Generator Ownership of Financial Transmission Rights and Market Power

Abstract

Game theory is well suited to analyze a situation with strategic interdependence of multiple decision makers. Electricity markets include both physical and operational attributes. Likewise, electricity markets are characterized by a relatively small number of large market players, limited competitiveness and strategic behavior. Cournot models compete in quantities while Bertrand models compete in prices. Supply function equilibrium function models assume market players compete both in quantity and price. These are realistic assumptions for electricity markets where market players submit a price-quantity schedule. However these models are complex to solve and may not incorporate all technical attributes of electricity markets. Cournot models are easily solvable and yield under reasonable conditions a unique Nash equilibrium. They are also more suitable for short term analysis.

Appendix

This appendix provides proofs of the Lemmas.

Proof of Lemma 1

First, we consider the direction of option share from market i to j. Suppose that there is no congestion. In this case, prices are equated across markets so each market gets half of the total output of both firms. On the other hand, if there is congestion, then the prices of the markets are different. Two congested situations can be differentiated: one is to import K MW with line congestion, and the other is to export K MW with line congestion. Here, I notice that line congestion can occur only when the output difference of both firms is greater than 2K MW. More precisely, market i imports with congestion when q _i  < q _j  − 2K, and exports with congestion when q _i  > q _j  + 2K. In this setting, the profit of firm i is represented by the profit function ${\pi_i}$:

$$ {\pi_i}=\left\{ \begin{array}{lll} P\left( {{q_i}-K} \right){q_i}+\eta_i^{ij }K\left( {P\left( {{q_j}+K} \right)-P\left( {{q_i}-K} \right)} \right)-C\left( {{q_i}} \right),\quad if\;{q_i}>{q_j}+2K, \hfill \\ P\left( {{q_i}+K} \right){q_i}-C\left( {{q_i}} \right),\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad\;\;\;if\;{q_i}<{q_j}-2K, \hfill \\ P\left( {\frac{{{q_i}+{q_j}}}{2}} \right){q_i}-C\left( {{q_i}} \right),\quad \quad \quad \quad \quad \quad \quad \quad \quad\;if\;{q_j}-2K\le {q_i}\le {q_j}+2K, \end{array} \right\} $$

(5.22)

where q _i is firm i’s output and q _j is firm j’s output.

Using the explicit forms of demand and costs from (5.1) to (5.2), the profit function of firm i is:

$$ {\pi_i}=\left\{ \begin{array}{lll} {q_i}\left( {-\alpha {q_i}+\alpha K+\beta } \right)+\eta_i^{ij }K\left( {\alpha \left( {{q_i}-{q_j}} \right)-2\alpha K} \right)-\frac{a}{2}q_i^2-b{q_i}-c,\ if\;{q_i}>{q_j}+2K, \hfill \\ {q_i}\left( {-\alpha {q_i}-\alpha K+\beta } \right)-\frac{a}{2}q_i^2-b{q_i}-c,\quad \quad \quad \quad \quad \quad \quad \quad \quad\quad\quad\quad\quad\quad\quad\quad\;\;if\;{q_i}<{q_j}-2K, \hfill \\ {q_i}\left( {-\alpha \frac{{{q_i}+{q_j}}}{2}+\beta } \right)-\frac{a}{2}q_i^2-b{q_i}-c,\quad \quad \quad\quad\quad\quad\quad\quad\quad\;\;if\;{q_j}-2K\le {q_i}\le {q_j}+2K. \end{array} \right\} $$

(5.23)

From (5.23) and the definition, firm i’s optimal aggressive and passive outputs and Cournot best response output, which are denoted by $q_i^{{u{o_{ij }}+}}\left( {K,\eta_i^{ij }} \right)$, $q_i^{{u{o_{ij }}-}}(K)$, and $q_i^{obC}\left( {{q_j}} \right)$, respectively, are obtained by (5.24), (5.25), and (5.26):

$$ \begin{array}{lll} q_i^{{u{o_{ij }}+}}\left( {K,\eta_i^{ij }} \right) =\mathop{{\arg \max }}\limits_{{{q_i}}}\cr \quad\times\left[ {{q_i}\left( {-\alpha {q_i}+\alpha K+\beta } \right)+\eta_i^{ij }K\left( {\alpha \left( {{q_i}-{q_j}} \right)-2\alpha K} \right)-\frac{a}{2}q_i^2-b{q_i}-c} \right],\end{array} $$

(5.24)

$$ q_i^{{u{o_{ij }}-}}(K)=\mathop{{\arg \max }}\limits_{{{q_i}}}\left[ {{q_i}\left( {-\alpha {q_i}-\alpha K+\beta } \right)-\frac{a}{2}q_i^2-b{q_i}-c} \right], $$

(5.25)

$$ q_i^{{u{o_{ij }}C}}\left( {{q_j}} \right)=\mathop{{\arg \max }}\limits_{{{q_i}}}\left[ {{q_i}\left( {-\alpha \frac{{{q_i}+{q_j}}}{2}+\beta } \right)-\frac{a}{2}q_i^2-b{q_i}-c} \right]. $$

(5.26)

By solving (5.24), (5.25), and (5.26), the optimal aggressive and passive outputs and the Cournot best response output can be explicitly expressed as (5.27), (5.28), and (5.29):

$$ q_i^{{u{o_{ij }}+}}\left( {K,\eta_i^{ij }} \right)=\frac{{\beta +\left( {1+\eta_i^{ij }} \right)\alpha K-b}}{{2\alpha +a}}, $$

(5.27)

$$ q_i^{{u{o_{ij }}-}}(K)=\frac{{\beta -\alpha K-b}}{{2\alpha +a}}, $$

(5.28)

$$ q_i^{{u{o_{ij }}C}}\left( {{q_j}} \right)=-\frac{\alpha }{{2\left( {\alpha +a} \right)}}{q_j}+\frac{{\beta -b}}{{\alpha +a}}. $$

(5.29)

By comparing (5.27), (5.28), and (5.29) with (5.3), (5.4), and (5.5), we can easily observe that (5.6), (5.7), and (5.8) hold.

Similarly, for the case in which firm i possesses an $\eta_i^{ji }$ FTR option in the other direction, the following results are obtained:

$$ q_i^{{u{o_{ji }}+}}(K)=\frac{{\beta +\alpha K-b}}{{2\alpha +a}}, $$

(5.30)

$$ q_i^{{u{o_{ji }}-}}\left( {K,\eta_i^{ji }} \right)=\frac{{\beta -\left( {1+\eta_i^{ji }} \right)\alpha K-b}}{{2\alpha +a}}, $$

(5.31)

$$ q_i^{{u{o_{ji }}C}}\left( {{q_j}} \right)=-\frac{\alpha }{{2\left( {\alpha +a} \right)}}{q_j}+\frac{{\beta -b}}{{\alpha +a}}, $$

(5.32)

Therefore, we also observe that (5.9), (5.10), and (5.11) hold.

Q.E.D.

Proof of Lemma 3

In the FTR obligation model, the profit of each firm i is represented by the profit function ${\pi_i}$:

$$ {\pi_i}=\left\{ \begin{array}{lll} P\left( {{q_i}-K} \right){q_i}+{\gamma_i}K\left( {P\left( {{q_j}+K} \right)-P\left( {{q_i}-K} \right)} \right)-C\left( {{q_i}} \right),\quad if\;{q_i}>{q_j}+2K, \hfill \\ P\left( {{q_i}+K} \right){q_i}-{\gamma_i}K\left( {P\left( {{q_i}+K} \right)-P\left( {{q_j}-K} \right)} \right)-C\left( {{q_i}} \right),\quad if\;{q_i}<{q_j}-2K, \hfill \\ P\left( {\frac{{{q_i}+{q_j}}}{2}} \right){q_i}-C\left( {{q_i}} \right),\quad \quad \quad \quad \quad \quad \quad \quad \quad if\;{q_j}-2K\le {q_i}\le {q_j}+2K, \end{array} \right\} $$

(5.33)

where q _i is firm i’s output and q _j is firm j’s output.

Using the explicit forms of demand and costs of (3.1) and (3.2), the profit function of firm i is rewritten such that:

$$ {\pi_i}=\left\{ \begin{array}{lll} {q_i}\left( {-\alpha {q_i}+\alpha K+\beta } \right)+{\gamma_i}K\left( {\alpha \left( {{q_i}-{q_j}} \right)-2\alpha K} \right)-\frac{a}{2}q_i^2-b{q_i}-c,\quad if\;{q_i}>{q_j}+2K, \hfill \\ {q_i}\left( {-\alpha {q_i}-\alpha K+\beta } \right)-{\gamma_i}K\left( {\alpha \left( {{q_j}-{q_i}} \right)-2\alpha K} \right)-\frac{a}{2}q_i^2-b{q_i}-c,\quad if\;{q_i}<{q_j}-2K, \hfill \\ {q_i}\left( {-\alpha \frac{{{q_i}+{q_j}}}{2}+\beta } \right)-\frac{a}{2}q_i^2-b{q_i}-c,\quad \quad \quad \;\;if\;{q_j}-2K\le {q_i}\le {q_j}+2K. \end{array} \right\} $$

(5.34)

From (5.34) and the definition, firm i’s optimal aggressive and passive outputs and Cournot best response output, which are denoted by $q_i^{ob+}\left( {K,{\gamma_i}} \right)$, $q_i^{ob-}\left( {K,{\gamma_i}} \right)$, and $q_i^{obC}\left( {{q_j}} \right)$ respectively, are obtained by (5.35), (5.36), and (5.37).

$$ q_i^{ob+}\left( {K,{\gamma_i}} \right)=\mathop{{\arg \max }}\limits_{{{q_i}}}\left[ {{q_i}\left( {-\alpha {q_i}+\alpha K+\beta } \right)+{\gamma_i}K\left( {\alpha \left( {{q_i}-{q_j}} \right)-2\alpha K} \right)-\frac{a}{2}q_i^2-b{q_i}-c} \right], $$

(5.35)

$$ q_i^{ob-}\left( {K,{\gamma_i}} \right)=\mathop{{\arg \max }}\limits_{{{q_i}}}\left[ {{q_i}\left( {-\alpha {q_i}-\alpha K+\beta } \right)-{\gamma_i}K\left( {\alpha \left( {{q_j}-{q_i}} \right)-2\alpha K} \right)-\frac{a}{2}q_i^2-b{q_i}-c} \right], $$

(5.36)

$$ q_i^{obC}\left( {{q_j}} \right)=\mathop{{\arg \max }}\limits_{{{q_i}}}\left[ {{q_i}\left( {-\alpha \frac{{{q_i}+{q_j}}}{2}+\beta } \right)-\frac{a}{2}q_i^2-b{q_i}-c} \right]. $$

(5.37)

By solving (5.35), (5.36), and (5.37), the optimal aggressive and passive outputs and the Cournot best response output can be explicitly expressed as (5.16), (5.17), and (5.18).

Q.E.D.

Proof of Lemma 4

The profit of firm i, $\pi_i^{{\eta_i^{ij }}}$, with an FTR $\eta_i^{ij }$ is represented as:

$$ \pi_i^{{\eta_i^{ij }}}=\left\{ \begin{array}{lll} -\left( {\frac{c}{2}+\alpha } \right)q_i^2-\left( {b+\alpha K-\beta } \right){q_i}-a,\quad \quad \quad if\;{q_i}<\frac{1}{{c+\alpha }}\left( {\beta -b-\left( {2c+\alpha } \right)K} \right), \hfill \\ -\left( {\frac{c}{2}+\frac{{\alpha c}}{{2c+\alpha }}} \right)q_i^2-\left( {b-\frac{1}{{2c+\alpha }}\left( {2c\beta +\alpha b} \right)} \right){q_i}-a, \hfill \\ \quad \quad \quad \quad\;if\;\frac{1}{{c+\alpha }}\left( {\beta -b-\left( {2c+\alpha } \right)K} \right)\le {q_i}\le \frac{1}{{c+\alpha }}\left( {\beta -b+\left( {2c+\alpha } \right)K} \right), \hfill \\ -\left( {\frac{c}{2}+\alpha } \right)q_i^2-\left( {b-\alpha \left( {1+\eta_i^{ij }} \right)K-\beta } \right){q_i}-a-\frac{{\alpha \eta_i^{ij }K}}{{c+\alpha }}\left( {\beta -b+\left( {2c+\alpha } \right)K} \right), \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad if\;{q_i}>\frac{1}{{c+\alpha }}\left( {\beta -b+\left( {2c+\alpha } \right)K} \right). \end{array} \right\} $$

(5.38)

The profit of firm i, $\pi_i^{{\eta_i^{ji }}}$, with an FTR $\eta_i^{ji }$ is represented as:

$$ \pi_i^{{\eta_i^{ji }}}=\left\{ \begin{array}{lll} -\left( {\frac{c}{2}+\alpha } \right)q_i^2-\left( {b+\alpha \left( {1+\eta_i^{ji }} \right)K-\beta } \right){q_i}-a-\frac{{\alpha \eta_i^{ji }K}}{{c+\alpha }}\left( {\beta -b-\left( {2c+\alpha } \right)K} \right), \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad if\;{q_i}<\frac{1}{{c+\alpha }}\left( {\beta -b-\left( {2c+\alpha } \right)K} \right), \hfill \\ -\left( {\frac{c}{2}+\frac{{\alpha c}}{{2c+\alpha }}} \right)q_i^2-\left( {b-\frac{1}{{2c+\alpha }}\left( {2c\beta +\alpha b} \right)} \right){q_i}-a, \hfill \\ \quad \quad \quad \quad\;if\;\frac{1}{{c+\alpha }}\left( {\beta -b-\left( {2c+\alpha } \right)K} \right)\le {q_i}\le \frac{1}{{c+\alpha }}\left( {\beta -b+\left( {2c+\alpha } \right)K} \right), \hfill \\ -\left( {\frac{c}{2}+\alpha } \right)q_i^2-\left( {b-\alpha K-\beta } \right){q_i}-a,\quad \quad \quad if\;{q_i}>\frac{1}{{c+\alpha }}\left( {\beta -b+\left( {2c+\alpha } \right)K} \right). \end{array} \right\} $$

(5.39)

Since there is no strategic response from market j, firm i faces the above profit function to maximize. Each of $\pi_i^{{\eta_i^{ij }}}$ and $\pi_i^{{\eta_i^{ji }}}$ has three different regions with respect to ${q_i}$ and we need to compare the maximum profit in each region to identify firm i’s optimal output. Since we can observe that the possession of FTRs only affects the third row of (5.38) and the first row of (5.39), we need to consider only these two rows in order to assess the effect of FTR rights. So, suppose that the maximum profit is obtained by the third row of (5.38) or the first row of (5.39) with the FTR option $\eta_i^{ij }$ and $\eta_i^{ji }$, respectively. Then, firm i’s optimal output $q_i^{{\eta_i^{ij }}}$ ($q_i^{ji }$) with $\eta_i^{ij }$ ($\eta_i^{ji }$) will be (5.20) ((5.21)).

Q.E.D.

Proof of Lemma 5

We denote by $\pi_i^{{{\gamma_i}}}$ firm i’s profit with an FTR obligation ${\gamma_i}$. It is given by:

$$ \pi_i^{{{\gamma_i}}}=\left\{ \begin{array}{lll} -\left( {\frac{c}{2}+\alpha } \right)q_i^2-\left( {b+\alpha \left( {1-{\gamma_i}} \right)K-\beta } \right){q_i}-a+\frac{{\alpha {\gamma_i}K}}{{c+\alpha }}\left( {\beta -b-\left( {2c+\alpha } \right)K} \right), \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad if\;{q_i}<\frac{1}{{c+\alpha }}\left( {\beta -b-\left( {2c+\alpha } \right)K} \right), \hfill \\ -\left( {\frac{c}{2}+\frac{{\alpha c}}{{2c+\alpha }}} \right)q_i^2-\left( {b-\frac{1}{{2c+\alpha }}\left( {2c\beta +\alpha b} \right)} \right){q_i}-a, \hfill \\ \quad \quad \quad \quad\;if\;\frac{1}{{c+\alpha }}\left( {\beta -b-\left( {2c+\alpha } \right)K} \right)\le {q_i}\le \frac{1}{{c+\alpha }}\left( {\beta -b+\left( {2c+\alpha } \right)K} \right), \hfill \\ -\left( {\frac{c}{2}+\alpha } \right)q_i^2-\left( {b-\alpha \left( {1+{\gamma_i}} \right)K-\beta } \right){q_i}-a-\frac{{\alpha {\gamma_i}K}}{{c+\alpha }}\left( {\beta -b+\left( {2c+\alpha } \right)K} \right), \hfill \\ \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad if\;{q_i}>\frac{1}{{c+\alpha }}\left( {\beta -b+\left( {2c+\alpha } \right)K} \right). \end{array} \right\} $$

(5.40)

To examine the effect of FTR obligations, we suppose that the maximum profit is obtained either by the first row or by the third row of (5.40). Then, firm i’s optimal output will be either $\displaystyle\frac{{\beta -b+\alpha \left( {1+{\gamma_i}} \right)K}}{{c+2\alpha }}$ or $\displaystyle\frac{{\beta -b-\alpha \left( {1-{\gamma_i}} \right)K}}{{c+2\alpha }}$.

Q.E.D.