Structural Uncertainty and Pollution Control: Optimal Stringency with Unknown Pollution Sources

Abstract

We relax the common assumption that regulators know the structural relationship between emissions and ambient air quality with certainty. We find that uncertainty over this relationship can manifest as a unique form of multiplicative uncertainty in the marginal damages from emissions. We show how the optimal stringency of environmental regulation depends on this structural uncertainty. We also show how new information, like the discovery of previously unknown emission sources, can counterintuitively lead to increases in both optimal emissions and ambient pollution levels.

Appendices

Appendix 1

Proposition 1

An increase in the set of known emitters may decrease the expected level of ambient pollution if (1) marginal physical effects are sufficiently biased upward and (2) the set of new emitters is sufficiently small.

Proof

Let a subset of emitters $K = \alpha _{0}N$ initially be used to estimate marginal effects of emissions on ambient pollution levels where $\alpha _0<1$. Assume all emitters are homogeneous except that some are known and some are unknown. Let estimated transfer coefficients conditional on $\alpha _0$ be $\hat{\beta _0}$ and assume that $\hat{\beta _0}>\beta$.

Expected ambient pollution, conditional on regulation, is $E [y^{*}|\alpha _{0}] = \tilde{X*}'\hat{\beta _0} = \alpha _0 N {\tilde{x}}^{*}_{0} \hat{\beta _0}$ where ${\tilde{x}}^{*}_{0}$ is the regulated level of emissions of known emitters conditional on $\alpha _{0}$ and $\hat{\beta _0}$ given by the first order conditions in Eq. (4).

Given the homogeneity assumption, the new class of unregulated emissions that were emitting at a level ${\tilde{x}}^{*}_{un}$ where ${\tilde{x}}^{*}_{un} > {\tilde{x}}^{*}_{0}$. Assume there are $(\alpha _{1}-\alpha _{0})N$ emitters discovered (e.g., $\alpha _{1}>\alpha _{0}$). Taken together, this implies that total discovered emissions could be either larger or smaller than the existing set of emissions. Lastly, assume that the new estimated transfer coefficient is $\hat{\beta _1} <\hat{\beta _0}$ so that transfer coefficient estimates are revised downward.

The new expected level of ambient pollution before any changes in emissions of regulated and unregulated emitters is $E [y|\alpha _{1}] = (\alpha _{1} -\alpha _{0} )N {\tilde{x}}^{*}_{un} \hat{\beta _1} + \alpha _0 N {\tilde{x}}^{*}_{0} \hat{\beta _1}$.

The sign of the difference between expected emissions ex ante and ex post of discovery can be expressed as:

$$\begin{aligned} sign[E [y|\alpha _{1}] - E [y^{*}|\alpha _{0}]]= & {} sign [(\alpha _{1} -\alpha _{0} )N {\tilde{x}}^{*}_{un} \hat{\beta _1} + \alpha _0 N {\tilde{x}}^{*}_{0} \hat{\beta _1} - \alpha _0 N {\tilde{x}}^{*}_{0} \hat{\beta _0}] \nonumber \\= & {} sign [\alpha _{0}{\tilde{x}}^{*}_{0}(\hat{\beta _1} - \hat{\beta _0})+{\tilde{x}}^{*}_{un}\hat{\beta _1}(\alpha _1 - \alpha _0) ]. \end{aligned}$$

(7)

There are two terms in Eq. (7). The first term describes the decrease in expected emissions from previously identified emitters when the regulator revises their transfer coefficient estimate downward. This term is negative since transfer coefficients are initially biased upward by assumption. The second term is the additional emissions expected from the newly discovered emitters. If the second term dominates expected ambient pollution increases. However, if the first term dominates the second, expected ambient pollution decreases giving the desired result. $\square$

Proposition 2

If expected ambient pollution decreases upon the discovery of new emitters, then previously known emitters’ optimal emission levels strictly increase. Conversely, if expected ambient pollution increases, previously known emitters’ optimal emission levels strictly decrease.

Proof

To determine the impact of new emission sources on known emitter’s optimal pollution levels, Proposition 1 shows we must address both increases and decreases in expected ambient pollution levels. Let initial estimated transfer coefficients be $\hat{\beta _0}$ and the estimated transfer coefficient after new emissions are discovered be $\hat{\beta _1}<\hat{\beta _0}$. Assume that $\alpha _{0}$ increases to $\alpha _{1}$ causing $E [y|\alpha _{1}]< E [y^{*}|\alpha _{0}]$ and damages are convex, $E [D'(y)|\alpha _{1}] < E [D'(y^{*})|\alpha _{0}]$. By construction, the marginal expected damages attributable to previously known emitters in the first order condition after the discovery of new emitters, shown in Eq. (4), is smaller than the marginal expected damages before the discovery:

$$\begin{aligned} E[D'(y|\alpha _{1})]\hat{\beta _1} < E[D'(y^{*}|\alpha _{0})]\hat{\beta _0}. \end{aligned}$$

(8)

The previous level of regulated emissions, therefore, cannot be optimal:

$$\begin{aligned} \pi _{i}'({\tilde{x}}^{*}_{i}|\alpha _{0}) = D'(E[y^{*}|\alpha _{0}])\hat{\beta _0} >D'(E[y|\alpha _{1}])\hat{\beta _1}. \end{aligned}$$

(9)

Concavity of the profit function dictates that the marginal benefit of emissions is too high and it must be that ${\tilde{x}}^{*}_{i}|\alpha _{1}>{\tilde{x}}^{*}_{i}|\alpha _{0}$. By inspection, the converse is also true giving the desired result. $\square$

Proposition 3

An increase in the set of known emitters can decrease the optimal level of ambient pollution if benefits of emissions for newly discovered emissions is sufficiently large (e.g., decreasing emissions is sufficiently costly).

Proof

The objective function of the regulator in this model is to maximize expected social welfare. If the regulator is fully informed, their maximization problem is:

$$\begin{aligned} \max _{\{x\}} E \left[ \Sigma _{i=1}^{N} \pi _{i}(x_{i}) - D(y) \right] , \end{aligned}$$

where $y = \Sigma x_i'\beta _i + \epsilon$ and x is the N x 1 vector of emissions. Differentiating this equation with respect to the N control variables gives the N first order conditions ($E[\pi _i'(x_i^*)] = E[D'(y^*)\beta _i] \; \forall \;i$) and implicitly define the set of optimal emission levels $\{ x^{*} \}$.

Take a simplified case where there are only two emitters i and j. Without loss of generality, assume that $\beta _i=\beta _j = \beta =1$. Assume that both $\pi _i(\cdot )$ and $\pi _j(\cdot )$ are strictly concave and that $\pi _i''(\cdot ) = -k_i$ and $\pi _j''(\cdot ) = -k_j$ so that k indexes the magnitude of the second derivative for each emitter’s profit function. Assume that the regulator only knows of emitter i and has formed an upwardly biased estimate the marginal physical effect of emitter i of ${\hat{\beta }}>1$. Leveraging the linear marginal damage assumption to drop the expectations operator, the regulator will let the following single first order condition set emitter’s i’s:

$$\begin{aligned} \pi _i'(x_{i0}^*) = D'(x_{i0}^*){\hat{\beta }}. \end{aligned}$$

(10)

Note that emitter j will emit at the level defined by $\pi _i'(x_{j0}^*) = 0$ since they are unidentified and therefore unregulated. As a result, before being regulated emitter j’s total emissions are $x_{i0}^* +x_{j0}^*$.

Assume that after identifying emitter j, the regulator correctly identifies marginal physical effects $\beta = 1$. The new optimal levels of regulated emissions are given by

$$\begin{aligned} \pi _i'(x_{i0}^*+ \Delta x_i) = D'(x_{i0}^*+ \Delta x_i + x_{j0}^* - \Delta x_j) \nonumber \\ \pi _j'(x_{j0}^*- \Delta x_j) = D'(x_{i0}^*+ \Delta x_i + x_{j0}^* - \Delta x_j) \end{aligned}$$

(11)

where $\Delta x_i$ and $\Delta x_j$ represent the increase and decrease in emissions after emitter j is identified and $x_{i0}^*+ \Delta x_i + x_{j0}^* - \Delta x_j$ the associated level of emissions. Since $D'(x_{i0}^*+ \Delta x_i + x_{j0}^* - \Delta x_j)>0$ there must be a decrease in emitter j’s emissions due to the concavity of $\pi _j(\cdot )$.

To determine how discovering new emission source j impacts optimal ambient pollution, Proposition 1 and Proposition 2 show we must analyze three cases: no change in emissions after discovery of j, an increase in emissions and finally an decrease in emissions. Assume first that $\Delta x_i - \Delta x_j=0$. In this case there is no change in total emissions after j is discovered and the regulator sets emissions according to Eq. (11). This would occur if ${\hat{\beta }}$ was sufficiently high so that when the marginal damage curve rotates down, the increase in i’s emissions exactly offset the decrease in j’s. This would occur despite emitter j’s emissions ($x_{j0}^* - \Delta x_j$) entering the damages function. Hence, emissions could remain unchanged after the discover of an additional emitter.

Now augment the profit function for emitter j such that $\tilde{\pi _j}'(x_{j0}^*)=0$ at the same point as previous but that $\tilde{\pi _j}''(\cdot ) = -\tilde{k_j}$ where $\tilde{k_j}>k_j$. By assumption, $\tilde{\pi _j}'(x_{j0}^* - \Delta x_j)>\pi _j'(x_{j0}^*- \Delta x_j)$. This cannot be an equilibrium. As a result, emissions from j must increase and emissions from i must decrease thereby leading to an increase in emissions after the discovery of emitter j, correctly revising estimates of marginal physical effects of emissions and subsequently revising regulated emission levels. Note that to arrive at a decrease in emissions let $\tilde{k_j}<k_j$ and the converse is true. This completes the proof. $\square$

Proposition 4

If only a subset of homogeneous emitters is accounted for by the regulator but marginal emissions are estimated unbiasedly, it is optimal to over-regulate known emitters when accounting for structural uncertainty with Bayesian priors relative to when it is not accounted for.

Proof

The first order condition in the full information case is $\pi '(x^{*}) = D'(Nx^{*}\beta )$ which implicitly defines the optimal level of emissions for all emitters. The first order condition when accounting for structural uncertainty, Eq. (6), can be rewritten as $\pi '({\tilde{x}})=E[D'(\alpha N {\tilde{x}}\beta + (1-\alpha )N x^{*}_{un}\beta )\beta |\mu (\alpha )]$. The term $(1-\alpha )N x^{*}_{un}$ will be positive for non-degenerate beliefs. Now assume that ${\tilde{x}} = x^{*}$. Because the term $(1-\alpha )N x^{*}_{un}$ will be positive and the profit function, $\pi (\cdot )$ is strictly concave, it must be the case that $x^{*}_{un}>{\tilde{x}}$, therefore this cannot be an equilibrium. By concavity of the profit function, allowed emissions of known emitters, ${\tilde{x}}$, should be reduced to less than emissions of emitters in the full information case, $x^*$, to satisfy the first order condition (6), thereby completing the proof. $\square$

Proposition 5

For linearly increasing marginal damages and upwardly biased estimates, it is not optimal to regulate known emitters more stringently when accounting for structural uncertainty with Bayesian priors relative to when it is not accounted for if expected marginal damages when accounting for structural uncertainty are lower than when not.

Proof

Assume that the marginal damages from ambient pollution increase at a constant rate c. The first order condition when accounting for structural uncertainty, Eq. (6), can be rewritten as $\pi '({\tilde{x}})=E[c(\alpha N {\tilde{x}}{\hat{\beta }} + (1-\alpha )N x^{*}_{un} {\hat{\beta }}){\hat{\beta }}|\mu (\alpha )]$. Assuming that that bias can be corrected (e.g., $E[{\hat{\beta }}|\alpha ] = \beta$) and simplifying terms, we can further reduce the expression to $\pi '({\tilde{x}})=E[c( \alpha N {\tilde{x}}\beta + (1-\alpha )N x^{*}_{un} \beta )\beta |\mu (\alpha )]$. The term $(1-\alpha )N x^{*}_{un}\beta$ will be positive for non-degenerate beliefs. Take ${\tilde{x}}$ to be the optimal level of emissions of identified emitters when accounting for structural uncertainty.

When structural uncertainty is not accounted for assume that ${\hat{\beta }}>\beta$, the first order condition for the $\alpha N$ known emitters is $\pi '(x^{*}) = E[c(\alpha N x^{*}{\hat{\beta }}){\hat{\beta }}]$ which implicitly defines the optimal level of emissions, $x^*$ for all known emitters.

Further, temporarily assume that ${\tilde{x}} = x^{*}$ so that $\pi '({\tilde{x}})=\pi '(x^{*})$. We can compare the expected marginal damages across regulatory regimes by signing the expression:

$$\begin{aligned} E[c(\alpha N x^{*}{\hat{\beta }}){\hat{\beta }}] - E[c( \alpha N {\tilde{x}}\beta + (1-\alpha )N x^{*}_{un} \beta )\beta |\mu (\alpha )] \end{aligned}$$

(12)

Taking advantage of the linearity and ${\tilde{x}} = x^{*}$ assumptions, Eq. (12) simplifies to $x^*\alpha ({\hat{\beta }}^2 - \beta ^2) - (1-\alpha )x^*_{un}\beta ^2$. The first component of this expression is positive since ${\hat{\beta }}^2 - \beta ^2>0$ by assumption. Note further that positivity requires only that the incorrectly estimated coefficient, ${\hat{\beta }}$, be larger than the corrected coefficient, $\beta$. The second component of this expression is negative since $(1-\alpha )x^*_{un}\beta ^2>0$. As a result, the sign of this expression is determined by the relative magnitude of the bias in $\beta$, the relative size of identified versus unidentified emitters and the relative slope of the profit function (e.g., the difference between ${\tilde{x}}$ and $x^*_{un}$). If the expression is positive, the known emitters must be less intensely regulated when not accounting for bias. The converse is also true. This completes the proof. $\square$

Appendix 2

Regulator Techniques to Account for Incomplete Emission Inventories

In practice, the set of emitters that affect ambient pollution levels at a particular time and place are imperfectly observed. For both uniformly mixing pollutants like methane and non-uniformly pollutants like heavy metals the full set of emissions is unknown (Pacyna et al. 2010; Miller et al. 2013). Regulators sometimes, but not always, acknowledge these imperfections by allowing modelers to use estimates of “fugitive emissions” to account for emission inventories from unmeasurable sources.

In addition to having an incomplete set of emitters, the regulator must also estimate the marginal physical effect of emissions on ambient pollution levels from an incomplete set of emissions. The U.S. EPA and its counterparts in other industrialized countries acknowledge the imperfections of their theoretically driven air dispersion models in some cases. Implemented in 2006, “Response Surface Modeling” uses maximum likelihood statistical techniques to apportion known emissions to ambient pollution levels where atmospheric dispersion models are inaccurate (U.S. EPA 2006; Wang et al. 2011).

This situation is likely the case for many pollutants: in the United States, the U.S. EPA implicitly assumes that their models correctly identify the relationship between historical emissions and current ambient pollution levels. For example, in order to perform forecasts of the effect of new pollution sources on ambient pollution levels, the U.S. EPA typically calibrates its air quality models to the ambient pollution readings at any given monitoring station. The estimated or calibrated parameters are then to forecast the effect of reducing emissions from existing sources or the effect of additional emissions on ambient air quality from siting a new point source at a particular location. This problem may be getting worse rather than better over time: as the number of regulated air and water pollutants increases, the need for additional U.S. EPA air and water dispersion modelers also increases but it is unclear if funding for these modelers increases as well. Conversations with two different economists and two different air modelers at the U.S. EPA suggest that there is a bottleneck at the U.S. EPA with respect to their capacity to develop high quality emission and pollution models despite the best efforts of the atmospheric chemists and modelers currently on staff.

We believe that estimating the relationship between emissions and ambient levels of pollution will often lead biased estimates of the marginal physical effect $\beta$. Assume, for example, that the regulator does not account for the fact that they only observe a subset of the true set of emitters when attributing known emissions to a ambient pollution levels. Intuitively, the regulator will over-attribute known emitters to ambient levels of pollution, leading to upwardly biased estimates of the contribution of emissions to ambient pollution levels.