%Paper: ewp-it/9502001
%From: Uday Rajan <urajan@leland.Stanford.EDU>
%Date: Mon, 13 Feb 1995 22:21:17 -0800

\documentstyle[11pt]{article}
\title{Refutable Implications of the Heckscher-Ohlin Model
 }
\author{Uday Rajan\thanks{I am grateful to Don Brown,
Alan Deardorff, Julie Schaffner and 
Sue Snyder for helpful
conversations and to seminar participants at Stanford University and the 
Midwest International Economics
Meetings, 1994, for their comments.} \\ Department of Economics \\
        Stanford University \\ Stanford, CA 94305 \\ 
E-mail: urajan@leland.stanford.edu \\ \\
First Version: July 25, 1994 \\
This Version: \ \ \ October 31, 1994
}
\date{ }

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\begin{document}

\maketitle

\begin{abstract}

Previous empirical tests have found that, contrary to the conclusions
of the Heckscher-Ohlin model, the factor composition of traded goods
fails to reveal relative factor abundance rankings. Using a
nonparametric approach, this paper
discusses the refutability of two of the assumptions of the HO model:
that countries have identical homothetic preferences and identical
constant returns to scale production functions. We find that, for two
countries, the
assumption on preferences cannot be refuted with expenditure data
alone. However, the assumption on technologies is refutable even when
some factor prices (such as the rental rate on capital) are
unobserved. Finally, we consider the refutability of Deardorff's
(1982) more general HO model, the main result of which is that the
value of net factor exports at (intrinsically unobservable) autarky
factor prices is negative. We show that, in the $2 \times 2$ case,
this model can be refuted using just observed data, i.e. data from the
observed situation under trade.

\end{abstract}

\etpage

\setlength{\baselineskip}{18pt} 

\section{Introduction}

The main conclusion of the Heckscher-Ohlin-Vanek (HOV) model is that,
through trade in commodities, countries export factors they are
relatively abundant in, and import scarce inputs. Further, ranking
factors by net factor exports reveals their relative abundance. 
Previous empirical
tests have tended to reject this conclusion. For example, Maskus
(1985) finds that factor abundance rankings implied by U.S. trade in
1972 do not correspond to the rankings in terms of endowment shares.
Bowen, Leamer and Sveikauskas (1987) test the HOV model under
different assumptions about preferences, technology, and measurement
errors, and also conclude that trade does not reveal relative factor
abundance. Staiger (1988) finds evidence of misspecification of the HOV
model, and concludes that endowments affect trade in important ways
not captured by the HOV relationship.

The conclusions of the HOV model are driven by four sets of
assumptions---identical homothetic preferences across countries,
identical constant returns to scale production functions across
countries, perfect competition in commodity and factor markets, and 
equal commodity and factor prices across countries\ft{If there exactly
as many goods as factors, factor price equalization can be derived
from the other assumptions.}. 
Commodity and factor prices are, in fact, not equal across countries,
as a result of both trade barriers (such as tariffs) and domestic
policies such as taxes and subsidies. Since the failure of any one of
the assumptions may invalidate the conclusions of the model, it is
natural to ask whether there is any support for the remaining
assumptions, or for more general versions of the HOV model that do not
rely on commodity and factor price equalization. 

This paper discusses nonparametric tests of the assumptions of
identical homothetic preferences and identical constant returns to
scale production functions across countries. One reason for being
interested in these questions is the lowering of trade barriers in
pockets across the globe (such as in Europe and North America), which
will lead to commodity prices across certain countries being
roughly similar. Predicting trade patterns in such a world depends on
the accuracy of the assumptions used by the modeller. 
We also investigate 
the refutability of Deardorff's (1982) model, the main result of which
is that, for each country, the value of net factor exports at
(unobservable) autarky factor prices is negative.

The mathematical principle underlying the tests discussed is the
Tarski-Seidenberg Theorem (Tarski, 1951, and Seidenberg, 1954), which
states that, given a system of polynomial inequalities 
$\Gamma(x,y) \leq 0$ in
observed ($x$) and unobserved ($y$) variables, there exist values for
the unobserved variables such that the system of inequalities is
satisfied {\em if and only if\/} the observed variables, $x$, satisfy
a corresponding system of inequalities, $\Sigma(x) \leq 0$. An early
application of this principle in economics was in the area of revealed
preference, with Afriat (1967) deriving conditions on observed prices
and quantities consumed that implied the existence of unobserved
utility levels that could have come from a well-behaved utility function.

Extending Afriat's (1967) work, Varian (1982,1983,1984) proposed
conditions under which there exist well-behaved homothetic utility
functions or constant returns to scale production functions consistent
with observed data on prices and quantities of goods and factors.
Again, no functional form assumptions were made.
Brown and Matzkin (1993) further extended this approach to general
equilibrium, when quantities consumed by individual agents are
unobservable. Snyder (1994), also in a general equilibrium framework,
derives conditions for a more general class of utility functions, and
also examines the testability of Pareto optimality.

Two important features about this approach are that, firstly, 
{\em no\/} 
parametric assumptions need to be made about the utility (or
production, as the case may be) function, and
secondly, the conditions on observed data exhaust {\em all\/} testable
implications of the model. Clearly, to reject the model, it is
sufficient to reject any implication of the model. However, if the
observed data satisfy the proposed tests, no other tests on the same
set of data can reject the model.

Varian's tests for homothetic preferences depend on the observability
of both prices and quantities consumed. Country level consumption
data tends to be in the form of expenditures (since the data generally
refer
to aggregate commodities), usually with price indices (relative to
some base year) for each aggregate commodity. In Section 2 of this
paper, we show that, for two
countries, if, on even one commodity, only expenditure data are
available, the hypothesis of a common homothetic utility function
across the two countries is irrefutable. 

However, if across the two countries, we observe relative price and
quantity ratio (i.e. the ratio of price in country 1 to price in
country 2), Varian's test can be implemented. This is the case with
the Kravis, Heston, and Summers (1982) data set, which quantities prices
and quantities (relative to U.S. levels) for aggregate commodities
across 34 countries. This data set was used by Hunter and Markusen
(1988), who rejected the hypothesis of a common homothetic utility
function across all 34 countries. However, Hunter and Markusen tested
for a specific functional form of the utility function (one that gives
rise to linear expenditure systems). The nonparametric approach avoids
this pitfall. Further, it can be used on as few as two countries,
unlike the regression approach of Hunter and Markusen. For example, it
may be used to test the hypothesis that countries that are
geographically close have the same homothetic preferences. To
illustrate this, we apply this test to the pairs (U.S. and India),
(India and Malawi), and (U.S. and India), and find that {\em no\/}
homothetic utility function is consistent with the data from U.S. and
India. However, for the pairs (India and Malawi) and (U.S. and India),
the data are consistent with common homothetic utility functions.

We examine tests on technologies in Section 3. Even when some factor
prices are unobserved\ft{For example, the return to capital or land
may not be observed.}, we find restrictions that observed quantities
of factors and the prices that are observed must satisfy in order to
be consistent with common constant returns to scale production
functions across two countries. We show that Helpman's (1984)
post-trade restrictions on data are necessary, but not sufficient, for
the existence 
common constant returns to scale production functions across two
countries. 

In Section 4, we consider a generalized Heckscher-Ohlin model, due to
Deardorff (1984). The main result of this model is that, under the
maintained assumptions, the value of net factor exports at autarky
factor prices is negative. Since autarky factor prices are
intrinsically unobservable, it may appear that this model cannot be
refuted. For the $2$ goods, $2$ factors case, we show that the model
can be refuted using only data from the (observed) trade situation.

\section{Testing the Assumptions on Preferences}

Suppose that at some point of time, we have (cross-sectional) data for
$K$ countries and $N$ commodities. For each country $k$, we observe 
${\bf y}^k$ (the $N$-vector of 
commodity outputs), and ${\bf p}^k$ (the $N$-vector of commodity prices).
The test proposed by Varian (1983) may then be used to check whether 
the data support the assumption that 
all countries have identical, homothetic utility functions.

\bline\noindent{\bf Theorem} (Varian, 1983, Theorem 2) \eol
{\em Suppose that we observe $({\bf p}^k,{\bf y}^k)$ for 
$k=1,\ldots,K$. Then the data are consistent with utility maximization
given a monotonic, concave, homothetic utility function if and only if }
\beqa
(\bfp^a \bfy^b)(\bfp^b \bfy^c) \ldots (\bfp^d \bfy^a) & \geq & 
(\bfp^a \bfy^a) \ldots (\bfp^d \bfy^d)
\label{tcon}
\eeqa
{\em 
for all choices of  indices $a,b,c,\ldots,d \in [1,\ldots,K]$.} \eol
\bline

Varian proves the ``if'' part of this theorem by actually constructing a 
utility function with the required properties. Therefore, when (\ref{tcon})
holds, we can find a homothetic utility function consistent with the data.

This test requires that both prices and quantities consumed be
observed. Country level data generally entails some level of
aggregation, and is often in expenditure terms. The question we pose
is: are expenditure data sufficient to refute the homotheticity
hypothesis? 

Suppose, then,
that for $N_1 \geq 1$ commodities, we observe only aggregate
expenditures $\bar{y}_{i}^{k}$, where the subscript denotes the commodity
and the superscript the country. For all other $N - N_1$ commodities, both 
prices and 
quantities are observed. Suppose further that $K=2$, i.e. there are only
two countries. In this context, we cannot refute the hypothesis that 
the countries have identical homothetic preferences.

\bthm
Suppose that we observe $(p_{j}^k, y_j^k, \bar{y}_i^k)$ for $k=1,2$,
$i = 1,\ldots,N_1$, and 
$j=N_1+1,\ldots,N$, with some
$\bar{y}_i^k > 0$ for $k=1,2$. 
Then there always exists
a monotonic, concave, 
homothetic utility function consistent with the data.
\label{t1}
\ethm
\bproof
We prove the theorem for the case of $N_1 = 1$. The case $N_1 > 1$ follows
readily from this by just fixing $p_i^k,y_i^k$ such that $p_i^k y_i^k 
= \bar{y}_i^k$ for $k=1,2$ and $i=1,\ldots,N_1-1$. 

Suppose, then, that we have prices and quantities for $N-1$ commodities
and just expenditures for the $N^{th}$ commodity.

Define 
\beqb
A^k & = & \sum_{i=1}^{N} p_i^k y_i^k \quad\mbox{ for } k=1,2,  \\
B^1 & = & \sum_{i=1}^{N-1} p_i^1 y_i^2 \\
B^2 & = & \sum_{i=1}^{N-1} p_i^2 y_i^1 
\eeqb
Then, by Varian (1983) Theorem 2, the data are 
consistent with a utility function satsifying the requisite properties
if and only if there exist $p_N^k, y_N^k$ for $k=1,2$ such that:
\beqa
(B^1 + p_N^1 y_N^2)(B^2 + p_N^2 y_N^1) &  \geq & A^1 A^2 \label{e1}\\
p_N^k y_N^k & = & \bar{y}_N^k \quad \mbox{ for } k = 1,2 \label{e2}
\eeqa

If 
\beqb
B^1 B^2 + \bar{y}_N^1\bar{y}_N^2 \geq A^1 A^2
\eeqb
any choice of 
$p_N^k, y_N^k$ that satisfies $p_N^k y_N^k = {\bar{y}_N^k}$ for $k=1,2$
will work. 

Suppose, therefore, that 
\beqb
B^1 B^2 + {\bar{y}_N^1}{\bar{y}_N^2} 
< A^1 A^2 
\eeqb
Fix $p_N^1 = q$, and set $y_N^2 = z$, where 
\beqb
z & \geq & \frac{A^1 A^2 - B^1 B^2 - {\bar{y}_N^1}{\bar{y}_N^2} } {q B^2} 
\eeqb
Now set 
\beqb
y_N^1 & = & \frac{ {\bar{y}_N^1} }{q} \\
p_N^2 & = & \frac { {\bar{y}_N^2} }{z}
\eeqb
Clearly, these choices satisfy (\ref{e1}) and (\ref{e2}).
\eproof

In practice, national level input-output data often contain
price information in the form of price indices, so that for each
commodity aggregate, the price in the current year relative to the
price in some base year is observed. This is still not enough, since
the relative prices across commodities (i.e. of one commodity to
another) remain unobserved. 

Let $q_i^k$, for $i = 1, \ldots,N$ indicate the price index associated
with good $i$ in country $k$, relative to the price of the same good
in the same country in some previous year. Let $\lambda_i^k$ denote
this unobserved base price\ft{ If both the base price and
the price index are observed, the current price can be computed, and
(\ref{tcon}) is implementable.}.

\bthm
Suppose that we observe $(p_{j}^k, y_j^k, \bar{y}_i^k, q_i^k)$ for $k=1,2$,
$i = 1,\ldots,N_1$, and 
$j=N_1+1,\ldots,N$, with some
$\bar{y}_i^k > 0$ for $k=1,2$. 
Then there always exists
a monotonic, concave, 
homothetic utility function consistent with the data.
\label{t2}
\ethm
\bproof
Again, we prove the theorem for the case $N_1 = 1$. Let the
$N^{\mbox{th}}$ good be the good for which only expenditure data and a
price index are available. Let $B^1 =
\sum_{i=1}^{N-1} p_i^1 y_i^2$, $B^2 = \sum_{i=1}^{N-1} p_i^2 y_i^1$, and
$A^k = \sum_{i=1}^{N} p_i^k y_i^k$ for $k = 1,2$. 
Then, (\ref{tcon}) reduces to
\beqa
(B^1 + \lambda_N^1 q_N^1 y_N^2) (B^2 + \lambda_N^2 q_N^2 y_N^1) \geq
A^1 A^2
\label{xx}
\eeqa

As with Theorem (\ref{t1}), if $B^1 B^2 + \bar{y}^1 \bar{y}^2 \geq A^1
A^2$, any choice of $\lambda_N^k, y_N^k$ such that $\lambda_N^k q_N^k
y_N^k = \bar{y}_N^k$ leads to (\ref{xx}) being satisfied. Suppose,
then, that 
\beqb
B^1 B^2 + \bar{y}^1 \bar{y}^2 < A^1 A^2
\eeqb

Fix $\lambda_N^1 = \mu$, and set $y_N^2 = z$, where
\beqb
z \geq \frac{A^1 A^2 - B^1 B^2 - \bar{y}_N^1 \bar{y}_N^2}
{\mu q_N^1 B^2} 
\eeqb
Now set $y_N^1 = \bar{y}_N^2 / \mu q_N^1$ and $\lambda_N^2 =
\bar{y}_N^2 / q_N^2 y_N^2$. Clearly, these choices lead to (\ref{xx})
being satisfied.
\eproof

Note that the two theorems above hold if there is just one commodity
for which we have only expenditure and price index data\ft{Typically, 
national level consumption data on most commodities is in 
expenditure form.}. Further, it is clear from the proof of Theorem
\ref{t2} that having a quantity index in addition to a price index for
commodities on which only expenditure data are available is
of no benefit. In equation (\ref{xx}) above, interpreting $q_N^k$ as
the product of a price index in country $k$ and a quantity index in
country $\ell$ creates no difficulty whatsoever. 

Additional observations (i.e. data from additional countries) increase
both the number of unknowns and the number of inequalities in the system.
In general, if there are $q$ commodities for which only aggregate expenditures
are observed, and we have data from $r$ countries, we have a system with
$2qr$ unknowns, $qr$ equations, and $\sum_{j=2}^{r} \frac{r!}{j!(r-j)!}$
inequalities. In priniciple, the Tarski-Seidenberg algorithm (outlined
in the proof of the Tarski-Seidenberg theorem, by Tarski, 1951, and 
Seidenberg, 1954) could be used
to reduce this system of nonlinear inequalities to an equivalent 
system in the observables.
However, this algorithm is doubly exponential in the 
unobservables. There is no known practical method for effecting this 
reduction.

The implication of these theorems is that, for two countries, 
 we cannot test the assumptions on 
preferences made by the Heckscher-Ohlin model with expenditure data alone. 
There are at least three alternatives available. One is to try and 
bound relative or absolute prices. Notice that the proof assumes that 
we are free to pick prices and quantities on the $i^{th}$ commodity.
Every bound obtained increases the number of inequalities in the system.
This creates no theoretical difficulty, but may make it harder to determine
whether or not there is a feasible solution. 

Secondly, we could make additional assumptions
and test a joint hypothesis. A famous example of this is, of course, the 
HOV model itself.

Thirdly, one could try and find data on
prices and quantities separately. This data is not easily
available. Any test must be at a highly disaggregated level
(since aggregation is necessarily by value) and will therefore involve 
hundreds of commodities, if not more. This will further mean that several
sources will have to be consulted to gather complete data. For example, 
the {\em Statistical Abstract of the United States\/} contains price and
quantity data for the U.S.A. for several non-manufacturing commodities. 
Obtaining manufacturing data may involve consulting several specialized
industry bulletins.

Finally, we consider the data provided by Kravis, Heston, and Summers
(1982), who do provide comparable prices for commodity aggregates, for
a sample of 34 countries, and for the year 1975.
Both prices and quantities of commodities across countries
are provided relative to their levels in the U.S. However, commodity 
prices in the U.S. are not quoted separately. In other words, we can 
compare prices across countries for the same commodity, but not 
across commodities for the same country. Total 
expenditure on each commodity category for each country is also shown.

Hunter and Markusen (1988)
test for identical homothetic preferences across
countries, using this data set. They 
perform their test by assuming a linear 
expenditure system, derived from a utility function of the form 
\beqb
U(c) & = & \prod_{i=1}^N (c_i - \bar{c}_i)^{\beta_i}
\eeqb
and estimate it with and without restrictions imposed by
homotheticity. They find that there are strong grounds for rejecting 
homotheticity of preferences across countries, in favour of common but
non-homothetic preferences. However, their test clearly
fails to rule out any other functional form for the utility function.
Further, like any regression-based analysis, it cannot be carried out
with a small sample of countries\ft{ The data contains 34
observations, and the regression estimated has 12 coefficents.}.

In this situation, we observe all we need to directly check (\ref{tcon}). 
The data for country $k$ include $(p_i^k y_i^k, \alpha_i^k, \beta_i^k)$, 
where 
\beqb 
\alpha_i^k & = & \frac{p_i^k}{p_i^{US}} \\
\beta_i^k & = & \frac{y_i^k}{y_i^{US}}
\eeqb

Define $z_i = p_i^{US} \; y_i^{US}$. 
It is easy to see that there exists a monotonic, concave, homothetic
utility function consistent with the data if and only if 
\beqa
(\sum_{i=1}^N \alpha_i^a \beta_i^b \; z_i) 
\ldots
(\sum_{i=1}^N \alpha_i^c \beta_i^a \; z_i)
& \geq &
(\sum_{i=1}^N \alpha_i^a \beta_i^a \; z_i)
\ldots
(\sum_{i=1}^N \alpha_i^c \beta_i^c \; z_i)
\label{hpref}
\eeqa
for all  choices of indices $a,b,\ldots,c \in [1,\ldots,K]$.

Unlike a regression-based method, this test can be carried out with 
data from as few as two countries (i.e. just two observations). It is 
therefore possible to check whether subsets of countries have common 
homothetic preferences. To
illustrate this method, 
we used the Kravis, Heston and Summers (1982) data set to test whether
the U.S.A., India and Malawi could have had common homothetic preferences. 
The details of the test are reported in appendix A.1.
The results 
indicate that there exist homothetic utility functions consistent with the 
data from the U.S.A. and Malawi, and India and Malawi, but not the U.S.A. 
and India.

We next ask whether there exist {\em any\/} well-behaved (i.e.
continuous, monotonic, concave) utility functions consistent with the
data from the three countries. This test is reported in Appendix A.2.
We find that we cannot refute the hypothesis that the three countries
had a common well-behaved, non-homothetic utility function.

\section{Testing the Assumptions on Technologies}

Suppose that we observe
data on prices and quantities, and on 
$M$ factors of production, i.e. for each country $k$ we observe
$p_i^k$, $y_i^k$, 
$\wk_i$ (the $M$-vector of factor prices in the $i^{th}$ sector) and 
$\xk_i$ (the $M$-vector of factor quantities used to produce the $i^{th}$
commodity). Suppose further 
that producers in each sector are competitive 
profit maximizers. This implies that, regardless of the price paid by 
consumers in each country, the producers of country $k$ always receive 
the same price for their output\ft{In other words, the benefits of price 
differentials are captured by governments, say in the form of tariffs.}. 
Varian (1984) has proposed a test for the assumption of a 
common, constant returns to scale production function across all 
countries. Again, given data that satisfy the conditions of his
theorem, we can actually construct a production function with the 
requisite properties.

\bline \noindent
{\bf Theorem} (Varian, 1984, Theorem 6) \eol
{\em 
Suppose that for each country $k=1,\ldots,K$ and for some commodity 
$i$, we observe $(p_i^k,y_i^k,$ ${\bf w}_i^k, {\bf x}_i^k)$. Then the data
are consistent with profit maximization given a monotonic, concave, CRS
production function if and only if } 
\beqa
 0 \; = \; p_i^k y_i^k - {\bf w}_i^k {\bf x}_i^k \; \geq \;
p_i^k y_i^j - {\bf w}_i^k {\bf x}_i^j
\label{e3}
\eeqa
{\em for all $k \in \{1,\ldots,K\}$ and 
for all $j \neq k$.} \eol
\bline

In the case where only commodity expenditures are observed, but not prices
or quantities, the condition above can be further simplified. The following
lemma shows that, if there are two countries and only commodity 
expenditures are observed, the inequality in (\ref{e3}) reduces to a 
condition analogous to the one for well-behaved 
homothetic preference functions. 

\blem
Suppose that $K = 2$, and that for each country $k$ and some commodity
$i$, we observe 
$(\bar{y}_i^k, {\bf w}_i^k, {\bf x}_i^k)$. Then the data
are consistent with profit maximization given 
a monotonic, concave, CRS
production function if and only if \eol
(i) $p_i^k y_i^k = {\bf w}_i^k {\bf x}_i^k$, for $k=1,2$, and \eol
(ii) 
$(\wone_i \xtwo_i) (\wtwo_i \xone_i)  \geq 
(\wone_i \xone_i) (\wtwo_i \xtwo_i) $
\label{plem2}
\elem
\bproof
{}From Varian (1984) Theorem 6,  we need that there exist $p^1, p^2, y^1, y^2$
such that (dropping the $_i$ subscript for convenience)
\beqa
p^1 y^1 - \wone \xone & = & 0 \label{e4} \\
p^1 y^2 - \wone \xtwo & \leq & 0 \label{e5} \\
p^2 y^2 - \wtwo \xtwo & = & 0 \label{e6} \\
p^2 y^1 - \wtwo \xone & \leq & 0  \label{e7} \\
p^1 y^1 & = & \bar{y}^1 \\
p^2 y^2 & = & \bar{y}^2
\eeqa
(\ref{e4}) and (\ref{e6}) are restated as condition (i) of the lemma.

Transfer the observables $({\bf w}, {\bf x})$ to the right-hand side of the 
first
four equations above, and then divide (\ref{e4}) by (\ref{e5}) and 
(\ref{e7}) by (\ref{e6}) to get
\beqa
\frac{ \wone \xone }{ \wone \xtwo } \; \leq \; \frac{y^1}{y^2} \; \leq \;
\frac{ \wtwo \xone }{ \wtwo \xtwo }
\label{e8}
\eeqa

To see sufficiency of this condition, consider the first inequality 
in (\ref{e8})
and multiply the RHS above and below by $p^1$. (\ref{e4}) now implies
(\ref{e5}). To get (\ref{e7}), consider the second inequality in (\ref{e8})
and multiply
the LHS above and below by $p^2$. 

Condition (ii) of the lemma follows
immediately from (\ref{e8}).
\eproof

In general, factors used to produce a commodity will include primary 
factors (labor, capital, land) and other (intermediate) commodities. 
However, in
principle, input-output data can be used to solve out for the outputs as a 
function of the 
primary inputs alone.

Brecher and Choudhri (1992) implement this test with production data for
several commodities from
the U.S. and Canada, and find support for the hypothesis that each commodity
is produced with a common, CRS production function. The basic test they 
propose is 
\beqa
\frac{\wone \xtwo}{p^2 y^2} \frac{\wtwo \xone}{p^1 y^1} \geq
\frac{\wone \xone}{p^1 y^1} \frac{\wtwo \xtwo}{p^2 y^2} = 1
\eeqa
which is equivalent to condition (ii) of Lemma \ref{plem2} above. 

Even this test is difficult to implement due to lack of data (the 
rental rate on capital is not directly observable). Brecher and 
Choudhri 
propose different methods for inferring the rental rate. One is to assume
perfect competition in commodity markets, and that capital is the only factor 
with an unobserved price. Since perfect competition implies that returns to 
factors exhaust the value of the output, the rental rate can be inferred 
(given the value of the capital stock). Another is to use the long-term
interest rate as a proxy. Neither is entirely satisfactory, but there are
yet further complications. 

Firstly, there may be more than one kind of capital, or other primary
inputs (factors or commodities) with unobserved prices.
Brecher and Choudhri use data on two kinds of capital (equipment and 
structures), whereas
Maskus (1985) examines a third, human capital. The returns to different 
kinds of capital may well differ, rendering estimation of each rental rate
impossible. Secondly, Brecher and Choudhri mention that observed wage rates 
differ across industries within the same country. This implies that prices
of other factors may also exhibit inter-sectoral differences, so that it
is impossible to use a single proxy variable to approximate them.
Brecher and Choudhri circumvent these problem by using different methods to 
infer the rental rate on capital, and assuming a price of one for 
primary commodity inputs with unobserved prices.

In this context, if we observed commodity expenditures, quantities of all 
factors and prices of 
all but one, the assumptions of constant returns to scale and competitive,
frictionless markets allow us to compute the single unobserved price
(since the value of the output must equal the total cost of production). This
would enable the above test to be carried out.

However, even if there is more than one factor whose return is observed, 
we can derive a test for CRS production functions.
In general, suppose that\ft{Again, for some commodity $i$. The $_i$ subscript
is dropped for convenience.}
 we observe quantities of all $M$ factors, and prices 
of just $M_1 < M$. Denote the observed prices and quantities of these 
$M_1$ factors by $\wk, \xk$, and the unobserved prices and 
observed quantities of the remaining factors by $\vk, \zk$. Suppose
further that there is data for just two countries. Then the following theorem
provides a necessary and sufficient condition for the data to be consistent 
with a well-behaved CRS production function.

\bthm
Suppose that $K=2$, and that, for the $i^{th}$ commodity, 
we observe \break $(p^k, y^k, \wk, \xk, \zk)$
for $k=1,2$, where $\wk, \xk \in \rx^{M_1}_+$, with $M_1 < M$, and 
$\zk \in \rx^{M-M_1}_{++}$. 
Then the data is consistent with profit maximization given some
monotonic, concave, CRS production function if and 
only if:\eol
(i) $p^k y^k \geq \wk \xk$, and \eol
(ii) $ (p^k y^k - \wk \xk) \; \max_{j \in \{M_1+1,\ldots,M\}} 
\left(\frac{z_j^\ell}{z_j^k} \right) \; \geq \; 
p^\ell y^k - {\bf w}^\ell \xk
\quad \mbox{ for } k=1,2 \mbox{ and } \ell \neq k$.
\ethm
\bproof
Consider the problem 
\beqa
\min_{\vone} & p^1 y^2 - \wone \xtwo - \vone \ztwo
\label{e10} \\
\mbox{subject to } &
p^1 y^1 - \wone \xone - \vone \zone = 0
\label{e12}
\eeqa
Clearly, the minimum value of (\ref{e10}) is attained by setting 
\beqb
      v_k^1  = \left\{ \begin{array}{ll}
                        \frac{p^1 y^1 - \wone \xone }{z_k^1} 
                & \mbox{ if } k = 
                     \mbox{ Arg max }_{j \in \{M_1+1,\ldots,M\}}
                              \frac{z_j^2}{z_j^1} \\
                        0 & \mbox{ otherwise}
                        \end{array}
                        \right.
\eeqb
Now we just need the minimized value of (\ref{e10}) to be non-positive,
from which part (ii) of the Lemma readily follows.
\eproof

Finally, suppose now that for some commodity $i$, in each of two countries
all we observe is the expenditure on the commodity (but not price or quantity),
quantities of all $M$ factors, and prices of $M_1 < M$ factors. Further, 
define 
\[ \alpha^k  =  \max_{j \in \{M_1+1, \ldots, M\} } ( z_j^\ell / z_j^k )
\quad \mbox{ for } k=1,2 \mbox{ and } \ell \neq k
\]
The hypothesis of a CRS production function is still refutable, as the
next theorem shows.

\bthm
Suppose that $K=2$, and that, for the $i^{th}$ commodity, we observe \break
$(\bar{y}^k,\wk, \xk, \zk)$ for k=1,2, where $\wk,\xk
\in \rx^{M_1}_+$, with $M_1 < M$, and $\zk \in \rx^{M-M_1}_{++}$.
Then there exists a monotonic, concave, CRS
production function consistent with the data if and only if: \eol
(i) $\bar{y}^k \geq \wk \xk$ for $k=1,2$, and \eol
(ii) $
\bigl( \wone \xtwo + \alpha_1 (\bar{y}^1 - \wone \xone) \bigr)
\bigl( \wtwo \xone + \alpha_2 (\bar{y}^2 - \wtwo \xtwo) \bigr)
\geq \bar{y}^1 \bar{y}^2 $
\ethm
\bproof
Such a production function exists if and only if 
there exist $\vk$ such that part (ii) of Lemma \ref{plem2} above is satisfied.
In this case part (ii) of Lemma \ref{plem2} is written as
\beqa
(\wone \xtwo + \vone \ztwo) (\wtwo \xone + \vtwo \zone) \geq
\bar{y}^1 \bar{y}^2
\eeqa

Consider the terms on the left-hand side. Since the data from each country 
must also satisfy (\ref{e12}), it is clear that the maximum value of 
$(\vk {\bf z}^\ell)$ is attained by setting
\beqb
v_j^k = \left\{ \begin{array}{ll}
                        \frac{\bar{y}^k - \wk \xk}{z_j^k} 
               & \mbox{ if } j = \mbox{ Arg max } \frac{z_j^\ell}{z_j^k} \\
                        0 &  \mbox{ otherwise}
                        \end{array}
                        \right.
\eeqb
leading to part (ii) of the theorem.
\eproof

Helpman's (1984) post-trade restrictions on data can be seen to be
necessary but not sufficient conditions for the existence of common CRS
production functions for each commodity. For two countries, $k$ and $\ell$, 
Helpman's condition is stated as
\beqa
({\bf w}^\ell - {\bf w}^k) \; \sum_{i=1}^N (T^{\ell k}_i {\bf a}_i^k - 
                                           T^{k \ell}_i {\bf a}_i^\ell) 
\geq 0
\eeqa
where ${\bf a}_i^k$ is the cost-minimizing quantity of inputs necessary to 
produce one unit of commodity $i$ in country $k$, and $T^{\ell k}_i$ is the 
gross exports of commodity $i$ from $k$ to $\ell$. If commodity $i$
is exported from $\ell$ to $k$, $T^{k \ell}_i > 0$ and $T^{\ell k}_i = 0$.

This condition is necessary for the existence of common CRS production 
functions across the two countries. To see this, note that under
CRS, ${\bf a}^k_i = {\bf x}^k_i / y_i^k$, so that from 
equation (\ref{e3}) we have $p_i^k = {\bf w}^k {\bf a}_i^k$ and 
$p_i^\ell \leq {\bf w}^\ell {\bf a}_i^k$.
Now, setting $p_i^k = p_i^\ell$ yields 
$({\bf w}^\ell - {\bf w}^k) {\bf a}_i^k \geq 0$. Similarly, we have
$({\bf w}^k - {\bf w}^\ell) {\bf a}_i^\ell \geq 0$. 
Now, multiplying the first inequality  by $T_i^{\ell k}$ and the 
second by $T_i^{k \ell}$, and then summing across all 
commodities yields Helpman's result.

This result fails to be sufficient on two counts. Firstly, sufficiency 
requires that 
$p_i^k = {\bf w}^k {\bf a}_i^k$, and, secondly, the result must hold for 
each commodity, and not just in the aggregate. Since the result boils down
to a test of CRS production functions for each commodity, the results of 
this section go through completely in terms of deriving necessary and 
sufficient conditions for this model to be consistent with the data in the
presence of unobserved variables. 

\section{A Generalized Heckscher-Ohlin Model}

In this section, we consider the model of Deardorff (1982), who relaxes two of 
the three assumptions of the
basic HOV model. There is no assumption on commodity or factor price 
equalization, and the only restriction on preferences is that they satisfy
WARP (the Weak Axiom of Revealed Preference). On the production side, 
constant returns to scale are assumed for each good, along with frictionless
domestic markets and competitive behavior\ft{The lack of commodity or 
factor price equalization, therefore, is assumed to arise due to trade 
policies alone.}. For the main theorem of the paper, technology is not 
assumed to be identical across countries\ft{However, identical technologies
are assumed for the two corollaries that follow.}.
This theorem 
states that $\wa {\bf S} < 0$, where $\wa$ is a vector of factor
prices for the country under autarky, and $S_j$ represents the country's
net exports (positive or negative) of factor $j$, as embodied in the 
commodities traded by the country.

Autarky factor prices inversely reflect factor abundance in the sense of Ohlin,
so that this result says that countries tend to export their more abundant 
factors (those with lower prices) and import their less abundant ones. 
In the absence of joint production, and assuming CRS production functions, 
${\bf S}$ can be directly
computed from observed data for one country alone, by imputing factors to 
goods on the basis of domestic factors of production\ft{As computed by this 
method, ${\bf S}$
satisfies Deardorff's Assumption 11, which is needed to prove his main theorem,
but not his Assumption 12, which is used only for the corollaries.}. 
Autarky factor prices, however, are intrinsically unobservable, and are a 
natural candidate for elimination from the system of inequalities that defines
this model. 

To illustrate the refutable implications of this model, 
we restrict the analysis to the $2 \times 2$ case (i.e. 2 commodities
and 2 factors). Further, we assume that only the commodities, and not the 
factors, are directly traded. 

Suppose, then, that over a single time period, for some country, we 
observe $\bpt$, the vector of commodity prices under trade, 
$\yp$, the produced vector of goods, $\yc$, the consumed
vector of goods, $\wt$, the vector of factor prices, and 
$\{x^t_{ij}\}_{i,j=1,2}$, the quantities of factors used in the production of 
each good\ft{Note that, as mentioned in earlier sections, this is more than
is readily observed with available data.}. 
The net factor exports vector, 
${\bf S}$ is then computed as:
\beqa
S_j & = & \sum_{i=1}^2 x_{ij} (y_i^p - y_i^c) \quad\mbox{ for } j=1,2
\eeqa

Let $\pa, \ya, \wa$ denote the unobserved 
commodity prices,
the produced (and consumed) commodity bundle, and factor prices under autarky.
Following Deardorff, we assume that both the observed trade quantities 
and the unobserved autarky quantities imply full employment of factors. For
each country, we
can now write down a system of polynomial inequalities in the observed and 
unobserved variables that describes the assumptions and conclusion of the 
model:
\beqa
\sum_{i=1}^2 \xaij & = & \sum_{i=1}^2 \xtij \quad \mbox{ for } j=1,2 
\label{fendow} \\
\pti \ypi - \sum_{j=1}^2 \wtj \xtij & = & 0 \quad\mbox{ for } 
i=1,2 \label{gcrs1} \\
\pti \yai - \sum_{j=1}^2 \wtj \xaij & \leq & 0 \quad\mbox{ for } i=1,2
\label{gwapm1} \\
\pai \yai - \sum_{j=1}^2 \waj \xaij & = & 0 \quad\mbox{ for } 
i=1,2 \label{gcrs2} \\
\pai \ypi - \sum_{j=1}^2 \waj \xtij & \leq & 0 \quad\mbox{ for } i=1,2
\label{gwapm2} \\
\bpt \yc & \geq & \bpt \yp \label{tariff} \\
\pa \yc & > & \pa \ya \label{warp} \\
\wa {\bf S} & < & 0  \label{dorf}
\eeqa

Equation (\ref{fendow}) above merely states that there is full employment of 
factors under both trade and autarky (consistent with a monotonic production 
function). (\ref{gcrs1})-(\ref{gwapm2}) are necessary and sufficient for 
the existence of monotonic, concave CRS production functions for each good. 
(\ref{tariff}) is an assumption of the model (it is a natural assumption; 
importantly, for our purposes, it is a restriction only on observed data). 

Note that (\ref{fendow})-(\ref{gwapm2}) together imply 
\beqa
\bpt \yp & \geq & \bpt \ya \label{wapm1} \\
\pa \ya & \geq & \pa \yp \label{wapm2}
\eeqa
Now, (\ref{wapm1}) and (\ref{tariff}) lead to $\bpt \yc \geq \bpt \ya$. 
The bundle consumed under autarky is therefore affordable at trade prices, so 
that WARP (necessary and sufficient for 
the existence of a monotonic, concave utility function, as shown by Varian, 
1982) now implies (\ref{warp}).

Notice that (\ref{dorf}) in this case holds if and only if 
\beqa
\frac{w_1^a}{w_2^a} S_1 + S_2 < 0
\label{dorf1}
\eeqa
Hence, the (unobserved) factor price ratio under autarky, $\frac{w_1^a}{w_2^a}$
is of immediate interest. We approach the problem of deriving restrictions on 
observables such that the above system of equations is satisfied by first 
ignoring (\ref{dorf}), and trying to determine restrictions on
$\frac{w_1f^a}{w_2^a}$. Once we have those, we can then re-impose (\ref{dorf}).

First, we define some restrictions that we will need observed data to 
satisfy. Of these, OC 2 and OC 3 below merely offer notational convenience:
OC 2 defines good 1 to be the import good, and OC 3 states that good 1 
is relatively intensive in factor 2. OC 4 is necessary for the existence of 
CRS production functions. OC 1 states that the country has balanced trade 
at its own prices. The natural assumption here is $\pf \yp = \pf \yc$, 
where $\pf$ represents world prices for commodities. OC 1 will hold for 
a country that either practices free trade or has a uniform tariff on all 
commodities. 

\bdefn
The following are referred to as {\em Observable Conditions\/}:
\begin{list}{OC \arabic{oc}.}{\usecounter{oc}}
\item $\bpt \yp = \bpt \yc$.
\item $y_1^c > y_1^p$ and $y_2^c < y_2^p$.
\item $\frac{x_{11}^t}{x_{12}^t} < \frac{x_1}{x_2} 
< \frac{x_{21}^t}{x_{22}^t}$.
\item $p_i^t y_i^p = \sum_{j=1}^2 w_i^t x_{ij}^t$ for $i=1,2$.
\end{list}
\edefn

\bdefn
An economy is a {\em Heckscher-Ohlin Production Economy\/} (HOPE) if it 
has a 
monotonic, concave utility function and 
monotonic, concave CRS production functions for each good. 
\edefn

\blem
Suppose that, for some country,  the observed data 
$(\bpt,\yp,\yc)$ satisfy OC 1 and OC 2.
Then the economy can be a HOPE only if (i) $y_1^a \geq y_1^p$ and (ii)
$\frac{p_1^a}{p_2^a} > \frac{p_1^t}{p_2^t}$. 
\label{ya}
\elem
\bproof
$\bpt \yp = \bpt \yc$ implies that $\frac{p_1^t}{p_2^t} = 
\frac{y_2^p - y_2^c}{y_1^c - y_1^p}$. Further, from (\ref{wapm2}) and 
(\ref{warp}), we have
 $\pa \yc > \pa \yp$, so that $\frac{p_1^a}{p_2^a} 
> \frac{y_2^p - y_2^c}{y_1^c - y_1^p}$, leading to part (ii) of the 
Lemma. 

Now, (\ref{wapm1}) and (\ref{wapm2}) further imply that 
\beqa
\frac{p_1^t}{p_2^t} \; (y_1^a - y_1^p) & \leq & y_2^p - y_2^a \\
\frac{p_1^a}{p_2^a} \; (y_1^a - y_1^p) & \geq & y_2^p - y_2^a
\eeqa
We therefore have $(\frac{p_1^a}{p_2^a} - \frac{p_1^t}{p_2^t}) 
(y_1^a - y_1^p) \geq 0$, and, given part (ii), part (i) of the Lemma 
now follows.
\eproof

This intuition is expressed in Figure 1 below, which exhibits the budget
line under trade. (\ref{wapm1}) implies that $\ya$ lies inside the triangle
AOB. Further, the budget line is 
steeper under autarky than trade, and, along with 
(\ref{wapm2}), this further restricts the autarky bundle $\ya$ to lie
in the shaded region.

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\put(0,0){\vector(0,1){150}}
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\put(70,65){\circle*{5}}
\put(120,40){\circle*{5}}
\put(-10,-10){O}
\put(-15,95){A}
\put(200,-15){B}
\put(275,-15){$y_1$}
\put(-25,140){$y_2$}
\put(80,65){${\bf y^p}$}
\put(120,50){${\bf y^c}$}
\multiput(70,65)(0,-8){8}{\line(0,-1){4}}
\multiput(80,60)(0,-8){7}{\line(0,-1){4}}
\multiput(90,55)(0,-8){7}{\line(0,-1){4}}
\multiput(100,50)(0,-8){6}{\line(0,-1){4}}
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\multiput(120,40)(0,-8){5}{\line(0,-1){4}}
\multiput(130,35)(0,-8){4}{\line(0,-1){4}}
\multiput(140,30)(0,-8){4}{\line(0,-1){4}}
\multiput(150,25)(0,-8){3}{\line(0,-1){4}}
\multiput(160,20)(0,-8){3}{\line(0,-1){4}}
\multiput(170,15)(0,-8){2}{\line(0,-1){4}}
\multiput(180,10)(0,-8){1}{\line(0,-1){4}}
\multiput(190,5)(0,-8){1}{\line(0,-1){4}}


\put(0,-30){Figure 1: Region in which $\ya$ must lie}
\end{picture}

The next theorem provides a necessary and sufficient condition for 
autarky factor prices to be consistent with a HOPE.

\bthm
Suppose that for some country the observed data $(\bpt, \yp, \yc, \wt,
\{x_{ij}^t\}_{i,j=1,2})$ satisfy conditions OC 1-4. Then, for given 
autarky factor prices $\wa$, there exist commodity prices $\pa$ and 
quantities $\ya$, and factor quantities $\{x_{ij}^a\}_{i,j=1,2}$ such 
that all observed and unobserved values are consistent with a HOPE 
if and only if 
$\frac{w_1^a}{w_2^a} < \frac{w_1^t}{w_2^t}$.
\label{autarky}
\ethm
\bproof
\noindent \underline {``Only if'' part}:

{}From Lemma \ref{ya}, it must be the case that $\frac{p_1^a}{p_2^a} 
> \frac{p_1^t}{p_2^t}$. The rest now 
follows from a standard Stolper-Samuelson argument. Consider the unit
output isoquant for each good, and let $z_{ij}$ denote the optimal quantities 
of factors along these isoquants, given $\frac{w_1}{w_2}$. We have
\beqa
\frac{p_1}{p_2} & = & \frac{z_{11} + (w_2/w_1) z_{12}}
                        {z_{21} + (w_2/w_1) z_{22}}
\eeqa
so that, at given $\{z_{ij}\}_{i,j=1,2}$, $\frac{w_2}{w_1}$ is an increasing
function of $\frac{p_1}{p_2}$. 

\par
\noindent \underline {``If'' part}:

Suppose that $\frac{w_1^a}{w_2^a} 
< \frac{w_1^t}{w_2^t}$. We will choose $\pa,\ya,x_{11}^a,x_{21}^a,$ $x_{12}^a,
x_{22}^a$ such that (\ref{fendow})-(\ref{warp}) are satisfied.

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\put(85,150){$\bf \omega$}
\put(50,-25){Figure 2: Edgeworth Box for Production}
\multiput(150,225)(3,-0.5){50}{.}
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\put(135,234){$\bf \delta$}
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\put(223,220){$\bf \beta$}
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\put(190,185){$\bf \gamma$}
\put(167.5,208.75){\circle*{4}}
\put(175,205){$\bf \lambda$}
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\put(140,285){\line(1,-6){25}}
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\put(350,-15){$x_{11}$}
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\put(407,180){$x_{12}$}
\put(415,190){\vector(0,1){10}}
\put(140,310){$x_{21}$}
\put(133,312){\vector(-1,0){10}}
\put(-20,130){$x_{22}$}
\put(-12,120){\vector(0,-1){10}}

\end{picture}

Consider the Edgeworth box for production, depicted in Figure 2. Let 
${\bf \omega}$ denote the production point under trade.
Choose any $y_1^a$ such that $y_1^p < y_1^a < y_1^c$. Let 
${\bf \delta} = (\delta_1, \delta_2) = 
\frac{y_1^a}{y_1^p}(x_{11}^t, x_{12}^t)$. Now, define ${\bf \alpha},{\bf 
\beta} \in \rx_{++}^2$ such that
\beqb
(\wt {\bf \alpha} = \wt {\bf \delta}) & \mbox{ and } & 
(\frac{x_1 - \alpha_1}{x_2 - \alpha_2} = \frac{x_{21}^t}{x_{22}^t} ) \\
({\bf w}^a {\bf \beta} = {\bf w}^a {\bf \delta}) & \mbox{ and } &
(\frac{x_1 - \beta_1}{x_2 - \beta_2} = \frac{x_{21}^t}{x_{22}^t} \} 
\eeqb

Notice that $\alpha_1 < \beta_1 $ and $\alpha_2 < \beta_2$, so that 
$\wt {\bf \beta} > \wt {\bf \alpha}$.
Choose $\rho \in (0,1)$ and define ${\bf \gamma} = \rho {\bf \alpha}
+ (1 - \rho) {\bf \beta}$. Set 
\beqb
y_2^a = y_2^p \; \frac{x_1 - \gamma_1}{x_1 - x_{21}^t} = 
y_2^p \; \frac{x_2 - \gamma_2}{x_2 - x_{22}^t}
\eeqb

Finally, choose $\mu \in (0,1)$, and define ${\bf \lambda} = 
\mu {\bf \delta} + (1 - \mu) {\bf \gamma}$. ${\bf \lambda}$ is a candidate 
production point under autarky.

These choices obviously satisfy (\ref{fendow}), (\ref{gcrs1}) and 
(\ref{gcrs2}) above. To see
that (\ref{gwapm1}) is satisfied, notice that 
\beqb
p_1^t y_1^a = (\frac{y_1^a}{y_1^p}) (p_1^t y_1^p) = 
(\frac{y_1^a}{y_1^p})  (w_1^t x_{11}^t + w_2^t x_{12}^t)
= \wt {\bf \alpha}
\eeqb
But, since $\wt {\bf \beta} > \wt {\bf \alpha}$, we have
$\wt {\bf \lambda} > \wt {\bf \alpha}$, proving that 
(\ref{gwapm1}) holds for good 1. A similar argument shows that it holds for 
good 2 as well. 

Next, consider (\ref{gwapm2}) for good 1. From the definition of $p_1^a$, 
we have
\beqb
p_1^a y_1^p - w_1^a x_{11}^t - w_2^a x_{12}^t < 0 & \iff & 
(\frac{y_1^p}{y_1^a})(w_1^a x_{11}^a + w_2^a x_{12}^a) <
                     w_1^a x_{11}^t + w_2^a x_{12}^t \\
& \iff & {\bf w}^a {\bf \lambda} < {\bf w}^a {\bf \delta}
\eeqb
where the last inequality follows from the construction of ${\bf\lambda}$. 
Again, a similar argument proves the case for good 2. 

Finally, (\ref{gwapm1}) and (\ref{gwapm2}), together with $y_1^a > y_1^p$, 
imply that $\frac{p_1^a}{p_2^a} > \frac{p_1^t}{p_2^t}$, which along with
$\bpt \yc > \bpt \ya$ and $y_1^c > y_1^a$ implies (\ref{warp}).
\eproof

Now, we are in a position to re-impose (\ref{dorf}). We say that 
a set of observed data $(\bpt, \yp, \yc, \wt, \{x_{ij}^t\}_{i,j=1,2})$ 
is consistent with the generalized Heckscher-Ohlin model of Deardorff
if there exists some HOPE that is both consistent with the observed data
and satsifies (\ref{dorf}).

\bdefn
The observed data $(\bpt,\yp,\yc,\wt, \{x_{ij}^t\}_{i,j=1,2})$ 
are {\em GHO-consistent} if there exists some HOPE such that:\eol
(i) all observed and unobserved values are consistent with that HOPE, and \eol
(ii) $\wa {\bf S} < 0$.
\edefn

\bthm
Suppose that for some country the observed data $(\bpt, \yp, \yc, \wt,
\{x_{ij}^t\}_{i,j=1,2})$ satisfy conditions OC 1-4.
Then the data are GHO-consistent if and only if at least one of the 
following two conditions holds: \eol
(i) $S_2 < 0$ or (ii) $\wt {\bf S} < 0$.
\label{dorf22}
\ethm
\bproof
\noindent \underline {``If'' part}:

Suppose that $S_2 < 0$. Then either $S_1 < 0$, in which case we are done, 
or $S_1 \geq 0$, in which case there exists an $\epsilon > 0$ such that 
\beqb
\delta S_1 + S_2 < 0 \quad \mbox{ for all } \delta < \epsilon
\eeqb
Choose $\frac{w_1^a}{w_2^a} \in (0, \min(\frac{w_1^t}{w_2^t},\epsilon) )$. 
The rest now follows from Theorem \ref{autarky}.

Suppose now that $S_2 \geq 0$ but $\wt {\bf S} < 0$. This can happen only if
$S_1 < 0$. Further $\wt {\bf S} < 0 \iff \frac{w_1^t}{w_2^t} \; S_1 
+ S_2 < 0$. Then there exists a $\rho \in (0, \frac{w_1^t}{w_2^t})$ such that
$\rho S_1 + S_2 < 0$. Choose $\frac{w_1^a}{w_2^a} = \rho$; once again, the 
rest follows from Theorem \ref{autarky}.

\noindent \underline {``Only if'' part}:

Suppose that neither (i) nor (ii) holds; i.e. $S_2 \geq 0$ and 
$\wt {\bf S} \geq 0$. Since $\frac{w_1^a}{w_2^a} < \frac{w_1^t}{w_2^t}$,
we have $\wa {\bf S} \geq 0$. 
\eproof

The observed data are therefore consistent with the model if and only
if at least one of the following hold: the country is a net importer
of the factor that the import good is relatively intensive in, or the
value of net factor exports at factor prices under trade is negative.
Each of these reflects a more traditional interpretation of the HOV model.

Finally, we consider the case where $\bpt \yc > \bpt \yp$. This is the 
normal case when there is  a tariff on the import good. The country trades on
the world markets at world prices, $\pf$, so that balanced trade implies
that $\pf \yp = \pf \yc$. In this case, significantly weaker restrictions on 
observed data are derived. The intuition here is that the observed situation
under tariffs allows us to make an inference about the unobserved situation
under free trade, from which a further inference about autarky must be drawn.

\bdefn
The observed data $\yp, \yc, \pf, \{x_{ij}\}_{i,j=1,2}$ satisfy {\em 
Observable Condition 5\/} (OC 5) if :\eol
(i) $\pf \yp = \pf \yc$, \eol
(ii) $\bpt \yc > \bpt \yp$. 
\edefn

Let $\wf$ denote the factor prices, $\yfp$ the production 
bundle, and $\yfc$ the consumption bundle of the economy in a hypothetical
free trade situation.

\blem
Suppose that the observed data $(\pf,\bpt,\yp,\yc,\wt)$ satisfy OC 2-5. 
Then, given $\wf$, there exist $\yfp, \yfc$ such that all observed and 
unobserved values are consistent with some HOPE if and only if 
$\frac{w_1^f}{w_2^f} > \frac{w_1^t}{w_2^t}$. 
\elem
\bproof
Here $\frac{p_1^f}{p_2^f} < \frac{p_1^t}{p_2^t}$. It is straightforward
to modify the proof of Theorem \ref{autarky} to this case.
\eproof

\blem
Suppose that the observed data $(\pf,\bpt,\yp,\yc,\wt)$ satisfy OC 2-5. 
Then, for all $\wa \in \rx^2_{++}$, 
there exist $\wf,\yfp,\yfc,\pa,\ya$ such that all observed and 
unobserved values are consistent with some HOPE.
\label{lem45}
\elem
\bproof
As in the proof of Theorem \ref{autarky}, given a $\wa \in \rx^2_{++}$, 
choose a production point  under autarky, and $\pa,\ya$ such that 
(\ref{fendow})-(\ref{warp}) are satisifed. Now choose any $\wf, y_1^f$ such 
that 
\beqb
\frac{w_1^f}{w_2^f} & > & \max \left\{ \frac{w_1^t}{w_2^t}, 
                                     \frac{w_1^a}{w_2^a} \right\} \\
y_1^f & < & \min \{ y_1^t, y_1^a \}
\eeqb

Now, as in the proof of Theorem \ref{autarky}, choose a production 
point under free trade, such that $(\pf,\yfp,\{x_{ij}^f\}_{i,j=1,2})$ and
$(\bpt,\yp,\{x_{ij}^t\}_{i,j=1,2})$ are consistent with CRS production 
functions for each good. Clearly, it is possible to choose $\yfc$ such that 
$\pf \yfc = \pf \yfp$ and the bundles $\yc,\ya,\yfc$ satisfy WARP pairwise.

It is now straightforward to check that the production points 
under autarky
and free trade chosen above satisfy the additional inequalities
\beqb
p_i^f y_i^a - \sum_{j=1}^2 w_i^f x_{ij}^a & \leq & 0 
                        \quad\mbox{ for } i=1,2 \\
p_i^a y_i^f - \sum_{j=1}^2 w_i^a x_{ij}^f & \leq & 0 
                        \quad\mbox{ for } i=1,2 
\eeqb

Hence, all observed and unobserved values are consistent with some HOPE.
\eproof


The next theorem describes necessary and sufficient conditions for the 
data to be consistent with this model under the new set of assumptions 
about observed data. These conditions are, of course, significantly weaker 
than those of Theorem \ref{dorf22}; in this case, the model is
rejected if and only if a country is a net exporter of both factors. 

\bthm
Suppose that the observed data $(\pf,\bpt,\yp,\yc,\wt,
\{x_{ij}^t\}_{i,j=1,2})$ satsify conditions OC 2-5.
Then the data are GHO-consistent if and only if $\min \{S_1, S_2\} < 0$.
\ethm 
\bproof
The ``only if'' part is obvious. Consider the ``if'' part.
By Lemma \ref{lem45}, 
 $\wa$ could lie anywhere in $\rx^2_{++}$ to be consistent with some 
HOPE. Clearly, $\wa$ can be chosen to satisfy the inequality $\wa {\bf S} < 0$
whenever at least one of $S_1,S_2$ is strictly negative.
\eproof

\section{Conclusion}

Previous tests of the HOV model have tended to reject the conclusions of 
the model. Effectively, these tests amount to a joint hypothesis test
of all the assumptions of the model, and the failure of any one
assumption may render the conclusions invalid. Commodity and factor
prices are known to differ across countries. This paper discusses
nonparametric tests of two of the remaining assumptions of the
model---the assumptions of identical homothetic
preferences and identical constant returns to scale production
fucntions across countries, as well as the refutability of Deardorff's
version of the HOV model, which does not assume commodity or factor
price equalization, and which asserts that net factor exports,
valued at autarky factor prices, have a negative value.

The nonparametric tests we investigate are based on earlier work on
revealed preference and revealed cost theory by Afriat (1967) and
Varian (1982,1983,1984). Importantly, 
this technique leads to restrictions that are necessary and sufficient for
observed data to be consistent with a proposed model, and that
therefore exhaust {\em all\/} refutable implications of a given set of data. 

With respect to consumption data, we find that it is impossible
to refute the hypothesis that two countries have identical homothetic 
utility functions unless data on prices 
and quantities of all goods are available, at least in the form of
ratios across countries. It is not sufficient to have, on even {\em one\/}
commodity, only expenditure
data and price or quantity indices for each country.

We illustrate the use of the nonparametric test for a common
homothetic utility function using the Kravis, Heston 
and Summers (1982) data set. We find that there does not exist a 
homothetic utility function consistent with the data from India and the U.S.
However, there do exist such functions for the U.S. and Malawi, and 
Malawi and India. The nonparametric approach contrasts strongly with
the assumed functional form approach of Hunter and Markusen (1988),
who use the same data set and reject the assumption of a common
homothetic utility function across all 34 countries in the sample. An
additional virtue of the nonparametric approach is that it can be used
on as few as two countries.

On the production side, we investigate the refutability of the
assumption of an identical constant returns to scale production
function for some good across two countries, under different
assumptions about what data are observed. If all factor prices and
factor quantities and the total value of the good produced are
observed, but the price and quantity of the good remain unobserved,
the only restriction on the observed data is analogous to that imposed
by any homothetic production function. We further show that even when
some factor prices are unobserved, the constant returns to scale
assumption imposes substantive restrictions on factor quantities and
the observed factor prices. These restrictions are weaker the greater
the number of factors with unobserved prices.


Finally, we illustrate how refutable propositions may be derived in the 
more general Heckscher-Ohlin model of Deardorff (1982),
whose result states that the value of net factor exports at (clearly
unobservable) autarky factor prices is negative. We show that, in the
case of $2$ goods and $2$ factors, and assuming balanced trade (which
is a restriction only on observed data), observed data on commodity
and factor prices and quantities (from the situation under trade) are
consistent with the model if and only if either (i) the country in
question is a net importer of  the factor that the imported good is 
intensive in, or
(ii) the value of net factor exports at factor prices under trade is
negative. 

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\epage
\section*{Appendix}
\appendix
\section{Testing for Common Preferences---Malawi, India, U.S.A.}

\subsection{Homothetic Preferences}

The Kravis, Heston, and Summers data set reports, for 34 countries and
several commodity categories, the per capita expenditure on that commodity
category, a purchasing power parity (PPP) index relative to the U.S., and
a per capita quantity index relative to the U.S. All data are for the year
1975. The PPP index can be 
converted into a relative price index by dividing by the exchange rate. 
However, for our purposes, there is no need to do so. 

Notice that (\ref{hpref}), the inequality we wish to test, is invariant
to the scale of the price and quantity indices, $\alpha_i^k$ and $\beta_i^k$. 
Hence, we can directly use the PPP index numbers to represent the $\alpha$ 
values, and similarly the per capita quantity indices (rather than aggregate 
national consumption) to represent the $\beta$ values. 

For this illustration, we examined the eight major commodity categories
reported by Kravis, Heston, and Summers under consumption. The data 
we use, therefore, do not consider the consumption versus saving 
decision.

The raw data are shown in Tables 1 and 2, with the results of the calculations
performed for the test are shown in Table 3. 

\begin{table} \centering
\begin{tabular}{|l|c|c|c|}\hline
Category & U.S. (dollars) & India (rupees) & Malawi (kwacha) \\ \hline
    Food, beverages, tobacco &   818.24 &  565.34 & 55.543 \\
    Clothing, footwear       & 336.33   &  66.32  & 6.704 \\
    Gross rent, fuel      & 903.84    & 53.80 & 5.916 \\
    House furnishings, operations        & 343.14    & 24.43  &    9.000 \\
    Medical care        &  653.69    & 28.40  & 1.834 \\
    Transport, communications    &  692.17    &  57.20 & 7.194 \\ 
    Recreation, education   &  783.74   & 46.48 & 6.267 \\
    Other expenditures      &  628.47   & 39.74 & 3.206 \\ \hline
\end{tabular}
\caption{Expenditures by Commodity Category (in Local Currency)}
\end{table}

\begin{table} \centering
\begin{tabular}{|l|cc|cc|cc|}\hline
Category & \multicolumn{2}{c|}{U.S.} & \multicolumn{2}{c|}{India}
& \multicolumn{2}{c|}{Malawi} \\ \hline
&  PPP & Quantity & PPP & Quantity & PPP & Quantity \\ \hline 
Food, beverages, tobacco  & 1.00     & 1.00 &  3.78 & 0.183  & 0.373 & 0.182 \\
Clothing, footwear   & 1.00     & 1.00 &  4.54 & 0.043  & 0.606 & 0.033 \\
Gross rent, fuel         & 1.00     & 1.00 &  1.85 & 0.032  & 0.672 & 0.010 \\
House furnishings, operations 
& 1.00     & 1.00 &  3.53 & 0.020  & 0.508 & 0.052 \\
Medical care            & 1.00     & 1.00 &  1.05 & 0.041  & 0.112 & 0.025 \\
Transport, communications
        & 1.00     & 1.00 &  2.59 & 0.032  & 1.020 & 0.010 \\
Recreation, education   & 1.00     & 1.00 &  0.69 & 0.086  & 0.128 & 0.062 \\
Other expenditures    & 1.00     & 1.00 &  1.83 & 0.035  & 0.391 & 0.013 \\
\hline
\end{tabular}
\caption{PPP and Quantity Indices}
\end{table}

\begin{table} \centering
\begin{tabular}{|c|c|c|c|} \hline
Country $k$ & Country $\ell$ & $(\bfp^k \bfy^\ell) (\bfp^\ell \bfy^k)$ &
$(\bfp^k \bfy^k) (\bfp^\ell \bfy^\ell)$ \\ \hline
U.S. & Malawi & 645,263 & 493,590 \\
India & Malawi & 101,187 & 84,348 \\
U.S & India & 3,950,008 & 4,549,288 \\ \hline
\end{tabular}
\caption{Calculations for  Homotheticity Test}
\end{table}

The calculations indicate that we can reject the hypothesis that the 
U.S. and India had the same monotonic, concave, homothetic utility 
function. For the pairs U.S. and Malawi, and Malawi and India, this 
hypothesis cannot be rejected. 

Finally, we use the method of Varian (1982, Theorem 2) to construct a 
homothetic utility function consistent with the data from the U.S. and 
Malawi. Let the superscript $M$ denote values for Malawi and $U$ those
for the U.S. Define
\beqb
V^M & = & \min\{1,\frac{\bfp^U \bfy^M}{\bfp^U \bfy^U}, \frac{\bfp^M \bfy^U}
{\bfp^M \bfy^M} \frac{\bfp^U \bfy^M}{\bfp^U \bfy^U} \} \\
& = & 0.052 \\
V^U & = & \min\{1,\frac{\bfp^M \bfy^U}{\bfp^M \bfy^M}, \frac{\bfp^U \bfy^M}
{\bfp^U \bfy^U} \frac{\bfp^M \bfy^U}{\bfp^M \bfy^M} \} \\
    & = & 1
\eeqb

To define the utility function, we further need to know commodity 
prices in the U.S., $\bfp^U$. Since that information is not reported in the
Kravis, Heston, and Summers data set, we can pick any values of $p_i^U, y_i^U$
consistent with the reported expenditures. 
Now, define $U(\bfy)$ as
\beqa
U(\bfy) &  = & \min\{V^M \frac{\sumc \alpha_i^M p_i^U y_i}{\bfp^M \bfy^M}, \;
                  V^U \frac{\bfp^U \bfy}{\bfp^U \bfy^U} \}
\eeqa
This function is monotonic, concave, homothetic, and consistent with the 
observed data for Malawi and the U.S.

\subsection{Any Common Preferences}

Since the data from the U.S.A., India, and Malawi are not consistent
with a common homothetic utility function, we next test for whether they
are consistent with {\em any\/} well-behaved (i.e. continuous, concave,
monotonic) utility function. 

We say that bundle 1 is revealed preferred to bundle 2 ($y^1 \rp y^2$)
if 
\beqa
{\bf p}^1 {\bf y}^1 \geq {\bf p}^1 {\bf y}^2 
\eeqa
By Afriat's Theorem (Afriat, 1967) the
data are consistent in this manner if and only if the Generalized
Axiom of Revealed Preference (see Varian, 1982) is satisfied, i.e. 
if and only if 
\beqa
x^i \rp x^j \ldots \rp x^\ell \implies \mbox{ not } x^\ell \rp x^i
\label{rp}
\eeqa

Notice that the relationship (\ref{rp}) is invariant to the scale of
the prices or quantities. Hence, we can again directly use the price
and quantity indices provided by Kravis, Heston and Summers.

Let ${\bf p}^k = {\bf \alpha}^k {\bf p}^{US}$ for $k=U,I,M$, where
$U,I,M$ represents the U.S.A., India, and Malawi respectively.
Similarly, let ${\bf y}^k = {\bf \beta}^k {\bf y}^{US}$ for $k=U,I,M$.
The following table indicates the revealed preferred relationships.

\begin{table} \centering
\begin{tabular}{|c|c|} \hline
${\bf p}^U {\bf y}^U$ & $ 5,159.62$ \\
${\bf p}^I {\bf y}^I$ & $   881.71$ \\
${\bf p}^M {\bf y}^M$ & $    95.66$ \\
${\bf p}^U {\bf y}^I$ & $   338.38$ \\
${\bf p}^U {\bf y}^M$ & $   267.08$ \\
${\bf p}^I {\bf y}^U$ & $11,673.24$ \\
${\bf p}^I {\bf y}^M$ & $   776.17$ \\
${\bf p}^M {\bf y}^U$ & $ 2,415.99$ \\
${\bf p}^M {\bf y}^I$ & $   130.37$ \\ \hline
\end{tabular}
\caption{Calculations for Common Utility Function}
\end{table}

The calculations show that ${\bf y}^U \rp {\bf y}^I \rp {\bf y}^M$.
Hence, there does exist a well-behaved common utility function
consistent with the data from the three countries. Following Varian
(1982), one such utility function is given by:
\beqb
U^U & = & 1 \\
\lambda^U & = & 1 \\
U^I & = & \min\{U^U + \lambda^U p^U( y^I - y^U, U^U \} = -4,820.24 \\
\lambda^I & = & \max\{(U^U - U^I) / p^I (y^U - y^I), 1\} = 0.4468 \\
U^M & = & \min_{x=U,I}\min\{U^x + \lambda^x p^x( y^x - y^M), U^x\} 
     = -4,891.54 \\
\lambda^M & = & \max_{x=U,I} \max \{ (U^x - U^M) / p^M (y^x - y^M), 1
\} = 1 \\
U (y) & = & \min_{x = U,I,M} \{U^x + \lambda^x p^x (y - y^x) \}
\eeqb 

\end{document}

