McClelland, John W.

Musser, Wesley N.

Wetzstein, Michael E.

English

Research Papers in Economics

Returns to Scale  and Size  in
Agricultural Economics:  Reply
John W. McClelland,  Michael  E. Wetzstein,  and Wesley  N. Musser
We  appreciate  the  opportunity  that  Fare's
comment provides to further discuss the topic
of returns  to  scale  and size.  His  summary  of
literature on ray homothetic functions is useful
for  establishing  further  linkages  between  re-
turns to scale and size and associated economic
theory.  This literature and his comment  reit-
erate several points made in our paper. First,
the careless use of size and scale concepts, which
we discuss in agricultural economics,  is also a
problem  in the  mining  economics  literature
(Boyd). His summary further demonstrates the
relevance  of returns  to  scale  in  applied  pro-
duction economics and the influence  of tech-
nology characteristics on long-run cost. We are
pleased that the importance  of this subject  is
emphasized by Fare's comment.
Fare's summary  does introduce  a potential
ambiguity.  He states that ray homothetic  and
ray homogenous  functions  fail  to satisfy  free
disposability and quasi-concavity of inputs. In
particular,  the  latter property  limits  his  con-
tention that  "this  property  makes them suit-
able for the study of returns to scale."  How-
ever,  Fare, Jansson,  and  Lovell (p. 625) state
"Although  neither  of these  two  strong prop-
erties  is  imposed  globally  by  the ray-homo-
thetic function, either may be satisfied locally.
.. " Thus, ray-homothetic functions may have
applications where environmental constraints
preclude  standard regularity conditions hold-
ing globally.
Fare correctly points out an additional lim-
iting condition  on  ray  production  technolo-
gies.  He states that "the class of functions that
satisfies (3) consists of  the ray-homothetic pro-
duction  functions  alone."  We  agree  that the
John W.  McClelland is  an agricultural economist,  Economic Re-
search Service,  U.S. Department of Agriculture; Michael E.  Wetz-
stein is  an  associate  professor,  Department  of Agricultural  Eco-
nomics, University of Georgia;  Wesley  N. Musser is an associate
professor,  Department  of Agricultural  Economics  and Rural So-
ciology, Pennsylvania  State University.
general statement concerning expansion paths
for (3)  is misleading.  A linear expansion  path
is characteristic of a technology if and only if
the input structure of the technology is homo-
thetic.  However,  equation (3)  is a basic equa-
tion corresponding to the class of ray produc-
tion  functions  including  homogenous,
homothetic,  ray homogenous, and ray homo-
thetic functions.  Appropriate modifications of
this general form result in these four particular
specifications  and provide a link between  our
propositions  1 and  2  and  previous  research.
Proposition  1 establishes  a  relation  between
elasticity  of size and  scale  for  a homothetic
production  technology.  We  noted  that  this
proposition does not hold for ray homogenous
or ray homothetic technologies (p.  131).
Fare's comment did not address proposition
2, which is the main contribution of  our article.
Proposition  2  establishes  elasticity  of size as
the envelope  of the elasticity  of scale, or  the
long-run average cost curve as the envelope of
scale  average  cost  curves.  For a  homothetic
technology,  the  scale  average  cost curve  cor-
responding  to the expansion path is the long-
run average cost curve. Otherwise the long-run
average cost curve is the envelope of scale av-
erage cost curves.  Subsequently, Revier  inde-
pendently derived these results.  Investigating
additional implications  associated with prop-
osition  2  appears  more  fruitful  than further
attention to the taxonomy of production  and
cost functions.
[Received March 1988;
final revision received March 1988.]
References
Boyd,  G.  "Factor Intensity  and Site  Geology as  Deter-
minants  of Returns  to  Scale in  Coal  Mining."  Rev.
Econ. and  Statist. 69(1987):18-23.
Western Journal  of Agricultural  Economics,  13(1):  151-152
Copyright  1988 Western Agricultural Economics  AssociationWestern Journal of  Agricultural  Economics
Fire,  R.,  L.  Jansson,  and  C.  A.  K.  Lovell.  "Modeling
Scale  Economies  with  Ray-Homothetic  Production
Function."  Rev. Econ. and  Statist. 67(1985):624-29.
McClelland,  J.  W., M.  C.  Wetzstein,  and W. N. Musser.
"Returns  to Size and  Scale in Agricultural  Econom-
ics."  West. J. Agr. Econ. 11(1986):129-33.
Revier, C.  F.  "The Elasticity of Scale, the  Shape of Av-
erage Costs, and the Envelope Theorem." Amer. Econ.
Rev.  77(1987):486-88.
152  July 1988

RETURNS TO SCALE AND SIZE IN AGRICULTURAL ECONOMICS: REPLY

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