Incorporating learning-by-doing into mixed complementarity equilibrium models

Abstract

Market equilibrium models are often specified and solved as mixed complementarity problems (MCPs). These formulations combine the Karush–Kuhn–Tucker (KKT) optimality conditions of the optimization problems faced by multiple strategic players with market-clearing conditions, such that the solution to this system provides the Nash equilibrium prices and quantities. MCPs are widely applied to energy markets including those for electricity, oil, and natural gas. While researchers have made substantial progress on expanding the model features included in MCPs and on solving these problems, a limitation of existing MCPs is that they treat costs as exogenous input parameters. Therefore, MCPs have not been able to capture learning-by-doing (LBD), the empirically observed phenomenon whereby production costs tend to decline as a function of cumulative production experience. In this paper, we demonstrate the incorporation of LBD into a mixed complementarity equilibrium model. We consider two closely related, but nevertheless distinct, LBD formulations: one with discrete changes in cost from period to period, and another where cost declines continuously. Through theoretical analysis and numerical exploration, we establish the conditions under which these LBD formulations lead to convex optimization problems. Confirming convexity is important because it guarantees that their KKT conditions are sufficient for optimality. Then, we demonstrate the practical application of a mixed complementarity equilibrium model with LBD using the North American natural gas market as an example. When LBD is incorporated into the cost of liquefaction, North America exports more liquefied natural gas, which raises prices and reduces domestic consumption.