Abstract
Effective endowments and virtual endowments are insightful ideas to incorporate different technologies with the Heckscher-Ohlin-Vanek framework. This study shows that trade patterns are localized when countries have different technologies, while factor prices are non-equalized (or localized). The Leontief paradox is a localized trade phenomenon or pattern conceptually. It demonstrates that both effective endowments and virtual endowments can explain and present the trade pattern observed in the Leontief test. The core idea of the effective endowment (and the virtual endowment) is that a country exports its effective-abundant factor referred to by its productivity. This paper illustrates that the Leontief paradox arises naturally if a country’s actual factor abundance is inconsistent with its effective factor abundance. The study presents the price-trade equilibrium with non-equalized factor prices to help view trade patterns as trade consequences. The study shows that the Leontief paradox phenomenon may occur even in the absence of factor intensity reversal. The study proposes the factor price definition of trade patterns that mirrors trade patterns defined by factor abundances.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
Feenstra and Taylor (2012, p. 102) first used the term actual factor endowment as distinct from effective endowments.
Virtual endowment is deep or full expression of the productivity-equivalent unit for different technologies.
Wong (1995, p. 129) talked about this issue and said “The theoretical implications of trade surplus in the Heckscher-Ohlin framework and theorem have not been well investigated in the literature.”
Fisher (2011) proposed the cone of goods price diversification.
Using \({\mathrm\Pi V}^F=V^{hF}\) , rewrite Eq. (3) as \(A^HX^F=V^{hF}\) . Rewrite Eq. (5) by \(W_{}^F=\mathrm\Pi W^H\) as \(P^F{=(\;{A^H)}^\text{'}W}^H\). Both the production constraint and the unit cost function in country F are expressed by the technology matrix \(A^H\) . Those two equations, along with Eqs. (2), and (4), compose an effective endowment system under the same technologies matrix \(A^H\). It is then a Heckscher-Ohlin model mathematically.
The solution for Eqs. (9), (10), and (11) can be extended to multi-country equilibrium if supposing h = (1, 2, …, m) and if assuming that trade for a country is a transaction of goods between this country and the rest of the world. Leamer (1984, preface p. xiii) addressed this issue clearly “This theorem, in its most general form, states that a country’s trade relations with the rest of the world depend on its endowments of productive factors…”. The solution for multiple country’s trade will be:
$$\frac{{w}^{*}}{{r}^{*}}=\frac{{K}^{W}}{{L}^{W}},\frac{{p}_{1}^{*}}{{p}_{2}^{*}}=\frac{{a}_{K1}{L}^{W}+{a}_{L1}{K}^{W}}{{a}_{K2}{L}^{W}+{a}_{L2}{K}^{W}},{s}^{h}=\frac{1}{2}\left(\frac{{K}^{h}}{{K}^{W}}+\frac{{L}^{h}}{{L}^{W}}\right),(h=\mathrm{1,2},..,m).$$The parallelogram is identified by the ray of the diversification cone of factor endowments of country H since the factor endowment of country F is measured by the productivity of country H.
One can also assume \({r}^{*H}=1\) then \({w}^{*H}=\frac{{K}^{hW}}{{L}^{hW}}\). However, the relative goods price is still \(\frac{{p}_{1}^{*}}{{p}_{2}^{*}}=\frac{{a}_{K1}^{h}{L}^{hW}+{a}_{L1}^{h}{K}^{hW}}{{a}_{K2}^{h}{L}^{hW}+{a}_{L2}^{h}{K}^{hW}}\).
The numerator, such as in Eq. (16), is the trade volume of factor content, which is the value of \(\overline{EG }\) in Fig. 1. It is the difference in local factor composition from world effective consumption. Helpman and Krugman (1980, p. 24) called it the sole basis of trade in the Heckscher-Ohlin model. It is also true when countries have different productivities. Guo (2023) specified it as the sole source of comparative advantage.
The solution for two factors, two goods, and multiple countries will be
$$\frac{{w}^{*}}{{r}^{*}}=\frac{{K}^{hW}}{{L}^{hW}}, \frac{{p}_{1}^{*}}{{p}_{2}^{*}}=\frac{{a}_{K1}^{h}{L}^{hW}+{a}_{L1}^{h}{K}^{hW}}{{a}_{K2}^{h}{L}^{hW}+{a}_{L2}^{h}{K}^{hW}},{s}^{h}=\frac{1}{2}\left(\frac{{K}^{h}}{{K}^{hW}}+\frac{{L}^{h}}{{L}^{hW}}\right),\left(h=\mathrm{1,2},..,m\right).$$It is related to a country’s factor price pattern and trade pattern before trade (see next section).
It is always true for \(\frac{{K}^{H}}{{L}^{H}}<\frac{{K}^{hW}}{{L}^{hW}}=\frac{{K}^{H}+{{\pi }_{K}K}^{F}}{{L}^{H}+{{\pi }_{L}L}^{F}} <\frac{{{\pi }_{K}K}^{F}}{{{\pi }_{L}L}^{F}}\).
It is always true for \(\frac{{K}^{F}}{{L}^{F}}>\frac{{K}^{fW}}{{L}^{fW}}=\frac{{K}^{H}/{\pi }_{K}+{K}^{F}}{{L}^{H}/{\pi }_{L}+{{\pi }_{L}L}^{F}}>\frac{{K}^{H}/{\pi }_{K}}{{L}^{H}/{\pi }_{L}}\).
In Fig. 1 of Trefler (1993), trades from the developed countries are in the top-right corner. For the developing countries, they are in the lower-left corner. They are sorted by \(\frac{{\pi }_{K}}{{\pi }_{L}}\). Mutual Leontief trade may occur if \(\frac{{K}^{H}{L}^{F}}{{L}^{H}{K}^{F}}<\frac{{\pi }_{K}}{{\pi }_{L}}\). For the multiple-country case, refer to the multiple-country criteria in footnote 10.
Trefler (1995) proposed a country-specific productivity-difference model to solve the issue of missed trade. The model consists of a single cone of factor diversification and a single cone of goods price diversification. The relative prices among countries are the same so that only the Heckscher-Ohlin trade patterns can be presented. Only the differences in relative factor prices among countries can lead to non-Heckscher-Ohlin trade. The author thought that checking Eq. (26) might help to fix the issue of missed trade.
Constant-elasticity-of-substitution (CES) production can specify the FIRs. When both goods have Cobb-Douglas production functions, the FIR is impossible (Bhagwati et al. (1998, p. 61). However, when assuming technology differences and using them in production functions (even by Cobb-Douglas functions), the FIRs will be more significant.
Deardorff (1986) studied the FIRs. To present a model with FIR, he used the approach of switching goods in the production function. It is the same logic as we use, an anti-diagonal matrix to switch technology input coefficients.
It is always true for \(\frac{{K}^{H}}{{L}^{H}}<\frac{{K}^{H}+{\theta }_{L}{L}^{F}}{{L}^{H}+{\theta }_{K}{K}^{F}}<\frac{{\theta }_{L}{L}^{F}}{{\theta }_{K}{K}^{F}}\).
This result remains the Heckscher-Ohlin general theory that a country exports a good it produced with a comparative advantage and imports another good that the foreign country with a comparative advantage produces. Homogenous products or no product differentiation is assumed. A country exporting both goods and importing both goods has no conceptual background.
Jones (1956) stated that the Heckscher-Ohlin theorem would not hold in one country under a two-country economy if the Leontief paradox occurs. He thought that it violated the Heckscher-Ohlin theorem and denied it. This study shows that Jones’ description is true for the models with different productivities.
It is assumed that country H is plentiful in good 1 if \(\frac{x_1^H}{x_2^H}>\frac{x_1^F}{x_2^F}\), based on the assumption that the tastes or preferences of both countries are homothetic. A plentiful good is a good that is produced by using its effective abundant factor intensively.
Samuelson (1949, p.194-195) indirectly alluded to this logic by his argument on equilibrium, wherein the world price is essentially the autarky price of the world isolated from the outside. Samuelson employed a clever analogy by introducing an angle recording geographer device, demonstrating that world prices remain unaffected regardless of how the two countries (America and Europe) are geographically labeled within the world. Expanding upon this narrative, consider the discovery of new lands, such as Australia. If America and the European Union wish to engage in trade with Australia, what would be the autarky prices for America and the European Union? It is beyond dispute that the autarky price of the America-European Union would be equivalent to the world price within that specific union (e.g., Guo 2023).
The factor price definition of trade patterns was applied only for effective endowments, because we cannot express localized factor prices in virtual endowments yet.
It is assumed that two countries engaged in free trade immediately form a trade pattern in Eq. (41). This may not be easy to find in the real world, but shows relationships among variables within the price-trade system.
The localized factor prices are also satisfied with Helpman (1984) restrictions between factor price differences and factor content of trade
$$\begin{array}{c}({{w}^{j}-{w}^{i})}^{\prime}{F}^{ij}>0\\ ({{w}^{j}-{w}^{i})}^{\prime}\left({F}^{ij}-{F}^{ji}\right)>0\end{array}$$where \(w^j\) is the vector of factor payment in country j and \(F^{ij}\) is the vector of factor content of trade exported from country j to country i, i = 1,2, and j = 1,2. This can be displayed numerically for the three trade patterns.
In the original notation, it is considered to be the indirect primary factors as intermediate inputs in their empirical analysis, such as \(A=B(I-\widetilde{A})\), where B is the input-output matrix, \(\widetilde{A}\) is the direct factor requirement matrix. We simplify it to illustrate trade patterns only.
The equilibrium solution indicated by \({s}^{H}\) to virtual endowments to Country H normally may be different from the solution for virtual endowments to the technology of Country F, i.e., \({s}^{H}+{s}^{F}\ne 1\).
Fisher (2011) suggested using the middle of the intersection cone of two goods price diversification cones as the price solution. The author suggested using the average boundaries of two trade boxes to derive a country’s size to determine world prices.
It can be shown that the trade directions are the same for any prices that fall within the intersection, numerically.
Equation (63) is still not a theoretical equilibrium price. It deals only with factor cost requirements, no matter of factor endowment differences across countries. It should be related to world virtual factor endowments or quantities of world goods outputs also.
This paper provides two explanations for the Leontief (1953) test now. One is that the trade pattern for the U.S. and the rest of the world is mutual Leontief trade. Another is that the U.S. trade pattern is Leontief trade and the trade pattern for the rest of the world is Heckscher-Ohlin trade. Both sides export labor services from the viewpoint of the factor content of trade.
The empirical test in the Leontief paradox will no longer be a binary test for which trade pattern is correct, the Leontief trade pattern or the Heckscher-Ohlin trade pattern. Research will focus on which country is under what trade pattern rather than which pattern is right.
References
Acemoglu, D. (2003). Cross-country inequality trends. The Economic Journal, 113(485), 121–149.
Behrman, J. R., Birdsall, N., & Szekely, M. (2007). Economic policy and wage differentials in Latin America. Economic Development and Cultural Change, 46(1), 57–91.
Bhagwati, J., Panagariya, A., & Srinivasan, T. N. (1998). Lectures on International Trade (2nd ed.). MIT Press Books.
Chipman, J. S. (1969). Factor price equalization and the Stolper-Samuelson theorem. International Economic Review, 10(3), 399–406.
Deardorff, A. V. (1986). FIRLESS FIRWOES: How preferences can interfere with the theorems of international trade. Journal of International Economics, 20(1–2), 131–142.
Dixit, A. K., & Norman, V. (1980). Theory of International Trade. England, Cambridge University Press.
Feenstra, C. R., & Taylor, A. M. (2012). International Economics, Worth (2nd edn).
Fisher, E. O. (2011). Heckscher-Ohlin theory when countries have different technologies. International Review of Economics and Finance, 20(April), 202–210.
Fisher, E. O., & Marshall, K. G. (2011). The structure of the American economy. Review of International Economics, 19(February), 15–31.
Fisher, E. O., & Marshall, K. G. (2016). Leontief was not right after all. Journal of Productivity Analysis, 46(February), 15–24.
Guo, B. (2005). Demand-oriented trade equilibrium of multinational economies. The Journal of Economics & Economic Education Research, 6(2), 33–60.
Guo, B. (2023). General trade equilibrium of integrated world economy. International Advances in Economic Research, 29(August), 193–205.
Helpman, E. (1984). Factor content of foreign trade. Economic Journal, 94(373), 84–94.
Helpman, E., & Krugman, P. (1985). Market structure and foreign trade. MIT Press.
Jones, R. W. (1956). Factor proportions and the Heckscher-Ohlin theorem. The Review of Economic Studies, 24(1), 1–10.
Kiyota, K. (2020). The Leontief paradox redux. Review of International Economics, 29(2), 296–313.
Kozo, K., & Yoshinori, K. (2022). Factor intensity reversals redux: Feenstra is right! Review of International Economics, 30(4), 885–914.
Krugman, P. R. (2000). Technology, trade and factor prices. Journal of International Economics, 50(1), 51–71.
Leamer, E. E. (1980). The Leontief paradox, reconsidered. Journal of Political Economy, 88(3), 495–503.
Leamer, E. (1984). Sources of international comparative advantage theory and evidence. The MIT Press.
Leontief, W. (1953). Domestic production and foreign trade; The American capital position re-examined. Proceedings of the American Philosophical Society, 97(4), 332–349.
Rassekh, F., & Thompson, H. (1993). Factor price equalization: Theory and evidence. Journal of Economic Integration, 8(1), 1–32.
Reshef, A. (2007). Heckscher-Ohlin and the global increase of skill premia: Factor intensity reversals to the rescue. Working Paper, Paris School of Economics. Available at: https://www.parisschoolofeconomics.com/reshef-ariell/papers/tradeNwages.pdf. Accessed Aug 2023.
Samuelson, P. A. (1949). International factor-price equalization once again. Economic Journal, 59(234),181-197. Reprinted in ed. JE Stiglitz (1966), 869-885.
Sampson, T. (2016). Assignment reversals: Trade, skill allocation and wage inequality. Journal of Economic Theory, 163(May), 365–409.
Trefler, D. (1993). International factor price differences: Leontief was right! Journal of Political Economy, 101(6), 961–987.
Trefler, D. (1995). The case of the missing trade and other mysteries. The American Economic Review, 85(5), 1029–1046.
Wong, K.-Y. (1995). International Trade in Goods and Factor Mobility. MIT Press.
Acknowledgements
I sincerely thank the editors and anonymous reviewers for their comments, encouragement, insight and broad perspective. Any remaining errors are my responsibility.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
Below is the link to the electronic supplementary material.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Guo, B. Leontief Paradox vs. Leontief Trade and Localized Factor Prices vs. Localized Trade Patterns. Int Adv Econ Res 30, 83–105 (2024). https://doi.org/10.1007/s11294-024-09886-1
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11294-024-09886-1
Keywords
- Leontief paradox
- Factor price localization
- Factor price non-equalization
- General trade equilibrium
- Integrated world equilibrium
- Factor intensity reversal
- Effective endowments
- Sign prediction