Abstract
The paper considers an industry where production costs rise due to pollution, but where this effect can be partially offset 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 underadapation. 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.
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Notes
An alternative would be to compare the market equilibrium with a secondbest solution where the social planner only decides about the number of firms (as, e.g. Mankiw and Whinston 1986).
The following proposition also holds when the Cournot equilibrium is compared with a secondbest solution. Mankiw and Whinston (1986) show under fairly general conditions that the secondbest number of firms \({\bar{n}} < n^+\). This can again be reduced to a counterfactual social planner solution as above by setting \(\alpha =(1+1/({\bar{n}} \epsilon _p))^{1}\) and \({\tilde{q}} = \alpha q n\). It follows that \(1<\alpha <(1\epsilon _a)\). When \({\bar{n}}>1\), the secondbest resembles a counterfactual social optimum with more pollution and more expensive adaptation than in the first best, but with less pollution and cheaper adaptation than in the Cournot equilibrium.
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Acknowledgments
The author wants to thank Heinz Welsch for a valuable hint. This paper is a work of the Chameleon Research Group (www.climatechameleon. de), funded by the German Ministry for Education and Research under grant 01UU0910.
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Appendix
Appendix
1.1 Comparative Statics of Social Planner
The social planner solution is determined by Eqs. (2), (3), here stated again as
since \(x_i^*=x^*, a_i^*=a^*\). The total differential is
It follows from Eq. (38) that
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
Equation (42) together with Eq. (40) yields
I now turn to the effect of ceteris paribus changing pollution, i.e. \(dq=0\). It follows from Eq. (39) that
and equating with Eq. (41) yields
and by analogue calculations
These expressions are now simplified using elasticities. Due to Eq. (2)
The (identical) denominator in Eqs. (42)–(45) is thus equal to
This can now be applied to all four equations. Define \(u:=(\epsilon _a \epsilon _p + \epsilon _a 1)\). Equation (42) boils down to
is obtained. With the pollution elasticity of costs \(\epsilon _k = c_k\frac{k}{c} > 0\), the numerator of Eq. (45)
and
By Eq. (46), the numerator of Eq. (44) equals
yielding
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
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
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)
Underadaptation 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
Equation (35) can be written as
Now use the elasticities and the comparative statics Eqs. (10), (25) to determine
with
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
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
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.
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Eisenack, K. The Inefficiency of Private Adaptation to Pollution in the Presence of Endogenous Market Structure. Environ Resource Econ 57, 81–99 (2014). https://doi.org/10.1007/s1064001396676
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DOI: https://doi.org/10.1007/s1064001396676