2.1.1. Competitive labor “packer” firm

Differentiated wages, $W_{jt}$ , are set as in Erceg et al. (Reference Erceg, Henderson and Levin2000), assuming the existence of a labor packer firm. Total labor input is given by:

(1) \begin{equation} N_t=\left (\int _{0}^{1} N_{jt}^{\frac {\theta ^h-1}{\theta ^h}}dj\right )^{\frac {\theta ^h}{\theta ^h-1}} \end{equation}

where $\theta ^h$ denotes the elasticity of substitution across labor types. As in standard New Keynesian models with wage rigidities, the demand for a particular labor type $j$ is:

(2) \begin{equation} N_{jt}=\left (\frac {W_{jt}}{W_t}\right )^{-\theta ^h}N_t \end{equation}

where the aggregate wage index, $W_t$ , is defined as:

(3) \begin{equation} W_t = \left (\int _{0}^{1} W_{jt}^{1-\theta ^h} dj\right )^{\frac {1}{1-\theta ^h}} \end{equation}

2.1.2. A representative infinitely-lived agent

A representative household maximizes expected lifetime utility:

(4) \begin{equation} \mathbb{E}_t \sum _{i=0}^\infty \beta ^t \left (\frac {C_{t}^{1-\sigma }}{1-\sigma }-\mu _N \frac {N_{jt}^{1+\eta }}{1+\eta }\right ) \end{equation}

subject to the period-by-period budget constraint:

(5) \begin{align} (1+\tau _t^C)P_tC_t + Q_{t,t+1}^N \tilde {B}_{t+1} &= (1-\tau _t^N)W_{jt}N_{jt}+D_t + \tilde {B}_t + T_t \end{align}

(6) \begin{align} N_{jt} &= \left (\frac {W_{jt}}{W_t}\right )^{-\theta ^h}N_t \end{align}

where $\beta$ is the discount factor, $\sigma$ is the coefficient of relative risk aversion, $\eta$ is the inverse Frisch elasticity of labor supply, and $\mu _N$ governs the disutility of labor. $P_t$ is the price of the consumption good, $\tilde {B}_{t+1}$ designates risk-free bond demand and $Q_{t,t+1}^N$ is the price of bonds, $C_t$ is consumption, $D_t$ are dividends from firms, and $T_t$ denotes lump-sum transfers. $\tau _t^C$ and $\tau _t^N$ are the tax rates on consumption and labor income, respectively.

The first-order condition for consumption yields the standard Euler equation:

(7) \begin{equation} 1=\beta (1+i_t) \mathbb{E}_t \frac {C_{t+1}^{-\sigma }}{C_t^{-\sigma }} \left (\frac {1+\tau _{t}^C}{1+\tau _{t+1}^C}\right ) \frac {P_t}{P_{t+1}} \end{equation}

where $i_t$ is the nominal interest rate.

To introduce wage stickiness, I assume that households supply differentiated labor services and set wages in a staggered fashion, following the Calvo (Reference Calvo1983) mechanism. That is, each household can reset its nominal wage with probability $1 - \alpha _W$ in any given period. With probability $\alpha _W$ , the wage remains fixed at its previously set level. This friction implies that not all households can optimally adjust their wages in response to economic conditions each period. Let $w^*_t$ denote the optimal real wage chosen by a household when it has the opportunity to reset.Footnote ⁶ The optimal wage $w^*_t$ satisfies the following condition:

(8) \begin{equation} {w_t^*}^{1+\theta ^h \eta }=\frac {\theta ^h}{\theta ^h-1} \frac {\mathbb{E}_t \sum _{i=0}^\infty \beta ^i \alpha _W^i \mu _N w_{t+i}^{\theta ^h(1+\eta )}\Pi _{t+i}^{\theta ^h(1+\eta )}N_{t+i}^{1+\eta }}{\mathbb{E}_t \sum _{i=0}^\infty \beta ^i \alpha _W^i C_{t+i}^{-\sigma } \left (\frac {1-\tau _{t+i}^N}{1+\tau _{t+i}^C}\right ) w_{t+i}^{\theta ^h}\Pi _{t+i}^{\theta ^h-1}N_{t+i}} \end{equation}

where $w_t = W_t/P_t$ is the real wage and $\Pi _{t+i} = P_{t+i}/P_t$ is the gross inflation rate.Footnote ⁷ In the special case where wages are fully flexible ( $\alpha _W = 0$ ), the household can adjust its wage every period. In this case, the optimal wage simplifies to a static markup condition:

(9) \begin{equation} w_t=\underbrace {\frac {\theta ^h}{\theta ^h-1}}_{\text{markup}}\underbrace {\frac {\mu _N N_t^\eta }{C_t^{-\sigma }}}_{\text{MRS}} \underbrace {\frac {1+\tau _t^C}{1-\tau _t^N}}_{\text{tax distortion}} \end{equation}

This expression shows that the real wage is a markup over the marginal rate of substitution (MRS) between consumption and labor, adjusted for tax distortions. The MRS reflects the household’s willingness to trade off consumption for leisure, while the markup arises from the household’s market power in setting wages.

2.2.1. Final good

The final good is produced by a representative firm operating under perfect competition. This firm aggregates a continuum of differentiated intermediate goods, $Y_{it}$ , using a CES technology:

(10) \begin{equation} Y_t=\left (\int _{0}^{1} Y_{it}^{\frac {\theta ^f-1}{\theta ^f}}di\right )^{\frac {\theta ^f}{\theta ^f-1}} \end{equation}

where $\theta ^f \gt 0$ denotes the elasticity of substitution across intermediate varieties. The firm minimizes the cost of acquiring intermediate inputs, taking prices as given. The demand for each intermediate good $i$ is:

(11) \begin{equation} Y_{it}=\left (\frac {P_{it}}{P_t}\right )^{-\theta ^f}Y_t \end{equation}

where $P_{it}$ is the price of variety $i$ and $P_t$ is the aggregate price index, defined as:

(12) \begin{equation} P_t = \left (\int _{0}^{1} P_{it}^{1-\theta ^f} di\right )^{\frac {1}{1-\theta ^f}} \end{equation}

2.2.2. Intermediate goods

Intermediate goods are produced by a continuum of monopolistically competitive firms. Each firm uses labor as the sole input and generates emissions as a by-product of production along the lines of Annicchiarico and Di Dio (Reference Annicchiarico and Di Dio2017). The production function is given by:

(13) \begin{equation} Y_{it}=\Lambda _t A_t N_{it} \end{equation}

where $N_{it}$ is labor input, $A_t$ is total factor productivity,Footnote ⁸ and $\Lambda _t$ captures climate-related damages. Total factor productivity evolves according to a stationary AR(1) process:

(14) \begin{equation} \log A_t = (1-\rho _A) \log \bar {A} + \rho _A \log A_{t-1} + \epsilon _{A_t} \end{equation}

with $\rho _A \in (0,1)$ and $\epsilon _{A_t}$ an i.i.d. shock with zero mean. Climate damages are modeled as in Golosov et al. (Reference Golosov, Hassler, Krusell and Tsyvinski2014), with $\Lambda _t$ decreasing in the stock of pollution $M_t$ relative to its pre-industrial level $\overline {M}$ :

(15) \begin{equation} \Lambda _t = \exp (\!-\chi (M_t-\overline {M})) \end{equation}

where $\chi \gt 0$ measures the sensitivity of productivity to environmental degradation.

Firms generate emissions $Z_{it}$ proportionally to output, but can reduce emissions by allocating resources $X_{it}$ to abatement. As in Annicchiarico and Di Dio (Reference Annicchiarico and Di Dio2017, Reference Annicchiarico and Di Dio2015) and Heutel (Reference Heutel2012), emissions are given by:

(16) \begin{equation} Z_{it}=(1-X_{it})\psi Y_{it} \end{equation}

where $\psi \gt 0$ is the emissions intensity of output. The cost of abatement, expressed in units of output, is:

(17) \begin{equation} \mathbb{C}_{X_{it}}=\phi _1 B_t X_{it}^{\phi _2}Y_{it} \end{equation}

where $\phi _1, \phi _2 \gt 0$ and $B_t$ captures abatement cost uncertainty. As in Annicchiarico and Diluiso (Reference Annicchiarico and Diluiso2017), $B_t$ follows:

(18) \begin{equation} \log B_t = \rho _B \log B_{t-1} + \epsilon _{B_t} \end{equation}

with $\rho _B \in (0,1)$ and $\epsilon _{B_t}$ an i.i.d. shock with zero mean.

The law of motion for the stock of pollution is:

(19) \begin{equation} M_{t+1}=(1-\delta _M)M_{t}+Z_{t}+\overline {Z} \end{equation}

where $0 \lt \delta _M \lt 1$ is the natural decay rate of pollution, $Z_t$ is aggregate industrial emissions, and $\overline {Z}$ denotes constant non-industrial emissions.

Each firm minimizes costs subject to its production function and the demand for its variety. The first-order conditions for labor and abatement are:

(20) \begin{align} \varphi _{t}\Lambda _t A_t &= \frac {W_t}{P_t} \end{align}

(21) \begin{align} \frac {\phi _1 \phi _2}{\psi } B_t X_{t}^{\phi _2-1} &= \frac {P_{Zt}}{P_t} \end{align}

where $\varphi _t$ is the marginal cost of labor, and $P_{Zt}$ is the price of emissions (or the permit price under cap-and-trade).

Since all firms face the same prices, the marginal cost and abatement effort are identical across firms. Real profits are:

(22) \begin{equation} D_{it}=\left [\frac {P_{it}}{P_t}-MC_t\right ]Y_{it} \end{equation}

with total marginal cost:

(23) \begin{equation} MC_t=\varphi _t+\frac {P_{Zt}}{P_t}(1-X_t)\psi +\phi _1 B_tX_t^{\phi _2} \end{equation}

The first term reflects labor costs, the second captures emissions costs, and the third represents abatement costs.

To introduce price stickiness, I assume that firms can reset prices with probability $1-\alpha _P$ each period, as in Calvo (Reference Calvo1983). The optimal reset price $P_t^*$ satisfies:

(24) \begin{equation} \frac {P_t^*}{P_t}=\frac {\theta ^f}{\theta ^f-1} \frac {\mathbb{E}_t \sum _{j=0}^\infty \alpha _P^j Q_{t,t+j}^R MC_{t+j}\left (\frac {P_{t+j}}{P_t}\right )^{\theta ^f}Y_{t+j}}{\mathbb{E}_t \sum _{j=0}^\infty \alpha _P^j Q_{t,t+j}^R \left (\frac {P_{t+j}}{P_t}\right )^{\theta ^f-1} Y_{t+j}} \end{equation}

In the absence of price rigidities ( $\alpha _P = 0$ ), the optimal price is a constant markup over marginal cost:

(25) \begin{equation} \frac {P_t^*}{P_t}=\underbrace {\frac {\theta ^f}{\theta ^f-1}}_{\text{markup}} MC_t \end{equation}

2.6.1. Price and wage dynamics

The evolution of the aggregate price level follows from the Calvo pricing mechanism:

(34) \begin{eqnarray} P_t &=& \left (\int _{0}^{1} P_{it}^{1-\theta ^f} di\right )^{\frac {1}{1-\theta ^f}} \nonumber \\ &=& \left [(1-\alpha _P){P_t^*}^{1-\theta ^f} + \alpha _P P_t^{1-\theta ^f}\right ]^{\frac {1}{1-\theta ^f}} \end{eqnarray}

Dividing by $P_{t-1}$ and rearranging yields the inflation dynamics:

(35) \begin{equation} 1 = \alpha _P \Pi _t^{\theta ^f - 1} + (1 - \alpha _P){p_t^*}^{1 - \theta ^f} \end{equation}

where $p_t^* = P_t^*/P_t$ is the relative reset price. The law of motion for price dispersion is:

(36) \begin{equation} D_{p,t} = (1 - \alpha _P){p_t^*}^{-\theta ^f} + \alpha _P \Pi _t^{\theta ^f} D_{p,t-1} \end{equation}

Similarly, the evolution of the aggregate wage index, $W_t$ , implies the following dynamics for real wages:

(37) \begin{equation} 1 = (1 - \alpha _W){\Pi _{w,t}^*}^{1 - \theta ^h} + \alpha _W \Pi _t^{\theta ^h - 1} \Pi _{w,t}^{\theta ^h - 1} \end{equation}

where $\Pi _{w,t}^* = w_t^*/w_t$ is the relative reset wage and $\Pi _{w,t} = w_t/w_{t-1}$ is the gross wage inflation rate.

4.2.1. Carbon tax policy

Figure 1 displays the impulse responses to an abatement cost shock under a carbon tax regime. Since the emissions price is fixed, firms respond by reducing abatement effort and increasing emissions, thereby raising environmental tax revenues. This substitution effect is consistent with cost-minimizing behavior, as firms reallocate resources away from abatement toward production.

Figure 1. Dynamic responses to an abatement cost shock under a carbon tax.

Note: Each column reports the dynamic paths for selected variables under different fiscal policy specifications. The blue line with circles represents the model with flexible wages, the black line with squares represents the extension with sticky wages. Horizontal axis displays the time which is measured in quarters. Vertical axis values refer to deviations from steady state in percentage.

Figure 2. Dynamic responses to an abatement cost shock under a cap-and-trade system.

Note: Each column reports the dynamic paths for selected variables under different fiscal policy specifications. The blue line with circles represents the model with flexible wages, the black line with squares represents the extension with sticky wages. Horizontal axis displays the time which is measured in quarters. Vertical axis values refer to deviations from steady state in percentage.

Nominal rigidities play a central role in the transmission of the shock. When both prices and wages are sticky (black line with squares), the economy experiences a pronounced contraction in output and consumption. Firms are unable to adjust prices in response to higher marginal costs, and households face rigid nominal wages that prevent real wage adjustment. In contrast, when wages are flexible (blue line with circles), the real wage adjusts downward, partially absorbing the shock and mitigating the decline in employment and output. This result echoes the findings of Blanchard and Galí (Reference Blanchard and Galí2007), who show that wage rigidity amplifies the real effects of supply shocks in New Keynesian models.

The fiscal use of environmental revenues significantly affects the propagation of the shock. When revenues are recycled through labor tax reductions, the contraction is dampened, particularly under wage rigidity. This occurs because lower labor taxes reduce the wedge between the MRS and the real wage, encouraging labor supply (see Figure http://doi.org/10.1017/S1365100525100424A.1). The most effective stabilization, however, arises when revenues are used to cut consumption taxes. This policy directly lowers the effective price of consumption, stimulating demand and offsetting the contractionary effects of the shock. These results provide strong support for the double dividend hypothesis (Bovenberg and Goulder, Reference Bovenberg, Goulder, Bovenberg and Goulder2002), especially when tax cuts target intertemporal margins.

The interaction between fiscal policy and nominal rigidities is particularly important. Under sticky prices, the reduction in consumption taxes has a stronger effect on output because it boosts demand in a context where firms cannot adjust prices downward. In contrast, if lump-sum transfers increase via fiscal adjustment, reflecting the endogenous rise in emissions and tax revenues, which are redistributed to households, this channel is less effective in stimulating demand compared to targeted tax cuts. Overall, the results highlight the importance of coordinating environmental and fiscal policy in the presence of nominal frictions, as emphasized by Annicchiarico and Di Dio (Reference Annicchiarico and Di Dio2015) and Roach (Reference Roach2021).

4.2.2. Cap-and-trade policy

Figure 2 presents the impulse responses of the economy to a one percent increase in abatement costs under a cap-and-trade regime. Unlike the carbon tax case, where the emissions price is fixed, the cap-and-trade system imposes a quantity constraint on emissions. Firms must purchase permits at a market-determined price, which adjusts endogenously to clear the allowances market. When abatement costs rise, firms reduce their mitigation efforts and increase their demand for permits, driving up the emissions price. This mechanism amplifies the cost shock, as the higher emissions price enters the marginal cost function and tightens the supply side of the economy.

The rigidity of the emissions cap implies that firms cannot increase emissions in response to the shock. Instead, they must either pay more for permits or reduce output. This constraint exacerbates the contractionary effects of the shock, particularly in the presence of nominal rigidities. When both prices and wages are sticky (black line with squares), firms are unable to adjust nominal variables to absorb the increase in marginal costs, resulting in a sharper decline in output and consumption. These dynamics are consistent with the findings of Annicchiarico and Diluiso (Reference Annicchiarico and Diluiso2017), who show that cap-and-trade systems can be more distortionary in the short run when abatement cost uncertainty is high.

Figure 2 also explores the role of fiscal policy in this context. When environmental revenues from permit sales are used to reduce labor income taxes, the contraction in output and consumption is partially mitigated. However, the most effective stabilization again arises when revenues are used to reduce consumption taxes. This policy lowers the effective price of consumption, stimulating demand and partially offsetting the negative supply shock. The model highlights that under a cap-and-trade regime, the intertemporal distortion from consumption taxes is particularly costly, as it compounds the rigidity imposed by the emissions cap. This finding aligns with Roach (Reference Roach2021), who emphasizes the importance of revenue recycling design in enhancing the macroeconomic resilience of climate policy.

The interaction between nominal rigidities and fiscal policy is especially important under cap-and-trade. With sticky prices, the increase in the emissions price cannot be passed on to consumers immediately, compressing firm profits. Wage stickiness further exacerbates the downturn by preventing real wage adjustment, leading to a sharper decline in labor supply and consumption (see Figure http://doi.org/10.1017/S1365100525100424A.2). These results emphasize the importance of flexible fiscal instruments in buffering the economy against environmental cost shocks. In particular, using environmental revenues to reduce consumption taxes provides a countercyclical stimulus that is both timely and efficient, especially when monetary policy is constrained or focused on inflation stabilization.

4.3.1. Carbon tax policy

Figure 3 shows the impulse responses of the economy to a government spending shock under a carbon tax regime. The shock is modeled as a temporary increase in exogenous public consumption, financed through different fiscal instruments. In the baseline case, the government adjusts lump-sum transfers to satisfy its budget constraint. The increase in government spending raises aggregate demand, leading to an expansion in output and labor. This response is consistent with standard New Keynesian dynamics, where fiscal multipliers are positive in the presence of nominal rigidities (Christiano et al., Reference Christiano, Eichenbaum and Rebelo2011; Drautzburg and Uhlig, Reference Drautzburg and Uhlig2015).

Figure 3. Dynamic responses to a government spending shock under a carbon tax.

Note: Each column reports the dynamic paths for selected variables under different fiscal policy specifications. The blue line with circles represents the model with flexible wages, the black line with squares represents the extension with sticky wages. Horizontal axis displays the time which is measured in quarters. Vertical axis values refer to deviations from steady state in percentage.

The magnitude of the expansion depends critically on the degree of nominal rigidity. When both prices and wages are sticky (black line with squares), the fiscal multiplier is largest. Firms cannot adjust prices upward in response to higher demand, so they increase production and employment instead. Wage rigidity prevents nominal wages from rising, keeping real labor costs low and encouraging hiring (see also Figure http://doi.org/10.1017/S1365100525100424A.3). In contrast, when wages are flexible (blue line with circles), the real wage increases in response to higher labor demand, partially offsetting the expansion. This mechanism is embedded in the wage-setting equation, where the optimal wage reflects expected future labor disutility and consumption paths.

The increase in output also raises emissions, which in turn boosts environmental tax revenues. However, since the carbon tax is fixed, firms do not adjust their abatement effort, and the emissions intensity of output remains unchanged. As a result, the environmental externality worsens during the expansion, highlighting a potential trade-off between short-run stabilization and long-run environmental goals. The additional revenues are not sufficient to fully finance the increase in government spending, so lump-sum transfers must fall. This reduction in household income contributes to the observed decline in consumption, which is further exacerbated by the rise in the nominal interest rate. The Taylor rule responds aggressively to inflation, crowding out private demand and reinforcing the contraction in consumption (Leeper et al., Reference Leeper, Traum and Walker2017).

Figure 3 also explores alternative fiscal financing schemes. When the government uses labor income taxes, the expansionary effect is smaller than under lump-sum financing. This is because labor taxes increase after-tax wages, raising the marginal cost of labor and dampening the employment response. Moreover, increasing consumption taxes has a less favorable impact. It raises the effective price of consumption, discouraging demand and partially exacerbating the contractionary effects of monetary tightening. This result sheds light on the relevance of the tax instrument used to finance public spending. In line with the findings of Furlanetto (Reference Furlanetto2011), the model shows that fiscal multipliers are larger when financing is non-distortionary and when nominal rigidities are present.

4.3.2. Cap-and-trade policy

Figure 4 presents the impulse responses of the economy to a one percent increase in government spending under a cap-and-trade regime. As in the carbon tax case, the shock is modeled as a temporary increase in exogenous public consumption, financed through alternative fiscal instruments. However, the emissions constraint imposed by the cap introduces a distinct transmission mechanism that interacts with nominal rigidities and fiscal policy in important ways.

Figure 4. Dynamic responses to a government spending shock under a cap-and-trade system.

Note: Each column reports the dynamic paths for selected variables under different fiscal policy specifications. The blue line with circles represents the model with flexible wages, the black line with squares represents the extension with sticky wages. Horizontal axis displays the time which is measured in quarters. Vertical axis values refer to deviations from steady state in percentage.

The increase in government spending stimulates aggregate demand, raising output and labor. However, since emissions are fixed by the cap, firms cannot respond by increasing emissions. Instead, they must either purchase additional permits at a higher market price or intensify abatement efforts. This leads to an endogenous increase in the emissions price, which enters the marginal cost function and acts as a supply-side constraint (see also Figure http://doi.org/10.1017/S1365100525100424A.4). The resulting rise in marginal costs partially offsets the expansionary effects of the fiscal shock, especially when nominal rigidities are present. This mechanism is consistent with the findings of Annicchiarico and Diluiso (Reference Annicchiarico and Diluiso2017), who show that cap-and-trade systems can amplify the macroeconomic effects of cost shocks due to the endogenous adjustment of permit prices.

Nominal rigidities significantly shape the propagation of the shock. When both prices and wages are sticky (black line with squares), firms are unable to adjust nominal variables to accommodate the higher emissions price. This results in a larger increase in real marginal costs, compressing firm profits. Wage rigidity further exacerbates the downturn by preventing real wage adjustment, leading to a sharper decline in labor supply and consumption as emphasized by Blanchard and Galí (Reference Blanchard and Galí2007) and Christiano et al. (Reference Christiano, Eichenbaum and Rebelo2011).

The fiscal instrument used to finance the increase in government spending plays a critical role in shaping the macroeconomic response. Under lump-sum taxation, the expansionary effects are strongest, as the fiscal stimulus is not offset by distortionary tax increases. The weakest expansion occurs when consumption taxes are used to finance the spending increase. Higher consumption taxes raise the effective price of consumption goods, reducing household demand and amplifying their contractionary effects. This result again is consistent with the findings of Furlanetto (Reference Furlanetto2011) and Drautzburg and Uhlig (Reference Drautzburg and Uhlig2015).