrint requires tht the trget level of utility U (ω) hs slope of u (x (ω)+ω) =p sinceeveryoneconsumesx (ω) +ω. Nmely, U (ω) hs to be prllel upwrd shift of U (ω) by the mount of (1 + δ)(u (1) U (1)). By this construction, for ll ω the trget level of utility is bove the utility without trnsfer. Therefore, the government hs to give positive trnsfers to everyone to chieve the trget level of utility. The sum of the trnsfers is δz (ω) dω = U (ω) dω +(1+δ)(U (1) U (1)) U (x (ω),p, ω) dω. On the other hnd, becuse x (ω) >x (ω), in ggregte there is imports of good x so tht the government erns triff revenue R 1 (p p ) x (ω). Thus, the overll government expenditure needed to implement the trnsctiontrnsfer schedule is = δz (ω) dω (p p ) x (ω) U (ω) dω +(1+δ)(U (1) U (1)) = (1+δ)(U (1) U (1)). U (x,p ) dω (p p ) x (ω) So, when B =(1+δ)(U (1) U (1)), the government hs just enough budget to implement x (ω) with the IR constrint binding only t w =1. In other words, when the 13
14 government hs lrge enough budget such tht B (1 + δ)(u (1) U (1)), themost efficient trnsction x (ω) is implemented, with the GB constrint not binding. As B flls below (1 + δ)(u (1) U (1)), letting everyone consume x (ω)+ω is no longer comptible with the IR constrint. To stisfy the IR constrint with the smller mount of the budget vilble to the government, now the trnsfer hs to be mde smller for the individuls with low ω (since they re the ones who gin from free trde, there is room to shve the trnsfer for them). Then, with smller trnsfer for low ω, if x (ω) =x (ω) were offered, the individuls with low ω would hve n incentive to pretend tht his ω is higher thn wht ctully is in order to get lrge trnsfer. To prevent it, trnsction schedule x (ω) hs to be distorted in such wy tht consumption x (ω) +ω is no longer the sme for everyone but smller for high ω. Thisisseenby eqution (11): consumption x (ω) +ω is equl to x n (ω) +ω, which is distorted from x (ω)+ω by mω. Put differently, we cn explin the following trdeoffs of distorting consumption. Tke prticulr individul with < ω 1. As consumption is distorted to be smller thn the free trde level, his period utility without trnsfer is decresed. This mkes trnsfer needed to be given to this prticulr individul lrger. However, such distortion steepens the slope of the trget utility u (x n (ω)+ω), which lowers the trget utility levelofthell individuls below ω. This mkes trnsfers needed to be given to those individuls smller. In ggregte, the ltter effect outweighs the former effect, nd thus the government expenditure for trnsfers is mde smller by such distortion. In prticulr, for the individuls with lrge ω, the ltter effect is dominting. This is why distortion is lrge for lrge ω, s illustrted in Figure 4. The trget utility level is depicted in Figure 2. It is now steeper thn the one when the GB constrint is not binding, but the IR constrint is still binding only t ω =1. As the government budget gets more stringent, consumption of good x is more distorted nd the slope of the trget level utility gets steeper. Eventully, the trget utility becomes s steep s the right hnd side (RHS) of the IR constrint t ω =1. Then, consumption of good x cnnot be lowered without violting the IR constrint for high 14
15 ω. In this cse, the IR constrint becomes binding not only t ω =1but lso for lrge ω such tht ω ω < 1, where ω is threshold vlue. Nmely, for ω ω < 1, the trnsction schedule is chosen so s to mke the trget utility s steep s the RHS of the IR constrint: u (x (ω)+ω) =p + δ (p p ) (see Figure 3). Letting x b (ω) (b for binding ) denote the trnsction schedule stisfying this equlity, we hve x b (ω) = p b δ (p p ) b ω for ω ω < 1. (13) Since the trget utility is U (ω)+δ (U (ω) U (ω)), the trnsction schedule for ω ω < 1 is given by δz b (ω) U (ω)+δ (U (ω) U (ω)) U(x b (ω),p, ω). (14) On the other hnd, for ω < ω, the trnsction schedule is the sme s (11), while the trget utility is modified to U ( ω)+δ (U ( ω) U ( ω)) R ω ω u (x n (t)+t) dt. Therefore, the trnsction schedule for ω < ω is Z ω δ z n (ω) U ( ω)+δ (U ( ω) U ( ω)) u (x n (t)+t) dt U (x n (ω),p, ω). (15) ω The threshold vlue ω is determined by equting x n (ω) nd x b (ω). Thtis, ω solves the equlity p b (m +1) ω = p b δ (p p ) b ω, (16) tht is, ω =(1+δ)(p p ) /bm. (17) The more stringent the government budget constrint becomes, m rises, nd ω flls, mening tht the more individuls re mde just indifferent between utrky nd free trde. Then, nturlly, the following question rises. No mtter how smll the budget B is,sy,evenwhenb =, cn some individuls be mde strictly better off by free trde? Proposition 1 clims tht the nswer is no: when B =, the gins from free trde 15
16 cn be distributed to mke someone strictly better off nd no one worse off only when δ is sufficiently smll. If δ is lrge enough nd B is smll enough, there is no solution to problem (7). See Figure 6 for grphicl presenttion. Proposition 1 There is vlue δ (, 1) such tht for δ δ problem (7) hs solution for ny B. On the other hnd, for δ > δ, there is threshold function B (δ) > such tht B (δ) > nd for B< B (δ) problem (7) hs no solution. The proof is given in the ppendix. Here, we provide intuitive explntion in the following three steps. First, we show tht in problem (7), even when mking everyone just indifferent between utrky nd free trde, the government needs positive budget. With the trnsctiontrnsfer pir x b (ω),z b (ω) ª for ll ω, the government expenditure for individul ω is written s δz b (ω) (p p ) x b (ω) =U (ω)+δ (U (ω) U (ω)) U(x b,p ) Adding nd subtrcting U (ω), wehve ³ δz b (ω) (p p ) x b (ω) =(1+δ)(U (ω) U (ω)) + U (ω) U(x b,p ) The first term is negtive when it is ggregted, while the second term is positive since U (ω) =U (x,p ). By decomposing the first term by dding nd subtrcting U (x,p ), we hve δz b (ω) (p p ) x b (ω) ³ = (1+δ)(U (ω) U (x,p )+U (x,p ) U (ω)) + U (ω) U(x b,p ) (18) ³ = (1 + δ)(p p ) x (1 + δ)(u (x,p ) U (x,p )) + U (x,p ) U(x b,p ). The first term disppers when ggregted over ω 1. The second term is negtive: it represents, in terms of per cpit, the gins from free trde llowed in the both periods 16
17 (with the negtive sign in front of it). By giving everyone the IRbinding level of utility insted of the freetrde level of utility, the government cn collect this mount of tx revenue. The third term is positive, which is interpreted s the loss of the individuls induced to consume distortedly smll mount of x b +ω due to the IC constrint, insted of consuming the free trde level of x + ω. So, the government hs to compenste this loss to the individuls. Grphiclly (see Figure 5), the third term is big tringle representing the chnge in the consumer surplus by consuming x b + ω t p rther thn consuming x + ω t p.for the second term, (U (ω) U (x,p )) is smll tringle representing the chnge in the consumer surplus by consuming x +ω t p rther thn x +ω t p. Multiplied by (1 + δ), it becomes the second term. Since (x + ω) x b + ω =(1+δ)((x + ω) (x + ω)), the big tringle representing the third term is lwys lrger thn the tringles representing the second term. Therefore, expression (18) is positive, mening tht it requires positive government budget to mke everyone just indifferent between utrky nd free trde (i.e., mking the IR constrint binding for ll ω). Of course, mking the IR constrint binding for ll ω my not minimize the government expenditure. In fct, strting from the IR constrint binding for everyone (i.e., strting from ω =), the government cn improve its budget by giving someone strictly positive gins from trde (i.e., by incresing ω). So, second, we explin this. Tke prticulr individul with ω very smll. As ω increses from zero so tht his ω becomes below ω, his consumption is now less distorted, nd thus his utility level without trnsfer increses. By this, the government cn collect more txes from him (or needs to give less subsidy to him), with the government expenditure getting smller. On the other hnd, however, n increse in ω rises his trget utility level. This hppens becuse due to the less distorted (incresed) consumption, the slope of the trget utility of, not only him, but lso ll individuls between his ω nd ω, getsfltter. By this, the government cn collect less txes from him (or needs to give more subsidy to him), with the government expenditure getting lrger. Since the ltter effect is smll when ˆω is smll, the former effect is dominting so tht the government expenditure flls by n increse in ω from ω =. Eventully, 17
18 however, s ω is incresed sufficiently lrge, the government expenditure hits the bottom, since the ltter effect is getting bigger nd comes to outweigh the former. Third, we explin tht (18) is lrge when δ is lrge. This is seen by inspecting Figure 5. When δ is lrge, the distnce between p + δ(p p ) nd p is lrge, nd so is the distnce between x + ω nd x b + ω. Therefore, the re of the big tringle not covered by the smll tringles gets lrger. This mens tht (18) is lrge. Then, lthough the government expenditure decreses s ω increses, the lowest government expenditure is still positive if δ is lrge. So, when B is smller thn the lowest government expenditure, the government cnnot implement the trnsfer policy to redistribute the gins from free trde. 4.3 Should the trnsfers be dependent on trnsctions under utrky or under free trde? So fr, we hve considered trnsfer rules under which the mount of trnsfer is dependent on the trnsction mde in period. Tht is, the government offers trnsctiontrnsfer menu x (ω),z (ω) ª,wherex (ω) is the trnsction in period nd z (ω) is the trnsfer given in period 1. In our nlysis, the resulting intertemporl utility cn be written s U x (ω),p, ω + δz (ω)+δu (ω). (19) On the other hnd, in order to redistribute the gins from free trde, we could consider trnsfer rules dependent on trnsctions mde in period 1. Nmely, the government could offer trnsctiontrnsfer menu x 1 (ω),z 1 (ω) ª,wherex 1 (ω) is the trnsction in period 1 nd z 1 (ω) is the trnsfer given in period 1. In fct, this trnsfer rule is the one studied by FL (1991). Formlly, menu x 1 (ω),z 1 (ω) ª is solution to the 18
19 welfremximiztion problem formulted s follows: mx x(ω),z(ω) [U (x (ω),p, ω)+z (ω)] dω + B z (ω) dω (2) s.t. U (x (ω),p, ω)+z (ω) U x ω,p, ω + z ω for ll ω, ω, U (x (ω),p, ω)+z (ω) U (ω) for ll ω. z (ω) dω B. The solution to this problem, denoted by x 1 (ω),z 1 (ω) ª, is qulittively similr to x (ω),z (ω) ª, the solution to problem (7) we derived in the previous section. Nmely, when B is sufficiently lrge, the GB constrint is not binding nd the trnsction schedule is efficient: x 1 (ω) =x (ω). As B flls, the GB constrint becomes binding nd the trnsction is distorted so tht consumption is smller for lrge ω, whiletheirconstrint is binding only t ω =1. Then, s B flls further, eventully the IR becomes binding for some ω < 1. However, different from problem (7), there is lwys solution to problem (2), s shown by FL (1991). This is becuse, in problem (2), mking the IR constrint binding for ll ω (tht is, forcing everyone consume x + ω) cn be implemented by blnced budget (i.e., with B =). Therefore, strting from the IR binding for everyone, the government cn let some individuls with smll ω consume more thn x + ω to mke them strictly better off by trde, nd still the GB constrint is stisfied even when B = (See Proposition 4 of FL (1991)). By the trnsfer rule x 1 (ω),z 1 (ω) ª, the intertemporl utility is given by U (ω)+δz 1 (ω)+δu x 1 (ω),p, ω. (21) Compring (19) nd (21), we cn see tht in (19) the government does not intervene trnsctions in period 1, leving the period1 utility U (ω) intct, while in (21) the government does not intervene trnsctions in period with leving the period utility U (ω) intct. Then, we sk the following question: by which trnsfer rules cn the welfre be mde higher? In other words, to redistribute the gins from free trde to everyone 19
20 with stisfying the IC constrint, which is less hrmful, distorting period trnsctions, or distorting period1 trnsctions? The proposition below shows tht the nswer depends onthesizeofb. If the government hs lrge budget for trnsfer, the former is better, nd if it hs smll budget, the ltter is better. Proposition 2 If B (1 + δ)(u (1) U (1)), the trnsfer rule dependent on trnsctions in period gives higher welfre thn the trnsfer rule dependent on trnsctions in period 1. If B< B (δ), the trnsfer rule dependent on trnsctions in period 1 gives higher welfre thn the trnsfer rule depending on trnsctions in period. Proof. When B (1 + δ)(u (1) U (1)), the GB constrint is not binding for problem (7) nd x = x. Letting W denote the resulting intertemporl welfre of problem (7), W = = U x (ω),p, ω + δz (ω)+δu (ω) δz (ω)+(p p ) x (ω)+δb dω (U (ω)+δu (ω)+δb) dω. For problem (2), note tht the GB constrint is not binding if B U (1) U (1) (The reson comes from the sme logic discussed in Section 4.2). Therefore, when B (1 + δ)(u (1) U (1)), the government cn implement x 1 = x. Let W 1 denote the resulting welfre for problem (2). We hve W 1 = = U (ω)+δz 1 (ω)+δu x 1 (ω),p, ω δz 1 (ω)+δb dω (U (ω)+δu (ω)+δb) dω. Since R 1 U (ω) dω > R 1 U (ω) dω, it holds tht W >W 1. If B< B (δ), there is no solution to problem (7). So, the best thing the government cn do is to sty utrky in both periods, thus W = R 1 (U (ω)+δu (ω)+δb) dω. On the other hnd, for problem (2), the government cn give some individuls strictly 2
21 positive gins from trde. Therefore, W 1 = U (ω)+δu x 1 (ω),p, ω + δb dω > (U (ω)+δu (ω)+δb) dω = W. Moreover, since W nd W 1 re continuous in B B (δ),evenwhenb<(1 + δ)(u (1) U (1)), s long s B is lrge enough, W stys bove W 1. This cn be seen in Figure 6, where W is higher thn W 1 even when B is fr below (1 + δ)(u (1) U (1)). 5 Concluding remrks We hve considered twoperiod model where in period the utrky price is mintined ndinperiod1freetrdeisllowed. Tochieve Preto gins from trde, the government designs trnsfer rule to collect txes from giners nd give subsidies to losers from free trde. By considering trnsfer rules tht mke the mount of trnsfer dependent on trnsctions in the utrky period, we hve shown tht there is no incentive comptible trnsfer rules tht successfully redistribute the gins from free trde if the discount fctor is sufficiently lrge or if the government budget vilble for redistribution is limited. Nmely, our finding tells tht redistributing the gins from free trde is often very costly under incomplete informtion; redistribution uses up ll gins from free trde nd still not ble to mke everyone better off. 21
22 Appendix A. Solution to problem (7). (i) When the IR constrint is binding only t ω =1: From eqution (11) nd (12), the government expenditure is clculted s follows: δz (ω) dω (p p ) x (ω) dω =(1+δ)(U (1) U (1)) mb (2 m) 6 where U (1) U (1) = µ 1 2 (p p ) (p p ) 2b nd m = λ/ (1 + λ). The GB constrint is thus (1 + δ)(u (1) U (1)) mb (2 m) 6 = δb. Solving this for m, weget m = b p b 2 6b ((1 + δ)(u (1) U (1)) δb). b Note tht m is decresing in B. So, s B flls, m increses, x (ω) decreses, nd therefore the trget utility gets steeper. As B flls enough tht the trget utility is s steep s the RHS of the IR constrint t ω =1, the IR constrint becomes binding other thn ω =1.ThebordervlueofB below which this hppens is derived by u x (1) + 1 = p + δ (p p ) p + mb = p + δ (p p ). Solving this for B, we hve the border vlue of B: µ B = 1 (2 δ) (p p ) (1 + δ)(p p ). b 6δ From our ssumption tht ( p ) /b =1/2 < ( p ) /b < 1, wehve(p p ) /b < 1/2. So, the B bove is lwys positive, mening tht for ny prmeter vlues, if B is 22
23 smll enough, the IR constrint becomes binding for ω ω 1. (ii) When the IR constrint is binding for ω ω 1: For ω ω 1, the trnsction schedule nd the trnsfer schedule re respectively given by eqution (13) nd (14). For ω < ω, since the IR constrint is not binding, nd γ () = by trnsverslity condition, the trnsction schedule is given by (11), nd thus the trnsfer schedule is by (15). From these, the government expenditure is clculted s ³ δz b (ω) (p p ) x b (ω) ω µ ω = (1+δ)(p p 2 ) Z ω dω + 2 ((1 + δ) ω δ)(p p ) 2b (δ z n (ω) (p p ) x n (ω)) dω µ bm 3 bm2 ω 3. (22) 6 The threshold vlue ω is determined s follows. For ω ω 1, x b (ω) should stisfy (1): u (x b (ω)+ω) p = R ω ω μ (t) dt + λω b. 1+λ Here, we used tht γ () = nd μ (ω) =for ω < ω (the IR constrint is not binding for this rnge of ω). At the sme time, by the IC constrint, we hve u (x b (ω) +ω) = p + δ (p p ).Thus, Z ω ω μ (t) dt = λω (1 + λ) (1 + δ)(p p ). b Especilly, when ω = ω, it holds tht Z ω ω μ (t) dt = λ ω (1 + λ) (1 + δ)(p p ) b =. Solving this for ω gives ω =(1+δ)(p p ) /bm. Substituting this into (22), nd rrnging, we hve the following GB constrint: (1 + δ)(p p ) ω 2 (1 + δ)2 (p p ) 2 ω + δ (1 + δ)(p p ) 2 6 3b 2b = δb. (23) Combined with ω =(1+δ)(p p ) /bm, thisissolvedfor ω nd m. 23
24 B.ProofofProposition1. Consider the determinnt of the qudrtic eqution (23). It is nonnegtive if B (1 + δ)(p p ) 2 6bδ Ã! 3δ (1 + δ)2 (p p ). (24) b Define the RHS of (24) s B (δ). If B< B (δ), there is no solution to problem (7). Note tht B (δ) is negtive if 3δ (1 + δ)2 (p p ) b <. (25) Since the inequlity (25) holds when δ =, violted when δ =1, nd the LHS of (25) is incresing in δ, there is unique δ (cll it δ) belowwhich B (δ) <. Thus, for δ δ, there is solution to problem (7) for ny B. Forδ > δ, B (δ) > nd B (δ) = B (δ) (1 + δ) + (1 δ)(1+δ)2 (p p ) 3 6b 2 δ 2 >. 24
25 References [1] Che, Jihu nd Giovnni Fcchini, Dul Trck Reforms: with nd without Losers, Journl of Public Economics 91 (27): [2] Dixit, Avinsh nd Victor Normn, Gins from Trde without LumpSum Compenstion, Journl of Interntionl Economics 21 (1986): [3] Dong, Bomin nd Lsheng Yun, The Loss from Trde under Interntionl Cournot Oligopoly with Cost Asymmetry, Review of Interntionl Economics 18 (21): [4] Feenstr, Robert C. nd Trcy R. Lewis, Distributing the Gins from Trde with Incomplete Informtion, Economics nd Politics 3 (1991): [5] Grndmont, JenMichel nd Dniel McFdden, A Technicl Note on Clssicl Gins from Trde, Journl of Interntionl Economics 2 (1972): [6] Ichino, Ysukzu, Trnsction Flsifiction nd the Difficulty of Achieving Pretoimproving Gins from Trde through Lumpsum Trnsfers, Review of Interntionl Economics 2 (212): [7] Kemp, Murry C., The Gin from Interntionl Trde, Economic Journl 72 (1962): [8] Kemp, Murry C., The Gins from Trde in CournotNsh Trding Equilibrium, Review of Interntionl Economics 18 (21): [9] Kemp, Murry C. nd Henry Y. Wn, Jr., The Gins from Free Trde, Interntionl Economic Review 13 (1972): [1] Wong, Kryiu, Interntionl Trde in Goods nd Fctor Mobility, Cmbridge: MIT Press, [11] Wong, Kryiu, Gins from Trde with LumpSum Compenstion. Jpnese Economic Review 48 (1997):
26 U U ( ) when the GB is not binding. U ( ) U *( ) (slope = p ) (slope = p* ) U ( x*( ), p, ) U ( ) ( U ( ) U * ( )) (slope = p ( p p*) ) 1 Figure 1: U (ω) when the GB constrint is not binding 26
27 U U ( U ( ) when the GB is binding nd the IR is binding only t 1. ) when the GB is not binding. U ( ) (slope = p ) U *( ) (slope = p* ) U ( ) ( U ( ) U * ( )) (slope = p ( p p*) ) 1 Figure 2: U (ω) when the GB constrint is binding nd the IR constrint is binding only t ω =1 27
28 U U ( ) when the IR is binding for some 1. U ( ) U *( ) (slope = p ) (slope = p* ) U ( ) ( U ( ) U * ( )) (slope = p ( p p*) ) 1 Figure 3: U (ω) when the IR constrint is binding for some ω < 1 28
29 x( ) x * ( ) x ( ) b x ( ) x ( ) when the IR is binding only t 1. x ( ) when the IR is binding for some 1. 1 Figure 4: Consumption of good x 29
30 p b U ( x*, p*) U ( x, p*) p ( p p*) p ( U( x*, p*) U( x, p*)) U( x*, p*) U( x, p*) p * u ( x ) b x x x * x Figure 5: 3
31 B.3.25 * B (1 )( U (1) U (1)) : bove this curve, the GB constrint is not binding Below this curve, the IR constrint is binding for some 1. Above this curve, W >W 1 : the trnsfer rule in our model gives higher welfre thn FL(1991) s..5 B( ): no solution below this curve A numericl exmple: when =2, b=1, p*=5/4 Figure 6: 31
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