The welfare ranking of prices and quantities under noncompliance

Abstract

We study the welfare ranking of an emission tax and emissions trading when firms self-report their emissions to the regulator and may be noncompliant. We allow for the subjective probabilities of auditing and, using conventional assumptions, find that an emission tax produces a higher level of welfare than emissions trading under noncompliance. The main driver of the result is that the compliance pattern of the firms affects the marginal compliance cost (price of emissions) in the case of permits, thus affecting the level of emissions and welfare. The result also holds when enforcement and sanctioning costs are taken into account and differs from the result found by Montero (J Public Econ 85:435–454, 2002). We also show that the ranking may be reversed if these costs are taken into account and the regulator must audit at least one firm. We also analyze the welfare ranking when the expected penalties depend on relative violations.

Appendix

1.1 Proof of Lemma 1

Fix $k =0,\ldots ,n-1$ and define a function $F :\mathbb {R}_+ \rightarrow \mathbb {R}$ with

$$\begin{aligned} F(p)=\sum _{i=1}^{k} \hat{e}_i(p)+\sum _{i=k+1}^{n} e_i^h(p)-E^0. \end{aligned}$$

(38)

The function $F$ is strictly decreasing in $p$ since emission reports and actual emissions are strictly decreasing in $p$. Since $p_{n-k}$ is the equilibrium price with $k$ noncompliant firms, we have $F(p_{n-k})=0$.

We evaluate the difference $F(p_{n-k})-F(p_{n-(k+1)})$ as follows.

$$\begin{aligned} F\left( p_{n-k}\right) -F\left( p_{n-(k+1)}\right)&=\sum _{i=1}^{k} \hat{e}_i(p_{n-k})+\sum _{i=k+1}^{n}e_i^h\left( p_{n-k}\right) -E^0 \nonumber \\&\quad -\sum _{i=1}^{k} \hat{e}_i\left( p_{n-(k+1)}\right) -\sum _{i=k+1}^{n}e_i^h\left( p_{n-(k+1)}\right) +E^0 \nonumber \\&=-\sum _{i=1}^{k} \hat{e}_i\left( p_{n-(k+1)}\right) -\sum _{i=k+1}^{n}e_i^h\left( p_{n-(k+1)}\right) +E^0 \nonumber \\&<-\sum _{i=1}^{k+1} \hat{e}_i\left( p_{n-(k+1)}\right) -\sum _{i=k+2}^{n} e_i^h\left( p_{n-(k+1)}\right) +E^0=0, \end{aligned}$$

(39)

where the inequality follows from the definition of noncompliant behavior, that is, from the inequality $\hat{e}_i(p)<e_i^h(p)$ for any $p$ and for any noncompliant firm, and the last equality follows from the market clearing condition for $k+1$ noncompliant firms.

Now we show that $p_{n-k}>p_{n-(k+1)}$ by contradiction. Assume that $p_{n-k}\le p_{n-(k+1)}$ holds. If $p_{n-k}=p_{n-(k+1)}$, then $F(p_{n-k})=F(p_{n-(k+1)})$, contradicting (39), and if $p_{n-k}<p_{n-(k+1)}$, then $F(p_{n-k})>F(p_{n-(k+1)})$, since $F$ is strictly decreasing, which again contradicts (39). To conclude, the assumption $p_{n-k}\le p_{n-(k+1)}$ leads to a contradiction implying that the inequality $p_{n-k}> p_{n-(k+1)}$ holds.

1.2 Proof of Lemma 3

If $v_i=0$, we have from Eq. (28) that $S'_i(0)\frac{1}{e_i }=p+\lambda \ge p$. We give a proof for the other direction by showing that $v_i>0$ implies that $S'_i(0)\frac{1}{e_i }<p$. When $v_i>0$, we have $\lambda _i=0$. Therefore equality

$$\begin{aligned} S'_i\left( \frac{e_i-\hat{e}_i}{\hat{e}_i}\right) \frac{e_i}{\hat{e}_i^2}=p \end{aligned}$$

(40)

holds. Using the strict convexity of $S_i$ and the inequality $e_i>\hat{e}_i$, we obtain

$$\begin{aligned} p=S'_i\left( \frac{e_i-\hat{e}_i}{\hat{e}_i}\right) \frac{e_i}{\hat{e}_i^2}>S'_i\left( \frac{e_i-\hat{e}_i}{\hat{e}_i}\right) \frac{1}{e_i}>S'_i(0)\frac{1}{e_i}. \end{aligned}$$

(41)

1.3 Proof of Lemma 4

Part (i). (We assume differentiability where ever necessary.) We first use Theorem 5 (Topkis’s Theorem) in Milgrom and Shannon (1994) to show that the actual and reported emissions are nonincreasing in the permit price. The objective function $\Pi _i^p$ is supermodular in $(e_i,\hat{e}_i)$ and satisfies increasing differences in $(e_i,\hat{e}_i;-p)$, since

$$\begin{aligned} \frac{\partial ^2 \Pi _i^p}{\partial e_i \partial \hat{e}_i}&=S''_i\left( \frac{e_i-\hat{e}_i}{\hat{e}_i}\right) \frac{e_i}{\hat{e}_i^3}+S'_i\left( \frac{e_i-\hat{e}_i}{\hat{e}_i}\right) \frac{1}{\hat{e}_i^2}>0, \end{aligned}$$

(42)

$$\begin{aligned} \frac{\partial ^2 \Pi _i^p}{\partial e_i \partial (-p)}&=0, \quad \text { and } \quad \frac{\partial ^2 \Pi _i^p}{\partial \hat{e}_i \partial (-p)}=1>0. \end{aligned}$$

(43)

Therefore, $\mathop {\text {argmax}}_{(e_i,\hat{e}_i) \in X} \Pi _i^p(e_i,\hat{e}_i;-p)$, where $X$ is the feasible set $\{ (e_i,\hat{e}_i) \in \mathbb {R}^2 \ | \ e_i>0, \hat{e}_i>0, e_i-\hat{e}_i\ge 0 \}$, is nondecreasing in $-p$ or equivalently nonincreasing in $p$. In particular,

$$\begin{aligned} \frac{\partial e_i(p)}{\partial (-p)}\ge 0. \end{aligned}$$

(44)

Now we show that the reported emissions of a noncompliant firm are strictly decreasing in the permit price. We do this by applying the aggregation method described in Milgrom and Shannon (1994) together with Strict Monotonicity Theorem 1 (SMT1) in Edlin and Shannon (1998). Define

$$\begin{aligned} K_i\left( \hat{e}_i; -p\right) := \max _{e_i> \hat{e}_i} \Pi _i^p\left( e_i,\hat{e}_i;-p\right) , \end{aligned}$$

(45)

where $\hat{e}_i \in \mathbb {R}_{++}$. The function $K_i$ has increasing marginal returns (that is, $\frac{\partial K_i}{\partial \hat{e}_i}$ is increasing in $-p$), since

$$\begin{aligned} \frac{\partial ^2 K_i}{\partial \hat{e}_i \partial (-p)}=\frac{\partial ^2 \Pi _i^p}{\partial \hat{e}_i \partial e_i }\frac{\partial e_i}{\partial (-p)}+1 > 0 \end{aligned}$$

(46)

by (44). Let $-p'>-p^*$ (the other case is derived similarly) and define

$$\begin{aligned} \hat{e}_i^*&=\mathop {\text {argmax}}\limits _{\hat{e}_i \in \mathbb {R}_{++}}K_i(\hat{e}_i; -p^*) \quad \,\text { and } \end{aligned}$$

(47)

$$\begin{aligned} \hat{e}_i'&=\mathop {\text {argmax}}\limits _{\hat{e}_i \in \mathbb {R}_{++}}K_i(\hat{e}_i; -p'). \end{aligned}$$

(48)

From the beginning of the proof, we know that $\hat{e}_i'\ge \hat{e}_i^*$, which can be strengthened to $\hat{e}_i'> \hat{e}_i^*$ by (46) and by SMT1. In other words, we have that $p'<p^*$ implies $\hat{e}_i'> \hat{e}_i^*$, as desired.

Part (ii). We can use the same reasoning as in the proof of Lemma 1, since the reported emissions (and the actual) are strictly decreasing in the permit price.

1.4 Proof of Lemma 5

A noncompliant firm chooses its emissions and reported emissions such that

$$\begin{aligned} \pi _i'(e_i^d)-t+S_i'\left( \frac{e_i-\hat{e}_i}{\hat{e}_i}\right) \left[ \frac{e_i^d}{\hat{e}_i^2}-\frac{1}{\hat{e}_i}\right] =0. \end{aligned}$$

(49)

The term in the brackets is positive, since $e_i^d>\hat{e}_i$. When the firm is compliant, equation $\pi _i'(e_i^h)=t$ holds. Since $\pi _i''<0$, we have $e_i^d>e_i^h$.