Modelling Bidding Behaviour in Electricity Auctions : Supply Function Equilibria with Uncertain Demand and Capacity Constraints

Abstract: In most electricity markets, producers submit supply functions to a procurement uniform-price auction under uncertainty before demand has been realized. In the Supply Function Equilibrium (SFE), every producer commits to the supply function that maximises his expected profit given the bids of competitors. The presence of multiple equilibria is a basic weakness of the SFE framework. Essay I shows that with (i) symmetric producers, (ii) perfectly inelastic demand, (iii) a reservation price (price cap), and (iv) capacity constraints that bind with a positive probability, a unique symmetric SFE exists. The equilibrium price reaches the price cap exactly when capacity constraints bind.Another weakness is difficulty finding a valid asymmetric SFE with non-decreasing supply functions. Essay II shows that for firms with asymmetric capacity constraints but identical constant marginal costs there exists a unique and valid SFE. Equilibrium supply functions exhibit kinks as well as vertical and horizontal segments. The price at which the capacity constraint of a firm binds is increasing in the firm’s share of market capacity. The capacity constraint of the second largest firm binds when the market price reaches the price cap. Thereafter, the largest firm supplies its remaining capacity with a perfectly elastic segment at the price cap. Essay III presents a numerical algorithm that calculates a similar SFE for asymmetric firms with increasing marginal costs. Essay IV derives the SFE of a pay-as-bid auction such as the balancing market for electric power in Britain. A unique SFE always exists if the demand’s hazard rate is monotonically decreasing, as for a Pareto distribution of the second kind. Assuming this probability distribution, the pay-as-bid procurement auction is compared to the SFE of a uniform-price procurement auction. Two theorems in Essay V prove that the demand-weighted average price is (weakly) lower in the pay-as-bid procurement auction.