Silviase/jgov_v1.4_full_ann huggingface dataset:
Then, the (gross) balanced growth rate can be expressed as $ \mu_{z^{+},t} \equiv \frac{z_{t}^{+}}{z_{t-1}^{+}} $ . The growth rates of output, consumption and real wage become $ \mu_{z^{+},t} $ . Meanwhile, previous papers point out that investment specific technological (IST) progress has an important role for variations of output (e.g. Greenwood et al. 2000). This implies private capital and investment have a different growth rate from the above balanced growth rate. The IST progress is captured by a scaling variable, $ \Psi_{t} $ , and the scaling variable for private capital and investment can be expressed as $ z_{t}^{+}\Psi_{t} $ .

We can remove trend components using these scaling variables. The detrended production function of the intermediate good can be expressed as follows:

$$ y_{j,t}=\epsilon_{t}\left(\frac{1}{\mu_{z^{+},t}\mu_{\Psi,t}}\right)^{\alpha}\left(\tilde{k}_{j,t-1}\right)^{\alpha}H_{j,t}^{1-\alpha}-\Theta, $$

where $ y_{j,t} \equiv \frac{Y_{j,t}}{z_{t}^{+}}, \tilde{k}_{j,t} \equiv \frac{\tilde{K}_{j,t}}{z_{t}^{+} \Psi_{t}} $ .

Hereafter, model variables in scaled form are denoted by small letters.

Given the capital rental price, $ R_{t}^{k} $ , and nominal wage, $ W_{t} $ , intermediate good firms solve the cost minimization problem. Then, we obtain the following real marginal cost, $ mc_{t} $ in scaled form:

$$ \begin{aligned}mc_{t}&=\frac{1}{\epsilon_{t}}\frac{\left(r_{t}^{k}\right)^{\alpha}w_{t}^{1-\alpha}}{\alpha^{\alpha}(1-\alpha)^{1-\alpha}}\\ where\quad w_{t}&\equiv\frac{W_{t}}{z_{t}^{+}P_{t}},\quad r_{t}^{k}\equiv\frac{\Psi_{t}R_{t}^{k}}{P_{t}}.\end{aligned} $$

The FOC to the cost minimization problem gives:

$$ \frac{\tilde{k}_{j,t-1}}{H_{j,t}}=\mu_{z^{+},t}\mu_{\Psi,t}\frac{\alpha}{1-\alpha}\frac{w_{t}}{r_{t}^{k}}. $$

From this result, the capital-labor ratio does not depend on the subscript j. Therefore, real marginal cost and optimal capital-labor ratio are the same across intermediate goods firms.

Since intermediate goods firms have some market power in the intermediate goods market, they set prices to maximize their profits. We introduce Calvo-type nominal rigidities for price settings of intermediate goods firms. Intermediate goods firms can optimize their prices with a probability $ 1 - \xi_{p} $ , and the firms that cannot revise their price index those prices partially with an inflation rate at the previous period, $ \Pi_{t-1} $ , and with the trend inflation rate, $ \bar{\Pi}_{t} $ , controlled by the central bank. Therefore, given $ P_{t} $ and $ Y_{t} $ , the intermediate good firm j sets price so as to maximize the following objective function:

$$ \begin{aligned}\max_{P_{j,t}}\quad&\mathrm{E}_{t}\sum_{i=0}^{\infty}\left(\beta\xi_{p}\right)^{i}\Lambda_{t,t+i}\left[\left(P_{j,t+i}-P_{t+i}m c_{t+i}\right)Y_{j,t+i}-z_{t+i}^{+}P_{t+i}\Theta\right]\\ where\quad&P_{j,t}=\Pi_{\kappa,t}P_{j,t-1},\\&\Pi_{\kappa,t}\equiv\left(\Pi_{t-1}\right)^{\kappa}\left(\bar{\Pi}_{t}\right)^{1-\kappa}\\&Y_{j,t}=\left(\frac{P_{j,t}}{P_{t}}\right)^{\frac{\lambda}{1-\lambda}}Y_{t}\end{aligned} $$

$ \beta \in (0,1) $ stands for the subjective discount factor, and $ \Lambda_{t,t+i} $ stands for the stochastic
are derived as follows:

$$ \Lambda_{t}P_{t}\left(1+\tau_{t}^{c}\right)=\frac{\zeta_{t}^{c}}{\tilde{C}_{t}-\theta\tilde{C}_{t-1}}, $$

$$ \Lambda_{t}=\beta\mathrm{E}_{t}R_{t}\Lambda_{t+1}, $$

$$ \Lambda_{t}P_{t}^{i}=\zeta_{t}^{i}Q_{t}\left[1-S\left(\frac{I_{t}}{I_{t-1}}\right)-S^{\prime}\left(\frac{I_{t}}{I_{t-1}}\right)\frac{I_{t}}{I_{t-1}}\right]+\beta\mathrm{E}_{t}Q_{t+1}\zeta_{t+1}^{i}S^{\prime}\left(\frac{I_{t+1}}{I_{t}}\right)\left(\frac{I_{t+1}}{I_{t}}\right)^{2}, $$

$$ Q_{t}=\beta\mathrm{E}_{t}\left[(1-\delta)Q_{t+1}+\Lambda_{t}(1-\tau_{t}^{k})\left(P_{t+1}^{k}u_{t+1}-P_{t+1}^{i}a(u_{t+1})\right)\right] $$

$$ P_{t}^{k}=P_{t}^{i}a^{\prime}(u_{t}) $$

From equation (A.21), the Lagrange multiplier is marginal utility of consumption. Using equations (A.21) and (A.22), we can derive a consumption Euler equation. Let the stochastic discount factor be defined as the following equation:

$$ \mathrm{E}_{t}\Lambda_{t,t+i}\equiv\mathrm{E}_{t}\frac{\Lambda_{t+i}}{\Lambda_{t}}. $$

$ \Lambda_{t,t+i} $ represents the marginal rate of intertemporal substitution between consumption at period t and period $ t+i $ .

FOCs in scaled forms of households are expressed as follows:

$$ \varphi_{t}^{c}(1+\tau_{t}^{c})=\frac{\zeta_{t}^{c}}{\tilde{c}_{t}-\theta\tilde{c}_{t-1}\frac{1}{\mu_{z^{+},t}}}, $$

$$ \beta\mathbf{E}_{t}\Lambda_{t,t+1}=\frac{1}{R_{t}}, $$

$$ 1=\zeta_{t}^{i}\left[1-S\left(\mu_{z^{+},t}\mu_{\Psi,t}\frac{i_{t}}{i_{t-1}}\right)-S^{\prime}\left(\mu_{z^{+},t}\mu_{\Psi,t}\frac{i_{t}}{i_{t-1}}\right)\mu_{z^{+},t}\mu_{\Psi,t}\frac{i_{t}}{i_{t-1}}\right]q_{t} $$

$$ +\beta\mathrm{E}_{t}\Lambda_{t,t+1}\frac{\Pi_{t+1}}{\mu_{\Psi,t+1}}\zeta_{t+1}^{i}S^{\prime}\left(\mu_{z^{+},t+1}\mu_{\Psi,t+1}\frac{i_{t+1}}{i_{t}}\right)\left(\mu_{z^{+},t+1}\mu_{\Psi,t+1}\frac{i_{t+1}}{i_{t}}\right)^{2}q_{t+1} $$

$$ q_{t}=\beta\mathrm{E}_{t}\Lambda_{t,t+1}\frac{\Pi_{t+1}}{\mu_{\Psi,t+1}}\left[(1-\delta)q_{t+1}+(1-\tau_{t+1}^{k})\left(p_{t+1}^{k}u_{t+1}-a(u_{t+1})\right)\right] $$

$$ p_{t}^{k}=a^{\prime}(u_{t}), $$

$$ where\quad\varphi_{t}^{c}\equiv z_{t}^{+}P_{t}\Lambda_{t},\tilde{c}_{t}\equiv\frac{\tilde{C}_{t}}{z_{t}^{+}},q_{t}\equiv\frac{\Psi_{t}Q_{t}}{P_{t}\Lambda_{t}}. $$

The stochastic discount factor in scaled form can be derived as follows:

$$ \mathrm{E}_{t}\Lambda_{t,t+1}=\mathrm{E}_{t}\frac{\varphi_{t+1}^{c}}{\Pi_{t+1}^{d}\mu_{z^{+},t+1}\varphi_{t}^{c}}. $$

The budget constraint in scaled form can be obtained as the following equation:

$$ \begin{align*}&(1+\tau_{t}^{c})c_{t}+i_{t}+b_{t}+\tau_{t}\\=&(1-\tau_{t}^{h})w_{t}H_{t}+(1-\tau_{t}^{k})\left[\left(p_{t}^{k}u_{t}-a(u_{t})\right)\frac{k_{t-1}}{\mu_{z^{+},t}\mu_{\Psi,t}}+v_{t}+v_{t}^{k}\right]+\frac{1}{\mu_{z^{+},t}}\frac{R_{t-1}}{\Pi_{t}}b_{t-1}\end{align*} $$
where

$$ b_{t}\equiv\frac{B_{t}}{z_{t}^{+}P_{t}},w_{t}\equiv\frac{W_{t}}{z_{t}^{+}P_{t}},v_{t}\equiv\frac{V_{t}}{z_{t}^{+}P_{t}},\tau_{t}\equiv\frac{T_{t}}{z_{t}^{+}P_{t}}. $$

$ b_{t} $ stands for bond holdings, $ w_{t} $ stands for real wage, $ v_{t} $ stands for real dividend and $ \tau_{t} $ stands for real lump-sum transfer from government. Finally, households satisfy the following transversarity conditions:

$$ \lim_{i\rightarrow\infty}\beta\Lambda_{t,t+i}\mu_{z^{+},t+i}\Pi_{t+i}\left(b_{t+i}+k_{t+i}\right)=0. $$

## Wage Setting

Following Erceg et al. (2000), employment agencies bundle household's differentiated skill $ H_{j,t} $ into a homogenous aggregate labor $ H_{t} $ , and sell the bundled labor to intermediate goods firms. Given the nominal wage $ W_{t} $ and the differentiated skill j's nominal wage $ W_{j,t} $ , the employment agency maximizes the following profit:

$$ \begin{aligned}&\max_{H_{j,t}}W_{t}H_{t}-\int_{0}^{1}W_{j,t}H_{j,t}dj,\\&\text{s.t.}H_{t}=\left[\int_{0}^{1}H_{j,t}^{\frac{1}{\lambda_{w}}}dj\right]^{\lambda_{w}}.\end{aligned} $$

FOC leads to the demand function for differentiated labor $j$:

$$ H_{j,t}=\left(\frac{W_{j,t}}{W_{t}}\right)^{\frac{\lambda_{w}}{1-\lambda_{w}}}H_{t}. $$

Substituting this into bundling technology, the nominal wage level is derived by

$$ W_{t}=\left[\int_{0}^{1}W_{j,t}^{\frac{1}{1-\lambda_{w}}}d j\right]^{1-\lambda_{w}}. $$

Household j can revise its nominal wage with a probability $ 1 - \xi_{w} $ , otherwise it indexes its wage with past wage inflation rate and trend inflation rate. Household j's objective function is as follows:

$$ \begin{aligned}\max_{W_{j,t}}\quad&E_{t}\sum_{i=0}^{\infty}(\beta\xi_{w})^{i}\left[-\zeta_{t+i}^{h}\chi_{t+i}^{h}A_{H}\frac{H_{j,t+i}^{1+\sigma_{H}}}{1+\sigma_{H}}+\Lambda_{j,t+i}(1-\tau_{t+i}^{h})W_{j,t+i}H_{j,t+i}\right]\\ where\quad&W_{j,t}=\Pi_{\kappa,t}^{w}W_{j,t-1},\\&\Pi_{\kappa,t}^{w}\equiv\left(\Pi_{t-1}^{w}\right)^{\kappa_{w}}\left(\bar{\Pi}_{t}\right)^{1-\kappa_{w}}.\\&H_{j,t}=\left(\frac{W_{j,t}}{W_{t}}\right)^{\frac{\lambda_{w}}{1-\lambda_{w}}}H_{t}.\end{aligned} $$

where $ \bar{\Pi}_{t} $ is trend inflation and $ \kappa_{w}\in(0,1) $ is wage inflation persistency. Letting $ w_{t}^{opt} $ denote optimal wage and $ \Pi_{t}^{w,opt} $ denote optimal inflation rate, we can derive the following FOCs in
scaled forms:

$$ w_{t}^{opt}=\left\{A_{H}\lambda_{w}\frac{K_{t}^{w}}{F_{t}^{w}}\right\}^{\frac{1-\lambda_{w}}{1-\lambda_{w}-\lambda_{w}\sigma_{h}}} $$

$$ K_{t}^{w}=\zeta_{t}^{h}\chi_{t}^{h}w_{t}^{-\frac{\lambda_{w}(1+\sigma_{H})}{1-\lambda_{w}}}H_{t}^{1+\sigma_{H}}+\beta\xi_{w}(\Pi_{\kappa,t+1}^{w})^{\frac{\lambda_{w}(1+\sigma_{H})}{1-\lambda_{w}}}(\mu_{t+1}^{z+}\Pi_{t+1})^{-\frac{\lambda_{w}(1+\sigma_{H})}{1-\lambda_{w}}}K_{t+1}^{w} $$

$$ \begin{array}{r}{F_{t}^{w}=\varphi_{t}^{c}(1-\tau_{t}^{h})w_{t}^{-\frac{\lambda_{w}}{1-\lambda_{w}}}H_{t}+\beta\xi_{w}(\Pi_{\kappa,t+1}^{w})^{\frac{1}{1-\lambda_{w}}}(\mu_{t+1}^{z+}\Pi_{t+1})^{-\frac{1}{1-\lambda_{w}}}F_{t+1}^{w}}\end{array} $$

$$ where\quad w_{t}^{opt}\equiv\frac{W_{t}^{opt}}{z_{t}^{+}P_{t}},\quad w_{t}\equiv\frac{W_{t}}{z_{t}^{+}P_{t}}. $$

The endogenous preference shifters in scaled forms can be written by

$$ \chi_{t}^{h}=\frac{z_{\chi,t}}{\tilde{c}_{t}-\theta\tilde{c}_{t-1}\frac{1}{\mu_{z^{+},t}}} $$

$$ z_{\chi,t}=\left(\frac{z_{\chi,t-1}}{\mu_{z^{+},t}}\right)^{1-v}\left(\tilde{c}_{t}-\theta\tilde{c}_{t-1}\frac{1}{\mu_{z^{+},t}}\right)^{v} $$

where $ z_{\chi,t} \equiv \frac{Z_{\chi,t}}{z_{t}^{+}}. $

Nominal wage level and wage inflation rate can be derived as follows:

$$ \begin{align*}W_{t}&=\left[\left(1-\xi_{w}\right)\left(W_{t}^{opt}\right)^{\frac{1}{1-\lambda_{w}}}+\xi_{w}\left(\Pi_{\kappa,t}^{w}W_{t-1}\right)^{\frac{1}{1-\lambda_{w}}}\right]^{1-\lambda_{w}}.\\\Longleftrightarrow\Pi_{t}^{w}&=\left[\left(1-\xi_{w}\right)\left(\Pi_{t}^{w,opt}\right)^{\frac{1}{1-\lambda_{w}}}+\xi_{w}\left(\Pi_{\kappa,t}^{w}\right)^{\frac{1}{1-\lambda_{w}}}\right]^{1-\lambda_{w}}.\end{align*} $$

The wage-type new Keynesian Phillips curve consists of equations (A.34), (A.35), (A.36), (A.37) and (A.40).

## Unemployment

Following GSW, a member h in household j has a differentiated skill j and the labor disutility $ \zeta_{t}^{h}\chi_{t}^{h}A_{H}h^{\sigma_{H}} $ . Given the real wage $ w_{j,t} $ , we consider the desirable labor supply $ L_{j,t} $ of the member h within the household j. As long as the real wage is greater than the MRS between labor and consumption, the member is willing to work:

$$ (1-\tau_{t}^{h})w_{t}\geq\frac{\zeta_{t}^{h}\chi_{t}^{h}A_{H}h^{\sigma_{H}}}{\varphi_{j,t}^{c}} $$

$ \tau_{t}^{h} $ stands for labor income tax rate. The left-hand side indicates the marginal benefit of labor supply after tax adjustment. The right-hand side is MRS, which corresponds to reservation wage of member h.

Thanks to the consumption sharing assumption, the marginal utility of consumption becomes common across members represented as $ \varphi_{t}^{c} $ . Letting the marginal supplier of union j's member be denoted by $ L_{j,t} $ , we have:

$$ (1-\tau_{t}^{h})w_{j,t}=\frac{\zeta_{t}^{h}\chi_{t}^{h}A_{H}L_{j,t}^{\sigma_{H}}}{\varphi_{t}^{c}} $$

It is worth noting that the above labor supply condition is explicitly derived as the FOC wrt. $ H_{j,t} $ given nominal wage, since household members are assumed to be price takers.
Then, using an endogenous preference shifter (A.20), household j's labor supply $ L_{j,t} $ is derived as follows:

$$ (1-\tau_{t}^{h})w_{j,t}=(1+\tau_{t}^{c})\frac{\zeta_{t}^{h}}{\zeta_{t}^{c}}z_{\chi,t}A_{H}L_{j,t}^{\sigma_{H}} $$

Let aggregate employment be denoted by $ H_{t} $ and let aggregate labor supply be denoted by $ L_{t} $ . Then, unemployment rate $ U_{t} $ is defined as the following equation.

$$ U_{t}\equiv\frac{L_{t}-H_{t}}{L_{t}} $$

#### A.1.3 Fiscal and Monetary Authorities

## Fiscal Authority

Government purchases final goods at price $ P_{t} $ . $ G_{t}^{c} $ and $ G_{t}^{i} $ are government consumption and investment, respectively. Let $ B_{t} $ denote the government nominal bond issued at the beginning of the period t. Then, government budget constraint can be written by

$$ \begin{align*}&P_{t}G^{c}_{t}+P^{i}_{t}G^{i}_{t}+R_{t-1}B_{t-1}+T_{t}\\&=\tau^{c}_{t}P_{t}C_{t}+\tau^{h}_{t}W_{t}H_{t}+\tau^{k}_{t}\left[\left(P^{k}_{t}u_{t}-P^{i}_{t}a(u_{t})\right)K_{t-1}+V_{t}+V^{k}_{t}\right]+B_{t}.\\\Longleftrightarrow\quad&g^{c}_{t}+g^{i}_{t}+\frac{1}{\mu_{z^{+},t}}\frac{R_{t-1}}{\Pi_{t}}b_{t-1}+\tau_{t}\\&=\tau^{c}_{t}c_{t}+\tau^{h}_{t}w_{t}H_{t}+\tau^{k}_{t}\left[(p^{k}_{t}u_{t}-a(u_{t}))\frac{k_{t-1}}{\mu_{z^{+},t}\mu_{\Psi,t}}+v_{t}+v^{k}_{t}\right]+b_{t}.\end{align*} $$

$$ where\quad g_{t}^{c}\equiv\frac{G_{t}^{c}}{z_{t}^{+}},g_{t}^{i}\equiv\frac{G_{t}^{i}}{z_{t}^{+}\Psi_{t}}. $$

Public capital is accumulated by government investment as follows:

$$ \begin{aligned}K_{t}^{g}&=(1-\delta_{g})K_{t-1}^{g}+\zeta_{t}^{g,i}G_{t}^{i}.\\\Longleftrightarrow\quad k_{t}^{g}&=\frac{1}{\mu_{z^{+},t}}(1-\delta_{g})k_{t-1}^{g}+\zeta_{t}^{g,i}g_{t}^{i}\\ where\quad k_{t}^{g}&\equiv\frac{K_{t}^{g}}{z_{t}^{+}\Psi_{t}}.\end{aligned} $$

where $ \delta_{g} $ is depreciation rate of public capital and $ \hat{\zeta}_{t}^{g,i} $ is government investment specific technology shock. It should be noted that government investment and public capital have the same growth rate as private investment and private capital, i.e. the growth rate is $ z_{t}^{+}\Psi_{t} $ . Following Corsetti et al. (2012) and Iwata (2013), government spending rules are given by

$$ \ln\frac{g_{t}^{c}}{g^{c}}=\rho_{g^{c}}\ln\frac{g_{t-1}^{c}}{g^{c}}-(1-\rho_{g^{c}})\phi_{g^{c}}\ln\frac{b_{t}}{b}+\epsilon_{t}^{g^{c}} $$

$$ \ln\frac{g_{t}^{i}}{g^{i}}=\rho_{g^{i}}\ln\frac{g_{t-1}^{i}}{g^{i}}-(1-\rho_{g^{i}})\phi_{g^{i}}\ln\frac{b_{t}}{b}+\epsilon_{t}^{g^{i}} $$

In case of $ \phi_{g,c}\;(\phi_{g,i})>0 $ , government consumption (government investment) is determined so as to smooth deficits (the so-called “spending reversals” rule). $ \varepsilon_{t}^{g,c} $ and $ \varepsilon_{t}^{g,i} $ are government consumption and investment shocks, respectively. Furthermore, tax rules are assumed to be represented by the following AR(1) processes:

$$ \tau_{t}^{c}-\tau^{c}=\rho_{\tau_{c}}(\tau_{t-1}^{c}-\tau^{c})+\epsilon_{t}^{\tau_{c}} $$
ESRI Discussion Paper Series No.329

" Impacts of Government Spending on Unemployment: Evidence from a Medium-scale DSGE Model"

$$ \tau_{t}^{k}-\tau^{k}=\rho_{\tau_{k}}(\tau_{t-1}^{k}-\tau^{k})+\epsilon_{t}^{\tau_{k}} $$

$$ \tau_{t}^{h}-\tau^{h}=\rho_{\tau_{h}}(\tau_{t-1}^{h}-\tau^{h})+\epsilon_{t}^{\tau_{h}} $$

$$ \tau_{t}-\tau=\rho_{\tau}(\tau_{t-1}-\tau)+\epsilon_{t}^{\tau} $$

## Monetary Authority

Following CTW, we assume that monetary authority controls nominal interest rate according to the following rule:

$$ \ln\frac{R_{t}}{R}=\rho_{R}\ln\frac{R_{t-1}}{R}+\left(1-\rho_{R}\right)\left[\ln\frac{\bar{\Pi}_{t}}{\bar{\Pi}}+\phi_{\Pi}\ln\frac{\Pi_{t}}{\bar{\Pi}_{t}}+\phi_{y}\ln\frac{y_{t}}{y}\right]+\epsilon_{t}^{R} $$

$ \phi_{II} $ and $ \phi_{y} $ are Taylor parameters and $ \rho^{R} \in (0,1) $ stands nominal interest rate smoothing parameter. $ \varepsilon_{t}^{R} $ is monetary policy shock and assumed as iid shock with zero mean.

#### A.1.4 Aggregation and Resource Constraint

Aggregating the production function of the differentiated intermediate good, we derive aggregate supply of homogenous good as the following equation:

$$ y_{t}=\frac{1}{\tilde{p}_{t}}\left[\epsilon_{t}\left(\frac{1}{\mu_{z^{+},t}\mu_{\Psi,t}}\right)^{\alpha}\left(\tilde{k}_{t-1}\right)^{\alpha}\left(\frac{H_{t}}{\tilde{w}_{t}}\right)^{1-\alpha}-\Theta\right] $$

where

$$ \tilde{p}_{t}\equiv\int_{0}^{1}\left(\frac{P_{j,t}}{P_{t}}\right)^{\frac{\lambda}{1-\lambda}}d j\quad\tilde{w}_{t}\equiv\int_{0}^{1}\left(\frac{W_{j,t}}{W_{t}}\right)^{\frac{\lambda_{w}}{1-\lambda_{w}}}d j $$

$ \tilde{p}_{t} $ and $ \tilde{w}_{t} $ are price and wage dispersions, respectively. Price and wage dispersions satisfy the following recursive equation:

$$ \tilde{p}_{t}=\left(1-\xi_{p}\right)\left(\frac{1-\xi_{p}\left(\frac{\Pi_{\kappa,t}}{\Pi_{t}}\right)^{\frac{1}{1-\lambda}}}{1-\xi_{p}}\right)^{\lambda}+\xi_{p}\left(\frac{\Pi_{\kappa,t}}{\Pi_{t}}\right)^{\frac{\lambda}{1-\lambda}}\tilde{p}_{t-1}. $$

$$ \tilde{w}_{t}=\left(1-\xi_{w}\right)\left(\frac{1-\xi_{w}\left(\frac{\Pi_{\kappa,t}^{w}}{\Pi_{t}^{w}}\right)^{\frac{1}{1-\lambda_{w}}}}{1-\xi_{w}}\right)^{\lambda_{w}}+\xi_{w}\left(\frac{\Pi_{\kappa,t}^{w}}{\Pi_{t}^{w}}\right)^{\frac{\lambda_{w}}{1-\lambda_{w}}}\tilde{w}_{t-1}. $$

Using household and government budget constraints ((A.32) and (A.42)), we have the following resource constraint equation:

$$ c_{t}+i_{t}+g_{t}^{c}+g_{t}^{i}=w_{t}H_{t}+(p_{t}^{k}u_{t}-a(u_{t}))\frac{k_{t-1}}{\mu_{z^{+},t}\mu_{\Psi,t}}+v_{t}+v_{t}^{k} $$

The real profit of intermediate goods firms can be derived as:

$$ v_{t}=y_{t}-\left(w_{t}H_{t}+r_{t}^{k}u_{t}\frac{\tilde{k}_{t-1}}{\mu_{z^{+},t}\mu_{\Psi,t}}\right) $$

The real profit of capital producing firm can be expressed as:

$$ v_{t}^{k}=\alpha_{g}\frac{\tilde{k}_{t-1}}{\mu_{z^{+},t}\mu_{\Psi,t}}r_{t}^{k} $$

Substituting those profits into the resource constraint equation, and using the FOC of capital-producing firm, $ p_{t}^{k}u_{t}k_{t-1} = (1 - \alpha_{g})r_{t}^{k}\tilde{k}_{t-1} $ , we can obtain the following resource constraint
of goods market:

$$ y_{t}=c_{t}+i_{t}+g_{t}^{c}+g_{t}^{i}+a(u_{t})\frac{k_{t-1}}{\mu_{z^{+},t}\mu_{\Psi,t}}. $$

### A.2 Summary of the Model

This subsection summarizes specifications of functional forms on adjustment costs, nonlinear simultaneous difference equations and structural shocks.

#### A.2.1 Specifications of Functional forms

Capital utilization adjustment cost :

$$ \begin{aligned}&a(u_{t})=\frac{1}{2}\sigma_{b}\sigma_{a}u_{t}^{2}+\sigma_{b}(1-\sigma_{a})u_{t}+\sigma_{b}\left(\frac{\sigma_{a}}{2}-1\right)\\&a^{\prime}(u_{t})=\sigma_{b}\sigma_{a}u_{t}+\sigma_{b}(1-\sigma_{a})\\ \end{aligned} $$

Private investment adjustment cost :

$$ S(x_{t})=\frac{1}{2}\left[\exp\left(\sqrt{S^{\prime\prime}}(x_{t}-\mu_{z^{+},t}\mu_{\Psi,t})\right)+\exp\left(-\sqrt{S^{\prime\prime}}(x_{t}-\mu_{z^{+},t}\mu_{\Psi,t})\right)-2\right] $$

$$ S^{\prime}(x_{t})=\frac{1}{2}\sqrt{S^{\prime\prime}}\left[\exp\left(\sqrt{S^{\prime\prime}}(x_{t}-\mu_{z^{+},t}\mu_{\Psi,t})\right)-\exp\left(-\sqrt{S^{\prime\prime}}(x_{t}-\mu_{z^{+},t}\mu_{\Psi,t})\right)\right] $$

$$ where\quad x_{t}=\mu_{z^{+},t}\mu_{\Psi,t}\frac{i_{t}}{i_{t-1}} $$

#### A.2.2 Firms

Real marginal cost:

$$ m c_{t}=\frac{\left(r_{t}^{k}\right)^{\alpha}w_{t}^{1-\alpha}}{\epsilon_{t}\alpha^{\alpha}(1-\alpha)^{1-\alpha}} $$

Cost minimization condition:

$$ \frac{\tilde{k}_{t-1}}{H_{t}}=\mu_{z^{+},t}\mu_{\Psi,t}\frac{\alpha}{1-\alpha}\frac{w_{t}}{r_{t}^{k}} $$

New Keynesian Phillips Curve (price):

$$ \frac{\Pi_{t}^{opt}}{\Pi_{t}}=\lambda\frac{K_{t}}{F_{t}} $$

$$ K_{t}=m c_{t}y_{t}+\beta\xi_{p}\mathrm{E}_{t}\Lambda_{t,t+1}\mu_{z^{+},t+1}\Pi_{t+1}^{1-\frac{\lambda}{1-\lambda}}\Pi_{\kappa,t+1}^{\frac{\lambda}{1-\lambda}}K_{t+1} $$

$$ F_{t}=y_{t}+\beta\xi_{p}\mathrm{E}_{t}\Lambda_{t,t+1}\mu_{z^{+},t+1}\Pi_{t+1}^{-\frac{\lambda}{1-\lambda}}\Pi_{\kappa,t+1}^{1+\frac{\lambda}{1-\lambda}}F_{t+1} $$

$$ \Pi_{t}=\left[\left(1-\xi_{p}\right)(\Pi_{t}^{o p t})^{\frac{1}{1-\lambda}}+\xi_{p}\Pi_{\kappa,t}^{\frac{1}{1-\lambda}}\right]^{1-\lambda} $$

$$ \Pi_{\kappa,t}=\Pi_{t-1}^{\kappa}\bar{\Pi}_{t}^{1-\kappa} $$

Production function for capital goods:

$$ \tilde{k}_{t-1}=\left(k_{t-1}^{g}\right)^{\alpha_{g}}\left(u_{t}k_{t-1}\right)^{1-\alpha_{g}} $$
Rental price of private capital goods:

$$ p_{t}^{k}=(1-\alpha_{g})\frac{\tilde{k}_{t-1}}{u_{t}k_{t-1}}r_{t}^{k} $$

#### A.2.3 Household

Edgeworth complementarities between private and government consumption:

$$ \tilde{c}_{t}=c_{t}+\nu_{g}g_{t}^{c} $$

Marginal utility of consumption:

$$ \varphi_{t}^{c}(1+\tau_{t}^{c})=\frac{\zeta_{t}^{c}}{\tilde{c}_{t}-\theta\tilde{c}_{t-1}\frac{1}{\mu_{z+,t}}} $$

Euler equation on consumption:

$$ \beta\mathrm{E}_{t}\Lambda_{t,t+1}=\frac{1}{R_{t}} $$

Stochastic discount factor:

$$ \mathrm{E}_{t}\Lambda_{t,t+1}=\mathrm{E}_{t}\frac{\varphi_{t+1}^{c}}{\Pi_{t+1}\mu_{z^{+},t+1}\varphi_{t}^{c}} $$

Capital accumulation equation:

$$ k_{t}=\frac{1}{\mu_{z^{+},t}\mu_{\Psi,t}}(1-\delta)k_{t-1}+\zeta_{t}^{i}\left[1-S\left(\mu_{z^{+},t}\mu_{\Psi,t}\frac{i_{t}}{i_{t-1}}\right)\right]i_{t} $$

FOC wrt. private investment:

$$ 1=\zeta_{t}^{i}\left[1-S\left(\mu_{z^{+},t}\mu_{\Psi,t}\frac{i_{t}}{i_{t-1}}\right)-S^{\prime}\left(\mu_{z^{+},t}\mu_{\Psi,t}\frac{i_{t}}{i_{t-1}}\right)\left(\mu_{z^{+},t}\mu_{\Psi,t}\frac{i_{t}}{i_{t-1}}\right)\right]q_{t} $$

$$ +\beta\mathrm{E}_{t}\Lambda_{t,t+1}\frac{\Pi_{t+1}}{\mu_{\Psi,t+1}}\zeta_{t+1}^{i}S^{\prime}\left(\mu_{z^{+},t+1}\mu_{\Psi,t+1}\frac{i_{t+1}}{i_{t}}\right)\left(\mu_{z^{+},t+1}\mu_{\Psi,t+1}\frac{i_{t+1}}{i_{t}}\right)^{2}q_{t+1} $$

FOC wrt. capital:

$$ q_{t}=\beta\mathrm{E}_{t}\Lambda_{t,t+1}\frac{\Pi_{t+1}}{\mu_{\Psi,t+1}}\left[(1-\delta)q_{t+1}+\left(1-\tau_{t+1}^{k}\right)\left(p_{t+1}^{k}u_{t+1}-a(u_{t+1})\right)\right] $$

FOC wrt. capital utilization rate:

$$ p_{t}^{k}=a^{\prime}(u_{t}) $$
New Keynesian Phillips curve (wage):

$$ w_{t}^{opt}=\left\{A_{H}\lambda_{w}\frac{K_{t}^{w}}{F_{t}^{w}}\right\}^{\frac{1-\lambda_{w}}{1-\lambda_{w}-\lambda_{w}\sigma_{h}}} $$

$$ K_{t}^{w}=\zeta_{t}^{h}\chi_{t}^{h}w_{t}^{-\frac{\lambda_{w}(1+\sigma_{h})}{1-\lambda_{w}}}H_{t}^{1+\sigma_{h}}+\beta\xi_{w}(\Pi_{\kappa,t+1}^{w})^{\frac{\lambda_{w}(1+\sigma_{h})}{1-\lambda_{w}}}(\mu_{z^{+},t+1}\Pi_{t+1})^{-\frac{\lambda_{w}(1+\sigma_{h})}{1-\lambda_{w}}}K_{t+1}^{w} $$

$$ F_{t}^{w}=\varphi_{t}^{c}(1-\tau_{t}^{h})w_{t}^{-\frac{\lambda_{w}}{1-\lambda_{w}}}H_{t}+\beta\xi_{w}(\Pi_{\kappa,t+1}^{w})^{\frac{1}{1-\lambda_{w}}}(\mu_{z^{+},t+1}\Pi_{t+1})^{-\frac{1}{1-\lambda_{w}}}F_{t+1}^{w} $$

$$ w_{t}=\left\{(1-\xi_{w})(w_{t}^{o p t})^{\frac{1}{1-\lambda_{w}}}+\xi_{w}\left(\frac{\Pi_{k,t}^{w}w_{t-1}}{\mu_{z^{+},t}\Pi_{t}}\right)^{\frac{1}{1-\lambda_{w}}}\right\}^{1-\lambda_{w}}. $$

$$ \Pi_{\kappa,t}^{w}=\left(\Pi_{t-1}^{w}\right)^{\kappa_{w}}\left(\mu_{z^{+},t}\bar{\Pi}_{t}\right)^{1-\kappa_{w}}. $$

Endogenous preference shifter:

$$ \chi_{t}^{h}=\frac{z_{\chi,t}}{\tilde{c}_{t}-\theta\tilde{c}_{t-1}\frac{1}{\mu_{z^{+},t}}} $$

$$ z_{\chi,t}=\left(\frac{z_{\chi,t-1}}{\mu_{z^{+},t}}\right)^{1-v}\left(\tilde{c}_{t}-\theta\tilde{c}_{t-1}\frac{1}{\mu_{z^{+},t}}\right)^{v} $$

Desirable labor supply:

$$ (1-\tau_{t}^{h})w_{t}=(1+\tau_{t}^{c})\left(\frac{\zeta_{t}^{h}}{\zeta_{t}^{c}}\right)z_{\chi,t}L_{t}^{\sigma_{h}} $$

Unemployment rate:

$$ U_{t}=\frac{L_{t}-H_{t}}{L_{t}} $$

#### A.2.4 Fiscal and Monetary Authorities

Public capital accumulation equation:

$$ k_{t}^{g}=\frac{1}{\mu_{z^{+},t}\mu_{\Psi,t}}(1-\delta_{g})k_{t-1}^{g}+\zeta_{t}^{g,i}g_{t}^{i} $$

Fiscal policy rule:

$$ \ln\frac{g_{t}^{c}}{g^{c}}=\rho_{g^{c}}\ln\frac{g_{t-1}^{c}}{g^{c}}-(1-\rho_{g^{c}})\phi_{g^{c}}\ln\frac{b_{t}}{b}+\epsilon_{t}^{g^{c}} $$

$$ \ln\frac{g_{t}^{i}}{g^{i}}=\rho_{g^{i}}\ln\frac{g_{t-1}^{i}}{g^{i}}-(1-\rho_{g^{i}})\phi_{g^{i}}\ln\frac{b_{t}}{b}+\epsilon_{t}^{g^{i}} $$

Monetary policy rule:

$$ \ln\frac{R_{t}}{R}=\rho_{R}\ln\frac{R_{t-1}}{R}+\left(1-\rho_{R}\right)\left[\ln\frac{\bar{\Pi}_{t}}{\bar{\Pi}}+\phi_{\Pi}\ln\frac{\Pi_{t}}{\bar{\Pi}_{t}}+\phi_{y}\ln\frac{y_{t}}{y}\right]+\epsilon_{t}^{R} $$

#### A.2.5 Aggregation and Resource Constraint

Aggregate supply:

$$ y_{t}=\epsilon_{t}\left(\frac{1}{\mu_{z^{+},t}\mu_{\Psi,t}}\right)^{\alpha}\tilde{k}_{t-1}^{\alpha}H_{t}^{1-\alpha}-\Theta $$