CHAPTER 4. FACTOR PRICE EQUALIZATION and we would have the desired result. But the last two inequalities need not hold in general, so we cannot conclude that commodity trade leads to dimin ishing factor price differences. Although the notion is intuitively appealing trade in goods and factor mobility are not always substitutes. Thus policy arguments based on that notion have no solid theoretical foundations 4.2 Factor price equalization The question of factor price equalization(FPE) is related to the previous discussion. When factor prices are equalized through trade, they are obvi- ously closer together than in autarchy. We know that FPE is not a general under what circumstances it can result. There are two reasons for such. o property of a free trade equilibrium, but it is nevertheless important to terest in fpe. first, if there is fpe. there are no incentives for factors to move and trade in goods is a perfect substitute for trade in factors. Second when FPe prevails it is much easier to describe trade patterns. But when are factor prices indeed equalized? We assume no joint production, identical technologies and constant re- turns to scale. so that we are able to use the unit cost functions derived earlier. Let w and w be the equilibrium factor price vectors, then the free trade equilibrium conditions are as follows b(u)≥ p and a≥0 b(W)≥ p and X≥0 )x=0 a(W)X r+x=∑d(,w)+∑D"(p,WV) FPE means that the two factor prices, w and W are identical. This means that unit costs are the same for each good in the two countries. Assuming that all goods are essential and thus have to be produced somewhere, for eachi the nonpositive profit condition has to hold with equality. Let us use w and p for the common factor price and price vectors, i for total production (i.e. i=a+X), and let us add up the two factor market clearing conditions
CHAPTER 4. FACTOR PRICE EQUALIZATION 18 and we would have the desired result. But the last two inequalities need not hold in general, so we cannot conclude that commodity trade leads to diminishing factor price differences. Although the notion is intuitively appealing, trade in goods and factor mobility are not always substitutes. Thus policy arguments based on that notion have no solid theoretical foundations. 4.2 Factor price equalization The question of factor price equalization (FPE) is related to the previous discussion. When factor prices are equalized through trade, they are obviously closer together than in autarchy. We know that FPE is not a general property of a free trade equilibrium, but it is nevertheless important to see under what circumstances it can result. There are two reasons for such interest in FPE. First, if there is FPE, there are no incentives for factors to move and trade in goods is a perfect substitute for trade in factors. Second, when FPE prevails it is much easier to describe trade patterns. But when are factor prices indeed equalized? We assume no joint production, identical technologies and constant returns to scale, so that we are able to use the unit cost functions derived earlier. Let w and W be the equilibrium factor price vectors, then the free trade equilibrium conditions are as follows: b(w) ≥ p and x ≥ 0 b(W) ≥ p and X ≥ 0 a(w)x = v a(W)X = V x + X = X h d h (p, wvh ) +X H D H(p, W V H). FPE means that the two factor prices, w and W are identical. This means that unit costs are the same for each good in the two countries. Assuming that all goods are essential and thus have to be produced somewhere, for each j the nonpositive profit condition has to hold with equality. Let us use wˆ and ˆp for the common factor price and price vectors, ˆx for total production (i.e. ˆx = x+X), and let us add up the two factor market clearing conditions
CHAPTER 4. FACTOR PRICE EQUALIZATION Then we get that b(a)=p a()=+V =∑(m)+∑D(mv If you look at the second set of equalities, you can see that these would b the equilibrium conditions for a world where both factors and goods are mo- bile. in other words where there are no countries. We will call this construct the integrated world equilibrium. Thus, in essence we have proved that when factor prices are equalized, the world can achieve the integrated equilibrium through trade in goods alone. Thus even if factor movements were possible they would not take place when FPe prevails. The construct of integrated quilibrium also shows us when factor price equalization will occur. The first set of equations(no pure profits)must hold in a free-trade equilibrium with equal factor prices. The last set of equations(goods markets clear) is also the same in the integrated equilibrium and in free trade. The only difference is that with two countries factor markets have to clear separately, with a and X between 0 and i. Thus a trade equilibrium with equal factor prices in the two countries exists when a(0)x=t,x∈[0., has a solution. In words, if using the techniques of production that preva the integrated equilibrium(a) we can split production into two nonnega tive parts that exhaust factor supplies in both countries, we can have FPe Otherwise. we cannot Formally, the condition for FPE is a condition on the distribution of factor endowments. Assuming the integrated equilibrium choices of P and i are unique, the set of endowments that are consistent with FPe is given by 业={vl=a()x,x∈0,} Of course if v is in 4, the equivalent condition on the foreign country's endowment is also satisfied. Thus FPe depends on the likelihood of v fallin nto y. In the next chapters we will look at that likelihood in different cases
CHAPTER 4. FACTOR PRICE EQUALIZATION 19 Then we get that b( ˆw) = ˆp a( ˆw)ˆx = v + V xˆ = X h d h (ˆp, wvˆ h ) +X H D H(ˆp, wVˆ H). If you look at the second set of equalities, you can see that these would be the equilibrium conditions for a world where both factors and goods are mobile, in other words where there are no countries. We will call this construct the integrated world equilibrium. Thus, in essence we have proved that when factor prices are equalized, the world can achieve the integrated equilibrium through trade in goods alone. Thus even if factor movements were possible, they would not take place when FPE prevails. The construct of integrated equilibrium also shows us when factor price equalization will occur. The first set of equations (no pure profits) must hold in a free-trade equilibrium with equal factor prices. The last set of equations (goods markets clear) is also the same in the integrated equilibrium and in free trade. The only difference is that with two countries factor markets have to clear separately, with x and X between 0 and ˆx. Thus a trade equilibrium with equal factor prices in the two countries exists when a( ˆw)x = v, x ∈ [0, xˆ] has a solution. In words, if using the techniques of production that prevail in the integrated equilibrium (a[ ˆw]) we can split production into two nonnegative parts that exhaust factor supplies in both countries, we can have FPE. Otherwise, we cannot. Formally, the condition for FPE is a condition on the distribution of factor endowments. Assuming the integrated equilibrium choices of ˆw, ˆp and ˆx are unique, the set of endowments that are consistent with FPE is given by: Ψ = {v|v = a( ˆw)x, x ∈ [0, xˆ]}. Of course if v is in Ψ, the equivalent condition on the foreign country’s endowment is also satisfied. Thus FPE depends on the likelihood of v falling into Ψ. In the next chapters we will look at that likelihood in different cases
CHAPTER 4. FACTOR PRICE EQUALIZATION 4.2.1 More factors than good In this case fpe is a measure zero event To see this. note that the di mensionality of a(w) is at most n, the number of goods. Then y will be a subspace of the n dimensional space, whereas the factor endowment space has a dimension of m>n. Thus it is very unlikely that factor endowments will fall into y. and we can rule out fpe as accidental in this case See graph at lecture 4.2.2 At least as many goods as factors n this case the dimensionality of y will be m, assuming that technologies for producing different goods are different, that is a(u)a(w). We will assume this to be the case. Then FPE will have positive measure, and its numerical probability will depend on details of technology. The graphs in DN are very instructive! An interesting problem emerges when n >m. In this case there are m equations in a(w) U, which means that the production plan is not unique. Thus many production vectors are compatible with the same distribution of endowments. On the other hand, world output is uniquely determined by demand, so the integrated equilibrium is unique. There is discussion in Dn about the effect of adding more goods, you can read it there n general, adding more goods might increase or decrease the likelihood of FPE. HK has a chapter on adding non-traded goods, the main point is that we need at least as many traded goods as factors for the FPe set to have positive measure 4.3 The pattern of trade under FPE We saw earlier that in general we can only show a correlation result between autarchy prices and trade pattern, and the link between factor endowment and prices is even weaker. We will now show that with Fpe we have much stronger results. To focus on endowments, we will have no joint production identical technologies and identical homothetic preferences. Since FPE is unlikely when there are more factors than goods, we will only look at the other case, that is n m. Since when n>m production patterns are inde- terminate, it is futile to have results on commodity trade. Even when m=n there is no strong relationship between factor endowments and commodity
CHAPTER 4. FACTOR PRICE EQUALIZATION 20 4.2.1 More factors than goods In this case FPE is a measure zero event. To see this, note that the dimensionality of a( ˆw) is at most n, the number of goods. Then Ψ will be a subspace of the n dimensional space, whereas the factor endowment space has a dimension of m > n. Thus it is very unlikely that factor endowments will fall into Ψ, and we can rule out FPE as accidental in this case. See graph at lecture! 4.2.2 At least as many goods as factors In this case the dimensionality of Ψ will be m, assuming that technologies for producing different goods are different, that is a l ( ˆw) 6= a j ( ˆw). We will assume this to be the case. Then FPE will have positive measure, and its numerical probability will depend on details of technology. The graphs in DN are very instructive! An interesting problem emerges when n > m. In this case there are m equations in a( ˆw)x = v, which means that the production plan is not unique. Thus many production vectors are compatible with the same distribution of endowments. On the other hand, world output is uniquely determined by demand, so the integrated equilibrium is unique. There is a discussion in DN about the effect of adding more goods, you can read it there. In general, adding more goods might increase or decrease the likelihood of FPE. HK has a chapter on adding non-traded goods, the main point is that we need at least as many traded goods as factors for the FPE set to have positive measure. 4.3 The pattern of trade under FPE We saw earlier that in general we can only show a correlation result between autarchy prices and trade pattern, and the link between factor endowments and prices is even weaker. We will now show that with FPE we have much stronger results. To focus on endowments, we will have no joint production, identical technologies and identical homothetic preferences. Since FPE is unlikely when there are more factors than goods, we will only look at the other case, that is n ≥ m. Since when n > m production patterns are indeterminate, it is futile to have results on commodity trade. Even when m = n there is no strong relationship between factor endowments and commodity
CHAPTER 4. FACTOR PRICE EQUALIZATION trade patterns, unless n m= 2. On the other hand, we have very nice results on the factor content of trade, and this is what we will look at now With identical homothetic preferences, consumers will spend a share of their income on each good, where the share only depends on relative prices Let th be the vector of factors embodied in country k's imports. This is the difference between the factor content of consumption and the factor content of production in country k. The latter, of course, is just Uk, the factor endowment of country k. For the factor content of consumption, we know that spending on each good is a function only of the equilibrium prices, p Since preferences are identical, each country will spend the same share of its income on a particular good. Then, for a particular good 3, market clearing implies the following Pics PcI where c, is world consumption of good j and v is world endowment of fac- tors(and hence wv is world income). Then the factor content of country k consumption is given as follows a()4=∑(uo)4 ∑()n/ a) Using the notation s=wu/(v) for country k's share of world income, we have that The equation tells us that a country exports the services of factors with which it is relatively well endowed compared to the world. If there is balanced trade, then some elements of the net factor import vector will be positive and others negative. If we rank factors by their relative endowment size (i.e. uF/ui), there will be a cutoff such that all factors above the cutoff are
CHAPTER 4. FACTOR PRICE EQUALIZATION 21 trade patterns, unless n = m = 2. On the other hand, we have very nice results on the factor content of trade, and this is what we will look at now. With identical homothetic preferences, consumers will spend a share of their income on each good, where the share only depends on relative prices. Let t k v be the vector of factors embodied in country k’s imports. This is the difference between the factor content of consumption and the factor content of production in country k. The latter, of course, is just vk, the factor endowment of country k. For the factor content of consumption, we know that spending on each good is a function only of the equilibrium prices, p. Since preferences are identical, each country will spend the same share of its income on a particular good. Then, for a particular good j, market clearing implies the following: X k pj c k j = X k sjwvk ⇒ sj = pj cj wv , where cj is world consumption of good j and v is world endowment of factors (and hence wv is world income). Then the factor content of country k consumption is given as follows: a(w)c k = X j a j (w)c k j = X j a j (w)sjwvk /pj = X j a j cj wvk wv = wvk wv v. Using the notation s k = wvk/(wv) for country k’s share of world income, we have that t k v = s k v − v k . The equation tells us that a country exports the services of factors with which it is relatively well endowed compared to the world. If there is balanced trade, then some elements of the net factor import vector will be positive and others negative. If we rank factors by their relative endowment size (i.e. v k i /vi), there will be a cutoff such that all factors above the cutoff are
CHAPTER 4. FACTOR PRICE EQUALIZATION exported and the others imported. This is the famous Vanek chain argument for the factor content of trade. Notice that you can construct such a chain even if trade is not balanced, but then we have to use the country's share in world spending, and it is possible that a country exports or imports all factor services
CHAPTER 4. FACTOR PRICE EQUALIZATION 22 exported and the others imported. This is the famous Vanek chain argument for the factor content of trade. Notice that you can construct such a chain even if trade is not balanced, but then we have to use the country’s share in world spending, and it is possible that a country exports or imports all factor services