Finance and Credit in a Model of Monetary Policy

Abstract

The question facing many emerging economies is the extent to which a workhorse advanced economy model can yield important insights for monetary policymaking. We note that the standard sticky price, monopolistically competitive model does not allow analysis of money and credit dynamics and led to a concentration of research on simple interest rate reaction functions. Time-varying financial frictions tend to act as a tax on intermediation activities and so can vary output in a significant manner. In this paper, we consider the implications of financial frictions for baseline monetary policy using a model calibrated on Indian data and find that a simple interest rate reaction function may not be welfare maximizing.

We are grateful for helpful conversations and comments from Philip Arestis, Chetan Ghate, Mike Joyce, Ken Kletzer, Jack Meaning, James Warren and participants at the BIS-OECD Workshop Panel on Policy Interaction: Fiscal Policy, Monetary Policy and Debt Interaction, the Reserve Bank of India. We also thanks Luisa Corrado, Germana Corrado, Sean Holly, Philip Tuner, Alex Waters and Fabrizio Zampolli for permission to draw on joint work. Any remaining errors are our own.

7 Appendix

1.1 7.1 First-Order Condition

1.1.1 7.1.1 Households and Banking Sector

$$\begin{aligned} c_{t}:\frac{\phi }{c_{t}}-\lambda _{t}-\mu _{t}=0 \end{aligned}$$

(45)

$$\begin{aligned} B_{t+1}:\frac{\mu _{t}}{\lambda _{t}} \Omega _{t}-1+\beta E_{t}\left( \frac{\lambda _{t+1}}{\lambda _{t}}\frac{P_{t}}{ P_{t+1}}(1+R_{t}^{B})\right) =0 \end{aligned}$$

(46)

$$\begin{aligned} K_{t+1}:\frac{\mu _{t}}{\lambda _{t}} kA3_{t}q_{t}\Omega _{t}-q_{t}+\beta E_{t}\left( \frac{\lambda _{t+1}}{ \lambda _{t}}\frac{\xi _{t+1}}{\lambda _{t+1}}\eta \left( \frac{K_{t+1}}{ A1_{t+1}n_{t+1}}\right) ^{(\eta -1)}+\frac{\lambda _{t+1}}{\lambda _{t}} q_{t+1}(1-\delta )\right) =0 \end{aligned}$$

(47)

where $\frac{\mu _{t}}{\lambda _{t}}=\frac{\phi }{c_{t}\lambda _{t}}-1$ measures households’ marginal utility relative to shadow value of funds. $\Omega _{t}=\frac{\partial y_{t}}{\partial b_{t+1}}=\frac{\alpha y_{t}}{ b_{t+1}+A3_{t}kq_{t}K_{t+1}}$ is marginal value of bonds as collateral.

$$\begin{aligned} I_{t}:-\lambda _{t}+\lambda _{t}q_{t}(1-S(\frac{I_{t}}{ I_{t-1}})-S^{\prime }(\frac{I_{t}}{I_{t-1}})\frac{I_{t}}{I_{t-1}} )+\beta \lambda _{t+1}q_{t+1}S^{\prime }(\frac{I_{t+1}}{I_{t}})\left( \frac{ I_{t+1}}{I_{t}}\right) ^{2}-\mu _{t}=0 \end{aligned}$$

$$\begin{aligned} n_{t}^{s},m_{t}^{s}:\frac{1-\phi }{ 1-n_{t}^{s}-m_{t}^{s}}=w_{t}\lambda _{t} \end{aligned}$$

(48)

$$\begin{aligned} m_{t}:w_{t}=\frac{\mu _{t}}{\lambda _{t}}(1-\alpha ) \frac{y_{t}}{m_{t}} \end{aligned}$$

(49)

1.1.2 7.1.2 Firms (Wholesale Good Producing Firm and Retailer)

$$\begin{aligned} n_{t}:w_{t}=\frac{\xi _{t}}{\lambda _{t}}(1-\eta )\frac{ y_{t}}{n_{t}} \end{aligned}$$

(50)

$$\begin{aligned} mc_{t}=\frac{\epsilon -1}{\epsilon }\frac{\xi _{t}}{\lambda _{t}} \end{aligned}$$

$$\begin{aligned} P_{t}^{f}:\sum _{s=0}^{\infty }\theta ^{s}E_{t}\left[ \Lambda _{t,s}Y_{t,t+s}\left( \frac{P_{t}^{f}}{P_{t-1}}-\mathrm { X_{\psi }mc_{t}\pi _{t-1,t+s}}\right) \right] =0 \end{aligned}$$

(51)

where $P_{t}^{f}$ is forward-looking firms’ optimal price (resetting price), $\Lambda _{t,s}=\beta ^{s}\left( \frac{c_{t+s}}{c_{t}}\right) ^{-1}\left( \frac{ P_{t}}{P_{t+s}}\right)$ is the households’ stochastic discount factor, $X_{\psi }=1/mc$ is desired markup. $mc_{t}$ is real marginal cost, $\pi _{t-1,t+s}=\frac{P_{t+s}}{P_{t-1}}$.

Aggregate price level is given by

$$\begin{aligned} P_{t}:P_{t}=\theta P_{t-1}+(1-\theta )\bar{P}_{t}^{*} \end{aligned}$$

(52)

where $\bar{P}_{t}^{*}$ is an index of prices set in period t based on the forward-looking and backward-looking price setting behavior such that

$$\begin{aligned} \bar{P}_{t}^{*}=\omega P_{t}^{b}+(1-\omega )P_{t}^{f} \end{aligned}$$

(53)

where $P_{t}^{b}$ is the price set by the backward-looking rule of thumb ($P_{t}^{b}=\bar{P}_{t-1}^{*}+\pi _{t-1}$), $P_{t}^{f}$ is the price set by forward-looking firms, and $\omega$ is the degree of backward-lookingness and $1-\omega$ is forward-lookingness.

Hybrid New Keynesian Philips Curve is given by

$$\begin{aligned} \widehat{\pi }{}_{t}=\kappa \widehat{mc}_{t}+\gamma _{f}E_{t}(\widehat{\pi }{} _{t+1})+\gamma _{b}\widehat{\pi }{}_{t-1}+a5_{t} \end{aligned}$$

(54)

1.2 7.2 Market Clearing

Resource Constraint

$$\begin{aligned} y_{t}=c_{t}+I_{t} \end{aligned}$$

(55)

Financial Market (Deposits)

$$\begin{aligned} D_{t}^{s}=D_{t}^{d}=\frac{P_{t}y_{t}}{v_{t}} \end{aligned}$$

(56)

Balanced Government Budget

$$\begin{aligned} T_{t}=g_{t}-tax_{t}=\frac{B_{t+1}}{P_{t}(1+R_{t}^{B})}-\frac{B_{t}}{ P_{t}^{A} }-\frac{r_{t}}{P_{t}}R_{t}^{IB} \end{aligned}$$

(57)

1.3 7.3 Steady States

For the productivity and monitoring shocks we assume a trend growth rate equal to $A2_{t}=A1_{t}=(1+\varrho )^{t}$. In steady state $q=1$, $A2=A1=(1+\varrho )$, $\lambda$ shrinks at rate $\varrho$ so $\frac{ \lambda _{t+1}}{\lambda _{t}}=\frac{1}{1+\varrho }$. There is no inflation so $P=1$ while capital grows at rate $\varrho$ in steady state. We use the constant steady-state bonds to output ($boy=\frac{\text{ B }}{(1+R^{B})y}$). That is, we assume that the fiscal authority’s policy is set in order to stabilize $boy_{t}$ at an exogenous policy-determined value.

1.3.1 7.3.1 The Core Steady States

The following 10 equations give the steady-state value for y, c, I, K, m, n, w, $\lambda$, $\mu$, $\Omega$:

$$\begin{aligned} y=c+I \end{aligned}$$

(58)

$$\begin{aligned} 1=\frac{vF}{1-rr}\left( boy+\frac{kK}{y}\right) ^{\alpha }\left( \frac{m}{y} \right) ^{1-\alpha } \end{aligned}$$

(59)

$$\begin{aligned} \Omega =\frac{\alpha }{\left( boy+\frac{kK}{y}\right) } \end{aligned}$$

(60)

$$\begin{aligned} \frac{1-\phi }{1-n-m}=w\lambda \end{aligned}$$

(61)

$$\begin{aligned} w=\frac{\mu }{\lambda }\frac{(1-\alpha )y}{m} \end{aligned}$$

(62)

$$\begin{aligned} w=\frac{\epsilon -1}{\epsilon }(1-\eta )\left( \frac{K}{n(1+\varrho )} \right) ^{\eta } \end{aligned}$$

(63)

$$\begin{aligned} \frac{\mu }{\lambda }k\Omega -1+\beta \left[ \frac{1-\delta }{1+\varrho }+\eta \frac{ \epsilon -1}{\epsilon }\left( \frac{K}{n(1+\varrho )}\right) ^{\eta -1}\right] =0 \end{aligned}$$

(64)

$$\begin{aligned} \frac{1}{mc}=\left( \frac{K}{y(1+\varrho )}\right) ^{\eta }\left( \frac{n}{y} \right) ^{(1-\eta )} \end{aligned}$$

(65)

$$\begin{aligned} I=\frac{\rho +\delta }{1+\varrho }K \end{aligned}$$

(66)

$$\begin{aligned} \frac{\mu }{\lambda }=\frac{\phi }{c\lambda }-1 \end{aligned}$$

(67)

1.3.2 7.3.2 Other Variables

$$\begin{aligned} D=\frac{y}{v} \end{aligned}$$

(68)

$$\begin{aligned} r=rrD=rr\frac{y}{v} \end{aligned}$$

(69)

$$\begin{aligned} L=(1-rr)D=(1-rr)\frac{y}{v} \end{aligned}$$

(70)

$$\begin{aligned} T=g-tax=b-b(1+R^{B})-rR^{IB} \end{aligned}$$

(71)

$$\begin{aligned} b=boy\times y \end{aligned}$$

(72)

$$\begin{aligned} R^{T}-R^{B}=LSY^{B}&=\Omega \end{aligned}$$

(73)

$$\begin{aligned} LSY^{K}=k\times LSY^{K} \end{aligned}$$

(74)

$$\begin{aligned} EFP&=\frac{vmw}{(1-\alpha )(1-rr)y}=\frac{\mu }{\lambda }\frac{v}{(1-rr)} \end{aligned}$$

(75)

$$\begin{aligned} CEFP=(1-\alpha )\times EFP=\frac{vmw}{(1-rr)y}=(1-\alpha )\frac{\mu }{\lambda } \frac{v}{(1-rr)} \end{aligned}$$

(76)

$$\begin{aligned} \tau =R^{L}-R^{IB} \end{aligned}$$

(77)

Accordingly spreads between interest rates can be written as follows:

$$\begin{aligned} R^{T}&=\frac{1+\varrho }{\beta }-1 \end{aligned}$$

(78)

$$\begin{aligned} R&=R^{T}-EFP \end{aligned}$$

(79)

$$\begin{aligned} R^{L}&=R^{IB}+CEFP \end{aligned}$$

(80)

$$\begin{aligned} R^{B}&=R^{T}-LSY^{B} \end{aligned}$$

(81)

$$\begin{aligned} R^{D}&=R(1-rr) \end{aligned}$$

(82)

1.4 7.4 Log-linearization

Consumption

$$\begin{aligned} c_{t}=-c(\lambda \lambda _{t}+\mu \mu _{t}) \end{aligned}$$

(83)

Supply of labor

$$\begin{aligned} w_{t}=\frac{n}{1-n-m}n_{t}+\frac{m}{1-n-m}m_{t}-\lambda _{t} \end{aligned}$$

(84)

Demand for labor in the goods sector

$$\begin{aligned} w_{t}=mc_{t}+y_{t}-n_{t} \end{aligned}$$

(85)

Demand for monitoring work

$$\begin{aligned} w_{t}=\mu _{t}-\lambda _{t}+y_{t}-m_{t} \end{aligned}$$

(86)

Broad liquidity problem

$$\begin{aligned} y_{t}=v_{t}+rr_{t}+(1-\alpha )(a2_{t}+m_{t})+\alpha \left( \frac{b}{boy+kK} b_{t+1}+\frac{kK}{boy+kK}(a3_{t}+q_{t}+K_{t+1})\right) \end{aligned}$$

where $boy=\frac{B}{P(1+R^{B})y}$ and $b_{t+1}=\frac{B_{t+1}}{ P_{t}(1+R_{t}^{B})}=boy_{t}+y_{t}$

Marginal value of bond as collateral

$$\begin{aligned} \Omega _{t}=\frac{kK}{boy+kK/y}(y_{t}-q_{t}-K_{t+1}-a3_{t})+\frac{b}{boy+kK/y} b_{t+1} \end{aligned}$$

(87)

Asset price

$$\begin{aligned} \left( 1-k\Omega \frac{\mu }{\lambda }\right) q_{t}=&\frac{\beta }{1+\varrho } \left( 1-\delta +\eta mc\left( \frac{K}{n(1+\varrho )}\right) ^{\eta -1} \right) (E_{t}\lambda _{t+1}-\lambda _{t}) \\&\quad +\frac{\beta }{1+\varrho }(1-\delta )E_{t}q_{t+1}+k\Omega \frac{\mu }{\lambda } (\mu _{t}-\lambda _{t}+\Omega _{t}+a3_{t}) \\&\qquad \, +\frac{\beta }{1+\varrho }\eta mc\left( \frac{K}{n(1+\varrho )} \right) ^{\eta -1}E_{t}\left( mc_{t+1}+(1-\eta )(n_{t+1}+a1_{t+1})\right) \end{aligned}$$

Deposit-in-advance constraint (DIA)

$$\begin{aligned} P_{t}+y_{t}=D_{t}+v_{t} \end{aligned}$$

(88)

production function

$$\begin{aligned} y_{t}=\eta K_{t}+(1-\eta )(a1_{t}+n_{t}) \end{aligned}$$

(89)

Goods Market Clearing Condition

$$\begin{aligned} y_{t}=\frac{c}{y}c_{t}+\frac{I}{y}I_{t} \end{aligned}$$

(90)

Law of Motion of Capital

$$\begin{aligned} K_{t+1}=\frac{1-\delta }{1+\varrho }K_{t}+\frac{\varrho +\delta }{1+\varrho }I_{t} \end{aligned}$$

(91)

Investment

$$\begin{aligned} q_{t}=(1+\varrho )^{2}s\left[ (I_{t}-I_{t-1})-\beta \left( E_{t}(I_{t+1})-I_{t} \right) \right] +\frac{\mu }{\lambda }\mu _{t} \end{aligned}$$

(92)

Inflation

$$\begin{aligned} \pi _{t}=p_{t}-p_{t-1} \end{aligned}$$

(93)

Philips curve

$$\begin{aligned} \pi _{t}=\gamma _{b}\pi _{t-1}+\gamma _{f}E_{t}(\pi _{t+1})+\kappa mc_{t}+a5_{t} \end{aligned}$$

(94)

Government Budget Constraint

$$\begin{aligned} TT_{t}=b\left( b_{t+1}-b_{t}-R^{B}(b_{t}+R_{t}^{B}) \right) -rR^{IB}(r_{t}+R_{t}^{IB}) \end{aligned}$$

(95)

Riskless Interest Rate (Benchmark Interest Rate)

$$\begin{aligned} R_{t}^{T}=\lambda _{t}-E_{t}(\lambda _{t+1})+E_{t}(\pi _{t+1}) \end{aligned}$$

(96)

Liquidity Service Yield ($LSY_{t}$)

$$\begin{aligned} R_{t}^{T}-R_{t}^{B}=\mu _{t}-\lambda _{t}+\Omega _{t} \end{aligned}$$

(97)

External Finance Premium ($EFP_{t}$)

$$\begin{aligned} EFP_{t}=v_{t}+w_{t}+m_{t}-y_{t}+rr_{t}=\mu _{t}-\lambda _{t}+v_{t}+rr_{t} \end{aligned}$$

(98)

Interbank Interest Rate (Policy Rate)

$$\begin{aligned} R_{t}^{IB}=R_{t}^{T}-EFP_{t} \end{aligned}$$

(99)

Loan Interest Rate

$$\begin{aligned} R_{t}^{L}=R_{t}^{IB}+(C)EFP_{t} \end{aligned}$$

(100)

Bond Interest Rate

$$\begin{aligned} R_{t}^{B}=R_{t}^{T}-LSY_{t}^{B} \end{aligned}$$

(101)

Deposit Interest Rate

$$\begin{aligned} R_{t}^{D}=R_{t}^{IB}-\frac{rr}{1-rr}rr_{t} \end{aligned}$$

(102)

Monetary Policy Rule (Taylor Rule)

$$\begin{aligned} R_{t}=\rho R_{t-1}+(1-\rho )(\phi _{\pi }\pi _{t}+\phi _{y}mc_{t}+\phi _{f}f_{t})+a4_{t} \end{aligned}$$

(103)

OMO Policy Rule (Fiscal Policy Rule)

$$\begin{aligned} boy_{t}=a6_{t}=\rho _{boy}boy_{t-1}+\epsilon _{t}^{boy} \end{aligned}$$

(104)

Velocity

$$\begin{aligned} v_{t}=a7_{t} \end{aligned}$$

(105)

Liquidity Preference

$$\begin{aligned} \tau _{t}=a8_{t} \end{aligned}$$

(106)

Endogenous Reserves

$$\begin{aligned} r_{t}=\frac{1}{rR^{T}}\left( R^{IB}R_{t}^{IB}-R^{L}R_{t}^{L}+\tau \tau _{t} \right) \end{aligned}$$

(107)

Loans

$$\begin{aligned} L_{t}=D_{t}-\frac{1}{1-rr}rr_{t} \end{aligned}$$

(108)