The Inefficiency of Private Adaptation to Pollution in the Presence of Endogenous Market Structure

Abstract

The paper considers an industry where production costs rise due to pollution, but where this effect can be partially off-set by investing in adaptation as a private good. The focus is not on external effects, but industries where economies of scale are introduced from adapting to pollution. The structure of the resulting oligopolistic market is endogenous, since the level of adaptation is chosen by the firms. The analysis of externalities usually disregards defensive or adaptation measures, with a few exceptions that point to considerable complications. The present debate on adaptation to climate change shows the importance of understanding defensive measures. I show that the market failure caused by economies of scale leads to production costs above the social optimum, i.e. to under-adapation. When pollution increases, adaptation only increases if demand is price inelastic. Otherwise, welfare loss from market failure decreases with pollution. The total welfare loss is only convex if demand is price inelastic and the influence of pollution on production costs is stronger than the influence of adaptation. Concave welfare loss has crucial implications for abatement policies.

Appendix

1.1 Comparative Statics of Social Planner

The social planner solution is determined by Eqs. (2), (3), here stated again as

$$\begin{aligned}&\displaystyle p(x^*) = \alpha c(a^*,k), \end{aligned}$$

(36)

$$\begin{aligned}&\displaystyle - \alpha c_a(a^*,k) \; x^* = q, \end{aligned}$$

(37)

since $x_i^*=x^*, a_i^*=a^*$. The total differential is

$$\begin{aligned}&\displaystyle p^{\prime } dx = \alpha ( c_a da + c_k dk), \end{aligned}$$

(38)

$$\begin{aligned}&\displaystyle -\frac{1}{\alpha } dq = c_a dx + x c_{aa} da + x c_{ak} dk. \end{aligned}$$

(39)

It follows from Eq. (38) that

$$\begin{aligned} \frac{dx}{da}&= \frac{\alpha c_a}{p^{\prime }}+\frac{\alpha c_k}{p^{\prime }}\frac{dk}{da}, \end{aligned}$$

(40)

$$\begin{aligned} \frac{dx}{dk}&= \frac{\alpha c_a}{p^{\prime }}\frac{da}{dk}+\frac{\alpha c_k}{p^{\prime }}. \end{aligned}$$

(41)

First consider the case where the unit cost of adaptation $q$ changes ceteris paribus, i.e., $dk=0$. It then follows from substituting Eqs. (40) into (39) that

$$\begin{aligned} \frac{dx}{dq}=-\frac{c_a}{\alpha c_a^2 + x p^{\prime } c_{aa}}. \end{aligned}$$

(42)

Equation (42) together with Eq. (40) yields

$$\begin{aligned} \frac{da}{dq}=\frac{dx/dq}{dx/da}=-\frac{1}{\alpha }\frac{p^{\prime }}{\alpha c_a^2 + x p^{\prime } c_{aa}}. \end{aligned}$$

(43)

I now turn to the effect of ceteris paribus changing pollution, i.e. $dq=0$. It follows from Eq. (39) that

$$\begin{aligned} \frac{dx}{dk}=-\frac{x c_{aa}}{c_a}\frac{da}{dk}-\frac{x c_{ak}}{c_a}, \end{aligned}$$

and equating with Eq. (41) yields

$$\begin{aligned} \frac{da}{dk}=-\frac{xp^{\prime }c_{ak}+\alpha c_a c_k}{xp^{\prime }c_{aa}+\alpha c_a^2}, \end{aligned}$$

(44)

and by analogue calculations

$$\begin{aligned} \frac{dx}{dk}=\frac{\alpha x (c_k c_{aa} - c_a c_{ak})}{xp^{\prime }c_{aa}+\alpha c_a^2}. \end{aligned}$$

(45)

These expressions are now simplified using elasticities. Due to Eq. (2)

$$\begin{aligned} x p^{\prime } = \frac{p}{\epsilon _p}=\frac{\alpha c}{\epsilon _p}. \end{aligned}$$

(46)

The (identical) denominator in Eqs. (42)–(45) is thus equal to

$$\begin{aligned} \alpha \frac{c^2}{a^2} \frac{\epsilon _a}{\epsilon _p}(\epsilon _a \epsilon _p + \epsilon _a -1). \end{aligned}$$

This can now be applied to all four equations. Define $u:=(\epsilon _a \epsilon _p + \epsilon _a -1)$. Equation (42) boils down to

$$\begin{aligned} \frac{dx}{dq}=-\frac{\epsilon _p}{\alpha u}\frac{a}{c}. \end{aligned}$$

(47)

With Eqs. (46) and (5),

$$\begin{aligned} \frac{da}{dq}=\frac{a}{q u} \end{aligned}$$

(48)

is obtained. With the pollution elasticity of costs $\epsilon _k = c_k\frac{k}{c} > 0$, the numerator of Eq. (45)

$$\begin{aligned} \alpha x (c_k c_{aa} - c_a c_{ak})=-\alpha \epsilon _a \epsilon _k \frac{c^2 x}{a^2 k} > 0, \end{aligned}$$

and

$$\begin{aligned} \frac{dx}{dk}= - \frac{\epsilon _k \epsilon _p}{u} \frac{x}{k}. \end{aligned}$$

(49)

By Eq. (46), the numerator of Eq. (44) equals

$$\begin{aligned} \alpha \epsilon _a \epsilon _k \frac{1+\epsilon _p}{\epsilon _p} \frac{c^2}{a k}, \end{aligned}$$

yielding

$$\begin{aligned} \frac{da}{dk}=-\frac{\epsilon _k (1+\epsilon _p)}{u}\frac{a}{k}. \end{aligned}$$

(50)

1.2 Comparison of Market and Social Optimum

This section shows that $x_i^+ < x_i^* \Leftrightarrow a_i^+ < a_i^*$.

The inequality $x_i^+ < x_i^*$ implies that

$$\begin{aligned} -\frac{q}{x_i^+}<-\frac{q}{x_i^*}. \end{aligned}$$

Consequently, due to Eqs. (20) and (3), $c_a(a_i^+,k) < c_a(a_i^*,k),$ such that the convexity of $c$ implies $a_i^+ < a_i^*$, being the first direction of the proposition.

Now assume that $a_i^+ < a_i^*$, such that the monotonicity of $c$ results in

$$\begin{aligned} c(a_i^+,k) > c(a_i^*,k) > 0. \end{aligned}$$

Thus also $(1-\epsilon _a) c(a_i^+,k) > c(a_i^*,k)$, since the first term is greater than one. Then Eqs. (19) and (2) imply $p(n^+ x_i^+) > p(n^* x_i^*)$. Since $n^+ > 1 = n^*$, the monotonicity of $p$ implies that $x_i^+ < x_i^*$.

1.3 Proof of the Overall Effects of Increasing Pollution

Proof of Proposition 5.

• (i) The production of a single firm $x_i^+$ decreases with $k$ due to the comparative statics Eq. (24). Since the number of firms is independent of $k$ due to Eq. (18), total production $x^+$ decreases as well.

• (ii) Welfare decreases with pollution by Eq. (30).

• (iii) Under-adaptation for all cases is already stated in Proposition 4.

Proof of Proposition 6.

Adaptation: The difference between case (2) on the one hand, and case (1a), (1b) becomes obvious when comparing with Table 1. Recall that Eqs. (22)–(25) show that the comparative statics for the oligopoly solution have the same signs. Thus, adaptation is increasing with pollution in case (1a), (1b), while in case (2), the opposite holds.

Total welfare loss: Recall that the welfare loss is convex if Eq. (33) holds. In case (2), this is impossible since $\epsilon _p+1<0$, and $u<0$ by assumption. In cases (1a) and (1b) with $0<\epsilon _p+1$, Eq. (33) is simply equivalent to the condition $\epsilon _k<\frac{1}{\epsilon _p+1}-\epsilon _a$.

Welfare loss from market failure: By defining

$$\begin{aligned} v&:= \left( a_i^* - \frac{\epsilon _a-1}{u} n a_i^+ \right) ,\end{aligned}$$

(51)

$$\begin{aligned} \beta&:= \frac{\epsilon _k}{\epsilon _a}q < 0, \end{aligned}$$

(52)

Equation (35) can be written as

$$\begin{aligned} \frac{d \varDelta }{d k} = \beta \frac{v}{k}. \end{aligned}$$

(53)

Now use the elasticities and the comparative statics Eqs. (10), (25) to determine

$$\begin{aligned} \frac{dv}{dk} = -\frac{\epsilon _k(\epsilon _p+1)}{u} \left( a^* - \frac{\epsilon _a-1}{u} n a_i^+ \right) = \mu \frac{v}{k}, \end{aligned}$$

(54)

with

$$\begin{aligned} \mu := -\frac{\epsilon _k(\epsilon _p+1)}{u}. \end{aligned}$$

(55)

Since $u<0,\,\mu$ has the same sign as $(\epsilon _p+1)$. Equation (54) represents a differential equation for $v$ with respect to $k$ that is solved by

$$\begin{aligned} v=v_0 k^{\mu }, \end{aligned}$$

(56)

where $v_0$ is a constant that needs to be chosen properly. The welfare loss from market failure $\varDelta (k)>0$ in the presence of pollution $k$ can then be determined by integrating Eq. (53) with respect to $k$ as

$$\begin{aligned} \varDelta (k)=\int \limits _0^k \beta \frac{v_0 \kappa ^{\mu }}{\kappa } d\kappa = \frac{\beta }{\mu } v_0 k^{\mu }. \end{aligned}$$

(57)

In case (2), $\mu$ is negative, such that Eq. (57) shows that $\varDelta$ is convexly decreasing in $k$ as stated in Table 2. In case (1b), the condition $\epsilon _k<\frac{1}{\epsilon _p+1}-\epsilon _a$ is equivalent to $0 < \mu < 1$, making $\varDelta$ an increasing but concave function in $k$. By the same argument $1<\mu$ in case (1a), yielding a convex function.

It has thus been shown that all the properties given in Table 2 hold under the conditions given in the first row and the assumption that there is an interior solution for the oligopoly market.