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Controlling Episodic Air Pollution with a Seasonal Gas Tax: The Case of Cache Valley, Utah


Using daily data spanning 10 years, we establish a statistical relationship between episodic particulate-matter \((\hbox {PM}_{2.5})\) concentrations and vehicle trips in Cache Valley, Utah, and estimate an average gas-price elasticity for the region. We also estimate the benefits and costs associated with a seasonal gas tax set to reduce vehicle trips during the winter-inversion season and thereby lower health costs through concomitant decreases in the \(\hbox {PM}_{2.5}\) concentrations. We find a strong positive relationship between vehicle trips reduced and associated reductions in \(\hbox {PM}_{2.5}\) concentrations. Further, we estimate a mean gas price elasticity of approximately \(-\)0.3 in what we call a “high price variability environment.” Incorporating these results, cost-benefit analysis suggests that the social net benefit for Cache Valley associated with the imposition of a seasonal gas tax during the winter-inversion season is highly dependent upon the type of benefit estimation method used.

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  1. Logan is Cache Valley’s largest city. In 2009, Logan’s population consisted of 46,000 people residing in 16,000 households (U.S. Census Bureau, 2010).

  2. For example, a study conducted by Iowa State University shows that ammonia emissions can be reduced 40–50 % by using biofiltration in animal housing areas (Shih et al. 2006). A biofilter, which is a porous layer of organic material that supports a population of microbes, funnels dirty air from animal housing areas and converts the pollution to carbon dioxide and water (Nicolai 2011). At a cost of approximately $200 per filter, and annual operating and maintenance costs typically not exceeding $5–$10 per filter, biofiltration seems to be a cost-effective way to control ammonia emissions (Nicolai 2011).

  3. Cache Valley has been in “non-attainment status” with the EPA for \(\hbox {PM}_{2.5}\) concentrations since 2008 due to its winter-inversion problem (EPA 2014).

  4. This gas-price elasticity estimate corresponds closely to those reported in the literature. We briefly discuss this literature in Sect. 4.

  5. Because it provides an aggregate measure of trips per day across households, we acknowledge that our trip count measure is a better proxy for changes in vehicle miles traveled (VMT) on the extensive margin (e.g. a reduction in the number of trips taken very likely implies a reduction in VMT) than on the intensive margin (i.e., a reduction in VMT per trip only implies a reduction in trips if the now shortened trip no longer passes—and is therefore no longer registered by—one of the county’s ATR stations). As a result, our trip count measure likely underestimates the effect of the latter margin, and, therefore, to the extent that (1) a gas tax induces a requisite number of households to switch to more fuel-efficient vehicles, and (2) driving a more fuel-efficient vehicle reduces emissions per mile, the ultimate relationship that we estimate in this paper between changes in gasoline prices and changes in \(\hbox {PM}_{2.5}\) contributions (as linked by our estimated gas-price elasticity) should, all else equal, also be considered a lower-bound estimate. This in turn suggests that our gas tax estimate(s) are more likely upper-bound than lower-bound estimates. We thank an anonymous reviewer for sharing this insight.

  6. This creates a scenario in which there is a potential for endogeneity in our \(\hbox {PM}_{2.5}\) regression model. We explore this problem at length in the following section.

  7. Augmented Dickey–Fuller and Phillips–Perron tests indicate that each of the variables used in our regression analyses are stationary (Becketti 2013). This was expected given that the yearly timeframes of analysis are restricted solely to the winter months.

  8. Based on admittedly vague empirical evidence, the precursors for \(\hbox {PM}_{2.5}\) are, at least to some extent, cumulative during inversion conditions. Perrino (2010), Seinfeld and Pandis (2012), and Gillies and Wang (2014) estimate that particulates ranging in size from 0.1–1 \(\upmu \hbox {m}\) accumulate in the atmosphere over the course of weeks. Larger particulate sizes—such as \(\hbox {PM}_{2.5}\)—are generally associated with shorter residence times. More specifically, Schmel (2015) estimates average residence times of 25 h, three days, and 10 days for particulates consisting of \(\hbox {SO}_{2}\), sulfate, and ammonia, respectively. Taken together, these estimates suggest that particulate matter is indeed a cumulative problem in Cache Valley during the winter inversion season, i.e. a given winter day’s concentration level is at least partially the result of lagged emissions. Given the inherent variability associated with particulate matter’s atmospheric residence times in any given location at any given time, various lag lengths were tested for Trip_Count in the \(\hbox {PM}_{2.5}\) regressions. The results for these regressions were qualitatively similar to those reported in Sect. 4.

  9. We tested several alternative specifications of the model described in (1) using different combinations and transformations of the explanatory variables listed in Table 2. We did not include stationary-source emissions as a separate explanatory variable for two reasons. First and foremost, we were unable to obtain these emissions estimates on a daily basis from the UDEQ. Second, because PM2.5 is effectively incorporating the effects of stationary-source emissions (in terms of their contributions to concentration levels), the coefficient on our Trip_Count variable can effectively be interpreted as the percentage of \(\hbox {PM}_{2.5}\) concentrations explained by mobile sources, on average.

  10. Durbin’s alternative (Durbin 1970) and Breusch–Godfrey (Godfrey 1978) tests along with an auxiliary regression of the residuals indicated the existence of serial correlation (AR1) in the OLS version of Eq. (1).

  11. As discussed in footnote 7, standard statistical tests indicate that the variables used in the ensuing analysis are stationary. Thus, we set \(d = 0\) in each of our regressions (i.e., we do not difference any of the variables used in our regressions).

  12. We thank an anonymous reviewer for sharing this insight.

  13. As with the \(\hbox {PM}_{2.5}\) regression model, we estimated several alternative specifications of (2).

  14. Although explicit control for seasonality in PM2.5 [Eq. (1)] was unnecessary, we note that in our \(\hbox {PM}_{2.5}\) regression model moving average processes eliminate a potential repetitive seasonal component in the dependent variable (Maddala and Kim 1998). In the IV \(\hbox {PM}_{2.5}\) regression equation, we use a HAC weighting matrix to correct for autocorrelation, the main source of which is seasonality (Baum et al. 2007).

  15. A 4-month interval was necessary in order to obtain a large enough subsample for econometric estimation.

  16. Based on their assessment of the literature, Parry and Small (2005) adopt a central value of \(-\)0.4 and a range of \(-\)0.2 to \(-\)0.6 for the gas price elasticity measure used in their assessment of efficient gas tax rates in the US and Britain. See Parry et al. (2007) for a comprehensive listing of price-elasticity studies.

  17. Bento et al. (2009) argue that the main difference between their results and those of Austin and Dinan (2005) is based upon how the marginal costs of engineering improvements are calibrated. The latter study uses cost estimates provided by the National Research Council (NRC), while the former “calibrates....marginal a way that reconciles observed automobile choices with the assumptions of profit and utility maximization, [yielding] marginal costs of improving fuel economy that are larger than the marginal costs implied by the NRC study” (Bento et al. 2009).

  18. All regressions in this study were performed using Stata/IC 11.2 for Windows 64 bit x86-64.

  19. Bearing in mind that other than a scale factor for the constant term in the ARMAX(1,0,0) model, lagged-dependent variable (LDV) and ARMAX(1,0,0) models are equivalent when the LDV model includes an explicit control for serial correlation, we introduced a standard control into the LDV model via estimation of an equation of with two lags of \(\hbox {PM}_{2.5}\) and one lag of trip count (following the method proposed by Becketti 2013). Two key results emerged from this analysis of the LDV model. First, while the model fits the data quite well overall (Adj. \(\hbox {R}^{2} = 0.81\)) and coefficient estimates for the first lag of \(\hbox {PM}_{2.5}\) and contemporaneous trip count are statistically significant and aligned with theoretical expectations (0.94 and 0.0002, respectively), the coefficient estimates for the second lag of \(\hbox {PM}_{2.5}\) and first lag of trip count are not. The (statistically significant) estimate for second-lag \(\hbox {PM}_{2.5}\) is \(-\)0.21 and the (statistically insignificant) estimate for first-lag trip count is \(-\)0.00007. Second, while Durbin’s alternative and BG tests indicate the absence of an AR(1) process in the residuals, each test indicates the presence of an AR(2) process. Interestingly, a double-log version of the LDV model with single-lag, logged \(\hbox {PM}_{2.5}\) and contemporaneous logged trip count variables produces statistically significant and theoretically appealing coefficient estimates (0.56 and 0.33, respectively) with a good overall fit (quasi-\(\hbox {R}^{2} = 0.9\)). However, as with the non-logged model that corrects for AR(1) in the residuals, Durbin’s alternative and BG tests indicate the presence of an AR(2) process in the residuals of the logged equation.

  20. Both the OLS and ARMAX(1,0,0) models for the first-stage regression show relatively high levels of significance for the Monday, Tuesday, and Friday dummies in terms of explaining variation in trip count (the OLS regression also suggests a significant Thursday effect), which in turn—by virtue of their obvious exogeneity—suggest that days of the week are statistically valid instruments. We also tested an IV model using two lags of Trip_Count as instruments, following Davidson and MacKinnon (1993). This model yielded comparable first-stage results. We were precluded from using gas price as an instrument because estimating a gas-price elasticity using the daily observations without any time differential (in specific, gas-price changes greater than or equal to $1.00 per gallon over a 4-month interval) produced statistically insignificant results. This is because there is not enough variability in the per-gallon price of gasoline at a daily time step to adequately test for a response in trip count (which is an aggregated variable). Therefore, in order to be aligned measurement-wise with the \(\hbox {PM}_{2.5}\) regressions—which include variables measured at the daily time step—we would have needed to use as an instrument a gas-price variable that we know a priori is an unsuitable candidate (in a statistical sense) for the purpose.

  21. This time interval was not chosen arbitrarily, but rather out of statistical necessity. Shorter time differentials were too restrictive in terms of limiting the number of usable observations, which made estimating the effect of gas prices on Trip_Count untenable. As expected, estimating a gas-price elasticity using the daily observations without any time differential produced statistically insignificant results.

  22. Seasonal ARMAX \((1/7,0,0) \times (0,1,0)_{7}\) is used to account for the weekly (7-day) trend in Trip_Count. This specification applies the lag-7 seasonal difference operator to the dependent and independent variables, which removes the seasonal trend. Note that only lags 1 and 7 of the non-seasonal autoregressive terms of the structural model’s disturbance are included. This accounts for additive seasonal effects and corrects for autocorrelation (Becketti 2013).

  23. The average 2012 gas price in Cache Valley was $3.49 per gallon (GasBuddy 2013). Therefore, doubling gas price effectively satisfies the lower-bound condition placed on our regression analyses, where price increases brought about by a seasonal tax are no less than $1.00.

  24. Because they do not explicitly account for the social benefits arising from reductions in traffic congestion and accidents and reduced carbon emissions, the benefit estimates derived using these three methods (which are reported in Table 7), all else equal, understate the full benefits associated with reduced \(\hbox {PM}_{2.5}\) concentrations. We acknowledge an anonymous reviewer for sharing this insight.

  25. TSPs refer to airborne particulates 100 \(\upmu \text {m}\) in size or less. Hence, \(\hbox {PM}_{10}\) and \(\hbox {PM}_{2.5}\) concentrations can be thought of as special cases of (or nested) TSP concentrations. All MWTP estimates are subsequently converted to their 2014-year equivalents in Table 7.

  26. Chay and Greenstone’s (2005) $12 MWTP estimate corresponds to what they call a one-unit decline in TSPs that lasts one year. By comparison, their estimated MWTP for a permanent one-unit decline is $243.

  27. For seminal discussions on the theory and potential prevalence of benefit transfer error empirically see the Water Resources Research review of the topic (1992, Vol. 28, No. 3), Brookshire and Neill (1992), Bergstrom and Civita (1999), Bergland et al. (1999), and Desvousges et al. (1992).

  28. For the COBRA simulations, we reduce each pollutant found in the mobile source emissions inventory for Cache Valley per season (Table 6) by 51 %. Once done, these values (in tons) are used as inputs to COBRA for purposes of simulation.

  29. Given the size of the tax needed to induce the requisite reductions in vehicle trips, two potential social dilemmas present themselves. First, either gas prices outside of the valley itself, e.g., along the Wasatch Front to the south and into southern Idaho to the north, would need to rise in tandem with the valley’s prices, or quantity restrictions of some kind would need to be put in place outside of the valley (that apply solely to the valley’s residents) in order to prevent valley residents from “driving across the border” for cheaper prices. Second, some form of control would need to be enacted to prevent valley residents from hoarding gasoline during the non-inversion season for use during the inversion season, which is a form of the classic “midnight-dumping” problem investigated by Sigman (1995, 1998) and Fullerton and Kinnaman (1996). Unfortunately, exploring how these two potential dilemmas might best be controlled is outside the scope of this particular study.

  30. According to Redd (2014), approximately five Tier 3 vehicles collectively emit as much as a single Tier 2 vehicle, and approximately 30 Tier 3 vehicles emit as much as a single Tier 1 vehicle.

  31. This fuel-economy effect is also noted in Parry and Small (2005).

  32. According to Redd (2014), Tier 3 vehicle prices are expected to rise by roughly $135 per new car, and the EPA estimates that Tier 3 fuel will add an additional cent to the per-gallon cost of gasoline at the pump.

  33. Despite our acknowledgement of the potential correlation between enhanced fuel efficiency and Tier 3 technology, Parry et al. (2007) argue that a gas tax alone is ultimately inadequate since it does not directly encourage adoption of technologies that lower exhaust emissions per gallon of fuel combustion, e.g., Tier 3 technology itself.

  34. Our empirical estimation of the gas price regression effectively assumes that function \(U^{i}\) is additively separable in \(\hbox {Z}_{1}\) and \(\hbox {Z}_{2}\).


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Correspondence to Arthur J. Caplan.


Appendix 1

The theoretical framework for our gas-price regressions can be represented most conveniently by a variation of Becker’s (1965) household production model, where household i’s welfare in time period t is a function of a composite good obtained via vehicle trips \((\hbox {Z}_{1})\), a numeraire good \((\hbox {Z}_{2})\), and parameterized by the study area’s \(\hbox {PM}_{2.5}\) concentration level \((\overline{G})\), and a vector of seasonal variables proxied by temperature (\(\Theta \)) (time subscripts t are removed from the variables and functions for convenience),Footnote 34

$$\begin{aligned} U^{i}(Z_{1i} ,Z_{2i} ,\overline{{G}}, \Theta ) \end{aligned}$$

which we assume exhibits standard curvature conditions for \(Z_{1i}, Z_{2i},\) and \(\overline{G}\), in particular \(U_G^i <0\) and \(U_{GG}^i <0.\) Equation (3) can be re-written as:

$$\begin{aligned} U^{i}(F^{1i}(X_{1i} ,T_{1i} )Z_{2i} ;\overline{G} ,\Theta ) \end{aligned}$$

where \(X_{1i}\) represents total amount of gas used to obtain \(Z_{1i}, T_{1i} \) represents household i’s time spent obtaining \(Z_{1i}\), and \(F^{1i}\) is the household production function for \(Z_{1i}\). Similar to \(U^{i}\), we assume \(F^{1i}\) exhibits standard production-function curvature conditions.

Household i maximizes (4) subject to its budget constraint,

$$\begin{aligned} P_1 X_{1i} +Z_{2i} +T_{1i} \overline{w}_i =\hbox {T}\overline{w}_i , \end{aligned}$$

where \(P_1\) represents the per-unit price of \(Z_{1i},\overline{w}_i\) represents household i’s composite wage rate, and T represents total work time available to household. Solving the corresponding maximization problem yields the following four first order conditions (FOC’s),

$$\begin{aligned} U_{Z1i}^i F_{X1i}^{1i} -\lambda _i P_1 =0 \end{aligned}$$
$$\begin{aligned} U_{Z2i}^i -\lambda _i =0 \end{aligned}$$
$$\begin{aligned} U_{Z1i}^i F_{T1i}^{1i} -\lambda \overline{{w}}_i =0 \end{aligned}$$
$$\begin{aligned} \hbox {T}\overline{{w}}_i -P_1 X_{1i} -Z_{2i} -T_{1i} \overline{w}_i =0 \end{aligned}$$

where \(\uplambda \) represents the problem’s Lagrangian multiplier (marginal utility of income). FOC’s (6)–(9) can be solved for household i’s fuel demand, the numeraire good, and time spent obtaining \(Z_{1i} \), i.e., \(X_{1i} (\overline{w}_i , P_1 , \Theta , \overline{G} ), Z_{2i} (\overline{w}_i , P_1 , \Theta , \overline{G})\), and \(T_{1i} (\overline{w}_i , P_1 ,\Theta , \overline{G})\), respectively. The demand for goods obtained via vehicle trips can then be written as \(Z_{1i} =F^{1i}(X_{1i} (\cdot ), T_{1i} (\cdot ))\). For future reference, the household’s gas price elasticity is shown by (10),

$$\begin{aligned} \varepsilon _{X_{1i} P_1}^i =\frac{\partial X_{1i} }{\partial P_1 }\frac{P_1 }{X_{1i} (\cdot )} \end{aligned}$$

To establish the benchmark, socially optimal allocation of household demand, assume a social planner maximizes the sum of all individual’s utilities subject to an economy-wide resource constraint. Hence, the benchmark optimization problem may be written as,

$$\begin{aligned} \mathop {{\max }}\limits _{\left\{ {X_{1i} ,T_{1i} ,Z_{2i} } \right\} } {\sum }_i U^{i}(F^{1i}(X_{1i} ,T_{1i} ),Z_{2i} ,{\sum }_i F^{1i}(X_{1i} ,T_{1i} );\theta ) \end{aligned}$$

where now \(\overline{G}\) is explicitly recognized by the planner as endogenous variable \(G={\upalpha }{\sum }_i F^{1i}(X_{1i} , T_{1i})\), where \({\upalpha }\) represents an emissions factor, subject to the economy-wide resource constraint,

$$\begin{aligned} {\sum }_i \left( {-P_1 X_{1i} -Z_{2i} -T_{1i} \overline{w}_i +T\overline{w}_i } \right) =0 \end{aligned}$$

Solving this maximization problem yields the following FOC’s for households \(i=1\),...,N, where \(\mu \) is this problem’s Lagrangian multiplier,

$$\begin{aligned}&U_{Z1i}^i F_{X1i}^{1i} +{\upalpha }{\sum }_i U_G^i F_{X1i}^{1i} -\mu P_1 =0 \end{aligned}$$
$$\begin{aligned}&U_{Z2i}^i -\mu =0 \end{aligned}$$
$$\begin{aligned}&U_{Z1i}^i F_{T1i}^{1i} +{\upalpha }{\sum }_i U_G^i F_{T1i}^{Ti} -\mu \overline{w}_i =0 \end{aligned}$$
$$\begin{aligned}&{\sum }_i \left( {T\overline{w}_i -P_1 X_{1i} -Z_{2i} -T_{1i} \overline{w}_i } \right) =0 \end{aligned}$$

The social planner is tasked with implementing a socially optimal (per-unit) gas tax. To accomplish this, the planner sets the tax according to (17), where \(t_{X1i}^*\) represents the optimal (Pigovian), individualistic tax rate on fuel used to obtain \(Z_1 \).

$$\begin{aligned} t_{X1i}^*=-\frac{{\upalpha }{\sum }_i U_G^i F_{X1i}^{1i}}{\lambda _i}. \end{aligned}$$

The optimal gas tax is added to \(P_1 \), and the social planner further normalizes \(P_1\) by \(\frac{\mu }{\lambda _i}\) so that (6) becomes \(U_{Z1i}^i F_{X1i}^{1i} - \lambda _i \big [\big (\frac{\mu }{\lambda _i}\big )P_1 +\frac{{\upalpha }{\sum }_i U_G^i F_{X1i}^{1i}}{\lambda _i}\big ]\) which collapses to (13). Similarly, the social planner sets a socially optimal tax on individual i’s time spent obtaining \(Z_1\), shown by (18), also normalizing \(\overline{w}_i\) by \(\frac{\mu }{\lambda _i}\) so that (8) becomes (15):

$$\begin{aligned} t_{T1i}^*=-\frac{{\upalpha }{\sum }_i U_G^i F_{T1i}^{1i}}{\lambda _i} \end{aligned}$$

Finally, to transform (7)–(14) the social planner normalizes each individual’s adjusted net income by \(\frac{\mu }{\lambda _i}\) . In determining each tax rate, the social planner evaluates both \(t_{X1i}^*\) and \(t_{T1i}^*\) at the solution’s optimal values. This implies that, in a world with perfect information, the social planner would assign a unique tax to each individual. Thus, there would be no uniform gas tax, but instead one tailored to an individual’s “adjusted” marginal utility. “Smart” gas-pumps are therefore required to identify each (type of) individual and change the pump price accordingly.

Appendix 2

To illustrate its seasonality, consider a graph of Trip_Count’s autocorrelations,

figure a

Note the seasonal (weekly) trend repeating every 7th observation, as well as the fact that each of the autocorrelation values lie outside the shaded region demarcating no statistical difference at the 5 % level. To mitigate the adverse effects that this trend may have on our gas price regression model, we lag-7 seasonally difference Trip_Count (along with all other explanatory variables) to remove the trend (Becketti 2013). The following figure shows the transformation of Trip_Count after seasonal adjustment, denoted as S7Trip_Count,

figure b

Comparing the two figures, we conclude that seasonal-differencing has mitigated the potential confounded marginal effect of GPrice’s on Trip_Count due to seasonality, since statistical differences between the autocorrelations at the various lag lengths no longer exist beyond the fourth lag (our chosen seasonal lag-length is seven).

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Moscardini, L.A., Caplan, A.J. Controlling Episodic Air Pollution with a Seasonal Gas Tax: The Case of Cache Valley, Utah. Environ Resource Econ 66, 689–715 (2017).

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  • Episodic air pollution
  • Seasonal gas tax
  • Gas price elasticity
  • \(\hbox {PM}_{2.5}\) concentrations

JEL Classification

  • Q53
  • Q58