### 3.1 Equilibrium conditions from the housing/investment sector

The Euler equation for consumption is as follows:

$$ \hat{c}_{t}=\frac{h}{1+h}\hat{c}_{t-1}+\frac{1}{1+h}E_{t}\hat{c}_{t+1}-\frac{1-h}{(1+h)\sigma_{c}}(\hat{R}_{t}-E_{t}\hat{\pi}_{t+1})+\frac{1-h}{(1+h)\sigma_{c}}(u_{t}^{c}-E_{t}u_{t+1}^{c}), $$

where an additional assumption is that the external habit stock $ H_{t} $ is proportional to aggregate past consumption: $ H_{t} = h c_{t-1} $ , is introduced where h denotes the habit persistence parameter. When h = 0, equation (3.1) reduces to the traditional forward-looking consumption equation. With external habit formation, consumption depends on a weighted average of past and expected future consumption. Note that in this case the interest elasticity of consumption depends not only on the intertemporal elasticity of substitution ( $ \sigma_{c} $ ), but also on the habit persistence parameter. A high degree of habit persistence will tend to reduce the impact of the real rate on consumption for a given elasticity of substitution. A persistent shock $ u_{t}^{c} $ is an AR(1) process with the coefficient $ \rho^{c} $ and therefore the expected value $ E_{t}u_{t+1}^{c} $ can be rewritten as $ \rho^{c}u_{t}^{c} $ .

The Euler equation for investment is as follows:

$$ \widehat{i n v}_{t}=\frac{1}{1+\beta}\widehat{i n v}_{t-1}+\frac{\beta}{1+\beta}E_{t}\widehat{i n v}_{t+1}+\frac{\varphi}{1+\beta}\hat{q}_{t}+\frac{\beta}{1+\beta}(E_{t}u_{t+1}^{i n v}-u_{t}^{i n v}), $$

where we set $ \varphi = 1/\bar{S}'' $ and the inverse, $ 1/\varphi $ , implies the elasticity of investment on the price of capital. Modeling the capital adjustment cost as a function of the change in investment rather than its level introduces additional dynamics into the investment equation, which is useful in capturing the hump-shaped response of investment to various shocks including monetary policy shocks. A positive shock to the adjustment cost function, $ u_{t}^{inv} $ , temporarily reduces investment. The expected value $ E_{t}u_{t+1}^{inv} $ is set as $ \rho^{inv}u_{t}^{inv} $ using the AR(1) coefficient $ \rho^{inv} $ .

The Euler equation for asset pricing is written as:

$$ \hat{q}_{t}=-(\hat{R}_{t}-E_{t}\hat{\pi}_{t+1})+\frac{1-\tau}{1-\tau+\overline{r}^{k}}E_{t}\hat{q}_{t+1}+\frac{\overline{r}^{k}}{1-\tau+\overline{r}^{k}}E_{t}\hat{r}_{t+1}^{k}+\varepsilon_{t}^{q}, $$

where $ \beta = 1/(1 - \tau - \bar{r}_{k}) $ , $ \tau $ is the depreciation rate, and $ \bar{r}_{k} $ is the steady-state rental rate. The current value of the capital stock, Tobin's q, depends negatively on the ex ante real interest rate, and positively on its expected future value and the expected rental rate. The introduction of a white noise shock to the required rate of return on equity investment, $ \varepsilon_{t}^{q} $ , is meant as a shortcut to capture changes in the cost of capital that may be due to stochastic variations in the external finance premium.

The real wage law of motion is expressed as:

$$ \begin{align*}\hat{w}_{t}&=\frac{\beta}{1+\beta}E_{t}\hat{w}_{t+1}+\frac{1}{1+\beta}\hat{w}_{t-1}+\frac{\beta}{1+\beta}E_{t}\hat{\pi}_{t+1}-\frac{1+\beta\gamma_{w}}{1+\beta}\hat{\pi}_{t}+\frac{\gamma_{w}}{1+\beta}\hat{\pi}_{t-1}\\&-\frac{\lambda_{w}\big(1-\beta\xi_{w}\big)\big(1-\xi_{w}\big)}{(1+\beta)\big(\lambda_{w}+(1+\lambda_{w})\sigma_{L}\big)\xi_{w}}\left[\hat{w}_{t}-\sigma_{L}\hat{l}_{t}-\frac{\sigma_{c}}{1-h}(\hat{c}_{t}-h\hat{c}_{t-1})-u_{t}^{L}-\varepsilon_{t}^{w}\right],\end{align*} $$