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(2015). "International Trade. Theory and Policy", 10th Edition. ''Pearson''</ref> and ''factor proportions'' [[Heckscher–Ohlin model]]s. Subsequent developments in the [[new trade theory]], motivated in part by the empirical shortcomings of the H–O model and its inability to explain [[intra-industry trade]], have provided an explanation for aspects of trade that are not accounted for by comparative advantage.<ref>{{cite book|last1=Maneschi|first1=Andrea|title=Comparative Advantage in International Trade: A Historical Perspective|year=1998|publisher=Elgar|location=Cheltenham|pages=6–13}}</ref> Nonetheless, economists like [[Alan Deardorff]],<ref>{{cite journal|last1=Deardorff|first1=Alan|title=The General Validity of the Law of Comparative Advantage|journal=Journal of Political Economy|date=Oct 1980|volume=88|issue=5|pages=941–57|doi=10.1086/260915|s2cid=58917101}}</ref> [[Avinash Dixit]], [[Gottfried Haberler]], and [[Victor D. Norman]]<ref>{{cite book|last1=Dixit|first1=Avinash|last2=Norman|first2=Victor|title=Theory of International Trade: A Dual, General Equilibrium Approach|year=1980|publisher=Cambridge University Press|location=Cambridge|pages=93–126}}</ref> have responded with weaker generalizations of the principle of comparative advantage, in which countries will only ''tend'' to export goods for which they have a comparative advantage. ===Dornbusch et al.'s continuum of goods formulation=== In both the Ricardian and H–O models, the comparative advantage theory is formulated for a 2 countries/2 commodities case. It can be extended to a 2 countries/many commodities case, or a many countries/2 commodities case. Adding commodities in order to have a smooth continuum of goods is the major insight of the seminal paper by Dornbusch, Fisher, and Samuelson. In fact, inserting an increasing number of goods into the chain of comparative advantage makes the gaps between the ratios of the labor requirements negligible, in which case the three types of equilibria around any good in the original model collapse to the same outcome. It notably allows for transportation costs to be incorporated, although the framework remains restricted to two countries.<ref>{{Cite journal | last1 = Dornbusch | first1 = R. | last2 = Fischer | first2 = S. | last3 = Samuelson | first3 = P. A. | year = 1977 | title = Comparative Advantage, Trade, and Payments in a Ricardian Model with a Continuum of Goods | journal = [[American Economic Review]] | volume = 67 | issue = 5 | pages = 823–39 | jstor = 1828066 }}</ref><ref>{{Cite journal | doi = 10.2307/1885496| jstor = 1885496| title = Heckscher-Ohlin Trade Theory with a Continuum of Goods| journal = The Quarterly Journal of Economics| volume = 95| issue = 2| pages = 203–224| date = 1980| last1 = Dornbusch | first1 = R. | last2 = Fischer | first2 = S. | last3 = Samuelson | first3 = P. A. }}</ref> But in the case with many countries (more than 3 countries) and many commodities (more than 3 commodities), the notion of comparative advantage requires a substantially more complex formulation.<ref>{{Cite journal | doi = 10.1111/j.1467-9396.2005.00552.x| title = How Robust is Comparative Advantage?| journal = Review of International Economics| volume = 13| issue = 5| pages = 1004–16| date = 2005| last1 = Deardorff | first1 = A. V.| url = https://deepblue.lib.umich.edu/bitstream/2027.42/73670/1/j.1467-9396.2005.00552.x.pdf| hdl = 2027.42/73670| s2cid = 29501533| hdl-access = free}}</ref> ===Deardorff's general law of comparative advantage=== Skeptics of comparative advantage have underlined that its theoretical implications hardly hold when applied to individual commodities or pairs of commodities in a world of multiple commodities. Deardorff argues that the insights of comparative advantage remain valid if the theory is restated in terms of averages across all commodities. His models provide multiple insights on the correlations between vectors of trade and vectors with relative-autarky-price measures of comparative advantage. "Deardorff's general law of comparative advantage" is a model incorporating multiple goods which takes into account tariffs, transportation costs, and other obstacles to trade. ===Alternative approaches=== Recently, Y. Shiozawa succeeded in constructing a theory of international value in the tradition of Ricardo's [[cost-of-production theory of value]].<ref>Y. Shiozawa, A new construction of Ricardian trade theory / A many-country, many commodity case with intermediates goods and choice of production techniques, Evolutionary and Institutional Economics Review '''3'''(2): 141–87, 2007.</ref><ref>Y. Shiozawa, A Final Solution of the Ricardo Problem on International Values, Iwanami Shoten, 2014.</ref> This was based on a wide range of assumptions: Many countries; Many commodities; Several production techniques for a product in a country; Input trade ([[International trade#Traded intermediate goods|intermediate goods]] are freely traded); Durable capital goods with constant efficiency during a predetermined lifetime; No transportation cost (extendable to positive cost cases). In a famous comment, McKenzie pointed that "A moment's consideration will convince one that Lancashire would be unlikely to produce cotton cloth if the cotton had to be grown in England."<ref>L. W. McKenzie Specialization and Efficiency in World Production, Review of Economic Studies 21(3): 165–80. Citation from p.179.</ref> However, McKenzie and later researchers could not produce a general theory which includes traded input goods because of the mathematical difficulty.<ref name = economics>Appendix A: Previous Literature in A. Deardorff, Ricardian Comparative Advantage with Intermediate Inputs, The North American Journal of Economics and Finance '''16'''(1): 11–34, March 2005. http://fordschool.umich.edu/rsie/workingpapers/Papers501-525/r501.pdf</ref> As John Chipman points it, McKenzie found that "introduction of trade in intermediate product necessitates a fundamental alteration in classical analysis."<ref>Chipman, John S. (1965). "A Survey of the Theory of International Trade: Part 1, The Classical Theory". Econometrica 33 (3): 477–519. Section 1.8, p. 509.</ref> Durable capital goods such as machines and installations are inputs to the productions in the same title as part and ingredients. In view of the new theory, no physical criterion exists. Deardorff examines 10 versions of definitions in two groups but could not give a general formula for the case with intermediate goods.<ref name = economics/> The competitive patterns are determined by the traders trials to find cheapest products in a world. The search of cheapest product is achieved by world optimal procurement. Thus the new theory explains how the global supply chains are formed.<ref>Y. Shiozawa (2016) The revival of classical theory of values, in Nobuharu Yokokawa et als. (Eds.) The Rejuvenation of Political Economy, May 2016, Oxon and New York: Routledge. Chapter 8, pp. 151–72.</ref><ref>Y. Shiozawa, The New Interpretation of Ricardo's Four Magic Numbers and the New Theory of International Values / A Comment on Faccarello's "Comparative advantage"). 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