(Un)conventional policy and the effective lower bound.
Date: 2018-10-23
Institutions: European Central Bank
Subjects: EU policy, financial market, macroeconomics, monetary policy

Fiorella De Fiore, Oreste Tristani 

Working Paper Series 

(Un)conventional policy and the 
effective lower bound 

No 2183 / October 2018 

Disclaimer: This paper should not be reported as representing the views of the European Central Bank 
(ECB). The views expressed are those of the authors and do not necessarily reflect those of the ECB. 

Abstract

We study the optimal combination of conventional (interest rates) and unconventional

(credit easing) monetary policy in a model where agency costs generate a spread between

deposit and lending rates. We show that unconventional measures can be a powerful

substitute for interest rate policy in the face of certain …nancial shocks. Such measures

help shield the real economy from the deterioration in …nancial conditions and warrant

smaller reductions in interest rates. They therefore lower the likelihood of hitting the

lower bound constraint. The alternative option to cut interest rates more deeply and avoid

deploying unconventional measures is sub-optimal, as it would induce unnecessarily large

changes in savers’intertemporal consumption patterns.

Keyworks: optimal monetary policy, unconventional policies, zero-lower bound, asym-

metric information

JEL codes: E44, E52, E61

ECB Working Paper Series No 2183 / October 20181Non-Technical Abstract

After the …nancial crisis of 2008-09, central banks have aggressively cut monetary policy

rates, in many cases all the way to their lower bound. At the same time, many central banks

have implemented so-called "non-standard" or "unconventional" monetary policy measures.

However, standard and non-standard measures have been combined in di¤erent ways. Non-

standard measures were introduced in late 2008 and then implemented for a number of years,

both in the US and in the euro area. As far as standard monetary policy is concerned, the

Federal Reserve cut its interest rates to near zero almost at the same time. The European

Central Bank, on the contrary, did not immediately cut its main policy interest rate to zero.

The rate on the main re…nancing operations (MRO) was reduced sharply at the end of 2008

but it bottomed at 1% in May 2009 and did not fall below that threshold until mid-2012.

Which considerations should determine the mix of standard and non-standard policy re-

sponses? This paper presents a theoretical analysis of this issue. It speci…cally seeks to provide

answers to the following questions. Do non-standard measures reduce the likelihood that in-

terest rates reach the e¤ective lower bound (ELB)? Should interest rate policy be used at all,

once unconventional measures have been deployed? Looking forward to the return to normal

conditions, the so-called “exit”, how long should non-standard policies be optimally kept in

place?

We attempt to provide an answer to these questions in the context of a simple dynamic,

general equilibrium model that features both sticky prices and …nancial frictions in the form

of asymmetric information between borrowers (…rms) and lenders (banks). Following a well-

established literature, we assume that banks must pay a monitoring cost to audit borrowers

that do not repay their loans. The monitoring cost captures all bankruptcy costs, including

legal expenses and any losses associated with asset liquidation. We assume that monitoring

costs are subject to stochastic shocks over time. We speci…cally have in mind time variations in

losses associated with asset liquidation, which may increase markedly during crises for example

due to …re sales. We interpret adverse …nancial shocks as increases in monitoring costs, which

lead to higher bank lending rates and a reduction in lending.

In this model environment, we de…ne non-standard measures in a stylised fashion. Rather

than letting the central bank purchase assets from commercial banks, we assume that it can

directly lend to …rms, also subject to asymmetric information and a monitoring cost. The key

di¤erence is that the monitoring cost of the central bank is higher on average–hence the central

ECB Working Paper Series No 2183 / October 20182bank is normally less e¢ cient–but it does not increase with adverse …nancial shocks. Given

these assumptions, central bank intermediation is not desirable under normal circumstances.

Following an adverse …nancial shocks, however, commercial banks’monitoring e¢ ciency dete-

riorates. If the shock is su¢ ciently severe, private bank lending rates may increase up to the

point where central bank intermediation becomes competitive.

Our main result is that, in the face of adverse shocks of a …nancial nature–shocks which

reduce banks’ monitoring e¢ ciency–non-standard measures can be powerful substitutes of

standard interest rate policy. Non-standard measures can mitigate the repercussions of the

…nancial shock on the real economy and on in‡ation. They therefore reduce the need for

interest rate cuts.

The alternative option to cut interest rates and avoid deploying non-standard measures

would be suboptimal. Reductions in policy rates would at the same time dampen the increase

in …nancing costs for borrowers and reduce asset returns for lenders. By contrast, non-standard

measures can shield borrowers from unwarranted increases in lending rates without a¤ecting

lenders. Interest rate cuts remain however optimal in reaction to other demand-type shocks

which do not directly arise from the …nancial sector.

We illustrate our results in a "…nancial crisis" scenario in which the economy is hit by a

highly persistent …nancial shock. In this scenario, interest rate cuts and non-standard measures

must be used in combination, because the crisis creates downward pressure on prices. In the

normalization phase, interest rates are increased once de‡ationary pressures abate, but non-

standard measures remain in place for a prolonged period of time due to the persistent e¤ects

of the crisis.

1 Introduction

In response to the …nancial and economic crisis of 2008-09, central banks have aggressively

cut monetary policy rates, in many cases all the way to the e¤ective lower bound (henceforth

ELB), namely the rate below which it becomes pro…table for …nancial institutions to exchange

ECB Working Paper Series No 2183 / October 20183central bank reserves for cash.1 At the same time, all central banks have implemented so-called

"non-standard" or "unconventional" monetary policy measures.

However, standard and non-standard measures have been combined in di¤erent ways by

di¤erent central banks (for a cross-country comparison see e.g. Lenza, Pill and Reichlin,

2010). Taking the expansion of the central banks’balance sheets as an indicator, non-standard

measures were implemented in late 2008, after the failure of Lehman Brothers, both in the US

and in the euro area. As far as standard monetary policy is concerned, the Federal Reserve

cut its interest rates to near zero almost at the same time: the Federal funds rate reached 1%

at the end of October and the 0.00-0.25% range in December. The European Central Bank,

on the contrary, did not immediately cut its main policy interest rate to zero. The rate on the

main re…nancing operations (MRO) was reduced sharply at the end of 2008 but it bottomed

at 1% in May 2009 and did not fall below that threshold until mid 2012.2

Some guidance on the sequencing of standard and non-standard measures can be obtained

from the ELB literature which predates the …nancial crisis (see e.g. Reifschneider and Williams,

2000, Eggertsson and Woodford, 2003, Adam and Billi, 2006, and Nakov, 2008). The tenet of

that literature is that standard interest rate policy is the best monetary policy tool in response

to shocks leading to a fall in the natural rate of interest. Any other type of policy response

should only be considered as a substitute for standard interest rate policy, once the latter is

no longer available because the ELB constraint is binding. Indeed any non-standard measures

involving the central bank balance sheet are ine¤ective in the standard new-Keynesian model.

A number of recent papers have reconsidered this issue and demonstrated that certain

non-standard measures can be an e¤ective tool in the presence of distortions which prevent

the e¢ cient allocation of …nancial resources – see e.g. Gertler and Karadi (2010), Gertler

and Kiyotaki (2010), Del Negro et al. (2010), Eggertsson and Krugman (2010), Cúrdia and

Woodford (2011) and Correia et al. (2016). Such measures have been described as "credit

policy", i.e. measures aimed at o¤setting impairments to the process of credit creation. As

1 The recent experience of many developed countries has shown tht the lower bound for nominal interest rates

is not zero, as previously assumed, but negative due to cash storage costs. This is why we refer to the "e¤ective"

lower bound (rather than the more traditional "zero" lower bound). In our theoretical model, however, cash

storage costs are ignored, thus the e¤ective lower bound is equal to zero.

2 The interest rates were further reduced after the intensi…cation of the sovereign debt crisis and during the

following economic crisis. The rate on the depost facility reached zero in July 2012, before entering negative

territory. The MRO rate was cut to zero in March 2016.

ECB Working Paper Series No 2183 / October 20184such, standard and non-standard measures can be complementary to each other. Their optimal

mix and sequencing are no longer straightforward to determine.

Our paper analyses the optimal combination of standard and non-standard policies and

seeks answers to the following questions.

If non-standard measures can be targeted to the

prevailing source of …nancial ine¢ ciency, do they reduce the likelihood that interest rates

reach the ELB? Should interest rate policy be used at all, once unconventional measures have

been deployed? Looking forward to the return to normal conditions, the so-called “exit”, how

long should non-standard policies be optimally kept in place?

We attempt to answer these questions within a simple dynamic, general equilibrium model

that features both sticky prices and …nancial frictions in the form of asymmetric information

between borrowers and lenders. As in Bernanke, Gertler and Gilchrist (1999), we also assume

that lenders must pay a monitoring cost to audit borrowers that do not repay their loans.

The monitoring cost captures all bankruptcy costs, including legal expenses and any losses

associated with asset liquidation. We however deviate from Bernanke, Gertler and Gilchrist

(1999) in assuming that lending can potentially be provided by both commercial banks and

the central bank. Private lending is normally more e¢ cient, because commercial banks have a

superior loan monitoring technology. As a result, only commercial banks will provide credit to

the economy under normal conditions. In a crisis, however, commercial banks monitoring costs

rise due to the increase in losses associated with asset liquidation. If the crisis is su¢ ciently

severe, the central bank becomes a competitive lender and can replace commercial banks in

providing loans to …rms.3

Our main result is that under certain circumstances –notably in reaction to a speci…c type

of …nancial shocks which reduce banks’monitoring e¢ ciency –credit policy may be a strictly

more e¢ cient tool than policy interest rates.

Credit policy is desirable because it contributes to insulate the real economy from the

increase in credit spreads caused by deterioration in banks’e¢ ciency. The more e¤ective the

insulation, the lower the need to lower policy rates. In an illustrative example, we show that

it can be optimal for the central bank not to cut rates to zero, and to implement non-standard

measures instead. By contrast, reductions in policy rates without deploying non-standard

3 Our modelling of unconventional policy mirrors the early phase of purchase of private sector securities in the

US (the so-called QE1, which was implemented from December 2008 to March 2010) and the recent Corporate

Bonds Purchase Programme in the euro area (started in June 2016).

ECB Working Paper Series No 2183 / October 20185measures would be ine¢ cient. While they can (away from the ELB) insulate borrowers from

the increase in …nancing costs induced by the …nancial shock, lower policy rates also induce

savers to change their intertemporal consumption patterns. Given the …nancial nature of the

shock, such changes are ine¢ cient.

In general, the exact timing of implementation of standard and non-standard measures

depends on the size of the monitoring advantage of commercial banks over the central bank –

an object which is di¢ cult to calibrate. Non-standard measures are more likely to be deployed

in response to large …nancial shocks, while there is no scope for using them in reaction to

demand, or technology shocks.

To develop an intuition for what optimal policy ought to do in reaction to …nancial shocks

that increase credit spreads, we derive in closed form the target rule which would implement the

Ramsey allocation under the timeless perspective in our simple model. We focus for simplicity

on the case when the ELB is ignored. Compared to the model with frictionless …nancial

markets, the target rule implies a stronger mean reversion of the price level. In response to a

shock which increases the price level on impact, the price level falls over time and eventually

returns to a value lower than its initial level –and viceversa.

In our simple model, …nancial shocks a¤ect …rms’ marginal costs and have a cost-push

component. As a result, while typically lowering interest rates on impact to cushion the

adverse e¤ects on the real economy, optimal policy also requires stabilizing in‡ation through

a commitment to increase rates relatively quickly thereafter – notably increasing them long

before non-standard measures are reabsorbed.

We test the robustness of this conclusion in a richer model with capital, where an increase

in credit spreads directly a¤ects aggregate demand, notably by depressing investment.

In

this case, interest rates are optimally increased much more slowly than in the simple model.

However it remains true that non-standard measures tend to remain in place long after the

policy interest rate has returned to its long run level.

Finally, we revisit the prescription of the simple new Keynesian literature that the likelihood

of being at the ELB and the severity of the ensuing recession can be reduced by an appropriate

policy commitment. More speci…cally the central bank should promise to keep interest rates

low in the future for a longer period than optimal in the absence of the ELB. Such a promise,

if credible, generates high in‡ation expectations, reduces the current real interest rate and

stimulates the economy. When non-standard measures are ruled out, or unwarranted because

ECB Working Paper Series No 2183 / October 20186adverse shocks are not of a …nancial nature, these prescriptions remain valid in our model.

When non-standard measures are e¤ectively deployed, however, keeping the policy rate low for

an extended period of time may no longer be necessary.

This paper contributes to a recent literature that explains the lack of de‡ationary pressure

during the Great Recession. In our model, because …rms need to borrow in advance of produc-

tion and debt is nominal, increases in the policy rate exert upward pressure on marginal costs

and in‡ation. This e¤ect has been named in the literature the "cost" channel (Ravenna and

Walsh, 2010). Increases in credit spreads also rise the costs of …nance and exert similar in‡a-

tionary pressure through the so called "credit" channel (De Fiore and Tristani, 2012). Recent

empirical evidence is consistent with these channels being active during the …nancial crisis. For

instance, Gilchrist et al. (2015) analyses good-level transaction prices and …rms’income and

balance sheet data during the U.S. Great Recession. It …nds that …rms with limited internal

liquidity and high leverage signi…cantly increased their prices in 2008, a period characterized

by disruptions in credit markets and a sharp contraction in output. Abbate et al.

(2016)

uses a VAR analysis where …nancial shocks are identi…ed through sign restrictions to assess

the extent to which these shocks account for the "missing disin‡ation" during the U.S. Great

Recession. It …nds that adverse …nancial shocks helped preventing de‡ation during the crisis

and that the response of in‡ation can mainly be attributed to the cost channel.

Our work relates to Curdia and Woodford (2015), which analyses the optimal conduct of

monetary policy in a New Keynesian (NK) model where credit spreads arise from a di¤erent

type of …nancial friction. Curdia and Woodford (2015) derives an analytical target rule for the

special case when credit spreads do not depend on the volume of loans. In this case, a main

…nding is that the target rule is identical to the one that arises in the standard NK model

with frictionless …nancial markets. We also derive the optimal target rule but …nd that the

presence of the credit channels does modify it from the benchmark NK case, creating di¤erent

implications for the price level. The main reason for this di¤erence is that in our model credit

spreads depend on the corporate sector’s leverage and on the resources needed to conduct

banks’ monitoring activity.

In the numerical analysis of a more general case where credit

spreads are allowed to depend on the volume of loans, Curdia and Woodford (2015) also …nds

that the standard NK target rule does not perfectly approximate the optimal policy, in line

with our results.

ECB Working Paper Series No 2183 / October 20187The paper also relates to Harrison (2017), which studies the optimal combination of interest

rate policy and quantitative easing in a New Keynesian model with portfolio adjustment costs.

Under those conditions non-standard measures are implemented only when the interest rate

hits the zero-lower bound.

The paper is structured as follows. In section 2, we describe the model and characterize

the equilibrium. We also derive a system of log-linear equilibrium conditions, which we later

use to develop an intuition for our numerical results.

In section 3, we present the welfare

analysis. We derive a second-order approximation to the welfare function and the …rst order

conditions of the Ramsey allocation. This allows us to derive in closed form the target rule

which, absent the ELB constraint, would implement the Ramsey allocation. In section 4, we

outline the procedure we use to solve the model under the ELB constraint and we present

our numerical results. In this section, we also show that our main results extend to a model

where credit spreads directly a¤ect aggregate demand by depressing investment. Section 5

o¤ers some concluding remarks.

2 The model

The economy is inhabited by a representative in…nitely-lived household, wholesale …rms owned

by risk-neutral entrepreneurs, monopolistically competitive retail …rms owned by the house-

holds, zero-pro…t …nancial intermediaries, a government and a central bank. We describe in

turn the problem faced by each class of agents.

2.1 Households

At the beginning of period t; households receive interest payments from the nominal …nancial

assets acquired at time t

(cid:0)

1 . The households, holding an amount Wt of nominal wealth,

choose to allocate it among existing nominal assets, namely money Mt ; a portfolio of nominal

state-contingent bonds At+1; and one-period deposits denominated in units of currency, Dt.

In the second part of the period, the goods market opens. Households’ money balances

are increased by the nominal amount of their revenues and decreased by the value of their

expenses. Taxes are also paid or transfers received. The amount of nominal balances brought

into period t + 1; M

t; is given by

f

M

t

(cid:17)

f

Mt + Ptwtht + Zt

Ptct + Tt;

(cid:0)

(1)

ECB Working Paper Series No 2183 / October 20188where ht is hours worked, wt is the real wage, Zt are nominal pro…ts transferred from retail

producers to households, and Tt are lump-sum nominal transfers from the government. ct de-

note a CES aggregator of a continuum (cid:17)
"

2

by retail …rms, ct =

1
0 ct ((cid:17))

"

1
(cid:0)
" d(cid:17)

(0; 1) of di¤erentiated consumption goods produced

"

(cid:0)

1 ; with " > 1: Pt is the price of the CES aggregator.

Nominal wealth at the beginning of period t + 1 is given by

i

hR

Wt+1 = At+1 + Rd

t Dt + Rm
t

Mt;

(2)

where each of the state-contingent bonds in the portfolio At+1 pays one unit of currency in

f

a particular state in period t + 1; and Rd
t

is the gross interest paid on deposits at the end

of period t. As in Woodford (2003), we allow end-of-period private money holdings to be

remunerated at the rate Rm

t : In particular, we assume that monetary policy sets Rm

t at a level

that is proportional (and possibly equal) to the risk-free rate paid on central banks’reserves,

Rd
t :

The household’s problem is to maximize preferences, de…ned as

Eo

1

(

0
X

(cid:12)t [u (ct) + (cid:20) (mt)

v (ht)]

)

;

(cid:0)

where uc > 0; ucc < 0; (cid:20)m

(cid:21)

0; (cid:20)mm < 0; vh > 0; vhh > 0; and mt

Mt=Pt denotes real

(cid:17)

balances, subject to the budget constraints

together with (1) and (2).

Mt + Dt + Et [Qt;t+1At+1]

Wt;

(cid:20)

(3)

De…ne (cid:3)m;t

(cid:17)

Rd

Rm
t

t (cid:0)
Rd
t

: Under our assumption of proportionality between the remuneration

of cash and the risk-free rate, (cid:3)m;t = (cid:3)m for all t: The households’optimality conditions are

then given by

1

Rt = Rd

t = Et [Qt;t+1](cid:0)
vh (ht)
uc (ct)

= wt;

(cid:0)

uc (ct) = (cid:12)RtEt

(cid:25)t+1 (cid:27)
: The money demand is given residually by

(cid:26)

uc (ct+1)

where (cid:25)t

(cid:17)

Pt

Pt

1

(cid:0)

;

(4)

(5)

(cid:0)
The optimal allocation of expenditure between the di¤erent types of goods is given by ct ((cid:17)) =

(cid:20)m (mt) =

uc (ct) :

(cid:3)m

1

(cid:3)m

"

(cid:0)

Pt((cid:17))
Pt

(cid:16)

(cid:17)

ct; where Pt ((cid:17)) is the price of good (cid:17).

ECB Working Paper Series No 2183 / October 201892.2 Wholesale …rms

Wholesale …rms, indexed by i; are competitive and owned by in…nitely lived entrepreneurs.

Each …rm i produces the amount yi;t of an homogeneous good, using a linear technology

yi;t = !i;tli;t:

(6)

Here !i;t is an iid productivity shock with distribution function (cid:8) and density function (cid:30);

which is observed at no cost only by …rms.

At the beginning of the period, each …rm receives an exogenous endowment (cid:28) , which can

be used as internal funds. Since these funds are not su¢ cient to …nance the …rm’s desired level

of production, …rms need to raise external …nance. Before observing !i;t; …rms sign a contract

with a …nancial intermediary to raise a nominal amount Pt (xi;t

(cid:28) ) ; where

(cid:0)

xi;t

(cid:21)

wtli;t.

(7)

Each …rm i’s demand for labor is derived by maximizing …rm’s expected pro…ts, subject to the

…nancing constraint (7).

Let P t be the price of the wholesale homogenous good, P t
Pt

1

= (cid:31)(cid:0)
t

the relative price of

wholesale goods to the aggregate price of retail goods, and (qt

1) the Lagrange multiplier on

(cid:0)

the …nancing constraint. Optimality requires that

implying that

qt =

1
wt(cid:31)t
xi;t = wtli;t

(yt) = (cid:31)tqtxt;

E

(8)

(9)

(10)

where

[

] is the expectation operator at the time of the factor hiring decision.
(cid:1)

E

Equation (10) states that wholesale …rms must sell at a mark-up (cid:31)tqt over …rms’production

costs to cover for the presence of credit frictions and for the monopolistic distortion in the retail

sector. Notice that all …rms are ex-ante identical. Hence, we drop below the subscripts i.

The assumption that …rms receive an endowment from the government at the beginning

of the period is made for simplicity, in order to facilitate the analytical characterization of

the optimal monetary policy and the computation of the numerical non-linear solution of

the model. The absence of accumulation of …rms’ net worth implies that the persistence of

ECB Working Paper Series No 2183 / October 201810the endogenous variables merely re‡ects the persistence of the exogenous shocks. Nonetheless,

…nancial frictions provide an important transmission channel in our economy, through the credit

constraint faced by …rms and the endogenous spread charged by …nancial intermediaries. As

documented in De Fiore and Tristani (2012), up to a linear approximation, the model with and

without capital accumulation delivers qualitatively similar responses to both real and …nancial

shocks. The characterization of optimal monetary policy is also broadly similar in these two

cases.

2.3 The …nancial contract

In writing the …nancial contract we need to be explicit about what constitutes unconventional

policy in our model. We will focus on an interpretation of non-standard measures in which the

central bank replaces the private banking sector and does direct intermediation to …rms.

Direct lending is closest to the Federal Reserve facilities set up for direct acquisition of high

quality private securities (see also Gertler and Kiyotaki, 2010) and to the Corporate Bonds

Asset Purchase Programme implemented by the Eurosystem since June 2016. As in both the

Fed and the ECB cases, in our model the central bank lending program is …nanced though an

increase in interest bearing banks’reserves. As a result, non-standard measures lead to a large

increase in the central bank’s balance sheet.

Direct lending in our model is entirely demand determined: central bank intermediation

is chosen endogenously when it can be performed at a lower cost (spread) than private bank

intermediation.

Finally, we design credit policy in such a way that the central bank takes on no credit risk.

Together with the assumption that reserves are remunerated, this implies that the expansion

of the central bank’s balance sheet has no in‡ationary consequences, nor any implications for

government …nances.

The …nancial contract is structured as follows. External …nance takes the form of either

bank loans or direct lending from the central bank. Firms face the idiosyncratic productivity

shock !i;t, whose realization is observed at no costs only by the entrepreneur. If the realization

of the idiosyncratic shock !i;t is su¢ ciently low, the value of …rm production is not su¢ cient

to repay the loans and the …rm defaults.

The …nancial intermediaries (banks or the central bank) can monitor ex-post the realization

of !i;t, but a fraction of …rm’s output is consumed in the monitoring activity. These monitoring

ECB Working Paper Series No 2183 / October 201811costs are associated with legal fees and asset liquidation in case of bankruptcy. We assume that

commercial banks are on average more e¢ cient monitors than the central bank, i.e. (cid:22)c > (cid:22)b;

where (cid:22)c and (cid:22)b denote the steady state fraction of the …rm output lost in monitoring by the

central bank and by commercial banks, respectively.

Commercial banks collect deposits Dt from households. Deposits are the only funds avail-

able to …nance loans in the economy. Each representative commercial bank uses a fraction (cid:13)t

of deposits to …nance loans to …rms, and deposits the remaining fraction, 1

at the central bank. These reserves are remunerated at the market rate Rd

(cid:13)t; as reserves

(cid:0)

t and used in turn

by the central bank to …nance …rms. The fraction (cid:13)t of deposits lent by each commercial bank

is then combined with a fraction (cid:13)t of the …rms’ internal funds to …nance the production of

(cid:13)tqt(cid:31)txi;t units of wholesale goods.

interest rate Rb

The contract stipulates a loan amount xi;t
i;t on bank loans and Rc
Rb

…rms. The total repayment is Pt

(cid:28) : The …rm needs to pay back a unit gross

(cid:0)

i;t on loans from the central bank, which can di¤er across
i;t(cid:13)t + Rc

(cid:28) ). The …rm is able to meet

i;t (1

(xi;t

(cid:13)t)

!i;t, where !i;t is the minimum productivity level such that the

(cid:16)

(cid:0)

(cid:17)

(cid:0)

those payments when !i;t

(cid:21)

…rm can pay the agreed return and is implicitely de…ned by

P t!i;tli;t = Pt

(cid:16)

Rb

i;t(cid:13)t + Rc

i;t (1

(cid:0)

(cid:13)t)

(xi;t

(cid:17)

(cid:28) )

(cid:0)

(11)

When !i;t < !i;t, the …rm goes bankrupt, and hands out all the production P tAt!i;tli;t, in units
of currency. In this case, a constant fraction (cid:22)j

t of the …rm’s output is destroyed in monitoring.

The bank obtains (cid:13)t

1

(cid:22)b
t

P tAt!i;tli;t and the central bank (1

(cid:0)

(cid:13)t) (1

(cid:0)

(cid:0)

(cid:22)c

t ) P tAt!i;tli;t.

Using the equilibrium relationshis P t
(cid:0)
Pt

(cid:1)

= (cid:31)(cid:0)
t

1

; qt = 1
wt(cid:31)t

and xi;t = wtli;t we can then rewrite

!i;t as

De…ne

and

!i;t =

Rb

i;t(cid:13)t + Rc
i;t (1
qt

(cid:16)

(cid:0)

(cid:13)t)

(cid:17)

1
(cid:18)

(cid:0)

(cid:28)
xi;t (cid:19)

f (!)

1

(cid:17)

!
Z

!(cid:8) (d!)

! [1

(cid:0)
(cid:0)
!(cid:8) (d!) + Rb 1
q

(cid:8) (!)] ;

(cid:28)
x

1

(cid:0)

(cid:16)

(cid:17)

(cid:8) (!)]

[1

(cid:0)

(cid:22)b

1

(cid:0)

(cid:16)

(cid:17)

(cid:22)c) !(cid:8) (d!) + Rc 1
q

(1

(cid:0)

(cid:28)
x

1

(cid:0)

(cid:16)

(cid:17)

(cid:8) (!)]

[1

(cid:0)

!

!

gb

!; (cid:22)b

(cid:16)

(cid:17)

0
Z

(cid:17)

gc (!; (cid:22)c)

(cid:17)

0
Z

(12)

(13)

(14)

(15)

ECB Working Paper Series No 2183 / October 201812as the expected unit shares of output accruing respectively to the …rm, the commercial bank

and the central bank, after stipulating a …nancial contract that sets a lending rate Rb or Rc.

Notice that

f (!i;t) + (cid:13)tgb

!i;t; (cid:22)b
t

+ (1

where G (!i;t) =

(cid:16)
!i;t
0

monitoring.

R

(cid:13)t) gc (!i;t; (cid:22)c

t ) = 1

(cid:0)

(cid:0)

(cid:17)

!(cid:8) (d!). On average,

t + (1

(cid:13)t(cid:22)b
h

(cid:0)

(cid:13)t) (cid:22)c
t

G (!i;t)

(16)

i

G (!i;t) of output is lost in

(cid:13)t(cid:22)b

t + (1

(cid:13)t) (cid:22)c
t

(cid:0)

(cid:2)

(cid:3)

The budget constraint for the commercial bank is

(cid:13)t) Rd

t Pt (xi;t

(1

(cid:0)

(cid:0)

(cid:28) ) + (cid:13)tP tqt(cid:31)tgb

!i;t; (cid:22)b
t

xi;t

(cid:16)

(cid:17)

Rd

t Pt (xi;t

(cid:28) ) :

(cid:0)

(cid:21)

The …rst term on the left-hand side is the amount of reserves held at the central bank, gross

of their remuneration, in units of currency. The second term is the gross nominal return to

banks from extending credit of (cid:13)tPt (xi;t

(cid:0)

the cost of funds for the bank.

(cid:28) ) units of money to …rms. The righ-hand side is

The central bank uses all its funds (reserves) to satisfy the demand for credit by …rms. Its

budget constraint is

P tqt(cid:31)tgc (!i;t; (cid:22)c

t ) xi;t

Rd

t Pt (xi;t

(cid:28) ) :

(cid:0)

(cid:21)

The constraint says that the return to the central bank from lending Pt (xi;t

(cid:28) ) units of

(cid:0)

money to …rms must be su¢ cient to cover for the costs of funds (the remuneration of reserves).

Each …rm stipulates a contract that sets a …xed repayment on each unit of debt of P t(cid:31)tqt!i;t:

The contract also sets the fraction of deposits the commercial bank needs to devote to loans and

the fraction to be held as reserves at the central bank. The informational structure corresponds

to a standard costly state veri…cation (CSV) problem (see e.g. Gale and Hellwig (1985)). The

problem is

subject to

max
!i;t;xi;t;(cid:13)t

f (!i;t)qtxi;t

qtgb

!i;t; (cid:22)b
t

xi;t

(cid:16)

(cid:17)
qtgc (!i;t; (cid:22)c
t ) xi;t

f (!i;t) + (cid:13)tgb

!i;t; (cid:22)b
t

+ (1

(cid:16)

(cid:17)

(cid:13)t) gc (!i;t; (cid:22)c
t )

(cid:0)

qtxtf (!i;t)

0

(cid:21)

(cid:21)

(cid:20)

(cid:21)

(cid:20)

Rd

t (xi;t

(cid:28) )

(cid:0)

Rd

t (xi;t

(cid:28) )

(cid:0)
(cid:13)t(cid:22)b
t + (1

1

(cid:28)

(cid:0)

h

1:

(cid:13)t (cid:20)

(cid:13)t) (cid:22)c
t

(cid:0)

i

(17)

(18)

(19)

(20)

(21)

ECB Working Paper Series No 2183 / October 201813The optimal contract is the set

xi;t; !i;t; (cid:13)tg

f

that maximizes the entrepreneur’s expected

pro…ts, subject to the pro…ts of the private bank and those of the central bank being su¢ cient

to cover their respective repayment on deposits, (17) and (18), the feasibility condition, (19),

the entrepreneur being willing to sign the contract, (20), and the share (cid:13)i;t being between
zero and one. De…ne (cid:21)1i;t and (cid:21)2i;t as the lagrangean multipliers associated to (cid:13)i;t (cid:21)
(cid:13)i;t (cid:20)

1, respectively.

0 and to

The optimality conditions include (16) and

Rd
t
(cid:13)t)(cid:22)c
(cid:8)(!i;t)]

t]!i;t(cid:30)(!i;t)

(cid:28)
xi;t

f (!i;t)qt =

1

(cid:0)

xi;t = 1 +

1

[(cid:13)t(cid:22)b

t +(1
(cid:0)
[1
(cid:0)
f (!i;t)

(cid:0)
f (!i;t)

(cid:13)t(cid:22)b
((cid:13)t(cid:22)b

t + (1
t +(1
(cid:0)
1
(cid:0)

(cid:13)t) (cid:22)c
t
t)!i;t(cid:30)(!i;t)
(cid:1)

(cid:0)
(cid:13)t)(cid:22)c
(cid:8)(!i;t)

(cid:0)
1

(cid:0)
(cid:0)

G (!i;t)

(cid:21)

(cid:20)
(xi;t

Rc

i;t (cid:0)
1

(cid:28) ) 1

Rb
i;t
(cid:30)(!i;t)!i;t[(cid:13)t(cid:22)b
[1

(cid:28) [1
t +(1
(cid:8)(!i;t)]

(cid:0)

(cid:17)

(cid:0)

(cid:0)

(cid:0)

(cid:8) (!i;t)]

(cid:0)
t]
(cid:13)t)(cid:22)c

(cid:17)1;t (cid:0)

(cid:17)2;t =

(cid:16)

(cid:17)1;t(cid:13)t = 0

(cid:17)2;t (1

(cid:0)

(cid:13)t) = 0

(22)

(cid:28)

(23)

(24)

(25)

(26)

together with (cid:21)1i;t

0 and (cid:21)2i;t

(cid:21)

(cid:21)

can be found in appendix A.

0: A formal derivation of the solution to the optimal contract

Notice that equation (23) expresses the size of production of each …rm, xi;t; as a function

of the threshold !i;t and of aggregate variables only. Plugging this expression into equation

(22), it becomes clear that threshold !i;t is itself a function of aggregate variables only. Hence,

!i;t = !t; and xi;t = xt; for all i:

The solution to the optimal contract is such that (cid:13)t takes the value of either zero or one.
i;t > Rb

0: Now, if both (cid:17)1;t and (cid:17)2;t are strictly positive, at least

If Rc

i;t; then (cid:17)1;t > (cid:17)2;t (cid:21)

one of conditions (25)-(26) cannot be veri…ed: It follows that the only solution to the optimal

contract is (cid:13)t = 0 and (cid:17)2;t = 0: If Rc

t < Rb

t ; then 0

(cid:17)1;t < (cid:17)2;t; implying that (cid:13)t = 1 and

(cid:20)

(cid:17)1;t = 0: Only when the interest rates on commercial bank loans and on central bank loans are
the same, Rc

i;t; any value of (cid:13)t satis…es the optimality conditions of the contract. It is

i;t = Rb

irrelevant which share of intermediation is conducted by commercial banks or by the central

bank:

Given the solution to the CSV problem, the gross interest rate on loans extended to …rms

by the commercial bank, Rb

i;t; and the one extended to …rms by the central bank, Rc

i;t, can be

ECB Working Paper Series No 2183 / October 201814backed up from the condition for debt repayment, (11). They are implicitely given by

when (cid:13)t = 1; and by

P t!t(cid:31)tqtxt = Rb

i;tPt (xt

(cid:28) ) ;

(cid:0)

P t!t(cid:31)tqtxt = Rc

i;tPt (xt

(cid:28) ) ;

(cid:0)

(27)

(28)

when (cid:13)t = 0: From conditions (27) and (28), it follows that both commercial banks and the
central bank charge a single loan rate to all …rms, Rb

i;t = Rb

t and Rc

i;t = Rc

t ; for all i:

From (17) and (18), which hold as equality in equilibrium, we know that gc (!i;t; (cid:22)c
!i;t; (cid:22)b
t

: It follows from (14) and (15) that

t ) =

gb

(cid:0)

(cid:1)

Therefore, Rc

[1

1
1
qt (cid:18)
t > Rb

(cid:28)
xt (cid:19)
t implies that (cid:22)c

(cid:0)

(cid:0)

(cid:8) (!t)]

(cid:16)
t > (cid:22)b
t:

Rc

t (cid:0)

Rb
t

!t

=

(cid:17)

0
Z

(cid:16)

(cid:22)c
t (cid:0)

(cid:22)b
t

(cid:17)

!t(cid:8) (d!) :

The optimal …nancial contract thus has to satisfy the conditions (22) and (23), together

with

1 if (cid:22)c

0 if (cid:22)c

(cid:22)b
t (cid:21)
t
t < (cid:22)b
t

(cid:13)t =

8
<

:

(29)

De…ne now the spread between loan rates and the risk-free rate as (cid:3)j

:

t = Rj

t
Rt

; for j = b; c:

We can now use expressions (17)-(18) and (27)-(28) to relate those spreads to the thresholds
for the idiosyncratic productivity shocks, !j
t ,

(cid:3)j

t =

!j
t
t ; (cid:22)j
g(!j
t )

:

(30)

2.4 Entrepreneurs

Entrepreneurs die with probability (cid:13)t. They have linear preferences over the same CES basket
of di¤erentiated consumption goods as households, with rate of time preference (cid:12)e. This latter

is su¢ ciently high so that the return on internal funds is always larger than the rate of time

preference, 1
(cid:12)e

(cid:0)

1; and entrepreneurs postpone consumption until the time of death.

As in De Fiore, Teles and Tristani (2011), we assume that the government imposes a tax (cid:23)

on entrepreneurial consumption. It follows that

1

(1 + (cid:23))

Pt ((cid:17)) et ((cid:17)) d(cid:17) = P t (!t

0
Z

!t) (cid:31)tqtxt;

(cid:0)

ECB Working Paper Series No 2183 / October 201815where et ((cid:17)) is the …rm’s consumption of good (cid:17): Notice that

1
0 Pt ((cid:17)) et ((cid:17)) = Ptet;where et is

the demand of the …nal consumption good. We can then write
R

(1 + (cid:23)) et = f (!t)qtxt:

We consider the case where (cid:23) becomes arbitrarily large.4 Consumption of the bankers ap-

proaches zero, et

!

0, and the consumption tax revenue,

T e
t =

(cid:23)
1 + (cid:23)

(cid:18)

(cid:19)

f (!t)qtxt;

(31)

approaches the total funds of the bankers that die.

2.5 Government

Revenues from taxes on entrepreneurial consumption are used by the government to …nance

the transfer (cid:28) : Funds below (in excess of) (cid:28) are supplemented through (rebated to) households

lump-sum taxes (transfers), T h

t . The budget constraint of the government is

T e
t = (cid:28)

T h
t :

(cid:0)

(32)

2.6 Retail …rms

As in Bernanke, Gertler and Gilchrist (1999), monopolistic competition occurs at the retail

level. A continuum of monopolistically competitive retailers buy wholesale output from en-

trepreneurs in a competitive market and then di¤erentiate it at no cost. Because of product

di¤erentiation, each retailer has some market power. Pro…ts, Zt, are distributed to the house-

holds, who own …rms in the retail sector.

Output sold by retailer (cid:17), Yt ((cid:17)) ; is used for households’and entrepreneurs’consumption.

Hence, Yt ((cid:17)) = ct ((cid:17)) + et ((cid:17)) : The …nal good Yt is a CES composite of individual retail goods

Yt =

1
0 Yt ((cid:17))

"

1
(cid:0)
" d(cid:17)

"

"

(cid:0)

1 ; with " > 1:

We assume that each retailer can change its price with probability 1

i

hR

(cid:18); following Calvo

(cid:0)

(1983). Let Pt(cid:3) ((cid:17)) denote the price for good (cid:17) set by retailers that can change the price at

4 The reason for this assumption is that, with et > 0; it would be optimal for policy to generate a redistribution

of resources between households and entrepreneurs. This would enable to exploit the risk-neutrality of the latter

to smooth out consumption of the former. Since risk neutrality of entrepreneurs is a simplifying assumption

needed to derive debt as an optimal contract, we eliminate this type of incentives for monetary policy by

completely taxing away entrepreneurial consumption. Allowing entrepreneurs to consume would also require

arbitrary choices on the weight of entrepreneurs to be given in the social welfare function.

ECB Working Paper Series No 2183 / October 201816time t; and Y (cid:3)t ((cid:17)) the demand faced given this price. Then each retailer chooses its price to

maximize expected discounted pro…ts. The optimality conditions are given by

1 = (cid:18)(cid:25)"

1
t + (1
(cid:0)

(cid:0)

(cid:2)1;t =

1
(cid:31)t

Yt + (cid:18)Et

1

"
(cid:0)

(cid:18))

"

"

(cid:2)1;t
(cid:2)2;t (cid:19)
(cid:0)
t+1Qt;t+1(cid:2)1;t+1

(cid:18)

1

(cid:25)"

(cid:2)2;t = Yt + (cid:18)Et

1

(cid:25)"
(cid:2)
t+1 Qt;t+1(cid:2)2;t+1
(cid:0)

(cid:3)

;

where Qt;t+k = (cid:12)k

uc(ct+k)
uc(ct)

:

(cid:2)

(cid:3)

Recall that the aggregate retail price level is given by Pt =

h

i

the relative price of di¤erentiated good (cid:17) as pt ((cid:17))

express everything in terms of relative prices, 1 =

price dispersion term as

Pt((cid:17))
Pt
(cid:17)
1
0 (pt ((cid:17)))1
R
" d(cid:17):

1

st

(cid:17)

0
Z

(pt ((cid:17)))(cid:0)

(33)

(34)

(35)

1

0 Pt ((cid:17))1

" d(cid:17)

(cid:0)

1

(cid:0)

1

" : De…ne

hR
and divide both sides by Pt to
" d(cid:17): Now de…ne the relative

i

(cid:0)

This equation can be written in recursive terms as

st = (1

(cid:18))

(cid:0)

"

1

(cid:0)

(cid:0)

"

1

(cid:0)
1

1

(cid:18)(cid:25)"
(cid:0)
t
(cid:18)

(cid:0)

(cid:18)

(cid:19)

+ (cid:18)(cid:25)"

t st:

(36)

2.7 Monetary policy

We characterize "standard" monetary policy as one where the central bank uses the nominal

interest rate to implement the desired allocation, subject to a non-negativity constraint on the

nominal interest rate

Rt

0:

(cid:21)

(37)

We de…ne as "non-standard" monetary policy the ability of the central bank to a¤ect

allocations by intermediating credit directly. Commercial banks deposit part of their funds

(households’ deposits) at the central bank as reserves. These latter are remunerated at the

risk-free rate Rd

t and used by the central bank to provide direct credit to …rms.

The central bank also remunerates households’money holdings at a rate Rm
t

that is pro-

portional to the risk-free rate:

ECB Working Paper Series No 2183 / October 2018172.8 Market clearing

Market clearing conditions for money, bonds, labor, loans, wholesale goods and retail goods

are given, respectively, by

Mt = M s
t ;

At = 0;

ht = lt;

Dt = Pt (xt

(cid:28) ) ;

1

yt =

(cid:0)
Yt ((cid:17)) d(cid:17);

0
Z

Yt ((cid:17)) = ct ((cid:17)) + et ((cid:17)) ; for all (cid:17):

(38)

(39)

(40)

(41)

(42)

(43)

2.9 Log-linearization

We log-linearize the system of equilibrium conditions. This helps us to characterize the main

channels of transmission of shocks in our economy. It also enable us to show analytically how

the presence of …nancial frictions a¤ects the optimal monetary policy.

We log-linearize the equilibrium conditions around a steady state where (cid:13)t = pt ((cid:17)) = st =

1, assuming the functional form for utility u (ct)

(cid:0)
We de…ne the e¢ cient equilibrium as one where all …nancial frictions, as well as nominal

(cid:0)

v (ht) = c1

t
1

(cid:27)

(cid:0)

  h1+'
1+' :

t

(cid:27) (cid:0)

price stickiness, are absent. Denote variables in such equilibrium with the e superscript, and a

variable with a tilde hat as the log deviation of the variable from its steady state level. Because

…nancial shocks are absent in the e¢ cient equilibrium,

Y e
t =

re
t = 0; where

re
t is the e¢ cient

real interest rate:

b

b

b

The system of log-linearized equilibrium conditions can be simpli…ed to

((cid:11)3

(cid:11)1)

(cid:0)

(cid:3)j

t = (1 + (cid:27) + ') xt + ((cid:11)2 + (cid:11)4)

(cid:22)j
t

xt = Etxt+1
b

1

(cid:27)(cid:0)

(cid:0)
(cid:16)
((cid:27) + ') xt +

Rt

Et

(cid:25)t+1
b
(cid:17)
t + (cid:11)2

(cid:0)
(cid:3)j
Rt + (cid:11)1
b
b

(cid:25)t = (cid:21)

where

b

h

j =

b

;

b
(cid:22)c
t (cid:21)
(cid:22)c
t <
b

b

(cid:22)b
t
(cid:22)b
t
b

b

b if

c if

8
<

:

(cid:22)j
t

b

i

+ (cid:12)Et

(cid:25)t+1

b

(44)

(45)

(46)

(47)

ECB Working Paper Series No 2183 / October 201818and where xt =

Yt

(cid:0)

Y e
t denotes the output gap. The coe¢ cients (cid:11)1; (cid:11)2; (cid:11)3 and (cid:11)4 are de…ned

in appendix B, and (cid:21)
b

b

(1

(cid:0)

(cid:17)

(cid:18)) (1

(cid:0)

(cid:12)(cid:18)) =(cid:18). Notice that (cid:11)1 and (cid:11)3 can be signed and are

always positive. Under our calibration, the coe¢ cients (cid:11)2; (cid:11)4 and (cid:11)5 also take positive values.

t < !b

(cid:13)t = 0; or if !c
values of !j

t: From equation (30), it can be shown that @(cid:3)j

To understand condition (47), notice that j = b if (cid:13)t = 1; or if !c

t; while j = c if
@!j can be negative either for
t close to zero, or for values falling in the right tail of the distribution of !: Under
parameterizations that delivers reasonable default rates, !j always lie in the left tail of the
distribution, so that @(cid:3)j
(cid:22)c
t (cid:21)

@!j > 0: At the same time, we know from equation (44) that (cid:3)c
t if

(cid:22)b
t, and (cid:3)c

t < (cid:3)b

(cid:22)c
t <

t (cid:21)

t (cid:21)

(cid:22)b
t:

t if

(cid:3)b

!b

Equation (44) shows that the spread between the loan rate and the policy rate,

b

b
with the output gap, xt. A larger demand for retail goods (and thus for wholesale goods to be

b

b

(cid:3)j

t , increases

b

used as production inputs) tightens the credit constraint of …rms, since they need to …nance

a higher level of debt given the same amount of internal funds. The increased default risk

generates a larger spread. The spread is also positively related to the shock to monitoring

costs,

(cid:22)j
t : The reason is that intermediaries need to set a higher repayment threshold to cover

for increased monitoring costs, which results in larger credit spreads.

b

Equation (45) is a standard forward-looking IS-curve describing the determinants of the

gap between actual output and its e¢ cient level.

Equation (46) represents an extended Phillips curve. The …rst determinant of in‡ation

in this equation is the output gap. Ceteris paribus, a higher demand for retail goods, and

correspondingly for intermediate goods, implies that wholesale …rms need to pay a higher real

wage to induce workers to supply the required labor services. The second determinant is the

nominal interest rate, whose increase also pushes up marginal costs due to the presence of the

cost channel. The third term is the credit spread,

(cid:3)j

t . A higher spread implies a higher cost of

external …nance for wholesale …rms and therefore exerts independent pressure on in‡ation.

b

As in De Fiore and Tristani (2012), the credit spread and the nominal interest rate act as

endogenous "cost-push" terms in our model. While raising marginal costs and in‡ation, an

increase in either term also exerts downward pressure on economic activity. A higher nominal

interest rate determines an output contraction through the ensuing increase in the real interest

rate, which induces households to postpone their consumption to the future. An increase in

the credit spread contracts activity through the increase in the …nancial markup qt and the

consequent fall in the real wage.

ECB Working Paper Series No 2183 / October 201819The shock to monitoring costs acts as an exogenous "cost-push" factor in the New-Phillips

curve, as it creates in‡ationary pressures independently from those exerted by the output gap.

In our model, a positive shock to monitoring costs raises the cost of external …nance and

depresses economic activity. At the same time, it increases the spread that banks charge over

the risk-free rate, and thus …rms’marginal costs, which are passed through to higher prices for

…nal consumption goods. In equilibrium, in‡ation rises in spite of the fall in the output gap.

As a result, this shock does not lead the economy to hit the ELB under a simple Taylor-type

of monetary policy rule. The central bank would react to such a shock by raising the policy

instrument.

3 Welfare analysis

We characterize analytically the solution to the Ramsey problem under committment.

The welfare criterion in our analysis is the utility of the economy’s representative household

Wt0 = Et0

1

(

t=t0
X

(cid:12)tUt

;

)

where temporary utility is given by Ut = c1

(cid:27)

(cid:0)

t
1

(cid:27) (cid:0)

(cid:0)

h1+(cid:30)
1+(cid:30) .
t

We can provide an analytic approximate characterisation of optimal policy using the log-

linear model conditions. Speci…cally, under the functional form for household’s utility de…ned

above, appendix C shows that the present discounted value of social welfare can be approxi-

mated to second order by

Wt0 '

(cid:27)

c1

(cid:0)

{

"

(cid:0)

1
2

Et0

1

t=t0
X

(cid:12)t

t0Lt

(cid:0)

#

+ t:i:p:;

where t:i:p: denotes terms independent of policy,

Lt

(cid:17)

(cid:20)(cid:25)(cid:25)2

t + ((cid:27) + ') xt;

(48)

(49)

(cid:21) [

(cid:27) + (cid:11)1(cid:11)5 (1 +

(cid:27))] : The planner maximizes (49)

(cid:17)

(cid:20)(cid:25) =

(1

"(cid:18)
(cid:18))(1

(cid:0)
De…ne

(cid:0)

(cid:27)

(cid:12)(cid:18)) and { =
(cid:21)
(cid:27) + ',

1

1
1+(cid:30)

.

1

(cid:27) (cid:0)
(cid:0)
(cid:17)
(cid:11)
(cid:21)(cid:11)1(cid:11)5 and

(cid:17)

e

(cid:16)
(cid:17)

e

subject to the linearized equilibrium condition (45), the New-Phillips curve rewritten as

e

e

e

(cid:25)t = (cid:12)Et

(cid:25)t+1 +

(cid:11)xt + (cid:21)

Rt +

(cid:21) ((cid:11)2 + (cid:11)4) + (cid:21)(cid:11)2

(cid:13)t

(cid:22)b
t + (1

the ELB constraint

b

b

e

h

e

b

i h

b

(cid:0)

(cid:13)t)

(cid:22)c
t

;

i

b

R
t

(cid:21)

ln (cid:12);

b

ECB Working Paper Series No 2183 / October 201820and the restriction

(cid:22)c
t (cid:0)
Notice that the social planner does not choose (cid:13)t: Equation (50) is a restriction to the

b
Ramsey problem which ensures that the optimal allocation satis…es the optimality conditions

(cid:13)t = (cid:9)

(50)

(cid:22)b
t

(cid:17)

(cid:16)

b

:

of the CSV problem.

((cid:27) + ') xt

The …rst-order conditions of the Ramsey problem can be written as
1 t (cid:0)
(cid:21)(cid:0)
(cid:12)(cid:0)
1(cid:27)(cid:0)
1
1
(cid:0)
1 (cid:12) + (cid:21)
(cid:12)

(cid:25)t + (cid:30)t
e
(cid:0)

1 + (cid:12)(cid:0)
(cid:11)(cid:21)(cid:0)

1 + (cid:27)(cid:0)

(cid:0)
(cid:11)(cid:21)(cid:0)

 t =

1 t
1

(cid:30)t =

(cid:11)(cid:21)(cid:0)

e
 t

1(cid:30)t

(cid:11)(cid:30)t

1 +

(cid:0)

"

(cid:0)

(cid:0)

(cid:0)

1

1

0 =

(cid:16)

Rt
b

(cid:0)

ln (cid:12)

(cid:30)t

(cid:17)

e

e

((cid:27) + ') xt

1 (cid:0)
(cid:27)

(cid:0)
(cid:0)

where  t and (cid:30)t are the lagrangean multipliers on the Euler equation and the ELB constraint,

b

respectively (the New-Phillips curve multiplier, (cid:23)t, has been substituted out).

3.1 Target rule without ELB and non-standard measures

We provide some intuition on what monetary policy ought to do in our model by abstract-

ing from the ELB constraint and from the possibility that the central bank intervenes with

non-standard policy measures. The aim is to disentangle the consequences of the nominal

denomination of debt (the cost channel) and the costly state veri…cation environment (the

existence of endogenous credit spreads) for the optimal monetary policy.

Under the assumption that the ELB constraint can be ignored, and when (cid:13)t = 1, the

optimality conditions of the Ramsey problem can be rewritten in terms of the following target

rule

(cid:1)xt =

(cid:0)

"

1 +

(cid:20)

(cid:11)3

(cid:11)1

(cid:0)

1 +

(cid:11)1 (cid:18)

1
(cid:27) + '

(cid:19)(cid:21)

(cid:25)t +

(cid:27)
(cid:27) + '

"

(cid:25)t

(cid:18)

1

(cid:25)t
(cid:0)
(cid:12)

(cid:0)

+

(cid:21)
(cid:12)

xt

1

(cid:0)

(cid:19)

(51)

Equation (51) nests the target rule which implements optimal policy in the New Keynesian
b

b

b

model, given by (cid:1)xt =

(cid:0)

"(cid:25)

t (see eg Woodford, 2003). In that model, the target rule can be

interpreted as the simple prescription to keep contracting the output gap as long as in‡ation

is positive (and viceversa for negative in‡ation).

b

The introduction of the cost channel in the model is responsible for the last two terms in

equation (51). In fact, when monitoring costs are zero, (cid:11)1 = 0: To realize the implications of the

cost channel for optimal policy, consider the prescription of the target rule in the …rst period

after a shock has hit the economy. Because in steady state x = (cid:25)

= 0; in the …rst period

b

ECB Working Paper Series No 2183 / October 201821(cid:1)xt =

(cid:0)

"

1

(cid:27)
(cid:27)+'

(cid:0)

(cid:16)

(cid:17)

b

(cid:25)t: In response to a certain increase in in‡ation, the last two terms

suggest that the initial contraction in the output gap should be smaller than in the model

with frictionless …nancial markets.

Intuitively, these terms take into account the cost-push

in‡ationary e¤ects of the increase in the nominal interest rate, which have to be implemented

to induce a contraction of the output gap.

Finally, the existence of asymmetric information and credit spreads calls for a more aggres-

sive policy response to current in‡ation –the coe¢ cient is higher than in the frictionless case by

the positive amount (cid:11)1= ((cid:11)3

(cid:0)

(cid:11)1) (1 + 1= ((cid:27) + ')). This is necessary to contain any additional

in‡ationary pressures coming from credit spreads.

Equation (51) can also be written di¤erently to highlight its implications on the price level.

We then have

pt = pt

1

(cid:0)

(cid:0)

(cid:12)

' + (cid:27)
(cid:27)

1
"

(cid:20)

(cid:1)xt + (cid:21)

' + (cid:27)
(cid:27)

("(cid:27)

1) xt

1 + (cid:21)

(cid:0)

Rt

(cid:0)

(cid:0)

1 + (cid:21)(cid:11)1

(cid:3)t

1 + (cid:12)Et

(cid:25)t

1

(cid:0)

(cid:0)

(52)

(cid:21)

where

" is a positive reaction coe¢ cient given by

e

"

Note that the NK model would require pt = pt
e

e

b
1
"(cid:12)(cid:27)(cid:0)

b
' + (cid:11)1
(cid:11)1
(cid:0)

(cid:11)3

b
(1 + (cid:27) + ')

.

(1=") (cid:1)xt. Should a positive output

i

h

(cid:17)
1

(cid:0)

(cid:0)

gap, and thus de‡ation, ever occur, the central bank should generate an in‡ationary period

until output gap is stabilised and (cid:1)xt = 0. The price level should return to its initial point.

In the case of our model, a return to the original price level is not su¢ cient. Note that all

terms inside the square brackets on the right-hand side of equation (52) are positive. Should

a positive output gap ever occur, the central bank should not only generate in‡ation until the

output gap is stabilized and (cid:1)xt = 0. Some additional upward pressure on prices may be

warranted as long as lagged policy rates and credit spreads remain above their steady state

levels. As a result, the price level will eventually return to a higher point than where it started.

The price level will nevertheless remain, as in the NK model, trend stationary.

4 Numerical results

For our numerical analysis, we …rst derive analytically the …rst order conditions of the nonlinear

Ramsey problem of an optimal planner. We then solve the resulting model using nonlinear,

deterministic simulation methods. Given initial conditions for pre-determined variables and

ECB Working Paper Series No 2183 / October 201822terminal conditions for non-predetermined variables, the path of all endogenous variables can

be found as the solution of a large system of nonlinear equations at all simulation dates.5

A more complete solution to the system would include stochastic terms, e.g. using the

collocation method as suggested by Judd (1998) or Miranda and Fackler (2002). A stochastic

solution would in principle allow for precautionary policy motives, e.g. the possibility to target

a slightly positive in‡ation rate in order to reduce the likelihood of hitting the ELB. Such pre-

cautionary e¤ects, however, have been found to be negligible in the new Keynesian literature.

Since, as we illustrate below, non-standard measures reduce the likelihood of reaching the ELB,

precautionary e¤ects are likely to be small also in our model. There should therefore be no

loss of accuracy in our deterministic nonlinear solution, which is much simpler to compute and

feasible also for relatively large models.

To simplify the solution procedure, we smooth out the two kinks in (cid:13)t through a simple

approximation. Speci…cally, we replace equation (29) with

(cid:13)t = (cid:9)

(cid:22)c

(cid:22)b
t (cid:0)

(cid:16)

(cid:17)

(53)

where we also assumed that (cid:22)c is not a¤ected by …nancial shocks that increase (cid:22)b

t. The

idea is that ‡uctuations in (cid:22)b

t are mainly related to changes in asset liquidation costs, which

can increase markedly during …nancial crises, for example due to …re sales. By contrast,

central banks are not subject to liquidity constraints and can always perform an orderly asset

liquidation. Hence (cid:22)c remains constant over time.

The functional form for the function (cid:9) (x) is (cid:9) (x) = 1
2

2 where (cid:20) is a parameter
which can be tuned to improve the accuracy of the approximation at the points of discontinuity.

(cid:20)x) + 1

(cid:0)

(cid:0)

e((cid:20)x)
e(
e((cid:20)x)+e(

(cid:0)

(cid:20)x)

Parameter values are in line with the literature. More speci…cally, we set the elasticity of

intratemporal substitution " = 11 and the Calvo parameter (cid:18) = :66. The discount factor is set

as (cid:12) = 0:995, to mimic the low interest rates environment which prevailed over the years before

the …nancial crisis. For the utility parameters, we use standard values: (cid:27) = 1:0, (cid:30) = 0:0. The

contract parameters (cid:28) and (cid:27)! are set consistently with the parametrization used in De Fiore

and Tristani (2012), which matches US data on the average annual spread between lending

and deposit rates (approximately 2%) and on the quarterly bankruptcy rate (around 1%).
1 = 0:008. Consistently with actual …nancial

These values imply that (cid:11)1 = 4:7 and ((cid:11)3

(cid:11)1)(cid:0)

(cid:0)

5 In practice we use Newton methods as implemented in the Dynare command "simul".

ECB Working Paper Series No 2183 / October 201823developments over the past 5 years, we assume very persistent monitoring cost shocks: they

have a serial correlation coe¢ cient equal to 0:95.

A new coe¢ cient which we need to calibrate is (cid:22)c, the monitoring cost of central bank

lending activities. To gauge a value for this parameter, we draw from the euro area experience

during the …nancial crisis, when the ECB intervened to o¤set impairments in the interbank

market.

In the …rst phase of the crisis, in 2008-09, lack of trust emerged between banks

concerning each other’s ability to repay interbank loans. Possibly as a consequence of higher

interbank market rates, many banks chose to borrow directly from the ECB at the rate on the

main re…nancing operations, rather than from other banks at the prevailing interbank overnight

rate (the EONIA). The spread between MRO and overnight rates, which was essentially zero

on average in pre-crisis times, hovered between 50 and 70 basis points in 2009.

We interpret this hike in the MRO-overnight spread as due to an increase in the ine¢ ciency

of commercial banks in monitoring their loans to other banks, or equivalently as an increase

in their monitoring costs. Only when the spread reached 50 basis points, the ECB became

a competitive …nancial intermediary, i.e.

it became more e¢ cient than commercial banks at

monitoring credit worthiness. We therefore interpret this spread level as a measure of the

ECB’s lower monitoring e¢ ciency under normal circumstances. We set (cid:22)c so as to imply a

steady state credit spread between ECB loans and banks’loans of 50 basis points.

Finally, in line with a large fraction of the standard new-Keynesian literature, we study

economic dynamics around an e¢ cient steady state. The steady state is e¢ cient thanks to a

constant subsidy which removes the steady state wedge between marginal rate of substitution

and marginal rate of transformation.

Figures 1 and 2 display the impulse responses to a (cid:22)b

t shock under optimal policy in the

simple model. As already discussed above, this shock generates an immediate increase in the

loan-deposit rate spread, which pushes up …rms’marginal costs and thus generates in‡ationary

pressure. The increase in marginal costs also generates a persistent increase in the mark-up qt

and downward pressure on wages, inducing a reduction in both labour supply and the demand

for consumption goods. Hence, the spread moves anti-cyclically.

Figures 1 focuses on the case in which the central bank responds solely with the standard

policy instrument. The shock is such that spreads increase by approximately 100 basis points.

Optimal policy requires a cut in interest rates, in spite of the in‡ationary pressure created by

the increase in spreads. The main reason for this policy response is that the …nancial shock

ECB Working Paper Series No 2183 / October 201824is ine¢ cient, hence the fall in households’consumption is undesirable. The expansion in the

monetary policy stance helps smooth the adjustment of households’ consumption after the

shock, at the cost of producing a short in‡ationary episode. As already apparent from the

target rule, at the end of the adjustment period the price level reverts back to the original

level and then crosses it to eventually end up below the starting value. The promise of a future

fall in the price level keeps expectations of future in‡ation down. It ensures that only a short

in‡ationary episode follows an in‡ationary shock, in spite of the impact fall in the policy rate

when the shock hits. If we ignore the ELB constraint, the nominal rate falls to almost

3%,

(cid:0)

before returning relatively quickly towards the steady state to react to the increase in in‡ation.

Once we impose the ELB constraint, the nominal rate is zero for two periods and then rises

faster than in the absence of the ELB. Figures 1 thus illustrates a well-known property of the

optimal interest rate policy: policy rates remain "low for longer" in the presence of the ELB

–they increase later and faster than they would in the absence of that constraint.

Figure 2 displays impulse responses to a shock of the same size as in …gure 1 when non-

standard policy is also available (solid line), and compares it to the case when only standard

policy can be used (dotted line). In the case where non-standard policy is implemented, the

shock does not require the policy rate to reach the zero lower bound. The smaller response

of the policy rate is nonetheless able to substantially reduce the volatility of both the output

gap and in‡ation. Also, a lower fall in the future price level is necessary to limit the initially

in‡ationary consequences of the shock.

Non-standard measures – i.e. central bank intermediation – are deployed as soon as the

credit spread on banks’ loans increases above 50 basis points. Given the persistence of the

shock, this is the case for the entire period considered. Non-standard measures are therefore

implemented irrespectively of the level of the policy rate. They can optimally be deployed

when the interest rate has not yet reached the zero bound. Once the shock hits, the central

bank starts providing loans to the economy at the same time as it lowers interest rates. This

direct intermediation activity continues long after interest rates have essentially returned to

their steady state. Standard monetary policy is again tightened quickly few quarters after the

shock hits, because in‡ation is stabilised in spite of the fact that …nancial market conditions

remain impaired.

ECB Working Paper Series No 2183 / October 201825The optimal combination of standard and non-standard measures delivers a superior out-

come to the case in which non-standard measures are unavailable and the policy rate is kept

"lower for longer": in‡ation is better stabilised and the output gap is smaller.

4.1 Robustness: a model with capital

In our model, a shock to monitoring costs acts as a purely "cost-push" factor, as it creates

in‡ationary pressures independently from those exerted by the output gap. The increase

in credit spreads induced by the shock exerts upward pressure on in‡ation by raising …rms’

marginal costs, without directly a¤ecting aggregate demand. The contraction in real activity

occurs because the increase in credit spreads raises the …nancial markup qt; and consequently

lowers the real wage and labor supply.

The pure cost-push nature of the …nancial shock in our simple model could have an impact

on our conclusions concerning the optimal mix of standard and non-standard measures. In

particular, policy interest rates are optimally increased quickly after a very persistent …nan-

cial shock, because the ensuing increase in spreads puts upward pressure on marginal costs.

We therefore check robustness of this conclusion to a model similar to the one described in

section 2, the key di¤erence being the presence of competitive …rms operating an investment

sector. These …rms are endowed with a technology which transforms …nal consumption goods

into capital goods, and experience an idiosyncratic productivity shock. As …rms need to raise

external …nance to buy consumption goods, they need to stipulate a contract with …nancial

intermediaries. The …nancial friction takes the form of asymmetric information on the idio-

syncratic productivity shock and of monitoring costs, as in our simple benchmark model. We

provide the details of that framework in appendix D.

Figures 3 presents the impulse responses to a (cid:22)b

t shock under the optimal policy in the model

with capital, when the interest rate is constrained by the ZLB and non-standard measures are

implemented. The shock is calibrated to generate the same increase in market spreads as in

…gure 2. The …gure shows the spreads o¤ered by both banks (solid line) and the central bank

(dashed line). Notice that in‡ation, the inverse of leverage, zt; investment and output are

plotted in percentage deviations from the steady state.

In this model, the exogenous increase in monitoring costs depresses investment through a

sharp and persistent rise in the price of capital, which contributes to drive output down. The

interest rate is reduced to limit the fall in output but less than in the model without capital, and

ECB Working Paper Series No 2183 / October 201826it is later increased much more gradually. The reason is that the upward pressure on in‡ation

exerted through the cost channel by a relatively higher interest rate on impact is compensated

by the strong de‡ationary pressures induced by the severe reduction in investment and output.

As a consequence, the overall reaction of in‡ation is similar to the one arising in the model

without capital.

Our robustness analysis shows that, when credit spreads have a direct e¤ect on some

components of the aggregate demand (in this case, investment), the optimal reaction of the

interest rate is milder on impact and its convergence to the steady state takes longer.

The main conclusions obtained in the benchmark model without capital, however, are

una¤ected. In reaction to …nancial shocks that negatively a¤ect banks’monitoring e¢ ciency,

it is optimal to use unconventional policies. Once these measures have been deployed, it is

sub-optimal to lower policy rates further. Moreover, under the type of shocks we consider, it

is desirable to increase interest rates before unconventional measures are discontinued.

5 Conclusion

We have presented a microfounded model with credit market imperfections and nominal price

rigidities, which we use to analyse the response of monetary policy to …nancial shocks in

the presence of the ELB. The model can also sheed light on the role of non-standard policy

measures both at the ELB and away from it.

We …nd that adverse …nancial shocks (notably a shock that increases banks’ monitoring

costs) can lead the economy to the ELB under optimal policy. Non-standard meaures can

be e¤ective in these situations. When adverse …nancial shocks impair the e¢ ciency of pri-

vate banks in intermediating …nance, the ability of the central bank to provide direct credit

to the economy mitigates the negative consequences of the shock on in‡ation and real activ-

ity. Cutting policy rates to zero may be unnecessary after non-standard measures have been

implemented.

ECB Working Paper Series No 2183 / October 201827Figure 1: Response to a shock to (cid:22) under the optimal monetary policy: with ZLB (solid blue

line) and without ZLB (dotted blue line). All variables are in levels.

24681012 202rd246810120123Spread2468101200.20.4Inflation24681012 0.4 0.200.20.4P level24681012 202real rate24681012 1.5 1 0.50Output gapECB Working Paper Series No 2183 / October 201828Figure 2: Response to a shock to (cid:22) under the optimal monetary policy: with ZLB and

non-standard measures (solid blue line) and with ZLB only (dotted blue line). All variables

are in levels.

2468101200.511.52rd24681012012Spreads2468101200.20.4Inflation24681012 0.4 0.200.20.4P level2468101200.511.52real rate24681012 1.5 1 0.50Output gapprivate spreadCB spreadECB Working Paper Series No 2183 / October 201829Figure 3: Impulse responses to an AR(1) (cid:22) shock with non-standard measures in the model

with capital. The nominal interest rate and the spreads are plotted in levels. All other

variables are in percentage deviations from the steady state.

24681012%00.511.5rd24681012%0123SpreadsCBmarket24681012% from SS 0.0200.020.040.06Inflation24681012% from SS 1.5 1 0.50investment24681012% from SS02468z24681012% from SS 0.25 0.2 0.15 0.1 0.05OutputECB Working Paper Series No 2183 / October 2018306 Appendix

A. The …nancial contract

The optimal …nancial contract solves a standard costly state veri…cation problem (see e.g. Gale

and Hellwig (1985)). The problem is

subject to

max
!i;t;xi;t;(cid:13)t

f (!i;t)qtxi;t

qtgb

!i;t; (cid:22)b
t

xi;t

(cid:16)

(cid:17)
qtgc (!i;t; (cid:22)c
t ) xi;t

f (!i;t) + (cid:13)tgb

!i;t; (cid:22)b
t

+ (1

(cid:16)

(cid:17)

qtxtf (!i;t)

(cid:13)t) gc (!i;t; (cid:22)c
t )

(cid:0)

0

(cid:21)

(cid:21)

(cid:21)

(cid:20)

(cid:20)

(cid:28)

1

(cid:0)
h
(cid:13)t (cid:20)

Rd

t (xi;t

Rd

t (xi;t

(cid:28) )

(cid:28) )

(cid:0)

(cid:0)

(54)

(55)

(56)

(cid:13)t(cid:22)b

t + (1

(cid:13)t) (cid:22)c
t

(cid:0)

1:

G (!i;t) (57)

i

(58)

The functions f (!i;t); gb

!i;t; (cid:22)b
t

and gc (!i;t; (cid:22)c

t ) are de…ned in equations (13)-(15) and

denote the expected unit shares of output accruing respectively to the …rm, the commercial

(cid:1)

(cid:0)

bank and the central bank, after stipulating a …nancial contract that sets a lending rate Rb or

Rc; respectively.

Denote with (cid:21)b

1;t; (cid:21)c

1;t and (cid:21)2;t the lagrangean multipliers associated to constraints (13),

(14) and (15), respectively, and (cid:17)1i;t and (cid:17)2i;t those associated to (cid:13)i;t (cid:21)
Condition (57) is used with equality to replace the gb
and gc

!i;t; (cid:22)b
t

0 and to (cid:13)i;t (cid:20)
!i;t; (cid:22)b
t

out in the

1.

…rst-order conditions of the problem.

(cid:0)

(cid:1)

(cid:0)

(cid:1)

De…ne also zi;t

(cid:28)
xi;t

(cid:17)

as the share on internal funds over the size of production. We

conjecture (and later verify) that (cid:21)2;t = 0. The …rst-order conditions of the problem are

qt
zi;t
qt
zi;t
qt
zi;t

)

)

(cid:1)

+ (cid:21)b
1;t

gb
z (

) qt + Rd
t

+ (cid:21)c
1;t

gc
z (

(cid:1)

+ (cid:21)b

(cid:16)
1;tgb

Rb (

(cid:17)
1;tgc
) qt + (cid:21)c

Rb (

(cid:16)
) qt

(cid:1)

(cid:1)

+ (cid:21)b

1;tgb

Rc (

) qt + (cid:21)c

1;tgc

Rc (

(cid:1)

) qt

(cid:1)

f (

)

(cid:1)

qt
z2
i;t

= fz(

)

(cid:1)

0 = fRb(

0 = fRc(

(cid:1)
t (1

Rd

Rd

t (1

gb (

gc (

(cid:17)1;t (cid:0)

) qt

(cid:1)

(cid:21)

) qt

(cid:1)
(cid:21)
(cid:17)2;t = f(cid:13)(

zi;t)

zi;t)

+ (cid:21)b

1;tg(cid:13)

b (

(cid:1)

) qt + (cid:21)c

1;tg(cid:13)

c (

) qt

(cid:1)

(cid:0)

(cid:0)
qt
zi;t

)

(cid:1)

) qt + Rd
t

(cid:1)

(cid:17)

(59)

(60)

(61)

(62)

(63)

(64)

ECB Working Paper Series No 2183 / October 201831Use the de…nitions (13), (14) and (15) to obtain partial derivatives of the functions f (!i;t);
!i;t; (cid:22)b
t

and to rewrite equation (59) as

!i;t; (cid:22)b
t

and gc

gb

(cid:13)t)

f! (!i;t)

(65)

(cid:0)

(cid:1)

(cid:0)

f (!i;t)

(cid:1)
=

qt
zi;t

(cid:0)

h

Rb

t (cid:13)t + Rc

t (1

(cid:0)

i
Rb
t [1

(cid:0)
(cid:30) (!i;t)

Rd

t (cid:0)
(cid:13)t)

(cid:8) (!i;t)]

+(cid:21)b

1;tzi;t

+(cid:21)c

1;tzi;t

equation (60) as

0

@

0

@

(cid:0)

(cid:0)

(cid:2)

(cid:0)

Rb

t (cid:13)t + Rc

t (1

Rd

t + Rc
t (cid:13)t + Rc

Rb

t (1

(cid:0)
(cid:3)
t [!i;t(cid:30) (!i;t)

(cid:0)
(1

(cid:13)t)

(cid:0)

(cid:1)

(cid:22)b
t

!i;t + Rb 1

z

(cid:0)
q

1

(cid:0)

h(cid:0)
1 + (cid:8) (!i;t)]

(cid:1)

;

(cid:0)

t ) !i;t(cid:30) (!i;t) 1
(cid:22)c
A

0 = (cid:13)t

(1

zi;t)

(cid:0)
qt

f! (!i;t)

qt
zi;t

+(cid:21)b

1;t (1

+(cid:21)c

1;t (1

zi;t)

zi;t)

(cid:0)

(cid:0)

1

(cid:13)t
(cid:20)
(cid:16)
(cid:13)t (1
(cid:20)

(cid:0)

(cid:0)

equation (61) as

(cid:22)b
t

!i;t(cid:30) (!i;t) + 1

(cid:8) (!i;t)

(cid:13)Rb

t (cid:30) (!i;t)

(cid:0)

(cid:17)
t ) !i;t(cid:30) (!i;t)

(cid:22)c

(cid:13)Rc

t (cid:30) (!i;t)

(cid:0)

(cid:0)
1

(cid:0)
qt

zi;t

;

(cid:21)

0 = (1

(cid:13)t)

(cid:0)

1

zi;t

(cid:0)
qt

f! (!i;t)

qt
zi;t

1

i

A

1

zi;t

(cid:0)
qt

(cid:21)

(66)

(67)

+(cid:21)b

1;t (1

+(cid:21)c

1;t (1

(cid:0)

(cid:0)

(cid:13)t) (1

(cid:0)

zi;t) (cid:30) (!i;t)

1

(cid:22)b
t

!i;t

(cid:0)

1

Rb
t

zi;t

(cid:0)
qt

(cid:0)

(cid:20)(cid:16)
t ) (1

(cid:22)c

(cid:17)

(1

(cid:0)
(cid:8) (!i;t)

(cid:13)t) !i;t(cid:30) (!i;t)

(cid:0)
Rc
t (cid:30) (!i;t) (1

(cid:13)t) 1

zi;t
(cid:0)
qt

(cid:0)

(cid:0)

(cid:21)

;

3

5

zi;t)

+1

(cid:0)

2

4

and equation (64) as

(cid:17)1;t (cid:0)
+(cid:21)b
1;t

(cid:17)2;t =

Rb

t (cid:0)

1

Rc
t

zi;t

(cid:0)
zi;t

f!i;t (!i;t)

(68)

(cid:16)
Rc
t

Rc
t

(1

(1

(cid:17)

(cid:0)

(cid:0)

Rb

t (cid:0)

Rb

t (cid:0)

(cid:16)

+(cid:21)c
1;t

(cid:17)
zi;t) (cid:30) (!i;t)

zi;t) (cid:30) (!i;t)

(1

(1

(cid:20)

zi;t

(cid:22)b)!i;t

(cid:22)c) !i;t

(cid:0)

(cid:0)

Rb 1

1

Rc
t

(cid:0)

(cid:0)

(cid:0)
qt

(cid:0)
qt

(cid:21)
zi;t

:

(cid:20)
Rearranging equations (66) and (67), we get that

(cid:16)

(cid:17)

From equations (66) and (69), it follows that

(cid:13)t) (cid:21)b

1;t = (cid:13)t(cid:21)c

1;t:

(1

(cid:0)

1
(cid:13)t

(cid:21)b
1;tzi;t =

1

(cid:0)

1
(cid:13)t)(cid:22)c
(cid:8)(!i;t)

t)!i;t(cid:30)(!i;t)

:

((cid:13)t(cid:22)b

t +(1
(cid:0)
1
(cid:0)

(cid:21)

(69)

ECB Working Paper Series No 2183 / October 201832This latter condition, together with (69), ensures that constraints (54) and (55) are binding at

the optimum.

We can therefore simplify equation (65) to get

f (!i;t)

qt
zi;t

=

1

(cid:0)

Rd
t
(cid:13)t)(cid:22)c
(cid:8)(!i;t)]

t]!i;t(cid:30)(!i;t)

;

[(cid:13)t(cid:22)b

t +(1
(cid:0)
[1
(cid:0)

and equation (68) to get

(cid:17)1;t (cid:0)

(cid:17)2;t =

1

zi;t
(cid:0)
zi;t (cid:19)

(cid:0)

(cid:18)

(cid:8) (!i;t)

1

(cid:0)

(cid:0)

Rc
t

Rb
t (cid:0)
(cid:13)t(cid:22)b
t + (1
(cid:0)

(cid:13)t) (cid:22)c
(cid:1)
(cid:0)

:

!i;t(cid:30) (!i;t)

Now use the de…nition zi;t

(cid:28)
xi;t

(cid:17)

and condition (57) to rewrite the optimality conditions as

(cid:3)

(cid:2)

(cid:28)
xi;t

[(cid:13)t(cid:22)b

t]!i;t(cid:30)(!i;t)

Rd
t
(cid:13)t)(cid:22)c
(cid:8)(!i;t)]

t +(1
(cid:0)
[1
(cid:0)
(cid:28)
xi;t (cid:19)
xi;t
Rb
i;t
(cid:28) (cid:0)
(cid:30)(!i;t)!i;t[(cid:13)t(cid:22)b
[1

(cid:17) (cid:0)

(cid:18)
Rc

1

(cid:0)

i;t (cid:0)
1

(cid:0)

1

[1

(cid:8) (!i;t)]

(cid:0)
t]
(cid:13)t)(cid:22)c

t +(1
(cid:1)
(cid:8)(!i;t)]

(cid:0)

(cid:0)

f (!i;t)qt =

1

(cid:0)
qt = Rd
t

gb

!i;t; (cid:22)b
t

(cid:16)

(cid:17)

(cid:17)1;t (cid:0)

(cid:17)2;t =

(cid:16)

(cid:17)1;t(cid:13)t = 0

(cid:17)2;t (1

(cid:0)

(cid:13)t) = 0

(70)

(71)

(72)

(73)

(74)

together with (cid:17)1;t (cid:21)
veri…es our conjecture that (cid:21)2;t = 0:

0 and (cid:17)2;t (cid:21)

0: Notice that, from condition (70), f (!i;t)qtxi;t > Rd

t (cid:28) ; which

Substituting the expression for qt obtained from equation (70) in (71), and using condition

(57) and (69), we can rewrite equation (71) as

1

= 1 +

xi;t
(cid:28)

f (!i;t)

(cid:0)
f (!i;t)

(cid:13)t(cid:22)b
((cid:13)t(cid:22)b

t + (1
t +(1
(cid:0)
1
(cid:0)

(cid:13)t) (cid:22)c
t
t)!i;t(cid:30)(!i;t)
(cid:1)

(cid:0)
(cid:13)t)(cid:22)c
(cid:8)(!i;t)

(cid:0)
1

(cid:0)
(cid:0)

G (!i;t)

(cid:21)

(cid:20)

:

(75)

The optimality conditions can therefore be written as the equations (22)-(26) in the main text.

ECB Working Paper Series No 2183 / October 201833B. Coe¢ cients

The coe¢ cients of the system of log-linearized equilibrium conditions are given by

(cid:11)1 =

q
R

(cid:0)

(cid:11)2 = (cid:22)

(cid:17)

(cid:30)2
(cid:22) f !
(cid:30)! (cid:0)
f!
f!
(cid:16)
g!(cid:3))
(1
(cid:0)
f !
f! (cid:18)
(cid:30)! (cid:0)

(cid:22) f
f!

(cid:8) +

q
R "

(cid:30)! (cid:0)

(cid:30)2
f!
(cid:17)
f + (cid:22)f (cid:30)
f!

(cid:11)3 =

(cid:0) 0

(cid:16)

(cid:22)(cid:8)
g
g!(cid:3)) (cid:0)

f (cid:30)
f! #

(cid:30)2
f! (cid:19)
+ (f! + (cid:22)(cid:30))

(1

(cid:0)

!
g!(cid:3))

(cid:0)

1

(1

A

(cid:11)4 =

@
(cid:22)(cid:8)
g

(cid:11)3 +

(cid:22) f (cid:30)
f!
f + (cid:22)f (cid:30)
f!

(cid:11)5 = ((cid:11)3

1
(cid:11)1)(cid:0)

(cid:0)

C. Welfare approximation

Welfare is

Wt0 = Et0

be approximated as

Ut

'

U + ucc

ct +

+u(cid:24)

(cid:18)
1
b
(cid:24)t +
2

(cid:18)

(cid:18)

b

1
2

(cid:18)
1 +

1 +

uccc
uc (cid:19)
(cid:24)

2
t

(cid:19)

b

u(cid:24)(cid:24)
u(cid:24) (cid:19)

where households’ temporary utility is given by Ut = u (ct; (cid:24)t)

v (ht) : This latter can then

(cid:0)

1

(cid:12)tUt

;

)

(

t=t0
X

(cid:0)

(cid:19)

c2
t

b

vhh

ht +

(cid:18)

b

1
2

1 +

(cid:18)

vhhh
vh (cid:19)

h2
t

b

(cid:19)

+ uc(cid:24)c

ct

(cid:24)t

b

b

where hats denote log-deviations from the deterministic steady state and c and h denote steady

state levels.

Under the functional form Ut = (cid:24)t

(cid:27)

(cid:0)

c1
t
1

(cid:27) (cid:0)

(cid:0)

h1+(cid:30)
1+(cid:30) , and assuming that in steady state (cid:24) = 1,
t

households’temporary utility can be rewritten as

Ut

'

(cid:27)

(cid:0)

c1
t
1

(cid:0)
+c1

(cid:27) (cid:0)

(cid:27)

ct

(cid:0)

(cid:24)t +

h1+(cid:30)
t
1 + (cid:30)
c1
1

b

b

+ c1

(cid:27)

(cid:0)

ct

(cid:0)

(cid:0)

(cid:27)

(cid:27)

b
(cid:24)t +

(cid:18)

b

(cid:0)
1
2

(cid:24)

b

 h1+(cid:30)

ht +

1
2

c1

(cid:0)

(cid:27) (1

(cid:27))

c2
t (cid:0)

(cid:0)

1
2

 h1+(cid:30) (1 + (cid:30))

h2
t

b

:

2
t

(cid:19)

b

b

ECB Working Paper Series No 2183 / October 201834We can now express hours and households’consumption as ht = styt
At

so that

ht =

st +

yt

at:

(cid:0)

Using this expression together with ct = yt; we can write utility as

b

b

b

b

Ut
c1

(cid:0)

(cid:27) '

1

+

+

1

(cid:27) (cid:0)
(cid:0)
 h1+(cid:30)
c1
(cid:0)
1

1

h1+(cid:30)
t
(cid:27)
c1
t

(cid:0)

+

1
(cid:18)

(cid:0)

1 + (cid:30)

(cid:27) (1 + ')
1
2

(cid:24)t +

(cid:27)

yt

at +

+

2
(cid:24)
b
t

b
(cid:19)

yt

(cid:24)t
(cid:0)
 h1+(cid:30)
b
b
(cid:27)
c1
(cid:0)

(cid:18)
or, given that st is of second order, as

(cid:0)

b

b

(cid:0)

yt

 h1+(cid:30)
(cid:27)
c1
(cid:0)
(cid:19)
 h1+(cid:30)
b
(cid:27) (1 + ')
c1
 h1+(cid:30)
1
c1
2

at

(cid:0)

(cid:0)

(cid:0)

h1+(cid:30)
(cid:27)
c1

(cid:0)

st

st

b
yt +

 h1+(cid:30)
c1

1
2
(cid:0)
(cid:18)
 h1+(cid:30)
(cid:27) (1 + ')
c1

(cid:0)

(cid:0)

(cid:27) (1 + ')

(cid:0)

y2
t

(1

(cid:27))

(cid:0)
 h1+(cid:30)
c1

(cid:0)

(cid:19)
b
(cid:27) (1 + ')

s2
t

b

st

at

1
2

(cid:0)

b

b

(cid:27) (1 +  )
b

b

a2
t

b

b

h1+(cid:30)
t
(cid:27)
c1
t

(cid:0)

1

(cid:0)

1

(cid:27) (cid:0)

1 + (cid:30)

(cid:27) (1 + ')

(1

(cid:0)

(cid:0)

(cid:27))

y2
t

(cid:19)

(cid:27) '

Ut
c1
(cid:0)
 h1+(cid:30)
c1

(cid:0)

1
2

(cid:0)

(cid:18)

Assume a subsidy such that  h1+(cid:30)

b
(cid:27) = 1. Then

c1

(cid:0)

+

+

1

 h1+(cid:30)
(cid:27)
c1

(cid:0)

(cid:0)
(cid:18)
 h1+(cid:30)
c1

(cid:0)

(cid:19)
(cid:27) (1 + ')

yt

(cid:0)

h1+(cid:30)
(cid:27)
c1

(cid:0)

st

b
yt

at +

(cid:24)t

b

yt + t:i:p:s

b

b

b

b

Ut
c1

(cid:0)

1

1

(cid:27) '

1

(cid:27) (cid:0)

1 + (cid:30) (cid:0)

1
2

st

(cid:0)

(' + (cid:27))

y2
t +

(1 + ')

at +

(cid:24)t

yt + t:i:p:s:

Now recall that

(1 + ') at +

b

(cid:24)

: Then

b

(cid:0)
t = 1
ye

((cid:27)+')

h

i

b

b

b

Ut
b
c1

(cid:0)

(cid:27) '

h

1

(cid:27) (cid:0)

1 + (cid:30) (cid:0)

1

(cid:0)

1

This can be rewritten as

i
b
st
(cid:0)

1
2

b

((cid:27) + ')

y2
t + ((cid:27) + ')

ye
t

yt + t:i:p:s

b

b

b

Ut
c1

(cid:0)

(cid:27) (cid:0)

1

1

(cid:18)

(cid:0)

(cid:27) (cid:0)

1
1 + (cid:30)

1
2

(1

(cid:0)

"(cid:18)
(cid:18)) (1

(cid:0)

(cid:25)2
t (cid:0)

(cid:12)(cid:18))

1
2

' (cid:0)

(cid:19)

((cid:27) + ') x2

t + t:i:p:s:

D. A model with capital

b

We analyse in this section the robustness of our results to a richer model. We use a version of the

model described in section 2. The key di¤erence is the presence of competitive …rms operating

an investment sector. We describe here the various sectors, highlighting the di¤erences relative

to the model of section 2. The problems of the households and of the …nal production sector

are unchanged, as well as those of the government and the central bank.

D.1 The capital producing sector

There is a continuum of competitive capital producing …rms who produce at time t

1

(cid:0)

new capital to be used in production at t; kt: In order to produce new capital, they need to

ECB Working Paper Series No 2183 / October 201835 
 
 
 
acquire old capital kt

(cid:0)

1; whose price in terms of consumption good is qt

1: The …rms have the

(cid:0)

following production function

kt = (1

(cid:14)) kt

(cid:0)

(cid:0)

1 +

1

(cid:20)

'

(cid:0)

(cid:18)

It
It

1
2 (cid:19)(cid:21)

(cid:0)

(cid:0)

It

1

(cid:0)

where (cid:14) denotes the depreciation rate, It
1 ; and ' (

where It =

1
(cid:0)
" dj

1
0 It (j)

(cid:0)

"

"

"

1 is investment in the composite …nal good in t

(cid:0)
) is an adjustment cost function,which is increasing

1;

(cid:0)

and convex. The price of the new capital in terms of consumption good is denoted with qt:

hR

i

(cid:1)

Firms’pro…ts at time t are given by

(cid:5)k

t = Pt (qtkt+1

It

(cid:0)

(cid:0)

qtkt) :

The …rst order conditions of the pro…t maximization problem are

qt

1
(cid:20)

'

(cid:0)

(cid:18)

It
It

(cid:0)

(cid:0)

1 (cid:19)

'0

It
It

(cid:0)

(cid:18)

1 (cid:19)

It
It

(cid:0)

1 (cid:21)

= 1

qt = (1

(cid:14)) qt

(cid:0)

D.2 The entrepreneurs

There is a continuum of risk-neutral, in…nitely lived entrepreneurs, denoted with i; who

die with probability (cid:24)t. At the end of period t

(cid:0)

1; each entrepreneur buys new capital ki;t

from capital producers at price qt and transforms it into capital services ki;t using a linear

technology

ki;t = !iki;t;

where the random variable ! is i.i.d. across time and across entrepreneurs, with distribution

(cid:8); density (cid:30) and mean unity. The shock ! is drawn after the capital ki;t is bought and is

private information. Its realization can be observed by the …nancial intermediary at the cost

of (cid:22)tki;t units of capital.

During period t, entrepreneurs rent the capital to intermediate goods producing …rms,

earning a real return (cid:26)t; and then sell the undepreciated capital to capital producers at the

end of the goods market, at price qt: Hence, …rm i gross real return on capital at time t is

Rk

t =

(cid:26)t + qt (1
qt

(cid:14))

:

(cid:0)

The …rm’s expected revenue is given by

Pt [(cid:26)t + qt (1

(cid:0)

(cid:14))] !iki;t = PtqtRt

k!iki;t:

ECB Working Paper Series No 2183 / October 201836In order to dispose of the capital necessary for production at time t, the …rm needs to raise

external …nance. At at the end of period t

(cid:0)

1; in the …nancial market, …rm i disposes of a

nominal amount of internal funds Zi;t

(cid:0)

1: It therefore needs to raise additional funds in the

amount Xi;t

1

Zi;t

1; for total funds at hand Xi;t

(cid:0)
stock of capital such that

(cid:0)

(cid:0)

1: Each …rm is thus restricted to buy a

(cid:0)

Ptqtki;t

Xi;t.

(cid:20)

(76)

D.3 The …nancial contract

The derivation of the loan contract is una¤ected, once we replace the exogenous internal

funds (cid:28) t in the benchmark model without capita with the accumulated internal funds Ni;t;
and the markup qt with the return on capital Rk

t : The …rst-order conditions are then given by

equations (22)-(26).

D.4 Entrepreneurial net worth

At the beginning of t; …rms’ pro…ts net of debt repayment are allocated to either entre-

prenurial consumption (gross of consumtpion taxes) or to the accumulation of …rms’nominal

funds,

Rk

t f (!i;t) Xi;t = Zi;t + Pt (1 + (cid:28) ) ei;t:

Here ei;t =

1
0 ei;t ((cid:17))

"

1
(cid:0)
" d(cid:17)

"

"

(cid:0)

1 ; with " > 1:

Since the entrepreneurs postpone consumption to the time of death, aggregate consumption

i

hR

is given by

(1 + (cid:28) ) Ptet = (cid:24)tRk

t f (!t) Xt

and the aggregate accumulation of internal funds by

Zt = (1

(cid:24)t)

(cid:0)

Rk
t
zt

1

f (!t) Zt:

(77)

(cid:0)
We consider the limiting case where consumption of entrepreneurs is fully taxed. As the

tax rate is made arbitrarily large, the consumption of the entrepreneurs approaches zero,
t = (cid:24)tRk

0, and the consumption tax revenue, T e

, approaches the total funds of the

t f (!t)Xt
1+(cid:28)

et

!

entrepreneurs that die.

D.5 The intermediate goods sector

Firms in the intermediate goods sector are monopolistically competitive. They produce

intermediate good j; with j

(0; 1) ; using the technology

2

yt (j) = Atlt (j)(cid:11) kt (j)1

(cid:11) ;

(cid:0)

(78)

ECB Working Paper Series No 2183 / October 201837where (cid:11)

2

(0; 1) denotes the labor share; lt (j) and kt (j) denote the amount of labor and capital

services rented on the market by …rm j, while At is an aggregate exogenous productivity shock.

Because of product di¤erentiation, each intermediate good …rm has some market power.

We assume that each retailer can change its price with probability 1

(cid:18); following Calvo (1983).

(cid:0)

The optimality conditions for price setting are given by (33)-(36) in the main text.

D.6 Market clearing

Market clearing conditions for money, bonds, labor, loans, wholesale goods and retail goods

are given, respectively, by

Mt = M s
t ;

Bt;t+1 = Bs

t;t+1;

ht = lt;

Zt;

Dt = Xt
(cid:0)
1
yt (j) dj;

yt =

0
Z

yt (j) = Atlt (j)(cid:11)

t k (j)1

(cid:11) ;

(cid:0)

yt (j) = ct (j) + et (j) + It (j) ; for all j;

where

Zi;tdi = Zt and

Xi;tdi = Xt:

R

R

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(81)

(82)

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(84)

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ECB Working Paper Series No 2183 / October 201841Acknowledgements 

All opinions expressed are personal and do not necessarily represent the views of the European Central Bank or the European System 
of Central Banks. We wish to thank Kosuke Aoki, Gabriele Camera and Jinill Kim for useful discussions. We also thank participants in 
the SNB Research Conference on "Policy Challenges and Developments in Monetary Economics" and the HKMA-FRBNY Conference 
on "Domestic and International Dimensions of Unconventional Monetary Policy" for useful comments on previous versions of the paper. 

Fiorella De Fiore (corresponding author) 
European Central Bank, Frankfurt am Main, Germany; CEPR, London, United Kingdom; email: fiorella.de_fiore@ecb.europa.eu 

Oreste Tristani 
European Central Bank, Frankfurt am Main, Germany; email: oreste.tristani@ecb.europa.eu 

© European Central Bank, 2018 

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electronically, in whole or in part, is permitted only with the explicit written authorisation of the ECB or the authors.  

This paper can be downloaded without charge from www.ecb.europa.eu, from the Social Science Research Network electronic library or 
from RePEc: Research Papers in Economics. Information on all of the papers published in the ECB Working Paper Series can be found 
on the ECB’s website. 

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