Discriminatory Price Mechanism for Smart Grid

03/30/2020 ∙ by Diptangshu Sen, et al. ∙ Indian Institute of Technology Delhi 0

We consider a scenario where the retailers can select different prices to the users in a smart grid. Each user's demand consists of an elastic component and an inelastic component. The retailer's objective is to maximize the revenue, minimize the operating cost, and maximize the user's welfare. The retailer wants to optimize a convex combination of the above objectives using a price signal. The discriminations across the users are bounded by a parameter η. We formulate the problem as a Stackelberg game where the retailer is the leader and the users are the followers. However, it turns out that the retailer's problem is non-convex and we convexify it via relaxation. We show that even though we use discrimination the price obtained by our method is fair as the retailers selects higher prices to the users who have higher willingness for demand. We also consider the scenario where the users can give back energy to the grid via net-metering mechanism.

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I Introduction

I-a Motivation

The traditional power grid is becoming smarter where users are now equipped with advanced metering infrastructure. The advent of home automation has enabled users to control their consumption depending on the prices. The retailer or utility company111We use terms retailer and utility company interchangeably can also control the total consumption by setting the price. For example, when the demand is high, the retailer can set high prices in order to de-incentivize the users to consume less. The retailer needs to be profitable, otherwise, she cannot maintain the transmission lines and distribution lines. However, electricity is essential for sustainability, so a high price is not only detrimental to users but it can also push the economy of a country down. Hence, we need to maximize the users’ welfare simultaneously. Thus, we need to develop a pricing mechanism which will try to maximize the retailer’s profit along with user’s payoffs.

Due to the advent of smart meters, the retailer can now charge different prices to different users. Such discriminatory price mechanisms may increase the users’ payoff without reducing the retailer’s profit. Several forms of discriminatory price mechanisms can be observed in practice. For example in India, tariffs vary depending on the consumption level of the users. Further, researchers have argued that different prices to different users can in fact increase the efficiency[sean]. We seek to answer the question whether allowing prices to vary within a certain limit across different users can result in gains in the users’ welfare or retailer’s profit.

Users now have distributed energy resources such as solar panels and wind energy generators. These users can also feed back energy to the grid. Such users are better known as ’prosumers’. Net-metering is a widely adopted technique where the retailer buys energy from prosumers at the retail rate. Thus, the retailer now needs to set prices judiciously depending on whether prosumers are giving back or consuming energy at a certain time instance. We need to determine optimal price mechanisms for such scenarios.

I-B Our approach

We consider a stylized model where a retailer sets a price for each consumer in each time period. First, we consider the case where no consumer can feed back energy to the grid. We formulate the problem as a Stackelberg game where the retailer selects a price, each user selects how much to consume in each period by maximizing its own payoff. The retailer’s optimization problem involves a weighted average of the retailer’s profit and the users’ welfare. We show that the optimization problem of the retailer is non-convex even when the user’s optimization problem is convex. Subsequently, we convexify the problem by introducing three different types of modifications. The retailer can discriminate among the users by charging different prices to different users. However, we restrict the discrimination by an amount . Numerically, we evaluate how can impact the retailer’s profits and users’ payoffs.

Subsequently, we consider the scenario where the user can also feed back energy to the grid. We investigate the net-metering price mechanism where the selling price and the buying price remain the same. Thus, if the retailer selects a higher price, users can be incentivized to sell back more, hence, it is not apriori clear which price will maximize the retailer’s objective. We formulate the problem of determining the optimal price of the retailer as an optimization problem and convexify it with the methods described in the last paragraph. The impact of this kind of pricing scheme on different stakeholder metrics is evaluated.

I-C Literature Review

Load profiling in the smart grid through design of demand response programs has seen a lot of research effort over the last decade. In this subsection, we will attempt to provide the reader an overview of the research that already exists in this area. The organization of this subsection is as follows : we will start off by referring to a few review papers followed by a summary of recent work in Demand Response through real-time pricing schemes. This will be followed by a short review of game-theoretic techniques commonly adopted in literature. The last segment will be dedicated to discussion about distributed generation and how it has affected design of DR programs.

[wang2015load] provides a comprehensive review of data mining techniques that are useful for load profiling and customer segmentation and how they have been used for designing price-based and incentive-based DR programs. [aghaei2013demand] summarises some of the work that has been done with respect to DR programs in smart grids equipped with renewable energy resources(RERs). Successful DR implementations around the world have also been analyzed. Readers can refer to [shariatzadeh2015demand] for further reading.

[ma2014distributed] is an important paper when it comes to Real-time pricing. It shows that under a Real-Time Pricing scheme, the price function will be linear in consumption and rotationally symmetric. [li2011optimal] proves that a dynamic pricing strategy can be designed in such a way that consumers’ selfish utility maximization aligns with social utility maximization. A distributed algorithm for joint computing of prices and usage schedules is proposed. [liu2014pricing] deals in data centre demand response, designing an appropriate prediction-based pricing scheme that is robust to prediction errors.

The Stackelberg framework is one of the most common game theoretic approaches adopted in literature. [wei2014energy] develops a 2-stage problem for both pricing and energy dispatch. Supplier-enduser interaction has been modelled by a Stackelberg game, while robust optimization techniques take care of market uncertainty. [chen2011innovative]

explores the Stackelberg formulation to solve an energy scheduling game. It builds on a Day-Ahead pricing scheme, but also has a notion of a price gap that gets updated on the basis of the real-time load vector. This paper presents encouraging results on the the Peak-to-Average Load ratio and the problem of ’rebound’ peaks.

[saad2012game]

provides a very good review on other popular game theory techniques in DR-related problems.

Lately, the focus has shifted to distributed generation capabilities. One such possibility is consumers who can produce, or ’prosumers’. [lampropoulos2010methodology] conducts a simulation analysis of the effects of distributed generation on the grid as a whole. [zafar2018prosumer] and [kanchev2011energy] discuss energy management systems for microgrids involving prosumers. Optimal pricing schemes under net-metering have been discussed in [brown2016design]. With this basic overview, we proceed to discuss how our model contributes to the rich existing literature.

I-D Original Contributions

To summarize, the main contributions of our paper are the following–

  • We consider a discriminatory price scenario where the retailer can charge different prices to different users. Even though the price is discriminatory, we show that the price mechanism is fair as the users who have higher valuation for demand, are priced higher. We empirically evaluate the impact of the level of discrimination on the users’ welfare and the retailer’s revenue.

  • In the proposed formulation, we consider that the retailer’s objective is to maximize the profit, minimize the cost to serve the user’s demand, and maximize the user’s welfare. We,numerically, evaluate how the price mechanism impacts each of the objectives.

  • We also consider the scenario where the users may have renewable resources and can sell back energy to the grid. We formulate a net metering scenario where the selling price and the buying price are the same for each user. We numerically evaluate the prices and show the impact of discrimination on the amount sold to the retailer, the consumption of the users, users’ welfare, and the retailer’s revenue.

Ii System Model

Ii-a Entities

There is a retailer or utility company who purchases electricity from the wholesale Day-Ahead market and supplies to a community of consumers. Time is slotted. The retailer selects a price for each period by anticipating the amount of energy that will be consumed in that period. Note that the duration of the period can be of any magnitude, however, if the duration is small, a retailer may need to compute prices a large number of times within a given day. In every period , she communicates her price to the consumers based on which they choose their elastic demands for the said period.

Every household also has an inelastic demand in each period which needs to be satisfied. The examples of inelastic loads are electricity required to switch on lights or TV. Total demand for a household in a period consists of both elastic and inelastic demand. An user may choose the temperature setting of its household. Further, it can also choose how much to use for charging the batteries of electric vehicles. Those are a few examples of elastic demand.

Ii-B Game Definition

Since all users and retailer are interested in optimizing their own payoff, we formulate the problem as a game-theoretic problem. The retailer, first, selects a price for a period and the users then decide how much to consume in that period . Thus, we formulate the game as a sequential game. There is a ’leader’ (retailer) who takes the first turn at playing the game (by setting price) and the ’followers’ (consumers) respond accordingly (by deciding their consumption). Due to hierarchy of players, qualifies as a Stackelberg game which can be solved by Backward Induction. We assume that all players are rational and the game is one with complete information.

We now define the strategy space of the players. The retailer selects prices in each time period. The users decide how much to consume in each time period. Note that the retailer can charge different prices to different users. The users optimize payoff functions in order to decide how much to consume. We define the payoff functions of the users in the subsequent section.

Ii-C Notation Key

In this segment, we will define the notations elaborately. Unless specified otherwise, notations will bear the same meaning across the paper. Subscript and superscript mean that the quantity pertains to the household in the period of the day.
- price per unit electricity charged by retailer
- base price charged by retailer per unit inelastic demand
- additional price above base price charged by retailer per unit elastic demand
- energy buyback rate for retailer
- purchase made from grid by consumer
- energy sold back by consumer to retailer
- net energy transaction from grid
- inhouse solar generation
- elastic demand of consumer
- inelastic demand of consumer
- level of price discrimination allowed

Iii Optimal Pricing With no renewable resources

First, we consider the scenario where users do not have renewable resources. However, users are equipped with smart devices and can optimize their consumption for a given price. In the next section, we consider the scenario where users have renewable resources. We, first, define the users’ objective and subsequently, we define the objective of the retailer.

Iii-a Decision of Consumers

For a given price , consumers decide how much to consume. The decision is based on the convenience function and the price . Each user derives some comfort from consuming energy. We model this comfort level in monetary terms using a convenience function. Convenience function is used in Economics as well as for modeling the convenience of users in the power grid([samadi2010optimal], [fahrioglu1999designing] and [fahrioglu2001using]). There is no comfort obtained from inelastic demand because that is the bare essential. So, convenience function is dependent only on elastic demand. The convenience function must have the following nice properties :

  • = 0, i.e., the function has a fixed point at the origin. If elastic demand is zero, convenience derived is zero.

  • 0. Convenience should be an increasing function of elastic demand. If the demand is high, the convenience of a user should be higher.

  • 0. The higher the consumption of elastic demand, the lower the marginal convenience derived from it.

  • The convenience saturates once marginal convenience goes to zero. Thus, if a user’s demand exceeds a certain threshold, the demand will not fetch any additional convenience to the users.

  • is continuous and at least twice differentiable over . This is for the analysis.

Taking all the above into consideration, we define our convenience function as :

is the consumer preference factor in period and varies across consumers, while is a predetermined constant. This form of quadratic convenience functions are common in the smart grid literature([samadi2010optimal], [fahrioglu1999designing]).

Note that convenience function is also time dependent. A user may be willing to consume more at some specific time periods compared to other time periods. Hence, convenience function may also vary over time. We have assumed the convenience function is not correlated across different time periods. The characterization of the price when the convenience function is correlated across different time periods is left for the future.

Definition 1

The consumer utility is defined as the difference between the convenience derived from the elastic demand consumption and the total price paid for the consumption. Hence, mathematically, the utility function is

(1)
Observation 1

The optimal user-end elastic demand consumption in the period in response to price charged by retailer is

(2)

If is higher, the consumption will be higher. On the other hand, if the price is higher the consumption will be smaller. The total consumption is scaled by . Higher means that users are more likely to be satisfied with smaller level of consumption, hence, optimal consumption is also smaller.

Iii-B Retailer’s Decision

The retailer charges a price to consumer in the period for any consumption beyond the inelastic demand. is a base price which accounts for the cost to sustain the minimum consumption of the users. is not a decision variable, rather, it is fixed. The reason behind fixing is that the users need to consume the minimum amount regardless of the value of . Thus, it would not be fair to the users if retailer optimizes over .

Note that we consider that the retailer can charge different prices to different users, This is a discriminatory pricing model. Several kinds of discriminatory pricing models can be seen in practice. For example, in India, people who consume more pay larger prices compared to the ones who consume less. Further, discriminatory pricing models are also proposed by academics in order to achieve better efficiency [sean]. We also show that if a user consumes less its price will be smaller at the same time period compared to the one who consumes more.

The retailer decides over across the users and over different time periods. In order to select prices, we assume the following

Assumption 1

We assume that smart meters installed in the households can accurately measure ’s and communicate that intelligence to the retailer. This assumption helps in making the game a complete information one.

Since the user’s convenience function is known to the retailers, she also knows the optimal consumption for a given price (Observation 1).

Retailer’s objectives: The retailer will obviously try to maximize her profit which consists of the revenue . The retailer also incurs a cost for serving the consumption . Generally, the cost is quadratic, we also assume the same. Additionally, the retailer needs to ensure that the user’s welfare is maintained. In other words, the user’s consumption should not be very far from the optimal consumption level when the price is zero. The above may be imposed by the government as part of a regulation since electricity is an essential commodity.

Thus we have the following optimization problem for the retailer :

Formulation 0:

(3)
(4)

are the weight factors. Those weights must be chosen judiciously depending on the need. The first term in the objective corresponds to the revenue, the second term corresponds to the cost of serving the consumption. The third term in the objective represents a penalty if the consumption is far away from the consumption of a user when the price is .

The first term in the constraint denotes the fact that user’s consumption is given by the expression in Observation 1. The second constraint denotes that even though we have used discriminatory pricing we have limited the discrimination to . The last constraint gives an upper and lower limit of the decision variable price.

Limiting the Discrimination: Note that the prices differ between two users by at most amount. If , we revert to the scenario where there is no discrimination. On the other hand, if we have we revert to the scenario where the retailer is not bounded by any discrimination level. is a policy choice for the social planner. We, numerically, show the impact of on each of the objectives.

Formulation 0 is not convex since the first constraint is a non-linear equality constraint. Thus, it is difficult to obtain an optimal price. In the following, we relax the constraint and reformulate the problem as a convex one.

Iii-B1 Reformulations

We propose three modifications of the original problem Formulation 0.

Formulation 1:

(5)

If the reader observes the first constraint, the reader will discern that we do away with the max term of the original formulation. Hence, the equality constraint becomes linear and the overall problem becomes convex. Note that here we have introduced another constraint where , thus, the price is further restricted from the original formulation.

The above formulation can be alternatively written by replacing with the first constraint as the following

(6)

Formulation 2:

(7)

We can reformulate the above as the following:

(8)

This formulation is again convex. Note that compared to Formulation 1, in this formulation, we do not put the hard constraint of rather we put a penalty if is negative. Thus, this formulation does not restrict the price unlike in formulation 1. Unlike in formulation 1, in formulation 2, we need to compute separately using Observation 1 after obtain optimal price.

For both formulations 1 and 2,we observe the following

Theorem 1

If , in an optimal price for both formulations 1 and 2. Further, if , .

Obviously, note that if , . The above result shows that if , can be higher than if . Thus, Theorem 1 ensures fairness in the discriminatory setting. Even though the prices are different, the retailer sets a higher price to the users who have higher willingness to consumer more.

Note that Formulation 2 may have negative which is not possible in reality. We, thus, have the last modification.

Formulation 3:

(9)

Compared to the first two formulations, the retailer here obtains both and the corresponding . The first two constraints provide the upper and lower bounds on respectively. The fourth term in the objective will penalize if is far from . Thus, instead of the hard constraints in the first two formulations, here, the retailer relaxes it and adds a penalty in the objective. Thus, compared to the first two formulations, this formulation provides a higher price.

Unlike in Theorem 1 we can not conclusively say whether the formulation 3 gives prices which are fair. This is because in this formulation, the retailer here decides over both and unlike in formulations 1 and 2.

Iii-C Extension

Iii-C1 Optimal

Throughout this section, we assume that is a parameter. However, alternatively, we can consider as a decision variable. All the reformulated versions would still remain convex if we make as a decision variable. The optimal would provide the optimal level of discrimination necessary to achieve optimal price for the retailer.

Iii-C2 Different across the users

Throughout this paper, we assume that is the same across the users. However, our analysis will go through even when

is different across the users. A retailer can estimate

for a user using a regression model by observing the response of a user following a price signal. The details have been omitted here owing to the space constraint.

Iii-C3 Different s across the users

We have also assumed that the minimum inelastic demand requirement is the same for each user. Our analysis will go through even when s are different across the users since the reformulated problems would remain convex.

Iv Optimal Pricing when users have renewable resources

In this section, we consider the scenario where each consumer has renewable energy generation capabilities. The renewable energies can range from solar, biomass, to wind energies. Note that when a user is equipped with renewable energies, it may feed back energy to the grid. We assume the popular net-metering mechanism. Thus, the energy which is fed back is compensated at the same buying price. Thus, effectively, the consumer only pays for the net energy purchased from the grid. Since a user can technically produce energy, we denote it as a prosumer (producer+consumer).

Iv-a Decision of Prosumers

In the period, consumer has an solar energy generation amounting to . This is complemented by a purchase of amount from the retailer at rate . In case, is beyond what is required in the household, it sells back at same retail rate. is the net energy transaction made, i.e., . We prove that a prosumer does not purchase and sell-back in the same period222See Appendix for proof.

Recall from Definition 1 that the prosumer’s utility is defined as the difference between the convenience derived from the elastic demand consumption and the price paid for the net purchase from the grid. is the total demand consumption by the prosumer in the period, hence is the corresponding elastic demand consumption. Mathematically, thus the utility function is

(10)

Recall that is the inelastic demand which is required to be satisfied at any cost. Hence,similar to Observation 1, we obtain that

Observation 2

The net optimal grid purchase in the period in response to price set by the retailer for both retail and sell-back is given by

(11)

Positive indicates that renewable energy generation was insufficient and purchase was made from the grid to meet residual demand. While negative indicates that the renewable energy generated exceeds the requirement or it is more profitable to sell-back energy by consuming less. Note that when the grid is congested, the grid can select higher prices to incentivize the prosumers to sell back more. Thus, the prosumers may find it more profitable to sell back when the grid is congested.

Iv-B Retailer’s Decision

The retailer sets price for the prosumer in the period. The same price is applicable for both purchase and sell-back. The prosumer again employs discrminatory price setting. We, numerically, evaluate the impact of this price mechanism on the revenue of the retailer and the user’s utilities in this scenario.

In addition to the assumptions in Section III, we have the following:

Assumption 2

We assume that the prosumer can accurately predict and communicate it to the retailer for each .

Note that a prosumer can predict this value fairly accurately close to the realization time. Since we are employing a real time price mechanism, it is expected that a prosumer will inform the estimated value to the retailer minutes before the start of the period, the retailer will then update the prices to everyone. We assume that the prosumer will inform the exact estimated value. With the knowledge of and and the form of the convenience function already known, the retailer also knows the net optimal purchase amount for that user using Observation 2.

Retailer’s Objectives : As mentioned in Section III, the retailer will try to maximize her own revenue, minimize the cost, and maximize the user’s welfare. Thus, the retailer’s optimization problem is

Formulation 4:

(12)
(13)

The first term in the objective corresponds to the revenue, the second term corresponds to the cost of serving the consumption. Note that even when is negative, the retailer needs to dispatch this additional energy which incurs a cost. This is because the balance needs to maintained between the supply and demand, and even when supply exceeds the demand the retailer pays a penalty for the imbalance. The third term in the objective represents a penalty if the consumption is far away from the consumption of a user when the price is .

The first term in the constraint denotes the fact that user’s consumption is given by the expression in Observation 2. The second constraint indicates that the retailer should be able to sell a net positive amount of energy to the users which will result in her revenue. The last constraint gives an upper and lower limit of the decision variable price.

Formulation 4 is not convex since the first constraint is a non-linear equality constraint. Thus, it is difficult to obtain an optimal price. So, we relax the constraint and reformulate the problem as a convex one. The reformulations are exactly identical in structure to the ones provided in Section III and thus, we omit them here.

V NUMERICAL EXPERIMENTS : Simulations and Results

In this section, we numerically validate the formulations that have been provided above. For sake of simplicity, we assume across all consumers and across all periods of the day. Consumer preference parameter is drawn uniformly from [3, 7]. depends on the inelastic demand requirement in a particular period and the following relationship is assumed to hold: .

V-a Optimal Pricing with no renewable resources

  • : This translates to the case when the pricing is non-discriminatory in nature. The same price is charged to all users in a given period. In this segment, we will investigate the effect of weights , and on the final prices, retailer revenues, elastic load and consumer welfare. We will also compare results across our 3 formulations to identify which of them reflects reality the closest.
    As we increase , prices decrease, but individual elastic demand consumption increases. This leads to overall increase in retailer revenues and average consumer convenience values. Similarly, with increase in , prices increase, hence total elastic load decreases. Retailer revenue and consumer welfare also go down. The pattern of variation with is identical to . The same trends are observed across all formulations except for formulation 1 which is apparently insensitive to variation. However, when we decrease by an order of magnitude, the same trends are observed again.

    Figure 1: Variation of different metrics with

    In general, formulation 1 predicts the lowest prices because it tries to ensure that all consumers have positive elastic demands. So, total elastic loads are very high which leads to high retailer revenue and high consumer welfare. However, such high values of elastic load are not admissible, so formulation 1 is not a very realistic model. Formulation 2 predicts the highest prices among all formulations. This was expected because the retailer has the choice to dissatisfy consumers who have very low values. Prices predicted by formulation 3 are slightly lower compared to formulation 2, but it errs on the lower side in calculating consumption, so despite having lower prices, formulation 2 shows slightly lower total elastic loads and lower consumer welfare as compared to formulation 2.

  • : When , our pricing model becomes discriminatory in nature, charging different prices to different users. All our formulations converge in this setting. We also make several interesting observations as we vary .

    Figure 2: Variation of revenue and elastic load with eta
    Figure 3: Variation of prices and consumer welfare with eta

    With increase in

    , retailer revenues increase gradually, which means that this pricing scheme is lucrative to her. However, total elastic load and average consumer convenience roughly remains constant. Rather, the variation (standard deviation) in consumer elastic consumption gradually goes down.

    Figure 4: Variation of std. deviation of elastic demand consumption with eta

    This means that an energy redistribution is taking place, where users who were earlier consuming less, are charged lower prices and hence are able to consume more. While high-end consumers are being charged high prices, bringing down their elastic consumption. This means that discriminatory pricing leads to a fairer distribution of energy in the community where high-end consumers no longer have an upper hand.

    Figure 5: Price and energy distribution among consumers for discriminatory pricing

V-B Optimal Pricing when users have renewable resources

Now, we discuss the scenario where users have inhouse renewable energy generation capabilities. We restrict our analysis to solar energy only. According to our time-slotting choice, solar power is generated only during periods 2 through 5. Hence, we assume that the retailer uses the standard model in the first and last periods and reverts to the net-metering model during the other periods.
We observe that the retailer chooses to offer significantly high prices during the daytime. This incentivizes her consumers to sell back huge amounts of energy to the grid by sacrifising convenience, thereby lowering her wholesale-market purchases. As a result, a significant flattening of the load-curve is achieved. Retailer revenues also increase significantly during the daytime due to high prices. Thus, net-metering is a viable strategy for the retailer also.

Figure 6:
Figure 7:
Figure 8:

Vi Conclusions

In this paper, we developed optimal pricing strategies for retailers in the scenario where users do not have renewable energy resources. The pricing model was formulated as a Stackelberg game and solved by Backward Induction. The consumer objective was individual utility maximization while retailer objective was a weighted average of her revenues, cost of generation and consumer welfare. We showed that by appropriately varying weights, the retailer can prioritize any of the objectives according to necessity. We also investigated the impact of discriminatory pricing on the different stakeholders. It was shown that discriminatory pricing is profitable to the retailer because it leads to higher revenues, at the same time, it helps in fairer distribution of energy in the community. In the last segment of the paper, we extended our model to include the scenario when users have inhouse renewable energy generation capabilities. With net-metering in place, it was found that consumers can be incentivized to sell back large amounts of energy to the grid, even at the cost of individual convenience, if the prices are sufficiently high. Significant sell-back flattens the load curve and minimizes risk of outages.

References

Vii Appendix

Claim : If a consumer has a higher , then the price charged to that consumer under the discriminatory pricing regime will be higher.
Proof : Let us start by constructing the Lagrangian to the constrained retailer-end optimization problem. If we recall, the constraint set(presented in general form) is as follows:

The Lagrangian is given by the following :

(14)

Using KKT conditions for stationarity and complementary slackness, we have the following :

(15)
(16)
(17)

Now, = Min(, P) and = 0 . Since and , we can safely say that . Again, if is a loose bound, for all practical purposes, . If , goes to zero, which is undesirable, so we search for potential maximizer candidates by putting .
Since our aim is to find which maximizes the objective, we set so that is never active. Therefore, and go to zero. We now proceed to solve for using the linear system in (15).
We obtain the following :

(18)

Since varies directly as , we conclude that our claim is justified. Now, let us try to find an expression for . is given by the difference between the maximum and minimum prices when prices are unconstrained. Hence,

(19)

Alternate Proof(Shorter, Simpler and Elegant) : Let be the optimal price vector obtained from the discriminatory pricing model. Let if possible, there exist a pair such that , but .

Since p is the optima, .

Now, let us consider a slightly modified price vector where prices and are interchanged. We will refer to this new price vector as q.

Therefore, . This contradicts our initial assumption that p is the optimal price vector. Hence, .
To extend the above proof to formulation 2, we need to prove additionally that :

(21)

when and . , , and can be related in 24 ways. Because of the already assumed inequalities, there are 6 possible ways of arrangement. They are as follows :

  • :

    (22)
  • :

    (23)
  • :

    (24)
  • :

    (25)
  • :

    (26)
  • :

    (27)

Thus, there is a contradiction again and .