Adoption incentives and environmental policy timing under asymmetric information and strategic firm behaviour

Abstract

We consider the incentives of a single firm to invest in a cleaner technology under quotas and emission taxation. We assume asymmetric information about the firm’s cost of employing the new technology. Policy is set either before the firm invests (commitment) or after (time consistency). Contrary to the conventional wisdom, we find that with commitment (time consistency), quotas give higher (lower) investment incentives than taxes. With quotas (taxes), commitment generally leads to higher (lower) welfare than time consistency. Under commitment with quadratic abatement costs and environmental damages, a modified Weitzman rule applies and quotas usually lead to higher welfare than taxes.

Appendices

Appendix 1: Proofs

Lemma 1. Recall that $a_{\phi }^{*},\ \phi =l,h,1,$ is defined by (2). For any $a\ge a_{l}^{*}$ (with the new or the current technology), normalize the firm’s payment T to the regulator to zero. For any a with $a_{h}^{*}\le a<a_{l}^{*}$ (again regardless of the technology choice), the firm pays $T^{*}$ with

$$\begin{aligned} l\left[ C(a_{l}^{*})-C(a_{h}^{*})\right]<T^{*}<h\left[ C(a_{l}^{*})-C(a_{h}^{*})\right] . \end{aligned}$$

For any a with $0\le a<a_{h}^{*}$ and investment in the new technology, the firm pays $\hat{T}$ with $\hat{T}>T^{*}+hC(a_{h}^{*})$. For any a with $0\le a<a_{h}^{*}$ without investment in the new technology, the firm pays $\bar{T}$ with $\bar{T}>T^{*}+hC(a_{h}^{*})+F$. The firm will then invest, regardless of its type, and firm ${\theta }$ will abate $a_{{\theta } }^{*}$.

Lemma 2. The lemma holds if and only if $F<F_{h}(a_{m}^{*})$ as given by (7) and (10). We find from (5) and (7):

$$\begin{aligned} F_{h}(a_{m}^{*})&= \left( 1-h\right) C(a_{m}^{*})>(1-h)C(a_{h}^{*})=C(a_{h}^{*})+D(e-a_{h}^{*})-\left[ hC(a_{h}^{*})+D(e-a_{h}^{*})\right] \\& > C(a_{1}^{*})+D(e-a_{1}^{*})-\left[ hC(a_{h}^{*})+D(e-a_{h}^{*})\right] =F_{h}^{*}. \end{aligned}$$

The first inequality follows from (11) and $C^{\prime }>0$. The second inequality follows from the fact that $a_{1}^{*}$ minimizes ${{SC}}_{1}$. Since we have assumed that $F<F_{h}^{*},$ we also have $F<F_{h}(a_{m}^{*}),$ and both types of firm will invest when the abatement target is $a_{m}^{*}$.

Lemma 3. We see from (8) and (12) that $E[{{SC}}_{m}(a_{m}^{*})]-E[{{SC}}_{1l}(a_{1l})]$ is increasing in F. Thus, if $E[{{SC}}_{m}(a_{m}^{*})]-E[{{SC}}_{1l}(a_{1l})]<0$ for $F=F_{h}^{*},$ then the inequality holds for all $F<F_{h}^{*}$. Let us now determine the sign of $E[{{SC}}_{m}(a_{m}^{*})]-E[{{SC}}_{1l}(a_{1l})]$ for $F=F_{h}^{*}$.

First, define $\bar{a}_{1m}>a_{1}^{*}$ implicitly by ${{SC}}_{1}(\bar{a} _{1m})={{SC}}_{h}(a_{m}^{*}),$ or using (5) for $F=F_{h}^{*}$:

$$\begin{aligned} {{SC}}_{1}(\bar{a}_{1m})-{{SC}}_{1}(a_{1}^{*})={{SC}}_{h}(a_{m}^{*})-{{SC}}_{h}(a_{h}^{*}). \end{aligned}$$

(38)

We shall now see that $\bar{a}_{1m}<a_{m}^{*}$. Suppose that $\bar{a} _{1m}\ge a_{m}^{*}$. Then we can write the LHS of (38) as

$$\begin{aligned} {{SC}}_{1}(\bar{a}_{1m})-{{SC}}_{1}(a_{1}^{*})=\left[ {{SC}}_{1}(\bar{a} _{1m})-{{SC}}_{1}(a_{m}^{*})\right] +\left[ {{SC}}_{1}(a_{m}^{*})-{{SC}}_{1}(a_{h}^{*})\right] +\left[ {{SC}}_{1}(a_{h}^{*})-{{SC}}_{1}(a_{1}^{*})\right] . \end{aligned}$$

(39)

The first term between square brackets on the RHS is nonnegative, because $\bar{a}_{1m}\ge a_{m}^{*}>a_{1}^{*}$ (the second inequality follows from (3) and $m<1)$ and ${{SC}}_{1}^{\prime }(a)>0$ for $a>a_{1}^{*}$. The third term between square brackets on the RHS of (39) is positive, because $a_{1}^{*}$ minimizes ${{SC}}_{1}(a)$. Thus we have

$$\begin{aligned} {{SC}}_{1}(\bar{a}_{1m})-{{SC}}_{1}(a_{1}^{*})>{{SC}}_{1}(a_{m}^{*})-{{SC}}_{1}(a_{h}^{*}). \end{aligned}$$

(40)

Comparing the RHS of (40) to the RHS of (38) yields

$$\begin{aligned} {{SC}}_{1}(a_{m}^{*})-{{SC}}_{1}(a_{h}^{*})-\left[ {{SC}}_{h}(a_{m}^{*})-{{SC}}_{h}(a_{h}^{*})\right] =(1-h)\left[ C(a_{m}^{*})-C(a_{h}^{*})\right] >0. \end{aligned}$$

(41)

The inequality follows from $h<1,$ (11) and $C^{\prime }>0$. It follows from (40) and (41) that $\bar{a}_{1m}\ge a_{m}^{*}$ cannot hold. Thus $\bar{a}_{1m}<a_{m}^{*}$.

We can now write

$$\begin{aligned}& \left[ p{{SC}}_{h}(a_{m}^{*})+(1-p){{SC}}_{l}(a_{m}^{*})\right] -\left[ p{{SC}}_{h}(a_{h}^{*})+(1-p){{SC}}_{l}(a_{l}^{*})\right] \nonumber \\&\quad = p{{SC}}_{1}(\bar{a}_{1m})+(1-p){{SC}}_{l}(a_{m}^{*})-\left[ p{{SC}}_{1}(a_{1}^{*})+(1-p){{SC}}_{l}(a_{l}^{*})\right] \nonumber \\ &\quad <\left[ p{{SC}}_{1}(a_{1l})+(1-p){{SC}}_{l}(a_{1l})\right] -\left[ p{{SC}}_{1}(a_{1}^{*})+(1-p){{SC}}_{l}(a_{l}^{*})\right] . \end{aligned}$$

(42)

The equality follows from (38) and $F=F_{h}^{*}$ in (5). The inequality follows from $\bar{a}_{1m}<a_{m}^{*},$ which implies $\bar{a}_{1m}<a_{1l}<a_{m}^{*}$. Then ${{SC}}_{1}(\bar{a} _{1m})<{{SC}}_{1}(a_{1l})$, since $a_{1}^{*}<\bar{a}_{1m}<a_{1l}$ and ${{SC}}_{1}^{\prime }(a)>0$ for $a>a_{1}^{*},$ and ${{SC}}_{l}(a_{m}^{*})<{{SC}}_{1}(a_{1l})$, since $a_{l}^{*}>a_{m}^{*}>a_{1l}$ and ${{SC}}_{l}^{\prime }(a)<0$ for $a<a_{l}^{*}$.

Since the second term in square brackets in the first line of (42) cancels out against the corresponding term in the last line by (5) with $F=F_{h}^{*}$, the Lemma follows.

Lemma 4. $F_{m}^{h}$ in (14) is decreasing in $a_{m}^{*}$ (since $C^{\prime }>0)$ and thus by (3) increasing in m. Since by (9), the highest value of m is h for $p\rightarrow 1,$ the highest possible value of $F_{m}^{h}$ is

$$\begin{aligned} F_{h}^{h}\equiv C(a_{1}^{*})-hC(a_{h}^{*}). \end{aligned}$$

(43)

We thus need to prove that $F_{h}^{h}<F_{l}^{l}$ which implies from (15) and (43):

$$\begin{aligned} hC(a_{h}^{*})>lC\left( a_{l}^{*}\right) . \end{aligned}$$

(44)

Since $F_{l}^{l}>0$ in (15), so that $C(a_{1}^{*})>lC(a_{l}^{*})>0,$ and also $l<1$, $lC(a_{l}^{*})$ must be on the increasing branch of $\phi C(a_{\phi }^{*})$: $l<\widetilde{\phi }$ by Assumption 1. Then, if h is between l and $\widetilde{\phi },$ it is also on the increasing branch and (44) holds. If h is between $\widetilde{\phi }$ and 1,  it is on the decreasing branch of $\widetilde{\phi }C(a_{\phi }^{*}),$ so that $hC(a_{h}^{*})>C(a_{1}^{*})>lC(a_{l}^{*})$ and (44) also holds.

Lemma 5. Suppose $t_{hl}\le t_{l},$ then by (23), we would have $a_{l}(t_{hl})\le a_{l}(t_{l})$ and $a_{h}(t_{hl})<a_{h}(t_{h}),$ so that by $D^{\prime \prime }>0$ and (26),  $D^{\prime }\left[ e-a_{l}(t_{hl})\right] \ge t_{hl}$ and $D^{\prime }\left[ e-a_{h}(t_{hl})\right] >t_{hl}$. Thus, the first term in curly brackets on the LHS of (29) is negative and the second term is nonpositive. This, combined with (23) means that the LHS of (29) is positive, so that (29) cannot hold. In the same way, we can show that $t_{hl}<t_{h}$.

Proposition 4. In the first equilibrium, for very low values of F, in stage 1, the regulator sets the optimal tax rate of $t_{hl}$ (defined by (29)), given that both types of firm will invest, and in stage 2, both types of firm will invest. At $t_{hl},$ firm h will invest for all $F<F_{h}(t_{hl})$ given by (31). As we have seen following (32), $F_{h}(t_{hl})>0$.

In the second equilibrium, while firm h would no longer invest at $t_{hl},$ the regulator would still like to induce it to invest. Thus, the regulator sets the tax rate at $\hat{t}_{h}(F)+\varepsilon ,$ with $\hat{t} _{h}(F)$ given by (30).

As F keeps rising, there comes a point at which the regulator prefers to see only firm l investing. In the third equilibrium, the regulator sets the optimal tax rate of $t_{1l}$ (defined by (29)) given that firm l will invest, but firm h will not. At $F=\bar{F}_{h}$ in (35), the regulator is indifferent between the second and the third equilibrium.

When F grows larger, it will reach $F_{l}(t_{1l}),$ defined by (31) as the point beyond which firm l no longer wants to invest at tax rate $t_{1l}$. However, in the fourth equilibrium, the regulator still induces firm l to invest by setting the tax rate at $\hat{t} _{l}(F)+\varepsilon ,$ with $\hat{t}_{l}(F)$ given by (30).

For even higher F,  the regulator no longer wishes firm l to invest. When neither type of firm invests, the regulator would ideally like to set the tax rate at $t_{1}$. However, if in stage 1 the regulator sets $t_{1}$, firm h will invest in stage 2. Fig. 3 illustrates this: Firm h invests at $t_{1}$ for $F<F_{h}(t_{1})={\text {OBN}},$ which exceeds the maximum F of $F_{h}^{*}=\text{OBH}$.

Formally, comparing $F_{h}(t_{1})$ to $F_{h}^{*}$ in (5), we find

$$\begin{aligned} F_{h}(t_{1})-F_{h}^{*}= & {} \left\{ t_{1}(a_{h}^{*}-a_{1}^{*})-\left[ D(e-a_{1}^{*})-D(e-a_{h}^{*})\right] \right\} \\&\quad +\left\{ t_{1}[a_{h}(t_{1})-a_{h}^{*}]-h\left[ C[a_{h}(t_{1})]-C(a_{h}^{*}) \right] \right\} >0 \end{aligned}.$$

The first term in curly brackets on the RHS is positive by (4), $t_{1}=D^{\prime }(e-a_{1}^{*})$ by (26) and $D^{\prime \prime }>0$. This term is given by area BVH in Fig. 3. The second term in curly brackets is positive, because $a_{h}(t_{1})>a_{h}(t_{h})=a_{h}^{*}$ as $t_{1}>t_{h}$ by (27) and by (23), $hC^{\prime }[a_{h}(t_{1})]=t_{1}$ by (22), and $C^{\prime \prime }>0$. This term is given by area VNH in Fig. 3.

Since setting the tax rate at $t_{1}$ would not have the desired result, the fifth equilibrium features the regulator setting the tax rate as high as possible, while still discouraging firm l from investing. This means that the tax rate will be $\hat{t}_{l}(F)-\varepsilon ,$ with $\hat{t} _{l}(F)$ given by (30). We have defined $\bar{F}_{l}$ in (34) as the level of fixed cost at which the regulator is indifferent between inducing firm l to invest and discouraging investment. We have assumed $\bar{F}_{l}<$ $F_{h}^{*},$ in order for all five equilibria to occur for $F\in (0,F_{h}^{*})$.

Lemma 6. From (5) and (36), we find

$$\begin{aligned} F_{h}^{hl}-F_{h}^{*}= & {} \left\{ (t_{1}-t_{hl})(e-a_{h}^{*})\right\} +\left\{ (t_{1}-t_{h})(a_{h}^{*}-a_{1}^{*})-\left[ D(e-a_{1}^{*})-D(e-a_{h}^{*})\right] \right\} \\&\quad +\left\{ (t_{h}-t_{hl})\left[ a_{h}^{*}-a_{h}(t_{hl})\right] -h\left( C(a_{h}^{*})-C[a_{h}(t_{hl})]\right) \right\} >0 \end{aligned}$$

The first term in curly brackets is positive, because $t_{1}>t_{h}>t_{hl}$ by (27) and Lemma 5. This is area ${\text V}t_{1}t_{hl}{\text X}$ in Fig. 3. The second term is positive by (4), $t_{h}=D^{\prime }(e-a_{h}^{*})$ in (26), $D^{\prime \prime }>0$, and $t_{1}>t_{h}$ by (27). This is area BVH in Fig. 3. The third term is positive, because $t_{h}>t_{hl}$ by Lemma 5; $a_{h}^{*}=a_{h}(t_{h})>a_{h}(t_{hl})$ by (23), (26) and $C^{\prime \prime }>0$. This is area ZHX in Fig. 3.

Proposition 6. The first commitment equilibrium listed in Proposition 4 is the same as the only equilibrium under time consistency (Proposition 5) and thus yields the same expected social costs. The second equilibrium under commitment yields higher expected social costs than this, because while both equilibria have both types of firm investing, $t_{hl}$ is the optimal tax rate given that they do. At $F= \bar{F}_{h}$ under commitment,  the regulator is indifferent between the tax rates of $\hat{t}_{h}+\varepsilon$ (the second equilibrium) and $t_{1l}$ (the third equilibrium). Thus, by continuity, the third equilibrium under commitment also yields higher expected social costs than the time consistency equilibrium for F close to the lower bound of $\bar{F}_{h}$. For F close to the higher bound of $F_{l}(t_{1l})$, however, social costs could be lower under commitment. Since $\hat{t}_{l}[F_{l}(t_{1l})]=t_{1l},$ social cost could also be lower in the fourth commitment equilibrium for F close to the lower bound of $F_{l}(t_{1l})$. However, at the higher bound of $F=\bar{F }_{l},$ social cost is higher in commitment. This is because in the fifth commitment equilibrium, social cost is higher under commitment:

$$\begin{aligned} {{SC}}_{1}(\hat{t}_{l}+\varepsilon )>{{SC}}_{1}(t_{1})={{SC}}_{1}(a_{1}^{*})>{{SC}}_{h}(a_{h}^{*})={{SC}}_{h}(t_{h})>E\left[ {{SC}}_{hl}(t_{hl})\right] . \end{aligned}$$

The first inequality follows from the fact that $t_{1}$ minimizes social cost, given that the firm does not invest. The second inequality follows from $F<F_{h}^{*}$ in (5). The third inequality follows from the fact that $E\left[ {{SC}}_{hl}(t_{hl})\right]$ is decreasing in l with $l<h$.

Lemma 7. Part 1. A linear marginal abatement cost curve $C^{\prime }(a)$ with slope 1 implies $C(a)=\frac{1}{2}a^{2},\ C^{\prime }(a)=a$. A linear marginal environmental damage curve $D^{\prime }(e-a)$ with slope d implies $D(e-a)=\frac{d}{2}(e-a)^{2},$ $D^{\prime }(e-a)=d(e-a)$. From (10), the quota $a_{m}^{*}$ is given by

$$\begin{aligned} a_{m}^{*}=\frac{de}{d+ph+(1-p)l}. \end{aligned}$$

(45)

The corresponding expected social costs are

$$\begin{aligned} E\left[ {{SC}}_{m}(a_{m}^{*})\right] =F+\frac{1}{2}\frac{de^{2}\left[ ph+(1-p)l\right] }{d+ph+(1-p)l}. \end{aligned}$$

(46)

From (22), the firm with technology $\phi$ responds to a tax rate of t by setting:

$$\begin{aligned} a_{\phi }(t)=\frac{t}{\phi }. \end{aligned}$$

(47)

From (29), the first-order condition for social cost minimization with respect to t implies

$$\begin{aligned} t_{hl}=\frac{dehl\left[ \left( 1-p\right) h+pl\right] }{\left( 1-p\right) h^{2}\left( d+l\right) +pl^{2}\left( d+h\right) }. \end{aligned}$$

(48)

The corresponding (minimum) expected social costs at $t_{hl}$ are

$$\begin{aligned} E\left[ {{SC}}_{hl}(t_{hl})\right] =F+\frac{1}{2}de^{2}\frac{dp(1-p)\left( h-l\right) ^{2}+hl\left[ (1-p)h+pl\right] }{d\left[ (1-p)h^{2}+pl^{2}\right] +hl\left[ (1-p)h+pl\right] }. \end{aligned}$$

(49)

The differential gain in favour of quotas is from (46) and (49):

$$\begin{aligned} E\left[ {{SC}}_{hl}(t_{hl})\right] -E\left[ {{SC}}_{m}(a_{m}^{*})\right] =\frac{ \frac{1}{2}d^{2}e^{2}\left( h-l\right) ^{2}p(1-p)\left[ d-(1-p)h-pl\right] }{ \left[ d+ph+(1-p)l\right] \left[ \left( 1-p\right) h^{2}\left( d+l\right) +pl^{2}\left( d+h\right) \right] } \end{aligned}$$

which is positive if and only if (37) holds.

Part 2. (2) and (15) imply, in our quadratic setting:

$$\begin{aligned} F_{l}^{h}=\frac{1}{2}\left( \frac{de}{d+1}\right) ^{2}-\frac{h}{2}\left( \frac{de}{d+l}\right) ^{2}>0. \end{aligned}$$

As $F_{l}^{h}$ is increasing in $l<h,$ a necessary condition for the inequality to hold is $(d+h)^{2}>h(d+1)^{2}$ which can be rewritten as $d^{2}(1-h)>h(1-h),$ or $d^{2}>h,$ since $h<1$. If $d>1,$ then $d>h,$ so that (37) always holds. If $d<1,$ then $d>d^{2}>h$ and again (37) always holds.

Appendix 2: Additional assumptions for Proposition 4

To make sure that the second equilibrium in Proposition 4 exists, we must examine the regulator’s and the firm’s incentives at $F=F_{h}(t_{hl})$. If $t_{1l}>t_{hl},$ firm h would invest at a tax rate of $t_{1l},$ so that even if the regulator would prefer $t_{1l}$ and no investment by firm h,  this outcome is not feasible. If $t_{1l}<t_{hl},$ firm h would not invest at $t_{1l}$ and the regulator might prefer to set $t_{1l}$. In that case, there would not be an equilibrium with $\hat{t}_{h}+\varepsilon$. To ensure existence of this equilibrium, we must thus assume:

Assumption 2

If $t_{1l}<t_{hl},$ with $t_{1l}$ and $t_{hl}$ given by (29), then $E \left[ { {SC}}_{hl}(t_{hl})\right] <E\left[ {{SC}}_{1l}(t_{1l})\right]$ at $F=F_{h}(t_{hl}),$ with $E\left[ {{SC}}_{hl}(t)\right]$ and $E\left[ {{SC}}_{1l}(t)\right]$ given by (28), and $F_{h}(t)$ by (31).

From (28), $E\left[ {{SC}}_{hl}(t_{hl})\right] -E\left[ {{SC}}_{1l}(t_{1l}) \right]$ is increasing in F. Thus, Assumption 2 means that $E\left[ {{SC}}_{hl}(t_{hl})\right] <E\left[ {{SC}}_{1l}(t_{1l})\right]$ for all $F\le F_{h}(t_{hl})$.

As for the third equilibrium, the switch from $\hat{t}_{h}$ to $t_{1l}$ at $\bar{F}_{h}$ can only occur when firm h will not invest, but firm l will. We shall assume that this is the case:

Assumption 3

$\hat{t}_{l}(\bar{F}_{h})<t_{1l}<\hat{t }_{h}(\bar{F}_{h}),$ with $t_{\theta}(F)$ given by (30), $\bar{F}_{h}$ by (35), and $t_{1l}$ by (29).