Entry, Accuracy of Beliefs and Long-Run Survival in Cournot Games Patrick L. Leoni∗

Abstract In a model that encompasses a general equilibrium framework, we consider a monopolist (a producer) with subjective beliefs that endogenously hedges against fluctuations in input prices in a complete market. Allowing for entries and Cournot competition in this economy, we show that if at least one entrant makes accurate predictions, in a sense introduced here, the monopolist must also make accurate predictions or else its long-run profit will converge to zero for almost every path. In the latter case, the whole market power switches to the entrant making the most accurate predictions.

Keywords:

Market selection hypothesis, survival, entrants, heterogenous beliefs.

JEL codes: ∗

G3, D82, D84

University of Southern Denmark, Department of Business and Economics, Campusvej

55 DK-5230 Odense M, Denmark. E-mail: [email protected], and EUROMED School of Management, Domaine de Luminy - BP 921, 13 288 Marseille cedex 9, France.

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1

Introduction

The Market Selection Hypothesis states that economic agents must make accurate predictions to survive in the long-run, or else market pressure will eventually drive out (Alchian [1] and Friedman [7] for instance). The domain of validity of this intuitive conjecture has received much attention, and this paper follows this line of literature. Despite some counterexamples (Blume and Easley [2, 3, 4] among others), Sandroni [10] provides a sound theoretical foundation for the Hypothesis in competitive settings with complete markets. Leoni [9] extends this result to the case where one agent behaves strategically and influences market prices, but the validity of the Hypothesis to purely strategic settings remain an open question that we partially address here. In a model that encompasses a general equilibrium framework, we consider a monopolist (a producer) with subjective beliefs that endogenously hedges against fluctuations in input prices in a complete market. Allowing for entries and Cournot competition in this economy, we show that if at least one entrant makes accurate predictions, in a sense introduced here, the monopolist must also make accurate predictions or else its long-run profit will converge to zero for almost every path. In the latter case, the whole market power switches to the entrant making the most accurate predictions. Beyond the extension of the Market Selection to this type of interaction, we implicitly show that the minimal cost of entry for which there is no possible entry is exactly the intertemporal profit of being a monopolist at

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some point in the future under accurate beliefs (in our sense). If the cost of entry is below this value, then an entrant making accurate prediction will eventually dominate the market against an inaccurate monopolist. In more details, we consider a monopolist with subjective beliefs and facing uncertainty about consumer demand and/or cost of processing, in a framework that encompasses a general equilibrium model. Input prices also depend on those shocks, and the monopolist has access to a complete financial market to purchase future contracts on input delivery to hedge against those shocks. We introduce a notion of entropy of beliefs, which can be regarded as a measure of accuracy of beliefs. We allow for entries in the first period, we consider one entrant only but the result readily extends to a finite number of entrants. A firm may decide to enter in a Cournot competition (without collusion), and the entrant differs in beliefs and gross profit functions. We use the notion of making accurate profits along an infinite history introduced in Sandroni [10], and we characterize this notion of accuracy of predictions using our concept of entropy. We show that, in every Nash equilibrium of this game, when the entrant makes accurate predictions the monopolist must also make accurate predictions or otherwise its profit will converge to 0 almost surely. The paper is organized as follows. In Section 2 we described the general model and we introduce our notions of entropy beliefs, in Section 3 we analyze the possible transfer of market power to entrants, and finally Section 4 contains some concluding remarks. The technical proofs are given in the Appendix. 3

2

The model

In this section, we formalize the model and we define the relevant notion of accuracy of beliefs and entropy. Time is discrete and continues forever. In every period t ∈ N+ , a state is drawn by nature from a set S = {1, ..., L}, where L is strictly greater than 1. Before defining how nature draws the states, we first need to introduce some notations. Denote by S t (t ∈ N ∪ {∞}) the t−Cartesian product of S. For every history st ∈ S t (t ∈ N ), a cylinder with base on st is defined to be the set C(st ) = {s ∈ S ∞ | s = (st , ...)} of all infinite histories whose t initial elements coincide with st . Define the set Γt (t ∈ N ) to be the σ−algebra which consists of all finite unions of cylinders with base on S t .1 The sequence (Γt )t∈N generates a filtration, and define Γ to be the σ−algebra generated by ∪t∈N Γt . Given an arbitrary probability measure M on (S ∞ , Γ), we define dM0 ≡ 1 and dMt to be the Γt −measurable function defined for every st ∈ S t (t ∈ N+ ) as dMt (s) = M (C(st )) where s = (st , ...). Given data up to and at period t − 1 (t ∈ N ), the probability according to M of a state of nature at period t, denoted by Mt , is Mt (s) =

dMt (s) for every s ∈ S ∞ , dMt−1 (s)

with the convention that if dMt−1 (s)=0 then Mt (s) is defined arbitrarily. 1

The set Γ0 is defined to be the trivial σ−algebra, and Γ−1 = Γ0 .

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Finally, we define the posterior probability in history s to be Ms (A) =

M (C(A)) for every A ∈ Γ. M (C(s))

In every period and for every finite history, nature draws a state of nature according to an arbitrary probability distribution P on (S ∞ , Γ).

2.1

The agents

We now formally describe the interaction of the agents. In the next section, we explain how our model can re-interpreted in terms of a General Equilibrium, even if our analysis goes beyond this framework. There are two goods available in every period, an output good x ∈ R and an input good y ∈ R. There is a producer that lives forever and produces the output good in every period. The producer is in situation of monopoly for its production. In every period, an arbitrary number of consumers is born and will live for this period only; consumers own the input good y and seek the output good x. The assumption that consumers live for one period only simplifies the analysis by avoiding the problem of commitment to future prices as in Gul et al. [5]; it can also be justified as a one-time buy on the consumer side. One can also easily extend the framework to an overlapping generation model, this issue is omitted to simplify the analysis and similar results obtain in the later case. In period 0, There is one producer that owns a quantity y0 of input good, and it is monopolist on the output market. In every period t, a new market opens with an arbitrary number of new consumers who live for this period 5

only. For a given level of input good, the producer produces the output good in this same period, and the output good is delivered to current consumers. We thus avoid without loss of generality the issue of delay in production. In any history st , an entry may occur and the entrant stays on the same market as the monopolist. The entrant is identical to the monopolist up to its beliefs and profit functions described next, and both producers engage in a duopolistic Cournot game. Once entered, we do not need to assume that there is a possible exit as explained later. Similar qualitative results obtain when considering multiple entries, we leave this issue aside to simplify the exposition. We consider the following class of gross profit functions for the entrant and the monopolist that satisfies standard textbook assumptions. The upperscript 1 will denote the producers or first in place, and the upper-script 2 will denote the entrant. For a net of input y = (y 1 , y 2 ) chosen by a the firms in any history s, we will focus on gross profit functions for any producer i of the form
Qis (ysi , ys−i ) = fsi (ysi )ps ysi , ys−i ,

(1)

where fsi is the production function (we will drop the upper-script from now on to simplify notations) and ps is the price function given the choice of input by the firms. This price function incorporates various issues about price formation given the output level, in particular it may go beyond the direct and usual dependency on aggregate output. We assume that the partial derivatives of p are negative, an assumption consistent with textbooks

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approach. When an entry has not occurred yet, the gross profit function of the monopolist is naturally derived from above by setting y 2 = 0. We assume for every s that fs is differentiable, concave and increasing, with fs0 (0) = ∞. Any standard production function of the form fs = ln or fs (y) = y α · l1−α for some α ∈ (0, 1) and any given labor input l will satisfy those assumptions. For every s, the price function is assumed to be decreasing in every variable and differentiable. Input is purchased by the producer in the current period for the next period, taking as given the price of the input good. We assume that the producer has access to a complete market where she can purchase futures contracts on input delivery next period. The price of a future contract purchased in history st and paying off one unit of input if event i occurs next period, and 0 otherwise, is denoted by qst ,i . We denote the quantity of this contract held by the monopolist by θist .2 The monopolist is a price-taker on the futures market, and we assume that its volume of trade does not affect prices to simplify the analysis. This last assumption is not restrictive since our results hold at any given system of prices, in particular at equilibrium prices; however, the price-taker assumption on the input market is essential for our result. We also assume that the net (qs,i )s is bounded away from 0. The financial market is not modelled to simplify the exposition, and without loss of generality since the monopolist is assumed to be a price-taker and our results hold for arbitrary positive nets of security prices. Those prices can 2

Without loss of generality, We can restrict our analysis to this type of contracts, also

known as Arrow securities, since markets are complete.

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stem from, albeit without being restricted to, market clearing conditions on demand functions consistent with rational investors endowed with standard von Neuman-Morgenstern utility and facing standard budget constraints. To simplify notations and avoid confusion, we assume without loss of generality that the entry occurs in period 0. The producers purchase those contracts through retained earnings, while seeking to extract dividends from the proceeds. A strategy for producer i (i = 1, 2) is then the choice of a net of future contracts θi = (θis )s for every possible finite history. Let eist denote the dividends extracted by producer i in history st . The dividends depend on the strategy of producer −i. Formally, the dividends in history st = (st−1 , j) to producer i implementing a strategy θi and facing a strategy θ−i satisfy, or equivalently can defined as est +

X

θist · qsi t ≤ Qist (θist−1 , θ−i st−1 )

(2)

i

Producer i (i = 1, 2) has subjective beliefs about the uncertainty in the economy, which is denoted by the probability measure M i defined on (S ∞ , Γ). As standard in finance, we assume that the monopolist seeks to maximize the (subjective) expected net present value of the firm; i.e., producer i seeks to maximize the expression ! i

−i

P V i (θ , θ ) ≡ E i

X

t

i

−i

β · et (θ , θ ) ,

(3)

t

where β i ∈ (0, 1) is the intertemporal discount factor (the upper-script will be removed from now to simplify the notations), and where E i (.) is the expectation operator associated with the probability measure M i . We could have 8

assumed that the producers are risk-averse on dividends payments without changing the qualitative nature of our results. We consider the concept of Nash equilibrium for the game above; that is, i −i i we focus on couples (¯θ , ¯θ ) such that ¯θ maximizes Eq. (3) taking as given

¯θ−i for every i. We focus on this concept instead of Subgame Perfect Nash equilibrium for tractability, and we conjecture that our results extend to this notion. Our model encompasses a standard general equilibrium model, with an infinitely-lived producers and consumers living for one-period only with standard preferences. Supply (of the input good) and demand (for the production good) functions from the consumers’ side, and stemming from standard maximization problem, can be regarded as already embedded in the gross profit function of the monopolist. We can easily extend those demand functions to be consistent with other settings such as the emergence of monopsonist behavior from consumers. The arbitrary prices that we will allow throughout the paper can be chosen so as to be market-clearing prices both on financial markets and goods markets for those supply and demand functions, as long as equilibrium market prices remain positive as above.

2.2

Accuracy of beliefs

Our analysis relies on a notion of accuracy of beliefs (or predictions) described next. We introduce two concepts of entropy, well fitted for long-run analysis. First, we need to ensure that both nature and the monopolist’ beliefs assigns

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strictly positive probability to every event. Definition 1 The entropy of the belief of the monopolist at period t (t ∈ N ) along a path s ∈ S is defined by Pt (s) Mt (s)

Πt (s) =

if Mt > 0 and an arbitrary finite real otherwise. We next introduce two notions capturing the long-run evolution of the above entropy. Definition 2 The upper entropy of beliefs of the monopolist along a path s ∈ S is the function Π defined by Π(s) = lim Πt (s) t

The lower entropy of beliefs along this path s is the function Π defined by Π(s) = lim Πt (s) t

The basic motivation for introducing two distinct notions of entropy, involving both the lim inf and sup of the ratio above, is that learning processes used to form individual beliefs may not converge or may also display erratic behavior around accurate beliefs. Given so, the ratio of beliefs may not have a limit for every leaning process forcing us to make this distinction. It is also important to notice that, in the above definition, the evolution of long-run beliefs only matter. Any particular belief formed early in the past does not

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influence the entropy. This represents an important departure from the concept of entropy introduced in Lehrer and Smorodinsky [8], which considers a weighted average of all previous entropy at any point in time (the entropy at any point in time differs from ours in this last concept). We will also use a notion of accuracy of predictions when analyzing the effects of entrants of monopolistic power; this notion was already introduced in Sandroni [10]. Definition 3 An agent with beliefs M makes accurate predictions on a path s ∈ S ∞ if kMst − Pst k → 0. Definition 3 says that a firm makes accurate prediction on a given path if the posterior subjective probabilities along this path become arbitrarily close to those of nature for the sup-norm. The following lemma makes the link between this notion and our concept of entropy. Lemma 4 P -almost surely, an agent makes accurate predictions on a path s if and only if ∞ > Π(s) = Π(s) > 0 The above lemma states that, for P -almost every path, a firm makes accurate predictions along a path if and only the upper and lower entropies coincide and are finite.

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3

Entry and loss of market power

We now analyze how bad beliefs may affect the monopolist when there are potential entrants. In particular, we show that when at least one entrant makes P -almost surely accurate predictions the monopolist must also make accurate predictions or otherwise vanquish almost-surely. The notion of vanquishing means that the long-run profits of the monopolist will converge to 0 P -almost surely. We consider now a finite number of potential entrants. Entries occur in period 0 without the possibility of exit to simplify matters. Competitors will play a Cournot game over the infinite horizon; that is, they will compete with the monopolist in quantity. Every entrant seeks to maximize its subjective expected profit as before. We focus on any Nash equilibrium of this game. We can now state our result about the survival of a monopolist that does not make accurate predictions. Proposition 5 Assume that the entrant makes accurate predictions P -almost surely. In every equilibrium, If the monopolist does not make accurate predictions then its profit will converge to 0 P -almost surely. Proof. See Appendix. Proposition 5 states that it is enough to have one entrant making accurate predictions to force the monopolist to have accurate predictions. In this case, failure to accurately forecast will drive away the monopolist from the market. Eventually, the domination of the market will switch to this entrant

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with superior predictions. It is easy to see that the same survival result can rephrased in terms of entropy of beliefs by Lemma 4.

4

Conclusion

We have studied a general situation where a monopolist must make accurate or else an entrant making accurate will eventually drive it out of market and set itself as a new monopoly. A corollary of this result is that the minimal cost of entry for which there is no possible entry is exactly the intertemporal profit of being a monopolist at some point in the future. If the cost of entry is below this value, then an entrant making accurate prediction will eventually dominate the market against an inaccurate monopolist. Our work can easily be extended to oligopolistic games where collusion is not possible, showing that the Market Selection Hypothesis extends to this class of economic situations.

A

Appendix

We now prove all the results stated earlier.

A.1

Proof of Lemma 4

This technical lemma making the link between making accurate predictions and our notion of entropy is a direct consequence of Proposition B.3 in Sandroni [10]. It is shown there that, P -almost surely, making accurate predic13

tions along a path s is equivalent to ∞ > lim t

dMt (s) > 0. dPt (s)

Since we have by definition that  −1 dMt (s) dPt−1 (s) dPt dMt−1 Πt (s) = · = · , dPt−1 (s) dMt−1 (s) dMt dPt−1 the above result is equivalent to saying that the upper and lower entropies along this path s coincide and are finite. The proof is complete.

A.2

Proof of Proposition 5

We now prove all the results stated earlier. We first derive a fundamental equation describing the evolution of equilibrium variables along a given infinite path, as a function of individual beliefs. By rearranging terms, the program consisting of maximizing (3) subject to (2) can rewritten as X

Max

β t · dMsit · [Qist (θst−1 ,j , θ−i st−1 ,j ) −

X

θst ,i qst ]

(4)

i

st =(st−1 ,j)

over the strategy set consisting of all θ and taking as given the strategy of the other firm θ−i , and where θst−1 ,j is the vector of security purchased in history st−1 with payoff in history (st−1 , j). Following a standard variational as in Fudenberg and Tirole [6] p. 216 for instance, it must be true that in every Nash equilibrium (θ1 , θ2 ) (the upper bar is remove to simplify notations) directly gives that j β · Mti · (Qist )0 (θist−1 ,j , θ−i st−1 ,j ) = qst ,

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(5)

for every i, j and every history st . Denote now by ˜θ = (˜θst )st ,t the optimal hedging plan when the monopolist has correct beliefs P . Taking the ratio yields immediately −i i Mti (Qist )0 (θst−1 ,j , θst−1 ,j ) = 1 for every j and every st . · i 0 −i Mt−i (Q−i st ) (θ st−1 ,j) , θ st−1 ,j )

(6)

Rearranging terms and removing some upper script to simplify notations (and without loss of generality) yields ∂p

0

st 2 2 1 2 2 1 Mt2 fst (θst )pst (θst , θst ) + fst (θst ) ∂θ2st (θst , θst ) · = 1. Mt1 fs0 (θ1s )pst (θ1s , θ2s ) + fst (θ1s ) ∂p1st (θ1s , θ2s ) t ∂θ t t t t t t

(7)

st

Assume now that the entrant i makes accurate predictions, and that the monopolist does not make accurate prediction. Then by Lemma 4 the fraction

Mt2 Mt1

must converge to +∞ for P −almost every path. ∂p

0

We first claim that the numerator fst (θ2st )pst (θ2st , θ1st )+fst (θ2st ) ∂θ2st (θ2st , θ1st ) st

is bounded away from 0. Indeed, assume by way of contradiction that it converges to 0. Since the term β · dMs2t is bounded above, the left-hand side of Eq. (5) converges to 0. The sequence (qst )t is bounded away from 0 by assumption, so there exists a time t0 after which Eq. (5) cannot hold. This is a contradiction, and thus the numerator must be bounded away from 0. Therefore, it must be true for Eq. (7) to hold that 0

M 2 M fst (θM st )pst (θ st , θ st ) + fst (θ st )

Since

∂pst (θist , θ1st ) ∂θist

∂pst j 2 (θst , θst ) →t +∞. ∂θM st

(8)

0

M 2 < 0 by assumption, it must be true that fst (θM st )pst (θ st , θ st ) 0

converges to +∞. Since the price function is always finite, and since fst (0) = +∞, it must be true for Eq. (7) to hold that (θ1st ) → 0. 15

In other words, the profits of the monopolist converges to 0 P -almost surely, and the proof is now complete.

References [1] Alchian, A. (1950) “Uncertainty, Evolution and Economic Theory,” Journal of Political Economy, 58: 211-221. [2] Blume, E., and D. Easley (1992) “Evolution and Market Behavior,” Journal of Economic Theory, 58: 9-40. [3] Blume, E., and D. Easley (2002) “Optimality and Natural Selection in Markets,”Journal of Economic Theory, 107: 95-135. [4] Blume, E., and D. Easley (2007) “If You’re So Smart, Why Aren’t You Rich? Belief Selection in Complete and Incomplete Markets,” Econometrica 74, 929–966. [5] Gul, F., H. Sonnenschein and R. Wilson (1986) “Foundations of Dynamic Monoply and the Coase Conjecture,”Journal of Economic Theory, 39: 155-190. [6] Fudenberg, D. and J. tirole (1996) Game Theory, MIT Press. [7] Friedman, M. (1953) Essays in Positive Economics. Chicago: University of Chicago Press.

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[8] Lehrer, E., and R. Smorodinski (1996) “Compatible Measure and Merging,” Mathematics of Operations Research, 21: 697-706. [9] Leoni, P. (2008) “Market Power, Survival and Accuracy of Predictions in Financial Markets,” Economic Theory 34: 189-206. [10] Sandroni, A. (2000) “Do Markets Favor Agents Able To Make Accurate Predictions?” Econometrica, 68: 1303-1341.

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