A dual definition for the factor content of trade and its effect on factor rewards in us manufacturing sector.

Dells, Agelos ; Mamuneas, Theofanis P.

I. INTRODUCTION

The possible relationship between international trade and wage inequality in developed countries has been a very important and regularly debated topic for both academics and politicians in the last two decades. Unskilled workers in many developed countries and especially in the United States have seen a significant decline in their relative wages, while at the same time international trade increased considerably. Some have argued that the increase of international trade is likely to explain this decline of relative wages. Trade economists have approached this question using the Heckscher-Ohlin model, from various angles. The first is based on the Factor Content of Trade (FCT) theorem of Vanek (1968) and the work of Deardorff and Staiger (1988), where changes in the volume of net exports are transformed (via an input-output matrix) into changes in relative factor rewards (Borjas et al. 1992; Katz and Murphy 1992; Wood 1995); the second is based on the traditional Stolper-Samuelson theorem, where changes in product prices cause changes in factor rewards (Leamer 1998, 1994; Baldwin and Cain 2000; Harrigan and Balaban, 1999).

Furthermore, to analyze the increasing wage inequality in the United States, Feenstra and Hanson (1996, 1999) and Sitchinava (2007), among others, introduce outsourcing in a Stolper-Samuelson framework, while Michaels (2008), relying on the Heckscher-Ohlin framework, assesses the effects of trade on wage inequality in the U.S. states, by investigating the effects of highway infrastructure. Finally, there are recent papers that follow different theoretical frameworks in order to analyze the increasing wage inequality. For instance, Blum (2008) uses a Ricardo-Viner model, while Zhu and Trefler (2005) use a model that combines Heckscher-Ohlin and Ricardian environments.

The FCT approach has been heavily criticized on the ground that it lacks a solid theoretical foundation and especially that FCT is not related with factor prices. For instance, Panagariya (2000), Learner and Levinsohn (1995) and Learner (2000) argue that FCT calculates quantities of indirectly exported and imported factors via international trade, but according to the Stolper-Samuelson theorem, it is product prices and not factor quantities that are related with factor prices. Yet, by introducing the concept of the Equivalent Autarkic Equilibrium (EAE), Deardorff and Staiger (1988) provide the theoretical foundation and show under which assumptions the FCT and relative wages are related (see also, Deardorff 2000; Krugman 2000; Wood 1995).

In this paper, in contrast to all previous FCT studies which rely on the use of input-output matrices to calculate the FCT (see Borjas et al. 1992; Katz and Murphy 1992; Wood 1995), we calculate the FCT by directly estimating the endowments required to achieve the EAE. This is accomplished by estimating a revenue function similar to Harrigan and Balaban (1999). We assume the revenue function to be of the Symmetric Normalized Quadratic functional form, which is more attractive to other functional forms (like the Translog that has been used extensively), because it has the important property of flexibility when convexity and concavity are imposed. We also allow for a more general technology that is joint in output quantities. Under such technology the analysis departs from the hypothesis of Factor Price Equalization (FPE). (1) We find that the FCT for capital is positive, the FCT for skilled labor is negative, but quite close to zero, while the FCT of unskilled labor is negative and large in magnitude. Hence, there is no Leontief Paradox in the United States for the period 1965-1991 in our framework. This result is consistent with the findings of Bowen et al. (1987), Davis and Weinstein (2001), and Feenstra and Hanson (2000) in terms of relative factor abundance.

Then, by using the quadratic approximation lemma (Diewert 1976, 2002), we are able to decompose the growth rate of factor rewards of trade equilibria (TE) to the growth rate of FCT, the growth rate of endowments and technological change. We find that the growth rate of the rewards for both types of labor gains from FCT Effect, while the rewards to capital have losses. The endowment effect is positive for the growth of the wages of unskilled workers and negative for the wages of skilled workers and the rewards to capital. Lastly, technological change has a positive effect in all factor rewards with capital experiencing the highest gains and unskilled labor the least. Finally, it seems that technological change is the most important determinant for the decline in relative factor rewards for unskilled workers in the United States from 1967 to 1991. This is in accordance with most studies of all different approaches with the exception of Wood (1995), Learner (1998), Bivens (2007), and Sitchinava (2007).

The rest of the paper is organized into six sections. Section II develops the theoretical model and provides a dual definition of the factor content of trade. Section III contains a discussion of the empirical specification and estimation of the revenue function. Section IV presents the FCT for each factor and discusses the Leontief Paradox. In Section V, we decompose the growth rate of factor rewards into an FCT effect, an endowment effect and a technology effect and present the results based on this decomposition. Finally, the last section concludes the paper.

II. THE MODEL

In this section, we develop a general equilibrium model for a trading economy using duality (Dixit and Norman 1980). The production side of the economy is described by a revenue function, while the consumption side is described by an expenditure function. The use of duality, and more specifically the implementation of a revenue function, is preferred because it complies with the standard assumptions made in international trade theory that product prices and endowments are given, while factor prices and outputs are the endogenous variables to be determined. (2)

Let F (y, v, t) = 0 be a transformation function for an economy with a linearly homogeneous technology, which produces y = ([y.sub.1], ..., [y.sub.n]) goods with the use of v = ([v.sub.1], ..., [v.sub.m]) inputs (n [greater than or equal to] m) in a perfect competitive environment where t is a time index that captures technological change. Then, at given international prices, p = ([p.sub.1], ..., [p.sub.n]) and domestic inputs v, there exists a competitive production equilibrium. In such equilibrium, we can think of the economy as one that maximizes the value of total output subject to the technological and endowment constraints. In other words, there is a revenue or Gross Domestic Product (GDP) function such that:

(1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The revenue function has the usual properties, that is, it is increasing, linearly homogeneous and concave in v and nondecreasing, linearly homogeneous and convex in p. In addition, if R(p, v, t) is differentiable then from Hotteling's Lemma (Diewert 1974) the equilibrium output and factor rewards are:

(2) y(p, v, t) = [R.sub.p](p, v, t)

(3) w(p, v, t) = [R.sub.v](p, v, t)

where [R.sub.p] and [R.sub.v] are the vectors of first partial derivative of the revenue function with respect to product prices and endowments, respectively.

On the consumption side the economy's preferences defined over the n goods are represented by an expenditure function, which is continuous and twice differentiable on prices:

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where u is the level of utility and x = ([x.sub.1], ..., [x.sub.n]) is the consumption bundle. The expenditure function is nondecreasing, linear homogenous and concave in prices and increasing in u. From Shepherd's Lemma (Diewert 1974) the consumption vector of the economy is:

(5) x(p, u) = [E.sub.p](p, u)

where [E.sub.p] is the vector of first partial derivative of the expenditure function with respect to product prices.

The trade equilibrium is defined as

(6a) R(p, v, t) = E(p,u)

(6b) T = [R.sub.p](p, v, t) - [E.sub.p](p, u)

that is the total value of production should be equal to the total expenditure for the economy, which implies trade balance and the difference between production and consumption gives the economy's vector of net exports, T.

Consider now a hypothetical equilibrium, the Equivalent Autarky Equilibrium introduced by Deardorff and Staiger (1988), where production equals consumption, at the same product prices and at the same utility level as in the trading equilibrium. This equilibrium can be achieved by changing the initial endowment of the economy in such a way that the economy is producing what it desires to consume, having no incentive to trade with other countries. Hence, the vector of net exports is going to be a vector of zeros and trade is by definition balanced

(7a) R(p, [v.sup.e], t) = E(p, u)

(7b) [R.sub.p](p, [v.sup.e], t) = [E.sub.p](p, u)

where [v.sup.e] is the Equivalent Autarky Equilibrium endowments vector and p, u the price vector and utility level, respectively, as in the trade equilibrium.

[FIGURE 1 OMITTED]

In Figure 1, following Krugman (2000), we depict the trading and Equivalent Autarky Equilibria. In the Trade Equilibrium, the economy is producing where the production possibilities frontier DE is tangent to the relative product prices line AB, at P, while the economy is consuming at C where the relative product prices line is tangent to the indifference curve [U.sup.o]. The economy is exporting [Y.sub.1] - [X.sub.1] units of good 1 and imports [X.sub.2] - [Y.sub.2] units of good 2. The Equivalent Autarky Equilibrium is depicted at C. There, the economy is endowed with the necessary inputs that allow the production of its consumption bundle at the trade relative product prices AB. At the EAE, the production possibilities frontier is FG, both consumption and production takes place at C and therefore the trade volume is zero. Note that at the trading equilibrium P and at the EAE C the level of utility is the same and because product prices are also unchanged the vector of consumption is unaltered. Under the assumption of balanced trade, (3) GDP and the economy's total expenditure would be identical in both equilibria.

As consumption is the same in both equilibria then from Equations (6b) and (7b) we have

(8) Rp(p, [v.sup.e], t) = [R.sub.p](p, v, t) - T

and therefore we can explicitly solve from Equation (8) for the EAE endowments vector [v.sup.e] by knowing the net exports and the revenue function of the economy. (4) Assuming that the implicit function theorem holds, [absolute value of [R.sub.pv](p, [v.sup.e], t)] [not equal to]0, (5) we can solve for the EAE endowment vector [v.sup.e](p, v, t; T) which is going to depend on the trade equilibrium prices, initial endowment, technology, and the net export vector. Then, the factor content of trade is defined as the difference between the actual endowments in a trading equilibrium and the endowments at the Equivalent Autarky Equilibrium,

(9) f = v - [v.sup.e](p, v, t; T)

In the literature, the usual definition of FCT is just the product of an input requirement matrix, [GAMMA], times the trade vector T (see for example Deardorff and Staiger 1988). Harrigan (2003) has shown that if there is nonjointness in output quantities, the input requirement matrix [GAMMA] is equal to [R.sup.-1.sub.pv] and therefore the factor content of trade will be equal to [R.sup.-1.sub.pv]T. It is not difficult to show that our definition of FCT is identical to [R.sup.-.sub.pv]T under the nonjointness assumption. Under this assumption a revenue function can be written as R(p, v, t) = r(p, t)v, then the vector of outputs is [R.sub.p] = [r.sub.p]v, where [r.sub.p] is the vector of partial derivatives of r(p, t) with respect to product prices and [R.sub.pv] = [r.sub.p] which is independent of the endowment vector. From Equation (8) we have that T = [R.sub.p](p, v, t) - [R.sub.p](p, [v.sup.e], t) = [r.sub.p]v - [r.sub.p][v.sup.c] = [r.sub.p](v - [v.sup.e]) = [R.sub.pv] f, and therefore f = [R.sup.-1.sub.pv]T. Therefore our definition of FCT given by Equation (9) is equivalent to the usual definition appearing in the literature under the assumption of nonjointness. However, it is a generalization to wider technologies even in cases where jointness in output quantity is present. For instance, the proposed dual definition of FCT allows for a general technology, in contrast to the standard definition which implicitly assumes Leontief technology. In addition, the dual definition of FCT is well defined in the case of jointness in output quantity while the trade vector input requirement definition of FCT becomes empirically intractable. Furthermore, we define the FCT using net output and this takes into consideration the potential problem that arises in the presence of nontraded intermediate inputs when the usual definition is used. (6)

III. ECONOMETRIC SPECIFICATION AND ESTIMATION

The revenue function is assumed to have the Symmetric Normalized Quadratic (SNQ) functional form as discussed in the work of Kohli (1991, 1993):

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where p and v are the product prices and input endowment vectors, respectively, and t is an index of exogenous technological change. There are N(N -1) + M(M - 1) + (N x M) + 2 unknown parameters [a.sub.ih], [b.sub.jl], [c.sub.ij], [d.sub.i], [e.sub.j], [h.sub.t], and [h.sub.tt], where i, h = 1, ..., N and j, l = 1, ..., M. There are also N + M predetermined parameters [[theta].sub.i] and [[psi].sub.j]. In particular, [[theta].sub.i] and [[psi].sub.j] are set equal to the share value of each product and input, respectively, at the base year.

By construction the SNQ function is linearly homogeneous in p and v. Symmetry conditions are imposed [a.sub.ih] = [a.sub.hi]; [b.sub.jl] = [b.sub.lj] and normalization requires some additional restrictions:

(11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This functional form is attractive because it is a flexible functional form that retains its flexibility under global imposition of convexity and concavity in prices and endowments, respectively. (7) The necessary and sufficient condition for global concavity in inputs is that the matrix B = [[b.sub.jl]] is negative semidefinite and for global convexity that the matrix A = [[a.sub.ih]] is positive semidefinite. If these are not satisfied then they are imposed following Diewert and Wales (1987) without removing the flexibility properties of the revenue function.

On the basis of Equation (10), the rewards of the jth factor become:

(12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Similarly the output supply of the ith good becomes:

(13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The estimating model is the Equation sets (12) and (13) together with the parameter restrictions (11). The errors related to Equations (12) and (13) are assumed to be identically and independently distributed with zero expected value and a positive definite covariance matrix. As the United States is a large open economy, we consider product prices to be endogenous in the estimating model as in the works of Kohli (1993) and Harrigan and Balaban (1999). Equations (12) and (13) are jointly estimated by the Iterative Three Stages Least Square (13SLS) estimator applied to data for the U.S. manufacturing sector over the period from 1965 to 1991. As instruments one year lagged values of the product prices and endowments are used. There are six equations, three relating to outputs and three relating to factor rewards. The goods are exportable, importable, and nontradeable, and the three factors of production are capital, skilled, and unskilled labor. We use data for the value and price of capital and aggregate labor from Dale Jorgenson's 35 KLEM dataset. In order to decompose labor into skilled and unskilled, we have used the NBER Mare-Winship Data. Because of limitations of obtained from the Centre for international Data at the University of California Davis.

The assumption of balanced trade is not satisfied by the data. For that reason, the actual trade volumes for each good are adjusted according to the share of output relative to total revenue in the economy in order to guarantee balanced trade. (8) Finally, data for the output deflators are used from the Bureau of Economic Analysis.

Table 1 shows the estimated parameters and the [R.sup.2] for the system of the six equations. The revenue function is linearly homogeneous in prices and inputs, but initially convexity in prices and concavity in inputs were not satisfied. Following the method proposed by Diewert and Wales (1987), we impose convexity for product prices and concavity for input quantities. The hypothesis of convexity and concavity cannot be rejected at a 5% level of significance (Wald test statistic(4)= 32.7). The jointness in output quantities cannot be rejected at a 5% level of significance (Wald test statistic(2) = 29.1), which is in accordance with the more general technology used above. In addition, the hypothesis of non-technological change is rejected (Wald test statistic(6) = 534). Furthermore, following Diewert and Ostensoe (1988), we test for the assumption of constant returns to scale (CRS) by estimating an augmented SNQ revenue function that includes an additional term, ([[summation].sup.N.sub.i=1] [[theta].sub.i] [p.sub.i]) ([[summation].sup.M.sub.j=1] [u.sub.j] [v.sub.j]) [([[summation].sup.M.sub.j=1] [[psi].sub.j] [v.sub.j]).sup.-1], where [u.sub.j] are unknown parameters. Using a quasilikelihood ratio test CRS cannot be rejected at 5% level of significance (QLR (3) statistic is 7.17 < [X.sup.2.sub.c] is 7.82).

In Table 2, we report the estimated price and endowment elasticities for all goods and factors. All own price elasticities of output are positive and well below unity, suggesting that the output supplies are inelastic. In addition, an increase in the price of exportables reduces the quantity produced for both importable and nontradable goods. While an increase on importable goods price increases the output of nontradable goods.

An increase of capital endowment leads to a decline in the output of both the importable and the nontradable goods, while it increases the output of the exportable goods. An increase in the endowment of skilled labor increases the output of all goods, while an increase in the endowment of unskilled labor increases the output of the importable and exportable goods, but it reduces the output of nontradable goods. Technological change has a positive effect on the production of the exportable goods and a negative effect on the production of other goods.

The rewards of all factors gain from an increase in exportable goods price. An increase in importable goods price decreases capital rewards, while it increases the wages of both types of labor. An increase in nontradable goods price reduces the rewards to both capital and unskilled labor, while it increases the rewards to skilled labor. All own inverse factor price elasticities are negative and inelastic; the only exception being the capital price elasticity (-1.14%). Moreover, capital is a gross-substitute with skilled and unskilled labor while skilled and unskilled labor are gross-complements. Finally, technological change appears to enhance the rewards to both capital and skilled labor, but reduce the rewards to unskilled labor.

Overall these results are consistent with the studies of Harrigan (2000) and Kohli (1993) in terms of the sign of the elasticities, but in some cases differ in magnitude. However, it should be noted that the specification of the model, aggregation of inputs and outputs and time period are different among these studies.

IV. FACTOR CONTENT OF TRADE

The estimated parameters of the revenue function are used in order to calculate the FCT for each input. In particular, solving Equation (8) for [v.sup.e] and then using Equation (9), allow us to obtain the factor content of trade, [f.sub.j], for each input for the period 1965-1991. The FCT for all three factors are plotted in Figure 2. We observe that FCT of capital, [f.sub.K], is positive and generally increasing throughout our sample period. The FCT of both skilled, [f.sub.S], and unskilled, [f.sub.U], labor is negative and declines until 1986 and then increases until 1991. The FCT of skilled labor has relatively the smallest magnitude compared with the other inputs. Hence, the U.S. economy was exporting the services of capital and importing the services of both types of labor for all the years between 1965 and 1991. The net exports of capital services in 1965 were 16.34 billion dollars, (9) these reached a maximum of 62 billion dollars in 1986 and fell to 54.30 billion dollars in 1991. The net imports of skilled labor services rose from 9.89 billion dollars in 1965 to 44.04 billion dollars in 1986 and then were reduced to 32.50 billion dollars in 1991. Similarly, the net imports of unskilled labor increased from 20.45 billion dollars in 1965 to 96.88 billion dollars in 1986 and then decreased to 68.48 billion dollars in 1991.

Thus, it is evident that, for this period, there is no Leontief Paradox in the U.S. economy. An explanation of the absence of the Leontief Paradox could be the disaggregation of labor input to skilled and unskilled which is consistent with some of the early explantions in the literature (Kenen 1965; Baldwin 1971; Winston 1979). An alternative explanation for such absence could be that no FPE is assumed in our analysis, as in the work of Davis and Weinstein (2001), who also find that there is no Leontief Paradox in this case.

[FIGURE 2 OMITTED]

Our finding is consistent with the analysis of Leamer (1980). Learner showed that in a multifactor, multiproduct H-O-V environment, a country is revealed by trade to be relatively abundant in a particular factor compared to any other factor, if the FCT of this factor is positive and the FCT of the other is negative. Hence, capital is revealed by trade to be relatively abundant compared to either types of labor in the U.S. economy for the period 1965-1991. In addition, Leamer (1980) showed under which condition a country with negative FCT for two inputs is revealed by trade to be relatively abundant in one of them. A country is revealed by trade to be relatively more abundant in an input if the ratio of the FCT of that input to the FCT of the other input is smaller than their ratio used in the production. Hence, we find that trade reveals that skilled labor is relatively abundant to unskilled labor, because the share of skilled labor imported is less than the share of unskilled labor imported in the U.S. economy between 1965 and 1991.

In comparison, our findings are very similar with Bowen et al. (1987) and Feenstra and Hanson (2000). However, for some categories of skilled labor we find an opposite sign from Bowen et al. (1987); nevertheless, we employ a different definition of skilled labor from them. We also find overall similar results to Feenstra and Hanson (2000) regarding the ordering of relative abundance for capital, skilled, and unskilled labor even though they have used a finer level of aggregation than ours.

To summarize, for all of the years in the sample period more unskilled and skilled labor could have been employed in a hypothetical EAE relative to capital, but more unskilled labor could have been employed relative to skilled labor. Therefore, in the U.S. manufacturing sector, there is a clear ordering of factor abundance revealed by trade. Capital is the most abundant factor relative to both types of labor, while skilled labor is relatively more abundant when compared with unskilled labor between 1965 and 1991. And therefore, we find no evidence of a Leontief Paradox.

V. FACTOR REWARDS DECOMPOSITION

So far we have discussed the definition of the Equivalent Autarky Equilibrium, the estimation of the revenue function for the United States and the calculation of the FCT using duality in the case of jointness in output quantities. In this section, our goal is to establish a general relationship between changes in factor prices in one side and changes of endowments, FCT, and technology in the other. For this reason, we first show how the difference between the factor rewards in the two equilibria can be approximated.

[FIGURE 3 OMITTED]

In Figure 3, we portray two TE and also their respective EAE at time periods t and s. For each TE, [P.sup.t] and [P.sup.s], the factor rewards are given by [w.sub.t] = [R.sub.v] ([p.sub.t], [v.sub.t], t) and [w.sub.s] = [R.sub.v] ([p.sub.s], [v.sub.s], s), respectively. Recall that from Equation (8), we can obtain the endowments vector at the two EAE, [C.sup.t] and [C.sup.s]. Hence, the factor rewards at the EAE are given by [w.sup.e.sub.t] = [R.sub.v] ([p.sub.t], [v.sup.e.sub.t], t) and [w.sup.e.sub.s] = [R.sub.v] ([p.sub.s], [v.sup.e.sub.s], s), respectively. Our objective is to find the effect of FCT changes on changes of rewards over time. Instead of comparing the factor rewards between equilibria [P.sup.t] and [P.sup.s] directly, we go through the Equivalent Autarky Equilibria [C.sup.t] and [C.sup.s]. In other words, the difference in factor rewards between periods t and s is given by the difference between the TE and EAE for period t minus the difference between TE and EAE for period s plus the difference between the EAE in t and s. This enables us to link factor rewards changes with changes in endowments, FCT and technology. This approach is novel, because we explicitly model for and incorporate in the decomposition the effect of technological change on factor rewards changes. This is in contrast to most such studies where the effects of technology are estimated residually.

By using the quadratic approximation lemma (Diewert 1976, 2002) the TE factor rewards [w.sub.t] = [R.sub.v] ([p.sub.t], [v.sub.t], t) at period t, evaluated at the EAE endowments [v.sup.e.sub.t] are

(14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where the matrix [[bar.R].sub.vv] = 1/2([R.sub.vv] + [R.sup.e.sub.vv]) has a typical entry [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] that is the mean effect of a change in the lth endowment on the rewards of the jth factor evaluated at the trade and equivalent autarky equilibrium at period t. Totally differentiating Equation (14) with respect to time t, we get:

(15) [dw.sub.t]/dt = [dw.sup.e.sub.t]/dt + [[bar.R].sub.vv] d[f.sub.t]/dt

Therefore Equation (15) relates the change in factor rewards at the trade equilibrium with the change of factor rewards at the EAE plus the changes of factor content of trade. (10)

Consider now the rewards at the Equivalent Autarky Equilibrium and note that because the equilibrium price is endogenous it would be a function of endowments and exogenous technical change that is [p.sub.t] = p([v.sup.e.sub.t]. t) and hence the factor rewards at EAE can be written as

(16) [w.sup.e.sub.t] = [R.sub.v](p([v.sup.e.sub.t], t), [v.sup.e], t)

Totally differentiating Equation (16) with the respect to t we get:

(17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Substituting Equation (17) in Equation (15), noting that from the definition of factor content of trade [dv.sup.e.sub.t]/dt = [dv.sub.t]/dt - d[f.sub.t]/dt and collecting terms we have that

(18) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Expression (18) relates the changes of the observed rewards at trade equilibrium to the changes of FCT of all factors, [f.sub.t], endowments, [v.sub.t] and exogenous technical change, t. It is a generalization of Deardorff and Staiger (1988) and also of Learner (1998). If we assume no technological change and that the endowments remain constant, the change in factor rewards will be just a function of the change of the FCT. In addition, if there is nonjointness in output quantities or [R.sub.pv] is locally independent of v, factor rewards and consequently their changes between the trade and the equivalent autarky equilibrium will be identical. Then the change of factor rewards will collapse to [dw.sub.t]/dt = -[R.sup.e.sub.vp]([[partial derivative]p.sub.t]/[[partial derivative]v.sup.e.sub.t])(d[f.sub.t]/dt) similar as in Deardorff and Staiger (1988).

However, decomposition in Equation (18) depends on the demand side of the economy and in particular on [partial derivative]p/[[partial derivative]v.sup.e] and [partial derivative]p/[partial derivative]t. From Equation (7b), the matrix of first partial derivatives of product prices with respect to EAE endowments is [partial derivative]p /[[partial derivative]v.sup.e] = -[([R.sub.pp] - [E.sub.pp]).sup.-1] [R.sub.pv] and the vector of first partial derivatives of product prices with respect to time is [partial derivative]p /[partial derivative]t = -[([R.sub.pp] - [E.sub.pp]).sup.-1] [R.sub.pt]. Therefore Equation (18) depends on the second derivatives of the expenditure function with respect to prices. Instead of making assumptions for the second derivatives of the expenditure function, in the empirical part of this section, we estimate directly [partial derivative]p /[[partial derivative]v.sup.e] and [partial derivative]p/[partial derivative]t by using a Seemingly Unrelated Regression Estimator and assuming that the relationship between the growth rate of prices, the growth rate of EAE endowments and technological change is given by,

(19) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where a [??] over a variable means growth rate, [a.sub.it] = ([[partial derivative]p.sub.t]/[partial derivative]t)(L/[p.sub.i] is the effect of technical change on price and [[beta].sub.ij] = ([[partial derivative]p.sub.i]/[[partial derivative]v.sup.e.sub.j]])/([v.sup.e.sub.j]/[p.sub.i]) is the elasticity of price with respect to EAE endowments.

Using Equations (19) and (18), we can write the rewards to the lth factor in growth form as

(20) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [[epsilon].sup.e.sub.li], [[eta].sup.e.sub.lj], [[eta].sup.e.sub.lt] are the elasticity of the factor rewards with respect to price, endowments, and time, respectively, and [[bar.[epsilon]].sup.e.sub.lj] is the weighted mean elasticity of the factor rewards with respect to endowments between the TE and EAE (see Appendix B for details). Equation (20) decomposes the growth rate of factor rewards into three terms. The first term is the change of the factor content of trade, the second term is the effect of the change of endowments and the last term the technological change effect.

Table 3 reports all factor rewards elasticities evaluated at EAE and Table 4 the parameter estimates from the price Equation (19). These elasticities are used to calculate the decomposition given by Equation (20). From Table 3, it is clear that an increase in the price of the exportable leads to a rise in the rewards for capital and unskilled labor and a decline for skilled labor's rewards. An increase in the price of the importable or nontradable goods increases the rewards of capital and skilled labor, while it reduces the rewards of unskilled labor, respectively. All own inverse factor price elasticities are negative as expected. Capital is a gross-substitute with skilled and unskilled labor, while skilled and unskilled labor are gross-complements. Technological change increases the rewards to capital and skilled labor and reduces the rewards to unskilled labor. The parameter estimates of Table 4 show that an increase in the EAE endowments of capital and unskilled labor reduces the equilibrium price of all goods while that of skilled labor works in the opposite direction. Finally, technology has a positive effect on the equilibrium price of exportable, importable, and nontraded goods.

In Table 5, the factor rewards decomposition of U.S. manufacturing is presented for the period 1967-1991. For this period, the factor rewards of capital, skilled and unskilled labor have increased, on average, by 2.4%, 7%, and 6%, respectively. The pattern that emerges is that the rewards growth differ according to the type of factor. In the case of capital and skilled labor, this can be mostly attributed to the effect of technological change, while in the case of unskilled labor to the factor content of trade and endowments changes. For both types of labor the FCT Effect has a positive impact on the growth of their factor rewards. On average for the period 1967-1991, the FCT Effect is 2.5% and 3.3% for skilled and unskilled labor, respectively, while the FCT Effect on the growth of the rewards of capital is negative, -1.8%. (11)

The Endowments Effect is negative for both capital and skilled labor's rewards, -13.35% and -1.27%, respectively, and positive for the growth rate of unskilled labor rewards, 2.10%. Capital is the factor with the highest growth in its endowments, followed by skilled labor and this growth had affected adversely the rewards for each of these two factors. On the opposite side, unskilled labor endowments have declined over the period of investigation and such decline in the supply of unskilled labor has caused, ceteris paribus, an increase on the rewards of unskilled labor.

The last column of Table 5 presents the Technology Effect. This effect is positive on average for the growth rate of factor rewards for all three inputs. The technological effect on the growth of capital rewards is the highest in magnitude, an average of 17.52%, followed by skilled labor's growth, 5.68%. For the same period the Technology Effect on the growth of unskilled labor rewards is only 0.50%.

Furthermore, in Table 5, we report the average growth rate of factor rewards and their decomposition for the periods 1967-1981 and 1982-1991. As the second column shows, the growth rate of the rewards to all factors has decreased significantly from the first to the second subperiod. But the ranking of the growth rates among the three factors remains unchanged, skilled labor experiences the highest growth and capital the lowest in both subperiods. Looking at the decomposition, the FCT Effect for capital and skilled labor rewards increases over time, while it decreases for unskilled labor. It is important to stress that in the second subperiod the FCT Effect is the highest for capital and the lowest for unskilled labor. This could be seen as evidence that for the period 1982-1991 international trade has benefited the growth of capital rewards the most and the one of unskilled labor the least. The Endowment Effect decreases over time for all three factors and is one of the reasons of the lower growth rates of factor rewards in the last subperiod. Similarly, the Technology Effect decreases for all three factors of production between the two subperiods. But while it remains positive for the rewards to capital and skilled labor, it becomes negative for unskilled labor in the last subperiod. This seems to suggest not only that technical change favors the rewards to capital and skilled labor but that it causes a decline in absolute terms for the growth of unskilled labor rewards.

It is clear from Table 5 that the difference between the rewards of capital and the two types of labor has narrowed, but that the wage inequality between the two types of workers has increased at a rate of slightly above 1% on average. This seems to be attributed to technological change that has considerably favored skilled labor much more in relation to unskilled labor. For the period 1967-1991, the FCT and the Endowment Effects are higher for unskilled labor than to skilled labor. The Technology Effect is positive for both types of labor over the whole sample period, but skilled labor's magnitude is much higher relative to unskilled labor's. Consequently, the observed increasing wage inequality between skilled and unskilled workers can be attributed to the Technology Effect. Hence, the widening on relative wages between skilled and unskilled workers seems to be the result of technological change that is biased toward skilled labor.

To summarize, our results indicate that the most important effect for the widening of the wage inequality is the Technology Effect, followed by the Endowment Effect, while the FCT Effect has the least impact. In particular, the Technology Effect contributes by 80% to the growth of skilled labor rewards and by 5% to the growth of unskilled labor rewards. This result is qualitative similar to Katz and Murphy (1992), Feenstra and Hanson (1996,1999), Baldwin and Cain (2000), Canals (2006), Bloom et al. (2008), Lawrence (2008), and Blum (2008), where they found evidence that skilled biased technological change is the most significant factor explaining the widening of U.S. wage inequality.

Regarding the FCT Effect, our results indicate that it is responsible for 37% of the growth of skilled labor rewards and for 56% of the growth of unskilled labor. Similar results were found by Borjas et al. (1992), Katz and Murphy (1992), Feenstra and Hanson (1996, 1999), Baldwin and Cain (2000), Canals (2006), Lawrence (2008), Michaels (2008) and Blum (2008). In particular, they concluded that trade contributed a little to the increase of wage inequality in the United States. On the other hand, our results differ to the ones found by Learner (1998), Bivens (2007), and Sitchinava (2007), where they argue that international trade played a crucial role for the opening of wage inequality in the United States.

Finally, we find that the Endowment Effect reduced the growth of skilled labor rewards and increased the growth of unskilled labor rewards by 18% and 35%, respectively. This result is similar in terms of direction with the findings of Baldwin and Cain (2000).

It is worth noticing that although our approach finds a result that seems to be the consensus in the literature for the period that we investigate, it is the first to our knowledge that allows for a decomposition of the growth of factor rewards in to three components that are explicitly modeled. That is the factor rewards growth is decomposed into a FCT Effect, an Endowment Effect and a Technology Effect.

VI. CONCLUSION

In this paper, we provide a dual definition for the factor content of trade based on the Equivalent Autarky Equilibrium introduced by Deardorff and Staiger (1988). This new definition of FCT allows for a more general technology that permits the existence of jointness in output quantities. This implies that in our analysis, we depart from FPE and that changes in input endowments cause changes in factor rewards. By estimating a symmetric normalized quadratic revenue function we calculate the FCT of capital, skilled and unskilled labor for the U.S. manufacturing sector for the period 1965-1991. Moreover by applying the quadratic approximation lemma to the difference of factor rewards between the trading equilibrium and EAE, we are able to link the observed growth of factor rewards to the growth of FCT, endowments and technological change for 1967-1991.

We find that the FCT of capital is positive while the FCT of skilled and unskilled labor are negative. Hence, for the period of investigation, the level of aggregation and under the technological specification of our model, it appears that there is no Leontief Paradox. This suggests that if the economy was at EAE, less capital would have been employed relative to skilled and unskilled labor. The positive sign of capital's FCT and the negative sign of the FCT of both types of labor implies that US manufacturing sector was a net exporter of goods that were more capital intensive between 1965 and 1991 and that capital was revealed by trade to be relatively more abundant to the two types of labor. In addition, following Leamer (1980) we show that skilled labor is revealed by trade to be relatively more abundant to unskilled labor, because the ratio of factor content of skilled labor to factor content of unskilled labor is smaller than the ratio of skilled to unskilled labor used in the production.

Overall factor rewards between the two types of labor and capital have narrowed but within labor wage inequality has increased. We find that the FCT Effect on factor rewards, for the period considered, is positive for the two types of labor (37% and 56% increase in the rewards to skilled and unskilled labor, respectively) and negative for capital. This is probably the result of the more general technology used in the analysis as the decomposition of the FCT Effect indicates in Table 5. The Endowments Effect is negative for the growth of capital's and skilled labor's rewards and positive for unskilled labor. Suggesting that the increasing endowments of capital and skilled labor have suppressed their rewards, ceteris paribus, while the opposite happened for unskilled labor. Technological change has benefited mainly the rewards to capital, but also skilled labor's rewards to a smaller magnitude. On the contrary, the rewards to unskilled labor had almost no gains arising from technological change. Finally, the increasing inequality between skilled and unskilled labor's rewards seems to be the cause of technological change that was biased in favor of skilled labor's rewards.

APPENDIX A

There are three inputs in our model, capital, [v.sub.K], skilled labor, [v.sub.S], and unskilled labor, [v.sub.U]. Data for the value and price of capital and aggregate labor, at a 2-digit SIC87 analysis are obtained from Dale Jorgenson's database for the period 1963 1991. (12) We construct the value added for capital and aggregate labor and also the price of capital and labor. In particular, the price of inputs is a weighted average of their prices in each 2-digit industry with weights the share of each input in every 2-digit industry. We get the quantity of capital and aggregate labor by dividing their value added by their price, respectively.

The division of aggregate labor into skilled and unskilled labor is implemented by using data from the NBER collection of Mare-Winship Data, 1963-1991. We get data on educational levels, weekly wages, status, and weeks worked for full time workers in 2-digit SIC industries. We divide workers into skilled and unskilled following Katz and Murphy (1992), a worker is treated as skilled if he or she spent at least 12 years in education. Our sample contains only full-time workers, aged 16-45, that have completed their educational grade and are working in the private sector. First, we calculate the total number of weeks worked per year and also the annual wages and salaries for skilled and unskilled workers. (13) Then we divide the annual value of wages and salaries by the corresponding total weeks worked in order to calculate the full-time weekly wage for each group respectively. After that we calculate the share of weeks worked for skilled and unskilled workers relative to the total hours worked of all workers. Similarly, we find the shares of wages for each occupational group in the sample. Finally, these shares are multiplied with the total quantity and total wages of aggregate labor, respectively, obtained from Jorgenson's data set in order to get the quantity and wages for skilled and unskilled workers in the United States. We should note that the last year in our sample is 1991 because of limitations of the NBER Mare-Winship Data. This data set is drawn from the CPS March data set that it has changed significantly the way that defines a lot of variables after 1991. Our goal is to have a consistent way of defining skilled and unskilled labor and for this we decided to make use only of the Mare-Winship NBER data set that covers the period 1965-1991.

In our model there are three aggregate products, exportable. [y.sub.E], importable, [y.sub.t], and nontradable, [y.sub.N]. Initially the products are divided into tradeable and nontradeables. A 2-digit industry is termed tradable if the ratio of its exports plus imports divided by its revenue is above 10%, otherwise it is termed as nontradable. (14) Then tradable industries are grouped to exportable and importable depending on whether their net exports are positive or negative, respectively.

For the calculation of value added of the three aggregate products we again use Jorgenson's data set. Data for output deflators are obtained from the Bureau of Economic Analysis at a 2-digit SIC level. As these are available from 1977 onwards, the values of output deflators for years before 1977 are obtained by interpolation assuming a constant growth rate equal to the growth rate between 1977 and 1978. The aggregation of the three goods is achieved in three stages. (15) First, we calculate the value added for each aggregate good, then an aggregate price is constructed for each of them. This aggregate price is a weighted average of the prices of all 2-digit industries that belong to an aggregate good, with weights the share of each 2-digit industry. The aggregate quantity of output is calculated by dividing the value of each aggregate good by its aggregate price. Similarly, the volume of net exports is calculated by dividing the value of net exports for each aggregate good by its corresponding aggregate price.

The assumption of balanced trade is not satisfied by the data. For that reason, the actual trade volumes for each good are adjusted according to the revenue shares in order to guarantee balanced trade. That is,

[summation over (i)) [p.sub.i] [T.sup.*.sub.i] = B

where [p.sub.i], [T.sup.*.sub.i] and B are the product price for the ith good, the unadjusted volume of net exports for the ith good and the trade imbalance, respectively. In order to impose trade balance, we calculate the adjusted value of net exports by subtracting from the value of net exports for every good i a fraction of B equal to its revenue share,

[p.sub.i][T.sub.i] = [p.sub.i][T.sup.*.sub.i] - [[p.sub.i][y.sub.i]]/[[[summation].sup.N.sub.i=1][p.sub.i][y.sub.i]] B

where [T.sub.i] is the adjusted volume of net exports and therefore

[summation over (i)][p.sub.i][T.sub.i] = 0

TABLE A1
SIC Codes for Aggregate Goods

Aggregate
Good SIC Code Category

Exportable Food & Kindred Products (SIC 20)
 Chemicals & Allied Products (SIC 28)
 Industrial & Commerce Machinery &
 Computer Equipment (SIC 35)
 Electronic & Other Electric Equipment
 (SIC 36)

 Transportation Equipment (SIC 37)
 Instruments, Photographic, Medical &
 Optical Goods (SIC 38)

Importable Textile Mill Products (SIC 22)
 Apparel & Other Finished Products (SIC 23)
 Lumber & Wood Products (SIC 24)
 Paper & Allied Products (SIC 26)
 Petroleum Refining & Related Industries
 (SIC 29)

 Leather & Leather Products (SIC 31)
 Primary Metal Industries (SIC 33)
 Miscellaneous Manufacturing Industries
 (SIC 39)

Nontradable Tobacco Products (SIC 21)
 Furniture & Fixtures (SIC 25)
 Printing, Publishing & Allied Industries
 (SIC 27)
 Rubber & Miscellaneous Plastic Products
 (SIC 30)
 Stone, Clay, Glass & Concrete Products
 (SIC 32)
 Fabricated Metal Products, Except
 Machinery (SIC 34)

APPENDIX B

We define [R.sup.e.sub.vp] as the matrix of the second partial derivatives of the revenue function with respect to prices and endowments evaluated at the Equivalent Autarky Equilibrium at period t with a typical entry [[partial derivative]w.sup.e.sub.lt]/[[partial derivative]p.sub.it]. Similarly, [R.sub.vv] and [R.sup.e.sub.vv] are the matrices of the second partial derivatives of the revenue function with respect to endowments evaluated at the trade and Equivalent Autarky Equilibrium at period t and have as typical entries [[partial derivative]w.sub.lt]/[[partial derivative]v.sub.jt] and [[partial derivative]w.sup.e.sub.lt]/[[partial derivative]v.sup.e.sub.jt], respectively. While [R.sup.e.sub.vt] is the vector of the second partial derivatives of the revenue function with respect to endowments and time evaluated at the Equivalent Autarky Equilibrium at period t, with a typical entry [[partial derivative]w.sup.e.sub.lt]/[partial derivative]t. Using the above definitions we can write Equation (18) for the lth factor as:

(B1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

We proceed by dividing both sides of Equation (B1) by 1/[w.sub.lt] in order to obtain the growth rate of factor reward for the lth factor on the left hand side

(B2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Then we multiply and divide by [v.sup.e.sub.jt]/[w.sup.e.sub.lt] the first three lines on the RHS of Equation (B2), while we multiply and divide by [w.sup.e.sub.lt] the last line on the RHS of Equation (B2)

(B3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In order to obtain the growth of factor content of trade and the growth of endowments, we multiply and divide by [f.sub.jt] and [v.sub.jt] the first four lines of Equation (B3), respectively. We also multiply and divide by [v.sub.jt]/[w.sub.lt] the first term inside the brackets in the second line of Equation (B3)

(B4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

Finally, recall [[epsilon].sup.e.sub.li], [[eta].sup.e.sub.lj] [[eta].sup.e.sub.lt] are the elasticities of the factor rewards with respect to price, endowments, and time, respectively, at the Equivalent Autarky Equilibrium and [[eta].sub.lj] is the elasticity of the factor rewards with respect to endowments at the trade equilibrium. While from Equation (19), we know that [[beta].sub.ij] = [[[partial derivative]p.sub.it]/[[partial derivative]p.sup.e.sub.jt]]/ [[v.sup.e.sub.jt]/[p.sub.it]] is the elasticity of price with respect to EAE endowments and [a.sub.it] = ([[partial derivative]p.sub.it]/[partial derivative]t)/[p.sub.it] is the effect of technical change on price. After collecting terms we reach

(B5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

This is Equation (20) in the main text, where we define 1/2 ([[eta].sub.lj]([w.sub.lt]/[v.sub.jt])([v.sup.e.sub.jt]/[w.sup.e.sub.lt]) + [[eta].sup.e.sub.lj]) to be [[bar.[eta]].sub.lj], the weighted mean elasticity of the factor rewards with respect to endowments between the TE and EAE. It involves on the first line on the RHS the growth rate of the FCT for all factors that we call the FCT Effect. The expression on the next line incorporates the growth rate of TE endowments and is called the Endowment Effect. Finally, the expression on the last line is the Technology Effect.

ABBREVIATIONS

CRS: Constant Returns to Scale

EAE: Equivalent Autarkic Equilibrium

FCT: Factor Content of Trade

FPE: Factor Price Equalization

GDP: Gross Domestic Product

QLR: Quasilikelihood Ratio

SNQ: Symmetric Normalized Quadratic

TE: Trade Equilibria

doi: 10.1111/j.1465-7295.2010.00349.x

REFERENCES

Baldwin, R. "Determinants of the Commodity Structure of US Trade." American Economic Review, 61 (1), 1971, 126-46.

Baldwin, R., and G. G. Cain. "Shifts in Relative Wages: The Role of Trade, Technology and Factor Endowments." The Review of Economics and Statistic, 82(4), 2000, 580-95.

Bivens J. "Globalization, American Wages, and Inequality: Past, Present, and Future." Economic Policy Institute, Working Paper No. 279, 2007.

Bloom N., M. Draca, and J. Van Reenen. "Trade Induced Technical Change: The Impact of Chinese Imports on IT and Innovation." Manuscript, University of Stanford, 2008.

Blum B. S. "Trade, Technology, and the Rise of the Service Sector: An Empirical Assessment of the Effects on US Wage Inequality." Journal of International Economics, 74(2), 2008, 441-58.

Borjas G., R. Freeman, and L. Katz. "On the Labor Market Effects of Immigration and Trade," in Immigration and the Work Force, edited by G. Borjas and R. Freeman. Chicago: University of Chicago and NBER, 1992, 213-44.

Bowen H., E. Learner, and L. Sveikauskas. "Multicounty, Multifactor Tests of the Factor Abundance Theory." American Economic Review, 77(5), 1987, 791-809.

Canals C. "What Explains the Widening Wage Gap? Outsourcing vs. Technology." La Caixa Working Paper No 1. 2006.

Davis D. R., and D. E. Weinstein. "An Account of Global Factor Trade." American Economic Review, 91(5), 2001, 1423-53.

"The Factor Content of Trade," in The Handbook of International Trade, edited by E. K. Choi and J. Harrigan. London: Blackwell, 2003, 119-45.

Deardorff A. "Factor Prices and Factor Content of Trade Revisited: What is the Use?" Journal of International Economics, 50(1), 2000, 73-90.

Deardorff A., and R. Staiger. "'An Interpretation of the Factor Content of Trade." Journal of International Economics, 24(12), 1988. 93-107.

Diewert W. E. "Applications of Duality Theory," in Frontiers of Quantitative Economics, vol. 2, edited by M. Intriligator and D. Kendrick. Amsterdam: North-Holland, 1974, 106-71.

--. "Exact and Superlative Index Numbers." Journal of Econometrics, 4(2), 1976, 114-45.

--. "The Quadratic Approximation Lemma and Decomposition of Superlative Indexes." Journal of Economic and Social Measurement, 28(12), 2002, 63-88.

Diewert W. E., and L. Ostensoe. "Flexible Functional Forms for Profit Functions and Global Curvature Conditions," in Dynamic Econometric Modeling, edited by W. A. Barnett, E. R. Berndt, and H. White. Cambridge: Cambridge University Press, 1988, 43-52.

Diewert W. E., and T. J. Wales. "Flexible Functional Forms and Global Curvature Conditions." Econometrica, 55(1), 1987, 43-68.

Dixit A., and V. Norman. Theorv of International Trade, Cambridge: Cambridge University Press, 1980.

Feenstra R. C. "NBER Trade Database, Diskl: U.S. Imports, 1972-1994: Data and Concordances." National Bureau of Economic Research Working Paper No. 5515, 1996.

Feenstra R. C., and G. H. Hanson. "Globalization, Outsourcing, and Wage Inequality." American Economic Review, 86(2), 1996, 240-45.

--. "The Impact of Outsourcing and High-Technology Capital on Wages: Estimates for The United States, 1979-1990." Quarterly Journal of Economics, 114(3), 1999, 907-40.

--. "'Aggregation Bias in the Factor Content of Trade: Evidence from U.S. Manufacturing." American Economic Review, 90(2), 2000, 155-60.

Harrigan J. "International Trade and American Wages in General Equilibrium, 1967-1995," in The Impact of International Trade on Wages, edited by R. C. Feenstra. Chicago: University of Chicago Press, 2000, 171-93.

--. "Specialization and the Volume of Trade: Do the Data Obey the Laws?" in The Handbook of International Trade, edited by E. K. Choi and J. Harrigan. London: Blackwell, 2003, 85-118.

Harrigan J., and R. A. Balaban. "U.S. Wages in General Equilibrium: The Effects of Prices, Technology, and Factor Supplies, 1963-1991." Federal Reserve Bank of New York Staff Report No. 64, 1999.

Jorgenson D. W., and K. J. Stiroh. "Raising the Speed Limit: U.S. Economic Growth in the Information Age." Brookings Papers on Economic Activity 1,2000, 125-212.

Katz L., and K. Murphy. "Changes in Relative Wages, 1963-1987: Supply and Demand Factors." Quarterly Journal of Economics, 107(1), 1992, 36-78.

Kenen P. "Nature, Capital and Trade." Journal of Political Economy, 73(5), 1965, 437-60.

Kohli U. Technology, Duality, and Foreign Trade: The GNP Function Approach to Modelling Imports and Exports, Ann Arbor, MI: University of Michigan Press, 1991.

--. "A Symmetric Normalised Quadratic GNP Function and the US Demand for Imports and Supply of Exports." International Economic Review, 34(1), 1993, 243-55.

Krugman P. "Technology, Trade and Factor Prices." Journal of International Economics, 50(1), 2000, 51-71.

--. "Trade and Wages, Reconsidered." Brookings Papers on Economic Activity 1, 2008.

Lawrence R. Z. Blue-Collar Blues: Is Trade to Blame for Rising US Income Inequality? Washington: Peterson Institute for International Economics, 2008.

Learner E. "The Leontief Paradox, Reconsidered." Journal of Political Economy, 88(3), 1980, 495-503.

--. "Trade, Wages and Revolving Door Ideas." National Bureau of Economic Research Working Paper No. 4716, 1994.

--. "In Search of Stolper-Samuelson Effects on U.S. Wages," in Imports, Exports and the American Worker, edited by S. Collins. Washington DC: Brookings Institution Press, 1998, 141-214.

--. "What's the Use of Factor Contents?" Journal of International Economics, 50(1), 2000, 17-49.

Learner E., and J. Levinsohn. "International Trade Theory: The Evidence," in Handbook of International Economics, edited by G. Grossman and K. Rogoff. Amsterdam: North-Holland, 1995, 1339-94.

Michaels G. "The Effect of Trade on the Demand for Skill-Evidence from the Interstate Highway System." The Review of Economics and Statistics, 90(4), 2008, 683-701.

National Bureau of Economic Research Mare-Winship Data, 1964-1992.

Panagariya A. "Evaluating the Factor-Content Approach to Measuring the Effect of Trade on Wage Inequality." Journal of International Economics, 50(1), 2000, 91-116.

Samuelson P. A. "Factor-Price Equalization by Trade in Joint and Non-joint Production." Review of International Economics, 1(1), 1992, 1-9.

Sitchinava N. "Trade, Technology, and Wage Inequality: Evidence from U.S. Manufacturing, 1989-2004." University of Oregon Working Paper, 2007.

Vanek J. "The Factor Proportions Theory: The N-Factor Cases." Kyklos, 21(4), 1968, 749-56.

Winston G. "On Measuring Factor Proportions in Industries with Different Seasonal and Shift Patterns or Did the Leontief Paradox Ever Exist." Economic Journal, 89(December), 1979, 897-904.

Wood A. "How Trade Hurt Unskilled Workers." Journal of Economic Perspectives, 9(Summer), 1995, 57-80.

Zhu S. C., and D. Trefler. "Trade and Inequality in Developing Countries: A General Equilibrium Analysis." Journal of International Economics, 65(1), 2005, 21-48.

(1.) For a discussion about jointness and FPE, see Samuelson (1992).

(2.) In this part, we follow the standard Heckscher-Ohlin assumption of a small open economy. Later in the empirical part, we relax the assumption of exogenous product prices, as in the work of Kohli (1993) and Harrigan (2000).

(3.) The analysis can be extended to the case of non-balanced trade, by assuming homothetic preferences and that capital inflows effects on demand are equivalent to a transfer payment to consumers (see for instance, Krugman (2008) and Panagariya (2000)).

(4.) In order to have a unique solution for [v.sup.e], it is necessary that the number of goods (n) are equal to the number of inputs (v). With a higher number of goods relative to the number of inputs (n > m), it is still possible to have a unique solution for [v.sup.e] with the additional assumption of nonjointness.

(5.) The determinant of matrix [R.sub.pv] is different from zero, where [R.sub.pv] is the matrix of the second partial derivatives of the revenue function with respect to product prices and endowments.

(6.) See the discussion of Davis and Weinstein (2003) for the problems that arise with FCT, when there are traded and nontraded intermediate inputs.

(7.) This is in contrast to other functional forms, like the widely used translog, that cease to be flexible when concavity and convexity are globally imposed.

(8.) We could relax the assumption of trade balance. This requires estimation of an expenditure function, but because we focus on the production side of the economy, we consider this as part of potential future research. In Appendix A, we provide a detailed construction and sources of the data.

(9.) All net trade services of factors are measured in constant 1970 prices and it is assumed that the economy is in a balanced trade equilibrium (see more in Appendix A).

(10.) Notice that when there is nonjointness in output quantities, [R.sub.vv] = 0, and therefore [w.sub.r] = [w.sup.e.sub.t] and [dw.sub.t]/dt = [dw.sup.e.sub.t]/dt.

(11.) Note that the overall sign and magnitude of the FCT Effect for each factor reward depends on all inverse factor price elasticities, equilibrium product price elasticities, and the FCT growth of all factors and therefore the direction of the effect is ambiguous.

(12.) http://www.economics.harvard.edu/faculty/jorgen son./files/35klem.html. See Jorgenson and Stiroh (2000).

(13.) Following Katz and Murphy (1992), we include only full time workers that have worked more than 39 weeks in that year. Also, top code wage and salaries were multiplied by 1.45.

(14.) Trade data at a 2-digit SIC87 level were obtained online from the Centre for International Data at the University of California Davis. See Feenstra (1996).

(15.) Table Al shows the SIC categories that are included in each aggregate good.

AGELOS DELLS and THEOFANIS P. MAMUNEAS *

* We would like to thank the editor and two anonymous reviewers for helpful comments and suggestions.

Dells: Lecturer, Department of Economics, University of Cyprus, Nicosia, P.O. Box 20537, CY 1678, Cyprus; Leverhulme Centre for Research on Globalisation and Economic Policy, University of Nottingham, UK. Phone 0035722893682, Fax 0035722895028, E-mail dells. agelos@ucy.ac.cy

Mamuneas: Professor, Department of Economics, University of Cyprus, Nicosia, P.O. Box 20537, CY 1678, Cyprus. Phone 0035722893705, Fax 0035722895028, E-mail tmamuneas@ucy.ac.cy

TABLE 1
Parameter Estimates-Revenue Function

Parameter Estimate t-stat.

[a.sub.EE] 47085.9 0.286
[a.sub.EI] -31871.6 -0.394
[a.sub.EN] -15214.3 -0.171
[a.sub.II] 21573.3 0.521
[a.sub.IN] 10298.3 0.213
[a.sub.NN] 4916 0.120
[e.sub.K] 2184.5 1.333
[e.sub.S] -620.7 -1.003
[e.sub.U] -1563.7 -1.224
[C.sub.EK] 64498 2.044
[C.sub.ES] -11935.4 -0.420
[C.sub.EU] 64737.3 3.018
[C.sub.IK] -13286.6 -0.607
[C.sub.IS] 72514 3.714
[C.sub.IU] 6805.5 0.428
[C.sub.NK] -2048 -0.851
[C.sub.NS] 61639.2 4.617
[C.sub.NU] -3075.6 -0.243
[b.sub.KK] -68690.5 -2.394
[b.sub.KS] 29583.7 2.294
[b.sub.KU] 39106.7 1.779
[b.sub.SS] -12741.2 -1.515
[b.sub.SU] -16842.6 -2.523
[b.sub.UU] -22264.2 -1.303
[d.sub.E] 1557.5 0.607
[d.sub.I] -948.9 -0.639
[d.sub.N] -608.6 -0.452
[h.sub.t] 1146.6 0.808
[h.sub.tt] 42.2 0.386
Sy.st. [R.sup.2] 0.980

Hypothesis Testing Test Statistic [chi].sup.2
 .sub.0.5]

No convexity & concavity Wald(4) = 32.7 9.488
Nonjointness: Wald(2) = 29.1 5.991
No technological change Wald(6) = 534 12.590

TABLE 2
Trade Equilibrium Elasticities
(Mean values, Std. Dev in parenthesis)

 Output Price

 Exportable Importable Nontradable
Output Supply [MATHEMATICAL [MATHEMATICAL [MATHEMATICAL
 EXPRESSION NOT EXPRESSION NOT EXPRESSION NOT
 REPRODUCIBLE IN REPRODUCIBLE IN REPRODUCIBLE IN
 ASCII] ASCII] ASCII]

Exportable 0.326 -0.223 -0.103
 (0.034) (0.024) (0.010)
Importable -0.528 0.361 0.166
 (0.027) (0.017) ((1.012)
Nontradable -0.253 0.173 0.079
 (0.025) (0.016) (0.009)

Factor Reward [MATHEMATICAL [MATHEMATICAL [MATHEMATICAL
 EXPRESSION NOT EXPRESSION NOT EXPRESSION NOT
 REPRODUCIBLE IN REPRODUCIBLE IN REPRODUCIBLE IN
 ASCII] ASCII] ASCII]

Capital 1.254 -0.232 -0.022
 (0.108) (0.073) (0.035)
Skilled labor 0.041 0.516 0.441
 (0.090) (0.055) (0.037)
Unskilled labor 1.017 0.061 -0.079
 (0.031) (0.010) (0.021)

 Endowment

 Capital Skilled labor Unskilled labor
Output Supply [MATHEMATICAL [MATHEMATICAL [MATHEMATICAL
 EXPRESSION NOT EXPRESSION NOT EXPRESSION NOT
 REPRODUCIBLE IN REPRODUCIBLE IN REPRODUCIBLE IN
 ASCII] ASCII] ASCII]

Exportable 0.621 0.048 0.330
 (0.046) (0.091) (0.137)
Importable -0.284 1.239 0.044
 (0.117) (0.122) (0.012)
Nontradable -0.027 1.094 -0.067
 (0.043) (0.016) (0.040)

Factor Reward [MATHEMATICAL [MATHEMATICAL [MATHEMATICAL
 EXPRESSION NOT EXPRESSION NOT EXPRESSION NOT
 REPRODUCIBLE IN REPRODUCIBLE IN REPRODUCIBLE IN
 ASCII] ASCII] ASCII]

Capital -1.149 0.678 0.470
 (0.132) (0.217) (0.128)
Skilled labor 0.324 -0.199 -0.1
 (0.082) (0.090) (0.016)
Unskilled labor 0.759 -0.454 -0.305
 (0.100) (0.163) (0.064)

 Techn Change

Output Supply [MATHEMATICAL
 EXPRESSION NOT
 REPRODUCIBLE IN
 ASCII]

Exportable 0.019
 (0.002)
Importable -0.004
 (0.005)
Nontradable -0.001
 (0.004)

Factor Reward [MATHEMATICAL
 EXPRESSION NOT
 REPRODUCIBLE IN
 ASCII]

Capital 0.044
 (0.006)
Skilled labor 0.001
 (0.000)
Unskilled labor -0.017
 (0.002)

TABLE 3
Equivalent Autarky Equilibrium Elasticities
(Mean values, Std. Dev in parenthesis)

 Output Price

Factor Reward Exportable Importable Nontradable
 [MATHEMATICAL [MATHEMATICAL [MATHEMATICAL
 EXPRESSION NOT EXPRESSION NOT EXPRESSION NOT
 REPRODUCIBLE IN REPRODUCIBLE IN REPRODUCIBLE IN
 ASCII] ASCII] ASCII]

Capital 0.841 0.042 0.115
 (0.030) (0.018) (0.012)
Skilled labor -0.056 0.569 0.487
 (0.058) (0.040) (0.023)
Unskilled labor 1.997 -0.329 -0.668
 (0.698) (0.279) (0.420)

 Endowment

Factor Reward Capital Skilled Labor Unskilled Labor
 [MATHEMATICAL [MATHEMATICAL [MATHEMATICAL
 EXPRESSION NOT EXPRESSION NOT EXPRESSION NOT
 REPRODUCIBLE IN REPRODUCIBLE IN REPRODUCIBLE IN
 ASCII] ASCII] ASCII]

Capital -0.397 0.062 0.334
 (0.051) (0.043) (0.045)
Skilled labor 0.039 -0.008 -0.030
 (0.029) (0.009) (0.020)
Unskilled labor 0.831 -0.150 -0.681
 (0.407) (0.139) (0.295)

 Techn. Change

Factor Reward
 [MATHEMATICAL
 EXPRESSION NOT
 REPRODUCIBLE IN
 ASCII]

Capital 0.019
 (0.001)
Skilled labor 0.002
 (0.001)
Unskilled labor -0.049
 (0.017)

TABLE 4
Parameter Estimates--Price Growth Equations

Parameter Estimate t-stat

[a.sub.ET] 0.069 6.607
[[beta].sub.EK] -0.208 -1.743
[[beta].sub.ES] 0.288 2.108
[[beta].sub.EU] -0.274 -1.889
[a.sub.IT] 0.076 5.011
[[beta].sub.IK] -0.227 -1.312

[[beta].sub.IS] 0.446 2.260
[[beta].sub.IU] -0.308 -1.473
[a.sub.NT] 0.071 6.670
[[beta].sub.NK] -0.191 -1.562
[[beta].sub.NS] 0.269 1.919
[[beta].sub.NU] -0.238 -1.601

Syst. [R.sup.2] 0.99

TABLE 5
Factor Rewards Decomposition
(Annual growth rates %)

 Growth of Tech.
 Factor FCT Endowment Change
Period Reward Effect Effect Effect

Capital
1967-1991 2.38 -1.79 -13.35 17.52
1967-1981 3.63 -4.84 -10.01 18.48
1982-1991 0.53 2.79 -18.34 16.08

Skilled Labor
1967-1991 6.95 2.54 -1.27 5.68
1967-1981 9.17 2.75 -0.11 6.53
1982-1991 3.62 2.23 -3.00 4.39

Unskilled Labor
1967-1991 5.93 3.33 2.10 0.50
1967-1981 8.44 4.47 2.46 1.51
1982-1991 2.16 1.62 1.57 -1.03