Pollution Under Imperfect Competition in Vertical Related Energy Markets

Abstract

This chapter aims at identifying the mechanisms through which imperfect competition in energy markets can affect the performance of environmental policy. Vertical related markets are investigated: (1) the electricity market (output market); (2) the market for natural gas (input market). Two supply scenarios are simulated: (1) Scenario 1 (highly polluting supply) where power generation is based on more polluting technologies; (2) Scenario 2 (cleaner supply) where power generation is based on cleaner technologies. Finally, two regulatory scenarios are explored: (a) Scenario A where firms can maximize profits; (b) Scenario B when, because of the regulatory pressure exerted by competition and sector-specific authorities, firms pursue strategies besides the profit maximization, namely by pursuing a short- or long-term profit target. The analysis shows that imperfect competition may lessen the performance of environmental policy, that is its ability to reduce pollution and/or its ability to minimize the cost of meeting environmental targets, in either Scenario 1 or Scenario 2 and in either Scenario A or Scenario B. However, in Scenario 2 relatively low pollution prices are necessary. Furthermore, this effect seems to be more likely when firms can maximize profits (Scenario A) rather than under regulatory pressure (Scenario B) unless environmental regulation is based on ETS with benchmarking and firms pursue a long-term profit target. Finally, looking at the entire demand cycle, increasing pollution in the short run is virtually excluded or very unlikely in all cases.

Appendix

1.1 Proof of Lemma 2

Let ${\bar{K}_e} = {D_{e_M }} - \bar{K}_e^{d_e }$ be the peak demand minus the dominant firm's capacity in least efficient plants ($\bar{K}_e^{d_e }$) with $\bar{H} = {D^{- 1 }}({\bar{K}_e})$. It is immediately intuitive that when ${D_e} \ge {\bar{K}_e}$ the system marginal price equals the price threshold, ${\hat{p}_e}$. When ${D_e} \le {\mathop {\underline K}}_e^{f_e }$, pure Bertrand equilibria (first marginal cost pricing) arise and prices equal the marginal cost of the most efficient plants (${\underline {{\it MC}}_e}$).

In fact, on the one hand, whenever the demand is so high that both leader’s and the fringe’s least efficient units can enter the market, the dominant firm would not gain any advantage by competing à la Bertrand, i.e., by attempting to undercut the rivals. Therefore, it will maximize its profit by bidding the residual monopoly price.

On the other hand, whenever the power demand is lower than the fringe's power capacity in most efficient plants, competing à la Bertrand is the only leader's available strategy in order to have a positive probability of entering the market. As a consequence, prices will converge to the marginal cost of the most efficient plants.

It remains to identify the leader's optimal choice on ${D_e} \in \left] {{{\bar{K}}_e};\underline K_e^{f_e }} \right]$.¹² Under the assumptions of the model, each firm in the competitive fringe has a unique dominant strategy regardless of the market demand: bidding according to its own marginal cost of production. By converse the best choice of the dominant firm might consist in (i) bidding the price threshold (${\hat{p}_e}$) or in (ii) bidding ${\overline {{\it MC}}_e}$.¹³

Let $\pi_A^{d_e }$ and $\pi_B^{d_e }$ be the profits corresponding to the first and second strategies above, respectively. Whenever the least efficient units could enter the market (i.e., ${D_e} \ge {{\mathop {\underline K}}_e}$), the profit the dominant firm earns by choosing the first strategy (i.e., $\forall H \in \left] {\bar{H};{\mathop {\underline H}}} \right]$) is

$$ \pi_A^{d_e } = ({\hat{p}_e} - {\underline {{\it MC}}_e})\left[ {D_e - (1 - \delta ){K_{e_T }}} \right] + \sum\nolimits_{i = 1}^z {\sum\nolimits_{j = a,b } { au \cdot {r_{bn }} \cdot \bar{q}_j^i} } $$

(4.8)

Where

${r_{bn }}$ =emission rate of the technology chosen as the benchmark

$\bar{q}_j^i$ = reference production for the free allocation of allowances to the i-th unit belonging to the group j of plants. Obviously $\bar{q}_j^i = 0$ with full auctioning.

If the dominant firm chooses the second strategy, he earns

$$ \pi_B^{d_e } = ({\overline {{\it MC}}_e} - {\underline {{\it MC}}_e}){\mathop {\underline K}}_e^{d_e } + \sum\nolimits_{i = 1}^z {\sum\nolimits_{j = a,b } { au \cdot {r_{bn }} \cdot \bar{q}_j^i} } $$

(4.9)

Let ${C_{e_d }}$ the dominant firm’s total cost. When the dominant firm maximizes profits, ${\hat{p}_e}$ is the residual monopoly price with

$$ {\hat{p}_e} = p_e^r = {\arg}{\mathrm{ ma}}{{\mathrm{ x}}_{p_e }}\left\{ {p_e \left[ {D_e ({p_e}) - {K^{f_e }}} \right] - {C_{e_d }}} \right\} = \frac{1}{\beta}\left[ {D_e - (1 - \delta ){K_{e_T }}} \right] + {\underline {{\it MC}}_e} $$

if the share of most efficient plant operated by the dominant firm is large enough. It is to be noted that by differentiating this equation with respect to $au$ and when $au < { au^s}$, then $\displaystyle\frac{\partial p_e^r }{{\partial au }} = \displaystyle\frac{r_a }{2}$.

Consequently $\pi_A^{d_e } \ge \pi_B^{d_e }$ if and only if

$$ {D_e} \ge {\hat{D}_{e_1 }} = \left[ {{{\underline \mu}^{d_e }}\delta \zeta + (1 - \delta )} \right]{K_{e_T }} $$

(4.10)

Where $\zeta = \displaystyle\frac{{{{\overline {{\it MC}}}_e} - {{\underline {{\it MC}}}_e}}}{{p_e^r - {{\underline {{\it MC}}}_e}}}$ and $\displaystyle\beta = - \frac{{{{\it d}}{D_e}}}{{{{\it d}}{p_e}}}$

When ${D_e} \in \left] {{{\mathop {\underline K}}_e};{\mathop {\underline K}}_e^{f_e }} \right]$ (i.e., $H \in \left] {{\mathop {\underline H}},{{\mathop {\underline H}}^{f_e }}} \right]$) the profit the dominant firm earns by choosing the first strategy is

$$ \pi_C^{d_e } = (p_e^r - {\underline {{ MC}}_e})\left[ {D_e - (1 - \delta ){K_{e_T }}} \right] + \sum\nolimits_{i = 1}^z {\sum\nolimits_{j = a,b } { au \cdot {r_{bn }} \cdot \bar{q}_j^i} } $$

(4.11)

and by choosing the second strategy the profit is

$$ \pi_D^{d_e } = ({\overline {{\it MC}}_e} - {\underline {{\it MC}}_e})\left[ {D_e - {{\mathop {\underline \mu}}^{f_e }}(1 - \delta ){K_{e_T }}} \right] + \sum\nolimits_{i = 1}^z {\sum\nolimits_{j = a,b } { au \cdot {r_{bn }} \cdot \bar{q}_j^i} } $$

(4.12)

Thus, the dominant firm will choose the first strategy if and only if $\pi_C^{d_e } \ge \pi_D^{d_e }$, i.e., if and only if

$$ {D_e} \ge {\hat{D}_{e_2 }} = (1 - \delta )\left[ {\frac{{1 - \zeta {{\underline \mu}^{f_e }}}}{{1 - \zeta }}} \right]{K_{e_T }} $$

(4.13)

Finally by differentiating ${\hat{D}_e}$ with respect to ${{\mathop {\underline \mu}}^{d_e }}$ and ${{\mathop {\underline \mu}}^{f_e }}$ we get

$$ \frac{{\partial {{\hat{D}}_{e_1 }}}}{{\partial {{\mathop {\underline \mu}}^d}}} = \delta { \ }\zeta { \ }{K_{e_T }} > 0 $$

and

$$ \frac{{\partial {{\hat{D}}_{e_2 }}}}{{\partial {{\mathop {\underline \mu}}^f}}} = - {K_{e_T }}(1 - \delta )\frac{\zeta }{{1 - \zeta }}{} < 0 $$

1.2 Proof of Lemma 3

The derivative of ${\hat{D}_e}$ with respect to $au$ can be written as

$$ \frac{{\partial {{\hat{D}}_e}}}{{\partial au }} = \frac{{\partial {{\hat{D}}_e}}}{{\partial \zeta }}\frac{{\partial \zeta }}{{\partial au }} $$

(4.14)

Since [from (4.10) and (4.13)]

$$ \frac{{\partial {{\hat{D}}_{e_1 }}}}{{\partial \sigma }} = {{\mathop {\underline \mu}}^{d_e }} \cdot \delta \cdot {K_{e_T }} > 0 $$

(4.15a)

and

$$ \frac{{\partial {{\hat{D}}_{e_2 }}}}{{\partial \zeta }} = \frac{(1 - \delta )(1 - {{\underline \mu}}^{f_e })}{{{{(1 - \zeta )}^2}}}{K_{e_T }} > 0 $$

(4.15b)

then the degree of market power is a decreasing function of $\zeta$.

By differentiating $\zeta$ with respect to $au$ and given that ${\hat{p}_e}$ and ${c_{e_g }}$ depend on $au$, we get

$$ \begin{array}{clclcl}\frac{{\partial \zeta }}{{\partial au }} = \frac{1}{{{{(p_e^r - {c_{e_a }} - {r_a} au )}^2}}}\left[ {({r_{{\it AF}}} - {r_a})(p_e^r(0) - {c_{e_a }}) - ({\partial p_e^r \left/ {{\partial au }} \right.} - {r_a})({c_{e_g }}(0) - {c_{e_a }})} \right]\end{array} $$

(4.16)

if $au < { au^s}$.

and

$$ \begin{array}{clclcllc}\frac{{\partial \zeta }}{{\partial au }} = \frac{1}{{{{(p_e^r - {c_{e_g }} - {r_g} au )}^2}}}\left[ {({r_a} - {r_{{\it AF}}})(p_e^r(0) - {c_{e_g }}(0)) - ({\partial p_e^r \left/ {{\partial au }} \right.} - {r_{{\it AF}}})({c_{e_a }} - {c_{e_g }}(0))} \right]\end{array} $$

(4.17)

if $au \ge { au^s}$.

From (4.16), since $\displaystyle\frac{\partial p_e^r }{{\partial au }} = \frac{r_a }{2}$ then $\displaystyle\frac{{\partial \zeta }}{{\partial au }} > 0$ and $\displaystyle\frac{{\partial {{\hat{D}}_e}}}{{\partial au }} > 0$, if

$$ {r_{{ AF}}} > \hat{r} = {r_a}{ \ }\left( {1 - \frac{1}{2}\frac{{D_e - (1 - \delta ){K_{e_T }}}}{K_a^d }} \right) $$

Vice versa, if ${r_{{ AF}}} < \hat{r}$ then $\displaystyle\frac{{\partial {{\hat{D}}_e}}}{{\partial au }} < 0$.

From (4.17), since $\displaystyle\frac{\partial p_e^r }{{\partial au }} = \displaystyle\frac{{{r_{{\it AF}}}}}{2}$ then $\displaystyle\frac{{\partial \zeta }}{{\partial au }} > 0$ and $\displaystyle\frac{{\partial {{\hat{D}}_e}}}{{\partial au }} > 0$, if

$$ {r_a} > \hat{r} = {r_{{\it AF}}}{ \ }\left( {1 - \frac{1}{2}\frac{{D_e - (1 - \delta ){K_{e_T }}}}{K_g^d }} \right) $$

Vice versa, if ${r_a} < \hat{r}$ then $\displaystyle\frac{{\partial {{\hat{D}}_e}}}{{\partial au }} < 0$.

In conclusion $\displaystyle\frac{{\partial {{\hat{D}}_e}}}{{\partial au }} > 0$ if

$$ \bar{r} > \hat{r} = {\mathop {\underline r}}{ \ }\left( {1 - \frac{1}{2}\frac{{D_e - (1 - \delta ){K_{e_T }}}}{K_g^d }} \right) $$

When the natural gas market is fully competitive then $\bar{r} = {r_g}$ and ${\mathop {\underline r}} = {r_a}$ if $au < { au^s}$ while $\bar{r} = {r_a}$ and ${\mathop {\underline r}} = {r_g}$ if $au \ge { au^s}$. When the natural gas market is imperfectly competitive then $\bar{r} = {r_{{ AF}}}$ and ${\mathop {\underline r}} = {r_a}$ if $au < { au^s}$ while $\bar{r} = {r_a}$ and ${\mathop {\underline r}} = {r_{{\it AF}}}$ if $au \ge { au^s}$.

It is to be noted that when $au \ge { au^s}$ then ${\left( {\displaystyle\frac{{\partial {{\hat{D}}_e}}}{{\partial au }}} \right)_{{ au < { au^s}}}} > 0$ always. Then, since $\Delta {\hat{D}_e}$ can be expressed as

$$ \Delta {\hat{D}_e} = \int_0^{ au^s } {{{\left( {\frac{{\partial {{\hat{D}}_e}}}{{\partial au }}} \right)}_{{ au < { au^s}}}}{{\it d}} au } + \int_{ au^s}^{ au } {{{\left( {\frac{{\partial {{\hat{D}}_e}}}{{\partial au }}} \right)}_{{ au \ge { au^s}}}}{{\it d}} au } $$

For a given $au$ and a given ${ au^s}$, $\Delta {\hat{D}_e} > 0$ when ${\left( {\displaystyle\frac{{\partial {{\hat{D}}_e}}}{{\partial au }}} \right)_{{ au < { au^s}}}} < 0$ and ${\left| {\displaystyle\frac{{\partial {{\hat{D}}_e}}}{{\partial au }}} \right|_{{ au < { au^s}}}}$ is sufficiently high and/or ${\left| {\displaystyle\frac{{\partial {{\hat{D}}_e}}}{{\partial au }}} \right|_{{ au \ge { au^s}}}}$ is sufficiently low. This can occur when the dominant firm holds a low share of gas units (high ${{\mathop {\underline \mu}}^{d_e }} = \mu_a^{d_e }$ when $au < { au^s}$; low ${{\mathop {\underline \mu}}^{d_e }} = \mu_g^{d_e }$ when $au \ge { au^s}$). Finally, note that $\left| {\displaystyle\frac{{\partial {{\hat{D}}_e}}}{{\partial au }}} \right|$ is an increasing function of $au$.

1.3 Proof of Lemma 2bis

From (4.8) and (4.9)

$$ {D_e} \ge {\hat{D}_{e_1 }} = \left[ {{{\mathop {\underline \mu}}^{d_e }}\delta { \ }\sigma + (1 - \delta )} \right]{K_{e_T }} $$

(4.18)

and from (4.11) and (4.12)

$$ {D_e} \ge {\hat{D}_{e_2 }} = (1 - \delta )\left[ \frac{{1 - \sigma {\underline \mu}}^{f_e }}{{1 - \sigma }} \right]{K_{e_T }} $$

(4.19)

where $\displaystyle\sigma = \frac{{{{\overline {{\it MC}}}_e} - {{\underline {{\it MC}}}_e}}}{{{{\hat{p}}_e} - {{\underline {{\it MC}}}_e}}}$

1.4 Proof of Lemma 3bis

The derivative of ${\hat{D}_e}$ with respect to $au$ can be written as

$$ \frac{{\partial {{\hat{D}}_e}}}{{\partial au }} = \frac{{\partial {{\hat{D}}_e}}}{{\partial \sigma }}\frac{{\partial \sigma }}{{\partial au }} $$

(4.20)

Since (from (4.18) and (4.19))

$$ \frac{{\partial {{\hat{D}}_{e_1 }}}}{{\partial \sigma }} = {{\mathop {\underline \mu}}^{d_e }} \cdot \delta \cdot {K_{e_T }} > 0 $$

(4.21a)

and

$$ \frac{{\partial {{\hat{D}}_{e_2 }}}}{{\partial \sigma }} = \frac{{(1 - \delta )(1 - {{\mathop {\underline \mu}}^{f_e }})}}{{{{(1 - \sigma )}^2}}}{K_{e_T }} > 0 $$

(4.21b)

then the degree of market power is a decreasing function of $\sigma$.

By differentiating $\sigma$ with respect to $au$ and given that ${\hat{p}_e}$ and that ${c_{e_g }}$ depends on $au$, we get

$$ \begin{array}{clcllclc}\frac{{\partial \sigma }}{{\partial au }} = \frac{1}{{{{({{\hat{p}}_e} - {c_{e_a }} - {r_a} au )}^2}}}\left[ {({r_{{\it AF}}} - {r_a})({{\hat{p}}_e}(0) - {c_{e_a }}) - ({{{\partial {{\hat{p}}_e}}} \left/ {{\partial au }} \right.} - {r_a})({c_{e_g }}(0) - {c_{e_a }})} \right] \end{array}$$

(4.22)

if $au < { au^s}$.

and

$$\begin{array}{clclclc} \frac{{\partial \sigma }}{{\partial au }} = \frac{1}{{{{({{\hat{p}}_e} - {c_{e_g }} - {r_g} au )}^2}}}\left[ {({r_a} - {r_{{\it AF}}})({{\hat{p}}_e}(0) - {c_{e_g }}(0)) - ({{{\partial {{\hat{p}}_e}}} \left/ {{\partial au }} \right.} - {r_{{\it AF}}})({c_{e_a }} - {c_{e_g }}(0))} \right] \end{array}$$

(4.23)

if $au \ge { au^s}$.

From (4.22)

$\displaystyle\frac{{\partial \sigma }}{{\partial au }} > 0$ and $\displaystyle\frac{{\partial {{\hat{D}}_e}}}{{\partial au }} > 0$, if ${{{\partial {{\hat{p}}_e}}} \left/ {{\partial au }} \right.} < \hat{r} = {r_a} + \lambda ({r_{{\it AF}}} - {r_a})$

with $\lambda = {{{({{\hat{p}}_e}(0) - {c_{e_a }})}} \left/ {(} \right.}{c_{e_g }}(0) - {c_{e_a }})$. Vice versa, if ${{{\partial {{\hat{p}}_e}}} \left/ {{\partial au }} \right.} > \hat{r}$ then $\displaystyle\frac{{\partial {{\hat{D}}_e}}}{{\partial au }} < 0$.

From (4.23)

$\displaystyle\frac{{\partial \sigma }}{{\partial au }} > 0$ and $\displaystyle\frac{{\partial {{\hat{D}}_e}}}{{\partial au }} > 0$, if ${{{\partial {{\hat{p}}_e}}} \left/ {{\partial au }} \right.} < \hat{r} = {r_{{\it AF}}} + \lambda ({r_a} - {r_{{\it AF}}})$ with $\lambda = {{{({{\hat{p}}_e}(0) - {c_{e_g }})}} \left({c_{e_a }}(0) - {c_{e_g }}\right)}$. Vice versa, if ${{{\partial {{\hat{p}}_e}}} \left/ {{\partial au }} \right.} > \hat{r}$ then $\displaystyle\frac{{\partial {{\hat{D}}_e}}}{{\partial au }} < 0$.

1.5 Proof of Corollary 3

The following procedure is based on the assumption that the dominant firm considers the profit in each hour of the year separately. In other words, the dominant firm looks at the target in each hour of the year regardless of what may happen in the remaining hours.

When ${D_e} \in \left] {{{\bar{K}}_e};\underline K_e^{f_e }} \right]$ the profit that the dominant firm earns by setting the price threshold before the implementation of the ETS is

$$ {\hat{\pi}_e}(0) = ({\hat{p}_e}(0) - {c_{e_a }}){{ R}}{{{ D}}_e}({\hat{p}_e}(0)) $$

(4.24)

with ${{ R}}{{{ D}}_e} = {D_e} - {K^{f_e }}$.

After the ETS the profit is

$$ {\hat{\pi}_{e_1 }}( au ) = ({\hat{p}_{e_1 }}( au ) - {{ M}}{{{ C}}_{e_a }}){{ R}}{{{ D}}_e}({\hat{p}_{e_1 }}( au )) + au \cdot {r_{bn }} \cdot {\bar{q}^{d_e }} $$

(4.25)

Then, ${\hat{\pi}_{e_1 }}( au ) = {\hat{\pi}_e}(0)$ implies that

$$ \begin{array}{clclcl}({{\hat{p}}_{e_1 }}( au ) - {c_{e_a }}) = ({{\hat{p}}_{e_1 }}(0) - {c_{e_a }})\frac{{{{ R}}{{{ D}}_e}({{\hat{p}}_e}(0))}}{{{{ R}}{{{ D}}_e}({{\hat{p}}_{e_1 }}( au ))}} + \hfill \\ - au \cdot {r_{bn }} \cdot \frac{{\bar{q}_e^{d_e }}}{{{{ R}}{{{ D}}_e}({{\hat{p}}_{e_1 }}( au ))}} + au \cdot {r_a} \end{array}$$

(4.26)

Under full auctioning (${r_{bn }} = 0$), (4.26) implies that $({\hat{p}_e}( au ) - {\hat{p}_e}(0)) > {r_a} \cdot au$. Under benchmarking (${r_{bn }} > 0$), then $({\hat{p}_{e_1 }}( au ) - {\hat{p}_e}(0)) < {r_a} \cdot au$

If the profit target is a short-term one, under auctioning market power never decreases. In fact, in order to maintain the profit constant, the price threshold has to increase more than the increase in marginal cost of the more efficient (more polluting) plants ($\Delta {\hat{p}_e} > \Delta {{ M}}{{{ C}}_a} = {r_a} au$).

Finally it remains to analyze the case in which the dominant firm pursues a long-term profit target (e.g., multiperiod) with the limit to ensure some stability in prices. Consider three periods: the initial one, before the ETS ($au = 0$); the period $t = 1$, after the implementation of the ETS, in which the carbon price increases by $au$ and the subsequent period $t = 2$ in which the carbon price decreases by $au$. The dominant firm aims at maintaining the average profit (over the two periods after the ETS) equal to the initial profit, ${\hat{\pi}_{e_1 }} + {\hat{\pi}_{e_1 }} = 2{\hat{\pi}_e}(0)$, at the same time reducing the price volatility.

If, in the first period, the dominant firm sets a price lower than the price corresponding to the profit invariance, $\hat{p}_1^{*} < {\hat{p}_1}$, the profit loss will be

$$ \begin{array}{clclcl} \Delta {{\hat{\pi}}_{e_1 }}( au ) = \left[ {\hat{p}_{e_1}^{*} - ({c_{e_a }} + {r_a} au )} \right]{{ R}}{{{ D}}_e}(\hat{p}_{e_1}^{*}) + \hfill \\ - \left[ {{{\hat{p}}_e}(0) - {c_{e_a }}} \right]{{ R}}{{{ D}}_e}({{\hat{p}}_e}(0)) < 0 \end{array} $$

In order to keep constant the average profit, in the second period the dominant firm will choose a price, $\hat{p}_{e_2}^{*}$, such that

$$ \begin{array}{clclclclclcl}{ \left[ {\hat{p}_{e_2}^{*} - {c_a}}\right]{{ R}}{{{ D}}_e}(\hat{p}_{e_2}^{*}) + \left[{\hat{p}_{e_1}^{*}( au ) - ({c_{e_a }} + {r_a} au )} \right]{{R}}{{{ D}}_e}(\hat{p}_{e_1}^{*}) =} \\ {= 2\left[ {{\hat{p}}_e}(0) -{c_a}\right]{{ R}}{{{ D}}_e}({\hat{p}_e}(0)) - 2{r_{bn }}{ \ } au{ \ }\bar{q}_e^{d_e }} \end{array}$$

(4.27)

From (4.27), $\hat{p}_{e_2}^{*} < \hat{p}_{e_1}^{*}$ if

$$ \begin{array}{clclcl} \left[ {\hat{p}_{e_1}^{*}( au ) - {c_{e_a }}} \right] > {r_a} au \frac{{{{ R}}{{{ D}}_e}(\hat{p}_{e_1}^{*})}}{{{{ R}}{{{ D}}_e}(\hat{p}_{e_1}^{*}) + {{ R}}{{{ D}}_e}(\hat{p}_{e_2}^{*})}} +\\ +2\left[ {({{\hat{p}}_e}(0) - {c_{e_a }}) - {r_{bn }} au } \right]\frac{{{{ R}}{{{ D}}_e}({{\hat{p}}_e}(0))}}{{{{ R}}{{{ D}}_e}(\hat{p}_{e_1}^{*}) + {{ R}}{{{ D}}_e}(\hat{p}_{e_2}^{*})}} \end{array}$$

(4.28)

Now, by subtracting the expression ${\hat{p}_e}(0) - {c_{e_a }}$ from either the right side or the left side of (4.28), in infinitesimal terms we get

$$ {\left( {\frac{{\partial {{\hat{p}}_e}}}{{\partial au }}} \right)_{t = 1 }} = \mathop{\lim}\limits_{{ au \to 0}} \frac{{\left[ {\hat{p}_{e_1}^{*}( au ) - {{\hat{p}}_e}(0)} \right]}}{ au } > \frac{1}{2}{r_a} - {r_{bn }} $$

Therefore, since ${\hat{p}_e}(0) < \hat{p}_{e_2}^{*}( au ) < \hat{p}_{e_1}^{*}( au )$, in infinitesimal terms if ${\left( {\displaystyle\frac{{\partial {{\hat{p}}_e}}}{{\partial au }}} \right)_{t = 1 }} > \displaystyle\frac{1}{2}{r_a}$ or ${\left( {\displaystyle\frac{{\partial {{\hat{p}}_e}}}{{\partial au }}} \right)_{t = 1 }} > \frac{1}{2}{r_a} - {r_{bn }}$, under auctioning and benchmarking, respectively, then ${\left( {\displaystyle\frac{{\partial {{\hat{p}}_e}}}{{\partial au }}} \right)_{t = 2 }} > - {\left( {\displaystyle\frac{{\partial {{\hat{p}}_e}}}{{\partial au }}} \right)_{t = 1 }}$ .

If firms hold both kinds of plants, it is to be noted that when ${D_e} \in \left] {{D_{e_M }};{{\bar{K}}_e}} \right]$ both kinds of dominant firm’s plants are dispatched: kind $a$ with capacity $K_a^{d_e }$; kind $g$ with capacity ${D_e} - (K_a^{d_e } - {K^{f_e }})$. Therefore, the conditions above become:

$$ {\left( {\frac{{\partial {{\hat{p}}_e}}}{{\partial au }}} \right)_{t = 1 }} > \frac{1}{2}{r_m} $$

or

$$ {\left( {\frac{{\partial {{\hat{p}}_e}}}{{\partial au }}} \right)_{t = 1 }} > \frac{1}{2}{r_m} - {r_{bn }} $$

with

$$ {r_m} = \frac{{r_a K_a^{d_e } + {r_g}\left[ {D_e - (K_a^{d_e } + {K^{f_e }})} \right]}}{{K_a^{d_e } + \left[ {D_e - (K_a^{d_e } + {K^{f_e }})} \right]}} $$

Obviously when the dominant firm holds only plants $a$ then ${r_m} = {r_a}$.

1.6 Proof of Corollary 4

When the dominant firm pursues a short-term target, since $\displaystyle\frac{{\partial {{\hat{p}}_e}}}{{\partial au }} > {r_a}$, the condition for decreasing market power (see Lemma 3bis), $\displaystyle\frac{{\partial {{\hat{p}}_e}}}{{\partial au }} < \hat{r} = {r_a} + \lambda ({r_{{ AF}}} - {r_a})$, becomes

$$ {r_{{ AF}}} > \hat{r} = {r_a}{ \ }\left[ {1 + (\rho - 1)\frac{{D_e - {K^f}}}{K_a^d }} \right] $$

with $\rho > 1$, under taxation or ETS with auctioning.

Conversely, with ETS and benchmarking, the change in the price cap is net of the carbon cost of the benchmark technology, $\displaystyle\frac{{\partial {{\hat{p}}_e}}}{{\partial au }} > {r_a} - {r_{bn }}$. As a consequence the condition for decreasing market power is

$$ {r_{{ AF}}} > \hat{r} = {r_a}{ \ }\left[ {1 + (\rho - 1)\frac{{D_e - {K^f}}}{K_a^d }} \right] - \rho { \ }{r_{bn }}\frac{{D_e - {K^f}}}{K_a^d } $$

1.7 Proof of Corollary 5

When the dominant firm pursues a long-term target then $\displaystyle\frac{{\partial {{\hat{p}}_e}}}{{\partial au }} > \frac{1}{2}{r_a}$. In this case the condition for decreasing market power, $\displaystyle\frac{{\partial {{\hat{p}}_e}}}{{\partial au }} < \hat{r} = {r_a} + \lambda ({r_{{ AF}}} - {r_a})$, becomes

$$ {r_{{ AF}}} > \hat{r} = {r_a}{ \ }\left[ {1 + \left( {\frac{\rho }{2} - 1} \right)\frac{{D_e - {K^f}}}{K_a^d }} \right] $$

with ETS and auctioning.

If allowances are benchmarked then $\displaystyle\frac{{\partial {{\hat{p}}_e}}}{{\partial au }} > \frac{1}{2}{r_a} - {r_{bn }}$. The condition for decreasing market power becomes:

$$ {r_{{ AF}}} > \hat{r} = {r_a}{ \ }\left[ {1 + \left( {\frac{\rho }{2} - 1} \right)\frac{{D_e - {K^f}}}{K_a^d }} \right] - \rho { \ }{r_{bn }}\frac{{D_e - {K^f}}}{K_a^d } $$

1.8 Proof of Equation (4.5)

Given the load duration curves in Figs. 4.4 and 4.5, the total amount of pollutant emissions, $E$, under imperfect competition is

$$ \begin{array}{clclcl}{E_{ic}} = {r_g}\left[ {\int_0^{\hat{H}} {D_e (H,{{\hat{p}}_e}){d}H} } \right] + {r_g}\left[ {\int_{\hat{H}}^{\underline H}} {D_e (H,{{\hat{p}}_e}){{ d}}H} \right] + ({r_g} - {r_a})({K_{e_T }}\hat{H} - {K^{d_e }}{\mathop {\underline H}}) +\cr \quad+{r_a}\left[ {\int_{{{{H}}( au )}}^{{H_L ( au )}} {D_e (H,{{\hat{p}}_e}){{ d}}H} } \right] \end{array}$$

And under full competition

$$ {E_{{ fc}}} = {r_g}\int_0^{{\underline H}} {D_e {{ d}}H} + {r_a}\left[ {\int_{{\underline H}}^{H_L } {D_e {{ d}}H} } \right] $$

By differentiating ${E_{{ ic}}}$ and ${E_{{ fc}}}$ with respect to $au$ and given that

$$ \begin{array}{clclcl}\frac{\partial }{{\partial au }}\left[ {\int_{{H_i ({p_e}( au ))}}^{{H_j ({p_e}( au ))}} {D_e (H,{p_e}( au )){{ d}}H} } \right] = {D_e}({H_j},{p_e}( au ))\frac{{\partial {H_j}}}{{\partial au }} - {D_e}({H_i},p( au ))\frac{{\partial {H_i}}}{{\partial au }} +\\ + \int_{{H_i ({p_e}( au ))}}^{{H_j ({p_e}( au ))}} {\frac{{\partial {D_e}(H,p( au ))}}{{\partial au }}{{ d}}H} \end{array}$$

and that $\beta = - \displaystyle\frac{{\partial {D_e}}}{{\partial {p_e}}} = - {{{\frac{{\partial {D_e}}}{{\partial au }}}} \left/ {{\frac{{\partial {p_e}}}{{\partial au }}}} \right.}$ we get

$$ \begin{array}{clclcl}\frac{{\partial {E_{{ ic}}}}}{{\partial au }} = {r_a}\left[ { - \beta \frac{{\partial {{\hat{p}}_e}}}{{\partial au }}\int_0^{\hat{H}} {D_e {{ d}}H} } \right] + ({r_g} - {r_a})\left[ {({K_{e_T }} - {{\hat{D}}_e})\frac{{\partial \hat{H}}}{{\partial au }}} \right] +\\ \quad+ {r_g}\left[ { - \beta { \ }{r_{{ AF}}}\int_{\hat{H}}^{\mathop {\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{\underline H}} {D_e {{ d}}H} } \right] + {r_a}\left[ { - \beta { \ }{r_{{ AF}}}\int_{\mathop {\raise0.3em\hbox{$\smash{\scriptscriptstyle}$}}{\underline H}}^{H_L } {D_e {{ d}}H} } \right] \end{array}$$

(4.29)

and

$$ \frac{{\partial {E_{{ fc}}}}}{{\partial au }} = {r_g}\left[ { - \beta { \ }{r_g}\int_0^{\underline H}} {D_e {{ d}}H} \right] + {r_a}\left[ { - \beta { \ }{r_a}\int_{\underline H}}^{H_L } {D_e {{ d}}H} \right] $$

(4.30)

Where $\displaystyle\frac{{\partial \hat{H}}}{{\partial au }}$ includes two components: $\displaystyle\frac{\partial H }{{\partial {D_e}}}\frac{{\partial {{\hat{D}}_e}}}{{\partial au }}$ which is the change in $\hat{H}$ due to the change in ${\hat{D}_e}$ (see Lemma 3 and Lemma 3bis); $\displaystyle\frac{\partial H }{{\partial {D_e}}}\frac{{\partial {D_e}}}{{\partial {p_e}}}\frac{{\partial {p_e}}}{{\partial au }} = - \beta \frac{\partial H }{{\partial {D_e}}}\frac{{\partial {p_e}}}{{\partial au }}$ which is the change in $\hat{H}$ to the change in demand.

By replacing these components in (4.29) and by subtracting $\displaystyle\frac{{\partial {E_{{ fc}}}}}{{\partial au }}$ from $\displaystyle\frac{{\partial {E_{{ ic}}}}}{{\partial au }}$ we get the difference in marginal emissions between imperfect and full competition [(4.5) in Section 5].