Modeling Strategic Investment Decisions in Spatial Markets

2 Institute of Energy Econoics at the University of Cologne (EWI) Alte Wagenfabrik Vogelsanger Straße Köln Gerany Tel.: +49 () Fax: +49 () ISSN: The responsibility for working papers lies solely with the authors. Any views expressed are those of the authors and do not necessarily represent those of the EWI.

3 The influence of arket structure on production capacity investents in resource arkets Stefan Lorenczik a, Raiund Malischek b, Johannes Trüby c a Institute of Energy Econoics at the University of Cologne; Vogelsanger Strasse 321, 5827 Cologne, Gerany; stefan.lorenczik@ewi.uni-koeln.de b Institute of Energy Econoics at the University of Cologne; Vogelsanger Strasse 321, 5827 Cologne, Gerany; raiund.alischek@ewi.uni-koeln.de c International Energy Agency (IEA); 9 rue de la Fédération, Paris Cedex 15, France, johannes.trueby@iea.org Abstract Markets for natural resources and coodities are often oligopolistic. In these arkets, production capacities are key for strategic interaction between the oligopolists. We analyze how different arket structures influence oligopolistic capacity investents and thereby affect supply, prices and rents in spatial natural resource arkets using atheatical prograing odels. The odels coprise an investent period and a supply period in which players copete in quantities. We copare three odels, one perfect copetition and two Cournot odels, in which the product is either traded through long-ter contracts or on spot arkets in the supply period. Tractability and practicality of the approach are deonstrated in an application to the international etallurgical coal arket. Results ay vary substantially between the different odels. The etallurgical coal arket has recently ade progress in oving away fro long-ter contracts and ore towards spot arket-based trade. Based on our results, we conclude that this regie switch is likely to raise consuer rents but lower producer rents. The total welfare differs only negligibly. Keywords: Spatial Natural Resource Markets, Capacity Investent, Oligopoly, Equilibriu Proble with Equilibriu Constraints (EPEC), Coal JEL classification: L13, L72, C61, C7 1. Introduction Markets for natural resources and coodities such as iron ore, copper ore, coal, oil or gas are often highly concentrated and do not appear to be copetitively organized. In these arkets, large copanies run ines, rigs or gas wells and trade their product globally. In the short ter, arginal production costs and capacities are given and deterine the copanies copetitive position in the oligopolistic arket. However, in the longer ter, copanies can choose their capacity and consequently alter their copetitive position. Investing in production capacity is a key anagerial challenge and deterining the right aount of capacity is rarely trivial in oligopolistic arkets. Suppliers have to take copetitors reactions into account not only when deciding on the best supply level but also when choosing the best aount of capacity. In this paper, we introduce three different odels to address this capacity expansion proble in oligopolistic natural resource arkets under varying assuptions of arket structure and conduct. Moreover, we pursue the question as to how different arket structures influence capacity investents, supply, prices and rents. The odels coprise two periods: an investent period and a supply period in which players copete in quantities. We explicitly account for the spatial structure of natural resource arkets, i.e., The contents of this paper reflect the opinions of its authors only and not those of the EWI or the International Energy Agency. Although this paper was written when Johannes Trüby was eployed by the IEA, the views expressed in this paper do not necessarily reflect the views or policy of the IEA Secretariat or of its individual eber countries.

4 deand and supply regions are geographically separated and arket participants incur distance-dependent transportation costs. The first odel assues arkets to be contestable; hence investent follows copetitive logic. Solving this odel yields the sae result as would be given by a perfectly copetitive arket. The second odel assues the product to be sold through long-ter contracts under iperfect copetition. Even though supply takes place in period two, the supply and investent decisions are ade siultaneously in period one. The outcoe is tered open-loop Cournot equilibriu and corresponds to the result of a static one-period Cournot gae (accounting for investent costs). The third odel assues that investent and supply decisions are ade consecutively: In the first period, oligopolists choose their capacity investent, anticipating the best-supply response of their rivals on a spot arket in the second period. The resulting equilibriu is tered closed-loop Cournot equilibriu and ay differ fro the open-loop outcoe. As discussed for instance in Fudenberg and Tirole (1991) in a ore general context, each player in the closed-loop odel has a strategic incentive to deviate fro his first period open-loop action as he can thereby influence the other players second period action. Applying this general econoic fraework to the capacity expansion proble exained in this paper, tends to lead to higher investent and supply levels in the closed-loop odel and hence to lower prices. Coputing open-loop gaes is relatively well understood, and existence and uniqueness of the equilibriu can be guaranteed under certain conditions. The open-loop Cournot odel can be solved via the Karush-Kuhn-Tucker conditions as a ixed copleentarity proble. This approach has been widely deployed in analyzing spatial arket equilibria without investents, e.g., for stea coal arkets (Kolstad and Abbey, 1984; Haftendorn and Holz, 21; Trüby and Paulus, 212), etallurgical coal arkets (Graha et al., 1999; Trüby, 213), natural gas arkets (Gabriel et al., 25; Holz et al., 28; Growitsch et al., 213), wheat arkets (Kolstad and Burris, 1986), oil arkets (Huppann and Holz, 212) or for iron ore arkets (Hecking and Panke, 214). The closed-loop odel is coputationally challenging and due to its highly non-linear nature, existence and uniqueness of (pure strategy) equilibria cannot be guaranteed. Previous closed-loop odels in energy arket analysis have priarily been used to study restructured (single node) electricity arkets, e.g., by Murphy and Seers (25) and Wogrin et al. (213a,b). Aong others, this research strea has analyzed the iplications of closed- and open-loop odeling on arket output and social welfare as well as characterized conditions under which closed- and open-loop odel results coincide. Our closed-loop odel, which is forulated as an Equilibriu Proble with Equilibriu Constraints (EPEC), is ipleented using a diagonalization approach (see, e.g., Gabriel et al., 212). In doing so, we reduce the solution of the EPEC to the solution of a series of Matheatical Progras with Equilibriu Constraints (MPEC). Concerning the solution of the MPECs we ipleent two algoriths, grid search along the investent decisions of the individual players and a Mixed Integer Linear Progra reforulation following Wogrin et al. (213a). We deonstrate the tractability and practicality of our investent odels in an application to the international etallurgical (or coking) coal trade. Metallurgical coal is, due to its special cheical properties, a key input in the process of steel-aking. The arket for this rare coal variety is characterized by a spatial oligopoly with producers ainly located in Australia, the United States and Canada copeting against each other and providing the bulk of the traded coal (Bowden, 212; Trüby, 213). The players hold existing ining capacity and can invest into new capacity. Investent and ining costs differ regionally. Key uncertainties in this arket are deand evolution and price responsiveness of deand. We therefore copute sensitivities for these paraeters to deonstrate the robustness of our results. Our findings are generally in line with previous results found in the literature, i.e., we find that prices and supply levels in the closed-loop gae fall between those in the perfect copetition and the open-loop gae (see, e.g., Murphy and Seers, 25). If investent costs are low copared to variable costs of supply, the strategic effect of the bilevel optiization in the closed-loop gae diinishes. With investent costs approaching zero, the closed-loop result converges to the open-loop result. Hence, the closed-loop odel is particularly useful for capital-intensive natural resource industries in which the product is traded on spot arkets. The nuerical results for supply levels, prices and rents in the etallurgical coal arket analysis differ 2

5 arkedly between the three odels. Consistent with actual industry investent pipelines, our odel suggests that the bulk of the future capacity investent coes fro copanies operating in Australia followed by Canadian and US firs. Starting in 21, the etallurgical coal arket has undergone a paradig shift, oving away fro long-ter contracts and ore towards a spot arket-based trade with siilar tendencies being observed in other coodity arkets such as the iron ore trade. In light of our findings, this effect is detriental to the copanies profits but beneficial to consuer rents. The effect on welfare is negligible: Gains in consuer rents and losses in producers profits are of alost equal agnitude. The contribution of this paper is threefold: First, by extending the ulti-stage investent approach to the case of spatial arkets, we introduce a novel feature to the literature on Cournot capacity expansion gaes. Second, we outline how our odeling approach can be ipleented and solved to analyze capacity investents in natural resource arkets. We thereby extend previous research on natural resource arkets, which has typically assued capacities to be given. Finally, we illustrate and discuss the odel properties on the basis of a real-world application to the international etallurgical coal trade and draw conclusions for this arket. By coparing open- and closed-loop odel results, we illustrate possible consequences of the ongoing regie switch fro long-ter contracts to a ore spot arket-based trade in the international etallurgical coal arket. The reainder of the paper is structured as follows: Section 2 describes the odels developed in this paper and Section 3 provides details about their ipleentation. The data is outlined in Section 4, results are presented in Section 5. Section 6 discusses coputational issues and Section 7 concludes. 2. The Model We introduce three different approaches to the capacity expansion proble two open-loop odels and a closed-loop odel. In the open-loop odels, players decide siultaneously on their investent and production levels, whereas in the closed-loop odel players first decide on their investent levels and then, based on observed investent levels, decide on their production levels. The two open-loop odels vary in their underlying arket structure: one odel assues perfect copetition, the other odel assues Cournot copetition with a copetitive fringe. The closed-loop odel also assues Cournot copetition with a copetitive fringe General Setting and Notations We assue a spatial, hoogeneous good arket consisting of producers i I, production facilities M and deand regions j J. Each producer i owns production facilities M i M. Furtherore, we assue that M i M j = for i j, i.e., production facilities are exclusively owned by one producer. Producers decide on both their investent in production facilities as well as on their supply levels, with supply taking place at tie points t T. 1 The supply fro production facility to arket j at tie t is given by x t,j. Total production of production facility at tie t is hence given by j xt,j. It is liited by the facilities capacity cap + y, where cap is the initial production capacity and y denotes the capacity investent. Capacity investents y are non-negative and liited by y ax. Capacity investents in an existing production facility (i.e., cap ) can be interpreted as capacity expansions, and investents in the case of cap = as newly built production facilities. Investent expenditures for facility are given by C inv the investent level y, with k denoting arginal investent costs, i.e., C inv (y ) = k y.. We assue that C inv is a linear function in Variable costs C var,t at tie t are specific to the production facility. They are coposed of transportation costs τ,j t per unit delivered fro to arket j as well as the variable production costs vt. We 1 In our application to the etallurgical coal arket, we consider only one tie point; in this section, however, we present the odel in a ore general for. 3

6 assue that v t is a linear function in the total production of the facility. Total variable costs of facility at tie t therefore aount to C var,t (x t ) = j (x t,j τ t,j) + v t ( j x t,j), with x t = (x t,j ) j denoting the production vector of facility at tie t. Market prices Pj t in arket j at tie t are given by a linear inverse deand function, i.e., P t j = a t j b t j x t,j. Abbreviation Model sets M j J i I t T Model paraeters k v t τ,j t a t j b t j cap y ax Model variables C var,t C inv x t,j Pj t y Table 1: Model sets, paraeters and variables Description Production facilities Markets Players Tie Marginal investent costs [US$ per unit per year] Variable production costs [US$ per unit] Transportation costs [US$ per unit] Reservation price [US$ per unit] Linear slope of deand function Initial production capacity [units per year] Maxiu capacity expansion [units per year] Total variable production costs [US$] Investent expenditures [US$] Supply [units] Market price [US$ per unit] Capacity investents [units per year] 2.2. Model 1: The Open-Loop Perfect Copetition Model In the open-loop perfect copetition odel (in the following siply tered perfect copetition odel ), each producer i I solves the optiization proble ( ax P x t j t x t,j C var,t (x t ) ) C inv (y ),y: Mi t T j J M i subject to M i P t j = a t j b t j (X t i,j + X t i,j), j, t cap + y x t,j, M i, t (λ t ) j y ax y, M i (θ ) x t,j, M i, j, t y, M i 4

7 while taking the supplies X i,j t of the other producers ( i) as given. Here and in the following, we use the abbreviation XI t = 1,j i I 1 M i x t,j for soe I 1 I. Hence, in the perfect copetition odel, each producer siultaneously akes his ( long-ter ) investent and ( short-ter ) production decisions in order to axiize profits. In doing so, each producer takes capacity restrictions into account. However, players do not take into account their influence on price. Any solution to the above optiization proble has to satisfy the short-ter Karush-Kuhn-Tucker (KKT) conditions Cvar,t (x t ) x t [a t j b t j (Xi,j t + X i,j)] t + λ t x t,j, i, M i, j, t,j cap + y j x t,j λ t, i, M i, t as well as the long-ter KKT conditions k t T λ t + θ y, i, M i y ax y θ, i, M i. In equilibriu, all KKT conditions have to hold siultaneously. Uniqueness of the solution is guaranteed due to the quasi-concave objective function and the convexity of the restrictions. The derived KKT conditions are thus necessary and sufficient for obtaining the solution Model 2: The Open-Loop Cournot Model with Copetitive Fringe As in the perfect copetition odel, in the open-loop Cournot odel with copetitive fringe (in the following siply tered open-loop odel ), each producer i I solves the optiization proble subject to ax x t,y: Mi t T M i ( j J Pj t x t,j C var,t (x t ) ) C inv (y ) M i P t j = a t j b t j (X t i,j + X t i,j), j, t cap + y x t,j, M i, t (λ t ) j y ax y, M i (θ ) x t,j, M i, j, t y, M i while taking the supplies X i,j t of the other producers ( i) as given. Each producer siultaneously akes his investent and production decisions with the objective to axiize profits. In doing so, each producer takes into account his capacity restrictions and his influence on price. We assue this price influence to be represented by a conjectural variation paraeter ψ i, where P t j x t,j = ψ i b t j for all M i. This allows us to odel Cournot behavior with a copetitive fringe (ψ i = 1 for the Cournot players and ψ i = for the copetitive fringe). Any solution to the above optiization proble satisfies the short-ter Karush-Kuhn-Tucker (KKT) conditions Cvar,t (x t ) x t [a t j b t j (Xi,j t + X i,j)] t + ψ i b t j Xi,j t + λ t x t,j, i, M i, j, t,j cap + y j x t,j λ t, i, M i, t 5

8 as well as the long-ter KKT conditions k t T λ t + θ y, i, M i y ax y θ, i, M i. In equilibriu, all KKT conditions have to hold siultaneously. As in the perfect copetition case, uniqueness of the solution is guaranteed due to the quasi-concave objective function and the convexity of the restrictions. The derived KKT conditions are thus necessary and sufficient for obtaining the solution Model 3: The Closed-Loop Model In the closed-loop odel, producers play a two-stage gae: In the first stage, lead (oligopolistic) producers l (l L I) decide on their investent levels. In the second stage, they choose, based on observed investent decisions of the other lead producers, their production and supply levels. In addition, in the second stage, a further player, the fringe player (F ), akes his investent and supply decisions siultaneously The Short-Run Proble For a given investent vector (y l, y l ) of the lead producers, let the short-run (Cournot) proble of lead producer l be given by ( ax Pj t x t,j C var,t (x t ) ) subject to x t,j : M l t T M l j J P t j = a t j b t j (X t l,j + X t l,j + X t F,j), j, t cap + y j x t,j, M l, j, t. x t,j, M l, t (λ t ) As in the open-loop odel, lead producer l decides on his supplies while taking the supplies of the other lead producers ( l) and of the fringe player (F ) as given. We further assue Cournot behavior, i.e., we assue ψ l = 1. The corresponding KKT conditions to this proble are then given by Cvar,t (x t ) x t [a t j b t j (Xl,j t + X l,j t + XF,j)] t + b t j Xl,j t + λ t x t,j, M l, j, t,j cap + y j x t,j λ t, M l, t. In addition, for a given investent vector (y l, y l ), the (copetitive) fringe player faces the decision proble ( ax P x t,j,y: M j t x t,j C var,t (x t ) ) C inv (y ) F j J M F subject to t T M F P t j = a t j b t j (X t F,j + X t F,j), j, t cap + y j x t,j, M F, t (λ t ) y ax y, M F (θ ax,f ) x t,j, M F, j, t y, M F, 6

9 i.e., the fringe producer F akes his investent and supply decisions while taking the supply decisions of the other producers ( F ) as given. We assue that the fringe player does not take into account his influence on price, i.e., we assue ψ F =. The corresponding KKT conditions to this proble are then given by Cvar,t (x t ) x t [a t j b t j (XF,j t + X F,j)] t + λ t x t,j, M F, j, t,j k t T λ t + θ ax,f y, M F cap + y j x t,j λ t, M F, t y ax y θ ax,f, M F. In the short-run equilibriu, the KKT conditions of fringe and lead producers have to hold siultaneously. In the following, let x t,j (y l, y l ) denote the short-run production equilibriu for a given investent vector (y l, y l ) The Long-Run Proble The long-run proble for lead producer l L is given by ( ax P j t x t y : M,j(y l, y l ) C var,t ( x t (y l, y l ) ) C inv (y ) l t T j J M l subject to M l P t j = a t j b t j ( X t l,j(y l, y l ) + X t l,j(y l, y l ) + X t F,j(y l, y l )), j, t y, M l y, M l, y ax i.e., lead producer l chooses his investent levels in order to axiize profits for a given investent strategy of the other lead producers (y l ) under consideration of the resulting short-run equilibriu outcoe. Cobining the short-run and the long-run proble, we obtain the following MPEC for producer l, hereafter referred to as MPEC l : subject to ax Ω l y ax t T M l ( j J y, M l y, M l (a t j b t j (X t l,j + X t l,j + X t F,j)) x t,j C var,t (x t ) ) M l C inv (y ) Cvar,t (x t ) x t [a t j b t j (Xi,j t + X i,j)] t + [b t j Xi,j] t i L + λ t x t,j, M i, j, t,j cap + y j x t,j λ t, M i, t k t T λ t + θ ax,f y, M F y ax y θ ax,f, M F given the investent vector (y l ) of the other lead producers. Here, Ω l is given by Ω l = {(y ) Ml ; (x t,j, λt ) M,j J,t T, (y, θ ax,f ) MF }. 2 2 Note that the leader s decision variable is separated fro the followers decision variables by a seicolon. The latter are indirectly deterined by the leader s choice. 7

10 An investent strategy (ỹ l, ỹ l ) is a closed-loop equilibriu if for all l L, ỹ l solves l s MPEC proble MPEC l given ỹ l. The proble of finding a closed-loop equilibriu is hence of EPEC type (Gabriel et al., 212), and therefore existence and uniqueness of equilibria typically is non-trivial and paraeter dependent Discussion of the Models and Equilibriu Concepts strategies allow players to condition their actions on actions taken in previous tie periods; in open-loop strategies, this is not possible. Thus, equilibria in the closed-loop odel are by definition subgae perfect, whereas open-loop equilibria are typically erely dynaically (tie) consistent. The latter is a weaker equilibriu concept than subgae perfection. It requires only that no player has an incentive at any tie to deviate fro the strategy he announced at the beginning of the gae, given that no player has deviated in the past and no agent expects a future deviation (see Karp and Newbery, 1992). Therefore, with subgae perfect equilibria requiring actions to be optial in every subgae of the gae, i.e., requiring that no player has an incentive to deviate fro his strategy regardless of any deviation in the past, an equilibriu of the open-loop odel ay fail to be an equilibriu in the closed-loop gae. 3 Fudenberg and Tirole (1991) and the literature cited therein generally address the issue of diverging results of open-loop odels in coparison to closed-loop odels and provide intuition for the divergence: In the closed-loop odel, in contrast to the open-loop odel, a player s influence via its own actions in the first stage on the other players actions in the second stage is taken into account. Applying this intuition to the special case of the capacity expansion proble, Murphy and Seers (25) show that in the closed-loop equilibriu, arginal investent costs ay be higher than the su of the short-ter arginal value iplied by the KKT conditions. In particular, they note that the difference between the two characterizes the value for the player of being able to anipulate the short-ter arket by its first stage investents. This ay lead to higher investents and supplies and hence lower prices in the closed-loop odel copared to the open-loop odel. The existing literature on the subject, in particular the above entioned Murphy and Seers (25) as well as Wogrin et al. (213b), provides general properties of closed-loop and open-loop odels and conditions for diverging and non-diverging results between the two odels, assuing siplified settings (e.g., ignoring existing capacities). We conjecture that in a spatial application with non-generic data and existing capacities available to the players, equilibria are likely to deviate between the two odeling approaches, which is confired by our application to the etallurgical coal arket (see Sections 4 and 5). Analytical analysis is no longer available in this setting due to increased coplexity and thus akes a nuerical analysis necessary. The nuerical approach is also suitable to address an issue which to our knowledge has not yet been coprehensively touched upon in previous literature: a quantification of the agnitude of the divergence between closed-loop and open-loop odel results. 3. Ipleentation 3.1. Model 1: The Open-Loop Model Both open-loop odels introduced in Section 2, i.e., the open-loop perfect copetition odel and the open-loop Cournot copetition odel with copetitive fringe, are ipleented as ixed copleentarity probles Model 2: The Closed-Loop Model We solve the closed-loop odel using diagonalization (see for instance Gabriel et al., 212): 1. Set starting values for the investent decisions y l of all lead producers l L, a convergence criterion ɛ, a axiu nuber of iterations N and a learning rate R 2. n = 1 3 See Selten (1965) for the first foralization of the concept of subgae perfect equilibria and, e.g., Karp and Newbery (1989) for a general account on dynaic consistency. 8

11 3. Set yl n = y n 1 l 4. Do for all l L (a) Fix the investent decisions y l n of l (b) Solve player l s MPEC proble MPEC l to obtain an optial investent level y l (c) Set yl n 5. If yl n y n 1 l equal to R y l + (1 R) yl n < ɛ for all producers l L: quit 6. If n = N: quit 7. n = n + 1 and go back to step 3 Diagonalization thus reduces the closed-loop proble to a series of MPEC probles. Concerning the solution of the MPECs, we ipleent two procedures: grid search along the investent decision y l and a reforulation of the MPEC as a Mixed Integer Linear Progra (MILP). Ipleenting both the grid search and MILP reforulation allows for the coparison of the coputer run-ties of the two odels, with grid search typically being faster for reasonable grid sizes (see Section 6 for details on this issue) Grid Search When applying grid search along the investent decision y l, MPEC l siplifies to a sequence of copleentarity probles. In our ipleentation, the grid width in the grid search is the sae for all producers; the nuber of steps for a producer is thus dependent on his capacity expansion liit MILP Reforulation In addition to grid search, we ipleent a MILP reforulation of the MPEC. Non-linearities arise in the MPEC due to the copleentarity constraints and the non-linear ter in the objective function. The forer are replaced by their corresponding disjunctive constraints (see Fortuny-Aat and McCarl, 1981), e.g., we replace cap + y x t,j λ t j by M λ b λ,t λ t M λ (1 b λ,t) cap + y j x t,j for soe suitably large constant M λ and binary variables b λ,t. For the discretization of the non-linear ter in the objective function, we proceed following Pereira et al. (25) using a binary expansion of the supply variable. The binary expansion of x t,j is given by x t,j = x + x 2 k b x k,,j,t, where x is the lower bound, x the stepsize, k the nuber of discretization intervals and b x k,,j,t binary variables. Substituting Pj t x + x k 2k zk,,j,t x for P j t xt,j, we have to ipose the additional constraints for soe suitably large constant M x. z x k,,j,t M x b x k,,j,t k P t j z x k,,j,t M x (1 b x k,,j,t) 9

12 4. Data Set The odels are paraetrized with data for the international etallurgical coal arket (see Table 1 and the Appendix). Yet, as the structure of the international etallurgical coal trade is (fro a odeling perspective) siilar to that of other coodities, the odel could easily be calibrated with data for other arkets. Metallurgical coal is used in steel-aking to produce the coke needed for steel production in blast furnaces and as a source of energy in the process of steel-aking. Metallurgical coal is distinct fro theral coal, which is typically used to generate electricity or heat. Currently around 7% of the global steel production crucially relies on etallurgical coal as an input. 4 International trade of etallurgical coal aounted to 25 illion tonnes (Mt) in International trade is predoinantly seaborne, using dry bulk vessels. Up until 21, etallurgical coal was alost exclusively traded through long-ter contracts. Since then, the arket has begun to ove away fro this syste towards ore spot arket-based trading. While the share of spot arket activity has increased rapidly, a substantial aount of etallurgical coal is still traded through long-ter contracts. Key players in this arket are large ining copanies such as BHP-Billiton, Anglo-Aerican, Glencore and Rio Tinto. These copanies produce ainly in Australia and, together with Peabody Energy s Australian operations, control ore than 5% of the global export capacity. In addition, adding to this the arket share of the Canadian Teck consortiu and the two key etallurgical coal exporters fro the United States, Walter Energy and Xcoal, results in alost three quarters of the global export capacity, arketed by an oligopoly of eight copanies. For the sake of siplicity and coputational tractability, we aggregate these players existing ines into one ining operation per player. Saller exporters fro Australia, the United States, Russia, New Zealand, Indonesia and South Africa are aggregated into three players: one Cournot player fro Australia (AUS6), one Cournot player fro the United States (USA1) and one copetitive fringe player that coprises all other regions (Fringe). This results in eleven asyetric players who differ with respect to their existing production capacity and the associated production and transport costs (see Table 2). 6 Players Table 2: Existing Capacity, Variable and Investent Costs Existing Capacity [Mtpa] Variable Costs [US$/t] Investent Costs [US$/tpa] USA USA USA AUS AUS AUS AUS AUS AUS CAN Fringe Max. Investent [Mtpa] We assue that the three players representing the saller exporters, i.e., USA1, AUS6 and Fringe, cannot invest in additional capacity. Hence, only the largest eight copanies can endogenously expand their supply capacity. The investent decision, ade in period one, is based on the players capacities and costs in 211. We consider one investent cycle with capacities becoing available after six years (i.e., 4 See WCA (211). 5 See IEA (213). 6 Data on capacities and costs are taken fro Trüby (213). 1

13 in 217) serving one deand period. Investent costs per tonne of annual production capacity (tpa) are broken down into equal annual payents based on an annuity calculation using an interest rate of 1% and a depreciation tie of 1 years. Note that production cost of new ines correspond to the production cost of the respective player s existing ine. The two largest iporters of etallurgical coal are Europe and Japan, followed by India, China and Korea. These key iporters account for ore than 8% of the trade. We aggregate these and the reaining saller countries into two deand regions: Europe-Atlantic and Asia-Pacific. 7 The forer also includes the Mediterranean s neighboring countries and iporters fro the Atlantic shores of the Aericas. The latter includes iporters with coastlines on the Pacific or the Indian Ocean. Exporters fro the United States have a transport cost advantage in the Europe-Atlantic region, while Canadian and Australian exporters are located closer to the consuers in the Asia-Pacific region (see Table 5 in the Appendix). We assue the inverse iport deand function for etallurgical coal to be linear. The function can be specified using a reference price and a corresponding reference quantity in cobination with a pointelasticity eta. Investors in production capacity face deand evolution as a key uncertainty. We therefore run sensitivities in which we vary the point-elasticity paraeter eta across the range -.2 to -.5 (see Figure 1). 8 This bandwidth is generally considered reasonable in the etallurgical coal arket (see Trüby (213) and the literature cited therein). Furtherore, we vary the reference deand quantity (see Table 6 in the Appendix) fro 6% to 14% to account for different deand evolution trajectories Price [US$/t] Supply [Mt] Price [US$/t] 8 eta2 6 eta3 4 eta4 eta5 2 Reference Supply [Mt] eta -.2 eta -.3 eta -.4 eta -.5 Reference Figure 1: Deand functions for Europe-Atlantic and Asia-Pacific regions with varying elasticity 5. Results 5.1. Variation of Deand Elasticity Production is highest in all odel runs in the perfect copetition odel followed by the closed-loop and the Cournot open-loop odel (see Figure 2). Accordingly, prices are highest in the open-loop odel and lowest in the perfect copetition odel. With decreasing eta (i.e., consuers are ore price responsive), which results in a flatter gradient of the deand function (see Figure 1), total production increases. Average prices decrease in the Cournot odels and increase in the perfect copetition odel. Note that in the perfect copetition case, the aggregate supply and aggregate deand curves intersect below the reference point (explaining the increase in production), whereas for the open-loop and closed-loop odel the inverse elasticity rule applies. investents are higher than open-loop investents in all odel runs (see Figure 3 on the left). Model results (assuing eta to be -.3) indicate a cobined increase in production capacity in Australia and Canada of about 38 Mtpa and 64 Mtpa for the Cournot open-loop and the closed-loop odels, respectively 7 Our approach covers 1% of the global seaborne etallurgical coal iports and exports (based on data fro 211). 8 For eta saller than -.4, closed-loop odel runs did not converge. Therefore, the results presented in Section 5 only coprise the range -.2 to -.4. For a discussion on coputational issues, see Section 6. 11

14 Production [Mt] Avg. price [US$/t] Perfect copetition 1 5 Perfect copetition eta eta Figure 2: Total production (left) and average arket price (right) for varying deand elasticity (see Figure 4). As for a coparison of odel results to actual industry data, the total su of additional production capacities in Australia and Canada is projected for 217 to lie between 39 Mtpa (only copleted and coitted projects in Australia) and 63 Mtpa (with projects in feasibility stage). 9 Investent [Mtpa] Idle capacity [Mtpa] Perfect copetition Perfect copetition eta eta Figure 3: Capacity investents (left) and idle capacity (right) for varying deand elasticity odel investents are higher than open-loop investents, reflecting the value of influencing other players production decisions by one s own investents. In the open-loop odel, players invest until long-ter arginal costs equal arginal revenue, whereas in the perfect copetition odel investents take place until long-ter arginal costs equal arket price. In our setting, perfect copetition investents lie between closed-loop and open-loop investents. In the odel setup used in Murphy and Seers (25), investents in the perfect copetition case generally exceed those in the closed-loop odel. Our results are due to existing capacities, that are withheld in the closed-loop odel (and thus enable profitable investents into new capacities) but are ostly utilized in the copetitive case (see Figure 3 on the right). Note that withholding (or idle capacity) refers, here and in the following, only to existing capacities. Each player fully exhausts existing capacities before investing in new capacities. Newly built capacities are always fully utilized in equilibriu as otherwise players could increase their profit by reducing investents. Interestingly, the increase in production with decreasing eta is not accopanied by an increase in investents in capacities in all odels (see Figure 3 on the left). In the perfect copetition and the closed-loop odels available production capacities are constant and increase, respectively, with decreasing deand elasticity in accordance with increasing supply. In the open-loop odel, investents in new capacities decrease despite increasing production. This is due to a deviation in capacity withholding: In the closed-loop and perfect copetition odels, the aount of idle capacity decreases only slightly, whereas in the open-loop 9 Based on Australian Governent (212) and McCloskey (213). 12

15 6 6 Investent [Mtpa] Investent [Mtpa] AUS1 AUS2 AUS3 AUS4 AUS5 USA2 1 1 USA3 Australia United States Canada Australia United States Canada CAN Figure 4: Capacity investents for the closed-loop (left) and open-loop odel (right) (eta =.3) odel withheld capacity rapidly declines and overcopensates for decreasing capacity additions. In the perfect copetition case, idle capacity is due to excess existing capacities: Players produce until (short-ter) arginal costs equal arket price. Here, soe high-cost capacity is not utilized in the United States. In the Cournot odels, players withhold when (short-ter) arginal costs exceed arginal revenue. In both Cournot odels, capacity is exclusively withheld by the two largest players in Australia and in the United States, for each country respectively. Consuer rent and profits for different levels of deand elasticity are depicted in Figure 5: Profits are lowest in the perfect copetition case and highest in the open-loop odel. The existence of profits in the perfect copetition odel is due to capacity restrictions of existing ines and liited expansion potential for new ines. Profit [bn US$] eta Consuer rent [bn US$] Perfect copetition eta Perfect copetition Figure 5: Accuulated profits (left) and consuer rent (right) with varying deand elasticity Total welfare is siilar in all odels: in a perfectly copetitive arket welfare is slightly higher than in the Cournot odels. Welfare is lowest in the open-loop odel (see Figure 6). Thus, the different underlying assuptions concerning the prevailing arket structure (long-ter contracts versus spot arket) priarily influence the surplus distribution rather than its su: In the open-loop case, in which the product is traded through long-ter contracts, copanies can earn significantly higher profits, while consuer surplus is higher in arkets with spot arket-based trade. The corresponding iplications are especially interesting, when taking into account that, in coodity arkets such as the etallurgical coal, copper or iron ore arkets, consuers and producers are located in different countries and hence consuer and producer rents arise in different legislations. 13

16 Welfare [bn US$] eta Perfect copetition Welfare difference [bn US$] eta Perfect copetition - Perfect copetition - Figure 6: Overall welfare (left) and welfare differences (right) 5.2. Variation of Reference Deand For the variation of reference deand, consuers deand elasticity eta has been fixed to a value of -.3; thus the case of 1% reference deand corresponds to the depicted results of the previous subsection with the sae deand elasticity. Production quantities under perfect copetition are highest in all odel runs, and the open-loop odel yields the lowest production. Accordingly, prices are highest in the open-loop case followed by closed-loop and perfect copetition (see Figure 7). In addition, production and average prices increase with increasing reference deand. Production [Mt] % 8% 1% 12% 14% Reference deand Avg. price [US$/t] Perfect copetition 1 5 6% 8% 1% 12% 14% Reference deand Perfect copetition Figure 7: Total production (left) and average arket price (right) for varying reference deand The results in the open- and closed-loop Cournot odel alost coincide for very low reference deand as only few investents are carried out. Investents in new production capacities are onotonously increasing for growing reference deand (see Figure 8). As in the case of varying deand elasticity, investents are constantly lower in the open-loop than in the closed-loop odel. For low deand, investents in the copetitive odel are below those in the strategic odels as existing capacities suffice to serve deand and thus render new investents unprofitable. In the Cournot odels, investents are profitable for sall players due to withholding by larger players. For higher levels of deand, investents in the copetitive odel exceed those in the closed-loop odel. with investents being lowest in the open-loop odel. The order of idle capacity is siilar to the case of varying deand elasticity: Idle capacity is highest in the closed-loop odel followed by the open-loop case (both due to strategic considerations) and the perfect copetition odel (due to arket prices below arginal costs). With increasing deand, profits as well as consuer rents increase (see Figure 9). Again, results for the open-loop and closed-loop odel alost coincide in the case of very low deand as investents are of 14

17 Investent [Mtpa] % 8% 1% 12% 14% Reference deand Idle capacity [Mtpa] Perfect copetition 6% 8% 1% 12% 14% Reference deand Perfect copetition Figure 8: Capacity investents (left) and idle capacity (right) for varying reference deand inor relevance. In the case of higher reference deand, profits in the open-loop odel exceed those in the closed-loop odel. Results for consuer rent are vice versa. Profit [bn US$] % 8% 1% 12% 14% Reference deand Consuer rent [bn US$] Perfect copetition % 8% 1% 12% 14% Reference deand Perfect copetition Figure 9: Accuulated profits (left) and consuer rent (right) with varying reference deand Overall welfare turns out to be quite siilar for all odels, with the highest welfare occuring in the copetitive odel followed by the closed-loop and open-loop odels (see Figure 1). Welfare [bn US$] % 8% 1% 12% 14% Reference deand Perfect copetition Welfare difference [bn US$] % 8% 1% 12% 14% Reference deand Perfect copetition - Perfect copetition - Figure 1: Overall welfare (left) and welfare difference (right, open-loop inus closed-loop) 5.3. Suary The results concerning production, price, consuer rent and profits for the three odels are in line with the expectations based on previous work (see, e.g., Murphy and Seers, 25). Total production as well 15

18 as consuer rent are highest in the case of a perfectly copetitive arket followed by the closed-loop and open-loop odels. Profits, on the other hand, increase fro the perfect copetition to the closed- and open-loop odels. Interestingly, capacity expansions in the closed-loop odel typically exceed those in the perfect copetition case but are accopanied by the withholding of existing production capacities. Those who invest are different fro those who withhold: Investent coes ostly fro saller players while withholding is done by the players who already have large existing capacities. This result is driven by the asyetric endowent with existing production capacity of the players. Players that are already big in ters of capacity have a lower incentive to grow while saller players expand their capacity. The agnitude of result deviations between the different odels, and thus the iplications for arket participants, are quite significant. The odels of iperfect copetition differ, for instance, in capacity expansions between 19% and up to 33% (low and high deand elasticity, respectively). Even though social welfare only deviates slightly between the open-loop and closed-loop odels in the applied dataset, the difference ay be higher for different arkets and different odel paraeters. In addition, the welfare distribution between consuer rent and profits differ significantly and thus ay raise covetousness, especially considering regional differences between consuption and production. 6. Coputational Issues Equilibria in a closed-loop odel, if any exist, do not necessarily have to be unique. Therefore, we perfor a robustness check for our closed-loop results by using different starting values for capacity investents. Starting values are randoly drawn fro a reasonable range of possible investents, with the axiu investent of each player as given in Table 2. Liiting the range of possible investents drastically reduces coputer run-ties and increases the probability of finding equilibria. In addition, calculations are ade with starting values set to zero and to the open-loop results. The algorith terinates if overall adjustents of investents δ are less than ɛ =.1 Mtpa copared to the previous iteration. We use a learning rate paraeter R for the adoption rate of new investents in order to avoid cycling behavior. The learning rate paraeter is randoly set between.6 and 1. (see Gabriel et al., 212). Calculations have been done on a 16 core server with 96 GB RAM and 2,67 GHz using CPLEX Table 3 shows calculation statistics when using the MILP version of our odel (see Subsection 3.2.2). We perfor six runs per paraeter setting using rando start values. Most runs converged to an equilibriu before the axiu nuber of iterations was reached. With increasing deand elasticity, the algorith had difficulties to converge. In the case of eta =.4, only every third run converged to an equilibriu; for eta <.4, no equilibriu could be found at all. Using either zero investents or open-loop results as starting values, a closed-loop equilibriu was found, except for eta <.4. Table 3: Coputation tie and convergence to equilibriu - MILP version (rando, zero, open-loop starting values) Scenario Convergence (ax. 1 iterations) Iterations until convergence (only converged runs, ax. 1) Calculation tie (only converged runs) [h] reference case 6/6, yes, yes 6-7 (avg. 6.8), 7, (avg. 12.4), 7.1, 5.2 (eta -.3, de 1.) eta -.2 6/6, yes, yes 7-8 (avg. 7.3), 6, (avg. 11.), 5.7, 4.1 eta /6, yes, yes 7-1 (avg. 8.2), 7, (avg. 12.8), 6.9, 5.5 eta /6, yes, yes 6-8 (avg. 7.2), 6, (avg. 12.7), 5.1, 5.6 eta -.4 2/6, yes, yes 7-8 (avg. 7.5), 9, (avg. 12.4), 5.6, 7.8 eta -.45 /6, no, no -, -, - -, -, -. eta -.5 /6, no, no -, -, - -, -, - de.6 5/6, yes, yes 7-9 (avg. 7.4), 7, (avg. 2.2),.1,.2 de.8 6/6, yes, yes 7-8 (avg. 7.5), 6, (avg. 7.1), 2., 2.3 de 1.2 6/6, yes, yes 6-9 (avg. 7.8), 7, (avg. 11.9), 8.3, 6. de 1.4 6/6, yes, yes 6-1 (avg. 8.3), 7, (avg. 8.7), 9.7,

19 Figure 11 illustrates the iterative solution process for a single odel run for eta =.5 using rando starting values. The odel run did not converge to an equilibriu. 1 After initial adjustents of investents in the first iterations, investents start to cycle in a rather sall range. Total investents fro iteration 5 to 1 vary between 89 Mtpa and 97 Mtpa. This range is typical for all runs regardless of the starting values. The axiu range for a single player s investent deviations is 3 Mtpa. Thus, even if no equilibriu is reached, analyzing the solution process ay hint to possible arket developents Investent [Mtpa] Delta [Mtpa] iter1 iter2 iter3 iter4 iter5 iter6 iter7 iter8 iter9 iter1 iter1 iter2 iter3 iter4 iter5 iter6 iter7 iter8 iter9 iter1 Figure 11: Course of investents of single players during solution process (eta =.5) Using zero investents or open-loop equilibriu results as starting values led to a significant reduction of coputer run-ties copared to rando starting values. This is probably due to the rather large range of rando starting values and the (coparably) rather sall equilibriu investents. Thus, starting fro zero investents in ost cases is closer to the equilibriu values than starting with rando values. In suary, using reasonable starting values can support the solution process significantly. If the algorith converged, odel results were identical for all runs with the sae paraeters concerning deand level and deand elasticity. Thus, even if the existence of ultiple equilibria cannot be excluded, equilibria appear to be stable. Calculations using the MILP version of our odel usually took several hours to converge to an equilibriu. Applying the grid search approach (see Subsection 3.2.1) and thus discretizing the search space reduced coputer run-ties significantly. The sae calculations as in the MILP version have been done using grid search with investent steps of.1 Mtpa and the sae convergence criterion as in the MILP version (ɛ =.1 Mtpa). The odel was ipleented in GAMS using GUSS (see Bussieck et al., 212). Applying grid search, the solution process took only several inutes to converge. Thus, reducing the optiization process fro a series of coputationally challenging MPECs to coparably easy-to-solve copleentarity probles reduced overall coputer run-tie significantly. As for the MILP version, all odel runs converged to the sae equilibriu (for eta.4) or did not converge at all (for eta <.4). Aggregated absolute deviations of investents between the MILP and the grid search version of our odel vary between.3% and 3.7%. Thus, in our paraeter setting, only inor differences in the results occurred. 7. Conclusions We presented three investent odels for oligopolistic spatial arkets. Our approach accounts for different degrees of copetition and as to whether the product is sold through long-ter contracts or on spot arkets. The odels are particularly suited for the analysis of investents in arkets for natural resources and inerals. We applied the odels to the international etallurgical coal trade, which features characteristics siilar to those of other coodity arkets. 1 In our iterative approach, convergence depends on the choice of (an arbitrarily sall) ɛ. 17

20 Table 4: Coputation tie and convergence to equilibriu - Grid Search (rando, zero, open-loop starting values) Scenario Convergence (ax. 1 iterations) Iterations until convergence (only converged runs, ax. 1) Calculation tie (only converged runs) [in] Accuulated absolute difference between investents in MILP and grid version [%] reference case 6/6, yes, yes 6-7 (avg. 6.3), 7, (avg. 9.3), 2.2, ,.8,.8 (eta -.3, de 1.) eta -.2 6/6, yes, yes 5-7 (avg. 6.3), 7, (avg. 1.2), 2.3, 2. 1., 1., 1. eta /6, yes, yes 6-7 (avg. 6.7), 7, (avg. 9.4), 2.2, 2.4.8,.7,.8 eta /6, yes, yes 6-8 (avg. 7.), 7, (avg. 1.4), 2.2, , 1.2,.8 eta -.4 6/6, yes, yes 6-7 (avg. 6.5), 7, (avg. 9.3), 2.2, , 1.5, 1.5 eta -.45 /6, no, no -, -, - -, -, -. -, -, - eta -.5 /6, no, no -, -, - -, -, - -, -, - de.6 6/6, yes, yes 6-8 (avg. 7.), 5, (avg. 9.7), 2., , 3.1, 3.3 de.8 6/6, yes, yes 6-7 (avg. 6.7), 6, (avg. 9.6), 2.6, 2.4.9, 1.,.9 de 1.2 6/6, yes, yes 5-7 (avg. 6.8), 7, (avg. 1.1), 2.3, 2.5.3,.3,.3 de 1.4 6/6, yes, yes 5-6 (avg. 6.3), 7, (avg. 9.7), 2.3, ,.4,.4 Results ay differ substantially between the different odels. The closed-loop odel, which is coputationally challenging, is particularly well suited for when the product is traded on a spot arket and the investent expenditure is large copared to production costs. The open-loop odel is appropriate for arkets with perfect copetition or iperfectly copetitive arkets on which the product is traded through long-ter contracts. Moreover, the open-loop odel approxiates the closed-loop outcoe when investent costs are inor. Over the last several years, progress has been ade in the etallurgical coal and iron ore arkets to ove away fro long-ter contracts and introduce spot arkets in coodity trade. Siilarly, efforts are being ade to introduce spot arket-based pricing between European natural gas iporters and the Russian gas exporting giant Gazpro. Such developents can have a ultitude of effects positive or negative. However, with respect to investents, our results suggest that oving away fro long-ter contracts in oligopolistic arkets is likely to stiulate additional investent and consequently reduce profits and increase consuer rents. The overall effect on welfare is negligible. However, in natural resource arkets, export revenues and consuer rents fro iports are typically accrued in different legislations. Hence, policy akers fro exporting and iporting countries are likely to have differing views on how coodity trade should be organized. Further research is needed to iprove ethods for solving coplex bilevel probles. In addition, further research could apply the odels presented here to other oligopolistic ining industries such as the copper or iron ore trade. Given that static pricing odels tend to give unsatisfactory results for the oil arket, in which variable costs are low but capital expenditure is very high, the closed-loop approach ay provide interesting insights into the oligopolistic pricing when accounting for investents in capacity. References References Australian Governent, October 212. Resources and energy ajor projects. Tech. rep., Bureau of Resources and Energy Econoics. Bowden, B., 212. A history of the Pan-Pacific coal trade fro the 195s to 211: Exploring the long-ter effects of a buying cartel. Australian Econoic History Review 52 (1), Bussieck, M. R., Ferris, M. C., Lohann, T., 212. GUSS: Solving Collections of Data Related Models Within GAMS. In: Kallrath, J. (Ed.), Algebraic Modeling Systes. Vol. 14 of Applied Optiization. Springer Berlin Heidelberg, pp Fortuny-Aat, J., McCarl, B., A representation and econoic interpretation of a two-level prograing proble. The Journal of the Operational Research Society 32 (9), Fudenberg, D., Tirole, J., Gae Theory. The MIT Press. 18