3.1 Households

Households consume $c$ and supply labor $l$ . They provide homogeneous labor services, which are sold to entrepreneurs. Entrepreneurs set the nominal wage $w$ . The representative household maximizes the following utility function:

(1) \begin{align} \max _{\{c_t,l_t\}} \mathbb{E}_t \sum _{s=0}^{\infty } \beta ^{s} \left [ \ln (c_{t+s}) + \phi _l \ln (1-l_{t+s}) \right ] \end{align}

which depends on consumption $c_{t+s}$ , labor supply $l_{t+s}$ , and the relative utility weight of leisure $\phi _l$ , with $\beta \in (0,1)$ denoting the household’s subjective discount factor. The household’s maximization problem is subject to the following budget constraint in real terms:

(2) \begin{align} c_t + d_t = \dfrac {R_{t-1}^{d} d_{t-1}}{\pi _t} + w_t l_t + \Gamma ^{CP}_{t} + \Gamma ^{R}_{t} + div_{t}^{B} \end{align}

where $\pi _t \equiv \frac {p_t}{p_{t-1}}$ denotes the inflation rate, $d_t$ represents households’ deposits, $R^{d}_{t}$ is the interest rate on deposits, and $\Gamma ^{CP}_{t}$ and $\Gamma ^{R}_{t}$ are lump sum aggregate profits from the capital producers and retailers sectors, respectively. Dividends $div_{t}^{B}$ are paid by CBs. The first-order conditions of this problem can be expressed as:

(3) \begin{align} \lambda _t &= \dfrac {1}{c_t} \\[-12pt] \nonumber \end{align}

(4) \begin{align} \lambda _t w_t &= \dfrac {\phi _l}{(1-l_t)} \\[-12pt] \nonumber \end{align}

(5) \begin{align} \lambda _t &= \beta \mathbb{E}_t \left [ \lambda _{t+1} \dfrac {R_{t}^{d}}{\pi _{t+1}} \right ] \\[12pt] \nonumber \end{align}

where (5) represents the intertemporal Euler equation, which can also be written as:

(6) \begin{align} 1 = \mathbb{E}_t \left [\Lambda _{t,t+1}\dfrac {R_{t}^{d}}{\pi _{t+1}}\right ] \end{align}

where $ \Lambda _{t,t+1} = \beta \frac {\lambda _{t+1}}{\lambda _t} = \beta \frac {u'(c_{t+1})}{u'(c_t)} = \beta \frac {c_t}{c_{t+1}}$ is the stochastic discount factor in period $t$ for real payoffs in period $t+1$ , with $u(c_t) = \ln (c_t)$ .

3.2 Entrepreneurs

Entrepreneurs derive utility from consumption $c^{E}_{t}$ :

(7) \begin{align} \mathbb{E}_t \sum _{s=0}^{\infty } (\beta ^E)^s \ln \left(c_{t+s}^{E} \right) \end{align}

where $\beta ^E$ is the discount factor. Entrepreneurs maximize their lifetime utility subject to the following budget constraint:

(8) \begin{align} c^{E}_{t} + w_t l_{t} + q_t k_{t} + \dfrac {R^{b}_{t-1}b_{t-1}}{\pi _t} = \dfrac {y_{t}^{w}}{x_t} + q_t(1-\delta )k_{t-1} + b_t \end{align}

where $\delta$ is the depreciation rate of physical capital, $q_t$ is the price of physical capital, $R^{b}_{t}$ is the interest rate on loans $b_t$ , and $\frac {1}{x_t}$ denotes the price of the wholesale good $y^{w}_{t}$ produced in period $t$ using Cobb-Douglas production technology with constant returns to scale (CRS):

(9) \begin{align} y^{w}_{t} = z_t k_{t-1}^{\alpha } l_{t}^{1-\alpha } \end{align}

where productivity $z_t$ follows an autoregressive AR(1) process:

(10) \begin{align} \ln (z_t) = \psi _z \ln (z_{t-1}) + \varepsilon _{t}^{z} \end{align}

where $\psi _z \in (0,1)$ is the persistence of process $z_t$ , and $\varepsilon _{t}^{z}$ is a productivity shock with variance $\sigma ^{2}_{z}$ .

Entrepreneurs are also subject to the borrowing constraint:

(11) \begin{align} b_t \leq m^{k}_{t} \mathbb{E}_t \left [ \dfrac {q_{t+1} (1-\delta ) k_{t} \pi _{t+1} }{R^{b}_{t}} \right ] \end{align}

where $m^{k}_{t}$ is the pledgeability ratio, and the term in brackets represents the value of the entrepreneurs’ collateral given by the market value of physical capital $k_t$ . Assuming that physical capital $k_t$ can be used as collateral, let $m^{k}_{t} \in (0,1)$ denote the fraction of physical capital that banks can seize if entrepreneurs default on their loans. The pledgeability ratio $m_{t}^{k}$ is subject to a collateral shock and follows an AR(1) process:

(12) \begin{align} \ln \big(m_{t}^{k}\big) = \psi _{m^{k}} \ln \left(m^{k}_{t-1}\right) + \varepsilon ^{m^{k}}_{t} \end{align}

where $\psi _{m^{k}}$ indicates the persistence of process $m_{t}^{k}$ , and $\varepsilon ^{m^{k}}_{t}$ is a collateral shock with variance $\sigma ^{2}_{m^k}$ . The first-order conditions of this problem are:Footnote ⁸

(13) \begin{align} \lambda _{1,t}^{E} &= \dfrac {1}{c_{t}^{E}} \\[-12pt] \nonumber \end{align}

(14) \begin{align} w_t &= (1-\alpha ) \dfrac {y^{w}_{t}}{x_t l_{t}} \\[-12pt] \nonumber \end{align}

(15) \begin{align} \lambda _{2,t}^{E} &= \lambda _{1,t}^{E} - \beta ^E \mathbb{E}_t \left [\lambda _{1, t+1}^{E} \dfrac {R^{b}_{t}}{\pi _{t+1}}\right ] \\[-12pt] \nonumber \end{align}

(16) \begin{align} \lambda _{1,t}^{E} q_{t} &=\beta ^E \mathbb{E}_t \left [\lambda _{1,t+1}^{E} \left (\alpha \dfrac {y^{w}_{t+1}}{x_{t+1} k_{t}} + (1-\delta ) q_{t+1}\right )\right ] \notag \\ &\quad + \lambda _{2,t}^{E} \mathbb{E}_t \left [\dfrac {m^{k}_{t}(1-\delta ) q_{t+1} \pi _{t+1}}{R^{b}_{t}}\right ] \\[12pt] \nonumber \end{align}

Combining (13) and (15), we obtain the following expression in the steady state:

(17) \begin{align} \lambda _{2}^{E} = \dfrac {1}{c^E}\left ( 1-\beta ^E \dfrac {R^{b}}{\pi } \right ) \end{align}

To ensure that the borrowing constraint is always binding in the steady state, $\lambda _{2}^{E}$ must be positive, which implies that $\beta ^{E} \lt \beta$ . The heterogeneity in $\beta$ and $\beta ^E$ guarantees that entrepreneurs are net borrowers in the steady state.Footnote ⁹

The entrepreneur’s net worth $n_t$ in period $t$ , after the productivity shock has been realized and the output $y^{w}_{t}$ produced, is defined as:

(18) \begin{align} n_t = \dfrac { y^{w}_{t}}{x_t} - w_t l_t + q_t(1-\delta )k_{t-1} - \dfrac {R_{t-1}^{b}b_{t-1}}{\pi _t} \end{align}

where $q_t(1-\delta )k_{t-1}$ is the total value of the capital stock and $\frac {R_{t-1}^{b}b_{t-1}}{\pi _t}$ is the loan interest payment at the beginning of period $t$ . Then, $c^{E}_{t}$ can be expressed in terms of $n_t$ :

(19) \begin{align} c_{t}^{E} + q_t k_{t} = n_t + b_t \end{align}

which implies that entrepreneurs finance consumption $c^{E}_{t}$ and the purchase of new capital $k_t$ through bank loans $b_t$ and retained earnings $n_t$ . Under the assumption of log utility, $c_{t}^{E}$ is a fixed proportion of accumulated profits $n_t$ :

(20) \begin{align} c_{t}^{E} = (1-\beta ^E)n_t \end{align}

Note that the binding borrowing constraint (11) determines entrepreneurs’ loan demand, implying an inverse relationship between the equilibrium loan rate $R^{b}_{t}$ and loan quantity $b_t$ . In a perfectly competitive banking environment, the loan rate $R^{b}_{t}$ equals the gross deposit rate $R^{d}_{t}$ , thus $b_t$ is determined accordingly. Under IBC, each bank determines its loan quantity and thereby influences the loan rate $R^{b}_{t}$ . In particular, for given asset prices $q_{t+1}$ and $\pi _{t+1}$ , a higher loan rate $R^{b}_{t}$ corresponds to a lower loan quantity $b_t$ .

3.3 Capital producers

The perfectly competitive capital producers purchase non-depreciated capital from entrepreneurs and also purchase final consumption good $i_t$ from retail firms to produce new capital $k_t$ at the end of period $t$ :

(21) \begin{align} k_t = (1-\delta )k_{t-1} + i_t \end{align}

where $i_t$ is also gross investment. The activity of capital producers increases the physical capital stock $k_t$ , which is then sold back to entrepreneurs at price $q_t$ at the end of the period. Following Christiano et al. (Reference Christiano, Eichenbaum and Evans2005), it is assumed that old capital can be converted into new capital at a one-to-one rate subject to a quadratic investment adjustment cost $f\big(\frac {i_t}{i_{t-1}}\big) = \frac {\chi }{2}\big(\frac {i_t s^{qk}_{t}}{i_{t-1}} -1\big)^2$ . Moreover, $\chi \gt 0$ reflects the magnitude of the adjustment cost, and $s_{t}^{qk}$ is the total factor productivity of the investment $i_t$ that follows an AR(1) process:

(22) \begin{align} \ln \left(s_{t}^{qk}\right) = \psi _{s^{qk}}\ln \left(s_{t-1}^{qk}\right)+ \varepsilon _{t}^{qk} \end{align}

where $\psi _{s^{qk}}$ measures the degree of persistence of $s_{t}^{qk}$ and $\varepsilon _{t}^{qk}$ is an investment-productivity shock with variance $\sigma ^{2}_{s^{qk}}$ . The capital producers’ optimization problem can be written as:

(23) \begin{align} \begin{aligned} \max _{\{i_t,k_t\}} \quad & \mathbb{E}_t \sum _{s=0}^{\infty } \Lambda _{t,t+s} \left [ q_t k_t - q_t(1-\delta ) k_{t-1} - i_t - \dfrac {\chi }{2} \left ( \dfrac {i_t s_{t}^{qk} }{i_{t-1}}-1\right )^2 i_t\right ] \\ \textrm {s.t.}\quad & k_t = (1-\delta )k_{t-1} + i_t \end{aligned} \end{align}

where $\Lambda _{t,t+s} \equiv \beta ^s \frac {u'(c_{t+s})}{u'(c_t)}$ is the stochastic discount factor since households own capital producers. The capital producers’ problem yields the following relation to the capital price $q_t$ , taking the first-order condition with respect to $i_t$ :

(24) \begin{align} q_t &= 1 + \frac {\chi }{2}\left (\frac {i_t s_t^{qk}}{i_{t-1}}-1\right )^2 + \chi \left (\frac {i_t s_t^{qk}}{i_{t-1}}-1\right )\left (\frac {i_t}{i_{t-1}}\right )s_t^{qk} \notag \\ &\quad - \chi \mathbb{E}_t \left [ \Lambda _{t,t+1}\left (\frac {i_{t+1}s_{t+1}^{qk}}{i_t}-1\right ) \left (\frac {i_{t+1}}{i_t}\right )^2 s_{t+1}^{qk} \right ] \end{align}

All profits $\Gamma _{t}^{CP}$ earned outside the steady state ( $q \neq 1$ ) by the capital producers’ sector are returned to households, where $\Gamma _{t}^{CP} = (q_t -1)i_t - \frac {\chi }{2}\left (\frac {i_t s_{t}^{qk}}{i_{t-1}}-1\right )^2 i_t$ .

3.4 Retailers

The retailers are assumed to be monopolistically competitive. They buy wholesale good $y^{w}_{t}(i)$ from entrepreneurs at wholesale price $p^{w}_{t}(i)$ and differentiate it without cost. Each retailer $i$ then sells its unique variety $y_t(i)$ at nominal price $p_t(i)$ , applying a markup $x_t(i) = \frac {p_t(i)}{p^{w}_{t}(i)}$ on the wholesale price. The output of the final consumption good $y_t$ is a constant elasticity of substitution (CES) composite of all the different varieties produced by the retailers (using the Dixit and Stiglitz (Reference Dixit and Stiglitz1977) framework):

(25) \begin{align} y_t = \left [ \int _{0}^{1} y_{t} (i)^{\frac {\epsilon -1}{\epsilon }} di\right ]^{\frac {\epsilon }{\epsilon -1}} \end{align}

where $\epsilon \gt 1$ is the elasticity of substitution across different varieties. Retailers’ prices are sticky and indexed to a combination of past and steady-state inflation. Then, retailers must choose $\{p_t(i)\}_{t=0}^{\infty }$ to maximize profits given by:

(26) \begin{align} \Gamma ^{R} &= \mathbb{E}_t \sum _{s=0}^{\infty } \Lambda _{t,t+s} \Big [p_t(i) y_t(i) - p_t^{w}(i) y_t(i) - \frac {\kappa _{\pi }}{2}\Big (\frac {p_t(i)}{p_{t-1}(i)} - \pi _{t-1}^{\iota _p} \overline {\pi }^{\,1-\iota _p}\Big )^2 p_t y_t \Big ] \end{align}

subject to a downward-sloping demand coming from consumers’ maximization of a consumption aggregator:

(27) \begin{align} y_t(i) = y_t \left ( \dfrac {p_t(i)}{p_t}\right )^{-\epsilon } \end{align}

where $\kappa _{\pi }$ is a quadratic adjustment cost for changing prices beyond the indexation rule. In symmetrical equilibrium, $p_t(i) = p_t$ , the first-order conditions imply the Phillips curve, given by:

(28) \begin{align} \frac {\epsilon }{x_t} - \kappa _{\pi }\big (\pi _t - \pi _{t-1}^{\iota _p} \overline {\pi }^{1-\iota _p}\big )\pi _t + \mathbb{E}_t\!\left [ \Lambda _{t,t+1} \kappa _{\pi } \big (\pi _{t+1} - \pi _t^{\iota _p} \overline {\pi }^{1-\iota _p}\big ) \pi _{t+1}^2 \frac {y_{t+1}}{y_t}\right ] &= \epsilon - 1 \end{align}

where $\Lambda _{t,t+s} = \beta ^s \frac {u'(c_{t+s})} {u'(c_t)}$ is the stochastic discount factor since households own retail firms, and $x_t = \frac {p_t(i)}{p_{t}^{w}(i)} = mc_t(i)$ is the markup of the final good price.

3.5 Banks

The Cournot banking sector is used to characterize oligopolistic competition and capture banks’ market power, as the banking sector tends to be dominated by a few prominent players that accumulate capital.Footnote ¹⁰ The model’s framework does not allow new banks to enter the banking sector in response to profits, allowing the effects of Cournot competition to be analyzed without the dynamics of market entry. Stricter capital requirements can act as a barrier to entry for new banks, since these regulations require banks to maintain a considerable amount of capital, serving as a safety net that protects them from unexpected losses and ensures solvency during periods of financial stress. Thus, while stricter capital requirements promote financial stability, they may simultaneously deter new banks from entering the market.Footnote ¹¹

Assume there are $N$ banks in the economy, indexed by $j$ . Banks have market power in conducting their intermediation activity, which allows them to adjust loan rates in response to adverse shocks. CBs play an essential role in the model, as they act as intermediaries for all financial transactions between agents in the economy and must satisfy the following balance-sheet identity:

(29) \begin{align} b_t(j) = d_t(j) + k^{B}_{t}(j) \end{align}

where loans $b$ equal deposits $d$ plus bank capital $k^{B}$ .Footnote ¹² Equation (29) shows that banks use household deposits together with the capital accumulated over time to issue loans in the economy. However, the model does not consider factors that might discourage individuals from depositing their money in banks, such as low interest rates or lack of trust in the banking system. If such issues arose, banks would have to rely exclusively on liquidating their accumulated capital to provide loans, which could eventually lead to solvency problems.

Banks aim to keep the capital-to-loans ratio $\frac {k^B}{b}$ close to an optimal target $\tau ^{B}_{t}$ , which can be interpreted as a capital requirement imposed by the regulatory authority.Footnote ¹³ The capital requirement is essential for protecting banks from financial crises, ensuring that they have sufficient capital to meet deposit withdrawals and fulfill other obligations. Acting as an effective buffer, the capital requirement is a key element in the system’s resilience, absorbing financial shocks and enhancing stability.Footnote ¹⁴ The time-varying capital requirement ratio $\tau ^{B}_{t}$ will be the main instrument of macroprudential policy:

(30) \begin{align} \Omega _{t}^{B} (j) = \dfrac {\kappa _{k^B}}{2}\left (\dfrac {k^{B}_{t}(j)}{b_t(j)} - \tau ^{B}_{t} \right )^2 k^{B}_{t} (j) \end{align}

where $\kappa _{k^B}$ is the magnitude of the cost $\Omega ^{B}_{t}$ that banks incur if they deviate from $\tau ^{B}_{t}$ .Footnote ¹⁵ Thus, bank capital in period $t$ is accumulated from retained earnings $\Gamma ^{B}_{t}$ , after accounting for dividend payments $div_{t}^{B}$ , plus the bank capital $k_{t-1}^{B}$ from the previous period:

(31) \begin{align} k^{B}_{t} (j) = (1-\delta ^B)\dfrac {k^{B}_{t-1}(j)}{s^{k^B}_{t}} + \Gamma _{t}^{B}(j) - div^{B}_{t} (j) \end{align}

where $\delta ^B$ is the bank operation cost, and the retained earnings $\Gamma ^{B}_{t}$ can be written as:

(32) \begin{align} \Gamma _{t}^{B}(j) = \dfrac {1}{\pi _t}\left [ R_{t-1}^{b} \left (b_{t-1} (j) + \sum _{m \neq j} b_{t-1}(m) \right )b_{t-1}(j) - R_{t-1}^{d} d_{t-1}(j) - \Omega _{t-1}^{B}(j) \right ] \end{align}

The dependance of the loan rate $R^b$ on loans $b$ means that each bank $j$ has a certain degree of control over the equilibrium gross loan interest rate by adjusting its own quantity $b_t(j)$ given the amounts of $b_t(m)$ granted by the other banks ( $m \neq j$ ) in a banking system with a Cournot structure. Bank capital accumulation (31) is subject to an unexpected financial shock $s^{k^B}_{t}$ , which follows an AR(1):

(33) \begin{align} \ln \left(s^{k^B}_{t} \right) = \psi _{s^{k^B}} \ln \left(s^{k^B}_{t-1} \right) + \varepsilon ^{k^B}_{t} \end{align}

where $\varepsilon ^{k^B}_{t}$ is the financial shock with variance $\sigma ^{2}_{k^B}$ . In each bank’s optimization problem, bank $j$ maximizes the present discounted value of future dividends, subject to the bank capital accumulation law and $b(j), div(j) \geq 0$ :Footnote ¹⁶

(34) \begin{align} \max _{\{b_t(j), k^{B}_{t}(j), div_{t}^{B}(j)\}} \mathbb{E}_t \sum _{s=0}^{\infty } \Lambda _{t,t+s} \big[\ln \big(div_{t+s}^{B} (j)\big) \big] \end{align}

where $\Lambda _{t,t+s} = \beta ^s\frac {u'(c_{t+s})}{u'(c_t)}$ is the stochastic discount factor, as households own the banks. Solving the banks’ optimization problem with respect to $b(j)$ , this yields:

(35) \begin{align} \mathbb{E}_t \Lambda _{t,t+1} \left \{\dfrac {\lambda _{t+1}^{B}(j)}{{\pi _{t+1}}} \left [ \dfrac {\partial \Omega _{t}^{B}(j)}{\partial b_t(j)} - \left (\dfrac {\partial R_{t}^{b}}{ \partial b_t(j)} b_t(j) + R_{t}^{b} - R_{t}^{d} \right ) \right ]\right \} = 0 \end{align}

where $\lambda _{t}^{B}(j)$ is the Lagrange multiplier of the banks’ optimization problem. In Cournot equilibrium, the total optimal loan quantity is $b_t = b_t(j) + \sum _{m\neq j} b_t(m)$ , and the total optimal bank capital is $k_{t}^{B} = k_{t}^{B}(j) + \sum _{m\neq j} k_{t}^{B}(m)$ . Assuming banks are identical, $b(j) = \frac {b_t}{N}$ , $k_{t}^{B}(j) = \frac {k_{t}^{B}}{N}$ , and $\frac {\partial R_{t}^{b}}{\partial b_t(j)} = \frac {\partial R_{t}^{b}}{\partial b_t } \frac {\partial b_t}{\partial b_t(j)} = \frac {\partial R_{t}^{b}}{\partial b_t }$ . In Cournot equilibrium, the first-order condition (35) can be written as:

(36) \begin{align} \mathbb{E}_t \Lambda _{t,t+1} \left \{\dfrac {\lambda _{t+1}^{B}}{{\pi _{t+1}}} \left [ \dfrac {\partial \Omega _{t}^{B}}{\partial b_t} - \left (\dfrac {\partial R_{t}^{b}}{ \partial b_t} \dfrac {b_t}{N} + R_{t}^{b} - R_{t}^{d} \right ) \right ]\right \} = 0 \end{align}

The loan rate directly affects market loan demand $b_t$ because an increase in $R^{b}_{t}$ reduces entrepreneurs’ borrowing capacity. Furthermore, the loan rate has an indirect effect on loans taken by borrowers, influencing entrepreneurs’ demand for physical capital.Footnote ¹⁷ When bank $j$ chooses $b_t(j)$ to maximize dividends, it needs to consider how entrepreneurs would respond by changing their demand for physical capital, $\frac {\partial k_t}{\partial R_{t}^{b}}$ , which affects the level of investment in the economy. Entrepreneurs’ demand for physical capital decreases with the loan rate because $\frac {\partial k_t}{\partial R_{t}^{b}} \lt 0$ , and the interest rate elasticity of capital demand, $PEK_t \equiv -\frac {\partial k_t}{\partial R_{t}^{b}}\frac {R^{b}_{t}}{k_t}$ , monotonically decreases with the expected marginal product of capital:

(37) \begin{align} PEK_t = \dfrac {1}{1-\alpha } \left ( \dfrac { m^{k}_{t}\mathbb{E}_t \left [ \dfrac {q_{t+1}(1-\delta )\pi _{t+1}}{R^{b}_{t}} \right ] }{ \mathbb{E}_t \left [\Lambda _{t,t+1}^{E} MPK_{t+1} \right ]} \right ) \end{align}

where $MPK_{t+1} \equiv \frac {\alpha z_{t+1} (k_t)^{\alpha -1} (l_{t+1})^{1-\alpha }}{x_{t+1}}$ is the marginal product of capital in real terms. The entrepreneurs’ loan demand elasticity $PED_t$ captures their dependency on the capital demand elasticity $PEK_t$ , which is given by:Footnote ¹⁸

(38) \begin{align} PED_t \equiv - \dfrac {\partial b_t} {\partial R_{t}^{b}} \dfrac {R_{t}^{b}}{b_t} = 1 + PEK \gt 0 \end{align}

Solving (36), the following expression for the loan interest rate is obtained, assuming $\Lambda _{t,t+1} \gt 0$ and $\pi _{t+1} \equiv \frac {p_{t+1}}{p_t} \gt 0$ :

(39) \begin{align} R_{t}^{b} = \dfrac {R_{t}^{d} - \kappa _{k^B}\left (\dfrac {k_{t}^{B}} {b_{t}} - \tau ^{B}_{t} \right ) \left (\dfrac {k_{t}^{B}}{b_t} \right )^2}{\left (1 - PED_{t}^{-1} \dfrac {1}{N} \right )} \end{align}

where $N$ is the number of banks, $\kappa _{k^B}$ is the bank capital adjustment cost, and the spread, given by $R^b - R^d$ , is the difference between the rate charged on loans taken by entrepreneurs and the rate paid on households’ deposits.

From (39), $R_{t}^{b}$ decreases with the number of banks $N$ (more banking competition), and with the elasticity $PED_t$ . Entrepreneurs respond quickly to an increased loan interest rate and reduce the amount of loans $b_t$ demanded, which forces banks to lower $R_{t}^{b}$ . Modeling the leverage level of banks (and their capital accumulation) allows the introduction of shocks originating on the credit supply side. This framework enables the study of their effects on setting loan rates subject to capital requirements and their propagation in the real economy. In particular, it is possible to analyze the impact of a weakening in the balance sheet position of the Cournot banking sector, arising from an exogenous shock that destroys banking capital.

In an environment characterized by Cournot competition with bank capital adjustment cost $\kappa _{k^B}$ , if banks maintain a $\frac {k^B}{b}$ ratio above the minimum requirement imposed by the macroprudential authority, the excess capital would have several implications for the equilibrium loan rate $R^b$ . Specifically, higher capital ratios could lead to a situation where the interest rate on deposits exceeds the interest rate on loans. This misalignment would result in significant financial losses for banks, as they pay higher rates to depositors while receiving lower rates on loans. Consequently, banks would have no incentive to accumulate capital beyond what the macroprudential authority mandates.

3.6 Equilibrium

Equilibrium is imposed on the deposit and loan markets, and the aggregate resource constraint is:

(40) \begin{align} c_t + c^{E}_{t} + i_t + div_{t}^{B} + \dfrac {\kappa _{\pi }}{2} (\pi _t - \overline {\pi })^2 y_t + \frac {\chi }{2}\left (\frac {i_t}{i_{t-1}}-1\right )^2 i_t + \dfrac {\kappa _{k^B}}{2}\left (\frac {k_{t}^{B}}{b_{t}} - \tau ^{B}_{t} \right )^2 k_{t}^{B} = y_t \end{align}

The new capital supplied by capital producers equals entrepreneurs’ capital demand, and the labor supplied by households equals entrepreneurs’ labor demand. The Cournot equilibrium of the banking sector can be written as:

(41) \begin{align} b_t^{B} = \sum _{j=1}^{N} b_t(j), \quad d_t^{B} = \sum _{j=1}^{N} d_t(j), \quad k_t^{B} = \sum _{j=1}^{N} k_t^{B}(j) \end{align}

where the banking sector’s loan supply $b^{B}_{t}$ equals market loan demand $b_{t}$ ; the banking sector’s demand for deposits $d^{B}_{t}$ equals households’ deposit supply $d_{t}$ ; and the banking system capital $k_{t}^{B}$ is equal to the sum of the $N$ banks’ capital. From (29), the total loan supply $b_{t}^{B}$ equals the total deposits $d^{B}_{t}$ held in the banking system plus the total capital accumulated $k_{t}^{B}$ in the banking sector:

(42) \begin{eqnarray} b_{t}^{B} = d^{B}_{t} + k_{t}^{B} \end{eqnarray}

4.1 Monetary policy

The central bank instrument is modeled using the Taylor rule, specified as:

(43) \begin{align} R_{t} = (1-\rho _r)\overline {R} + (1-\rho _r) [\phi _{\pi }(\pi _t - \overline {\pi }) + \phi _{y}(y_t - \overline {y})] + \rho _r R_{t-1} \end{align}

where $\phi _{\pi }$ and $\phi _{y}$ measure the responses to inflation and output deviations from their targets, and $\rho _r$ captures the persistence in interest rate adjustments. In this economy, the central bank controls the gross nominal interest rate on deposits $R^{d}$ paid by CBs to households, according to the Taylor rule. In this way, $R^{d}_{t} = R_t$ , and deviations in inflation and output from their respective targets affect the deposit rate $R^d$ .

4.2 Macroprudential policy

In line with Angelini et al. (Reference Angelini, Neri and Panetta2014), it is assumed that macroprudential authorities react to abnormal credit behavior, as one of the objectives of macroprudential policy is to ensure the financial system’s resilience. Time-varying capital requirements are chosen as the macroprudential instrument for two main reasons. First, banking capital has been central in the debate on regulatory reform, and the Basel III package has recently introduced a countercyclical capital requirement. Second, systemic financial crises directly or indirectly affect banking capital and credit supply. Thus, the model directly addresses how capital buffers influence the banking sector’s ability to withstand shocks and continue lending. The macroprudential authority is assumed to define a time-varying capital requirement rule, as follows:

(44) \begin{align} \tau ^{B}_{t} = (1-\rho _{\tau ^B})\overline {\tau }^B + (1-\rho _{\tau ^B}) \left [ \chi _{\tau ^B} \left ( \dfrac {b_t}{y_t} - \dfrac {\overline {b}}{\overline {y}} \right )\right ] + \rho _{\tau ^B} \tau ^{B}_{t-1} \end{align}

where $\overline {\tau }^B$ is the steady-state value of $\tau ^{B}_{t}$ . In the baseline specification of (44), the macroprudential authority is assumed to adjust capital requirements only in response to deviations in the loans-to-output ratio.Footnote ¹⁹

The parameter $\chi _{\tau ^B}$ in macroprudential policy measures the response of capital requirements to deviations of the loans-to-output ratio from its target $\frac {\overline {b}}{\overline {y}}$ . This parameter captures how aggressively the macroprudential authority reacts to discrepancies between the actual loan-to-output ratio and its target. If $\chi _{\tau ^B} \gt 0$ , an increase in $\frac {b}{y}$ above its target causes the macroprudential authority to raise capital requirements $\tau ^B$ , making policy countercyclical. If $\chi _{\tau ^B} \lt 0$ , policy is procyclical, and an increase in $\frac {b}{y}$ above its target reduces capital requirements. If $\chi _{\tau ^B} = 0$ , no macroprudential policy is applied. As the macroprudential instrument affects lending rates, policymakers can directly offset fluctuations in the spread caused by adverse shocks.

6.1 Productivity shock

Figure 1 shows the impulse responses to an adverse productivity shock $z$ for loans, inflation, the spread, and other main variables. The scenario with only monetary policy amplifies the negative shock $z$ ’s effects on output $y$ in the initial quarters. Productivity $z$ falls at the beginning of period one, reducing the expected Marginal Product of Capital (MPK). The decline in MPK lowers entrepreneurs’ demand for physical capital and causes a drop in its price $q$ . The reduction in entrepreneurs’ collateral value makes them more financially constrained with inelastic demand for loans. Under Cournot competition, banks respond to the more inelastic loan demand by reducing their loan supply to achieve a higher loan rate, which leads to a rise in the spread.

A higher spread allows banks to accumulate more capital, increasing the $\frac {k^B}{b}$ ratio in both policy scenarios. However, the reduction in loans provided by banks leads to a fall in the loans-to-output ratio, triggering a response from the macroprudential authority, which reduces the capital requirement $\tau ^{B}$ . The movement of $\tau ^{B}$ in the opposite direction of the $\frac {k^B}{b}$ ratio mitigates the effect of the productivity shock through the BS channel. Consequently, the spread increase is more significant when only monetary policy is active, compared to the scenario with active macroprudential policy.

The differential responses of output and other macroeconomic variables under the two policy scenarios can be explained by the difference in spread fluctuations when capital requirements vary over time. The lower cost of loans under active macroprudential policy impacts entrepreneurs’ accumulation of physical capital $k$ and the level of investments. The decline in $k$ is larger in the scenario with only monetary policy. Entrepreneurs’ capital begins to recover around 20 quarters after the initial shock in the scenario with macroprudential policy. This coincides with the period when the spread returns to its initial equilibrium and fluctuations are eliminated. The fall in output $y$ under active macroprudential policy is smaller than in the scenario with only monetary policy.

Given that entrepreneurs use bank loans $b$ to finance their activities, reducing the cost of loans in the presence of macroprudential policy allows them to face a smoother budget constraint, thereby raising their consumption level $c^{E}$ compared to the scenario with only monetary policy. A similar pattern is observed for household consumption $c$ , where the smaller decline in accumulated output $y$ under macroprudential policy results in higher household income and, consequently, higher consumption. With a smaller drop in $y$ and other productive sectors being less affected, the decline in labor $l$ hired by entrepreneurs is also smaller compared to the scenario with only monetary policy, with employment returning to its initial equilibrium after about 20 quarters following the negative productivity shock. Figure 1 shows that the larger decline in accumulated output in the only monetary policy scenario leads to higher inflation, which reduces households’ purchasing power. In this way, the central bank increases the policy rate to restore households’ consumption.

Figure 1.

Impulse responses to a negative productivity shock.

Note: IRFs to a negative $z$ shock under the baseline (TR) and time-varying capital requirements (MP) scenarios.

Figure 2.

Impulse responses to a negative collateral shock.

Note: IRFs to a negative $m^k$ shock under the baseline (TR) and time-varying capital requirements (MP) scenarios.

6.2 Collateral shock

Figure 2 shows the effects of a negative collateral shock $m^k$ that reduces the guarantees entrepreneurs can pledge as collateral to obtain loans.Footnote ²⁰ This adverse shock affects entrepreneurs’ loan demand elasticity (PED), making them more financially constrained. Banks respond by reducing loan supply to raise the loan rate, given the more inelastic demand and entrepreneurs’ reliance on credit. However, even if banks set a higher loan rate, the reduction in entrepreneurs’ collateral prevents an increase in bank capital accumulation $k^B$ , which directly lowers the capital-to-loans ratio. The decline in $\frac {k^B}{b}$ generates a higher amplification of the spread if capital requirements $\tau ^B$ were fixed over time. Under Cournot competition and time-varying capital requirements, banks adjust by reducing the loans-to-output ratio $\frac {b}{y}$ . In response, the macroprudential authority lowers capital requirements, according to (44). This policy intervention mitigates the amplification of the spread through the bank-stress channel, leading to a more favorable economic outcome under active macroprudential policy.

Investment recovers rapidly when macroprudential policy is active, taking about five quarters to return to its initial equilibrium. With only monetary policy, a return to the initial equilibrium occurs about 20 quarters after the shock. The rapid recovery of entrepreneurs allows physical capital $k$ to return to its initial equilibrium within a few periods under active macroprudential policy. In contrast, the fall in physical capital lasts around 15 quarters with only monetary policy, and full recovery is achieved only after 40 quarters. Therefore, the quick recovery of investment by entrepreneurs affects accumulated output in the economy, preventing a decline in output as severe as in the scenario with only monetary policy. Thus, the output decline under active macroprudential policy is smaller than in the scenario with only monetary policy.

Figure 2 also shows that the agents most affected by a negative collateral shock are entrepreneurs, who experience a sharp decline in consumption following the reduction in credit granted by oligopolistic banks due to the loss of part of their collateral. With only monetary policy, entrepreneurs’ consumption $c^E$ takes around 20 quarters to return to its initial equilibrium, whereas the introduction of macroprudential policy reduces this time to approximately 10 quarters, due to the lower cost of credit. The decline in entrepreneurs’ assets caused by the negative collateral shock also affects labor, resulting in a sharp drop when there is no regulation on CBs. Employment recovery takes approximately 40 quarters. For this negative shock, the central bank responds to the increase in inflation by raising the policy rate.

6.3 Financial shock

Figure 3 shows the effects of a bank capital loss resulting from an unexpected shock that destroys bank capital $k^B$ . It can be observed that macroprudential policy is more effective in reducing the spread amplification for this type of shock because it acts through the BS channel. A sudden drop in bank capital (31) reduces the capital-to-loans ratio $\frac {k^B}{b}$ and increases the loan rate in (39). The increase in the loan rate $R^b$ is a mechanism that banks use to recover their losses in the oligopolistic banking sector, which directly raises the spread. In the model, $\chi _{\tau ^B} = 0.5$ ensures that $\tau ^B$ reacts countercyclically to the loans-to-output ratio and remains highly persistent ( $\rho _{\tau ^B} = 0.9$ ).

Figure 3.

Impulse responses to a negative financial shock.

Note: IRFs to a negative $s^{k^B}$ shock under the baseline (TR) and time-varying capital requirements (MP) scenarios.

Figure 4.

Impulse responses to a negative investment shock.

Note: IRFs to a negative $s^{qk}$ shock under the baseline (TR) and time-varying capital requirements (MP) scenarios.

Comparing both policy scenarios reveals that monetary policy alone is insufficient to stabilize the economy under an adverse financial shock. When time-varying capital requirements are implemented, spread fluctuations are significantly lower. The reduction in the loans-to-output ratio $\frac {b}{y}$ due to the decline in loans, triggers an adjustment in capital requirements (44), lowering them. This reduces the gap between the capital-to-loans ratio $\frac {k^B}{b}$ and the capital requirements $\tau ^B$ . With the smaller increase in the spread, entrepreneurs are less financially constrained, mitigating the fall in investment when macroprudential policy is active. The lower cost of credit also reduces the decline in physical capital $k$ compared to the scenario with only monetary policy. Consequently, the mitigated effect on the spread results in a smaller contraction of accumulated output.

The implementation of a macroprudential policy to minimize the effects of a negative financial shock directly impacts entrepreneurs’ consumption, causing $c^E$ to rapidly recover to equilibrium due to the lower cost of bank credit. Consequently, entrepreneurs can maintain higher leverage, and their consumption returns to equilibrium after roughly 25 quarters. A similar pattern is observed for employment, which returns to its initial equilibrium within 20 quarters as investment by entrepreneurs recovers in the presence of macroprudential policy.

However, households experience a sharp drop in consumption $c$ in the absence of macroprudential policy, as they receive lower dividends from oligopolistic banks that incur costs due to deviation from the optimal target $\tau ^B$ . Additionally, households face reduced hiring opportunities from entrepreneurs and lower profits from the capital-producing and retail sectors, given the decline in productive activity. Figure 3 shows that in an adverse financial shock scenario, the central bank responds to the smaller increase in inflation under active macroprudential policy with a correspondingly lower increase in the policy rate, compared to the scenario with only monetary policy.

6.4 Investment shock

Figure 4 shows the effects of a negative investment shock, which increases the cost of transforming old physical capital into new capital for the production of wholesale goods. This shock explicitly affects capital producers in the economy. The investment shock reduces the price $q$ of collateral that entrepreneurs can offer to banks to obtain loans, making them more financially constrained (inelastic PED). However, the impact of the investment shock on the spread is not mitigated by macroprudential policy at time zero; the spread trajectories start to diverge only after a few quarters. It can be observed that, following a negative investment shock, the spread returns to its initial equilibrium after approximately 40 quarters, regardless of whether macroprudential policy is active. Figure 4 also shows that macroprudential policy is not more effective than monetary policy in reducing fluctuations in output, investment, or physical capital.

The banks’ decision to reduce loans $b$ and consequently increase the spread leads to a fall in the loans-to-output ratio $\frac {b}{y}$ , which in turn reduces capital requirements set by the macroprudential authority. After a negative investment shock, the capital-to-loans ratio $\frac {k^B}{b}$ increases significantly, indicating a rapid recovery of bank capital. This immediate response reflects the market power of the Cournot banking system. Opposite movements in $\frac {k^B}{b}$ and $\tau ^B$ reduce the spread amplifying effect after a few periods following the initial shock when macroprudential policy is active. Approximately 10 quarters after the shock, the initial recovery of investment and physical capital begins to influence accumulated output, with $y$ returning to its initial level roughly 30 quarters after the investment shock in both policy scenarios.

Negative investment shock also has a negative impact on the productive activity of entrepreneurs, who face higher production costs and consequently begin to hire fewer workers. Regarding employment, it is noticeable that in the absence of macroprudential policy, the decline becomes more pronounced due to the elevated cost of credit, which slows down entrepreneurs’ capacity to invest and hire new labor. It can be observed that the recovery of employment occurs approximately 30 quarters after the shock. The reduction in productive activity also makes it more difficult for entrepreneurs to sell their old capital at price $q$ and to sell the wholesale good $y^w$ to retailers, thereby reducing their consumption $c^E$ .

Figure 4 shows that the return of $c^E$ to its initial equilibrium in the scenario with macroprudential policy occurs almost simultaneously with the return in the scenario with only monetary policy. However, the negative investment shock does not affect households’ consumption $c$ immediately, and after about 10 quarters, household consumption recovers more quickly in the presence of macroprudential policy. With only monetary policy, household consumption takes more than 40 quarters to return to its initial equilibrium, whereas with active macroprudential policy, recovery occurs in roughly 30 quarters.

7.1 Negative productivity shock

Table 3 presents the welfare gains of households, entrepreneurs, and oligopolistic banks resulting from the introduction of macroprudential policy, while keeping the baseline calibration for policy parameters, in the event of a negative productivity shock. Entrepreneurs benefit from time-varying capital requirements as a macroprudential instrument, mainly due to the lower cost of credit. As the economy’s borrowers, entrepreneurs are affected by the BS channel when adverse shocks cause fluctuations in the capital-to-loans ratio and increase the loan rate, even when these shocks originate in the IBC channel. In this context, a reduction in capital requirements narrows the gap $\big(\frac {k^B}{b} - \tau _{t}^{B}\big)$ , which otherwise elevates the loan rate after an adverse productivity shock. This lowers credit costs and helps guide the economy toward a low-inflation, high-output equilibrium relative to a scenario with only monetary policy, as illustrated in Figure 1.

Table 3.

Welfare gains with baseline parameters – negative productivity shock

$^*$ Welfare gains compared with the baseline policy regime (only TR).

Figure 5.

Welfare gains with $\chi _{\tau ^B} \in [0,5]$ – negative productivity shock.

Note: Welfare gains from introducing the macroprudential regulation, given monetary policy (different values of the loans-to-output feedback parameter in the time-varying capital requirement rule).

Entrepreneurs can leverage their investment with a lower cost of credit, which directly increases the level of accumulated output $y$ in the economy and, consequently, their consumption $c^E$ and welfare $W^E$ with a gain of 0.22%. Moreover, macroprudential policy can enhance the welfare of households, as the growth in accumulated output resulting from higher investment activity by entrepreneurs tends to bring the economy to an equilibrium with higher aggregate consumption and lower inflation. This positively affects households’ real earnings, allowing them to smooth their consumption over time. In contrast, macroprudential policy reduces the welfare gains of banks. The amplification effect of the productivity shock is smaller because the BS channel has less impact on the increase of the loan rate. With a lower loan rate $R^b$ and a smaller spread, oligopolistic banks accumulate less capital $k^B$ and consequently pay lower dividends $div^B$ . Table 3 shows that, under the baseline calibration, macroprudential policy results in a -0.40% change in banks’ welfare gains.

Figure 5 shows the scenario in which macroprudential policy is active and $\chi _{\tau ^B}$ varies in the interval $[0,5]$ . If $\chi _{\tau ^B} = 0$ , there is no macroprudential policy. The monetary policy parameters are set to their baseline values: $\phi _{\pi } = 1.56$ and $\phi _{y} = 0.33$ . Higher values of $\chi _{\tau ^B}$ correspond to larger welfare gains for households and entrepreneurs, and to welfare losses for banks. Entrepreneurs’ welfare gains increase more than households’ gains as $\chi _{\tau ^B}$ rises. A stronger macroprudential response (a larger adjustment of $\tau ^B$ following a negative productivity shock) reduces the ability of Cournot banks to amplify shocks through the BS channel, thereby mitigating spread responses. Figure 1 shows that spread and accumulated output fluctuations are smaller when macroprudential policy is active.Footnote ²⁶

7.2 Negative collateral shock

This section analyzes the effects on agents’ welfare resulting from a reduction in entrepreneurs’ collateral for bank loans. Table 4 reports the welfare gains across agents from the introduction of macroprudential policy. The gains are higher for banks than for households and entrepreneurs. Banks experience an improvement in welfare $W^B$ from the implementation of time-varying capital requirements. Importantly, macroprudential policy helps mitigate the costs of deviating from the optimal target in a scenario where banks are unable to accumulate capital.

Table 4.

Welfare gains with baseline parameters – negative collateral shock

$^*$ Welfare gains compared with the baseline policy regime (only TR).

Figure 6.

Welfare gains with $\chi _{\tau ^B} \in [0,5]$ – negative collateral shock.

Note: Welfare gains from introducing the macroprudential regulation, given monetary policy (different values of the loans-to-output feedback parameter in the time-varying capital requirement rule).

An adverse collateral shock tightens entrepreneurs’ borrowing constraints, leading banks under Cournot competition to reduce loan supply in order to raise the loan rate. However, even if banks charge a higher loan rate, loan demand falls, which directly contributes to a decline in bank capital $k^B$ and hence to a lower capital-to-loans ratio $\frac {k^B}{b}$ (see Figure 2). The reduction in bank lending also lowers the loans-to-output ratio $\frac {b}{y}$ , and the macroprudential authority responds by lowering capital requirements, resulting in a welfare gain for banks of 0.31%.

Introducing a macroprudential policy also generates welfare gains for entrepreneurs due to lower credit costs. This gain equals 0.01% when maintaining the baseline calibration for monetary policy and introducing macroprudential policy with $\chi _{\tau ^B} = 0.5$ . Entrepreneurs’ welfare gains are smaller compared to those in the other shock scenarios. Figure 6 shows that entrepreneurs’ gains can be higher if the value of $\chi _{\tau ^B}$ is increased. The recovery of economic activity after an adverse collateral shock involves the recovery of investments made by entrepreneurs, which contributes to a faster recovery of accumulated output.

7.3 Negative financial shock

In this section, I analyze the effects of an unexpected financial shock on the welfare of the main agents of the model. Figure 7 shows that all agents benefit from the introduction of macroprudential policy, with banks benefiting particularly due to lower costs from deviating from their optimal leverage target. When a financial shock destroys the capital accumulated by banks, there is an immediate fall in the capital-to-loans ratio, which generates deviation costs (30) from the optimal target. If macroprudential policy is active, capital requirements can vary over time and reduce the difference $\left (\frac {k^B}{b} - \tau ^{B}_{t}\right )$ , which would otherwise increase costs following the negative financial shock.

Figure 7.

Welfare gains with $\chi _{\tau ^B} \in [0,5]$ – negative financial shock.

Note: Welfare gains from introducing the macroprudential regulation, given monetary policy (different values of the loans-to-output feedback parameter in the time-varying capital requirement rule).

Table 5.

Welfare gains with baseline parameters – negative financial shock

$^*$ Welfare gains compared with the baseline policy regime (only TR).

The reduction in the target allows banks to recover the capital loss more quickly and avoid charging a high loan rate. With lower credit costs, entrepreneurs can increase their investment levels, which mitigates the decline in accumulated output compared to the scenario with only monetary policy. A macroprudential policy with a strong response to deviations in the loans-to-output ratio $(\chi _{\tau ^B})$ leads to higher welfare gains for all agents, as shown in Figure 7.

The positive effect of macroprudential policy on banks is not observed in adverse shocks during which they accumulate additional capital, such as productivity and investment shocks. In these cases, macroprudential policy reduces the spread and the accumulation of bank capital, resulting in lower dividends $div^B$ and diminished welfare $W^B$ . By contrast, under an adverse financial shock, macroprudential policy enhances bank welfare, thereby mitigating the shock’s impact. Table 5 shows that the welfare gain with macroprudential policy is 0.29% higher than under monetary policy alone, when the economy faces an adverse financial shock of standard deviation 0.01 and macroprudential policy is calibrated at $\chi _{\tau ^B}=0.5$ . Households also benefit from macroprudential policy, as stabilizing output fluctuations reduces inflation and increases their real gains relative to the only monetary policy scenario.

7.4 Negative investment shock

This section analyzes the effects of an adverse shock that increases the cost of transforming old capital into new capital in the production of the wholesale good. This shock reduces the price of capital, making entrepreneurs more financially constrained and creating incentives for the Cournot banking system to raise the loan rate by reducing credit supply. As shown in Figure 4, banks are able to accumulate more capital under this shock.

Macroprudential policy plays a dual role. On the one hand, it mitigates the amplifying effect of the BS channel on the loan rate by narrowing the gap $\left (\frac {k^B}{b} - \tau ^{B}_{t}\right )$ in the presence of a negative investment shock, while also improving the welfare of households and entrepreneurs by reducing fluctuations in accumulated output. On the other hand, it worsens banks’ position: the spread declines, capital accumulation $k^B$ falls, and dividends decrease. Table 6 shows that banks experience a welfare loss of –0.40% following the introduction of macroprudential policy under a negative investment shock.

Table 6.

Welfare gains with baseline parameters – negative investment shock

$^*$ Welfare gains compared with the baseline policy regime (only TR).

Figure 8.

Welfare gains with $\chi _{\tau ^B} \in [0,5]$ – negative investment shock.

Note: Welfare gains from introducing the macroprudential regulation, given monetary policy (different values of the loans-to-output feedback parameter in the time-varying capital requirement rule).

Figure 8 shows that increasing the macroprudential policy feedback parameter $\chi _{\tau ^B}$ within the range $[0,5]$ further reduces the welfare gains of banks, an effect not observed for households and entrepreneurs, who benefit from the intervention of the macroprudential authority. Welfare gains can reach 0.05% for households and 0.23% for entrepreneurs if $\chi _{\tau ^B}$ equals the maximum value in the given range. In contrast, the increase in $\chi _{\tau ^B}$ implies a higher welfare loss for banks, reaching -0.48% when $\chi _{\tau ^B} = 5$ .

7.5 Increasing banking competition

Table 7 shows the effects of increased banking competition on agents’ welfare gains when macroprudential policy is active. Two scenarios are considered: (i) the baseline scenario with only monetary policy, and (ii) the alternative scenario with macroprudential policy active. Monetary policy feedback parameters are kept at $\phi _{\pi } = 1.56$ and $\phi _{y} = 0.33$ , while the macroprudential policy feedback parameter is set to $\chi _{\tau ^B} = 0.5$ . The baseline scenario also assumes the number of banks $N$ is five, corresponding to the five largest Brazilian banks. Tables 7 and 8 also show total welfare, an aggregate measure of the agents’ welfare in the model (households, entrepreneurs, and banks), for $N=15$ and $N=30$ . This measure is used to assess the efficiency and effectiveness of economic policies (monetary and macroprudential) and their implications for overall societal welfare.

Table 7.

Welfare gains with more banking competition (macroprudential policy active)

$^{*}$ Welfare gains compared with the alternative policy regime formed by monetary and macroprudential policies with $N=5$ .

Table 8.

Welfare gains with more banking competition (only monetary policy)

$^{*}$ Welfare gains compared with the baseline policy regime formed by only monetary policy with $N=5$ .

The welfare gains for households and entrepreneurs resulting from the introduction of macroprudential policy are amplified under higher banking competition, as shown in Figure 9. If entry barriers for new banks are removed in the presence of a productivity shock, allowing competition to increase to $N=15$ , welfare gains are 5.38% for households and 0.33% for entrepreneurs compared to the scenario with $N=5$ . Further increasing banking competition to $N=30$ raises the welfare gains to 6.77% for households and 0.42% for entrepreneurs (Table 7).

In contrast, in the presence of a negative $z$ shock, banks lose welfare under increased competition when macroprudential policy is active because they accumulate less capital due to the lower spread, which in turn reduces dividends. When the loan-to-output ratio falls, the macroprudential authority can intervene by reducing capital requirements $\tau ^{B}$ , limiting banks’ ability to amplify the spread through the BS channel. As a result, banks’ welfare gains are lower. A similar outcome occurs for an investment shock, where banks are also unable to charge a higher spread under active macroprudential policy, leading to a welfare loss compared to the scenario with only monetary policy.

Figure 9.

Welfare gains with $N \in [5,30]$ .

Note: Welfare gains from introducing the macroprudential rule, given monetary policy (different values of the number of banks).

Figure 9 shows a contrasting effect on banks’ welfare gains when banking competition increases in the presence of financial shocks. Higher competition reduces banks’ ability to adjust the loan rate by cutting loan supply when the economy is hit by shocks transmitted mainly through the IBC channel. As a result, banks accumulate less capital and pay lower dividends, thereby reducing their welfare gains. The introduction of macroprudential policy, however, helps reduce the costs that penalize banks for deviating from the optimal leverage target when an unexpected financial shock diminishes bank capital. More banking competition also implies lower profits and smaller capital buffers, which weakens banks’ capacity to absorb financial shocks. In this scenario, macroprudential policy becomes crucial for supporting banks’ financial recovery. Table 7 shows that, under macroprudential policy active, banks experience welfare losses of −1.43% when the number of banks increases to $N=15$ , and −1.85% when $N=30$ . Under the only monetary policy, welfare losses amount to −1.76% when $N$ rises from 5 to 15, and −2.70% when $N=30$ .

Macroprudential policy has a similar effect on bank welfare $W^B$ under an adverse collateral shock because of the associated bank capital loss. However, it is less effective in limiting the decline in bank welfare when competition increases, as shown in Figure 9. This lower effectiveness arises because the collateral shock affects the IBC channel through PED, whereas macroprudential policy acts on the BS channel. Table 7 shows that banks face a welfare loss of −1.15% when the number of banks rises to $N=15$ , and −1.54% when $N=30$ . In the only monetary policy scenario, the entry of new banks also increases households’ and entrepreneurs’ welfare gains. Equation (39) shows that more banks reduce the adverse effects of shocks transmitted through the IBC channel, thereby lowering the loan rate under imperfect competition. With cheaper credit, entrepreneurs can increase investment, raising output $y$ and directly improving households’ and entrepreneurs’ welfare. However, these welfare gains are smaller than under the scenario with active macroprudential policy.

In the presence of a productivity shock, increasing banking competition from $N=5$ to $N=15$ yields welfare gains of 1.58% for households and 0.15% for entrepreneurs. With further competition up to $N=30$ , gains increase to 1.90% for households and 0.19% for entrepreneurs (Table 8). Higher banking competition also improves $W^H$ and $W^E$ under collateral and investment shocks. However, monetary policy alone is insufficient to stabilize banks’ welfare $W^B$ in the face of financial shocks.

9.1 Investment adjustment cost $\chi$

Figure 12 shows that variations in the adjustment cost $\chi$ significantly affect the model’s main variables under negative collateral, financial, and investment shocks. In contrast, fluctuations are minimized when a negative productivity shock occurs. The adjustment cost $\chi$ represents the cost of transforming old capital into new capital and is the responsibility of the capital producers.

The largest divergence in the trajectories of the analyzed variables following an initial negative shock arises from an investment shock affecting capital producers. For a higher value of $\chi$ ( $\chi _3 = 10$ ), a negative investment shock leads to a substantial fall in the price of capital $q$ , constraining entrepreneurs financially since capital serves as collateral. Banks respond to this tightening of financial conditions by raising loan interest rate $R^b$ , which increases the spread. Under these circumstances, banks are able to accumulate more capital.

Banks can also accumulate more capital following a negative productivity shock, but increasing $\chi$ does not amplify fluctuations in the spread. A negative collateral shock reduces the collateral that entrepreneurs can provide to obtain loans, and higher $\chi$ further limits collateral availability, resulting in fewer guarantees for banks. Banks exploit this situation by charging a higher spread, pushing the economy toward a worse equilibrium with a decline in accumulated output. Finally, in the event of a financial shock affecting banks, they can charge higher interest rates if entrepreneurs are financially constrained, which occurs in scenarios with a high $\chi$ value.

Figure 12.

Impulse responses to different negative shocks with variation in $\chi$ .

Note: The horizontal axis shows quarters after a negative shock occurring at the beginning of period one. The vertical axis shows the percentage deviation from the steady state of the analyzed variable. Macroprudential policy is active.

Figure 13.

Impulse responses to different negative shocks with variation in $\kappa _{k^B}$ .

Note: The horizontal axis shows quarters after a negative shock occurring at the beginning of period one. The vertical axis shows the percentage deviation from the steady state of the analyzed variable. Macroprudential policy is active.

9.2 Bank capital adjustment cost $\kappa _{k^B}$

Figure 13 shows the effects of different shocks that hit the economy as the bank capital adjustment cost $\kappa _{k^B}$ varies. The parameter $\kappa _{k^B}$ plays a crucial role in determining the cost $\Omega _B$ , as given by (30). A higher bank capital adjustment cost $\kappa _{k^B}$ amplifies fluctuations in the loan rate $R^b$ when adverse shocks are transmitted through the BS channel. These shocks directly or indirectly lead to deviations of bank capital from the optimal target set by the macroprudential authority.

An unexpected negative financial shock affects the BS channel, causing an immediate decline in the capital-to-loans ratio $\frac {k^B}{b}$ , which leads to an increase in the spread. When banks face higher adjustment costs ( $\kappa _{k^B,3} = 30$ ), the rise in the spread becomes more pronounced and intensifies fluctuations in real variables affected by it. As a result, a higher loan rate and reduced borrowing capacity impose financial constraints on entrepreneurs (characterized by inelastic PED), contributing to a fall in accumulated output $y$ as entrepreneurs slow down productive activities, hire fewer workers $l$ , and acquire less capital $k$ .

Regarding the negative investment shock, an increase in $\kappa _{k^B}$ also affects spread fluctuations. A higher value of $\kappa _{k^B}$ leads to significant fluctuations in the spread, which take more than 40 quarters to return to the initial equilibrium. In this scenario, there is a sharp decline in accumulated output, and banks accumulate more capital as the loan rate increases. Higher bank capital raises the capital-to-loans ratio $\frac {k^B}{b}$ during a negative investment shock. A similar pattern in the capital-to-loans ratio is observed under a negative productivity shock. In this case, banks accumulate more capital by charging higher interest rates on loans, as entrepreneurs face financial difficulties and have limited bargaining power. Accumulated output also takes approximately 40 quarters to return to its initial equilibrium after a negative productivity shock when $\kappa _{k^B}$ is high.

Lastly, a negative collateral shock amplifies the spread when the bank capital adjustment cost $\kappa _{k^B}$ is high. As entrepreneurs have less physical capital to offer as collateral, their loan demand becomes inelastic, incentivizing banks in an imperfectly competitive environment to raise the spread. However, the reduction in loan demand from entrepreneurs limits banks’ ability to expand their capital $k^B$ . Consequently, bank capital accumulation declines, leading to a lower capital-to-loans ratio that takes approximately 25 quarters to return to its steady state after the negative collateral shock. Moreover, the more pronounced slowdown in productive activity when $\kappa _{k^B}$ is high results in a sharper decline in accumulated output $y$ , which takes approximately 40 quarters to return to its initial equilibrium.