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GHG Emissions Control and Monetary Policy

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Abstract

This paper examines the optimal environmental and monetary policy mix in a New Keynesian model embodying pollutant emissions, abatement technology and environmental damage. The optimal response of the economy to productivity shocks is shown to depend crucially on the instruments policy makers have available, the intensity of the distortions they have to address (i.e. imperfect competition, costly price adjustment and negative environmental externality) and the way they interact.

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Notes

  1. According to observers, in fact, the recent 2030 Climate and Energy Policy Framework of the European Union would represent a sort of compromise between countries that rely heavily on coal and those willing to push even further the process of decarbonization. See European Council (2014) and European Commission (2014).

  2. Starting from the seminal paper by Weitzman (1974) the debate on which environmental policies would best serve the goal of cutting GHG emissions has been going on in different modelling settings. See Goulder et al. (1999), Parry and Williams (1999), Dissou (2005), Hoel and Karp (2002), Kelly (2005), Newell and Pizer (2008), Quirion (2005), Jotzo and Pezzey (2007).

  3. On the macroeconomic approach to environmental policy issues, see the survey by Fischer and Heutel (2013).

  4. See e.g. Leeper (1991), Schmitt-Grohé and Uribe (2004) and Schmitt-Grohé and Uribe (2007).

  5. Nonetheless, the specific institutional set-up of the euro area, with centralized monetary policy and decentralized fiscal policy, renders the central bank commitment to control inflation even more challenging.

  6. For the sake of parsimony, in fact, here we deliberately abstract from capital accumulation and opt to model nominal rigidities by introducing quadratic adjustment costs as an alternative to the Calvo-pricing, as instead was done in Annicchiarico and Di Dio (2015).

  7. In this way, the costs introduced by price adjustments are neutralized. See e.g. Clarida et al. (2000), Woodford (2003) and Schmitt-Grohé and Uribe (2004). Exceptions are given by Faia (2008, 2009) who show that the optimality of fully inflation targeting does not hold when the basic New Keynesian model is extended along different dimensions.

  8. See the “Appendix” for details.

  9. Denoting by \(\lambda _{t}\) the marginal utility of real wealth, (10) can be expressed as \(\frac{1}{R_{t}}=\beta E_{t}\frac{1}{\Pi _{t+1}}\frac{ \lambda _{t+1}}{\lambda _{t}}\). The discount factor \(\beta \frac{\lambda _{t+1}}{\lambda _{t}}\) corresponds to \(Q_{t,t+1}^{R}\) in (7).

  10. In addition, when allowing for government borrowing and in the impossibility of using lump-sum taxes, it might be optimal for the government to use fiscal deficits (i.e. adjustments in the level of public debt) as shock absorber and, therefore, the welfare implications might differ. Although this is an important issue, we leave this aspect for future research.

  11. The carbon cycle described by Eq. (15) in assuming that all existing stock of pollution depreciates at a constant rate and abstracting from carbon-circulation models simplifies the one adopted by Golosov et al. (2014), who instead assume a permanent and a temporary component of emissions along with a specific rule regarding the evolution of pollution from both components.

  12. Notice that in the absence of environmental damage \(\lambda _{t}^{M}=0\) and \(U_{t}=0\), while condition (18) boils down into the familiar efficient condition equalizing the marginal rate of substitution between consumption and labor, \(\mu _{L}L_{t}{}^{\eta }C_{t},\) to the marginal rate of transformation, \(A_{t},\) implying that \(L_{t}=\left( \frac{1}{\mu _{L}}\right) ^{\frac{1}{1+\eta }},\) that is, at the optimum, labor is constant.

  13. The model has been solved in Dynare. For details see http://www.cepremap.cnrs.fr/dynare/ and Adjemian et al. (2011)

  14. For further details, see the Technical Appendix available from the authors upon request.

  15. The RICE-2010 model is available for download at http://www.econ.yale.edu/~nordhaus/homepage/RICEmodels.htm. For a detailed description of the model, see Nordhaus and Boyer (2000).

  16. In the absence of a damage function, in fact, under Pareto efficiency hours worked are constant, while consumption and output move proportionally to productivity.

  17. See, e.g., Heutel (2012).

  18. The steady state has been computed numerically using the ordinary least square approach proposed by Schmitt-Grohé and Uribe (2012). For further details, see the Technical Appendix.

  19. We perform this sensitivity analysis so as to keep the steady-state level of the abatement cost per unit of output constant by changing the scale parameter \(\phi _{1}\) accordingly.

  20. In this setting, under a Ramsey environmental policy, the existence of a unique stable equilibrium requires that the policy parameter governing the reaction of the nominal interest rate to inflation, \(\rho _{\Pi },\) is less than one. For values larger than one, in fact, the model becomes unstable.

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Correspondence to Barbara Annicchiarico.

Additional information

We are grateful to two anonymous referees for their insightful comments and suggestions on an earlier version of the paper. The views expressed herein are those of the authors and not necessarily reflect those of Sogei S.p.A.. This paper has previously circulated under the title “Ramsey Monetary Policy and GHG Emissions Control”.

Appendix

Appendix

1.1 Derivation of the New Keynesian Phillips Curve

The Lagrangian function of the generic j firm’s problem is

$$\begin{aligned} \mathcal {L}_{0}=E_{0}\sum _{i=0}^{\infty }Q_{t,t+i}^{R}\left\{ \begin{array}{c} \left[ \begin{array}{c} \frac{P_{j,t}}{P_{t}}Y_{j,t}-\frac{W_{t}}{P_{t}}L_{j,t}-\phi _{1}U_{j,t}^{\phi _{2}}Y_{j,t}-p_{Z,t}\left( 1-U_{j,t}\right) \varphi Y_{j,t}+ \\ -\frac{\gamma }{2}\left( \frac{P_{j,t}}{P_{i,t-1}}-1\right) ^{2}Y_{t}\end{array} \right] + \\ +\Psi _{j,t}\left( \Lambda _{t}A_{t}L_{j,t}-Y_{j,t}\right) \end{array} \right\} \end{aligned}$$
(27)

where \(Q_{t,t+i}^{R}\) is real stochastic discount factor, which is related to the household discount factor \(\beta \) and the marginal utility of real wealth \(\lambda _{t}\) by the formula \(Q_{t,t+i}^{R}=\beta ^{i}\frac{\lambda _{t}}{\lambda _{t+i}},\) and \(\Psi _{j,t}\) is the Lagrange multiplier of constraint (1). Differentiating the above expression with respect to \(L_{j,t} \), \(U_{j,t}\) and \(P_{j,t},\) using the fact that \(Y_{j,t+i}=\left( \frac{P_{j,t}}{P_{t}}\right) ^{-\theta }Y_{t+i}\) for \(i=0,1,2,3,\dots ,\) gives:

$$\begin{aligned}&\Lambda _{t}A_{t}\Psi _{j,t}=\frac{W_{t}}{P_{t}}, \end{aligned}$$
(28)
$$\begin{aligned}&\quad \varphi p_{Z,t}Y_{j,t}=\phi _{1}\phi _{2}U_{j,t}^{\phi _{2}-1}, \end{aligned}$$
(29)
$$\begin{aligned}&\quad \frac{1}{P_{t}}Y_{j,t}-\theta Y_{j,t}\frac{1}{P_{t}} +\theta \left( \frac{ P_{j,t}}{P_{t}}\right) ^{-\theta -1}Y_{t+i}\frac{1}{P_{t}}\phi _{1}U_{j,t}^{\phi _{2}}+\nonumber \\&\qquad +\,\theta p_{Z,t}\left( 1-U_{j,t}\right) \varphi \left( \frac{P_{j,t}}{P_{t}}\right) ^{-\theta -1}Y_{t+i}\frac{1}{P_{t}}+ \nonumber \\&\qquad +\,\theta \Psi _{j,t}\left( \frac{P_{j,t}}{P_{t}}\right) ^{-\theta -1}Y_{t+i} \frac{1}{P_{t}}-\gamma \left( \frac{P_{j,t}}{P_{i,t-1}}-1\right) \frac{1}{ P_{i,t-1}}Y_{t}+\nonumber \\&\qquad +\,\gamma E_{t}Q_{t,t+1}^{R}\left( \frac{P_{j,t+1}}{P_{i,t}} -1\right) \frac{P_{j,t+1}}{P_{i,t}^{2}}Y_{t+1}=0. \end{aligned}$$
(30)

This last condition can be re-written as

$$\begin{aligned}&\frac{1}{P_{t}}Y_{j,t}-\theta Y_{j,t}\frac{1}{P_{t}}+MC_{j,t}\theta \left( \frac{P_{j,t}}{P_{t}}\right) ^{-\theta -1}Y_{t+i}\frac{1}{P_{t}}+ \nonumber \\&\qquad -\,\gamma \left( \frac{P_{j,t}}{P_{i,t-1}}-1\right) \frac{1}{P_{i,t-1}} Y_{t}+\gamma E_{t}Q_{t,t+1}^{R}\left( \frac{P_{j,t+1}}{P_{i,t}}-1\right) \frac{P_{j,t+1}}{P_{i,t}^{2}}Y_{t+1}=0. \end{aligned}$$
(31)

where \(MC_{j,t}=\phi _{1}U_{j,t}^{\phi _{2}}+p_{Z,t}\left( 1-U_{j,t}\right) \varphi +\Psi _{j,t}.\)

Imposing symmetry across firms and simplifying, (5)–(7) immediately follow.

1.2 Ramsey Monetary and Environmental Policy

The Lagrangian function associated to the Ramsey problem is

$$\begin{aligned} \mathcal {L}_{0}= & {} E_{0}\left\{ \sum _{t=0}^{\infty }\beta ^{t}\left[ \log \left( \Lambda _{t}A_{t}L_{t}\left( 1-\phi _{1}U_{t}^{\phi _{2}}-\frac{ \gamma }{2}\left( \Pi _{t}-1\right) ^{2}\right) \right) -\mu _{L}\frac{ L_{t}{}^{1+\eta }}{1+\eta }\right. +\right. \nonumber \\&-\lambda _{t}^{\Pi }\left( \begin{array}{c} \frac{1-\theta -\gamma \left( \Pi _{t}-1\right) \Pi _{t}}{1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}}+\theta \mu _{L}L_{t}^{\eta +1}+\theta \frac{\phi _{1}U_{t}^{\phi _{2}}+\phi _{1}\phi _{2}U_{t}^{\phi _{2}-1}\left( 1-U_{t}\right) }{1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}} \\ +\gamma \beta E_{t}\frac{\left( \Pi _{t+1}-1\right) \Pi _{t+1}}{1-\phi _{1}U_{t+1}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t+1}-1\right) ^{2}} \end{array} \right) + \nonumber \\&+\lambda _{t}^{M}\left( M_{t}-(1-\delta _{M})M_{t-1}-\left( 1-U_{t}\right) \varphi \Lambda _{t}A_{t}L_{t}\right) + \nonumber \\&\left. \left. +\lambda _{t}^{C}\left( \begin{array}{c} \frac{1}{\Lambda _{t}A_{t}L_{t}\left( 1-\phi _{1}U_{t}^{\phi _{2}}-\frac{ \gamma }{2}\left( \Pi _{t}-1\right) ^{2}\right) }\frac{1}{R_{t}}+ \\ -\beta E_{t}\frac{1}{\Pi _{t+1}}\frac{1}{\Lambda _{t+1}A_{t+1}L_{t+1}\left( 1-\phi _{1}U_{t+1}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t+1}-1\right) ^{2}\right) } \end{array} \right) \right] \right\} , \end{aligned}$$
(32)

where \(\lambda _{t}^{\Pi }\), \(\lambda _{t}^{M}\) and \(\lambda _{t}^{C}\) are the Lagrange multipliers.

The first-order conditions (FOC) relative to the variables \(R_{t},\) \(L_{t}\), \(U_{t}\), \(\Pi _{t}\), \(M_{t}\) are the following.

$$\begin{aligned} \text {FOC }R_{t}:-\lambda _{t}^{C}\frac{1}{\Lambda _{t}A_{t}L_{t}\left[ 1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2} \right] }\frac{1}{R_{t}^{2}}=0, \end{aligned}$$
(33)

which implies that \(\lambda _{t}^{C}=0\). Therefore, we can neglect this constraint which is never binding when the Ramsey planner controls \(R_{t}.\)

$$\begin{aligned}&\text {FOC }L_{t}:\frac{1}{L_{t}}-\mu _{L}L_{t}{}^{\eta }-\lambda _{t}^{\Pi }\left( \eta +1\right) \theta \mu _{L}L_{t}^{\eta }-\lambda _{t}^{M}\left( 1-U_{t}\right) \varphi \Lambda _{t}A_{t}=0, \end{aligned}$$
(34)
$$\begin{aligned}&\text {FOC }U_{t}:-\frac{\phi _{1}\phi _{2}U_{t}^{\phi _{2}-1}}{1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}}-\theta \lambda _{\Pi ,t}\frac{(\phi _{2}-1)\left( 1-U_{t}\right) \phi _{1}\phi _{2}U_{t}^{\phi _{2}-2}}{1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2} \left( \Pi _{t}-1\right) ^{2}}+ \nonumber \\&\qquad +\,\frac{\phi _{1}\phi _{2}U_{t}^{\phi _{2}-1}}{\left[ 1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}\right] ^{2}}\lambda _{t}^{\Pi }\left\{ \theta -1-\theta \left[ \phi _{1}U_{t}^{\phi _{2}}+\phi _{1}\phi _{2}U_{t}^{\phi _{2}-1}\left( 1-U_{t}\right) \right] \right\} +\nonumber \\&\qquad +\,\frac{\phi _{1}\phi _{2}U_{t}^{\phi _{2}-1}}{\left[ 1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}\right] ^{2}}\gamma \left( \lambda _{t}^{\Pi }-\lambda _{t-1}^{\Pi }\right) \left( \Pi _{t}-1\right) \Pi _{t}+\lambda _{t}^{M}\varphi \Lambda _{t}A_{t}L=0, \end{aligned}$$
(35)
$$\begin{aligned}&\quad \text {FOC }\Pi _{t}:.-\gamma \left( \Pi _{t}-1\right) + \nonumber \\&\qquad +\,\left( \lambda _{t}^{\Pi }-\lambda _{t-1}^{\Pi }\right) \gamma \frac{\left( 2\Pi _{t}-1\right) \left[ 1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2} \left( \Pi _{t}-1\right) ^{2}\right] +\gamma \left( \Pi _{t}-1\right) ^{2}\Pi _{t}}{1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}} + \nonumber \\&\qquad +\,\gamma \lambda _{t}^{\Pi }\frac{\left( \Pi _{t}-1\right) \left( \theta -1\right) }{1-\phi _{1}U^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}}+\nonumber \\&\qquad -\,\theta \gamma \lambda _{t}^{\Pi }\frac{\left( \Pi _{t}-1\right) \left[ \phi _{1}U_{t}^{\phi _{2}}+\phi _{1}\phi _{2}U_{t}^{\phi _{2}-1}\left( 1-U_{t}\right) \right] }{1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}}=0, \end{aligned}$$
(36)
$$\begin{aligned}&\quad \text {FOC }M_{t}:-\chi +\lambda _{t}^{M}-\beta (1-\delta _{M})E_{t}\lambda _{t+1}^{M}+\chi \left( 1-U_{t}\right) \varphi \Lambda _{t}A_{t}L_{t}=0. \end{aligned}$$
(37)

It should be noted that the above first-order conditions are optimal from a “timeless perspective”, rather than from the perspective of the particular date at which the policy is actually adopted.

In steady state the FOC with respect to inflation \(\Pi _{t}\) (36) simplifies to:

$$\begin{aligned}&\gamma \left( \Pi -1\right) \left[ -1+\lambda ^{\Pi }\frac{\theta -1}{1-\phi _{1}U^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi -1\right) ^{2}}\right. +\nonumber \\&\quad \left. -\theta \gamma \lambda ^{\Pi }\frac{\phi _{1}U^{\phi _{2}}+\phi _{1}\phi _{2}U^{\phi _{2}-1}\left( 1-U\right) }{1-\phi _{1}U^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi -1\right) ^{2}}\right] =0. \end{aligned}$$
(38)

Since \(\lambda ^{\Pi }>0\) it must be that in steady state \(\Pi =1,\) that is Ramsey optimal steady-state inflation rate is zero.

1.3 Ramsey Monetary Policy and Environmental Regulations

When GHG emissions control policy is given by a carbon tax, and the Ramsey planner controls monetary policy, the first-order conditions relative to the variables \(R_{t},\) \(L_{t},\) \(\Pi \) \(_{t},\) \(M_{t}\) are the following:

$$\begin{aligned} \text {FOC }R_{t}:-\lambda _{t}^{C}\frac{1}{\Lambda _{t}A_{t}L_{t}\left[ 1-\phi _{1}U^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2} \right] }\frac{1}{R_{t}^{2}}=0, \end{aligned}$$
(39)

which implies that \(\lambda _{t}^{C}=0\). Thefore, we can neglect this constraint which is never binding when the Ramsey planner controls \(R_{t}\).

$$\begin{aligned}&\text {FOC }L_{t}:\frac{1}{L_{t}}-\mu _{L}L_{t}{}^{\eta }-\lambda _{t}^{\Pi }\left( \eta +1\right) \theta \mu _{L}L_{t}^{\eta }-\lambda _{t}^{M}\left( 1-U\right) \varphi \Lambda _{t}A_{t}=0, \end{aligned}$$
(40)
$$\begin{aligned}&\text {FOC }\Pi _{t}:.-\gamma \left( \Pi _{t}-1\right) + \nonumber \\&\qquad +\,\left( \lambda _{t}^{\Pi }-\lambda _{t-1}^{\Pi }\right) \gamma \frac{\left( 2\Pi _{t}-1\right) \left[ 1-\phi _{1}U^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}\right] +\gamma \left( \Pi _{t}-1\right) ^{2}\Pi _{t}}{ 1-\phi _{1}U^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}}+\nonumber \\&\qquad +\,\gamma \lambda _{t}^{\Pi }\frac{\left( \Pi _{t}-1\right) \left( \theta -1\right) }{1-\phi _{1}U^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}}+ \nonumber \\&\qquad -\,\theta \gamma \lambda _{t}^{\Pi }\frac{\left( \Pi _{t}-1\right) \left[ \phi _{1}U^{\phi _{2}}+\phi _{1}\phi _{2}U^{\phi _{2}-1}\left( 1-U\right) \right] }{1-\phi _{1}U^{\phi _{2}}-\frac{\gamma }{2} \left( \Pi _{t}-1\right) ^{2}}=0, \end{aligned}$$
(41)
$$\begin{aligned}&\quad \text {FOC }M_{t}:-\chi +\lambda _{t}^{M}-\beta (1-\delta _{M})E_{t}\lambda _{t+1}^{M}+\chi \left( 1-U\right) \varphi \Lambda _{t}A_{t}L_{t}=0. \end{aligned}$$
(42)

Under a cap-and-trade scheme the first-order conditions with respect to \( R_{t}\) and \(\Pi _{t}\) are equivalent to (33) and (41), while the remaining optimal conditions relative to \(L_{t},\) \(U_{t},\) \(M_{t}\) read as follows

$$\begin{aligned}&\text {FOC }L_{t}:\frac{1}{L_{t}}-\mu _{L}L_{t}{}^{\eta }-\lambda _{t}^{\Pi }\left( \eta +1\right) \theta \mu _{L}L_{t}^{\eta }-\lambda _{t}^{Z}\left( 1-U_{t}\right) \varphi \Lambda _{t}A_{t}=0, \end{aligned}$$
(43)
$$\begin{aligned}&\text {FOC }U_{t}:-\frac{\phi _{1}\phi _{2}U_{t}^{\phi _{2}-1}}{1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}}-\theta \lambda _{t}^{\Pi }\frac{(\phi _{2}-1)\left( 1-U_{t}\right) \phi _{1}\phi _{2}U_{t}^{\phi _{2}-2}}{1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2} \left( \Pi _{t}-1\right) ^{2}} + \nonumber \\&\quad +\,\frac{\phi _{1}\phi _{2}U_{t}^{\phi _{2}-1}}{\left[ 1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}\right] ^{2}}\lambda _{t}^{\Pi }\left\{ \theta -1-\theta \left[ \phi _{1}U_{t}^{\phi _{2}}+\phi _{1}\phi _{2}U_{t}^{\phi _{2}-1}\left( 1-U_{t}\right) \right] \right\} + \nonumber \\&\quad +\,\frac{\phi _{1}\phi _{2}U_{t}^{\phi _{2}-1}}{\left[ 1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}\right] ^{2}}\gamma \left( \lambda _{t}^{\Pi }-\lambda _{t-1}^{\Pi }\right) \left( \Pi _{t}-1\right) \Pi _{t}+\lambda _{t}^{Z}\varphi \Lambda _{t}A_{t}L=0, \end{aligned}$$
(44)
$$\begin{aligned}&\quad \text {FOC }M_{t}:-\chi +\lambda _{t}^{M}-\beta (1-\delta _{M})E_{t}\lambda _{t+1}^{M}+\lambda _{t}^{Z}\chi \left( 1-U_{t}\right) \varphi \Lambda _{t}A_{t}L_{t}=0, \end{aligned}$$
(45)

with \(\lambda _{t}^{Z}\) being the Lagrange multiplier associated to the constraint \(Z=\left( 1-U_{t}\right) \varphi A_{t}L_{t}.\)

1.4 Ramsey Environmental Policy and Interest-Rate-Rules

When then Ramsey planner controls environmental policy, but takes as given monetary policy which is conducted according to an interest-rate rule of the type (24), the optimal conditions read as follows:

$$\begin{aligned}&\text {FOC }R_{t}:\frac{\lambda _{t}^{R}}{R}-\beta \rho _{R}E_{t}\lambda _{t+1}^{R}\frac{R_{t+1}}{RR_{t}}-\lambda _{t}^{C}\frac{1}{\Lambda _{t}A_{t}L_{t}\left[ 1-\phi _{1}U^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}\right] }\frac{1}{R_{t}^{2}}=0, \end{aligned}$$
(46)
$$\begin{aligned}&\quad \text {FOC }L_{t}:\frac{1}{L_{t}}-\mu _{L}L_{t}{}^{\eta }-\lambda _{t}^{\Pi }\left( \eta +1\right) \theta \mu _{L}L_{t}^{\eta }-\lambda _{t}^{M}\left( 1-U_{t}\right) \varphi \Lambda _{t}A_{t}+ \nonumber \\&\qquad -\,\frac{1}{L_{t}^{2}A_{t}\Lambda _{t}\left( 1-\phi _{1}U_{t}^{\phi _{2}}- \frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}\right) }\left( \frac{\lambda _{t}^{C}}{R_{t}}-\frac{\lambda _{t-1}^{C}}{\Pi _{t}}\right) + \nonumber \\&\qquad -\,\lambda _{t}^{R}\rho _{Y}\left( 1-\rho _{R}\right) \frac{R_{t}}{R}\frac{1}{L_{t}}=0, \end{aligned}$$
(47)
$$\begin{aligned}&\text {FOC }U_{t}:-\frac{\phi _{1}\phi _{2}U_{t}^{\phi _{2}-1}}{1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}}-\theta \lambda _{t}^{\Pi }\frac{(\phi _{2}-1)\left( 1-U_{t}\right) \phi _{1}\phi _{2}U_{t}^{\phi _{2}-2}}{1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2} \left( \Pi _{t}-1\right) ^{2}} + \nonumber \\&\qquad +\,\frac{\phi _{1}\phi _{2}U_{t}^{\phi _{2}-1}}{\left[ 1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}\right] ^{2}}\lambda _{t}^{\Pi }\left\{ \theta -1-\theta \left[ \phi _{1}U_{t}^{\phi _{2}}+\phi _{1}\phi _{2}U_{t}^{\phi _{2}-1}\left( 1-U_{t}\right) \right] \right\} + \nonumber \\&\qquad +\,\frac{\phi _{1}\phi _{2}U_{t}^{\phi _{2}-1}}{\left[ 1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}\right] ^{2}}\gamma \left( \lambda _{t}^{\Pi }-\lambda _{t-1}^{\Pi }\right) \left( \Pi _{t}-1\right) \Pi _{t}+\lambda _{t}^{M}\varphi \Lambda _{t}A_{t}L + \nonumber \\&\qquad +\,\frac{\phi _{2}\phi _{1}U_{t}^{\phi _{2}-1}}{\Lambda _{t}A_{t}L_{t}\left[ 1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2} \right] ^{2}}\left( \lambda _{t}^{C}\frac{1}{R_{t}}-\lambda _{t-1}^{C}\frac{1 }{\Pi _{t}}\right) =0, \end{aligned}$$
(48)
$$\begin{aligned}&\quad \text {FOC }\Pi _{t}:.-\gamma \left( \Pi _{t}-1\right) \nonumber \\&\qquad +\,\left( \lambda _{t}^{\Pi }-\lambda _{t-1}^{\Pi }\right) \gamma \frac{\left( 2\Pi _{t}-1\right) \left[ 1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2} \left( \Pi _{t}-1\right) ^{2}\right] +\gamma \left( \Pi _{t}-1\right) ^{2}\Pi _{t}}{1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}} + \nonumber \\&\qquad +\,\gamma \lambda _{t}^{\Pi }\frac{\left( \Pi _{t}-1\right) \left( \theta -1\right) }{1-\phi _{1}U^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}} + \nonumber \\&\qquad -\,\theta \gamma \lambda _{t}^{\Pi }\frac{\left( \Pi _{t}-1\right) \left[ \phi _{1}U_{t}^{\phi _{2}}+\phi _{1}\phi _{2}U_{t}^{\phi _{2}-1}\left( 1-U_{t}\right) \right] }{1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}} + \nonumber \\&\qquad +\,\lambda _{t}^{C}\left( \frac{\gamma \left( \Pi _{t}-1\right) \Lambda _{t}A_{t}L_{t}}{\left[ \Lambda _{t}A_{t}L_{t}\left( 1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}\right) \right] ^{2}} \frac{1}{R_{t}}\right) + \nonumber \\&\qquad -\,\lambda _{t-1}^{C}\left( \frac{\gamma \left( \Pi _{t}-1\right) \Lambda _{t}A_{t}L_{t}}{\left[ \Lambda _{t}A_{t}L_{t}\left( 1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}\right) \right] ^{2}} \frac{1}{\Pi _{t}}\right. +\nonumber \\&\qquad \left. -\frac{1}{\Lambda _{t}A_{t}L_{t}\left( 1-\phi _{1}U_{t}^{\phi _{2}}-\frac{\gamma }{2}\left( \Pi _{t}-1\right) ^{2}\right) } \frac{1}{\Pi _{t}^{2}}\right) + \nonumber \\&\qquad -\,\lambda _{t}^{R}\rho _{\Pi }(1-\rho _{R})\frac{R_{t}}{R}\frac{1}{\Pi _{t}} =0, \end{aligned}$$
(49)
$$\begin{aligned}&\quad \text {FOC }M_{t}:-\chi +\lambda _{t}^{M}-\beta (1-\delta _{M})E_{t}\lambda _{t+1}^{M}+\chi \left( 1-U_{t}\right) \varphi \Lambda _{t}A_{t}L_{t} + \nonumber \\&\qquad +\,\lambda _{t}^{R}\chi \rho _{Y}\left( 1-\rho _{R}\right) \frac{R_{t}}{R} +\chi \frac{1}{\Lambda _{t}}\left( \lambda _{t}^{C}\frac{1}{R_{t}}-\lambda _{t-1}^{C}\frac{1}{\Pi _{t}}\right) =0, \end{aligned}$$
(50)

where \(\lambda _{t}^{R}\) is the Lagrange multiplier associated to the interest-rate rule (24).

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Annicchiarico, B., Di Dio, F. GHG Emissions Control and Monetary Policy. Environ Resource Econ 67, 823–851 (2017). https://doi.org/10.1007/s10640-016-0007-5

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