# Strategic investment in electricity generation with renewable entrants: The case of French nuclear power

Abdel Mokhtari, Richard Ruble

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# Strategic investment in electricity generation with renewable entrants: The case of French nuclear power

Abdel Mokhtari, Richard Ruble

**Abstract:**

We study incentives to invest in electricity generation capacity if an incumbent using nuclear power competes with an endogenous number of entrants using intermittent renewable energy sources. The intermittence of renewables makes the incumbent less aggressive, and the incumbent accommodates if the efficiency difference between technologies is not too large. We analyze France's long-running subsidy scheme, the ARENH, through the prism of our model. This policy achieves its aim of making product market outcomes more competitive through an endogenous entry channel, but if investments in nuclear power are restarted then pursuing such a scheme would run the risk of facilitating deterrence.

**Keywords:**

Free entry, intermittence, Renewable energy, Stackelberg leadership

**JEL codes:**

D24, D61, L13, Q41
# Strategic investment in electricity generation with renewable entrants: The case of French nuclear power

Abdel Mokhtari* Richard Ruble†

December 10, 2024

## Abstract

We study incentives to invest in electricity generation capacity if an incumbent using nuclear power competes with an endogenous number of entrants using intermittent renewable energy sources. The intermittence of renewables makes the incumbent less aggressive, and the incumbent accommodates if the efficiency difference between technologies is not too large. We analyze France's long-running subsidy scheme, the ARENH, through the prism of our model. This policy achieves its aim of making product market outcomes more competitive through an endogenous entry channel, but if investments in nuclear power are restarted then pursuing such a scheme would run the risk of facilitating deterrence.

JEL Classification: D24, D61, L13, Q41

Keywords: Free entry, Intermittence, Renewable energy, Stackelberg leadership

<footnote>*emlyon business school, 144 avenue Jean Jaurès, 69007, Lyon, France. E-mail address: mokhtari@em-lyon.com.</footnote>

<footnote>†emlyon business school, CNRS, Université Lumière Lyon 2, Université Jean Monnet Saint-Etienne, GATE, 69007, Lyon, France. E-mail address: ruble@em-lyon.com. Tel +33 4 78 33 78 00. (corresponding author)</footnote>
A growing share of electricity worldwide is produced from renewable energy sources whose supply is abundant but intermittent, raising the question of how these energy sources can effectively rival established technologies with higher incremental cost but whose availability is more readily controlled. The technological asymmetry is compounded by institutional conditions in many countries whereby renewable producers compete with historical operators that are often former public monopolies and likely to wield residual market power. This is particularly true in France where the incumbent electricity producer Electricité de France (EDF) runs one of the world's largest fleets of nuclear power plants, compelling policy makers there to strike a balance between the twin objectives of allowing users to benefit from an efficient established technology with high irreversibility while promoting competition and the entry of renewable generation capacity. Such questions are moving to the forefront once again as French policy shifts towards restarting investments in nuclear capacity.<fnref n="1" />

To address these issues, we develop a stylized model of France's electricity sector. An incumbent relying on a “sticky” production technology faces potential entrants that use a renewable technology which generates electricity intermittently. These firms make sequential choices in a Stackelberg framework with free entry. This theoretical setup allows us to characterize investment incentives in a framework where an incumbent behaves strategically and entrants earn quasi-rents from electricity generation, in a market where the degree of competition is endogenous. We complete the model by incorporating a policy measure whereby a portion of incumbent capacity is made available to entrants at a regulated price, in line with France's current subsidy scheme.

Our analysis leads to a several main findings regarding investment incentives and policy implications. First of all, stickiness of the incumbent's technology implies that the intermittence of entrant output favors their penetration of the market, rationalizing observed renewable entry patterns. Intuitively, this shift towards accommodation arises because the incumbent is compelled otherwise to carry excessive capacity in states where entrants are inactive, making

<footnote n="1">¹This is according to the France 2030 plan recently announced by the country's economic ministry, see https://www.economie.gouv.fr/france-2030-plan-ambitieux-nucleaire-demain.</footnote>
deterrence more costly. Secondly, we establish that regulated electricity transfers under France's subsidy mechanism lead to more competitive outcomes all else equal, suggesting that this policy may have been a more reasonable measure for France to adopt than some economists originally feared. Intuitively, this result is due to the endogenous entry channel in our model, which implies that higher follower profit levels induce increased competition. Finally, France's policy has recently veered towards maintaining or expanding the share of nuclear power in the energy mix, and our model shows that in this new context of increasing investments in nuclear power it is likely wise not to prolong the policy of regulated purchases beyond its expiration in 2025. In addition we establish several secondary results. First, outcomes are more efficient if the incumbent's technology is sticky rather than flexible, and second comparing equilibrium outcomes with a second-best welfare benchmark suggests that a profit-maximizing incumbent actually underprovides deterrence in certain cases.

Our paper contributes to a research area studying strategic interactions in the power sector in the context of energy transition, of which Fabra (2021) gives a comprehensive overview highlighting the imperfectly competitive nature of wholesale electricity markets. Our work seeks to bring a complementary perspective to the competitive paradigm she describes, e.g. in Fabra and Llobet (2023), by representing price determination and the price-depressing effect of renewables synthetically and focusing instead on strategic investment incentives given long-run price levels. Our framework is perhaps closest to the model of Acemoglu et al. (2017), but in contrast with them we consider a single incumbent with whom renewable-only producers compete in quantities rather than as a competitive fringe.

To represent investment incentives consistently with key features of France's electricity sector, we build on the imperfectly competitive market structure pioneered by Etro (2006) and Etro (2008), and later applied to policy settings like mixed public and private oligopoly (Ino and Matsumura 2010). With regard to this stream of work, our paper can be seen as an extension of existing results regarding aggressive leader behavior in the face of free entry, to a setting where entrants have a seeming technological disadvantage insofar as their output is not readily
controllable. Our results also highlight a specific normative feature of this kind of market struc-
ture, whereby the endogenous level of competition among entrants may constitute an objective
for the regulator.

To model renewable energy production, we use the approach of developed by Ambec and Crampes (2012) and Ambec and Crampes (2021) in the case of perfectly competitive firms. These articles identify the efficient energy mix in a stylized setting with a deterministic fossil-based technology and a stochastic renewable technology, and demonstrate that this mix is achieved by a competitive industry facing reactive consumers and under real-time pricing (RTP). The results these authors obtain provide a valuable benchmark for our own results, which can be viewed as extending their work to a setting with imperfect competition. It bears mention that more recent research on intermittence has highlighted the interaction between intermittent production and storage technologies, finding that the two are not always complementary albeit for reasons other than those that we identify in our framework (Andres-Cerezo and Fabra 2024).

In addition to information available from public sources,<fnref n="2" /> the stylized features of the French electricity sector that we identify draw on recent surveys (e.g. Lebrouhi et al. 2022) and some more technical studies (Cany et al. 2018a, Cany et al. 2018b). The original reception given to France's subsidy scheme by prominent energy economists was guarded at best (Crampes et al. 2009). A more recent discussion of institutional features of the French market and their underlying economics can also be found in Percebois (2019). In addition, Creti et al. (2013) develop a model of short-run price determination under France's subsidy scheme suggesting positive effects of the regulation on market outcomes.

The rest of the paper is organized as follows. Section 1 gives an overview of the French electricity sector highlighting the specific features that motivate our model, and Section 2 lays out the assumptions we make in light of these features. Section 3 characterizes equilibrium in the absence of regulation and establishes our results regarding intermittence and accommodation. Section 4 develops our main insights with respect to regulation accounting for leader strategy

<footnote n="2">²Particularly from via the data portal provided by France's electricity network operator at
analysesetdonnees.rte-france.com.</footnote>
and the follower entry channel. We study the second-best welfare benchmark in Section 5, and round off the theoretical analysis with an example in Section 6. Section 7 concludes.

# 1 The electricity sector in France

France operates one of the world's largest fleets of nuclear power plants, comprising fifty-six second-generation nuclear reactors. A third-generation evolutionary power reactor (EPR) is scheduled to come online in summer 2024, with six additional reactors now being planned for 2035 and possibly eight more after that. The capacity and production data in Figure 1 illustrate the dominant role that nuclear power has played in the country's energy mix, covering about three-quarters of national electricity production historically. But against a backdrop of generally stable shares of nuclear, fossil-based and hydroelectric generation capacities, like many other countries France has also experienced a significant rise in non-hydroelectric renewable energy sources (NRES) over the last two decades, to the point where these now represent an unprecedented share of non-nuclear capacity and, to a lesser extent, production.

We have followed convention in grouping all NRES together, but most of this electricity is produced from two intermittent energy sources, wind and solar. According to France's electricity network operator Réseau de Transports de l'Electricité (RTE), these two sources accounted for 15% of France's electricity production in 2023, against 14% for other renewables (mainly hydroelectric, but also biofuels, geothermal, and waste) and 6% for fossil fuels, and they represent the fastest-growing of all these segments. The two panels of Figure 1 also show there is a very large gap between installed capacity and effective production for NRES. This gap reflects the overall impact of the intermittent availability of solar and wind-based electricity generation. This intermittence is commonly measured by the ratio or load factor, which has been estimated by RTE to be 14% for solar and 25% for wind for 2023.

For an electricity sector seeking to maintain stability objectives, such intermittence represents a technical challenge if NRES are to continue growing. This is because nuclear electricity production is generally viewed as inflexible, for both economic and technical reasons. Looking
<figure><img src="image_1.png" /><figcaption>(a) Installed capacity (GW)</figcaption></figure>

<figure><img src="image_2.png" /><figcaption>(b) Production (TWh)</figcaption></figure>

Figure 1: French mix of nuclear (black), non-hydroelectric renewable (dotted black), fossil (dotted grey), and hydroelectric (grey) electricity generation. (source: IEA)
<figure><img src="image_3.png" /><figcaption>(a) Week of Jan 18</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>(b) Week of Apr 26</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(c) Week of Aug 9</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(d) Week of Oct 4</figcaption></figure>

Figure 2: Percentage daily variation from weekly production of nuclear (black), wind (dark grey), and solar (light grey) electricity for four weeks in 2021. (source: RTF)
at the long run to begin with, nuclear capacity has a longer operational lifetime than other
generation technologies, as evidenced by the extended usage of France's current fleet which was
commissioned in the 1970s and 80s. In the short run, nuclear capacity arguably can and to some
extent is operated in load-following mode, but even so there are significant limitations to fully
flexible operations, particularly as the share of intermittent renewables increases (Cany et al.
2018a, Cany et al. 2018b). The intermittence of NRES is predictable at some frequencies (e.g.,
no solar electricity is produced at night), but exhibits considerable random daily variation. To
illustrate the relative stickiness of nuclear electricity generation compared with wind and solar,
Figure 2 plots daily output variations over four randomly chosen weeks in 2021. The percentage
daily variation is orders of magnitude greater for wind power in particular, and the absolute
variation remains significantly larger than for nuclear power.

Alongside other energy sources, France's historic electricity producer EDF has a monopoly
over nuclear power. Complementary activities, such as high voltage electricity transmission,
have been spun off into separate entities like RTE, of which EDF retains majority ownership.
Having been delisted from Euronext in 2023, EDF is now once again wholly owned by the French
government. In France, meeting CO2 emissions targets has been a less pressing motivation for
investing in renewables because the share of nuclear power implies electricity production is
already largely decarbonated. There are however other motivations for promoting renewable
entry into the sector, such as compliance with EU objectives to liberalize electricity markets.
Entrants now represent significant shares of different stages in the electricity supply chain. These
entrants include some vertically integrated firms with sizeable renewable energy operations such
as CNR, Engie, and TotalEnergies, along with foreign companies. As an indicator of the sector's
moderately fragmented market structure, in 2023 the top ten operators held sixty percent of the
country's twenty-two megawatts of wind generation capacity.<fnref n="3" />

Over the past fifteen years, the French government has broadly sought to allow users to
benefit from the low amortized cost of existing French nuclear power while promoting entry into

<footnote n="3">³france-renouvelables.fr.</footnote>
the electricity sector (Percebois 2019). The balance of these objectives is reflected in the 2010 NOME (“Nouvelle organisation du marché de l’électricité”) law which liberalized French electricity markets and included a specific provision, the ARENH (“accès au nucléaire historique”), whose intent is to facilitate entry. This provision originally stipulated that entrants can acquire up to a set amount of existing nuclear production (100 TWh) on a basis related to their existing client portfolios at a wholesale price of roughly forty euros per MWh which was intended to reflect the operating and maintenance costs of existing nuclear generation capacity. The primary rationale of this mechanism, up until it is phased out at the end of 2025, is to endow entrants with a “virtual” nuclear generation capacity allowing them to compete more effectively with the dominant incumbent. Looking ahead, the French energy commission recently revised its statutory estimate of nuclear energy costs upward to about sixty euros per MWh,<fnref n="4" /> and the French government has reached a tentative agreement with EDF to replace the ARENH starting in 2026 with a price cap measure of approximately seventy euros per MWh, so as to stabilize end user prices while allowing the company to fund planned investments in its future nuclear generation capacity.

On the demand side, electricity distribution has been generally deregulated in France leading to entry of a host of alternative providers. Roughly a third of France's electricity is consumed by households who subscribe to a variety of contracts, ranging from uniform pricing to variable-price contracts, such as the Tempo contract which half million households subscribe to. Ninety percent of households are equipped with a smart meters allowing real-time electricity pricing in principle, though adoption has so far been limited. Aside from some specific uses like transportation, the rest of consumption is split evenly between industrial and tertiary uses with heterogeneous contractual terms and demand patterns.

## 2 Assumptions

We model an electricity sector along the lines described in the preceding section.

<footnote n="4">⁴www.ecologie.gouv.fr.</footnote>
Because electricity is a homogeneous product, final (inverse) demand is a function of total output $Q$, which we denote by $P(Q)$. Total output per period depends on the output realization of intermittent producers, which is uncertain. We suppose that $P(Q)$ is decreasing and differentiable over its range, and that industry marginal revenue is decreasing. For a constant unit cost $c$, denote the output and the profit of a monopoly firm in such a market by $q_M(c)$ and $\pi_M(c)$ respectively.

There is a single incumbent electricity producer. The incumbent chooses an output strategy $q_L$ ahead of other firms, so we refer to this incumbent as the leader henceforth. The leader has constant unit cost $c_L$, with $c_L < P(0)$.

There is a large number of identical potential entrants. These firms decide independently whether or not to enter by paying a positive entry cost $K$. Those that do enter act as followers with respect to the leader. They engage in Cournot competition by choosing individual output strategies $q_F$ with respect to the residual demand $P(Q_F + q_L)$, where $Q_F$ denotes total follower output.

Followers have an intermittent production technology, so their effective level of output per period (e.g., on a given day) is a random variable. We model intermittence by supposing like Ambec and Crampes (2012) that there are two states of nature, $\bar{\omega}$ and $\underline{\omega}$. With positive probability $\nu$, state $\bar{\omega}$ occurs and follower output is $q_F > 0$, and with probability $1 - \nu$, state $\underline{\omega}$ occurs and follower output is zero. Followers have constant unit cost $c_F$, with $c_F/\nu < P(0)$. For the solar and wind technologies that we consider, marginal cost is deemed to be negligible so this unit cost can be viewed entirely as a capacity cost. We denote the total number of followers by $n_F$. As is standard in the entry literature we require that $n_F \in \{0\} \cup [1, \infty)$. In order for entry to be viable, we assume that $\nu\pi_M(c_F/\nu) > K$.

Throughout the model, we refer to the comparative efficiency of followers, which we denote by $\Delta_c = c_L - c_F/\nu$. Negative values of $\Delta_c$ indicate that the incumbent is more efficient than entrants, and if $\Delta_c = 0$ we say that both kinds of firms are equally efficient.

We assume that there is a competitive electricity retailing sector. Prices therefore adjust to
current electricity output. In determining current electricity supply as we do, we abstract away
from electricity production technologies other than those of the leader and followers, as well as
from storage technologies and foreign interconnections. Under these assumptions, the electricity
market operates with two state-dependent prices, $\bar{P} = P(q_L + Q_F)$ in state $\bar{\omega}$ and $\underline{P} = P(q_L)$ in state $\underline{\omega}$.

We start by examining a baseline version of our model with the following timing:

1) the incumbent chooses $q_L$;

2) potential entrants decide whether to enter;

3) followers choose $q_F$;

4) nature determines the state;

5) firms produce and outcomes are realized.

In addition to this baseline version, we also study an alternative game that reflects the entry
subsidy embodied by France's ARENH mechanism. In the alternative game, a set share $\alpha$ of
the leader's electricity output is available to entrants at a regulated tariff, which we denote by
$w$. Under the original ARENH mechanism, the regulated tariff is set to be no higher than the
incumbent's unit cost, i.e. $w \le c_L$. To reflect how this electricity is allocated on the basis of
existing client portfolios, we suppose entrants can purchase this electricity up to their capacity
$q_F$. ARENH purchases are not intended to be used for arbitrage profits in the spot market and
monitored to a certain extent, so in our model we restrict these purchases to occur in state $\underline{\omega}$. If
total demand for the leader's electricity exceeds the allotment $\alpha q_L$, we assume that it is rationed
proportionally to capacity shares.

Incorporating the ARENH regulation therefore adds a stage to the game, once firms have
chosen capacities and the state of nature is known, where followers decide how much electricity
to purchase. The timing in this alternative model is as above, except for the additional electricity
purchase stage:

1) - 4) as above;

5) followers determine electricity purchases $\tilde{q}_F$;

6) firms produce and outcomes are realized.

We next turn to the analysis of the baseline case.

## 3 Entry and intermittence

We begin our analysis by characterizing equilibrium investment in a baseline scenario where followers can enter freely but have no regulated access to the leader's output. We reason by backward induction to find the equilibrium, and therefore start with the follower problem.

After entering, followers compete in quantities given the leader's output. The net profit of an individual follower i consists of expected revenue net of cost,

$$ \pi_F (q_i, Q_{-i}) = \nu P (q_i + Q_{-i}) q_i - c_F q_i - K, \quad (1) $$

where $Q_{-i} = q_L + \sum_{j \neq i} q_j$ is the aggregate output of all rival firms.

Because industry marginal revenue is decreasing (by assumption), the quantity competition stage has a unique and symmetric equilibrium. Moreover if the leader's output is not prohibitively large, entry occurs and this equilibrium is interior.

If the equilibrium is interior, each follower chooses a positive output $\hat{q}_F$ which solves the symmetric first-order condition

$$ P(\hat{q}_F + \hat{Q}_{-F}) + \hat{q}_F P'(\hat{q}_F + \hat{Q}_{-F}) = \frac{c_F}{\nu}. \quad (2) $$

At the entry stage, substituting the first-order condition into the net profit expression gives $\pi_F (\hat{q}_F, \hat{Q}_{-F}) = -\nu P' (\hat{q}_F + \hat{Q}_{-F}) \hat{q}_F^2 - K$. The number of firms adjusts so net profit is zero
in a free entry equilibrium, implying

$$-P'(\hat{q}_F + \hat{Q}_{-F}) \hat{q}_F^2 = \frac{K}{\nu}. \qquad (3)$$

Equations 2 and 3 jointly determine the follower output $\hat{q}_F$ and corresponding rival output level $\hat{Q}_{-F}$. The assumptions $P(0) > c_F/\nu$ and $\nu\pi_M(c_F/\nu) > K/\nu$ ensure that this system of equations has a positive solution. These equations also indicate that the main effect of introducing intermittence at the follower stage is simply to scale up the absolute cost parameters $c_F$ and $K$ by a factor $1/\nu$, i.e. to their effective per MWh levels.

If the equilibrium is interior, the solution to equations 2 and 3 and therefore the total output $\hat{q}_F + \hat{Q}_{-F}$ are independent of leader output. Where the leader output does play a role is in determining whether or not there is an interior solution to begin with. More precisely, by choosing capacity $q_L^D = \hat{Q}_{-F}$, the leader ensures that the symmetric equilibrium at the follower stage involves no entry.

We next turn to the leader's problem. The leader's net profit is defined piecewise, depending upon whether its output level induces an equilibrium with entry or not. If there is entry, then the leader's profit incorporates the expected revenue over the two states, i.e. those where followers operate so total output is $\hat{Q}_{-F} + \hat{q}_F$ as determined by equations 2 and 3, and where they do not. Otherwise, the leader's net profit is just its monopoly profit. Formally,<fnref n="5" />

$$\pi_L(q_L) = \begin{cases} \nu P(\hat{Q}_{-F} + \hat{q}_F) q_L + (1-\nu) P(q_L) q_L - c_L q_L - K, & \text{if } q_L < q_L^D \\ (P(q_L) - c_L) q_L - K, & \text{if } q_L \ge q_L^D. \end{cases} \qquad (4)$$

Each piece of the leader's net profit is a quasiconcave function, and there is an upward jump at $\hat{Q}_{-F}$ amounting to $\nu(P(\hat{Q}_{-F}) - P(\hat{Q}_{-F} + \hat{q}_F))\hat{Q}_{-F}$. The first piece of the profit function is nonlinear because of intermittence. Without intermittence, it would be linear and have an extremum at either 0 or $q_L^D$. Intermittence therefore allows the leader's profit function to have

<footnote n="5">⁵We include the leader's entry cost here for consistency with followers, but it plays no further role in the analysis.</footnote>
an interior maximum on the first piece where the leader accommodates follower entry, and which we denote $q_L^A$.

The following proposition characterizes equilibrium leader behavior with respect to comparative efficiency.

**Proposition 1 (Equilibrium with free entry and intermittence)** If the entry cost is not too large there is a unique Stackelberg equilibrium with free entry and there exist comparative costs $\Delta_0 < \Delta_1 < \Delta_2$ such that

$$q_L^* = \begin{cases} q_M, & \text{if } \Delta_c \le \Delta_0 \\ q_L^D, & \text{if } \Delta_0 < \Delta_c \le \Delta_1 \\ q_L^A, & \text{if } \Delta_1 < \Delta_c < \Delta_2 \\ 0, & \text{if } \Delta_c \ge \Delta_2. \end{cases}$$

In addition $0 \in (\Delta_1, \Delta_2)$ so the leader accommodates if the efficiency difference is not too large.

**Proof.** See Appendix 8.1.

Proposition 1 describes how the spectrum of equilibrium outcomes relates to the comparative efficiency of the two types of firms in our model, as illustrated in Figure 3. It should first of all be noted that the critical levels $\Delta_0$, $\Delta_1$, and $\Delta_2$ are themselves functions of $c_F$ so the terms on either side of the inequalities are not entirely independent, though these critical levels are independent of $c_L$. The main insight does not lie so much in these levels however, as in establishing that accommodation arises at all. The possibility of an accommodation outcome is noteworthy here because without intermittence, the accommodation region vanishes and the leader either blockades, deters or shuts down (with deterrence arising in the equal efficiency case). Our finding that the leader accommodates entrants with intermittent production and similar efficiency therefore contrasts sharply with the situation without intermittence (see Proposition 4 of Etro 2008), where entrants with a deterministic technology are shut out in equilibrium unless they are significantly more efficient than the incumbent. In addition, Proposition 1 also informs with regards to how imperfect competition can influence the energy mix. In a perfectly
<figure><img src="image_7.png" /><figcaption>Figure 3: By proposition 1, as follower relative efficiency $\Delta_c$ increases, the leader shifts from deterrence to accommodation, with accommodation arising in the equal efficiency case.</figcaption></figure>

competitive setting, Ambec and Crampes (2012) show that with similar enough efficiencies fossil and renewable technologies operate jointly. Such a mix of energy sources also arises in our model, but if it does this happens is as a result of the leader's strategic behavior. We return to this aspect further below in our discussion of welfare in Section 5.

Our model therefore shows that intermittence leads to accommodation and a mix of energy sources for similar enough efficiencies. A natural follow-up question is to ask if introducing more intermittence in an appropriately defined sense has the same effect. To this end, consider decreasing the availability of the intermittent energy source $v$ while holding its effective cost $c_F/v$ constant. In practical terms this exercise amounts to comparing how different types of producers, e.g. solar and wind, enter in equilibrium under the assumption that their costs per megawatt hour are similar. Intuitively, increasing intermittence has the same effect on the follower equilibrium as raising the entry cost (see equations 2 and 3). Greater intermittence therefore leads to less entry and competition, which lowers equilibrium output in state $\bar{\omega}$. That is, holding $c_F/v$ constant, $d\hat{Q}_F/dv > 0$, which facilitates deterrence by lowering $q_L^D$, but also increases the leader's expected monopoly profit, reducing the leader's incentive to deter. The following corollary establishes that it is the latter of these two effects that dominates.

**Corollary 2 (Increasing intermittence)** If the entry cost is not too large, the leader is more likely to accommodate energy sources that are more intermittent, i.e. with a lower load factor.

**Proof.** See Appendix 8.2.

We can compare the outcome described in Proposition 1 with a similar industry where there is free entry but the incumbent does not have the possibility of moving first. In terms of
interpretation, such a comparison amounts to seeing how allowing the historical operator behave strategically as a private profit-maximizing firm compares with a more level playing field with new entrants where the incumbent does not have a sticky technology. Intuitively, the comparison of the sequential and simultaneous output choice situations runs as follows. First observe that as long as some entry occurs, equations 2 and 3 imply that total electricity output is the same in state $\bar{\omega}$ in both situations. In addition to this, free entry implies that entrants earn zero profit. From a welfare standpoint therefore, the difference between the scenarios with simultaneous and sequential moves hinges on the comparison of just two welfare components: the incumbent's profit and the level of consumer surplus in state $\omega$. By a revealed preference argument, the incumbent obtains at least as much profit under sequential moves as under simultaneous moves. In addition, the incumbent chooses a higher output as a leader so consumer surplus in state $\omega$ is higher. We can therefore conclude that welfare is higher if the incumbent exercises market leadership. Note that this reasoning holds even if the leader has higher costs (positive $\Delta_c$), so the relative efficiency of the Stackelberg outcome is not as much of a foregone conclusion as it may first seem.

**Corollary 3 (Leadership and welfare)** *Assuming entry occurs, welfare is higher if the incumbent acts as a leader than with simultaneous moves.*

**Proof.** See Appendix 8.3.

# 4 Subsidizing entry

In this section we introduce the possibility that followers purchase a set amount of electricity from the incumbent at a regulated price, in line with France's ARENH policy. These purchases lead to transfers of electricity from the incumbent to entrants, so total output and the final price in each state are unaffected all else equal. Followers prefer to purchase and resell the incumbent's output in state $\omega$ where the final price and margin are the highest. In the long-run, the additional profits that followers obtain encourage entry, so we expect total output to
increase in state $\bar{\omega}$ leading to a lower price, i.e. the policy should lead to more competitive market outcomes via the follower entry channel.

To verify this intuition, first observe that the follower net profit with regulated purchases is

$$ \pi_F(\tilde{q}_i, q_i, Q_{-i}) = \nu P(q_i + Q_{-i}) q_i + (1 - \nu) P(q_L) \tilde{q}_i - c_F q_i - (1 - \nu) w \tilde{q}_i \quad (5) $$

where $\tilde{q}_i$ is the amount purchased at the regulated wholesale price $w$. In any equilibrium where the leader actively produces, it must be that $P(q_L) > c_L \ge w$. It is therefore optimal for a follower to purchase as much as possible up to their allotment. If regulated purchases do not exhaust the allotment, i.e. in a candidate equilibrium with $\hat{Q}_F \le \alpha q_L$ where $\hat{Q}_F = \hat{Q}_{-F} + q_L - \hat{q}_F$ denotes total follower output, the reduced form net follower profit is

$$ \pi_F(q_i, Q_{-i}) = \nu \left( P(q_i + Q_{-i}) - \frac{c_F}{\nu} \right) q_i + (1 - \nu) (P(q_L) - w) q_i. \quad (6) $$

We focus for now on the case of unconstrained purchases and discuss the case of constrained purchases at the end of the section.<fnref n="6" />

Suppose tentatively therefore that regulated purchases are unconstrained in equilibrium ($\hat{Q}_F \le \alpha q_L$). The follower first-order condition is

$$ \nu \left( P(\hat{q}_F + \hat{Q}_{-F}) + \hat{q}_F P'(\hat{q}_F + \hat{Q}_{-F}) - \frac{c_F}{\nu} \right) + (1 - \nu) (P(q_L) - w) = 0. \quad (7) $$

In equation 7 the first set of terms is the first-order condition from the preceding section, scaled by the state probability $\nu$. The second set of terms is the additional expected follower marginal revenue from ARENH purchases from exercising the regulated purchase option in state $\omega$. Using the first-order condition to substitute for $\hat{q}_F$ yields the free-entry condition, which is identical to equation 3. Intuitively, the effect of regulated purchases at the follower stage is the same as the effect of a reduction in $c_F$. Followers therefore choose quantities more aggressively in equilibrium,

<footnote n="6">⁶In addition we verify in Section 6 that the unconstrained case arises as an equilibrium if the leader and followers are equally efficient.</footnote>
and it is straightforward to check that total output increases as a result of introducing regulated purchases, and that output increases all the more the lower is the regulated tariff $w$.

The case of constrained follower purchases is more involved because of an additional market share effect as we show further below, because followers have an additional incentive to raise capacity so as to increase their allotment of the incumbent's output on the margin.<fnref n="7" /> It is straightforward to check though that the overall effect of regulated purchases is qualitatively the same.

To sum up,

**Proposition 4 (Regulated purchases and competition)** For given $q_L$, introducing regulated purchases leads to higher output and a lower price. A lower transfer price $w$ leads to a more competitive outcome.

**Proof.** See Appendix 8.4.

For most of the past decade, the stated policy of successive French governments had been to progressively phase out nuclear power by shutting down nuclear plants as they aged (a notorious example was the Fessenheim plant near France's border with Germany, which was shut down in 2020).<fnref n="8" /> Proposition 4 establishes that in this context, i.e. ruling out any increase in nuclear generation capacity, within our framework the ARENH policy achieves its aims of making electricity markets more competitive.

Taking a more long-run perspective however, regulated purchases also have implications for leader behavior which can result in very different equilibrium outcomes. Observe first that the leader's capacity enters the follower first-order condition equation 7 in the same way as the transfer price does, so a direct implication of Proposition 4 is that higher leader capacity lowers total output. Regulated purchases therefore provide the leader with a channel through which to depress follower profitability and discourage entry, by lowering the margin that followers obtain from regulated sales. More broadly, the profitable sales lost through the transfer to followers as

<footnote n="7">⁷Creti et al. (2013) find the same effect in their model.</footnote>

<footnote n="8">⁸This was one of the provisions of a 2015 energy transition law, see www.ecologie.gouv.fr/loi-transition-energetique-croissance-verte.</footnote>
well as the increased competitiveness of market outcomes raise the attractiveness of deterrence for the leader, but deterrence is also costlier because regulated purchases make entry more attractive for followers.

To assess the overall effect of regulated purchases on equilibrium, we again take the limiting competitive fringe case by letting $K$ approach zero. With unconstrained purchases, equation 7 implies that the deterrence output with regulated purchases is $q_L^{D'} = P^{-1}(c_F + (1 - \nu)w)$. This output is lower than the deterrence output $q_L^D$ obtained in the preceding section if $c_F + (1 - \nu)w > c_F/\nu$, i.e. if $w > c_F/\nu$. In the case of constrained purchases, followers cannot use regulated purchases as extensively as they would like and equation 10 implies that the deterrence output is somewhat lower, i.e. $q_L^{D'} = P^{-1}(c_F + \alpha(1 - \nu)w)$.

Continuing under the assumption that regulated purchases are unconstrained, denote total capacity by $\hat{Q}$. The leader therefore sells $\hat{Q} - q_L$ to followers at the regulated price $w$, and $2q_L - \hat{Q}$ directly. The net payoff of the leader is then

$$ \pi_L (q_L) = \begin{cases} \nu P (\hat{Q}) q_L + (1 - \nu) (P(q_L) (2q_L - \hat{Q}) + w (\hat{Q} - q_L)) - c_L q_L - K, & \text{if } q_L < q_L^{D'} \\ (P(q_L) - c_L) q_L - K, & \text{if } q_L \ge q_L^{D'} \end{cases} \quad (8) $$

where $q_L^{D'} = \hat{Q}_{-F}$ solves equations 3 and 7.

Reasoning similar to the preceding section establishes:

**Proposition 5 (Regulated purchases and leader investment)** For small enough entry cost and high enough allotment, with a policy of regulated sales at $w = c_L$ entry is deterred if $\Delta_c \le 0$. A lower transfer price $w$ raises the likelihood of deterrence.

**Proof.** See Appendix 8.4.
## 4.1 Constrained purchases

For completeness we round off the analysis of this section by presenting the main elements of the constrained purchases case ($\hat{Q}_F > \alpha q_L$).

To begin with, if purchases exceed the aggregate allotment we assume they are rationed among follower firms according to their capacity shares, so that

$$ \pi_F (q_i, Q_{-i}) = \nu \left( P(q_i + Q_{-i}) - \frac{c_F}{\nu} \right) q_i + (1 - \nu) (P(q_L) - w) \frac{q_i}{q_i + Q_{-i}} \alpha q_L. \quad (9) $$

Starting with the follower stage, with constrained regulated purchases the follower first-order condition is

$$ \nu \left( P(\hat{q}_F + \hat{Q}_{-F}) + \hat{q}_F P'(\hat{q}_F + \hat{Q}_{-F}) - \frac{c_F}{\nu} \right) + (1 - \nu) (P(q_L) - w) \frac{\hat{Q}_{-F}}{(\hat{q}_F + \hat{Q}_{-F})^2} \alpha q_L = 0. \quad (10) $$

Like in the unconstrained case regulated purchases raise marginal revenue, but they do so here to a lesser extent. This follower first-order condition implies a more involved zero-profit condition,

$$ -P' (\hat{q}_F + \hat{Q}_{-F}) \hat{q}_F^2 + \frac{1 - \nu}{\nu} (P(q_L) - w) \frac{\hat{q}_F^2}{(\hat{q}_F + \hat{Q}_{-F})^2} \alpha q_L = \frac{K}{\nu}. \quad (11) $$

Finally, the leader's net profit expression is analogous to equation 8,

$$ \pi_L (q_L) = \begin{cases} \nu P (\hat{Q}) q_L + (1 - \nu) ((1 - \alpha) P (q_L) + \alpha w) q_L - c_L q_L - K, & \text{if } q_L < q_L^{D'} \\ (P (q_L) - c_L) q_L - K, & \text{if } q_L \ge q_L^{D'}. \end{cases} \quad (12) $$

Although conditions for an equilibrium could be developed further, the main insight from the constrained purchases case does not come from characterizing the leader's problem further. Rather, consider the limiting case of very small allotments. As $\alpha$ goes to zero, the leader's problem just approaches the unregulated case characterized in Proposition 1. A sufficiently moderate policy therefore would lead to the same spectrum of equilibrium outcomes as in the
unregulated case.

## 5 Welfare objective

As stated in our overview of Section 1, the historic French operator EDF was re-nationalized in 2023 after operating as a listed firm for nearly twenty years. Up to now we have taken its (former) privatized status at face value and assumed an objective of profit-maximization. Looking to the future, it is all the more reasonable to consider other objectives, and in particular welfare maximization.

With welfare maximization it is the social costs of the electricity generation technologies we consider that are relevant. However, none of the technologies that we study have sizeable carbon emissions. This is not to say that these technologies might not exert other externalities, like unaccounted environmental and safety risks or nuisance costs. But absent compelling evidence of the relative magnitude of externalities exerted by the leader and follower technologies, we prefer to remain agnostic and suppose that $c_L$, $c_F$, and $K$ measure actual social costs.

We focus on second-best welfare, i.e. where the regulator sets the incumbent's output given the entry behavior of followers. Under our assumptions, social welfare is

$$ W(q_L) = \begin{cases} \nu \int_0^{\hat{q}_F + \hat{Q}_{-F}} (P(s) - P(\hat{q}_F + \hat{Q}_{-F})) ds + \nu (P(\hat{q}_F + \hat{Q}_{-F}) - c_L) q_L & \text{if } q_L < \hat{Q}_{-F} \\ +(1 - \nu) \int_0^{q_L} (P(s) - c_L) ds - K, & \text{if } q_L \ge \hat{Q}_{-F} \\ q_L (P(s) - c_L) ds - K, & \text{if } q_L \ge \hat{Q}_{-F} \end{cases} \tag{13} $$

In equation 13, the first line uses the follower zero-profit condition to express welfare in state $\bar{\omega}$ as the sum of the consumer surplus and leader profit, and the second line is the sum of consumer surplus and leader profit in state $\omega$.

If the leader accommodates, differentiating the first piece establishes that the optimal capacity $q_L^A$ solves $\nu P(\hat{q}_F + \hat{Q} - F) + (1 - \nu)P(q_L) = c_L$. Then,
**Proposition 6 (Second-best welfare)** If the entry cost is not too large, a welfare-maximizing incumbent sets capacity $q_L^D$ or $q_M$ if $\Delta_c \le 0$, and $q_L^A$ if $\Delta_c > 0$.

**Proof.** See Appendix 8.6.

If we compare the leader's behavior in a welfare second-best with Proposition 1, the industry outcome is the same for comparative costs below $\Delta_1$ insofar as no entry occurs. For comparative costs above zero there is accommodation in both cases, though the leader's output with the welfare-maximization objective is larger. Finally the outcomes diverge if comparative costs lie in $(\Delta_1, 0]$. In this last case, the leader's slight cost advantage implies that it should deter follower entry, and a profit-maximizing leader strays from the welfare optimum by underproviding deterrence.

# 6 Example

Many of the arguments made so far rely on continuity around the limiting case of a competitive fringe ($K = 0$). In this section we complement this local analysis by positing a demand specification which yields explicit solutions, so that we can gauge the predictions of the model more broadly.

Suppose that inverse demand is given by

$$P(Q) = 1 - Q. \tag{14}$$

The cost parameters must then satisfy $c_L < 1$, $c_F < \nu$, and $K/\nu < (1 - c_F/\nu)^2/4$.

We first verify that the equilibrium characterization in Proposition 1 extends to entry costs within the admissible parameter range. Starting with the follower decision stages, solving equations 2 and 3 yields

$$\hat{q}_F = \sqrt{\frac{K}{\nu}}, \hat{Q}_{-F} = 1 - \frac{c_F}{\nu} - 2\sqrt{\frac{K}{\nu}} \tag{15}$$

provided that there is entry, i.e. $q_L < \hat{Q}_{-F}$. Next we turn to the leader's decision. The leader's
net payoff is

$$ \pi_L (q_L) = \begin{cases} \left( (1-\nu)(1-\frac{c_F}{\nu}) + \nu\sqrt{\frac{K}{\nu}} - (1-\nu)q_L - (c_L - \frac{c_F}{\nu}) \right) q_L - K, & \text{if } q_L < 1 - \frac{c_F}{\nu} - 2\sqrt{\frac{K}{\nu}} \\ (1-q_L - c_L)q_L - K, & \text{if } q_L \ge 1 - \frac{c_F}{\nu} - 2\sqrt{\frac{K}{\nu}}. \end{cases} \tag{16} $$

The first piece has a stationary point $q_L^A$.<fnref n="9" /> If this stationary point is interior, accommodation is feasible and yields a profit of $(1 - \nu)(q_L^A)^2$. If $q_L^A \le 0$ then the leader prefers to shut down rather than accommodate. The upper cutoff for accommodation with respect to comparative cost is therefore

$$ \Delta_c \ge (1 - \nu) \left(1 - \frac{c_F}{\nu}\right) + \nu \sqrt{\frac{K}{\nu}}. \tag{17} $$

Conversely, the leader produces the monopoly output rather than deterring actively if $q_M \ge \hat{Q}_F$, which holds if

$$ \Delta_c \le -\left(1 - \frac{c_F}{\nu}\right) + 4\sqrt{\frac{K}{\nu}}. \tag{18} $$

The right-hand sides of equations 17 and 18 generalize the critical thresholds $\Delta_2$ and $\Delta_0$ respectively in Proposition 1 to positive levels of the entry cost $K$. The remaining critical threshold $\Delta_1$ is obtained by equating the profits from accommodation and deterrence, i.e. setting

$$ (1 - \nu) (q_L^A)^2 = \left( 2\sqrt{\frac{K}{\nu}} + \frac{c_F}{\nu} - c_L \right) \left( 1 - \frac{c_F}{\nu} - 2\sqrt{\frac{K}{\nu}} \right). \tag{19} $$

Substituting for $q_L^A$ and solving for $\Delta_c$ yields that $\Delta_1$ is the upper root of a quadratic expression $P(x)$.<fnref n="10" /> With respect to the intermittence effect described in Corollary 2, it is straightforward to check that $\Delta_2$ is increasing in intermittence (i.e. decreasing in $\nu$, holding $c_F/\nu$ constant) but that this upper root $\Delta_1$ is increasing only if $\sqrt{K/\nu} < (1 - c_F/\nu)/4$. For higher entry costs it is negative, implying that the effect of greater intermittence is non-monotonic in general.

<footnote n="9">⁹Specifically, $q_L^A = \frac{(1-\nu)(1-\frac{c_F}{\nu})+\nu\sqrt{\frac{K}{\nu}}-(c_L-\frac{c_F}{\nu})}{2(1-\nu)}$.</footnote>

<footnote n="10">¹⁰This quadratic expression is $P(x) = x^2 + 2 \left( (1 - \nu)(1 - \frac{c_F}{\nu}) - (4 - 3\nu)\sqrt{\frac{K}{\nu}} \right) x + (1 - \nu)^2 \left( 1 - \frac{c_F}{\nu} \right)^2 - 2(v^2 - 5v + 4) \left( 1 - \frac{c_F}{\nu} \right) \sqrt{\frac{K}{\nu}} + (v^2 - 16v + 16) \frac{K}{\nu}$.</footnote>
Figure 4 illustrates equilibrium outcomes with respect to the entry cost K and state probability $\nu$ assuming the effective follower cost $c_F/\nu$ is constant. The figure illustrates two cases, that of an equally efficient leader and followers in black, and a case where the leader has a small cost advantage in grey. Beyond $K = \nu/16$, entry in unprofitable even for a monopoly follower. Up to this line, the figure successively plots the boundaries between accommodation, deterrence, and monopoly (or blockade) outcomes. In both the equal efficiency and small cost advantage cases, the accommodation region is non-empty for $\nu \in (0, 1)$, in line with Proposition 1. Confering a cost advantage on the leader predictably shrinks the accommodation region. Because $c_F/\nu$ is held constant, we can visualize the effect of increasing intermittence. In this regard the figure shows that intermittence has a non-monotonic effect leading first to accommodation but subsequently to further deterrence and blockade, as deterrence becomes relatively easier for the leader for small enough values $\nu$.

Next we turn to the analysis of regulated purchases. In the unconstrained case, solving equations 7 and 3 yields

$$ \hat{q}_F = \sqrt{\frac{K}{\nu}}, \hat{Q}_{-F} = \frac{1}{\nu} \left( 1 - \nu \sqrt{\frac{K}{\nu}} - c_F - (1 - \nu)w - (1 - \nu)q_L \right). \quad (20) $$

It follows directly that total output $\hat{Q}_{-F} + \hat{q}_F$ is decreasing in $w$.

The leader's deterrence output follows by solving $q_L^D = \hat{Q}_{-F}$, which yields $q_L^D = 1 - \nu\sqrt{K/\nu} - c_F - (1 - \nu)w$. To verify that the leader deters, we determine the payoff under accommodation and verify that followers are unconstrained. Differentiating the first piece of the leader payoff (equation 8) yields a value for $q_L^A$.<fnref n="11" /> The accommodation output is consistent with unconstrained follower purchases if $\hat{Q}_F \le \alpha q_L^A$.<fnref n="12" /> In the equal efficiency case, we have $q_L^A = 1 - c_F - (1 - \nu)w >$

$$ \alpha \ge \frac{1 - c_F - (1 - \nu)w - (1 - \nu)q_L^A}{\nu q_L^A}. $$

The right hand side is equal to one in the equal efficiency case, and lower than one for $\Delta_c < 0$.

<footnote n="11">¹¹Specifically, $q_L^A = 1 - c_F - (1 - \nu)w - \frac{1}{2}\frac{\nu}{1-\nu}(c_L - \frac{c_F}{\nu})$. Given that $w \le c_L$, a sufficient condition for $q_L^A$ to be positive is if $1 - \frac{c_F}{\nu} - \frac{2-\nu}{\nu(1-\nu)}(c_L - \frac{c_F}{\nu}) > 0$, and this condition is met in the equal efficiency case.</footnote>

<footnote n="12">¹²From equation 7, $\hat{Q}_F = (1 - c_F - (1 - \nu)w - (1 - \nu)q_L)/\nu$. The condition for unconstrained purchases is met therefore if</footnote>
<figure><img src="image_8.png" /><figcaption>Figure 4: Equilibrium outcomes for equal efficiency ($\Delta_c = 0$, in black) and small leader cost advantage ($\Delta_c = -1/8$, in grey) in terms of $\nu$ and $K$, for $c_F/\nu = 1/2$.</figcaption></figure>
$q_L^D$ implying that the leader deters for all admissible $K$, which extends the result in Proposition 6.

# 7 Conclusion

In this article, we have developed a stylized model of France's electricity sector so as to highlight several consequences of its specific structure, involving an incumbent producing a significant amount of nuclear power facing a wave of entry from renewable energy producers, and its institutional features, particularly a policy of regulated electricity transfers to entrants. Our analysis has shown first of all how the intermittence of renewables production promotes accommodation by the incumbent, as the combination of entrant intermittence and a sticky incumbent technology makes deterrence more costly by forcing the incumbent to carry excessive capacity in states where renewable producers are idle. From a policy perspective, our model lends support the idea that France's ARENH policy has made electricity market outcomes more competitive given the context of constant or decreasing nuclear capacity in past decades. This effect arises because of an endogenous entry channel which is specific to our model. Our analysis also suggests that pursuing such policies in the future if France sets itself on a course for further nuclear generation capacity investments would provide an incentive for deterrence and a less balanced energy mix.

Our analysis rests on several simplifying assumptions to which we now return. To begin with, we posited a strategic interaction involving sequential output choices and free entry which seems appropriate to us for the French context and which delivers crisp results regarding accommodation and deterrence. If the number of entrants were fixed or limited, a sequential choice model would still be appropriate and would arguably yield similar insights, without being as informative with respect to the level of competition among followers which in our model is endogenous. Another important assumption has to do with underlying technologies. To begin with, we simplify the analysis considerably by restricting attention to just two types of electricity production, with firms entirely specialized in one technology or the other. Some producers operate a mix of generation technologies in fact, which could have an impact on how they compete.
At the same time, the historic stability of non-NRES shares in France suggests to us that we are
not overlooking a significant strategic interaction or dynamic in the sector with respect to other
energy sources. More significantly perhaps, we simplify renewable electricity generation to a
considerable extent by supposing that there are only two states. Whereas this is a more innocu-
ous assumption in the perfectly competitive case, introducing a richer distribution of outcomes
in our model makes the system of equations characterizing the outcome of the follower stage less
transparent. The key insight with respect to the leader's deterrence incentives remains relevant
in such a setting, as it is still costly for the leader to maintain excess output in states where
followers have lower production. Finally we have assumed that followers are symmetric, which
in practice seems less and less likely to hold as available sites for solar and wind farms get filled.

The main goal of our analysis has been to shed light on some of the strategic interactions
arising in France's electricity sector, and perhaps in a few other countries with similar features.
In time, the greatest limitations to our analysis could be entirely different in nature from those
described above. If progress in storage technologies leads to significant enough cost decreases,
intermittent generation might find itself to be much less of a constraint on firms than it is today.
In addition, not only is technology constantly evolving but so are institutions, as it seems like
electricity generation in France is set to return to a situation not so different from the regulated
utility of its not too distant past.

**Acknowledgements.** We are grateful to Bruno Versaevel, Dimitrios Zormpas, and participants at the 12th Mannheim Conference on Energy and the Environment, the 2024 Public Economic Theory Conference in Lyon, France, and the 45th International Association for Energy Economics Conference in Istanbul, Turkey for valuable comments. All remaining errors are ours.

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# 8 Appendix

## 8.1 Proof of Proposition 1

The solutions $\hat{q}_F$ and $\hat{Q}_{-F}$ to Eqs. 2 and 3 are continuous functions of $K$, which implies that $\pi_L(q_L)$ is also continuous in $K$. We characterize the leader's optimum if the entry cost is zero and can therefore argue that the solution extends over a neighborhood of positive entry costs.

Set $K = 0$. Equations 2 and 3 imply that in an interior equilibrium with entry, $\hat{q}_F = 0$ and $\hat{Q}_{-F} = P^{-1}(c_F/\nu)$, so $P(\hat{q}_F + \hat{Q}_{-F}) = c_F/\nu$. The leader's net profit in this case is a continuous function

$$ \pi_L (q_L) = \begin{cases} (c_F + (1 - \nu)P(q_L) - c_L)q_L - K, & \text{if } q_L < P^{-1}\left(\frac{c_F}{\nu}\right) \\ (P(q_L) - c_L)q_L - K, & \text{if } q_L \ge P^{-1}\left(\frac{c_F}{\nu}\right). \end{cases} \quad (21) $$

If $\pi'_L(P^{-1}(c_F/\nu)) \ge 0$ the leader's net profit has a monopoly optimum, which must lie over the second piece. Evaluating the derivative gives the condition for such a monopoly optimum $q^M$ to exist,

$$ \frac{c_F}{\nu} - c_L + P'\left(P^{-1}\left(\frac{c_F}{\nu}\right)\right) P^{-1}\left(\frac{c_F}{\nu}\right) \ge 0. \quad (22) $$
Let $\Delta_0 := P'(P^{-1}(c_F/\nu))P^{-1}(c_F/\nu)$. If $\Delta_c \le \Delta_0$, then $q_M$ is well-defined and we need only verify that it dominates any choice on the first piece. Suppose that there is an interior accommodation optimum, so $q_L^A < P^{-1}(c_F/\nu)$. Then $P(q_L^A) > c_F/\nu$ and the profit under accommodation satisfies $\pi_L(q_L^A) = (c_F + (1-\nu)P(q_L^A) - c_L)q_L^A < (P(q_L^A) - c_L)q_L^A < \pi^M(c_L)$. $q^M$ is therefore a global maximum, implying that the leader blocks entry naturally if Eq. 22 holds.

Next, suppose that there is no accommodation optimum, i.e. $\pi'_L(0) \le 0$. Evaluating the derivative $\pi'_L(0)$ gives i.e.

$$c_F + (1 - \nu)P(0) - c_L \le 0. \qquad (23)$$

Let $\Delta_2 := (1 - \nu)(P(0) - c_F/\nu)$, which is positive because $c_F/\nu < P(0)$ (by assumption). If $\Delta_c \ge \Delta_2 > 0$, then Eq. 22 is violated implying $\pi'_L(P^{-1}(c_F/\nu)) < 0$. Because $\Delta_2 > \Delta_0$, there is no maximum on the second piece either and the global optimum for the leader is to set zero output.

Suppose that the comparative cost satisfies $\Delta_c \in (\Delta_0, \Delta_2)$. Then the leader must either deter or accommodate. Because the net profit is continuous (given $K = 0$), the choice between deterrence and accommodation depends on whether there is an interior optimum under accommodation, that is to say whether $q_L^A$ is well-defined with $q_L^A < P^{-1}(c_F/\nu)$. The condition for this to be true is if $\lim_{q_L \to [P^{-1}(c_F/\nu)]^{-}} \pi'_L(q_L) < 0$, i.e.

$$\frac{c_F}{\nu} - c_L + (1 - \nu)P'\left(P^{-1}\left(\frac{c_F}{\nu}\right)\right)P^{-1}\left(\frac{c_F}{\nu}\right) < 0. \qquad (24)$$

Letting $\Delta_1 := (1 - \nu)P'(P^{-1}(c_F/\nu))P^{-1}(c_F/\nu)$, the leader therefore deters if $\Delta_c \le \Delta_1$ and accommodates otherwise.

Finally it is straightforward to check that $\Delta_0 < \Delta_1 < 0 < \Delta_2$, establishing the last part of the proposition. $\square$
## 8.2 Proof of Corollary 2

From the expressions of $\Delta_1$ and $\Delta_2$ in the preceding proof, it follows directly that

$$ \left. \frac{d\Delta_1}{d\nu} \right|_{\frac{c_F}{\nu}=cst} = -P'\left(P^{-1}\left(\frac{c_F}{\nu}\right)\right) P^{-1}\left(\frac{c_F}{\nu}\right) > 0, $$

and

$$ \left. \frac{d\Delta_2}{d\nu} \right|_{\frac{c_F}{\nu}=cst} = -(P(0) - \frac{c_F}{\nu}) > 0, $$

which establishes the result.

Going a little further, we establish the result alluded to in the text, i.e. that $\hat{Q}_F$ is increasing in $\nu$ holding $c_F/\nu$ constant. First totally differentiate equations 2 and 3 holding $c_F/\nu$ fixed to get

$$ \left( \begin{array}{cc} 2P' + \hat{q}_F P'' & P' + \hat{q}_F P'' \\ -2\hat{q}_F P' - \hat{q}_F^2 P'' & -\hat{q}_F^2 P'' \end{array} \right) \left( \begin{array}{c} d\hat{q}_F \\ d\hat{Q}_{-F} \end{array} \right) = \left( \begin{array}{c} 0 \\ -\frac{K}{\nu^2} \end{array} \right) d\nu \Big|_{\frac{c_F}{\nu}=cst}. \quad (25) $$

Applying the implicit function theorem and using Cramer's rule yields

$$ \frac{d\hat{Q}_{-F}}{d\nu} = - \frac{K}{\nu^2 \hat{q}_F P' (\hat{q}_F + \hat{Q}_{-F})} \quad (26) $$

which is positive as desired. $\square$

## 8.3 Proof of Corollary 3

Express welfare as the sum of expected consumer surplus and profits. Because free entry drives entrant profit to zero,

$$ \hat{W}(q_L) = \nu \int_0^{\hat{Q}_F + q_L^A} (P(s) - P(\hat{Q}_F + q_L^A)) ds + (1 - \nu) \int_0^{q_L^A} (P(s) - P(q_L^A)) ds + \pi_L(q_L^A). \quad (27) $$
in the sequential moves case, where $\hat{Q}_F = \hat{q}_F + \hat{Q}_{-F} - q_L^A$ denotes state $\bar{\omega}$ total output. In the simultaneous moves case, welfare is

$$W^C(q_L^C) = \nu \int_0^{\hat{Q}_F^C + q_L^C} (P(s) - P(\hat{Q}_F^C + q_L^C)) ds + (1-\nu) \int_0^{q_L^C} (P(s) - P(q_L^C)) ds + \pi_L(q_L^C) \quad (28)$$

where $\hat{Q}_F^C$ and $q_L^C$ denote total entrant and incumbent outputs respectively. Because total output is the same under both scenarios, the welfare difference $\hat{W}(q_L) - W^C(q_L^C)$ is

$$(1 - \nu) \left( \int_0^{q_L^A} (P(s) - P(q_L^A)) ds - \int_0^{q_L^C} (P(s) - P(q_L^C)) ds \right) + (\pi_L(q_L^A) - \pi_L(q_L^C)) \quad (29)$$

as claimed in the text. The leader could choose $q_L^C$ under sequential moves but opts for $q_L^A$ instead, so $\pi_L(q_L^A) \ge \pi_L(q_L^C)$. To establish that $\hat{W}(q_L) - W^C(q_L^C)$ is positive, it is therefore sufficient to verify that $q_L^A \ge q_L^C$ so the consumer surplus difference is positive. Because equations 2 and 3 imply constant total output, the aggregate entrant best response $\hat{Q}_F(q_L)$ is decreasing. $\hat{Q}_F$ and $q_L$ are therefore strategic substitutes, implying that the incumbent sets a higher output in the sequential move case so $q_L^A > q_L^C$. $\square$

## 8.4 Proof of Proposition 4

In the unconstrained case, the free entry condition (equation 3) yields an expression for individual output in terms of the total output $\hat{Q} = \hat{q}_F + \hat{Q}_{-F}$, $\hat{q}_F = \sqrt{K/(-\nu P'(\hat{Q}))}$. This expression can in turn be substituted into the first-order condition 7 to obtain a single condition in terms of total output only,

$$P(\hat{Q}) - \sqrt{\frac{K}{\nu}} \sqrt{-P'(\hat{Q})} + \frac{1-\nu}{\nu} (P(q_L) - w) = \frac{c_F}{\nu}. \quad (30)$$
Differentiating then gives

$$ \frac{d\hat{Q}}{dw} = 2 \frac{1 - \nu}{\nu} \frac{1}{2P'(\hat{Q}) + \hat{q}_F P''(\hat{Q})} < 0. \quad (31) $$

If regulated purchases are constrained, define $A = (1 - \nu)(P(q_L) - w)\alpha q_L / \nu$ so as to write the follower first-order condition and the zero-profit condition in terms of $\hat{q}_F$ and total output $\hat{Q}$,

$$ \left\{ \begin{array}{l} P(\hat{Q}) + \hat{q}_F P'(\hat{Q}) + A \frac{\hat{q}_F + \hat{Q}}{\hat{Q}^2} = \frac{c_F}{\nu} \\ -P'(\hat{Q}) \hat{q}_F^2 + A \frac{\hat{q}_F^2}{\hat{Q}^2} = \frac{K}{\nu}. \end{array} \right. \quad (32) $$

Differentiate totally to get

$$ \left( \begin{array}{cc} P'(\hat{Q}) + \hat{q}_F P''(\hat{Q}) - A \frac{2\hat{q}_F + \hat{Q}}{\hat{Q}^3} & P'(\hat{Q}) + \frac{A}{\hat{Q}^2} \\ -P''(\hat{Q}) + 2\frac{A}{\hat{Q}^3} \hat{q}_F^2 & 2\hat{q}_F(-P'(\hat{Q}) + \frac{A}{\hat{Q}^2}) \end{array} \right) \left( \begin{array}{c} d\hat{Q} \\ d\hat{q}_F \end{array} \right) = - \left( \begin{array}{c} \frac{\hat{q}_F + \hat{Q}}{\hat{Q}^2} \\ \frac{\hat{q}_F^2}{\hat{Q}^2} \end{array} \right) dA. \quad (33) $$

Applying the implicit function theorem and Cramer's rule, $\frac{d\hat{Q}}{dA} = \frac{N}{D}$ where $N = \begin{vmatrix} -\frac{\hat{q}_F + \hat{Q}}{\hat{Q}^2} & P' + \frac{A}{\hat{Q}^2} \\ -\frac{\hat{q}_F^2}{\hat{Q}^2} & 2\hat{q}_F(-P' + \frac{A}{\hat{Q}^2}) \end{vmatrix} = -\hat{q}_F \left( \left( 2\hat{Q} + \hat{q}_F \right) A - \left( 2\hat{Q} + 3\hat{q}_F \right) P'(\hat{Q}) \hat{Q}^2 \right) / \hat{Q}^4 < 0$, and

$$ D = -\hat{q}_F \left( 2 \left( \hat{Q} + 3\hat{q}_F \right) A^2 - \hat{Q}^2 \left( \hat{q}_F \left( 2P' + \hat{Q}P'' \right) + 2\hat{Q} \left( 2P' + \hat{q}_F P'' \right) \right) A + \left( 2P' + \hat{q}_F P'' \right) P' \hat{Q}^5 \right) / \hat{Q}^4 < 0 \quad (34) $$

implying that $d\hat{Q}/dw = -((1 - \nu)/\nu)d\hat{Q}/dA < 0$. $\square$

## 8.5 Proof of Proposition 5

Under accommodation, the leader's first-order condition with unconstrained purchases is

$$ 2 \left( \nu P(\hat{Q}) + (1 - \nu) P(q_L) - c_L \right) - (w - c_L) + (1 - \nu) P'(q_L) \left( 2q_L - \hat{Q} \right) + \frac{d\hat{Q}}{dq_L} \left( \nu P'(\hat{Q}) \left( 2q_L - \hat{Q} \right) - \left( \nu P(\hat{Q}) + (1 - \nu) P(q_L) - c_L \right) + (w - c_L) \right). \quad (35) $$
From equation 30,

$$ \frac{d\hat{Q}}{dq_L} = -2 \frac{1-\nu}{\nu} \frac{P'(q_L)}{2P'(\hat{Q}) + \hat{q}_F P''(\hat{Q})} < 0. \qquad (36) $$

For $K = 0$, evaluating at the left limit of $\hat{Q}_{-F}$ gives $d\hat{Q}/dq_L = -(1-\nu)/\nu$. After simplifying, the first-order condition becomes

$$ \frac{1+\nu}{\nu} (P(\hat{Q}) - c_L) - \frac{1}{\nu} (w - c_L) = \frac{c_F}{\nu} - c_L + c_F - \nu w \qquad (37) $$

i.e. if the regulated price is set at $w = c_L$ then there is deterrence if $\Delta_c < 0$, whereas setting $w$ lower than $c_L$ raises the likelihood of deterrence. $\square$

## 8.6 Proof of Proposition 6

As in the preceding proofs we argue by continuity around $K = 0$. For $K = 0$, $\hat{q}_F = 0$ and $\hat{Q}_{-F} = P^{-1}(c_F/\nu)$ implying that the welfare objective equation 13 reduces to

$$ W(q_L) = \begin{cases} \nu \int_0^{P^{-1}\left(\frac{c_F}{\nu}\right)} \left(P(s) - \frac{c_F}{\nu}\right) ds + \nu \left(\frac{c_F}{\nu} - c_L\right) q_L & \text{if } q_L < P^{-1}\left(\frac{c_F}{\nu}\right) \\ + (1-\nu) \int_0^{q_L} (P(s) - c_L) ds & \text{if } q_L \ge P^{-1}\left(\frac{c_F}{\nu}\right). \end{cases} \qquad (38) $$

$W(q_L)$ is continuous and evaluating $W'(q_L)$ gives $\lim_{q_L \to (P^{-1}(\frac{c_F}{\nu}))^{-}} W'(q_L) = c_F/\nu - c_L$, so there is an interior maximum involving accommodation if $\Delta_c > 0$. $\square$