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The Supply Function Equilibrium Economics Essay

Supply Function Equilibrium (SFE) is a better model of competition in oligopoly as it includes both Cournot and Bertrand models as one of its special cases. It helps ISO (Independent System Operator) and allows other genco’s to build optimal bidding strategies. Most of the earlier researches of SFE deal with single period and single market model which did not consider the facts that the genco’s hold a combination of shares or other investments in electricity and fuel markets. In this paper a proper SFE has been recognized which can be applied to multiple-period situation, then by using Two-Settlement Approach maximization of genco’s social welfare and pay off of each genco for a specified period is done.

The project work comprises of five chapters:

Chapter 1 a brief introduction of market competition has been provided with some introduction to the important characters playing role in the marketing environment which makes the understanding simple.

Chapter 2 specifies role of economics in the electricity market with different ways through which bidding takes place has been specified, this chapter smoothens the path to understand the project work easily.

Chapter 3 is the backbone of the project work, it deals with SFE and its evolution and role in the project work and it further incorporates two-settlement approach used in LSFE so as to obtain equilibrium parameters of the market.

Chapter 4 deals with the simulation and results of the project work and Chapter 5 Concludes the work with a section of Appendix to follow.



What is Competition:

Competition leads to the improvement of commodity in many fields of human endeavor. It is pervasive; every organization needs a strategy to deliver superior value to its customer. Today the competition has intensified dramatically over last several decades in almost all domains.

Competition in electric industry generally means competition only in the production (generation) of electricity and in the commercial functions of wholesaling and retailing.

Fig.1.1. Physical functions of electricity market

The transportation function (transmission and distribution) cannot be competitive as they are natural monopolies, because it does not make economic sense to build up multiple sets of competing transmission system. Moreover, the system operations also have to be a monopoly, since the system operator has to control all the plants in the control area.

Dramatis Personae

Generation Companies (Gencos) produce and sell electrical energy, they may also sell services such as regulation, voltage control and reserve. These services are required by the system operator to maintain the quality and security of electrical supply. A generating company can own a single plant or portfolio of plants of different technologies.

Distribution Companies (Discos) own and operate distribution networks, in a traditional environment they have a monopoly for the sale of electrical energy to all consumers connected to their network.

Retailers buy electrical energy from the wholesale market and resell it to consumers who do not wish or are not allowed to participate in wholesale market. They do not have to own any power generation, transmission and distribution assets.

Market Operator (MO) typically runs a computer system that matches the bids and offers that buyer and seller of electrical energy have submitted. It also takes care of the settlement of the accepted bids and offers; this means that it forwards payments from buyer to sellers following delivery of the energy. The Independent system operator (ISO) is usually responsible for running the market of last resort, which is the market in which load and generation are balanced in real time.

Independent System Operator (ISO) has the primary responsibility of maintaining the security of the power system, it is called independent because in a competitive environment the system must be operated in a manner that does not favor or penalize one market participant over another. An ISO would normally own only the computing and communication assets required to monitor and control the power system.

Transmission Companies (TransCo) own transmission assets such as lines, cables, transformers and reactive compensation devices, they operate this equipment according to the instructions of the independent system operator. Transmission companies are sometimes subsidiaries of companies that also own generating plants.

Regulator is the government body responsible for ensuring the fair and efficient operation of the electricity sector; it determines or approves the rules of the electricity market and investigates suspected cases of abuse of market power.

Small Consumers buy electrical energy from a retailer lease a connection to the power system from their local distribution company, their participation in the electricity market usually amounts to no more than choosing one retailer among others when they have this option.

Large Consumers on the other hand will often take an active role in electricity markets by buying their electrical energy directly through the market, some of them may offer their ability to control their load as a resource that the ISO can use to control the system; the largest consumers are sometimes connected directly to the transmission system.

Models of Competition:

1.3.1 Monopoly: The figure shown here is a model of a monopoly utility. It deals with the case where integration of generation, transmission and distribution of electricity takes place through a single utility. In this the generation and transmission are handled by one utility, which sells the energy to local monopoly distribution companies.

Fig.1.2. Monopoly model of electricity market

Purchasing Agency: A possible first step towards the introduction of competition in the electricity supply industry has been shown in (fig.1.2). In case of purchasing agency the integrated utility no longer owns all the generation capacity (Fig.1.3). Here in (Fig.1.4) Independent power producers (IPP) are the connected to the network, they sell their output to the utility that acts as a purchasing agent.

Fig.1.3. Integrated version of utility

Fig.1.4. Disaggregated version of utility

In this model the utility no longer owns any generation capacity and purchases all its energy from the IPP’s, the distribution and retail activities are also disaggregated. Discos then purchase the energy consumed by their customer from the wholesale purchasing agency.

Wholesale Competition: In this model (Fig.1.5) disco purchase the electrical energy directly from generating companies. These transactions take place in a wholesale electricity market; this market can take the form of pool or bilateral transactions. At the wholesale level the only function that remain centralized are the operation of the spot market and the operation of transmission network. At the retail level the system remains centralized because each disco not only operate the distribution network in its area but also purchase electrical energy on behalf of the consumer located in its service territory.

Fig.1.5. Wholesale competition market

Retail Competition: In this form of competitive electricity market (Fig.1.6) all consumers can choose their supplier, only large consumer chooses wholesale market as supplier. Most small and medium consumers purchase energy from the retailers. The retailers buy energy from the wholesale market and supply it to the consumers. In this the distribution companies are normally separated from their retail activities because they no longer hold monopoly for the supply of electrical energy in the area covered by the network. The only monopoly held in this competition is by transmission and distribution networks.

When such a market gets established then no further regulation of retail price takes place, because small consumers can change their retailer when offered better price.

Fig.1.6. Retail Competition model of electricity market

From economics perspective this model is most satisfactory because in this energy prices are set through interactions. Implementing this model however requires considerable amount of metering, communication and data processing.



2.1. Supply and Demand: Supply and Demand is an economic model of price determination in the market, it includes that in a competitive market the unit price of a particular good will vary until it settles at a point where the quantity demanded by the consumer is equal to the quantity supplied by the producer, resulting to economic equilibrium between price and quantity.

Graphical Representation:

Fig.2.1. Supply-Demand Curve

Here the graph shown is the supply demand curve it is observable from the above graph that when demand increases from D1 to D2 then price and quantity both increase P1 to P2 and Q1 to Q2 respectively, supply remaining same.

When supply curve increases from S1 to S2 then price decreases from P2 to P2’ and quantity increases from Q2 to Q2’ conversely is also true for both cases.

Supply schedule:

Graph depicted above also has supply curve, which represents the amount of some goods that the producer is willing and able to sell at various prices assuming ceteris paribus, that is assuming all determinants of supply other than the price of goods in question remaining same. Under the assumption of perfect competition, supply is determined by marginal cost, the firms will produce additional output as long as cost of producing an extra unit of output is less than the price they will receive.

Demand schedule:

Graph depicted above also has demand curve; it represents the amount of some goods that the buyers are willing and able to purchase at various prices. Assuming all determinants of demand other than the price of the good in question, such as income, personal taste, remain the same. Following the law of demand that if the price will decrease the purchaser will buy more goods, hence always a downward sloping.


Elasticity refers to how strongly the quantity supplied and demanded respond to various factors including price and other determinants. It can also be stated as the percentage change in one variable (quantity supplied or demanded) to percentage change in causative variable in simple words it is measure of relative changes, it is measured as percentage change in quantity to the percent change in price.

If the quantity demanded or supplied change by a larger percentage than the change in price then the demand or supply is said to be elastic, conversely inelastic, in case of zero elasticity that is no percentage change in quantity supplied or demanded then supply is perfectly inelastic.

Here we model an oligopoly facing uncertain demand in which each firm chooses as its strategy a “supply function” relating its quantity to its price, by forcing each firm’s supply function to be optimal against a range of possible residual demand curves.

There are two kinds of modeling which were used for choosing a supply function that is

Cournot modeling in this model quantity were fixed and price was varied (steep supply function)

Bertrand modeling in this model quantity were varied and price was fixed (horizontal supply function)

Further it has been analyzed that the firm may achieve higher profit by committing to a supply function than by committing to fixed price or fixed quantity, because a supply function allows better adaptability to the uncertainty [1].

In this we develop a formal model of supply function competition in oligopoly under demand uncertainty, before a demand shock is realized, each firm commits to a function specifying the quantity it will produce as a function of its price. After the shock is realized, all markets clear: each firm produces at the point on its supply function which intersects its realized residual demand curve (intersection of demand curve and supply function) [1].

An electricity market has its own characteristics which are different from other commodity markets, electricity cannot be stored and its demand is almost inelastic and varies with seasons and daily weather conditions. Supply of electricity also varies with time as a result of planned maintenance and forced outages, previous research has shown that existing electricity markets are not perfectly competitive and that generators have some market power.

Previous studies concerning the market power issues provide several indices, such as the Lerner index and HHI index to identify the existence of market power. The generators can exert their market power in advance by taking into consideration both the available generation capacity or total supply and the demand level. It is suggested that a demand supply ratio will indicate such conditions. Once this condition does exist, generators might be able to strategically bid into market and reap their extra profits in daily trading. The term ‘market power’ refers to an ability of a firm to raise price above the competitive level without a rapid loss of ability to sell. Similarly, market power is an ability to profitably maintain prices above competitive level by restricting output above competitive level.

There are two principal mechanisms, i.e. strategic bidding and capacity withholding by which generating firms may exercise market power in the widely used bid based pool markets, both of these strive to force up the market clearing price. The key factors that determine the extent of market power include supplier’s concentration, demand elasticity and style of competition.

In recent years many equilibrium models have been used in the analysis of strategic interaction between participants in an electricity market, including oligopoly models of Cournot, Betrand, Stackelberg, Supply function equilibrium(SFE) and Collusion. Among them the Cournot and SFE models are the most extensively used models for analyzing pool- based electricity markets. The general SFE model was introduced by Klemperer and Meyer [1] and the first analyzed by Green and Newbery [2], in which each firm chooses as its strategy a “supply function” relating its quantity to its price. The effect of three policies that could increase the amount of competition has been modeled in the electricity spot market in England and Wales through SFE approach [3]. The (MPEC) has been used to formulate the problem of calculating SFE in the presence of transmission constraints [4], it presents an analysis that estimates the price of electricity dispatched and sold using a closed firm mathematical formula derived from the analytical concept of SFE. An example to compare Cournot and SFE models of bid based electricity markets with and without transmission constraints has been done and a demonstration of the effect by parameterization of SFE on the calculated results has been shown [5]. A conjectured supply function (CSF) model of competition among power generators on a linearized dc network been presented in [6]. Coevolutionary Computation approach has been used to obtain close form solutions when practical issues in electricity market are considered such as non convexity and discontinuity of cost function and inter temporal scheduling of generators [7]. Another new approach has been presented using agent based market simulation technique [8,9].

The exercise of market power may be facilitated by some characteristic of electricity markets including inelastic demand, limited transmission capacity, and the requirements that supply and demand of the power systems must balance continuously. A lot of work has been done on market power analyzing different economic models; a survey of electricity market modeling especially the equilibrium models has been presented. Non-cooperative game theoretical approaches such as the Cournot and SFE models are widely used for power market simulation. The electricity spot markets modeled by SFE and Cournot models have been extended to include contract markets in, the English electricity market is modeled by SFE model with a contract market, and the entry condition of the contract market. It has been shown that competition in contract market could lead the generators to sell contracts and increase their outputs and also hedge the spot market price in England and Wales. Moreover it proposes an asymmetric linear supply function equilibrium (LSFE) model to develop firm’s optimal bidding strategies given their forward contracts and market power mitigation effect of forward contracts have also been evaluated [10]. To study the interaction between the spot market and forward markets, it is commonly assumed that the Gencos are risk neutral two settlement electricity market with transmission line constraints is studied and compared with a single settlement market. It is shown that spot market prices will decrease when the supplies enter forward contracts.

Market Equilibrium: In (fig.2.2) consumer and producer surplus has been shown when a market reaches equilibrium.

Fig.2.2. Consumer & Producer surplus in market equilibrium

Consumer surplus equals the area above the price and under demand curve, where as Producer surplus equals the below the price and above supply curve. The total area between the supply and demand up to the point of equilibrium represents the total surplus in the market. In (fig.2.3) if the quantity is less than the equilibrium quantity such as Q1, the value to buyer exceeds the cost to sellers.

Fig.2.3. Efficiency of equilibrium quantity

If the quantity is greater than equilibrium quantity such as Q2, the cost to sellers exceeds the value to buyers. Therefore, the market equilibrium maximizes the sum of producer and consumer surplus.

Bidding Strategy:

Bidding strategies deals with set of plans leading to offering of a price one is willing to pay for something so as to maximize its profit or fulfill its requirement. In electricity market bidding plays a vital role, where a set of supply functions are offered as bids by number of Genco’s so as to meet the demand, accordingly each Genco receives its profit by the use of the concept of supply function equilibrium which helps them in making their bids, by the use of Two-settlement approach the amount of quantity, price and profit of each Genco can be obtained.

Previous researches on bidding strategies can be divided into three sections.

Optimization model:

This model pays attention over a single player that is the player under study in which number of mathematical programming models has been developed in this section to find optimal bidding strategy; some of them are Fuzzy Linear Programming, Dynamic Programming, and Stochastic Dynamic Programming etc. The proposed bidding strategy like Markov Decision Process (MDP) in which effects of market share and production limit has been discussed on optimal bidding strategy, peak/off-peak load; peak/off-peak price has been used to reduce the number of states [11]. It has been proposed so as to support in decision making for hedging and scheduling in power portfolio optimization. In this model inputs such as electricity demand, electricity forward price, gas forward price and electricity spot price are done through several stochastic processes, the draw back in this model is that it does not model behavior aspect of players.

2.3.2 Game Theory Model:

In this model bidding strategies has been discussed which incorporates the feature of interaction between players. The whole purpose of this model is to analyze economic equilibria of the system hence often called as equilibrium model. The mutual interaction is represented by Game Theory. It is further divided into two areas- Cooperative Game Theory and Non-cooperative Game Theory. In another model named, Stackelberg game assumes that the firm with the largest market power can manipulate prices but firms having less market power cannot affect prices. This game can be modeled through mathematical program with equilibrium constraints (MPEC) problem [4]. There are more competitive models like Cournot and Bertrand but these models also have a draw back as Cournot deals with fixed quantity and variable price and Bertrand deals with fixed price and variable quantity far different from actual market supply functions where both quantity and price can be varied, it gave rise to a much better model SFE which has been used in this project work.

Agent and heuristic Model:

This model incorporates computer science techniques to model human being intelligence to simulate optimal bidding strategies [12], Genetic Algorithm based framework has been proposed in a double side auction market place through Pascal language [13]. Bidding strategy has also been discussed in an evolutionary programming [14]. Simulation in Genco’s bidding has been analyzed and compared via pros and cons of genetic algorithm, evolutionary programming and particle swarm optimization has discussed issues on modeling electricity market as Multi-Agent System (MAS) both practical and theoretical aspects.

This entire chapter dealt with the involvement of economics in the electricity market, it proposes the concepts of supply and demand which is necessary in understanding the subject matter. It further specifies market equilibrium conditions and move on to discuss different bidding strategies which are taken up by gencos in the competitive market.

Now, the next chapter deals with SFE models which are useful in spot markets, this model has also been incorporated in this project work, a brief knowledge of this concept has been discussed in this chapter itself.



3.1 Review:

The general supply function equilibrium (SFE) model was introduced by Klemperer and Meyer and applied to the electricity industry reform in England & Wales (E&W) [1,2]. A linear SFE model has been used to evaluate the previous market performance of ERCOT BES. A study has been made on bidding strategy of market participants in ERCOT within the same period and showed that several major participants with largest market share behave close to what a SFE predicts. These examples prove that SFE model is a valuable tool to simulate current electricity market. In SFE model, functional forms such as demand function, quadratic cost function, and linear supply function are specified. It is simpler to assume a linear demand function, quadratic cost function, and linear supply function. Assuming SFE with linear functional forms is more advantageous for analytical solvability.

Suppose there are Gencos (suppliers) in the electricity market where each Genco is risk neutral and has a generator characterized by following cost function


and be the quantity generated by genco ; and are the coefficients of the generator cost function, the marginal cost function of genco is as follows:

, where

When there is negligible transmission loss the aggregate demand which is inverse of linear demand function will be equal to the total output of gencos (market clearing condition), that is (3.3)

where is the spot market price; and coefficients of demand function and , where is the slope of system demand function.

Also assuming that each genco bid a linear increasing supply function, having two strategic parameters; intercept () and slope () where .

Supply function bided by genco be

In this project work the affect of transmission network has not been considered as a result the market clearing price is same for all gencos. Each genco changes its parameters in form of intercept and slope so as to maximize its profit. Supply function equilibrium means that no genco can increase its profit by unilaterally changing its bid supply function.

3.2 Multiple Period SFE:

It is assumed that demand curve can experience any random shock, which meant that the bid function needed to be optimal for any realization of the demand [1]. It is assumed that the bid function is consistent across all time periods. A single period, single market model restricts the flexibility of bidding strategy compared to true flexibility in these markets. A totally different outlook has been proposed in which supply function is not assumed to be same across multiple pricing periods; they can only be changed by varying the intercept of the bid function not their slope. The advantage of using intercept as strategic parameter is that it leads to a linear equation whose equilibrium can be easily proved in terms of existence, uniqueness, and stability.

In this report two-settlement game model has been used to formulate the two settlement market consisting of a spot market. The Linear Supply Function Equilibrium (LSFE) models have been used in spot market.

The optimization problem faced by each genco is to maximize its expected total profit, where it is equal to

LSFE Model: Genco participate in the spot market by submitting their bids in form of LSF , through different parameterization studied in [ 5 ] which can be represented as:

Different parameterizations of LSF are:

-parameterization in this the genco can choose in (3.4) arbitrary but is required to specify a fixed , usually .

-parameterization in this the genco can choose in (3.4) arbitrary but is required to specify a fixed , usually .

The optimal value of can be obtained by differentiating (3.5) with respect to and using equations (3.4) and (3.3) the following equation is obtained

By equating above equation to zero optimal value of is obtained as [16]

3.3 SFE with Resource Constraint:

Considering its position in the fuel market genco makes a bidding decision in a day a head type electricity market. At a particular day genco submits a series of bidding functions for the following T periods on the next day, there by maximizing social welfare of ISO and its own pay off. If transmission congestion is considered then market clearing price is not same for all gencos.

In this we are designing a model which is good to be analyzed in biding strategy with multiple period constraints, which genco face in the real world such as fuel inventory, energy limit group, volume of reservoir etc. Genco’s normally hold a portfolio of assets in both electricity and fuel markets, a single market model severely restricts the flexibility of bidding strategy compared to the true flexibility in these markets.

The linear SFE model requires players to fix a parameter of bidding function in order to solve a unique equilibrium, the main advantage to choose slope as strategy parameter is that most electricity markets worldwide allow genco to bid a different supply function at each period and the slope parameterization model cannot be applied to multiple period situation. Another key advantage is that since intercept parameterization leads to a linear equation system, the existence, uniqueness, and stability of the equilibrium are easy to prove, and a lot of computational difficulties will be reduced compared to the slope parameterization model.

It is assumed that a genco makes bidding decision in a day a head type electricity market considering its position in a fuel market. The day a head market structure is assumed to have a uniform non discriminatory pricing rule. Every morning on day D, genco are required to submit a series of bidding functions for the following T periods on the next day D+1. After a market clearing mechanism, ISO maximizes its social welfare. Each genco is informed of the market prices and awarded MW quantities for every period “t”, then gencos can settle with ISO on their profits/ losses for the next day.

Multiple time bidding can be applied to two separate systems

1) the system that deals with single genco multiple time bidding, in this we use PIPA (Penalty Interior Point Algorithm) to solve for single genco

2) the system in which multiple genco can be dealt according to different time periods, this is the case we are going to solve.

Fig.3.1. Two-level optimization

The -parameters can be obtained through the equation , the derivation of this equation has been shown in the appendix.



To understand the proposed concepts a case study dealing with 3 genco and 5 genco has been done. In this section and , where is the intercept and is the slope of the inverse demand function. Here costate variable and reciprocal of fuel cost of genco has been provided as which is fixed for all gencos.

Here Table 1 presents which denotes intercept of inverse demand function, slope of inverse demand function, reciprocal of fuel cost function, costate variable respectively.


Constants for Gencos








Cost Coefficients of 3 Gencos

Genco no.




Cost parameter($/MWh)




Cost parameter ($/MWh)




Table II deals with coefficients of cost function for 3 genco, where as Table III shows the simulation results for 3 genco, Table IV deals with coefficients of cost function for 5 genco and Table V shows the simulation results for 5 genco.

Strategic parameters set up to individual genco have been obtained using the concept of multiple genco bidding strategy. Moreover, power supplied, profit earned, and price per MW has been obtained for individual genco using the concept of two-settlement model discussed above.

Costate variable proposed in this model are constants, because fuel price is estimated by the gencos before submitting a set of bids in a multi period, day a head market. The estimated forward price does affect gencos bidding strategies. The impacts are represented by the parameter which is included into .

Some important observations can be made from the Table III and Table V, i.e. in their respective cases the strategic parameter of SFE obtained is nearly equal to other gencos bidding strategic parameter. It confirms that the electricity market proposed here is an oligopoly market having few numbers of large firms.

Table III

Simulation Results for 3 Genco







( GWh)


( GWh)


( GWh)


1.0e+005 *1.4224

1.0e+005 *1.0296

1.0e+005 *0.7982




Cost Coefficients of 5 genco

genco no.






Cost parameter ($/MWh)






Cost parameter ($/MWh)







Simulation Results for 5 Genco











( GWh)


( GWh)


( GWh)


( GWh)


( GWh)


1.0e+003 *1.9681

1.0e+003 *1.1696

1.0e+003 *3.0621

1.0e+003 *2.8649

1.0e+003 *1.1696





In this project a proper SFE model has been discussed which can be applied to multiple period and multiple market situation, compared to the slope parameterization the intercept parameterization require less computational efforts, moreover genco’s decisions in both fuel and electricity market has been incorporated.

A Two-settlement approach has been successfully applied to determine market equilibrium parameters in form of price, profit, power delivered by each of the genco. It has been found that the proposed approach is very effective in determining the market equilibrium in all the cases. Simulation results further confirms that there is no effect of forward contract in the intercept parameterization as incentives in intercept parameterization are zero. It has also been justified from the simulation results that electricity market are oligopoly market consisting of few large firms as the strategic parameters are very close to each other in both the cases of 3 genco and 5 genco.

Optimal bidding strategy is determined considering parameters at particular time. The parameters so obtained depend on other rival genco decision at time. It is noted that the work done employs the commonly used assumption of risk neutrality on all gencos and sufficient arbitrators in the market.

There could be a possibility to investigate the bidding behaviors of gencos when the risk aversion for gencos is considered. The work may be extended by considering risk management in gencos decision involving a portfolio of management theory.



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