Can a LumpSum Transfer Make Everyone Enjoy the Gains. from Free Trade?


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1 Can a LumpSum Transfer Make Everyone Enjoy te Gains from Free Trade? Yasukazu Icino Department of Economics, Konan University June 30, 2010 Abstract I examine lumpsum transfer rules to redistribute te gains from free trade. Wen individuals anticipate tat free trade policy will come wit a lumpsum transfer, tey are going to cange teir beaviors under autarky in order to get larger transfers. In spite of tis falsi cation, can a lumpsum transfer still make everyone enjoy te gains from free trade? In a model of simple excange economy, I found a condition under wic a lumpsum transfer can make it, but in general it is di cult to acieve Pareto gains from free trade. Key Words: Gains from trade, lumpsum transfer, redistribution JEL Classi cation Numbers: F11, H23 *Address: Okamoto, Higasinadaku Kobe , Japan. 1
2 1 Introduction It is well known tat free trade creates winners and losers. However, wit an appropriate lumpsum transfer, Pareto improvement is acieved as te country moves from autarky to free trade. For example, see Kemp (1962), and Grandmont and McFadden (1972). Essentially, te idea of Paretoimproving lumpsum transfer rules is to let wat eac individual consumed under autarky a ordable under free trade, too. 1 Later, Dixit and Norman (1986) pointed out practical di culties of implementing suc lumpsum transfer rules. In implementing lumpsum transfer to make everyone better o after free trade, te government as to gure out not only wo are taxed and wo are subsidized, but also ow muc is taxed or subsidized for eac individual. To do so, te government needs to collect a lot of information about te caracteristics of individuals. Moreover, wen asked by te government, eac individual may not reveal is caracteristics tat te government is going to use to calculate te amount of transfer. As Dixit and Norman (1986) writes, individuals ave an incentive to manipulate teir beavior so as to mislead te planner about tese caracteristics and secure a larger net transfer. How do tey manipulate teir beavior? Does tat manipulation matter? Wong (1997) studies tese questions in a general setting. His answer to te questions is, it may not matter very muc: e argues tat a lumpsum transfer rule tat allows eac individual to consume under free trade wat tey used to consume under autarky can acieve Pareto gains from free trade, in spite tat consumers manipulate teir beavior. However, partly because e worked on a very general setting, Wong did not analyze weter and ow consumers manipulation of teir beavior a ects te autarkic equilibrium price, and ow te cange in te autarkic equilibrium price will a ect te performance of te lumpsum transfer. In tis paper, I work on a simpler, more speci c setting tan Wong: two goods, quasilinear utility. By doing so, I can analyze weter and ow consumers manipulation of teir beavior in uences te autarkic equilibrium price. Ten, I ask te questions Wong 1 Capter 8 of Wong (1995) as a detailed discussion on tis topic. 2
3 asked: How do consumers manipulate teir beavior? Does tat manipulation matter? 2 Contrary to Wong s, my answer is, it does matter quite. I found tat it is in general di cult to redistribute te gains from free trade to everyone as a country moving from autarky to free trade, if te government cannot observe individuals preferences and if individuals anticipate a lumpsum transfer under free trade. Te reason is as follows. Knowing tat te lumpsum transfer is based on ow muc tey buy and sell under autarky, individuals distort teir consumption under autarky, in order to increase te compensation tey receive from te government or to decrease te lumpsum tax tey ave to pay to te government. Tis results in a drop of te autarkic equilibrium price, wic is likely to make te lumpsum transfer ine ective. Te rest of te paper is organized as follows. In Section 2 we set up te model. In Section 3, we investigate weter te traditional lumpsum transfer, a transfer tat guarantees autarkic consumption bundle a ordable under free trade, can redistribute te gains from free trade so as to acieve Pareto improvement even wen consumers are to manipulate teir beavior under autarky. We ten consider a wider range of lumpsum transfer rules in Section 4. Section 5 concludes te paper. 2 Model We consider an excange economy wit two goods, good x and good y, were good y is a numeraire. Tere are H individuals: = 1; 2; :::H. Individual s excess demand for two goods are respectively denoted by x and y. Individual s endowment of good x is denoted by! and tat of good y is denoted by y. In tis model, we ave two periods: in period 0, te economy is under autarky; in period 1, te economy is under free trade. We assume tat te utility function is quasilinear. Individual s utility maximization 2 Ce and Faccini (2007) studies tis kind of manipulation in a context of market liberalization, wile we study te manipulation in te context of trade liberalization. 3
4 problem in period 0 is given by max x ;y u (x +! ) + y + y s.t. px + y 0, were u () is individual s subutility function, wit u 0 () > 0 and u00 () < 0, and p is te price of good x. Solving te rstorder conditions gives te excess demand functions x (! ; p) = u 0 1 (p)! and y (! ; p) = px (! ; p). From te excess demand function, te autarkic equilibrium price p a is de ned by P x (! ; p a ) = 0. Let x a x (! ; p a ) denote te excess demand (or, we call it transaction ) of individual in te autarkic equilibrium. Ten, we can de ne te indirect utility of individual in te autarkic equilibrium: U a U (x a ; pa ) u (x a +! ) p a x a + y. In period 1, wen te country allows free trade, te price of good x is equal to te world price p (we assume te economy is small). Ten, transaction and indirect utility of individual in te freetrade equilibrium are respectively denoted by x x (! ; p ) and U U (x ; p ) u (x +! ) p x + y. We assume tat p < p a. Namely, te world price of good x is smaller tan te autarky price. 2.1 Gains from trade falls unequally In aggregate, tere are gains from free trade: P U P U a = P = P [U (x ; p ) U (x a ; p )] + P [U (x a ; p ) U (x a ; pa )] [U (x ; p ) U (x a ; p )] + (p a p ) P x a > 0 since te rst term is positive (given p, te utility maximizing consumption is x ) and te second term is zero because P xa = 0. However, te gains from trade do not fall equally to everyone. Te buyers of good x 4
5 gain from free trade: U U a = [U (x ; p ) U (x a ; p )] + [U (x a ; p ) U (x a ; pa )] = [U (x ; p ) U (x a ; p )] + (p a p ) x a > 0, provided x a > 0 (i.e., individual is a buyer under autarky). On te oter and, te sellers of good x lose from free trade: U U a = [U (x ; p ) U (x ; pa )] + [U (x ; pa ) U (x a ; pa )] = (p a p ) x + [U (x ; pa ) U (x a ; pa )] < 0, provided tat x < 0 (i.e., individual is a seller under free trade). 3 Traditional lumpsum transfer Let te government give some lumpsum transfer in period 1 to make sure tat everyone can enjoy te gains from free trade. Suppose tat te government gives a lumpsum transfer z a = (pa p ) x a to individual in period 1 (note tat za > 0 is a subsidy and z a < 0 is a tax). Ten, te budget constraint of individual in period 1 becomes p x + y z a. It is easy to sow tat tis transfer makes (xa ; ya ) a ordable under p : p x + y z a, p x + y (p a p ) x a, p x + y p x a + ya : Terefore, wit tis transfer, everyone gets better o by free trade. Tat is, U +za > U a for all. In addition, wit tis lumpsum transfer, te government budget is balanced since P za = (pa p ) P x = Consumption Falsi cation Te discussion above tat te free trade policy wit te lumpsum transfer z a makes everyone better o, owever, depends on an implicit assumption tat te government 5
6 can gure out wat z a s are. For te government to calculate za, it as to know eac individual s utility function and endowments. But tis is not likely te case. Witout knowing preferences and endowments of eac individual, te government is not able to calculate x a, and tus cannot gure out za. Ten, te only way for te government to nd te amount of lumpsum transfer for individual is to observe te transaction made by individual in period 0 (under autarky). However, if eac individual knows tat te government is going to observe is transaction in period 0 to determine te amount of transfer for im, ten e may want to cange is transaction from x a in order to get larger transfer in period 1 (under free trade). We call tis transaction falsi cation. Given tis, te question we want to ask in tis paper is as follows: ow does eac individual falsify is transaction in period 0, and ow does it a ect te equilibrium price in period 0? If te individuals falsify teir transaction in period 0, does tis mean tat Pareto gains from free trade cannot be acieved? To examine tese questions, let us now clarify our setting. As we ave mentioned, in te following analysis te government knows neiter endowments nor preferences of eac individual; te government just observes transaction of eac individual in period 0 (under autarky). In period 0, all individuals know tat free trade will be allowed in period 1. Tey also know tat te lumpsum transfer z = (p p ) x will be given in period 1, were x is te observed transaction of individual in period 0 and p is te prevailing price in period 0. We refer tis form of transfer, z = (p p ) x, as te traditional lumpsum transfer. Notice tat te traditional transfer z = (p p ) x as te same spirit as z a = (pa p ) x a : z makes wat individual consumed in period 0 (under autarky) a ordable in period 1 (under free trade). Also, note tat if tere were no transaction falsi cation, te traditional lumpsum transfer would be identical to z a in equilibrium. We are going to investigate weter tis traditional lumpsum transfer can redistribute te gains from free trade so as to acieve Pareto improvement, even wen tere is transaction falsi cation. 6
7 Now, te utility maximization problem of individual sould be written as follows: max u (x +! ) + y + y + u (x 1 x ;y ;x 1 +! ) + y 1 + y ;y1 s.t. px + y 0, and p x 1 + y1 (p p ) x were x and y are period0 transaction of good x and y; x 1 and y1 are period1 transaction of good x and y, respectively, and is a discount factor for individual, 0 1 for all. Solving te rstorder conditions, we ave x 1 = x and x = u 0 1 (p + (p p ))!. Here, te period1 transaction is te same as te one we saw in Section 2. But te period0 transaction is now di erent: anticipating tat te transfer will be given in period 1, individual regards te cost of consuming one unit of good x in period 0 not as p but as p + (p p ). Now, let p 0 denote te equilibrium price in period 0 under transaction falsi cation. It is determined by P u0 1 p 0 + p 0 p = P!. How is p 0 di erent from p a? Tat is, ow does transaction falsi cation a ect te equilibrium autarky price? We found te following result. Lemma 1 p < p 0 < p a. Proof. First, compare te individual demand function under nonfalsi cation u 0 1 (p) and under falsi cation u 0 1 (p + (p p )). For any given p, u 0 1 (p) > u 0 1 (p + (p p )), p p > 0, and u 0 1 (p) = u 0 1 (p + (p p )), p p = 0. Since tis is true for all, P u0 1 (p) > P u0 1 (p + (p p )), p p > 0, and P u0 1 (p) = P u0 1 (p + (p p )), p p = 0. Grapically, te demand curve of nonfalsi cation is steeper tan tat of falsi cation, and tey intersect at p (see Figure 1). In oter words, at tose quantities less tan te quantity suc tat P u0 1 (p) = P u0 1 (p + (p p )), te demand curve of nonfalsi cation is above tat of falsi cation. Te equilibrium nonfalsi cation price p a is determined by P u0 1 (p a ) = P! and by assumption p < p a. Tus, te intersection of te aggregate falsi cation demand curve P u0 1 (p + (p p )) and te aggregate supply P!, at wic p 0 is determined, is below te intersection of 7
8 P u0 1 (p) and P!. Terefore, p < p 0 < p a. Wit transaction falsi cation, te equilibrium price in period 0 falls. Intuitively, tis is because sellers try to sell more in period 0 in order to get more subsidy in period 1, and buyers try to buy less in order to lower tax tey ave to pay. Having found te period0 equilibrium price under transaction falsi cation, now we can de ne te period0 equilibrium transaction as x 0 u0 1 p 0 + p 0 p!. Is x 0 larger or smaller tan xa? Namely, wen anticipating te transfer of period 1, is individual going to consume more or less in period 0? Tere are two opposing e ects. To get larger transfer of period 1, an individual wants to buy less (or sell more). Tis e ect tends to make x 0 less tan xa. On te oter and, since good x is now ceaper (p 0 < p a ), an individual wants to buy more (or sell less). Tis e ect tends to make x 0 more tan x a. Wic e ect is dominating depends on. In fact, if everyone as te same discount factor, tese two e ects just cancel eac oter, and te falsi ed transaction x 0 is just equal to te nonfalsi ed transaction xa for all. Lemma 2 x 0 = xa for all, = for all. Proof. (() Suppose tat x 0 < xa for some. Ten, p0 + p 0 p > p a. Since P x0 = P xa, tere must be anoter agent 0 suc tat x 0 0 > x a 0. Tus p p 0 p < p a. However, since = 0, tis is a contradiction. ()) x 0 = xa for all implies p0 + p 0 p = p a for all. Ten, s must be te same for all. Now, let s consider te case were s are di erent. For tose individuals wit large, it olds tat p 0 + p 0 p > p a, and tus x 0 < xa. Tis means tat for tose individuals wit large, securing larger transfer in period 0 is more important tan utilizing te ceaper price in period 0. On te oter and, for tose individuals wit small, it olds tat p 0 + p 0 p < p a, and tus x 0 > xa. 8
9 3.2 Can everyone gain from free trade? We are now ready to answer our main question. Even wen te individuals falsify teir transaction in period 0 in order to a ect te transfer tey receive in period 1, can te traditional lumpsum transfer, wic guarantees te period0 consumption bundle affordable in period 1, still make everyone enjoy te gains from free trade? To see tis, we are going to compare U 0 + (U + z ) wit U a + U a, were U 0 U x 0 ; p0 u (x 0 +! ) p 0 x 0 + y, and z = p 0 p x 0 (ereafter, z is to denote te traditional lumpsum transfer in equilibrium wen tere is transaction falsi cation). Tat is, we are comparing te welfare wen free trade policy comes in period 1 wit te traditional lumpsum transfer under transaction falsi cation, wit te welfare wen te economy stays in autarky in period 1. First, we consider te special case were all individuals ave te same discount factor Wen = for all Proposition 1 If = for all, ten U 0 + (U + z ) = U a + U for all. Proof. [U 0 + (U + z )] [U a + U ] = (U 0 + z ) U a = U (x 0 ; p0 + p 0 p ) U (x a ; pa ) = 0. Te last equality comes from Lemma 2. Terefore, wen =, te traditional lumpsum transfer is completely neutralized. Tat is, even toug te free trade policy in period 1 is accompanied by te lumpsum transfer, te transaction falsi cation in period 0 makes te resulting intertemporal welfare just equal to te one wit free trade policy not accompanied by any transfers. Because of Proposition 1, wen = for all, comparing U 0 + (U + z ) wit U a + U a is reduced to comparing U wit U a. Hence, we readily ave te following corollary. 9
10 Corollary 1 If = for all, ten U 0 + (U + z ) > U a + U a for suc tat x a 0, and U 0 + (U + z ) < U a + U a for suc tat x 0. From free trade, te buyers of good x still gain and te sellers of good x still lose. Wen = for all, due to transaction falsi cation, te traditional lumpsum transfer z = p 0 p x 0 fails to redistribute te gains from free trade to everyone Di erent Next, we consider te cases were s are di erent. Let us examine buyers rst. As we con rm below, in spite of transaction falsi cation, buyers of good x are made better o by free trade wit te lumpsum transfer z = p 0 p x 0. Proposition 2 U 0 + (U + z ) > U a + U a for suc tat xa 0. Proof. [U 0 + (U + z )] [U a + U a ] = U 0 + z U a + (U U a ) = U x 0 ; p0 + p 0 p U x a ; p0 + p 0 p + U x a ; p0 + p 0 p U (x a ; pa ) + [U (x ; p ) U (x a ; p ) + U (x a ; p ) U (x a ; pa )] = U x 0 ; p0 + p 0 p U x a ; p0 + p 0 p + [U (x ; p ) U (x a ; p )] + p a p 0 + p 0 p x a + (p a p ) x a = U x 0 ; p0 + p 0 p U x a ; p0 + p 0 p + [U (x ; p ) U (x a ; p )] > 0. + (1 + ) p a p 0 x a How about sellers? Wit te lumpsum transfer, can sellers of good x be made better o by free trade? As we sow in Proposition 3, due to transaction falsi cation, some sellers may be made worse o by free trade even toug tey receive te lumpsum transfer. 10
11 Proposition 3 For suc tat x 0, if is small enoug suc tat p a p 0 + p 0 p, ten U 0 + (U + z ) < U a + U a. Proof. U 0 + (U + z ) [U a + U a ] = U 0 + z U a + (U U a ) = U x 0 ; p0 + p 0 p U x 0 ; pa + U x 0 ; pa U (x a ; pa ) + [U (x ; p ) U (x ; pa ) + U (x ; pa ) U (x a ; pa )] = p a p 0 + p 0 p x 0 + U x 0 ; pa U (x a ; pa ) + (p a p ) x + [U (x ; pa ) U (x a ; pa )] < 0. Note tat x 0 0 for tose suc tat x 0 since p0 > p. Intuitively, tis proposition is explained as follows. Wen a seller of good x as small, because te ceaperprice e ect outweigs te securinglargertransfer e ect, e sells less in period 0 under transaction falsi cation tan under nonfalsi cation. Tus, te lumpsum transfer e receives in period 1 will be smaller. Furtermore, te fall in period 0 price due to transaction falsi cation makes is period0 welfare smaller (because e is a seller of good x). As a result, wen tere is transaction falsi cation, e is made worse o by free trade policy in period 1 even toug te lumpsum transfer is given. Proposition 3 gives us a necessary condition under wic te lumpsum transfer can make everyone better o : te traditional lumpsum transfer successfully redistributes te gains from trade to everyone only if all sellers of good x ave large discount factor so tat p 0 + p 0 p > p a : wit all sellers aving large s, tey sell more in period 0 so as to secure large amount of lumpsum transfer in period 1. As a result tey can be enoug compensated under free trade. In Appendix A, we present an example to demonstrate tat tere is actually a case were everyone is made better o. 11
12 3.3 Discussion So, after all, ow do we evaluate te performance of te traditional lumpsum transfer rule z = p 0 p x 0? Can it successfully redistribute te gains from free trade to everyone, in spite of transaction falsi cation? Our answer is, yes, it can, but not always. Weter tis lumpsum transfer can make everyone better o depends on te distribution of individuals caracteristics. Speci cally, it can make everyone better o only wen all of te sellers of good x (i.e., tose wo ave larger endowment of good x) care about te future more tan te buyers do. How is tis likely to appen? Casually speaking, tere seems no special reason to believe tat te individuals aving larger endowment of good x is more patient tan tose aving smaller endowment. Ten, te situation under wic tis transfer rule makes everyone better o is not specially likely to appen. Moreover, wit te assumption tat te government does not know te preferences of eac individual, it is natural to consider tat te government does not know te discount factor of eac individual eiter. So, te government will not be able to identify te situation were te lumpsum transfer rule works ne. Overall, our conclusion is tat te traditional lumpsum transfer rule z = p 0 p x 0 does not perform very well wen tere is transaction falsi cation. 4 Any linear transfer rules Tus far, we ave examined a particular form of lumpsum transfer, z = p 0 p x 0, because it is a transfer rule tat guarantees te period0 consumption bundle a ordable in period 1. We ave sown tat tis traditional transfer rule does not perform well to redistribute te gains from free trade to everyone. A reason wy tis transfer rule does not work well is tat te amount of transfer is a ected by te autarkic price drop from p a to p 0 due to transaction falsi cation. To avoid tis, we may want to look for oter rules of lumpsum transfer. For example, ow about z = (p a p ) x? Wit tis rule, te amount of transfer is not directly a ected by te autarkic price cange. To analyze suc a transfer rule, in tis section, we consider a more 12
13 general class of lumpsum transfer rules. Speci cally, we examine a class of transfer rules tat are budget balancing and linear in x. A budgetbalancing, linear lumpsum transfer rule is denoted by z (x ) = sx, were s can be a constant, or it can be a function of te autarkic price p, as in te case of te traditional transfer rule. So, wen necessary, we write s (p). Are tere any linear transfer rules tat outperform te traditional transfer rule z = p 0 p x 0? Tis is te question we ask in tis section. As we will see below, te answer to tis question is essentially no. Basically, any linear transfer rules ave te similar caracteristics to te traditional lumpsum transfer rule we ave studied in te last section. Wit a general linear lumpsum transfer, te utility maximization problem is written as follows: max u (x +! ) + y + y + u (x 1 x ;y ;x 1 +! ) + y 1 + y ;y1 s.t. px + y 0, and p x 1 + y1 sx Tis setting is exactly te same as te one we ad in Section 3.1, except (p p ) x is replaced by sx. Solving te rstorder conditions, we derive x 1 = x and x = u 0 1 (p s)!. Te equilibrium autarky price p 0 is determined by P u0 1 p 0 s p 0 = P!. We let x 0 denote te equilibrium transaction in period 0: x0 = u0 1 p 0 s p 0!. Ten, in equilibrium, te lumpsum transfer is z x 0 = s p 0 x Wen = for all Now, we can sow tat te same lemma olds as in te case of te traditional transfer. Lemma 3 Let z x 0 = sx 0. x 0 = xa for all, = for all. Proof. (() Suppose tat x 0 < xa for some. Ten, p0 s p 0 > p a. Since P x0 = P xa, tere must be anoter agent 0 suc tat x 0 0 > x a 0. Tus p 0 0s p 0 < p a. However, since = 0, tis is a contradiction. 13
14 ()) x 0 = xa for all implies p0 s p 0 = p a for all. Ten, it must be tat are te same for all. Using tis lemma, again, we can sow tat any linear transfer rules are neutralized wen = for all. Proposition 4 Let z x 0 = sx 0. If = for all, ten U 0 + (U + z ) = U a + U for all. Proof. U 0 + U + z x0 = U 0 + z x0 U a (U a + U ) = U x 0 ; p0 s U (x a ; pa ) = 0. Terefore, any linear and budgetbalancing transfer rules cannot make everyone better o wen everyone as te same discount factor. 4.2 Wen s are di erent So, again, to see if linear transfers work well, we ave to look at te cases were s are di erent. Before doing tis, let us take a closer look at ow te caracteristics of te linear lumpsum transfer rules z x 0 = s p 0 x 0 a ect te equilibrium autarkic price p0. Lemma 4 sows tat as long as te transfer is positive for sellers (subsidy for sellers) and negative for buyers (tax on buyers), i.e., s p 0 0, te equilibrium autarkic price falls due to transaction falsi cation. Ten, Lemma 5 sows tat as long as te size of transfer is not too large, te equilibrium autarkic price p 0 is iger tan te world price. Lemma 4 p 0 p a, s p
15 Proof. (() Suppose p 0 > p a. s p 0 0 implies p 0 s p 0 > p 0 for all. Ten, p 0 s p 0 > p 0 > p a for all. Tis implies x 0 < xa for all. But tis is a contradiction since P x0 = P xa. ()) Suppose p 0 p a. s p 0 > 0 implies p 0 > p 0 s p 0. Ten, p a p 0 > p 0 s p 0 for all. Tis implies x a < x0 for all, but tis is a contradiction. Lemma 5 If 0 s p 0 p a p, ten p p 0 p a. Proof. Note tat s p 0 0 implies p 0 p a from Lemma 4. Given tat p a p s p 0, we ave p 0 p p 0 s p 0 p a p 0 s p 0 p a, were = max. Since p 0 s p 0 p a (oterwise, x 0 > xa for all, wic cannot appen), it olds tat p 0 p. Having tese lemmas, we can sow te following: te results we ave derived in te case of te traditional transfer (Section 3) is generalized to te case of any linear transfer rules, provided tat te transfer is positive (subsidy) for sellers and negative (tax) for buyers, and its size is not too large. Proposition 5 Let z x 0 = sx 0 and suppose tat 0 s p 0 p a p. (1) For suc tat x a > 0, U 0 + (U + z ) > U a + U a. (2) For suc tat x 0, if pa p 0 s (i.e., if is relatively small), ten U 0 + (U + z ) < U a + U a. Proof. (1) For tose individuals suc tat x a > 0, [U 0 + (U + z )] [U a + U a ] = U x 0 ; p0 s U x a ; p0 s + U x a ; p0 s U (x a ; pa ) + [U (x ; p ) U (x a ; p ) + U (x a ; p ) U (x a ; pa )] = U x 0 ; p0 s U x a ; p0 s + [U (x ; p ) U (x a ; p )] > 0 + p a p 0 + ((p a p ) + s) x a 15
16 since p a p s by assumption and p a p 0 by Lemma 4. (2) For tose individuals suc tat x 0, U 0 + (U + z ) [U a + U a ] = U x 0 ; p0 s U x 0 ; pa + U x 0 ; pa U (x a ; pa ) + [U (x ; p ) U (x ; pa ) + U (x ; pa ) U (x a ; pa )] = p a p 0 s x 0 + U x 0 ; pa U (x a ; pa ) + (p a p ) x + [U (x ; pa ) U (x a ; pa )] < 0 since p a > p 0 s, and since x 0 x < 0 because of p0 p, wic is from Lemma 5. Tus, te discussion we ad in Section 3.3 applies to any linear transfer rules suc tat 0 s p 0 p a p. Tat is, tese linear transfer rules do not perform well in making everyone enjoy te gains from free trade. It does so only wen all sellers of good x are more patient tan te buyers of good x. In Appendix B, we consider te oter linear transfer rules, suc tat 0 < p a p < s p 0, and tat s p 0 > 0. For tose lumpsum transfer rules, too, weter tey can make everyone better o depends on te distribution of individuals discount factors. Terefore, practically speaking, tose transfer rules, eiter, do not perform well in making everyone enjoy te gains from free trade. 5 Concluding remarks In tis paper, we examined lumpsum transfer rules to redistribute te gains from free trade to everyone. Speci cally, we considered a transfer rule tat makes wat eac individual consumed under autarky a ordable under free trade. Wen individuals anticipate tat free trade policy will come wit tis lumpsum transfer, tey are going to falsify teir transaction under autarky in order to get larger transfer. In spite of tis transaction falsi cation, can tis lumpsum transfer successfully redistribute te gains from free trade 16
17 to everyone? Our answer is, yes, it can, but not always. Weter tis lumpsum transfer can make everyone better o depends on te distribution of individuals caracteristics. More speci cally, it can make everyone better o only wen all of te sellers of good x (i.e., tose wo ave larger endowment of good x) care more about te future tan te buyers do. Te similar result will old for any linear, budgetbalancing transfer rules. Appendix A. An example tat everyone is made better Here we provide an example tat everyone is made better o by free trade policy wit te traditional lumpsum transfer z = p 0 p x 0. Let te subutility function u (! + x ) be quadratic: u (! + x ) = 1 (! + x ) (! + x ) 2. Suppose tat 2 tere are two individuals. Let 1 = 0:05, 2 = 1, 1 = 2 = 1, 1 = 2, 2 = 1,! 1 = 0:249,! 2 = 0:751, and p = 1=4. We can calculate te equilibrium prices as follows: p a = ( ) (! 1 +! 2 ) = 1=3; and ( ) p 0 = ( ) (! 1 +! 2 ) + ( ) p = 0: ( ) + ( ) Te equilibrium consumptions of good x are as follows: x a 1 +! 1 = 1=3; x a 2 +! 2 = 2=3; x 1 +! 1 = 1=2; x 2 +! 2 = 3=4; x 0 1 +! 1 = 0:37195; and x 0 2 +! 2 = 0: Te indirect utilities are: U a 1 = 0: y 1; U a 2 = 0: y 2; U 1 = 0: y 1; U 2 = 0:469 + y 2; U z 2 1 = 0: y 1; and 17
18 U z 2 2 = 0: y 2. In tis case everyone is made better o since: U (U1 + z 1) (U1 a + 1U1 a ) = 0:00270 > 0; and U (U2 + z 2) (U2 a + 2U2 a ) = 0:00044 > 0. B. General linear transfer rules: z = s p 0 x 0 B1. Wen 0 < p a p < s p 0 Since s p 0 < 0, in tis case it olds tat p a > p 0 (See, Lemma 4). For buyers, [U 0 + (U + z )] [U a + U a ] = U x 0 ; p0 s U x a ; p0 s + [U (x ; p ) U (x a ; p )] + (p a p ) p 0 s p a x a. Wen x a 0, a su cient condition for tis to be positive is (p a p ) p 0 s p a. Tis condition is satis ed wen is not too large. For sellers, U 0 + (U + z ) [U a + U a ] = p a p 0 s x 0 + U x 0 ; pa U (x a ; pa ) + (p a p ) x + [U (x ; pa ) U (x a ; pa )]. Wen x 0, a su cient condition for tis to be negative is pa p 0 s x 0 0. Tis is, p a p 0 s and x 0 0, or pa p 0 s and x 0 0. However, te latter one, p a p 0 s and x 0 0, can t appen since pa p 0 s, x a x0 and since 0 > x a. Terefore, a su cient condition for tis to be negative is pa p 0 s and x 0 0. In terms of, tis is tat is small enoug to satisfy p a p 0 s, but not too small to violate x
19 B2. Wen s p 0 > 0 In tis case p a < p 0 by Lemma 4. For buyers, [U 0 + (U + z )] [U a + U a ] = U x 0 ; p0 s U x a ; p0 s + [U (x ; p ) U (x a ; p )] + (p a p ) + p a p 0 s x a. Wen x a 0, a su cient condition for tis to be positive is pa p 0 s. Tis condition is satis ed wen is large enoug. For sellers, U 0 + (U + z ) [U a + U a ] = p a p 0 s x 0 + U x 0 ; pa U (x a ; pa ) + (p a p ) x + [U (x ; pa ) U (x a ; pa )]. Wen x 0, a su cient condition for tis to be negative is pa p 0 s x 0 0. Tis is p a p 0 s and x 0 0, or pa p 0 s and x 0 0. However, again, te latter one, p a p 0 s and x 0 0, can t appen since pa p 0 s, x a x0 and since 0 > xa. Terefore, a su cient condition for tis to be negative is p a p 0 s and x 0 0. In terms of, tis is tat is large enoug to satisfy p a p 0 s, but not too large to violate x
20 References [1] Ce, Jiaua and Giovanni Faccini (2007), Dual Track Reforms: wit and witout Losers, Journal of Public Economics, 91, pp [2] Dixit, Avinas and Victor Norman (1986), Gains from Trade witout LumpSum Compensation, Journal of International Economics, 21, pp [3] Grandmont, J. M., and D. McFadden (1972), A Tecnical Note on Classical Gains from Trade, Journal of International Economics, 2, pp [4] Kemp, Murray C. (1962) Te Gain from International Trade, Economic Journal, 72, pp [5] Wong, Karyiu (1995), International Trade in Goods and Factor Mobility, Cambridge, Mass.: MIT Press. [6] Wong, Karyiu (1997), Gains from Trade wit LumpSum Compensation. Japanese Economic Review, 48, pp
21 p ω a p 0 p * p 1 * u ( p + ρ ( p p )) u 1 ( p ) quantity Figure 1: Lemma 1: te equilibrium price in Period 0 (under autarky) 21
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