CHAPTER 3. EQUILIBRIUM AND THE GAINS FROM TRADE 13 in the chain of argument. The first is the gain from having able to consume at different prices and the second is the gain from having able to produce at different prices. If one of the inequalities is strict, so will be the comparison of utilities An extension of the argument above adds tariffs(or subsidies )with a net revenue of T. In this case the home price vector (p) will be different from the rest of the world's and there is a net revenue(or loss) generated by the tariffs Thus the national income identity has to be modified to e(p, u)=r(p, u)+T It is easy to see that as long as T>0, managed trade is preferable to autarchy, since the inequalities above do not change. This is true regardless of the fact that home faces different prices than the rest of the world. As long as trade subsidies are not very large, home will benefit from trade Now we introduce heterogeneity. In this case it can obviously happen that some people are better off with trade but others are hurt. Thus the only thing we can hope for is the existence of a compensating mechanism through which a Pareto-improvement can be achieved. The most powerful such tools are lump-sum transfers, and we can show that if the government can redistribute income, every body can be made better off. One way to do that is to show that a scheme that makes the autarchy consumption level just affordable for all consumers generates positive revenue. Let Th stand for the lump-sum transfers and let p be the resulting equilibrium price vector Th is defined as and it is easy to see that where the second equality uses the autarchy budget constraint. Thus the autarchy consumption vector satisfies the budget constraint at the free trade prices p and transfers Th. We only need to see that the government generates non-negative revenue ≤p
CHAPTER 3. EQUILIBRIUM AND THE GAINS FROM TRADE 13 in the chain of argument. The first is the gain from having able to consume at different prices and the second is the gain from having able to produce at different prices. If one of the inequalities is strict, so will be the comparison of utilities. An extension of the argument above adds tariffs (or subsidies) with a net revenue of T. In this case the home price vector (ˆp) will be different from the rest of the world’s and there is a net revenue (or loss) generated by the tariffs. Thus the national income identity has to be modified to e(ˆp, u) = r(ˆp, v)+T. It is easy to see that as long as T ≥ 0, managed trade is preferable to autarchy, since the inequalities above do not change. This is true regardless of the fact that home faces different prices than the rest of the world. As long as trade subsidies are not very large, home will benefit from trade. Now we introduce heterogeneity. In this case it can obviously happen that some people are better off with trade but others are hurt. Thus the only thing we can hope for is the existence of a compensating mechanism through which a Pareto-improvement can be achieved. The most powerful such tools are lump-sum transfers, and we can show that if the government can redistribute income, everybody can be made better off. One way to do that is to show that a scheme that makes the autarchy consumption level just affordable for all consumers generates positive revenue. Let τ h stand for the lump-sum transfers and let p be the resulting equilibrium price vector. τ h is defined as: τ h = (p − p a )c ah + (w a − w)v h , and it is easy to see that wvh + τ h = w a v h − p a c ah + pcah = pcah , where the second equality uses the autarchy budget constraint. Thus the autarchy consumption vector satisfies the budget constraint at the free trade prices p and transfers τ h . We only need to see that the government generates non-negative revenue: X h τ h = p X h c ah − w X h v h = pxa − wv ≤ px − wv = 0
CHAPTER 3. EQUILIBRIUM AND THE GAINS FROM TRADE 14 Thus the transfers are feasible, and the consumers are at least as well off as in autarchy(possibly better if they choose a different consumption vector) ump-sum transfers are usually not politically possible, so it is interesting to ask whether some other type of taxes can achieve the desired result. As DN show, commodity and income taxes can also be used. The proof is similar to the one above except that now we guarantee people their autarchy utility levels and show that the government can achieve positive revenue. The idea is that the government will set taxes in such a way that prices and factor rewards equal the autarchy levels for consumers, pa and w. Facing the same prices, they will make the same choices as in autarchy. On the other hand, producers' decision will be based on the world equilibrium prices, so the country is able to reap the gains from trade on the production side Formally, let T be the government's tax revenue, (p, w) the equilibrium price and factor price vectors and a the equilibrium output vector -p)∑e+(-2)∑ But this is exactly the same revenue as above, which we know is non-negative The difference between this outcome and the one above is that now consumers will not consume a different bundle, because we changed not only their in- come but the prices they face. Thus the only gains come from the production side, as government revenue There is one question that you should ask yourself, what happens with the surplus in the two cases? DN is quite sloppy about this, and in the lump sum case I think they are not quite correct. This is why I used the proof in Feenstra, which show that even if the government dumps the proceeds people are likely to be better off. In the commodity tax case, you cannot argue the same way, but dn shows in a paper that under some conditions you can redistribute the revenue and make everyone strictly better of IDixit-Norman: Gains from trade without lump-sum compensation, Journal of Inter ational Economics, August 1986
CHAPTER 3. EQUILIBRIUM AND THE GAINS FROM TRADE 14 Thus the transfers are feasible, and the consumers are at least as well off as in autarchy (possibly better if they choose a different consumption vector). Lump-sum transfers are usually not politically possible, so it is interesting to ask whether some other type of taxes can achieve the desired result. As DN show, commodity and income taxes can also be used. The proof is similar to the one above, except that now we guarantee people their autarchy utility levels and show that the government can achieve positive revenue. The idea is that the government will set taxes in such a way that prices and factor rewards equal the autarchy levels for consumers, p a and w a . Facing the same prices, they will make the same choices as in autarchy. On the other hand, producers’ decision will be based on the world equilibrium prices, so the country is able to reap the gains from trade on the production side. Formally, let T be the government’s tax revenue, (p, w) the equilibrium price and factor price vectors and x the equilibrium output vector: T = (p a − p) X h c ah + (w − w a ) X h v h . But this is exactly the same revenue as above, which we know is non-negative. The difference between this outcome and the one above is that now consumers will not consume a different bundle, because we changed not only their income but the prices they face. Thus the only gains come from the production side, as government revenue. There is one question that you should ask yourself, what happens with the surplus in the two cases? DN is quite sloppy about this, and in the lumpsum case I think they are not quite correct. This is why I used the proof in Feenstra, which show that even if the government dumps the proceeds, people are likely to be better off. In the commodity tax case, you cannot argue the same way, but DN shows in a paper1 that under some conditions you can redistribute the revenue and make everyone strictly better off. 1Dixit-Norman: Gains from trade without lump-sum compensation, Journal of International Economics, August 1986
Chapter 4 Factor price equalization 4.1 General results It is time now to try to see what kind of general results emerge from our model. We will look at comparative advantage, factor proportions and factor reward 4.1.1 Comparative advantage Generalizing comparative advantage is quite easy, given the properties of trade equilibrium articular. we have that pr2≤r(p2,t),pc≥e(p2,u)ande(p2,u2)≥e(p,u2) p(c Now we can use the facts that a similar inequality holds for the foreign country, and that in equilibrium m M. Combining these with the above. we have that (P0-P)m2≥0 Thus, on average, a country will import a good for which it had a higher autarchy price. Notice, however, that this does not have to be true for a particular good and dn gives a counterexample
Chapter 4 Factor price equalization 4.1 General results It is time now to try to see what kind of general results emerge from our model. We will look at comparative advantage, factor proportions and factor rewards. 4.1.1 Comparative advantage Generalizing comparative advantage is quite easy, given the properties of trade equilibrium. In particular, we have that: p ax t ≤ r(p a , v), pa c t ≥ e(p a , ut ) and e(p a , ut ) ≥ e(p a , ua ) ⇓ p a (c t − x t ) = p amt ≥ 0. Now we can use the facts that a similar inequality holds for the foreign country, and that in equilibrium mt = −Mt . Combining these with the above, we have that: (p a − P a )mt ≥ 0. Thus, on average, a country will import a good for which it had a higher autarchy price. Notice, however, that this does not have to be true for a particular good and DN gives a counterexample. 15
CHAPTER 4. FACTOR PRICE EQUALIZATION 4.1.2 Factor proportions What about explanations for comparative advantage? We will get back to technology when we discuss the generalized ricardian model, so let us for now focus on the factor proportions explanation. As we discussed earlier, we need to assume identical technologies and uniform homothetic preferences Then we can write the expenditure functions as e(p)u. Since we can choose an arbitrary normalization of prices, it is convenient to have e(pa)=e(pa)=1 Then from the autarchy equilibrium conditions we have that u=r(pa, u) But we saw that for an arbitrary price vector utility is higher than in autarchy, so in particular we have r(p,V)≥r(P° Combining these, we get r(p,v)-r(P,)-{r(p,V)-r(P,V)≤0. This is a general result about the connection between autarchy prices and factor endowments. If r(p, u)was linear in(p, u), we could get a correlation imilar to comparative advantage above. In the absence of linearity, we can approximate the above inequality when(Pa, V)and(p, v) are sufficiently close together. Then the inequality can be rewritten as follows (p-P)rm(v-V)≤0 where rpu is the matrix of cross-derivatives of the revenue function. To prove this, just note that because r(p, u) is homogenous of degree one in(p, u) if technology is CRs You can relate r to the notion of factor intensities we discussed earlier. see DN for more details. Thus for small changes we have a negative correlation between autarchy prices and factor endowments, when we relate the two with the concept of factor intensities
CHAPTER 4. FACTOR PRICE EQUALIZATION 16 4.1.2 Factor proportions What about explanations for comparative advantage? We will get back to technology when we discuss the generalized Ricardian model, so let us for now focus on the factor proportions explanation. As we discussed earlier, we need to assume identical technologies and uniform homothetic preferences. Then we can write the expenditure functions as e(p)u. Since we can choose an arbitrary normalization of prices, it is convenient to have e(p a ) = e(P a ) = 1. Then from the autarchy equilibrium conditions we have that u a = r(p a , v) and U a = r(P a , V ). But we saw that for an arbitrary price vector utility is higher than in autarchy, so in particular we have r(P a , v) ≥ r(p a , v) and r(p a , V ) ≥ r(P a , V ). Combining these, we get [r(p a , v) − r(P a , v)] − [r(p a , V ) − r(P a , V )] ≤ 0. This is a general result about the connection between autarchy prices and factor endowments. If r(p, v) was linear in (p, v), we could get a correlation similar to comparative advantage above. In the absence of linearity, we can approximate the above inequality when (P a , V ) and (p a , v) are sufficiently close together. Then the inequality can be rewritten as follows: (p a − P a )rpv(v − V ) ≤ 0, where rpv is the matrix of cross-derivatives of the revenue function. To prove this, just note that because r(p, v) is homogenous of degree one in (p, v) if technology is CRS, r(p, v) = p rpv(p, v) v. You can relate rpv to the notion of factor intensities we discussed earlier, see DN for more details. Thus for small changes we have a negative correlation between autarchy prices and factor endowments, when we relate the two with the concept of factor intensities
CHAPTER 4. FACTOR PRICE EQUALIZATION 4.1.3 Factor prices We can say something about factor prices if we assume identical technologies and rule out joint production. An immediate result comes from the property of the revenue function that factor demand curves must be downward slopin Applying this to the factor endowments and prices in Home and Foreign (noting that goods prices are equalized through trade), we get that (-W)t-V)≤0. Thus a country will have on average lower factor prices for factors it is rela- tively well endowed with Another important question concerns factor rewards in free trade vs autarchy. Given that goods prices are equalized in the free trade equilib- rium, we would expect factor prices to move closer together. Unfortunately this need not be the case. We would like to show that(u-v)(wa-wa)< (U-V(w-W), that is factor prices at free trade are "closer"than they were in autarchy. Since free trade is preferable to autarchy, using the homothetic equilibrium conditions above we have that a wv<wv Moreover, both w and w satisfy the constraint in the alternative definition of the revenue function, since the output price vector is the same in the two countries. This gives us and WV<wv two chains of inequalities that complete the argumell. could write down the If we new that wu≤ wav and that wV≤uav,we a,< wu< wu< wa and WaV<Wv≤uV≤uaV
CHAPTER 4. FACTOR PRICE EQUALIZATION 17 4.1.3 Factor prices We can say something about factor prices if we assume identical technologies and rule out joint production. An immediate result comes from the property of the revenue function that factor demand curves must be downward sloping. Applying this to the factor endowments and prices in Home and Foreign (noting that goods prices are equalized through trade), we get that (w − W)(v − V ) ≤ 0. Thus a country will have on average lower factor prices for factors it is relatively well endowed with. Another important question concerns factor rewards in free trade vs autarchy. Given that goods prices are equalized in the free trade equilibrium, we would expect factor prices to move closer together. Unfortunately this need not be the case. We would like to show that (v − V )(w a − Wa ) ≤ (v−V )(w−W), that is factor prices at free trade are “closer” than they were in autarchy. Since free trade is preferable to autarchy, using the homothetic equilibrium conditions above we have that w a v ≤ wv and WaV ≤ W V. Moreover, both w and W satisfy the constraint in the alternative definition of the revenue function, since the output price vector is the same in the two countries. This gives us wv ≤ W v and W V ≤ wV. If we new that W v ≤ Wa v and that wV ≤ w aV , we could write down the two chains of inequalities that complete the argument: w a v ≤ wv ≤ W v ≤ Wa v and WaV ≤ W V ≤ wV ≤ w aV