Optimal Climate Change Policies When Governments Cannot Commit

Abstract

We analyse the optimal design of climate change policies when a government wants to encourage the private sector to undertake significant immediate investment in developing cleaner technologies, but the relevant carbon taxes (or other environmental policies) that would incentivise such investment by firms will be set in the future. We assume that the current government cannot commit to long-term carbon taxes, and so both it and the private sector face the possibility that the government in power in the future may give different (relative) weight to environmental damage costs. We show that this lack of commitment has a significant asymmetric effect: it increases the social benefits of the current government to have the investment undertaken, but reduces the private benefit to the private sector to invest. Consequently the current government may need to use additional policy instruments—such as R&D subsidies—to stimulate the required investment.

Appendix

In this appendix we set out the results for the special case of our model in which the elasticity of demand, $\varepsilon (x)$ is a constant, $\varepsilon$. Thus we assume that the benefit function takes the form:

$$\begin{aligned} B(x)\equiv B_0 +\frac{B_1 }{(1-\beta )}x^{1-\beta }\;\quad B_0 >0,\;B_1 >0,\beta >1 \end{aligned}$$

where $\beta =-\frac{x{B}^{\prime \prime }(x)}{{B}^{\prime }(x)}=\frac{1}{\varepsilon }$ is the constant elasticity of marginal benefits (inverse of the elasticity of demand) and $B_0$ is assumed to be sufficiently large that $B(x)\ge 0$ all relevant values of $x$. Clearly $\beta >1\Leftrightarrow 0<\varepsilon <1$ (demand is inelastic). ${B}^{\prime }(x)=B_1 x^{-\beta }$.

1.1 Second Period

From (3) in the text, optimal output in the second period, $\hat{{x}}(e,\omega )$ is given by:

$$\begin{aligned} {B}^{\prime }(\hat{{x}}(e,\omega ))=B_1 \hat{{x}}(e,\omega )^{-\beta }=e\cdot \omega \cdot \delta \end{aligned}$$

(26)

$$\begin{aligned} \text{ so } :\quad \hat{{x}}(e,\omega )=\left( \frac{B_1 }{\delta }\right) ^{\varepsilon }(e\cdot \omega )^{-\varepsilon } \end{aligned}$$

(27)

1.2 First Period

$\Pi ^{n}(e,\omega )=B_0 +\frac{B_1 }{1-\beta }\cdot \hat{{x}}(e,\omega )^{1-\beta }-\omega \cdot \delta \cdot e\cdot \hat{{x}}(e,\omega ).$ Substituting (27) and rearranging:

$$\begin{aligned} \Pi ^{n}(e,\omega )=B_0 +\left( \frac{1}{(\varepsilon -1}\right) B_1 ^{\varepsilon }\delta ^{1-\varepsilon }(e\cdot \omega )^{1-\varepsilon } \end{aligned}$$

(28)

and

$$\begin{aligned} \Delta \Pi ^{n}(\omega )=\Pi ^{n}(e_L ,\omega )-\Pi ^{n}(e_H ,\omega )=\eta (\varepsilon )\omega ^{1-\varepsilon } \end{aligned}$$

(29)

where

$$\begin{aligned} \eta (\varepsilon )\equiv \left\{ \frac{1}{1-\varepsilon }B_1 ^{\varepsilon }\delta ^{1-\varepsilon }\left( e_H ^{1-\varepsilon }-e_L ^{1-\varepsilon }\right) \right\} \end{aligned}$$

(30)

$W^{n}(e,\omega )=B_0 +\frac{B_1 }{(1-\beta )}\cdot \hat{{x}}(e,\omega )^{1-\beta }-\delta \cdot e\cdot \hat{{x}}(e,\omega ).$ Substituting (27) and rearranging:

$$\begin{aligned} W^{n}(e,\omega )=B_0 +\frac{1}{\varepsilon -1}B_1 ^{\varepsilon }\delta ^{1-\varepsilon }e^{1-\varepsilon }[\varepsilon \omega ^{1-\varepsilon }+(1-\varepsilon )\omega ^{-\varepsilon }] \end{aligned}$$

(31)

and:

$$\begin{aligned} \Delta W^{n}(\omega )&= W^{n}(e_L ,\omega )-W^{n}(e_H ,\omega ) \nonumber \\&= \eta (\varepsilon )[\varepsilon \omega ^{1-\varepsilon }+(1-\varepsilon )\omega ^{-\varepsilon }] \end{aligned}$$

(32)

From (30) and (32) it is straightforward to see that:

$$\begin{aligned}&\Delta \Pi ^{n}(0)=0;\;\Delta \Pi ^{n}(1)=\eta (e);\Delta \Pi ^{{n}^{\prime } }(\omega )>0;\quad \Delta \Pi ^{{n}^{\prime \prime }}(\omega )<0;\\&\omega \rightarrow 0\Rightarrow \Delta W^{n}(\omega )\rightarrow \infty ;\quad \Delta W^{n}(1)=\eta (\varepsilon )\\&\Delta W^{{n}^{\prime }}(\omega )=\eta (\varepsilon )(1-\varepsilon )\varepsilon \omega ^{-(1+\omega )}(\omega -1)\le 0\Leftrightarrow \omega \le 1;\\&\Delta W^{{n}^{\prime \prime }}(\omega )=\eta (\varepsilon )\varepsilon (1-\varepsilon )\omega ^{-(\varepsilon +2)}[(1+\varepsilon )-\varepsilon \omega ]\ge 0\Leftrightarrow \omega \le \tilde{\omega }\equiv (1+\varepsilon )/\varepsilon \ge 2\\&\Delta W^{n}(\omega )-\Delta \Pi ^{n}(\omega )=\eta (\varepsilon )(1-\varepsilon )\omega ^{-\varepsilon }(1-\omega )\ge 0\Leftrightarrow \omega \le 1 \end{aligned}$$

1.3 Expected Profit and Welfare Differences

We suppose that $\omega$ is distributed over the range [0, 2] with either a uniform density function:

$$\begin{aligned} f^{U}(\omega )\equiv 0.5 \end{aligned}$$

(33)

or a triangular density function:

$$\begin{aligned} f^{T}(\omega )\equiv \omega \quad \quad \quad 0\le \omega \le 1\end{aligned}$$

(34a)

$$\begin{aligned} f^{T}(\omega )\equiv 2-\omega \quad \quad 1\le \omega \le 2 \end{aligned}$$

(34b)

We chose these two functions because they differ with respect to the weight in the tails of the two density functions, with the uniform density having more weight in the tails than the triangular density function.

Then for a function of the form: $G(\omega )\equiv \omega ^{\mu }$ we have the following expected values:

$$\begin{aligned} \text{ Uniform } : E[G(\omega )\left| \mu \right. ]&= \frac{2^{\mu }}{(1+\mu )}\end{aligned}$$

(35a)

$$\begin{aligned} \text{ Triangular } : E[G(\omega )\left| \mu \right. ]&= \frac{2(2^{1+\mu }-1)}{(1+\mu )(2+\mu )} \end{aligned}$$

(35b)

Applying (35) to the expressions in (29) and (32) we have:

$$\begin{aligned} \text{ Uniform } : E[\Delta \Pi ^{n}(\omega )]=\eta (\varepsilon )\frac{2^{1-\varepsilon }}{(2-\varepsilon )};\quad E[\Delta W^{n}(\omega )]=\eta (\varepsilon )\frac{2^{-\varepsilon }(2+\varepsilon )}{2-\varepsilon }; \end{aligned}$$

(36)

Triangular:

$$\begin{aligned} E[\Delta \Pi ^{n}(\omega )]&= \frac{2\eta (\varepsilon )}{(2-\varepsilon )(3-\varepsilon )}[2^{2-\varepsilon }-1]; \nonumber \\ E[\Delta W^{n}(\omega )]&= \frac{2\eta (\varepsilon )}{(2-\varepsilon )(3-\omega )}[2^{1-\varepsilon }(3+\varepsilon )-3]. \end{aligned}$$

(37)

1.4 Calculation of Required Subsidy

Since expected profits are below expected welfare, we are interested in the subsidy required to align private and social incentives for R&D. For given values of the parameters $\bar{{B}}, \delta , e_{H}$, and $e_{L}$ the required (absolute) subsidy is $AS(\varepsilon )\equiv \{E[\Delta W^{n}(\varepsilon )]-E[\Delta \Pi ^{n}(\varepsilon )]\}$.

We are interested in how the required subsidy varies with different values of the demand elasticity $\varepsilon$, holding constant the remaining parameters $\bar{{B}}, \delta , e_{H}$, and $e_{L}$. Note that the two terms in the formula for AS($\varepsilon )$ have a common factor $\eta (\varepsilon )$, which is also equal to the gain in profits from R&D with commitment, $\Delta \Pi ^{c}$. So for simplicity we shall focus on the relative subsidy, $S\equiv \{AS(\varepsilon )/\Delta \Pi ^{c}\}=\{E[\Delta W^{n}(\varepsilon )]-E[\Delta \Pi ^{n}(\varepsilon )]\}/\Delta \Pi ^{c}\}$, the absolute R&D subsidy relative to the gain in profits from R&D with commitment, which does not depend on $\eta (\varepsilon )$. Using the formulae (36) and (37) for the uniform and triangular density functions, the required relative subsidies as a proportion of expected profit gain from R&D with commitment are:

$$\begin{aligned} \text{ Uniform } : S^{U}(\varepsilon )&\equiv \frac{2^{-\varepsilon }\varepsilon }{2-\varepsilon };\quad S^{U}(0)=0;\quad S^{U}(1)=0.5;\\ \text{ Triangular } : S^{T}(\varepsilon )&= \frac{2}{(2-\varepsilon )(3-\varepsilon )}[(1+\varepsilon )2^{1-\varepsilon }-2];\quad S^{T}(0)=0;\quad S^{T}(1)=0 \end{aligned}$$

In the Table 1 we present values for the key variables for values of the demand elasticity $\varepsilon$ lying between 0 and 1. Clearly the size of the relative subsidy depends critically on the weight in the tails of the two density functions.

Table 1 Values of key variables for different values of elasticity of demand