Credit search and credit cycles

Abstract

The supply and demand of credit are not always well aligned, as is reflected in the countercyclical excess reserve-to-deposit ratio and interest spread between the lending rate and the deposit rate. We develop a search-based theory of credit allocations to explain the cyclical fluctuations in both bank reserves and interest spread. We show that search frictions in the credit market can naturally explain the countercyclical bank reserves and interest spread, as well as generate endogenous business cycles driven primarily by the cyclical utilization rate of credit resources, as long conjectured by the Austrian school of the business cycle. In particular, we show that credit search can lead to endogenous local increasing returns to scale and variable capital utilization in a model with constant returns to scale production technology and matching functions, thus providing a microfoundation for the indeterminacy literature of Benhabib and Farmer (J Econ Theory 63(1):19–41, 1994) and Wen (J Econ Theory 81(1):7–36, 1998).

Appendix: Proofs

Proof of Proposition 1

Since each match between the household and the banking sector utilizes $\widetilde{S}_{t}=e_{t}S_{t}$ units of household capital (savings), and each match between the banking sector and the production sector (firms) utilizes $K_{t}=u_{t}\widetilde{S}_{t}$ units of bank capital (deposits), given the total initial available credit resources $S_{t}$ in the economy, the fraction of aggregate credit resources ultimately utilized in production is hence given by

$$\begin{aligned} K_{t}=u_{t}e_{t}S_{t}. \end{aligned}$$

(31)

As each matched firm employs $n_{t}$ units of labor, the labor market equilibrium then requires

$$\begin{aligned} N_{t}=V_{t}q_{t}n_{t}=V_{t}q_{t}\left[ A_{t}\left( \frac{1-\alpha }{W_{t}} \right) \right] ^{\frac{1}{\alpha }}\widetilde{S}_{t}. \end{aligned}$$

(32)

Finally, it is easy to show that the total production output of all firms is given by

$$\begin{aligned} Y_{t}=V_{t}q_{t}y_{t}=V_{t}q_{t}A_{t}\widetilde{S}_{t}^{\alpha }n_{t}^{1-\alpha }=A_{t}\left( V_{t}q_{t}\widetilde{S}_{t}\right) ^{\alpha }N_{t}^{1-\alpha }=A_{t}K_{t}^{\alpha }N_{t}^{1-\alpha }, \end{aligned}$$

(33)

where $K_{t}$ is determined by Eq. (31). Since $K_{t}=e_{t}u_{t}S_{t}$, the aggregate production function in Eq. (33) can also be written as in Eq. (18).$\square$

Proof of Proposition 2

Substituting Eqs. (21) and (23) into Eq. (18) yields Eq. (24).

$\square$

Proof of Proposition 3

Equations (12) and (32) together imply

$$\begin{aligned} W_{t}=\left( 1-\alpha \right) \left( \frac{Y_{t}}{N_{t}}\right) . \end{aligned}$$

(34)

Substituting Eq. (34) into Eqs. (4), (6), and (7) yields

$$\begin{aligned} \dot{S}_{t}= & {} \left( 1-\alpha \eta \right) Y_{t}-\delta \left( e_{t}\right) S_{t}-C_{t}, \end{aligned}$$

(35)

$$\begin{aligned} \frac{\dot{C}_{t}}{C_{t}}= & {} \left( 1-\eta \right) \alpha \left( \frac{Y_{t} }{S_{t}}\right) -\delta \left( e_{t}\right) -\rho , \end{aligned}$$

(36)

$$\begin{aligned} \psi N_{t}^{\xi }= & {} \left( 1-\alpha \right) \left( \frac{Y_{t}}{N_{t}} \right) \left( \frac{1}{C_{t}}\right) . \end{aligned}$$

(37)

Consequently, we can reduce the dynamic system $\{C_{t},S_{t},N_{t},W_{t},R_{t}^{d},R_{t}^{l},\pi _{t},K_{t},e_{t},u_{t},q_{t},\theta _{t},Y_{t},V_{t}\}$ to a simplified one with fewer variables in $\{C_{t},S_{t},e_{t},u_{t},Y_{t},N_{t}\}$ with Eqs. (18), (21), (23), (35), (36), and (37), where $\delta \left( e\right)$ is defined in Eq. (3). The FOCs indicate $\delta ^{\prime }\left( e_{t}\right) =\left( 1+\kappa \right) \left( \frac{\delta \left( e_{t}\right) }{e_{t}}\right) =R_{t}^{d}$. Thus,

$$\begin{aligned} \delta \left( e_{t}\right) =\frac{R_{t}^{d}e_{t}}{1+\kappa }=\varepsilon _{H}\alpha \left( 1-\eta \right) \left( \frac{Y_{t}}{S_{t}}\right) . \end{aligned}$$

Log-linearizing the above simplified transition dynamics yields

$$\begin{aligned} \dot{c}_{t}= & {} \left( 1-\varepsilon _{H}\right) \left( \frac{Y}{S}\right) \left( 1+\widehat{y}_{t}-\widehat{s}_{t}\right) -\rho \\ \dot{s}_{t}= & {} \left[ \left( 1-\alpha \eta \right) -\varepsilon _{H}\alpha \left( 1-\eta \right) \right] \left( \frac{Y}{S}\right) \left( 1+\widehat{y}_{t}-\widehat{s}_{t}\right) -\left( \frac{C/Y}{S/Y}\right) \left( 1+\widehat{ c}_{t}-\widehat{s}_{t}\right) \\ \widehat{y}_{t}= & {} \alpha \left( \widehat{e}_{t}+\widehat{u}_{t}+\widehat{s} _{t}\right) +\left( 1-\alpha \right) \widehat{n}_{t} \\ \widehat{e}_{t}= & {} \varepsilon _{H}\left( -\widehat{s}_{t}\right) \\ \widehat{u}_{t}= & {} \varepsilon \widehat{y}_{t} \\ \left( 1+\xi \right) \widehat{n}_{t}= & {} \left( 1-\alpha \right) \left( \widehat{y}_{t}-\widehat{c}_{t}\right) . \end{aligned}$$

Consequently, we obtain the simplified dynamic system on $\left( s_{t},c_{t}\right)$ as

$$\begin{aligned} \left[ \begin{array}{c} \dot{s}_{t} \\ \dot{c}_{t} \end{array} \right] =J\cdot \left[ \begin{array}{c} \widehat{s}_{t} \\ \widehat{c}_{t} \end{array} \right] , \end{aligned}$$

where

$$\begin{aligned} J\equiv \delta \cdot \left[ \begin{array}{ll} \left( \frac{1+\kappa }{\alpha }-1\right) \left( \frac{1}{1-\eta }\right) \lambda _{s} &{}\quad \left( \frac{1+\kappa }{\alpha }-1\right) \quad \left( \frac{1}{1-\eta }\right) \left( \lambda _{c}-1\right) \\ \kappa \left( \lambda _{s}-1\right) &{}\quad \kappa \lambda _{c} \end{array} \right] . \end{aligned}$$

$\kappa \equiv \frac{1}{\varepsilon _{H}}-1$, $\alpha _{s}\equiv \frac{ \alpha \left( 1-\varepsilon _{H}\right) }{1-\alpha \left( \varepsilon +\varepsilon _{H}\right) }$, $\alpha _{n}\equiv \frac{1-\alpha }{1-\alpha \left( \varepsilon +\varepsilon _{H}\right) }$, $\lambda _{s}\equiv \frac{ \alpha _{s}\left( 1+\xi \right) }{1+\xi -\alpha _{n}}$, and $\lambda _{c}\equiv \frac{-\alpha _{n}}{1+\xi -\alpha _{n}}.$

Note that the local dynamics around the steady state is then determined by the eigenvalues of J. If both eigenvalues of J are negative, then the model is indeterminate. As a result, the model can experience endogenous fluctuations driven by sunspots. The eigenvalues of J, $x_{1}$ and $x_{2}$ satisfy

$$\begin{aligned} x_{1}+x_{2}= & {} \hbox {Trace}(J)=\delta \left[ \left( \frac{1+\kappa }{\alpha } -1\right) \left( \frac{1}{1-\eta }\right) \lambda _{s}+\kappa \lambda _{c} \right] , \\ x_{1}x_{2}= & {} \hbox {Det}(J)=\delta ^{2}\left( \frac{1+\kappa }{\alpha }-1\right) \left( \frac{\kappa }{1-\eta }\right) \left( \lambda _{s}-\lambda _{c}-1\right) . \end{aligned}$$

Therefore, indeterminacy emerges if and only if $\hbox {Trace}(J) <0$ and $\hbox {Det}(J)>0$. We can prove that $\hbox {Trace}(J)<0$ and $\hbox {Det}(J) >0$ if and only if the following four conditions hold, in addition to the restriction that $\varepsilon ,\varepsilon _{H}\in [0,1]$:

$$\begin{aligned} \varepsilon +\varepsilon _{H}< & {} \frac{1}{\alpha }, \end{aligned}$$

(38)

$$\begin{aligned} \varepsilon +\varepsilon _{H}> & {} \left( \frac{1}{\alpha }\right) \left( \frac{\alpha +\xi }{1+\xi }\right) , \end{aligned}$$

(39)

$$\begin{aligned} \varepsilon _{H}< & {} 1-\frac{\left( 1-\eta \right) \left( 1-\alpha \right) \kappa }{\left( 1+\kappa -\alpha \right) \left( 1+\xi \right) }, \end{aligned}$$

(40)

$$\begin{aligned} \varepsilon< & {} \frac{1}{\alpha }-1. \end{aligned}$$

(41)

First, since $\varepsilon _{H}\in [0,1]$, comparing Conditions (38) and (41) suggests that the former is never binding. Second, note that $\kappa \equiv \frac{1}{\varepsilon _{H}}-1$. Thus, Condition (40) can be rewritten as

$$\begin{aligned} \varepsilon _{H}<\left[ \frac{1+\xi -\left( 1-\eta \right) \left( 1-\alpha \right) }{1+\xi }\right] \left( \frac{1}{\alpha }\right) . \end{aligned}$$

Since $\xi \ge 0,$ we have $[\frac{1+\xi -(1-\eta ) (1-\alpha )}{1+\xi }] ( \frac{1}{\alpha }) >[ 1-( 1-\eta )(1-\alpha )](\frac{1}{\alpha })>1$, and therefore, we know that Condition (40) is not binding. Finally, if $\alpha \in [\frac{1}{2},1)$, then we know that $\frac{1}{\alpha }-1\in (0,1]$, and we must have $0\le \varepsilon < \frac{1}{\alpha }-1$. Besides, we know that $\widetilde{\varepsilon }\equiv (\frac{1}{\alpha }) (1-\frac{1-\alpha }{1+\xi }) >2$ when $\alpha \in [\frac{1}{2},1)$. Therefore, Condition (39) always holds in this case. In contrast, when $\alpha \in (0,\frac{1}{2})$, we have $\frac{1}{\alpha }-1>1>\varepsilon$, and thus, Condition (41) always holds. Meanwhile, since $\varepsilon +\varepsilon _{H}\le 2$, to guarantee that Condition (39) can be satisfied, we must have $\widetilde{\varepsilon }\equiv ( \frac{1}{\alpha }) (1-\frac{1-\alpha }{1+\xi }) <2$, i.e., $\xi \in [0, \frac{\alpha }{1-2\alpha })$.$\square$

Proof of Proposition 4

The FOCs are given by

$$\begin{aligned} \delta ^{0}e_{t}^{\kappa }= & {} \frac{\alpha Y_{t}}{e_{t}S_{t}}, \end{aligned}$$

(42)

$$\begin{aligned} \varDelta ^{0}u_{t}^{\lambda }= & {} \frac{\alpha Y_{t}}{u_{t}}. \end{aligned}$$

(43)

Substituting Eqs. (42) and (43) into Eq. (18) yields

$$\begin{aligned} Y^\mathrm{SP}=\widetilde{Y}^\mathrm{SP}A_{t}^{\tau }S_{t}^{\alpha _{s}}N_{t}^{\alpha _{n}}, \end{aligned}$$

(44)

where $\widetilde{Y}^{SP}=[(\frac{\alpha }{\delta ^{0}})^{\varepsilon _{H}}(\frac{\alpha \eta }{\varDelta ^{0}})^{\varepsilon }] ^{\frac{\alpha }{1-\alpha ( \varepsilon +\varepsilon _{H}) }}$. Dividing Eq. (24) by Eq. (44) yields Eq. (27). Then the FOC of Eq. (27) yields $\eta ^{*}$.$\square$