Alchian and Allen hypothesized that a country will export high rather than low quality goods. Here we show that if the inherent attributes of the commodities are identical the country will only export the high-quality good.transport costs, patterns of trade, product quality, International Relations/Trade, F1,

Smith, Vincent H.

Vedrashko, Alexander

English

Research Papers in Economics

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Alexander Vedrashko, Research Assistant
University of California, Berkeley
Vincent H. Smith, Professor
Montana State University
Research Discussion Paper No. 22
November 1998
The purpose of research discussion papers is to make research findings available to researchers and the public
before they are available in professional journals.  Consequently, they are not peer reviewed.Perfect Substitutes:
 A Note on the Alchian-Allen Hypothesis
Abstract
Alchian and Allen hypothesized that a country will export high rather than low quality goods. 
Here we show that if the inherent attributes of the commodities are identical the country will only
export the high-quality good.   Perfect Substitutes: A Note on the Alchian-Allen Hypothesis 1
Perfect Substitutes: A Note on the Alchian-Allen Hypothesis
Introduction
In 1964, Alchian and Allen postulated that a country or region would export relatively
high valued products. They argued that when identical per unit fees such as transportation costs
are added to the prices of both high and low quality goods, the relative price of the high quality
product declines, and its relative consumption increases in markets in which those fees must be
paid. Borcherding and Silberberg (1978) argued that when markets are competitive and both
commodities are exported from region A to region B, relative exports of the high quality good
increase as transportation costs increase. They also found that this effect becomes stronger when
the two goods are closer substitutes. In re-examining the Alchian-Allen proposition, Umbeck
(1980) focused on the quality attributes of the two competing commodities to obtain his results. 
In this note, Umbeck’s discussion of quality attributes is elaborated by introducing the
concept of hedonic prices into the analysis. We show that, when the two commodities are
effectively perfect substitutes in terms of their mix of quality attributes, trade will take place only
in one commodity—the “high quality” good—if both commodities are consumed in both
countries. Alternatively, if both commodities are shipped from the exporting country to the
importing country then only one commodity—the “low quality” good—will be consumed in the
exporting country. We believe that neither of these findings has been previously reported in the
literature. 
 Perfect Substitutes: A Note on the Alchian-Allen Hypothesis 2
The Model
Lancaster (1966) and Rosen (1974) developed the concept of implicit or hedonic prices to
account for the fact that consumers do not derive utility from a commodity itself, but from its
quality-determining characteristics or attributes. Here, we consider two goods that have two common
quality attributes q and r, but the high quality good, h, has more units of each quality attribute than
the low quality good, l.  Both commodities are produced in each of two countries or regions, Country
1 and Country 2.  In each country, buyers have the same perceptions of and preferences with respect
to the quality attributes of the two commodities.  In addition, markets are assumed to be competitive.
Suppose that each commodity has the two quality attributes, q and r, and that units of their
attributes are identical.  In general, given competitive markets, the relationships between the
observed market prices and the hedonic prices for the quality attributes of each commodity in each
country will then be as follows: 
, (1) 
h
i
h
i
h
i r v q w p + =
and 
,  (2)
l
i
l
i
l
i r v q w p + =
where   and   are the prices of the high and low quality goods,   and   are the hedonic prices
h
i p
l
i p i w i v
of the quality attributes,   and   are the quantities of the quality attributes in each commodity,
j q
j r
where  i = 1, 2  denotes the country of interest and  j= h, l  denotes the commodity. Thus, when both
commodities are consumed in each country, in equilibrium the relative market prices of the two
commodities will be:Perfect Substitutes: A Note on the Alchian-Allen Hypothesis 3
, where i =1, 2. (3) l
i
l
i
h
i
h
i
l
i
h
i
r v q w
r v q w
p
p
+
+
=
In the absence of trade, there is no link between prices in Country 1 and prices in Country
2.  However, if trade is feasible, then that is no longer the case. Following Alchian and Allen, we
assume that a fixed per-unit transport cost t is incurred when either commodity is shipped from
Country 1 to Country 2. A competitive equilibrium that allows for trade then implies that:
, for  j = h, l. (4) t p p
j j + £ 1 2
If both goods are traded internationally, then equations (4) hold as equalities, and it follows that:
. (5)
p
p
pt
pt
h
l
h
l
2
2
1
1
=
+
+
Since  t>0, it must be the case that:
. (6)
p
p
p
p
h
l
h
l
2
2
1
1
<
This is the key Alchian-Allen result which states that the relative price of the high quality
commodity is lower in the importing country, Country 2, than in the exporting country, Country 1.
It holds if both commodities are exported from Country 1 to Country 2.
We now show that this cannot be the case if the commodities are perfect substitutes. The
proof is straightforward. Given that attributes q and r are each measured in the same units, if Perfect Substitutes: A Note on the Alchian-Allen Hypothesis 4
, where a >1,( 7 ) a = = l
h
l
h
r
r
q
q
then one unit of the high quality good and a  units of the low quality good are perfect substitutes.
This interpretation of perfect substitutes is slightly different from the usual textbook definition,
which implies that individual units of each commodity are perfect substitutes, that is, =1. If
, then the two commodities are not perfect substitutes because they contain different l
h
l
h
r
r
q
q
¹
proportions of the two attributes.  From equations (7) and (3), we obtain:
, i =1, 2. (8) a = l
i
h
i
p
p
This equation implies that   must be equal to  . However, this contradicts the inequality in l
h
p
p
1
1
l
h
p
p
2
2
equation (6), which must hold if both goods are being exported from Country 1 to Country 2. Thus,
when the two commodities are perfect substitutes and both are consumed in both countries, only one
of the two commodities can be traded.
The above result also implies that if both commodities are to be exported from Country 1 to
Country 2 then the two commodities must be imperfect substitutes for one another. When one unit
of the high quality good and a units of the low quality good are close substitutes, but not perfect
substitutes,  . Thus, based on equation (3), the price ratios may be different across the l
h
l
h
r
r
q
q
¹
countries because the hedonic prices (which are determined by consumer tastes and producer costs
[Rosen, 1974]) are not identical in the two countries, so that corresponding TERMS in (3) do not
cancel. As condition (6) may hold in equilibrium, both goods may be traded internationally. Perfect Substitutes: A Note on the Alchian-Allen Hypothesis 5
Is the Alchian/Allen Proposition Correct in the Case of Perfect Substitutes?
If the two commodities are perfect substitutes in the sense defined above, and both
commodities are consumed in each country, does the Alchian/Allen proposition hold?  The answer
is yes. When all local markets are in equilibrium from (8) it follows that:
. (9) a = =
l
h
l
h
p
p
p
p
1
1
2
2
According to inequality (6), however, if  both goods are traded internationally, and in Country 1,
 is equal to a, then in Country 2,   must be less than a.  As we have seen, this condition l
h
p
p
1
1
l
h
p
p
2
2
cannot be satisfied if both commodities are consumed in Country 2. Therefore, it follows that both
commodities cannot be exported from Country 1 to Country 2.  
To determine which commodity will be exported from Country 1, we consider the two
possible cases: (1) Country 1 exports only the low quality commodity, and (2) Country 1 exports
only the high quality commodity. If only the low quality commodity is exported, then the following
commodity price relationships must hold:
, and  . (10) t p p
h h + < 1 2 t p p
l l + = 1 2
This implies that 
(11)
t p
t p
p
p
l
h
l
h
+
+
<
1
1
2
2Perfect Substitutes: A Note on the Alchian-Allen Hypothesis 6
Given that both commodities are being consumed in both countries, equation (9) must be satisfied.
However, substituting equation (9) into equation (11) implies that     . 0 ) ( 1 1 < -
l h p p
This is clearly not feasible; the higher quality product cannot sell at a discount relative to the low
quality commodity. When Country 1 exports only the high quality commodity, then it follows that:
, and  .  (12) t p p
h h + = 1 2 t p p
l l + < 1 2
Thus,  , and it can also be shown that  , which is perfectly feasible.
t p
t p
p
p
l
h
l
h
+
+
>
1
1
2
2 0 ) ( 1 1 > -
l h p p
It is also straightforward to show that if both commodities are perfect substitutes and both
are exported from Country 1 to Country 2 then only the low quality commodity will be consumed
in Country 1. Given that both commodities are exported from Country 1 and, consequently,
consumed in Country 2, it follows from equation (8) and equation (5) that:
, and  . (13) a = l
h
p
p
2
2 p
p
pt
pt
h
l
h
l
2
2
1
1
=
+
+
This implies that   and, therefore, that  .  This result implies that in Country 1 the l
h
l
h
p
p
p
p
1
1
2
2 < a >
l
h
p
p
2
2
hedonic prices of the quality attributes in the high quality commodity are greater than in the low
quality commodity.  This is only feasible if the high quality commodity is not consumed in Country
1. Thus, Country 1 must export all its production of the high quality commodity to Country 2.Perfect Substitutes: A Note on the Alchian-Allen Hypothesis 7
Conclusion
We have shown that when two commodities are perfect substitutes in the sense that one unit
of the high quality good and more than one unit of the low quality good are perfect substitutes, and
if both countries consume both commodities, then only the high quality commodity will be traded.
Borcherding and Silberberg relied on an assumption that both goods are exported to obtain their
comparative statics result that an increase in transport costs would lead to an increase in the relative
exports of the high quality commodity. Their statement that as the two goods “become closer and
closer substitutes, … we should expect to see Alchian and Allen’s hypothesis confirmed” should be
interpreted carefully. When the two goods become closer substitutes, a corner solution certainly
becomes more likely. However, when they become perfect substitutes, a corner solution becomes
a certainty; that is, either exports of the low quality good will cease or the exporting country will
cease to consume the high quality commodity. Perfect Substitutes: A Note on the Alchian-Allen Hypothesis 8
References
Alchian, A., and W. Allen, 1964.  University Economics (Wadsworth Publishing Co., Belmont, 
CA)
Borcherding, T., and E. Silberberg, 1978. Shipping the Good Apples Out: The Alchian and Allen
Theorem Reconsidered, Journal of Political Economy 86 (11), 131-138.
Lancaster, K. J., 1966.  A New Approach to Consumer Theory, Journal of Political Economy 74 
(2), 132-157.
Rosen, S., 1974.  Hedonic Prices and Implicit Markets: Product Differentiation in Pure 
Competition, Journal of Political Economy 82 (1), 34-55.
Umbeck, J., 1980.  Shipping the Good Apples Out: Some Ambiguities in the Interpretation of 
“Fixed Charge,” Journal of Political Economy, February 88 (1), 199-208.

PERFECT SUBSTITUTES: A NOTE ON THE ALCHIAN-ALLEN HYPOTHESIS

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