Factor Endowments, Relative Commodity Prices, and Anything Goes (Summary) by Hugo Sonnenschein University of Chicago

Fifty years ago T. M. Rybczynski [3] published a frequently referenced note in which he inaugurated a systematic investigation of the comparative statics associated with a change in the endowment of a factor. The questions that he posed are broad and fundamental: How do the prices and consumptions of final goods depend on factor endowments? How do factor prices and consumer wealth vary with changes in factors endowments? What are the welfare impli- cations of these changes? Rybczynski considered a closed economy with two factors of production A and B and two consumption goods X and Y ,each produced according to constant returns to scale and perfect competition. He argued that if X is A intensive and Y is B intensive, then an increase in A leads to an increase in the equilibrium output of X and a decrease in the equilibrium output of Y , provided that the marginal rate of transformation in production between X and Y (MRT(X, Y )) does not change when one moves from the orig- inal to the new equilibrium. In other words, under Rybczynski’s assumptions, the expanded production possibilities associated with a factor increase, which allows for an increased production of both outputs, leads in equilibrium to an increase in just one of the two outputs (the one that is intensive in the factor that has increased) and a decrease in the other. This unambiguous compara- tive statics result has been applied to the case of international trade in which an economy is small (more properly infinitesimal), so that the MRT at every interior equilibrium is determined by international prices and does not change. Rybczynski understood that in a closed economy MRT(X, Y ) at equilibrium varies with factor endowments (this is a general equilibrium effect), and he pro- vided a rather complete analysis of how outputs (which are equal to consump- tions in a closed economy) and prices will change in a closed economy following an increase in A. In particular, Rybczynski argued that in the absence of inferior goods, an increase in the amount of factor A leads to an increase in X,butthat the effect on Y is ambiguous. Furthermore it is apparent that he understood that with X inferior, an increase in A can lead to an absolute decrease in X and an increase in Y . Wewillcallthisoutcome"reverseRybczynski". Itisanex- ample of "extreme comparative statics". The purpose of this note is to extend the Rybczynski analysis more fully to general equilibrium by considering economies with more than one consumer, and in particular, to extend his analysis to the 2 2 2 Heckscher-Ohlin model of international trade. The investigation is inspired× × by the work of Cheng, Sachs, and Yang [1] who conjecture that in the Heckscher-Ohlin model the relationship between factor endowments, outputs, and commodity prices can be quite arbi- trary. Their conjecture was based on the S-M-D Theorem on the structure of excess demand functions( which has been referred to as "Anything Goes", see [2]

1 and [4]). Cheng and her co-authors observe that the particular functional forms that they use in their presentation restricts their comparative statics. Here, by not restricting functional forms, we are able to obtain more extreme compara- tive statics, even while ruling out inferior goods, and thus we are able to come closer to establishing that "anything goes" in the Heckscher-Ohlin model. One should note that ambitions of this paper are in other ways significantly different than those in [1]; namely, here I faithfully follow the classical Rybczynski pro- gram: transaction costs are ignored and the extreme comparative statics that are illustrated take place at interior equilibria. In the Heckscher-Ohlin model that we consider here production functions are CRS and identical across countries. Factor endowments are different in the two countries and we make the strong assumption that demand in each country is generated by a single consumer with homothetic preferences; in particular, no goods are inferior. (It is this final assumption that gives our result weight.) As before,weassumethatX is A intensive. Nevertheless, we show that in general equilibrium it is possible for an increase in the amount of factor A in the first country to lead to a decrease in the equilibrium output of X in that country (and the other country as well.) In other words, in general equilibrium, and without the small country assumption, the output implications of an increase in a factor endowment can be the reverse of what is established in the Rybczynski analysis; that is, "reverse Rybczynski", even with no inferior goods.1 At the heart of the demonstration is the fact that in general equilibrium an increase in the amount of A in the home country can lead to an arbitrarily large rate of decrease in the relative price of X, and thus a decline in the amount of X produced. The proof uses a homothetic preferences version of "anything goes" (see [4]). Three other comparative statics results are observed. First, if preferences are identical and homothetic in the Heckscher-Ohlin model, then reverse Rybczynski is not possible (since the demand sector of the economy is "as if" there was a single consumer with homothetic preferences: this is Eisenberg’s Theorem, see [4]). Second (with M. Opp), reverse Rybczynski can be obtained in a closed economy with two consumers, each with homothetic preferences.2 To state this more finely: the reversal that Rybczynski observed is possible in a closed economy with an inferior good does not require inferiority. Instead, with non identical preferences, the reversal can be driven by changes in the distribution of income associated with the change in factor prices. Finally, in a closed economy with a single consumer and homothetic preferences, Rybczynski observed that his result is moderated in the sense that following the factor increase, the relative price of the good that is intensive in the factor that is increased must decrease, and so there is a tendency for the output (and consumption) of both goods

1 Furthermore, "reverse Rybczynski" can be demonstrated in an economy in which equi- librium is unique both before and after the increase in A. 2 The closed economy result follows directly from the previous result; however, the con- struction that we use to prove the closed economy result is not sufficient to prove our main theorem.

2 to rise. However, even with homothetic preferences, with two consumers it is possible that the price of the good that is intensive in the factor that is increased will increase, in which case the Rybczynski effect is amplified: the output of Y falls even more.

References [1] Cheng, W. L.., Sachs, J., Yang, X.(2000), "A General Equilibrium Re- appraisal of the Stolper-Samuelson Teorem", Journal of Economics, pp.1-18. [2] Mas-Colell, A. et al(1995), Microeconomic Theory,OxfordUniversity Press, Section 17E. [3] Rybczynski, T. M.,(1955), "Factor Endowments and Relative Commodity Prices", Economica, pp. 336-41. [4] W. Shafer and H. Sonnenschein(1982), "Market Demand and Excess De- mand Functions", Handbook of Mathematical Economics,vol.2,K.Arrowand M. Intriligator, eds., Chapter 14.

Note: See my web page at the Department of Economics, University of Chicago (http://home.uchicago.edu/~hfsonnen/) for a sketch of the proof of the main result and for various diagrams that are used in the presentation.

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