Prices Versus Quantities: Technology Choice, Uncertainty and Welfare

Abstract

Technological improvements have proven essential in mitigating environmental problems such as climate change, depletion of the ozone layer and acid rain. While it is well-known that price- and quantity-based regulatory instruments provide different investment levels, the effects on the choice between different technologies have received scant attention. This paper expands on the prices versus quantities literature by investigating firms’ technology choice in the face of demand and supply side uncertainty. I show that the regulator can not design tradable emissions permits and an emissions tax such that the two regimes are equivalent, even in terms of expected values. Moreover, a tax encourages the most flexible abatement technology if and only if stochastic costs and the equilibrium permit price have sufficiently strong positive covariance, compared with the variance in consumer demand for the good produced. Finally, the firms’ technology choices are socially optimal under tradable emissions permits, but not under an emissions tax.

Appendix

Proof of Lemma 1. I first prove that $\rho _{\eta }\in \left[ -1/(n-1),1\right]$. A matrix is a valid covariance matrix if and only if it is positive semi-definite. With $n^{\prime }$ identical firms and $\rho _{\eta }=E\left( \eta _{i}\eta _{i^{\prime }}\right) /\sigma _{\eta }^{2}$  the covariance matrix is given by the following $n^{\prime }\times n^{\prime }$ matrix:

$$\begin{aligned} \left( \begin{array}{cccc} 1 &{}\quad \rho _{\eta } &{}\quad \cdots &{}\quad \rho _{\eta }\\ \rho _{\eta } &{}\quad 1 &{}\quad \cdots &{}\quad \rho _{\eta }\\ \vdots &{}\quad \vdots &{}\quad \ddots &{}\quad \vdots \\ \rho _{\eta } &{}\quad \rho _{\eta } &{}\quad \cdots &{}\quad 1 \end{array}\right) . \end{aligned}$$

The determinant of this matrix is given by $\left( 1-\rho _{\eta }\right) ^{n^{\prime }-1}\left( 1+\rho _{\eta }\left( n^{\prime }-1\right) \right)$. It can then be shown that the principal minors of the $n\times n$ covariance matrix satisfy the criteria necessary for positive semi-definiteness if and only if $\rho _{\eta }\in \left[ -1/(n-1),1\right]$ (use the determinant criteria for positive semi-definiteness with the given formula for $n^{\prime }=1,n^{\prime }=2,...,n^{\prime }=n$). The proof that $\rho _{\varepsilon }\in \left[ -1/(m-1),1\right]$ is similar.

Proof of Corollary 2 and derivation of Eqs.  (11), (12) and (14) to (17): In the following, I solve the model under the general cost function $\widetilde{c}(q_{i},a_{i},{\varvec{\alpha }}_{i},{\varvec{\eta }}_{i})$ in order to prove Corollary 2. The needed equations are derived on the way by inserting specification (1) in the general solutions. Firm $i\in N$ solves:

$$\begin{aligned} \widetilde{\Pi }_{i}=\max _{\alpha _{i}}E\left[ \max _{q_{i},a_{i}} \left[ pq_{i}-\widetilde{c}(q_{i},a_{i}, {\varvec{\alpha }}_{i} ,{\varvec{\eta }}_{i})-w(q_{i}-a_{i})\right] -\widetilde{k}\left( {\varvec{\alpha }}_{i}\right) \right] . \end{aligned}$$

The solution to inner maximization problem (with ${\varvec{\alpha }}_{i}$ given) yields the firm’s first order conditions in period 3:

$$\begin{aligned} p-\widetilde{c}_{q}-w&= 0,\\ -\widetilde{c}_{a}+w&= 0. \end{aligned}$$

Let ${\varvec{\alpha }}_{i}=\left( \alpha _{1i},\alpha _{2i},...,\alpha _{Vi}\right)$. By the envelope theorem, the first order conditions in period 2 of firm $i\in N$ are:

$$\begin{aligned} \frac{d\widetilde{\Pi }_{i}}{d\alpha _{vi}}=E\left( \frac{d\widetilde{c}}{d\alpha _{vi}}\right) +\frac{d\widetilde{k}}{d\alpha _{vi}}=0, \forall v\in \left\{ 1,2,...,V\right\} . \end{aligned}$$

(18)

In the particular case with $\widetilde{c}(q_{i},a_{i},{\varvec{\alpha }}_{i},{\varvec{\eta }}_{i})=c(q_{i},a_{i})$ (see Eq. 1) and $\widetilde{k}\left( \alpha _{i}\right) =k(\alpha _{i},\beta _{i})$, Eq. (18) yields:

$$\begin{aligned} -\frac{dk}{d\alpha }&= E\left( \frac{dc}{d\alpha }\right) =E(a), \\ -\frac{dk}{d\beta }&= E\left( \frac{dc}{d\beta }\right) = \frac{1}{2}E\left( a_{i}^{2}\right) =\frac{1}{2} \left( var(a)+\left( E(a)\right) ^{2}\right) , \end{aligned}$$

which are the firms’ first order conditions (11) and (12) given in the text.

The social planner solves:

$$\begin{aligned} \widetilde{W}=\max _{{\varvec{\alpha }}_{i}}E\left[ \sum _{j\in M}\left( bq_{j}-\frac{d}{2}q_{j}^{2}+\varepsilon _{j}q_{j}\right) \!-\!\sum _{i\in N}\left( \widetilde{c}\left( q_{i},a_{i},{\varvec{\alpha }}_{i}, {\varvec{\eta }}_{i}\right) +\widetilde{k}\left( {\varvec{\alpha }}_{i}\right) \right) -G(Z)+X\right] , \end{aligned}$$

with

$$\begin{aligned} X=p\left( \sum _{i\in N}q_{i}-\sum _{j\in M}q_{j}\right) +w\sum _{i\in N}\left( q_{i}-q_{i}+a_{i}-a_{i}\right) =0. \end{aligned}$$

Differentiating wrt. ${\varvec{\alpha }}_{i}$, we get:

$$\begin{aligned} \frac{d\widetilde{W}}{d\alpha _{i}}&= E\left[ \sum _{j\in M}(b-dq_{j}+\varepsilon )\frac{dq_{j}}{d\alpha _{vi}}-\sum _{i\in N} \left( \widetilde{c}_{q}\frac{dq_{i}}{d\alpha _{vi}}+\widetilde{c}_{a} \frac{da_{i}}{d\alpha _{vi}}+\frac{d\widetilde{c}}{d\alpha _{vi}}+ \frac{d\widetilde{k}}{d\alpha _{vi}}\right) \right. \\&\left. \qquad -\frac{dG}{dZ}\frac{dZ}{d\alpha _{vi}}+\frac{dX}{d\alpha _{vi}}\right] , \end{aligned}$$

for all $i\in N$ and $v\in \left\{ 1,2,...,V\right\}$. Calculating $\frac{dX}{d\alpha _{vi}}$ and rearranging yields:

$$\begin{aligned} \frac{d\widetilde{W}}{d\alpha _{i}}&= E\left[ \sum _{j\in M}(b-dq_{j}+\varepsilon -p)\frac{dq_{j}}{d\alpha _{vi}}+\sum _{i\in N}\left( \left( p-\widetilde{c}_{q}-w\right) \frac{dq_{i}}{d\alpha _{vi}}+ \left( w-\widetilde{c}_{a}\right) \frac{da_{i}}{d\alpha _{vi}}\right. \right. \\&\left. \left. \qquad -\frac{d\widetilde{c}}{d\alpha _{vi}} -\frac{d\widetilde{k}}{d\alpha _{vi}}\right) \right] +E\left[ \frac{dG}{dZ}\frac{dZ}{d\alpha _{vi}}+w\sum _{i\in N} \left( \frac{dq_{i}}{d\alpha _{vi}}-\frac{da_{i}}{d\alpha _{vi}}\right) \right] \\&= E\left[ \sum _{i\in N}\left( -\frac{d\widetilde{c}}{d\alpha _{vi}} -\frac{d\widetilde{k}}{d\alpha _{vi}}\right) +\left( w-\frac{dG}{dZ}\right) \frac{dZ}{d\alpha _{vi}}\right] , \end{aligned}$$

where I used the firm’s first order conditions in period 3 given above and Eq. (6). The social planner’s first order conditions $\frac{d\widetilde{W}}{d\alpha _{i}}=0$ then become:

$$\begin{aligned} -\frac{d\widetilde{k}}{d\alpha _{vi}}=E\left[ \frac{d\widetilde{c}}{d\alpha _{vi}}\right] +\frac{1}{n}E\left( \frac{dG}{dZ}-w\right) \frac{dZ}{d\alpha _{vi}}, \forall v\in \left\{ 1,2,...,V\right\} , \end{aligned}$$

(19)

for all $i\in N$. Under emissions trading, aggregate emissions $Z=S$ is exogenous to the technology choice parameter vector ${\varvec{\alpha }}_{i}$. Therefore, $\frac{dZ}{d\alpha _{vi}}=0$ in Eq. (19). It follows that the first order conditions of the social planner (19) are equal to the first order conditions of the firms given in Eq. (18). The same is not true under an emissions tax, because emissions are endogenous and $\frac{dZ}{d\alpha _{vi}}\ne 0$ in general. Corollary 2 follows.

In the particular case with $\widetilde{c}(q_{i},a_{i},{\varvec{\alpha }}_{i},{\varvec{\eta }}_{i})=c(q_{i},a_{i})$ (see Eq. 1) and $\widetilde{k}\left( {\varvec{\alpha }}_{i}\right) =k(\alpha _{i},\beta _{i})$, Eq. (19) becomes:

$$\begin{aligned} -\frac{ourdk}{d\alpha }&= E(a)+\frac{1}{n}E\left( \frac{dG}{dZ}-w\right) \frac{dZ}{d\alpha }, \\ -\frac{dk}{d\beta }&= \frac{1}{2}E\left( a_{i}^{2}\right) +\frac{1}{n}E \left( \frac{dG}{dZ}-w\right) \frac{dZ}{d\beta }. \end{aligned}$$

Under emissions trading we have $\frac{dZ}{d\alpha }=\frac{dZ}{d\beta }=0$, while an emissions tax yields $\frac{dZ}{d\alpha }=0$ and $\frac{dZ}{d\beta }=\frac{-1}{\beta ^{2}}\sum _{i\in N}\eta _{i}$ (cf. Table 1). Hence, $\frac{1}{n}E\left[ \left( \frac{dG}{dZ}-\tau \right) \frac{dZ}{d\beta }\right] =\frac{-1}{n\beta ^{2}}E\left[ \frac{dG}{dZ}\sum _{i\in N}\eta _{i}\right] =\frac{-1}{n\beta ^{2}}cov\left[ \frac{dG}{dZ},\sum _{i\in N}\eta _{i}\right]$. The Eqs. (14)–(17) follows (remember that $E(a_{i}^{2})=var(a_{i})+(E(a_{i}))^{2}$).

Proof of Proposition 1. I first derive the second order conditions to the maximization problem (10). The Hessian is given by:

$$\begin{aligned} \left( \begin{array}{cc} \Pi _{\alpha \alpha } &{} \Pi _{\alpha \beta }\\ \Pi _{\alpha \beta } &{} \Pi _{\beta \beta } \end{array}\right) =\left( \begin{array}{cc} -k_{\alpha \alpha }-\frac{dE(a)}{d\alpha } &{}\quad -k_{\alpha \beta }-\frac{dE(a)}{d\beta }\\ -k_{\alpha \beta }-\frac{1}{2}\frac{dE(a^{2})}{d\alpha } &{}\quad -k_{\beta \beta }-\frac{1}{2}\frac{dE(a^{2})}{d\beta } \end{array}\right) , \end{aligned}$$

where $\frac{1}{2}\frac{dE(a^{2})}{d\alpha }=\frac{dE(a)}{d\beta }=-E(a)V_{3}$, cf. Table 2. The conditions for the Hessian to be negative semi-definite are $\Pi _{\alpha \alpha }\le 0$ and $\Pi _{\alpha \alpha }\Pi _{\beta \beta }-\Pi _{\alpha \beta }^{2}\ge 0$. I assume the last equation holds with strict inequality.

We observe from Eqs. (11), (12) and Table 2 that the regulatory regimes induce equal technology if and only if $Var(a_{\sigma })=Var(a_{\tau })$ when $S_{\sigma }=S_{\tau }$. Differentiating Eqs. (11) and (12) wrt. $z\in \left\{ \sigma _{\eta },\sigma _{\varepsilon },\rho _{\eta },\rho _{\varepsilon }\right\}$, i.e., increasing $Var(a)$ using exogenous parameters, we get:

$$\begin{aligned} -k_{\alpha \alpha }\frac{d\alpha }{dz}-\frac{dE(a)}{d\alpha }\frac{d\alpha }{dz}-k_{\alpha \beta }\frac{d\beta }{dz}-\frac{dE(a)}{d\alpha }\frac{d\beta }{dz}&= 0, \\ -k_{\alpha \beta }\frac{d\alpha }{dz}-\frac{1}{2}\frac{dE(a^{2})}{d\alpha }\frac{d\alpha }{dz}-k_{\beta \beta }\frac{d\beta }{dz}-\frac{1}{2}\frac{dE(a^{2})}{d\beta }\frac{d\beta }{dz}&= \frac{1}{2}\frac{dE(a^{2})}{dz}. \end{aligned}$$

Or, equivalently, using matrix notation and the definitions from the Hessian:

$$\begin{aligned} \left( \begin{array}{cc} \Pi _{\alpha \alpha } &{} \Pi _{\alpha \beta }\\ \Pi _{\alpha \beta } &{} \Pi _{\beta \beta } \end{array}\right) \left( \begin{array}{c} \frac{d\alpha }{dz}\\ \frac{d\beta }{dz} \end{array}\right) =\left( \begin{array}{c} 0\\ \frac{1}{2}\frac{dVar(a)}{dz} \end{array}\right) , \end{aligned}$$

where I also used $\frac{dE(a^{2})}{dz}=\frac{dVar(a)}{dz}$ (cf. Table 2). Solving for the changes in the technology parameters, we get:

$$\begin{aligned} \left( \begin{array}{c} \frac{d\alpha }{dz}\\ \frac{d\beta }{dz} \end{array}\right) =\frac{1}{\Pi _{\alpha \alpha }\Pi _{\beta \beta }-\Pi _{\alpha \beta }^{2}}\left( \begin{array}{c} -\frac{1}{2}\frac{dVar(a)}{dz}\Pi _{\alpha \beta }\\ \frac{1}{2}\frac{dVar(a)}{dz}\Pi _{\alpha \alpha } \end{array}\right) . \end{aligned}$$

It then follows from the second order conditions that $\frac{d\alpha }{dz}\le (\ge )0\Leftrightarrow \Pi _{\alpha \beta }\ge (\le )0$ and that $\frac{d\beta }{dz}<0$. Hence, $\beta _{\sigma }\le \left( \ge \right) \beta _{\tau }$ if and only if $Var(a_{\sigma })\ge \left( \le \right) Var(a_{\tau })$. This proves the proposition.

Proof of Corollary 1: Without loss of generality, specify $\widetilde{c}(q_{i},a_{i},\alpha _{\mathbf{i}},\eta _{i})=\gamma \widehat{c}(q_{i},a_{i},\alpha _{\mathbf{i}},\eta _{i})$ $+c(q_{i},a_{i})$, where $\gamma \in \left[ 0,\infty \right)$ is a constant and $c(q_{i},a_{i})$ is given by Eq. (1). The proof of Proposition 1 proves Corollary 1 in the particular case with $\gamma =0$. The Corollary follows because ambiguity in the particular case with $\gamma =0$ implies ambiguity in the general case $\gamma \in \left[ 0,\infty \right)$.