Market Equilibrium Models with Continuous and Binary Variables by J. David Fuller Department of Management Sciences University of Waterloo 25 June 2008

Abstract Social welfare maximizing equilibrium models with continuous and binary variables usually have no solutions if prices are related only to the continuous variable commodities. Previous research has shown that introduction of prices related to the binary variables, a type of multipart pricing, can produce an equilibrium. However, no previous research has considered the introduction of binary variables into nonintegrable models or Cournot models, which are not based on social welfare maximization. This paper proposes a general definition of equilibrium, in a mathematical programming framework, that allows for continuous variable commodity prices and prices related to binary variables, and that is applicable to social welfare maximization models, to nonintegrable models, and to Cournot models.

Some properties of the definition

are explored. Another definition of equilibrium, called “strategic equilibrium” is also given for single-part pricing — prices are only related to continuous variable commodities. Algorithms for the solution of the equilibrium models are presented. Solutions of several small examples show that multipart pricing can induce a multiplicity of equilibria, and that there can be multiple strategic equilibria, even with asymmetry in the data for producers and consumers.

1. Introduction For several decades, economic equilibrium models have been expressed and solved by mathematical programming techniques involving only continuous variables — see, e.g., [21], [1], [13], [8].

These models usually involve convexity assumptions, corresponding to production with

constant or decreasing returns to scale.

Some of these models assume price-taking market

participants and “integrability” (see [1] for a definition of integrability of demand), and they can therefore be expressed as constrained mathematical programs having a single objective 1

representing social welfare, usually consumers’ plus producers’ surplus. However, nonconvexities, such as integer valued activities, are abundant in the real world ([23], [22], [4]), but attempts to define equilibrium prices in such situations have been fraught with difficulties including the frequent non-existence of equilibrium (e.g. [12], [23], [16]). Recent research has explored ways to price binary valued activities such as the startup of an electric generator or the decision to build a production facility, in order to guarantee existence of equilibrium prices.

For a social welfare maximization model expressible as a mixed integer

linear program with binary variables, O’Neill et al. [20] define an auxiliary linear program in which the binary variables are fixed at their optimal values in extra constraints whose dual variables give the prices associated with the binary variables. In applications, the scheme in [20] is a type of two-part pricing:

a price for the commodity itself, and a separate, binary-

related price for making capacity available.

Potential drawbacks in [20] include: the binary

related price is different for every producer; it is possible for some capacities to have a negative price while others have a positive price which means that money flow related to these prices can be into some producers and out of other producers; and the total amount of money paid to producers does not necessarily equal the total amount collected from consumers. In recent work, Sen and Genc [24] define binary-related prices that are all non-negative and that are zero for binary variables fixed at zero. The present paper also considers multipart pricing as a means to find an equilibrium, but the prices need not be different for every producer — e.g., a single price can be paid to all producers, in proportion to the capacity turned on (binary variable equals one). In the present paper, the binary-related prices can be positive or negative, but it is possible to ensure, in model formulation, that the direction of flow of money related to these prices is either all from consumers to producers, or all from producers to consumers (a rebate). Furthermore, in this paper, the equality of money paid to that collected is enforced in the model formulation. In some electricity markets, charges to consumers that are over and above the commodity price are called “uplift.” Hogan and Ring [14] pointed out that the binary-related payments in [20] would in practice come from uplift charges on consumers, and they proposed a method to adjust commodity prices in order to minimize the uplift, while maintaining the social welfare optimizing settings of the continuous and binary variables.

2

In [9] and [18], prices related

to binary variables and producer cost adjustment parameters are determined by the market operator in a “generalized uplift” scheme to ensure that all producers are satisfied that the social welfare maximizing outputs also maximize their profits; however, payments to some producers are funded by charges to other producers. The present paper generalizes the idea of minimizing uplift by allowing prices (for continuous and binary variables) and quantities to change in the minimization, thus removing the assumption that the social welfare maximum is the only potential equilibrium to consider. One of the problems with the social welfare maximum for models containing binary variables is that some producers can earn negative profits; this is clearly not an equilibrium, because these producers would prefer to produce nothing, at zero cost and zero profit. Several papers ([11], [10], [19]) have proposed methods for electricity market operators to modify the social welfare maximization approach in order to require that all producers achieve at least a minimum specified profit (e.g, profit at least zero).

However, a state of disequilibrium can occur even

if all producers achieve positive profits at the social welfare maximum, as illustrated in the present paper.

This paper defines equilibrium as a set of prices for continuous commodities

and binary variables, and quantities (continuous and binary) such that no agent (producer or consumer) can do any better, whether “better” means escaping negative profits or finding higher positive profits. This paper also defines “strategic equilibrium,” a Nash equilibrium in which the producers choose their binary variables with the knowledge of the commodity prices and profits that will arise due to their choices, given the binary variables chosen by the other producers; there are no binary-related prices, and consequently, negative profits are avoided by restricting the set of producing facilities so that commodity prices are high enough to cover all costs. To the best of this author’s knowledge, the attempts to define equilibrium prices in the presence of integer or binary variables have been confined to social welfare maximization models, which are usually single commodity models with price-taking participants. There have been no attempts to extend nonintegrable multicommodity models (e.g., [1]), or models with non-pricetaking behaviour (e.g., Cournot models as in [13]) from the realm of all-continuous variable models to include integer or binary variables. The first step to extend an all-continuous social welfare maximization model is obvious:

include integer or binary variables, as appropriate,

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and move from linear or nonlinear programming solvers to mixed integer linear or nonlinear programming solvers, whose capabilities have improved greatly in recent years. But for nonintegrable or Cournot models, there is no obvious first step. For example, if one formulates a variational inequality or complementarity problem [5] for an all-continuous variable model to be solved by PATH [6] in GAMS [2], then the inclusion of binary variables in the model brings the modelling effort to a halt because there is no solver for a mixed integer complementarity problem, and worse, there is not even an established definition of equilibrium in mathematical programming terms.

In addition to social welfare maximization models, this paper includes

nonintegrable multicommodity models and Cournot models in the definitions of equilibrium and in the algorithms to solve the models. The main contributions of this paper are as follows: (1) a definition of equilibrium prices and quantities in a mathematical programming framework that - applies to models with continuous variables and binary variables - coincides with well established complementarity or optimization models of equilibrium if there are no binary variables - provides a measure of disequilibrium when no equilibrium exists - can incorporate multipart pricing that has commodity prices and additional payments that are proportional to binary variables, with varying degrees of price discrimination - applies not only to one commodity price-taker models, but also to multicommodity pricetaker models and to Cournot models; (2) an algorithm that solves for equilibrium prices and quantities, or determines that none exist; (3) to show, theoretically and by examples, that equilibria can exist for a multipart pricing model even when no equilibrium exists for the single-part pricing version of the same model; (4) to show, for the single-part pricing maximum social welfare model, that an equilibrium exists if and only if the duality gap equals zero; (5) to show, by examples, that with multipart pricing for the one commodity price-taker model, there can be several equilibria that are not the social welfare maximum; (6) a definition and algorithm for solution of “strategic equilibrium,” a single-part pricing equilibrium in which agents foresee the new equilibrium prices that will arise due to unilateral

4

changes to their binary variables; (7) to show, by examples, that there can be radically different multiple equilibria for the multipart pricing models and for the single-part strategic equilibrium model even with asymmetric data for the producers and consumers; and (8) to show, by examples, that a multipart pricing equilibrium can have infinitely many solutions for the binary-related prices. Section 2 motivates the work with several simple examples of equilibrium models involving both continuous and binary variables.

Section 3 presents the general model, definitions of

equilibrium, some theoretical properties, and algorithms to solve for equilibrium or determine that none exist. Section 4 discusses solutions to the simple models of section 2, and section 5 suggests avenues for future research.

2. Motivation: Several Simple Examples of Equilibrium Models Several simple models are presented in this section, to motivate the general form discussed in section 3. A model of one commodity is presented first, with single-part pricing and two-part pricing variants, followed by a two commodity model, also with single- and two-part pricing. The third subsection presents a Cournot model with one commodity and single-part pricing. Much of the notation for all models is introduced in the discussion of the one commodity model; extra symbols or different interpretations of symbols are defined as needed in the second and third subsections. Some computed solutions are discussed in section 4.

2.1 One Commodity Price-Taker Models The single-part pricing model is based loosely on the price-taker electricity market model of Hobbs [13] but with the transmission system removed, and without arbitraging agents. In the present paper, unlike Hobbs [13], the model contains binary variables to represent the on/off state of the firms’ production units.

Demand curves are assumed to be linear, and variable

production costs are constant within the production capacities of the units. The symbols in the single- and two-part pricing models are as follows. Indices and Sets f = firm, f ∈ {f 1, f 2} h = production unit, h ∈ Hf = {h1, h2} j = consumer, j ∈ {j1, j2, j3} 5

Parameters Cf h = variable production cost of unit h, for firm f Kf h = fixed cost of unit h, for firm f XKf h = capacity of unit h, for firm f αj = price intercept of inverse demand curve for consumer j β j = absolute value of slope of inverse demand curve for consumer j M = large positive number (only in two-part pricing model) Variables zf h = binary variable for state of unit h, for firm f (0 for off, 1 for on) qfSh = nonnegative continuous variable of output of unit h, for firm f qjD = nonnegative continuous variable of demand by consumer j p = nonnegative continuous variable of market price zjD = binary variable; 1 if qjD > 0; 0 otherwise (only in two-part pricing model) pzD = access fee paid by any consumer j if zjD =1 (only in two-part pricing model) pzS = capacity price per unit of capacity with zf h =1 (only in two-part pricing model)

2.1.1 Single-Part Pricing: One Commodity, Price-Taker At an equilibrium, if it exists, the values of production, consumption and market price would have these properties: (A) each firm’s profit and each consumer’s surplus would be maximized, given the price (see (1) - (4) below); (B) the total amount produced by all firms exceeds or equals the total amount demanded by all consumers, at the equilibrium price (see (5) below); (C) the total revenue collected from all consumers equals the total revenue received by all producers (see (6) below). Property C is not enforced by O’Neill et al. [20]. It is in fact a complementarity condition, but it is important to note that the motivation to include it is based on a simple economic idea, and not on abstract mathematical requirements — the only revenue available to producers comes from sales at the market price, and all money paid by consumers makes its way to producers. The price-taker equilibrium model can be summarized as optimization models for firms and consumers, where each market participant takes the market price as a “given,” beyond its

6

control, together with the market clearing and complementarity conditions: Firm (for f ∈ {f 1, f 2}) : max

qfSh ,zf h

X

h∈Hf

{pqfSh − Cf h qfSh − Kf h zf h }

s.t. 0 ≤ qfSh ≤ XKf h zf h , zf h ∈ {0, 1}, ∀h ∈ Hf

(1) (2)

Consumer (for j ∈ {j1, j2, j3}) : 1 max αj qjD − β j (qjD )2 − pqjD D 2 qj

(3)

s.t. qjD ≥ 0

(4)

Market Clearing and Complementarity: X

X j

j

qjD −

pqjD −

X X f

h∈Hf

X X

f h∈Hf

qfSh ≤ 0

pqfSh = 0, p ≥ 0

(5) (6)

If there were no binary variables, or if they were known, then because the model would have only continuous variables, the Karush-Kuhn-Tucker (KKT) conditions for the firms’ and consumers’ optimization models could be written, and when combined with the market clearing and complementarity conditions, a Complementarity Problem (CP) would be defined. It would be a simple matter to code the CP in software such as GAMS [2], and call a solver such as PATH [6] to find a solution. Furthermore, unless the model is infeasible, a solution is guaranteed to exist, under quite general conditions. However, because the binary variables are not known in advance, there is no CP problem that is equivalent to the model. It is not immediately obvious how to solve the equilibrium model (1) - (6) with binary variables in the formulation, or even to determine whether or not an equilibrium exists.

2.1.2 Social Welfare Maximization Model For models in which all variables are continuous and demand is integrable (as it is for the simple model in this section), a common approach to calculation of price-taker, single-part 7

pricing equilibrium is to formulate the social welfare maximization problem as a constrained optimization problem; the idea is usually attributed to Samuelson [21]. The objective function is the sum of all firms’ and consumers’ objective functions (in which the revenue terms of producers and payments by consumers cancel), and the constraints are the collection of all constraints in the firms’ and consumers’ models, plus market clearing.

When a solution is

found, the market price is recovered as the dual variable of the market clearing constraint, and complementarity ensures that the total of payments by consumers equals total revenue to producers. The objective function is often described as “consumers’ plus producers’ surplus,” or “social welfare.” When binary variables are present in a model, there is no impediment to formulation of a mixed integer optimization problem, with fixed producer cost terms in the objective and binary variables in the constraints. The social welfare maximization model with binary variables can be coded in software such as GAMS, and solved by one of several available solvers for mixed integer nonlinear programs; prices are dual variables of the market clearing constraints in the nonlinear program derived by fixing binary variables at their optimal values. Unfortunately, social welfare maximization frequently gives a solution that is not an equilibrium — e.g., firms can earn negative profits, less than the zero profit that they would earn if they produced no output. Section 4 includes an illustration of negative profits, and another illustration of disequilibrium even though both firms enjoy positive profits at the social welfare maximum solution. For the simple price-taker model of this section, the social welfare maximization model is detailed in (7) - (10) below. Social Welfare Maximization Model X X X 1 {αj qjD − β j (qjD )2 } − {Cf h qfSh + Kf h zf h } 2 qjD ,qfSh ,zf h j f h∈Hf X X X qjD − qfSh ≤ 0 s.t. max

j

0≤

qfSh

f

(7) (8)

h∈Hf

≤ XKf h zf h ∀f, ∀h ∈ Hf

qjD ≥ 0 ∀j, zf h ∈ {0, 1} ∀f, ∀h ∈ Hf

8

(9) (10)

2.1.3 Two-Part Pricing: One Commodity, Price-Taker The price-taker equilibrium model (1) - (6) is reformulated to include an “access fee” which a consumer must pay in order to be allowed to take any positive amount of the continuous variable commodity; the continuous commodity is paid for by the usual per-unit price, as in the single-part pricing model. The total of all access fees is distributed among all producers in proportion to the capacities of their units that are turned on, via a capacity price. Other two-part pricing schemes are possible — e.g., to have each consumer pay an access fee which is proportional to its “size,” as measured by the main circuit breaker rating in amperes at the consumer’s electrical supply panel. The two-part pricing model formulation is as follows.

Note that no restriction is placed

on the sign of pzD , or of pzS . Both must have the same sign, due to (17) below. The model is set up so that positive values mean that consumers pay an access fee, and producers receive a capacity price. Negative values mean that consumers receive a rebate which is funded by producers paying a capacity price for each unit of capacity that is turned on. Firm (for f ∈ {f 1, f 2}) : max

qfSh , zf h

X

h∈Hf

{pqfSh + pzS XKf h zf h − Cf h qfSh − Kf h zf h }

s.t. 0 ≤ qfSh ≤ XKf h zf h , zf h ∈ {0, 1}, ∀h ∈ Hf

(11) (12)

Consumer (for j ∈ {j1, j2, j3}) : 1 max αj qjD − β j (qjD )2 − pqjD − pzD zjD D D 2 qj , zj

(13)

s.t. 0 ≤ qjD ≤ M zjD , zjD ∈ {0, 1}

(14)

9

Market Clearing, Complementarity, Distribution of Access Fees: X

X j

X

j

pqjD

j

pzD zjD −

qjD − −

X X f

X X f

h∈Hf

X X

f h∈Hf

qfSh ≤ 0

(15)

pqfSh = 0, p ≥ 0

pzS XKf h zf h = 0

(16) (17)

h∈Hf

2.2 Two Commodity Price-Taker Models Much of the notation of the one commodity models is reused and reinterpreted for the two commodity model; new notation and definitions are given below.

The two commodities are

indexed by j, and each inverse demand function depends not only on its “own” quantity, but also on the the quantity demanded of the other commodity. Firm f 1 produces commodity j1, and f 2 produces commodity j2. Firm f 2 can produce directly from unit h1, but to produce from unit h2, it must purchase some of commodity j1 and convert it to commodity j2. For a concrete example, commodity j1 could be natural gas, and j2 could be electricity produced from a hydro source (h1) or a natural gas turbine (h2). New Notation and New Definitions of Symbols Indices and Sets j = commodity, j ∈ {j1, j2} Parameters δ j = coefficient of qjD0 in inverse demand for j, j 0 6= j ε = efficiency of conversion of commodity j1 into j2 (by firm f 2 in unit h2) Variables pj = price for commodity j (a continuous variable) If the coefficients δ j are different for the two commodities, then, as is well known, demand is not integrable, and so even if all variables are continuous, the equilibrium model cannot be formulated as a social welfare maximization model.

However, it is still possible to write an

expression for the consumers’ surplus of a commodity, contingent upon the value of demand for the other commodity; at a simultaneous equilibrium in both markets, the contingent consumers’ surplus in a market is maximized. 10

2.2.1 Single-Part Pricing: Two Commodities, Price-Taker Firm f 1 : max

qfS1,h , zf 1,h

X

h∈Hf 1

{pj1 qfS1,h − Cf 1,h qfS1,h − Kf 1,h zf 1,h }

s.t. 0 ≤ qfS1,h ≤ XKf 1,h zf 1,h , zf 1,h ∈ {0, 1}, ∀h ∈ Hf 1

(18) (19)

Firm f 2 : max

qfS2,h , zf 2,h

X

h∈Hf 2

{pj2 qfS2,h − pj1 (qfS2,h /ε) − Cf 2,h qfS2,h − Kf 2,h zf 2,h }

s.t. 0 ≤ qfS2,h ≤ XKf 2,h zf 2,h , zf 2,h ∈ {0, 1}, ∀h ∈ Hf 2

(20) (21)

Consumers’ Surplus Maximization (for j ∈ {j1, j2}): 1 max (αj + δ j qjD0 6=j )qjD − β j (qjD )2 − pj qjD D 2 qj

(22)

s.t. qjD ≥ 0

(23)

Market Clearing and Complementarity: D + (qfS2,h /ε) − qj1 D − qj2 D + pj1 (qfS2,h /ε) − pj1 qj1 D pj2 qj2

−

X

h∈Hf

X

h∈Hf

X

h∈Hf

X

h∈Hf

qfS1,h ≤ 0

(24)

qfS2,h ≤ 0

(25)

pj1 qfS1,h = 0, pj1 ≥ 0

(26)

pj2 qfS2,h = 0, pj2 ≥ 0

(27)

2.2.2 Two-Part Pricing: Two Commodities, Price-Taker For simplicity of the illustration, the same form of capacity price and access fee is introduced as in the one commodity model:

a single capacity price pzS applies to producers of both

commodities, and one access fee pzD is paid twice, for separate access to each commodity. To have a capacity price and fee structure common to two commodity markets may be unlikely to

11

arise in the real world, but the structure serves to illustrate the point that multipart pricing can lead to a binary state being an equilibrium even when it is not an equilibrium under single part pricing — see section 4 for an illustration. Firm f 1 : max

qfS1,h , zf 1,h

X

h∈Hf 1

{pj1 qfS1,h + pzS XKf 1,h zf 1,h − Cf 1,h qfS1,h − Kf 1,h zf 1,h }

s.t. 0 ≤ qfS1,h ≤ XKf 1,h zf 1,h , zf 1,h ∈ {0, 1}, ∀h ∈ Hf 1

(28) (29)

Firm f 2 : max

qfS2,h , zf 2,h

s.t. 0 ≤

X

h∈Hf 2

qfS2,h

{pj2 qfS2,h + pzS XKf 1,h zf 1,h − pj1 (qfS2,h /ε) − Cf 2,h qfS2,h − Kf 2,h zf 2,h }(30)

≤ XKf 2,h zf 2,h , zf 2,h ∈ {0, 1}, ∀h ∈ Hf 2

(31)

Consumers’ Surplus Maximization, Commodity j : 1 max (αj + δ j qjD0 6=j )qjD − β j (qjD )2 − pj qjD − pzD zjD D 2 qj

(32)

s.t. 0 ≤ qjD ≤ M zjD , zjD ∈ {0, 1}

(33)

Market Clearing, Complementarity, Distribution of Access Fees: D + (qfS2,h /ε) − qj1 D − qj2 D pj1 qj1

+ pj1 (qfS2,h /ε) − D pj2 qj2

X j

pzD zjD −

−

X X f

X

h∈Hf

X

h∈Hf

X

h∈Hf

X

h∈Hf

qfS1,h ≤ 0

(34)

qfS2,h ≤ 0

(35)

pj1 qfS1,h = 0, pj1 ≥ 0

(36)

pj2 qfS2,h = 0, pj2 ≥ 0

(37)

pzS XKf h zf h = 0

h∈Hf

12

(38)

2.3 Cournot Model It is assumed here that there are two firms that supply a single commodity to three customer classes, under single part pricing. As has been discussed in many textbooks and papers (e.g., [13]) each firm is aware of the relationship between price and total sales of all firms in each customer class, and each firm assumes that the other firm’s sales quantities are “given,” i.e. beyond its control. Consumers merely react to prices as given on the demand curves; only the firms make decisions, in order to maximize their profits. The notation here is the same as for the one commodity model, except that in a Cournot model, it is necessary to distinguish total demand quantity of consumers by the various firms which supply them. New Variables for Cournot Models sf j = sales by firm f to consumer j The variables qjD do not appear in the Cournot model, but in the expression for inverse P demand, qjD corresponds to f sf j , the total sales of all firms to j. There is no explicit market

clearing constraint, but for each firm, total output must exceed or equal the total sales to all customer classes. Prices may be different for the different customer classes; price for a customer class is represented in the model by the inverse demand expression as a function of total sales by all firms to that customer class.

Firm (for f ∈ {f 1, f2}) : max

sf j ,qfSh ,zf h

X X X {[αj − β j (sf j + sgj )]sf j } − {Cf h qfSh + Kf h zf h } j

g6=f

s.t. 0 ≤ qfSh ≤ XKf h zf h , ∀h ∈ Hf X X sf j − qfSh ≤ 0 j

(39)

h∈Hf

(40) (41)

h∈Hf

sf j ≥ 0, ∀j, zf h ∈ {0, 1} , ∀h ∈ Hf

(42)

It is possible to define two part pricing within a Cournot framework, as with the previous models, but this would require a substantially revised formulation, in which each consumer class becomes a player in the game, deciding upon whether or not to pay the access fee, and how much to purchase. In order to conserve space here, this topic is left for future work.

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3. General Model, Definitions of Equilibrium, and Properties This section presents a model of equilibrium that is general enough to include all of the examples of section 2. The general term “agent” can be a firm or a consumer (or consumer class). The general term “linking constraint” includes the market clearing constraints of section 2, but it may also apply to any other constraint which links the decisions on continuous variable values by different agents, e.g. government imposed limits on emissions of pollutants. The first definition of equilibrium, used in sections 3.1 and 3.2, is a set of values of continuous quantities, binary variables and prices with two properties: (a) feasibility with respect to all constraints, including the constraints that require equality of revenues collected and distributed; and (b) no agent can do better by unilateral deviation from these continuous quantities and binary variables, given the prices and continuous variable values of other agents.

This definition

is formulated as a constrained optimization problem. The second equilibrium definition, called strategic equilibrium, is presented in section 3.4. The symbols for the general model are defined below. Indices and Sets i = agent, i ∈ I K = set of feasible solutions to equilibrium problem Ki = set of feasible solutions to agent i optimization problem Parameters Ai = matrix of coefficients of continuous variables controlled by i, in the linking constraints a = vector of right sides of linking constraints Bi = matrix of coefficients of continuous variables controlled by i, in the constraints of agent i optimization problem Di = matrix of coefficients of binary variables controlled by i, in the constraints of agent i optimization problem Ei = matrix defining allocation of binary related payments or charges for agent i bi = vector of right sides of the constraints of agent i optimization problem di = vector of coefficients of binary variables controlled by i, in the objective function of agent i optimization problem Variables and Functions

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xi = vector of continuous variables controlled by agent i x = vector which is concatentation of xi for all i x−i = continuous variables not controlled by i, i.e., vector x except for xi wi = alternate symbol for xi zi = vector of binary variables controlled by agent i; zi ∈ {0, 1}ni z = vector which is concatentation of zi for all i yi = alternate symbol for zi p = vector of continuous prices corresponding to linking constraints pz = vector of continuous prices related to binary variables ri = vector of continuous dual variables for constraints of agent i optimization problem Fi (xi ; x−i ) = xi -dependent term in objective function of agent i optimization problem (possibly parameterized by x−i ); assumed continuously differentiable in xi vi (x, zi , p, pz ) = maximum improvement in objective value of i by unilaterally deviating from (x, z) under prices (p, pz ) V = total of profit improvements of all agents i In the models of section 2, the indices f and j are examples of the general index i. The variables qjD , qfSh , and sf j are examples of components of the general vector x (or w), and zf h and zjD are examples of components of the general vector z (or y).

The general symbol p

appears as a scalar variable in the one commodity price-taker models of section 2, and as a two dimensional vector in the two commodity model. The general vector pz has the two components pzS and pzD in the two-part pricing models of section 2. Note that with different choices of the dimensions of pz and of Ei , the modeller can represent differing degrees of price discrimination, between the complete discrimination of [20] and the single price per unit of capacity in the examples of section 2.

3.1 Equilibrium with Multipart Pricing The feasible region of the optimization problem of the first equilibrium definition is given in (43) - (48) below, and summarized in (49):

15

P

i∈I

Ai xi ≤ a

(43)

Bi xi + Di zi ≤ bi , ∀i ∈ I P pT (a − Ai xi ) = 0 pzT

P

(44) (45)

i∈I

Ei zi = 0

(46)

i∈I

x ≥ 0, p ≥ 0, pzT unrestricted in sign

(47)

zi ∈ {0, 1}ni , ∀i ∈ I

(48)

K = {(x, z, p)|(43), (44), (45), (46), (47), (48)}

(49)

The definition of equilibrium, introduced below, requires an evaluation, for every agent, of the maximum objective value that the agent can achieve, given a set of prices and continuous variables of other agents, and ignoring linking constraints. The feasible region for each of these optimization problems is given in (50) - (52) and summarized in (53); the alternate symbols wi and yi are used, in order to distinguish them from the symbols xi and zi in the equilibrium problem itself.

Bi wi + Di yi ≤ bi

(50)

wi ≥ 0

(51)

yi ∈ {0, 1}ni

(52)

Ki = {(wi , yi )|(50), (51), (52)}

(53)

The maximum value achievable by agent i, given the prices and other agents’ continuous variable values x−i , is determined by solving the following problem Pi (x−i , p, pz ) for optimal (wi∗ , yi∗ ), and the maximum improvement in value for i of the deviation to (wi∗ , yi∗ ) instead of (xi , zi ) is defined in (54). A key property of this improvement is given in Lemma 1. 16

Pi (x−i , p, pz ):

max

(wi ,yi )∈Ki

Fi (wi ; x−i ) + dTi yi − pT Ai wi − pzT Ei yi

vi (x, zi , p, pz ) = Fi (wi∗ ; x−i ) + dTi yi∗ − pT Ai wi∗ − pzT Ei yi − [Fi (xi ; x−i ) + dTi zi − pT Ai xi − pzT Ei zi ] (54) Lemma 1: vi (x, zi , p, pz ) ≥ 0 for every i ∈ I. Proof: (xi , zi ) is feasible in Pi (x−i , p, pz ), and so the value of an optimal solution (wi∗ , yi∗ ) exceeds or equals the value of (xi , zi ). QED The total improvement in value due to deviations of all agents is minimized over feasible (x, z, p, pz ) in the Minimum Deviation Problem (MDP): P MDP: min V = vi (x, zi , p, pz ) (x,z,p, pz )∈K i∈I ∗ ∗ Lemma 2: Let (x , z , p∗ ,

pz∗ ) be an optimal solution to MDP with optimal value V ∗ . Then

V ∗ = 0 if and only if vi (x∗ , zi∗ , p∗ , pz∗ ) = 0 for all i ∈ I. P vi (x∗ , zi∗ , p∗ , pz∗ ) = 0, and so by Lemma 1, vi (x∗ , zi∗ , p∗ , pz∗ ) = 0 Proof: If V ∗ = 0, then i∈I P for all i ∈ I. If vi (x∗ , zi∗ , p∗ , pz∗ ) = 0 for all i ∈ I, then V ∗ = vi (x∗ , zi∗ , p∗ , pz∗ ) = 0. QED i∈I

Definition 3:

(x∗ , z ∗ , p∗ ,

pz∗ )

is an equilibrium if it solves MDP with value V ∗ = 0.

If the values of the binary variables were known to be z = z, then an equilibrium in the continuous variables could be found by well established methods, i.e., by solving the following complementarity problem CP(z). This problem plays a key role in the theory and in algorithms to find an equilibrium or to determine that none exist. The symbol Oi represents the gradient with respect to continuous variables xi (excluding derivatives with respect to x−i ). CP(z): P

i∈I

Ai xi ≤ a ⊥ p ≥ 0

(55)

Bi xi + Di z i ≤ bi ⊥ ri ≥ 0, ∀i ∈ I

(56)

Oi Fi − pT Ai − riT Bi ≤ 0 ⊥ xi ≥ 0, ∀i ∈ I

(57)

Theorem 4: If an equilibrium exists, (x∗ , z ∗ , p∗ , pz∗ ), then there exists a vector r∗ composed of subvectors ri∗ ∈ R|bi | , i ∈ I, such that (x∗ , p∗ , r∗ ) solves CP(z ∗ ). Proof: By feasibility of (x∗ , z ∗ , p∗ , pz∗ ) in MDP, (55) holds. 17

The remaining conditions in

CP(z ∗ ), (56) and (57), are shown to hold by the following argument.

By Definition 3,

V ∗ = 0. Therefore, by Lemma 2, vi (x∗ , zi∗ , p∗ , pz∗ ) = 0 for every i ∈ I. Consequently, (x∗i , zi∗ ) achieves the optimal value in Pi (x∗−i , p∗ , pz∗ ), and so x∗i achieves the optimal value in the same

problem with yi fixed at the optimal value zi∗ . Because Fi is continuously differentiable in xi

and the constraints are linear, the Karush-Kuhn-Tucker conditions hold (see, e.g., [17], Lemma 5.2 and Theorem 5.1), i.e., (56) and (57) with zi∗ in place of z i . QED This theorem establishes that equilibrium commodity prices p relate to the continuous variables x just as in all-continuous variable models, given the equilibrium values of the binary variables, z. For example, average cost pricing is not, in general, an equilibrium state. Also, the costs di , related to the binary variables, bear no relation to the equilibrium commodity prices p.

Therefore, it should come as no surprise if revenue to firms from commodity sales

alone is sometimes insufficient to cover all costs; revenues related to binary variables can remedy this problem, as will be illustrated in section 4. In light of Theorem 4, it is clear that the search for any equilibria can be restricted to the solutions of CP(z) — (55), (56), (57) — for all z that lead to feasible solutions x ≥ 0 of (43), (44). The following developments pursue this idea to define a computational procedure to find all equilibria, or to determine that none exist, by enumerating all possible values of the binary vector z. Since it is possible, in general, that there could be multiple solutions (x, p) of CP(z), or no solutions, the following assumption is made, to simplify the algorithm. Assumption 5. For any feasible z, there is a unique solution (x, p) of CP(z). To justify Assumption 5, first note that CP(z) is equivalent to the following variational inequality problem ([5], volume I, Proposition 1.2.1): P Find x ∈ L = {x| Ai xi ≤ a , Bi xi + Di z i ≤ bi , xi ≥ 0 , ∀i ∈ I} such that VI(z): i∈I P Oi Fi (xi ; x−i )T (xi − xi ) ≤ 0 ∀x ∈ L. i∈I

The existence part of the assumption can be justified simply by assuming that the mapping (Oi Fi , ∀i) is continuous and the set L is bounded ([5], volume I, Corollary 2.2.5). Uniqueness in x can be assured by assuming that the mapping (−Oi Fi , ∀i) is strictly monotone ([5], volume

I, Theorem 2.3.3).

Uniqueness in p occurs for models such that (57) allows one to solve for

p in terms of a unique x (e.g., the single and multicommodity price-taker models of section 18

2 yield formulas that say that price is equal to the inverse demand function evaluated at the equilibrium demands). The examples of section 2 do not satisfy strict monotonicity due to cost terms in the firms’ objective functions that are linear in the continuous variables. However, a less restrictive assumption that applies to all the examples is discussed after Algorithm 6 below. Now to return to the development of the algorithm, the minimization problem MDP is first summarized as a min-max problem, as follows.

∗

V = min z x,z,p, p

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

max

P

wi ,yi i∈I

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

Fi (wi ; x−i ) + dTi yi

− pT Ai wi

− pzT Ei yi

s.t. Bi wi + Di yi ≤ bi , ∀i ∈ I

⎤ ⎥ ⎥ ⎥ ⎥ ⎦

wi ≥ 0, yi ∈ {0, 1}ni , ∀i ∈ I ∙ ¸ P T T zT − Fi (xi ; x−i ) + di zi − p Ai xi − p Ei zi i∈I

s.t.

P

i∈I

Ai xi ≤ a

Bi xi + Di zi ≤ bi , ∀i ∈ I P pT (a − Ai xi ) = 0, p ≥ 0 i∈I P zT p Ei zi i∈I

=0

xi ≥ 0, zi ∈ {0, 1}ni , ∀i ∈ I

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(58)

Suppose that an equilibrium exists, (x∗ , z ∗ , p∗ , pz∗ ). Restricting the search for the minimum in (58) by fixing (x, z, p) = (x∗ , z ∗ , p∗ ) defines a problem that will determine the remaining variables in the vector pz∗ . This problem can be simplified by adding and subtracting pT a P in the two main terms of the objective function of (58), by using p∗T (a − Ai x∗i ) = 0 and i∈I P pzT Ei zi∗ = 0 to simplify the second main term of the objective function, and by dropping i∈I

all constraints in x, z and p except the one involving pz (because (x∗ , z ∗ , p∗ ) satisfy these

constraints). The result is:

19

V ∗ = min z p

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

max

P

wi ,yi i∈I

Fi (wi ; x∗−i ) + dTi yi − p∗T Ai wi − pzT Ei yi s.t. Bi wi + Di yi ≤ bi , ∀i ∈ I wi ≥ 0, yi ∈ {0, 1}ni , ∀i ∈ I P s.t. pzT Ei zi∗ = 0 i∈I

⎤ ⎫ ⎪ ⎪ ⎪ ⎥ ⎪ ⎪ ⎥ ⎪ ⎪ ⎥ ⎪ ¸ ∙ ⎥ ⎬ ∗T P ∗ ∗ T ∗ ⎦ +p a− Fi (xi ; x−i ) + di zi ⎪ i∈I ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭ (59)

Since any equilibrium must satisfy V ∗ = 0, and by Theorem 4 it must occur at the solution

(x, p) of CP(z) for a feasible z, an algorithm can now be defined that will find all equilibria or determine that none exist. For every feasible z, the algorithm calculates a solution (x, p) of CP(z), and then calculates a solution of:

W = min z p

If W + pT a −

∙

P

i∈I

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

max

P

wi ,yi i∈I

Fi (wi ; x−i ) + dTi yi − pT Ai wi − pzT Ei yi s.t. Bi wi + Di yi ≤ bi , ∀i ∈ I wi ≥ 0, yi ∈ {0, 1}ni , ∀i ∈ I P s.t. pzT Ei z i = 0 i∈I

¸

⎤ ⎫ ⎪ ⎪ ⎪ ⎥ ⎪ ⎪ ⎥ ⎪ ⎪ ⎥ ⎪ ⎥ ⎬ ⎦ . ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(60)

Fi (xi ; x−i ) + dTi z i = 0, then (x, z, p, pz ) is an equilibrium, where pz solves

(60). The problem (60) is in general a bilevel optimization problem with binary variables in the lower level (inner maximization) — a difficult problem to solve — but note that the lower level decomposes into separate problems for the different agents, given a value of pz . However, some particular models can be solved easily — e.g. as a linear program in two models of section 4. The algorithm is outlined in detail, next. Algorithm 6: For each combination of z 1. if there is no feasible solution to (43), (44), x ≥ 0 with this z, then skip to the next combination; 2. solve CP(z) and record the solution (x, p); z 3. calculate W from ¸ solution p ; ∙ (60) and record the P Fi (xi ; x−i ) + dTi z i = 0, then record (x, z, p, pz ) as an equilibrium; else 4. if W + pT a − i∈I

skip to the next combination.

Strict monotonicity of the mapping (−Oi Fi , ∀i) may be relaxed for the examples of section 20

2. One need only assume that the mapping (−Oi Fi , ∀i) is: (a) separable as elements related to and depending on the consumer demand variables (or the firms’ sales variables, for the Cournot model) in one group, with all other continuous variables in the other group; (b) strictly monotone in the consumer demand variables (or firms’ sales); and (c) the gradient of a convex function of the other continuous variables. For example, in the two-commodity model of section 2.2, (−Oi Fi , ∀i) = (−αj + β j qjD − δ j qjD0 6=j , ∀j; Cf h , ∀f, h). The part of the mapping that ⎤ ⎡ β 1 −δ 1 ⎦ is positive is related to and depends on qjD is strictly monotone if the matrix ⎣ −δ 2 β 2 definite, which is true if the off-diagonal elements δ j are sufficiently smaller than the diagonal elements β j , as is the case for the data presented in section 4. The remaining elements Cf h form the gradient of a convex (linear) function of the variables xf h . In [7] (Theorem 9), it is shown that the solution to such a VI is unique in the consumer demand variables (or firms’ sales) and in the optimal continuous variable production costs; uniquess of price p for the examples of section 2 comes from the fact that at the solution, p equals the inverse demand function evaluated at the unique consumer demands.

Uniqueness in ∙ the consumer demand¸ variables, P Fi (xi ; x−i ) + dTi z i in step 4 the prices and production costs implies uniqueness of pT a − i∈I

of Algorithm 6. Also at step 4, W is uniquely defined by (60) for the examples of section 2

because p is unique, and because the only components of x−i are the unique consumer demands (or firms’ sales). Algorithm 6 relies on complete enumeration of all combinations of the binary variables z, and so it will be impractical to solve for models with a large number of binary variables. Future research may be able to devise more efficient algorithms. Nevertheless, the algorithm can be used for models with a small number of binary variables, as shown in section 4. If a given choice of dimensions of pz and of Ei produces a model with no equilibrium, then an increase in the dimensions (for more price discrimination) will give more degrees of freedom in the minimization and therefore may produce a model which has an equilibrium. This idea is explored in the next section, for the case of moving from single-part pricing (the dimension of pz is zero) to multipart pricing (the dimension of pz exceeds zero).

3.2 Equilibrium with Single-Part Pricing In section 4, several examples demonstrate that a single-part pricing model may have no

21

equilibrium, while a two-part pricing variant of the same model has at least one equilibrium. The theorem in this subsection provides a simple explanation of this phenomenon. Single-part pricing can be represented by following the development of the multipart pricing equilibrium model, but deleting all terms that involve pz∗ and deleting the constraint (46) from the definition of the feasible set K for the problem MDP. It is easy to see that Lemma 1, Lemma 2 and Theorem 4 are true for single-part pricing. An alternative, equivalent representation of single-part pricing equilibrium is to keep the entire multipart pricing model, but with the additional constraint that pz = 0. Introducing this restriction into the multipart model implies that the optimal value of V for single-part pricing is greater than or equal to the optimal value of V for multipart pricing. Thus, it is possible that a single part pricing model has no equilibrium, while the same model with multipart pricing does have an equilibrium. These ideas are stated precisely in the following definitions and in Theorem 7. e = {(x, z, p, pz )|(43), (44), (45), (46), (47), (48), pz = 0 } , and define the minimum Let K

deviation problem for single-part pricing as P ^ MDP: min V = vi (x, zi , p, pz ) . e (x,z,p, pz )∈K

i∈I

^ Theorem 7: If the optimal value of MDP is denoted by V ∗ and the optimal value of MDP by Vf∗ , then V ∗ ≤ Vf∗ .

Algorithm 6 can be used to find a single-part pricing equilibrium, or to determine that

none exist, but step 3 is considerably simplified because pz is fixed at zero.

Problem (60)

to determine W reduces to the inner maximization problem, which decomposes into separate maximization problems for each agent.

3.3 Social Welfare Maximization: Relation to Equilibria Many authors have noticed that the single-part prices for one commodity models with integer variables, produced by the solution of a maximum social welfare problem, frequently are not equilibrium prices, and that they can be equilibrium prices only if the duality gap is zero (e.g., [16], [18]). This subsection presents a theorem that establishes the latter for the definition of equilibrium in this paper. The maximum social welfare model can be written with the general symbols of this section as long as Fi (xi ; x−i ) does not depend on x−i ; the resulting model is denoted GMSW below 22

(G for general). GMSW: P max Fi (xi ) + dTi zi xi ,zi i∈I P s.t. Ai xi ≤ a

(61) (62)

i∈I

Bi xi + Di zi ≤ bi , ∀i ∈ I

(63)

xi ≥ 0, zi ∈ {0, 1}ni , ∀i ∈ I

(64)

The Lagrangean dual of GMSW can be written, by relaxing constraint (62) which links the agents. The next theorem shows that, when Fi (xi ; x−i ) does not depend on x−i , there is an equilbrium of the single-part pricing model if and only if the duality gap of GMSW is zero. Theorem 8:

Suppose that Fi (xi ; x−i ) does not depend on x−i , i.e. we write Fi (xi ) for all

i ∈ I. Then the following statements are equivalent: (a) (x∗ , z ∗ , p∗ ) is an equilibrium with single-part pricing; and (b) (x∗ , z ∗ ) solves GMSW, p∗ solves the dual of GMSW, and the duality gap of GMSW is zero. Proof: The duality gap of GMSW is denoted by γ here.

It is the difference between the

optimal values of the dual and the primal problems. For clarity in what follows, the alternate symbols w and y are used in place of x and z, respectively, in the definition of the dual problem. The definition of γ is as follows:

γ = min p≥0

⎧ P P ⎪ ⎪ max Fi (wi ) + dTi yi + pT (a − Ai wi ) ⎪ ⎪ ⎨ wi ,yi i∈I i∈I ⎪ ⎪ ⎪ ⎪ ⎩

s.t. Bi wi + Di yi ≤ bi , ∀i ∈ I wi ≥ 0, yi ∈

{0, 1}ni

, ∀i ∈ I

⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭

−

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

max

P

xi ,zi i∈I

s.t.

Fi (xi ) + dTi zi

P

i∈I

Ai xi ≤ a

⎪ ⎪ ⎪ ⎪ Bi xi + Di zi ≤ bi , ∀i ∈ I ⎪ ⎪ ⎪ ⎪ ⎩ x ≥ 0, z ∈ {0, 1}ni , ∀i ∈ I i i

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

(65)

^ may be written in the compact It is well known that γ ≥ 0. The optimal value Vf∗ of MDP form of (58), but without the pz terms or constraints:

23

Vf∗ = min

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

x,z,p ⎪ ⎪

⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

⎡ ⎢ ⎢ ⎢ ⎢ ⎣

⎫ ⎪ ⎪ ⎥ ∙ ⎪ ¸ ⎪ wi ,yi i∈I ⎪ ⎥ ⎪ P ⎪ T z − pT A x ⎥− ⎪ F (x ) + d ⎪ i i i i i i ⎥ s.t. Bi wi + Di yi ≤ bi , ∀i ∈ I ⎪ ⎪ i∈I ⎪ ⎦ ⎪ ⎪ n ⎪ i ⎪ wi ≥ 0, yi ∈ {0, 1} , ∀i ∈ I ⎪ ⎪ ⎬ P s.t. Ai xi ≤ a ⎪ i∈I ⎪ ⎪ ⎪ ⎪ ⎪ Bi xi + Di zi ≤ bi , ∀i ∈ I ⎪ ⎪ ⎪ P ⎪ ⎪ T ⎪ p (a − Ai xi ) = 0, p ≥ 0 ⎪ ⎪ ⎪ i∈I ⎪ ⎪ ⎪ ⎭ n i xi ≥ 0, zi ∈ {0, 1} , ∀i ∈ I

max

P

Fi (wi ) + dTi yi − pT Ai wi

⎤

(66)

First it is shown that (a) implies (b). In the above definition of Vf∗ , (66), the term pT a can

be added to the first term of the objective function (the max term) and subtracted from the

second term of the objective function, leaving the objective function unchanged in value. In P the altered second term of the objective function, the resulting expression pT (a − Ai xi ) = 0, i∈I

because of the constraint which enforces this, thus leaving the objective function independent of P p. Relaxation of the constraint pT (a − Ai xi ) = 0 decouples the p variable vector from the x i∈I

and z vectors, which gives a problem of the same form as (65); the same relaxation in the outer

minimization allows for a possibly smaller optimal value. Thus, Vf∗ ≥ γ ≥ 0. Therefore, if

(x∗ , z ∗ , p∗ ) is an equilibrium, then by Theorem 4, Vf∗ = 0, and so γ = 0. Furthermore, because

(x∗ , z ∗ ) is feasible in GMSW, p∗ is feasible in the dual of GMSW, and the gap between the values of these primal and dual solutions is zero, it follows that (x∗ , z ∗ ) solves GMSW, and p∗ solves the dual of GMSW.

The other half of the claim — that (b) implies (a) — is now shown. Application of Theorem 3 of Larsson and Patriksson [15] guarantees that if γ = 0, then for any solution (x∗ , z ∗ ) of P GMSW and corresponding dual solution p∗ , p∗T (a − Ai x∗i ) = 0. (In the notation of i∈I

Larsson and Patriksson [15], the duality gap Γ = ε + δ, where ε ≥ 0 and δ ≥ 0; therefore, P Γ = 0 implies that δ = 0, which here means that p∗T (a − Ai x∗i ) = 0.) Therefore, if γ = 0, i∈I

then a solution (x∗ , z ∗ ) of GMSW, together with the corresponding dual solution p∗ , provides

a feasible solution to (66), with a value of 0, implying that Vf∗ ≤ 0. However, by Lemma 1, Vf∗ ≥ 0. Therefore, Vf∗ = 0, and so (x∗ , z ∗ , p∗ ) is an equilibrium. QED

3.4 Strategic Equilibrium for Single Part Pricing

Section 3.2 and examples in section 4 establish that multipart pricing can sometimes lead to 24

an equilibrium when single-part pricing cannot. This subsection presents another definition of single-part pricing equilibrium that can sometimes exist even if there is no equilibrium solution in the sense of section 3.2. There is no theory to support this result, but there are examples in section 4. The basic idea is this: when an agent determines its optimal reaction to prices p, it may be realistic for the agent to suppose that small changes in its continuous variables will have negligible effect on the market price p, or on the actions x−i of other agents; but changes in the agent’s binary variables may cause or allow large changes in the agent’s continuous variables, and this may have a significant effect on p or x−i . Therefore, when contemplating a change in binary variables from zi to yi , agent i might more realistically evaluate the new state reached by this unilateral action by estimating its profit when p and x−i take on new values in the new state. An equilibrium exists if no agent can unilaterally reach any other binary state that improves its profit. This “strategic equilibrium” is defined formally as a Nash equilibrium in which the agents choose among a finite number of discrete actions. Definition 9: For any fixed z, denote the solution to CP(z), by (x, p). The profit of agent i is π i (z) = Fi (xi ; x−i ) + dTi zi − pT Ai xi . Definition 10: A unilateral deviation z\yi by agent i from z is the replacement of zi by yi in T , y T , z T , . . .)T . z, i.e. z\yi = (z1T , . . . , zi−1 i i+1

Definition 11: (x∗ , z ∗ , p∗ ) is a strategic equilibrium if max π i (z ∗ \yi ) ≤ π i (z ∗ ) for all i ∈ I. yi

A complete enumeration algorithm can be defined to find all strategic equilibria. Algorithm 12: For each combination of z 1. if there is no feasible solution to (43), (44), x ≥ 0 with this z, then skip to the next combination; 2. solve CP(z) and record solution (x, p) and profits for all agents π i (z). For each combination of z 3. if π i (z\yi ) ≤ π i (z) for every unilateral deviation z\yi and for every agent i, then record z as a strategic equilibrium.

25

4. Solutions to Several Simple Examples Solutions of the models of section 2 are presented in this section. All computations are done by GAMS [2] with calls to MIP, LP, NLP and complementarity problem solvers as appropriate. Computing times are negligible for these small models — about 20 seconds for each run, including all model generation time, on a modest notebook computer.

4.1 One Commodity Models In order to illustrate the various types of equilibria (and non-existence in some cases), three data sets, labelled A, B, and C, are devised for the one commodity models of Section 2: the price-taker models with single-part and two-part pricing, and the Cournot model with singlepart pricing. The inverse demand function parameters are the same in all three data sets (see Table 1); the production cost and capacity parameters differ among them (Table 2). Consumer

j1

j2

j3

Intercept αj

40

42

32

Slope β j 0.0800 0.0700 0.0516 Table 1: Parameters of Inverse Demands — One Commodity Models. Data

Set A

Firms

f1

Set B f2

f1

Set C f2

f1

f2

Units

h1

h2

h1

h2

h1

h2

h1

h2

h1

h2

h1

h2

Cf h

13

18

17

22

13

15

14

18

13

15

14

19

Kf h

2600

1400

2100

1100

2600

800

2100

250

2600

800

2100

250

XKf h 400 100 600 85 400 100 350 85 Table 2: Data Sets for Two Firms — One Commodity Models.

400

100

350

85

4.1.1 One Commodity, Single-Part, Price-Taker, and Social Welfare Maximization Results Algorithm 6 (the simplified form, with pz = 0 as discussed in section 3.2) was applied to each data set, and in addition, the social welfare maximum was calculated for each set.

No

single-part equilibria were found for sets A or B, but one equilibrium was found for set C, and it was also the social welfare maximum, as predicted by Theorem 8. The outputs of social welfare maximization are presented in Tables 3, 4 and 5. Table 6 shows the optimal unilateral actions that producers would take, given the price from the social welfare maximum solution (there is no need to show consumers’ unilateral actions, as their

26

optimal reaction to the price is to consume what is in the social welfare maximum solution), and the extra profit that could be earned with unilateral action, compared to the social welfare maximum. It can be seen that for set A, both firms have negative profits at the social welfare maximum, which is clearly not an equilibrium because producing nothing, for zero profit, would be better, as indicated in Table 6.

It is interesting that for set B, both firms have positive

profits, but the social welfare maximum is still not an equilibrium, because firm f 2 has an incentive to turn on its unit h2 as indicated in Table 6. For data set C, neither firm has an incentive to deviate from the social welfare optimum, which means that this is an equilibrium. Data

Set A

Firms

Set B

f1

f2

Set C

f1

f2

f1

f2

Units

h1

h2

h1

h2

h1

h2

h1

h2

h1

h2

h1

h2

zf h

1

0

1

0

1

0

1

0

1

0

1

0

qfSh

400

0

535.34

0

400

0

350

0

400

0

350

0

Profitf -1000 -2100 605.88 355.14 605.88 Table 3: Output of Social Welfare Maximization for Firms. Data

Set A

355.14

Set B

Set C

Consumers

j1

j2

j3

j1

j2

j3

j1

j2

j3

qjD

287.5

357.1

290.7

237.3

299.8

212.9

237.3

299.8

212.9

Cons.Surp.j 3306.3 4464.3 2180.2 2252.8 3145.6 1169.4 2252.8 3145.6 1169.4 Table 4: Output of Social Welfare Maximization — Demands and Consumers’ Surpluses. Data

Set A

Set B

Set C

Social Welfare

6850.8

7528.7

7528.7

Market Price p 17 21.01 21.01 Table 5: Output of Social Welfare Maximization — Social Welfare and Market Price. Data

Set A

Firms

f1

Set B f2

Set C

f1

f2

f1

f2

Units

h1

h2

h1

h2

h1

h2

h1

h2

h1

h2

h1

h2

zf h

0

0

0

0

1

0

1

1

1

0

1

0

qfSh

0

0

0

0

400

0

350

85

400

0

350

0

Profitf

0

0

605.88

361.39

605.88

355.14

Extra Profitf

1000

2100

0

6.25

0

0

27

Table 6: Optimal Unilateral Deviations from Social Welfare Maximum, Given Price. The total, across firms, of extra profit in Table 6 gives an upper bound on the minimum value of Vf∗ and can therefore be interpreted as a measure of disequilibrium. For data set A, the

disequilibrium is quite large, at 3100, but for data set B, disequilibrium is very small, at 6.25. Such a measure of disequilibrium could be of practical use in electricity pool markets — e.g., the single-part price of set A might be rejected as being too far from equilibrium, but the result of social welfare maximization for set B might be viewed as acceptably close to equilibrium. Strategic equilibria were found for all three data sets, using Algorithm 12.

Because

consumers do not control any binary variables in this single-part pricing model, the firms were the only agents considered in step 3 of Algorithm 12. Two equilibria were found for set A, and one equilibrium for each of sets B and C. None of the strategic equilibria corresponded to the social welfare maximum solution. The key results are presented in Table 7. For all three data sets, the strategic equilibria involve the withholding of capacity, compared to the social welfare maximum, which supports the price enough that there are no other binary states with better profits reachable by unilateral action of either firm. Data

Set A #1

Firms

f1

Set A #2

f2

f1

Set B

f2

f1

Set C f2

f1

f2

Units

h1

h2

h1

h2

h1

h2

h1

h2

h1

h2

h1

h2

h1

h2

h1

h2

zf h

0

0

1

0

1

0

0

0

1

0

0

1

1

0

0

1

Profitf Social Welfare

0

2258.3 6632.8

3638.4

0

2902.0

494.2

2902.0

409.2

5846.8

6419.3

6334.3

Market Price p 24.26 28.60 Table 7. Strategic Equilibria for Three Data Sets.

26.75

26.75

4.1.2 One Commodity, Two-Part, Price-Taker Results Algorithm 6 requires the calculation of W by solving the min-max problem (60) for each combination z of binary variables, after solving the complementarity problem CP(z) (55) (57). In the min-max problem (60), the commodity price is fixed at p as calculated by CP(z). Therefore, for the one commodity, two-part price-taker model, in the inner maximization problem for each consumer, the choice of qjD would be identical to that computed by CP(z), if zjD = 1, and the value of consumers’ surplus would be (consumers’ surplus as calculated by 28

CP(z)) - pzD . If zjD = 0, then the value of consumers’ surplus would be 0. For a given value of pzD , the maximum consumers’ surplus for the inner maximization of (60) would be the larger of these two values. Similarly, all possible values of profit for each firm can be specified for any given value of pzS . Because of the simple linear form of the model (11) - (12), it is possible to write expressions for the maximum profit to be earned on each unit h as a function of pzS and p, under the conditions that could prevail: the unit is off (zero profit); the unit is on but not producing output; or the unit is on and producing output at the maximum capacity. The maximum value of profit for a unit would be the largest of the profits under these three conditions, and the maximum profit for the firm would be the sum of the maxima for the two units. These considerations lead to the LP (67) - (72) for the calculation of W . The constraint on pz in (60), here relates pzD to pzS , and appears in the LP as the constraint (17), with z D j and z f h fixed as parameters with values set at the current combination being considered in the complete enumeration of Algorithm 6. The LP model contains new symbols, as defined below just before the statement of the model. New Parameters for LP to Calculate W CS j = consumers’ surplus for j from CP(z) New Variables for LP to Calculate W φf h = profit to firm f on unit h θj = consumer surplus for consumer j LP to determine W , pzS , pzD :

W =

min

φf h , θj , pzD , pzS

s.t. θj ≥ CS j − pzD , ∀j

X j

θj +

X

φf h

φf h ≥ pzS XKf h − Kf h , ∀f, h φf h ≥ (p − Cf h )XKf h + pzS XKf h − Kf h , ∀f, h X X X pzD z D = pzS XKf h z f h j j

f

(67)

f,h

(68) (69) (70) (71)

h∈Hf

θj ≥ 0, ∀j; φf h ≥ 0, ∀f, h 29

(72)

Algorithm 6 found multiple equilibrium states z for each data set: two for data set A, and four for each of data sets B and C. Moreover, for every equilibrium state found, the optimal value of pzS (and pzD ) was not unique — there was an interval range of optimal values. In order to determine this range, the following LP was solved twice (once for min and again for max) for the minimum and maximum optimal values of pzS . Two LPs to determine Range of Optimal pzS :

pzS

min (or max) pzD , pzS

φf h , θj ,

s.t. W

X

≥

(73) θj +

j

X

φf h

(74)

f,h

(68), (69), (70), (71), (72) The results for the three data sets are presented in Tables 8 and 9.

Note that, for data

set C, the second listed equilibrium state is the social welfare maximum state, and the range of optimal pzS values includes zero. This makes sense, because the social welfare maximum is a one-part pricing equilibrium for data set C, which implies that even with a zero capacity price, this state is an equilibrium. The same cannot be said for data sets A and B: the social welfare maximum is among the equilibria that are found, but the range of pzS is either all negative or all positive. Data

A

Firm

B

f1

f2

C

f1

f2

f1

f2

Equilibrium

Unit

h1

h2

h1

h2

h1

h2

h1

h2

h1

h2

h1

h2

#1

zf h

1

0

0

0

1

0

0

0

1

0

0

0

#2

zf h

1

0

1

0

1

0

1

0

1

0

1

0

#3

zf h

-

-

-

-

1

0

1

1

1

0

1

1

#4 zf h 1 1 1 1 1 1 1 1 Table 8: Equilibrium Binary Variables for One Commodity,Two-part Price-taker Model

30

Data & Equilibrium #

p

Min pzS

Max pzS

A #1

17

-9.10

-8.10

5846.8

A #2

17

3.50

6.54

6850.8 (*)

B #1

28.60

-9.10

-8.60

5846.8

B #2

21.01

-1.01

-0.07

7528.7 (*)

B #3

19.17

1.77

3.83

7456.7

B #4

18.00

5.00

6.09

6988.5

C #1

28.60

-9.10

-8.60

5846.8

C #2

21.01

-1.01

0.93

7528.7 (*)

C #3

19.17

2.77

3.83

7371.7

Social Welfare

C #4 18.85 4.15 5.38 6971.9 Table 9: Prices and Social Welfare for One Commodity,Two-part Price-taker Model (Social welfare maximum is indicated by (*). ) For data set B, recall that the social welfare maximum is close to a single-part pricing equilibrium (as measured by the profit improvement that the firms perceive for deviations) — this corresponds to the very small negative value of pzS , -0.07, that will turn the state into an equilibrium.

Contrast this with the social welfare maximum for data set A — it is far from

a single-part pricing equilibrium, and the smallest magnitude of pzS that will turn it into an equilibrium is the much larger 3.50.

4.1.3 One Commodity, One-Part, Cournot Results There was no equilibrium for any of the three data sets A, B or C. The incentive to deviate unilaterally from any of the fixed-binary equilibria was very large compared to the magnitudes of profits in the fixed-binary states.

On the other hand, all three data sets led to unique

strategic equilibria. Tables 10 and 11 give the results. Data

Set A

Firms

f1

Set B f2

Set C

f1

f2

f1

f2

Units

h1

h2

h1

h2

h1

h2

h1

h2

h1

h2

h1

h2

zf h

0

0

0

0

0

1

0

1

0

1

0

1

qfSh

0

0

0

0

0

61.18

0

52.18

0

61.35

0

49.35

Profitf

0

0

484.94

694.88 31

492.34

599.51

Table 10: Strategic Equilibria of One Commodity Cournot Model.

Data

Set A

Set B

Set C

Consumers P qjD = f sf j

j1

j2

j3

j1

j2

j3

j1

j2

j3

0

0

0

40.52

44.74

28.10

39.66

43.86

27.19

pj

—

—

—

36.76

38.87

30.55

36.83

38.93

30.60

Cons.Surp.j

0

0

0

65.67

70.05

20.37

62.90

67.33

19.08

Social Welfare 0 1335.91 1241.16 Table 11: Strategic Equilibria One Commodity Cournot Model — Demands, Prices, Consumers’ Surpluses and Social Welfare.

4.2 Two Commodity Price-Taker Models The two commodity model of section 2 is quite different in structure from the one commodity models. Therefore, quite different data were needed to give a meaningful illustration. Just one data set is given in Table 12. The W calculation at step 3 of Algorithm 6, and the pzS range calculation are done by the same methods as for the one commodity price-taker model, for single- and multi-part pricing. Commodity Data

Firm Data

j1

j2

Firm

f1

f2

αj

60

87

Unit

h1

h2

h1

h2

βj

0.08

0.07

Cf h

13

18

32

5

δj

0.002

0.003

Kf h

2600

1400

2100

600

XKf h

400

100

350

85

Conversion Efficiency ε 0.4 Table 12. Data for Two Commodity Price-Taker Models.

4.2.1 Two Commodity, Single-Part, Price-Taker Results One single-part pricing equilibrium was found — see Table 13. strategic equilibrium.

32

This was also the only

Demands & Prices

Firm Related Output

j1

j2

Firm

f1

f2

qjD

400

350

Unit

h1

h2

h1

h2

pj

28.70

63.70

zf h

1

0

1

0

xf h

400

0

350

0

Profitf 3680 8995 Table 13. Two Commodity, Single-Part Price-Taker Equilibrium; also Strategic Equilibrium.

4.2.2 Two Commodity, Two-Part, Price-Taker Results With two-part pricing, there are two equilibria, one of which is given in Table 13, with a range for pzS of -9.20 to 3.30, which includes zero, as expected because this is a single-part pricing equilibrium.

The second equilibrium is shown in Table 14.

If profits were instead

calculated with pzS = 0 at the second equilibrium, then both firms would have positive profits on their h1 units and negative profits on their h2 units, and so they would want to shut down their h2 units. Therefore, the capacity price is necessary to make the solution in Table 14 into an equilibrium. Demand & Prices

Firm Related Output

j1

j2

Firm

qjD

468.88

362.45

Unit

h1

h2

h1

h2

pj

23.21

63.04

zf h

1

1

1

1

xf h

400

100

350

12.45

Min pzS

8.79

f1

f2

Max pzS 9.84 Profitf 4999.9 ∗ 11984.1 ∗ Table 14. Additional Two Commodity, Two-Part Price-Taker Equilibrium. (* Profit is calculated using the minimum value of pzS .)

5. Directions for Future Research The simple illustrative models of this paper have only four binary variables and just a few continuous variables, and so the complete enumeration algorithms solve very quickly on a modest computer, because the search is only over 16 binary states. Models with N binary variables would require a search through 2N binary states with complete enumeration. Therefore, for models with large N , a more efficient method of search must be employed. Furthermore, the minimum-W problem (60) can be a difficult bilevel problem in general, and even in the case 33

when it is a LP, as in the illustrations in this paper, the size of the LP can become prohibitive when N is large. There is much work to be done to develop efficient algorithms for large N models. The ideas of this paper should be tested on realistic applications, e.g., short-term generator scheduling and dispatch models for competitive electricity markets, long-term generation investment planning models for competitive electricity markets, or multi-commodity models of energy markets. However, until efficient algorithms are available for large N models, applications will be restricted to models with small enough N to make computing times practical. Other types of multipart pricing should be explored.

In this paper, the “extra” prices

are paid in proportion to binary variables, but in practice, there are multipart schemes in which the “extra” prices are paid in proportion to the continuous variables.

For example,

in some electricity markets, “uplift payments” (which can include payments to generators for fixed costs) are billed to customers as an extra price per kwh, in addition to the commodity price. This line of research requires modification of (46) to include continuous variables xi , and the corresponding extra terms in (54). However, this complicates the definition of CP(z) by making its solution parameterized by pz . It is not immediately obvious how to proceed with development of any of the algorithms. In addition to Cournot models with binary variables, other models of imperfect competition should be explored, with single- and multipart pricing, e.g., the conjectured supply function model (Day et al. [3]). The strategic equilibrium concept can be extended from its present form as a finite game of discrete strategies to one with randomized strategies.

With this change, the existence of

equilibrium could be guaranteed under fairly general conditions.

34

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