The Micro and Macro of (Unconventional) Monetary Policy

Gregor Boehl
Uni Bonn

Motivation

Common shortcut:
central bank sets interest rate
Reality:
central bank supplies reserves & reserve rate to banks
banks set interest rate

Q: Banking sector important?
  • For unconventional monetary policy: Yes!
  • UMP potentially counterproductive

Data I

  • Deposit & lending rates not directly controlled
  • But rate central in macroeconomic theory: $\small c_t = E_t c_{t+1} - \frac{1}{\sigma} (\color{lightgreen}r_t \color{white}- E_t \pi_{t+1})$
  • After 2009: No clear pass-through to deposit rate

Data II

  • Reserves: funds of commercial banks stored at ECB
  • Clearly policy driven (via asset purchases)
  • No clear pass-through to HH deposits

This Paper

  • Develop an IO-model of banking sector with liquidity friction
  • Banks hold reserves to hedge liquidity risk
  • Embedded into a medium-scale NK framework
  • Mechanism:

Reserves up liquidity up liquidity costs down
credit volume up
lending rate down deposit rate ???

If credit demand very elastic:
credit volume up² lending rate const. deposit rate up
aggregate demand falls!

Policy Effectiveness

IOR policy reserves policy
MRR* binding low high
MRR* slack high low
sign of effect state dependend:
depends on supply of reserves
state dependend:
depends on investment elasticity
*Minimal reserve requirement

Literature

  • DSGE literature: Gertler & Karadi (2011,2013), Carstrom et al. (2013), Boehl et. al. (forthc.)
  • Banking literature: Corbae & D'Erasmo (2021), Drechsler et al. (2017), Poole (1968)
  • Monetary: Bianchi & Bigio (2022), Piazzesi et al. (2019)

Model:

An IO model of the banking sector

Banking sector: bank $i$'s problem



  • banks hold assets and reserves, backed by deposits:
    $$ \definecolor{sgreen}{RGB}{78, 126, 100} \definecolor{sred}{RGB}{231, 75, 75} \definecolor{sblue}{RGB}{112, 174, 255} A_{i,t} + J_{i,t} = D_{i,t} \color{sgreen} - \Delta D_{i,t} $$
    $$ \color{sred}A_{i,t} \color{white} + J_{i,t} = D_{i,t} \color{sgreen} - \Delta D_{i,t} $$
    $$ \color{sred}A_{i,t} \color{white} + J_{i,t} = D_{i,t} \color{sblue} - \Delta D_{i,t} $$
    $$ \color{sred}A_{i,t} \color{sblue} + J_{i,t} \color{white} = D_{i,t} - \Delta D_{i,t} $$
    $$ \color{sred}A_{i,t} \color{sblue} + J_{i,t} \color{white} = D_{i,t} - \Delta D_{i,t} $$
  • assets are fixed at the beginning of each period
  • deposits subject to random transfers (prob. $\chi$)
  • reserves $J_i > 0$ must compensate for $\Delta D_i$
  • any unit $\Delta D_i > J_i$ incurs a cost $\gamma$

Banking sector: bank $i$'s problem

$$ \small \begin{align} \text{Net outflow}& &\Delta {D}_{i} &\sim \mathcal{N} \left( 0, \frac{{D}_{i} D_{-i}}{\sum D_{j}} (2 \chi - \chi^2)\right)\\ \text{Exp. liquidity cost}& &g(J_{i}, D_{i}) &= E \Big[\Delta D_{i} - J_{i} | \Delta D_{i} > J_{i}\Big] \end{align} $$

Banking sector: bank $i$'s problem

  • Expected cost for each bank: $$ g(J_{i,t}, D_{i,t}) = h(D_{i,t}) f_h\left(J_{i,t}\right) - J_{i,t} \left[1-F_h\left(J_{i,t}\right)\right], $$
  • Banks are subject to a minimal reserve requirement (MRR) $$ \psi D_{i,t} \leq J_{i,t} $$
  • Each bank $i$ maximizes profits s.t. balance sheet, expected costs, MRR & competitors actions

Banking sector: aggregation

FOCs:
$$ \small \definecolor{sgreen}{RGB}{78, 126, 100} \begin{align} R = (1-\psi) R^b + \psi R^j &-\gamma \left( 0.5\hat{f} - \psi \left[1-\hat{F}\right] \right) \\ \color{sgreen} \underbrace{ \color{white} R^j - R^b + \gamma \left[1- \hat{F} \right] }_{MPJ} &= \min\left\{ 0, \; R^j - R^b + \gamma \left[1- \hat{F}_\psi \right] \right\} \end{align} $$
FOCs:
$$ \small \begin{align} R = (1-\psi) R^b + \psi R^j &-\gamma \left( 0.5\hat{f} - \psi \left[1-\hat{F}\right] \right) \\ \underbrace{ R^j - R^b + \gamma \left[1- \hat{F} \right] }_{MPJ} &= \min\left\{ 0, \; R^j - R^b + \gamma \left[1- \hat{F}_\psi \right] \right\} \end{align} $$
FOCs when MRR slack:
$$ \small \begin{align} R = \color{sgreen}(1-\psi)\color{white} R^b \color{sgreen}+ \psi R^j \color{white} &-\gamma \color{sgreen} \left( \color{white} 0.5\hat{f} \color{sgreen} - \psi \left[1-\hat{F}\right] \right) \\ \color{white} \underbrace{ R^j - R^b + \gamma \left[1-\hat{F}\right] }_{MPJ} &= \color{sgreen} \min\left\{ \color{white} 0 \color{sgreen}, \; R^j - R^b + \gamma \left[1-\hat{F}_\psi \right] \right\} \end{align} $$
FOCs at MRR:
$$ \small \begin{align} R = (1-\psi) R^b + \psi R^j &-\gamma \left( 0.5\hat{f} - \psi \left[1-\hat{F}\right] \right) \\ \color{sgreen} \underbrace{ R^j - R^b + \gamma \left[1- \color{white} \hat{F} \color{sgreen} \right]}_{MPJ} &= \color{sgreen} \min\left\{ 0, \; R^j - R^b + \gamma \left[1- \color{white} \hat{F}_\psi \color{sgreen} \right] \right\} \end{align} $$
  • IOR rate $R^j$, deposit rate $R$, lending rate $R^b$
  • $\hat{f} = f\left(\hat{J}| 0, \hat{D}\right)$, $\hat{f}_\psi = f\left(\psi\hat{D}| 0, \hat{D}\right)$, $\hat{F}=...$
  • Reserves & deposits scaled by liquidity risk $\hat{J} = \frac{J}{\nu}$

Banking sector: partial equilibrium

$$ \scriptstyle \begin{align} \text{demand:}& &{d_A}\left(R^b\right) \; &= \;\hat{D}, \qquad \left\{\hat{J}, R^j\right\} \text{ fixed}\\ \text{supply:}& &\gamma \left[1- \hat{F} \right] \; &= \; \min\left\{ R^b - R^j, \; \gamma \left[1- \hat{F}_\psi \right] \right\} \end{align} $$
$$ \scriptstyle \begin{align} \text{demand:}& &{d_A}\left(R^b\right) \; &= \; \hat{D}, \qquad \left\{\hat{J}, R^j\right\} \text{ fixed}\\ \text{supply:}& &\color{sgreen} \gamma \left[1- \color{white} \hat{J} \color{sgreen} \right] \; &= \; \color{sgreen} \min\left\{ R^b - R^j, \; \gamma \left[1- \color{white} \psi \hat{D} \color{sgreen} \right] \right\} \end{align} $$
$$ \scriptstyle \begin{align} \text{demand:}& &{d_A}\left(R^b\right) \; &= \; \hat{D}, \qquad \left\{\hat{J}, R^j\right\} \text{ fixed}\\ \text{supply:}& &\color{white} \gamma \left[1-\hat{F}\right] \; &= \; \color{sgreen}\min\left\{ \color{white} R^b - R^j \color{sgreen}, \; \gamma \left[1-\hat{F}_\psi \right] \right\} \end{align} $$
$$ \scriptstyle \begin{align} \text{demand:}& &{d_A}\left(R^b\right) \; &= \; \hat{D}, \qquad \left\{\hat{J}, R^j\right\} \text{ fixed}\\ \text{supply:}& &\gamma \left[1- \hat{F} \right] \; &= \; \min\left\{ R^b - R^j, \; \gamma \left[1- \hat{F}_\psi \right] \right\} \end{align} $$

Banking sector: spreads


  • Holding reserves & deposit constant
$$ \scriptstyle \begin{align} R^b - R^j \; &= \; \gamma \left[1- F\left(\frac{J}{\nu};\; 0, \frac{D}{\nu}\right) \right] &(MPJ = 0) \\ R^b - R \; &= \; 0.5 \gamma f\left(\frac{J}{\nu};\; 0, \frac{D}{\nu}\right) &(MPD = 0) \end{align} $$

Results:

partial equilibrium

partial equilibrium: reserves policy


$$ \scriptsize \begin{align} R &= (1-\psi) R^b + \psi R^j -\gamma \left( 0.5\hat{f} - \psi \left[1-\hat{F}\right] \right) \\ \gamma \left[1- \hat{F} \right] &= \min\left\{ R^b - R^j, \; \gamma \left[1- \hat{F}_\psi \right] \right\} \end{align} $$

partial equilibrium: higher elasticity

$$ \scriptsize \begin{align} R &= (1-\psi) R^b + \psi R^j -\gamma \left( 0.5\hat{f} - \psi \left[1-\hat{F}\right] \right) \\ \gamma \left[1- \hat{F} \right] &= \min\left\{ R^b - R^j, \; \gamma \left[1- \hat{F}_\psi \right] \right\} \end{align} $$

partial equilibrium: IOR policy

$$ \scriptsize \begin{align} R &= (1-\psi) R^b + \psi R^j -\gamma \left( 0.5\hat{f} - \psi \left[1-\hat{F}\right] \right) \\ \gamma \left[1- \hat{F} \right] &= \min\left\{ R^b - R^j, \; \gamma \left[1- \hat{F}_\psi \right] \right\} \end{align} $$

Results:

general equilibrium

general equilibrium: model

  • Embed banking sector into medium scale DSGE
  • Features:
    capital, capital adjustment, capital utilization, sticky prices á la Rotemberg, debt financed transfers
  • Fact: determined under IOR & reserves peg (no Taylor principle)
  • Nonlinear perfect foresight solution (using econpizza)

general equilibrium

reserves policy & the MRR

general equilibrium

reserves policy if MRR is slack

general equilibrium

reserves policy @ the MRR

general equilibrium

IOR policy: @ vs. slack MRR

Conclusion

  • Provide a unifying theory of the relationship of
    • bank deposits
    • central bank reserves
    • minimal reserves requirements
    • deposits, lending & reserves rate
  • Effect of UMP is highly state dependent