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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 implications 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 (M RT (X, Y )) does not change when one moves from the original 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 comparative statics result has been applied to the case of international trade in which
an economy is small (more properly infinitesimal), so that the M RT at every
interior equilibrium is determined by international prices and does not change.
Rybczynski understood that in a closed economy M RT (X, Y ) at equilibrium
varies with factor endowments (this is a general equilibrium effect), and he provided a rather complete analysis of how outputs (which are equal to consumptions 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, but that
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 . We will call this outcome "reverse Rybczynski". It is an example 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 arbitrary. 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 comparative 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 program: 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, we assume that X 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 equilibrium is unique both before and after the increase in A.
2 The closed economy result follows directly from the previous result; however, the construction 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 Reappraisal of the Stolper-Samuelson Teorem", Journal of Economics, pp.1-18.
[2] Mas-Colell, A. et al(1995), Microeconomic Theory, Oxford University
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 Demand Functions", Handbook of Mathematical Economics, vol. 2, K. Arrow and
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.
3
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