Energy technology investments in competitive and regulatory environments

Abstract

The goal of this paper was to develop a better understanding of how energy firms might respond to competitive pressures in the context of regulatory risk. We model how competitive pressures affect capacity investments that firms make into their portfolio of technologies. Using comparative statics, we characterize energy firms’ incentives to invest in different energy technologies under imperfect competition in outputs and prices and within different environmental regulatory regimes. We find that under Cournot competition, firms that invest in renewable technologies benefit from both strategic and spillover effects on their overall profits. Under Bertrand competition, these benefits are enjoyed only by firms that invest in conventional technologies. Our findings can provide guidance for policy makers. Notably, even relatively weak targets set by regulators are likely to spur additional investment into renewable technologies. Regardless of policy type, strategic interactions and spillover benefits drive the optimal management of energy technology R&D activities. Our results also suggest ways for regulators to exploit the inherent benefits of imperfections in competitive markets to stimulate firms’ efforts at improving on the technologies in their portfolios. Overall, our results describe how technology investment incentives are shaped by the strategic interactions between firms, market structures, and environmental policy choices.

Appendix

1.1 Two energy firms

In this section, we examine two energy firms in their primary states of jurisdiction—Xcel in Colorado and Florida Power and Light (FPL) in Florida—with respect to their energy technology portfolios and how their strategic investment plans are shaped by their emissions reduction plans. Figure 4 shows the current mix of technologies at the two firms. FPL generate their supplies primarily from natural gas plants while Xcel engages more coal plants. In terms of their renewable contents, Xcel has a significantly higher proportion than FPL. In order to implement Colorado’s Clean Air-Clean Jobs Act (ACT), Xcel examined three scenarios that would allow the firm to achieve its emissions reduction plan. Of the seven coal plants in Colorado, Xcel has scheduled one for retirement and aims to either retire or retrofit another four by implementing emissions control on them. The control is to replace the coal-fired generators with gas-fired technologies. Thus, in the context of this article, we interpret Xcel as the firm with investments into conventional resources.

Fig. 4

Technology portfolios of Xcel in Colorado and Florida power and light

FPL is an energy company with a carbon dioxide emissions rate that is 35 % lower than the industry average. With over 75 % of its generation capacity from clean-burning natural gas, FPL also generates a significant portion of power production from nuclear. FPL commenced operation of commercial-scale solar generation facilities in 2009 and 2010. Due to their focus on solar and other renewable technologies, we characterize FPL as the firm with investments into the renewable components of its portfolio.

1.2 Cournot equilibrium: optimal outputs and price

Taking the first-order conditions of the firms’ profit functions, we have

$$\begin{aligned} P(Q)+P_{i}^{\prime }(Q)q_{i}& = (1-\alpha _{i})C_{i}^{\prime }(q_{i}-\bar{e}_{i})\nonumber \\ P(Q)+P_{j}^{\prime }(Q)q_{j}& = C_{j}^{\prime }\left( \frac{q_{j}-\bar{e}_{j}-\alpha _{j}}{1-\alpha _{j}}\right) m \end{aligned}$$

(38)

Substituting the linear inverse demand function,

$$\begin{aligned} P(Q)=a-(q_{i}+\lambda q_{j}) \end{aligned}$$

(39)

where $\lambda =1$, we have

$$\begin{aligned} q_{i}= & \frac{1}{2}(a-q_{j}-(1-\alpha _{i})C_{i}^{\prime }(q_{i}-\bar{e}_{i})) \\ q_{j}= & \frac{1}{2}\left( a-q_{j}-C_{j}^{\prime }\left( \frac{q_{j}-\bar{e}_{j}-\alpha _{j}}{1-\alpha _{j}}\right) m\right) \end{aligned}$$

Solving both expressions simultaneously, we have,

$$\begin{aligned} q_{i}^{*}= & \frac{1}{3}\left( a+C_{j}^{\prime }\left( \frac{q_{j}^{*}- \bar{e}_{j}-\alpha _{j}}{1-\alpha _{j}}\right) m-2(1-\alpha _{i})C_{i}^{\prime }(q_{i}^{*}-\bar{e}_{i})\right) \nonumber \\ q_{j}^{*}= & \frac{1}{3}\left( a-2C_{j}^{\prime }\left( \frac{q_{j}^{*}-\bar{e}_{j}-\alpha _{j}}{1-\alpha _{j}}\right) m+(1-\alpha _{i})C_{i}^{\prime }(q_{i}^{*}-\bar{e}_{i})\right) \end{aligned}$$

(40)

Substituting these into the (39), we have the market price for the outputs as

$$\begin{aligned} P^{*}=\frac{1}{3}\left( a+C_{j}^{\prime }\left( q_{j}^{*}-\bar{e}_{j}-\alpha _{j}\right) +(1-\alpha _{i})C_{i}^{\prime }(q_{i}^{*}-\bar{e}_{i})\right) \end{aligned}$$

1.3 Bertrand equilibrium: optimal prices

$$\begin{aligned}&q_{i}+(p_{i}-(1-\alpha _{i})C_{i}^{\prime }(q_{i}-\bar{e}_{i}))\frac{{\mathrm{d}}q_{i}}{ {\mathrm{d}}p_{i}}=0 \nonumber \\&q_{j}+\left( p_{j}-C_{j}^{\prime }\left( \frac{q_{j}^{*}-\bar{e}_{j}-\alpha _{j}}{1-\alpha _{j}}\right) \right) \frac{{\mathrm{d}}q_{j}}{{\mathrm{d}}p_{j}}=0 \end{aligned}$$

(41)

Substituting $q_{i}=\frac{1}{1-\lambda ^{2}}\left( a(1-\lambda )-p_{i}+\lambda p_{j})\right)$ $i,j=1,2$ and $i\ne j$, and $\frac{ {\mathrm{d}}q_{i}}{{\mathrm{d}}p_{i}}=\frac{{\mathrm{d}}q_{j}}{{\mathrm{d}}p_{j}}=-\frac{1}{1-\lambda ^{2}}$ into these expressions, we have

$$\begin{aligned} \frac{1}{1-\lambda ^{2}}\left( a(1-\lambda )-p_{i}+\lambda p_{j})\right)= & \frac{1}{1-\lambda ^{2}}(p_{i}-(1-\alpha _{i})C_{i}^{\prime }(q_{i}-\bar{e} _{i})) \nonumber \\ p_{i}^{*}= & \frac{1}{2}(a(1-\lambda )+\lambda p_{j}^{*}+(1-\alpha _{i})C_{i}^{\prime }(q_{i}-\bar{e}_{i}) \end{aligned}$$

(42)

Similarly,

$$\begin{aligned} p_{j}^{*}=\frac{1}{2}(a(1-\lambda )+\lambda p_{i}^{*}+C_{j}^{\prime }\left( \frac{q_{j}^{*}-\bar{e}_{j}-\alpha _{j}}{1-\alpha _{j}}\right) \end{aligned}$$

(43)

The simultaneous solution to (42) and (43) gives the expressions in (24).

1.4 Signing the Hessian

$$\begin{aligned} \pi _{i}&=P(Q)q_{i}-(1-\alpha _{i})C_{i}(q_{i}-e_{i}) \end{aligned}$$

(44)

$$\begin{aligned} \pi _{j}&=P(Q)q_{j}-C_{j}\left( \frac{q_{j}-e_{j}-\alpha _{j}}{1-\alpha _{j} }\right) \end{aligned}$$

(45)

$$\begin{aligned} \frac{{\mathrm{d}}\pi _{i}}{{\mathrm{d}}q_{i}}&=P(Q)+P^{\prime }(Q)q_{i}-(1-\alpha _{i})C_{i}^{^{\prime }}(q_{i}-e_{i}) \end{aligned}$$

(46)

$$\begin{aligned} \frac{{\mathrm{d}}^{2}\pi _{i}}{{\mathrm{d}}q_{i}^{2}}&=2P^{\prime }(Q)+P^{\prime \prime }(Q)q_{i}-(1-\alpha _{i})C_{i}^{^{\prime \prime }}(q_{i}-e_{i})\le 0 \nonumber \\&\quad \quad (-) \quad \qquad (-)\qquad \qquad (-) \nonumber \\ \frac{\partial ^{2}\pi _{i}}{\partial q_{j}q_{i}}&=P^{\prime }(Q)+P^{^{\prime \prime }}(Q)q_{i}\le 0 \end{aligned}$$

(47)

We obtain similar results for $\frac{d^{2}\pi _{j}}{{\mathrm{d}}q_{j}^{2}}$ and $\frac{ \partial ^{2}\pi _{j}}{\partial q_{i}q_{j}}$. In absolute terms, $\left| \frac{d^{2}\pi _{i}}{{\mathrm{d}}q_{i}^{2}}\right| >\left| \frac{\partial ^{2}\pi _{j}}{\partial q_{i}q_{j}}\right|$, thus, in magnitude terms,

$$\begin{aligned} \frac{\partial ^{2}\pi _{i}}{\partial q_{i}^{2}}\frac{\partial ^{2}\pi _{j}}{ \partial q_{j}^{2}}-\frac{\partial ^{2}\pi _{i}}{\partial q_{j}q_{i}}\frac{ \partial ^{2}\pi _{j}}{\partial q_{i}q_{j}}>0 \end{aligned}$$

1.5 Proof of proposition 2

Proof

Consider a firm with a portfolio of technologies seeking to choose the outputs, $q_{i}$, from the independent sources to satisfy demand at minimum cost. We formulate this model as

$$\begin{aligned} \min \Sigma c_{i}q_{i} \end{aligned}$$

subject to $-\Sigma e_{i}q_{i}\ge -\bar{e}$, and $\Sigma q_{i}\ge \overline{q}$. $c_{i}$ is the output unit cost, $e_{i}$ represents the emissions coefficient of the output sources, $\bar{e}$ is the limit of aggregate emissions by the regulator, and $\overline{q}$ is the output demand to be satisfied. Dropping the summations, the Lagrangian for this problem can be written as

$$\begin{aligned} {\mathcal {L}} =c_{i}q_{i}-\omega _{1}(-e_{i}q_{i}+\bar{e})-\omega _{2}(q_{i}- \overline{q}) \end{aligned}$$

where $\omega 1$ is the marginal cost of abatement and $\omega 2$ is an output shadow price. The Kuhn–Tucker first-order conditions are

$$\begin{aligned} c_{i}+\omega _{1}e_{i}-\omega _{2}= & 0 \end{aligned}$$

(48)

$$\begin{aligned} q_{i}e_{i}-\bar{e}= & 0 \end{aligned}$$

(49)

$$\begin{aligned} \Sigma q_{i}-\overline{q}= & 0 \end{aligned}$$

(50)

To simplify, let $q_{i}=\frac{\overline{q}}{n}\ \equiv \hat{p}$ where n is the number of technologies in the portfolio. Thus, substituting (50) into (49), we have, $e_{i}=\frac{\bar{e}}{\hat{q}}$. Finally, putting this into (48), we have

$$\begin{aligned} \omega _{1}=\frac{\hat{q}}{\bar{e}}(\omega _{2}-c_{i}) \end{aligned}$$

The marginal abatement cost, $\omega _{1}$ increases as the regulator’s emissions target, $\bar{e}$ reduces, and thus influences the firm’s efficient frontier on the choice of technologies to satisfy the constraints. $\square$

1.6 Proof of proposition 3

Proof

Using stylized functions for linear marginal cost curves in Fig. 5, let $e^{\prime }$ be the standard abatement level that makes the firms indifferent in their marginal costs. This standard limit is:

$$\begin{aligned} be^{\prime }= & \max (ce^{\prime }-k,0) \\ e^{\prime }= & \frac{k}{c-b} \end{aligned}$$

Fig. 5

Stylized marginal cost curves

$e^{\prime }>0$ because $c>b$ (by definition). Shaded in Fig. 5 is the region where the emissions policy is higher than the ‘equilibrium’ limit, i.e., $e^{\prime }<e_{H}$, then $be_{H}>\max (ce_{H}-k,0)$. For $e^{\prime }>e_{L}, be_{L}<\max\,(ce_{L}-k,0)$. $e_{L}$ and $e_{H}$ are low and high abatement levels, respectively. Tying this with (6), we characterize the optimal output relationships as follows:

$$\begin{aligned} \mathrm{Output}\equiv \left\{ \begin{array}{c} q_{j}^{*}>q_{i}^{*} \ \ {\text{if}}\;\bar{e}<e^{\prime } \\ q_{j}^{*}<q_{i}^{*} \ \ {\text{if}}\;\bar{e}>e^{\prime } \end{array} \right. \end{aligned}$$

$\square$