## Objective Function of Household

At the beginning at period t, household j owns two types of assets: bond $ B_{j,t-1} $ and capital stock $ K_{j,t-1} $ . Household j maximizes the following discounted present value of her utility, controlling consumption $ C_{j,t} $ , bond holdings $ B_{j,t} $ , investment $ I_{j,t} $ , capital stock $ K_{j,t} $ and capital utilization rate $ u_{j,t} $ .

$$ \max\quad\mathrm{E}_{t}\sum_{i=0}^{\infty}\beta^{i}\left(\zeta_{t+i}^{c}\ln\left(\tilde{C}_{j,t+i}-\theta\tilde{C}_{t+i-1}\right)-\zeta_{t+i}^{h}\chi_{t+i}^{h}A_{H}\frac{\left(H_{j,t+i}\right)^{1+\sigma_{h}}}{1+\sigma_{h}}+V(G_{t}^{c})\right), $$

$$ \mathrm{s.t.}\quad\left(1+\tau_{t}^{c}\right)P_{t}C_{j,t}+P_{t}^{i}I_{t}+B_{j,t}+T_{t} $$

$$ =\left(1-\tau_{t}^{h}\right)W_{j,t}H_{j,t}+\left(1-\tau_{t}^{k}\right)\left[\left(P_{t}^{k}u_{j,t}-P_{t}^{i}a(u_{j,t})\right)K_{j,t-1}+V_{t}+V_{t}^{k}\right]+R_{t-1}B_{j,t-1}, $$

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

where $ \tilde{C}_{j,t}=C_{j,t}+\nu_{g}G_{t}^{c} $ ,

$$ \chi_{t}^{h}\equiv\frac{Z_{\chi,t}}{\tilde{C}_{t}-\theta\tilde{C}_{t-1}},\quad Z_{\chi,t}=Z_{\chi,t-1}^{1-v}\left(\tilde{C}_{t}-\theta\tilde{C}_{t-1}\right)^{v},\quad\tilde{C}_{t}\equiv\int_{0}^{1}\tilde{C}_{j,t}d j, $$

$ \tilde{C}_{j,t} $ consists of private consumption $ C_{j,t} $ and government consumption $ G_{t}^{c} $ . If $ \nu_{g} $ is negative, an increase in government consumption raises marginal utility of private consumption at the present period. As a result, households respond with an increase in private consumption for an increase in government consumption: A negative $ \nu_{g} $ causes a cyclical comovement between private consumption and government consumption, which is the so-called Edgeworth complementarities (hereafter, EC). If $ \nu_{g} $ is positive, a counter-cyclical comovement is shown, which implies government consumption is substitutes for private consumption. If $ \nu_{g} $ is zero, government consumption is independent of private consumption.

In addition, $ V(G_{t}^{c}) $ is assumed to be satisfied by $ \partial V/\partial G_{t}^{c} > 0 $ to ensure that the marginal utility of households is positive when $ \nu_{g} $ is negative, following Karras (1994), Ganelli and Tervala (2009) and Iwata (2013).

Equation (A.17) is household j's budget constraint at period t. The household gains labor income, dividends $ V_{j,t} $ and $ V_{j,t}^{k} $ from intermediate good firms and capital-producing firms, and interest payments from bonds and capital holdings. On the other hand, the household expends its income as consumption, investment and bond holdings. $ P_{t}^{i} $ is investment goods price, $ W_{j,t} $ is nominal wage and $ R_{t} $ is nominal interest rate. $ \tau_{t}^{c} $ , $ \tau_{t}^{h} $ , $ \tau_{t}^{k} $ , $ T_{t} $ are consumption tax rate, labor income tax rate, corporate income tax rate and (nominal) lump-sum transfer, respectively. (A.18) is the capital accumulation equation. $ \delta $ is depreciation rate, S is investment adjustment cost where $ S(\mu_{z^{+}}\mu_{\Psi}) = S'(\mu_{z^{+}}\mu_{\Psi}) = 0 $ . $ \zeta_{t}^{i} $ stands for the investment specific technology shock. The rental price of effective private capital is $ P_{t}^{i}a(u(j,t)) $ . $ a(u_{j,t}) $ stands for capital utilization cost where $ a(\cdot) = 0 $ in a steady state and $ a'(\cdot) > 0 $ .

## FOCs of Households

Let Lagrange multipliers for the budget constraint be denoted by $ \Lambda_{j,t} $ and for the capital accumulation equation denoted by $ Q_{j,t} $ . FOCs with respect to $ C_{j,t} $ , $ B_{j,t} $ , $ I_{j,t} $ , $ K_{j,t} $ and $ u_{j,t} $