Emission permits, innovation and sanction in an evolutionary game

Abstract

This paper studies the evolutionary dynamics of a market regulated by an auctioned emission trading system with a price floor in which there exist three populations of firms that interact strategically: (i) non-polluting, (ii) polluting and compliant, (iii) polluting but non-compliant. Firms that adopt a non-polluting technology need no permits to operate, while firms that use a polluting technology can either buy the required permits (and be compliant) or not (being non-compliant). The latter do not buy emission permits and face the risk to be sanctioned if discovered. From the analysis of the model emerges that all three types of firms coexist at the equilibrium only under specific parameter values. More precisely, it can generically be excluded the coexistence between non-polluting firms and non-compliant polluting ones. The regulatory authority can favor the extinction of non-compliant firms by increasing their probability of being discovered and/or the sanction level. Moreover, the regulatory authority can favor the diffusion of innovation by increasing the permits price floor since polluting-compliant firms exit the market and new non-polluting firms enter the market. However, this policy instrument should be used with caution because it tends to increase also the number of non-compliant firms.

Mathematical appendix

1.1 First order conditions

By imposing non negativity constraints $q_{P}\geqslant 0$, $q_{PP}\geqslant 0$, $q_{NP}\geqslant 0$, and applying Khun-Tucker conditions, we obtain that given the values of the variables m, n, s the quantities $q_{NP},q_{P},q_{PP}$ are chosen according to the first order conditions:

$$\begin{aligned} \frac{\partial \pi _{P}}{\partial q_{P}}= \,& {} p-C_{P}^{V}\cdot q_{P}-p_{T}\leqslant 0 \text {, }\frac{\partial \pi _{P}}{\partial q_{P}}\cdot q_{P}=0 \end{aligned}$$

(14)

$$\begin{aligned} \frac{\partial \pi _{PP}}{\partial q_{PP}}=\, & {} p-C_{P}^{V}\cdot q_{PP}\leqslant 0 \text {, }\frac{\partial \pi _{PP}}{\partial q_{PP}}\cdot q_{PP}=0 \end{aligned}$$

(15)

$$\begin{aligned} \frac{\partial \pi _{NP}}{\partial q_{NP}}=\, & {} p-C_{NP}^{V}\cdot q_{NP}\leqslant 0 \text {, }\frac{\partial \pi _{NP}}{\partial q_{NP}}\cdot q_{NP}=0 \end{aligned}$$

(16)

Remark 1

From (14) and (15) it follows that $q_{P}\leqslant q_{PP}$ always holds.

Remark 2

Consider the case in which there exist values of $q_{NP}>0$ and $q_{PP}>0$ that satisfy conditions (15), (16) and, therefore, equations:

$$\begin{aligned} \frac{\partial \pi _{PP}}{\partial q_{PP}}=\, & {} p-C_{P}^{V}\cdot q_{PP}=0 \\ \frac{\partial \pi _{NP}}{\partial q_{NP}}=\, & {} p-C_{NP}^{V}\cdot q_{NP}=0 \end{aligned}$$

From such equations, it follows:

$$\begin{aligned} p-C_{P}^{V}\cdot q_{PP}=p-C_{NP}^{V}\cdot q_{NP} \end{aligned}$$

and then:

$$\begin{aligned} q_{NP}=\frac{C_{P}^{V}}{C_{NP}^{V}}\cdot q_{PP} \end{aligned}$$

(17)

From Eq. (17) it results that if $q_{PP}$ becomes equal to zero, then also $q_{NP}$ becomes equal to zero, and vice-versa; moreover, from Remark 1, it occurs also to $q_{P}$. Consequently, taking into account that $p=Q-a\cdot \left( m\cdot q_{NP}+n\cdot q_{P}+s\cdot q_{PP}\right) >0$ when the quantities $q_{NP}$, $q_{P}$, $q_{PP}$ are sufficiently low, it occurs that PP-firms and NP-firms never choose $q_{PP}=0$ and $q_{NP}=0$, respectively.⁷ This implies that NP-firms and PP-firms always produce strictly positive quantities, regardless of the values of variables m, n, s.

1.2 Proof of Proposition 1

Taking into account Remark 2 and Eq. (17), in order to determine the values $q_{NP}$, $q_{P}$, $q_{PP}$, it is sufficient to analyze only conditions (14), (15). Differently from (15), (16), condition (14) can be satisfied also for $q_{P}=0$. Therefore, P-firms, which buy emission permits, could choose to produce a zero output, given m, n, s; such an event can occur only if $\overline{p_{T}}>0$, and, therefore, $p_{T}>0$ also for $q_{P}=0$.

Consider first the case in which (14) is satisfied for $q_{P}>0$. In such a context, conditions (14), (15) become:

$$\begin{aligned} p-C_{P}^{V}\cdot q_{P}-p_{T}= & {} 0 \end{aligned}$$

(18)

$$\begin{aligned} p-C_{P}^{V}\cdot q_{PP}= & {} 0 \end{aligned}$$

(19)

From (18), (19) it derives:

$$\begin{aligned} p-C_{P}^{V}\cdot q_{P}-p_{T}=p-C_{P}^{V}\cdot q_{PP} \end{aligned}$$

from which:

$$\begin{aligned} C_{P}^{V}\cdot q_{P}+p_{T}=C_{P}^{V}\cdot q_{PP} \end{aligned}$$

(20)

Substituting (5) in (20), it becomes:

$$\begin{aligned} C_{P}^{V}\cdot q_{P}+\overline{p_{T}}+d\cdot q_{P}\cdot n=C_{P}^{V}\cdot q_{PP} \end{aligned}$$

from which we obtain:

$$\begin{aligned} q_{PP}=\frac{\overline{p_{T}}}{C_{P}^{V}}+\left( 1+\frac{d}{C_{P}^{V}}\cdot n\right) \cdot q_{P} \end{aligned}$$

(21)

Finally, substituting (4) and (21) in (18), and taking account of (17), we obtain:

$$\begin{aligned}&Q-a\cdot \frac{\overline{p_{T}}}{C_{P}^{V}}\cdot \left( \frac{C_{P}^{V}}{ C_{NP}^{V}}\cdot m+s\right) -\overline{p_{T}}\nonumber \\=\, & {} q_{P}\cdot \left[ C_{P}^{V}+(a+d)\cdot n+a\cdot \left( \frac{C_{P}^{V}}{C_{NP}^{V}}\cdot m+s\right) \right. \nonumber \\&\left. \cdot \left( 1+\frac{d}{C_{P}^{V}}\cdot n\right) \right] \end{aligned}$$

(22)

Equation (22) determines the equilibrium value of $q_{P}$. Once determined $q_{P}^{*}$, the equilibrium values of $q_{PP}$ and $q_{NP}$ are respectively determined by Eqs. (21) and (17). This completes the proof of Proposition 1.

1.3 The separatrix between the regimes with $q_{P}>0$ and with $q_{P}=0$

Setting $q_{P}=0$, Eq. (22) becomes:

$$\begin{aligned} \frac{C_{P}^{V}}{C_{NP}^{V}}\cdot m+s=\frac{C_{P}^{V}}{a\cdot \overline{p_{T} }}\cdot \left( Q-\overline{p_{T}}\right) \end{aligned}$$

(23)

Equation (23) represents the plane (in the space (m, n, s)) that separates the region in which $q_{P}>0$ (below the plane) from that in which $q_{P}=0$ (above the plane). If the values of m and/or s are sufficiently high to satisfy conditions $\left( C_{P}^{V}/C_{NP}^{V}\right) \cdot m+s\geqslant \left( C_{P}^{V}/a\cdot \overline{ p_{T}}\right) \cdot \left( Q-\overline{p_{T}}\right)$, then the polluting-compliant firms choose $q_{P}=0$.

1.4 Proof of Proposition 2

To prove Proposition 2, notice that when $q_{P}=0$, the equilibrium values of $q_{PP}$ and $q_{NP}$ are determined by (19), taking into account Eq. (17), and that the demand function can be rewritten as $p=Q-a\cdot \left( \frac{C_{P}^{V}}{C_{NP}^{V}}\cdot m+s\right) \cdot q_{PP}$.

1.5 Proof of Proposition 3

Below the plane represented by Eq. (23), that is in the region where $q_{P}>0$, we have $\dot{s}=0$ and $\dot{m}=0$ if the following equations are respectively satisfied:

$$\begin{aligned} \pi _{PP}= & {} -C_{P}^{F}-T+p\cdot q_{PP}-\frac{C_{P}^{V}}{2}\cdot q_{PP}^{2}=0 \end{aligned}$$

(24)

$$\begin{aligned} \pi _{NP}= & {} -C_{NP}^{F}+p\cdot q_{NP}-\frac{C_{NP}^{V}}{2}\cdot q_{NP}^{2}=0 \end{aligned}$$

(25)

Using condition (17), Eq. (25) can be rewritten as:

$$\begin{aligned} -C_{NP}^{F}+p\cdot \frac{C_{P}^{V}}{C_{NP}^{V}}\cdot q_{PP}-\frac{C_{NP}^{V} }{2}\cdot \left( \frac{C_{P}^{V}}{C_{NP}^{V}}\right) ^{2}\cdot q_{PP}^{2}=0 \end{aligned}$$

Multiplying both sides for $C_{NP}^{V}/C_{P}^{V}$, we obtain:

$$\begin{aligned} \frac{C_{NP}^{V}}{C_{P}^{V}}\cdot \left[ -C_{NP}^{F}+p\cdot \frac{C_{P}^{V}}{ C_{NP}^{V}}\cdot q_{PP}-\frac{C_{NP}^{V}}{2}\cdot \left( \frac{C_{P}^{V}}{ C_{NP}^{V}}\right) ^{2}\cdot q_{PP}^{2}\right] =0 \end{aligned}$$

from which:

$$\begin{aligned} -C_{NP}^{F}\cdot \frac{C_{NP}^{V}}{C_{P}^{V}}+p\cdot q_{PP}-\frac{C_{P}^{V}}{ 2}\cdot q_{PP}^{2}=0 \end{aligned}$$

(26)

Notice that in Eqs. (25), (26), the coefficients of $q_{PP}^{2}$ and $q_{PP}$ coincide, therefore a value of $q_{PP}$ such that $\overset{\cdot }{s}=\overset{\cdot }{m}=0$ can exist only in the case in which the following equality

$$\begin{aligned} T=C_{NP}^{F}\cdot \frac{C_{NP}^{V}}{C_{P}^{V}}-C_{P}^{F} \end{aligned}$$

holds. This proves Proposition 3.

1.6 Stability and phase planes

Figure 8 shows three numerical examples where only one stationary state is attractive, $P_{2}$ in Fig. 8a, $P_{4}$ in Fig. 8b, and $P_{6}$ in Fig. 8c. In the figures, an empty dot “$\circ$” indicates a non attractive stationary state (that is, with at least one eigenvalue with positive real part), while a full dot “$\bullet$” denotes a locally attractive stationary state (that is, with 3 eigenvalues with negative real part).

The phase space corresponding to the numerical example in which $P_{6}$ (where PP-firms are not present) is attractive is shown in Fig. 9. Differently, Fig. 10 illustrates a non generic case (the equality condition (13) is satisfied) in which all three types of firms coexist and there exists a segment (the black line that links $P_{4}$ to $P_{6}$) constituted by (Lyapunov) stable stationary states. Conversely, the red line that links $P_{2}$ to $P_{5}$ is constituted by unstable stationary states located in the invariant plane $n=0$ where there are not P-firms.

Fig. 8

Numerical examples in which stationary states of type $P_{2}$ (Fig. 8a), $P_{4}$ (Fig. 8b) and $P_{6}$ (Fig. 8c) are attractive

Fig. 9

Phase space in which there is coexistence between NP-firms and PP-firms ($P_{6}$ is attractive)

Fig. 10

Phase space in which all three types of firms coexist