Optimal Output for the Regret-Averse
Competitive Firm Under Price Uncertainty
∗
Martı́n Egozcue
Department of Economics, Facultad de Ciencias Sociales
Universidad de la República
Department of Economics, University of Montevideo
Montevideo 11600, Uruguay
Wing-Keung Wong
Department of Economics
Hong Kong Baptist University
WLB, Shaw Campus, Kowloon Tong, Hong Kong
May 2, 2012
∗
The second author would like to thank Professors Robert B. Miller and Howard E. Thompson for their
continuous guidance and encouragement. This research is partially supported by grants from Universidad
de la República, University of Montevideo, Hong Kong Baptist University, and Research Grants Council
of Hong Kong.
Optimal Output for the Regret-Averse Competitive
Firm Under Price Uncertainty
Abstract. We study the optimal output of a competitive firm under price uncertainty.
Instead of assuming a risk-averse firm, we assume that the firm is regret-averse. We find
that optimal output under uncertainty would be lower than under certainty. We also
prove that optimal output could increase or decrease when the regret factor varies.
Keywords: competitive firm; risk aversion; regret aversion; decision making
JEL Classification: D00, D03, D21
1
Introduction
In a seminal paper, Sandmo (1971) studied the optimal output of a risk-averse firm that
faces uncertain prices. He showed that optimal output under uncertainty would be lower
than under certainty in an equivalent case. Sandmo also showed that optimal output under uncertainty satisfied the expectation of the von Neumann-Morgenstern (1944) utility
function maximization for risk averters.
Nonetheless, many studies have offered evidence of a violation of conventional expected
utility theory. For example, Kahneman and Tversky (1979) found from their experiments
that the theory of expectation utility maximization for risk averters is violated by the
following effects/paradox: the certainty effect or common ratio effect, the ‘Allais Paradox’
1
or ‘common consequences effect,’ and the ‘isolation effect.’ Sometimes, risk aversion and
risk loving are associated with problems involving increments/decrements of wealth. On
the other hand, Loomes and Sugden (1982) and others showed that regret aversion is an
alternative theory that not only explains the reflection effect and simultaneous gambling
and insurance effects, but also predicts the behaviors of the above-mentioned effects. They
also argued that such behavior can be defended as rational, and thus, regret theory is a
theory of rational choice under uncertainty.
There are alternative proposals for utility functions other than risk aversion. In particular, in an influential work, Savage (1951) suggested that decision makers may consider
the feeling of regret in their decision process, that is, the feeling of having made the wrong
decision. The seminal papers by Bell (1982) and Loomes and Sugden (1982) presented
a formal analysis of regret theory. Sugden (1993) took an axiomatic approach whereas
Quiggin (1994) extended the analysis to multiple choices. In their work, regret is defined
as the disutility of not having chosen the ex-post optimal alternative.
In recent years, there has been a growing literature on firm behavior that, instead of
assuming risk aversion, assumes that firms are regret-averse. For example, if the firm’s
prices turn out to be very high and sales turn out to be very good, firms might regret
not producing more. Conversely, if prices turn out to be low and sales are poor, firms
might regret over-producting. Paroush and Venezia’s paper (1979) is the first that applied a regret-averse model to the competitive firm. By modifying Sandmo’s model but
considering a firm with a regret utility function, they derived the conditions under which
the optimal output in this uncertain framework is lower than that under certainty in an
equivalent case. However, their result depends on the relative importance of the regret
2
term and the firm’s profits.
On the other hand, many studies have obtained a more specific and tractable regretaverse function than the one used by Paroush and Venezia (1979). One can find such a
function in Braun and Muermann (2004), Muermann et al. (2006), Mulandzi et al. (2008),
Michenaud and Solnik (2008), and others. For example, Muermann et al. (2006) used such
a function to study the portfolio optimal allocation problem for a regret-averse investor
that is confronted with a risky and a risk-free asset. Also, Mulandzi et al. (2008) studied
the optimal allocation between loans and Treasuries for a regret-averse bank. Laciana
and Weber (2008) estimated the set of parameter values that allow the regret function to
be consistent with the common consequence effect.
By adopting the more specific and tractable regret-averse function, we extend the
model developed by Paroush and Venezia (1979) to study the properties of firms’ behavior when optimal output is obtained for the regret-averse competitive firm under price
uncertainty. Our paper has contributed the following to the literature: First, our paper
is the first to introduce the two-attribute regret-averse function to study the properties of
cost and price when optimal output is reached for a regret-averse firm that faces uncertain
prices. Second, using a two-attribute regret-averse function, we show that a firm would
choose a smaller optimal output than under certainty. Third, we circumvent the limitation
of using the model developed by Paroush and Venezia (1979) that the regret term in their
model possesses the same regret even when the utility functions of the regret-averse firms
on regret are different while our model enables different firms to possess different regrets
if their utilities on the regret term are different. Four, our model not only represents
the production theory for risk-averse competitive firms, it could also represent the theory
3
for risk-averse competitive firms. Five, our model setting allows us to make comparative
statics of the optimal output by varying the parameters of the regret terms. Six, readers
may refer to our discussion in Section 2.4 for other advantages of using our model. The
rest of the paper is structured as follows. In the next section we present our main results,
then finish in Section 3 with concluding remarks.
2
The Theory
In this section we present the general framework of different models to obtain the properties of producing optimal output for competitive firms. To distinguish the well-known
results in the literature from the ones derived in this paper, all cited results will be called
Propositions and our derived results will be called Theorems.
The firm sells its output x at an uncertain price represented by a random variable
P . We first assume that the objective of the firm is to maximize the expected utility of
profits in which the firm’s utility function, u, is assumed to be a concave, continuous, and
twice differentiable function of profits such that
u0 (Π) > 0 and u00 (Π) < 0
(1)
to reflect risk aversion.
The total cost function of the firm is
T C(x) = C(x) + F ,
(2)
in which F is a fixed cost and C(x) is the variable cost function with C(0) = 0 and, for
4
any x > 0, C 0 (x) > 0 and C 00 (x) > 0. The firm’s profit function then becomes
Π(x) = P x − C(x) − F ,
(3)
where P is the price of output, assumed to be a random variable with density function
f (p) and expected value E(P ) = µ. Thus, the firm’s expected utility of profits is
£ ¡
¢¤
£ ¡
¢¤
E u Π(x) = E u P x − C(x) − F ,
(4)
where E is the expectation operator. The above notations and definitions will be used
throughout this paper. Based on this model setting, we first discuss the theory under
certainty in the next subsection.
2.1
The Theory Under Certainty
If there is no uncertainty, price is equal to µ; that is, P = µ, and, in this situation, we can
view the price P as the random variable with degenerate distribution at µ. Since there is
no uncertainty, the firm maximizes the following utility function of profits:
£
¤
£
¤
max u Π(x) = max u µx − C(x) − F .
x
x
(5)
The first-order condition of (5) is:
[µ − C 0 (x)] u0 (Π) = 0 .
5
(6)
£
¤
Let x∗ = arg maxx u Π(x) be the optimal output level that is the solution to the firstorder condition in (6). Then, we have
µ = C 0 (x∗ ) .
(7)
Since C 00 (x) > 0 and u00 (x) < 0 under the assumption, the second-order condition of (5)
satisfies:
£
¤2
u00 (Π) µ − C 0 (x) − u0 (Π)C 00 (x) < 0 .
(8)
From the first- and second-order conditions in (6) and (8), one could easily obtain the
solution of optimal output x∗ . We summarize the result in the following proposition:
Proposition 1 For any risk-averse firm with utility function u satisfying (1) that will
maximize the utility function u as shown in (5) and face a certain price P = µ, it will
choose an optimal output x∗ by setting x such that its marginal revenue (= P ) equals its
marginal cost, C 0 , such that
C 0 (x∗ ) = µ .
(9)
We will construct an example to illustrate this finding in Section 3.
2.2
The Theory Under Uncertainty for Risk-Averse Competitive Firms
We now discuss the theory of optimal output by adopting the theory under uncertainty
for risk-averse competitive firms. This theory was developed by Baron (1970), Sandmo
(1971) and others. In order to make the notations consistent with other sections in this
paper, our notations may differ from those in Sandmo (1971) and others.
6
The utility function, u, of the firm is defined in (1) to reflect risk aversion. The total
cost function of the firm is defined in (2), the firm’s profit function is defined in (3), and
the firm’s expected utility of profits is defined in (4). In contract to the objective of the
firm in the theory under certainty, the objective of the risk-averse competitive firm using
the theory under uncertainty is to maximize the expected utility of profits such that
£ ¡
¢¤
£ ¡
¢¤
max E u Π(x) = E u P x − C(x) − F ,
x
(10)
where Π is the firm’s profit function defined in (3). The first- and second-order conditions
of (10) for the risk-averse competitive firm using the theory under uncertainty are
£
¤
E u0 (Π)(P − C 0 (x)) = 0 ,
(11)
£
¤
E u00 (Π)(P − C 0 (x))2 − u0 (Π)C 00 (x) < 0 .
(12)
These are the necessary and sufficient conditions for a maximum expected utility of equation (10) to hold. Let x∗ be the optimal output level that is the solution to the first-order
condition in (11). We re-write the result obtained by Sandmo (1971) in the following
proposition:
Proposition 2 For any risk-averse firm with utility function u satisfying (1) that will
maximize the expectation of u as shown in (10) and face an uncertain price P , it will
choose an optimal output x∗ characterized by marginal cost C 0 that is less than the expected
price E(P ) such that
C 0 (x∗ ) ≤ E(P ) .
(13)
We note that the optimal output x∗ obtained by Sandmo (1971) for the theory un7
der uncertainty for risk-averse competitive firms as shown in (13) is different from that
obtained by adopting the theory under certainty as shown in (7). In addition, Sandmo
(1971) showed that at the optimum, the expected price must be larger than the average
variable cost, so that the firm requires a positive expected profit in order to choose a
positive output level; that is
E(P ) ≥
2.3
C(x∗ )
.
x∗
(14)
The Theory Under Uncertainty for Regret-Averse Competitive Firms
We next discuss the model developed by Paroush and Venezia (1979). In order to make
the notations consistent with other sections in this paper, our notations may differ from
those used by Paroush and Venezia (1979). Define the regret function
R(P, x) = Πmax − Π(P, x) ,
(15)
where Π is the firm’s profit function defined in (3) and the ex-post optimal profit, denoted
by Πmax , is the optimal profit if there is no price uncertainty.
Let u be the utility function of the firm in the theory under uncertainty for regretaverse competitive firms and a function of both profits Π(P, x) and regret R(P, x) such
that
£
¤
u Π(P, x), R(P, x) ,
(16)
with uΠ = ∂u/∂Π > 0 and uR = ∂u/∂R < 0 where Π(P, x) is defined in (3) and R(P, x)
is defined in (15), respectively. The firm is assumed to maximize the expectation of u in
8
(16) such that
© £
¤ª
max E u Π(P, x), R(P, x) .
(17)
x
We note that the assumptions in the theory under uncertainty developed by Paroush
and Venezia (1979) for regret-averse competitive firms are different from those in the
theory under uncertainty for risk-averse competitive firms. The former assumes that
firms maximize the expectation of u for both profits Π(P, x) and regret R(P, x) as shown
in (17), whereas the latter assumes that firms maximize the expectation of u, which is a
function only of profits Π as shown in (4).
Assuming the price follows
P = µ + γε ,
(18)
³
where E(ε) = 0, V (ε) = 1, γ is the standard deviation of P , P ε >
− µγ
´
= 1, and
E(P ) = µ, one could easily show that the covariance between P and (uΠ − uR ) satisfies
£
¤
Cov(P, uΠ − uR ) = γE ε (uΠ − uR ) ,
(19)
where uΠ and uR are defined in (16) and other terms are defined in (18). We modify the
results from Paroush and Venezia (1979) to obtain the following proposition to state the
relationships of price and cost when obtaining optimal output:
Proposition 3 For any regret-averse firm with utility function u defined in (16) that will
maximize the expectation of u as shown in (17) and face an uncertain price P , it will
choose an optimal output x∗ that satisfies the following condition for the expected price
9
E(P ) and the marginal cost C 0 :
if Cov(P, uΠ − uR ) > (=, <)0, then E(P ) > (=, <)C 0 (x∗ ) ,
(20)
where uΠ and uR are defined as in (16).
Proposition 3 tells us that Cov(P, uΠ − uR ) could be zero and even negative. We also
note that the optimal output x∗ obtained by adopting the theory under uncertainty for
regret-averse competitive firms developed by Paroush and Venezia (1979) as shown in
(20) is different from that obtained by using the theory under uncertainty for risk-averse
competitive firms developed by Sandmo (1971) as shown in (13) and is different from that
obtained by adopting the theory under certainty as shown in (7). We obtain the inference
made from the theory under uncertainty for regret-averse competitive firms developed by
Paroush and Venezia (1979) as stated in the following corollary:
Corollary 2.1 Under the conditions and assumptions stated in Proposition 3,
1. if uΠ − uR is increasing with respect to P , then E(P ) > C 0 (x∗ );
2. if uΠ − uR is uncorrelated to P , then E(P ) = C 0 (x∗ ); and
3. if uΠ − uR is decreasing with respect to P , then E(P ) < C 0 (x∗ ).
We note that the inference from Part 3 of Corollary 2.1 with E(P ) < C 0 (x∗ ) usually
does not happen in practice. Is this result realistic? Some economists may believe that
this is a limitation of using the framework developed by Paroush and Venezia (1979).
To circumvent this limitation, we introduce a more specific and tractable regret-averse
function for regret-averse competitive firms as discussed in the next subsection.
10
2.4
The Theory Under Uncertainty for Regret-Averse Competitive Firms for a Tractable Regret-Averse Function
In this paper we extend Paroush and Venezia’s work by assuming the following twoattribute regret-averse utility function u on the profits Π for the regret-averse firm:
u (Π) = v(Π) − kg [v(Πmax ) − v(Π)] ,
(21)
in which the first attribute accounts for risk aversion and is characterized by the firm’s
risk-averse utility function v to reflect risk aversion with v 0 > 0 and v 00 < 0. The second
attribute relates to the fact that the firm is concerned about the prospect of regret. The
function g indicates the regret-averse attribute in which g(0) = 0 and, for any x > 0,
g 0 (x) > 0 and g 00 (x) > 0, the parameter k ≥ 0 measures the weight of the regret
attribute relative to the first risk-aversion attribute, Π is the firm’s profit function as
defined in (3), and the ex-post optimal profit, denoted by Πmax , is the optimal profit if
there were no price uncertainty. We note that although our paper is the first to introduce
the two-attribute regret-averse utility function u as defined in (21) to the theory of optimal
output for the regret-averse competitive firm under price uncertainty, this function has
been used in the literature; see, for example, Braun and Muermann (2004), Muermann et
al. (2006) and Mulaudzi et al. (2008) for more information.
We note that the function v is essentially what Bernoulli and Marshall describe as
the psychological experience of pleasure associated with the satisfaction of desire. We
note that the utility function v satisfying v 0 > 0 and v 00 < 0 is also called the Bernoulli
utility function. However, the two-attribute regret-averse utility function u defined in (21)
11
suggests that the pleasurable psychological experience of having Π will depend not only on
v(Π) but also on the nature of v(Πmax )−v(Π). Possessing the maximum profit Πmax is the
most desirable rather than having Π and the individual may experience regret. One may
reflect on how much better one’s position would have been had one chosen differently,
and this reflection may reduce the pleasure that one derives from Π. One could also
view possessing the maximum profit Πmax as rejoicing, the extra pleasure associated with
knowing that, as matters have turned out, one has taken the best decision. Thus, the
two-attribute regret-averse utility function u defined in (21) incorporates the concepts of
both regret and rejoicing. To formulate the sensation of regret and rejoicing in this way is
to assume that the degree to which a person experiences these sensations depends only on
the utility associated with the two consequences in question: ‘what is’ and ‘what might
have been.’ The regret-averse attribute g with g(0) = 0 and, for any x > 0, g 0 (x) > 0 and
g 00 (x) > 0 indicates that the more pleasurable the consequence that might have been, the
more regret and the less rejoicing will be experienced.
We further assume that the regret-averse firm obtains the optimal output x∗ by maximizing the following expected utility of profits and regret:
© £
¤ª
max E u Π(x) = max E {v(Π) − kg [v(Πmax ) − v(Π)]} .
x
x
(22)
There are many advantages to using this modeling setting. For example, (1) this
modeling setting covers both the theory for risk-averse competitive firms developed by
Sandmo (1971) and others when k = 0 and the theory for regret-averse competitive firms
when k > 0. (2) Paroush and Venezia considered regret as R(P, x) = Πmax − Π(P, x) as
12
shown in (16) while we consider the regret term to be
£
¤
v (Πmax ) − v Π(P, x) .
(23)
The problem of using (16) as the regret term is that all regret-averse firms possess
the same regret as shown in (16) even when their utility functions on regret are different.
Nonetheless, we circumvent this limitation by using the regret term defined in (23), which
enables different firms to possess different regrets if their utilities on the regret term are
different. (3) Moreover, we introduce the regret-averse attribute, g, and the weight of
the regret attribute, k, so that we can study the behavior of regret-averse competitive
firms with different values of g and/or k: the higher the value of k and/or g 0 , the stronger
the attitude of regret. Nevertheless, the utility function u proposed by Paroush and
Venezia (1979) could not be used to study the behavior of different types of regret-averse
competitive firms. Thus, our regret-averse function defined in (21) is a more specific and
tractable regret-averse function. (4) In our modeling setting, we allow random prices to
be any random variable, whereas the form of P is fixed as displayed in (18) for Paroush
and Venezia’s model. (5) Because of the above advantages, our model setting allows us
to make comparative statics of the optimal output by varying the regret term as well as
g and v, but the model developed by Paroush and Venezia cannot. (6) Last, we conclude
that our model setting has greater appeal to intuition than the one developed by Paroush
and Venezia.
Before we further develop our model, we first solve the maximization problem in (22)
13
to obtain its first-order condition such that
£
¤
¤£
¤ª
¤ £
© £
¤£
¤ª
© £
dE u(Π)
= E v 0 Π(x) P − C 0 (x) + kE v 0 Π(x) g 0 v(Πmax ) − v(Π(x)) P − C 0 (x)
dx
¤£
¤ª
£
¤ £
© £
¤
(24)
= E v 0 Π(x) + kv 0 Π(x) g 0 v(Πmax ) − v(Π(x)) P − C 0 (x)
= 0.
By the assumptions of k, g, v, and C, we obtain the second-order condition to be
£
¤
n£
¤2 £ 00
£ 0 ¤2 00 ¤o
d2 E u(Π)
0
00 0
=
E
P
−
C
(x)
v
+
kv
g
−
k
v g
− E {v 0 + kv 0 g 0 } C 00 (x) < 0 ,
dx2
(25)
£
¤
£
¤
where v = v Π(x) and g = g v(Πmax )−v(Π(x)) . Thus, there is a unique global optimum,
x∗ .
Proposition 3 and Corollary 2.1 tell us that E(P ) could be less than C 0 (x∗ ) if we apply
the theory developed by Paroush and Venezia (1979), which some economists believe is
not reasonable. Thus, our first objective is to see whether the more specific and tractable
regret-averse function stated in (21) could be used to circumvent the limitation. We find
that our proposed regret-averse function stated in (21) could be used for this purpose as
stated in the following theorem:
Theorem 2.1 For any regret-averse firm with utility function u defined in (16) that will
maximize the expectation of u as shown in (17) and face an uncertain price P , it will
choose an optimal output x∗ such that the expected price, E(P ), exceeds the marginal
14
costs, C 0 (x∗ ) when producing x∗ ; that is,
E(P ) > C 0 (x∗ ) .1
In contrast with the work of Paroush and Venezia (1979), if we adopt this specific
two-attribute utility function, the firm would definitely choose an output smaller than
that in the theory under certainty. This result is different from that obtained by Paroush
and Venezia (1979) but is similar to that obtained by the theory of the risk-averse firm.
One question is whether this optimal output increases or decreases as the regret-averse
attribute varies. One way to answer this question is to study the comparative statics of
this optimal output when the regret parameter k changes, as we do in the following
theorem:
Theorem 2.2 Under the conditions and assumptions stated in Theorem 2.1, we have
1. if µ − C 0 (x∗ ) ≥
2. if µ − C 0 (x∗ ) ≤
Cov[−P, v 0 (Π(x∗ ))]
dx∗
,
then
≤ 0, and
Ev 0 (Π(x∗ )
dk
Cov[−P, v 0 (Π(x∗ ))]
dx∗
,
then
≥ 0.
Ev 0 (Π(x∗ )
dk
As we have shown in the above proposition optimal output can go either way when
the second attribute varies. Importantly, this condition does not directly depend on the
second attribute, which accounts for the feeling of regret. The intuition behind this result
is as follows: By concavity of v, the sign of Cov[−P, v 0 (Π(x∗ ))] is positive. It means that
if the difference between the certain price and the marginal cost is large enough, then the
1
We recall that for a random variable X, we have Cov[f (X), g(X)] > 0 if both functions are strictly
increasing or strictly decreasing. The sign of the above inequality is reversed if one function is increasing
and the other is decreasing. One may refer to Egozcue et al. (2009, 2010, 2011a, 2012) and the references
therein for further results and applications on this topic.
15
regret factor would lower the optimal output. In this case, the regret factor amplifies the
decline in the optimal product. For the second case, the analysis is the opposite.
3
Illustrations
We first construct an example to illustrate the finding of Proposition 1 to state the relationship between price and cost when attaining optimal output for a risk-averse firm in
the theory under certainty as shown in the following:
Example 3.1 Let C(x) = x2 and u(x) =
√
x so that the firm maximizes:
£
¤1/2
max µx − x2 − F
,
x
in which µ = P . Its first-order condition is:
¢−1/2
1¡
µx − x2 − F
(µ − 2x) = 0 .
2
Hence, the optimal value is x∗ = µ2 . On the other hand, C 0 (x∗ ) = 2 × µ2 = µ(= P ). Thus,
the marginal revenue is equal to the marginal cost at x∗ .
We then continue from Example 3.1 to construct an example to illustrate the finding
of Proposition 2 to state the relationship between price and cost when attaining optimal output for a risk-averse firm under the theory under uncertainty. Without loss of
generality, we assume F = 0 as shown in the following:
Example 3.2 We let C(x) = x2 , u(x) =
√
x, and the price of output, P , to be a random
variable with P = $50 and P = $10 of equal probability so that its expected value E(P ) =
16
$30 and the firm maximizes:
¡
max E P x − x
¢
2 1/2
x
n p
o
p
2
2
= max 0.5 (10x − x ) + 0.5 (50x − x ) .
x
Its first-order condition is:
10x − 2x
50x − 2x
p
+ p
=0.
2
4 (10x − x ) 4 (50x − x2 )
Thus, the optimum is at x∗ = 8.33 when C 0 (x∗ = 8.33) = 2 ∗ 8.33 = 16.66 < 30 = E(P ).
Example 3.2 supports the finding in Proposition 2 that an optimal output x∗ characterized by marginal cost C 0 (x∗ ) is less than the expected price E(P ).
We turn to constructing an example to illustrate both Proposition 3 and Corollary
2.1 that E(P ) could be less than C 0 (x∗ ) based on the theory developed by Paroush and
Venezia (1979):
Example 3.3 We define the utility function of the firm in the theory under uncertainty
for regret-averse competitive firms developed by Paroush and Venezia (1979) to be a function of both profits Π(P, x) and regret R(P, x) such that
£
¤ p
u Π(x, p), R(x, p) = Π(x, p) − R(x, p)1.1 ,
(26)
where R(x, p) = Πmax − Π(x, p).
We note that this utility function satisfies all the assumptions required, for example,
17
uΠ > 0 and uR < 0. In addition, we assume that price follows the following distribution:
Price Probability
20
98%
48
1%
50
1%
(27)
Thus, we have E(P ) = 20.58. We further assume that the cost function is the same as
the one used in Example 3.1 such that C(x) = x2 . We first examine the situation under
certainty in which the firm will obtain optimal output by equating its marginal revenue
to its marginal costs such that 2x = 20.58 and thus we have x = 10.29 and the optimal
output of the firm is 10.29 if we assume that price is certain, that is, P = 20.58.
We turn to examine optimal output in the theory under uncertainty for regret-averse
competitive firms developed by Paroush and Venezia (1979). Under the assumption of
utility function defined in (26) and that price follows the distribution defined in (27), the
firm will maximize:
h√
£
¤
¡
¢ i
2 1.1
2
max E u (Π) = 0.98 20x − x − 100 − 20x + x
x
h√
¡
¢1.1 i
+0.01 48x − x2 − 576 − 48x + x2
h√
¡
¢1.1 i
+0.01 50x − x2 − 625 − 50x + x2
.
(28)
Maximizing (28) yields x = 10.52, which is larger than that under the certainty theory:
x = 10.29.
At last, we construct the following example to illustrate Theorems 2.1 and 2.2 for the
theory of the behavior of regret-averse firms developed in this paper:
18
Example 3.4 Let v(x) =
√
x, C(x) = x2 , g(x) = x1.5 , and P = 50 or P = 10 with equal
probability. Under the theory developed in this paper, regret-averse firms will produce the
optimal output by maximizing the following:
½ ·
³
´1.5 ¸
√
1 √
50x − x2 − k 25 − 50x − x2
x
2
·
´1.5 ¸¾
³
√
1 √
2
2
.
10x − x − k 5 − 10x − x
+
2
£
¤
max E u (Π) = max
x
We have the following
• if k = 0, then x∗ = 8.33 (risk-averse case),
• if k = 0.5, then x∗ = 8.83, and
• if k = 1, then x∗ = 8.92.
The above findings show that (1) the optimal output under the regret-averse theory
developed in this paper is still less than that under the certainty theory, in which we get
x = 10.29, and (2) the optimal output varies when k changes. For example, from the
£
¤
above, we have dx∗ /dk > 0. We plot the expected utility of profits, E u (Π) , for k = 1
in the following:
19
Figure 1: Expected Utility of Profits for k = 1
EuHPL
2
1
7
8
9
10
x
-1
-2
£
¤
Note: The expected utility of profits, E u (Π) , is defined in (22).
£
¤
Figure 1 confirms that the maximal value of E u (Π) for k = 1 is when x∗ = 8.92.
One could easily plot the figures for other values of k. Thus, we do not report the figures
for other values of k in this paper.
4
Concluding Remarks
Kahneman and Tversky (1979) and many others pointed out a number of cases in which
commonly observed patterns of choice violate conventional expected utility axioms. These
violations indicate that some important factors that affect many people’s choices have been
overlooked or mis-specified by the conventional expected-utility theory. This problem is
also reflected in the theory of production under uncertainty. Paroush and Venezia (1979)
improved the theory by introducing the regret function to production theory.
However, there are some limitations of the regret function they introduced. To circumvent their limitations, in this paper we introduce a more specific and tractable regretaverse function into the theory of production to anticipate feelings of regret and rejoicing.
Our approach offers an alternative model that takes those feelings into consideration.
20
This model yields a range of inferences in the theory of production consistent with the
behavior of regret-averse managers. For instance, we have shown that a regret-averse firm
with an uncertain output price would choose an optimal output that is smaller than under
certainty. We have also shown that this optimal output could increase or decrease as the
regret-averse attribute varies.
It is common in economic theory to assume that firms are risk-averse. However, this
assumption may not always be correct either. Thus, in this paper we extend the theory to
include firms that are regret-averse. One may think that firms may not be regret-averse.
In this situation, one could conduct experiments to test the behavior of firms (managers)
to see whether they are risk-averse or regret-averse. If they are neither risk-averse or
regret-averse, one could then extend the theory to cover other types of firm’s behaviors.
For example, one could assume that firms are risk seekers (Wong and Li, 1999; Li and
Wong, 1999; Wong, 2007) or that firms’ utility functions could be S-shaped or reverse
S-shaped (Wong and Chan, 2008; Fong, et al., 2008; Broll, et al., 2010; Egozcue, et al.,
2011b). One could also use the behavioral finance concept to make assumption about
firms’ behavior; see, for example, Matsumura, et al. and Lam, et al. (2010, 2012). At last,
we note that one could apply the theory developed in this paper to incorporate the riskregret idea to other theories in economics and finance like investment decision making,
see, for example, Fong, et al. (2005), Wong and Ma (2008), Bai, et al. (2009), Chan, et
al. (2011) and Egozcue (2012).
21
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