discount factor. The fixed cost, $ \Theta $ , is introduced to ensure that profits of the intermediate good firms are zero in a steady state. $ \kappa \in [0,1] $ represents the inflation indexation parameter.

Since the firms that can revise their prices set the same price among them, FOCs in scaled forms are derived as follows:

$$ \frac{\Pi_{t}^{opt}}{\Pi_{t}}\left(=\frac{P_{t}^{opt}}{P_{t}}\right)=\lambda\left(\frac{K_{t}^{p}}{F_{t}^{p}}\right) $$

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

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

$$ where\quad\Pi_{t}^{opt}\equiv\frac{P_{t}^{opt}}{P_{t-1}},\Pi_{t}\equiv\frac{P_{t}}{P_{t-1}},y_{t}\equiv\frac{Y_{t}}{z_{t}^{+}},\mu_{z^{+},t}\equiv\frac{z_{t}^{+}}{z_{t-1}^{+}}. $$

In a flexible price economy, i.e., $ \xi_{p}=0 $ , we can see the optimal flexible price equals to mark-up rate $ \lambda $ times real marginal cost. The price level of final goods and inflation rate can be derived as follows:

$$ P_{t}=\left[\int_{0}^{1}\left(P_{j,t}\right)^{\frac{1}{1-\lambda}}dj\right]^{1-\lambda}=\left[\left(1-\xi_{p}\right)\left(P_{t}^{opt}\right)^{\frac{1}{1-\lambda}}+\xi_{p}\left[\left(\Pi_{\kappa,t}\right)P_{t-1}\right]^{\frac{1}{1-\lambda}}\right]^{1-\lambda}. $$

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

The new Keynesian Phillips curve consists of equations (A.5)-(A.8).

## Capital-Producing Firms

There are two types of capital in this economy. One is the effective private capital $ u_{t}K_{t-1} $ where $ u_{t} $ is capital utilization rate. The other is the public capital $ K_{t-1}^{g} $ accumulated by government. Capital-producing firms produce a homogenous capital good $ \tilde{K}_{t-1} $ , bundling the effective private capital with public capital. Given the rental price of effective private capital, $ P_{t}^{k} $ , the profit maximization problem of capital producing firm can be expressed as:

$$ \max_{u_{t}K_{t-1}}R_{t}^{k}\tilde{K}_{t-1}-P_{t}^{k}u_{t}K_{t-1}-z_{t}^{+}\Psi_{t}\Theta^{k} $$

$$ \mathrm{s.t.}\quad\tilde{K}_{t-1}=\left(K_{t-1}^{g}\right)^{\alpha_{g}}\left(u_{t}K_{t-1}\right)^{1-\alpha_{g}} $$

$ \Theta^{k} $ stands for the fixed cost and (A.10) is the production function of capital where $ \alpha_{g} $ is the marginal productivity of public capital for the bundled capital, $ \tilde{K}_{t-1} $ . It should be noted that the productivity of the public capital for output can be expressed as $ \alpha \times \alpha_{g} $ from (A.1) and (A.10). If $ \alpha_{g} $ is positive, then the public capital accumulation by government operates positive externalities to capital-producing firms in the form of increasing their productivities exogenously. From the production function of output (A.1), the public capital accumulation indirectly increases the productivity for the intermediate goods firms. Therefore, if $ \alpha_{g} > 0 $ , we call $ k_{t}^{g} $ productive public capital.

The FOC in scaled form is derived as:

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

where $ p_{t}^{k}=\frac{\Psi_{t}P_{t}^{k}}{P_{t}} $