A note on equilibria for two-tier supply chains with a single manufacturer and multiple retailers

Operations Research Letters 39 (2011) 471–474

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Operations Research Letters journal homepage: www.elsevier.com/locate/orl

A note on equilibria for two-tier supply chains with a single manufacturer and multiple retailers George J. Kyparisis ∗ , Christos Koulamas Department of Decision Sciences and Information Systems, College of Business Administration, Florida International University, Miami, FL 33199, USA

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Article history: Received 26 June 2011 Accepted 13 September 2011 Available online 25 September 2011 Keywords: Supply chains Stackelberg leader–follower Nash–Cournot equilibria Oligopoly

abstract A well-studied problem in the supply chain management literature considers a two-tier supply chain for a homogeneous product with a single manufacturer, multiple retailers and a general inverse demand function. The problem has been analyzed in the literature without a formal mathematical treatment of the existence/uniqueness of equilibria. Furthermore, the existence/uniqueness results derived for related models are not extendable to our model. The objective of this paper is to derive sufficient conditions for the existence/uniqueness of Stackelberg–Nash–Cournot equilibria for the two-tier problem. © 2011 Elsevier B.V. All rights reserved.

1. Introduction We consider a two-tier supply chain for a homogeneous product with a single manufacturer at the upstream tier and multiple retailers at the downstream tier. We impose a structure in which the single (monopolist) manufacturer acting as a Stackelberg leader selects the profit maximizing wholesale price w taking into account the product quantities to be chosen by the retailers. Then, in response, the oligopolist retailers (acting as Stackelberg followers) engage in quantity Cournot competition and maximize their individual profits by taking the wholesale price w as given and selecting their individual product quantities qi (w), i = 1, . . . , n. The main objective of this paper is to derive sufficient conditions for the existence/uniqueness of equilibria for the above model. Our model is closely related to the model of Tyagi [11] who assumed symmetric retailers with zero marginal costs. The motivation for our paper stems from the observation that Tyagi [11] proceeded with a comparative statics analysis without formally deriving conditions for the existence/uniqueness of equilibria. This approach (without a formal proof of existence/uniqueness) may not reveal the true market behavior because the overall two-tier equilibrium may not exist and/or multiple retailer equilibria may exist. To the best of our knowledge, the well known results for the existence/uniqueness of (single-tier) Nash–Cournot equilibria for oligopolistic markets (see the monograph [12] and the survey by Cachon and Netessine [2]) have not been extended to the

∗

Corresponding author. Tel.: +1 305 348 3403; fax: +1 305 348 4126. E-mail address: [email protected] (G.J. Kyparisis).

0167-6377/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.orl.2011.09.005

two-tier successive oligopoly model. In a related paper, Sherali et al. [10] derived conditions for the existence/uniqueness of Stackelberg–Nash–Cournot equilibria in a market where all firms compete in a Cournot oligopoly with a homogeneous product and both the single Stackelberg leader (a dominant retailer) and the multiple Stackelberg followers (all other retailers) decide on the optimal quantity to supply to the market. Our model also belongs to the class of mathematical programming problems with equilibrium constraints (MPECs) defined as MPEC max f (x, y) subject to x ∈ X , y ∈ Y (x),

(1)

where X ⊂ R , Y : X → R is a point-to-set map, f : R × R → R1 is a function, and the map Y is the solution map to an equilibrium problem. There is a large body of literature on MPEC problems detailed in the surveys of Harker and Pang [4] and Dempe [3], and in the literature review of Pang [8]. However, general existence results for MPEC problems (see [6,7]) do not apply directly to our model because of the special form of the problem studied in this paper. The rest of the paper is organized as follows. Our two-tier model is formally presented in Section 2. The existence/uniqueness results for the two-tier model are presented in Section 3. l

m

l

m

2. The two-tier supply chain model In this section, we present a formal definition of our model. Consider a two-tier supply chain in which the demand for the (single) homogeneous product is characterized by a general nonlinear inverse demand function p(Q ), where Q ≥ 0 is the total quantity of the product supplied to the market and p(Q ) is the retail price. Suppose that there are n symmetric oligopolistic

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G.J. Kyparisis, C. Koulamas / Operations Research Letters 39 (2011) 471–474

retailers at the downstream tier (Tier 1) with an identical variable cost function v(qi ) (where qi ≥ 0 denotes the quantity supplied to the market by retailer i, i = 1, . . . , n) and a single monopolistic manufacturer at the upstream tier (Tier 2) supplying the product to the retailers. We assume, without loss of generality, that v(0) = 0 since if v(0) > 0 then we can redefine v(qi ) as v 0 (qi ) = v(qi ) − v(0) for qi ≥ 0 without affecting the structure of the problem. The gross profit of retailer i, i = 1, . . . , n, is

Πi (q1 , . . . , qn , w) = qi [p(Q ) − w] − v(qi ) = qi [p(qi + Q−i ) − w] − v(qi ), ∑n where Q−i = j=1,j̸=i qj is the total quantity supplied by all ∑n retailers other than retailer i, Q = j=1 qj is the total quantity supplied by all retailers and w ≥ 0 is the wholesale price charged by the manufacturer. The gross profit of the upstream manufacturer is Π0 (w) = Q (w)w , where w ≥ 0, qi (w), i = 1, . . . , n, ∑ is the quantity supplied by retailer i at price w and n Q (w) = j=1 qj (w) is the total quantity supplied by all retailers at price w (it is implied that the total quantity produced by the manufacturer, Q (w), equals the total quantity ∑n supplied to the market by the retailers, that is Q (w) = j=1 qj (w)). The manufacturer selects the wholesale price w and then, in response, the retailers maximize their individual profits Πi (q1 , . . . , qn , w) by taking w as given and selecting their individual quantities qi , i = 1, . . . , n. As a result, the manufacturer will select the price w which maximizes Π0 (w) taking into account the quantities qi (w), i = 1, . . . , n, chosen by the retailers. The retailers engage in quantity competition within a Cournot oligopoly framework, that is given w , each retailer i maximizes Πi (q1 , . . . , qn , w) over all quantities qi ≥ 0, treating the quantities qj , j ̸= i, of all other retailers as given. For w ≥ 0, the quantities [q1 (w), . . . , qn (w)] constitute a single-tier Nash–Cournot equilibrium if the quantities qi (w) ≥ 0, i = 1, . . . , n, solve the maximization problem max qi [p(qi + Q−i (w)) − w] − v(qi ),

(2)

q i ≥0

∗ ∗ ∗ where Q−i (w) = j=1,j̸=i qj (w). The quantities [w , q1 , . . . , qn ] constitute a two-tier Stackelberg–Nash–Cournot equilibrium if the quantity w ∗ ≥ 0 solves the maximization problem

∑n

max Q (w)w,

(3)

w≥0

and qi = qi (w ) ≥ 0, i = 1, . . . , n. The Nash–Cournot equilibrium [q1 (w), . . . , qn (w)] is called symmetric if qi (w) = q(w), i = 1, . . . , n. Similarly, the Stackelberg–Nash–Cournot equilibrium [w ∗ , q∗1 , . . . , q∗n ] is called symmetric if, for w ≥ 0, qi (w) = q(w), i = 1, . . . , n (thus, q∗i = q∗ = q(w ∗ ), i = 1, . . . , n). Observe that if, for all w ≥ 0, the quantities [q1 (w), . . . , qn (w)] are a symmetric Nash–Cournot equilibrium, then qi (w) = q(w), i = 1, . . . , n, and q(w) ≥ 0 solves the maximization problem ∗

∗

max q[p(q + (n − 1)q(w)) − w] − v(q).

(4)

q ≥0

Moreover, if the quantities [w ∗ , q∗1 , . . . , q∗n ] are a symmetric Stackelberg–Nash–Cournot equilibrium, then w ∗ ≥ 0 solves the maximization problem maxw≥0 nq(w)w , where q∗i = q∗ = q(w ∗ ) ≥ 0, i = 1, . . . , n. In view of (4), the gross profits of individual retailers at symmetric Nash–Cournot equilibria [q(w), . . . , q(w)] are identical and given by Πi (q(w), . . . , q(w), w) = q(w)[p(nq(w))− w] − v(q(w)). We also define Π (q, q(w), w) = q[p(q + (n − 1)q(w)) − w] − v(q). Tyagi [11] studied the above model when v(q) = 0 and assumed (without providing a formal mathematical proof for existence/uniqueness) that the symmetric Stackelberg–Nash–Cournot equilibria [w ∗ (n), q∗ (n), . . . , q∗ (n)] (parameterized as functions of the number of retailers n) exist for any n. In the sequel, we denote the first order, second order, and third order derivatives of a function f (x) with respect to x as f ′ (x), f ′′ (x) and f ′′′ (x) respectively.

3. Existence and uniqueness of two-tier Stackelberg–Nash– Cournot equilibria The next theorem proves the existence of Stackelberg–Nash– Cournot equilibria for our problem. Theorem 1. Suppose that p : R1+ → R1+ is thrice continuously differentiable, p′ (Q ) < 0 for all Q ≥ 0, v : R1+ → R1+ is thrice continuously differentiable and nondecreasing, −p′ (Q ) + v ′′ (q) > 0 for all q ≥ 0, Q ≥ q and that there exists Q0 > 0 such that Qp(Q ) − v(Q ) < 0 for Q > Q0 . Then, a symmetric Stackelberg–Nash–Cournot equilibrium

[w ∗ , q∗ , . . . , q∗ ] exists for the two-tier supply chain. The assumptions that v is nondecreasing and that −p′ (Q ) + ′′ v (q) > 0 for all q ≥ 0, Q ≥ q imply that the retailer cost function C (q) = w q + v(q) (defined for a given w ≥ 0) is nondecreasing in q and that the slope of the marginal cost function C ′ (q) exceeds the slope of the inverse demand function respectively. The assumption that there exists Q0 > 0 such that Qp(Q ) − v(Q ) < 0 for Q > Q0 and i = 1, . . . , n guarantees that the profit function in (4) becomes negative for sufficiently large quantities q which in turn restricts the domain in the maximization problem (4). Theorem 1 is stated under assumptions allowing for nonunique Nash–Cournot equilibria. In order to establish the uniqueness of two-tier Stackelberg–Nash–Cournot equilibria we restrict our attention to situations where unique symmetric single-tier Nash–Cournot equilibria [q(w), . . . , q(w)] exist for w ∈ [0, p(0)]. Let us define the function F (q) = p(nq) + qp′ (nq) − v ′ (q). The next theorem proves the uniqueness of the Stackelberg–Nash–Cournot equilibrium. Theorem 2. Suppose that the assumptions of Theorem 1 hold and that p(Q ) is log-concave, p(0) − v ′ (0) > 0, v(q) − v(0) ≥ v ′ (0)q for all q ∈ [0, Q0 ], F (Q0 ) < 0, F ′ (q) < 0 for all q ∈ (0, Q0 ) and F ′′ (q) ≤ 0 for all q ∈ (0, Q0 ). Then, a unique symmetric Stackelberg–Nash–Cournot equilibrium [w ∗ , q∗ , . . . , q∗ ] exists for the two-tier supply chain. The assumption that p(Q ) is log-concave (see [1]) weakens the requirement of concavity of p(Q ). The assumption that p(0) − v ′ (0) > 0 ensures that the reservation price, p(0), exceeds the marginal variable cost at zero, v ′ (0). Since v(0) = 0 (as assumed in Section 2), the assumption requiring that v(q) − v(0) ≥ v ′ (0)q for all q ∈ [0, Q0 ] implies that the average variable cost, v(q)/q, exceeds v ′ (0). The assumption that F (Q0 ) < 0 implies that the marginal change in q[p(q + (n − 1)Q0 ) − w] − v(q) evaluated at q = Q0 is negative. The assumption that F ′ (q) < 0 for all q ∈ (0, Q0 ) is needed to prove the existence of derivatives of q(w). For the case of v(q) = v0 q, v0 > 0, this assumption is equivalent to a condition utilized by Seade [9] to ensure the stability of the Nash–Cournot equilibrium. The assumption that F ′′ (q) ≥ 0 for all q ∈ (0, Q0 ) is needed to prove the uniqueness of symmetric Stackelberg–Nash–Cournot equilibria. Observe that the downstream oligopoly of symmetric retailers may, in general, also have asymmetric equilibrium solutions. Amir and Lambson [1] address this issue in more detail. We close by demonstrating our existence/uniqueness results for the inverse demand function p(Q ) = e−Q defined for Q ≥ 0. Suppose that v(q) = v0 q, where 0 < e−(n+1)/n < v0 < 1 and that Q0 = − ln v0 . Then, F (q) = e−nq − qe−nq − v0 . The assumptions of Theorem 2 hold since p′ (Q ) = −e−Q < 0 for Q ≥ 0, for Q > Q0 = − ln v0 , p(Q ) < p(Q0 ) = v0 , ln p(Q ) = −Q is concave, p(0) = 1 > v0 and v0 > e−(n+1)/n implies that F ′ (q) < 0 and F ′′ (q) ≥ 0 for q ∈ (0, Q0 ). Therefore, by Theorem 2, a unique symmetric Stackelberg–Nash–Cournot equilibrium [w ∗ , q∗ , . . . , q∗ ] exists.

G.J. Kyparisis, C. Koulamas / Operations Research Letters 39 (2011) 471–474

Appendix Lemma 1. Suppose that the assumptions of Theorem 1 hold. Then, (1) for all w ≥ 0, the quantities [q(w), . . . , q(w)] are a symmetric Nash–Cournot equilibrium if and only if q(w) ∈ [0, Q0 ] solves the single-tier retailer maximization problem max q[p(q + (n − 1)q(w)) − w] − v(q),

(5)

0≤q≤Q0

(2) the quantities [w ∗ , q∗ , . . . , q∗ ] constitute a symmetric two-tier Stackelberg–Nash–Cournot equilibrium if and only if w ∗ ∈ [0, p(0)] solves the manufacturer’s profit maximization problem max

0≤w≤p(0)

Q (w)w,

(6)

and q = q(w ) ∈ [0, Q0 ] solves the retailer’s problem (5), where, for w ∈ [0, p(0)], q(w) ∈ [0, Q0 ]. ∗

∗

Proof of Lemma 1. (1) Let w ≥ 0 be fixed and assume that the quantities [q(w), . . . , q(w)] are a Nash–Cournot equilibrium. Suppose that q(w) > Q0 . Then, since p′ (Q ) < 0 for all Q ≥ 0, p(q(w) + (n − 1)q(w)) ≤ p(q(w)) and since Qp(Q ) − v(Q ) < 0 for Q > Q0 , q(w)p(q(w)) − v(q(w)) < 0, which together imply that Π (q(w), q(w), w) < Π (0, q(w), w). The preceding inequality shows that q(w) > Q0 does not solve problem (4) contradicting the assumption that [q(w), . . . , q(w)] are a Nash–Cournot equilibrium. Therefore, for all w ≥ 0, q(w) ∈ [0, Q0 ] which proves part (1) of Lemma 1. (2) Let w > p(0) be fixed, where p(0) > 0, and assume that the quantities [q(w), . . . , q(w)] are a Nash–Cournot equilibrium. Suppose that q(w) > 0. Since p′ (Q ) < 0 for all Q ≥ 0, p(q(w) + (n − 1)q(w)) < p(0) < w and since v is nondecreasing, v(q(w)) ≥ v(0), which together imply that Π (q(w), q(w), w) < Π (0, q(w), w). The preceding inequality shows that q(w) > 0 does not solve problem (4) contradicting the assumption that [q(w), . . . , q(w)] are a Nash–Cournot equilibrium. Therefore, for all w > p(0), q(w) = 0. Consider the maximization problem given by (3). Since for all w > p(0), q(w) = 0, then Q (w)w = nq(w)w = 0 in this case. Thus, we can restrict the maximization in (3) to w ∈ [0, p(0)]. By part (1) of Lemma 1, we can also restrict the maximization in (2) to q ∈ [0, Q0 ]. Thus, part (2) of Lemma 1 follows.  For w ∈ [0, p(0)], the quantities [q1 (w), . . . , qn (w)] constitute a restricted single-tier Nash–Cournot equilibrium if qi (w) ∈ [0, Q0 ], i = 1, . . . , n, solves the retailers’ problem max qi [p(qi + Q−i (w)) − w] − v(qi ).

0≤qi ≤Q0

The quantities [w ∗ , q∗1 , . . . , q∗n ] constitute a restricted two-tier Stackelberg–Nash–Cournot equilibrium if w ∗ ∈ [0, p(0)] solves the manufacturer’s problem (6) and q∗i = qi (w ∗ ) ∈ [0, Q0 ], i = 1, . . . , n, where [q1 (w), . . . , qn (w)] are a restricted single-tier Nash–Cournot equilibrium. Consider the restricted two-tier Stackelberg–Nash–Cournot equilibrium problem defined above. This problem can be equivalently stated as the MPEC problem in (1) with X = ∑n defined  l m [0, p(0)] ⊂ Rl and f = Q w = → R1 , j=1 qj w : R × R where l = 1, m = n, x = w ∈ X , y = [q1 , . . . , qn ] ∈ Y (x). Moreover, Y (x) = Y (w) is defined as the solution set to the restricted Nash–Cournot equilibrium problem given by Y (w) = {[q1 (w), . . . , qn (w)] : qi (w) ∈ argmax{qi [p(qi

+ Q−i (w)) − w] − v(qi ) : qi ∈ [0, Q0 ]}, i = 1, . . . , n}. A point-to-set map M : D → Rp , where D ⊂ Rl , is called closed at a point x ∈ D if {xk } ⊂ D, xk → x, yk ∈ M (xk ), and yk → y imply that y ∈ M (x) [5]. M is closed on D if it is closed at all x ∈ D.

473

Lemma 2. Suppose that the assumptions of Theorem 1 hold. Then, the point-to-set solution map Y is (1) nonempty-valued on [0, p(0)], (2) closed on [0, p(0)]. Proof of Lemma 2. (1) Amir and Lambson [1] used the theory of supermodular games to prove, under the assumptions of Theorem 1, that there exists at least one symmetric single-tier Nash–Cournot equilibrium [q(w), . . . , q(w)] for all w ≥ 0. By part (1) of Lemma 1, q(w) ∈ [0, Q0 ] and Y (w) is nonempty for all w ∈ [0, p(0)]. (2) Let w 0 ∈ [0, p(0)] and suppose that {w k } ⊂ [0, p(0)], w k → w 0 , [qk , . . . , qk ] → [q0 , . . . , q0 ], and [qk , . . . , qk ] ∈ Y (w k ). Then, for all q ∈ [0, Q0 ], qk [p(qk + (n − 1)qk ) − w k ] − v(qk )

≥ q[p(q + (n − 1)qk ) − wk ] − v(q).

(7)

In view of our assumptions, inequality (7) implies (for any fixed q ∈ [0, Q0 ]) that as k → ∞, q0 [p(q0 + (n − 1)q0 ) − w 0 ] − v(q0 )

≥ q[p(q + (n − 1)q0 ) − w 0 ] − v(q).

(8)

Since inequality (8) is true for any fixed q ∈ [0, Q0 ], then [q , . . . , q0 ] ∈ Y (w 0 ). This proves that Y is closed at w 0 ∈ [0, p(0)]. Since w 0 was an arbitrary point in [0, p(0)], Y is closed on [0, p(0)].  0

Proof of Theorem 1. By Lemma 2, the solution map Y is nonemptyvalued and closed on [0, p(0)]. By the definition of Y (w), for all w ∈ [0, p(0)], Y (w) ⊂ [0, Q0 ]n . Thus, the MPEC problem ∑n defined  in (1) with X = [0, p(0)] ⊂ Rl and f = Q w = j=1 qj w : Rl × Rm → R1 , where l = 1, m = n, x = w ∈ X , y = [q1 , . . . , qn ] ∈ Y (x) = Y (w), has a compact feasible set. Since f (x, y) = f (w, q1 , . . . , qn ) = Q w is also continuous in (w, q1 , . . . , qn ), this implies the existence of a restricted Stackelberg–Nash–Cournot equilibrium [w ∗ , q∗ , . . . , q∗ ]. By part (2) of Lemma 1, the quantities [w ∗ , q∗ , . . . , q∗ ] are a Stackelberg–Nash–Cournot equilibrium if and only if they are a restricted Stackelberg–Nash–Cournot equilibrium which completes the proof.  Lemma 3. Suppose that the assumptions of Theorem 1 hold and that p is log-concave, p(0) − v ′ (0) > 0, v(q) − v(0) ≥ v ′ (0)q for all q ∈ [0, Q0 ], and F (Q0 ) < 0. Then, (1) for all w ∈ [0, p(0)], there exists a unique symmetric Nash–Cournot equilibrium [q(w), . . . , q(w)] such that q(w) is a continuous function on (0, p(0)) and q(w) ∈ (0, Q0 ) q(w) = 0

for w ∈ (0, p(0) − v ′ (0)),

for w ∈ [p(0) − v ′ (0), p(0)].

(9)

(2) q(w) ∈ (0, Q0 ) solves the nonlinear equation F (q) = p(nq) + qp′ (nq) − v ′ (q) = w.

(10)

Proof of Lemma 3. (1) By Lemma 2, Y is nonempty-valued and closed on [0, p(0)]. Amir and Lambson [1] also proved that if the assumptions of Theorem 1 hold and p(Q ) is log-concave, then, for w ≥ 0, there exists a unique symmetric Nash–Cournot equilibrium [q(w), . . . , q(w)]. Thus, Y is single-valued on [0, p(0)]. Since by the definition of Y (w), for all w ∈ [0, p(0)], Y (w) ⊂ [0, Q0 ]n , Y is uniformly compact near w for all w ∈ (0, p(0)). Thus, the proof of Corollary 8.1 in [5] implies that Y is a continuous function at w for all w ∈ (0, p(0)). Furthermore, by part (1) of Lemma 1, for all w ∈ [0, p(0)] there exists a unique symmetric Nash–Cournot equilibrium [q(w), . . . , q(w)] such that q(w) ∈ [0, Q0 ]. Let w ∈

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G.J. Kyparisis, C. Koulamas / Operations Research Letters 39 (2011) 471–474

(0, p(0) − v ′ (0)) be fixed, where p(0) − v ′ (0) > 0 by our assumption. Suppose that q(w) = 0. Then, for q ∈ (0, Q0 ], Π (q, q(w), w) − Π (q(w), q(w), w) = q[p(q) − (v(q) − v(0))/q − w].

(11)

Since by our assumptions, limq→0+ [p(q) − (v(q) − v(0))/q] = p(0) − v ′ (0) and w ∈ (0, p(0) − v ′ (0)), there exists q0 ∈ (0, Q0 ) such that p(q0 ) − (v(q0 ) − v(0))/q0 − w > 0 and thus expression (11) implies that Π (q0 , q(w), w) − Π (q(w), q(w), w) > 0. This contradicts the assumption that q(w) = 0 is a symmetric equilibrium and therefore q(w) ∈ (0, Q0 ] for all w ∈ (0, p(0) − v ′ (0)). Suppose that q(w) = Q0 and define the function h(q) = qp(q + (n − 1)Q0 ) − v(q). Then, for q ∈ (0, Q0 ),

Π (q, q(w), w) − Π (q(w), q(w), w) = h(q) − qw − h(Q0 ) + Q0 w.

(12)

By our assumptions, h(q) is continuously differentiable for q ∈ (0, Q0 ) and since F (Q0 ) < 0, h′ (Q0 ) = p(nQ0 ) + Q0 p′ (nQ0 ) − v ′ (Q0 ) < 0. Thus, for q ∈ (0, Q0 ), sufficiently close to Q0 , h(q) > h(Q0 ) and, in view of expression (12), we obtain Π (q, q(w), w) − Π (q(w), q(w), w) > 0. This contradicts the assumption that q(w) = Q0 and therefore q(w) ∈ (0, Q0 ) for all w ∈ (0, p(0) − v ′ (0)). Let w ∈ [p(0)−v ′ (0), p(0)] be fixed and suppose that q(w) = 0. Then, for all q ∈ [0, Q0 ], since p′ (Q ) < 0 for all Q ≥ 0 and v(q) − v(0) ≥ v ′ (0)q for all q ∈ [0, Q0 ], we have Π (q(w), q(w), w) − Π (q, q(w), w) ≥ q[p(0) − p(q)] + [v(q) − v(0) − v ′ (0)q] ≥ 0. This proves that the quantities [q(w), . . . , q(w)] = [0, . . . , 0] are the unique symmetric Nash–Cournot equilibrium for all w ∈ [p(0) − v ′ (0), p(0)]. (2) By part (1) of Lemma 1 and part (1) of Lemma 3, for all w ∈ (0, p(0) − v ′ (0)) there exists a unique symmetric Nash–Cournot equilibrium [q(w), . . . , q(w)] such that q(w) ∈ (0, Q0 ) solves problem (5). Consider the objective function Π (q, q(w), w) in (5) for any fixed w ≥ 0 and q(w). Since q(w) ∈ (0, Q0 ) maximizes Π (q, q(w), w) on (0, Q0 ), for all w ∈ (0, p(0) − v ′ (0)), the first order necessary condition

Π ′ (q, q(w), w) = p(q + (n − 1)q(w))

+ qp′ (q + (n − 1)q(w)) − w − v ′ (q) = 0 (13) holds at q(w). Since condition (13) holds at q(w) if and only if q(w) solves Eq. (10), the result follows.  Lemma 4. Suppose that the assumptions of Lemma 3 hold and that F ′ (q) < 0 for all q ∈ (0, Q0 ). Then, for all w ∈ (0, p(0) − v ′ (0)), there exists a unique symmetric Nash–Cournot equilibrium [q(w), . . . , q(w)] such that q(w) ∈ (0, Q0 ) and q′ (w) = F ′ (q1(w)) < F ′′ (q(w))

0, q′′ (w) = − [F ′ (q(w))]3 . Proof of Lemma 4. By Lemma 3, for all w ∈ (0, p(0) − v ′ (0)), there exists a unique symmetric Nash–Cournot equilibrium [q(w), . . . , q(w)] such that q(w) ∈ (0, Q0 ) solves the nonlinear equation (10). By our assumptions, F (q) is twice continuously

differentiable and F ′ (q) = (n + 1)p′ (nq) + nqp′′ (nq) − v ′′ (q) < 0 for all q ∈ (0, Q0 ). Thus, by the inverse function theorem, for all w ∈ (0, p(0)−v ′ (0)) Eq. (10) has a unique solution q0 (w) ∈ (0, Q0 ) which coincides with q(w). Moreover, q(w) is twice differentiable and its derivatives are given by q′ (w) = F ′ (q1(w)) < 0 and q′′ (w) = (w)) − [FF′ ((qq(w))] 3 respectively. ′′



Proof of Theorem 2. By Theorem 1, a symmetric Stackelberg– Nash–Cournot equilibrium [w ∗ , q∗ , . . . , q∗ ] exists. By Lemma 3, for all w ∈ [0, p(0)], there exists a unique symmetric Nash–Cournot equilibrium [q(w), . . . , q(w)] such that (9) holds. Since, by part (2) of Lemma 1, [w ∗ , q∗ , . . . , q∗ ] is also a restricted Stackelberg–Nash–Cournot equilibrium, the profit function Π0 (w) = Q (w)w = nq(w)w in (6) is maximized at w ∗ ∈ (0, p(0)]. Observe that Π0 (0) = 0 and Π0 (w) = 0 for w ∈ [p(0) − v ′ (0), p(0)]. On the other hand, for w ∈ (0, p(0) − v ′ (0)), q(w) ∈ (0, Q0 ) and Π0 (w) = Q (w)w = nq(w)w > 0. Thus, the maximization problem (6) achieves its maximum at w ∗ ∈ (0, p(0) − v ′ (0)). By Lemma 4, for all w ∈ (0, p(0) − v ′ (0)), q′ (w) = F ′ (q1(w)) < 0 F ′′ (q(w)) and q′′ (w) = − [F ′ (q(w))]3 . Thus, the assumption that F ′′ (q) ≤ 0 for all q ∈ (0, Q0 ) implies that q′′ (w) ≤ 0. Since Π0′′ (w) = 2nq′ (w) + nw q′′ (w), q′ (w) < 0 and q′′ (w) ≤ 0, then Π0′′ (w) < 0 for w ∈ (0, p(0) − v ′ (0)) and Π0 (w) is strictly concave for w ∈ (0, p(0) − v ′ (0)). Therefore, w ∗ is the unique solution of problem (6) and [w ∗ , q∗ , . . . , q∗ ] is the unique restricted Stackelberg–Nash–Cournot equilibrium. By part (2) of Lemma 1, the quantities [w ∗ , q∗ , . . . , q∗ ] are a Stackelberg–Nash–Cournot equilibrium if and only if they are a restricted Stackelberg–Nash–Cournot equilibrium which completes the proof. 

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