Supply Function Equilibria: Step Functions and Continuous Representations Pär Holmberg Department of Economics, Uppsala University David Newbery Faculty of Economics, University of Cambridge, EPRG Daniel Ralph Judge Business School, University of Cambridge, EPRG

SUMMARY Continuous representation of stepped offers in electricity auctions is convenient and results in pure-strategy supply function equilibria (SFE) (Green and Newbery, 92). von der Fehr and Harbord (93) derive mixed NE of game with step offers that are discrete in quantity that do not converge to continuous SFE for infinitely many steps. Market design has inherent price instability. We derive pure-strategy NE of game with step offers that are discrete in prices and show that they converge to continuous SFE. Prices are stable. S p p S S p

Related literature Convergence of equilibria Dasgupta & Maskin (1986) and Simon (1987) show that convergence of Nash equilibria is problematic if payoff functions are discontinuous and that convergence depend on how the strategy space of the limit game is approximated. (Not directly applicable to our case, where the strategy space of the limit game is infinitely-dimensional). Step offers with discrete prices Wolak (2004) and Anderson and Xu (2004) derive best responses for step offers with discrete prices for the Australian market, where bidders choose their own price grid (different to our set-up). They do not analyze convergence to continuous SFE. Empirical support of continuous SFE Three empirical papers on the balancing market in Texas, ERCOT: Niu et al. (2005), Hortascu & Puller (2007), and Sioshansi & Oren (2007). Conclusion: Large producers approximately bid in accordance with the first-order condition for continuous SFE.

Wholesale electricity markets Typically organised as uniform price auctions. Separate auction for each delivery period (0.5h/1h) Producers submit non-decreasing stepped offer curves to the auction. Offers are submitted under demand uncertainty. Production costs are well-known (  common knowledge) Electricity is a homogeneous good Auction is repeated times a day and individual offer/demand curves or aggregated offer/demand curves are often publicly available => Residual demand can be calculated almost exactly.

Example from the power exchange of Amsterdam Demand Supply Market price

Market Installed capacity Max steps Price range Price tick size Quantity multiple No. quantities/ No. prices Nord Pool spot 90,000 MW 64 per bidder 0-5,000 NOK/MWh0.1 NOK/MWh 0.1 MWh18 ERCOT balancing 70,000 MW 40 per bidder -$1,000/MWh- $1,000/MWh $0.01/MWh0.01 MWh35 PJM160,000 MW 10 per plant 0-$1,000/MWh$0.01/MWh0.01 MWh160 UK (NETA)80,000 MW 5 per plant -₤9,999/MWh- ₤9,999/MWh ₤0.01/MWh0.001 MWh 4 Spain Intra- day market 46,000 MW 5 per plant Yearly cap on revenues €0.01/MWh0.1 MWh— Offer constraints in wholesale electricity markets

Modelling bidding in electricity markets – continuous SFE Green & Newbery (1992) assume that stepped offers can be approximated by continuous supply functions. => They use the SFE model developed by Klemperer & Meyer (1989), in which the set of allowed prices is a continuum and goods are divisible NE: Given uncertain demand and competitors’ offer curves, an electricity producer chooses its offer curve in order to maximise its profit at each demand level. p S Supply curve Marginal cost SFE can be determined from a system of differential equations:

Multi-unit auction model (discrete quantities) Von der Fehr & Harbord (1993) model the electricity market as a multi-unit auction (=> stepped supply functions). The set of allowed prices is a continuum, but goods are indivisible, They show that pure-strategy equilibria do not exist (if demand variation is sufficiently large), even if the number of units is arbitrarily large. But there are mixed NE. Markets have an inherent price instability. Continuous SFE and NE in multi-unit model are fundamentally different also for arbitrarily many steps. They do not converge! One characteristic of the von der Fehr and Harbord model is that their payoff function is discontinuous; a firm can discontinuously increase its profit by slightly undercutting rival bids. S p

Step supply functions with discrete prices (our model) We consider a model with step supply functions. In contrast to von der Fehr and Harbord, we assume that the set of price levels is finite, and that goods are divisible, Rationing rule: Pro-rata on the margin => All offers below the market clearing price are accepted. Offers at the market clearing price are accepted in proportion to the firm’s incremental supply offered at the clearing price. We show that in the limit, when the number of allowed price levels goes to infinity, the set of Nash equilibria converges to the set of Nash equilibria for continuous supply functions. Intuition: If the number of price levels is finite it is no longer possible to slightly undercut steps in competitors’ supply curves => Pay-off functions are continuous. p S

The first-order condition when prices are discrete Neg. contribution from ε=τ j Pos. contribution from τ j-1 ≤ε<τ j Pos. contribution from τ j <ε≤τ j+1 Note that the derivative is well-defined and accordingly the expected payoff is continuous. Also note that optimal supply at the price level p j depends on the probability density of demand.

The convergence proof 1.The discrete first-order condition has a unique solution. 2.Convergence of solutions to discrete first-order condition and differential equation by Klemperer and Meyer (1989). A) Consistency: The difference equation converges to the differential equation in the limit. B) Stability: The solution of the difference equation remains bounded in the limit when the price tick-size approaches zero. 3.Solutions to the difference equation that are non-decreasing for all realized prices are NE if the price-grid is sufficiently fine. Assume that a market with concave demand has a continuous SFE with increasing supply functions and positive mark-ups for all realized prices. Then if the price grid is sufficiently fine, a discrete SFE with non-decreasing supply functions will exist and it will converge to the continuous SFE.

Example SiSi p Cournot schedule Marginal cost Max demand Most competitive SFE Least competitive SFE Min demand

Concluding remarks * Convergence of NE for stepped offer strategies to continuous SFE depend on the discretization. * Inherent price stability depend on market design. To get continuous pay-off functions we recommend piece-wise linear offer curves as in Nord pool instead of stepped offer curves. For wholesale electricity markets, we recommend large price tick-sizes and small quantity multiples (in contrast to Kremer & Nyborg (2004)). Preferably restrictions in the number of steps per bidder/plant should be abolished. * Even if market design is mainly discrete in the quantity. Pure-strategy NE may still exist, because of uncertainty in competitors’ costs (Parisio and Bosco, 2003). Alternatively, uncertainty because of mixed NE may reduce as the number of steps increases. * With significant discreteness, the probability distribution of demand influences strategic offer curves. Thus using discrete first-order conditions rather than continuous ones can enhance the accuracy in empirical work, as in Wolak’s (2004) model of the Australian market.

Notation p Supply Offer of producer i sijsij ∆p p Demand Aggregate demand pjpj dj+εdj+ε Demand shock ε Probability density of demand shock: g(ε) Let τ j =s j -d j (elastic net-supply at p j ) Market price is p j pjpj

All offers below the clearing price are accepted, while offers at exactly p j are rationed on a pro-rata basis. Thus for the accepted supply of a producer i is given by: Hence, the contribution to the expected profit of generator i from realizations is given by: The expected profit of firm i The total expected profit is: