Missing the Warning Signs? The Case of “Yellow Air Day” Advisories in Northern Utah

Abstract

Using a dataset consisting of daily vehicle trips, $PM_{2.5}$ concentrations, and a host of climactic control variables, we test the hypothesis that “yellow air day advisories” issued by the Utah Division of Air Quality resulted in subsequent reductions in vehicle trips taken during northern Utah’s winter-inversion seasons in the early 2000 s. Winter inversions occur in northern Utah when $PM_{2.5}$ concentrations (derived mainly from vehicle emissions) become trapped in the lower atmosphere, leading to unhealthy air quality over a span of time known colloquially as “red air day episodes”. When concentrations rise above 15 $\upmu \textrm{g}/\textrm{m}^3$ toward the National Ambient Air Quality Standard average daily threshold of 35 $\upmu \textrm{g}/\textrm{m}^3$, residents are informed via different media sources and road signage that the region is experiencing a yellow air day, and are urged to reduce their vehicle usage during the day. Our results suggest that the advisories have provided at best weak, at worst perverse, incentives for reducing vehicle usage on yellow air days and ultimately for mitigating the occurrence of red air day episodes during northern Utah’s winter inversion seasons.

Appendices

Technical Appendix

Mathematical Derivations for the Theoretical Model in Sect. 4

Consider myopic individual (or household) i in a given time period t, who derives benefit from making vehicle trips (e.g., commuting to work, shopping, traveling to recreation sites, etc.), but also incurs costs associated with the aggregate amount of trips taken in i’s community or region during time t (to which individual i contributes atomistically), e.g., in the form of elevated $PM_{2.5}$ concentrations.⁴³ We specify i’s benefit function in period t, $u_{it}$, as,

$$\begin{aligned} u_{it}=u_{it}\left( z_{it}\left( q_{it}\right) ,x_{it};\beta ^z_i\left( \theta _t\right) ,\beta ^x_i\left( \theta _t\right) \right) , i=1,\ldots ,I, t=1,\ldots ,T, \end{aligned}$$

(A.1)

where $z_{it}$ represents the amount of a composite good obtained as a function of vehicle usage, denoted as $q_{it}$, and $x_{it}$ denotes the composite amount of all other goods not obtained via vehicle usage, i.e., household-produced goods. Information-conditioned parameters $0<\beta ^z_i(\theta _t)<1$ and $0<\beta ^x_i(\theta _t)<1$, respectively, parameterize $z_{it}$ and $x_{it}$ in function $u_{it}$ such that $\beta ^x_i(\theta _t) \equiv 1-\beta ^z_i(\theta _t)$, and $\theta _t$ is an information parameter representing issuance of a yellow air day advisory when $PM_{2.5}$ concentrations rise above the 15 $\upmu /\textrm{m}^3$ threshold.⁴⁴ For ease of exposition and without loss of generality, we assume all variables $z_{it}$, $q_{it}$, and $x_{it}$, and parameters $\beta ^z_i(\theta _t)$, $\beta ^x_i(\theta _t)$, and $\theta _t$ are measured continuously. In particular, increases in $\theta _t$ imply that the region’s individuals are being supplied with more information (via an advisory) about the onset of a yellow air day.

In addition to standard curvature conditions specified for function $u_{it}$, i.e., $\partial u_{it} / \partial z_{it}>0$, $\partial ^2 u_{it}/\partial z^2_{it}\le 0$, $\partial u_{it}/\partial x_{it}>0$, $\partial ^2 u_{it}/\partial x^2_{it}\le 0$, and $\partial ^2 u_{it}/\partial z_{it}\partial x_{it}=\partial ^2 u_{it}/\partial x_{it}\partial z_{it}> 0$, and for function $z_{it}$, i.e., $\partial z_{it}/\partial q_{it}>0$ and $\partial ^2 z_{it}/\partial q^2_{it}\le 0$, we specify a key curvature condition for the ensuing analysis: $\partial \beta ^z_{i}/ \partial \theta _t>0$. This condition indicates that, all else equal, the marginal value of $z_{it}$ (relative to that of $x_{it}$) increases with the issuance of a yellow air day advisory, i.e., $\left( \partial ^2 u_{it}/\partial z_{it}\partial \beta ^z_i\right) \left( \partial \beta ^z_{i}/ \partial \theta _t\right) >0$. Note that identity $\beta ^x_i(\theta _t) \equiv 1-\beta ^z_i(\theta _t)$ in turn implies $\left( \partial ^2 u_{it}/\partial x_{it}\partial \beta ^z_i\right) \left( \partial \beta ^z_{i}/ \partial \theta _t\right) <0$. These conditions underlie the intuition expressed in Sect. 4 that, given the issuance of a yellow air day advisory, an individual derives added benefit from any given vehicle trip, since making the trip using the next-best alternative, e.g., walking or riding a bus, involves greater exposure to the yellow air. Furthermore, given that a yellow air day advisory signals the onset of a subsequent red air day episode, individuals could perceive added benefit associated with intertemporally substituting vehicle trips forward in time to reduce the need for making future vehicle trips during the episode itself.

Individual i forms an expectation over the health and environmental damages s/he suffers with respect to aggregate $PM_{2.5}$ concentrations accumulated in the atmosphere during period t. We represent these expected damages with function $E\left[ d_{it}\right]$,

$$\begin{aligned} E\left[ d_{it}\right] =\bar{d}_{it}\left( Q_t;\alpha _i\left( \theta _t\right) \right) , i=1,\ldots ,I, t=1,\ldots ,T, \end{aligned}$$

(A.2)

where region-wide vehicle trips $Q_t=\sum _i q_{it}$, $\alpha _i\left( \theta _t\right)$ is an information-conditioned parameter distinct from $\beta ^z_i$, and standard curvature conditions are specified for expected damage function $E\left[ d_{it}\right]$, i.e., $\partial \bar{d}_{it}/\partial Q_t >0$, $\partial ^2 \bar{d}_{it}/\partial Q^2_t \ge 0$, and $\partial \bar{d}_{it}/\partial \alpha _i >0$. Similar to the relationship between $\beta ^z_i$ and $\theta _t$ we assume $\partial \alpha _{i}/ \partial \theta _t>0$, which in turn indicates that, all else equal, perceived marginal damages suffered by each individual i in period t increase in response to the issuance of a yellow air day advisory, i.e., $\left( \partial ^2 \bar{d}_{it}/\partial Q_t \partial \alpha _i\right) \left( \partial \alpha _i/\partial \theta _t\right) >0$. This condition accounts for an overall increase in expected marginal damages to an individual’s health due to the issuance of a yellow air day advisory.

The individual’s budget constraint in any given period t is given by,

$$\begin{aligned} w_{it}=p^z_tz_{it}(q_{it})+p^q_tq_{it}+x_{it}, i=1,\ldots ,I, t=1,\ldots ,T, \end{aligned}$$

(A.3)

where $w_{it}$ represents individual i’s given wealth level in period t, and per-unit prices $p^z_t$ and $p^q_t$ are taken as given for good $z_{it}$ and vehicle trips $q_{it}$, respectively (the price of $x_{it}$ is normalized to one).⁴⁵

Next, we consider three cases reflecting three stylized types of individuals comprising the region.⁴⁶ Case 1 pertains to individuals who completely ignore the expected damages associated with region-wide vehicle trips in each period t, $Q_t$, even though $\partial \alpha _i/\partial \theta _t \ne 0$, i.e., even though they are informed about elevated $PM_{2}.5$ concentrations via yellow air day advisories. Case 2 pertains to individuals who account solely for the expected damages that they personally incur in period t, i.e., individual i dissects function $\bar{d}_{it}$ as $\bar{d}_{it}\left( q_{it}+Q_{-it};\alpha _i\left( \theta _t\right) \right)$, where $Q_{-it}$ represents the aggregate trip count across all individuals in the region except individual i, and thereby accounts soley for the $q_{it}$ in $\bar{d}_{it}(\cdot )$ in his decision problem. Case 3 pertains to altruistic individuals who account not only for the expected damages that their vehicle trips impose on themselves and all other individuals in the region, but also the expected benefits that all other individuals obtain as a result of increasing their vehicle trips in response to a yellow air day advisory, i.e., these individuals are “pure altruists” (c.f., Antweiler 2015; Ottoni-Wilhelm et al. 2017).

1.1 Case 1

An individual i who fits the description of Case 1 myopically chooses $q_{it}$ and $x_{it}$ to solve the following Lagrangian in each period t,

$$\begin{aligned}{} & {} u_{it}\left( z_{it}\left( q_{it}\right) ,x_{it};\beta ^z_i\left( \theta _t\right) ,\beta ^x_i\left( \theta _t\right) \right) -\bar{d}_{it}\left( Q_t;\alpha _i\left( \theta _t\right) \right) \\{} & {} +\lambda _{it}\left( w_{it}-p^z_tz_{it}(q_{it})-p^q_tq_{it}-x_{it}\right) \end{aligned}$$

where $\lambda _{it}>0$ represents i’s period t Lagrangian multiplier. First-order conditions for this problem result in,

$$\begin{aligned} \frac{\partial u_{it}}{\partial z_{it}}\frac{\partial z_{it}}{\partial q_{it}}=\frac{\partial u_{it}}{\partial x_{it}}\left( p^z_t\frac{\partial z_{it}}{\partial q_{it}}+p^q_t\right) , i=1,\ldots ,I, t=1,\ldots ,T. \end{aligned}$$

(A.4)

The left-hand side of (A.4) represents the marginal benefit of an additional vehicle trip and the right-hand side represents the corresponding marginal cost. Together with (A.3) and function $z_{it}\left( q_{it}\right)$, Eq. (A.4) solves for $q^*_{it}=q_{it}\left( w_{it}, p^z_t, p^q_t, \alpha _{i}\left( \theta _t\right) ,\beta ^z_{i}\left( \theta _t\right) ,\beta ^x_{i}\left( \theta _t\right) \right)$, $z^*_{it}=z_{it}\left( w_{it}, p^z_t, p^q_t, \alpha _{i}\left( \theta _t\right) ,\beta ^z_{i}\left( \theta _t\right) ,\beta ^x_{i}\left( \theta _t\right) \right)$, and $x^*_{it}=x_{it}\left( w_{it}, p^z_t, p^q_t, \alpha _{i}\left( \theta _t\right) ,\beta ^z_{i}\left( \theta _t\right) , \beta ^x_{i}\left( \theta _t\right) \right)$.

Substituting $q^*_{it}$, $z^*_{it}$, and $x^*_{it}$ into (A.4) and differentiating allows us to solve for the marginal effect of a change in $\theta _t$ on $q^*_{it}$ relative to $x^*_{it}$.⁴⁷ The expression for this marginal effect is,

$$\begin{aligned} \frac{\partial q^*_{it}}{\partial \theta _t}=-\frac{\Psi _1}{\Omega _1}>0, i=1,\ldots ,I, t=1,\ldots ,T, \end{aligned}$$

(A.5)

where

$$\begin{aligned} \Psi _1=\frac{\partial ^2 u_{it}}{\partial z^*_{it}\partial \beta ^z_i}\frac{\partial \beta ^z_i}{\partial \theta _t}\frac{\partial z^*_{it}}{\partial q^*_{it}}-\frac{\partial ^2 u_{it}}{\partial x^*_{it}\partial \beta ^z_i}\frac{\partial \beta ^z_i}{\partial \theta _t}\left( p^z_t\frac{\partial z^*_{it}}{\partial q^*_{it}}+p^q_t\right) >0 \end{aligned}$$

(A.6)

and

$$\begin{aligned} \Omega _1=\frac{\partial ^2 u_{it}}{\partial z^{*2}_{it}}\left( \frac{\partial z^*_{it}}{\partial q^*_{it}}\right) ^2+\frac{\partial u_{it}}{\partial z^{*}_{it}}\frac{\partial ^2 z^*_{it}}{\partial q^{*2}_{it}}-\frac{\partial ^2 u_{it}}{\partial x^*_{it} \partial z^*_{it}}\frac{\partial z^*_{it}}{\partial q^*_{it}}\left( p^z_t\frac{\partial z^*_{it}}{\partial q^*_{it}}+p^q_t\right) -\frac{\partial u_{it}}{\partial x^*_{it}}p^z_t\frac{\partial ^2 u_{it}}{\partial z^{*2}_{it}}<0. \end{aligned}$$

(A.7)

Note that $\Psi _1>0$ in (A.6) follows directly from the curvature conditions specified above for $u_{it}\left( \cdot \right)$. To see why $\Omega _1<0$ in (A.7), first rewrite (A.4) as,

$$\begin{aligned} \frac{\partial u_{it}}{\partial z_{it}}-\frac{\partial u_{it}}{\partial x_{it}}p^z_t=\frac{p^q_t}{\frac{\partial z_{it}}{\partial q_{it}}}>0, i=1,\ldots ,I, t=1,\ldots ,T. \end{aligned}$$

(A.8)

Now note from (A.7) that $\Omega _1<0$ when

$$\begin{aligned} \left( \frac{\partial u_{it}}{\partial z^{*}_{it}}-\frac{\partial u_{it}}{\partial x^*_{it}}p^z_t\right) \frac{\partial ^2 z^*_{it}}{\partial q^{*2}_{it}}<0 \Longrightarrow \frac{\partial u_{it}}{\partial z^*_{it}}-\frac{\partial u_{it}}{\partial x^*_{it}}p^z_t>0, \end{aligned}$$

which coincides with the result in (A.8). Thus, $\Omega _1<0$.

Clearly, the result in (A.5) is driven by the assumptions underlying our problem, in particular the separability of $u_{it}$ and $\bar{d}_{it}$ in individual i’s Lagrangian function. In a more general specification of i’s welfare, e.g., $u_{it}\left( z_{it}\left( q_{it}\right) ,x_{it};Q_t,\beta ^z_i\left( \theta _t\right) ,\beta ^x_i\left( \theta _t\right) ,\beta ^Q_i\left( \theta _t\right) \right)$, where $\beta ^Q_i\left( \theta _t\right) <0$ parameterizes $Q_t$ in $u_{it}$, we cannot definitively sign $\partial q^*_{it}/\partial \theta _t$ without specifying additional assumptions governing the tradeoff between $z_{it}$ and $x_{it}$ in response to an increase in $\theta _t$. As is, our result for Case 1 depicts the predilection of certain types of individuals who weight the private benefit associated with their vehicle trips during yellow air days more than the correlative public damages to which their trips contribute (which, according to our particular welfare specification, are completely ignored in this case).

1.2 Case 2

An individual i who fits the description of Case 2 myopically chooses $q_{it}$ and $x_{it}$ to solve the following Lagrangian in each period t,

$$\begin{aligned}{} & {} u_{it}\left( z_{it}\left( q_{it}\right) ,x_{it};\beta ^z_i\left( \theta _t\right) ,\beta ^x_i\left( \theta _t\right) \right) -\bar{d}_{it}\left( q_{it}+Q_{-it};\alpha _i\left( \theta _t\right) \right) \\{} & {} +\gamma _{it}\left( w_{it}-p^z_tz_{it}(q_{it})-p^q_tq_{it}-x_{it}\right) \end{aligned}$$

where $\gamma _{it}>0$ represents i’s period t Lagrangian multiplier. First-order conditions for this problem result in,

$$\begin{aligned} \frac{\partial u_{it}}{\partial z_{it}}\frac{\partial z_{it}}{\partial q_{it}}=\frac{\partial u_{it}}{\partial x_{it}}\left( p^z_t\frac{\partial z_{it}}{\partial q_{it}}+p^q_t\right) +\frac{\partial \bar{d}_{it}}{\partial Q_t}, i=1,\ldots ,I, t=1,\ldots ,T. \end{aligned}$$

(A.9)

As with Case 1, the left-hand side of (A.9) represents the marginal benefit of an additional vehicle trip and the right-hand side represents the corresponding marginal cost, which in this case now accounts for the individual’s expected marginal damage associated with an additional vehicle trip, $\partial \bar{d}_{it}/\partial Q_t$. Similar to Case 1, Eq. (A.3), function $z_{it}\left( q_{it}\right)$, and optimality condition (A.9) solve for $q^{**}_{it}$, $z^{**}_{it}$, and $x^{**}_{it}$, which when substituted back into (A.9) and differentiated allows us to solve for the marginal effect of a change in $\theta _t$ on $q^{**}_{it}$ relative to $x^{**}_{it}$. The expression for this marginal effect is,

$$\begin{aligned} \frac{\partial q^{**}_{it}}{\partial \theta _t}=-\frac{\Psi _2}{\Omega _2}, i=1,\ldots ,I, t=1,\ldots ,T, \end{aligned}$$

(A.10)

where

$$\begin{aligned} \Psi _2=\Psi _1-\frac{\partial ^2\bar{d}_{it}}{\partial Q^{**}_t \partial \alpha _i}\frac{\partial \alpha _i}{\partial \theta _t} \end{aligned}$$

(A.11)

and

$$\begin{aligned} \Omega _2=\Omega _1-\frac{\partial ^2\bar{d}_{it}}{\partial Q^{**2}_t}<0. \end{aligned}$$

(A.12)

Comparing (A.10)–(A.12) with (A.5)–(A.7) we see that,

$$\begin{aligned} \frac{\partial q^{**}_{it}}{\partial \theta _t}<\frac{\partial q^{*}_{it}}{\partial \theta _t}. \end{aligned}$$

(A.13)

Further, we find that,

$$\begin{aligned} \frac{\partial q^{**}_{it}}{\partial \theta _t} \gtrless 0 \, \text {as} \, \frac{\partial ^2\bar{d}_{it}}{\partial Q^{**}_t \partial \alpha _i} \lessgtr \frac{\partial ^2 u_{it}}{\partial z^{**}_{it}\partial \beta ^z_i}\frac{\partial \beta ^z_i}{\partial \theta _t}\frac{\partial z^{**}_{it}}{\partial q^{**}_{it}}-\frac{\partial ^2 u_{it}}{\partial x^{**}_{it}\partial \beta ^z_i}\frac{\partial \beta ^z_i}{\partial \theta _t}\left( p^z_t\frac{\partial z^{**}_{it}}{\partial q^{**}_{it}}+p^q_t\right) , \end{aligned}$$

(A.14)

where the term $\frac{\partial ^2\bar{d}_{it}}{\partial Q^{**}_t \partial \alpha _i}$ represents the change in individual i’s perceived marginal damage (from vehicle trips) associated with the change in information-conditioned parameter $\alpha _i$ as a result of the issuance of a yellow air day advisory (i.e., change in $\theta _t$). The term $\frac{\partial ^2 u_{it}}{\partial z^{**}_{it}\partial \beta ^z_i}\frac{\partial \beta ^z_i}{\partial \theta _t}\frac{\partial z^{**}_{it}}{\partial q^{**}_{it}}-\frac{\partial ^2 u_{it}}{\partial x^{**}_{it}\partial \beta ^z_i}\frac{\partial \beta ^z_i}{\partial \theta _t}\left( p^z_t\frac{\partial z^{**}_{it}}{\partial q^{**}_{it}}+p^q_t\right)$ represents the corresponding change in individual i’s marginal benefit associated with the change in information-conditioned parameter $\beta ^z_i$. Our result for Case 2 therefore depicts a different type of individual than Case 1. In this case, the individual explicitly accounts for the (private effect of) the public damage to which his trips contribute, which leads to a lower increase in vehicle usage in response to a yellow air day advisory than for Case 1 individuals, all else equal. As Eqs. (A.13) and (A.14) demonstrate, Case 2 individuals may choose to decrease the number of their vehicle trips in response to a yellow air day advisory.

1.3 Case 3

An individual i who fits the description of Case 3 myopically chooses $q_{it}$ and $x_{it}$ to solve the following Lagrangian in each period t,

$$\begin{aligned}{} & {} u_{it}\left( z_{it}\left( q_{it}\right) ,x_{it}, \sum _{j\ne i}\bar{u}_{jt}\left( z_{jt}\left( q_{jt}\right) ,x_{jt};\beta ^z_j\left( \theta _t\right) ,\beta ^x_j\left( \theta _t\right) \right) ; \beta ^z_i\left( \theta _t\right) ,\beta ^x_i\left( \theta _t\right) \right) \\{} & {} \quad -\bar{d}_{it}\left( q_{it}+Q_{-it};\alpha _i\left( \theta _t\right) \right) \\{} & {} \quad -\sum _{j\ne i}\bar{d}_{jt}\left( q_{it}+Q_{-it};\alpha _j\left( \theta _t\right) \right) \\{} & {} \quad +\phi _{it}\left( w_{it}-p^z_tz_{it}(q_{it})-p^q_tq_{it}-x_{it}\right) \end{aligned}$$

where $\phi _{it}>0$ represents i’s period t Lagrangian multiplier. An altruistic individual i therefore fully accounts for the effect of a yellow air day advisory on the expected benefits that all other individuals j, $j \ne i, i,j=1,\ldots ,I$ obtain from their vehicle usage, represented by inclusion of the term $\sum _{j\ne i}\bar{u}_{jt}\left( z_{jt}\left( q_{jt}\right) ,x_{jt};\beta ^z_j\left( \theta _t\right) ,\beta ^x_j\left( \theta _t\right) \right)$ in i’s own utility function $u_{it}$. Altruistic individual i also fully accounts for the effects of both the yellow air day advisory and her vehicle usage on the expected damages incurred by all other individuals, represented by inclusion of the separate term $\sum _{j\ne i}\bar{d}_{jt}\left( q_{it}+Q_{-it};\alpha _j\left( \theta _t\right) \right)$ in her Lagrangian function. First-order conditions for this problem result in,

$$\begin{aligned} \frac{\partial u_{it}}{\partial z_{it}}\frac{\partial z_{it}}{\partial q_{it}}=\frac{\partial u_{it}}{\partial x_{it}}\left( p^z_t\frac{\partial z_{it}}{\partial q_{it}}+p^q_t\right) +\frac{\partial \bar{d}_{it}}{\partial Q_t}+\sum _{j \ne i}\frac{\partial \bar{d}_{jt}}{\partial Q_t}, i,j=1,\ldots ,I, t=1,\ldots ,T. \end{aligned}$$

(A.15)

where $\partial \bar{d}_{jt}/\partial Q_t>0 \forall j \ne i$, i.e., individual i perceives all other members of the region as suffering positive marginal damages from additional vehicle trips made within the region.

As with Cases 1 and 2, the left-hand side of (A.15) represents the marginal benefit of an additional vehicle trip and the right-hand side represents the corresponding marginal cost, which in this case now accounts for i’s expected private marginal damage associated with taking an additional vehicle trip as well as i’s expectation of the impact that that additional vehicle trip has on the damages incurred by all other individuals in the region, represented by the term $\sum _{j \ne i}\frac{\partial \bar{d}_{jt}}{\partial Q_t}$. Similar to Cases 1 and 2, Eq. (A.3), function $z_{it}\left( q_{it}\right)$, and optimality condition (A.15) solve for $q^{***}_{it}$, $z^{***}_{it}$, and $x^{***}_{it}$, which when substituted back into (A.15) and differentiated allows us to solve for the marginal effect of a change in $\theta _t$ on $q^{***}_{it}$ relative to $x^{***}_{it}$. The expression for this marginal effect is,

$$\begin{aligned} \frac{\partial q^{***}_{it}}{\partial \theta _t}=-\frac{\Psi _3}{\Omega _3}, i=1,\ldots ,I, t=1,\ldots ,T, \end{aligned}$$

(A.16)

where

$$\begin{aligned} \Psi _3= & {} \Psi _2+\sum _{j \ne i}\left( \frac{\partial ^2 u_{it}}{\partial z^{***}_{it} \partial \bar{u}_{jt}}\frac{\partial \bar{u}_{jt}}{\partial \beta ^z_j} \frac{\partial \beta ^z_j}{\partial \theta _t}\frac{\partial z^{***}_{it}}{\partial q^{***}_{it}}\right. \nonumber \\{} & {} \left. -\frac{\partial ^2 u_{it}}{\partial x^{***}_{it} \partial \bar{u}_{jt}}\frac{\partial \bar{u}_{jt}}{\partial \beta ^z_j}\frac{\partial \beta ^z_j}{\partial \theta _t}\left( p^z_t \frac{\partial z^{***}_{it}}{\partial q^{***}_{it}}+p^q_t\right) -\frac{\partial ^2\bar{d}_{jt}}{\partial Q^{***}_t \partial \alpha _j}\frac{\partial \alpha _j}{\partial \theta _t}\right) \end{aligned}$$

(A.17)

and

$$\begin{aligned} \Omega _3=\Omega _2-\sum _{j \ne i}\frac{\partial ^2\bar{d}_{jt}}{\partial Q^{***2}_t}<0. \end{aligned}$$

(A.18)

We note that $\frac{\partial ^2 u_{it}}{\partial z^{***}_{it} \partial \bar{u}_{jt}}>0$ and $\frac{\partial ^2 u_{it}}{\partial x^{***}_{it} \partial \bar{u}_{jt}}>0$ across all individuals j as a reflection of individual i’s altruism, and $\frac{\partial \bar{u}_{jt}}{\partial \beta ^z_j}\frac{\partial \beta ^z_j}{\partial \theta _t}\le 0$, which reflects the fact that before any given yellow air day advisory individuals j are assumed to have optimally set their respective $\beta ^z_j\left( \theta _t \right)$ parameter values.

Comparing (A.10)–(A.12) with (A.16)–(A.18) leads to a sufficient condition governing the relationship between $\partial q^{***}_{it}/\partial \theta _t$ and $\partial q^{**}_{it}/\partial \theta _t$ across all $i,j=1,\ldots ,I$, and $t=1,\ldots ,T$,⁴⁸

$$\begin{aligned} \frac{\partial q^{***}_{it}}{\partial \theta _t}< & {} \frac{\partial q^{**}_{it}}{\partial \theta _t} \quad \text {if} \quad \sum _{j \ne i}\left( \frac{\partial ^2 \bar{d}_{jt}}{\partial Q^{***}_t \partial \alpha _j} \frac{\partial \alpha _j}{\partial \theta _t}\right) >\nonumber \\&\quad&\sum _{j \ne i}\left( \frac{\partial ^2 u_{it}}{\partial z^{***}_{it} \partial \bar{u}_{jt}}\frac{\partial \bar{u}_{jt}}{\partial \beta ^z_j}\frac{\partial \beta ^z_j}{\partial \theta _t}\frac{\partial z^{***}_{it}}{\partial q^{***}_{it}}\right) \nonumber \\{} & {} \quad -\sum _{j \ne i}\left( \frac{\partial ^2 u_{it}}{\partial x^{***}_{it} \partial \bar{u}_{jt}}\frac{\partial \bar{u}_{jt}}{\partial \beta ^z_j}\frac{\partial \beta ^z_j}{\partial \theta _t}\left( p^z_t \frac{\partial z^{***}_{it}}{\partial q^{***}_{it}}+p^q_t\right) \right) . \end{aligned}$$

(A.19)

The left-hand side of the second inequality in (A.19) represents the change in individual i’s perceived marginal damage associated with the added aggregate damage suffered by individuals $j \ne i$ (from their vehicle trips) brought about by the respective changes in their information-conditioned parameters $\alpha _j$ as a result of the issuance of a yellow air day advisory (i.e., change in $\theta _t$). The right-hand side of the second inequality represents the corresponding change in i’s perceived marginal benefit associated with the added aggregate benefit obtained by individuals $j \ne i$ brought about by the respective changes in their information-conditioned parameters $\beta ^z_j$.

Similarly, comparing (A.5)–(A.7) with (A.16)–(A.18) leads to a sufficient condition governing the relationship between $\partial q^{***}_{it}/\partial \theta _t$ and $\partial q^{*}_{it}/\partial \theta _t$ across all $i,j=1,\ldots ,I$, and $t=1,\ldots ,T$,

$$\begin{aligned} \frac{\partial q^{***}_{it}}{\partial \theta _t}< & {} \frac{\partial q^{*}_{it}}{\partial \theta _t} \quad \text {if} \quad \frac{\partial ^2 \bar{d}_{it}}{\partial Q^{***}_t \partial \alpha _i}\frac{\partial \alpha _j}{\partial \theta _t} +\sum _{j \ne i}\left( \frac{\partial ^2 \bar{d}_{jt}}{\partial Q^{***}_t \partial \alpha _j} \frac{\partial \alpha _j}{\partial \theta _t}\right) >\nonumber \\{} & {} \sum _{j \ne i}\left( \frac{\partial ^2 u_{it}}{\partial z^{***}_{it} \partial \bar{u}_{jt}}\frac{\partial \bar{u}_{jt}}{\partial \beta ^z_j}\frac{\partial \beta ^z_j}{\partial \theta _t}\frac{\partial z^{***}_{it}}{\partial q^{***}_{it}}\right) \nonumber \\{} & {} -\sum _{j \ne i}\left( \frac{\partial ^2 u_{it}}{\partial x^{***}_{it} \partial \bar{u}_{jt}}\frac{\partial \bar{u}_{jt}}{\partial \beta ^z_j}\frac{\partial \beta ^z_j}{\partial \theta _t}\left( p^z_t \frac{\partial z^{***}_{it}}{\partial q^{***}_{it}}+p^q_t\right) \right) , \end{aligned}$$

(A.20)

where the left-hand and right-hand sides of the second inequality in (A.20) have the same interpretations as those in the second inequality in Eq. (A.19). However, in this case the sufficient condition is now more likely to hold because of the addition of the $\frac{\partial ^2 \bar{d}_{it}}{\partial Q^{***}_t \partial \alpha _i}\frac{\partial \alpha _j}{\partial \theta _t}>0$ term on the left-hand side of the second inequality.

Coefficient Plots for the Empirical Analysis in Sect. 6.2

Fig. 6

Coefficient plots for $YellowAdvisoryPlus1_t$

Fig. 7

Coefficient plots for $YellowAdvisory_{t-1}$

Fig. 8

Coefficient plots for $YellowAdvisoryPlus1_{t-1}$

Fig. 9

Source https://onlinelibrary.utah.gov/utah/counties/ and https://www.freeworldmaps.net/united-states/utah/location.html

Location of Cache Valley, Utah

Model 1 includes only contemporaneous effects associated with $YellowAdvisory_t$.

Model 2 includes both contemporaneous and single-day lag effects associated with $YellowAdvisory_t$.

Model 3 adds a second-day lag effect to Model 2.

Model 4 is the quadratic model with two-day lag effects.

Model 5 is the quadratic model with three-day lag effects.

Model 1 includes only contemporaneous effects associated with $YellowAdvisoryPlus1_t$.

Model 2 includes both contemporaneous and single-day lag effects associated with $YellowAdvisoryPlus1_t$.

Model 3 adds a second-day lag effect to Model 2.

Model 4 is the quadratic model with two-day lag effects.

Model 5 is the quadratic model with three-day lag effects.

Model 1 includes only single-day lag effects associated with $YellowAdvisory_{t-1}$.

Model 2 includes both contemporaneous and single-day lag effects associated with $YellowAdvisory_t$.

Model 3 adds a second-day lag effect to Model 2.

Model 4 is the quadratic model with two-day lag effects.

Model 5 is the quadratic model with three-day lag effects.

Model 1 includes only single-day lag effects associated with $YellowAdvisoryPlus1_{t-1}$.

Model 2 includes both contemporaneous and single-day lag effects associated with $YellowAdvisoryPlus1_t$.

Model 3 adds a second-day lag effect to Model 2.

Model 4 is the quadratic model with two-day lag effects.

Model 5 is the quadratic model with three-day lag effects.