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Does a Recession Call for Less Stringent Environmental Policy? A Partial-Equilibrium Second-Best Analysis

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Abstract

This paper analyses second-best optimal environmental policy responses to real and financial shocks in a two-period partial equilibrium model with heterogeneous firms, an environmental externality, and credit constraints. We show that, to alleviate credit constraints and encourage investment, the second-best optimal emission tax falls short of marginal emission damages. The optimal response to shocks depends on how the shock affects the size of the environmental and credit market failures and the effectiveness of the tax in alleviating these market failures. Under mildly restrictive assumptions on functional forms, the optimal response to a (persistent) negative productivity shock or a tightening of credit is to reduce the emission tax. Our results are informative for how climate change policy should optimally change with the business cycle.

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Notes

  1. See Fischer and Heutel (2013).

  2. See for instance Gertler and Kiyotaki (2010), Bassetto et al. (2015) and Khan and Thomas (2013) for evaluations of the effect of economic shocks in a model with credit market imperfections and heterogeneous firms. Earlier references include Bernanke and Gertler (1989), Kiyotaki and Moore (1997) and Bernanke et al. (1999).

  3. The inequality condition, which implies each firm i has a unique z, facilitates the exposition of results. All results generalize to \(z'(i)\ge 0\), or discontinuous distributions.

  4. The credit constraint is equivalent to those adopted in Bassetto et al. (2015) and Khan and Thomas (2013), and an approximation of the endogenous constraints derived in for example Bernanke and Gertler (1989).

  5. We present the full solution of the alternative case in “Appendix 1”, and also provide a short discussion of this case in Sect. 7.

  6. Since we already wrote e as a function of i above, this is a slight abuse of notation, but no confusion between the two functions will arise. Subscripts to the function symbols e and E will uniquely refer to derivatives of the functions just introduced.

  7. For the sake of brevity, we only refer to the derivatives of the first period. All relationships carry over to the second period variables (e.g. if \(e_{t}<0\), also \(E_{T}<0\)). An overview of all notation and derivatives can be found in “Appendix 2”.

  8. This effect is also known as the energy rebound effect. Over the past years, a literature has emerged that explores under what conditions improvements in green technology reduces harmful emissions, and in what cases stricter environmental policy leads to green technology adoption to begin with. See for example Gil-Molto and Dijkstra (2011), Bréchet and Meunier (2014), Gans (2012), Perino and Requate (2012), Di Maria and Smulders (2017) and also Gillingham et al. (2016).

  9. From (2) and (5), we derive \(e_{z}=t/(-y_{ee}z^{2})>0\) and \(\pi _{z}=te/z^{2}>0\).

  10. We can show that under Assumption 1 \(Z_{T}^{U}>0\). First, let \(E=\breve{E}(\tilde{T})\) with \(\tilde{T}\equiv T/Z\) solve (5). Since \(M=E/Z\), we find \(M_{T}=\breve{E}_{\tilde{T}}/Z^{2}\) and \(M_{Z}=-(T/Z)\breve{E}_{\tilde{T}}/Z^{2}-M/Z=-(M_{T}T+M)/Z\). Second, from (2) and (5), we find \(\varPi _{Z}=MT/Z>0\), \(\varPi _{ZT}=(M+TM_{T})/Z\), and \(\varPi _{ZZ}=(TM_{Z}-\varPi _{Z})/Z<0\). Finally, from (6) we find \(Z_{T}^{U}=-\varPi _{ZT}/(\varPi _{ZZ}-C_{ZZ})\). Combining results, we find \(Z_{T}^{U}=-M_{Z}/(C_{ZZ}-\varPi _{ZZ})>0\).

  11. See for instance Bassetto et al. (2015), Buera and Shin (2013) and Khan and Thomas (2013).

  12. Note that whenever the solution is interior, n is not only the share of constrained firms, but also the index of the smallest unconstrained firm.

  13. To be more precise, the marginal return to investing in Z reads \(\varPi _{Z}=TEZ^{-1}.\) For given E, an increase in T increases \(\varPi _{Z}\). However \(E_{T}<0\): additionally, an increase in T reduces firm energy use, which reduces \(\varPi _{Z}\). As long as Assumption 1 applies we can show the former effect dominates, so \(\varPi _{ZT}>0\).

  14. This assumption concerning \(\Delta \) is further discussed in Sect. 7.

  15. Note that maximizing the discounted sum of firm value added, net of environmental damages, is equivalent to minimizing the sum of firm abatement cost and environmental damages, which is the standard social planner optimization problem in a partial equilibrium framework (see for instance Requate and Unold (2003)). More specifically, let \(\pi _{0}\) and \(\Pi _{0}\) be a firm’s first and second-period profits in the absence of environmental policy, respectively. Abatement cost then equal \(\left( \pi _{0}-\pi \right) +\left( \varPi _{0}-\varPi \right) +C\) and the two-stage problem reads \(\underset{{\scriptstyle t}}{\min }\ \int _{0}^{1}\left[ \left( \pi _{0}-\pi \right) -s+\Delta m\right] di+L\) \(\text{ subject } \text{ to } (5) \text{ and } (6) \text{ for } {i\ge n}, \text{ and } (4) \text{ for } {i<n}, \) where \(L=\underset{T}{\min }\left\{ \int _{0}^{1}\left[ \left( \varPi _{0}-\varPi \right) +C-S+\Delta M\right] di\right\} \). As the \(\pi _{0}\) and \(\varPi _{0}\) are independent of taxes, this problem is equivalent to (8).

  16. As explained, we solve for T under the assumption that the regulator cannot commit to T before firms invest. One can however show that the result \(T^{*}=\Delta \) extends to the case where the regulator would be able to commit.

  17. For expositional purposes, the discussion below will assume that, initially, a subset of firms are constrained (\(n\in \left( 0,1\right) \)).

  18. In the macro literature, this effect is known as the “financial accelerator” (Bernanke et al. 1999).

  19. More precisely, we have \(B\equiv -\left[ \int _{0}^{1}z^{-1}\left[ e_{t}+\left( t^{*}-\Delta \right) e_{tt}\right] di+\int _{0}^{n}\left[ \left[ \varPi _{ZZ}-C_{ZZ}\right] Z_{t}^{R}Z_{t}^{R}+\left[ \varPi _{Z}-C_{Z}\right] Z_{tt}^{R}\right] di\right] \). As \(t^{*}\) maximizes \(v\equiv \int _{0}^{1}\left[ \pi +s+\varPi -C+S-\Delta \left( m+M\right) \right] di\) subject to (4) we must have that at \(t=t^{*}\), \(v_{t}=0\) and \(v_{tt}<0\). Here one can show \(B=-v_{tt}\).

  20. A positive superelasticity means that marginal cost \(C_{Z}\) rises faster with Z than average cost C / Z.

  21. Moreover, \(M_{Z}<0\) follows from Assumption 1. If we were to relax this assumption, and if in the left tail of the distribution a significant share of firms would have \(M_{Z}>0\), the tax might become countercyclical.

  22. Note we again use the result that either \(\Pi _{Z}=C_{Z}\) for firm \(i=n\), or \(n_{a}=0\).

  23. For the sake of brevity, the equations are expressed in terms of first-period variables. All equations equally apply for the second-period variables. For example, the second-period equivalent of (17) is \(Y\left( A,E\left( i\right) \right) =\exp \left( A\right) ^{1-\beta }E(i)^{\beta }\).

  24. Alternatively, by (18), we have \(ZC_{Z}/C=\gamma \). It then directly follows from Proposition 2 that \(t_{\xi }>0\).

  25. The existing literature on business cycles and environmental policy mostly abstracts from the question of damage cyclicality by focusing on the cost effectiveness of policies reducing emissions to a certain level (e.g. Angelopoulos et al. 2010; Fischer and Springborn 2011; Dissou and Karnizova 2017). An exception is Heutel (2012), who models the emission of \({\hbox {CO}}_{2}\), which cause damages that are quadratic in the current emission stock. This damage specification, as we explain below, may lead to an overstatement of the (pro-)cyclicality marginal damages.

  26. Even if one would assume marginal emission damages increase in cumulative emissions, the effect of a positive productivity shock on marginal damages is not immediate. Due to credit constraints, the positive productivity shock does not only increase emissions, but also investment in emission efficiency. Because of these efficiency improvements, and corresponding emission reductions, the effect of a productivity shock on cumulative emissions is ambiguous.

  27. In more detail, we have \(Z_{ta}^{R}=-C_{Z}^{-1}\left[ \xi e_{a}z^{-1}+\left[ C_{ZZ}Z_{a}^{R}+C_{Za}\right] Z_{t}^{R}\right] \). As \(Z_{t}^{R}<0\) and \(C_{Z}>0\), \(Z_{ta}^{R}\) is increasing in \(C_{Za}\).

  28. Note that as the lump sum can be used for loan collateral, this is a more ’generous’ redistribution scheme than a direct investment subsidy that reduces investment cost to \(C-s\).

  29. For details, see proof to Proposition 4.

  30. For details, see proof to Proposition 5.

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Correspondence to Inge M. van den Bijgaart.

Appendices

Appendix 1: Environmental Policy with Lump-Sum Recycling

In the main part of the paper, we assume the tax recycling scheme leaves the credit constraint, (4), unaffected. This assumption is justified if the tax revenues are a direct benefit to the government, or the rebate only occurs once the investment is made (and/or the loan is repaid). Alternatively, if only a small subset of firms are subject to the tax, whereas the returns are spread across a large group of firms, the rebate is of negligible size and can hence be ignored in (4). In any case, this setup greatly simplifies the analysis as it allows us to abstract from the redistributional effect of the tax; as a firm’s tax payment is a function of its emission efficiency, z, not all firms are taxed equally, and the lump sum rebate constitutes a redistribution of funds across firms. In this appendix, we re-establish Propositions 1, 4 and 5 for the case in which the lump sum can be used to invest and obtain credit. More specifically, the constraint defined by (4) will be replaced by

$$\begin{aligned} C\left( z(i),Z(i)\right) \le \xi \left[ \pi (i)+s\right] , \end{aligned}$$
(21)

where the lump sum rebate

$$\begin{aligned} s=t\int _{0}^{1}e(i)z(i)^{-1}di \end{aligned}$$
(22)

is considered exogenous by the firm.Footnote 28 This appendix is structured similar to the main text. We first solve for the firm optimization problem for given t and T. Next, we determine \(t^{*}\) and \(T^{*}\), as well as the response of the optimal tax to credit, \(\xi \), and productivity, a, for both the general and specific functional form.

1.1 Equilibrium

The firms’ optimization problem now reads:

$$\begin{aligned} \begin{array}{c} \underset{{\scriptstyle e,E,Z}}{\max }\ \pi +s+\varPi -C+S,\end{array} \end{aligned}$$
(23)

subject to (21). Output, profits and investment costs are defined as in (1)–(3). Firms take the lump sum as exogenous, so the first order conditions in (5) still apply: \(\pi _{e}=0\) and \(\varPi _{E}=0\). Also \(C_{Z}=\varPi _{Z}\) still holds for unconstrained firms (see (6)). For constrained firms we now have \(C=\xi \left[ \pi +s\right] \). From the above, we can directly conclude that the lump-sum rebate alleviates the credit constraint. As all firms receive the same lump sum, Corollary 1 still applies: firms \(i\in \left[ 0,n\right) \) are credit constrained, and firms \(i\in \left[ n,1\right] \) are unconstrained. With the tax recycling scheme in place, an additional separation across firms becomes relevant: firms to whom an increase in the first-period tax is a net benefit, versus firms to whom it is a net cost. The most straightforward way to see this is if we consider only 2 (types of) firms: one with zero emissions (i.e. \(z=\infty \)) and one with positive emissions (i.e. some \(z<\infty \)). Now the introduction of an emission tax imposes a cost on the latter group only. However, both firms receive the rebate. It must thus follow that the tax introduction is a net gain to firms with \(z=\infty \), and a net cost to firms with \(z<\infty \). This rationale holds for any first-period tax increase as long as total tax payments, and hence the size of the rebate, is increasing in the tax rate. The following Lemma establishes that this is indeed the case:

Lemma 3

\(s_{t}>0\)

Proof

By (22) we have \(s_{t}=\int _{0}^{1}z^{-1}\left[ e+te_{t}\right] di\). By (2) and (5) we have \(e_{t}=\left[ zy_{ee}\right] ^{-1}\) and \(e_{z}=-z^{-2}y_{ee}^{-1}t\) which implies \(te_{t}=-ze_{z}\). Next we use \(m_{z}=z^{-2}\left[ ze_{z}-e\right] \), so \(s_{t}=-\int _{0}^{1}zm_{z}di\). Now by Assumption 1, \(m_{z}<0\), so \(s_{t}>0\).

The above result can be explained as follows. The effect of an increase of z on emissions is twofold. On the one hand, a greater emission efficiency reduces emissions, given energy use. On the other hand, a higher z reduces the marginal cost of energy use, which increases firms’ choice of e. For emissions to fall in z we thus need e to be relatively insensitive to changes in the tax component of energy costs, t / z. In a similar manner, the effect of an increase in t on tax revenue, s, can be separated in two effects. One the one hand, given emissions, an increase in t increases tax revenue. On the other hand, the increase in t reduces energy use, reducing emissions and tax revenues. By assuming \(m_{z}<0\), we implicitly assume e is relatively insensitive to changes in t / z, and as a consequence we also find \(s_{t}>0\). Note that this does not mean the downward-sloping part of the Laffer curve does not exist in our specification. Instead, it implies that we restrict our analysis to the case with \(m_{z}<0\) and thereby \(s_{t}>0\).

Next we define

Definition 3

Let \(\tilde{z}\) be the first-period efficiency such that for a firm with \(z=\tilde{z}\), a marginal change in t leaves maximal investment unaffected, i.e. \(Z_{t}^{R}(\tilde{z},t,a,\xi )=0\).

Lemma 3 then allows us to prove

Lemma 4

\(\tilde{z}\) is unique and strictly larger than \(\underline{z}\). For firms with \(z<\tilde{z}\), we have \(Z_{t}^{R}<0\), while for firms with \(z>\tilde{z}\), \(Z_{t}^{R}>0\).

Proof

First we use (21) to establish \(Z_{t}^{R}=C_{Z}^{-1}\xi \left[ \pi _{t}+s_{t}\right] \). Next, by \(\pi _{t}=-m\), we have \(\pi _{tz}=-m_{z}>0\). With s and thus \(s_{t}\) common across firms, this implies that \(\left[ \pi _{t}+s_{t}\right] \) is more likely negative for low z firms. As \(\int _{0}^{1}\left[ \pi _{t}+s_{t}\right] di=\int _{0}^{1}z^{-1}\left[ te_{t}\right] di<0\), \(Z_{t}^{R}\) must be negative for the firm with the lowest z: \(i=0\). For \(z=\infty \), emissions are zero, so \(Z_{t}^{R}=C_{Z}^{-1}\xi s_{t}>0\). So by continuity, there must exist some unique emission efficiency \(\tilde{z}<\infty \), which satisfies \(Z_{t}^{R}=0\).

Corollary 2 then follows directly from Lemma 4:

Corollary 2

Let g be the the share of firms whose maximal investment falls in t, such that for firms \(i\in \left[ 0,g\right) \), \(Z_{t}^{R}<0\), while for firms \(i\in \left( g,1\right] \), \(Z_{t}^{R}>0\). Then, if \(\tilde{z}>\overline{z}\), \(g=1\), otherwise \(z(i)=\tilde{z}\) for \(i=g\).

Since emissions are falling in emission efficiency z, the least efficient firms pay the most taxes. As profits are rising in z, this implies that the tax scheme is regressive: it harms high profit (high z) firms less than low profit (low z) firms. With a lump sum recycling scheme, we thus find that more efficient firms are more likely to see \(\pi +s\) increase with tax increases. Hence, for these firms, \(Z^{R}\) may be increasing in the first-period tax rate, t.

1.2 Optimal Environmental Policy

The regulator still solves (8), yet now subject to (21). In line with the main text, we use (5) and \(S=T\int _{0}^{1}Z^{-1}Edi\), to reduce the first order condition with respect to T to \(\left( T^{*}-\Delta \right) \int _{0}^{1}Z^{-1}E_{T}di=0\). Hence, we can conclude that still \(T^{*}=\Delta \). Also for \(t^{*}\), we again arrive at (10). To make the distinction between firms for whom \(Z_{t}^{R}<0\) and those who have \(Z_{t}^{R}>0\) explicit, we rewrite (10) to

$$\begin{aligned} -\left( t^{*}-\Delta \right) \int _{0}^{1}\frac{e_{t}}{z}di=\int _{\min \left\{ n,g\right\} }^{n}\left[ \varPi _{Z}-C_{Z}\right] Z_{t}^{R}di+\int _{0}^{\min \left\{ n,g\right\} }\left[ \varPi _{Z}-C_{Z}\right] Z_{t}^{R}di. \end{aligned}$$
(24)

Because \(Z_{t}^{R}\) is positive for firms \(i\in \left( g,n\right] \) and negative for firms \(i\in \left[ 0,g\right) \), the sign of the RHS of (24) is not directly obvious. Hence we can wonder whether the fact that some constrained firms gain from tax increases may imply that the optimal tax exceeds marginal damages. By some tedious algebra, we can however show that Proposition 1 continues to apply:

Proposition 6

As long as some firms are constrained, the optimal first-period tax falls short of the Pigouvian tax (\(t^{*}<\Delta \)).

Proof

Two cases can be distinguished. First, if \(n\le g\), \(Z^{R}\) is decreasing in t for all constrained firms. In this case, the proof to Proposition 1 applies. If \(g<n\), \(Z^{R}\) is increasing in t for some constrained firms and the proof runs as follows.

  1. 1.

    As \(g<n\), we must have \(\varPi _{Z}\left( z(g),\cdot \right) -C_{Z}\left( z(g),\cdot \right) >0\). Next, we have \(\int _{0}^{1}Z_{t}^{R}di=\xi \int _{0}^{1}C_{Z}^{-1}\left[ \pi _{t}+s_{t}\right] di\). Here we know \(\int _{0}^{1}\left[ \pi _{t}+s_{t}\right] di=\int _{0}^{1}te_{t}di<0\) and \(\left[ \pi _{t}+s_{t}\right] \) is smaller (more negative) the smaller z(i). As \(C_{ZZ}>0\) and \(Z_{z}^{R}>0\), \(C_{Z}^{-1}\) is larger the smaller z(i). Hence, we must have \(\int _{0}^{1}Z_{t}^{R}di<0\) and \(\left[ \varPi _{Z}\left( z(g),\cdot \right) -C_{Z}\left( z(g),\cdot \right) \right] \int _{0}^{n}Z_{t}^{R}(i)di<0\).

  2. 2.

    Using \(\varPi _{ZZ}-C_{ZZ}<0\), we have \(\varPi _{Z}\left( z(g),\cdot \right) -C_{Z}\left( z(g),\cdot \right) \le \varPi _{Z}\left( z(i),\cdot \right) -C_{Z}\left( z(i),\cdot \right) \) for \(i\le g\). By Lemma 4, we have \(Z_{t}^{R}<0\) for \(i<g\) which gives \(\left[ \varPi _{Z}\left( z(g),\cdot \right) -C_{Z}\left( z(g),\cdot \right) \right] \int _{0}^{g}Z_{t}^{R}di\ge \int _{0}^{g}\left[ \varPi _{Z}-C_{Z}\right] Z_{t}^{R}di\). In a similar manner, we have \(\varPi _{Z}\left( z(g),\cdot \right) -C_{Z}\left( z(g),\cdot \right) \ge \varPi _{Z}\left( z(i),\cdot \right) -C_{Z}\left( z(i),\cdot \right) \) and \(Z_{t}^{R}>0\) for \(i\ge g\) which implies \(\left[ \varPi _{Z}\left( z(g),\cdot \right) -C_{Z}\left( z(g),\cdot \right) \right] \int _{g}^{n}Z_{t}^{R}di\ge \int _{g}^{n}\left[ \varPi _{Z}-C_{Z}\right] Z_{t}^{R}di\), so we have \(\left[ \varPi _{Z}\left( z(g),\cdot \right) -C_{Z}\left( z(g),\cdot \right) \right] \int _{0}^{n}Z_{t}^{R}di\ge \int _{g}^{n}\left[ \varPi _{Z}-C_{Z}\right] Z_{t}^{R}di+\int _{0}^{g}\left[ \varPi _{Z}-C_{Z}\right] Z_{t}^{R}di\).

  3. 3.

    Combining the results from step 1 and 2 we find \(\int _{g}^{n}\left[ \varPi _{Z}-C_{Z}\right] Z_{t}^{R}di+\int _{0}^{g}\left[ \varPi _{Z}-C_{Z}\right] Z_{t}^{R}di<0\). Then by (24) and \(e_{t}<0\) this implies \(t^{*}<\Delta \).

Even if, under the lump-sum recycling scheme, a tax increase allows some constrained firms to increase investment, this is no rationale for increasing the emission tax above the marginal emission damages. Even if there exist constrained firms that can increase investment following an increase in t, there are always constrained firms that are forced to reduce their investment due to a tax increase. One can then show that the aggregate cost of this reduction in investment by the latter outweighs the benefits of increased investment opportunities of the former. Low taxes continue to favor constrained firms’ investment in general, which implies that the optimal emission tax, \(t^{*}\), falls short of marginal emission damages, \(\Delta \).

1.3 Environmental Policy and Economic Shocks

By \(T^{*}=\Delta \), \(T^{*}\) is independent of \(\xi \), a and A. Accordingly, shocks to credit and productivity do not affect the second-period optimal emission tax. The response of \(t^{*}\) to the credit shock is again governed by the investment sensitivity and investment value effects. For the productivity shock, we in addition again identify the emission sensitivity and persistence effects. In “Equilibrium” section, we established that, because of the recycling scheme, the tax affects firms \(i\in \left[ 0,\,g\right) \) differently than firms \(i\in \left[ g,\,n\right] \). As a consequence, we must now further separate the effects across groups. We first evaluate the response of the optimal first-period tax to a change in \(\xi \). Taking the total derivative of (24), we find

(25)

with \(B>0\) defined as in the main text. If \(g\ge n\), the result collapses to (14). Note however that the partial derivatives of Z now include the effect of the shock though the lump sum rebate. For \(g<n\), we find that for firms who benefit from tax increases and for firms to whom a marginal tax increase is a net cost, the investment sensitivity effect continues to be ambiguous and may be of opposite signs for both groups. The investment value effect is still positive for the most constrained group, yet turns negative for firms \(i\in \left[ g,\,n\right] \). This can be explained as follows: an negative shock to credit reduces investment by constrained firms and thereby increases the marginal benefit of investment for these firms. As investment is increasing in the tax for firms \(i\in \left[ g,\,n\right] \), to benefit from this rise in the return to investment, they call for higher taxes following a drop in \(\xi \).

In a similar manner, we evaluate \(t_{a}^{*}\). Taking the total derivative of (24):

(26)

Again, if \(g\ge n\), the result collapses to (11). The investment sensitivity effect continues to be ambiguous and may be of opposite signs for both groups. As above, the fact that higher taxes increase investment for firms \(i\in \left[ g,\,n\right] \) while reducing it for firms \(i\in \left[ 0,\,g\right] \), is the reason behind the opposite signs of the investment value effect across groups. The same mechanism causes the signs of the persistence effect to be opposite. A drop in A reduces the benefit of investment and thus allow for a fall in investment in the optimum. For firms \(i\in \left[ 0,\,g\right] \), this fall is accomplished by an increase in the tax, for firms \(i\in \left[ g,\,n\right] \) this calls for a reduction in t. As in the main text, both \(t_{\xi }^{*}\) and \(t_{a}^{*}\) are ambiguous without further functional specification.

1.4 Example: Specific Functional Form

If we adopt the specific functional forms from Sect. 6, we again arrive at unambiguous results for the signs of \(t_{\xi }^{*}\) and \(t_{a}^{*}\). As before, both \(t_{\xi }^{*}\) and \(t_{a}^{*}\) are positive; Propositions 4 and 5 continue to hold. Although not formally proven below, also for the individual effects, all signs carry over for \(i\in \left[ 0,\,g\right) \). For \(i\in \left[ g,\,n\right] \), signs are opposite.

We use (18) and (20), to reduce (25) toFootnote 29

$$\begin{aligned} t_{\xi }^{*}=\left[ \gamma \xi B\right] ^{-1}\left[ \begin{array}{l} \int _{\min \left\{ n,g\right\} }^{n}Z_{t}^{R}\left[ TM_{Z}-\gamma C_{Z}\right] di\\ +\int _{0}^{\min \left\{ n,g\right\} }Z_{t}^{R}\left[ TM_{Z}-\gamma C_{Z}\right] di \end{array}\right] , \end{aligned}$$
(27)

and we can re-establish Proposition 4:

Proposition 7

Under specifications (17)–(18), as long as some firms are constrained, the optimal first-period tax falls in response to an adverse credit shock.

Proof

First of all, from the proof of Proposition 6, we know \(\int _{0}^{1}Z_{t}^{R}di<0\) with \(\int _{0}^{g}Z_{t}^{R}di<0\) and \(\int _{g}^{n}Z_{t}^{R}di>0\). Now, if \(M_{Z}\) is smaller (more negative) for low z firms, we must have that \(T\int _{0}^{n}Z_{t}^{R}M_{Z}di>0\). Since we have \(M_{ZZ}=Z^{-1}\left[ E_{ZZ}-2M_{Z}\right] >0\), by \(E_{ZZ}>0\) and \(Z_{z}^{R}>0\), this is indeed the case. Also, we know \(C_{Z}Z_{t}^{R}=\xi \left[ \pi _{t}+s_{t}\right] \), which gives \(\int _{0}^{n}C_{Z}Z_{t}^{R}di=\xi \int _{0}^{n}\left[ \pi _{t}+s_{t}\right] di<0\) This implies we must have \(t_{\xi }^{*}>0\).

Following the same procedure as for the credit shock, we use (18), (20) and (26) to findFootnote 30

$$\begin{aligned} t_{a}^{*}=\left[ \gamma B\right] ^{-1}\left[ \begin{array}{c} \int _{\min \left\{ n,g\right\} }^{n}\left[ \gamma \varPi _{Z}\left( \mu -1\right) +TM_{Z}\right] Z_{t}^{R}di\\ +\int _{0}^{\min \left\{ n,g\right\} }\left[ \gamma \varPi _{Z}\left( \mu -1\right) +TM_{Z}\right] Z_{t}^{R}di \end{array}\right] . \end{aligned}$$
(28)

This allows us to prove the following:

Proposition 8

Under specifications (17)–(18), as long as some firms are constrained, the optimal first-period tax falls in response to an adverse productivity shock.

Proof

First, we know \(M_{Z}=Z^{-2}\left[ ZE_{Z}-E\right] <0\) and \(M_{ZZ}=Z^{-1}\left[ E_{ZZ}-2M_{Z}\right] \). We have \(E_{ZZ}>0\), which implies \(M_{ZZ}>0\). Hence, \(T\int _{0}^{n}Z_{t}^{R}M_{Z}di>0\). Next, we have \(\varPi _{Z}=TMZ^{-1}>0\) and \(\varPi _{ZZ}=Z^{-2}T\left[ ZM_{Z}-M\right] <0\). Hence, \(\int _{0}^{n}\varPi _{Z}Z_{t}^{R}<0\). By \(\mu \le 1\), we must thus have \(t_{a}^{*}>0\).

Appendix 2: Notation and Signs of Derivatives

Tables 1 and 2 present an overview of the model variables and the signs of derivatives of the model laid out in Sect. 2 and solved in Sect. 3. Lower case letters refer to first-period variables while upper case letters denote second-period variables, and the i indicates that we are dealing with firm-specific variables.

Table 1 Model variables
Table 2 Model derivatives

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van den Bijgaart, I.M., Smulders, S. Does a Recession Call for Less Stringent Environmental Policy? A Partial-Equilibrium Second-Best Analysis. Environ Resource Econ 70, 807–834 (2018). https://doi.org/10.1007/s10640-017-0157-0

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