Short CommunicationEndogenous production capacity investment in natural gas market equilibrium models
Introduction
The natural gas model applied in Egging (2013) is one of the latest in a series of equilibrium models for this particular fuel. The interest stems from several potentially game-changing trends: liberalization of natural gas markets, carbon dioxide emission constraints and an expected replacement of coal by relatively clean natural gas in Europe (EC Energy Roadmap, 2011); unconventional reserves in North America and other regions (IEA, 2011); and frequent concerns regarding supply security and European dependence on a small number of suppliers (cf. Leveque et al., 2010).
Hence, a number of equilibrium models have been developed over the past decade to provide numerical analysis of different scenarios regarding future supply and demand patterns, environmental regulation and infrastructure investment options. Two large-scale natural gas equilibrium models are the GASTALE model, developed by ECN (Lise & Hobbs, 2008), and the World Gas Model (WGM), joint work by the University of Maryland and DIW Berlin (Egging, Holz, & Gabriel, 2010).1 These models share a number of characteristics, similar to the model in Egging (2013): they are spatial partial equilibrium models with a detailed geographic disaggregation, allowing for analysis and comparison of different pipeline and LNG export/import options; they consider seasonality within a year and explicitly model storage to shift natural gas between low- and high-demand seasons; they are multi-period models and endogenously determine optimal investment in infrastructure2; and they allow for oligopolistic behavior by (a subset of) suppliers, i.e., Cournot competition. Furthermore, all these models apply a logarithmic cost function, as first proposed by Golombek, Gjelsvik, and Rosendahl (1995), in order to capture the specific characteristics of natural gas production: sharply increasing costs when producing close to full capacity.
However, none of these models allows for endogenous investment in production capacity; instead, the production capacity in future periods is defined exogenously. Given that production capacity is a significant determinant of results and that these models simulate price and quantity trajectories for several decades into the future, this omission is certainly a major drawback. It is owed, in all likelihood, to the rather complicated functional form when including investment decision variables in the logarithmic cost function. This paper provides the proof that this extension yields a convex problem, which is a prerequisite for solving this problem as an equilibrium model.
Let me also mention one other recent natural gas model: the GaMMES model was developed by EDF and IFPEN (Abada, Gabriel, Briat, & Massol, 2013). In contrast to the models presented above, it distinguishes between spot market sales and long-term contracts. It also assumes a slightly different formulation of the logarithmic cost function: production costs are not increasing relative to capacity utilization, as in the other models, but relative to remaining reserves. This is an interesting approach, but differs from what is discussed here.
All these models are formulated as Mixed Complementarity Problems (MCP). The optimization problems of different players subject to engineering and other constraints are solved simultaneously by deriving their respective Karush–Kuhn–Tucker (KKT) conditions, combined with market clearing constraints. The MCP framework is convenient for this type of exercise, as it allows to include Cournot market power for certain suppliers, in contrast to welfare maximization or cost minimization problems. In addition, these models can easily be extended to include stochasticity (e.g., Gabriel, Zhuang, & Egging, 2009) or two-level problems such as Stackelberg competition (e.g., Siddiqui & Gabriel, 2013).
Section snippets
Mathematical formulation
Assume a supplier with decision variables qy (production quantity) and ey (production capacity expansion/investment). The periods are denoted by . In order to keep the notation concise, y denotes both a period as well as its position in the set. Hence, stands for both the last period as well as the number of periods in the set. Following this logic, I use y′ < y for “all periods y′ prior to period y” in sums and indices, and y′ > y for the inverse statement. The price at which the
Conclusions and outlook
This short note points out the omission of endogenous investment decisions in production capacity in the natural gas equilibrium model applied in Egging (2013) and similar models. Instead, they rely on exogenously determined capacity increases for future periods. I propose a mathematical formulation to incorporate capacity investments into the state-of-the-literature equilibrium models with logarithmic cost functions and show that this formulation is indeed a convex problem.
Nevertheless,
Acknowledgement
The author thanks lbrahim Abada, Franziska Holz and Philipp M. Richter as well as two anonymous referees for their helpful comments and feedback.
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