Can a Lump-Sum Transfer Make Everyone Enjoy the Gains. from Free Trade?

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1 Can a Lump-Sum Transfer Make Everyone Enjoy te Gains from Free Trade? Yasukazu Icino Department of Economics, Konan University June 30, 2010 Abstract I examine lump-sum transfer rules to redistribute te gains from free trade. Wen individuals anticipate tat free trade policy will come wit a lump-sum transfer, tey are going to cange teir beaviors under autarky in order to get larger transfers. In spite of tis falsi cation, can a lump-sum transfer still make everyone enjoy te gains from free trade? In a model of simple excange economy, I found a condition under wic a lump-sum transfer can make it, but in general it is di cult to acieve Pareto gains from free trade. Key Words: Gains from trade, lump-sum transfer, redistribution JEL Classi cation Numbers: F11, H23 *Address: Okamoto, Higasinada-ku Kobe , Japan. 1

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 rst-order 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 free-trade 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 lump-sum transfer Let te government give some lump-sum transfer in period 1 to make sure tat everyone can enjoy te gains from free trade. Suppose tat te government gives a lump-sum 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 lump-sum 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 lump-sum transfer z a makes everyone better o, owever, depends on an implicit assumption tat te government 5

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 period-0 transaction of good x and y; x 1 and y1 are period-1 transaction of good x and y, respectively, and is a discount factor for individual, 0 1 for all. Solving te rst-order conditions, we ave x 1 = x and x = u 0 1 (p + (p p ))!. Here, te period-1 transaction is te same as te one we saw in Section 2. But te period-0 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 non-falsi 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 non-falsi 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 non-falsi 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 period-0 equilibrium price under transaction falsi cation, now we can de ne te period-0 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 non-falsi 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

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 ceaper-price e ect outweigs te securing-larger-transfer e ect, e sells less in period 0 under transaction falsi cation tan under non-falsi cation. Tus, te lump-sum transfer e receives in period 1 will be smaller. Furtermore, te fall in period- 0 price due to transaction falsi cation makes is period-0 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 lump-sum transfer is given. Proposition 3 gives us a necessary condition under wic te lump-sum transfer can make everyone better o : te traditional lump-sum 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 lump-sum 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 lump-sum 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 lump-sum 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 lump-sum transfer rule works ne. Overall, our conclusion is tat te traditional lump-sum 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 lump-sum transfer, z = p 0 p x 0, because it is a transfer rule tat guarantees te period-0 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 lump-sum 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 lump-sum transfer rules. Speci cally, we examine a class of transfer rules tat are budget balancing and linear in x. A budget-balancing, linear lump-sum 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 lump-sum transfer rule we ave studied in te last section. Wit a general linear lump-sum 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 rst-order 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 lump-sum 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 budget-balancing 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 lump-sum 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 lump-sum 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 lump-sum 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 lump-sum 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 lump-sum transfer successfully redistribute te gains from free trade 16

17 to everyone? Our answer is, yes, it can, but not always. Weter tis lump-sum 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, budget-balancing 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 lump-sum 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 B-1. 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 B-2. 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 Lump-Sum 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, Kar-yiu (1995), International Trade in Goods and Factor Mobility, Cambridge, Mass.: MIT Press. [6] Wong, Kar-yiu (1997), Gains from Trade wit Lump-Sum 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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