Silviase/jgov_v1.4_full_ann huggingface dataset:
## 2 The Model

The model is an extended variant of the medium-scale estimated DSGE model developed by Smets and Wouters (2003). The Smets and Wouters (2003) model features various real and nominal rigidities: habit formation, investment adjustment cost, variable capital utilization, sticky price and wage à la Calvo (1983), and indexation in prices and wages. Coenen and Straub (2005) also introduce non-Ricardian households and distortionary taxes to the Smets and Wouters (2003) model, but they model distortionary taxes in a time-invariant manner and the model is estimated without using fiscal data. Therefore their model essentially captures the dynamics of the hypothetical lump-sum tax only. Below I introduce three distortionary taxes (consumption, labor and capital income taxes) and feedback policy rules for each, in addition to the non-Ricardian households. I also introduce a feedback rule for a government spending. I believe that these settings contribute to the proper identification of the fiscal authority's behavior with the help of fiscal data in estimation.

### 2.1 Households

There is a continuum of households indexed by $ n \in [0,1] $ . A fraction $ 1 - \omega $ of this continuum of households indexed by $ i \in [0,1 - \omega) $ has access to financial markets and acts $ Ricardian,\ i.e., $ the households maximize their lifetime utilities by choosing consumption, investment, financial assets in the form of government bonds, capital stock, and utilization rate of capital stock. The remaining households, indexed by $ j \in [1 - \omega,1] $ do not have access to financial markets and simply consume all of their current disposable income. The latter households are dubbed non-Ricardian. Modeling these two extremely different households in their time horizon is rather ad-hoc but it is considered parsimonious, since the model has been proven to provide a better description of aggregate consumption behavior (See Mankiw (2000) and Galí et al. (2007)). Recall that the international evidence reported in the original work by Campbell and Mankiw (1989) includes that from Japan.

#### 2.1.1 Ricardian households

Each member of Ricardian households i maximizes its lifetime utility by choosing consumption $ C_{t}^{R}(i) $ , investment $ I_{t}(i) $ , government bonds $ B_{t}(i) $ , next period's capital stock $ K_{t}(i) $ , and intensity of the capital stock utilization $ z_{t}(i) $ , given the following lifetime utility function that is additively separable between consumption and labor:
$$ E_{t}\sum_{t=0}^{\infty}\beta^{t}\varepsilon_{t}^{b}\left(\frac{1}{1-\sigma_{c}}\left(C_{t}^{R}(i)-h C_{t-1}^{R}\right)^{1-\sigma_{c}}-\frac{\varepsilon_{t}^{l}}{1+\sigma_{l}}L_{t}^{R}(i)^{1+\sigma_{l}}\right), $$

where, $ \beta $ is the discount factor, $ \sigma_{c} $ denotes the inverse of the intertemporal elasticity of substitution and $ \sigma_{l} $ is the inverse of the elasticity of work effort with respect to real wages. $ L_{t}^{R}(i) $ represents the labor supply of the Ricardian household i. h measures the degree of habit formation in consumption. $ C_{t-1}^{R} $ is lagged aggregate per capita Ricardian consumption. Two serially correlated shocks, a preference shock $ \varepsilon_{t}^{b} $ and a labor supply shock $ \varepsilon_{t}^{l} $ are considered and are assumed to follow a first-order autoregressive process with an i.i.d.-normal error term: $ \varepsilon_{t}^{b} = \rho_{b} \varepsilon_{t-1}^{b} + \eta_{t}^{b} $ and $ \varepsilon_{t}^{l} = \rho_{l} \varepsilon_{t-1}^{l} + \eta_{t}^{l} $ .

The Ricardian household faces a flow budget constraint (expressed in real terms):

$$ \begin{aligned}&(1+\tau_{t}^{c})C_{t}^{R}(i)+I_{t}(i)+\Psi(z_{t}(i))K_{t-1}(i)+\frac{B_{t}(i)}{R_{t}P_{t}}\\&=(1-\tau_{t}^{d})w_{t}(i)L_{t}^{R}(i)+(1-\tau_{t}^{k})r_{t}^{k}z_{t}(i)K_{t-1}(i)+(1-\tau_{t}^{k})\frac{D_{t}(i)}{P_{t}}+\frac{B_{t-1}(i)}{P_{t}},\\ \end{aligned} $$

where $ \Psi(z_{t}(i)) $ is the cost associated with variations in the degree of capital utilization $ z_{t}(i) $ . $ \tau_{t}^{c} $ , $ \tau_{t}^{d} $ , $ \tau_{t}^{k} $ denote consumption, labor and capital income tax rates, respectively. $ D_{t}(i) $ denotes dividends distributed by firms to the Ricardian household i. $ P_{t} $ is aggregate price level, $ R_{t} $ is riskless return on government bonds, $ w_{t}(i) $ is real wage income, and $ r_{t}^{k} $ is real rental rate of capital. Notice that capital stock and government bonds of the current period are denoted here as $ K_{t-1}(i) $ and $ B_{t-1}(i) $ meaning that their decisions are made at time t-1. For simplicity, I assume that a consumption tax is levied on private consumption expenditure alone, following Coenen and Straub (2005). A lump-sum tax or transfer is omitted since my interest focuses on examining the role of distortionary tax rule combination in fiscal policy effectiveness.

The physical capital accumulation law for the Ricardian household is expressed as:

$$ K_{t}(i)=(1-\delta)K_{t-1}(i)+\left[1-S\left(\frac{\varepsilon_{t}^{i}I_{t}(i)}{I_{t-1}(i)}\right)\right]I_{t}(i), $$

where $ \delta $ is the depreciation rate, $ S(\cdot) $ represents the adjustment cost function in investment. $ \varepsilon_{t}^{i} $ is a shock to investment cost function and is assumed to follow a first-order autoregressive process with an i.i.d.-normal error term: $ \varepsilon_{t}^{i} = \rho_{i} \varepsilon_{t-1}^{i} + \eta_{t}^{i} $ . Following Smets and Wouters (2003), I assume the capital utilization rate in steady state as $ \bar{z} = 1 $ , and the corresponding cost as $ \Psi(\bar{z}) = 0 $ . Moreover, the investment adjustment cost function is assumed to satisfy $ S(1) = S'(1) = 0 $ .
Letting $ \Lambda_{t} $ and $ \Lambda_{t}Q_{t} $ denote the Lagrange multipliers, the first-order conditions with respect to $ C_{t}^{R}(i) $ , $ B_{t}(i) $ , $ I_{t}(i) $ , $ K_{t}(i) $ , and $ z_{t}(i) $ are expressed as follows:

$$ (1+\tau_{t}^{c})\Lambda_{t}=\varepsilon_{t}^{b}\left(C_{t}^{R}(i)-h C_{t-1}^{R}\right)^{-\sigma_{c}}, $$

$$ \beta R_{t}E_{t}\left[\frac{\Lambda_{t+1}}{\Lambda_{t}}\frac{P_{t}}{P_{t+1}}\right]=1, $$

$$ \begin{align*}&Q_{t}\left[1-S\left(\frac{\varepsilon_{t}^{i}I_{t}(i)}{I_{t-1}(i)}\right)\right]-Q_{t}S^{^{\prime}}\left(\frac{\varepsilon_{t}^{i}I_{t}(i)}{I_{t-1}(i)}\right)\frac{\varepsilon_{t}^{i}}{I_{t-1}(i)}I_{t}(i)\\\end{align*}\begin{align*}=&-\beta E_{t}\left[\frac{\Lambda_{t+1}}{\Lambda_{t}}Q_{t+1}S^{^{\prime}}\left(\frac{\varepsilon_{t+1}^{i}I_{t+1}(i)}{I_{t}(i)}\right)\frac{\varepsilon_{t+1}^{i}I_{t+1}(i)}{I_{t}(i)^{2}}I_{t+1}(i)\right]+1,\end{align*} $$

$$ Q_{t}=\beta E_{t}\left[\frac{\Lambda_{t+1}}{\Lambda_{t}}\left((1-\delta)Q_{t+1}+(1-\tau_{t+1}^{k})r_{t+1}^{k}z_{t+1}(i)-\Psi(z_{t+1}(i))\right)\right]+\eta_{t}^{q}, $$

$$ \left(1-\tau_{t}^{k}\right)r_{t}^{k}=\Psi^{^{\prime}}(z_{t}(i)). $$

Here, $ Q_{t} $ represents the shadow price of additional unit of capital. Letting an over-bar denote a steady-state value, it can be shown that $ 1/\beta = \bar{R} = 1 - \delta + (1 - \bar{\tau}^{k})\bar{r}^{k} + \delta\bar{\tau}^{k} $ and $ \bar{Q} = 1 $ .

#### 2.1.2 Non-Ricardian households

Non-Ricardian households are modeled as non-optimizing agents following the original assumption in Campbell and Mankiw (1989) and Galí et al. (2007). Since the members of non-Ricardian households j do not have access to financial markets, they simply consume all of their after-tax disposable income. Denoting consumption and labor input of non-Ricardian households as $ C_{t}^{NR}(j) $ and $ L_{t}^{NR}(j) $ , the period-by-period budget constraint they face is given by (expressed in real terms):

$$ (1+\tau_{t}^{c})C_{t}^{N R}(j)=(1-\tau_{t}^{d})w_{t}(j)L_{t}^{N R}(j). $$

#### 2.1.3 Wage setting

Following Erceg et al. (2006), FMS, and essentially Coenen and Straub (2005), I assume that the members of Ricardian households act as wage setters for their differentiated labor services $ L_{t}^{R}(i) $ in monopolistically
competitive markets. The nominal wages for differentiated labor services $ W_{t}^{R}(i) $ are determined by staggered contracts à la Calvo (1983). On the other hand, the members of non-Ricardian households are assumed to set their wages $ W_{t}^{NR}(j) $ for their differentiated labor services $ L_{t}^{NR}(j) $ to be equal to the average wage of Ricardian households. Because all households face the same labor demand schedule, both wages and labor hours will be equal for every household, i.e., $ W_{t}^{R}(i) = W_{t}^{NR}(j) = W_{t}(n) $ and $ L_{t}^{R}(i) = L_{t}^{NR}(j) = L_{t}(n) $ . Notice that this assumption implies that labor hour response to a government spending shock in this economy is the same as those in an economy in which all households are Ricardian.

An independent and perfectly competitive employment agency bundles differentiated labor $ L_{t}(n) $ into a single type of effective labor input $ L_{t} $ using the following technology:

$$ L_{t}=\left[\int_{0}^{1}L_{t}(n)^{\frac{1}{1+\lambda_{w,t}}}d n\right]^{1+\lambda_{w,t}}, $$

where an i.i.d.-normal shock $ \eta_{t}^{w} $ is assumed for the wage markup $ \lambda_{w,t} = \lambda_{w} + \eta_{t}^{w} $ . The employment agency solves:

$$ \max_{L_{t}(n)}W_{t}\left[\int_{0}^{1}L_{t}(n)^{\frac{1}{1+\lambda_{w,t}}}d n\right]^{1+\lambda_{w,t}}-\int_{0}^{1}W_{t}(n)L_{t}(n)d n, $$

where $ W_{t} \equiv w_{t} P_{t} $ is aggregate nominal wage index. The labor demand schedule for each differentiated labor service is then expressed as:

$$ L_{t}(n)=\left(\frac{W_{t}(n)}{W_{t}}\right)^{-\frac{1+\lambda_{w,t}}{\lambda_{w,t}}}L_{t}. $$

Putting the labor demand to the bundler technology of the employment agency gives:

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

With probability $ 1 - \xi_{w} $ , each Ricardian household i is assumed to be allowed to reset its wage optimally, unless otherwise it adjusts its wage partially according to the following indexation scheme:

$$ W_{t}^{R}(i)=\left(\frac{P_{t-1}}{P_{t-2}}\right)^{\gamma_{w}}W_{t-1}^{R}(i), $$

where $ \gamma_{w} $ measures the degree of indexation. The Ricardian household i, which is allowed to optimally reset its wage, is assumed to maximize its lifetime utility taking aggregate nominal wage $ W_{t} $ and effective
labor $ L_{t} $ as given. Since the household knows the probability $ \xi_{w}^{s} $ that the wage it chooses in this period will still be in effect s periods in the future, the optimal wage $ W_{t}^{R*}(i) $ is given by:

$$ W_{t}^{R*}(i)\equiv\arg\max_{W^{*}(i)}E_{t}\sum_{s=0}^{\infty}(\beta\xi_{w})^{s}\left\lceil\frac{1}{1-\sigma_{c}}\left(C_{t+s}^{R}(i)-h C_{t+s-1}^{R}\right)^{1-\sigma_{c}}-\frac{\varepsilon_{t}^{l}}{1+\sigma_{l}}\left(\left(\frac{W_{t}^{R*}(i)}{W_{t+s}}\right)^{-\frac{1+\lambda_{w,t+s}}{\lambda_{w,t+s}}}L_{t+s}\right)^{1+\sigma_{l}}\right\rceil, $$

subject to

$$ (1+\tau_{t+s}^{c})C_{t+s}^{R}(i)+I_{t+s}(i)+\Psi(z_{t+s}(i))K_{t+s-1}(i)+\frac{B_{t+s}(i)}{R_{t+s}P_{t+s}} $$

$$ =(1-\tau_{t+s}^{d})\frac{W_{t}^{R*}(i)}{P_{t+s}}\left(\frac{W_{t}^{R*}(i)}{W_{t+s}}\right)^{-\frac{1+\lambda_{w,t+s}}{\lambda_{w,t+s}}}L_{t+s}+(1-\tau_{t+s}^{k})r_{t+s}^{k}z_{t+s}(i)K_{t+s-1}(i)+(1-\tau_{t+s}^{k})\frac{D_{t+s}(i)}{P_{t+s}}+\frac{B_{t+s-1}(i)}{P_{t+s}}. $$

Since we know that $ W_{t}^{R}(i) = W_{t}^{NR}(j) = W_{t}(n) $ , aggregate nominal wage law of motion is then expressed as:

$$ W_{t}=\left[\left(1-\xi_{w}\right)\left(W_{t}^{*}(n)\right)^{-\frac{1}{\lambda_{w,t}}}+\xi_{w}\left(\left(\frac{P_{t-1}}{P_{t-2}}\right)^{\gamma_{w}}W_{t-1}(n)\right)^{-\frac{1}{\lambda_{w,t}}}\right]^{-\lambda_{w,t}}, $$

where Wt*(n) = WtR*(i).

### 2.2 Firms

There are two types of firms: perfectly competitive final-good firms and monopolistically competitive intermediate-good firms indexed by $ f \in [0,1] $ . The final-good firm produces the good $ Y_{t} $ combining the differentiated intermediate goods $ y_{t}(f) $ produced by the firm f.

#### 2.2.1 Final-good firms

The final-good producing firm combines intermediate goods using the following bundler technology:

$$ Y_{t}=\left[\int_{0}^{1}y_{t}(f)^{\frac{1}{1+\lambda_{p,t}}}d f\right]^{1+\lambda_{p,t}}, $$

where an i.i.d.-normal shock $ \eta_{t}^{p} $ is assumed for the price markup $ \lambda_{p,t} = \lambda_{p} + \eta_{t}^{p} $ . The final-good firm solves:

$$ \max_{y(f)}P_{t}\left[\int_{0}^{1}y_{t}(f)^{\frac{1}{1+\lambda_{p,t}}}df\right]^{1+\lambda_{p,t}}-\int_{0}^{1}p_{t}(f)y_{t}(f)df, $$
where $ p_{t}(f) $ is the price of the intermediate good $ y_{t}(f) $ . Then, the demand function for the intermediate good is expressed as:

$$ y_{t}(f)=\left(\frac{p_{t}(f)}{P_{t}}\right)^{-\frac{1+\lambda_{p,t}}{\lambda_{p,t}}}Y_{t}. $$

Putting this demand to the bundler technology of the final-good firm gives a pricing rule:

$$ P_{t}=\left[\int_{0}^{1}p_{t}(f)^{-\frac{1}{\lambda_{p,t}}}d f\right]^{-\lambda_{p,t}}. $$

#### 2.2.2 Intermediate-good firms

Each intermediate-good firm f produces its differentiated output using an increasing-returns-to-scale Cobb-Douglas technology:

$$ y_{t}(f)=\varepsilon_{t}^{a}\tilde{k}_{t-1}(f)^{\alpha}l_{t}(f)^{1-\alpha}-\Phi, $$

where $ \tilde{k}_{t-1}(f) $ is the effective capital stock at time t given by $ \tilde{k}_{t-1}(f)=z_{t}k_{t-1}(f) $ . $ l_{t}(f) $ is the effective labor input bundled by the employment agency, and $ \Phi $ represents a fixed cost. $ \varepsilon_{t}^{a} $ is a technology shock assumed to follow a first-order autoregressive process with an i.i.d.-normal error term: $ \varepsilon_{t}^{a}=\rho_{a}\varepsilon_{t-1}^{a}+\eta_{t}^{a} $ .

Taking the real rental cost of capital $ r_{t}^{k} $ and aggregate real wage $ w_{t} $ as given, cost minimization subject to the production technology yields marginal cost and labor demand:

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

$$ \frac{w_{t}}{r_{t}^{k}}=\frac{1-\alpha}{\alpha}\frac{z_{t}k_{t-1}(f)}{l_{t}(f)}. $$

The latter immediately gives the labor demand function at the aggregate level:

$$ L_{t}=\frac{1-\alpha}{\alpha}\frac{r_{t}^{k}}{w_{t}}z_{t}K_{t-1}. $$

Nominal profits $ d_{t}(f) $ for the intermediate-good firm are given by:

$$ d_{t}(f)=p_{t}(f)y_{t}(f)-P_{t}m c_{t}(y_{t}(f)+\Phi), $$

which are distributed to Ricardian households as dividends. Aggregation gives:
$$ D_{t}=P_{t}Y_{t}-P_{t}m c_{t}(Y_{t}+\Phi). $$

#### 2.2.3 Price setting

As in the case of wage setting, I assume sluggish price adjustment due to the staggered price contracts à la Calvo (1983). A fraction $ 1-\xi_{p} $ of intermediate-good firms can re-optimize their prices, unless otherwise they follow the price indexation scheme:

$$ p_{t}(f)=\left(\frac{P_{t-1}}{P_{t-2}}\right)^{\gamma_{p}}p_{t-1}(f), $$

where $ \gamma_{p} $ measures the degree of indexation.

An intermediate-good firm f, which is allowed to re-optimize, knows the probability $ \xi_{w}^{s} $ that the price it chooses in this period will still be in effect s periods in the future. Taking aggregate nominal price index $ P_{t} $ and output $ Y_{t} $ as given, the optimal price $ p_{t}^{*}(f) $ is chosen as:

$$ p_{t}^{*}(f)\equiv\arg\max_{p^{*}(f)}E_{t}\sum_{s=0}^{\infty}(\beta\xi_{p})^{s}\left[(p_{t}^{*}(f)-P_{t+s}mc_{t+s})\left(\frac{p_{t}^{*}(f)}{P_{t+s}}\right)^{-\frac{1+\lambda_{p,t+s}}{\lambda_{p,t+s}}}Y_{t+s}-P_{t+s}mc_{t+s}\Phi\right]. $$

Aggregate price law of motion is then expressed as:

$$ P_{t}=\left[\left(1-\xi_{p}\right)\left(p_{t}^{*}(f)\right)^{-\frac{1}{\lambda_{p,t}}}+\xi_{p}\left(\left(\frac{P_{t-1}}{P_{t-2}}\right)^{\gamma_{p}}p_{t-1}(f)\right)^{-\frac{1}{\lambda_{p,t}}}\right]^{-\lambda_{p,t}}. $$

### 2.3 Fiscal and Monetary Authorities

#### 2.3.1 Fiscal policy

The fiscal authority purchases final goods $ G_{t} $ , issues bonds $ B_{t} $ , and levies a consumption tax, a labor income tax, and a capital income tax at rates $ \tau_{t}^{c} $ , $ \tau_{t}^{d} $ , and $ \tau_{t}^{k} $ , respectively. The real flow budget constraint for the fiscal authority is expressed as follows:

$$ G_{t}+\frac{B_{t-1}}{P_{t}}=\tau_{t}^{c}C_{t}+\tau_{t}^{d}w_{t}L_{t}+\tau_{t}^{k}r_{t}^{k}z_{t}K_{t-1}+\tau_{t}^{k}\frac{D_{t}}{P_{t}}+\frac{1}{R_{t}}\frac{B_{t}}{P_{t}}. $$

Notice that the budget constraint itself is a fiscal rule. Since we have five fiscal policy instruments $ \tau_{t}^{c} $ , $ \tau_{t}^{d} $ , $ \tau_{t}^{k} $ , $ G_{t} $ , and $ B_{t} $ , we need as many as four fiscal rules other than the budget constraint $ ^{10} $ . Although $ {}^{10} $ If we consider only one type of tax, fiscal rules are to be formulated for the tax and government spending other than the budget constraint. See, for example, Bilbiie and Straub (2004) and Galí et al. (2007). Although Coenen and Straub
little consensus exists in the literature on the formulation of fiscal rules, the rules in practice are typically designed to guarantee fiscal solvency in the model. Imposing feedback from government debt to taxes is quite popular in medium-scale DSGE models, especially in those at policy institutions such as SIGMA, NAWM, and QUEST III. Here, I consider three feedback rules for each tax and a government spending rule in log-linearized form following FMS. The tax rates $ ^{11} $ are supposed to positively respond to a debt-to-output ratio $ ^{12} $ :

$$ \hat{\tau}_{t}^{c}=\rho_{t c}\hat{\tau}_{t-1}^{c}+\left(1-\rho_{t c}\right)\phi_{t c b}(\hat{b}_{t-1}-\hat{Y}_{t-1})+\eta_{t}^{t c}, $$

$$ \hat{\tau}_{t}^{d}=\rho_{t d}\hat{\tau}_{t-1}^{d}+\left(1-\rho_{t d}\right)\phi_{t d b}(\hat{b}_{t-1}-\hat{Y}_{t-1})+\eta_{t}^{t d}, $$

$$ \hat{\tau}_{t}^{k}=\rho_{t k}\hat{\tau}_{t-1}^{k}+\left(1-\rho_{t k}\right)\phi_{t k b}(\hat{b}_{t-1}-\hat{Y}_{t-1})+\eta_{t}^{t k}, $$

where the hats above variables denote log-deviations from steady state. $ b_{t} \equiv B_{t}/P_{t} $ denotes government bonds in real terms. $ \eta_{t}^{g}, \eta_{t}^{tc}, \eta_{t}^{td} $ , and $ \eta_{t}^{tk} $ are i.i.d.-normal errors. It should be noted that the fiscal policy rules described here allow partial debt finance, while the debt is to be repaid through tax revenue over time. The speed of repayment is determined by a combination of the coefficients on the debt-to-output ratio of the tax rules, namely by the set of parameters $ \rho_{tc}, \phi_{tcb}, \rho_{td}, \phi_{tdb}, \rho_{tk} $ , and $ \phi_{tkb} $ .

Government spending is assumed to follow a feedback rule that responds to output gap in log-linearized form:

$$ \hat{G}_{t}=\rho_{g}\hat{G}_{t-1}+\left(1-\rho_{g}\right)\phi_{g y}\hat{Y}_{t-1}+\eta_{t}^{g}. $$

(2005) introduce multiple distortionary taxes to the Smets and Wouters (2003) model, the taxes are assumed to be fixed over time. Therefore fiscal policy instruments in their model are essentially three: government spending, lump-sum tax, and debt. Accordingly, their tax policy rule is formulated only for the lump-sum tax, following Galí et al. (2007).

$ ^{11} $ The aggregate tax rates faced by the representative agent in macroeconomics can be appropriately approximated by the aggregate effective tax rates, which are computed using macroeconomic data, such as national accounts and revenue statistics. The method for computing the effective tax rates was originally proposed by Lucas (1990), and was successfully extended by Mendoza et al. (1994).

$ ^{12} $ Although no formal budget rule has been legislated in Japan, fiscal consolidation has been a major priority for the government. The government is hence assumed to follow “passive fiscal policy” in the sense of Leeper (1991). Leeper and Yang (2008) also employ similar tax rules in log-linearized form for three different taxes (labor and capital income taxes, and a lump-sum tax) with positive coefficients on debt-to-output ratio.
#### 2.3.2 Monetary policy

The monetary authority sets nominal interest rates according to a simple feedback rule in log-linearized form:

$$ \hat{R}_{t}=\rho_{r}\hat{R}_{t-1}+\left(1-\rho_{r}\right)\phi_{r\pi}\hat{\pi}_{t-1}+\left(1-\rho_{r}\right)\phi_{r y}\hat{Y}_{t}+\eta_{t}^{R}, $$

where $ \pi_{t-1} \equiv \log(P_{t-1}/P_{t-2}) $ denotes inflation rate. An i.i.d.-normal shock $ \eta_{t}^{R} $ to the interest rate is assumed.

### 2.4 Aggregation and Market Clearing

Aggregate consumption $ C_{t} $ and labor hour $ L_{t} $ in per-capita term are given by a weighted average of the corresponding variables for each consumer type:

$$ C_{t}=(1-\omega)C_{t}^{R}(i)+\omega C_{t}^{N R}(j), $$

$$ L_{t}=(1-\omega)L_{t}^{R}(i)+\omega L^{N R}(j), $$

and again, since all households supply the same amount of labor by assumption, aggregate labor hour is given by:

$$ L_{t}=L_{t}^{R}(i)=L_{t}^{N R}(j). $$

Because only Ricardian households have access to financial markets, aggregate government bonds $ B_{t} $ , investment $ I_{t} $ , physical capital $ K_{t} $ , and dividends $ D_{t} $ distributed by firms are expressed as:

$$ B_{t}=(1-\omega)B_{t}^{R}(i), $$

$$ I_{t}=(1-\omega)I_{t}^{R}(i), $$

$$ K_{t}=(1-\omega)K_{t}^{R}(i), $$
$$ D_{t}=(1-\omega)D_{t}^{R}(i). $$

Finally, aggregate production equation and the final-goods market equilibrium condition are given by:

$$ Y_{t}=\varepsilon_{t}^{a}z_{t}K_{t-1}{}^{\alpha}L_{t}{}^{1-\alpha}-\Phi, $$

$$ Y_{t}=C_{t}+I_{t}+G_{t}+\Psi\left(z_{t}\right)K_{t-1}. $$

Notice that equation (22) can be derived from the budget constraints of Ricardian households (1), non-Ricardian households (8), and the government (14). It follows that one of them is redundant.

## 3 Bayesian Estimation of the Model

In estimating the model parameters, I first log-linearize the model around the deterministic steady state and conduct Bayesian inference using a Markov Chain Monte Carlo (MCMC) method, which is now a standard technique in estimating DSGE models. The log-linearized form of the model is reported in Appendix B.

### 3.1 Preliminary Setting

The model parameters are estimated on the Japanese data for the period starting from 1980:Q1 to 1998:Q4. The end period is determined following Iiboshi et al. (2008) not to include the period during which the zero-interest-rate policy was adopted. In estimating the model, I use fiscal data on government spending (calculated as a sum of government consumption and investment) and aggregate effective tax rates on consumption and labor income, in addition to the ordinary seven series in the literature, i.e., output, private consumption, investment, labor hour, wage, inflation rate, and interest rate. The aggregate effective tax rates are calculated following the method of Mendoza et al. (1994) as FMS. In contrast with FMS, only quarterly data are utilized for the tax rate calculation, which is available from the National Accounts of Japan from 1980:Q1. The series of effective tax rates on capital income is also calculated. However, the obtained data series is too volatile and hence is discarded in the estimation. Appendix A gives the details of the calculation.

All the variables are detrended using the Hodrick-Prescott filter. Since the model is an extended variant of Smets and Wouters (2003) and accordingly of its application to the Japanese economy of Iiboshi