On output price uncertainty and the comparative statics of industry equilibrium.
Horbulyk, Theodore M.
1. Introduction
The analysis of firm response to output price variability has,
until recently, been incomplete due to a failure to consider explicitly
the effect of firm response on the attainment of competitive equilibrium at the industry level. A recent paper by Appelbaum and Katz [1] has
addressed the issue of industry equilibrium under output price
uncertainty. They show how the optimal number of firms might change
through the process of exit and entry, and they characterize the
collective influence of firms' decisions on expected market price.
However, their analysis focuses on firms with a single-stage production
process which allows no production flexibility (in the form of
adjustment of output levels) after output price is revealed.
An earlier paper by Hartman [3] has shown that, at the firm level,
the optimal response to output price uncertainty will depend on whether
production flexibility exists. Specifically, if firms employ a two-stage
production process which is characterized by the presence of a
quasi-fixed factor, then firm-level output may increase or decrease in
response to changes in output price variability even if firms are risk
neutral. The assumption that production flexibility exists changes those
results initially developed by Sandmo [7], and applied more recently by
Appelbaum and Katz [1], which are based on a single-stage production
process.
This paper provides an extension of the Appelbaum and Katz analysis
to admit production flexibility. Equivalently, it extends the analysis
of Hartman to consider explicitly the effect of firm response on the
attainment of competitive equilibrium at the industry level.
Appelbaum and Katz show that for an industry of identical
risk-averse firms with a single-stage production technology, the effect
of a mean-preserving increase in price uncertainty is to decrease
industry output and to raise market price. A more recent comment by
Ishii [5] shows that output per firm will also fall, yet none of these
authors shows unambiguously whether the number of firms increases or
decreases. Although not explicitly stated, the analysis in Appelbaum and
Katz suggests there would be no firm or industry response to a
mean-preserving change in price spread if firms were risk-neutral (expected-profit maximizers).
The model employed in this paper is analogous to that of Appelbaum
and Katz with two principal exceptions: (i) here an industry is composed
of firms all of which are either risk neutral or risk averse,(1) and
(ii) firms employ a two-stage production process such as the one
analyzed by Hartman. There are two parts to the first-stage decision:
whether to produce at all, and if so, how much of a quasi-fixed (or
capital) input to commit before output price is revealed. Once output
price is revealed, the second-stage decision determines the level of
variable input use, and thus the output level. The effects of price
uncertainty on industry equilibrium are shown to vary according to the
assumptions which are made about the firms' preferences and
technology.
In particular, it is shown for an industry of identical risk-averse
firms which employ a two-stage production technology that, in long-run equilibrium, industry output might rise or fall in response to a
mean-preserving increase in the dispersion of output price. This is in
contrast to the result of Appelbaum and Katz who show that when
risk-averse firms employ a single-stage production technology, the
industry output will decrease unambiguously. in the special case of
risk-neutral firms, conditions are given here under which industry
output could increase or decrease in response to a mean-preserving
increase in output-price uncertainty.
It is implicit in this result, as in the work of Epstein [2],
Hartman [3] Turnovsky [8]and Wright [9], that under a two-stage
production technology firms may have an affinity for price variability.
Such an affinity may be present even where firms are averse to
variability of profits.
For example, suppose the technology at the firm level is such that,
under price uncertainty, increased use of a quasi-fixed factor ex ante
increases a firm's short-run elasticity of supply ex post. An
expected-profit-maximizing firm would choose to increase its use of the
quasi-fixed factor in response to greater output price uncertainty.
Provided that the factors used in each stage of production are technical
complements, this will result in increased output as well.(2) For firms
which are risk averse, there will be an offsetting tendency, described
by Sandmo [7], to produce less output when price uncertainty increases.
The optimal (short-run) firm response has thus been described in the
literature in terms of the firm's preferences and technology. This
paper extends that analysis by characterizing the long-run industry
equilibrium. The direction of firm response will depend on the
offsetting forces described above, as well as on adjustments in expected
price which result from changes in output per firm and the number of
firms.
Section II characterizes industry demand and the behavior of
individual firms. The following two sections describe two industry
equilibria and provide comparative-static analyses of industry responses
to a mean-preserving change in the dispersion of output prices. Section
III considers an industry composed of identical firms which are
risk-averse whereas section IV considers an industry of fisk-neutral
firms.
II. Industry Demand and Individual Firm Behavior
Expected price will be inversely related to industry output and
thus is influenced by entry and exit. Industry demand is stochastic, as
given by
[Mathematical Expression Omitted] (1)
where [micro] is expected price and Q is the industry output of a
homogeneous good such that [micro](Q) < 0, and where [epsilon] is a
random variable such that E[epsilon] = 0 and E[epsilon.sub.2] = 1. When
the positive constant, [gamma], increases there is a mean-preserving
spread in output price, p, signifying increased output price
uncertainty.
Consider an industry composed of representative (identical) firms
which produce a homogeneous good, the demand for which is variable in
successive periods. Production follows a two-stage process which
requires some commitment of resources by a firm before its uncertainty
about output prices is resolved. The firm employs two factors which may
be denoted capital, K, and labor, L, and which have the following
properties. Capital decisions must be made before a random output price
is revealed, and, once made, represent sunk costs to the firm. Labor
decisions are not binding until after the output price has been
revealed, but before production is in fact completed. Production is a
single-period process, with capital decisions made before the start of a
period and labor inputs chosen at the start of the period.
With such a two-stage production process, there are two parts to
the first-stage decision: whether to produce at all, and if so, how much
of the capital or quasi-fixed input to commit before output price is
revealed. Once output price is revealed, the second-stage decision
determines the level of variable input use, and thus the output level.
Let the number of identical firms, n, be continuous, so that the
output per firm, q, is given by Q/n where q = f(K, L) and where f is
strictly concave. A firm which chooses not to produce in some period
incurs fixed cost T but can avoid costs, c and w, associated with K and
L, respectively. Thus, a firm's profit per period will be [pi] = pq
- wL - cK - T = ([micro](Q) + [gamma][epsilon])f(K, L) - wL - cK - T.
For firms which are risk averse, let U([pi]) be the von
Neumann-Morgenstem utility function of each, such that U'([pi])
> 0 and U"([pi]) < 0. Let b denote the benchmark expected
utility level below which existing firms will exit and switch to their
next best activity, and above which new firms will be drawn into the
industry. For firms which are risk-neutral, U'([pi]) > 0 and
U"([pi]) = 0. Assuming U(0) = 0, the benchmark expected utility
level for risk-neutral firms will be b = 0, which implies the
expectation of (positive) economic profit shall signal entry of firms.
III. Industry Equilibrium with Risk-Averse Firms
Following Appelbaum and Katz, risk-averse firms will choose to
enter the industry (or to stay in the industry) and to produce some
output provided the expected utility gained from so doing is no worse
than in the next best alternative activity. Entry and exit are not
instantaneous, so the industry need not always be in equilibrium.
However, in industry equilibrium, this condition must hold with
equality. If firms choose to produce, they will commit capital to that
level where the expected marginal utility from its product, net of
marginal cost, is zero. With two-stage decision making this optimal
level of capital use is conditional on choosing the optimal level of the
variable input, L*, in the second stage once output price is revealed.
These two conditions are given by
E[U(pi)][greater than or equal to]b, (2)
and
[Mathematical Expression Omitted] (3)
By assumption, the decision to produce some positive quantity and
the decision of how much capital to employ are made simultaneously,
based on the same information and expectations.
To facilitate the exposition of comparative-static responses, the
analysis of equilibrium adjustment will focus on mean price, [micro],
and the level of capital commitment by firms, K. Graphical
representation of the relations (2) and (3) would be accomplished by
representing each as a locus of points in (K,[micro]) space. Using (3)
and (2), it is useful to define each such locus and to determine its
slope. From (3) one may define
[Mathematical Expression Omitted] (4)
Here, g (K, p, w) represents the maximized value of restricted
profit given by the solution to the firm's second-stage choice of
L. From (2), one gets
[Mathematical Expression Omitted] (5)
Thus, G (K, [micro]) = 0 will be a locus of points in (K, [micro])
space consistent with optimal use of K, ex ante. The locus H(K,[micro])
= b will represent the set of points where firms would expect to achieve
exactly their benchmark level of utility, b, such that there would be no
incentive for further exit or entry.(4) Therefore, long-run equilibrium
will be described by that expected peice and output level
3. With risk aversion, the firm's problem is to
The second stage (choice of L) is solved first, which defines the
optimal level of the variable input, L*, in terms of any positive values
of p and K which might occur. The second stage of the maximization
problem occurs under certainty, and is given by
[Mathematical Expression Omitted]
the solution of which gives, L* = L* (K, p, w). The maximized
short-run profit function i when L* is employed will be denoted by g (K,
p, w), where
[Mathematical Expression Omitted]
The capital decision is based on
[Mathematical Expression Omitted]
such that
[Mathematical Expression Omitted]
which satisfies both G = 0 and H = b. In the short run, firms will
always strive to satisfy G = 0 through their expected-utility-maximizing
choice of K each period. Attainment of H = b might only occur in
long-run equilibrium once firms have entered or exited causing expected
price to fall or rise, respectively.(5)
The slope of the G = 0 locus may be determined as follows
[Mathematical Expression Omitted] (6)
where
[Mathematical Expression Omitted] (7)
It is shown in section a of the Appendix that the first term on the
right-hand side will be positive (non-negative) when firms'
preferences exhibit decreasing (or non-increasing) absolute risk
aversion, provided [f.sub.12][greater than or equal to][0.sup.6] It will
be indeterminate in sign otherwise. It is also shown that the second
term on the right-hand side will be strictly positive when fixed and
variable inputs are not technically competitive. If these inputs are
technically competitive ([f.sub.12] < 0), the sign of this second
term is shown to be indeterminate. Thus, sufficient conditions for
G[micro] > 0 will be non-increasing absolute risk aversion and
technical complementarity (or independence) of fixed and variable inputs
([f.sub.12][greater than or equal to]0).
Consider next
[Mathematical Expression Omitted] (8)
The first term on the right-hand side of (8) is always negative (by
U"(lt) < 0) whereas the second term is always negative by
concavity of the restricted profit function in K.
Therefore, from (6), the slope of the G = 0 locus in (K,[micro])
space will take the sign of [G.sub.micro], which implies that it is not
determinate but will be positive, for example, when firms exhibit
decreasing absolute risk aversion and employ a technology with
[f.sub.12] [greater than or equal to]0.
The slope of the H = b locus may be determined similarly, using
[Mathematical Expression Omitted] (9)
with
[Mathematical Expression Omitted] (10)
and
[Mathematical Expression Omitted] (11)
Since H[micro] is strictly positive, the slope of the H = b locus
will be opposite in sign to [H.sub.K]. By (4) and (11), [H.sub.K] is
equivalent to G (K,[micro]), so [H.sub.K] Will equal zero when G
(K,[micro]) = 0, which
will occur when the H = b and G = O loci coincide. From the
definition of G(K,[mu]) = O given in (4), it is evident that if K <
[K.sup.*], then G(K, [mu]) = [H.sub.K] > O. Conversely, for any K
> [K.sup.*], [H.sub.K], < O. Thus, from (9) the H = b locus will
be U-shaped: negatively sloped if K < [K.sup.*] (that is, to the left
of the G = O locus in (K, [mu]) space) and positively sloped if K >
[K.sup.*].
The two possible configurations of the G = O and H = b loci in
Figure 1 are the same as given in Appelbaum and Katz [1, 525], except
that here the horizontal axis measures K instead of q. The intersection of the two loci indicates the level of capital use per firm and the
expected price level which define industry equilibrium. Given any price
expectation, [[mu].sub.o], firms will decide whether to produce, and
will choose their optimal level of K, which is indicated by the
coordinates on the G = O locus. If these coordinates are below (above)
the H = b locus, this will induce some firms to exit (enter) which will
have the effect of raising (lowering) the expected industry price in the
long run.
Following Appelbaum and Katz [1, 526], it can readily be shown that
the system of industry equilibrium conditions will be stable under these
assumptions about the dynamic response by firms when the industry is not
in equilibrium.
Having described the industry equilibrium, one may examine the
comparative-static effect of changes in various system parameters. Of
particular interest here is the effect on the expected price and on the
level of use of the quasi-fixed input of a mean-preserving change in
price variability. For any equilibrium value of [K.sup.*], one can
compare the relative magnitudes of the vertical shifts in the two loci
to assess how equilibrium values of [mu], q, and n will change.
Proposition 1. (a) In an industry composed of identical risk-averse
firms which employ a two-stage production technology, a mean-preserving
increase in price uncertainty may increase or reduce industry output.
(b) The effect of a mean-preserving increase in price uncertainiy on the
output of individual firms and on the number of firms appears to be
ambiguous.
Proof. Considering first the effect of the increase in the shift
parameter, [gamma], on the position of the H = b locus gives
[Mathematical Expression Omitted]
The sign of [H.sub.[mu] is strictly positive so the sign of
d[mu]/d[gamma] will be opposite to the sign of [H.sub.[gamma]. Section b
of the Appendix shows that positive or negative values are possible for
the sign of [H [gamma]] depending in a specific way on properties of the
production function and of the firm's preferences. A positive value
of [H [gamma]] would correspond to a price decrease in long-run
equilibrium with increased industry output.
This result may be contrasted with the Appelbaum and Katz result
for a single-stage production technology. In that case, [H [gamma] was
unambiguously negative and d [mu]/d [[gamma]\.sub.H=b] was was
unambiguously positive. This allowed those authors to conclude that a
mean-preserving increase (decrease) in price dispersion reduces
(increases) industry output. With a two-stage technology, however,
industry output changes are not governed by risk aversion alone. Here
one finds an offsetting tendency to increase output which is due to the
increased short-run elasticity of supply in which the firm has invested.
Either effect may dominate.
To sign the direction of change in q and n, one could examine the
shift of the G = 0 locus. Appelbaum and Katz were able to sign the shift
in their G = 0 locus as positive in the case of decreasing absolute risk
aversion but the change in output per firm and in the number of firms
was ambiguous.
For the two-stage production process employed above, the shift of
the G = O curve will be given by
[Mathematical Expression Omitted]
From (7), G [mu] is indeterminate in sign but will be positive, for
example, in the case of decreasing absolute risk aversion with
[f.sub.12] [is greater than or equal to] O. The numerator, G [gamma]
will be given by
[Mathematical Expression Omitted]
As shown in section of the Appendix, it is not apparent that this
can be signed unambiguously even with the adoption of specific
assumptions about the degree of absolute or relative risk aversion. In
particular, the value of G [gamma] is shown to depend on properties of
the firm's preferences [Mathematical Expression Omitted] about
which no general assumptions have been made.(7)
To compare the relative magnitudes of the vertical shifts in the
two loci, one would need to evaluate the sign of the difference d [mu]/
d [[gamma]\.sub.H=b] - d [mu]/d [[gamma]\.sub.G=O]. Given the
indeterminacy of the sign of G [gamma], it does not appear that this
difference can be signed unambiguously.
To recap, expected industry price, [gamma], and output, Q, can
increase or decrease under risk aversion. The direction of response in
output per firm, q, and number of firms, n, apparently cannot be signed.
Depending on the technology, there may be opposing influences on the
optimal output level: increases due to a firm's affinity for price
variability which is independent of its preferences, and decreases due
to the firm's aversion to variability of profits. The optimal
output levels will also depend on the equilibrium adjustments in
expected output price.
IV. Industry Equilibrium with Risk-Neutral Firms
When the industry is composed of representative firms which are
risk-neutral, the foregoing analysis is altered by setting U"([pi])
= O and re-interpreting the above expressions.
The G = O locus will again be of indeterminate slope under risk
neutrality, but will be positively sloped when [f.sub.l2] [is greater
than or equal to] O (as a sufficient but not necessary condition). This
is seen by evaluating G [mu] in (7) with U ([mu]) = O. In a similar
manner, re-evaluation of (8) under risk neutrality shows [G.sub.K] is
again unambiguously negative. These results imply the slope of the G = O
locus will take either sign (according to the sign of G [mu]) but that
under risk neutrality, knowledge that [f.sub.12] [is greater than or
equal] O is sufficient to ensure G = O is upward sloping.
Assuming U(O) = O, the equilibrium condition H = b becomes H = b =
O. The slope of the H = b locus is not influenced by the firm's
preferences towards risk, and will be U-shaped, as given by (9) through
(11).
Thus, the possible configurations of the two loci and of industry
equilibrium under risk neutrality are generally the same as for risk
aversion.
Proposition 2. (a) In an industry composed of identical
risk-neutral firms which employ a two-stage production technology, a
mean-preserving increase in price uncertainty may increase or reduce
industry output. (b) The effect of a mean-preserving increase in price
uncertainty on the output of individual firms and on the number of firms
will depend on specific characteristics of the production technology.
In examining a change in the value of the shift parameter, [gamma],
when U"([mu]) = O, it is evident from the earlier analysis (see
section b of the Appendix) that [H [mu] will now take the opposite sign
of [f1/fl2 + f2/f22]. Thus, for example, if (f1/f12]) < \f2/f22/ and
if f12 > O, then H [gamma] will be strictly positive, and by (12) the
H = b locus will shift down in response to an increase in the parameter
[gamma]. If f12 < O, these results also hold regardless of the
specific magnitudes of the other terms shown. In such a case, a
mean-preserving increase in the variability of output prices would cause
industry output to-rise and expected industry price to fall in
equilibrium. Conversely, if (f1/fl2) > \f2/f22\ and fl2 > O, then
H [gamma] will be strictly negative, the H = b locus will shift up, and
equilibrium output would fall.
Again, the response of q and n is less clear. In the case of the G
= O locus under risk neutrality, the sign of G [gamma] is no longer
indeterminate, but can be evaluated using the second term of (14). It is
shown in the Appendix ((A. 15), section c) that G [gamma] will take the
sign of [Mathematical Omitted] which, from Hartman [3, 677] is a
critical value in determining optimal firm response in the absence of
adjustment to industry equilibrium. Evaluating (7) with U"([pi]) =
O shows that G [mu] can take either sign, where f12 [is greater than or
equal to] O is a sufficient condition for G [mu] > O.
Using (13), the shift in the G = O locus can be up or down
depending on the signs of G [gamma] and G [mu]. To construct an example,
when [Mathematical Expression Omitted] > O and fl2 [is greater than
or equal to] O one obtains
[Mathematical Expression Omitted]
which signifies the (upward sloping) G = O locus will shift down when
price uncertainty is increased. These are conditions (from Hartman [3,
677]) under which, in the short run, a risk-neutral firm would have an
affinity for price variability and would increase output as variability
increased. This example is consistent with conditions under which H = b
could also be shifting down.
To summarize the movements in each locus associated with an
increase in price variability under risk neutrality, the direction of
change will be ambiguous for each, but will depend on various parameters
of the firm's production technology. Specific parameters such as
f12 and [Mathematical Expression Omitted] describe the technology per
se, and can, in principle, be estimated empirically [4]. The shifts in
the two curves are given by
[Mathematical Expression Omitted]
For the specific example above, where [Mathematical Expression
Omitted], the (determinate) signs are given by (13') and
[Mathematical Expression Omitted]
Although industry output will rise unambiguously, the effect on
output per firm in the above example will depend on the choice of K per
firm (given fl2 [is greater than or equal to] O) which will in turn
depend on the relative magnitudes of the shifts in the two loci. This
difference can be represented by d [mu]/d [[gamma]\.sub.H=b] - d [mu]/d
[[gamma]\.sub.G=O] which can be evaluated, in general, using the above
expressions.
If the two loci move the same distance vertically, the difference
will be zero and [K.sup.*] will be constant. If the H = b locus shifts
down further, the difference will be negative, and [K.sup.*] will fall.
If the G = O locus shifts further, the difference will be positive, and
[K.sup.*] will increase.
It does not appear to be possible to sign the difference between
the vertical shifts in the two loci under risk neutrality. This is so
even with the specific set of conditions which make up the above
example. This contrasts with the result of Appelbaum and Katz which
suggests there would be no firm or industry response to a
mean-preserving change in the price spread if firms employ a
single-stage productiqn technology and are risk neutral.
V. Concluding Comments
These results demonstrate that, in equilibrium, a competitive
industry's investment and output response to increased output price
uncertainty (i.e., a mean-preserving spread in output price) may be
positive or negative. In general, the direction of response depends on
specific attributes of the firms' preferences and technology. In
the special case of firms which are risk-neutral with respect to
profits, the direction of response is shown to depend on properties of
the production technology which, in principle, can be estimated
empirically.
An implication of these results is that industry response to
uncertainty may not be predictable in the manner suggested previously in
the literature. Tastes, technology and the timing of information used by
firms will each condition industry response. Capital investment, and the
relation between current aggregate output and previous factor choices,
may respond to output price volatility in ways beyond those suggested by
the risk preferences of firms.
Appendix
a. Evaluation of the Sign of (7)
[Mathematical Expression Omitted]
It is possible to evaluate the sign of the two right-hand-side
terms separately, and by providing conditions under which they will each
be positive, to show when their sum, [G.sub.[mu]] will be positive.
[Mathematical Expression Omitted] To sign the first term in (7) it is
noted that output, f(K,[L.sup.*]) will be strictly positive, so that it
remains to sign the expected product of the remaining two parts of the
first term. This is done following the approach of Sandmo [7, 68] who
analyzes firms with a single-stage production process.
This expression is shown to be positive under certain conditions
which employ the Arrow-Pratt measure of absolute risk aversion
[Mathematical Expression Omitted]
It will be shown for firms with decreasing (non-increasing)
absolute risk aversion that the sign of [Mathematical Expression
Omitted] will be non-negative provided f12 [is greater than or equal to]
O (as a sufficient but not necessary condition). Recalling that f
(K,[L.sup.*]) will be positive, the entire first expression of (7) will
then be non-negative.
Whereas Sandmo [7] shows similar results for a single-stage
production process, it is now evident that they are also valid for a
two-stage production process, where f12 [is greater than or equal to] O
is a sufficient condition for this to be so. Start with the first-order condition
[Mathematical Expression Omitted]
Hartmant [3,677] shows that
[Mathematical Expression Omitted]
Noting f1 > O, f2 > O, and f22 < O, the sign of the entire
expression will be strictly positive when fl2 [is greater than or equal
to] O and indeterminate when f12 < O. None of the assumptions made
about f puts a general restriction on fl2. Technical complementarity or
independence of fixed and variable factors ([f12 [is greater than or
equal to] O) represents a sufficient condition for [Mathematical
Expression Omitted] to be strictly positive. This says that in periods
when the realization of output price is higher than its expected value,
the marginal value product of capital will also be higher. This will be
true, even though the level of K is fixed in the short run, because the
level of use of variable inputs is increased to accommodate the higher
output price. It may not be true if K and L are technically competitive.
As a result, under the sufficient condition f12 [is greater than or
equal to] O, realizations of output price above (below) the expected
value will result in higher (lower) values of profit and higher (lower)
values of [Mathematical Expression Omitted] (marginal value product of
capital) than if the expected price were realized.
Denote the value of [pi] when [Mathematical Expression Omitted] as
[Mathematical Expression Omitted] and denote the value of [pi] when the
realized output price is higher than expected as [Mathematical
Expression Omitted] which from the foregoing would have
[Mathematical Expression Omitted]
at [Mathematical Expression Omitted] provided f2 [is greater than or
equal to] O. For a firm with decreasing (non-increasing) absolute risk
aversion
[Mathematical Expression Omitted]
or equivalently
[Mathematical Expression Omitted]
or
[Mathematical Expression Omitted]
Using (A.3) gives
[Mathematical Expression Omitted]
This will hold for [Mathematical Expression Omitted] for in the
latter case, (A.4) becomes
[Mathematical Expression Omitted]
which would leave (A.5) unchanged.
Taking expected values of (A.5) gives
[Mathematical Expression Omitted]
where [Mathematical Expression Omitted] is a certain value and where
the last term on the right-hand side equals zero by (3'), the
firm's first-order condition.
Thus the left-hand-side expression will be zero or positive, with
non-increasing or decreasing absolute risk aversion and fl2 [is greater
than or equal] O as a sufficient condition.
Recalling that f (K, [L.sup.*]) will be positive, the entire first
expression of (7) will be non-negative under the conditions given.
Whereas Sandmo [7] shows similar results for a single-stage production
process, it is now evident that they are also valid for a two-stage
production process, where f12 [is greater than or equal to] O is a
sufficient condition for this to be so.
ii. Evaluation of [Mathematical Expression Omitted]. The sign of the
second right-hand-side term of (7) depends on U'([pi]) which is
strictly positive, and on [Mathematical Expression Omitted] which can be
shown to be positive under prescribed conditions. Recall from (A.2) that
[Mathematical Expression Omitted] will be strictly positive when fl2 [is
greater than or equal to] O and indeterminate when fl2 < O. Technical
complementarity or independence of fixed and variable factors (f12 [is
greater than or equal to] O represents a sufficient condition for the
entire second term of (7) to be strictly positive.
b. Evaluation ofthe Sign of [H.sub.[gamma]]
From (12), [Mathematical Expression Omitted]. Proceed by re-defining
[H.sub.[gamma]] where
[Mathematical Expression Omitted]
It follows that [H.sub.[gamma]] = cov(A,[epsilon]), given E[epsilon]
= O, using
[Mathematical Expression Omitted]
The sign of cov(A, [epsilon]) will be the sign of [Mathematical
Expression Omitted], which measures whether larger [epsilon] values are
associated with larger values of A. Using (A. 7)
[Mathematical Expression Omitted]
where
[Mathematical Expression Omitted]
Implicit differentiation of the first-order condition
[pf2(K,[L.sup.*] - w = O] gives: [Mathematical Expression Omitted], and
[Mathematical Expression Omitted], so that
[Mathematical Expression Omitted]
Therefore, to evaluate the sign of [H.sub.[gamma]] which is the
sign of [Mathematical Expression Omitted], use
[Mathematical Expression Omitted]
The first term will be negative unambiguously by risk aversion. The
first part of the second term will be strictly positive, i.e.,
[U'([pi])(f2 [gamma]/mu]. > Therefore, when the second part of
the second term in (A.9') (in square brackets) is positive or zero,
[Mathematical Expression Omitted]. When the tertn in square brackets is
negative, the sign of [Mathematical Expression Omitted] may be positive
or negative.
By inspection, [Mathematical Expression Omitted] and f2/f22 < O,
so [f1/f12 + f2/f22] [is greater than or equal to] O and (f1/f12) [is
greater than or equal to] /f2/f22/. If fl2 < O, then the term in
square brackets is negative, and if f12 = O, then it is undefined. Thus,
under risk aversion, [H.sub.[gamma]], will be negative when fl2 > O
and (f1/fl2) [is greater than or equal to] /f2/f22/, where these
conditions are sufficient but not necessary. When the entire second term
of (A.9') is negative and exceeds the first term in absolute value,
[H.sub.[gamma]] will be positive. Using (A.9'), a set of sufficient
conditions for [Mathematical Expression Omitted] would be f12 > O,
[Mathematical Expression Omitted].
c. Evaluation of the Sign of G [gamma]
[Mathematical Expression Omitted]
Given the complexity of the two terms in (14), it is necessary to
evaluate each separately, using the methods employed to sign H [gamma]
[Mathematical Expression Omitted
It is clear from the first of these terms that the sign of
[Mathematical Expression Omitted] will depend, in part, on the sign of
U'''([pi]) None of the assumptions about U([pi]) places
any restriction on the sign of this term, so the value of [Mathematical
Expression Omitted] and thus of G [gamma] will be indeterminate. When
one assumes that the Arrow-Pratt measure of absolute risk aversion (A.1)
is decreasing, one is implicitly assuming that U'''([pi])
> O. However, even under such an assumption it does not appear
possible to sign the remaining terms in (A. 13) which make up
[Mathematical Expression Omitted]
It will prove useful in the analysis of industry equilibrium with
risk neutrality (which follows) to know the sign of the second term in
(14), which is evaluated next.
ii. Sign of [Mathematical Expression Omitted] Define C [Mathematical
Expression Omitted] so the term of interest becomes E[C [epsilon]],
which will equal cov(C, [epsilon]) because E [epsilon] = O. As before,
the sign of cov(C, [epsilon] ) will be the sign of [Mathematical
Expression Omitted] where
[Mathematical Expression Omitted]
Recall
[Mathematical Expression Omitted]
so the first term on the right-hand side of (A.14) becomes
[Mathematical Expression Omitted] From section a.i. of this Appendix,
the sign of a [Mathematical Expression Omitted] will be positive when
fl2 [is greater than or equal to] and indeterminate otherwise. The
product of the other parts of this expression is unambiguously negative.
The second term on the right-hand side of (A. 14) gives
[Mathematical Expression Omitted]
where U'([pi]) > O, [gamma] > O and [Mathematical
Expression Omitted] The final right-hand-side term here is the same
critical value employed by Hartman [3, 677] to determine optimal firm
response under risk neutrality in the absence of adjustment to industry
equilibrium. A positive value corresponds to the case where ex ante
investments in K increase the short-run supply elasticity ex post.
As G [gamma] is expressed as the sum of two expressions in (14),
the sign of the first of these will depend on the sign of
U''' ([pi]) and is thus indeterminate. The sign of the
second of these can be shown to be positive or negative, depending on
the sign of fl2 and [Mathematical Expression Omitted] (among others).
References
[1.] Appelbaum, Elie and Eliakim Katz, "Measures of Risk
Aversion and Comparative Statics of Industry Equilibrium." American Economic Review, June 1986, 524-29. [2.] Epstein, Larry G.,
"Production Flexibility and the Behaviour of the Competitive Firm
under Price Uncertainty." Review of Economic Studies, June 1978,
251-61. [3.] Hartman, Richard, "Factor Demand with Output Price
Uncertainty." American Economic Review, September 1976, 675-81.
[4.] Horbulyk, Theodore M., "Functional Forms and Multivariate Risk
Independence." Economics Letters, September 1992, 67-70. [5.]
Ishii, Yasunori, "Measures of Risk Aversion and Comparative Statics
of Industry Equilibrium: Correction." American Economic Review,
March 1989, 285-86. [6.] Pazner, Elisha A. and Assaf Razin,
"Industry Equilibrium under Random Demand." European Economic
Review, October 1975, 387-95. [7.] Sandmo, Agnar, "On the Theory of
the Competitive Firm under Price Uncertainty." American Economic
Review, March 1971, 65-73. [8.] Turnovsky, Stephen J., "Production
Flexibility, Price Uncertainty, and the Behavior of the Competitive
Firm." International Economic Review, June 1973, 395-413. [9.]
Wright, Brian D., "The Effects of Price Uncertainty on the Factor
Choices of the Competitive Firm." Southern Economic Journal,
October 1984, 443-55.
(1.) Pazner and Razin [6] analyze the structure of an industry in
long-run equilibrium when the industry is composed of otherwise
identical firms which are not identical in risk preferences. With free
entry and an unlimited number of potential entrants, risk-averse firms
will be driven out by those which are risk neutral. The latter will in
turn be driven out by those which are risk-loving. In this paper, it is
assumed initially that all potential entrants are risk averse.
Subsequently, an unlimited number of risk-neutral potential entrants are
admitted to the analysis. (2.) Firms may invest in automation for a
production process ex ante in order to gain more latitude in the level
of variable inputs employed later. Cattle ranchers may initially invest
in a larger breeding herd in order to be more supply-responsive later as
(future) market prices become known with certainty. (4.) Appelbaum and
Katz label a curve such as the H (K,[micro]) locus an iso-expected
utility curve. (5.) These loci are analogous to those portrayed by
Appelbaum and Katz for a single-stage production process. Those authors
show the possible configurations of corresponding G - 0 and H = b loci
in (q,[micro]) space. (6.) Quasi-fixed and variable inputs are
hereinafter described as technically complementary, technically
independent, and technically competitive as [f.sub.12] > 0,
[f.sub.12] = 0, and [f.sub.12] < 0, respectively. There is no general
a prior assumption about the production technology which would restrict
the sign of [f.sub.12] (7.) Although Ishii [5] has provided an alternate
analysis of the comparative statics of the Appelbaum and Katz model,
that approach and those results apparently do not extend to the
two-stage model. Ishii concludes that output per firm will fall with the
single-stage technology, and he does not sign the direction of change in
the number of firms.
