Market uncertainty and competitive equilibrium entry

European Economic Review 35 (1991) 39-48. North-Holland

Market uncertainty equilibrium entry*

and competitive

Elie Appelbaum York Universiiy.

Downsview.

Ontario, Canada M3J IP3

Chin Lim National

University

of Singapore, Singapore 0511

Received November 1988, final version received February 1990

Firms facing an uncertain market environment can choose the optimal mix of early and late production by trading off informational advantages of late production against cost advantages of early production. Consequently, their choice of entry into the market, whether early and/or late, depends on the interplay of cost and stochastic demand conditions. This paper characterizes the competitive equilibrium conditions that determine the mode of market entry and derives comparative statics results to show how competitive equilibrium entry modes vary with changes in various market conditions.

1. Introduction

In most studies dealing with the competitive firm facing market uncertainty, the firm’s behavioral mode is assumed to be exogenously determined. For instance, the well-known models of Sandmo (1971), Tisdell (1963) and Baron (1970) assume that the firm makes only early (ex ante) decisions, whereas the models of Oi (1961), Dreze and Gabszecwicz (1967), Sheshinski and Dreze (1976), and Lippman and McCall (1981) assume that the firm makes only late (ex post) decisions. Turnovsky’s (1973) model does allow for both behavioral modes, but the analysis is limited to the firm leoel and is not aimed at studying the properties of equilibrium. The purpose of this paper is to synthesize these two types of models and provide a framework within which the firm’s behavioural mode is determined endogenously and is consistent with market equilibrium. We show that, in general, uncertainty introduces a trade-off between cost and informational *We wish to thank the editor and the two referees of this journal suggestions. 00142921/91/$03.50 0 1991-Elsevier

for their valuable

Science Publishers B.V. (North-Holland)

E. Appelbaum and C. L.im, Market uncertainty and entry

40

advantages, thus, determining the firm’s choice of behavioural mode. Given firms’ behaviour, we characterize industry equilibrium and discuss its entry patterns. Assuming rational expectations, the model explains the equilibrium modes of behaviour, market shares of early and late firms, equilibrium price expectations and distribution, and probability of late entry. We also show that the equilibrium is efficient and examine the effects of cost and demand conditions.

2. The firm 2.1. Demand and cost conditions Consider an industry whose demand function is uncertain and characterized by

where Q is aggregate output demand, p is price, and 8 E [@,fl is a continuous random variable with probability distribution $((8;k’,k2) where k’ and k2 are distribution shifting parameters.’ The market is assumed to be perfectly competitive; firms are atomistic and there is free entry and exit. At the firm level, the equilibrium price facing each firm is p(0) whose probability distribution depends on &6) and the various demand and cost parameters of the industry. Each firm may start its production process early (before 8 is revealed), late (after 0 is revealed), or it may engage in both early and late production processes. Let i=(O, 1) denote the early and late processes respectively. For any given level of output, q, we define the costs of the two production modes as

g’(q)= s’(q)+ c’(q),

(2)

where the set-up costs, si, satisfy

s’(q)=

s’>O

I

o

if if

q>O q=O,

and the variable costs, ci, are parameter&d

(3) to be given by

c’(q) = a’c(q), ‘For the sake of notational parameters are not discussed.

(4) brevity, we suppress

k’ and kZ and write 4(e), whenever

these

E. Appelbaum and C. Lim, Market uncertainty and entry

41

and are assumed to be increasing and convex in q, with a’>0 and c(0) =O. In general, the late production mode has to meet demand faster and will involve a tighter time constraint. Hence, it tends to be more costly relative to the early production mode. Thus, we take

s’zs”

and

a1 Lao.

(5)

The difference in set-up costs, s1 -so, may result from the greater organizational difficulty in implementing a speedier production mode; and the difference in variable costs, a’ -a’, may stem from higher overtime rates, or other operating costs connected with a speedier production mode.2 We can interpret the two production modes as two types of plants and the costs functions are, therefore, plant cost functions.

2.2. Entry and output decisions Each firm can choose the early, the late, or both production modes. A firm that chooses the early mode is said to enter the market early, whereas a lirm that chooses the late mode is said to enter late. It is of course possible for a given firm to enter both early and late. Let y and z be the early and late output levels. Assuming risk neutrality, each firm solves the problem:

maxE YZO

maxp(Q(y+z)-go(y)-g’(z)

(6)

Z&O

where levels y and z are chosen before and after the revelation of 8, respectively. This problem is solved in two stages. In the second stage, given y and after the observation of p(8), the firm chooses its ex post adjustment, z. Its choice of y is made in the first stage, conditional on the ex post adjustment and the expected price. Consider the choice of z. Let J’(p(f3)) =maxZZo p(8)z- a’c(z) be the variable profit function. If J’ Bsr, ex-post production will be undertaken, otherwise it will not. Thus, defining p1 as the cutoff price such that

.l’(p’) =sl

or

p’ = min [a’c(z) + sl]/z,

(7)

‘For example, if the production possibilities set is defined by input/output time distributions (an ‘Austrian technology’, see Appelbaum and Harris (1977)), then a tighter time constraint may imply that input/output time distributions cannot be chosen optimally, hence leading to higher costs. This is shown in Appelbaum and Lim (1985).

42

E. Appelbaum and C. Lim, Market uncertainty and entry

then optimal z (by Hotelling’s Lemma) is

z(6)=

AJ’(j$0))/ap 0

1

if p(e) 2~’ if p(@
(8)

The actual profits from late production are, therefore,

d = max [P(p(e)) - 2, 01,

(9)

so that the expected profits from late production are given by

(10) where 8’ is the cutoff state (corresponding to the cutoff price; p’=p(O’)), below which no ex post production occurs. Consider now the choice of y. Substituting z(6) into (6), the first stage problem is max YZO

q-pfe)ly -

a”c(y)

-so(y)

+ E[a’].

(11)

To characterize the solution, define p= Ecp(e)] as the expected price; Jo(p) =max py - a’c(y) as the variable expected profit; and E[rr’] 3 Jo(b) -so as the expected profit of early production. Then, y > 0, if E[a”] 2 0; otherwise, y = 0. The cutoff expected price for early production is, therefore, p” such that J”(po)

= so

or

p” =

min [aOc(y)

+ sol/y,

(12)

and the optimal y is

(13) The solution {y”,z(@}, characterized by (13) and (8), depends on the parameters of the model. In general, a firm can engage in either one or both modes of production. To highlight the advantages and disadvantages of the two production modes, we write the difference in the expected profits (after some manipulation) as,

E[n”] - E[n’] = [JO(p)- EIJO(p)-J] +

[

“s’ [J’(p) -21 d4(B) 1 e

E. Appelbaum and C. Lim, Market uncertainty and entry

+

43

[E[JO(p)-J’(p)] + [s’ -SO]].

The first two terms in (14) capture the informational advantages of late entry. The convexity of J implies that Jo@) SJZ[J’[p(0)]], indicating the advantages of flexibility. In addition, whereas early entry may yield a loss, late entry occurs only if profits are actually obtained, J’cp(@] -sl 50 for all OSO1. The second two terms in (14) capture the cost advantages of early entry; s1 2 so, and since tlr 2 a’, we have Jo(p) -J’(p) 2 0 for all prices. There is therefore a trade-off between the informational and cost advantages of the two processes, and the choice between them depends on the parameters of the model. By allowing the firm to choose from the two processes, our model provides a generalization of previous models, and therefore fills a gap in the literature. We conclude this section by noting that entry is also affected by the firm’s subjective price distribution. In the following section, the firm’s expectations are taken to be consistent with the equilibrium price distribution, so that it, indeed, has rational expectations. 3. Equilibrium entry For exposition, we simplify the analysis by linearizing the demand function (1) as

mm=e-kb, Equilibrium satisfies

p-0.

is defined by the vector {m, y*, n(e), z*(8), p(O), EM@]}

(9

8 - pp =

(ii)

E[?P] 5 0,

(iii)

d(e) go

my*

(14 which

+ n(e)z*(e), (15)

for all

8,

where y*= y”, z*(e) =z(e) (as defined in (8) and (13)) and where m and n(0) are the number of early and late firms respectively.3 The first condition requires that the market clears for every 8 and the last two conditions require that, in equilibrium, expected profits are non-positive for both the early and late modes of production. Consider first the nature of late entry. Condition (8) requires non-negative profits for late production, but (15) requires non-positive profits. Hence, ‘It should be noted that since some firms will engage in both early and late production, the total number of firms will, in general, be smaller than (m+n(O)).

E.

44

Appelbaum

and C. Litn, Market

uncertainty and

entry

every state 0 must yield zero profit, i.e. z’(@=O for all 0. This implies that the equilibrium price distribution must satisfy p(@=p’

for

8~[#,fl.

(16)

In other words, the equilibrium price is bounded from above by pi = min [a’c(z) +sl]/z. The cutoff state #, being that state below which z =O, can, therefore, be written as

el=my*+ppl,

(17)

so that the equilibrium price distribution becomes 1

if

1

")=

$o--my*)/8 if

8281

(18)

8181.

Using this in (8) we get the (state-dependent) z*(e)=[ 2 E a.P(pl)/ap

8 2 81

if if

0

late output as

(19)

8581,

(where J’ is the unit cost minimizing level of output) and the number of late producers as (using (15)(i) and (17))

n(e)=

(e-el)/rl o

i

if if

8281

(20)

esel.

Let us now consider the early production. From the equilibrium distribution, we can obtain the expected equilibrium price as4

price

B>l3’ p=

I

B CJWI - my*]/8

Since equilibrium must satisfy

if

(21)

8s81.

requires that E[n’] SO, the expected

equilibrium

price

(22) 4For B>#, +I$’ [(O-my*)/@ldqS(fl)+~~ p’d~#((B)=p’-$ and using 0’ = j?p’ + my*.

&(e)dfQ9 a!ler integrating by parts

E. Appelbaum and C. Lim, Market uncertainty and entry

45

Thus, from (13) we get the equilibrium level of early output as y* =

aJ”(~oYa~if P=P’ if p
(23)

and the number of early entrants as m=

(e’--BP’vY* [0

if if

y* = ~?J~(p~)/f?p y*=O.

(24)

Given the definitions of pl, p” and 13~(in (7), (12) and (17)) the system of eqs. (18x24) determines the equilibrium vector (m, y*, n(0), z*(e), p(8), jI) (which satisfies condition (15)). This system can be used to obtain the following possible equilibrium configurations; case 1: case2: caSe 3: case 4:

if p’ -p” = T(@), 0’ < 0 then m > 0 and n(0) for some 8; ifp’-p”Oforsome8; if E(0) = my0 + /?p”, O1 > B then m > 0 and n(e) = 0 for all & if E(8) < jp”, O1 > B then m = 0 and n(0) = 0 for all 8;

where T(@) =j:’ &0) de//J and 8, is defined in (17). Thus, whether the equilibrium supports early and/or late entry depends on the parameters of the model. The effects of parameter changes on the equilibrium vector are examined in the next section. It is, however, clear that, in general, various types of equilibria are possible. Before we conclude this section, it is worth noting that since fixed costs are incurred only when there is production, all firms (early and late) in our model operate at capacity. This is in contrast with the excess capacity result of Dreze and Gabszewicz (1967) and Sheshinski and Dreze (1976) where firms are committed to irreversible fixed costs whether or not there is production. 4. Comparative

statics

The comparative static analysis is carried out for case 1, where m> 0 and n(0) >O for some 0. The effects of changes in various parameters are summarized’ in table 1. We take dk, >O and dkz >O to denote an increase in the mean and the spread of the distribution of 0, respectively. In general, table 1 demonstrates the trade-off between cost efficiency and informational advantages facing the firm. Greater relative efficiency of the early production mode (i.e. lower levels of (~‘,a’) and/or higher levels of sInterested

readers may request the proofs of the results in table 1 from the authors.

46

E. Appelbaum

and C. Lim, Market

uncertainty and entry

Table 1 Comparative statics. a0

so

d

-

+

0’ 0’ 0’

-

-

+ + ?

0’ 0’ + + ?+ + + + + + 0+ + 0’ l +

s’

+ + -

+ 0 +

k’

+ + -

0

? ? ? 0 -

0

k2

0 -

0’ + + + 0’ 0’

(a’,~‘)) will tend to lower the probability of late entry and to decrease (increase) the number of late (early) entrants and their market shares. Looking at the effects of shifts in the demand distribution, it is clear from the condition pi -p” = T(8)’ that since C/Q~0 whereas yi c#+s-O,~ changes in k’ and k2 will have the opposite effect on (early and late) production. It is particularly interesting that, despite risk neutrality, increasing uncertainty has an impact on the equilibrium. To see this, note that since p(B) has an upper bound, an increase in k2 (which shifts the probability weights from the center to the upper and lower ranges of 0) will result in an initial decrease in b, hence decreasing expected profits. This leads to a decline in m, which in turn (increases and) restores p and the expected profits of the remaining firms to their original equilibrium levels. In the new equilibrium, m is smaller and so is my0 (since y” is determined solely by the minimum average cost). As for late producers, increasing uncertainty will not affect the plant size (z(O) is solely determined by the minimum average cost), but will affect the probability of late entry. Since total early production is now lower and probability weights have been shifted away from the center, we will now have a higher probability of late entry, 1 - &0’); and at every state 02 8r, the total production n(@z(e) is also higher. Thus, since z(e) is unaffected, the number of late entrants, n(e), will be higher at every late entry state. This, of course, implies that the expected number of entrants, E[n(@], and the expected total late output, E[n(B)z(B)], will also be higher. The new price distribution will continue to have the same upper bound (which now occurs for a larger set of states), but the price, p(B), for all non-late entry states, 6Note that dk’>O shifts the probability density rightward provided that dtl ~0 for all 0. On the other hand, dk’>O represents a mean-preserving increase in spread of the probability density provided that jf & d0 > 0 for all 0* < 8. [See. Diamond and Stiglitz (1974).]

E. Appelbaum

and C. Lim, Market

uncertainty and entry

47

8x@, will be higher, since early production share is lower for all states. The mean price, jY=EM@], however, remains unchanged. Thus, despite the assumption of risk neutrality, the increase in risk in this model will affect - in opposite directions - the equilibrium number and market shares of late and early firms. Finally, before we conclude, it is useful to note that although we did not explicitly discuss forward and spot markets, they are in fact implicit in our model. Late production can be interpreted to be sold on the spot market at the spot market price p(8), whereas early output can be interpreted to be sold either on the forward market at forward price pf, or on the spot market at price p(0). The market participants in the forward market are risk averse traders who are willing to pay a premium for forward contracts. In equilibrium where there is free entry into the forward and spot markets, the expected spot market price, Q(0)], will be equal to the forward market price pf. Our model can, therefore, be interpreted as characterizing the equilibrium of a competitive industry with spot and forward markets.’ 5. Conclusion

In this paper, we generalize earlier work on the theory of the firm under uncertainty in which the firm’s behavioural mode is taken as given. We provide a model in which firms can make their decision either ex-ante and/or ex-post, thus synthesizing earlier models in the literature. We show that, in general, uncertainty introduces a trade-off between efficiency and informational advantages, thus endogenously determining the firm’s behavioural mode. We characterize the industry equilibrium, discuss its entry patterns and show how it is affected by cost and demand conditions. ‘It is interesting price p3 is invariant demand distribution only affected by the

to note (see table 1) that the equilibrium forward price (equal to expected with respect to the mean of the demand distribution, the fluctuations in the and also the late production cost parameters. The equilibrium value of pf is early production cost parameters, a0 and so.

References Appelbaum, E. and R.J. Harris, 1977, Estimating technology in an intertemporal framework; A neo-Austrian approach, Review of Economics and Statistics 59, 161-170. Appelbaum, E. and C. Lim, 1985, Entry equilibrium and market contestability (York University discussion paper). Appelbaum, E. and C. Lim, 1985, Contestable markets under uncertainty, Rand Journal of Economics 16,2M. Baron, D., 1970, Price uncertainty, utility and industry equilibrium in pure competition, International Economic Review 2, 463-480. Diamond, P. and J. Stiglitz, 1974, Increases in risk and risk aversion, Journal of Economic Theory 8, 337-360. D&Z, J. and J.J. Gabszewicz, 1967, Demand fluctuations, capacity utilization and prices, Operations Research Verfahren 3, 119-141.

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E. Appelbaum and C. Lim, Market

uncertainty and entry

Lippman, S.A. and J. McCall, 1981, Competitive production and increases in risk, American Economic Review 71,207-211. Oi, W., 1961, The desirability of price instability under price uncertainty, Econometrica 29, 58-64. Rothschild, M. and J.E. Stiglitz, 1970, Increasing risk: I. A definition, Journal of Economic Theory 2,225-243. Sandmo, A., 1971, On the theory of the competitive firm under price uncertainty, American Economic Review 61, 65-73. Sheshinski, E. and J. D&e, 1976, Demand fluctuations, capacity utilization, and costs, American Economic Review 66,731-742. Tisdell, C., 1963, Uncertainty, instability, expected profit, Econometrica 31, 243-247. Turnovsky, S.J., 1973, Production flexibility, price uncertainty and the behaviour of the competitive firm, International Economic Review 14, 395-412.