The Design of Emission Taxes in Markets with New Firm Acquisitions

Abstract

In the 1990s there was a great deal of interest in the study of the role of endogenous market structure under oligopoly in the characterization of emission taxes. This interest was instrumental in providing policy guidance on the design of emission taxes based on market characteristics. However, the literature has been silent on offering policy recommendations on the design of emission taxes under endogenous market structure in the presence of new firm acquisitions. We build a model where new firms enter the market where some are acquired by an incumbent multi-plant firm, altering the initial market structure. In this framework, we characterize the second-best emission tax and examine the role of the resulting market structure, in particular the role of acquiring more/fewer of the new firms, in the optimal design of emission tax. We argue that, under certain conditions, the acquisition of new firms may lead to higher taxation consistent with the Pigouvian rule or even exceed marginal damages. Our contribution is at the intersection of emission tax design and M &A (new firm acquisition) literature.

Appendices

Appendices

Appendix A: Comparative Statics

We first characterize the equilibrium via backward induction. First, from Eqs. (1)–(4) we obtain $q^{o}(k,m,t)$, q(k, m, t). Second, from equation $\pi ={\hat{\pi }}$ and using $q^{o}(k,m,t)$, q(k, m, t), we characterize k(m, t). Third, from equation $\pi ^{o}=0$ along with $q^{o}(k,m,t)$, q(k, m, t), k(m, t) we characterize m(t). Finally, substituting m(t) back into k(m, t) yields k(t); and substituting m(t) and k(t) into $q^{o}(k,m,t)$, q(k, m, t) characterizes the equilibrium level of number of new firms, share of new firm acquisition and output of each insider and outsider.

With this equilibrium in mind, differentiation of (1)–(4), $\pi ={\hat{\pi }}$, $\pi ^{o}=0$, the two inverse demand functions under symmetry ($P=\alpha -\beta q (N+1+km)-\gamma q^{o}(M+m(1-k))$ and $P^{o}=\alpha -\beta q^{o}(M+m(1-k)) -\gamma q(N+1+km)$) and using the market clearing conditions, $D=q(N+1+km)$ and $D^{o}=q^{o}(M+(m(1-k))$, gives four equations in four unknowns, $dD^{o}$, dD, dm, dk:

$$\begin{aligned} -\beta ^{o}(\mu +1) dD^{o}- \gamma \mu dD + \beta ^{o}q^{o}(1-k)dm -\beta ^{o}q^{o}m dk&= \mu \sigma ^{o} dt \end{aligned}$$

(A.1)

$$\begin{aligned} -\gamma dD^{o}- 2\beta dD&= \sigma dt \end{aligned}$$

(A.2)

$$\begin{aligned} \beta ^{o}q^{o}(1-\mu ) dD^{o}- q^{o}\gamma \mu dD -\beta ^{o}(q^{o})^{2}(1-k)dm +\beta ^{o}(q^{o})^{2}m dk&= e^{o}\mu dt \end{aligned}$$

(A.3)

$$\begin{aligned} -\gamma q dD^{o}- \beta q^{2}kdm - \beta q^{2}m dk&= e dt + d{\hat{\pi }} \end{aligned}$$

(A.4)

where $\mu = M+(1-k)m$ represents the total number of outsiders.

Next, we offer a detailed derivation of (A.1)–(A.4). But first, consider $D=q(N+1+km)$ and $D^{o}=q^{o}(M+(m(1-k))$. Hence,

$$\begin{aligned} dD^{o}&=\mu dq^{o}+q^{o}(1-k)dm-q^{o}mdk \end{aligned}$$

(A.5)

$$\begin{aligned} dD&= qkdm+qmdk+(N+1+km)dq \end{aligned}$$

(A.6)

Hence,

$$\begin{aligned} \mu dq^{o}&= dD^{o}-q^{o}(1-k)dm+q^{o}mdk \end{aligned}$$

(A.7)

$$\begin{aligned} (N+1+km)dq&= -qkdm-qmdk+(N+1+km)dD \end{aligned}$$

(A.8)

Next, to derive (A.1) combine (1) and (2), and impose $D=q(N+1+km)$ and $D^{o}=q^{o}(M+(m(1-k))$. This gives $\alpha -\beta q^{o} -\beta D^{o}-\gamma D-\sigma ^{o}t=0$. Differentiation gives

$$\begin{aligned} -\beta dq^{o} -\beta dD^{o}-\gamma dD-\sigma ^{o}dt=0 \end{aligned}$$

(A.9)

where substituting (A.7) into (A.9) and collecting terms gives Eq. (A.1).

Next, to derive (A.2) combine (3) and (4), and impose $D=q(N+1+km)$ and $D^{o}=q^{o}(M+(m(1-k))$. This gives $\alpha -2\beta D -\gamma D^{o}-\sigma t=0$. Differentiation gives

$$\begin{aligned} -2\beta dD -\gamma dD^{o}-\sigma dt=0 \end{aligned}$$

(A.10)

This is (A.2).

Next, to derive (A.4) consider ${\hat{\pi }}=\pi$, where $\pi _{j}=\pi$, $\forall j$. Impose $D=q(N+1+km)$ and $D^{o}=q^{o}(M+(m(1-k))$ into profits $\pi$. Simplifying profits gives ${\hat{\pi }}=(\alpha -\beta D -\gamma D^{o}-\sigma t)q+t^2/2-F$. Differentiation gives

$$\begin{aligned} d{\hat{\pi }} =(-\beta dD -\gamma dD^{o}-\sigma dt)q +(\alpha -\beta D -\gamma D^{o}-\sigma t)dq +tdt \end{aligned}$$

(A.11)

where (i) imposing first-order-condition $\beta q(N+1+km) =\alpha -\beta D -\gamma D^{o}-\sigma t$, where $\beta q(N+1+km)=\beta D$; (ii) simplifying $-\sigma dt +tdt$ using $e=\sigma q -t$; and (iii) substituting (A.8) gives

$$\begin{aligned} d{\hat{\pi }} = -\gamma q dD^{o} -\beta q^{2} k dm -\beta q^{2}mdk-edt \end{aligned}$$

(A.12)

This is Eq. (A.4).

Next, to derive (A.3) consider $\pi ^{o}=0$. Impose $D=q(N+1+km)$ and $D^{o}=q^{o}(M+(m(1-k))$ into profits $\pi ^{o}$. Simplifying profits gives $\pi ^{o}=(P^{o}-c-\sigma ^{o}t)q^{o}+t^2/2-F$. Differentiation and imposing first-order condition $\beta ^{o}q^{o}$ gives

$$\begin{aligned} (-\beta ^{o} dD^{o} -\gamma dD)q^{o} +(\beta ^{o}q^{o})dq^{o} -e^{o}dt=0 \end{aligned}$$

(A.13)

where substituting (A.7) and simplifying gives (A.3).

We assume $d{\hat{\pi }}$ is zero; that is, each insider’s reservation profit is small relative to the market and thus ${\hat{\pi }}$ constant. Using (A.1)–(A.4) yields $\rho dD^{o}=[t\beta /q^{o}-(2\beta \sigma ^{o}-\gamma \sigma )]dt <0$, $\rho dD=[t\gamma /2q^{o}-(\beta ^{o}\sigma -\gamma \sigma ^{o})]dt<0$. In addition, $\rho dm = \left[ \left( \frac{-2(\beta ^{o}\sigma -\gamma \sigma ^{o})}{q} - \frac{(2\beta \sigma ^{o}-\gamma \sigma )}{q^{o}} \right) + t \eta _{m}\right] dt$, $\rho m dk=\left[ (2\beta \sigma ^{o}-\gamma \sigma )(qk-(1-k)q^{o})+t\eta _{k}\right] dt$ where $\rho =2\beta \beta ^{o}-\gamma ^{2}>0$, $\eta _{m}>0$ and $\eta _{k}>0$ are complicated expressions, which denote the abatement effect; for example, $\eta _{k} = 2\beta \beta ^{o}\left( 2\beta ^{o}q^{{o}^{2}}(1-k)-(\mu +1)\beta q^{2}k\right) -\gamma ^{2}\left( 2\beta ^{o}q^{{o}^{2}}(1-k)-\beta q^{2}k\right)$.

The effect of the tax on total emissions is given by $\frac{\partial E}{\partial t} = \sigma \frac{\partial D}{\partial t} + \sigma ^{o} \frac{\partial D^{o}}{\partial t} - (N+1+km) - (M+ (1-k)m) - t\frac{\partial m}{\partial t}<0$ by Assumption 2.1 (i.e., the last term is small).

We now turn to the comparative statics exercise for the case where the cost function is general. Consider a cost function C(q, e), which satisfies (subscripts denote partial derivatives) $C_{q}>0$, $C_{qq}>0$, $-C_{e}>0$, $C_{ee}>0$, $-C_{eq}=-C_{qe}>0$, $C_{qq}C_{ee}-C_{qe}C_{eq}\ge 0$. These are standard properties of the cost function with abatement (see Requate 2006, p. 126). Similar to the comparative statics exercise explained earlier (but now with a more general cost function) we obtain the following system of equations:

$$\begin{aligned} -\left( \beta ^{o}(\mu +1)+\lambda \right) dD^{o}- \gamma \mu dD + q^{o}(1-k)\left( \beta ^{o}+\lambda \right) dm -q^{o}m\left( \beta ^{o}+\lambda \right) dk&= \mu \sigma ^{o} dt\\ -\gamma dD^{o}- \left( 2\beta +\frac{\lambda }{N+1+km}\right) dD +qk\frac{\lambda }{N+1+km}dm +qm\frac{\lambda }{N+1+km}dk&= \sigma dt\\ \beta ^{o}q^{o}(1-\mu ) dD^{o}- q^{o}\gamma \mu dD -\beta ^{o}(q^{o})^{2}(1-k)dm +\beta ^{o}(q^{o})^{2}m dk&= e^{o}\mu dt\\ -\gamma q dD^{o}- \beta q^{2}kdm - \beta q^{2}m dk&= e dt + d{\hat{\pi }} \end{aligned}$$

where $\lambda =(C_{ee}C_{qq}-C_{eq}C_{qe})/C_{ee}$, $\sigma =-C_{eq}/C_{ee}$, $\lambda ^{o}=(C^{o}_{e^{o}e^{o}}C^{o}_{q^{o}q^{o}}-C^{o}_{e^{o}q^{o}}C^{o}_{q^{o}e^{o}})/C^{o}_{e^{o}e^{o}}$, $\sigma ^{o}=-C^{o}_{e^{o}q^{o}}/C^{o}_{e^{o}e^{o}}$. It is noteworthy that (i) in the case of end-of-pipe $\lambda =0$ in which case we obtain Eqs. (A.1) and (A.2), and (ii) the last two equations in the system are identical to (A.3) and (A.4) so the presence of a general cost function does not play any role. To illustrate the role of the general cost function in the comparative statics we show the expression for $dD^{o}$:

$$\begin{aligned} \frac{H}{\mu 2\beta ^{o}\beta q^{{o}^{2}} q^{2}m}\frac{dD^{o}}{dt}&= -(2\beta \sigma ^{o}-\gamma \sigma )+\beta t/q^{o}\nonumber \\&\quad +\frac{1}{2\beta ^{o}}\big (-\lambda ^{o} (2\beta \sigma ^{o}-\gamma \sigma +2\beta t/q^{o})\nonumber \\&\quad +\lambda (-2\beta \sigma ^{o}+t\beta ^{o}/q^{o})-e^{o}\lambda \lambda ^{o}/q^{o}\big )\nonumber \\&\quad -\frac{e\gamma \lambda \lambda ^{o}}{2\beta ^{o}\beta q^{o}q} \end{aligned}$$

(A.14)

where $H>0$ is the determinant of the coefficient matrix and $\mu =M+(1-k)m$ is defined as before. The first line in (A.14) is the same as in the case of an end-of-pipe cost function, which we consider to be negative to capture the standard case where the tax lowers output i.e., small abatement effect, $t\beta /q^{o}$. And the second and third lines capture the role of the general cost function, where the second and third lines are negative. The second and third lines vanish in the case of an end-of-pipe cost function since $\lambda =0$, $\lambda ^{o}=0$. As a result, the presence of a general cost function does not change the qualitative results of $dD^{o}/dt$. An analogous expression is obtained for dD/dt.

Appendix B: Derivation of Fig. 3

We first argue the condition under which $\partial m/\partial k>0$. This is the case we consider in Fig. 3 i.e., acquisitions generate profits so that more firms enter the market. This happens if $\pi ^{o^{\prime }}_{k}+\pi _{k}>0$. In other words, acquisitions of firms by the incumbent offsets any profits loss of the outsiders which induces more firms into the market. Formally, differentiation of $\pi (m,k) +\pi ^{o^{\prime }}(m,k)=0$ gives $dm/dk = -(\pi ^{o^{\prime }}_{k}+\pi _{k})/(\pi ^{o^{\prime }}_{m}+\pi _{m})>0$, where the denominator is negative.

Part (i)—Intersection point of solid lines at ${\hat{t}}$ in Fig. 3 requires $\partial k/\partial t$ to be bounded. That is, the effect of the tax on the acquisition of new firms can’t be too large. This is because with too large an effect the emission tax at the intersection point in the figure would not be possible since the number of firms determined via the zero-profit condition would be too large (too many firms would be attracted by $\partial k/\partial t$) relative to the second-best optimal one. First, we show that $\partial k/\partial t<\eta _{1}$, where $\eta _{1}$ is defined below. (a) Consider $\pi ^{o^{\prime }}(m,t,k)$, whence $-dm/dt=(\pi ^{o^{\prime }}_{t}/\pi ^{o^{\prime }}_{m})+(\pi ^{o^{\prime }}_{k}/\pi ^{o^{\prime }}_{m})k_{t}$, where $k_{t}=\partial k/\partial t$. (b) Consider $W^{\prime }_{m}(m,t,m(k))$, whence $-dm/dt=(W^{\prime }_{mt}/W^{\prime }_{mm})+(m_{k})k_{t}$. Then, intersection point in Fig. 3 requires the absolute value of dm/dt from (a) to exceed that from (b). That is,

$$\begin{aligned} \partial k/\partial t<\frac{(\pi ^{o^{\prime }}_{t}/\pi ^{o^{\prime }}_{m})-(W^{\prime }_{mt}/W^{\prime }_{mm}) }{m_{k}-(\pi ^{o^{\prime }}_{k}/\pi ^{o^{\prime }}_{m})}=\eta _{1} \end{aligned}$$

(B.1)

Part (ii)—The welfare-maximizing tax in the presence of acquisitions, $t^{*}$, lies between one (i.e., marginal damages, which is equal to one in the case of Fig. 3) and the tax that equates the welfare-maximizing number of firms and the free-entry number of firms, ${\hat{t}}$. The condition which ensures $t^{*}>1$ is given by (7) where $\varphi ^{\prime }=1$ and we label the RHS as $\eta _{2}$. Hence, $1<t^{*}<{\hat{t}}$, if $\eta _{2}<\partial k/\partial t<\eta _{1}$; that is, $\partial k/\partial t$ is large but not too large i.e., $\partial k/\partial t$ is bounded.

Appendix C: Illustration of the Condition in Lemma 2.2 and the Case Where Abatement Effects are Large

Consider parameter values i.e., $\alpha =1$, $c=0$, $\beta =\beta ^o=\gamma =1$, $M=4$, $N=1$, ${\hat{\pi }}=1/12$, $F=1/1000$. We use Mathematica to solve the model via backward induction (as explained at the beginning of Appendix A) and illustrate the condition in Lemma 2.2. For given range of the emission tax, m is a decreasing function of the emission tax (which is the case we focus on the paper, where abatement effects are small), while k can be an increasing or decreasing function of the emission tax for this same range. Using the solution of the model we derive the share $q^o/(q+q^o)$ as a function of the emission tax, which we find to be approximately $30\%$. This is our threshold output share referred to in the paper. With these in mind and using the condition $k> q^o/(q+q^o)$ in Lemma 2.2, our results indicate that for given range of the emission tax, k increases with the emission tax and so $k> q^o/(q+q^o) = 30$%. But k decreases with the emission tax and so $k< q^o/(q+q^o) = 30$%. See the Figure below.

Fig. 4

Number of new firms, m, share of new firm acquisitions, k and the emission tax, t: $t\in (.01,0.2)$

Figure 4 shows that for $t\in (.01,0.2)$, $\partial m/\partial t<0$ i.e., abatement effect is small. For this very same range of t, $\partial k/\partial t$ can be either positive of negative as discussed in the main body of the article.

Now, for $t\in (0.2,0.4)$, abatement effects are large enough and so $\partial m/\partial t>0$, $\partial k/\partial t>0$. A large enough abatement effect implies that additional firms enter the market via an increase in the emission tax and, also, an increase in the share of new firm acquisitions. The reason for this is that higher profits (via the large abatement effect) prompts firms to enter the market and the incumbent firm to acquire a larger share of the now more profitable firms.

Next, we describe the solution to the model we used to derive the above figures and share $q^o/(q+q^o) = 30\%$. First, we combine first-order conditions (1) and (2) into one equation; we also substitute the demand function, P, into this newly derived equation. Second, we do the same with Eqs. (3) and (4), where we substitute $P^{o}$. From these two equations we solve simultaneously for q(m, k, t), $q^{o}(m,k,t)$. Second, we substitute the expressions for q(m, k, t), $q^{o}(m,k,t)$ obtained in the previous step into equation ${\hat{\pi }}=\pi$ (where $\pi _{j}=\pi$, $\forall j$). This yields k(m, t). Third, we simplify the zero-profit condition, $\pi ^{o}=0$, using Eqs. (3) and (4); then, we substitute $q^{o}(m,k,t)$ and k(m, t), which yields m(t). Fourth, substituting m(t) back into k(m, t) gives k(t). Subsequently, we substitute m(t) and k(t) back into q(m, k, t), $q^{o}(m,k,t)$, which gives q(t) and $q^{o}(t)$. We then use q(t) and $q^{o}(t)$ to calculate the share $q^{o}/(q^{o}+q)$ for range $t\in (.01,0.3)$. This share is approximately at $30\%$.