## 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.

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## Notes

- 1.
Recent papers in this vein include Coria and Hennlock (2014), who focus on policy reactions to technological development in the presence of transaction and political costs and Ambec and Coria (2013) who analyze the control of two pollutants with asymmetric information about their interdependent abatement costs. Goodkind and Coggins (2015) take corner solutions into account. In a two-country model, Weitzel (2017) analyses how an abatement cost shock in one country affects both countries.

- 2.
- 3.
All figures in this paper assume that marginal abatement costs and MED are linear in abatement. However, our formal analysis is not limited to this case. The MAC and MED curves in the figures are drawn for illustrative purposes and do not always satisfy all conditions that we impose on them in the paper.

- 4.
- 5.
This can be seen as follows. Let \(C(a)=\frac{1}{2}a^{2}\) and \(D(e-a)=\frac{d }{2}(e-a)^{2},\) making MAC and MED linear in

*e*. By (2), \(a_{\phi }^{*}\) solves \(\phi a=d(e-a),\) so that \(a_{\phi }^{*}=de/(\phi +d)\) and$$\begin{aligned} \phi C(a_{\phi }^{*})=\frac{\phi }{2}\left( \frac{de}{\phi +d}\right) ^{2}. \end{aligned}$$Maximizing this with respect to \(\phi\) yields \(\phi =d\) and thus \(a_{\phi }^{*}=e/2\).

- 6.
- 7.
- 8.
- 9.
We will assume that if the regulator believes neither type of firm would have invested, she would set \(a_{l}^{*}\) in the out-of-equilibrium event that the firm did invest.

- 10.
There is no interval of

*F*for which there is a mixed strategy PBE with firm*l*randomizing and firm*h*not investing. In this case, the regulator would set \(a_{l}^{*}\) given by (2) after investment. Firm*l*would only be indifferent between investing and not investing for \(F=F_{l}^{l}\) given by (15). - 11.
We shall see in Sect. 3.3 that the regulator prefers the equilibrium where only firm

*l*invests. - 12.
We assume that there is always an interior solution for \(a_{\phi }(t)\). Goodkind and Coggins (2015) show that when corner solutions can occur (either complete abatement or no abatement at all), taxes have an advantage over quantity controls.

- 13.
In the following, we will often write \(\hat{t}_{{\theta } }(F)\) simply as \(\hat{ t}_{{\theta } }\).

- 14.
In Appendix 2, we present the additional assumptions that guarantee that the second and third equilibria coexist.

- 15.
- 16.
The argument is the same for any tax rate \(\bar{t}\) and abatement target \(a_{1}(\bar{t})\) given by (22).

- 17.
We know from Proposition 4 that with taxation, there is another equilibrium, where both types invest for \(F>F_{h}(t_{hl})\), with the regulator setting the tax rate \(\hat{t}_{h}(F)+\varepsilon\) which just induces investment by firm

*h*. However, we also know from Proposition 4 that this requires the regulator to distort the tax rate by so much that she decides to give up on investment by firm*h*for \(F>\bar{F}_{h}\). - 18.

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## Acknowledgements

We thank Marco Caffera, Matti Liski, Alex Possajennikov and Marko Terviö and seminar participants at the universities of Stirling and Helsinki for valuable comments.

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## 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

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):

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 \(F=F_{h}^{*}\):

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

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

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

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

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

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

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

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

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:

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

The corresponding expected social costs are

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

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

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

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

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

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

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).

## About this article

### Cite this article

D’Amato, A., Dijkstra, B.R. Adoption incentives and environmental policy timing under asymmetric information and strategic firm behaviour.
*Environ Econ Policy Stud* **20, **125–155 (2018). https://doi.org/10.1007/s10018-017-0187-4

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### Keywords

- Asymmetric information
- Commitment
- Time consistency
- Emission taxation
- Quotas

### JEL Classification

- D62
- D82
- Q28