A Scalar Parameterized Mechanism for Two-Sided Markets
AA Scalar-Parameterized Mechanism For Two-Sided Markets
Mariola Ndrio Khaled Alshehri Subhonmesh BoseMarch 4, 2020
Abstract
We consider a market in which both suppliers and consumers compete for a product viascalar-parameterized supply offers and demand bids. Scalar-parameterized offers/bids are ap-pealing due to their modeling simplicity and desirable mathematical properties with the mostprominent being bounded efficiency loss and price markup under strategic interactions. Ourmodel incorporates production capacity constraints and minimum inelastic demand require-ments. Under perfect competition, the market mechanism yields allocations that maximizesocial welfare. When market participants are price-anticipating, we show that there exists aunique Nash equilibrium, and provide an efficient way to compute the resulting market allo-cation. Moreover, we explicitly characterize the bounds on the welfare loss and prices observedat the Nash equilibrium.
The distinction between consumers and producers in marketplaces is increasingly fading. In theretail electricity sector, increased consumer participation—either as generation suppliers or price-responsive demanders—is driving the emergence of a digital platform marketplace where end-use customers can engage in transcations coordinated via a central entity or market manager asproposed in Tabors et al. (2016). Similar digital marketplaces have emerged in the areas of ride-sharing (Uber, Lyft), lodging (Airbnb), online retail and trading auctions (Amazon, Ebay) etc. Acommon feature of these multi-sided marketplaces is a collection of agents who can take up themantle of being suppliers or consumers while the market clears through a centralized mechanism,often operated by a market manager. Motivated by these transformations, in this paper we studya two-sided market with a finite number of suppliers that compete to supply a product to price-responsive consumers. Our focus is on uniform price markets that clear through a centralizedmechanism that sets a single per-unit price on the product for all participants. Every consumer(supplier) expresses her willingness to buy (offer) via a demand bid (supply offer) that fully char-acterizes her demand (supply) quantity at a given market price. We investigate the followingmarket design question:
What is the right mechanism that allows market actors sufficient flexibility todeclare their willingness to offer/buy such that it yields efficient allocations, i.e., an allocation that maxi-mizes social welfare?
The seminal work by Klemperer and Meyer (1989) demonstrated that in the absence of un-certainty there exist an enormous multiplicity of equilibria in supply functions. Hence, there is aneed to resort to stylized offer/bid functions that appropriately restrict the family of supply offersand demand bids which the market actors are allowed to utilize. The well-known Bertrand and1 a r X i v : . [ ec on . GN ] M a r ournot competition models are examples of simple (degenerate) supply offer strategies in mar-kets with uniform prices. However, the Bertrand model typically assumes that each participant iswilling to supply the entire demand, which may not be satisfied in a number of cases. Variations ofthe Bertrand model with capacity constraints have been proposed, however, in such settings pureNash equilibria may not exist as shown in Shubik (1959). The Cournot model has a number of ap-pealing properties when studying oligopolies in markets with relatively high demand elasticity.However, when demand elasticity is low Cournot competition may exhibit arbitrarily high wel-fare loss (see Day et al. (2002)). Furthermore, pure quantity or price competition cannot adequatelyrepresent markets with more complicated offer structures. An example of such markets are day-ahead wholesale electricity markets that operate either as pools or power exchanges. In thesemarkets, power producers submit offers to supply varying quantities at succesively higher pricesand the demand side specifies the quantity willing to purchase at succesively lower prices. Linearsupply functions is another candidate family of functions to model strategic interactions amongsuppliers. However, the work of Baldick et al. (2004) illustrates that it is not straightforward toincorporate capacity constraints into linear supply offers. Moreover, arbitrary high efficiency lossat the Nash equilibrium is possible, particularly when suppliers have highly heterogeneous costfunctions (see Li et al. (2015)).In this work, we restrict our attention on a specific family of supply offers and demand bids,referred to as scalar-parameterized supply functions, studied by Johari and Tsitsiklis (2003) andJohari and Tsitsiklis (2011) in markets with inelastic supply and demand respectively. The spe-cific family of offer/bid functions allows market actors to have one-dimensional action spaces,when faced with a single market price. Such market mechanisms are simple to implement and areconsidered to be fair among market participants. Moreover, the work of Johari et al. (2004) andJohari and Tsitsiklis (2011) showed that such supply offers possess a number of attractive proper-ties including bounded Price of Anarchy and price markups at the Nash equilibrium. The familyof supply functions considered here is a capacitated version of those introduced by Johari andTsitsiklis (2011) that have been studied by Xu et al. (2016) and Lin and Bitar (2017) under perfectlyinelastic demand. Such supply functions prohibit situations where firms can offer in the marketbeyond their means.In this paper, we aim to study the most general setting: a two-sided market where both supplyand demand compete through supply offers and demand bids, which we present in Section 2.In sections 3 and 4, we characterize the market outcome in situations when all market actorsare (i) pure price-takers and (ii) price-anticipating. We show that under perfect competition ourmarket mechanism yields allocations that maximize social welfare. When both sides of the marketare price-anticipating, the misrepresentation of private information has the potential to inducemarket allocations that are suboptimal to the efficient outcome. However, our analysis in Section5 indicates that both the welfare loss and the price markup at the Nash equilibrium are bounded.Numerical experiments in Section 6 illustrate the main insights of the analysis. Section 7 concludesthe paper. All proofs are provided in the Appendix. Notation:
Let R denote the set of real numbers and R + the set of non-negative real numbers.Denote the transpose of a vector x ∈ R n by x T . Let x − i = ( x , . . . , x i − , x i +1 , . . . , x n ) ∈ R n − bethe vector including all but the i th element of x . Finally, denote by the vector of all ones withappropriate size. 2
We consider a market with a finite number of M consumers and N firms competing for a product.Denote the set of consumers by M = { , , . . . , M } and the set of suppliers by N = { , , . . . , N } .Let d i denote consumer i ’s quantity demanded, which must be greater than a minimum inelasticdemand level denoted by d . Let s i denote the quantity supplied by firm i that must lie beloweach supplier’s maximum capacity limit denoted by κ . Each consumer receives utility U i ( d i ) forconsuming amount d i and each firm incurs costs C i ( s i ) for producing quantity s i . We make thefollowing assumption on the utility and cost functions. Assumption 1.
For each i ∈ M , U i ( d i ) is concave, strictly increasing and continuously dif-ferentiable for d i ≥ d with U i ( d ) = 0 . For each i ∈ N , C i ( s i ) is convex, strictly increasing andcontinuously differentiable with C i ( s i ) ≥ for s i ≥ . Over the domain s i ≤ , C i ( s i ) = 0 .The aggregate welfare maximization problem is given bymaximize d , s S ( d , s ) := M (cid:88) i =1 U i ( d i ) − N (cid:88) i =1 C i ( s i ) , (1a)subject to M (cid:88) i =1 d i = N (cid:88) i =1 s i , (1b) ≤ s i ≤ κ , ∀ i = 1 , . . . , N, (1c) d ≤ d i , ∀ i = 1 , . . . , M, (1d)Henceforth, we will refer to every allocation ( d , s ) that solves (1) as efficient . In effect, such allo-cations can be viewed as those determined by a central entity or market manager that has perfectknowledge on the market and all participants. However, U i and C i are generally not available tothe market manager. Hence, is there a mechanism that allows market actors to express their preferencesin a way that it yields efficient market allocations? We consider the following market mechanism forsupply and demand allocations. Let consumer i ∈ M provide to the market manager a parameter3 id ≥ . Given price p > , the consumer is willing to buy d i = D ( θ id , p ) , where D ( θ id , p ) := d + θ id p . (2)The expression in (2) represents the quantity the consumer is willing to buy, given the inelasticcomponent d , the market price p , and the parameter θ id . The inelastic demand d represents theminimum quantity the consumer must be supplied while θ id /p represents the price-responsiveportion of her demand. Note that the demand bid is decreasing in price, i.e., it is downwardsloping. For ease of exposition we consider equal minimum demand among consumers in themarket. The case with distinct d is straightforward to generalize. Note that this assumptiondoes not make the consumers homogeneous as each consumer is described by a different utilityfunction.Let firm i ∈ N submit to the market manager a parameter θ is ≥ . Given price p > , the firmis willing to supply s i = S ( θ is , p ) , where S ( θ is , p ) := κ − θ is p . (3)The supply offer (3) represents the quantity the firm is willing to supply as a function of price.The supply offer is further parameterized in the capacity κ , which represents the supplier’s max-imum production capacity. For ease of exposition we consider equal capacities among firms in themarket. We refer to Figure 1 for illustrations of how D ( θ id , p ) and S ( θ is , p ) vary with price. Observethat as the demand approaches d , the consumer’s willingness to buy approaches infinity. Simi-larly, as the supply quantity approaches the firm’s maximum capacity the requested market pricegrows large.Let θ d = (cid:0) θ d , . . . , θ Md (cid:1) and θ s = (cid:0) θ s , . . . , θ Ns (cid:1) represent the collection of demand bid and supplyoffer parameters, respectively. The market manager chooses price p ( θ d , θ s ) > to clear the marketsuch that supply equals demand, i.e., M (cid:88) i =1 D ( θ id , p ) = N (cid:88) i =1 S ( θ is , p ) . (4)Such choice is only possible when T θ d + T θ s > in which case the market price is given by p ( θ d , θ s ) = T θ d + T θ s N κ − M d . (5)Throughout the paper, we assume M d < N κ and thus the market price is well-defined. In thecase where T θ d + T θ s = 0 , i.e., every market participant submits a zero parameter, we adoptthe following conventions D (0 ,
0) = d and S (0 ,
0) = κ . For markets with a perfectly inelastic demand D , the residual supply index ( RSI ) is often adoptedas a suitable indicator of market power. Precisely, the
RSI of firm i measures the capability ofthe aggregate market capacity—excluding that of i —to meet demand D . In the model consideredhere, the inelastic portion of demand is M d . Mathematically, if RSI i := ( N − κ M d
4s strictly less than one, then firm i is said to be pivotal . See Newbery (2008) and Swinand et al.(2010) for further details. As we show in Section 4, the presence of pivotal suppliers is critical inthe analysis of the market outcome under strategic interactions. In this section, we study the market outcome assuming all market participants are pure price-takers. We aim to establish the existence and characterization of the competitive market equilib-rium taking into account the profit-maximizing nature of market actors. Given market price µ > each consumer maximizes the payoff π id ( θ id , µ ) = U i (cid:0) D (cid:0) θ id , µ (cid:1)(cid:1) − µD (cid:0) θ id , µ (cid:1) , i ∈ M . (6)Similarly, each supplier maximizes π is ( θ is , µ ) = µS ( θ is , µ ) − C i ( S ( θ is , µ )) , i ∈ N . (7)We now proceed with our first result which shows that when consumers bid in (2) and firms offerin (3) the market supports an efficient competitive equilibrium. Theorem 1
Suppose Assumption 1 is satisfied. Then, there exists a competitive market equilibrium ( θ ∗ d , θ ∗ s , µ ) satisfying: π id ( θ i ∗ d , µ ) ≥ π id ( θ id , µ ) , ∀ θ id ≥ and i ∈ M (8) π is ( θ i ∗ s , µ ) ≥ π is ( θ is , µ ) , ∀ θ is ≥ and i ∈ N (9) µ is given by (5) . (10) Moreover, the supply vector defined by s ∗ i = S ( θ i ∗ s , µ ) and the demand vector defined by d ∗ i = D (cid:0) θ i ∗ d , µ (cid:1) isan efficient allocation. According to Theorem 1, under perfect competition, suppliers and demanders maximize theirpayoffs and the resulting market allocation is efficient. This implies that given price µ , the firmshave no incentive to deviate from supplying s ∗ and consumers have no incentive to deviate frombuying d ∗ . Thus the competitive market allocation is efficient and the market clearing price isthe shadow value of the constraint (cid:80) Mi =1 d i = (cid:80) Ni =1 s i . In other words, at µ the marginal socialbenefit of additional output equals the marginal social cost. The preceding argument establishesthe first fundamental theorem of welfare economics : if the price µ and the allocation ( d ∗ , s ∗ ) constitutea competitive equilibrium, then this allocation is efficient. . In contrast to the price-taking model, we now consider a model where the market participants areprice-anticipating. Price-anticipating suppliers and consumers realize that the market price is a In economics the notion of efficiency is, typically, equivalent to that of Pareto optimality. In other words, a Paretooptimal market allocation means that there is no other allocation that can make any market actor better off withoutmaking another actor worse off. i ∈ M is π id ( θ id , θ − id , θ s ) = U i (cid:18) d + θ id p ( θ d , θ s ) (cid:19) − p ( θ d , θ s ) d − θ id . (11)Note that the payoff of each consumer now depends on the actions of all other market participants,that are collectively incorporated in the market price. Similarly, for each firm the payoff functiondepends on her action θ is and the actions of all other market participants. Therefore, the payoff offirm i ∈ N is given by π is ( θ is , θ − is , θ d ) = p ( θ d , θ s ) κ − θ is − C i (cid:18) κ − θ is p ( θ d , θ s ) (cid:19) . (12)We define the game G with M ∪ N denoting the set of players with strategy spaces Θ i = R + and payoffs given by (11) and (12). Our goal is to study the existence (and uniqueness) of theNash equilibria of G and provide an efficient way to compute the equilibrium. A bid/offer profile (cid:16) ˜ θθθ d , ˜ θθθ s (cid:17) constitutes a Nash equilibrium if π id (˜ θ id , ˜ θ − id , ˜ θ s ) ≥ π id ( θ id , ˜ θ − id , ˜ θ s ) , ∀ θ id ≥ and i ∈ M π is (˜ θ is , ˜ θ − is , ˜ θ d ) ≥ π is ( θ is , ˜ θ − is , ˜ θ d ) , ∀ θ is ≥ and i ∈ N . We begin with the following result that illustrates how certain market parameters influence theexistence of a Nash equilibrium of G . Lemma 1 G does not admit a Nash equilibrium if a pivotal supplier exists in the market. In effect, Lemma 1 implies that when N − firms cannot supply the entire inelastic demandin the market, then there exists a pivotal supplier that faces a non-zero inflexible demand thathas infinite willigness to pay. This makes the suppliers’ payoff grow unbounded with respecttheir action θ s . Hence, a Nash equilibrium cannot exist in this case. As a consequence of Lemma1, there cannot exist a Nash equilibrium with N = 1 since, by definition, the single supplier ispivotal. In view of the above Lemma, we impose the following assumption. Assumption 2.
RSI i > for each firm i ∈ N . Equipped with the previous observations, we present our main result that explicitly character-izes the unique Nash equilibrium of G . Theorem 2
Suppose Assumptions 1-2 hold. G admits unique Nash equilibrium in (cid:16) ˜ θ d , ˜ θ s (cid:17) . Moreover,the supply profile ˜ s i = S i (cid:16) ˜ θ is , p ( ˜ θ d , ˜ θ s ) (cid:17) , i ∈ N and the demand profile ˜ d i = D i (cid:16) ˜ θ id , p ( ˜ θ d , ˜ θ s ) (cid:17) , i ∈ M re given by the unique solution of the following convex programmaximize d , s ˜ S ( d , s ) := M (cid:88) i =1 ˜ U i ( d i ) − N (cid:88) i =1 ˜ C i ( s i ) , (13a) subject to M (cid:88) i =1 d i = N (cid:88) i =1 s i , (13b) ≤ s i ≤ κ , i ∈ N , (13c) d ≤ d i , i ∈ M , (13d) where ˜ U i ( d i ) := (cid:18) − d i N κ − ( M − d (cid:19) U i ( d i ) + 1 N κ − ( M − d (cid:90) d i d U i ( z ) dz, (14) ˜ C i ( s i ) := (cid:18) s i ( N − κ − M d (cid:19) C i ( s i ) − N − κ − M d (cid:90) s i C i ( z ) dz. (15)Computing Nash equilibria is, in general, hard as shown by Daskalakis et al. (2009). Theorem2 establishes the computation of the market allocation at the Nash equilibrium—and the Nashequilibrium itself—through the solution of a convex program in ( d , s ) instead of solving M+Nproblems in the actions ( θ d , θ s ) , which can be cumbersome depending on the structure of theutility and cost functions. The crux of Theorem 2 is the construction of an appropriate convexprogram that yields the market allocation at the Nash equilibrium—a technique closely relatedto the use of potential functions in characterizing Nash equilibria (Monderer and Shapley (1996)).However, the functions (14) and (15) are not potentials for G , since they depend on the allocationsand not on the players’ decisions. Hence, we cannot use these functions to conclude anythingabout convergence of best response dynamics to the Nash equilibrium. However, in the followingsection, we exploit the structure of ˜ U i and ˜ C i to find bounds on the efficiency loss and the markupof prices observed at the Nash equilibrium. The structure of the modified utility and cost functions allows us to make a number of interest-ing observations about the behavior of strategic market actors. First, note that since C i ( s i ) areassumed convex and increasing, it follows that ˜ C i ( s i ) ≥ C i ( s i ) , ∀ s i ≥ . Similarly, since U i ( d i ) areconcave and increasing, for each consumer we have ˜ U i ( d i ) ≤ U i ( d i ) , ∀ d i ≥ . In effect, strategicsuppliers misrepresent their costs functions through ˜ C i ( s i ) , which are greater than the true cost C i ( s i ) at every s i . On the other hand, strategic consumers misrepresent their utilities through ˜ U i ( d i ) , which are smaller than the true utility U i ( d i ) at every d i . Moreover, S (˜ d , ˜ s ) ≤ S ( d ∗ , s ∗ ) sincethe maximum value of S occurs at ( d ∗ , s ∗ ) . However, in our next result, we show that the socialwelfare at the Nash is bounded below and can be relatively close to the optimal value providedsome minimum available production capacity. In order to compute bounds on price markups atthe Nash equilibrium we utilize the Lerner index ( see Lerner (1934)), which we define as LI ( ˜ θ d , ˜ θ s ) := 1 − p ( ˜ θ d , ˜ θ s ) max i (cid:26) ∂∂s i C i (cid:16) S (˜ θ is , p ( ˜ θ d , ˜ θ s )) (cid:17)(cid:27) . (16)7he Lerner index measures a firm’s market power and it varies from zero to one, with highervalues indicating greater market power. The following result summarizes the efficiency loss at theNash equilibrium and the price markups. Theorem 3
Suppose Assumptions 1-2 hold. Let ( d ∗ , s ∗ ) be the socially optimal allocation from (1) and (˜ d , ˜ s ) be the market allocation at the Nash equilibrium of G . It follows that M (cid:88) i =1 U i ( ˜ d i ) − N (cid:88) i =1 C i (˜ s i ) ≥ M (cid:88) i =1 U i ( d ∗ i ) − (cid:18) − κ ζ (cid:19) − N (cid:88) i =1 C i ( s ∗ i ) , (17) where ζ := N κ − M d and ζ ∈ ( κ , ∞ ) . Moreover, when ζ ∈ [4 κ , ∞ ) we have M (cid:88) i =1 U i ( ˜ d i ) − N (cid:88) i =1 C i (˜ s i ) ≥ N (cid:88) i =1 U i ( d ∗ i ) − N (cid:88) i =1 C i ( s ∗ i ) . (18) Finally, the Lerner index at the Nash equilibrium satisfies LI ( ˜ θ d , ˜ θ s ) ≤ κ ζ < . (19)In effect, Theorem 3 provides a lower bound on the social welfare at the Nash equilibrium andan upper bound on the market price with respect to the true marginal cost of suppliers. Noticethat S (˜ d , ˜ s ) is in the worst case 3/4 of the aggregate utility less ζζ − κ of the aggregate costs at theefficient allocation. We do not claim this bound is tight; there may exist an even tighter bound onthe social welfare the computation of which we relegate to future work. Higher values of ζ yieldvalues of the social welfare at the Nash equilibrium closer to S ( d ∗ , s ∗ ) . The worst-case values for S (˜ d , ˜ s ) arise when ζ → κ , although it never reaches it. Intuitively, when the aggregate produc-tion capacity of supply is relatively close to the total inelastic demand, then firms’ market powerincreases over consumers, gradually inducing pivotalness as ζ → κ . Specifically, for ζ ∈ ( κ , κ ) the efficiency loss can be arbitrarily high, similar to that derived by Xu et al. (2016) for a marketwith capacity-constrained suppliers. When ζ ∈ [2 κ , ∞ ) the worst-case aggregate cost coefficientin (17) is equal to two and we recover the worst-case bound of Johari and Tsitsiklis (2011) derivedfor uncapacitated supply function competition. Moreover, (18) shows that provided some mini-mum available production capacity, the social welfare at the Nash equilibrium is no lower than3/4 of the aggregate utility less 4/3 of the aggregate cost at the efficient allocation, which is notmuch lower than S ( d ∗ , s ∗ ) . From (19) note that the Lerner index is strictly less than one due to thenon-pivotal supplier assumption. As ζ grows large, LI ( ˜ θ d , ˜ θ s ) goes to zero, indicating less marketpower on the supply side. As M d approaches N κ , the index grows large implying high mar-ket power since there is little available capacity to supply anything more than the total inelasticdemand. In this section we provide numerical experiments to illustrate the behavior of the social welfareunder perfect competition and strategic interactions with respect to specific problem parameters.As shown in Section 5, the key parameter that affects social welfare is the total flexible capacity inthe market ζ . 8
10 20 30 40
Total flexible capacity ( ) S o c i a l W e l f a r e ( d , s )( d * , s * ) (a) potential gains from price-responsive demand (b) ⇣ [4 , )
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A Proof of Theorem 1.
The crux of our derivations relies on Lagrangian duality to establish that the equilibrium con-ditions of (8)-(9) together with (10) are equivalent to the optimality conditions of (1). We beginwith the consumer’s problem. The payoff in (6) is concave in each player’s action θ id . Hence, theKarush-Kuhn-Tucker (KKT) optimality conditions are both necessary and sufficient. An optimalstrategy θ i ∗ d ≥ must satisfy ∂U i (cid:0) D ( θ i ∗ d , µ ) (cid:1) ∂d i = µ, if θ i ∗ d > (21a) ∂U i (cid:0) D ( θ i ∗ d , µ ) (cid:1) ∂d i ≤ µ, if θ i ∗ d = 0 (21b)Each supplier’s payoff is concave in the action θ is . Moreover, an optimal strategy θ i ∗ s must lie inthe closed interval [0 , µκ ] . If not, then it is easy to show that S i ( θ is , µ ) < for θ is > µκ . Therefore,such strategies cannot occur at the equilibrium since they yield negative payoff. As such, anoptimal strategy θ i ∗ s must satisfy ∂C i (cid:0) S ( θ i ∗ s , µ ) (cid:1) ∂s i ≤ µ, if ≤ θ i ∗ s < µκ (22a)11 C i (cid:0) S ( θ i ∗ s , µ ) (cid:1) ∂s i ≥ µ, if < θ i ∗ s ≤ µκ (22b)We now turn to problem (1) solved by the market manager. Associate Lagrange multiplier λ withthe equality constraint (1b). The objective function is continuous and concave over a compactconvex set. Therefore, there exists at least one optimal solution ( d ∗ , s ∗ ) and λ ∗ ≥ that satisfy ∂U i ( d ∗ i ) ∂d i = λ ∗ , if d ∗ i > d (23a) ∂U i ( d ∗ i ) ∂d i ≤ λ ∗ , if d ∗ i = d (23b)Similarly, the supply vector s ∗ must satisfy ∂C i ( s ∗ i ) ∂s i ≥ λ ∗ , if ≤ s ∗ i < κ (24a) ∂C i ( s ∗ i ) ∂s i ≤ λ ∗ , if < s ∗ i ≤ κ (24b)Primal feasibility requires M (cid:88) i =1 d ∗ i = N (cid:88) i =1 s ∗ i . (25)Note that λ ∗ > since U i and C i are strictly increasing and there exists at least one s ∗ i > . Ifthe pair ( s ∗ , λ ∗ ) satisfies (24) and we let θ is = λ ∗ ( κ − s ∗ i ) then ( θ s , λ ∗ ) satisfy (22) and θ s ≥ .In effect (24) become equivalent to (22). Similarly, if the pair ( d ∗ , λ ∗ ) satisifes (23) and we let θ id = λ ∗ ( d ∗ i − d ) then ( θ d , λ ∗ ) satisfy (21) and θ d ≥ . In this case, (23) become equivalent to (21).Finally, the market clearing condition in (25) yields λ ∗ = µ . Hence, ( θ d , θ s , µ ) is a competitiveequilibrium. Now suppose that ( θ ∗ d , θ ∗ s , µ ) satisfy (21),(22) and (5). Let s i = S ( µ, θ i ∗ s ) for i ∈ N and d i = D ( µ, θ i ∗ d ) for i ∈ M . Then, it is easy to verify that the vector ( d , s ) satisfies (23) and (24).Therefore, ( d , s ) is an efficient allocation. B Proof of Lemma 1.
Let firm i be a pivotal supplier. Then it must hold RSI i = ( N − κ M d < , (26)i.e., the total production capacity less that of i ’s is less than the total inelastic demand in the market.In this case, the first derivative of the supplier’s payoff becomes ∂π is ( θ is , θ − is , θ d ) ∂θ is = κ N κ − M d − N κ − M d ) ∂C i ∂s i (cid:18) T θ d + T θ − is ( T θ d + T θ s ) (cid:19) > . (27)Therefore, the payoff is strictly increasing in the action θ is and grows unbounded. A Nash equilib-rium does not exist. 12 Proof of Theorem 2.
We break the proof into five steps. First, we show that any Nash equilibrium has at least twopositive components and we derive the necessary and sufficient conditions for such equilibrium.Next we establish the existence and uniqueness of the market allocation at the Nash equilibriumand derive the optimality conditions for (13). We show that under the demand bid (2) and thesupply offer (3) the equilibrium conditions of all players become equivalent to the optimalityconditions of (13). Finally, we establish uniqueness of the Nash equilibrium.
Step 1. Any Nash Equilibrium Has at Least Two Positive Components.
First, it is straightforward to see that T θ d + T θ s = 0 cannot occur at the Nash equilibriumsince N κ > M d and therefore the market does not clear. Next, we consider two cases. First,assume that T θ d = 0 . Fix firm j and let T θ − js = 0 . Note that, in this case, θ js > is not possibleby the non-pivotal supplier assumption. A Nash equilibrium cannot exist with all consumersbidding zero and all but one supplier offering a strictly positive θ is . Second, assume T θ s = 0 . Fixconsumer j and let T θ − jd = 0 . Then, θ jd > implies d j > d . Hence, consumer j faces a totalavailable residual supply equal to N κ − M d . In this case, the payoff of consumer j is given by U j ( d + N κ − M d ) − d N κ − M d θ jd − θ jd , (28)which is strictly increasing as θ jd becomes small and attains its maximum when θ jd = 0 . Thusfor any θ jd > there exists an infinitesimally smaller and positive θ jd that yields higher payoff.Moreover, by definition U j (0 ,
0) = U ( d ) = 0 . A Nash equilibrium does not exist in this case.Hence, at the Nash equilibrium, the vector θ = ( θ d , θ s ) has at least two positive components. Step 2. Necessary and Sufficient Nash Equilibrium Conditions.
Having shown that anyNash equilibrium must have at least two positive components, we only focus in the region where T θ d + T θ s > . Note that, for each consumer (firm), her payoff is strictly concave in the action θ id ( θ is ). Hence, the KKT optimality conditions are both necessary and sufficient. Moreover, we musthave ≤ ˜ θ is ≤ θ i max := κ ( N − κ − M d M (cid:88) i =1 θ id + N (cid:88) j (cid:54) = i θ js , in order for S (cid:16) ˜ θ is , p ( θ ) (cid:17) ≥ . We have the following equilibrium conditions.A demand profile ˜ θ d = (cid:16) ˜ θ d , . . . , ˜ θ Nd (cid:17) is a Nash profile if and only if (cid:32) − D (˜ θ id , p ( ˜ θ d , ˜ θ s )) N κ − ( M − d (cid:33) ∂U i (cid:16) D (˜ θ id , p ( ˜ θ d , ˜ θ s )) (cid:17) ∂d i = p ( ˜ θ d , ˜ θ s ) , if ˜ θ id > (29a) (cid:32) − D (˜ θ id , p ( ˜ θ d , ˜ θ s )) N κ − ( M − d (cid:33) ∂U i (cid:16) D (˜ θ id , p ( ˜ θ d , ˜ θ s )) (cid:17) ∂d i ≤ p ( ˜ θ d , ˜ θ s ) , if ˜ θ id = 0 (29b) A supply profile ˜ θ is = (cid:16) ˜ θ s , . . . , ˜ θ Ns (cid:17) is a Nash equilibrium if and only if S (cid:16) ˜ θ is , p ( ˜ θ d , ˜ θ s ) (cid:17) ( N − κ − M d ∂C i (cid:16) S (˜ θ is , p ( ˜ θ d , ˜ θ s ))) (cid:17) ∂s i ≤ p ( ˜ θ d , ˜ θ s ) , if ≤ ˜ θ is < θ i max (30a) S (cid:16) ˜ θ is , p ( ˜ θ d , ˜ θ s )) (cid:17) ( N − κ − M d ∂C i (cid:16) S (˜ θ is , p ( ˜ θ d , ˜ θ s ))) (cid:17) ∂s i ≥ p ( ˜ θ d , ˜ θ s ) , if < ˜ θ is ≤ θ i max (30b) The equilibrium conditions (29) and (30) are derived from the KKT conditions of each player’spayoff maximization problem, where the payoff of each consumer and supplier is given by ex-pressions (11) and (12) respectively.
Step 3. Existence and Uniqueness of the Nash Market Allocation.
Equipped with the aboverelations we now proceed to the market manager’s problem. Note that ˜ U i ( d i ) is strictly concaveand ˜ C i ( s i ) is strictly convex. Hence, the objective function (cid:101) S ( d , s ) is continuous and strictly con-cave over a compact set. Specifically, the Hessian matrix H of ˜ S ( d , s ) has diagonal elements h ii = ∂ ˜ U i ( d i ) ∂d i < , for i = 1 , . . . , M − ∂ ˜ C i ( s i ) ∂s i < , for i = M + 1 , . . . , M + N (31)and h ij = 0 for i (cid:54) = j . Hence, H is negative definite and there exists a unique optimal solution (˜ d , ˜ s ) to (13). Step 4. Necessary and Sufficient Conditions for the Market Allocation.
Let (˜ d , ˜ s ) be theunique optimal solution to (13). There exists a Lagrange mutliplier ˜ λ such that (cid:32) − ˜ d i N κ − ( M − d (cid:33) ∂U i ( ˜ d i ) ∂d i = ˜ λ, if ˜ d i > d (32a) (cid:32) − ˜ d i N κ − ( M − d (cid:33) ∂U i ( ˜ d i ) ∂d i ≤ ˜ λ, if ˜ d i = d (32b) (cid:18) s i ( N − κ − M d (cid:19) ∂C i ( ˜ s i ) ∂s i ≥ ˜ λ, if ≤ ˜ s i < κ (32c) (cid:18) s i ( N − κ − M d (cid:19) ∂C i ( ˜ s i ) ∂s i ≤ ˜ λ, if < ˜ s i ≤ κ (32d) M (cid:88) i =1 ˜ d i = N (cid:88) i =1 ˜ s i . (32e) Note that since there is at least one ˜ s i > and U i and C i are strictly increasing, then ˜ λ > .Consider the action vectors ˜ θ id = ˜ λ ( ˜ d i − d ) for i ∈ M and ˜ θ is = ˜ λ ( κ − ˜ s i ) for i ∈ N . Note that θ id ≥ and θ is ≥ for every consumer and every firm respectively. Suppose now that d i > d and d j = d for j (cid:54) = i and let s i = κ for i ∈ N . This implies that d i = d + N κ − M d . Then from(32a) we have ˜ λ = 0 . However, we have ∂U j ( d ) ∂d j > for each j ∈ M . Therefore, (32b) cannothold for every j (cid:54) = i . Thus, the vector (cid:16) ˜ θ d , ˜ θ s (cid:17) cannot have all components zero except one θ id > .Similarly, (cid:16) ˜ θ d , ˜ θ s (cid:17) cannot have all components zero except one θ is > for some firm i ∈ N . This isobvious by the non-pivotal supplier assumption since it holds ( N − κ > M d for every supplier i . Hence, at least two components of (cid:16) ˜ θ d , ˜ θ s (cid:17) are positive. Moreover, since ˜ s i = κ if and only14f θ is = 0 , ˜ s i = 0 if and only if θ is = θ i max , then it is not hard to see that (32) become equivalent to(29)-(30). Hence, the action vector ( ˜ θ d , ˜ θ s ) is a Nash equilibrium. This also establishes existence ofthe Nash equilibrium.We now reverse the argument. Let ( ˜ θ d , ˜ θ s ) be a Nash equilibrium profile. That is, it satisfies(29)-(30). Therefore, it has at least two positive components and p ( ˜ θ d , ˜ θ s ) > . Define the demandallocation ˜ d i = d + ˜ θ id p ( ˜ θ d , ˜ θ s ) for i ∈ M and the supply allocation ˜ s i = κ − ˜ θ is p ( ˜ θ d , ˜ θ s ) for i ∈ N . Itfollows that (˜ d , ˜ s ) satisfy (32) with ˜ λ = p ( ˜ θ d , ˜ θ s ) . Step 5. Uniqueness of the Nash Equilibrium.
We have shown that all Nash equilibria yielda unique market allocation. Uniqueness of the Nash equilibrium follows from the fact that thetransformation from ( θ d , θ s ) to ( d , s , λ ) is one-to-one. D Proof of Theorem 3.
Step 1. Bounding the Price Markups
To derive the upper bound on the Lerner index we notethat at the Nash equilibrium there exists at least one firm such that S i (cid:16) p ( ˜ θ d , ˜ θ s ) (cid:17) < κ or ˜ θ is > .Therefore, p ( ˜ θ d , ˜ θ s ) ≤ S (cid:16) ˜ θ is , p ( ˜ θ d , ˜ θ s ) (cid:17) ζ − κ ∂C i (cid:16) S i (cid:16) ˜ θ is , p ( ˜ θ d , ˜ θ s ) (cid:17)(cid:17) ∂s i ≤ (cid:18) κ ζ − κ (cid:19) ∂C i (cid:16) S i (cid:16) ˜ θ is , p ( ˜ θ d , ˜ θ s ) (cid:17)(cid:17) ∂s i ≤ ζζ − κ max i ∂C i (cid:16) S i (cid:16) ˜ θ is , p ( ˜ θ d , ˜ θ s ) (cid:17)(cid:17) ∂s i . (33) Utilizing (33) and substituting in the expression of LI ( ˜ θ d , ˜ θ s ) yields the bound in (19). Step 2. Bounding the Social Welfare.
Let x = ( d , s ) and ˜ x = (˜ d , ˜ s ) . In this step we aim to15ound the social welfare at the Nash equilibrium, i.e., S (˜ x ) . Specifically, S (˜ x ) ≥ S (˜ x ) + M + N (cid:88) i =1 ∂ ˜ S i (˜ x i ) ∂x i ( x ∗ i − ˜ x i ) (34a) = S (˜ x ) + (cid:40) M (cid:88) i =1 ∂ ˜ U i ( ˜ d i ) ∂d i ( d ∗ i − ˜ d i ) − N (cid:88) i =1 ∂ ˜ C i (˜ s i ) ∂s i ( s ∗ i − ˜ s i ) (cid:41) (34b) = S (˜ x ) + M (cid:88) i =1 (cid:32) − ˜ d i ζ + d (cid:33) ∂U i ( ˜ d i ) ∂d i ( d ∗ i − ˜ d i ) − N (cid:88) i =1 (cid:18) s i ζ − κ (cid:19) ∂C i (˜ s i ) ∂s i ( s ∗ i − ˜ s i ) (34c) ≥ S (˜ x ) + M (cid:88) i =1 (cid:32) − ˜ d i ζ + d (cid:33) (cid:16) U i ( d ∗ i ) − U i ( ˜ d i ) (cid:17) − N (cid:88) i =1 (cid:18) s i ζ − κ (cid:19) ( C i ( s ∗ i ) − C i (˜ s i )) (34d) ≥ M (cid:88) i =1 U i ( ˜ d i ) − N (cid:88) i =1 C i (˜ s i ) + M (cid:88) i =1 (cid:32) − ˜ d i d ∗ i (cid:33) (cid:16) U i ( d ∗ i ) − U i ( ˜ d i ) (cid:17) − (cid:18) κ ζ − κ (cid:19) N (cid:88) i =1 ( C i ( s ∗ i ) − C i (˜ s i )) (34e) ≥ M (cid:88) i =1 (cid:32) ˜ d i d ∗ i (cid:33) + 1 − ˜ d i d ∗ i U i ( d ∗ i ) − (cid:18) ζζ − κ (cid:19) N (cid:88) i =1 C i ( s ∗ i ) (34f) ≥ M (cid:88) i =1 U i ( d ∗ i ) − (cid:18) ζζ − κ (cid:19) N (cid:88) i =1 C i ( s ∗ i ) . (34g) Inequality (34a) follows from the optimality conditions of (13) while (34c) from the definitions of ˜ U i and ˜ C i . Inequality (34d) follows from concavity of U i ( d i ) and convexity of C i ( s i ) . Step (34e)follows from the fact that d ∗ i < ζ + d for every i ∈ M and ˜ s i ≤ κ for every i ∈ N . Inequality (34f)follows from concavity of U i ( d i ) and that U i (cid:32)(cid:32) − ˜ d i − d d ∗ i − d (cid:33) d + ˜ d i − d d ∗ i − d d ∗ i (cid:33) ≥ (cid:32) − ˜ d i − d d ∗ i − d (cid:33) U i ( d ) + ˜ d i − d d ∗ i − d U i ( d ∗ i ) ⇒ U i ( ˜ d i ) (cid:38) ˜ d i d ∗ i U i ( d ∗ i ) . The last inequality follows from minimizing the expression y − y + 1 , which is minimized for y ∗ = 1 / , where y = ˜ d i /d ∗ i . Finally, note that (cid:18) ζζ − κ (cid:19) is a decreasing function of ζ . Hence, when ζ ∈ [4 κ , ∞ ) the highest value of (cid:18) ζζ − κ (cid:19)(cid:19)