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Exit with incomplete information about cost
Information Economics and Policy 3 (1988) 241-263. North-Holland
241
EXIT WITH I N C O M P L E T E INFORMATION ABOUT COST* Esther G A L - O R University of Pittsburgh, Pittsburgh, PA 15260, USA This paper presents a model of output decisions by competitive firms which have incomplete information about costs. Unlike other work on this topic, our model incorporates the possibility of exit from the market after a firm has observed its unit cost. At the equilibrium expected aggregate output and the likelihood of exit depend upon the opportunity cost of staying in the market, the number of informed firms, and the intensity of competition. Uninformed firms may be better off than informed firms if exit is feasible. As a result, even if all firms face the same cost of acquiring information, informed firms may coexist with uninformed firms.
Keywords:
Asymmetric information, exit theory, uncertain costs.
1. Introduction
Many studies of the consequences of incomplete information about technology implicitly assume that the costs of exit from an industry are prohibitively high. In these studies a firm remains in the market even if it observes a very high realization of its unit cost [see, for instance, Li (1985), Gal-Or (1986), and Shapiro (1986)]. In this paper we assume that a firm can stop production and leave the market if it observes an undesirable private signal about its technology. 1 , We obtain surpi'ising and sometimes counterintuitive results if firms can stop production depending u p o n the private signal that they observe. For instance, the expected output produced in the market depends upon the opportunity costs of staying in the market, and upon the n u m b e r of firms that have access to private information. Expected aggregate output may decrease if more firms are established in the market. Ex-ante uninformed firms may be better off than informed firms. This last result arises in our model because being uninformed may provide a mechanism whereby the firm can commit to production. Being informed provides the flexibility of exit if an unfavorable signal is observed. There are circumstances where being commited to production is advantageous to the firm. In our model firms may differ in their state of information about the realizations of their uncertain technologies. Specifically, some firms may be better informed than others. This asymmetry in the state of information may *This work was supported by NSF grant SES-8420492. I wish to acknowledge the helpful suggestions of an anonymous referee and the editor of this journal. ~Several papers have, recently, been written to provide a theory of exit in oligopolistic markets [i.e., Ghemawat and Nalebuff (1985), Fudenberg and Tirole (1986a), and Fine and Li (1986)]. Those models are dynamic models, where the decision variable of the firm is the timing of exit. The focus of the present paper is completely different from those earlier papers. Our model is static and firms exit simultaneously. We wish to generalize earlier static models with incomplete information about unit cost, by incorporating the possibility of non-production. 0167-6245/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland)
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arise, for instance, if firms become established in the market at different times. It is a reasonable assumption that firms which invest later are less familiar with the technology. We model the asymmetry in the state of information by assuming that the incumbent firms divide into two types of groups: informed and uninformed. The informed firms observe the true realization of their unit variable costs without any additional noise. Moreover, if they observe a high realization of unit cost they may decide to exit the market. Uninformed firms are only familiar with the properties of the prior distributions that determine their unit costs. They continue production as long as their prior expected profits are non-negative. We demonstrate that the expected output of an informed firm may be higher or lower than the expected output of an uninformed firm. Expected output of an informed firm is higher if the opportunity costs of staying in the market are relatively low, and vice-versa. When opportunity costs are low, uninformed firms err by producing too little output on average, while their output is too large when opportunity costs are high. The latter error has a strategic effect which is favorable to uninformed firms. As a result, uninformed firms may earn higher expected profits than informed firms when the opportunity costs to staying in the market are relatively high. Similar strategic externalities of incomplete information have been previously demonstrated in the literature [see, for example, Riordan (1985), Fudenberg and Tirole (1986b), and Gal-Or (1987)]. In these studies, however, the externalities arise because firms try to signal 'bad news' about an uncertain state of nature. In contrast, signalling is not an issue in the present paper because our model is static, and firms cannot draw inferences from the past behavior of rivals. Nevertheless, uninformed firms may still be at a competitive advantage, since the absence of information may provide them with an indirect mechanism of c o m m i t m e n t to production. Initially we assume that the identity of the privately informed firms is exogenously given; however, later we consider the case in which firms endogenously determine whether to become informed at a cost. Even if all firms are identical a priori, asymmetric informational equilibria may arise. At such equilibria, some firms may decide to acquire information and others may decide to remain uninformed. We conduct a comparative statics analysis and demonstrate the following results. Informed firms are more likely to exit, and to produce less in expected value terms if: (a) there are more firms in the market, (b) the opportunity cost for staying in the market is higher, and (c) the n u m b e r of informed rivals increases when the opportunity cost is relatively low (and decreases if the opportunity cost is relatively high). Uninformed firms produce greater quantities of output if the opportunity cost is highei:~ Any other changes in the parameters have an ambiguous effect on the behavior of an uninformed firm. In particular, it is completely possible that an uninformed producer expands production if additional firms participate in the market. This counterintuitive result can be explained by observing that with more competitors in the market, each informed firm is more likely to exit. Hence, the increase in the number of firms may result in a relief of the competitive pressures experienced by the ignorant (uninformed) firm. Our study is organized as follows. In the next section, we outline the main assumptions of the model. In section 3, we derive the equilibria when exit is
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costless, and in section 4 we derive, for comparison, the equilibria w h e n exit is prohibitively expensive. In section 5, we allow firms to d e t e r m i n e e n d o g e n o u s l y whether to b e c o m e informed, and in section 6 we conclude.
2. The model
We consider a m a r k e t that consists of n established firms. T h e m a r k e t d e m a n d is linear and takes the form: p = a - Q ,
a >0,
(1)
where p is the price and Q is aggregate output. T h e technology exhibits constant returns to scale, where the unit cost of p r o d u c t i o n is stochastically determined. We assume that the unit cost of p r o d u c t i o n of one firm is identically and i n d e p e n d e n t l y distributed with the unit cost of other firms. 2 T h e idea behind the i n d e p e n d e n c e assumption is that the noise in the p r o d u c t i o n process of the firm is g e n e r a t e d by varying the efficiency of firm-specific inputs. Hence, increased efficiency in one firm does not necessarily imply other firms are m o r e efficient as well. Each unit cost is distributed according to the distribution function G(c) [with density g(c)], over the compact interval [0, 6]. Its expected value is E(c) and its variance is var (c). We assume that a > E(c). The value of the p a r a m e t e r 'a' and the distribution function G(-) are a s s u m e d to be c o m m o n knowledge. D e n o t e by Cg the unit cost of firm i and by c , the vector of unit cost of all firms except i. Each firm can earn an alternative income of $ F by exiting from the market. We assume that exit is costless. Of the n firms, m firms observe private information about their technologies. More specifically, each of the privately i n f o r m e d firms observes, at no cost, a priyate signal that predicts without noise its own unit cost. (Subsequently, we allow firms to d e t e r m i n e endogenously w h e t h e r to acquire information at a cost.) T h e strategy of each firm has two c o m p o n e n t s . T h e first is a decision w h e t h e r to remain in the market. T h e second is the quantity of o u t p u t to produce if the firm decides to remain in the market. A n i n f o r m e d firm can select its strategy contingent u p o n the realization of its unit cost. All firms select their strategies simultaneously and independently. The strategy of the ith informed is d e n o t e d by (e~,ql), where e~: R+---> {0, 1} and q~: R + ---->R +" T h e number el(ci) is 0 if firm i will leave the m a r k e t after observing its unit cost q , and 1 otherwise. T h e n u m b e r qi(ci) is the o u t p u t to be p r o d u c e d if the firm remains in the m a r k e t after observing c/. T h e strategy of the j t h u n i n f o r m e d firm is d e n o t e d by (e j, q j), where ej E {0, 1} and q~ E R +. Note that the strategy of the u n i n f o r m e d firm is i n d e p e n d e n t of its cost because such a firm does not observe any private information prior to making its decision. Our objective is to derive the Bayesian Nash equilibria of the above game, and to conduct a comparative statics analysis to d e t e r m i n e how changes in the U
U
U
2This assumption can be relaxed to allow for correlation among costs of different firms. In this case when an informed firm observes its unit cost, it is able to draw some inferences about the costs of its rivals.
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parameters affect the behavior of an informed versus an uninformed firm. We also wish to compare the equilibria with costless exit to the equilibria of an alternative game where exit is prohibitively expensive, as is implicitly assumed in many models with incomplete information about cost.
3. Market equilibria with costless exit An informed firm chooses its strategy to maximize its expected payoff V~, where
V~={Ec_i[[a-~qle~- ~ q~e~-cilq~lcil-F}e~. i=l
(2)
j=rn+l
In (2), the expected value is over the random variables that remain unobservable when firm i makes its exit and quantity decisions. These r a n d o m variables are the unit costs of other firms as summarized by the vector c_ i. The expected value for firm i is conditional on the realization of its own unit cost (ci). u An uniformed firm chooses its strategy to maximize its expected payoff V j , where
Vju = {E(cs_?[
a-
~-'~ q ~ e ~ - fi-' g=l
q jue ju- c j
] q ju- F } e ~ .
(3)
j=m+l
The expectation is with respect to the vector of unit costs, including firm j's that are unobservable to j when its decisions are made. Because the only difference among firms is whether they are informed, all equilibria are symmetric in that all firms of the same information state follow identical decision rules. For such equilibria, the exit decision of an informed firm is determined by a single cut-off point (see Appendix A). If the privately observed unit cost is below this cut-off point, the firm remains in the market. Otherwise, the firm quits the market. A n uniformed firm remains established in the market provided that equilibrium expected profits are non-negative. The above exit rules together with the linearity of d e m a n d imply that informed firms follow linear decision rules in choosing the quantity of output (see Appendix A). To summarize, our search for an equilibrium will consist of exit and quantity decision rules of the following form: ql=Ao
+Alcg,
el=l
if
ci<~c*,
if
c i > c* ,
= 0 u
qi=B0 u
ej = 1 =0
Ao, A 1 independent of c;,
B 0 i n d e p e n d e n t Of Cg, if
ETr u i>0
if
E7rU<0.
i=l,n,
(4)
i=l,n,
where ETr u denotes the equilibrium expected payoff of an uninformed firm.
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Proposition 1 derives the coefficients A0, A1, and B 0 and the cutoff point c* as a function of the p a r a m e t e r s a, m, n, and F and the distribution function G(.). The underlying assumption behind this derivation is that uninformed firms have non-negative expected profits and so find it optimal to produce, which assures that both uniformed and informed firms can coexist in the market.
Proposition 1.
(a) A t the equilibrium c*
2m-n-1 2
a+
f
cg(c) dc + (n - m)E(c)
I
0
qi=
2(n-m+l)+G(c*)(2m-n-1)
I
ei = 1 =0
if
C i ~ C* ,
if
Ci > C* ,
Ci
2 '
i=l,m,
c*
a(2 - G(c*)) - [2 + (m - 1)G(c*)]E(c) + u
u
eg(c) dc
0
qi = q = j
mf
2(n-
m + 1) + G ( c * ) ( 2 m -
n - 1)
=m+l,n,
e~U=eU = 1
if
q u ~>V~,
where c* satisfies the following equation: c*
-1 a + 2 m - n-2H(c*, a, m, n) =
f cg(c) de + (n - m ) E ( e ) 0
2(n-
m + 1) + G ( c * ) ( 2 m -
n - 1)
C~
(b ) This equilibrium satisfies the Routh Hurwitz condition for reaction functions stability 3 if -[2(n
- (2m - n - 1 ) g ( c * ) V ~ m + 1) + G ( c * ) ( 2 m - n - 1)
+1] 0
(c) The equilibrium is interior (0 < c* < ~) if 2(n - m + 1 ) V ~ -
(n - m ) E ( c ) < a < V ~ ( n
n+l + 1) + ~
_ c
n-1
- -2
E(c),
where the lower inequality is required for c* > 0 and the upper for c* < ~. 3The stability condition is the condition for the overall reaction mapping to be a contraction mapping and therefore implies uniqueness of equilibrium [see Friedman (1977, ch. 7)]. Alternatively, this condition allows direct application of the Gale-Nikaido global univalence result to prove uniqueness.
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E. Gal-Or / Exit with incomplete information about costs
Proof.
See Appendix B.
From Proposition 1, an informed firm of type c* is indifferent between producing q~(c*) or leaving the market, since q I ( c * ) = V'--F. An uninformed firm participates in the market if qU ~>X/-~. Hence, an equilibrium where uninformed firms coexist in the market with informed firms is guaranteed if qU _ qI(¢,) i> 0. From the proposition, the latter inequality holds if T(c*) =-- [a + (m - 1)E(c) - c ' m i l l - G(c*)] C*
G(c*)c* - I cg(c) dc + (n + 1)
c*-
0
2
-
E(c)
/>0
(5)
The first term of T(.) is a decreasing function of c*, and the second is an increasing function of c*. Hence, the sign of the derivative of T(.) is ambiguous. The value of the function at the extremes of the support of the distribution function is: T(0) = a - (n - m + 2)E(c) and T(() = (n + 1 ) ( ( - E(c))/2. As a result, T(.) is definitely positive for large 4 enough values of c*, but it may have an ambiguous sign for small values of c*. The above discussion implies that if the informed firms find it optimal to produce for high realizations of cost, the uninformed definitely will find it optimal to produce, since q"~> qI(c*) in this case. However, if the informed firms produce only for very favorable realizations of cost (c*" is small), the uninformed may not stay in the market. It is possible, however, that informed firms rarely produce while uninformed firms stay in business. In particular, if c * = 0 and a > ( n - m + 2)E(c), the uninformed are the only ones that produce. The last inequality is more likely to hold if nearly all firms are informed. When most firms are informed and have a low probability of producing output (e.g., when opportunity costs are relatively high and c* is small), the uninformed may be at a strategic advantage, in spite of the experimental advantage of the informed firms. Because in this case expected aggregate output is low, an uninformed firm can expand its production level and earn sufficient profits to recover its opportunity costs. Because we wish to characterize equilibria where informed firms coexist in the market with uninformed firms, we assume that inequality (5) holds, which guarantees that uniformed firms expect non-negative profits (ex ante) and so find it optimal to produce. The stability condition of part (b) of Proposition 1 guarantees the existence and uniqueness of the equilibrium. Substituting for X/-F from the function H(.) into this condition yields the result that reaction function stability holds provided that (2m- n-
1)g(c*)A o + (n - m + 1)
+ (2m - n - 1) [G(c*) > 0. 2 4Since T(.) is continuous and T(() > 0, the function is positive for large c values.
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If the distribution G(.) is concave and the informed firms constitute more than fifty percent of the market (rn/> (n + 1)/2), this condition definitely holds, since A 0 > 0. Using Proposition 1, we can derive expected output and expected profit for each firm in the market. For the informed firm we derive ex ante expected output and expected profits, i.e., profits prior to the actual realization of its private information. Eq I = E(q~lno exit)prob(i does not exit) c*
a G ( c * ) + (n - m ) E ( c ) G ( c * ) - (n - m +
1)f
cg(c) dc
0
(6)
2(n - m + 1) + G ( c * ) ( 2 m - n - 1) C*
a(2 - G ( c * ) ) - [2 + (m - 1 ) G ( c * ) ] E ( c ) + qu =
mf
cg(c) dc
0
2 ( n - m + 1) + G ( c * ) ( 2 m -
E,ni
[EqI] 2 +
G(c*)
4
n-
1)
var(clc • c*) - F G ( c * ) ,
ETr u = [qU]2 _ F .
(7)
From (6) and (7) it is unclear whether an informed firm is better off than an uninformed firm. The ambiguity arises because the quantity of output produced by the uninformed firm may exceed the expected output produced by the informed firm. In Proposition 2, a comparison is made between those quantities o f output. ~'
Proposition 2. There exists a level o f f i x e d cost 0 < F * < Fmax, such that, if F < F*, then E q ~ >I qU and, if F > F*, then E q ~ < qU, where
Fmax Proof.
=
[a + (n - m ) E ( c ) ] 2 / 4 ( n - m + 1) 2 .
See Appendix B.
To guarantee the existence of an interior equilibrium at which c* > 0, that is, to guarantee that informed firms produce for some realizations of their private signals, the opportunity cost can never exceed Fmax. According to Proposition 2 if the opportunity costs are relatively low (less than F*) the informed firm produces greater expected quantities of output than the uninformed. However, if the opportunity costs are relatively high (more than F*, but less than Fmax) the uninformed firm produces more. The reason for this result is that, for high values of opportunity costs, the informed firm stays out of the market for a wider range of realizations of its private signal. Only if its unit cost is very low, does it decide to remain in the market. Under condition (5), the uninformed produces with certainty since it earns positive expected profits. The reduction in the probability of production by the informed firm reduces the quantity of output that it is likely to produce. From the
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E. Gal-Or / Exit with incomplete information about costs
derivation of expected profit in (7), and the above discussion it follows that the uninformed firm may be better off compared with the informed firm in our model. This may happen for relatively high values of opportunity costs (but not values so high that they eliminate the profitability of production for the uninformed). To demonstrate that the uninformed may indeed be better off than the informed, consider the case when F = Fmax, so that the informed firm chooses to stay out of the market (i.e, c* -- 0). Its profits are zero as a result. In addition assume that a > (n - m + 2)E(c). Substituting c* = 0 into the expression for qU yields that qU _- [a - E ( c ) ] / ( n - m + 1), which exceeds Fma x under the condition that a > ( n - m + 2)E(c). Hence the uninformed can expect strictly positive expected profits, and is better off than the informed. We summarize this finding in Observation 1. O b s e r v a t i o n 1. (a) If F<~ F*, where F* is defined in Proposition 2, then ETr ~>ETrU; and (b) If F > F* it is p o s s i b l e that ETr u > E T r ~.
We can further interpret the observation by noticing that being uninformed in our model provides a mechanism of commitment to production by the firm. Being informed provides the flexibility of exit if an unfavorable signal is observed. When the opportunity costs are relatively high, part (b) of Observation 1 indicates that being committed to production may be to the advantage of the firm. A n o t h e r interesting property of the equilibrium is that both the expected output and the expected profit of each firm depends u p o n the number of informed firms. We elaborate on this dependence when we conduct the comparative statics analysis. In addition, in contrast to the traditional results that opportunity costs do not affect the output decision of a prbfit-maximizing firm, in our model the expected output of both informed and uninformed firms depends upon the opportunity cost (through c*). In the next section, we demonstrate that when exit is prohibitively expensive, expected output is independent of the n u m b e r of informed firms and of opportunity costs. In tables 1 and 2 we summarize the results of a comparative statics analysis. In table 1 we indicate how changes in the parameters affect the behavior of the
Table 1 Comparative statics of the informed firm.
Number of firms - n Number of informed firms- m Opportunity costs-F Intercept of demand- a
Cutoff point for production c*
Expected output of informed Eq t.......
Expected profits of informed ETrI
(-) (-)
(-) (-)
(-) (-) if Eq I I> qU(F<~ F*)
(+)
(+)
(-)
(-)
(+) if Eq I < qU(E > E*) Ambiguous, but (-) for uniform distribution
(+)
(+)
(+)
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249
informed firm, and in table 2 we indicate how it affects the uninformed firm. In Appendix C we derive the results reported in the tables. As shown in table 1, the informed firm is less likely to produce if the pool of firms increases, if opportunity costs increase, or if the intercept of the demand declines. If the number of informed firms rises, each informed firm is less likely to produce if the expected output produced by an informed firm exceeds that of an uninformed firm. It is more likely to produce if the expected output produced by the uninformed exceeds that of the informed: According to Proposition 2, for low levels of opportunity costs (F~< F*), increasing the number of informed firms reduces the likelihood that an informed firm will produce. For high opportunity costs ( F > F * ) , increasing the number of informed firms increases their likelihood of production. The expected output of each informed firm and its expected profits move in the same direction as the cutoff point c*. Hence, if informed firms are less likely to produce, they also produce lower expected output and earn lower expected profits. The only exception is that if opportunity costs increase, expected profit may increase as well, even though an informed firm is less likely to produce. To obtain the results stated in table 2, the equilibrium must satisfy the following regularity condition:
1
C(c*) 2
> 0
(s)
Notice that this condition does not follow from the second-order condition or the stability condition. From Proposition 1 the stability condition can be written as
2m g(c*)V'-F +
-
2
+ (n + 1) 1 - g(c*)X/-F
-
G(c*) 2
>0.
!
Hence, the regularity condition coincides with it only when m = 0, namely only if all the firms are uninformed. In general, an equilibrium satisfying the stability condition may exist without satisfying the regularity condition. This happens, for instance, when V ' F > [ 1 - G ( c * ) / 2 ] / g ( c * ) and m is sufficiently large. Specifically, if opportunity costs are sufficiently high so that informed firms are less likely to produce, and if the number of informed firms is large enough, the regularity condition may fail. Under such circumstances the absence of information may provide a strategic advantage to the uninformed firm. Table 2 Comparative statics of the uninformed firms. Regularity condition holds
N u m b e r of firms - n N u m b e r of informed f i r m s - m
Opportunity c o s t s - F Intercept of demand - a
Regularity condition does not hold
output - q"
profits E ~ -u
output - qU
profits - ETr u
(-)
(-)
(+)
(+)
(-)
(-)
(+)
(+)
(+)
(-)
(+)
ambiguous
(+)
( + ) if E q I >I q"( F < F* ) ( - ) if E q l < qU( F > F* ) ambiguous
(+)
(+)
(-)
(-)
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E. Gal-Or / Exit with incomplete information about costs
According to table 2 only a change in opportunity costs has an unambiguous effect upon the uninformed firm. Regardless of whether the regularity condition holds, an uninformed firm produces more in response to an increase in opportunity costs. With higher opportunity costs informed firms are more likely to exit, thereby reducing competitive pressures on uninformed firms. Changes in all the remaining parameters have an ambiguous effect on the uninformed. However, with the regularity condition, results are obtained which are more in line with traditional microeconomic theory. Under (8) the uninformed firm produces less if more firms are established in the market, if more firms have access to private information while the opportunity cost is relatively low (F <~ F*), and if the value of the intercept of the demand is smaller. If the regularity condition does not hold the above predictions are reversed. For instance, the uninformed firm produces more if the number of established firms increases. As pointed out above, the regularity condition fails when the absence of information provides a significant strategic advangage to the uninformed. If the total number of established firms increases in this case, informed firms have a reduced likelihood of production. The uninformed firm can take advantage of this by increasing production. Hence, whenever the absence of information provides a strategic advantage to uninformed firms, their response to changes in the parameters of the model is opposite to the response of informed firms. Next, we can use the computations of Appendix C to predict how changes in the parameters affect expected aggregate output. Expected aggregate output is given by EQ = mEq I + (n - m)q u .
In table 3 we summarize comparative statics results concerning aggregate production. From table 3 it is possible that an increase in the number of established firms will cause aggregate production to d e c l i n e (if the regularity condition does not hold). This result can be explained as follows. When the number of established firms is larger, each informed firm is more likely to exit. Hence even though there is more potential competition (larger n), actual competition, as measured by ( n - m + m G ( c * ) ) , may decline. As a result, expected aggregate output may decline as well. It is worthwhile to elaborate on how a change in the amount of information affects the market outcome. Based upon table 3 an increase in the number of informed firms may result in a d e c l i n e of aggregate production (under the regularity condition when F > F*, and if the regularity
Table 3 Comparative statics of aggregate output.
Number of firms- n Number of informed firms- m Opportunity costs - F Intercept of demand - a
Regularity condition holds
Regularity condition does not hold
(+) (+) if F ~< F* (-) if F > F * (-) (+)
(-) (-) if F ~< F* (+) if F > F * (-) ambiguous
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251
condition does not hold when F ~< F*). Such a result implies that consumers may prefer that firms be less informed about their technologies. With less private information available to firms, expected output may increase and prices may decline.
4. Market equilibria when exit is prohibitively expensive When exit is prohibitively expensive and firms are already established in the market, both types of firms are committed to production. If both types of firms invest in the industry, the unconditional expected profits of each type has to be non-negative. The strategy of an informed firm is ql(ci) and of an uninformed is q~. In Proposition 3 we state the properties of the equilibrium with no exit. Proposition 3.
n-1 a + ---5-
I
qi = tl
q/
A t the equilibrium with no exit 5
--
E(c)
Ci
2 '
n+l a - E(c) n+l '
(9)
j=m+l,n,
ETrI = [ a - E ( c ) ] 2 va4(c) nT1 + ETrU = [ a - E ( c ) ] n+l
i = 1, m ,
F,
(10)
2 -F.
Both types of firms are in the market if a > (n + 1)X/F + E(c). The proof of Proposition 3 is in Appendix B. The properties of the equilibrium with no exit are completely different from the properties obtained in the previous section. In particular, output decisions are independent of the number of informed firms and of opportunity costs. In addition, the expected output of an informed firm coincides with the quantity of output produced by an uninformed firm (i.e., Eq I = qU). Expected aggregate output is a strictly increasing function of n and a, and is independent of m or F. When exit is costly, being informed always guarantees higher profits than remaining uninformed. The properties of the equilibria with prohibitively costly exit have been established in the literature [see Li (1985), Gal-Or (1986), and Shapiro (1986)]. Especially, it has been demonstrated that the expected profits of a firm are a strictly increasing function of the precision of its private information. This last property may fail if exit is costless, as demonstrated in the previous section. Comparing eqs. (6) and (9) demonstrates that expected aggregate output with costless exit may be higher or lower than the expected aggregate output produced with costly exit, depending upon opportunity costs. More explicitly, if E Q e and E Q ne denote the expected aggregate output produced with free and costly exit, respectively, then 5An additional condition that [a + ((n - 1)/2)E(c) - ((n + 1)/2)~] > 0 is required to guarantee that an informed firm always produces positive amounts of output [i.e., qe(?)> 0]. This condition implies that zero production is never profitable once the firm is already established in the market.
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E Q e - E Q ne = m[Eq' - qU], where Eq ~ and qU are given by (6). From Proposition 2 it follows that for relatively low levels of F(<~F*), aggregate output is higher if firms can freely exit the market. If, however, opportunity costs are relatively high (F > F*), the argument is reversed. Since consumer surplus is directly related to expected aggregate output, it follows that consumers prefer the first regime with free exit if F is low and the second regime with no exit if F is high. It is noteworthy that informed firms may be better off if exit is extremely expensive, namely, if they have less flexibility after observing their private signals. This conclusion follows by comparing eqs. (7) and (10), which gives the expected profits of the informed firm under both regimes. To demonstrate that reduced flexibility may indeed be advantageous, suppose that F = Fmax. With maximal opportunity costs and free exit, the informed firm stays out of the market for every possible realization of its unit cost ( c * = 0), and earns zero profits. When exit is impossible (or extremely expensive), but
a _- E(__c)] /'/ q- i
(11)
> Fmax,
the informed firm does not exit and earns positive expected profits. Inequality (11) holds, for instance, if a>
E(c)[(n + 3 ) ( n - m) + 2] ( n - 2 m + 1)
and
m<(n+l)/2.
(12)
The fact that informed firms prefer the no-exit regime when opportunity costs are large is not surprising. The basic point is that the possibility of exit generates a strategic asymmetry, and the class of firms which lose out would prefer to eliminate the possibility of exit. A similar result is also derived by Judd (1985), who demonstrates that exit barriers may be beneficial for the purpose of entry deterrence.
5. Endogenous acquisition of information Under both regimes, we have assumed that the identity of the informed firms is exogenously given. If we extend the model to incorporate the decision whether to become informed, different results are obtained, depending upon the ability of firms to exit. If exit is extremely,.e~xpensive, and each firm incurs the same cost to acquire its information, either all the firms will purchase the information or all will remain uninformed, depending upon the cost of information. More specifically, if K denotes the cost of information, all firms will purchase information (m = n) if K < var(c)/4. No firm will purchase information (m = 0) if K>var(c)/4. If exit is costless, it is possible that only some firms will become informed (i.e., 0 < m < n), even if each firm incurs the same cost of information. More specifically, the first regime may predict asymmetric informational equilibria even if all firms are a priori identical. At this asymmetric equilibrium some firms purchase information and others remain uninformed.
E. Gal-Or / Exit with incomplete information about costs
253
in orocr to o c m o n s t r a t c ti]at costicss exit may predict informatlonai ab~lllmetries, consider the case in which F = 0 and unit cost is uniformly distributed over the inverval [0, ~?]. For this case we derive a sufficient condition to guarantee that the benefit of becoming informed declines when more firms are already informed. More explicitly, we derive a condition that guarantees that a ( E r r I - E r r u) cgm
<0.
The condition is stated in Observation 2.
Observation 2.
If F = 0, g(c) is uniform and exit is costless, then
a(E r' - Err u) <0 3m
if
a<
(n + 11) 18
C.
The proof of Observation 2 is in Appendix B. According to the observation, the advantage of an informed firm over an u n i n f o r m e d firm declines when m o r e firms are informed, which m a y result in an equilibrium with only a subset of informed firms. M o r e explicitly, denote by E~r~m and E~r~ the expected equilibrium payoffs when m firms purchase information. A t such an equilibrium an informed firm would not benefit if it were to deviate unilaterally and to remain uninformed, and no uninformed firm can benefit from purchasing information. If K is the cost of purchasing information, then Errlm - K ~> Err~m_ 1 ErrS,/>ETrIm+l - K . The first inequality guarantees that an informed firm cannot benefit if it unilaterally decides to remain uninformed. The second inequality guarantees that an uniformed firm cannot benefit if it purchases information. Both inequalities can hold simultaneously if the sequence (ETr~m-E1rU_t) is a decreasing sequence of m. In particular, if [E,n-In - E'rrU_l] < K a n d [ E T r l l ETr0] > K, it is possible that only a subset of firms will purchase information even if all firms face the same cost to purchase it. The sequence [ETr~mETrU 1] can be d e c o m p o s e d as follows: ETr~, - E~',~ 1 = [ETr~ - ETr~,] + [ETr u _ ETrm_l]u .
(13)
U n d e r the condition in Observation 2, the first term is a decreasing sequence of m, and the second t e r m may be either an increasing or decreasing sequence of m. However, it is possible further to restrict the parameters of the problem in order to guarantee that the monotonicity of the first term in (13) is always dominant, so that the sequence [ETr~m-Err~m_l] is a m o n o t o n e sequence. Hence, when exit is free, it is possible that informed firms coexist in the market with uninformed firms, even if the cost of purchasing information is identical. More explicitly, given the option to purchase information, some firms may decide to purchase it, and others may decide to remain uninformed. The informed firms may subsequently decide to exit, depending upon their private
254
E. Gal-Or / Exit with incomplete information about costs
information. The uninformed stay in the market as long as their expected profits are positive.
6. Concluding remarks Incorporating the possibility of non-production in a model where technology is stochastic and a firm may be privately informed about its unit cost leads to equilibria that differ from the equilibrium when non-production is not possible. In particular, expected aggregate output depends u p o n the opportunity costs of staying in the market and the number of informed firms. Being informed may be disadvantageous to the firm. As a result, when firms endogenously select whether to purchase information at a cost, asymmetric informational equilibria may arise, even if initially all firms are identical. Even though our focus has been on exit, the results can be interpreted differently by considering a new industry where potential investors have to make entry decisions. If some potential investors have access to private information about their unit costs prior to entry, while the remaining firms are uninformed, our analysis and results are still valid. With the entry interpretation each firm decides whether to enter, and how much to produce if entry occurs. According to our derivation several conclusions about the entering firms can be stated. For instance, a potential investor may prefer to be uninformed. Increasing the pool of potential investors may result in a decline in the expected aggregate quantity produced, since informed investors stay out of the market for a larger range of realizations of their unit cost. In the paper we have only considered uncertainty about the technology. It is also possible to introduce uncertainty about d e m a n d without significantly affecting the analysis or results. We have restricted consideration to a linear demand function and a stochastic technology that exhibits constant returns to scale. These assumptions are standard in the literature, but most welcome generalization would be to consider different d e m a n d or cost functions. Another direction in which this study can be extended is to consider the possibility of information pooling among informed firms. In previous models, where 'non-production' is not explicitly incorporated, Cournot oligopolists have an incentive to share information about private costs. A n interesting question is whether similar incentives exist when non-production is explicitly modelled.
Appendix A.
Equilibrium decision rules
In this appendix, we demonstrate that an exit decision rule with a single cutoff point and a linear quantity decision rule are the only decision rules consistent with an equilibrium.
Lemma 1. For an identical exit decision rule used by each informed firm the quantity of output of an informed firm is a linear function of its unit cost. Proof. Let firm i be an informed firm. Suppose that each informed rival of firm i follows an identical exit decision rule of the form
E. Gal-Or / Exit with incomplete information about costs 1
ej=l
if
cj~CC[O,d],
=0
if
cjf~C.
255
Denote by G the probability that a single informed rival of i remains in the market; then
G = f g(c) d c . C
Notice that G is independent of the unit cost of i, by the independence assumption made concerning the unit cost of different firms. The payoff function of i is VIgil 1 = ei -
{a-q~-ci-Ec-i
+ j=m+l
qIeI
u
(A1)
qjejj~qi-F'
k#i
where at the symmetric equilibrium Ec . 2.,
qUjeUj= (n - m ) q u
and
(A2)
-~ j = m + l
Ec ~ k=l
q kI e kI
.~_
k#i
=
prob K i n f o r m e d r i v a l s } I kEcj(qjlj k=0 / remain established remains established) m- I ~
E
m,( ) Gk(1 ~, m - 1 k=O
(A3) - G)
m
1 kk E c ~ ( q j Il j
k
remains established),
any j = 1, m ;
j#i.
Eq. (A2) holds since we restrict consideration to circumstances where production is profitable for the uninformed. Eq. (A3) holds, since the number of informed rivals that continue production has a binomial distribution with parameters (G, m - 1). Hence, Ec
1
q kI e ,i = ( m _ l ) G E ( q i j l
] produces) a n y j = l , m " ,
j#i
•
(A4)
k=l k#i
Differentiating (A1) with respect to qi and substituting (A2) and (A4) into the first-order condition yields I a(n - m ) q u - ( m - 1 ) G E ( q I j I j produces) qi= 2
any i , j = l , n
j#i.
Ci
2'
(A5)
The first term of (A5) is independent of c;, since c i is independent of cj for i # j . Hence the linearity follows. Q.E.D.
E. Gal-Or / Exit with incomplete information about costs
256
L e m m a 2. The exit decision rule o f an informed f i r m consists o f a single cutoff point, such that if c i <<-c* e~ = 1, and if c i > c* e I = O. Proof. From (A1) and (A5) if follows that at the equilibrium, if informed firm i continues to produce its profits are
Vt
= (q~)2_ F ,
where q~ is given by (A5).
V] is a strictly decreasing function of c i. Hence the lemma follows.
Appendix B.
Proofs of Propositions 1-3 and Observation 2
From (2) if the ith informed firm decides to produce,
P r o o f o f Proposition 1. its payoff is V~
Q.E.D.
= a-ql-ci-E~
qke~+
i leli - I
k=l
qjej
qi-F.
(B1)
j=m+l
k#i
In (B1) we have used the assumption that c i and cj are independently distributed for i ~ j . Notice that the expectation included in (B1) corresponds to the expected aggregate output produced by the rivals of i. Since we consider equilibria where uninformed firms coexist in the market with informed firms, e~ = 1 for every j. From (4) therefore, the aggregate output produced by the uninformed rivals of i is (n - m ) B o. Hence (B2)
q~eU~ = (n - m ) B o . /=m+l
Also, since each informed firm follows an exit decision that consists of a single cutoff point c*, the number of informed rivals of i that remain in the market is distributed according to a binomial distribution with parameters (G(c*), m 1), where G(c*) is the probability of a 'success' (informed rival produces) and (m - 1) is the maximum possible number of 'successes'. If we denote by EQ~_; the expected aggregate output produced by informed rivals of i then m--1
EQI_i = ~'~ E ( Q I i l k informed rivals produce)prob(k informed rivals k=0
produce), = 2
k[Ao+A1E(cilci<~c *
m
1
a(c,)l,[l_C(c,)r
k=O £*
f cg(c) dc =
A ° + A t 0 G(c*)
(m - 1)G(c*)
where we have used the property that the expected value of a binomial distribution function with parameters ( m - 1) and G is equal to ( m - 1)G.
E. Gal-Or / Exit with incomplete information about costs
257
Hence, C*
f cg(c) dc E¢_ i
qkek = k=l k#i
A ° + A1
0
(m - 1)G(c*).
G(c*)
(B3)
Differentiating (B1) with respect to q~ and substituting (B2), (B3) and the suggested decision rule (4) into the derivative, yields the following first-order condition for an informed firm: £*
f cg(c)dc a - 2A 0 - 2AlC i - c i -
(n
-
m)B
o =
0
A0+A
1
(m - 1)G(c*)
G(c*)
(B4)
O.
First-order condition (B4) should hold for every possible realization of c i E [0, ~]. Hence it follows that A 1 = - 1 / 2 . From (3), if the jth uninformed firm decides to produce, its payoff is V?Jl~; ,=
u
I I
u
u
a - qj - E ( c j ) - E c JLi=I q~e~ + k=m+1 qkek
u
qj-F.
(B5)
k~j
The expected output produced by the uninformed rivals of j is (n - m - 1)B 0, if we assume that uninformed expect non-negative profits. Hence
Ec
(B6)
qkeku u = (n - m - 1)B 0 I k=m+l k#j
The expected output produced by informed rivals of j is derived similarly to the derivation of (B3). More specifically, let EQ~_j denote this expected output; then
f cg(c) dc EQ'_]
=
k=O ~
[G(c*)lk[1 -
G(c*)] m-~ k
A o+ A 1
G(c*)
Hence,
[ c; l cg(c) dc
Ec ~ ~'Z., [ i -
i=1
qiei :
A° + A1
0
G---~i
mG(c*) .
(B7)
The difference between (B3) and (B7) is that the informed firm has (m - 1) informed rivals and the uninformed has m informed rivals. Differentiating the payoff function of (B5) with respect to q~ and substituting in the derivative (B6), (B7) and the assumed decision rule (4), yields the first-order condition
E. Gal-Or / Exit with incomplete information about costs
258
for the uninformed as follows: c*
f cg(c) dc a - 2B 0 - E ( c ) - (n - m -
1)B 0 -
Ao+
A 1
o
G(c*)
mG(c*) = 0 .
(B8)
Substituting into the first-order conditions (B4) and (B8) A t = - 1 / 2 yields a system of two equations in the two unknowns (A 0 and B0) as follows: ¢*
lf a - A012 + ( m - 1 ) G ( c * ) ] - ( n - m ) B o + -m --"~
cg(c) dc = 0 , o
c*
mf
a - A o m G ( c * ) - (n - m + 1)B 0 + ~
(B9)
cg(c) d c - E(c) = 0 .
0
The unique solution of (B9) is ¢*
a+ A0=
2m-n-1 2
f
cg(c) dc + (n - m)E(c)
0
2 ( n - m + 1) + G ( c * ) ( 2 m -
n - 1)
' (B10)
C*
a[2 - G(c*)] + m f cg(c) dc - [2 + (m - 1)G(c*)]E(c) 0 n 0 :-
2(n - m + 1) + G ( c * ) ( 2 m -
n - 1)
From the first-order conditions it follows that at the equilibrium VI
= [ql] 2 - F
VU
= [qUj]2 _ F .
and
Hence production is profitable for the uninformed if q~ I> X/F and for the informed if q~/> X/F. In consequence of the derived values of the constants A 0 and A1, the informed should remain established in the market if c*
a + 2 m - n2- 1 I
q i =-
"
""
f cg(c) dc + (n - m)E(c) 0
2(n - m + 1) + G(c*)(2m - n - 1)
- 1/2c~ >t X / F .
(Bll)
Since ql is a strictly decreasing function of c i, there may exist a point c* such that, if it is exceeded by the unit cost, inequality ( B l l ) is violated. In particular, at a symmetric interior equilibrium 0 < c *i < ( a n d c *i = c * f o r e v e r y i = 1, m. Hence the cutoff point is determined by the equation H(c*, a, m, n) = O.
E. Gal-Or / Exit with incomplete information about costs
259
To guarantee the uniqueness of such a point we need that He. < 0. This condition is also required for stability (own cost effects on output exceed cross cost effects on output.) Requiring He. < 0 yields part (b) of the proposition. To guarantee that the equilibrium is interior ( 0 < c * < ( ) , we need that H ( ( , a, m, n ) < 0 and H(0, a, m, n ) > 0. These two inequalities imply part (c) of the proposition. Q.E.D.
P r o o f o f Proposition 2.
F r o m (16) £*
2a(1 - G ( c * ) ) - [2 + (n - 1 ) G ( c * ) ] E ( c ) + (n + 1) f cg(c) dc u
q -Eq I =
0
2(n - m + 1) + G ( c * ) ( 2 m - n - 1) ¢* t*
= 2(1 - G ( c * ) ) A o + ] cg(c) d e -
E(c),
0
where A 0 is given from Proposition 1. Based upon the same proposition A o = 1 / 2 c * + V ~ , thus c
M ( V ~ ) - q" - E q ~ = c*(1 - G ( c * ) ) + 2X/-F(1 - G ( c * ) ) - ~ cg(c) d e , ¢*
c g~
M(O) = c*(1 - G ( c * ) ) - t cg(c) de < O. c* i
F r o m the definition of the function H in Proposition 1 the value of Fma x is the lowest possible level of opportunity cost for which an informed firm exits for every possible realization of its unit cost, n a m e l y c * = O. Substituting the value of Fmax into M ( V ~ ) yields M(V~max) = a + (n - m ) E ( c ) _ E(c) - [a - E(c)] > 0 n-m+l n-m+l '
where the last inequality follows from the assumption that a > E(c). dM dc* dX/-p - 2(1 - G ( c * ) ) + [(1 - G ( c * ) ) - 2X/Fg(c*)] dX/-F" F r o m part (b) of Proposition 1, de*
1
/4,. [2(n - m + 1) + G ( c * ) ( 2 m - n - 1)]
G(¢*)
[(2m - n - 1 ) g ( c * ) V ~ + (n - m + 1) + - 2 <0.
(2m - n - 1)]
E. Gal-Or / Exit with incomplete information about costs
260
Substituting for dc*/dX/F into d M / d X / - F yields dM
2 X / F g ( c * ) ( n + 1)
dVF
(2m - n - 1)g(c*)X/-F + (n - m + 1) + G(c*_____~)(2m - n - 1)
Hence by the mean value theorem the proposition follows.
>0.
Q.E.D.
P r o o f o f Proposition 3.
The informed firm follows a linear decision rule for determining its quantity of output. Thus, I
qi = Ao + A l C i ,
i = 1, m ,
(B12) u
qj=Bo,
j=m+l,n.
Differentiating the objective function of the ith informed yields
[a- 2ql- 2 qI_ 2 quk-ci]= 0 , k=l k~i
(B13)
k=m+l
Differentiating the objective function of the jth uninformed yields a - 2qj -
qk -- E(C) = O.
qk -k=l
(B14)
k=m+l k ,~j
Substituting the suggested solution (B12) into (B13) and (B14) and requiring that (B13) holds for every possible realization of c i E [0, d] yields that a 0=
n-1 a+~E(c)
]
/(n+l),
B o = [ a - E(c)]/(n + 1), A 1= -1/2. Thus (9) follows. From the first-order conditions Errl = E ( q l ) 2 - F , u
Evrj=[q~]Z- F. ..
,.-
the above result yields (10). Requiring that unconditional expected profits are non-negative yields the condition for entry by both types of firms. Q.E.D. Using
P r o o f o f Observation 2.
From Appendix C,
(E~r x - E ~ -u)
(qU _ E q I )
Om
X
{(qU _ Eqi)
+ qU[1 - G ( c * ) - 2 g ( c * ) X / F ] } .
E. Gal-Or / Exit with incomplete information about costs
261
F r o m Proposition 2 E q ~ > qU when F = 0, thus the above derivative is negative if qU< E q I - q
u
1 - G(c*)
(B15)
"
From the derivations in the proof of Proposition 2, 5
qlU=o =
Eq'-
f cg(c) d c -
c*(1 -
G(c*))
>0
c*
and from Proposition 1, c*
' o= EqlF=
½c*G(c*)- i f cg(c)dc. 0
H e n c e (B15) holds provided that 5
u f cg(c) dc q < 1 =--G(-~)
c*.
(B16)
C*
However, qU< Eqi, which implies that (B16) holds if c*
c
1 7-~(-~) 0
c*.
c*
For a uniform distribution the above inequality is satisfied if c .2 < 2E(E - c*) which certainly holds if c * < 2 / 3 5 . F r o m the comparative statics results, 8c*/Orn < 0 if F = 0. H e n c e if c* < 2 / 3 5 w h e n m = 1, c* < 2 / 3 ( for any value of 0 < m ~ < n . F r o m Proposition 1 c* satisfies the equation H(c*, rn, n, a) = 0. Since He, < 0, if H ( 2 / 3 5 , m, n, a ) < 0 it follows that c* < 2 / 3 ( . Evaluating the function H at c* = 2 / 3 ( and m = 1 yields n+ll _ c 18 n)lc.=2/3e = 2n - 2 / 3 ( n - 1) ' m=l a
H(c*, a, m,
which is negative u n d e r the condition stated in the observation.
Appendix C. Comparative statics Let
C(c*)
X = ( 2 m - n - 1 ) g ( c * ) V ~ + ( n - m + 1) + - then X > 0 by the r e q u i r e m e n t that He. < 0.
(2m - n - 1 ) ,
Q.E.D.
262
E. Gal-Or / Exit with incomplete information about costs c~C*
On = - q U / X < 0 ' c9C*
Om = ( q U - E q i ) / x < ~ O >0 OC*
ov-p c)C*
Oa
-
1) +
[2(n - m +
for F < - F *, for F > F * , G(c*)(2m - n -
1)l/X<0,
- l/X>0,
c)Eq ~ _ [G(c*)__ ] Oc* Ot 2 +g(c*)V~ --~ c)Eq~ - - 2 g ( c * ) V ~ ( n - m +
OE~r I Oc*
0 E ' a -I Ot
Oc*
Ot
-
where
t=a,m,n,
1)/X
EqI Oc* at
t=a,m,n
o~ETri
c/
- X/F[(2m - n - 1)g(c*)c*G(c*) - (2m - n - 1)g(c*)
cg(c) d c , 0
m + 1 ) - GZ(c*)(2m - n -
-2G(c*)(nOqU Oa
=[1
G(c*)
1)]/X,
]
2
g(c*)X/-F / X ,
dq u - q U [ g ( c , ) V ~ _ ( 1 OqU _(qU [ Om -Eq') 1
G~*))]/X G(c*) 2
] g(c*)C-F / X ,
OqU - 2 V ~ g ( c * ) m / X > 0 OV'-F
0ETrU OqU Ot - 2qu Ot
where
0 E ~ r U - 2V'F[ 2qUg(c*)m aX/-F X
t = a, n , m
1] "
References Dixit, A. and C. Shapiro, 1986, Entry dynamics with mixed strategies, in: Lacy Glen Thomas, ed., The economics of strategic planning, Essays in honor of Joel Dean (Lexington Books, Lexington, MA). Fine, C. and L. Li, 1986, A stochastic theory of exit and stopping time equilibria, Working paper (Sloan School of Management, MIT, Cambridge, MA). Friedman, J., 1977, Oligopoly and the theory of games (North-Holland, Amsterdam). Fudenberg, D. and J. Tirole, 1986a, A theory of exit in duopoly, Econometrica 54, 943-960. Frudenberg, D. and J. Tirole, 1986b, A signal jamming theory of predation, Rand Journal of Economics 17, 366-376.
E. Gal-Or / Exit with incomplete information about costs
263
Gal-Or, E., 1986, Information transmission-Cournot and Bertrand equilibria, Review of Economic Studies 53, 85-92. Gal-Or, E., 1987, First mover disadvantages with private information, Review of Economic Studies 54, 279-292. Ghemawat, P. and B. Nalebuff, 1985, Exit, Rand Journal of Economics 16, 184-194. Judd, K., 1985, Credible spatial preemption, Rand Journal of Economics 16, 153-166. Li, L., 1985, Cournot oligopoly with information sharing, Rand Journal of Economics 16, 521-536. Nikaido, H., 1968, Convex structures and economic theory (Academic Press, New York). Riordan, M., 1985, Imperfect information and dynamic conjectural variations, Rand Journal of Economics 16, 41-50. Shapiro, C., 1986, Exchange of cost information in oligopoly, Review of Economic Studies 53, 433-446.
241
EXIT WITH I N C O M P L E T E INFORMATION ABOUT COST* Esther G A L - O R University of Pittsburgh, Pittsburgh, PA 15260, USA This paper presents a model of output decisions by competitive firms which have incomplete information about costs. Unlike other work on this topic, our model incorporates the possibility of exit from the market after a firm has observed its unit cost. At the equilibrium expected aggregate output and the likelihood of exit depend upon the opportunity cost of staying in the market, the number of informed firms, and the intensity of competition. Uninformed firms may be better off than informed firms if exit is feasible. As a result, even if all firms face the same cost of acquiring information, informed firms may coexist with uninformed firms.
Keywords:
Asymmetric information, exit theory, uncertain costs.
1. Introduction
Many studies of the consequences of incomplete information about technology implicitly assume that the costs of exit from an industry are prohibitively high. In these studies a firm remains in the market even if it observes a very high realization of its unit cost [see, for instance, Li (1985), Gal-Or (1986), and Shapiro (1986)]. In this paper we assume that a firm can stop production and leave the market if it observes an undesirable private signal about its technology. 1 , We obtain surpi'ising and sometimes counterintuitive results if firms can stop production depending u p o n the private signal that they observe. For instance, the expected output produced in the market depends upon the opportunity costs of staying in the market, and upon the n u m b e r of firms that have access to private information. Expected aggregate output may decrease if more firms are established in the market. Ex-ante uninformed firms may be better off than informed firms. This last result arises in our model because being uninformed may provide a mechanism whereby the firm can commit to production. Being informed provides the flexibility of exit if an unfavorable signal is observed. There are circumstances where being commited to production is advantageous to the firm. In our model firms may differ in their state of information about the realizations of their uncertain technologies. Specifically, some firms may be better informed than others. This asymmetry in the state of information may *This work was supported by NSF grant SES-8420492. I wish to acknowledge the helpful suggestions of an anonymous referee and the editor of this journal. ~Several papers have, recently, been written to provide a theory of exit in oligopolistic markets [i.e., Ghemawat and Nalebuff (1985), Fudenberg and Tirole (1986a), and Fine and Li (1986)]. Those models are dynamic models, where the decision variable of the firm is the timing of exit. The focus of the present paper is completely different from those earlier papers. Our model is static and firms exit simultaneously. We wish to generalize earlier static models with incomplete information about unit cost, by incorporating the possibility of non-production. 0167-6245/88/$3.50 © 1988, Elsevier Science Publishers B.V. (North-Holland)
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E. Gal-Or / Exit with incomplete information about costs
arise, for instance, if firms become established in the market at different times. It is a reasonable assumption that firms which invest later are less familiar with the technology. We model the asymmetry in the state of information by assuming that the incumbent firms divide into two types of groups: informed and uninformed. The informed firms observe the true realization of their unit variable costs without any additional noise. Moreover, if they observe a high realization of unit cost they may decide to exit the market. Uninformed firms are only familiar with the properties of the prior distributions that determine their unit costs. They continue production as long as their prior expected profits are non-negative. We demonstrate that the expected output of an informed firm may be higher or lower than the expected output of an uninformed firm. Expected output of an informed firm is higher if the opportunity costs of staying in the market are relatively low, and vice-versa. When opportunity costs are low, uninformed firms err by producing too little output on average, while their output is too large when opportunity costs are high. The latter error has a strategic effect which is favorable to uninformed firms. As a result, uninformed firms may earn higher expected profits than informed firms when the opportunity costs to staying in the market are relatively high. Similar strategic externalities of incomplete information have been previously demonstrated in the literature [see, for example, Riordan (1985), Fudenberg and Tirole (1986b), and Gal-Or (1987)]. In these studies, however, the externalities arise because firms try to signal 'bad news' about an uncertain state of nature. In contrast, signalling is not an issue in the present paper because our model is static, and firms cannot draw inferences from the past behavior of rivals. Nevertheless, uninformed firms may still be at a competitive advantage, since the absence of information may provide them with an indirect mechanism of c o m m i t m e n t to production. Initially we assume that the identity of the privately informed firms is exogenously given; however, later we consider the case in which firms endogenously determine whether to become informed at a cost. Even if all firms are identical a priori, asymmetric informational equilibria may arise. At such equilibria, some firms may decide to acquire information and others may decide to remain uninformed. We conduct a comparative statics analysis and demonstrate the following results. Informed firms are more likely to exit, and to produce less in expected value terms if: (a) there are more firms in the market, (b) the opportunity cost for staying in the market is higher, and (c) the n u m b e r of informed rivals increases when the opportunity cost is relatively low (and decreases if the opportunity cost is relatively high). Uninformed firms produce greater quantities of output if the opportunity cost is highei:~ Any other changes in the parameters have an ambiguous effect on the behavior of an uninformed firm. In particular, it is completely possible that an uninformed producer expands production if additional firms participate in the market. This counterintuitive result can be explained by observing that with more competitors in the market, each informed firm is more likely to exit. Hence, the increase in the number of firms may result in a relief of the competitive pressures experienced by the ignorant (uninformed) firm. Our study is organized as follows. In the next section, we outline the main assumptions of the model. In section 3, we derive the equilibria when exit is
E. Gal-Or / Exit with incomplete information about costs
243
costless, and in section 4 we derive, for comparison, the equilibria w h e n exit is prohibitively expensive. In section 5, we allow firms to d e t e r m i n e e n d o g e n o u s l y whether to b e c o m e informed, and in section 6 we conclude.
2. The model
We consider a m a r k e t that consists of n established firms. T h e m a r k e t d e m a n d is linear and takes the form: p = a - Q ,
a >0,
(1)
where p is the price and Q is aggregate output. T h e technology exhibits constant returns to scale, where the unit cost of p r o d u c t i o n is stochastically determined. We assume that the unit cost of p r o d u c t i o n of one firm is identically and i n d e p e n d e n t l y distributed with the unit cost of other firms. 2 T h e idea behind the i n d e p e n d e n c e assumption is that the noise in the p r o d u c t i o n process of the firm is g e n e r a t e d by varying the efficiency of firm-specific inputs. Hence, increased efficiency in one firm does not necessarily imply other firms are m o r e efficient as well. Each unit cost is distributed according to the distribution function G(c) [with density g(c)], over the compact interval [0, 6]. Its expected value is E(c) and its variance is var (c). We assume that a > E(c). The value of the p a r a m e t e r 'a' and the distribution function G(-) are a s s u m e d to be c o m m o n knowledge. D e n o t e by Cg the unit cost of firm i and by c , the vector of unit cost of all firms except i. Each firm can earn an alternative income of $ F by exiting from the market. We assume that exit is costless. Of the n firms, m firms observe private information about their technologies. More specifically, each of the privately i n f o r m e d firms observes, at no cost, a priyate signal that predicts without noise its own unit cost. (Subsequently, we allow firms to d e t e r m i n e endogenously w h e t h e r to acquire information at a cost.) T h e strategy of each firm has two c o m p o n e n t s . T h e first is a decision w h e t h e r to remain in the market. T h e second is the quantity of o u t p u t to produce if the firm decides to remain in the market. A n i n f o r m e d firm can select its strategy contingent u p o n the realization of its unit cost. All firms select their strategies simultaneously and independently. The strategy of the ith informed is d e n o t e d by (e~,ql), where e~: R+---> {0, 1} and q~: R + ---->R +" T h e number el(ci) is 0 if firm i will leave the m a r k e t after observing its unit cost q , and 1 otherwise. T h e n u m b e r qi(ci) is the o u t p u t to be p r o d u c e d if the firm remains in the m a r k e t after observing c/. T h e strategy of the j t h u n i n f o r m e d firm is d e n o t e d by (e j, q j), where ej E {0, 1} and q~ E R +. Note that the strategy of the u n i n f o r m e d firm is i n d e p e n d e n t of its cost because such a firm does not observe any private information prior to making its decision. Our objective is to derive the Bayesian Nash equilibria of the above game, and to conduct a comparative statics analysis to d e t e r m i n e how changes in the U
U
U
2This assumption can be relaxed to allow for correlation among costs of different firms. In this case when an informed firm observes its unit cost, it is able to draw some inferences about the costs of its rivals.
E. Gal-Or / Exit with incomplete information about costs
244
parameters affect the behavior of an informed versus an uninformed firm. We also wish to compare the equilibria with costless exit to the equilibria of an alternative game where exit is prohibitively expensive, as is implicitly assumed in many models with incomplete information about cost.
3. Market equilibria with costless exit An informed firm chooses its strategy to maximize its expected payoff V~, where
V~={Ec_i[[a-~qle~- ~ q~e~-cilq~lcil-F}e~. i=l
(2)
j=rn+l
In (2), the expected value is over the random variables that remain unobservable when firm i makes its exit and quantity decisions. These r a n d o m variables are the unit costs of other firms as summarized by the vector c_ i. The expected value for firm i is conditional on the realization of its own unit cost (ci). u An uniformed firm chooses its strategy to maximize its expected payoff V j , where
Vju = {E(cs_?[
a-
~-'~ q ~ e ~ - fi-' g=l
q jue ju- c j
] q ju- F } e ~ .
(3)
j=m+l
The expectation is with respect to the vector of unit costs, including firm j's that are unobservable to j when its decisions are made. Because the only difference among firms is whether they are informed, all equilibria are symmetric in that all firms of the same information state follow identical decision rules. For such equilibria, the exit decision of an informed firm is determined by a single cut-off point (see Appendix A). If the privately observed unit cost is below this cut-off point, the firm remains in the market. Otherwise, the firm quits the market. A n uniformed firm remains established in the market provided that equilibrium expected profits are non-negative. The above exit rules together with the linearity of d e m a n d imply that informed firms follow linear decision rules in choosing the quantity of output (see Appendix A). To summarize, our search for an equilibrium will consist of exit and quantity decision rules of the following form: ql=Ao
+Alcg,
el=l
if
ci<~c*,
if
c i > c* ,
= 0 u
qi=B0 u
ej = 1 =0
Ao, A 1 independent of c;,
B 0 i n d e p e n d e n t Of Cg, if
ETr u i>0
if
E7rU<0.
i=l,n,
(4)
i=l,n,
where ETr u denotes the equilibrium expected payoff of an uninformed firm.
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245
Proposition 1 derives the coefficients A0, A1, and B 0 and the cutoff point c* as a function of the p a r a m e t e r s a, m, n, and F and the distribution function G(.). The underlying assumption behind this derivation is that uninformed firms have non-negative expected profits and so find it optimal to produce, which assures that both uniformed and informed firms can coexist in the market.
Proposition 1.
(a) A t the equilibrium c*
2m-n-1 2
a+
f
cg(c) dc + (n - m)E(c)
I
0
qi=
2(n-m+l)+G(c*)(2m-n-1)
I
ei = 1 =0
if
C i ~ C* ,
if
Ci > C* ,
Ci
2 '
i=l,m,
c*
a(2 - G(c*)) - [2 + (m - 1)G(c*)]E(c) + u
u
eg(c) dc
0
qi = q = j
mf
2(n-
m + 1) + G ( c * ) ( 2 m -
n - 1)
=m+l,n,
e~U=eU = 1
if
q u ~>V~,
where c* satisfies the following equation: c*
-1 a + 2 m - n-2H(c*, a, m, n) =
f cg(c) de + (n - m ) E ( e ) 0
2(n-
m + 1) + G ( c * ) ( 2 m -
n - 1)
C~
(b ) This equilibrium satisfies the Routh Hurwitz condition for reaction functions stability 3 if -[2(n
- (2m - n - 1 ) g ( c * ) V ~ m + 1) + G ( c * ) ( 2 m - n - 1)
+1] 0
(c) The equilibrium is interior (0 < c* < ~) if 2(n - m + 1 ) V ~ -
(n - m ) E ( c ) < a < V ~ ( n
n+l + 1) + ~
_ c
n-1
- -2
E(c),
where the lower inequality is required for c* > 0 and the upper for c* < ~. 3The stability condition is the condition for the overall reaction mapping to be a contraction mapping and therefore implies uniqueness of equilibrium [see Friedman (1977, ch. 7)]. Alternatively, this condition allows direct application of the Gale-Nikaido global univalence result to prove uniqueness.
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E. Gal-Or / Exit with incomplete information about costs
Proof.
See Appendix B.
From Proposition 1, an informed firm of type c* is indifferent between producing q~(c*) or leaving the market, since q I ( c * ) = V'--F. An uninformed firm participates in the market if qU ~>X/-~. Hence, an equilibrium where uninformed firms coexist in the market with informed firms is guaranteed if qU _ qI(¢,) i> 0. From the proposition, the latter inequality holds if T(c*) =-- [a + (m - 1)E(c) - c ' m i l l - G(c*)] C*
G(c*)c* - I cg(c) dc + (n + 1)
c*-
0
2
-
E(c)
/>0
(5)
The first term of T(.) is a decreasing function of c*, and the second is an increasing function of c*. Hence, the sign of the derivative of T(.) is ambiguous. The value of the function at the extremes of the support of the distribution function is: T(0) = a - (n - m + 2)E(c) and T(() = (n + 1 ) ( ( - E(c))/2. As a result, T(.) is definitely positive for large 4 enough values of c*, but it may have an ambiguous sign for small values of c*. The above discussion implies that if the informed firms find it optimal to produce for high realizations of cost, the uninformed definitely will find it optimal to produce, since q"~> qI(c*) in this case. However, if the informed firms produce only for very favorable realizations of cost (c*" is small), the uninformed may not stay in the market. It is possible, however, that informed firms rarely produce while uninformed firms stay in business. In particular, if c * = 0 and a > ( n - m + 2)E(c), the uninformed are the only ones that produce. The last inequality is more likely to hold if nearly all firms are informed. When most firms are informed and have a low probability of producing output (e.g., when opportunity costs are relatively high and c* is small), the uninformed may be at a strategic advantage, in spite of the experimental advantage of the informed firms. Because in this case expected aggregate output is low, an uninformed firm can expand its production level and earn sufficient profits to recover its opportunity costs. Because we wish to characterize equilibria where informed firms coexist in the market with uninformed firms, we assume that inequality (5) holds, which guarantees that uniformed firms expect non-negative profits (ex ante) and so find it optimal to produce. The stability condition of part (b) of Proposition 1 guarantees the existence and uniqueness of the equilibrium. Substituting for X/-F from the function H(.) into this condition yields the result that reaction function stability holds provided that (2m- n-
1)g(c*)A o + (n - m + 1)
+ (2m - n - 1) [G(c*) > 0. 2 4Since T(.) is continuous and T(() > 0, the function is positive for large c values.
E. Gal-Or / Exit with incomplete information about costs
247
If the distribution G(.) is concave and the informed firms constitute more than fifty percent of the market (rn/> (n + 1)/2), this condition definitely holds, since A 0 > 0. Using Proposition 1, we can derive expected output and expected profit for each firm in the market. For the informed firm we derive ex ante expected output and expected profits, i.e., profits prior to the actual realization of its private information. Eq I = E(q~lno exit)prob(i does not exit) c*
a G ( c * ) + (n - m ) E ( c ) G ( c * ) - (n - m +
1)f
cg(c) dc
0
(6)
2(n - m + 1) + G ( c * ) ( 2 m - n - 1) C*
a(2 - G ( c * ) ) - [2 + (m - 1 ) G ( c * ) ] E ( c ) + qu =
mf
cg(c) dc
0
2 ( n - m + 1) + G ( c * ) ( 2 m -
E,ni
[EqI] 2 +
G(c*)
4
n-
1)
var(clc • c*) - F G ( c * ) ,
ETr u = [qU]2 _ F .
(7)
From (6) and (7) it is unclear whether an informed firm is better off than an uninformed firm. The ambiguity arises because the quantity of output produced by the uninformed firm may exceed the expected output produced by the informed firm. In Proposition 2, a comparison is made between those quantities o f output. ~'
Proposition 2. There exists a level o f f i x e d cost 0 < F * < Fmax, such that, if F < F*, then E q ~ >I qU and, if F > F*, then E q ~ < qU, where
Fmax Proof.
=
[a + (n - m ) E ( c ) ] 2 / 4 ( n - m + 1) 2 .
See Appendix B.
To guarantee the existence of an interior equilibrium at which c* > 0, that is, to guarantee that informed firms produce for some realizations of their private signals, the opportunity cost can never exceed Fmax. According to Proposition 2 if the opportunity costs are relatively low (less than F*) the informed firm produces greater expected quantities of output than the uninformed. However, if the opportunity costs are relatively high (more than F*, but less than Fmax) the uninformed firm produces more. The reason for this result is that, for high values of opportunity costs, the informed firm stays out of the market for a wider range of realizations of its private signal. Only if its unit cost is very low, does it decide to remain in the market. Under condition (5), the uninformed produces with certainty since it earns positive expected profits. The reduction in the probability of production by the informed firm reduces the quantity of output that it is likely to produce. From the
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E. Gal-Or / Exit with incomplete information about costs
derivation of expected profit in (7), and the above discussion it follows that the uninformed firm may be better off compared with the informed firm in our model. This may happen for relatively high values of opportunity costs (but not values so high that they eliminate the profitability of production for the uninformed). To demonstrate that the uninformed may indeed be better off than the informed, consider the case when F = Fmax, so that the informed firm chooses to stay out of the market (i.e, c* -- 0). Its profits are zero as a result. In addition assume that a > (n - m + 2)E(c). Substituting c* = 0 into the expression for qU yields that qU _- [a - E ( c ) ] / ( n - m + 1), which exceeds Fma x under the condition that a > ( n - m + 2)E(c). Hence the uninformed can expect strictly positive expected profits, and is better off than the informed. We summarize this finding in Observation 1. O b s e r v a t i o n 1. (a) If F<~ F*, where F* is defined in Proposition 2, then ETr ~>ETrU; and (b) If F > F* it is p o s s i b l e that ETr u > E T r ~.
We can further interpret the observation by noticing that being uninformed in our model provides a mechanism of commitment to production by the firm. Being informed provides the flexibility of exit if an unfavorable signal is observed. When the opportunity costs are relatively high, part (b) of Observation 1 indicates that being committed to production may be to the advantage of the firm. A n o t h e r interesting property of the equilibrium is that both the expected output and the expected profit of each firm depends u p o n the number of informed firms. We elaborate on this dependence when we conduct the comparative statics analysis. In addition, in contrast to the traditional results that opportunity costs do not affect the output decision of a prbfit-maximizing firm, in our model the expected output of both informed and uninformed firms depends upon the opportunity cost (through c*). In the next section, we demonstrate that when exit is prohibitively expensive, expected output is independent of the n u m b e r of informed firms and of opportunity costs. In tables 1 and 2 we summarize the results of a comparative statics analysis. In table 1 we indicate how changes in the parameters affect the behavior of the
Table 1 Comparative statics of the informed firm.
Number of firms - n Number of informed firms- m Opportunity costs-F Intercept of demand- a
Cutoff point for production c*
Expected output of informed Eq t.......
Expected profits of informed ETrI
(-) (-)
(-) (-)
(-) (-) if Eq I I> qU(F<~ F*)
(+)
(+)
(-)
(-)
(+) if Eq I < qU(E > E*) Ambiguous, but (-) for uniform distribution
(+)
(+)
(+)
E. Gal-Or / Exit with incomplete information about costs
249
informed firm, and in table 2 we indicate how it affects the uninformed firm. In Appendix C we derive the results reported in the tables. As shown in table 1, the informed firm is less likely to produce if the pool of firms increases, if opportunity costs increase, or if the intercept of the demand declines. If the number of informed firms rises, each informed firm is less likely to produce if the expected output produced by an informed firm exceeds that of an uninformed firm. It is more likely to produce if the expected output produced by the uninformed exceeds that of the informed: According to Proposition 2, for low levels of opportunity costs (F~< F*), increasing the number of informed firms reduces the likelihood that an informed firm will produce. For high opportunity costs ( F > F * ) , increasing the number of informed firms increases their likelihood of production. The expected output of each informed firm and its expected profits move in the same direction as the cutoff point c*. Hence, if informed firms are less likely to produce, they also produce lower expected output and earn lower expected profits. The only exception is that if opportunity costs increase, expected profit may increase as well, even though an informed firm is less likely to produce. To obtain the results stated in table 2, the equilibrium must satisfy the following regularity condition:
1
C(c*) 2
> 0
(s)
Notice that this condition does not follow from the second-order condition or the stability condition. From Proposition 1 the stability condition can be written as
2m g(c*)V'-F +
-
2
+ (n + 1) 1 - g(c*)X/-F
-
G(c*) 2
>0.
!
Hence, the regularity condition coincides with it only when m = 0, namely only if all the firms are uninformed. In general, an equilibrium satisfying the stability condition may exist without satisfying the regularity condition. This happens, for instance, when V ' F > [ 1 - G ( c * ) / 2 ] / g ( c * ) and m is sufficiently large. Specifically, if opportunity costs are sufficiently high so that informed firms are less likely to produce, and if the number of informed firms is large enough, the regularity condition may fail. Under such circumstances the absence of information may provide a strategic advantage to the uninformed firm. Table 2 Comparative statics of the uninformed firms. Regularity condition holds
N u m b e r of firms - n N u m b e r of informed f i r m s - m
Opportunity c o s t s - F Intercept of demand - a
Regularity condition does not hold
output - q"
profits E ~ -u
output - qU
profits - ETr u
(-)
(-)
(+)
(+)
(-)
(-)
(+)
(+)
(+)
(-)
(+)
ambiguous
(+)
( + ) if E q I >I q"( F < F* ) ( - ) if E q l < qU( F > F* ) ambiguous
(+)
(+)
(-)
(-)
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E. Gal-Or / Exit with incomplete information about costs
According to table 2 only a change in opportunity costs has an unambiguous effect upon the uninformed firm. Regardless of whether the regularity condition holds, an uninformed firm produces more in response to an increase in opportunity costs. With higher opportunity costs informed firms are more likely to exit, thereby reducing competitive pressures on uninformed firms. Changes in all the remaining parameters have an ambiguous effect on the uninformed. However, with the regularity condition, results are obtained which are more in line with traditional microeconomic theory. Under (8) the uninformed firm produces less if more firms are established in the market, if more firms have access to private information while the opportunity cost is relatively low (F <~ F*), and if the value of the intercept of the demand is smaller. If the regularity condition does not hold the above predictions are reversed. For instance, the uninformed firm produces more if the number of established firms increases. As pointed out above, the regularity condition fails when the absence of information provides a significant strategic advangage to the uninformed. If the total number of established firms increases in this case, informed firms have a reduced likelihood of production. The uninformed firm can take advantage of this by increasing production. Hence, whenever the absence of information provides a strategic advantage to uninformed firms, their response to changes in the parameters of the model is opposite to the response of informed firms. Next, we can use the computations of Appendix C to predict how changes in the parameters affect expected aggregate output. Expected aggregate output is given by EQ = mEq I + (n - m)q u .
In table 3 we summarize comparative statics results concerning aggregate production. From table 3 it is possible that an increase in the number of established firms will cause aggregate production to d e c l i n e (if the regularity condition does not hold). This result can be explained as follows. When the number of established firms is larger, each informed firm is more likely to exit. Hence even though there is more potential competition (larger n), actual competition, as measured by ( n - m + m G ( c * ) ) , may decline. As a result, expected aggregate output may decline as well. It is worthwhile to elaborate on how a change in the amount of information affects the market outcome. Based upon table 3 an increase in the number of informed firms may result in a d e c l i n e of aggregate production (under the regularity condition when F > F*, and if the regularity
Table 3 Comparative statics of aggregate output.
Number of firms- n Number of informed firms- m Opportunity costs - F Intercept of demand - a
Regularity condition holds
Regularity condition does not hold
(+) (+) if F ~< F* (-) if F > F * (-) (+)
(-) (-) if F ~< F* (+) if F > F * (-) ambiguous
E. Gal-Or / Exit with incomplete information about costs
251
condition does not hold when F ~< F*). Such a result implies that consumers may prefer that firms be less informed about their technologies. With less private information available to firms, expected output may increase and prices may decline.
4. Market equilibria when exit is prohibitively expensive When exit is prohibitively expensive and firms are already established in the market, both types of firms are committed to production. If both types of firms invest in the industry, the unconditional expected profits of each type has to be non-negative. The strategy of an informed firm is ql(ci) and of an uninformed is q~. In Proposition 3 we state the properties of the equilibrium with no exit. Proposition 3.
n-1 a + ---5-
I
qi = tl
q/
A t the equilibrium with no exit 5
--
E(c)
Ci
2 '
n+l a - E(c) n+l '
(9)
j=m+l,n,
ETrI = [ a - E ( c ) ] 2 va4(c) nT1 + ETrU = [ a - E ( c ) ] n+l
i = 1, m ,
F,
(10)
2 -F.
Both types of firms are in the market if a > (n + 1)X/F + E(c). The proof of Proposition 3 is in Appendix B. The properties of the equilibrium with no exit are completely different from the properties obtained in the previous section. In particular, output decisions are independent of the number of informed firms and of opportunity costs. In addition, the expected output of an informed firm coincides with the quantity of output produced by an uninformed firm (i.e., Eq I = qU). Expected aggregate output is a strictly increasing function of n and a, and is independent of m or F. When exit is costly, being informed always guarantees higher profits than remaining uninformed. The properties of the equilibria with prohibitively costly exit have been established in the literature [see Li (1985), Gal-Or (1986), and Shapiro (1986)]. Especially, it has been demonstrated that the expected profits of a firm are a strictly increasing function of the precision of its private information. This last property may fail if exit is costless, as demonstrated in the previous section. Comparing eqs. (6) and (9) demonstrates that expected aggregate output with costless exit may be higher or lower than the expected aggregate output produced with costly exit, depending upon opportunity costs. More explicitly, if E Q e and E Q ne denote the expected aggregate output produced with free and costly exit, respectively, then 5An additional condition that [a + ((n - 1)/2)E(c) - ((n + 1)/2)~] > 0 is required to guarantee that an informed firm always produces positive amounts of output [i.e., qe(?)> 0]. This condition implies that zero production is never profitable once the firm is already established in the market.
E. Gal-Or / Exit with incomplete information about costs
252
E Q e - E Q ne = m[Eq' - qU], where Eq ~ and qU are given by (6). From Proposition 2 it follows that for relatively low levels of F(<~F*), aggregate output is higher if firms can freely exit the market. If, however, opportunity costs are relatively high (F > F*), the argument is reversed. Since consumer surplus is directly related to expected aggregate output, it follows that consumers prefer the first regime with free exit if F is low and the second regime with no exit if F is high. It is noteworthy that informed firms may be better off if exit is extremely expensive, namely, if they have less flexibility after observing their private signals. This conclusion follows by comparing eqs. (7) and (10), which gives the expected profits of the informed firm under both regimes. To demonstrate that reduced flexibility may indeed be advantageous, suppose that F = Fmax. With maximal opportunity costs and free exit, the informed firm stays out of the market for every possible realization of its unit cost ( c * = 0), and earns zero profits. When exit is impossible (or extremely expensive), but
a _- E(__c)] /'/ q- i
(11)
> Fmax,
the informed firm does not exit and earns positive expected profits. Inequality (11) holds, for instance, if a>
E(c)[(n + 3 ) ( n - m) + 2] ( n - 2 m + 1)
and
m<(n+l)/2.
(12)
The fact that informed firms prefer the no-exit regime when opportunity costs are large is not surprising. The basic point is that the possibility of exit generates a strategic asymmetry, and the class of firms which lose out would prefer to eliminate the possibility of exit. A similar result is also derived by Judd (1985), who demonstrates that exit barriers may be beneficial for the purpose of entry deterrence.
5. Endogenous acquisition of information Under both regimes, we have assumed that the identity of the informed firms is exogenously given. If we extend the model to incorporate the decision whether to become informed, different results are obtained, depending upon the ability of firms to exit. If exit is extremely,.e~xpensive, and each firm incurs the same cost to acquire its information, either all the firms will purchase the information or all will remain uninformed, depending upon the cost of information. More specifically, if K denotes the cost of information, all firms will purchase information (m = n) if K < var(c)/4. No firm will purchase information (m = 0) if K>var(c)/4. If exit is costless, it is possible that only some firms will become informed (i.e., 0 < m < n), even if each firm incurs the same cost of information. More specifically, the first regime may predict asymmetric informational equilibria even if all firms are a priori identical. At this asymmetric equilibrium some firms purchase information and others remain uninformed.
E. Gal-Or / Exit with incomplete information about costs
253
in orocr to o c m o n s t r a t c ti]at costicss exit may predict informatlonai ab~lllmetries, consider the case in which F = 0 and unit cost is uniformly distributed over the inverval [0, ~?]. For this case we derive a sufficient condition to guarantee that the benefit of becoming informed declines when more firms are already informed. More explicitly, we derive a condition that guarantees that a ( E r r I - E r r u) cgm
<0.
The condition is stated in Observation 2.
Observation 2.
If F = 0, g(c) is uniform and exit is costless, then
a(E r' - Err u) <0 3m
if
a<
(n + 11) 18
C.
The proof of Observation 2 is in Appendix B. According to the observation, the advantage of an informed firm over an u n i n f o r m e d firm declines when m o r e firms are informed, which m a y result in an equilibrium with only a subset of informed firms. M o r e explicitly, denote by E~r~m and E~r~ the expected equilibrium payoffs when m firms purchase information. A t such an equilibrium an informed firm would not benefit if it were to deviate unilaterally and to remain uninformed, and no uninformed firm can benefit from purchasing information. If K is the cost of purchasing information, then Errlm - K ~> Err~m_ 1 ErrS,/>ETrIm+l - K . The first inequality guarantees that an informed firm cannot benefit if it unilaterally decides to remain uninformed. The second inequality guarantees that an uniformed firm cannot benefit if it purchases information. Both inequalities can hold simultaneously if the sequence (ETr~m-E1rU_t) is a decreasing sequence of m. In particular, if [E,n-In - E'rrU_l] < K a n d [ E T r l l ETr0] > K, it is possible that only a subset of firms will purchase information even if all firms face the same cost to purchase it. The sequence [ETr~mETrU 1] can be d e c o m p o s e d as follows: ETr~, - E~',~ 1 = [ETr~ - ETr~,] + [ETr u _ ETrm_l]u .
(13)
U n d e r the condition in Observation 2, the first term is a decreasing sequence of m, and the second t e r m may be either an increasing or decreasing sequence of m. However, it is possible further to restrict the parameters of the problem in order to guarantee that the monotonicity of the first term in (13) is always dominant, so that the sequence [ETr~m-Err~m_l] is a m o n o t o n e sequence. Hence, when exit is free, it is possible that informed firms coexist in the market with uninformed firms, even if the cost of purchasing information is identical. More explicitly, given the option to purchase information, some firms may decide to purchase it, and others may decide to remain uninformed. The informed firms may subsequently decide to exit, depending upon their private
254
E. Gal-Or / Exit with incomplete information about costs
information. The uninformed stay in the market as long as their expected profits are positive.
6. Concluding remarks Incorporating the possibility of non-production in a model where technology is stochastic and a firm may be privately informed about its unit cost leads to equilibria that differ from the equilibrium when non-production is not possible. In particular, expected aggregate output depends u p o n the opportunity costs of staying in the market and the number of informed firms. Being informed may be disadvantageous to the firm. As a result, when firms endogenously select whether to purchase information at a cost, asymmetric informational equilibria may arise, even if initially all firms are identical. Even though our focus has been on exit, the results can be interpreted differently by considering a new industry where potential investors have to make entry decisions. If some potential investors have access to private information about their unit costs prior to entry, while the remaining firms are uninformed, our analysis and results are still valid. With the entry interpretation each firm decides whether to enter, and how much to produce if entry occurs. According to our derivation several conclusions about the entering firms can be stated. For instance, a potential investor may prefer to be uninformed. Increasing the pool of potential investors may result in a decline in the expected aggregate quantity produced, since informed investors stay out of the market for a larger range of realizations of their unit cost. In the paper we have only considered uncertainty about the technology. It is also possible to introduce uncertainty about d e m a n d without significantly affecting the analysis or results. We have restricted consideration to a linear demand function and a stochastic technology that exhibits constant returns to scale. These assumptions are standard in the literature, but most welcome generalization would be to consider different d e m a n d or cost functions. Another direction in which this study can be extended is to consider the possibility of information pooling among informed firms. In previous models, where 'non-production' is not explicitly incorporated, Cournot oligopolists have an incentive to share information about private costs. A n interesting question is whether similar incentives exist when non-production is explicitly modelled.
Appendix A.
Equilibrium decision rules
In this appendix, we demonstrate that an exit decision rule with a single cutoff point and a linear quantity decision rule are the only decision rules consistent with an equilibrium.
Lemma 1. For an identical exit decision rule used by each informed firm the quantity of output of an informed firm is a linear function of its unit cost. Proof. Let firm i be an informed firm. Suppose that each informed rival of firm i follows an identical exit decision rule of the form
E. Gal-Or / Exit with incomplete information about costs 1
ej=l
if
cj~CC[O,d],
=0
if
cjf~C.
255
Denote by G the probability that a single informed rival of i remains in the market; then
G = f g(c) d c . C
Notice that G is independent of the unit cost of i, by the independence assumption made concerning the unit cost of different firms. The payoff function of i is VIgil 1 = ei -
{a-q~-ci-Ec-i
+ j=m+l
qIeI
u
(A1)
qjejj~qi-F'
k#i
where at the symmetric equilibrium Ec . 2.,
qUjeUj= (n - m ) q u
and
(A2)
-~ j = m + l
Ec ~ k=l
q kI e kI
.~_
k#i
=
prob K i n f o r m e d r i v a l s } I kEcj(qjlj k=0 / remain established remains established) m- I ~
E
m,( ) Gk(1 ~, m - 1 k=O
(A3) - G)
m
1 kk E c ~ ( q j Il j
k
remains established),
any j = 1, m ;
j#i.
Eq. (A2) holds since we restrict consideration to circumstances where production is profitable for the uninformed. Eq. (A3) holds, since the number of informed rivals that continue production has a binomial distribution with parameters (G, m - 1). Hence, Ec
1
q kI e ,i = ( m _ l ) G E ( q i j l
] produces) a n y j = l , m " ,
j#i
•
(A4)
k=l k#i
Differentiating (A1) with respect to qi and substituting (A2) and (A4) into the first-order condition yields I a(n - m ) q u - ( m - 1 ) G E ( q I j I j produces) qi= 2
any i , j = l , n
j#i.
Ci
2'
(A5)
The first term of (A5) is independent of c;, since c i is independent of cj for i # j . Hence the linearity follows. Q.E.D.
E. Gal-Or / Exit with incomplete information about costs
256
L e m m a 2. The exit decision rule o f an informed f i r m consists o f a single cutoff point, such that if c i <<-c* e~ = 1, and if c i > c* e I = O. Proof. From (A1) and (A5) if follows that at the equilibrium, if informed firm i continues to produce its profits are
Vt
= (q~)2_ F ,
where q~ is given by (A5).
V] is a strictly decreasing function of c i. Hence the lemma follows.
Appendix B.
Proofs of Propositions 1-3 and Observation 2
From (2) if the ith informed firm decides to produce,
P r o o f o f Proposition 1. its payoff is V~
Q.E.D.
= a-ql-ci-E~
qke~+
i leli - I
k=l
qjej
qi-F.
(B1)
j=m+l
k#i
In (B1) we have used the assumption that c i and cj are independently distributed for i ~ j . Notice that the expectation included in (B1) corresponds to the expected aggregate output produced by the rivals of i. Since we consider equilibria where uninformed firms coexist in the market with informed firms, e~ = 1 for every j. From (4) therefore, the aggregate output produced by the uninformed rivals of i is (n - m ) B o. Hence (B2)
q~eU~ = (n - m ) B o . /=m+l
Also, since each informed firm follows an exit decision that consists of a single cutoff point c*, the number of informed rivals of i that remain in the market is distributed according to a binomial distribution with parameters (G(c*), m 1), where G(c*) is the probability of a 'success' (informed rival produces) and (m - 1) is the maximum possible number of 'successes'. If we denote by EQ~_; the expected aggregate output produced by informed rivals of i then m--1
EQI_i = ~'~ E ( Q I i l k informed rivals produce)prob(k informed rivals k=0
produce), = 2
k[Ao+A1E(cilci<~c *
m
1
a(c,)l,[l_C(c,)r
k=O £*
f cg(c) dc =
A ° + A t 0 G(c*)
(m - 1)G(c*)
where we have used the property that the expected value of a binomial distribution function with parameters ( m - 1) and G is equal to ( m - 1)G.
E. Gal-Or / Exit with incomplete information about costs
257
Hence, C*
f cg(c) dc E¢_ i
qkek = k=l k#i
A ° + A1
0
(m - 1)G(c*).
G(c*)
(B3)
Differentiating (B1) with respect to q~ and substituting (B2), (B3) and the suggested decision rule (4) into the derivative, yields the following first-order condition for an informed firm: £*
f cg(c)dc a - 2A 0 - 2AlC i - c i -
(n
-
m)B
o =
0
A0+A
1
(m - 1)G(c*)
G(c*)
(B4)
O.
First-order condition (B4) should hold for every possible realization of c i E [0, ~]. Hence it follows that A 1 = - 1 / 2 . From (3), if the jth uninformed firm decides to produce, its payoff is V?Jl~; ,=
u
I I
u
u
a - qj - E ( c j ) - E c JLi=I q~e~ + k=m+1 qkek
u
qj-F.
(B5)
k~j
The expected output produced by the uninformed rivals of j is (n - m - 1)B 0, if we assume that uninformed expect non-negative profits. Hence
Ec
(B6)
qkeku u = (n - m - 1)B 0 I k=m+l k#j
The expected output produced by informed rivals of j is derived similarly to the derivation of (B3). More specifically, let EQ~_j denote this expected output; then
f cg(c) dc EQ'_]
=
k=O ~
[G(c*)lk[1 -
G(c*)] m-~ k
A o+ A 1
G(c*)
Hence,
[ c; l cg(c) dc
Ec ~ ~'Z., [ i -
i=1
qiei :
A° + A1
0
G---~i
mG(c*) .
(B7)
The difference between (B3) and (B7) is that the informed firm has (m - 1) informed rivals and the uninformed has m informed rivals. Differentiating the payoff function of (B5) with respect to q~ and substituting in the derivative (B6), (B7) and the assumed decision rule (4), yields the first-order condition
E. Gal-Or / Exit with incomplete information about costs
258
for the uninformed as follows: c*
f cg(c) dc a - 2B 0 - E ( c ) - (n - m -
1)B 0 -
Ao+
A 1
o
G(c*)
mG(c*) = 0 .
(B8)
Substituting into the first-order conditions (B4) and (B8) A t = - 1 / 2 yields a system of two equations in the two unknowns (A 0 and B0) as follows: ¢*
lf a - A012 + ( m - 1 ) G ( c * ) ] - ( n - m ) B o + -m --"~
cg(c) dc = 0 , o
c*
mf
a - A o m G ( c * ) - (n - m + 1)B 0 + ~
(B9)
cg(c) d c - E(c) = 0 .
0
The unique solution of (B9) is ¢*
a+ A0=
2m-n-1 2
f
cg(c) dc + (n - m)E(c)
0
2 ( n - m + 1) + G ( c * ) ( 2 m -
n - 1)
' (B10)
C*
a[2 - G(c*)] + m f cg(c) dc - [2 + (m - 1)G(c*)]E(c) 0 n 0 :-
2(n - m + 1) + G ( c * ) ( 2 m -
n - 1)
From the first-order conditions it follows that at the equilibrium VI
= [ql] 2 - F
VU
= [qUj]2 _ F .
and
Hence production is profitable for the uninformed if q~ I> X/F and for the informed if q~/> X/F. In consequence of the derived values of the constants A 0 and A1, the informed should remain established in the market if c*
a + 2 m - n2- 1 I
q i =-
"
""
f cg(c) dc + (n - m)E(c) 0
2(n - m + 1) + G(c*)(2m - n - 1)
- 1/2c~ >t X / F .
(Bll)
Since ql is a strictly decreasing function of c i, there may exist a point c* such that, if it is exceeded by the unit cost, inequality ( B l l ) is violated. In particular, at a symmetric interior equilibrium 0 < c *i < ( a n d c *i = c * f o r e v e r y i = 1, m. Hence the cutoff point is determined by the equation H(c*, a, m, n) = O.
E. Gal-Or / Exit with incomplete information about costs
259
To guarantee the uniqueness of such a point we need that He. < 0. This condition is also required for stability (own cost effects on output exceed cross cost effects on output.) Requiring He. < 0 yields part (b) of the proposition. To guarantee that the equilibrium is interior ( 0 < c * < ( ) , we need that H ( ( , a, m, n ) < 0 and H(0, a, m, n ) > 0. These two inequalities imply part (c) of the proposition. Q.E.D.
P r o o f o f Proposition 2.
F r o m (16) £*
2a(1 - G ( c * ) ) - [2 + (n - 1 ) G ( c * ) ] E ( c ) + (n + 1) f cg(c) dc u
q -Eq I =
0
2(n - m + 1) + G ( c * ) ( 2 m - n - 1) ¢* t*
= 2(1 - G ( c * ) ) A o + ] cg(c) d e -
E(c),
0
where A 0 is given from Proposition 1. Based upon the same proposition A o = 1 / 2 c * + V ~ , thus c
M ( V ~ ) - q" - E q ~ = c*(1 - G ( c * ) ) + 2X/-F(1 - G ( c * ) ) - ~ cg(c) d e , ¢*
c g~
M(O) = c*(1 - G ( c * ) ) - t cg(c) de < O. c* i
F r o m the definition of the function H in Proposition 1 the value of Fma x is the lowest possible level of opportunity cost for which an informed firm exits for every possible realization of its unit cost, n a m e l y c * = O. Substituting the value of Fmax into M ( V ~ ) yields M(V~max) = a + (n - m ) E ( c ) _ E(c) - [a - E(c)] > 0 n-m+l n-m+l '
where the last inequality follows from the assumption that a > E(c). dM dc* dX/-p - 2(1 - G ( c * ) ) + [(1 - G ( c * ) ) - 2X/Fg(c*)] dX/-F" F r o m part (b) of Proposition 1, de*
1
/4,. [2(n - m + 1) + G ( c * ) ( 2 m - n - 1)]
G(¢*)
[(2m - n - 1 ) g ( c * ) V ~ + (n - m + 1) + - 2 <0.
(2m - n - 1)]
E. Gal-Or / Exit with incomplete information about costs
260
Substituting for dc*/dX/F into d M / d X / - F yields dM
2 X / F g ( c * ) ( n + 1)
dVF
(2m - n - 1)g(c*)X/-F + (n - m + 1) + G(c*_____~)(2m - n - 1)
Hence by the mean value theorem the proposition follows.
>0.
Q.E.D.
P r o o f o f Proposition 3.
The informed firm follows a linear decision rule for determining its quantity of output. Thus, I
qi = Ao + A l C i ,
i = 1, m ,
(B12) u
qj=Bo,
j=m+l,n.
Differentiating the objective function of the ith informed yields
[a- 2ql- 2 qI_ 2 quk-ci]= 0 , k=l k~i
(B13)
k=m+l
Differentiating the objective function of the jth uninformed yields a - 2qj -
qk -- E(C) = O.
qk -k=l
(B14)
k=m+l k ,~j
Substituting the suggested solution (B12) into (B13) and (B14) and requiring that (B13) holds for every possible realization of c i E [0, d] yields that a 0=
n-1 a+~E(c)
]
/(n+l),
B o = [ a - E(c)]/(n + 1), A 1= -1/2. Thus (9) follows. From the first-order conditions Errl = E ( q l ) 2 - F , u
Evrj=[q~]Z- F. ..
,.-
the above result yields (10). Requiring that unconditional expected profits are non-negative yields the condition for entry by both types of firms. Q.E.D. Using
P r o o f o f Observation 2.
From Appendix C,
(E~r x - E ~ -u)
(qU _ E q I )
Om
X
{(qU _ Eqi)
+ qU[1 - G ( c * ) - 2 g ( c * ) X / F ] } .
E. Gal-Or / Exit with incomplete information about costs
261
F r o m Proposition 2 E q ~ > qU when F = 0, thus the above derivative is negative if qU< E q I - q
u
1 - G(c*)
(B15)
"
From the derivations in the proof of Proposition 2, 5
qlU=o =
Eq'-
f cg(c) d c -
c*(1 -
G(c*))
>0
c*
and from Proposition 1, c*
' o= EqlF=
½c*G(c*)- i f cg(c)dc. 0
H e n c e (B15) holds provided that 5
u f cg(c) dc q < 1 =--G(-~)
c*.
(B16)
C*
However, qU< Eqi, which implies that (B16) holds if c*
c
1 7-~(-~) 0
c*.
c*
For a uniform distribution the above inequality is satisfied if c .2 < 2E(E - c*) which certainly holds if c * < 2 / 3 5 . F r o m the comparative statics results, 8c*/Orn < 0 if F = 0. H e n c e if c* < 2 / 3 5 w h e n m = 1, c* < 2 / 3 ( for any value of 0 < m ~ < n . F r o m Proposition 1 c* satisfies the equation H(c*, rn, n, a) = 0. Since He, < 0, if H ( 2 / 3 5 , m, n, a ) < 0 it follows that c* < 2 / 3 ( . Evaluating the function H at c* = 2 / 3 ( and m = 1 yields n+ll _ c 18 n)lc.=2/3e = 2n - 2 / 3 ( n - 1) ' m=l a
H(c*, a, m,
which is negative u n d e r the condition stated in the observation.
Appendix C. Comparative statics Let
C(c*)
X = ( 2 m - n - 1 ) g ( c * ) V ~ + ( n - m + 1) + - then X > 0 by the r e q u i r e m e n t that He. < 0.
(2m - n - 1 ) ,
Q.E.D.
262
E. Gal-Or / Exit with incomplete information about costs c~C*
On = - q U / X < 0 ' c9C*
Om = ( q U - E q i ) / x < ~ O >0 OC*
ov-p c)C*
Oa
-
1) +
[2(n - m +
for F < - F *, for F > F * , G(c*)(2m - n -
1)l/X<0,
- l/X>0,
c)Eq ~ _ [G(c*)__ ] Oc* Ot 2 +g(c*)V~ --~ c)Eq~ - - 2 g ( c * ) V ~ ( n - m +
OE~r I Oc*
0 E ' a -I Ot
Oc*
Ot
-
where
t=a,m,n,
1)/X
EqI Oc* at
t=a,m,n
o~ETri
c/
- X/F[(2m - n - 1)g(c*)c*G(c*) - (2m - n - 1)g(c*)
cg(c) d c , 0
m + 1 ) - GZ(c*)(2m - n -
-2G(c*)(nOqU Oa
=[1
G(c*)
1)]/X,
]
2
g(c*)X/-F / X ,
dq u - q U [ g ( c , ) V ~ _ ( 1 OqU _(qU [ Om -Eq') 1
G~*))]/X G(c*) 2
] g(c*)C-F / X ,
OqU - 2 V ~ g ( c * ) m / X > 0 OV'-F
0ETrU OqU Ot - 2qu Ot
where
0 E ~ r U - 2V'F[ 2qUg(c*)m aX/-F X
t = a, n , m
1] "
References Dixit, A. and C. Shapiro, 1986, Entry dynamics with mixed strategies, in: Lacy Glen Thomas, ed., The economics of strategic planning, Essays in honor of Joel Dean (Lexington Books, Lexington, MA). Fine, C. and L. Li, 1986, A stochastic theory of exit and stopping time equilibria, Working paper (Sloan School of Management, MIT, Cambridge, MA). Friedman, J., 1977, Oligopoly and the theory of games (North-Holland, Amsterdam). Fudenberg, D. and J. Tirole, 1986a, A theory of exit in duopoly, Econometrica 54, 943-960. Frudenberg, D. and J. Tirole, 1986b, A signal jamming theory of predation, Rand Journal of Economics 17, 366-376.
E. Gal-Or / Exit with incomplete information about costs
263
Gal-Or, E., 1986, Information transmission-Cournot and Bertrand equilibria, Review of Economic Studies 53, 85-92. Gal-Or, E., 1987, First mover disadvantages with private information, Review of Economic Studies 54, 279-292. Ghemawat, P. and B. Nalebuff, 1985, Exit, Rand Journal of Economics 16, 184-194. Judd, K., 1985, Credible spatial preemption, Rand Journal of Economics 16, 153-166. Li, L., 1985, Cournot oligopoly with information sharing, Rand Journal of Economics 16, 521-536. Nikaido, H., 1968, Convex structures and economic theory (Academic Press, New York). Riordan, M., 1985, Imperfect information and dynamic conjectural variations, Rand Journal of Economics 16, 41-50. Shapiro, C., 1986, Exchange of cost information in oligopoly, Review of Economic Studies 53, 433-446.