The Auction of Contracts by Consumer Groups and the Effect on Market Power

Abstract

This article discusses the auctioning of financial contracts by aggregations of consumers who aim to reduce the spot price of a concentrated industry’s product; this is a frequent arrangement in electricity markets. The contracts' underlying asset is the product; the auctions' bidding variable is the strike price; and the bidders are the producers. Using a three-stage complete-information game, we show that when all consumers belong to some group, in the subgame perfect Nash equilibrium, each group fully hedges its consumption, and total output reaches its efficient level. Otherwise, each group over-hedges its consumption, and total production is below the efficiency level.

Appendices

Appendix 1

This appendix finds a Nash equilibrium for the spot market when $\hat{x}({\mathbf{x}}) > a + \hat{n}({\mathbf{x}})\cdot c.$ For this purpose, we define the series $\left\{ {t_{i} } \right\}_{i},$  where $t_{i} = i \cdot x_{i} + a - \sum\nolimits_{h = 1}^{i} {x_{h} } .$ This series is decreasing because $t_{i + 1} = t_{i} + i \cdot (x_{i + 1} - x_{i} )\,$ and the x_i are decreasing. In the following, $\tilde{n}({\mathbf{x}})$ denotes the highest index for which t_i is positive and x_i > c, and $\tilde{x}({\mathbf{x}})$ denotes the sum $\sum\nolimits_{i = 1}^{{\tilde{n}({\mathbf{x}})}} {x_{i} }$. The definitions of $\hat{n}({\mathbf{x}})$ and $\hat{x}({\mathbf{x}})$ in Sect. 3 imply that $t_{{\hat{n}({\mathbf{x}})}} = \hat{n}({\mathbf{x}})\cdot x_{{\hat{n}({\mathbf{x}})}} + a - \hat{x}({\mathbf{x}})$. Thus, $t_{{\hat{n}({\mathbf{x}})}}$ is positive if and only if $\hat{x}({\mathbf{x}}) < a + \hat{n}({\mathbf{x}})\cdot x_{{\hat{n}({\mathbf{x}})}} .$

Next, we show that assuming that $\hat{x}({\mathbf{x}}) > a + \hat{n}({\mathbf{x}})\cdot x_{{\hat{n}({\mathbf{x}})}}$ leads to a contradiction. By definition, $s_{{\hat{n}({\mathbf{x}})}} = (\hat{n}({\mathbf{x}}) + 1) \cdot x_{{\hat{n}({\mathbf{x}})}} + \hat{n}({\mathbf{x}})(q* - \hat{x}({\mathbf{x}})\,).$ Thus, the condition $\hat{x}({\mathbf{x}}) > a + \hat{n}({\mathbf{x}}) \cdot x_{{\hat{n}({\mathbf{x}})}}$ implies that $s_{{\hat{n}({\mathbf{x}})}} < (\hat{n}({\mathbf{x}}) + 1) \cdot x_{{\hat{n}({\mathbf{x}})}} + \hat{n}({\mathbf{x}})\cdot(q* - a - \hat{n}({\mathbf{x}})\cdot \,x_{{\hat{n}({\mathbf{x}})}} ).$ Reordering the latter expression results in $s_{{\hat{n}({\mathbf{x}})}} < - (\hat{n}({\mathbf{x}})^{2} - \hat{n}({\mathbf{x}}) - 1) \cdot x_{{\hat{n}({\mathbf{x}})}} - \hat{n}({\mathbf{x}})\cdot c$, which is negative. This latter result contradicts the definition of $s_{{\hat{n}({\mathbf{x}})}}$: the last positive element in the series $\left\{ {s_{i} } \right\}_{i}$. Consequently, we focus on the case $a + \hat{n}({\mathbf{x}}) \cdot c < \hat{x}({\mathbf{x}}) < a + \hat{n}({\mathbf{x}}) \cdot x_{{\hat{n}({\mathbf{x}})}}$, which implies that $t_{{\hat{n}({\mathbf{x}})}}$ is positive.

Since the series $\left\{ { {t}_{i} } \right\}_{i}$ is decreasing, $t_{{\hat{n}({\mathbf{x}})}} > 0$ implies that $\tilde{n}({\mathbf{x}}) \ge \hat{n}({\mathbf{x}}).$ Hence, $\tilde{x}({\mathbf{x}}) - a - \tilde{n}({\mathbf{x}}) \cdot c = \hat{x}({\mathbf{x}}) - a - \hat{n}({\mathbf{x}}) \cdot c - (\hat{n}({\mathbf{x}}) - \tilde{n}({\mathbf{x}})) \cdot c + \sum\nolimits_{{\hat{n}({\mathbf{x}}) + 1}}^{{\tilde{n}({\mathbf{x}})}} {x_{h} }$. Since x_i > c for $i \le \tilde{n}({\mathbf{x}})$, $\hat{x}({\mathbf{x}}) > a + \hat{n}({\mathbf{x}}) \cdot c,$ implies that $\tilde{x}({\mathbf{x}}) > a + \tilde{n}({\mathbf{x}})\cdot c$.

Lemma 2

When $\hat{x}({\mathbf{x}}) > a + \hat{n}({\mathbf{x}}) \cdot c$ the Nash equilibrium strategies for firms are to supply:

$$\tilde{q}_{i} ({\mathbf{x}}) = \left\{ \begin{gathered} x_{i} + \frac{{a - \tilde{x}({\mathbf{x}})}}{{\tilde{n}({\mathbf{x}})}}\quad \quad \quad i \le \tilde{n}({\mathbf{x}}) \hfill \\ 0\quad \quad \quad \quad \quad \quad \quad \quad \;i > \tilde{n}({\mathbf{x}}). \hfill \\ \end{gathered} \right.$$

(35)

Proof

Since by definition $t_{{\tilde{n}({\mathbf{x}})}} > 0$, $\tilde{q}_{i} ({\mathbf{x}})$ is greater than zero for $i \le \tilde{n}({\mathbf{x}})$ and is zero otherwise. Hence, $\tilde{n}({\mathbf{x}})$ represents the number of firms that produce, and the total output $\tilde{q}({\mathbf{x}})$ equals a. Thus, the derivative in Eq. 4 evaluated at $(q_{i} ,\tilde{q}_{ - i} ({\mathbf{x}}),x_{i} ,\overline{p}_{i}^{s} )$ equals:

$$\frac{{\partial \pi (q_{i} ,\tilde{q}_{ - i} ({\mathbf{x}}),x_{i} ,\overline{p}_{i}^{s} )}}{{\partial q_{i} }} = \left\{ \begin{gathered} 2(\tilde{q}_{i} ({\mathbf{x}}) - q_{i} ) - \tilde{q}_{i} ({\mathbf{x}}) + x_{i} - c\quad \quad if\;0 \le q_{i} \le \tilde{q}_{i} ({\mathbf{x}}) \hfill \\ - c\quad \quad \quad \quad \quad \;\quad \quad \quad \quad \quad \quad \,\,if\;q_{i} > \tilde{q}_{i} ({\mathbf{x}}). \hfill \\ \end{gathered} \right.$$

(36)

Since expression $2(\tilde{q}_{i} ({\mathbf{x}}) - q_{i} ) - \tilde{q}_{i} ({\mathbf{x}}) + x_{i} - c$ is decreasing in q_i, we focus on the case $q_{i} = \tilde{q}_{i} ({\mathbf{x}})$ for which the expression becomes $- \tilde{q}_{i} ({\mathbf{x}}) + x_{i} - c,$ which we rewrite as $(\tilde{x}({\mathbf{x}}) - a - \tilde{n}({\mathbf{x}})\cdot c)/\tilde{n}({\mathbf{x}})$ using the definition of $\tilde{q}_{i} ({\mathbf{x}})$. Given the initial assumption, the latter formula is positive, and so is the derivative in Eq. 36 for $\;q_{i} \le \tilde{q}_{i} ({\mathbf{x}})$. Thus, if a firm participates in the market, it produces $\tilde{q}_{i} ({\mathbf{x}})$.

In what follows, we prove that a firm for which $\tilde{q}_{i} ({\mathbf{x}}) > 0$ chooses to produce. Equation 1 implies that if the firm produces, its profits are given by:

$$\pi (\tilde{q}_{i} ({\mathbf{x}}),\tilde{q}_{ - i} ({\mathbf{x}}),x_{i} ,\overline{p}_{i}^{s} ) = (p(a) - c)\cdot\tilde{q}_{i} ({\mathbf{x}}) + \left( {\overline{p}_{i}^{s} - p(a)} \right)\cdot x_{i} .$$

(37)

Thus:

$$\pi (\tilde{q}_{i} ({\mathbf{x}}),\tilde{q}_{ - i} ({\mathbf{x}}),x_{i} ,\overline{p}_{i}^{s} ) = \left( {\overline{p}_{i}^{s} - c} \right)\cdot x_{i} - \frac{{a - \tilde{x}({\mathbf{x}})}}{{\tilde{n}({\mathbf{x}})}}\cdot c.$$

(38)

If the firm decides not to produce, its profits are:

$$\pi (0,\tilde{q}_{ - i} ({\mathbf{x}}),x_{i} ,\overline{p}_{i}^{s} ) = \left( {\overline{p}_{i}^{s} - \frac{{a - \tilde{x}({\mathbf{x}})}}{{\tilde{n}({\mathbf{x}})}}} \right)\cdot x_{i} .$$

(39)

A comparison of Eqs. 38 and 39 shows that a firm produces when x_i > c—a condition satisfied for all firms $i \le \tilde{n}({\mathbf{x}})$—which completes the proof.

Appendix 2

Lemma 3

The consumer group ℓ auctions a contract for $q* - \sum\nolimits_{h \ne \ell } {y^{h} }$ CfD units or less when the other groups auction contracts that total (1-α^ℓ)q* units or less.

Proof

If the consumer group ℓ auctions a contract for $q* - \sum\nolimits_{h \ne \ell } {y^{h} }$ CfD units, the total number of CfD units that are subscribed will equal q*. Then from Corollary 1 and Proposition 1, $\mathbf{ \hat{x}(x)} = q*$, p(x) = c, and p^s,ℓ = c. Then Eq. 21 implies that the benefits of consumer group ℓ equal α^ℓ(q*)²/2.

In turn, if $y^{\ell } > q* - \sum\nolimits_{h \ne \ell } {y^{h} }$, then X > q*. Since $\sum\nolimits_{h \ne \ell } {y^{h} < q*}$, $y^{\ell }$ meets the condition in Eq. 3. Consequently, Proposition 2 implies that the award strike price takes the value $\hat{p}^{s,\ell }$ that is defined in Eq. 15. Thus, substituting p^s,ℓ in Eq. 20 leads to:

$$u(q({\mathbf{x}}),\alpha^{\ell } ,p^{s,\ell } ) = \left( {\frac{{\hat{n}({\mathbf{x}})\cdot q* + \hat{x}({\mathbf{x}})}}{{\hat{n}({\mathbf{x}}) + 1}}} \right)^{2} \cdot\frac{{\alpha^{\ell } }}{2} + \left( {\frac{1}{{y^{\ell } }}\left ( {\frac{{q* - \hat{x}{\mathbf{(x}}))}}{{\hat{n}({\mathbf{x}}) + 1}}} \right)^{2} + \frac{{q* - \hat{x}({\mathbf{x}})}}{{\hat{n}({\mathbf{x}}) + 1}}} \right)\cdot y^{\ell } .$$

(40)

Reordering terms and omitting the argument x results in:

$$u(q,\alpha^{\ell } ,p^{s\ell } ) = (q*)^{2} \cdot\frac{{\alpha^{\ell } }}{2} + \left( {\frac{{\hat{x} - q*}}{{\hat{n} + 1}}} \right)^{2}\cdot \frac{{\alpha^{\ell } }}{2} + (\alpha^{\ell } q* - y^{\ell } )\cdot\frac{{\hat{x} - q*}}{{\hat{n} + 1}} + \left( {\frac{{\hat{x} - q*}}{{\hat{n} + 1}}} \right)^{2}.$$

(41)

Furthermore, since $\hat{x} = \sum\nolimits_{h} {y^{h} }$ and by assumption $\sum\nolimits_{h \ne \ell } {y^{h} } \le (1 - \alpha^{\ell } )q*,$ it follows that $\alpha^{\ell } q* - y^{\ell } \le q* - \hat{x}$. This last condition, together with Eq. 41, implies that:

$$u(q,\alpha^{\ell } ,p^{s\ell } ) \le (q*)^{2} \cdot\frac{{\alpha^{\ell } }}{2} + \frac{{\alpha^{\ell } }}{2}\cdot\left( {\frac{{\hat{x} - q*}}{{\hat{n} + 1}}} \right)^{2} - (\hat{x} - q*)\cdot\frac{{\hat{x} - q*}}{{\hat{n} + 1}} + \left( {\frac{{\hat{x} - q*}}{{\hat{n} + 1}}} \right)^{2} .$$

(42)

Reordering Eq. 42 leads to:

$$u(q,\alpha^{\ell } ,p^{s\ell } ) \le (q*)^{2} \cdot \frac{{\alpha^{\ell } }}{2} - \frac{{2\hat{n} - \alpha^{\ell } }}{2}\left( {\frac{{\hat{x} - q*}}{{\hat{n} + 1}}} \right)^{2}.$$

(43)

Thus, consumer group ℓ never auctions a contract for more than $q* - \sum\nolimits_{h \ne \ell } {y^{h} }$ CfD units when the other groups auction contracts that add up (1-α^ℓ)q* or less, which completes the proof.

Appendix 3

Lemma 4

The entry of producers harms consumer groups' members when some consumers do not belong to any group.

Proof

In equilibrium, X < q* and thus p^s,ℓ = c. Consequently, Eq. 21 becomes:

$$u(q({\mathbf{x}}),\alpha^{\ell } ,p^{s,\ell } ) = \frac{{q({\mathbf{x}})^{2} }}{2} \cdot\alpha^{\ell } + (q* - q({\mathbf{x}}))\cdot y^{\ell } ,\quad \ell = 1, \, 2, \, \ldots , \, m.$$

(44)

From Eq. 28 it follows that (omitting x):

$$u(q,\alpha^{\ell } ,p^{s,\ell } ) = \frac{{q^{2} }}{2} \cdot\alpha^{\ell } + (q* - q) \cdot(\alpha^{\ell } \cdot q + q* - X),\quad \ell = 1, \, 2, \, \ldots , \, m.$$

(45)

Hence, the impact of the entry of a new firm on the benefits of consumer group ℓ is:

$$\Delta u^{\ell } = - \left( {(1 - \alpha^{\ell } )\cdot q* + X_{n}^{e} - \left( {q_{n}^{e} + \frac{{\Delta q_{n}^{e} }}{2}} \right)\cdot \alpha^{\ell } } \right)\cdot \Delta q_{n}^{e} - (q* - q_{n}^{e} )\cdot \Delta X_{n}^{e} ,\quad \ell = 1,2, \, \ldots , \, m,$$

(46)

where $X_{n}^{e}$ denotes the equilibrium total CfD units auctioned and $q_{n}^{e}$ is the output total when the number of firms is n. Then, from Eq. 30, it follows that:

$$\Delta X_{n}^{e} \equiv X_{n + 1}^{e} - X_{n}^{e} = \frac{\alpha \cdot(1 - \alpha )}{{\left( {(m + 1)\cdot(n + 2) - \alpha } \right) \cdot\left( {(m + 1)\cdot (n + 1) - \alpha } \right)}}q*.$$

(47)

In addition, from Eq. 31 we obtain:

$$\Delta q_{n}^{e} \equiv q_{n + 1}^{e} - q_{n}^{e} = \frac{(1 - \alpha )\cdot(m + 1)}{{\left( {(m + 1)(n + 2) - \alpha } \right)\cdot \left( {(m + 1) \cdot(n + 1) - \alpha } \right)}}\cdot q*.$$

(48)

Then, $\Delta X_{n}^{e}$ and $\Delta q_{n}^{e}$ are positive when α < 1. Since $q* > q_{n}^{e} > X_{n}^{e}$, we conclude that $\Delta u^{\ell } > 0.$ This result completes the proof.