3.1 Households

Households, denoted by $j\in \left \{ s,\:m \right \}$ , can categorize as either patient ( $s$ ) or impatient ( $m$ ). The degree of patience is given by the parametrization of $\beta ^j$ . Patient households save in equilibrium, depositing their savings in banks and holding part of the stock of physical capital. In contrast, impatient households take out loans from the banking sector, $b_t^m$ , to finance the acquisition of housing units, $h_t^m$ , which are purchased at price $q_t^H$ . These loans carry a contractual gross interest rate, $R_{t-1}^m$ , agreed upon in the previous period. Aside from these differences, both types of households exhibit similar behavior—they consume ( $c_t^j$ ), invest in housing ( $h_t^j)$ , and participate in the production sector as workers ( $l_t^j$ )—maximizing their life time utility

(1) \begin{equation} \mathbb{E}_t\left [\sum ^\infty _{i=0}\left(\beta ^j\right)^{t+i}\left[\log \left(c_{t+i}^j\right)+\tau _t\upsilon ^j\log \left(h^j_{t+i-1}\right)-\frac {\phi ^j}{1+\eta }\left(l^j_{t+i}\right)^{1+\eta }\right]\right ], \end{equation}

where $\eta$ is the inverse of the Frisch elasticity of labor supply, $\tau _t$ is a housing preference shock that is common to both households dynasties, $\upsilon ^j$ is a housing preference parameter and $\phi ^j$ is a labor preference parameter.

At the beginning of each period $t$ , impatient households experience an idiosyncratic shock, $\omega _t^m$ , to the efficiency units of housing owned from period $t-1$ . This shock follows a lognormal distribution and is independently and identically distributed across impatient households, with density and cumulative distribution functions denoted by $f(\cdot )$ and $F(\cdot )$ , respectively. The shock impacts the effective resale value of previously acquired housing, given by $\tilde {q}_t^H = \omega _{t}^m q_t^H \left (1 - \delta _t^H\right )$ . As a result, ex-post default on the loan becomes optimal for an impatient household if $\tilde {q}_t^H h_{t-1}^m \lt R_{t-1}^m b_{t-1}^m$ . In other words, an impatient household defaults whenever the idiosyncratic shock $\omega _t^m$ falls below the following threshold

(2) \begin{equation} \overline {\omega }_t^m = \dfrac {x_{t-1}}{R_t^H}, \end{equation}

where $R_t^H = q_t^H\left ( 1-\delta _t^H \right )/q_{t-1}^H$ represents the ex-post realized average return on housing and $x_t^m = R_t^m b_t^m/\left ( q_t^H h_t^m \right )$ is a measure of the households’ leverage.

3.2 Entrepreneurs

Entrepreneurs, together with patient households, collectively owns the economy’s stock of physical capital. Entrepreneurs rent their capital directly to firms in the productive sector, whereas patient households entrust specialized capital management firms to handle their holdings in exchange for a fee. Entrepreneurs finance their physical capital acquisitions using their inherited net worth, $n_t^e$ , and loans from the banking sector, $b_t^e$ . Each period $t$ , their objective is to maximize expected wealth, $\mathbb{E}(W_{t+1}^e)$ , subject to the resource constraint, $q_t^K k_t - b_t^e = n_t^e$ and the nonnegativity constraint $W_{t+1}^e = \max \left \{ \omega _{t+1}^e \left (r_{t+1}^K + \left (1 - \delta _{t+1}^k \right )q_{t+1}^K \right ) k_t - R_t^F b_t^e,\: 0 \right \}$ , where $q_t^K$ represents the price of physical capital, $k_t$ is the physical capital held by the entrepreneurs, $r_{t+1}^K$ is the rental rate per efficiency unit of physical capital, $\delta _{t+1}^K$ is the time-varying depreciation rate of each efficiency unit of physical capital, $R_t^F$ is the contractual gross interest rate of the loans, and $\omega _{t+1}^e$ is an idiosyncratic shock to the efficiency units of physical capital which is independently and identically distributed across entrepreneurs. Entrepreneurs default on their loans whenever the gross returns on investment in physical capital undertaken in the previous period falls below the contractual repayments due to the bank. This occurs whenever the idiosyncratic shock $\omega _{t+1}^e$ exceeds the threshold

(3) \begin{equation} \overline {\omega }_{t+1}^e = \dfrac {R_t^F b_t^e}{R_{t+1}^Kq_{t}^K k_t}, \end{equation}

where $R_{t+1}^K = \left (r_{t+1}^K + \left (1 - \delta _{t+1}^k \right )q_{t+1}^K \right )/q_{t}^K$ .

3.3 Banks

There are two specialized banks in the model, denoted by $j \in \{ H, \:F\}$ . One type of bank, $H$ , provides credit to impatient households, $b_t^m$ , for investment in housing (mortgage loans) while the other type, $F$ , grants credit to entrepreneurs, $b_t^e$ , to their investment in physical capital (corporate loans). Banks use deposits, $d_t^j$ , and equity, $e_t^j$ , to finance themselves, which can be accumulated internally through retained earnings, and externally by raising equity from bankers.

Banks are subject to regulatory capital constraints and operate with limited liability. Specifically, banks face the capital constraint:

(4) \begin{equation} e_t^j \geq \phi _t^j b_t^j, \end{equation}

where $\phi _t^j$ is the bank’s capital ratio, and the constraint is always binding by definition. It is assumed that the bank’s capital ratio, $\phi _t^j = \Xi _t^j + \phi _{vol}^j$ , is composed of regulatory capital requirements, $\Xi _t^j$ , which may be constant or vary over time, and a time-invariant management buffer, $\overline {\phi }_{vol}^j$ , which reflects the portion of capital held above the minimum regulatory requirement. Unlike in reality, where banks often adjust their management buffers to accommodate regulatory changes, this model assumes that the buffer remains constant. As a result, any change in capital ratios directly translates into changes in lending capacity, which simplifies the model but may influence short-term dynamics. Nonetheless, capital requirements differ by loan type. In this model, mortgage loans carry a 50% risk weightFootnote ¹¹ while corporate loans are assigned a 100% risk weight.

Bank equity is supplied by bankers and must match the total demand for equity from both mortgage-lending banks, $e_t^H$ , and corporate-lending banks, $e_t^F$ . Similarly, the deposits provided by patient households, $d_t$ , must equal the total demand for deposit funding from both types of banks, $d_t^H$ and $d_t^F$ . Consequently, the banking sector’s balance sheet constraint is given by:

(5) \begin{equation} n_t^b + d_t = b_t^e + b_t^m, \text{ with } n_t^b = e_t^H + e_t^F, \end{equation}

This constraint determines the total amount of credit banks can issue. Additionally, the sector-wide capital constraint can be expressed as:

(6) \begin{equation} n_t^b = \phi _t^F b_t^e + \phi _t^H b_t^m. \end{equation}

These two constraints play a key role in determining the transition dynamics of the banking sector. The effects on mortgage and corporate credit depend on how shocks impact bankers’ net worth and deposit availability, as well as the regulatory response. On the one hand, the capital requirement constraint tends to favor mortgage lending due to its lower risk weight. On the other hand, the balance sheet constraint directs resources toward the more profitable lending sector. The final allocation of resources between mortgage and corporate credit depends on the relative strength of these opposing forces.

Bank failure occurs when the observed return on its credit portfolio is lower than the obligations arising from the repayment of deposits and associated interest. Banks may default due to both idiosyncratic and aggregate shocks, $\omega _{t+1}^j$ , affecting the performance of their loan portfolios. In each period, the equity payoff for bank $j$ is determined by the difference between the returns on its loans and the repayments required on its deposits:

(7) \begin{equation} \pi _{t+1}^j = \max {\left[\omega _{t+1}^j \tilde {R}_{t+1}^j b_t^j - R_t^D d_t^j, \: 0\right]}, \end{equation}

where $b_t^j$ and $d_t^j$ represent the loans and deposits on bank $j$ balance sheet at time $t$ , respectively, $R_t^D$ is the gross interest rate paid on deposits taken at $t$ , and $\tilde {R}_t^j$ is the realized return on the portfolio of loans of class $j$ . Bank’s default occur due to fluctuations in the aggregate loan return $\tilde {R}_{t+1}^j$ , which is driven by firms’ and households’ default rates, as well as the bank-idiosyncratic shock, $\omega _{t+1}^j$ . The threshold level of $\omega _{t+1}^j$ below which the bank of type $j$ defaults is given by

(8) \begin{equation} \overline {\omega }_{t+1}^j = \left(1-\phi _t^j\right)R_t^D/\tilde {R}_{t+1}^j. \end{equation}

The threshold for bank’s default relates to the bank’s capital ratio, $\phi _t^j$ , and the spread between the realized gross loan return and the gross deposit rate, $R_t^d/\tilde {R}_t^j$ . An increase in the bank’s capital ratio positively impacts the default threshold, enhancing the bank’s ability to withstand adverse idiosyncratic shocks to its portfolio returns.

The aggregate default rate for the banking system, $PD_t^b$ , depends on the distribution of banks’ portfolios and the default probabilities of each bank type. It is given by:

(9) \begin{equation} PD_t^b = \dfrac {d_{t-1}^H F\left ( \overline {\omega }_t^H \right ) + d_{t-1}^F F\left ( \overline {\omega }_t^F \right )}{d_{t-1}^H + d_{t-1}^F}, \end{equation}

where $F\left ( \overline {\omega }_t^j \right )$ is the probability of default of a bank of class $j\in \left \{H,\: F \right \}$ .

3.4 Deposit insurance agency

Deposits are formally guaranteed by a deposit insurance agency (DIA), which covers the amount deposited and the corresponding interest. When necessary, the deposit guarantee system is financed through a fixed tax levied on households. However, recovering the fully insured amount is assumed to be costly, implying that deposits may still carry a risk premium reflecting the average bank’s default risk. This assumption accounts for depositors’ limited ability to monitor individual banks, making deposit risk premia dependent on the overall health of the banking sector. Within this framework, households incur a linear transaction cost, $\gamma$ , when recovering their funds, representing the time and effort involved in reclaiming deposits and interest following a bank failure. This cost establishes a direct link between the probability of bank default and bank funding costs, thereby creating a wedge between the deposit return rate and the risk-free rate:

(10) \begin{equation} \tilde {R}_t^D = R_{t-1}^D\left (1 - \gamma PD_t^b \right ), \end{equation}

which, in turn, raises the banking sector’s funding costs when default risk is high.

The aggregated losses to the DIA by failing $H$ and $F$ banks are given by

(11) \begin{equation} T_t = T_t^H + T_t^F = \underbrace {\left (\Omega _t^H(\bar {\omega }_t^H)\dfrac {\tilde {R}_{t}^H }{1-\phi _{t-1}^H}\dfrac {d^H_{t-1}}{d_{t-1}} + \Omega _t^F(\bar {\omega }_t^F)\dfrac {\tilde {R}_{t}^F }{1-\phi _{t-1}^F}\dfrac {d^F_{t-1}}{d_{t-1}}\right )}_{\Omega \left (\bar {\omega }_t^H,\bar {\omega }_t^F \right ) := \text{Average default loss per unit of bank debt}} d_{t-1}, \end{equation}

where $d_{t-1} = d_{t-1}^H + d_{t-1}^F$ is the aggregated deposits in the banking system, and $\Omega _t^H(\bar {\omega }_t^j) = \left [ \bar {\omega }_t^j - \Gamma ^j\left ( \bar {\omega }_t^j \right ) + \mu ^jG\left ( \bar {\omega }_t^j \right ) \right ]$ , for $j\in \left \{ H, \: F \right \}$ . Here, $\Gamma _t^j$ represents the expected fraction of defaulting bank’s assets that can be recovered, while $\mu ^j G^j\left ( \bar {\omega }_t^j \right )$ captures the total cost of bank default as a faction of total bank assets.

The deadweight default loss parameter, $\mu ^j$ , plays a critical role in shaping the relationship between capital regulation and banks’ internalization of the broader economic consequences of default. A lower value reduces the financial burden on households in funding the DIA.

This feature of the model creates incentives for banks to engage in excessive risk-taking practices, as it allows banks with a business model more focused on granting credit to higher-risk borrowers to implicitly benefit from a subsidy against banks with a more conservative lending profile. However, the model assumes that patient households do not know the risk profile of each bank, observing only the average risk of the sector.

3.5 Production sector and market clearing

Perfectly competitive firms producing the final good, new physical capital, and housing close the model in the production sector. In equilibrium, the total output in the economy is equal to the aggregate statistics of consumption, capital investment, housing investment, and wealth of bankers before dividends.

5.1 The flexibility measure and the dividends pay-out restriction

This subsection presents the results of a simulation exercise of the joint effects from the combined implementation of the flexibility measure and the dividends pay-out restriction, compared to the no-policy scenario.

In the no-policy scenario, where the measures are assumed not to be available, the results obtained from the shocks applied in the model simulate an output path characterized by a sharp fall in the second quarter of 2020 followed by a gradual recovery, which is temporarily halted in the first quarter of 2021 as a result of new lockdown measures. Thereafter, a gradual recovery toward steady-state values is simulated (see Figure B3 in Appendix). The shock transmission mechanism initially leads to a decline in spending and production, which brings down the relative price between housing and physical capital. In the labor market, working hours and wages decline. The default of firms and households increases, which results in an increase in bank interest rates and a decrease in credit. The increase in banks’ default induces a higher cost of deposits, which in turn is passed through to households and firms via increased spreads in credit operations that further increase bank interest rates.Footnote ²³ These effects spill over to the economy in the following periods, as they have a negative impact on investment, thereby amplifying the initial impact of the shocks. The recovery process is slow, in particular for credit. The increase in banks’ default induces additional costs on deposit insurance, supported by households, while overall deterioration of wealth reduces patient households’ income from owning banks and firms. This set of occurrences frames a strong, and moderately persistent, contraction in consumption—which after 5 years is still below its steady-state level.

In the baseline scenario, banks use the flexibility measure and refrain from distributing part of their profits in the form of dividends. The dividends pay-out restriction allows banks to increase their management capital buffers, becoming more resilient to adverse shocks (Figure 3). Strengthening capital ratios in this manner allows banks to have additional resources that are at the outset fully used to finance the economy, which contributes positively to dampening the contraction in credit to households and firms that follows the shock. In the first three years following the implementation of the policy measures, the dividends pay-out restriction cumulatively reduces the decline in credit by 9%. When combined with the impact of the flexibility measure, this results in a total credit injection that is 48.4% higher than in the no-policy scenario. In other words, the two policy measures together halve the decline in credit.

Figure 3.

Impact of the measures to shocks simulating a contraction in GDP.

Note: The impact is measured as the difference between the baseline scenario and the no-policy scenario.

Banks apply a higher spread to corporate credit than household credit as, in our calibration, corporate loans are associated with a higher risk weight.Footnote ²⁴ This characteristic is reflected in the different magnitude of the impact of shocks (see Figure B1 in Appendix). This leads to the measures having a greater impact on smoothing the decline in corporate loans than in mortgage loans, that is, the reduction in the cost of financing for firms is greater than the reduction in the cost of financing for households.

The smaller contraction in corporate loans is passed on to the economy through higher investment in physical capital, mitigating the effects of shocks and actively contributing to the faster recovery of GDP against the no-policy scenario. Over a three-year period, the dividends pay-out restriction prevents a 1.4% GDP loss. Combined with the impact of the flexibility measure, this adds up to a total GDP preservation of 2.7%.

The results of this simulation exercise highlight the complementary effect of the dividends pay-out restriction on mitigating the spillover of risk events to the economy. The effects of the two measures analyzed in this paper are transmitted to the economy through the credit channel resulting in a faster economic recovery.

5.2 The flexibility measure only

In the previous subsection, results indicate that there are benefits to financial stability from combining the flexibility measure with the dividends pay-out restriction policy. Now, we question what would be the effects to financial stability if only the flexibility measure was in place. This assumption could be explained by, to a large extent, banks not being keen to comply with the dividends pay-out restriction measure, for example, or by the decision of policy authorities to only apply the flexibility measure. A comparison of the results (Figure 4) allows us to assess the measure’s effectiveness in promoting the financial intermediation function of the banking system and in stabilizing the economy compared to the other two scenarios.

In the counterfactual scenario, banks make use of the flexibility they are provided with, using the capital buffers in response to deviations of total credit from its initial preshock value. Compared to the no-policy scenario, the flexibility measure mitigates the reduction in credit to households and firms. The effect is more pronounced in corporate loans, as this segment has higher risk weights. The credit smoothing effect is mainly reflected in investment, particularly in investment in physical capital by firms whose decline is mitigated in the first quarters. The improvement in the capacity to finance the economy dampens the fall in GDP over the first 13 quarters, although to a lesser extent than in investment. After that, due to a lower recovery rate, the effect of the flexibility measure becomes negative. Nonetheless, the net effect of the flexibility measure on GDP is still positive and although the effects on GDP of a reduction in capital ratios are relatively subdued, given that the shock is exogenous to the banking system, these results as well as those related to credit smoothing are in line with the aim of the capital requirements. Such regulation should in fact improve ex-ante the resilience of banks to prevent the propagation of shocks in the financial sector to the real economy, and help to mitigate adverse outcomes in the financial system from stress events with outside origination, avoiding amplification effects. In the first three years, the flexibility measure cumulatively injects 39.4% more credit compared to the no-policy scenario, helping to avert a 1.2% GDP loss.

Figure 4.

Impact of the flexibility measure.

Note: The impact is measured as the difference between the counterfactual scenario and the no-policy scenario.

The higher leverage of banks due to the flexibility measure is perceived as an increased risk in the banking system which is translated into higher banks’ default rates. This increase in the probability of banks’ default imposes a higher cost on households with deposit insurance, which motivates a greater reduction in consumption. Consequently, the flexibility measure moderately intensifies the contraction in consumption that was registered under the no-policy scenario.

The introduction of capital requirements, together with the possibility of using part of them in a situation of stress in the financial system, reduces the materialization of risks from spreading into the economy, as they provide the banking system with greater capacity to absorb losses. However, the effectiveness in containing the transmission of disruptions that arise first in the economy and only then in the banking system is lower (see also Lima et al. (Reference Lima, Maia and Pereira2023)). It may thus be necessary, depending on the magnitude of the shock, to implement other support measures that complement each other and allow the economy to recover from the pandemic crisis.

For consumption, the impact of the two combined measures does not differ much from the case where we only consider the flexibility measure, meaning that the dividends pay-out restriction barely affects the consumption decisions of agents. The slightly worse combined effect of the measures on consumption between the third and the ninth quarter is motivated by a reduction in households’ income due to the dividends pay-out restriction. After this period, the combined effect becomes positive and contributes to a slightly faster convergence of consumption to its prepandemic level. The dividends pay-out restriction has also the additional effect of reducing the need for banks to use the flexibility measure to mitigate the effects of the shock and keep financing the economy. This outcome reflects how the two measures complement each other and shows that by restricting dividends pay-out, banks have to make a smaller effort to return to the level of capitalization that existed before the shock materialized. Although the flexibility measure was only partially used, its relative impact on all variables is decisive. The banking sector plays an active role in the economic recovery contributing to a less pronounced contraction in corporate investment compared to the no-policy scenario, which has an impact on the performance of economic activity.

To sum up, results indicate that a prudential measure that introduces flexibility in banks’ capital use in situations of stress is effective in promoting the financing of the economy, even if it is the sole measure in place. By being able to make full use of this flexibility, banks mitigate the transmission effect of the shock to credit granted to both firms and households. This measure is particularly beneficial for corporate loans, resulting in more favorable dynamics for corporate investment.

6.1 Linear path regime

The impulse response functions (thereafter IRFs) for an abrupt increase in capital buffers and for the linear replenishment paths are presented in Figure 5. Before analyzing the results for the different transition periods, we focus on the responses to an abrupt increase of 4 p.p. in capital buffers, which represent a relevant reference point for the following considerations.

Figure 5.

IRFs of capital buffers’ replenishment (linear path regime).

Note: Impulse responses from a $4$ p.p. capital buffers’ replenishment. Deviations from the initial steady state.

An abrupt adjustment in capital buffers has positive effects by reducing the probability of bank failure quickly, increasing the banking system resilience. However, it also induces high bank financing costs translating into higher spreads for borrowers.Footnote ²⁷ The significant increase in interest rates is associated with a considerable reduction in total credit granted to the economy, in particular, in loans to businesses, as they bear the brunt of higher interest rates. This behavior also stems from the risk weights for housing credit being smaller than for corporate loans. These effects spill over to the economy: on the one hand, a greater resilience of the banking system has positive effects on household income, consumption and output, while, on the other hand, the sharp fall in credit triggers a significant drop in investment.Footnote ²⁸ The effects of an abrupt increase in capital buffers unwind after a few quarters, and the economy quickly converges to a new equilibrium, where total credit and output are higher, and bank resilience is strengthened.

Results indicate that the permanent increase in capital buffers considered in the model, even if abrupt, has beneficial effects on financial stability and economic activity in the long run, but may trigger severe disruptions in banks’ financial intermediation function throughout the process of convergence to the new equilibrium. Therefore, we now consider longer transition periods in order to assess whether they allow the mitigation of the procyclical response of the banking system without significantly compromising the benefits arising from replenishing capital buffers.

In the linear replenishment paths, we observe how in general, predictable timing and speed of transition allow banks to anticipate the necessary adjustment in each period, smoothing the rise in spreads and the fall in lending, compared with an abrupt increase. The smoothing effect on corporate loans also leads to a less sharp decline in business investment. The longer the transition period, the more significant the smoothing effect. However, the fall in business investment is still more significant than the increase in residential investment, which benefits from house purchases by households that are not dependent on bank financing. This evidence determines that overall investment remains below—although closer to—its initial value throughout the convergence process to the new equilibrium. Longer transitions also mitigate the positive effects on the banking system resilience, if compared with an abrupt adjustment. Longer transition periods reduce the bank failure rate, but to a lesser extent than an abrupt adjustment. Moreover, they postpone the convergence of this variable to the new equilibrium.

The less favorable behavior of the bank failure rate has macroeconomic impacts due to a lower increase in consumption, as households must bear higher costs associated with deposit guarantees, and, consequently, in GDP. The speed of adjustment towards equilibrium is also slower. The choice of the transition path thus entails macroprudential policy trade-offs. The model favors shorter transition horizons if the macroprudential policy’s purpose is mainly to strengthen the banking system resilience. If the focus is mainly on maintaining the flow of credit to the economy, then longer time horizons are preferable.

In addition to the previous analysis, we compute volatility metrics for the variables considered relevant for macroprudential policy: bank failure rate, credit, and output. We calculate the volatility associated with each of the three variables on the basis of the IRFs arising from the abrupt adjustment and the four transition periods adopted considering a linear pace of capital buffer accumulation (Figure 6).

The analysis indicates that longer transition periods lead to a significant reduction in overall credit volatility, which can be considered a benefit associated with longer time horizons. On the contrary, GDP and bank failure rate take more time to converge to the new equilibrium when the replenishment period is longer. Thus, as the transition period is extended, the volatility of GDP and bank failure rate increases. This result can be perceived as a cost associated with longer transition periods.

Figure 6.

Volatility measures for the capital buffers’ replenishment paths (linear path regime).

Note: The volatility for each variable is obtained by calculating the standard deviation of the IRFs from their final steady-state values.

6.2 Nonlinear path regime

This section examines the effects of implementing nonlinear replenishment paths. In particular, we consider a decreasing rate for the required replenishment of the capital buffers (Figure 7).Footnote ²⁹ This scheme may prove more effective in booming periods in the business and/or credit cycle, where higher buffer requirements can be more easily met by banks, by retaining profits and/or issuing capital, without disrupting lending activity.

The main conclusion drawn from the previous exercise still holds. Considering longer transition periods—compared to an abrupt adjustment—mitigates the costs of increasing capital buffers, yet it also curtails the development of a more resilient banking system. In a nonlinear path regime of capital buffer accumulation, the differences in the responses of the variables for each transition period are evident for the shorter horizons but fade out as horizons lengthen.

Figure 7.

IRFs of capital buffers’ replenishment (nonlinear path regime).

Note: Impulse responses from a $4$ p.p. capital buffers’ replenishment. Deviations from the initial steady state.

Compared with the results of the linear path regime for the corresponding transition periods, the nonlinear path regime leads to higher credit volatility (Figure 8). On the contrary, the volatility of output and bank failure rate are lower in the nonlinear path regime. Moreover, comparing the two path regimes, we can observe that changing the replenishment periods has a smaller effect on bank failure rate in the nonlinear regime. Additional simulation exercises for nonlinear path regimes suggest no significant benefits from choosing transition periods longer than 16 quarters.Footnote ³⁰

Figure 8.

Volatility measures for the capital buffers’ replenishment paths (nonlinear path regime).

Note: The volatility for each variable is obtained by calculating the standard deviation of the IRFs from their final steady-state value.