Retail Equilibrium with Switching Consumers in Electricity Markets

Abstract

The ongoing transformations of power systems worldwide pose important challenges, both economic and technical, for their appropriate planning and operation. A key approach to improve the efficiency of these systems is through demand-side management, i.e., to promote the active involvement of consumers in the system. In particular, the current trend is to conceive systems where electricity consumers can vary their load according to real-time price incentives, offered by retailing companies. Under this setting, retail competition plays an important role as inadequate prices or services may entail consumers switching to a rival retailer. In this work we consider a game theoretical model where asymmetric retailers compete in prices to increase their profits by accounting for the utility function of consumers. Consumer preferences for retailers are uncertain and distributed within a Hotelling line. We analytically characterize the equilibrium of a retailer duopoly, establishing its existence and uniqueness conditions for a wide class of utility functions. Furthermore, sensitivities of the equilibrium prices with respect to relevant model parameters are also provided. The duopoly model is extended to a network that includes multiple retailers with capacity constraints for which we perform static and dynamic numerical simulations. Results indicate that, depending on the retailer costs, loyalty rewards and initial market shares, the resulting equilibrium can range from complete competition to one in which a retailer has a leading or even a dominant position in the market, decreasing the consumers’ utility significantly. Moreover, the retailer network configuration also plays an important role in the competitiveness of the system.

Appendices

Appendix

Proof of statements

Proof of Proposition 2

From (8) and (10) we can check that, under condition (11), it holdsthat

$$\theta_{ij}^{\ast}> 1 \text{ and }\ \theta_{ji}^{\ast} \leq 0, $$

implying${F}_{ij}^{\ast } = 1$ and${F}_{ji}^{\ast } = 0$,with satisfies Definition 2 of a dominant position of retailer$R_i$.

For retailer $R_i$, from (4)we get $\pi ^{\ast }_i(p_i,p_j)= 1$ andthus the optimal utility becomes

$$ u_i^r(p_i,p_j) = (p_i - c_i)v^{\ast}(p_i) $$

(34)

and attains its maximum for a price$p_i$ thatsatisfies

$$ (p_i - c_i) v^{\prime}(p_i) + v(p_i) = 0. $$

(35)

For retailer $R_j$,because $\pi ^{\ast }_j(p_j,p_i)= 0$,the optimal utility vanishes and thus the profit regardless of the price$p_j$.□

Proof of Proposition 3

From (8) and (10), we can check that under condition (13), wehave

$$\theta_{ij}^{\ast} \geq 1 \text{ and }\ 0<\theta_{ji}^{\ast} < 1, $$

implying${F}_{ij}^{\ast } = 1$ and${F}_{ji}^{\ast } =\theta _{ji}^{\ast }$.

Hence, from (4) we get, $\pi ^{\ast }_i(p_i,p_j)=\pi _{ij}+(1-\theta _{ji}^{\ast })\pi _{ji}$ forretailer $R_i$ and$\pi ^{\ast }_j(p_1,p_2)=\pi _{ji}\ \theta _{ji}^{\ast }$ forretailer $R_j$.The respective utilities become

$$\begin{array}{@{}rcl@{}} u_i^r(p_i,p_j) &=& (p_i - c_i)v^{\ast}(p_i)(\pi_{ij}+(1-\theta_{ji}^{\ast})\pi_{ji}) \end{array} $$

(36)

$$\begin{array}{@{}rcl@{}} u_j^r(p_i,p_j) &=& (p_j - c_j)v^{\ast}(p_j)(\pi_{ji}\theta_{ji}^{\ast}). \end{array} $$

(37)

By considering (8) and (10), the first order conditions associated to maximize (36) and (37), with respectto $p_i$ and$p_j$,respectively, yields

$$\begin{array}{@{}rcl@{}} &&\left( v(p_i) + (p_i - c_i) v^{\prime}(p_i) \right) \left( 2k\pi_{ij} + \pi_{ji} (\varphi(p_i) - \varphi(p_j) + k - \gamma_j) \right)\\ &&+ \pi_{ji} (p_i - c_i) v(p_i) \varphi^{\prime}(p_i) = 0\\ &&\left( v(p_j) + (p_j - c_j) v^{\prime}(p_j) \right) \left( 2k\pi_{ji} - \pi_{ji} (\varphi(p_i) - \varphi(p_j) + k - \gamma_j) \right)\\ &&+ \pi_{ji} (p_j - c_j) v(p_j) \varphi^{\prime}(p_j) = 0 . \end{array} $$

Using the definition (12) the above conditions can be rewritten as

$$\begin{array}{@{}rcl@{}} {\Gamma} (p_i;c_i) &=& - \varphi(p_j) + \frac{1 + \pi_{ij}}{\pi_{ji}} k - \gamma_j\\ {\Gamma} (p_j;c_j) &=& - \varphi(p_i) + k + \gamma_j . \end{array} $$

□

Proof of Proposition 4

Under condition (16) we have

$$0<\theta_{ij}^{\ast} < 1 \text{ and }\ 0<\theta_{ji}^{\ast} < 1, $$

implying${F}_{ij}^{\ast } =\theta _{ij}^{\ast }$ and${F}_{ji}^{\ast } =\theta _{ji}^{\ast }$.

By using (4) we get $\pi ^{\ast }_i(p_i,p_j)=\theta _{ij}^{\ast }\ \pi _{ij}+(1-\theta _{ji}^{\ast })\pi _{ji}$,and $\pi ^{\ast }_j(p_i,p_j)=\theta _{ji}^{\ast }\ \pi _{ji}+(1-\theta _{ij}^{\ast })\pi _{ij}$.Thus, the profit function for each retailer becomes

$$\begin{array}{@{}rcl@{}} u_i^r(p_i,p_j) &=& (p_i - c_i)v^{\ast}(p_i)(\theta_{ij}^{\ast}\ \pi_{ij}+(1-\theta_{ji}^{\ast})\pi_{ji}) \end{array} $$

(38)

$$\begin{array}{@{}rcl@{}} u_j^r(p_i,p_j) &=& (p_j - c_j)v^{\ast}(p_j)(\theta_{ji}^{\ast}\ \pi_{ji}+(1-\theta_{ij}^{\ast})\pi_{ij}). \end{array} $$

(39)

By considering (8) and (10), the first order conditions associated to maximize (38) and (39), with respect to$p_i$ and$p_j$,respectively, yield

$$\begin{array}{@{}rcl@{}} &&\left( v(p_i) + (p_i - c_i) v^{\prime}(p_i) \right) \left( \varphi(p_i) - \varphi(p_j) +k+ \pi_{ij} \gamma_i - \pi_{ji} \gamma_j \right)\\ &&+ (p_i - c_i) v(p_i) \varphi^{\prime}(p_i) = 0\\ &&\left( v(p_j) + (p_j - c_j) v^{\prime}(p_j) \right) \left( \varphi(p_j) - \varphi(p_i) + k + \pi_{ji} \gamma_j - \pi_{ji} \gamma_i \right)\\ &&+ (p_j - c_j) v(p_j) \varphi^{\prime}(p_j) = 0. \end{array} $$

Using definition (12) the above optimal conditions can be rewritten as

$$\begin{array}{@{}rcl@{}} {\Gamma} (p_i;c_i) &=& -\varphi(p_j) + k + \pi_{ij} \gamma_i - \pi_{ij} \gamma_j\\ {\Gamma} (p_j;c_j) &=& -\varphi(p_i) + k - \pi_{ij} \gamma_i + \pi_{ji} \gamma_j. \end{array} $$

□

Proof of Theorem 1

As ${\Gamma }(c;c) = -\varphi (c)$,under Condition C.4 and from Lemma 1 there always exists a value$p_i \in [c_i;\tau (c_i))$ satisfying(23) and $p_j \in [c_j;\tau (c_j))$ satisfying(24), given any value for $p_j$ or$p_i$ respectively.Furthermore, this value is unique.

For$q \in [-\varphi (c);\infty )$ define

$$G(q; c) \in {\Gamma}^{-1} (q;c) , \quad G(q; c) < \tau(c). $$

Lemma 1 implies G is a continuous and increasing function of q on$[-\varphi (c);\infty )$.

From this definition, a solution of

$$\begin{array}{@{}rcl@{}} p_i &=& G(-\varphi(p_j) + \kappa_i; c_i) \end{array} $$

(40)

$$\begin{array}{@{}rcl@{}} p_j &=& G(-\varphi(p_i) + \kappa_j; c_j) , \end{array} $$

(41)

is also a solution of (23)–(24).Define

$$\omega_i \equiv G(-\varphi(\tau(c_j)) + \kappa_i;c_i), \quad \omega_j \equiv G(-\varphi(\tau(c_i)) + \kappa_j;c^j) , $$

and note that for any$p = (p_i, p_j)$ it will hold that$p \in [c_i;\omega _i)\times [c_j;\omega _j)$. Hence, by considering theproperty that $-\varphi$ isan increasing function, $\omega _k < \tau (c_k)$,for $k=\{i,j\}$,and ${\Gamma }(c;c) = -\varphi (c)$,we have

$$\begin{array}{@{}rcl@{}} && -\varphi(c_i) \leq - \varphi(c_j) + \kappa_i \leq -\varphi(p_j) + \kappa_i < -\varphi(\tau(c_j)) + \kappa_i\\ && \Rightarrow c_i \leq G(-\varphi(p_j) + \kappa_i; c_i) < \omega_i , \end{array} $$

with an equivalent bound holding for $\omega _j$.

As a consequence, system (40)–(41) defines a fixed-point iteration$p = {\Phi }(p)$ where${\Phi } : [c_i;\omega _i)\times [c_j;\omega _j) \rightarrow [c_i;\omega _i)\times [c_j;\omega _j)$ andis continuous on that set. Brouwer’s fixed-point theorem then implies the existence of a fixed point for${\Phi }$, that will also be asolution for (23)–(24). □

Proof of Theorem 2

Consider the fixed-point dynamics for the solutiondefined by (40)–(41), which we write in compact form as$p = {\Phi }(p)$,and define the fixed-point iteration

$$ p = {\Phi}_j({\Phi}_i(p)) \equiv \psi (p) , $$

(42)

where ${\Phi } = ({\Phi }_i \ {\Phi }_j )^T$ and$\psi : [c_j ; \tau (c_j)) \rightarrow [c_j ; \tau (c_j))$. Thisiteration is equivalent to (40)–(41), as any solution of (42) provides a solution for (40)–(41) by letting$p_j = p$ and$p_i = {\Phi }_i(p)$,and a solution for (42) can be obtained from (40)–(41) by letting$p = p_j$.

G is differentiable wrt p on $[-\varphi (c_k);\infty )$,$k=i,j$, andwe have that

$$\begin{array}{@{}rcl@{}} G^{\prime} \left( -\varphi(p) + \kappa_i;c_i \right) & = & \frac{v (p)}{{\Gamma}^{\prime}(G(-\varphi(p) + \kappa_i; c_i); c_i)} \end{array} $$

(43)

$$\begin{array}{@{}rcl@{}} G^{\prime} \left( -\varphi({\Phi}_i(p)) + \kappa_j;c_j \right) & = & \frac{v ({\Phi}_i(p)) }{{\Gamma}^{\prime}(G(-\varphi({\Phi}_i(p)) + \kappa_j; c_j); c_j)} . \end{array} $$

(44)

Using the expression for ${\Gamma }^{\prime }$ in(22), we can write

$$ {\Gamma}^{\prime}(p;c) = v(p) (1 + \rho(p;c)) , $$

(45)

for $\rho$ definedin (25). We can rewrite (43)–(44) as

$$\begin{array}{@{}rcl@{}} G^{\prime} \left( -\varphi(p) + \kappa_i;c_i \right) & = & \frac{v(p)}{v(G(-\varphi(p) + \kappa_i; c_i))} \frac{1}{1 + \rho(G(-\varphi(p) + \kappa_i; c_i);c_i)}\\ \end{array} $$

(46)

$$\begin{array}{@{}rcl@{}} G^{\prime} \left( -\varphi({\Phi}_i(p)) + \kappa_j;c_j \right) & = & \frac{v({\Phi}_i(p))}{v(G(-\varphi({\Phi}_i(p)) + \kappa_j; c_j))}\\ & & \null \times \frac{1}{1 + \rho(G(-\varphi({\Phi}_i(p)) + \kappa_j; c_j);c_j)} . \end{array} $$

(47)

From (42) and using (40)–(41), replacing (46)–(47) and${\Phi }_i(p) = G(-\varphi (p) + \kappa _i; c_i)$ wehave

$$\begin{array}{@{}rcl@{}} \psi^{\prime}(p) & = & \frac{v(p)}{v(G(-\varphi({\Phi}_i(p)) + \kappa_j; c_j))}\\ & & \null \times \frac{1}{1 + \rho(G(-\varphi({\Phi}_i(p)) + \kappa_j; c_j);c_j)} \frac{1}{1 + \rho(G(-\varphi(p) + \kappa_i; c_i);c_i)} . \end{array} $$

From Condition C.3’, for all$p \in [ \bar c_k;\tau (c_k))$ and$k=i,j$ it holdsthat $1+\rho (p;c_k) > \sqrt {v(\bar c)/v(\bar \tau )}$, and as wealso have $v(\bar c) \geq v(\bar c_j) \geq v(p) \geq v(\tau (c_j)) \geq v(\bar \tau )$, it followsthat

$$\psi^{\prime}(p) < \frac{v(\bar c_j)}{v(\tau (c_j))} \frac{v(\bar \tau)}{v(\bar c)} < 1 , $$

implying$1 - \psi ^{\prime }(p) > 0$.

From this bound, $p - \varphi (p)$ isstriclty increasing on $[\bar c_j; \tau (c_j))$,there can only be one zero of the function in that interval and the fixed point for (42) is unique on$[\bar c_i;\tau (c_i))\times [\bar c_j;\tau (c_j) )$.□