Bank procyclicality, business cycles and capital requirements

Abstract

The present work seeks to rationalize the procyclical movements of credit, deposits and bank leverage, basic elements that justify the existence of macroprudential policies, and the effects of capital requirements on the business cycle and financial stability. For this purpose, we use an extension of the theoretical framework of Bernanke et al. [Handbook of Macroeconomics, Elsevier, New York, pp 1341–1393, 1999], which includes the presence of banks through the existence of a second asymmetry between banks and depositors. This extension allows us to understand the role of bank capital on the business cycle and, in this way, how credit channels and financial stability are affected by the inclusion of capital requirements as has been proposed in the Basel III Agreement. The results suggest that capital requirements act as a significant financial accelerator in the presence of productivity and monetary shocks. In addition, during expansions, banking capital regulation helps not only the real part of the economy perform better but also the financial part, through lesser leverage for entrepreneurs and banks. Our results also suggest that an asymmetric regulation constraint throughout the business cycle is preferable to a fixed one.

Appendices

Appendix 1: Evolution of the entrepreneurial net worth and bank capital

To derive the evolution of entrepreneurial net worth, it is important to note that in the optimum the incentive participations constraints of bankers and depositors should be binding. It implies:

$$ \left[ {1 - F\left( {\bar{\omega }} \right)} \right]\bar{\omega }R_{t}^{k} Q_{t - 1} K_{t} = R_{t}^{p} M_{t + 1} + \left( {1 - \mu } \right)\left[ {1 - F\left( {\omega^{*} } \right)} \right]\omega^{*} R_{t}^{k} Q_{t - 1} K_{t} - \left( {1 - \mu } \right)\mathop \int \limits_{{\omega^{*} }}^{{\bar{\omega }}} \omega f\left( \omega \right)d\omega R_{t}^{k} Q_{t - 1} K_{t} $$

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Now, the depositors expected profits should be equal at their oportunity cost:

$$ \left( {1 - \mu } \right)\left[ {1 - F\left( {\omega^{*} } \right)} \right]\omega^{*} R_{t}^{k} Q_{t - 1} K_{t} = R_{t}^{p} \left( {Q_{t - 1} K_{t} - N_{t} - M_{t} } \right) - \left( {1 - \gamma } \right)\left( {1 - \mu } \right)\mathop \int \limits_{0}^{{\omega^{*} }} \omega f\left( \omega \right)d\omega R_{t}^{k} Q_{t - 1} K_{t} $$

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Using both equations we find that:

$$ \left[ {1 - F\left( {\bar{\omega }} \right)} \right]\bar{\omega }R_{t}^{k} Q_{t - 1} K_{t} = R_{t}^{p} \left( {Q_{t - 1} K_{t} - N_{t} } \right) - \left[ {\left( {1 - \gamma } \right)\mathop \int \limits_{0}^{{\bar{\omega }}} \omega f\left( \omega \right)d\omega R_{t}^{k} Q_{t - 1} K_{t} - \gamma \left( {1 - \mu } \right)\mathop \int \limits_{0}^{{\omega^{*} }} \omega f\left( \omega \right)d\omega R_{t}^{k} Q_{t - 1} K_{t} } \right] $$

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Replacing this expression in the entrepreneur expected profits, we can writte:

$$ \varPi^{e} = R_{t + 1}^{k} Q_{t} K_{t + 1} - R_{t}^{p} \left( {Q_{t - 1} K_{t} - N_{t} } \right) - \mu \mathop \int \limits_{{\omega^{*} }}^{{\bar{\omega }}} \omega f\left( \omega \right)d\omega R_{t}^{k} Q_{t - 1} K_{t} - \left[ {\gamma + \mu \left( {1 - \gamma } \right)} \right]\mathop \int \limits_{0}^{{\omega^{*} }} \omega f\left( \omega \right)d\omega R_{t}^{k} Q_{t - 1} K_{t} $$

Finally, the net worth entrepreneurial profits can be written as:

$$ V_{t} = R_{t + 1}^{k} Q_{t} K_{t + 1} - \left\{ {R_{t}^{p} + \frac{{\left[ {\mu \mathop \int \nolimits_{{\omega^{*} }}^{{\bar{\omega }}} \omega f\left( \omega \right)d\omega - \left( {\gamma + \mu \left( {1 - \gamma } \right)} \right)\mathop \int \nolimits_{0}^{{\omega^{*} }} \omega f\left( \omega \right)d\omega } \right]R_{t}^{k} Q_{t - 1} K_{t} }}{{Q_{t - 1} K_{t} - N_{t} }}} \right\}\left( {Q_{t - 1} K_{t} - N_{t} } \right) $$

where $\left( {\frac{{\left[ {\mu \mathop \int \nolimits_{{\omega^{*} }}^{{\bar{\omega }}} \omega f\left( \omega \right)d\omega - \left( {\gamma + \mu \left( {1 - \gamma } \right)} \right)\mathop \int \nolimits_{0}^{{\omega^{*} }} \omega f\left( \omega \right)d\omega } \right]R_{t}^{k} Q_{t - 1} K_{t} }}{{Q_{t - 1} K_{t} - N_{t} }}} \right)$ is the financial premium that bankers charge to entrepreneurs due the existence of asymetric information.

In a similar way, it is possible to derive the expression for the evolution of bank capital. To do that, we start with the banker expected profits function:

$$ \left[ {1 - F\left( {\bar{\omega }} \right)} \right]\bar{\omega }R_{t}^{k} Q_{t - 1} K_{t} + \left( {1 - \mu } \right)\mathop \int \limits_{{\omega^{*} }}^{{\bar{\omega }}} \omega f\left( \omega \right)d\omega R_{t}^{k} Q_{t - 1} K_{t} - \left( {1 - \mu } \right)\left[ {1 - F\left( {\omega^{*} } \right)} \right]\omega^{*} R_{t}^{k} Q_{t - 1} K_{t} $$

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Using the depositors expected profits binding, we can writte:

$$ \left[ {1 - F\left( {\bar{\omega }} \right)} \right]\bar{\omega }R_{t}^{k} Q_{t - 1} K_{t} + \left( {1 - \mu } \right)\mathop \int \limits_{{\omega^{*} }}^{{\bar{\omega }}} \omega f\left( \omega \right)d\omega R_{t}^{k} Q_{t - 1} K_{t} + \left( {1 - \gamma } \right)\left( {1 - \mu } \right)\mathop \int \limits_{0}^{{\omega^{*} }} \omega f\left( \omega \right)d\omega R_{t}^{k} Q_{t - 1} K_{t} - R_{t}^{p} \left( {Q_{t - 1} K_{t} - N_{t} - M_{t} } \right) $$

Rewritting, we obtain the evolution of bank wealth as:

$$ V_{t}^{b} = \left\{ {\left[ {1 - F\left( {\bar{\omega }} \right)} \right]\bar{\omega } + \left( {1 - \mu } \right)\mathop \int \limits_{0}^{{\bar{\omega }}} \omega f\left( \omega \right)d\omega } \right\}R_{t}^{k} Q_{t - 1} K_{t} - \left\{ {R_{t}^{p} + \frac{{\gamma \left( {1 - \mu } \right)\mathop \int \nolimits_{0}^{{\omega^{*} }} \omega f\left( \omega \right)d\omega R_{t}^{k} Q_{t - 1} K_{t} }}{{Q_{t - 1} K_{t} - N_{t} - M_{t} }}} \right\}\left( {Q_{t - 1} K_{t} - N_{t} - M_{t} } \right) $$

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where ($\frac{{\gamma \left( {1 - \mu } \right)\mathop \int \nolimits_{0}^{{\omega^{*} }} \omega f\left( \omega \right)d\omega R_{t}^{k} Q_{t - 1} K_{t} }}{{Q_{t - 1} K_{t} - N_{t} - M_{t} }}$ is the financial premium that depositors charge at bankers due the asymetric information problem.

Appendix 2: Model equations system

Aggregate demand:

$$ Y_{t} = C_{t} + C_{t}^{e} + C_{t}^{b} + I_{t} + \mu \mathop \int \limits_{{\omega^{*} }}^{{\bar{\omega }}} \omega f\left( \omega \right)d\omega + \left[ {\gamma + \mu \left( {1 - \gamma } \right)} \right]\mathop \int \limits_{0}^{{\omega^{*} }} \omega f\left( \omega \right)d\omega $$

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$$ \frac{1}{{C_{t} }} = \beta {\mathbb{E}}_{t} \left[ {\frac{{R_{t + 1}^{d} }}{{C_{t + 1} }}} \right] $$

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$$ C_{t}^{e} = \left( {1 - \gamma^{e} } \right)V_{t} $$

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$$ C_{t}^{b} = \left( {1 - \gamma^{b} } \right)V_{t}^{b} $$

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$$ {\mathbb{E}}_{t} \left( {R_{t + 1}^{k} } \right) = {\mathbb{E}}_{t} \left( {\frac{{1/X_{t + 1} \left( {\alpha Y_{t + 1} /K_{t + 1} } \right) + \left( {1 - \delta } \right)Q_{t + 1} }}{{Q_{t} }}} \right) $$

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$$ {\mathbb{E}}_{t} \left( {\frac{{R_{t + 1}^{k} }}{{R_{t} }}} \right) = \frac{{n_{{\bar{\omega }}} \left( {\bar{\omega }_{t + 1} } \right)g_{{\omega^{ *} }} \left( {\omega_{t + 1}^{ *} ,\bar{\omega }_{t + 1} } \right)}}{{n\left( {\bar{\omega }_{t + 1} } \right)g_{{\bar{\omega }}} \left( {\omega_{t + 1}^{ *} ,\bar{\omega }_{t + 1} } \right)h_{{\omega^{ *} }} \left( {\omega_{t + 1}^{ *} } \right) - n_{{\bar{\omega }}} \left( {\bar{\omega }} \right)g\left( {\omega_{t + 1}^{ *} ,\bar{\omega }_{t + 1} } \right)h_{{\omega^{ *} }} \left( {\omega_{t + 1}^{ *} } \right) + n_{{\bar{\omega }}} \left( {\bar{\omega }_{t + 1} } \right)g_{{\omega^{ *} }} \left( {\omega_{t + 1}^{ *} ,\bar{\omega }_{t + 1} } \right)h\left( {\omega_{t + 1}^{ *} } \right)}} $$

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$$ {\mathbb{E}}_{t} \left( {\frac{{R_{t + 1}^{k} }}{{R_{t} }}} \right) = \frac{{Q_{t} K_{t + 1} - M_{t + 1} - N_{t + 1} }}{{h_{{\omega^{ *} }} \left( {\omega_{t + 1}^{ *} } \right)Q_{t} K_{t + 1} }} $$

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$$ \frac{{Q_{t} K_{t + 1} }}{{N_{t + 1} }} = {\mathbb{E}}_{t} \left( {\frac{{R_{t} }}{{R_{t + 1}^{k} }}} \right)\frac{{M_{t + 1} }}{{g\left( {\omega_{t + 1}^{ *} ,\bar{\omega }_{t + 1} } \right)N_{t} }} $$

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$$ Q_{t} = \frac{1}{\theta }\left( {\frac{{I_{t} }}{{K_{t} }}} \right)^{{1 - \alpha_{1} }} $$

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Aggregate supply:

$$ Y_{t} = A_{t} K_{t}^{\alpha } \left[ {H_{t}^{\varOmega } \left( {H_{t}^{e} } \right)^{{\varOmega_{1} }} \left( {H_{t}^{b} } \right)^{{1 - \varOmega - \varOmega_{1} }} } \right]^{1 - \alpha } $$

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$$ \left( {1 - \alpha } \right)\varOmega \frac{{Y_{t} }}{{H_{t} }} = X_{t} W_{t} $$

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$$ \left( {1 - \alpha } \right)\varOmega_{1} \frac{{Y_{t} }}{{H_{t}^{e} }} = X_{t} W_{t}^{e} $$

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$$ \left( {1 - \alpha } \right)(1 - \varOmega - \varOmega_{1} ) \frac{{Y_{t} }}{{H_{t}^{b} }} = X_{t} W_{t}^{b} $$

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$$ \frac{{W_{t} }}{{C_{t} }} = \xi \frac{1}{{1 - H_{t} }} $$

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State variables evolution:

$$ K_{t + 1} = \theta I_{t}^{{\alpha_{1} }} K_{t}^{{1 - \alpha_{1} }} + \left( {1 - \delta } \right)K_{t} $$

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$$ N_{t + 1} = \gamma^{e} V_{t} + W_{t}^{e} $$

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$$ V_{t} = R_{t}^{k} Q_{t - 1} K_{t} - \left\{ {R_{t}^{p} + \frac{{\left[ {\mu \mathop \int \nolimits_{{\omega^{*} }}^{{\bar{\omega }}} \omega f\left( \omega \right)d\omega - \left( {\gamma + \mu \left( {1 - \gamma } \right)} \right)\mathop \int \nolimits_{0}^{{\omega^{*} }} \omega f\left( \omega \right)d\omega } \right]R_{t}^{k} Q_{t - 1} K_{t} }}{{Q_{t - 1} K_{t} - N_{t} }}} \right\}\left( {Q_{t - 1} K_{t} - N_{t} } \right) $$

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$$ M_{t + 1} = \gamma^{b} V_{t}^{b} + W_{t}^{b} $$

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$$ V_{t}^{b} = \left\{ {\left[ {1 - F\left( {\bar{\omega }} \right)} \right]\bar{\omega } + \left( {1 - \mu } \right)\mathop \int \limits_{0}^{{\bar{\omega }}} \omega f\left( \omega \right)d\omega } \right\}R_{t}^{k} Q_{t - 1} K_{t} - \left\{ {R_{t}^{p} + \frac{{\gamma \left( {1 - \mu } \right)\mathop \int \nolimits_{0}^{{\omega^{*} }} \omega f\left( \omega \right)d\omega R_{t}^{k} Q_{t - 1} K_{t} }}{{Q_{t - 1} K_{t} - N_{t} - M_{t} }}} \right\}\left( {Q_{t - 1} K_{t} - N_{t} - M_{t} } \right) $$

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Monetary policy and shocks:

$$ P_{t}^{N} = \frac{{\theta_{p} }}{{\theta_{p} - 1}}\varLambda_{t} P_{t}^{{1 + \theta_{p} }} s_{t} y_{t} + \beta \chi {\mathbb{E}}_{t} \left[ {\pi_{t + 1}^{{\theta_{p} }} P_{t + 1}^{N} } \right] $$

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$$ P_{t}^{D} = \varLambda_{t} P_{t}^{{\theta_{p} }} y_{t} + \beta \chi {\mathbb{E}}_{t} \left[ {P_{t + 1}^{D} } \right] $$

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$$ R_{t}^{n} = \rho_{0} R_{t - 1}^{n} + \theta^{\pi } \left( {\pi_{t} - \pi_{ss} } \right) + \theta^{y} \left( {y_{t} - y_{ss} } \right) + \varepsilon_{t} $$

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$$ A_{t} = \rho_{A} A_{t - 1} + \varepsilon_{t}^{A} ;\quad \varepsilon_{t}^{A} \sim N\left( {0,\sigma_{A}^{2} } \right) $$

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$$ \varepsilon_{t} = \rho_{\varepsilon } \varepsilon_{t - 1} + u_{t}^{\varepsilon } ;\quad u_{t}^{\varepsilon } \sim N\left( {0,\sigma_{\varepsilon }^{2} } \right) $$

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Appendix 3: Calibration and steady state

See Table 3.

Table 3 Calibration non regulated models