Energy efficiency subsidies, additionality and incentive compatibility under hidden information

Abstract

This paper examines the ‘additionality’ of energy savings in energy efficiency subsidy programs. ‘Additionality’ refers to the energy savings caused by actions beyond what would have occurred in the absence of the policy program. We characterise energy consumers’ strategic response to the subsidies in a formal adverse selection model and show how the subsidy program may fail to satisfy the additionality criterion. This occurs when energy consumers, who partake in the program, have different preferences for energy efficiency technologies that are unobservable. To resolve this, we propose an incentive compatible solution within the subsidy program that can mitigate the non-additionality problem and improve the effectiveness of the scheme.

Appendices

Appendix 1. Proof of Proposition 2

The incentive compatible constraint (IC) would induce the consumer or project recipient to choose to implement energy efficiency projects according to his true type θ, which means that the first- and second-order condition for the consumer’s optimisation problem must be satisfied at $\widehat{\theta}=\theta$:

$$ {T}^{\prime}\left(\widehat{\theta}\right)+\theta {E}^{\prime}\left(\widehat{\theta}\right)-{I}^{\prime}\left(E\left(\widehat{\theta}\right)\right){E}^{\prime}\left(\widehat{\theta}\right)=0\ \mathrm{FOC} $$

(A1)

$$ {T}^{\prime \prime \left(\widehat{\theta}\right)}+\theta {E}^{\prime \prime}\left(\widehat{\theta}\right)-{I}^{\prime \prime}\left(E\left(\widehat{\theta}\right)\right){\left[{E}^{\prime}\left(\widehat{\theta}\right)\right]}^2-{I}^{\prime}\left(E\left(\widehat{\theta}\right)\right){E}^{\prime \prime}\left(\widehat{\theta}\right)\le 0\ \mathrm{SOC} $$

(A2)

If we further differentiate (A1) with respect to θ, we obtain:

$$ {T}^{\prime \prime \left(\widehat{\theta}\right)}+{E}^{\prime}\left(\theta \right)+\theta {E}^{\prime \prime}\left(\widehat{\theta}\right)-{I}^{\prime \prime}\left(E\left(\widehat{\theta}\right)\right){\left[{E}^{\prime}\left(\widehat{\theta}\right)\right]}^2-{I}^{\prime}\left(E\left(\widehat{\theta}\right)\right){E}^{\prime \prime}\left(\widehat{\theta}\right)=0 $$

(A3)

Taking the result from (A2), the equation of (A3) implies E^′(θ) > 0. To derive the function of the optimal subsidised project, we follow the standard procedure introduced by Mirrlees (1971).

Appling the envelop theorem on the consumer’s objective function (12), we obtain

$$ \frac{d U\left(\widehat{\theta},\theta \right)}{d\theta}=E\left(\widehat{\theta}\right)>0 $$

Therefore, by integrating,

$$ U\left(\theta \right)={\int}_{\theta_{\_}}^{\theta }E(z) dz+E\left({\theta}_{\_}\right) $$

At the optimum, the individual rationality constraint (IR) at the lowest type is binding; therefore,

$$ E\left({\theta}_{\_}\right)=0\ \mathrm{and}\ U\left(\theta \right)={\int}_{\theta_{\_}}^{\theta }E(z) dz. $$

(A4)

From the consumer’s objective function (12), the above equation implies that:

$T\left(\theta \right)=I\left(E\left(\theta \right)\right)-\theta E\left(\theta \right)+{\int}_{\theta_{\_}}^{\overline{\theta}}E(z) dz$. Substituting for T(θ) in the project developer’s objective function, the project developer’s problem becomes:

$$ \underset{E\left(\theta \right)}{\max }{\int}_{\theta_{\_}}^{\overline{\theta}}\left[V\left(E\left(\theta \right)\right)-I\left(E\left(\theta \right)\right)+\theta E\left(\theta \right)-{\int}_{\theta_{\_}}^{\overline{\theta}}E(z) d z\right]f\left(\theta \right) d\theta $$

After integration by parts, we obtain a modified problem

$$ \underset{E\left(\theta \right)}{\max }{\int}_{\theta_{\_}}^{\overline{\theta}}\left[V\left(E\left(\theta \right)\right)-I\left(E\left(\theta \right)\right)-\theta E\left(\theta \right)-\frac{1+F\left(\theta \right)}{f\left(\theta \right)}E\left(\theta \right)\right]f\left(\theta \right) d\theta $$

(A5)

The solution to the maximisation problem is given by the first-order condition with respect to E(θ) under the integral. Thus, we have:

$$ {V}^{\prime}\left(E\left(\theta \right)\right)={I}^{\prime}\left(E\left(\theta \right)\right)-\theta +\frac{1-F\left(\theta \right)}{f\left(\theta \right)} $$

(A6)

Thus, the project developer offers subsidised projects to the consumers such that the marginal benefit from the energy efficiency project equals the marginal net cost while including an informational rent. Because the project developer’s benefit function V is assumed concave in E, the second-order condition for the project developer’s maximisation problem is satisfied.

From Eq. (A6), we can immediately infer that the level of efficiency for the highest top consumer $\overline{\theta}$ is not distorted. However, for consumers whose type is below $\overline{\theta}$, $\frac{1-F\left(\theta \right)}{f\left(\theta \right)}$, it is positive. Since V is concave in E, their subsidised project will be distorted downwards.

Finally, we need to check that the optimal solution in (A6) satisfies the monotonicity constraint for E(θ) in (3). Differentiate (A6) with respect to θ, we get

$$ {E}^{\prime}\left(\theta \right)=\frac{1-\frac{d}{d\theta}\left(\frac{1-F\left(\theta \right)}{f\left(\theta \right)}\right)}{I^{\prime \prime}\left(E\left(\theta \right)\right)-{V}^{\prime \prime}\left(E\left(\theta \right)\right)} $$

(A7)

Since I is the assumed convex and V is the assumed concave in E, the monotonicity constraint E^′(θ) > 0 is satisfied as long as the $\frac{1-F\left(\theta \right)}{f\left(\theta \right)}$ is decreasing in θ.

Appendix 2

Table 1 Table of key variables