2.1 Patient households

There is a continuum of mass one of savers that choose consumption, $c_{s,t}$ , bonds, $b_{s,t}$ , housing, $h_{s,t}$ , and working hours, $n_{s,t}$ , to maximize their lifetime utility:Footnote ⁶

\begin{align*} {\textrm{E}}_0 \sum _{t=0}^{\infty } \beta _s^t \left [ \Gamma _{c, s} \cdot \frac {(c_{s,t} - \phi _c c_{s, t-1})^{1-\sigma _c}-1}{1-\sigma _c} + J \cdot \frac {h_{s,t}^{1-\sigma _h}-1}{1-\sigma _h} - \kappa \cdot \frac {n_{s,t}^{1+\eta }}{1+\eta }\right ]\text{,} \end{align*}

where ${\textrm{E}}_0$ represents the expectation operator conditional on information in period 0, $\beta _s$ is the savers’ discount factor, $\sigma _c$ , $\sigma _h$ and $\eta$ determine the curvature of the utility function with respect to consumption, housing, and labor hours, respectively. Parameters $J$ and $\kappa$ reflect the preferences associated with housing and work. Additionally, $\phi _c$ measures the strength of consumption habit, and $\Gamma _{c, s} \equiv (1-\phi _c)/(1-\beta _s\phi _c)$ is a scaling factor ensuring the patient households’ marginal utility of consumption is $1/c_s$ in the steady state.

Savers face a budget constraint given by:

(1) \begin{align} c_{s,t}+q_t h_{s,t}+b_{s,t} \leq w_{s,t} n_{s,t} +q_t h_{s,t-1} + \frac {R_{t-1}}{\Pi _{t}} \cdot b_{s,t-1} + div_t\text{,} \end{align}

where $b_{s,t}$ represents the bond holdings of savers, $w_{s,t}$ represents the real wage rate, $q_t$ represents the relative price of housing, $R_t$ is the nominal interest rate, $\Pi _t$ is the inflation rate, and $div_t$ is the dividend earned from owning retailers’ businesses.

The first-order conditions associated with consumption, labor hours, housing, and bond holdings are as follows:

(2) \begin{align} \lambda _{s,t} &= uc_{s,t}, \end{align}

(3) \begin{align} \lambda _{s,t} w_{s,t} &= -un_{s,t}, \end{align}

(4) \begin{align} \lambda _{s,t} q_{t} &= \beta _s {\textrm{E}}_t\left ( \lambda _{s,t+1} q_{t+1} \right ) + uh_{s,t}, \end{align}

(5) \begin{align} \lambda _{s,t} &= \beta _s {\textrm{E}}_t\left ( \lambda _{s,t+1}\frac {R_{t}}{\Pi _{t+1}}\right ), \end{align}

where $\lambda _{s,t}$ is the Lagrange multiplier associated with the savers’ budget constraints, Eq. (1), $uc_{s,t}$ , $uh_{s,t}$ , and $un_{s,t}$ are the first-order derivatives of the savers’ utility function with respect to $c_{s,t}$ , $h_{s,t}$ , and $n_{s,t}$ , respectively, which can be expressed as follows:

\begin{eqnarray*} uc_{s,t} &=& \Gamma _{c, s} \bigg \{(c_{s,t} - \phi _c c_{s, t-1})^{-\sigma _c} - \beta _s\phi _c {\textrm{E}}_t\left [(c_{s,t+1} - \phi _c c_{s, t})^{-\sigma _c}\right ]\bigg \} \text{,}\\ uh_{s,t} &=& J h_{s,t}^{-\sigma _h}\text{,}\\ un_{s,t} &=& -\kappa n_{s,t}^\eta . \end{eqnarray*}

Combing Eqs. (2) and (3) yields the optimal labor-leisure condition as follows:

(6) \begin{equation} uc_{s,t} w_{s,t} =-un_{s,t}. \end{equation}

Combing Eqs. (2) and (4), we obtain the optimal condition of housing for patient households as follows:

(7) \begin{equation} q_{t}= {\textrm{E}}_t(\Lambda ^s_{t+1, t}q_{t+1}) + mrs^s_{hc, t} \text{,} \end{equation}

where $\Lambda ^s_{t+1, t}\equiv \beta _s\lambda _{s, t+1}/\lambda _{s, t}$ is savers’ stochastic discount factor (SDF) and $mrs^s_{hc, t} \equiv uh_{s, t}/\lambda _{s, t}$ is savers’ marginal rate of substitution (MRS) for housing with respect to non-durable goods. Because housing is a durable good, patient households select their housing so that the current housing purchase cost equals the combined benefit of the expected discounted resale value and the MRS between housing and non-durable goods.

Since the patient household can utilize both housing and bonds as savings instruments, the no-arbitrage condition dictates that the one-period return on bonds must be equal to the return from holding housing, that is,

(8) \begin{equation} {\textrm{E}}_t\left ( \frac {\Lambda ^s_{t+1, t} R_{t}}{\Pi _{t+1}}\right ) = {\textrm{E}}_t\left (\frac {\Lambda ^s_{t+1, t}q_{t+1}}{q_t}\right ) + \frac {mrs^s_{hc, t}}{q_t} . \end{equation}

2.2 Impatient households

There is a continuum of impatient households with mass one who choose consumption, $c_{b,t}$ , bonds, $b_{b,t}$ , housing, $h_{b,t}$ , and working hours, $n_{b,t}$ , to maximize their expected lifetime utility:

\begin{align*} {\textrm{E}}_0 \sum _{t=0}^{\infty } \beta _b^t \left [ \Gamma _{c, b}\cdot \frac {(c_{b,t} - \phi _c c_{b, t-1})^{1-\sigma _c}-1}{1-\sigma _c} + J \cdot \frac {h_{b,t}^{1-\sigma _h}-1}{1-\sigma _h} - \kappa \cdot \frac {n_{b,t}^{1+\eta }}{1+\eta } \right ]\text{,} \end{align*}

where $\beta _b$ is the discount factor of impatient households. We assume $\beta _b \lt \beta _s$ to ensure that the credit constraints for impatient households are binding in equilibrium. In addition, $\Gamma _{c, b} \equiv (1-\phi _c)/(1-\beta _b\phi _c)$ denotes the scaling factor that ensures the impatient households’ marginal utility of consumption is $1/c_b$ in the steady state.

The budget constraint for impatient households is:

(9) \begin{equation} c_{b,t}+q_t h_{b,t}+b_{b,t}\leq w_{b,t} n_{b,t}+q_t h_{b,t-1}+\frac {R_{t-1}}{\Pi _{t}} \cdot b_{b,t-1}\text{,} \end{equation}

where $b_{b,t}$ represents the bond holdings of the impatient households, and $w_{b,t}$ is the wage rate for the impatient households.

Furthermore, houses have a dual role for impatient households, serving as both residences and collateral assets. Specifically, the impatient households face borrowing limits, which are related to a fraction $m_b \in [0, 1]$ of their housing value:

(10) \begin{equation} -b_{b,t}\leq m_b {\textrm{E}}_t \left ( \frac {\Pi _{t+1}}{R_t}\cdot q_{t+1} \right ) h_{b,t}\text{,} \end{equation}

where $m_b$ represents the loan-to-value (LTV) ratio.

We then derive the first-order conditions associated with impatient households’ problems for $c_{b,t}$ , $n_{b,t}$ , $h_{b,t}$ , and $b_{b,t}$ :

(11) \begin{align} \lambda _{b,t} &=uc_{b,t} , \end{align}

(12) \begin{align} \lambda _{b,t} w_{b,t} &=-un_{b,t}, \end{align}

(13) \begin{align} \lambda _{b,t} q_{t} &={\textrm{E}}_t (\beta _b\lambda _{b,t+1} q_{t+1} + \rho _{b,t} m_b q_{t+1} \Pi _{t+1})+uh_{b,t}, \end{align}

(14) \begin{align} \lambda _{b,t} &=\beta _b{\textrm{E}}_t \left (\lambda _{b,t+1} \frac {R_{t}}{\Pi _{t+1}} \right ) +\rho _{b,t} R_t , \end{align}

where $\lambda _{b,t}$ denotes the Lagrange multiplier associated with the budget constraint, Eq. (9), and $\rho _{b,t}$ is Lagrange multiplier associated with the collateral constraint, Eq. (10). $uc_{b,t}$ , $uh_{b,t}$ , and $un_{b,t}$ denote respectively the first derivatives of the impatient households’ utility with respect to $c_{b,t}$ , $h_{b,t}$ , and $n_{b,t}$ , which can be expressed as follows:

(15) \begin{align} uc_{b,t} &= \Gamma _{c,b} \bigg \{ (c_{b,t}-\phi _cc_{b, t-1})^{-\sigma _c} - \beta _b \phi _c {\textrm{E}}_t[(c_{b,t+1}-\phi _cc_{b, t})^{-\sigma _c}]\bigg \}, \end{align}

(16) \begin{align} uh_{b,t} &= J h_{b,t}^{-\sigma _h}, \end{align}

(17) \begin{align} un_{b,t} &= -\kappa n_{b,t}^\eta . \end{align}

The main difference between the optimal conditions of the patient households and the impatient households lies in their housing choices. Combining Eqs. (13) and (14), we can derive the optimal housing condition for impatient households as follows:

(18) \begin{equation} \begin{split} q_t & = {\textrm{E}}_t (\Lambda ^b_{t+1, t} q_{t+1})+ m_b {\textrm{E}}_t \left ( \frac {\Pi _{t+1}}{R_t} \cdot q_{t+1}\right ) \\ & \quad \quad - m_b{\textrm{E}}_t\left ( \frac {\Lambda ^b_{t+1, t}}{\Pi _{t+1}}\right ){\textrm{E}}_t(\Pi _{t+1}q_{t+1})+ mrs^b_{hc, t}\text{,} \end{split} \end{equation}

where $\Lambda ^b_{t+1, t} \equiv \beta _b\lambda _{b,t+1}/\lambda _{b,t}$ is impatient households’ SDF and $mrs^b_{hc, t} \equiv uh_{b,t}/\lambda _{b,t}$ is their MRS for housing with respect to non-durable goods. Using ${\textrm{E}}_t( \frac {\Lambda ^b_{t+1, t}}{\Pi _{t+1}}) {\textrm{E}}_t(\Pi _{t+1}q_{t+1}) \approx {\textrm{E}}_t(\Lambda ^b_{t+1, t}q_{t+1})$ , we can approximate Eq. (18) as:

(19) \begin{equation} q_t {\approx } {\textrm{E}}_t \left [\left ((1-m_b) \cdot \Lambda ^b_{t+1, t} + m_b \cdot \frac {\Pi _{t+1}}{R_t}\right ) \cdot q_{t+1}\right ] + mrs^b_{hc, t}.^{\textrm{7}} \end{equation}

Similar to the optimal housing condition for patient households in Eq. (7), impatient households choose housing such that the current price equals the sum of the discounted resale value and the MRS between housing and non-durable goods. However, unlike patient households, who rely solely on their SDF to evaluate the expected resale value, impatient households use a weighted average of their SDF and the inverse of the real interest rate. This distinction arises because impatient households purchase housing with only a down payment, using the property as collateral for borrowing. As a result, they receive only a fraction of the housing’s resale value in the next period, corresponding to their initial out-of-pocket contribution. In contrast, the saver does not borrow, so they receive the full amount of resale value for housing in the next period.

Combing Eqs. (12) and (11) yields the optimal labor-leisure condition as follows:

(20) \begin{equation} uc_{b,t} w_{b,t} =-un_{b,t}. \end{equation}

Although the borrowing constraint does not directly affect the optimal labor-leisure trade-off, it plays a significant role in shaping labor supply and non-durable consumption choices. When impatient households use housing as collateral to borrow, they are required to pay only a fraction of the housing price upfront, that is, the down payment. By imposing borrowing limits, the consolidated budget constraint for impatient households can be expressed as follows:

(21) \begin{equation} \begin{split} {c_{b,t}+} &{\underbrace {\left [q_t - m_b \cdot {\textrm{E}}_t \left ( \frac {\Pi _{t+1}}{R_t} \cdot q_{t+1} \right )\right ]\cdot h_{b,t}}_{\text{down payments for housing}}} \\ &{ \leq w_{b,t} n_{b,t}+ \underbrace {\left [q_t-m_b\cdot \frac {{\textrm{E}}_{t-1}(\Pi _tq_t)}{\Pi _t}\right ]\cdot h_{b,t-1}}_{\text{initial housing wealth}} \text{,}} \end{split} \end{equation}

where the “down payment” faced by impatient households is defined as the difference between the current housing price and the discounted value of the collateralized portion.Footnote ⁸ Changes in relative housing expenditures, which affect borrowing limits, lead to shifts in the consolidated budget constraint. These shifts impact decisions regarding labor and consumption, illustrating how financial frictions tied to housing propagate throughout the economy.

2.3 The entrepreneurs

There is a continuum of mass one of entrepreneurs who produce homogeneous wholesale goods according to the following Cobb–Douglas production function:

(22) \begin{equation} y_{t}=A_t k_{t-1}^\mu h_{e,t-1}^\nu n_t^{1-\mu -\nu }\text{,} \end{equation}

where $A_t$ represents the aggregate TFP, $k_{t-1}$ represents the capital stock, $h_{e,t-1}$ represents the entrepreneurs’ real estate holdings, $n_t$ denotes the aggregate labor input, $\mu$ measures the share of capital, $\nu$ measures the share of housing, and $1-\mu -\nu$ represents the share of aggregate labor hours in the production function. Furthermore, the aggregate labor input, $n_t = n_{s,t}^{\alpha } n_{b,t}^{1-\alpha }$ , is a combination of labor inputs from both patient households, $n_{s,t}$ , and impatient households, $n_{b,t}$ . Accordingly, the aggregate wage is given by: $w_t = \frac {w_{s,t}^{\alpha } w_{b,t}^{1-\alpha }}{\alpha ^{\alpha } (1 - \alpha )^{1 - \alpha }}$ .

Entrepreneurs encounter credit constraints similar to those faced by impatient households. Entrepreneurs have the flexibility to use both houses and physical capital as collateral, relative to impatient households. Moreover, we allow different LTV ratios for housing and capital, denoted as $m^h_e\in [0,1]$ for housing loans and $m_e^k\in [0,1]$ for capital loans. Consequently, the borrowing limit can be expressed as follows:

(23) \begin{align} -b^h_{e,t}&\leq m^h_e {\textrm{E}}_t \left ( \frac {\Pi _{t+1}}{R_t} \cdot q_{t+1} \right )h_{e,t}, \end{align}

(24) \begin{align} -b^k_{e,t}&\leq m^k_e {\textrm{E}}_t \left ( \frac {\Pi _{t+1}}{R_t}\right )k_{t}\text{.} \end{align}

In Eq. (23), $b^h_{e,t}$ represents the loans secured by the value of housing, while in Eq. (24), $b^k_{e,t}$ represents the loans secured by the value of capital. Entrepreneurs’ overall loan position is defined as $b_{e, t} \equiv b^h_{e,t} + b^k_{e,t}$ .Footnote ⁹ These borrowing constraints reflect the limitations on entrepreneurs’ borrowing capacity based on the expected discounted value of their collateral and the applicable LTV ratios.

The entrepreneur starts each period with an initial loan of $b_{e,t-1}$ and earns $\frac {P_{t}^w}{P_{t}} \cdot y_t$ , by selling their output to retailers. Entrepreneurs then acquire new debt of $b_{e,t}$ , consume $c_{e,t}$ , invest $i_t$ , adjust their real estate holdings $q_t(h_{e,t}-h_{e,t-1})$ , and repay their previous debt $(R_{t-1}/\Pi _t)\cdot b_{e,t-1}$ . Additionally, they pay wages to both patient households $w_{s,t}n_{s,t}$ and impatient households $w_{b,t}n_{b,t}$ . Therefore, entrepreneurs’ budget constraints can be expressed as:

(25) \begin{equation} c_{e,t} + i_t + q_t(h_{e,t} - h_{e,t-1}) + w_{s,t}n_{s,t} + w_{b,t}n_{b,t} + b_{e,t} = \frac {P_{t}^w}{P_{t}}\cdot y_t + \frac {R_{t-1}}{\Pi _t} \cdot b_{e,t-1}\text{,} \end{equation}

The law of motion for capital is:

(26) \begin{equation} k_t=i_t+(1-\delta ) k_{t-1}, \end{equation}

where $k_t$ represents the capital stock, and $\delta$ is the depreciation rate.

Let $\beta _e$ be the discount factor for entrepreneurs. We assume the entrepreneurs’ discount factor satisfies $\beta _e \lt \beta _s$ to ensure that the credit constraints for the entrepreneurs are binding in equilibrium. Additionally, their lifetime utility function is given by:

\begin{align*} {\textrm{E}}_0 \sum _{t=0}^{\infty } \beta _e^t \left [ \Gamma _{c, e} \cdot \frac {(c_{e,t} - \varepsilon _{c}c_{e, t-1})^{1-\sigma _c}-1}{1-\sigma _c} \right ]\text{.} \end{align*}

where $\Gamma _{c, e} \equiv (1-\phi _c)/(1-\beta _e\phi _c)$ is the scaling factor that ensures the marginal utility of consumption equals $1/c_e$ in the steady state.

Finally, entrepreneurs maximize their lifetime utility subject to the budget constraint, Eq. (25), the law of motion of the capital stock, Eq. (26), and the credit constraints, Eqs. (23)–(24). The associated first-order conditions are as follows:

(27) \begin{align} \lambda _{e,t} &= uc_{e,t}, \end{align}

(28) \begin{align} \lambda _{e,t} q_t &={\textrm{E}}_t \left [\beta _e\lambda _{e,t+1} \left ( \frac {\nu y_{t+1}}{X_{t+1} h_{e,t}}+q_{t+1} \right ) +m^h_e \rho ^h_{e,t} q_{t+1}\Pi _{t+1}\right ], \end{align}

(29) \begin{align} \lambda _{e,t}&={\textrm{E}}_t \left (\beta _e \lambda _{e,t+1} \frac {R_t}{\Pi _{t+1}}\right )+ \rho ^h_{e,t}R_t , \end{align}

(30) \begin{align} \lambda _{e,t}&=\beta _e {\textrm{E}}_t \left \{\lambda _{e,t+1} \left [\frac {\mu y_{t+1}}{X_{t+1} k_t} + (1-\delta )\right ] +m^k_e \rho ^k_{e,t} \Pi _{t+1}\right \} , \end{align}

(31) \begin{align} \lambda _{e,t}&={\textrm{E}}_t \left (\beta _e \lambda _{e,t+1} \frac {R_t}{\Pi _{t+1}}\right )+ \rho ^k_{e,t} R_t, \end{align}

(32) \begin{align} w_{s,t}&=\frac {\alpha (1-\mu -\nu )y_{t}}{X_{t}n_{s,t}}, \end{align}

(33) \begin{align} w_{b,t}&=\frac {(1-\alpha ) (1-\mu -\nu ) y_{t}}{X_{t}n_{b,t}}, \end{align}

where $\lambda _{e,t}$ denotes the Lagrange multiplier for budget constraint, Eq. (25). $\rho ^h_{e,t}$ and $\rho ^k_{e,t}$ stand for the Lagrange multipliers associated with credit constraints, Eqs. (23)–(24), respectively. $uc_{e,t}$ represents the marginal utility of consumption of entrepreneurs, which is defined as

\begin{equation*} uc_{e,t} \equiv \Gamma _{c,e} \bigg \{ (c_{e,t}-\phi _cc_{e, t-1})^{-\sigma _c} - \beta _e \phi _c {\textrm{E}}_t[(c_{e,t+1}-\phi _cc_{e, t})^{-\sigma _c}]\bigg \}\text{.} \end{equation*}

In contrast to a standard model without borrowing constraints, the first-order conditions for an entrepreneur’s housing and capital now include terms that account for the shadow value associated with relaxing the borrowing constraint through an additional unit of housing or capital, respectively. As a result, the entrepreneur’s optimal choice of capital and housing equates today’s prices of capital and housing to the sum of their discounted resale values and the discounted marginal product of each factor. Similar to impatient households, the entrepreneur can use housing and capital as collateral for borrowing. Consequently, the effective discounted resale value is influenced by both the SDF and the real interest rate, which determine the repayment amount. Specifically, the entrepreneur’s optimal conditions are approximately expressed as:

(34) \begin{align} q_t &{\approx }{\textrm{E}}_t \left \{\Lambda ^e_{t+1, t}\cdot \nu y_{t+1}/(X_{t+1} h_{e,t}) +\left [ \left(1-m^h_e \right)\cdot \Lambda ^e_{t+1, t} +m^h_e \cdot \frac {\Pi _{t+1}}{R_t}\right ]q_{t+1}\right \}, \\[-12pt] \nonumber \end{align}

(35) \begin{align} 1 &{\approx }{\textrm{E}}_t \left \{\Lambda ^e_{t+1, t}\cdot \mu y_{t+1}/(X_{t+1} k_{e,t}) + \left [ \left(1-\delta - m^h_e \right)\cdot \Lambda ^e_{t+1, t} +m^h_e \cdot \frac {\Pi _{t+1}}{R_t} \right ]\right \} , \end{align}

where $\nu y_{t+1}/(X_{t+1} h_{e,t})$ and $\mu y_{t+1}/(X_{t+1} k_{e,t})$ stand for the marginal product of housing and the marginal product of capital, respectively. $\Lambda ^e_{t+1, t} \equiv \beta _e\lambda _{e, t+1}/\lambda _{e, t}$ is entrepreneurs’ SDF.Footnote ¹⁰

The next subsection introduces the final goods producer and the monopolistically competitive retailers. The inclusion of monopolistically competitive retailers enables a seamless transition between a flexible-price and a sticky-price economy by adjusting the parameter associated with price stickiness. Although our model does not rely on price stickiness to address the co-movement puzzle, we consider it for the purpose of model comparison.

2.4 Final goods producer

The representative final goods producer purchases a continuum of differentiated intermediate goods from retailers and transforms them into final products. The production process follows a constant elasticity of substitution technology, represented as:

(36) \begin{equation} Y_t = \left (\int _0^1 Y_t(z)^{\frac {\eta _y - 1}{\eta _y}} \text{d}z \right )^{\frac {\eta _y}{\eta _y - 1}}, \end{equation}

where $ Y_t$ denotes the quantity of final goods, $ Y_t(z)$ represents the quantity of intermediate good $ z$ purchased from retailer $ z$ , and $ \eta _y$ is the elasticity of substitution between different intermediate goods.

Let $ P_t(z)$ denote the price of intermediate good $ z$ . The aggregate price level of the final good, $ P_t$ , can be expressed as:

(37) \begin{align} P_t = \left ( \int _0^1 P_t(z)^{1-\eta _y} \text{d}z \right )^{1/(1-\eta _y)}. \end{align}

Assuming that the market for final goods is competitive, the final goods producer solves the following problem:

\begin{equation*} \max _{Y_t(z)} P_t Y_t - \int _0^1 P_t(z) Y_t(z) \text{d}z, \end{equation*}

where the objective is to maximize profits.

The resulting optimality condition for the demand of intermediate goods is given by:

(38) \begin{equation} Y_t(z) = \left [\frac {P_t(z)}{P_t}\right ]^{-\eta _y} Y_t. \end{equation}

2.5 Retailers

In our benchmark model, a continuum of monopolistically competitive retailers, indexed by $ z \in [0,1]$ , each purchase homogeneous goods from entrepreneurs at price $ P^w_t \equiv P_t / X_t$ , transform them into differentiated goods $ Y_t(z)$ at no cost, and sell $ Y_t(z)$ to the final goods firm at price $ P_t(z)$ .

We assume that retailers can adjust their prices every period without incurring any costs. Later, when we extend the analysis to include price stickiness, we will introduce a quadratic adjustment cost. For now, retailer $ z$ ’s optimization problem is given by:

\begin{equation*} \max _{\{P_t(z)\}^\infty _{j=t}} \left [\frac {P_t(z) - P^w_t}{P_t} \cdot Y_t(z) \right ], \end{equation*}

subject to the demand function in Eq. (38).

The first-order condition for retailer $ z$ ’s problem is:

(39) \begin{equation} (1 - \eta _y) \cdot \left [\frac {P_t(z)}{P_t}\right ]^{1 - \eta _y} + \eta _y \cdot \frac {P^w_t}{P_t} \cdot \left [\frac {P_t(z)}{P_t}\right ]^{-\eta _y} = 0. \end{equation}

Since all retailers face the same profit maximization problem, they choose the same price, $ P_t(z) = P_t$ , and produce the same quantity, $ Y_t(z) = Y_t$ . This leads to the following condition:

(40) \begin{equation} 0 = (1 - \eta _y) + \frac {\eta _y}{X_t}. \end{equation}

2.6 Source of the uncertainty

We assume that the log of TFP shock $A_t$ follows an AR(1) process, which takes the form:

(41) \begin{align} \log A_t = \rho _a \log A_{t-1} + \exp (V_t) \sigma _A \varepsilon _{A,t}, \end{align}

where $\varepsilon _{A,t}\sim N(0,1)$ , $\rho _{a}$ is the persistence of stochastic process to $A_t$ , and $\sigma _A$ is the standard deviation of innovations to $A_t$ . Furthermore, the autoregressive process of productivity features time-varying volatility. In particular, the log of the standard deviation, $V_{t}$ , of the innovations to productivity follows the autoregressive process:

(42) \begin{align} V_t=\rho _v V_{t-1}+\sigma _V \varepsilon _{V,t}, \end{align}

where $ \varepsilon _{V,t}\sim N(0,1)$ , $\rho _v$ is the persistence of stochastic process to $V_t$ , and $\sigma _V$ is the standard deviation of innovations to $V_t$ .

Two independent innovations, $\varepsilon _{A,t}$ and $\varepsilon _{V,t}$ , affect the productivity. The first innovation denotes the productivity shocks, which change the productivity itself, while the second innovation denotes the volatility shock, which affects the spread of values for productivity. The volatility shock in the productivity implies that all firms are affected by more volatile shocks. Given the timing assumption, firms learn in advance the dispersion of shocks from which they will draw in the next period. This timing assumption captures the notion of uncertainty that firms face about future business conditions.

2.7 Monetary policy and market-clearing conditions

We assume that the monetary authority follows a simple interest rate rule, which responds to changes in inflation as follows:Footnote ¹¹

(43) \begin{align} \frac {R_t}{R}=\left ( \frac {\Pi _t}{\Pi } \right )^{\phi _{\Pi }}, \end{align}

where $R$ and $\Pi$ are, respectively, the steady-state nominal interest rate and inflation rate, and $\phi _{\Pi }$ is the policy parameter.

Housing market equilibrium requires

(44) \begin{align} h_{s,t}+h_{b,t}+h_{e,t}&=1, \end{align}

Bonds market equilibrium requires

(45) \begin{align} b_{s,t}+b_{b,t}+b_{e,t}=0, \end{align}

Goods market equilibrium requires

(46) \begin{align} c_t+i_t&=Y_t \end{align}

where $c_t=c_{s,t}+c_{b,t}+c_{e,t}$ denotes the aggregate consumption.

We define the gross domestic product of this economy, $\widetilde {Y}_t$ , as the sum of aggregate output and the imputed housing services of owner-occupied homes:

(47) \begin{equation} \widetilde {Y}_t \equiv Y_t + h_{s, t}\cdot mrs^s_{hc, t} + h_{b, t}\cdot mrs^b_{hc, t}\text{.} \end{equation}

3.1 Solution method

We employ the third-order perturbation approximation with the pruning method introduced by Andreasen et al. (Reference Andreasen, Fernández-Villaverde and Rubio-Ramírez2017) to address our dynamic stochastic general equilibrium model. This choice stems from the need for at least a third-order approximation of the policy functions to analyze the impulse response to a second-moment shock, as indicated by Fernández-Villaverde et al. (Reference Fernández-Villaverde, Guerrón-Quintana, Rubio-Ramírez and Uribe2011). In addition, we prune terms with higher-order effects beyond the third order to prevent higher-order approximations from generating explosive sample paths.

Following the approach of Basu and Bundick (Reference Basu and Bundick2017), we set the exogenous shocks to zero and stimulate the economy for a sufficiently long period until all endogenous variables have converged to their stochastic steady state. Then, we introduce the uncertainty shock and compute its impulse responses as a percentage deviation from the steady state. Specifically, we implement the simulation using Dynare software.

3.2 Estimation of exogenous processes

An important aspect of our analysis is the estimate of the key parameters associated with the stochastic process of uncertainty shocks, including $\rho _A$ , $\rho _V$ , $\sigma _A$ , and $\sigma _V$ . Following Cesa-Bianchi and Fernandez-Corugedo (Reference Cesa-Bianchi and Fernandez-Corugedo2018), we estimate the process of productivity and uncertainty shocks using the time series data of aggregate TFP for the U.S. business sector. The sample periods ranges from 1947Q1 to 2024Q3.Footnote ¹²

By fitting an AR(1) process to the log-deviations of TFP from a linear trend, we estimate the parameter that captures the persistence of the productivity process, $\rho _A$ , and the standard deviation of its innovations, $\sigma _A$ , which are 0.9780 and 0.0089, respectively. These values are in line with the findings of Cesa-Bianchi and Fernandez-Corugedo (Reference Cesa-Bianchi and Fernandez-Corugedo2018) and Fernández-Villaverde et al. (Reference Fernández-Villaverde, Guerrón-Quintana, Rubio-Ramírez and Uribe2011).

We then compute the standard deviations of the TFP innovations with an eight quarter rolling window to get the proxy for the time-varying volatility of the TFP innovations. By fitting an AR(1) process to the log of the standard deviations of the TFP innovations from a linear trend, we estimate the parameter that captures the persistence of the uncertainty process, which yields the value of $\rho _V$ being 0.9110 and $\sigma _V$ being 0.1643. These parameter values are similar to the estimation from Cesa-Bianchi and Fernandez-Corugedo (Reference Cesa-Bianchi and Fernandez-Corugedo2018) and Caldara et al. (Reference Caldara, Fernández-Villaverde, Rubio-Ramirez and Yao2012).Footnote ¹³

Figure 1 plots the standard deviation and cyclical component associated with $\sigma _t^{\textit{TFP}}$ , with NBER recession periods shown as shaded areas. The top panel of Figure 1 presents the standard deviations of $\sigma _t^{\textit{TFP}}$ , which tends to spike during recessionary periods. We use the log-deviations of $\sigma _t^{\textit{TFP}}$ from a linear trend as a proxy for $V_t$ . These cyclical components of $\sigma _t^{\textit{TFP}}$ are displayed in the lower panel of Figure 1.

Figure 1. Time-varying volatility of TFP innovations.

3.3 Calibration

We now calibrate the key parameters of our model. Since our model is quarterly, we choose a savers’ discount factor of $\beta _s=0.9925$ , which corresponds to a 3% annual real interest rate. The impatient households’ and entrepreneurs’ discount factors are set to $\beta _b=0.94$ and $\beta _e=0.94$ , respectively, as in Iacoviello (Reference Iacoviello2015). We also follow Choa and Francis (Reference Choa and Francis2011) in setting the parameters $\sigma _c=3.0$ and $\sigma _h=1.5$ to match the increasing ratio of housing to non-housing consumption as income increases. Furthermore, when $\sigma _h$ exceeds $\sigma _c$ , changes in the housing stock have a milder effect on utility compared to changes in consumption. Such a feature allows households use housing stocks as a buffer against unexpected shocks. For a detailed discussion, please refer to Zanetti (Reference Zanetti2014). Both the inverse of the Frisch elasticity of labor supply, $\eta$ , and the weight of work, $\kappa$ , are chosen as 1, based on Guerrieri and Iacoviello (Reference Guerrieri and Iacoviello2017).

The weight of housing, $J$ , is set to 0.0757, which gives a real estate wealth to annual output ratio of 3.1, consistent with Iacoviello (Reference Iacoviello2015). The maximum LTV ratios for borrowers and entrepreneurs are both set to 0.9, in line with Iacoviello (Reference Iacoviello2015), and Guerrieri and Iacoviello (Reference Guerrieri and Iacoviello2017). The strength of consumption habit, $\phi _c$ , is set to 0.6842, following the research of Guerrieri and Iacoviello (Reference Guerrieri and Iacoviello2017).

We choose a depreciation rate of $\delta =0.025$ , corresponding to an annual depreciation rate of 10%. The shares of capital and real estate to output are fixed at $\mu =0.3$ and $\nu =0.03$ , respectively, as in Iacoviello (Reference Iacoviello2005). Savers’ wage share, $\alpha$ , is set to $0.67$ , consistent with Iacoviello (Reference Iacoviello2015). Finally, the monetary policy rule is set to $\phi _{\Pi }=1.5$ , a standard choice in the literature akin to Monacelli (Reference Monacelli2009). To enhance clarity, we provide a summary of all parameter values in Table 1.

Table 1. Parameters values

3.4 Quantitative analysis

To analyze the impacts of uncertainty shocks, we follow Fernández-Villaverde et al. (Reference Fernández-Villaverde, Guerrón-Quintana, Rubio-Ramírez and Uribe2011) to compute impulse response functions (IRFs) to a mean preserving shock to the variance of TFP. IRFs are expressed as percentage deviations from the stochastic steady state in response to a one-standard deviation uncertainty shock. For nominal interest rates and inflation, we report deviations from their stochastic steady-state levels. Since we are interested in investigating which model ingredients are key to propagating the uncertainty shocks during the business cycles, we focus on the responses of output, consumption, investment, housing holdings, employment, real wage, and total loans throughout our analysis.

3.4.1 A frictionless economy with a representative agent

We first consider a frictionless economy consisting solely of patient households, where borrowing constraints are absent. This setup mirrors a representative agent model, with patient households owning a representative entrepreneur who produces homogeneous goods according to the technology outlined in Eq. (22). The entrepreneur accumulates capital and housing for production without collateral constraints, with market-clearing conditions given as: $h_{s,t} + h_{e,t} = 1$ , $b_{s,t} = 0$ , and $c_{s,t} + i_t = Y_t$ . For a detailed presentation of the model without collateral constraints, please refer to Appendix A.

In this economy, precautionary saving motives shape household responses to uncertainty shocks. When uncertainty rises, patient households reduce consumption and increase labor supply. Given fixed capital and unchanged TFP, the additional labor input raises aggregate production and investment while lowering real wages. Although the model predicts a decline in real wages alongside rising output and investment, this contrasts with empirical findings, which typically show that uncertainty shocks lead to an increase in real wages and broad-based declines in output, consumption, and investment. As a result, this alternative model without collateral constraints fails to replicate the co-movement patterns seen in the data. Figure 2 presents the IRFs for the frictionless economy following a one-standard-deviation increase in TFP uncertainty.

Figure 2. IRFs to a TFP uncertainty shock: baseline model versus frictionless economy. Notes: IRFs are expressed as percentage deviations from the stochastic steady state in response to a one-standard deviation uncertainty shock.

3.4.2 The baseline model with financial frictions

We then transition to our baseline model, which incorporates financial frictions, including collateral constraints on both impatient households and entrepreneurs, with LTV ratios set at $m_b = m^h_e = m^k_e = 0.9$ . This framework introduces both patient and impatient households, allowing us to explore how credit constraints amplify or dampen the effects of uncertainty shocks. Unlike the frictionless model, the baseline model captures co-movement across macroeconomic aggregates by accounting for the distinct behaviors of patient and impatient households.

In our benchmark model, the behavior of patient households largely mirrors that observed in a frictionless economy, driven primarily by precautionary saving motives. In response to uncertainty shocks, patient households reduce their non-durable consumption and increase their labor supply. In contrast, the optimal decisions regarding non-durable consumption and labor for impatient households are closely tied to their housing decisions, which are influenced by both their SDF and the prevailing real interest rate when selling their collateral, as shown in Eq. (19). Specifically, the SDF term reflects the precautionary saving motive, while the real interest rate captures the resale risk associated with housing. The relative importance of these factors is determined by their respective weights.

To further illustrate the risks associated with borrowing, we decompose Eq. (19) as follows:

(48) \begin{equation} \begin{split} q_t {\approx }&\bigg [(1-m_b){\textrm{E}}_t(\Lambda ^b_{t+1, t}) + m_b {\textrm{E}}_t(\Pi _{t+1})/R_t \bigg ]\cdot {\textrm{E}}_t(q_{t+1}) \\ &+ (1-m_b){\textrm{Cov}}_t(\Lambda ^b_{t+1, t}, q_{t+1}) + m_b {\textrm{Cov}}_t(\Pi _{t+1}, q_{t+1})/R_t + mrs^b_{hc, t}. \end{split} \end{equation}

This decomposition includes four main components of the risk associated with borrowing: (1) the product of the inverse of the expected real interest rate and the expected housing resale price, (2) the product of the expected SDF and the expected housing resale price, (3) the covariance terms between the housing resale price and the inverse of the real interest rate, and (4) the covariance terms between the housing resale price and the SDF. With a LTV ratio of $ m_b = 0.9$ , the precautionary saving motive has a relatively minor influence on housing decisions, as its weight is $ (1 - m_b) = 0.1$ . In contrast, the real interest rate plays a significantly greater role, with a weight of $ m_b = 0.9$ . This disparity in weights underscores that the housing decisions of impatient households are primarily driven by the resale risk associated with the real interest rate, rather than by precautionary saving motives. For the subsequent analysis, we disregard the expected and covariance terms associated with the precautionary saving motive (i.e., the SDF term), as their impacts, characterized by $ 1 - m_b$ , are comparatively smaller.

Based on the decomposition in Eq. (48), the housing decision is primarily influenced by the dynamics of the real interest rate (future inflation rate) and housing resale prices. In response to uncertainty shocks, future inflation rises, and housing resale prices decline. While inflation generally benefits borrowers by reducing the real value of repayments in terms of goods, this advantage diminishes when inflation and housing prices move in opposite directions. Specifically, if housing prices fall more than inflation rises, borrowers become less willing to take on debt, reducing their housing holdings and transitioning from larger to smaller homes. Additionally, the covariance term captures the risk premiums associated with housing purchases.Footnote ¹⁴ A decline in this covariance term reduces borrowers’ willingness to pay, thereby lowering housing demand.

To analyze this dynamic, we follow Koop et al. (Reference Koop, Pesaran and Potter1996) to simulate IRFs for the covariance between inflation and housing prices. The simulation results, shown in Figure 3, indicate that heightened uncertainty amplifies the negative correlation between future inflation and housing prices, leading impatient households to reduce their housing holdings. In contrast, patient households, viewing housing as both a durable good and an investment, are less sensitive to the timing of purchases due to the stable shadow value of durable goods. Consequently, when the relative price of housing declines, patient households increase their housing holdings. This divergence in behavior drives a significant reallocation of housing consumption from impatient to patient households, reflecting their distinct responses to changes in inflation, housing prices, and associated risks. To deepen our understanding of this redistribution, we examine the empirical relationship between inflation and housing prices, utilizing the All-Transactions House Price Index (USSTHPI) and the Personal Consumption Expenditures Chain-type Price Index (PCECTPI) for the period spanning 1959:Q1 to 2023:Q4. Following the FRED-QD methodology, we transform the USSTHPI using first differences of natural logarithms and the PCECTPI using second differences of natural logarithms. The resulting correlation coefficient of −0.055 highlights a negative relationship between inflation and housing prices, underscoring the role of changing economic conditions in shaping housing consumption patterns.

Figure 3. Percentage deviation in simulated covariance between inflation and housing prices over 25,000 simulations.

Housing prices decline following uncertainty shocks due to differences in financing methods and the redistribution of housing between impatient and patient households. Impatient households rely on leverage to finance housing purchases, making them more sensitive to risk. In contrast, patient households, who purchase housing without loans, are less constrained by borrowing costs. This redistribution leads to a sharper decline in housing demand among impatient households compared to the corresponding increase among patient households. With aggregate housing supply fixed, the overall reduction in demand drives housing prices downward.

Inflation rises in response to uncertainty shocks, driven primarily by increasing labor costs in production. These higher costs stem from a decline in labor supply among impatient households, which outweighs the modest increase from patient households. The reduction in labor supply among impatient households is closely linked to their housing demand. To lower their risk exposure, impatient households reduce their housing holdings, which alters their choice set of non-durable consumption and leisure, as captured in Eq. (21). This decrease in housing reallocates resources from down payments to non-durable consumption and leisure, ultimately leading to a reduction in their labor supply.

Despite the contraction in aggregate labor supply, labor demandlargely determined by the marginal product of laborremains stable due to fixed housing and capital stocks. This imbalance results in higher aggregate wages and fewer hours worked, consistent with the findings of Cross et al. (Reference Cross, Hou, Koop and Poon2023), which show that average hours decrease while hourly earnings rise modestly during periods of heightened uncertainty. The increase in aggregate wages further amplifies inflationary pressures. The modest increase in real wages is consistent with our supply-side mechanism, whereby tighter collateral constraints reduce labor supply, raising the marginal product of labor despite the broader contraction in hours and output.

This reduction in labor supply also explains the observed positive correlation between household debt and labor supply, as documented by Fortin (Reference Fortin1995) and Del Boca and Lusardi (Reference Del and Daniela2003). This relationship aligns with the financial labor supply accelerator proposed by Campbell and Hercowitz (Reference Campbell and Hercowitz2011), which links higher minimum down payments to reduced labor supply. While Campbell and Hercowitz (Reference Campbell and Hercowitz2011) examines this relationship within a representative agent framework, we extend the analysis to a two-agent household model. In our model, patient and impatient households differ in their financial approaches to housing purchases. Patient households buy houses without loans, requiring higher down payments, while impatient households use leverage, resulting in lower down payment requirements. Consequently, when housing transitions from impatient to patient households, the total down payment required for housing purchases increases. This mechanism highlights a positive relationship between leverage and labor supply and a negative relationship between down payments and labor supply.

With housing and capital stocks fixed, the reduction in aggregate labor hours leads to a decline in overall output. Furthermore, as heightened uncertainty is expected to persist over multiple periods, the reduction in labor supply is likely to continue. This sustained contraction in labor hours leads to a prolonged decline in the expected marginal productivity of capital and housing, resulting in an extended decrease in investment in both assets. Additionally, entrepreneurs reduce their demand for housing and physical capital as collateral values become increasingly uncertain due to uncertainty shocks. This further decreases investment and housing demand.

In summary, this increase in uncertainty leads to a decrease in consumption, investment, aggregate labor hours, and output. Consequently, our baseline model effectively reproduces the co-movement among key macroeconomic aggregates following heightened uncertainty. These findings are illustrated in Figure 2, where we also present the dynamics associated with the frictionless economy. It’s worth noting that our model with financial frictions can address the co-movement puzzle, even in the absence of nominal rigidities.

3.4.3 The importance of the risk premium channel

To highlight the importance of the risk premium in addressing the co-movement puzzle, we examine an alternative scenario where both patient households and entrepreneurs employ the steady-state housing price, denoted as $\overline {q}$ , to evaluate the collateral value. Consequently, the borrowing constraints associated with housing are modified as follows:

\begin{align*} b_{b, t} &\leq m_b {\textrm{E}}_t \left (\frac {\Pi _{t+1}}{R_t} \cdot \overline {q}\right ) h_{b, t}\text{,} \\ b^h_{e, t} &\leq m^h_e {\textrm{E}}_t \left (\frac {\Pi _{t+1}}{R_t} \cdot \overline {q}\right ) h_{e, t}\text{.} \end{align*}

When housing prices are held constant, the risk premium channel becomes effectively inactive, as ${\textrm{Cov}}_t(\Lambda ^b_{t+1, t},\overline {q}) = {{\textrm{Cov}}_t(\Lambda ^e_{t+1, t},\overline {q})} = {\textrm{Cov}}_t(\Pi _{t+1}, \overline {q})/R_t = 0$ . As the risk premium terms associated with housing resale prices vanish, the incentive for impatient households to decrease their demand for housing following uncertainty shocks also disappears. Consequently, their labor supply will not decrease as it did previously. Therefore, the increase in aggregate labor hours among patient and impatient households leads to an overall rise in aggregate labor hours, which in turn leads to the higher aggregate production, given that current technology and capital stock remain unchanged. This boost in output, combined with reduced consumption, results in an increase in investment, a pattern similar to the outcomes in the frictionless economy model. Figure 4 presents the simulation results of the case with and without risk premium in our baseline model.

Figure 4. IRFs to a TFP uncertainty shock in alternative credit constraint with a fixed price of $q$ . Notes: IRFs are expressed as percentage deviations from the stochastic steady state in response to a one-standard deviation uncertainty shock.

4.1 The relative importance of households’ and entrepreneurs’ credit constraints

In this subsection, we examine the relative importance of the financial frictions associated with impatient households and entrepreneurs. In particular, we examine scenarios where only one of these economic agents faces these constraints, as opposed to situations where both patient households and entrepreneurs encounter borrowing restrictions.

4.1.2 The economy with impatient households’ credit constraints only

Next, we shift our focus to an economy in which entrepreneurs’ credit constraints have been removed, while impatient households still encounter credit constraints. Here, the behavior of patient and impatient households is identical to our baseline model. However, the behavior of entrepreneurs is identical to our previous frictionless economy model setting. This setting assumes that entrepreneurs use the same technology described in Eq. (22) and are owned by the patient households. Furthermore, in this case, entrepreneurs are not subject to any borrowing constraints when accumulating capital and housing. Additionally, new market-clearing conditions now becomes $h_{s, t} + h_{b, t} +h_{e, t} = 1$ , $b_{s, t} + b_{b, t} = 0$ , and $c_{s, t} +c_{b, t} + i_t = y_t$ .

During times of heightened uncertainty, the resale housing value becomes more volatile, prompting impatient households to downsize their homes in an effort to reduce their housing exposure. This downsizing leads to a reduction in working hours among impatient households, which in turn causes a decrease in aggregate labor hours, resulting in a decline in aggregate production.

Since the fall in aggregate labor hours also leads to a decrease in the expected marginal productivity of capital, investment will fall in response. However, in the absence of the financial frictions associated with entrepreneurs, the decline in investment is less severe, leading to a milder recession. The simulation results for this scenario are depicted by the green lines in Figure 5.

In summary, financial frictions related to impatient households play a crucial role in addressing the co-movement issue following uncertainty shocks. In contrast, financial frictions associated with entrepreneurs only act as a mechanism that amplifies the reduction in investment, thereby intensifying the recession when compared to scenarios without these financial constraints.

Figure 5. IRFs to a TFP uncertainty shock under different types of collateral constraints. Notes: IRFs are expressed as percentage deviations from the stochastic steady state in response to a one-standard deviation uncertainty shock.

4.2 Varying degrees of financial frictions and their effects

In this exercise, we manipulate the stringency of borrowing constraints for both impatient households and entrepreneurs by adjusting their LTV ratios. Our objective is to gain insight into how these constraints influence the amplification of uncertainty shocks and their subsequent impact on macroeconomic dynamics. Without loss of generality, we examine a scenario in which LTV ratios are uniform across agents, denoted as $m_b = m_e^k = m_e^h = M$ . In this context, a higher $M$ value signifies more relaxed borrowing constraints, allowing both impatient households and entrepreneurs to access greater collateral with a relatively smaller initial down payment, thereby increasing leverage. Consequently, when impatient households respond to heightened uncertainty by reducing their housing holdings, the resulting decline in aggregate labor supply becomes more pronounced at higher LTV ratios, leading to a deeper recession.

Figure 6 presents the IRFs under three different levels of financial friction, specifically with $M$ values set at 0.875, 0.9, and 0.925. The simulation results demonstrate that a lower LTV ratio corresponds to smaller fluctuations in aggregate output, whereas a higher LTV ratio leads to a more substantial decline in aggregate production. From this perspective, deregulation amplifies fluctuations in aggregate production, aligning with prior studies by Liou (Reference Liou2013); Born (Reference Born2011), and Chang (Reference Chang2011). Consequently, policymakers may consider regulating LTV ratios to mitigate excessive fluctuations in aggregate production.

Figure 6. IRFs to a TFP uncertainty shock under varying $M$ . Notes: IRFs are expressed as percentage deviations from the stochastic steady state in response to a one-standard deviation uncertainty shock.

4.3 Targeted macroprudential policies

In this subsection, we further investigate a targeted macroprudential policy by selectively tightening LTV ratios for different types of collateral. Specifically, we focus on three distinct borrowing constraints: (1) households’ housing collateral ( $M_b$ ), (2) entrepreneurs’ housing collateral ( $M^h_e$ ), and (3) entrepreneurs’ capital collateral ( $M^k_e$ ). Since our previous findings suggest that reducing LTV ratios can stabilize the economy, we now assess which specific constraint should be tightened to achieve the most effective stabilization. To this end, we conduct a policy experiment in which we lower each LTV ratio individually to 0.875 while keeping the other two constant at 0.9. This approach allows us to isolate the effects of tightening borrowing constraints on different economic agents and identify the most effective policy intervention in stabilizing the economy.

Figure 7 illustrates the macroeconomic effects of targeted LTV tightening. Comparing the baseline case (blue solid line) to the scenario where $M_b = 0.875$ (red dotted line), we find that tightening borrowing constraints for households serves as the most effective macroprudential policy tool. By limiting household leverage, this policy dampens the amplification of uncertainty shocks, as the reduction in housing held by impatient households is smaller. Consequently, the decline in their labor supply is less pronounced, leading to a milder economic downturn relative to the baseline.

Figure 7. IRFs to a TFP uncertainty shock targeting different collateral constraints. Notes: IRFs are expressed as percentage deviations from the stochastic steady state in response to a one-standard deviation uncertainty shock.

In contrast, adjusting borrowing constraints for entrepreneurs has little to no stabilizing effect. Tightening $M^k_e$ , which governs entrepreneurs’ capital collateral, has virtually no impact on macroeconomic dynamics. Since the real price of capital remains fixed at 1, the risk premium term in Eq. (34) remains unchanged, preventing financial tightening from influencing the broader economy. As a result, IRFs remain nearly identical across scenarios, suggesting that adjusting $M^k_e$ is an ineffective macroprudential tool. Similarly, tightening $M^h_e$ has little effect on macroeconomic dynamics. This is because entrepreneurs primarily use housing as a factor of production and as collateral for borrowing. However, the share of housing in the production function is relatively small, accounting for only 3% of total inputs. As a result, regulating LTV ratios associated with entrepreneurial housing has only a mild impact on macroeconomic fluctuations.

These findings underscore that macroprudential policies targeting household borrowing constraints are the most effective in mitigating economic volatility. While tightening entrepreneurial borrowing constraints has limited or no effect, restricting household leverage directly curtails the transmission of financial shocks, reducing economic instability and preventing excessive fluctuations in aggregate output.

4.4 Comparison with sticky price model

In this subsection, our primary objective is to conduct a quantitative comparison between our baseline model and the conventional sticky price model concerning their respective responses to uncertainty shocks. To address this, we introduce a theoretical framework that encompasses nominal frictions in the context of representative households. This model also incorporates real frictions like consumption habits and investment adjustment costs, which are extensively discussed in the literature. Following this, the section delves into a comparison of the IRFs between our baseline model and the one featuring price stickiness.

4.4.1 The model with sticky price and investment adjustment cost

In the model with price stickiness, there is a representative household that maximizes its lifetime utility by choosing consumption, housing, labor hours and saving just like the patient households in our baseline model. Moreover, the setting of entrepreneur is almost identically to our frictionless economy model. However, due to the investment adjustment cost, the law of motion for capital, Eq. (26), now becomes:

\begin{equation*} k_t = (1-\delta )k_{t-1} + \left [1 - \frac {\phi _k}{2}\left (\frac {i_t}{i_{t-1}}-1\right )^2\right ]i_t\text{,} \end{equation*}

where $\phi _k \geq 0$ represents the investment adjustment cost parameter.

Regarding the behavior of retailers, they purchase wholesale goods at the price $P^w_t \equiv P_t/X_t$ from entrepreneurs. These goods are then transformed into various intermediate goods, which are subsequently sold to the final goods producer. However, in contrast to our baseline model setting where retailers can freely change their prices each period without incurring any expenses, they now encounter quadratic adjustment costs in their price-setting behavior, akin to Rotemberg (Reference Rotemberg1982). Specifically, these adjustment costs take the form $ \Phi _{y, t} \equiv \frac {\phi _p}{2}\left [\frac {P_{t}(z)}{P_{t-1}(z)}-\Pi \right ]^2 Y_{t}$ , where $\phi _p$ is the adjustment cost parameter that determines the level of nominal price rigidity, and $\Pi$ represents the steady-state inflation level. Therefore, retailer $z$ ’s problem now becomes:

\begin{equation*} \max _{\{P_{t+j}(z)\}^\infty _{j=t}} {\textrm{E}}_t \sum _{j=0}^\infty \frac {\beta ^j \lambda _{s, t+j}}{\lambda _{s, t}}\cdot \left [\frac {P_{t+j}(z)-P^w_{t+j}}{P_{t+j}}\cdot Y_{t+j}(z) - \Phi _{y, t} \right ]\text{,} \end{equation*}

subject to its demand function Eq. (38).

The first-order condition associated with retailer $z$ ’s problem is:

(49) \begin{eqnarray} (1-\eta _y)&&\cdot \left [\frac {P_{t}(z)}{P_t}\right ]^{1-\eta _y} + \eta _y \cdot \frac {P^w_t}{P_t}\cdot \left [\frac {P_t(z)}{P_t}\right ]^{-\eta _y} - \phi _p \left [\frac {P_t(z)}{P_{t-1}(z)}-\Pi \right ]\cdot \frac {P_t(z)}{P_{t-1}(z)} \nonumber\\[10pt] && +\, {\textrm{E}}_t \left \{ \frac {\beta _s \lambda _{s, t+1}}{\lambda _{s, t}}\cdot \frac {Y_{t+1}}{Y_t} \cdot \phi _p \left [\frac {P_{t+1}(z)}{P_{t}(z)} -\Pi \right ]\cdot \frac {P_{t+1}(z)}{P_{t}(z)} \right \} = 0 . \end{eqnarray}

Since all retailers face the same profit maximization problem, they all choose the same price, $P_t(z) = P_t$ , and produce the same quantity, $Y_t(z) = Y_t$ . Hence, we get:

(50) \begin{equation} \phi _p (\Pi _{t} - \Pi ) \cdot \Pi _{t} = (1-\eta _y) + \frac {\eta _y}{X_{t}} + {\textrm{E}}_t \left [ \frac {\beta _s \lambda _{s, t+1}}{\lambda _{s, t}} \cdot \frac {Y_{t+1}}{Y_t} \cdot \phi _p (\Pi _{t+1} - \Pi ) \cdot \Pi _{t+1} \right ]\text{.} \end{equation}

Furthermore, the wholesale goods market-clearing condition implies that

\begin{equation*} y_t = \int ^1_0 Y_t(z) dz = Y_t\text{.} \end{equation*}

Figure 8. IRFs to a TFP uncertainty shock—collateral effect versus nominal rigidity. Notes: IRFs are expressed as percentage deviations from the stochastic steady state in response to a one-standard deviation uncertainty shock.