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Abstract

Increasing (or eliminating) the caps on short-term wholesale prices is generally thought to promote long-term forward contracting for electricity. We find that a higher price cap typically enhances the incentives of electricity buyers (e.g., load-serving entities) to undertake forward contracting. However, a higher cap can diminish the incentives of electricity generators to engage in forward contracting. Consequently, higher wholesale price caps can reduce industry forward contracting.

JEL Classification: L13, L51, L94

Keywords

wholesale price caps forward contracting electricity markets

1. Introduction

To protect consumers against price shocks, the maximum price that can be charged for electricity is explicitly limited in many restructured wholesale markets.¹ Wholesale price caps are often criticized, in part on the grounds that they reduce incentives to secure electricity via long-term “forward” contracts.² Forward contracting is valued for at least three reasons. First, like wholesale price caps, forward contracting can help to counteract short-term price volatility. Such volatility seems likely to increase over time as larger portions of electricity supply are generated by intermittent renewable resources (Joskow 2019; Newbery et al. 2018; Wolak 2022). Second, expanded forward contracting can encourage generators to increase the amount of electricity they supply in wholesale markets, thereby reducing wholesale prices (Allaz and Vila 1993). Third, forward contracting can encourage expanded investment in generation capacity. It can do so by allowing generators to secure in advance a stable revenue stream for the output that will ultimately be produced by the expanded capacity.³

The reason why wholesale price caps are commonly believed to reduce incentives for forward contracting is straightforward. If buyers are protected against high prices in the wholesale market, they will be less inclined to sign forward contracts that offer protection against high wholesale prices for electricity. Wolak (2021, 86–7) observes that in the presence of a “limited prospect of very high prices because of [price] caps, retailers may decide not to sign fixed-price forward contracts. … The lower the [price] cap, the greater is the likelihood that the retailer will delay its electricity purchases to the short-term market.” Similarly, Mays and Jenkins (2022, 2) suggest that “[w]ithout the threat of high prices, consumers of energy have insufficient incentive to enter forward contracts with generators.”

The purpose of the present research is to assess this common wisdom formally, explicitly accounting for the strategic decisions of generators and the endogeneity of short-term and long-term electricity prices. We find that, although the common wisdom has considerable merit, it does not fully capture the effects of wholesale price caps on incentives for forward contracting for two reasons. First, although a higher wholesale price cap typically enhances an electricity buyer’s incentive for forward contracting, it does not necessarily do so. Second, and of greater empirical relevance, a higher price cap can reduce the incentives of generators to undertake forward contracting.

A higher price cap ( $\bar{w}$ ) can either enhance or diminish a generator’s incentive for forward contracting because an increase in $\bar{w}$ entails both a revenue enhancement effect that promotes forward contracting and a regime shifting effect that can diminish incentives for forward contracting. The revenue enhancement effect arises because an increase in $\bar{w}$ increases the payment (i.e., the wholesale price) a generator receives for each unit of electricity it sells when the price cap binds. The increased wholesale price enhances a generator’s incentive to increase its equilibrium output, which it achieves by expanding its forward contracting.⁴ Thus, the revenue enhancement effect of an increase in $\bar{w}$ strengthens a generator’s incentive to undertake forward contracting.

The regime shifting effect arises because an increase in $\bar{w}$ reduces the likelihood that the cap binds. Consequently, an increase in $\bar{w}$ can reduce a generator’s incentive for forward contracting if the rate at which expanded forward contracting enhances a generator’s profit is lower when the price cap does not bind than when it binds. This outcome can arise under plausible conditions, in part because expanded forward contracting reduces the equilibrium wholesale price when the price cap does not bind, but does not alter the wholesale price when the cap binds.⁵

To determine whether the regime shifting effect can ever outweigh the revenue enhancement effect, causing forward contracting to decline as $\bar{w}$ increases, we examine equilibrium outcomes in a stylized setting that reflects selected elements of actual electricity markets. We find that when generators choose their preferred levels of forward contracting (non-cooperatively) in this setting, the equilibrium level of aggregate forward contracting often declines as the wholesale price cap increases.

Our analysis contributes to the extensive formal literature on forward contracting, which establishes that forward contracting can induce generators to compete aggressively and expand their outputs in wholesale markets. Allaz and Vila’s (1993) seminal work documents these effects of forward contracting in a non-repeated setting with Cournot competition and publicly-observed levels of forward contracting. Subsequent studies consider alternative forms of competition (Holmberg 2011; Holmberg and Willems 2015), repeated interactions (Green and Le Coq 2010; Liski and Montero 2006), and settings where prevailing levels of forward contracting are not observed publicly (Hughes and Kao 1997). Empirical studies also document the competition-enhancing effects of forward contracts (e.g., Bushnell, Mansur and Saravia 2008; van Eijkel, Kuper and Moraga-González 2016; Wolak 2000). This extensive formal literature largely abstracts from the impact of wholesale price caps on forward contracting.⁶

This literature focuses on settings in which generators dictate the levels of forward contracting. In practice, large buyers of electricity are key counterparties in forward market transactions. Like our analysis, a few studies consider settings where buyers exercise some control over the extent of forward contracting (Anderson and Hu 2008; Schneider 2020; Brown and Sappington 2023a, 2023b). However, these studies do not consider the effects of wholesale price caps on equilibrium levels of forward contracting.

A distinct strand of the literature analyzes the effects of price caps in oligopoly markets, but does not consider forward contracting. Studies in this literature analyze the impact of price caps on output, consumer surplus, and welfare in settings with uncertain demand (Earle, Schmedders and Tatur 2007; Grimm and Zottl 2010; Reynolds and Rietzke 2018). Other studies identify conditions under which a wholesale price cap reduces incentives for capacity investment (Fabra, von der Fehr and de Frutos 2011; Zottl 2011).

We contribute to these strands of the literature by examining how wholesale price caps affect the incentives of both generators and large buyers of electricity to undertake forward contracting. We do so in a stylized model with linear demand and costs, a uniform density for demand uncertainty, and a regulated load serving entity that is required to serve the realized demand of its customers. We identify conditions under which the aforementioned common wisdom – that wholesale price caps reduce buyers’ incentives for forward contracting – prevails.⁷ We further demonstrate that the common wisdom about buyers’ incentives does not readily extend to generators’ incentives. Consequently, any attempt to assess how a change in a prevailing wholesale price cap will affect industry forward contracting should consider both the relative influence of buyers and sellers in determining the levels of forward contracting and the distinct ways in which a price cap affects their incentives for forward contracting.

The ensuing analysis proceeds as follows. Section 2 describes the key elements of our model. Section 3 examines the electricity buyer’s incentives for forward contracting. Section 4 analyzes electricity generators’ incentives for forward contracting. Section 5 provides concluding observations. The Appendix provides the proofs of all formal conclusions.

2. Model Elements

A large buyer of electricity (e.g., a load serving entity) is required to deliver all the electricity its retail customers demand. Realized retail demand is $\bar{Q} + η$ , where $\bar{Q}$ is expected demand and $η \in [\underline{η}, \bar{η}]$ is the realization of a mean-zero random variable. Thus, retail demand is stochastic and perfectly price inelastic. Retail demand is always positive, so $\bar{Q} + \underline{η} > 0$ .

Commercial and industrial customers also consume electricity, which they purchase in the wholesale market at unit price $w$ . Their demand for electricity is $Q^{I} (w) = a^{I} - b^{I} w$ , where $a^{I}$ and $b^{I}$ are positive constants. Aggregate demand for electricity – the sum of retail demand and industrial demand – is:

Q (\cdot) = a^{I} - b^{I} w + \bar{Q} + η .

(1)

The corresponding inverse demand curve is:

w (\cdot) = a + ε - bQ where a = \frac{a^{I} + \bar{Q}}{b^{I}}, ε = \frac{η}{b^{I}}, and b = \frac{1}{b^{I}} .

(2)

$h (ε)$ is the density function for the random demand parameter $ε \in [- \bar{ε}, \bar{ε}]$ . $H (ε)$ is the corresponding distribution function.

Electricity is supplied by $n \geq 2$ generators, $G 1, \dots, Gn$ . $Gi$ ’s cost of producing $q$ units of output is $c_{i} q$ , where $c_{i} > 0$ is a parameter, for $i = 1, \dots, n$ .

The large buyer ( $B$ ) can purchase electricity in the wholesale market and/or procure electricity via a long-term forward contract. A forward contract between $B$ and generator $Gi$ obligates $Gi$ to ensure that $B$ ultimately can purchase a unit of electricity in the wholesale market at net price $p^{f}$ . This net price is the difference between the realized wholesale price, $w$ , and the compensation that $Gi$ delivers to $B$ , which is $w - p^{f}$ .⁸ To illustrate, when $Gi$ and $B$ sign $F_{i}$ forward contracts and the realized wholesale price is $w > p^{f}$ , $Gi$ must pay $B$ the amount $F_{i} [w - p^{f}]$ to ensure that $B$ ’s net cost of securing the $F_{i}$ units of electricity is $F_{i} [w - (w - p^{f})] = p^{f} F_{i}$ .⁹ For expositional ease, the ensuing discussion will refer to $p^{f}$ as the price of a forward contract.

We adopt the standard assumption in the literature that the price of a forward contract, $p^{f}$ , is equal to the expected wholesale price of electricity, $E {w (ε)}$ , where $w (ε)$ is the wholesale price that prevails when realized retail demand is $\bar{Q} + ε$ . This equality will prevail in the following two stage game. In the first stage, either the buyer or the generators specify the number of forward contracts that will be signed.¹⁰ In the second stage, risk-neutral, non-strategic actors (e.g., financial traders) with rational expectations bid for the contracts. The bidding ensures that the price of a forward contract reflects its expected value, an outcome that is anticipated in the first stage of the game.¹¹

If $Gi$ signs $F_{i}$ forward contracts with $B$ , then $Gi$ ’s profit when it produces $q_{i}$ units of output and wholesale price $w$ prevails is:

π_{i}^{G} = w [q_{i} - F_{i}] + p^{f} F_{i} - c_{i} q_{i} .

(3)

Equation (3) reflects the fact that $Gi$ sells $q_{i} - F_{i}$ units of output in the wholesale market at unit price $w$ and effectively sells $F_{i}$ units of output via forward contract at unit price $p^{f}$ .

When realized retail demand is $\bar{Q} + b^{I} ε$ and the wholesale price is $w$ , $B$ ’s profit is:

π^{B} (ε) = R (ε) - w [\bar{Q} + b^{I} ε - F] - p^{f} F - K,

(4)

where $R (ε)$ denotes $B$ ’s revenue when $ε$ is realized,¹² $F \equiv \underset{j = 1}{\sum^{n}} F_{j}$ , and $K \geq 0$ denotes the (fixed) cost that $B$ incurs in addition to electricity procurement costs. Equation (4) reflects the fact that $B$ purchases $\bar{Q} + b^{I} ε - F$ units of electricity at unit price $w$ in the wholesale market and procures $F$ units of electricity at unit price $p^{f}$ via forward contract.

$\bar{w}$ denotes the wholesale price cap, which is the highest wholesale price that is permitted. The wholesale price that prevails is the minimum of $\bar{w}$ and the wholesale price that equates the aggregate demand and the aggregate supply of electricity. We assume $\bar{w} >$ maximum ${c_{1}, \dots, c_{n}}$ .

The timing in the model is as follows. After the regulator specifies the wholesale price cap $\bar{w}$ and the retail revenue function $R (\cdot)$ , the levels of forward contracting are determined.¹³ Then the realization of $ε$ is observed publicly. Next, the generators choose their outputs, simultaneously and non-cooperatively. After the wholesale price is determined, industrial customers decide how much electricity to purchase in the wholesale market. Finally, the terms of all forward contracts are fulfilled and $B$ purchases the amount of electricity required to satisfy the realized demand of its retail customers.

Before examining the levels of forward contracting that arise in equilibrium, it is helpful to examine the wholesale price and the generators’ outputs that arise in equilibrium, given the prevailing levels of forward contracting. Lemma 1 characterizes $Gi$ ’s equilibrium output ( $q_{i} (ε)$ ) when generator $Gj$ signs $F_{j}$ forward contracts with $B$ ( $j = 1, \dots, n$ ) and when realized retail demand is $\bar{Q} + b^{I} ε$ . Lemma 1 also characterizes the corresponding equilibrium wholesale price ( $w (ε)$ ). The lemma refers to $\hat{ε} \in (\underline{ε}, \bar{ε})$ , which is the smallest realization of $ε$ for which the wholesale price cap binds.¹⁴ The lemma also refers to $C \equiv \underset{j = 1}{\sum^{n}} c_{j}$ , $C_{- i} \equiv \underset{\begin{matrix} j = 1 \\ j \neq i \end{matrix}}{\sum^{n}} c_{j}$ , $F \equiv \underset{j = 1}{\sum^{n}} F_{j}$ , and $F_{- i} \equiv \underset{\begin{matrix} j = 1 \\ j \neq i \end{matrix}}{\sum^{n}} F_{j}$ .

Lemma 1. In equilibrium, given $ε$ and $F_{1}, \dots, F_{n}$ :

For all ε < \hat{ε} \equiv [n + 1] \bar{w} - [a + C - b F] :

(5)

w (ε) = \frac{1}{n + 1} [a + ε + C - b F] and

(6)

q_{i} (ε) = \frac{a + ε + C_{- i} - n c_{i} + b n F_{i} - b F_{- i}}{b [n + 1]} for i = 1, \dots, n .

(7)

For all ε \geq \hat{ε} : w (ε) = \bar{w} and \sum_{i = 1}^{n} q_{i} (ε) = \frac{1}{b} [a + ε - \bar{w}] .

(8)

Equation (7) implies that when $ε < \hat{ε}$ (so the price cap does not bind), $Gi$ ’s equilibrium output increases and the equilibrium outputs of rival generators decline as $Gi$ ’s forward contracting ( $F_{i}$ ) increases. $Gi$ ’s increased forward contracting reduces the amount of electricity it sells at the prevailing wholesale price, $w$ . This reduced exposure to the decline in $w$ caused by an increase in output implies that $Gi$ ’s profit increases more rapidly with its output, which induces $Gi$ to increase its output. The corresponding reduction in $w$ induces rival generators to reduce their equilibrium outputs (Allaz and Vila 1993). On balance, industry output increases, so the wholesale price declines as a generator increases its forward contracting (see equation (6)). The reduced wholesale price causes the set of the highest $ε$ realizations for which the price cap binds to contract, that is, $\hat{ε}$ increases (see equation (5)).

Equation (8) implies that multiple equilibria arise when $ε \geq \hat{ε}$ , as in Buehler, Burger and Ferstl (2010). When the price cap binds, the wholesale price does not decline as a generator increases its output. Consequently, $Gi$ ’s profit increases (at the rate $\bar{w} - c_{i} > 0$ ) as $q_{i}$ increases, so each generator finds it profitable to serve the entire excess demand or any portion thereof. Therefore, when $ε \geq \hat{ε}$ so the price cap binds, any set of outputs that sum to the realized aggregate demand when $w = \bar{w}$ constitute equilibrium outputs.¹⁵

It remains to characterize the levels of forward contracting that arise in equilibrium. These levels vary according to whether the extent of forward contracting is determined by the buyer or by generators. Section 3 examines the levels of forward contracting preferred by the buyer ( $B$ ). Section 4 examines the corresponding equilibrium levels that will be chosen (non-cooperatively) by generators.

3. The Buyer’s Preferred Level of Forward Contracting

To determine $B$ ’s preferred levels of forward contracting, consider the setting where $B$ sells electricity to its retail customers at fixed unit price

r = γ r_{0} + [1 - γ] E {w (ε)}

(9)

where $γ \in (0, 1]$ and $r_{0} > 0$ are parameters. $r_{0}$ is assumed to be large enough to ensure that $B$ secures nonnegative expected profit in equilibrium.¹⁶ The formulation in equation (9) allows the regulated unit price to increase as the equilibrium expected wholesale price increases, as will be the case if the regulator adjusts the retail price to reflect anticipated impacts of forward contracting on the expected wholesale price. The formulation also encompasses the setting where the retail price of electricity is set at a fixed level $(r_{0}$ ) that does not vary with the anticipated wholesale price.¹⁷

Equation (4) implies that $B$ ’s profit when $ε$ is realized is:

π^{B} (ε) = r [\bar{Q} + b^{I} ε] - w (ε) [\bar{Q} + b^{I} ε - F] - p^{f} F - K .

(10)

Equation (10) implies that, because $p^{f} = E {w (ε)}$ , $B$ ’s expected profit is:

E {π^{B} (ε)} = E {[r - w (ε)] [\bar{Q} + b^{I} ε]} - K .

(11)

Equation (11) implies that $B$ ’s expected profit can be written as the sum of: (i) $E {π^{BD} (ε)}$ , $B$ ’s expected profit from serving expected (or “deterministic”) demand, $\bar{Q}$ ; and (ii) $E {π^{BS} (ε)}$ , $B$ ’s expected profit from serving stochastic demand, $b^{I} ε$ .¹⁸ Formally:

\begin{matrix} E {π^{B} (ε)} = E {π^{BD} (ε)} + E {π^{BS} (ε)} - K, where \\ E {π^{BD} (ε)} = E {[r - w (ε)] \bar{Q}} and E {π^{BS} (ε)} = E {[r - w (ε)] b^{I} ε} . \end{matrix}

(12)

Lemma 2 characterizes the impact of expanded forward contracting on $B$ ’s expected profit and its components.

Lemma 2. Expanded forward contracting: (i) increases $E {π^{BD} (ε)}$ ; (ii) reduces $E {π^{BS} (ε)}$ ; and (iii) increases $E {π^{B} (ε)}$ if $γ$ is sufficiently close to 1. More precisely, for each $i = 1, \dots, n$ :

\begin{matrix} \frac{\partial E {π^{BD} (ε)}}{\partial F_{i}} = \frac{γ b \bar{Q} [\hat{ε} + \bar{ε}]}{2 \bar{ε} [n + 1]} > 0; \frac{\partial E {π^{BS} (ε)}}{\partial F_{i}} = - \frac{(\bar{ε}) - (\hat{ε})}{4 \bar{ε} [n + 1]} < 0; and \\ \frac{\partial E {π^{B} (ε)}}{\partial F_{i}} = \frac{\hat{ε} + \bar{ε}}{4 \bar{ε} [n + 1]} [2 γ b \bar{Q} - (\bar{ε} - \hat{ε})] ⋛ 0 \Leftrightarrow γ ⋛ \frac{\bar{ε} - \hat{ε}}{2 b \bar{Q}} . \end{matrix}

(13)

An increase in $F_{i}$ increases $E {π^{BD} (ε)}$ by increasing $B$ ’s expected profit margin, $E {r - w (\cdot)}$ . The higher expected profit margin arises because an increase in $F_{i}$ reduces the expected wholesale price more rapidly than it reduces the retail price.¹⁹

An increase in $F_{i}$ reduces $E {π^{BS} (ε)}$ for two reasons. First, the increase in $F_{i}$ increases $B$ ’s profit margin, $r - w (ε)$ , whenever the price cap does not bind. This is the case because $w (ε)$ declines more rapidly than $r$ declines as $F_{i}$ increases.²⁰ The increased margin is applied primarily (if not entirely) to negative stochastic demand, so $E {π^{BS} (ε)}$ declines.²¹ Second, the increase in $F_{i}$ reduces $B$ ’s profit margin, $r - \bar{w}$ , when the price cap binds. This is the case because $r$ declines (when $γ < 1$ ) but $w (ε) = \bar{w}$ does not change. Because this reduced margin is applied primarily (if not entirely) to positive stochastic demand, $E {π^{BS} (ε)}$ declines.²²

When $γ$ is close to 1, the retail price is relatively insensitive to the expected wholesale price. Consequently, the predominant effect of an increase in $F_{i}$ is to increase $E {π^{BD} (ε)}$ by increasing $B$ ’s expected profit margin.²³ Therefore, $E {π^{B} (ε)}$ increases when $γ$ is sufficiently close to 1. When $γ$ is sufficiently small, an increase in $F_{i}$ reduces $r$ at nearly the same rate it reduces $E {w (ε)}$ . The resulting limited impact on $B$ ’s expected profit margin implies that the increase in $F_{i}$ has little impact on $E {π^{BD} (ε)}$ . Therefore, the predominant effect of an increase in $F_{i}$ is to reduce $E {π^{BS} (ε)}$ , so $E {π^{B} (ε)}$ declines.²⁴

For expositional convenience, we will refer to $\frac{\partial E {π^{B} (ε)}}{\partial F_{i}}$ as $B$ ’s incentive for forward contracting. Proposition 1 explains how changes in the level of the wholesale price cap affect this incentive.

Proposition 1. As the price cap increases: (i) $\frac{\partial E {π^{BD} (ε)}}{\partial F_{i}}$ increases; and (ii) $B$ ’s incentive for forward contracting increases if $\hat{ε} > 0$ or $γ$ is sufficiently close to 1. Formally, for each $i = 1, \dots, n$ :

\begin{matrix} \frac{\partial}{\partial \bar{w}} (\frac{\partial E {π^{BD} (ε)}}{\partial F_{i}}) = \frac{γ b \bar{Q}}{2 \bar{ε}} > 0 and \\ \frac{\partial}{\partial \bar{w}} (\frac{\partial E {π^{B} (ε)}}{\partial F_{i}}) = \frac{γ b \bar{Q} + \hat{ε}}{2 \bar{ε}} ⋛ 0 \Leftrightarrow \hat{ε} ⋛ - γ b \bar{Q} . \end{matrix}

(14)

Proposition 1 indicates that an increase in $\bar{w}$ always increases $\frac{\partial E {π^{BD} (ε)}}{\partial F_{i}}$ for two reasons. First, $\hat{ε}$ increases as $\bar{w}$ increases. The increase in $\hat{ε}$ expands the $[\underline{ε}, \hat{ε}]$ region in which an increase in $F_{i}$ increases $B$ ’s profit margin by reducing $w (ε)$ more rapidly than it reduces $r$ . Second, the increase in $\hat{ε}$ reduces the $(\hat{ε}, \bar{ε}]$ region in which an increase in $F_{i}$ reduces $B$ ’s profit margin by reducing $r$ (when $γ < 1$ ) without altering $w (\cdot)$ .

Proposition 1 also reports that high values of $γ$ increase the rate at which an increase in $\bar{w}$ enhances $B$ ’s incentive for forward contracting. The larger is $γ$ , the less sensitive is $r$ to $E {w (ε)}$ , and thus the more rapidly $B$ ’s expected profit margin increases as $F_{i}$ increases. When $γ$ is sufficiently close to 1 (so $r \approx r_{0}$ ), an increase in $\bar{w}$ always enhances $B$ ’s incentive for forward contracting (i.e., $\frac{\partial}{\partial \bar{w}} (\frac{\partial E {π^{B} (ε)}}{\partial F_{i}}) > 0$ because $\lim_{γ \to 1} {γ b \bar{Q} + \hat{ε}} = b \bar{Q} + \hat{ε} > b \bar{Q} - \bar{ε} > 0$ ).

Proposition 1 also identifies conditions under which an increase in $\bar{w}$ could, in principle, reduce $B$ ’s incentive for forward contracting. This inverse relationship can prevail when $γ$ is relatively small (so an increase in $F_{i}$ reduces $r$ relatively rapidly) and when the price cap binds for an extensive set of demand realizations. Specifically, the cap must bind even when realized demand is less than its expected value (because $\hat{ε} < 0$ ). This condition is not empirically relevant. However, if a situation were to arise in which $γ$ is close to 0 and $\hat{ε} < 0$ , an increase in $\bar{w}$ could reduce $\frac{\partial E {π^{B} (ε)}}{\partial F_{i}}$ . It could do so primarily by expanding the $[\underline{ε}, \hat{ε}]$ region and thereby reducing expected stochastic demand in this region ( $\int_{\underline{ε}}^{\hat{ε}} b^{I} ε d H (ε)$ ) when $\hat{ε} < 0$ .

In summary, Proposition 1 implies that an increase in the wholesale price cap typically increases $B$ ’s incentive for forward contracting in settings that arise in practice.

4. Generators’ Preferred Levels of Forward Contracting

In practice, a buyer of electricity typically does not dictate prevailing levels of forward contracting unilaterally. Therefore, it is important to consider how a binding wholesale price cap affects generators’ incentives for forward contracting. To do so, let ${\bar{q}}_{i} (ε)$ denote $Gi$ ’s output when $ε > \hat{ε}$ is realized (so the price cap binds), and let $q_{i} (ε)$ denote $Gi$ ’s output when $ε \leq \hat{ε}$ is realized. Equation (3) implies that, because $p^{f} = E {w (ε)}$ , $Gi$ ’s expected profit is:

E {π_{i}^{G} (ε)} = \int_{\underline{ε}}^{\hat{ε}} [w (ε) - c_{i}] q_{i} (ε) d H (ε) + \int_{\hat{ε}}^{\bar{ε}} [\bar{w} - c_{i}] {\bar{q}}_{i} (ε) d H (ε) .

(15)

Equation (15) implies that the rate at which $Gi$ ’s expected profit increases as its forward contracting increases is:

\begin{array}{l} \frac{\partial E {π_{i}^{G} (ε)}}{\partial F_{i}} = \int_{\underline{ε}}^{\hat{ε}} {[w (ε) - c_{i}] \frac{\partial q_{i} (ε)}{\partial F_{i}} + q_{i} (ε) \frac{\partial w (ε)}{\partial F_{i}}} d H (ε) \\ + \int_{\hat{ε}}^{\bar{ε}} [\bar{w} - c_{i}] \frac{\partial {\bar{q}}_{i} (ε)}{\partial F_{i}} d H (ε) . \end{array}

(16)

Equation (16) implies that the impact of a change in $\bar{w}$ on $Gi$ ’s incentive for forward contracting (i.e., on $\frac{\partial E {π_{i}^{G} (ε)}}{\partial F_{i}}$ ) is:

\begin{array}{l} \frac{\partial}{\partial \bar{w}} (\frac{\partial E {π_{i}^{G} (ε)}}{\partial F_{i}}) = \int_{\hat{ε}}^{\bar{ε}} \frac{\partial {\bar{q}}_{i} (ε)}{\partial F_{i}} d H (ε) \\ + \frac{\partial \hat{ε}}{\partial \bar{w}} {[w (\hat{ε}) - c_{i}] {\frac{\partial q_{i} (ε)}{\partial F_{i}} |}_{ε = \hat{ε}} + q_{i} (\hat{ε}) {\frac{\partial w (ε)}{\partial F_{i}} |}_{ε = \hat{ε}} - [\bar{w} - c_{i}] {\frac{\partial {\bar{q}}_{i} (ε)}{\partial F_{i}} |}_{ε = \hat{ε}}} h (\hat{ε}) . \end{array}

(17)

Equation (17) identifies two effects of an increase in the price cap, $\bar{w}$ , on $Gi$ ’s incentive for forward contracting: a revenue enhancement effect and a regime shifting effect. The integral term in equation (17) reflects the revenue enhancement effect, which arises when $ε > \hat{ε}$ , so the price cap binds. When $ε > \hat{ε}$ , an increase in $\bar{w}$ increases the rate at which $Gi$ ’s revenue increases as increased forward contracting increases $Gi$ ’s equilibrium output. The enhanced revenue increases $Gi$ ’s incentive for forward contracting.

The second of the two terms in equation (17) captures the regime shifting effect. This effect arises because an increase in $\bar{w}$ reduces the likelihood that the price cap binds and increases the likelihood that the cap does not bind. The regime shifting effect diminishes (enhances) $Gi$ ’s incentive for forward contracting if the rate at which $Gi$ ’s profit increases with its forward contracting is higher (lower) when the price cap binds than when the cap does not bind.

To determine whether $\frac{\partial π_{i}^{G} (ε)}{\partial F_{i}}$ is higher when the price cap binds or when it does not bind, it is necessary to determine the rate at which $Gi$ ’s equilibrium output changes as $F_{i}$ changes when the price cap binds. Recall that multiple equilibria arise when the cap binds. Therefore, we must introduce an assumption about how $Δ (ε) \equiv Q (\bar{w}, ε) - Q (\bar{w}, \hat{ε})$ , the incremental aggregate demand that arises as $ε$ increases above $\hat{ε}$ when $w (ε) = \bar{w}$ , is allocated among the generators. For analytic tractability, the following assumption is maintained throughout the ensuing discussion in the text:

{\bar{q}}_{i} (ε) = q_{i} (\hat{ε}) + α_{i} [Q (\bar{w}, ε) - Q (\bar{w}, \hat{ε})] for all ε \in [\hat{ε}, \bar{ε}],

(18)

where $α_{i} \in [0, 1]$ is a constant for all $i = 1, \dots, n$ , and $\underset{j = 1}{\sum^{n}} α_{j} = 1$ . The formulation in equation (18) permits any allocation of $Δ (ε)$ among generators that is not affected by the prevailing levels of forward contracting. For example, $Δ (\cdot)$ might be allocated equally among the $n$ generators, so $α_{i} = \frac{1}{n}$ . Alternatively, $Δ (\cdot)$ might be allocated equally among (only) the generators known to employ the least-cost technology.²⁵ Part C of the Appendix demonstrates that the key qualitative conclusions drawn below are not an artifact of the simplifying assumption that the generators’ outputs and levels of forward contracting do not affect the allocation of $Δ (\cdot)$ .

Lemma 3 examines how a binding price cap affects the impact of expanded forward contracting on a generator’s profit.

Lemma 3. Expanded forward contracting increases a generator’s equilibrium profit more rapidly when the price cap binds than when it does not bind if the generator’s share of $Δ (ε)$ is not too pronounced. Formally, $\frac{\partial π_{i}^{G} (ε)}{\partial F_{i}}$ is higher when $ε \in (\hat{ε}, \bar{ε})$ than when $ε \in [\underline{ε}, \hat{ε})$ if $α_{i} \in [0, \frac{2}{n + 1})$ .

The conclusion in Lemma 3 reflects the following considerations. An increase in $F_{i}$ reduces the wholesale price when the price cap does not bind (recall equation (6)), but does not alter the wholesale price when the cap binds. Therefore, an increase in $F_{i}$ will increase $π_{i}^{G} (ε)$ more rapidly when the price cap binds than when it does not bind as long as $q_{i} (ε)$ does not increase much more rapidly than ${\bar{q}}_{i} (ε)$ increases as $F_{i}$ increases.

Equation (7) implies that the rate at which $q_{i} (ε)$ increases as $F_{i}$ increases is $\frac{\partial q_{i} (ε)}{\partial F_{i}} = \frac{n}{n + 1}$ . Equation (18) implies that the rate at which ${\bar{q}}_{i} (ε)$ increases with $F_{i}$ has two components. First, $q_{i} (\hat{ε})$ increases as $F_{i}$ increases because a generator’s equilibrium output increases as its forward contracting increases for all $ε \in [\underline{ε}, \hat{ε})$ (recall equation (7)). Second, $Δ (ε)$ declines as $F_{i}$ increases. The decline in $Δ (\cdot)$ reflects the increase in $Q (\bar{w}, \hat{ε})$ induced by an increase in forward contracting. The increase in $Q (\bar{w}, \hat{ε})$ arises because $\hat{ε}$ increases as expanded forward contracting increases industry output. The increased output reduces the wholesale price, which increases $\hat{ε}$ , the smallest $ε$ for which the price cap binds. It is apparent from equation (18) that the decline in $Δ (ε)$ reduces ${\bar{q}}_{i} (ε)$ relatively slowly when $α_{i}$ is small. Consequently, if $α_{i}$ is sufficiently small, $q_{i} (ε)$ does not increase much more rapidly than ${\bar{q}}_{i} (ε)$ increases as $F_{i}$ increases. Consequently, $\frac{\partial π_{i}^{G} (ε)}{\partial F_{i}}$ is higher when the price cap binds than when it does not bind in this case.

If $π_{i}^{G} (ε)$ increases more rapidly with $F_{i}$ when the price cap binds than when it does not bind, then the regime switching effect causes $\frac{\partial E {π_{i}^{G} (ε)}}{\partial F_{i}}$ to decline as $\bar{w}$ increases. In contrast, the revenue enhancement effect causes $\frac{\partial E {π_{i}^{G} (ε)}}{\partial F_{i}}$ to increase as $\bar{w}$ increases. These observations underlie Proposition 2.

Proposition 2. An increase in $\bar{w}$ can either enhance or diminish a generator’s incentive for forward contracting because $\frac{\partial}{\partial \bar{w}} (\frac{\partial E {π_{i}^{G} (ε)}}{\partial F_{i}}) = \frac{Ω}{2 \bar{ε}}$ , where

Ω \equiv [1 - α_{i}] [a + \bar{ε} + C_{- i} - n c_{i} - b F] - b F_{i} + [\bar{w} - c_{i}] [(n + 1) (2 α_{i} - 1) - 2] .

(19)

The following corollary to Proposition 2 identifies factors that enhance or diminish the impact of an increase in $\bar{w}$ on $Gi$ ’s incentive for forward contracting.

Corollary 1. $S u p p o s e$ $α_{i} \in [0, 1)$ . Then the rate at which an increase in $\bar{w}$ enhances a generator’s incentive for forward contract $(\frac{\partial}{\partial \bar{w}} (\frac{\partial E {π_{i}^{G} (ε)}}{\partial F_{i}}))$ increases: (i) as $a^{I}$ , $\bar{Q}$ , or $C_{- i}$ increases; (ii) as $c_{i}$ increases if $α_{i} < \frac{3}{2 + n}$ ; and (iii) as $\bar{w}$ declines if $α_{i} < \frac{n + 3}{2 [n + 1]}$ .

Condition (i) in Corollary 1 reports that an increase in $a^{I}$ , $\bar{Q}$ , or $C_{- i}$ increases the rate at which an increase in $\bar{w}$ enhances $Gi$ ’s incentive for forward contracting. Expanded demand ( $a^{I}$ or $\bar{Q}$ ) or higher costs of rival generators promote higher wholesale prices, thereby increasing the range of demand realizations for which the price cap binds (recall equations (5) and (6)). The expected impact of the revenue enhancement effect increases as the price cap becomes more likely to bind, so an increase in $\bar{w}$ becomes more likely to enhance $Gi$ ’s incentive for forward contracting.²⁶

The equilibrium wholesale price also increases as $c_{i}$ increases. Therefore, an increase in $c_{i}$ will also increase $\frac{\partial}{\partial \bar{w}} (\frac{\partial E {π_{i}^{G} (ε)}}{\partial F_{i}})$ as long as the increase in $c_{i}$ does not substantially enhance a regime switching effect that reduces $\frac{\partial E {π_{i}^{G} (ε)}}{\partial F_{i}}$ . As condition (ii) in Corollary 1 indicates, this will be the case if $α_{i}$ is sufficiently small, because $\frac{\partial q_{i} (ε)}{\partial F_{i}}$ and $\frac{\partial {\bar{q}}_{i} (ε)}{\partial F_{i}}$ are not too disparate when $α_{i}$ is small. Consequently, an increase in $c_{i}$ reduces $\frac{\partial π_{i}^{G} (ε)}{\partial F_{i}}$ at relatively comparable rates when the price cap binds and when it does not bind.²⁷

Condition (iii) in Corollary 1 holds because the price cap becomes less likely to bind as $\bar{w}$ increases. The associated reduction in the expected impact of the revenue enhancement effect reduces the rate at which an increase in $\bar{w}$ increases $\frac{\partial E {π_{i}^{G} (ε)}}{\partial F_{i}}$ when $α_{i}$ is sufficiently small (so the regime switching effect promotes a reduction in $\frac{\partial E {π_{i}^{G} (ε)}}{\partial F_{i}}$ as $\bar{w}$ increases).²⁸

Proposition 2 also permits the specification of conditions under which an increase in $\bar{w}$ enhances a generator’s incentive for forward contracting, as Corollary 2 reports.

Corollary 2. An increase in the price cap enhances a generator’s incentive for forward contracting (so $\frac{\partial}{\partial \bar{w}} (\frac{\partial E {π_{i}^{G} (ε)}}{\partial F_{i}}) > 0$ ) if $α_{i}$ is sufficiently close to 1 and $F_{i}$ is sufficiently small.

Corollary 2 reflects the following considerations. When $α_{i}$ is close to 1, an increase in $F_{i}$ has little impact on $Gi$ ’s output when the price cap binds.²⁹ Consequently, $\frac{\partial π_{i}^{G}}{\partial F_{i}}$ is relatively small when the price cap binds. Therefore, the regime switching effect of an increase in $\bar{w}$ complements the revenue enhancement effect as long as $F_{i}$ is sufficiently small (which ensures $\frac{\partial π_{i}^{G}}{\partial F_{i}} > 0$ when the price cap does not bind). The two effects together ensure that an increase in $\bar{w}$ increases $Gi$ ’s incentive for forward contracting.

Corollary 3 provides an additional conclusion when the $n$ generators are symmetric.

Corollary 3. Suppose the generators are symmetric, so $α_{i} = \frac{1}{n}$ and $c_{i} = c$ for $i = 1, \dots, n$ . Then an increase in $\bar{w}$ reduces each generator’s incentive for forward contracting (i.e., $\frac{\partial}{\partial \bar{w}} (\frac{\partial E {π_{i}^{G} (ε)}}{\partial F_{i}}) < 0$ ) if $\bar{w}$ is sufficiently large, that is, if $\bar{w} > \frac{1}{n + 2} [a + \bar{ε} + (n + 1) c]$ .

Corollary 3 reflects the fact that $α_{i} < \frac{2}{n + 1}$ when $α_{i} = \frac{1}{n}$ .³⁰ Therefore, $\frac{\partial π_{i}^{G} (ε)}{\partial F_{i}}$ is higher when the price cap binds than when it does not bind (recall Lemma 3). Consequently, the regime switching effect of an increase in $\bar{w}$ reduces $Gi$ ’s incentive for forward contracting. Corollary 3 reports that this effect outweighs the countervailing revenue enhancement effect of an increase in $\bar{w}$ (which is only relevant when the price cap binds) if $\bar{w}$ is relatively high, so the price cap binds relatively infrequently.³¹

Corollary 3 establishes that an increase in $\bar{w}$ can reduce a generator’s incentive for forward contracting, holding constant the forward contracting of other generators. We now consider how an increase in $\bar{w}$ affects the aggregate equilibrium level of forward contracting undertaken by all generators. To do so, we examine numerical solutions to our model for selected parameter values.

We select initial parameter values to reflect elements of actual electricity markets. However, the ensuing analysis does not necessarily reflect activities in any particular market because our model does not capture all relevant elements of actual electricity markets. In particular, for analytic tractability, our model abstracts from the sharply rising marginal cost a generator effectively experiences as its output approaches capacity. Our model of Cournot competition also does not account for the activities of fringe generators and must-run generation (e.g., wind and cogeneration). Furthermore, we take all demand realizations to be equally likely, whereas extreme deviations from expected demand typically are relatively unlikely in practice.

With this caveat in mind, we proceed to establish baseline parameter values (Part B of the Appendix demonstrates that the key qualitative conclusions drawn below persist as baseline parameter values change). The wholesale price cap is initially taken to be 1,000 (so $\bar{w} = 1, 000$ ), reflecting its value in Alberta, Canada.³² We initially consider an expected wholesale price of 35 and assume that aggregate expected demand at this price is 8,500. Formally:

a^{I} - 35 b^{I} + \bar{Q} = 8, 500 .

(20)

This expected wholesale price approximates the $33.92 quantity-weighted average wholesale price in the eight major U.S. electricity hubs in 2020.³³ The identified aggregate expected demand reflects the 8,487.88 MWh average hourly electricity consumption in a U.S. state in 2020.³⁴

We initially take the ratio of expected residential electricity consumption to industrial electricity consumption at the expected wholesale price to be 0.65, reflecting the corresponding ratio in the U.S. in 2020.³⁵ Therefore:

\frac{\bar{Q}}{a^{I} - 35 b^{I}} = 0.65 .

(21)

Each generator’s marginal cost of production is initially assumed to be 25 (so $c_{i} = 25$ for $i = 1, \dots, n$ ), reflecting the $24.55 average cost of generating electricity in the U.S. in 2020 using natural gas.³⁶ We also initially assume there are five generators (so $n = 5$ ) to reflect a setting with a moderate level of competition among generators.

We initially set $b^{I} = 0.4$ and assume the maximum variation in residential demand is 50 percent of expected residential demand (i.e., $\bar{η} = \frac{1}{2} \bar{Q}$ ). This calibration helps to ensure the price cap binds for some, but not all, demand realizations, given the equilibrium levels of forward contracting in the model. Equations (20) and (21) imply that $a^{I} = 5, 165.52$ and $\bar{Q} = 3, 348.48$ (and so $\bar{η} = \frac{1}{2} \bar{Q} = 1, 674.24$ ) when $b^{I} = 0.4$ .³⁷

Table 1 summarizes these baseline parameter values.³⁸

Table 1.

Baseline Parameter Values.

$a^{I}$	$\bar{Q}$	$b^{I}$	$\bar{η}$	$c_{i}$	$n$	$\bar{w}$
5,165.52	3,348.48	0.4	1,674.24	25	5	1,000

The middle row of data in Table 2 presents equilibrium outcomes in our model for the baseline parameter values. The first and third rows of data report the corresponding outcomes when the price cap ( $\bar{w}$ ) is reduced and increased by 20 percent, respectively, holding all other parameter values at their baseline levels. Table 2 reports each generator’s level of forward contracting ( $F_{i}$ ), expected output ( $E {q_{i}}$ ), and expected profit ( $E {π_{i}^{G}}$ ). The table also reports the expected wholesale price ( $E {w}$ ) and the smallest realization of $ε$ for which the wholesale price cap binds ( $\hat{ε}$ ).³⁹

Table 2.

Equilibrium Outcomes as $\bar{w}$ Varies.

$\bar{w}$	$F_{i}$	$E {q_{i}}$	$E {π_{i}^{G}}$	$E {w}$	$\hat{ε}$
800	1,415.2	1,660.9	875,523	523.9	1,080.5
1,000	1,393.3	1,653.3	1,036,418	618.3	2,006.3
1,200	1,365.5	1,646.3	1,182,306	706.0	2,859.1

Table 2 reports that each generator reduces its equilibrium level of forward contracting as the wholesale price cap is increased. The reduced forward contracting and the higher price cap increase the expected wholesale price and expected generator profit, while reducing expected generator output.⁴⁰ The numerical solutions presented in Part B of the Appendix indicate that corresponding effects of an increased wholesale price cap arise as parameter values diverge from their baseline levels.

For the reasons explained above, the outcomes reported in Table 2 and in the Appendix do not imply that a higher wholesale price cap will always (or even often) reduce the levels of forward contracting preferred by generators in practice. However, the outcomes do suggest that this potential preference merits consideration when attempting to predict the effect of a change in the level of a wholesale price cap on industry forward contracting.

5. Conclusions

We have examined how a cap on the wholesale price of electricity affects incentives for forward contracting, explicitly accounting for the effects of forward contracting on short-term and long-term electricity prices. Our findings support the common wisdom that a price cap reduces a buyer’s incentive for forward contracting. When the maximum price it can face in the wholesale market declines, a buyer derives less benefit from securing electricity at the expected wholesale price rather than facing the actual wholesale price.

In contrast, a binding wholesale price cap can increase a generator’s incentive for forward contracting. Consequently, an increase in a prevailing price cap can reduce a generator’s incentive for forward contracting. This is the case in part because a higher price cap renders the cap less likely to bind, thereby increasing the likelihood that expanded forward contracting will reduce a generator’s profit by reducing the prevailing wholesale price. This regime shifting effect of an increase in $\bar{w}$ can induce generators to prefer reduced levels of forward contracting.

Our findings suggest that policymakers should consider the incentives of both buyers and generators when attempting to assess the likely impact of a change in the prevailing level of a wholesale price cap on industry forward contracting. The common wisdom that a higher price cap will enhance a buyer’s incentive for forward contracting does not necessarily extend to generators. Consequently, the ultimate impact of a higher price cap on industry forward contracting likely will depend in part on the details of the buyer-generator negotiations that determine the prevailing levels of forward contracting.

Future research should model these negotiations formally, allowing buyers and generators to bargain over both the number and the prices of forward contracts. To facilitate comparison with other studies in the literature, we adopted the common assumption that the price of a forward contract ( $p^{f}$ ) is the expected wholesale price of electricity ( $E {w (ε)}$ ). Negotiated prices would change the details of our analysis.⁴¹ However, such prices seem unlikely to alter our main qualitative findings because the key considerations that underlie our findings (e.g., forward contracts reduce the expected wholesale price and thereby reduce the likelihood that the wholesale price cap binds) persist when $p^{f}$ differs from $E {w (ε)}$ .⁴² Non-constant marginal costs and non-uniform distributions of demand uncertainty also seem unlikely to alter our primary qualitative conclusions because these changes similarly will not alter the key forces that underlie the conclusions.⁴³

A broader investigation of whether a binding wholesale price cap is likely to diminish generators’ incentives to sign forward contracts in practice would be valuable. We found that these reduced incentives can arise in one particular environment. Corresponding investigations across a broad spectrum of environments that prevail in practice would be valuable.

Future research might also analyze the effects of risk aversion. Buyer risk aversion introduces an additional consideration that runs counter to the conventional wisdom. A higher price cap can reduce the variance of a buyer’s profit by reducing the buyer’s profit margin when the price cap binds.⁴⁴ The lower variance, in turn, can diminish the risk-reducing benefit a risk averse buyer derives from forward contracting.⁴⁵ Consequently, in principle, if this effect were sufficiently pronounced, an increase in $\bar{w}$ could induce a sufficiently risk averse buyer to prefer a reduced level of forward contracting. Numerical solutions suggest this outcome is relatively unlikely.⁴⁶ However, this effect merits consideration in a comprehensive assessment of the impact of a wholesale price cap on incentives for forward contracting.⁴⁷

Footnotes

Appendix

Part A of this Appendix provides the proofs of the formal conclusions in the text. Part B presents additional numerical solutions. Part C considers an alternative rule for allocating $Δ (ε)$ .

Acknowledgements

We thank the editor and two anonymous referees for very helpful comments and suggestions. Support from the Government of Canada’s Canada First Research Excellence Fund under the Future Energy Systems Research Initiative and the Social Sciences and Humanities Research Council’s Canada Research Chair program is gratefully acknowledged.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

1

To illustrate, the prevailing cap on the wholesale price of electricity is $999.99 per megawatt hour (MWh) in Alberta, Canada (Brown and Olmstead 2017) and $5,000 per MWh in Texas (Smith 2022). A lower cap is imposed in Texas if electricity generators are deemed to have secured sufficient profit from an extended period of high wholesale prices ().

2

Wholesale price caps are also criticized because they limit the profit that generators secure during periods of particularly high demand, and can thereby reduce investment in generator capacity (Hogan 2017; ).

3

In part to facilitate entry by new generators, Australia requires large buyers of electricity to secure a significant portion of their anticipated demand via long-term forward contracts ().

4

show that expanded forward contracting endows a generator with a credible commitment to expand its output in the wholesale market.

5

Expanded forward contracting increases equilibrium output, which reduces the wholesale price when the price cap does not bind.

6

Holmberg (2011) analyzes a model in which firms choose forward contracts before competing via supply functions in the wholesale market. The author conjectures (p. 187) that “reducing price caps … would stimulate strategic [forward] contracting.” However, he does not investigate this issue formally. Yao, Oren and Adler (2007) develop an optimization program in which generators choose forward contracts and engage in Cournot competition. A wholesale price cap reduces the generators’ incentives for forward contracting in the numerical example the authors analyze.

7

We show that a buyer’s incentive for forward contracting depends in part on the nature of the retail price regulation it faces. This incentive typically declines as the regulated retail price tracks the expected wholesale price more closely.

8

Thus, we consider fixed-price, fixed-quantity financial forward contracts that are settled at the prevailing wholesale price. Such contracts are common in electricity markets.

9

If $w < p^{f}$ after $Gi$ and $B$ sign $F_{i}$ forward contracts, then $B$ pays $Gi$ the amount $F_{i} [p^{f} - w]$ . This payment ensures that $B$ ’s net cost of securing the $F_{i}$ units of electricity is $w F_{i} + F_{i} [p^{f} - w] = p^{f} F_{i}$ .

10

When the generators choose the levels of forward contracting, they do so simultaneously and non-collusively.

11

See Allaz and Vila (1993, Appendix A) for a formal proof of this conclusion in a setting where generators determine the levels of forward contracting in the first stage of the game. Holmberg and Willems (2015) explain why the assumption that forward markets are efficient in this sense often constitutes a reasonable caricature of electricity forward markets. The concluding discussion considers alternative processes for determining the prices of forward contracts. This discussion refers to model in which the price of a forward contract is determined by a form of bargaining between a buyer and a generator of electricity.

12

For example, as explained further below, the regulator might specify a unit price, $r$ , that retail customers must pay for electricity. In this case, $R (ε) = r [\bar{Q} + b^{I} ε]$ .

13

The values of $\bar{w}$ and $R (\cdot)$ are taken as given. The regulator’s choice of these values is not modeled formally.

14

The maintained assumption that $\hat{ε} \in (\underline{ε}, \bar{ε})$ rules out uninteresting settings in which the price cap never binds or binds for all realizations of $ε$ .

15

For any set of such outputs, no generator has an incentive to reduce its output unilaterally because $\frac{\partial π_{i}^{G}}{\partial q_{i}} > 0$ . Furthermore, no generator can increase its output unilaterally because, by construction, the inital set of outputs meet prevailing demand exactly.

16

This will be the case if $r_{0}$ is sufficiently large to ensure that $B$ ’s expected profit is nonnegative even in the absence of forward contracting.

17

For expositional ease, the ensuing analysis focuses on the case where $γ > 0$ , so the retail price does not track the expected wholesale price exactly. The case where $γ = 0$ is discussed in footnote 24 below.

18

The labels “deterministic” and “stochastic” should not be taken literally. Expected retail demand, $\bar{Q}$ , is not certain to arise, and it is the entire retail demand, $\bar{Q} + b^{I} ε$ , that is stochastic.

19

Equation (6) implies that $\frac{\partial E {w (ε)}}{\partial F_{i}} < 0$ . Therefore, equation (9) implies that $\frac{\partial E {r}}{\partial F_{i}} < 0$ and $| \frac{\partial E {r}}{\partial F_{i}} | = [1 - γ] | \frac{\partial E {w (ε)}}{\partial F_{i}} | < | \frac{\partial E {w (ε)}}{\partial F_{i}} |$ . Equation (6) also implies that $\frac{\partial w (ε)}{\partial F_{i}} = \frac{\partial w (ε)}{\partial F_{j}}$ for all $i, j \in {1, \dots, n}$ . This symmetry reflects the maintained assumptions that each generator operates with constant marginal cost and that the price of a forward contract is equal to the expected wholesale price. The symmetry underlies the conclusion that expanded forward contracting with a generator increases $B$ ’s expected profit at the same rate, regardless of the identity of the generator. Formerly, as reflected in , $\frac{\partial E {π^{B} (ε)}}{\partial F_{i}} = \frac{\partial E {π^{B} (ε)}}{\partial F_{j}}$ for all $i, j \in {1, \dots, n}$ .

20

Equation (6) implies that $| \frac{\partial w (ε)}{\partial F_{i}} | = \frac{b}{n + 1}$ when $ε \in [\underline{ε}, \hat{ε})$ . Equation (33) in the proof of Lemma 2 in the reports that $| \frac{\partial E {w (ε)}}{\partial F_{i}} | = \frac{b}{n + 1} [\frac{\hat{ε} + \bar{ε}}{2 \bar{ε}}] < \frac{b}{n + 1}$ . $| \frac{\partial E {w (ε)}}{\partial F_{i}} | < {\frac{\partial w (ε)}{\partial F_{i}} |}_{ε < \hat{ε}}$ because $w (ε)$ does not decline as $F_{i}$ increases when $ε \in (\hat{ε}, \bar{ε}]$ .

21

When stochastic demand is negative, demand is relatively small. The price cap does not bind for the smaller demand realizations.

22

When stochastic demand is positive, demand is relatively large. The price cap binds for the largest demand realizations.

23

From equation (9), $\frac{\partial E {r - w (ε)}}{\partial F_{i}} = [1 - γ] \frac{\partial E {w (ε)}}{\partial F_{i}} - \frac{\partial E {w (ε)}}{\partial F_{i}} = - γ \frac{\partial E {w (ε)}}{\partial F_{i}} > 0$ . The inequality holds because implies that $\frac{\partial E {w (ε)}}{\partial F_{i}} < 0 .$

24

The conclusions reported in Lemma 2 (and in Proposition 1 below) also hold when $γ = 0$ if is replaced by $r = R_{0} + γ r_{0} + [1 - γ] E {w (ε)}$ , where $R_{0} > 0$ is a lump-sum component of the retail price that is sufficiently large to ensure $B$ ’s expected profit is nonnegative even when $γ = 0$ . Lemma 2 implies that increased forward contracting reduces $B$ ’s expected profit in this setting (i.e., $\frac{\partial E {π^{B} (ε)}}{\partial F_{i}} < 0$ ), so $B$ prefers not to engage in forward contracting. Increased forward contracting reduces $B$ ’s expected profit when $γ = 0$ in part because $B$ secures no direct benefit from a lower expected wholesale price (i.e., $\frac{\partial E {π^{BD} (ε)}}{\partial F_{i}} = 0$ ) when the retail price declines at exactly the same rate the wholesale price declines as forward contracting increases.

25

Formally, suppose generators $G 1, \dots, Gm$ employ the least-cost technology and generators $Gm + 1, \dots, Gn$ employ different technologies (where $m < n$ ). Then the allocation rule would be $α_{i} = \frac{1}{m}$ for $i \in {1, \dots, m}$ and $α_{i} = 0$ for $i \in {m + 1, \dots, n}$ .

26

Equation (17) and equation (54) in the proof of Proposition 2 in the demonstrate that the revenue enhancement effect of an increase in $\bar{w}$ is $\frac{1}{2 \bar{ε}} [1 - α_{i}] [\bar{ε} - (n + 1) \bar{w} + a + C - b F]$ . This term is increasing in $a = b [a^{I} + \bar{Q}]$ and $C = C_{- i} + c_{i}$ .

27

$\frac{\partial \hat{ε}}{\partial \bar{w}} = n + 1$ from equation (5). Therefore, equation (17) and equation (53) in the proof of Proposition 2 in the imply that the regime switching effect of an increase in $\bar{w}$ is $\frac{1}{2 \bar{ε}} {[\bar{w} - c_{i}] [α_{i} (n + 1) - 2] - b F_{i}}$ . The magnitude of this effect increases with $α_{i}$ , holding $F_{i}$ constant.

28

The expressions in the two preceding footnotes reveal that an increase in $\bar{w}$ reduces the revenue enhancement effect at the rate $\frac{1}{2 \bar{ε}} [1 - α_{i}] [n + 1]$ and increases the regime shifting effect at the rate $\frac{1}{2 \bar{ε}} [α_{i} (n + 1) - 2]$ . The former rate exceeds the latter rate if and only if $[1 - α_{i}] [n + 1] > α_{i} [n + 1] - 2 \Leftrightarrow α_{i} < \frac{n + 3}{2 [n + 1]}$ .

29

$q_{i} (\hat{ε})$ increases at the same rate that $Δ (ε)$ declines as $F_{i}$ increases ( $\frac{\partial q_{i} (\hat{ε})}{\partial F_{i}} = - \frac{\partial Δ (ε)}{\partial F_{i}} = 1$ from equations (5) and (7)). Consequently, $q_{i} (\hat{ε})$ increases at nearly the same rate that $α_{i} Δ (ε)$ declines as $F_{i}$ increases when $α_{i}$ is close to 1. Therefore, implies that $lim_{α_{i} \to 1} \frac{\partial {\bar{q}}_{i} (ε)}{\partial F_{i}} = 0$ .

30

$\frac{1}{n} < \frac{2}{n + 1} \Leftrightarrow 2 n > n + 1 \Leftrightarrow n > 1 .$

31

It is readily shown that if $α_{i} = \frac{1}{n}$ for $i = 1, \dots, n$ but $c_{i}$ can vary across generators, then $\frac{\partial}{\partial \bar{w}} (\frac{\partial E {π_{i}^{G} (ε)}}{\partial F_{i}}) < 0$ if $\bar{w} > \frac{1}{n + 2} [a + \bar{ε} + C_{- i} + 2 c_{i}]$ .

32

As noted in the Introduction, the wholesale price cap in Alberta is $999.99 per MWh ().

33

U.S. Energy Information Administration ().

34

Total U.S. electricity consumption in 2020 was 3,717,674 thousand megawatthours. Dividing this number by the 8,760 hours in a year, multiplying by 1,000 to convert to MWhs, and dividing by the 50 U.S. states provides an average state hourly consumption of 8,487.84 MWhs ().

35

The ratio of electricity purchased by residential consumers to electricity purchased by commercial and industrial consumers in the U.S. in 2020 was $\frac{1, 464, 605}{1, 287, 440 + 959, 082} \approx 0.65$ ().

36

U.S. Energy Information Administration ().

37

$a^{I} - 35 b^{I} = 8, 500 - \bar{Q}$ , from . Therefore, $\bar{Q} = 0.65 [8, 500 - \bar{Q}] \Rightarrow \bar{Q} = 3, 348.48$ , from equation (21). Consequently, $a^{I} = 8, 500 - 3, 348.48 + 0.4 [35] = 5, 165.52$ , from equation (20).

38

These baseline parameter values imply that $\bar{ε} = \frac{\bar{η}}{b^{I}} = \frac{1, 674.24}{0.4} = 4, 185.6$ .

39

The entries in are rounded.

40

The relatively high expected wholesale prices reported in reflect the aforementioned special features of our model. A model that included a fringe of competitive generators, must-run generation, sharply rising marginal cost near capacity, and infrequent demand realizations well above expected demand would permit substantially lower expected wholesale prices while allowing the price cap to bind for some, but not all, demand realizations.

41

Anderson and Hu (2008) analyze a model in which a buyer proposes a level of forward contracting and associated compensation. The generator can either accept the buyer’s proposal or decline to engage in forward contracting altogether. The buyer’s bargaining power gives rise to an equilibrium price of a forward contract that exceeds the expected spot price of electricity. Ruddell, Downward and Philpott (2018) analyze a model in which buyers of electricity have demand curves for forward contracts with suppliers and for spot purchases of electricity. In this model, the equilibrium price of a forward contract exceeds the expected spot price of electricity. This price premium prevails because each buyer recognizes that signing a forward contract with a supplier will increase the supplier’s aggression in the ensuing Cournot competition. The increased aggression reduces the expected spot price of electricity at which the buyer will ultimately purchase a portion of its electricity consumption. The price premium persists even in the presence of speculators because each buyer values a forward contract with a supplier more highly than it values a forward contract with a speculator (since the latter contract does not induce increased aggression in the ultimate Cournot competition, and therefore does not directly reduce the expected spot price of electricity).

42

Some of our less central findings likely would change if the prices of forward contracts were determined by bargaining between buyers and sellers. For example, the equilibrium price of a forward contract might vary with the characteristics of the relevant buyer and seller (e.g., their size and market power). In addition, a buyer’s incentive for forward contracting might vary across generators. Furthermore, as in Ruddell, Downward and Philpott (2018), the equilibrium price of a forward contract might exceed the expected wholesale price, even in the presence of speculators.

43

To facilitate a tractable analysis, we have assumed that generators choose output levels. Models in which generators choose supply functions introduce considerable analytic complexity, including multiple equilibria (see Green 1999; Holmberg 2011; Holmberg and Willems 2015; Klemperer and Meyer 1989; Newbery 1998). identifies conditions under which a generator’s forward contracting induces its rival to shift its supply function inward. The author also notes that a more binding wholesale price cap can render the specified conditions more likely to arise. However, Holmberg (2011) does not undertake a systematic examination of the impacts of a wholesale price cap on incentives for forward contracting. Such an examination awaits future research.

44

examine how forward contracting affects the variance of buyer profit and generator profit in a model with no wholesale price cap.

45

In contrast, a higher price cap can increase a generator’s profit margin when the price cap binds, thereby increasing the variance of the generator’s profit. The increased variance can induce a risk averse generator to prefer a higher level of (risk-reducing) forward contracting.

46

We have extended our analysis to allow the buyer and the generators to have mean-variance preferences (e.g., Rolfo 1980; , 154–5). In the presence of such risk aversion, tractable analytic characterizations of equilibrium outcomes are difficult to derive. However, numerical solutions suggest that the buyer’s expected utility often increases systematically as its forward contracting increases for any price cap that binds for some, but not all, demand realizations. Thus, a risk averse buyer often prefers the highest feasible levels of forward contracting, regardless of the level of the price cap.

47

As noted above, a higher price cap can increase the variance of a generator’s profit and thereby enhance a risk averse generator’s incentive to undertake forward contracting. However, our numerical solutions reveal that when risk averse generators choose their preferred levels of forward contracting (non-cooperatively), a higher price cap can induce lower equilibrium levels of forward contracting, just as it can when generators are risk neutral.

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The Impact of Wholesale Price Caps on Forward Contracting

Abstract

Keywords

1. Introduction

2. Model Elements

3. The Buyer’s Preferred Level of Forward Contracting

4. Generators’ Preferred Levels of Forward Contracting

5. Conclusions

Footnotes

Appendix

Acknowledgements

Declaration of Conflicting Interests

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