to the following equations:

$$ \left(r^{k}\right)^{(j+1)}\leftarrow\frac{1}{1-\alpha_{g}}\left[\left(\frac{\tilde{k}}{H}\right)^{(j)}\frac{H^{(j)}}{\left(k^{g}\right)^{(j)}}\right]^{\frac{\alpha_{g}}{1-\alpha_{g}}}p^{k} $$

$$ \left(g^{c}\right)^{\left(j+1\right)}\leftarrow\left(\frac{g^{c}}{y}\right)y^{\left(j\right)} $$

$$ \left(g^{i}\right)^{(j+1)}\leftarrow\left(\frac{g^{i}}{y}\right)y^{(j)} $$

$$ H^{(j+1)}\leftarrow L^{(j)}(1-U) $$

The first equation is derived from (A.59), (A.66) and (A.65). The second and third equations are derived by definitions. Note that $ (g^{c}/y) $ and $ (g^{i}/y) $ are given as the sample means. The fourth equation comes from (A.83). Note that U is given as the sample mean.

Given $ \left(r^{k}\right)^{(j)} $ , $ \left(g^{c}\right)^{(j)} $ , $ \left(g^{i}\right)^{(j)} $ and $ H^{(j)} $ , we have explicit solutions through (A.111)-(A.119). However, we cannot derive explicit solutions for $ r^{k} $ , $ g^{c} $ , $ g^{i} $ and H, since we face highly nonlinear simultaneous equations described as (A.120)-(A.123). Therefore, we have to rely on the following numerical calculation method:

1. Guess the initial values of $ (r^{k})^{(1)} $ , $ (g^{c})^{(1)} $ , $ (g^{i})^{(1)} $ and $ H^{(1)} $ and set j=0.

2. Calculate (A.111)-(A.119).

3. Derive $ \left(r^{k}\right)^{(j+1)} $ , $ \left(g^{c}\right)^{(j+1)} $ , $ \left(g^{i}\right)^{(j+1)} $ and $ H^{(j+1)} $ from (A.120)-(A.123).

4. Calculate $ \left|\Phi^{(j+1)}-\Phi^{(j)}\right|<\varepsilon $ where $ \Phi\in\left\{r^{k},g^{c},g^{i},H\right\} $ and $ \varepsilon>0 $ indicates a convergence criterion.

5. If the criteria are not satisfied, then set $ j = j + 1 $ and go back to the second step. Otherwise, we have numerical solutions on steady states of $ \Phi $ and endogenous varialbes from (A.111) to (A.119).

#### A.3.3 Auxiliary Variables and Endogenously Determined Parameters

Steady states on the rest of endogenous variables can be derived as follows. It should note that several parameters are determined endogenously.

From (A.68), $ \varphi^{c}=\left[(1+\tau^{c})\left(1-\frac{\theta}{\mu_{z^{+}}}\right)\tilde{c}\right]^{-1} $ . From (A.80) and (A.81), $ \chi^{h}=\mu_{z^{+}}^{-(1-v)/v} $ . By definition, we have the steady state value of bond from $ (b/y)\times y $ where debt to GDP ratio, $ (b/y) $ , is given as the sample mean. Then, the lump-sum transfer is determined from government budget constraint:

$$ \tau=g^{c}+g^{i}+\left(\frac{R}{\mu_{z^{+}}\bar{\Pi}}-1\right)b-\left(\tau^{c}c+\tau^{h}w H+\tau^{k}p^{k}\frac{k}{\mu_{z^{+}}\mu_{\Psi}}\right) $$

From (A.75)-(A.79), $\Pi^{w} = \Pi_{\kappa}^{w} = \mu_{z^{+}} \bar{\Pi}$ and $A_{H} = [w \varphi^{c}(1 - \tau^{h})]/[\lambda_{w} \chi^{h} H^{\sigma_{h}}]$. Thus, the scaling parameter, $A_{H}$, is endogenously determined. In our specification on the utilization cost function, $a'(1) = \sigma_{b}$. Then, (A.74) requires $\sigma_{b} = p^{k}$ to be satisfied. Since $p^{k}$ is given by (A.110), the private capital adjustment cost parameter, $\sigma_{b}$, is also endogenously determined. Finally, fixed costs on goods and capital production are also determined endogenously as follows: $\Theta = (\lambda - 1)y$ and $\Theta^{k} = \alpha_{g} \frac{r^{k} \tilde{k}}{\mu_{z^{+}} \mu_{\Psi^{k}}}$ from (A.91) and (A.92).