On the Equilibrium Uniqueness in Cournot Competition with Demand Uncertainty
aa r X i v : . [ ec on . T H ] M a r On the Equilibrium Uniqueness in Cournot Competition withDemand Uncertainty
Stefanos Leonardos , Costis Melolidakis Singapore University of Technology and Design, 8 Somapah Rd, 487372 Singapore National and Kapodistrian University of Athens, Panepistimioupolis, 15784 Athens, Greece
Abstract
We revisit the linear Cournot model with uncertain demand that is studied in Lagerl¨of (2006)and provide sufficient conditions for equilibrium uniqueness that complement the existing results.We show that if the distribution of the demand intercept has the decreasing mean residual demand(DMRD) or the increasing generalized failure rate (IGFR) property, then uniqueness of equilibriumis guaranteed. The DMRD condition implies log-concavity of the expected profits per unit of outputwithout additional assumptions on the existence or the shape of the density of the demand interceptand, hence, answers in the affirmative the conjecture of Lagerl¨of (2006) that such conditions maynot be necessary.
Keywords:
Cournot Model, Demand Uncertainty, Unique Equilibrium, Demand Distributions
JEL Classification: C7 Lagerl¨of (2006) considers a model of Cournot competition with linear demand – up to the non-negativity constraintfor the market price – in which the demand intercept is stochastic. Demand uncertainty can make the expecteddemand sufficiently convex which in turn may lead to a multiplicity of equilibria, (Vives, 2001, Lagerl¨of, 2007).Motivated by this, Lagerl¨of (2006) studies conditions on the distribution of the stochastic demand that guaran-tee uniqueness of the equilibrium. His main result is that to establish uniqueness, it is sufficient to assume thatthe distribution has monotone or first decreasing and then increasing (equivalently bathtub-shaped , see also Sec-tion 2.2) hazard rate and that the expected profit per unit of resource is log-concave as the total output approacheszero (Lagerl¨of, 2006, Proposition 1). The latter is achieved by restricting the value at zero of the density of thestochastic demand intercept.In this note, we extend these conditions. We utilize unimodality conditions that we derived in the more gen-eral setting of pricing problems with stochastic linear demand (Leonardos and Melolidakis, 2018), and show thatequilibrium uniqueness is achieved if the distribution of the demand intercept has the decreasing mean residualdemand (DMRD) property or the increasing generalized failure rate (IGFR) property. This is established in The-orem 3.2. The DMRD and IGFR conditions are not comparable, i.e., neither implies the other. Concerning theirrelationship to the conditions from Lagerl¨of (2006), the increasing hazard rate property does imply both DMRDand IGFR. However, this is not necessarily true for the decreasing hazard rate nor for the bathtub-shaped property(Gupta and Olcay, 1995). Hence, the sufficient DMRD and IGFR conditions generalize rather than substitute theexisting ones. Given the inclusiveness of the IGFR class of distributions (Paul, 2005, Banciu and Mirchandani,2013), the present conditions cannot be significantly extended. In case that the Cournot oligopolists face zeromarginal cost, the uniqueness result can be directly obtained for the class of distributions with the decreasinggeneralized mean residual demand (DGMRD) property and finite n + n is the number ofcompetitors. This is the statement of Corollary 3.3. The DGMRD class includes as a proper subset the IGFR class(Belzunce, Candel, and Ruiz, 1998, Leonardos and Melolidakis, 2018). Finally, as shown in Theorem 3.5, theDMRD condition implies log-concavity of the expected profit per unit of output without requiring the existenceof a density nor any restrictions on its value close to zero (if such a density exists). This answers in the affirmativethe conjecture of Lagerl¨of (2006) that the imposed restriction on the value at zero of the density of the stochasticdemand may not be necessary. .1 Outline The rest of the paper is structured as follows. In Section 2.1 we restate the model of Lagerl¨of (2006) and inSection 2.2, we define all related probabilistic notions and shortly discuss the relationship between the IFR, IGFR,DMRD and DGMRD classes. To make the exposition self-contained, we also provide the relevant results fromLeonardos and Melolidakis (2018). Our main contribution is presented Section 3.
Consider a Cournot market with n ≥ i ∈ { , , . . ., n } that produce a homogeneousgood. Each firm has the same constant marginal cost, denoted by c ≥
0. Firm i ’s production quantity is de-noted by x i and the total industry output is denoted by x : = ∑ ni = x i . Also, using standard notation, let x − i : =( x , x , . . . , x i − , x i + , . . . , x n ) for any i = , , . . . , n . The firms face a linear inverse demand function p ( x ) = ( α − x ) + = ( α − x , if x ≤ α , otherwise (1)where p ( x ) denotes the price for output level x ≥
0. In Lagerl¨of (2006), the inverse demand function also includesthe slope parameter b >
0, i.e., it has the form p ( x ) = ( α − bx ) + , with b >
0. However, the parameter b appears inthe analysis – and in particular, in the first order conditions that determine the Nash equilibrium, cf. equation (1)in Lagerl¨of (2006) – only as multiplied by x . Thus, without loss of generality, we assume throughout this paperthat b = x . See also Remark 2.2).The demand intercept or demand level α is a non-negative random variable that takes values in [ , H ] for somereal number H > [ , H ) , if H = + ∞ . We assume that α has an absolutely continuous distribution function F , density f = F ′ , and finite expectation E α < + ∞ , that also satisfies E α > c . We will write ¯ F ( x ) : = − F ( x ) todenote the tail of the distribution of F . Firms are risk neutral and maximize their expected payoffs π i ( x i ; x − i ) : = x i (cid:0) E ( α − x ) + − c (cid:1) (2)with respect to their own output x i , for i = , , . . . , n . A vector of outputs (cid:0) x ∗ i , x ∗− i (cid:1) is a pure Nash equilibrium if π i (cid:0) x ∗ i , x ∗− i (cid:1) ≥ π i (cid:0) x i , x ∗− i (cid:1) , for all x i ≥
0. A pure Nash equilibrium is called symmetric if x ∗ i ≡ x ∗ for some x ∗ > i = , , . . . , n . In this case, we will slightly abuse notation and denote the Nash equilibrium with x ∗ . Anysymmetric Nash equilibrium satisfies the first order condition Z Hnx ∗ α f ( α ) d α − c = ( n + ) x ∗ ¯ F ( nx ∗ ) , (3)cf. Lagerl¨of (2006), Lemma 1. Our scope is to determine conditions on the distribution F of α , so that (3) has aunique solution, or equivalently that the game has a unique Nash equilibrium in pure strategies. Lagerl¨of (2006)derives such conditions which are reported in Theorem 3.1. Let h ( x ) : = f ( x ) / ¯ F ( x ) , for x < H denote the hazard or failure rate of α and g ( x ) : = x h ( x ) , the generalized failurerate of α (Belzunce et al., 1998, Lariviere, 1999, Van den Berg, 2007). We will say that h is bathtub-shaped orsimply B-shaped if there exists an x > ( x ) is non-increasing for x < x andh ( x ) is non-decreasing for x > x (Gupta and Olcay, 1995, Nadarajah, 2009). Also, we will say that F has the increasing failure rate (IFR) property, if h ( x ) is non-decreasing for x < H . Similarly, we will say that F has the increasing generalized failure rate property or simply that F is IGFR, if g ( x ) is non-decreasing for x < H . Finally,let m ( x ) : = E ( α − x | α > x ) = F ( x ) Z Hx ¯ F ( u ) d u , if x < H , otherwise (4)denote the mean residual demand (MRD) function of α (Shaked and Shanthikumar, 2007, Lai and Xie, 2006), and ℓ ( x ) : = m ( x ) / x , for 0 < x < H , the generalized mean residual demand (GMRD) function (Leonardos and Melolidakis, Up to the non-negativity constraint. α is non-negative, we have that m ( ) = E α . We will say that α has the decreasing mean residualdemand property, or simply that α is DMRD, if m ( x ) is non-increasing for x < H and similarly, that α has the decreasing generalized mean residual demand property, or simply that α is DGMRD, if ℓ ( x ) is non-increasing for x < H . To derive the set of sufficient conditions of the present paper, we will make use of the following propertiesfrom Leonardos and Melolidakis (2018). Lemma 2.1.
Let α be a non-negative random variable with E α < + ∞ .(i). If α is DGMRD with lim x → + ∞ ℓ ( x ) = γ , and β ≥ is a constant, then γ < / β , if and only if E α β + < + ∞ .In particular, γ = if and only if E α β + < + ∞ for every β > .(ii). If α is IGFR, then α is DGMRD. The converse is not true in general. Remark 2.2.
The normalization of the slope parameter b to 1 in equation (1) does not alter the probabilisticproperties of interest of the random variable α . This follows from the fact that the IFR, DMRD, IGFR andDGMRD classes of random variables are all closed under multiplication with positive constants (in this case with1 / b ), cf. Paul (2005), Propositions 1 and 4, and Leonardos and Melolidakis (2018), Corollary 4.2 and Theorem4.3. Based on the above definitions, the IFR property trivially implies the IGFR property and similarly, the DMRD triv-ially implies the DGMRD property. The IFR property also implies the DMRD property (Bagnoli and Bergstrom,2005). However, the DMRD and IGFR properties are not comparable (Leonardos and Melolidakis, 2018). Finally,Lemma 2.1-(ii), which has been independently shown by Belzunce et al. (1998), establishes that the DGMRD classis a proper superset of all these classes. These relationships are illustrated in Figure 1.
IFR IGFRDMRD DGMRDFigure 1:
Venn diagram illustrating the relationship between the IFR, DMRD, IGFR and DGMRD families. TheIFR property (inner circle) implies both the DMRD (dotted ellipse) and the IGFR (dashed ellipse) properties. TheDMRD and IGFR classes do not satisfy an inclusion relationship, yet IGFR seems more inclusive in terms ofdistributions that are interesting for economic applications. DGMRD distributions (outer rectangle) form a propersuperset of all these families. Concrete examples of distributions in each family are given in the text.To get a better understanding of the distributions that are contained in each class, we note that the IFR prop-erty is already satisfied by many families of parametric distributions and is, thus, very important for economicapplications. As detailed in Ortega and Li (2019), the IFR class includes the (lef-truncated) normal distribution,the uniform, the Beta with parameters larger than 1, the Gamma with shape parameter larger than 1, the Weibullwith exponent larger than 1, the Log-normal on ( , ) , the Logistic, the Laplace, the truncated logistic, the Gumbel(both max and min), the Power function with parameter larger than 1, the chi-squared with degrees of freedommore than 2, the chi with degrees of freedom more than 1, and the L´evy model on ( , / ) .The IGFR and DMRD classes properly generalize the IFR class, since they also include distributions withdecreasing failure rates (DFR). Such examples are the Beta distribution for values of its parameters less than 1,the Gamma distribution for any positive value of its shape parameter and the Weibull distribution for any positivevalues of its exponent (Lariviere, 1999). Further examples of IGFR but not IFR distributions are the Powerfunction on [ , ] with parameter less than 1, the Lognormal on ( , + ∞ ) , the Student’s t for certain values of itsparameters and the F distribution (Bagnoli and Bergstrom, 2005, Banciu and Mirchandani, 2013).An important distribution which is IGFR – and hence DGMRD – but not DMRD is the heavy-tailed Paretodistribution (Lariviere, 2006, Leonardos and Melolidakis, 2018). In the other direction, i.e., to find examplesthat are DMRD but not IGFR, it suffices to look at distributions that are supported over disjoint intervals. Suchdistributions can be easily constructed to be DMRD (e.g., proper mixtures of uniform distributions) but which – nevitably – are not IGFR, since IGFR distributions need to be defined over continuous intervals (Lariviere, 2006).It is not clear whether DMRD implies IGFR, if we restrict attention to distributions over continuous intervals.Through the same construction, one obtains distributions that are DGMRD but not IGFR. A concrete example of adistribution that is defined over the whole positive half line and which is DGMRD but not IGFR is the Birnbaum-Saunders distribution (which is extensively used in reliability applications) for certain values of its parameters(Leonardos and Melolidakis, 2018).Finally, we note that for pricing and economic applications, the IGFR class seems to exhibit the best trade-offbetween generality (inclusiveness of distributions) and tractability (handy characterizations), see e.g., Lariviere(1999, 2006) and Colombo and Labrecciosa (2012) among others. The DGMRD property, although more gen-eral, is technically more involved, yet it naturally arises in applications with pricing under demand uncertainty(Leonardos, Melolidakis, and Koki, 2017). Using the notation and terminology of the previous section, the main results of Lagerl¨of (2006) for the gamedefined in Section 2.1 can be restated as follows.
Theorem 3.1. (Lagerl¨of, 2006) . There exists at least one symmetric Nash equilibrium and no asymmetric equi-libria. If f ( ) < [ E α − c ] − and h ( x ) is (i) monotone or (ii) B-shaped, then this Nash equilibrium is unique.The statement of Theorem 3.1 in Lagerl¨of (2006) does not use the B-shaped terminology, but rather states thatthe slope of h ( x ) needs to change sign exactly once starting from negative and turning to positive. It is immediateto see that B-shaped distributions satisfy this property. To proceed with the derivation of new conditions, we willuse an equivalent formulation of the first order condition in (3). For this, we define Λ ( x ) : = m ( x ) − c ¯ F ( x ) − − xn − , for x ≥ x ∗ > Λ ( x ∗ ) =
0. Since Λ ( ) = E α − c > x → H − Λ ( x ) = − ∞ , we have that Λ ( x ) starts positive and ends negative. This establishes the– already well known – existence of symmetric pure Nash equilibria in this model (Amir and Lambson, 2000).Concerning uniqueness, we have the following sufficient conditions. Theorem 3.2. If α is (i) DMRD or (ii) IGFR, then the symmetric pure Nash equilibrium is unique.Proof. Part (i) is obvious, since in this case Λ ( x ) is decreasing for x >
0. To prove part (ii), it suffices to showthat Λ ′ ( x ∗ ) < x ∗ . By continuity of Λ ( x ) , this implies that Λ ( x ) crosses the x -axis atmost once and hence, it establishes the claim. Taking the derivative of Λ ( x ) , we obtain that Λ ′ ( x ) = h ( x ) Λ ( x ) + n ( g ( x ) − ( n + )) . Since Λ ( x ∗ ) = Λ ′ ( x ∗ ) is determined by the sign of the term g ( x ) − ( n + ) . To proceed, we consider Λ ( x ) / x which is equal to Λ ( x ) / x = ℓ ( x ) − c ( x ¯ F ( x )) − − n − Since, by assumption, α is IGFR, Lemma 2.1-(ii) implies that ℓ ( x ) is decreasing for all x ≥
0, since α is DGMRDin this case. Moreover, ( x ¯ F ( x )) ′ = ¯ F ( x ) ( − g ( x )) which implies that − ( x ¯ F ( x )) − is decreasing for all x ≥ ( x ) >
1. Hence, as a sum of decreasing functions, Λ ( x ) / x is also decreasing for all x ≥ ( x ) >
1. Along with Λ ( x ∗ ) = x ∗ >
0, this implies that0 > ( Λ ( x ) / x ) ′ (cid:12)(cid:12)(cid:12) x = x ∗ = Λ ′ ( x ∗ ) / x ∗ and hence, that Λ ′ ( x ∗ ) < x ∗ > ( x ∗ ) >
1. If x ∗ is such that g ( x ∗ ) ≤
1, then trivially Λ ( x ∗ ) = n ( g ( x ) − ( n + )) < f ( ) < ( E α − c ) − which isnecessary for Theorem 3.1 to hold. However, the DMRD, IGFR, monotone decreasing or B-shaped hazard rateconditions are not comparable, since none implies the other (Gupta and Olcay, 1995), and hence Theorem 3.2should be interpreted as complementing rather than substituting Theorem 3.1. In the special case that c = Corollary 3.3.
If c = and α is DGMRD with finite ( n + ) -th moment, then the Nash equilibrium is unique. roof. In this case Λ ( x ) / x = ℓ ( x ) − n − and the conclusion follows from Lemma 2.1-(i). Remark 3.4.
The assumption that α has a density f is not necessary for the DMRD or DGMRD conditions,since they are defined in terms of the cdf F . They still apply if F is merely continuous. In this case, we needto rewrite Equation (3) in terms of F . Additionally, the assumption that α is supported on an interval is alsonot necessary for DMRD and DGMRD conditions (Leonardos and Melolidakis, 2018). However, for the IGFRto hold, it is necessary that α is supported on an interval (Lariviere, 2006). Since these assumptions increase thecomplexity without truly generalizing the results in economic terms, we restricted our presentation to distributionswith a density. Finally, although DGMRD distributions are a proper superset of IGFR distributions as stated inLemma 2.1-(ii), Corollary 3.3 cannot be viewed as a direct generalization of Theorem 3.2 due to the momentrestriction.The condition f ( ) < ( E α − c ) − which is necessary for Theorem 3.1 to hold, guarantees that ( P ( x ) − c ) islog-concave for values of x close to 0. Lagerl¨of (2006) conjectures that in view of Amir and Lambson (2000),Theorem 2.7, this assumption may not be necessary . Theorem 3.5 shows that this is indeed the case for distri-butions with the DMRD property which implies the required log-concavity of the expected per unit profit. TheDMRD condition does not assume the existence of a density and hence applies to a broader set of distributions. Theorem 3.5. If α ∼ F is DMRD and F is continuous, then P ( x ) − c is log-concave.Proof. P ( x ) − c is log-concave if and only if ( P ( x ) − c ) ′ / ( P ( x ) − c ) is decreasing (Bagnoli and Bergstrom, 2005).To calculate the derivative of P ( x ) , we rewrite P ( x ) as P ( x ) = Z Hx ( α − x ) f ( α ) d α = E ( α − x ) + = m ( x ) ¯ F ( x ) Since F is continuous with finite expectation, we have thatdd x E ( α − x ) + = dd x (cid:18) E α − Z x ¯ F ( u ) d u (cid:19) = − ¯ F ( x ) Hence, ( P ( x ) − c ) ′ ( P ( x ) − c ) = − b ¯ F ( x ) m ( x ) ¯ F ( x ) − c = − · ( x ) − c ¯ F ( x ) which implies that ( P ( x ) − c ) ′ / ( P ( x ) − c ) is decreasing if m ( x ) − c ¯ F ( x ) − is decreasing. Since − c ¯ F ( x ) − isdecreasing, the claim follows. Acknowledgements
The authors thank an anonymous reviewer for the quality of their reports and their invaluable feedback, com-ments and corrections which considerably improved the final version of the paper. Stefanos Leonardos gratefullyacknowledges support by a scholarship of the Alexander S. Onassis Public Benefit Foundation.
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