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May 19, 2025
Climate change and monetary policy: a Bayesian DSGE perspective
We develop a DSGE model, building on Tervala and Watson (2022) and Tervala and Watson (2024), with the incorporation of CO2 emissions linked to GDP levels and a CO2 stock that reflects the cumulative amount of emissions and the CO2 decay rate.
2.1 Households
Households, identified by z, are uniformly distributed across the interval from 0 to 1. A fraction \(1-\lambda \) of these households, known as Ricardian, optimize their consumption over time. The remaining fraction \( \lambda \), referred to as non-Ricardian households, are constrained by their liquidity and thus consume exclusively from their current income and endowments. Each household is characterized by the same utility function:
$$\begin{aligned} U_{t}\left( z\right) =E_{t}\sum _{s=t}^{\infty }\beta ^{s-t}\epsilon _{s}^{TP} \left[ \log C_{s}-\frac{(N_{s}(z))^{1+1/\varphi }}{1+1/\varphi } \right] , \end{aligned}$$
where E stands for the expectation operator, \(\beta \) denotes the discount factor, \(\epsilon _{t}^{TP}\) represents time preference, \(C_{t}\) denotes a composite index of private consumption, formulated as \(C_{t}=\left[ \int _{0}^{1}C_{t}^{ }(z)^{\frac{\theta -1}{\theta }}dz\right] ^{\frac{ \theta }{\theta -1}}\), with \(C_{t}(z)\) being the consumption of good z and \(\theta \) the elasticity of substitution between different goods. \(N_{t}(z)\) indicates the labor hours supplied, and \(\varphi \) is the Frisch elasticity of labor supply. The time-preference shock is a commonly employed method to model demand-side shocks in DSGE models. The time preference follows a log-linear AR(1) process: \(\hat{\epsilon }_{t}^{TP}=\rho ^{TP}\hat{\epsilon } _{t-1}+\hat{\varepsilon }_{t}^{TP}\), where \(\rho ^{TP}\) is the persistence parameter, \(\hat{\varepsilon }_{t}^{TP}\) is a normally distributed i.i.d. error term with mean zero. The notation with a hat signifies the percentage deviation from the steady state in a log-linear model. Ricardian households, receiving dividends from firms and facing government-imposed income and consumption taxes, adhere to the following resource constraint:
$$\begin{aligned} \frac{R_{t}^{-1}B_{t+1}}{1-\lambda }=\frac{B_{t}}{1-\lambda }+\left( 1-\tau _{t}^{y}\right) w_{t}N_{R,t}+\frac{\left( 1-\tau _{t}^{y}\right) }{ }\left( r_{t}^{K}K_{t}+v_{t}\right) – \end{aligned}$$
$$\begin{aligned} \left( 1+\tau _{t}^{c}\right) P_{t}C_{R,t}+\frac{1}{1-\lambda }P_{t}I+\frac{\phi }{2}\left( \frac{I_{t}}{K_{t}}-\delta \right) ^{2}.- \end{aligned}$$
\(N_{R,t}\) and \(C_{R,t}\) denote the labor supply and consumption of Ricardian households, respectively. The model also incorporates \(B_{t}\) which represents the nominal price of a government bond that pays off $1 upon maturity at \(t+1\) and \(r_{t}\) the nominal yield of the bond. The nominal wage is indicated by \(w_{t}\), while \(v_{t}\) captures the financial returns from firms that fully impute dividends, with these returns being subject to taxation in a manner analogous to household income. The model specifies tax rates on income and consumption with the symbols \(\tau ^{y}\) and \(\tau ^{c},\) respectively. The return on private capital is denoted by \( r_{t}^{k}\), and the general price level is represented by \(P_{t}\), which is
$$\begin{aligned} P_{t} = \left( \int _{0}^{1} P_{t}(z)^{1-\theta } \, dz \right) ^{\frac{1}{1-\theta }}, \end{aligned}$$
where \(P_{t}^{ }\left( z\right) \) signifies the price of individual good z. Investment by the private sector is indicated by \(I_{t}\) and it incurs quadratic adjustment costs modeled as \(\phi (\cdot )=\) \(\phi /2(I_{t}/K_{t}-\delta )^{2}\), where \(\delta \) represents the depreciation rate of private capital. The formula for the private capital stock is given by \(K_{t+1}=(1-\delta )K_{t}+I_{t}\).
The optimal behavior conditions for Ricardian households can be summarized as follows:
$$\begin{aligned} & \beta R_{t}E_{t}\bigg (\frac{\epsilon _{t+1}^{TP}\left( 1+\tau _{t}^{c}\right) P_{t}C_{R,t} }{\epsilon _{t}^{TP}\left( 1+\tau _{t+1}^{c}\right) P_{t+1}C_{R,t+1}}\bigg )=1, \end{aligned}$$
(1)
$$\begin{aligned} & N_{R,t}(z)=\bigg (\frac{\left( 1-\tau _{t}^{y}\right) w_{t}}{C_{R,t}\left( 1+\tau _{t}^{c}\right) P_{t}} \bigg )^{\varphi }. \end{aligned}$$
(2)
$$\begin{aligned} & q_{t}=1+\phi \bigg (\frac{I_{t}}{K_{t}}-\delta \bigg ), \end{aligned}$$
(3)
$$\begin{aligned} & q_{t}=E_{t}\bigg \{\Lambda _{t,t+1}\bigg [\left( 1-\tau _{t+1}^{y}\right) r_{t+1}^{K}+q_{t+1}(1-\delta )-\phi _{t+1}+\bigg (\frac{I_{t+1}}{ K_{t+1}}\bigg )\phi _{t+1}^{^{\prime }}\bigg ]\bigg \}, \end{aligned}$$
(4)
where \(\Lambda _{t,t+1}=\beta \bigg (\frac{C_{R,t}}{C_{R,t+1}}\bigg )\) is a factor adjusting for future utility from consumption, and this equation balances the cost and returns of capital investment, inclusive of taxes and depreciation. It is assumed that the log-linearized investment equation includes investment shocks (IS) that follow an AR(1) process similar to other shocks within the model.
For non-Ricardian households, who derive their income from employment in firms and government transfers and are subject to taxation, the following conditions dictate their optimal behavior:
$$\begin{aligned} \left( 1+\tau _{t}^{c}\right) P_{t}C_{N,t}= \left( 1-\tau _{t}^{y}\right) w_{t}N_{N,t}+\omega \frac{ G_{t}^{T}}{\lambda }, \end{aligned}$$
(5)
$$\begin{aligned} N_{N,t}(z)=\bigg (\frac{\left( 1-\tau _{t}^{y}\right) w_{t}}{C_{N,t}\left( 1+\tau _{t}^{c}\right) P_{t}} \bigg )^{\varphi }, \end{aligned}$$
(6)
The aggregate consumption and labor supply for the economy are then calculated by weighting the consumption and labor supply of both non-Ricardian and Ricardian households according to their respective proportions in the population, as: \(C_{t}=\lambda C_{N,t}+(1-\lambda )C_{R,t} \) and \(N_{t}=\lambda N_{N,t}+(1-\lambda )N_{R,t}\).
2.2 Firms
TFP exhibits a strong correlation with GDP. For instance, Fernald and Wang (2016) identify a correlation coefficient ranging between 0.64 and 0.74 for the relationship between TFP and output using quarterly data from the period 1984–2015. To incorporate endogenous and procyclical TFP within a DSGE model framework, one can employ a learning-by-doing equation, as demonstrated in the works of (Chang et al. 2002; Engler and Tervala 2018), and Watson and Tervala and Watson (2022):
$$\begin{aligned} Y_{t}(z)=K_{t}(z)^{\alpha }\left( N_{t}(z)A_{t}\right) ^{1-\alpha }K_{G,t}^{\phi _{kg}}. \end{aligned}$$
(7)
The notation \(Y_{t}(z)\) s used to represent the output produced by firm z, while \(K_{G,t}\) stands for public capital, with \(\phi _{kg}\) representing its output elasticity. The term \(A_{t}\) is used to describe TFP, capturing the combined level of workforce skill and other efficiency-enhancing factors. The evolution of TFP is attributed to a learning-by-doing mechanism, influenced by historical labor inputs, and is mathematically formulated as:
$$\begin{aligned} A_{t}=A_{t-1}^{\rho _{x}}N_{t-1}^{\mu _{l}}(z), \end{aligned}$$
(8)
where \(\rho _{x}\) captures the persistence of the TFP stock over time, and \( \mu _{l}\) quantifies the responsiveness of TFP to the labor hours worked in the previous period.
Minimizing costs results in a capital-to-labor ratio given by
$$\begin{aligned} \frac{K_{t}(z)}{N_{t}(z)}=\frac{\alpha }{1-\alpha }\frac{w_{t}}{r_{t}^{K}}. \end{aligned}$$
(9)
Firms strive to optimize the present value of their future profits, which is represented as \(v_{t}(z)\)
$$\begin{aligned} \max _{p_{t}(z)}{v_{t}(z)}=E_{t}\sum _{s=t}^{\infty }\gamma ^{s-t}Q_{t,s}\frac{ v_{s}(z)}{P_{s}}, \end{aligned}$$
(10)
with \(1-\gamma \) indicating the probability that a firm can revise its prices in any given period. Utilizing the stochastic discount factor \(\xi _{t,s}\) for the interval between t and s, the optimal price setting, \( p_{t}(z)\), is determined as
$$\begin{aligned} p_{t}(z)=\frac{\theta }{\theta -1}\frac{E_{t}\sum _{s=t}^{\infty }\gamma ^{s-t}\xi _{t,s}Q_{s}MC_{s}(z)}{E_{t}\sum _{s=t}^{\infty }\gamma ^{s-t}\xi _{t,s}Q_{s}}, \end{aligned}$$
(11)
where
$$\begin{aligned} Q=\left( \frac{C_{s}+I_{s}+\phi (\frac{I_{s}}{K_{s}} )K_{s}+G_{s}^{C}+I_{s}^{G}}{P_{s}}\right) . \end{aligned}$$
Applying log-linearization to Eq. (11) results in the pricing formula \(\hat{p}_{t}(z)=\beta \gamma E_{t}(\hat{p}_{t+1}(z))+(1-\beta \gamma )(\hat{MC}_{t}(z))+\epsilon _{t}^{CP}\), where \(\epsilon _{t}^{CP}\) denotes a cost-push shock, following an AR(1) process similar to other shocks in the model. The overall price level can be represented as \(\hat{p}_{t}=\gamma \hat{p}_{t-1}+(1-\gamma )\hat{p}_{t}(z)\).
2.3 Environment
The core theme of this study is the consequences of business cycles and monetary policy on emissions. Heutel (2012) represents the first empirical investigation into the response of CO2 to cyclical variations in GDP. He assumes that domestic emissions are a function of GDP and estimates the elasticity of emissions with respect to GDP to be 0.7. We follow this modeling approach, where the log-linearized version of the flow of emissions, \(F_{t}\), is characterized by the equation \(\hat{F}_{t}=\zeta \hat{ Y}_{t}^{ }+\epsilon _{t}^{F}\). Here, \(\zeta \) denotes the elasticity of emissions with respect to GDP, and \(\epsilon _{t}^{F}\) represents an emissions shock. It follows an AR(1) process similar to other shocks in the model. It can be interpreted as fluctuations in the level of emissions that are not attributable to the cyclical variations in GDP. These fluctuations may arise, for example, due to changes in the composition of GDP affecting emissions.
Drawing from insights in both chemistry and economics, our modeling approach for the stock of pollution follows (Heutel 2012; Khan et al. 2019). The log-linearized version of the pollution stock, denoted as \(X_{t}\), is represented by
$$\begin{aligned} \hat{X}_{t}=\eta \hat{X}_{t-1}+\hat{F}_{t}, \end{aligned}$$
where \(\eta \) represents the decay rate of CO2, calibrated based on the “half-life” of atmospheric CO2. The equation diverges from Heutel (2012) and Khan et al. (2019) by exclusively accounting for the emissions of a single country, omitting emissions from the rest of the world. Consequently, the stock of CO2 pollution should be interpreted as the quantity of emissions attributable to Australia.
Environmental economics research occasionally explores the bidirectional relationship between climate change and the global economy (e.g., Nordhaus (2014)). Global economic activity generates CO2 emissions, which influence climate change, while climate change in turn affects global economic activity. However, Australia’s CO2 emissions have a minimal impact on the global CO2 stock driving climate change. This limitation prevents the development of a feedback loop model that addresses the Australian economy.
2.4 Government
It is posited that public consumption indices parallel those of private consumption, with the demand functions for domestic goods by the public sector being structured analogously to those of the private sector. The equation describing the formation of public capital is identical to that governing the formation of private capital. The government’s budget constraint is outlined as follows:
$$\begin{aligned} \tau _{t}^{y}\left( w_{t}N_{t}+r_{t}^{K}K_{t}+v_{t}\right) +\tau _{t}^{c}P_{t}C_{t}=B_{t}-R_{t}^{-1}B_{t+1}+P_{t}\left( G_{t}^{C}+I_{t}^{G}+G_{t}^{T}\right) . \end{aligned}$$
(12)
The government adjusts the income tax rate in response to deviations in the previous period’s public debt from its initial steady state level:
$$\begin{aligned} \tau _{t}^{y}=\tau _{SS}^{y}\bigg (B_{t-1}-B_{SS}\bigg )^{\Phi _{d}}, \end{aligned}$$
(13)
where \(\Phi _{d}\) signifies the tax elasticity with respect to public debt. Government spending across various categories, denoted as \(\hat{g}\) (where g represents \(G^{C}\), \(I^{G}\), and \(G^{T})\) follows an AR(1) process: \( \hat{g}_{g,t}=\rho ^{g}\hat{g}_{g,t-1}+\varepsilon _{g,t}^{g}\). In this formula, \(\rho ^{g}\) varies between zero and one, \(\hat{g}_{g,t}\) is expressed as \((G_{g,t}-G_{g,SS})/Y_{SS}\), and \(\epsilon _{g,t}^{g}\) represents an independent and identically distributed spending shock with a mean of zero.
Monetary policy adheres to a (Taylor 1993) rule, which includes the aspect of interest rate smoothing. The log-linearized formulation of the monetary policy rule is given by:
$$\begin{aligned} \hat{\imath }_{t}=\mu _{s}\hat{\imath }_{t-1}+(1-\mu _{s})\left( \mu _{p}\Delta \hat{ P}_{t}+\mu _{y}\hat{Y}_{t}\right) +\hat{\varepsilon }_{t}^{r}, \end{aligned}$$
where \(\Delta \) signifies the first difference operator, and \(\hat{ \varepsilon }_{t}^{r}\) represents a monetary policy shock with a mean of zero.
