Monetary Policy and Inequality
when Aggregate Demand depends on LiquidityI
Florin O. BilbiieII
Xavier RagotIII
Paris School of Economics,
Sciences Po, CNRS, and OFCE
U. Paris 1 PanthÈon-Sorbonne, and CEPR
This draft January 2017 (First preliminary draft February 2016)
Abstract
Monetary policy design changes a great deal when inequality matters. In our New Keynesian model, aggregate demand depends on liquidity as heterogeneous consumers hold money
in face of uninsurable risk and participate infrequently in Önancial markets. Endogenous áuctuations in precautionary liquidity challenge central bankís aggregate demand management:
the Taylor coe¢cients required for determinacy are in the double digits, for moderate market incompleteness. Responding to inequality or liquidity can restore conventional wisdom. A
novel tradeo§ for Ramsey-optimal monetary policy arises between inequality and standardó
ináation and outputóstabilization objectives. Price stability has signiÖcant welfare costs that
are inequality-related: ináation volatility hinders volatility of constrained agentsí consumption.
JEL Codes: D14, D31, E21, E3, E4, E5
Keywords: inequality; optimal (Ramsey) monetary policy; heterogenous agents; incomplete
markets; liquidity constraints; limited participation; determinacy; interest rate rules; Taylor
principle; money.
I
We are grateful to Adrien Auclert, Paul Beaudry, Wei Cui, Ester Faia, Nobu Kiyotaki, Jordi Gali, Alejandro
Justiniano, Eric Leeper, Francesco Lippi, Ralph Luetticke, Roland Meeks, Ben Moll, Cyril Monnet, Morten Ravn,
Gilles Saint-Paul, Vincent Sterk, Paolo Surico, Gianluca Violante, Pierre-Olivier Weill, and Mirko Wiederholt for
comments and suggestions, as well as other participants in several conferences and seminars since March 2016. We
thank (without implicating) Banque de France for Önancial support through the eponymous Chair at the Paris School
of Economics.
II
Paris School of Economics, Centre díEconomie de la Sorbonne, 106-112 Boulevard de líHopital 75013 Paris. Email:
á[email protected]. URL: http://áorin.bilbiie.googlepages.com.
III
URL:http://www.parisschoolofeconomics.com/ragot-xavier/
1
1
Introduction
Inequality is not a word customarily encountered in the realm of research on monetary policy. So
much so that the last two Chairs of the Federal Reserve explicitly called for more research on monetary
policy and inequality: Bernanke (2007) and Yellen (2014); see also Bernanke (2015). Echoes come
from the other side of the Atlantic: the ECB President recently dedicated a whole speech to the issue
(Draghi, 2016, October), as did previously other board members (CúurÈ, 2012; Mersch, 2014).
A recent but already established literature responds to that call on the empirical front. Several
papers studied the impact of monetary policy on inequality (e.g. through the redistribution e§ects
of ináation), using a variety of methods.1 A general and robust conclusion seems to be that looser
monetary policy is associated with less inequality. And monetary policy has certainly been loose in
the aftermath of the 2008 Önancial crisis and the ensuing Great Recession, most notably through unprecedented liquidity expansion: to take one example, the year-on-year growth rate of M1 quadrupled
(from 2.5 to 11 percent on average) in the post-crisis period as compared to the 2000-2008 interval.2
On the theoretical front too, a new synthesis is under way at the time of our writing. A very recent
quantitative literature that we review below analyzes monetary policy transmission in incompletemarket, heterogeneous-agent New Keynesian modelsóabbreviated "HANK" by Kaplan, Moll and
Violante (2014). These contributions can speak to the aforementioned empirical Öndings, and are
also consistent with recent microeconometric evidence on the heterogeneity of marginal propensities
to consume MPCs, its relation to liquidity constraints, and income and wealth distributions.3 Our
goal is to contribute to the understanding of optimal monetary policy in a model that belongs to this
new vintage.
We thus revisit standard "New Keynesian" monetary policy analysisóincluding optimal policyó
in a tractable general equilibrium model with such features: aggregate demand depends on liquidity,
and inequality matters. Households are heterogeneous and subject to liquidity constraints, and
money is used to self-insure against uninsurable risk: Önancial markets are incomplete ‡ la Bewley
and participation is limited (infrequent) in the Baumol-Tobin tradition.
The consequences for monetary policy transmission and design are dramatic, and endogenous liquidity and inequality are the keystones.4 Endogenous movements in liquidity occurring as households
1
Starting from Doepke and Schneider (2006), these include i.a. Adam and Zhu (2014), Coibion et al (2013), Adam
and Tzamourani (2016), Deutsche Bundesbank (2016), and Furceri, Loungani, and Zdzienicka (2016).
2
Since nominal GDP has actually fallen during the crisis and growth thereafter does not nearly match that of
money, velocity sank during the crisis and kept falling.
3
Recent empirical evidence using micro data from various sources supports the hypothesis that high MPCs correspond to households who are liquidity constrained (rather than, say, income-poor); see Kaplan and Violante (2014),
Cloyne et al (2016), Jappelli and Pistaferri (2014), Misra and Surico (2014) and Surico and Trezzi (2016).
4
We use "inequality" here in a speciÖc, limited sense: imperfect consumption insurance. There are other channels
through which monetary policy and inequality interact, some of which operate in the richer models reviewed below.
2
seek to insure make it di¢cult for the central bank to control aggregate demand. The response to
ináation required to ensure equilibrium uniqueness with interest rate rules is very large (double-digits
large) for even moderate degrees of market incompleteness. The central bank can break this vicious
spiral and get back in control by including inequality or liquidity in its reaction function. The former
can be justiÖed on welfare groundsóindeed, inequality is one of the objectives of optimal policy.
To the best of our knowledge, this is a novel contributionóthe analysis of (Ramsey-)optimal
monetary policy in an economy with incomplete markets, endogenous liquidity (money), limited
participation, and sticky prices. Optimal policy in our economy includes a new trade-o§ between
reducing inequality, and stabilizing ináation and aggregate demand. This implies optimal deviations
from price stability, in the long and in the short run, and signiÖcant welfare losses of stabilizing
pricesóeven around the optimal long-run target. Such deviations and welfare e§ects are larger than
those encountered in existing monetary models with nominal rigidities.
More precisely, our Örst contribution is positive: we start by unveiling the determinants of aggregate demand in this economy. Liquidity (money) injection increases aggregate demand because
it relaxes the constraint of high-MPC agents (in line with the empirical evidence discussed above).
Ináation has an independent e§ect on aggregate demand over and above its standard, intertemporalsubstitution e§ect through the real interest rate; this e§ect is two-fold. First, the Pigou e§ect operates: todayís ináation erodes the purchasing power of constrained householdsí real balancesówhich,
with nominal rigidities, has an aggregate demand e§ect. Second, expected ináation increases demand
today because participants hold less currency to self-insure. These ingredients deliver an "aggregate
IS" curve in which money, interest and prices all matter for aggregate demand. Together with an
"LM" curve, a Phillips curve and a speciÖcation of monetary policy, this delivers a compact model
that is as simple (mathematically) as monetary NK models, or old-fashioned dynamic IS-LM models
‡ la Sargent and Wallace (1975). We solve this model in closed form and analyze the determinacy
properties of interest rate rules, obtaining the results hinted to previouslyóthe Taylor principle fails
dramatically with incomplete markets, but an appropriate response to inequality can restore it.
Our second contribution is normative: we studyóto the best of our knowledge, for the Örst time
in this frameworkóRamsey-optimal monetary policy. A second-order approximation to the welfare function reveals a new monetary policy trade-o§ between reducing inequality and the standard
objectives (of stabilizing ináation and consumption). There is scope for a planner to decrease inequality and provide consumption insuranceóobjective which is costly to achieve through ináation
when prices are sticky (and absent Öscal instruments). This trade-o§ operates in the long-run, like in
any monetary model, making deáation optimal. But more importantly, and unlike other monetary
sticky-price frameworks, it also operates in the short run: insofar as there is long-run inequality,
optimal policy requires volatile ináation. Moreover, this ináation volatility matters for welfare: a
But here we focus on the one that has been at the core of monetary policy since at least Friedman (1969).
3
central bank that stabilizes ináation, albeit around an optimal long-run target, incurs a large welfare costóconsumers would pay (around 0.1 percent of consumption) to live in the economy with
volatile ináation. Ináation volatility is beneÖcial because it dampens the consumption volatility of
constrained agents without a§ecting much unconstrained agents who can self-insure.5
1.1
Relation to Literature
Our model blends, and thus belongs to, several literatures. First, it integrates two streams of monetary economics that evolved divergently over the past two decades: New Keynesian (NK) models
with nominal rigidities, and microfounded models of money demand with áexible prices.6 Within
these frameworks, we connect their two subsets that focus on heterogeneity, market incompleteness
and limited participationóNK models with "hand-to-mouth" consumers (or limited asset markets
participation) and monetary theory models in the Bewley and Baumol-Tobin tradition. In our
model, money is used to self-insure against idiosyncratic shocks as in Bewley models, but only for
non-participating agents as in the Baumol-Tobin literature. Some of the key contributions that we
build upon, all with áexible prices, include Bewley (1983); Scheinkman and Weiss (1986); Lucas,
(1990); Kehoe, Levine, and Woodford (1992); Algan, Challe, and Ragot (2010); Alvarez and Lippi
(2014), Khan and Thomas (2015), Cao et al (2016), Gottlieb (2015), and Rocheteau, Weill and Wong
(2015, 2016). Recent empirical work argues that such frictions are needed to explain money demand,
including the distribution of money holdings across agents (i.a. Alvarez and Lippi 2009; Cao et al
2012; Ragot 2014).
A related subset of the NK literature studied aggregate demand with (simple) heterogeneityó
one could call this "Örst-generation HANK". Gali, Lopez-Salido and Valles (2004, 2007) and Bilbiie
(2004, 2008) are two early examples of such models where a subset of agents are employed handto-mouth and have unit MPC.7 Compared to these models, we allow for temporarily-binding credit
5
A simple rule that responds to expected inequality as well as ináation can implement price stability over the cycle
without having to rely on large ináation responses. A rule that responds to liquidity directly performs rather poorly;
the very feature that makes it beneÖcial to rule out self-fulÖlling liquidity áuctuations also makes it undesirable from
a welfare perspective: it constrains liquidity injection when it is most needed.
6
Money demand in the NK model is generically residual when money is introduced in the utility function, through
a cash-in-advance constraint, or through shopping-time distortions. This has nevertheless important consequences for
optimal policy, that we review in due course.
7
Bilbiie (2004, 2008) derives for the Örst time an analytical aggregate demand IS curve with heterogenous agents,
and optimal policy. Eggertsson and Krugman (2012) combine the same IS curve with a particular theory of the natural
interest rate in order to build a fascinating story of deleveraging, debt deáation, and the liquidity trap. Gali et al (2004,
2007) distinguish households according to whether they hold physical capital or not and solve the model numerically
to study determinacy and Öscal multipliers. Nistico (2015) allows households to switch stochastically between the
two states, and also computes optimal monetary policy. Yet another, separate but related stream studies "Önancial
accelerator" modelsósee Kiyotaki and Moore (1997) and Iacoviello (2005). Gertler and Kiyotaki (2010) review this
4
constraints and allow constrained agents to self-insure. So does the more recent literature referred to
as "HANK" above: quantitative models with household heterogeneity and incomplete markets that
replicate plausible distributions of wealth and marginal propensities to consume.
Kaplan, Violante and Moll (2014) revisit the transmission mechanism of monetary policy in such a
model with liquid and illiquid assets. In contrast to representative-agent NK models where monetary
policy works mainly through intertemporal substitution, monetary policy works in their model mainly
through what they label an "indirect e§ect" (the endogenous, general equilibrium response of output).
Ravn and Sterk (2013) analyze an incomplete-markets model where search and matching frictions
generate unemployment risk; job uncertainty in their model can generate deep and lasting recessions
through aggregate demand ampliÖcation.8 Gornemann, Kuester and Nakajima (2012) also study
monetary transmission when markets are incomplete and unemployment risk endogenous, focusing on
the distributional welfare e§ects on households with di§erent wealth levels. Challe, Matheron, Ragot,
Rubio-Ramirez (2015) estimate a model with endogenous unemployment risk (search and matching)
using Bayesian methods and assess the quantitative importance of the link between precautionary
saving and aggregate demand. McKay, Nakamura, and Steinsson (2015) use a similar model to
those mentioned above, but with exogenous unemployment risk, to show that forward guidance is
less powerful than in the standard modelómostly because an incomplete-markets model implies a
form of "discounting" of aggregate demand. Auclert (2015) analyzes the role of redistribution for
the transmission mechanism, and decomposes it into three channels that are related to householdsí
asset positions.
Very few models in this realm consider money demand at all. Bayer, Luetticke, Pham-Dao, and
Tjaden (2015) use a HANK model where money is used as precautionary liquidityótheir underlying
model of money demand thus being conceptually similar to ours; they show how higher idiosyncratic
uncertainty increases precautionary money demand, thus leading to a recession. They analyze the
e§ects of di§erent money-growth rules, including the di§erential welfare e§ects on di§erent groups
of agents, depending on their wealth. Den Haan, Rendhal and Riegler (2016) study a model with
uninsurable unemployment risk and sticky wages and show that these two features taken together
provide ampliÖcation and can cause a deáationary spiral; a key element is (precautionary) demand
for money, where money enters the utility function.9
literature.
8
In a recent follow-up paper, the same authors analyze determinacy properties of interest rate rules in a special
case of their model that is analytically solvable (Ravn and Sterk, 2016). Their analysis of determinacy with interest
rate rules, derived in paralell and independently from ours, can be seen as complementary to ours in Section 3ófor
the case where job Önding and separation rates are endogenous, but there is no liquidity and no intensive margin, the
two key elemens for our results.
9
They also argue that there is a signiÖcant role for unemployment insurance in their model; see also McKay and
Reis (2015).
5
Our framework captures some key features and mechanisms of this recent "HANK" literature:
quantitative models that are consistent with microeconomic heterogeneity and data on household
Önances. In building our simpliÖed framework, we trade o§ someórelevant and important, but thoroughly analyzed elsewhereócomplexity for analytical tractability, which allows us to analyze monetary policy transmission, determinacy properties of interest rate rule andóperhaps most importantlyó
optimal monetary policy. These dimensions of analysis are all integral part of the state-of-the-art NK
framework. To Öx ideas, one could argue that while existing literature in this realm puts more emphasis on the "heterogeneous-agent" part of HANK, our framework does the oppositeóit simpliÖes
heterogeneity to put more emphasis on the latter part of HANK.
To the best of our knowledge, our paper is the Örst to derive (Ramsey-)optimal monetary policy
in a tractable model that captures some of the key "HANK" mechanisms, including precautionary
liquidity demand in face of uninsurable unemployment risk, its interaction with aggregate demand,
and a role for inequality. We thus owe much debt to the literature that, building on the seminal
paper of Lucas and Stokey, 1983, shaped our understanding of optimal policy in NK models. Some
of the key contributions include Khan, King and Wollman (2003), Adao, Correia and Teles (2003),
Woodford (2003, Ch. 6), Benigno and Woodford (2005, 2012), and Schmitt-Grohe and Uribe (2004,
2007). In our framework, signiÖcant deviations from price stability are optimal, and not only in the
long runóthe cited papers also imply, when relying on (other, di§erent) money demand theories,
some convex combination between the Friedman rule and a zero ináation long-run prescriptions. Our
framework also gives rise, more surprisingly, to signiÖcant optimal deviations from price stability over
the cycleóin response to shock that in those frameworks do not generate such deviations. A welfaremaximizing central bank relies on ináation volatility optimally, as such ináation volatility provides
insurance and contributes to reducing inequality. Renouncing this volatility (by adopting a policy of
constant deáation at the optimal asymptotic rate) thus has a large welfare cost in our model, whereas
it is innocuous in the NK models with money demand reviewed above. The key to this di§erence is
inequality.
2
A Monetary NK Model with Heterogeneous Agents
We build a simple, tractable heterogeneous-agent New Keynesian model with moneyóheterogenous
households hold money to self-insure against unemployment risk, markets are incomplete, participation is limited (infrequent), and price adjustment is costly.
Households There is a mass 1 of households, indexed by j 2 [0; 1], who discount the future at
rate # and derive utility from consumption cjt and disutility from labor supply ltj . The period utility
function is:
! j "1+'
! j"
lt
u ct " '
;
1+'
6
with u (c) = (c1!$ " 1) = (1 " *). Households have access to three assets: money (with zero nominal
return), public debt (with nominal return it > 0), and shares in monopolistically competitive Örms.
Money is held despite being a dominated asset because Önancial frictions give it a consumptionsmoothing, insurance role. These frictions are: uninsurable idiosyncratic risks, and infrequent participation in Önancial markets. Such frictions customarily generate a large amount of heterogeneity:
the economy is characterized by a continuous distribution of wealth, which is very hard to study with
aggregate shocks and sticky prices.
To simplify the problem (and thus enable us to perform the analysis previewed in the Introduction), we use tools developed in the incomplete-markets literature to reduce the amount of heterogeneity. These simpliÖcations keep the essence of intertemporal trade-o§s and of redistributive
e§ects of monetary policy in general equilibrium, and can be viewed as a simple generalization of
the Lucas (1990) multiple-member household metaphor. As we shall see, in our economy the key
intertemporal trade-o§s are captured by agentsí Euler equations for money and other assets; at the
same time, a relevant but limited amount of heterogeneity captures the redistributive e§ects of ináation and money creation.10 The gain of this modeling strategy is that standard tools used in
New-Keynesian economics can be used; in particular, we can compute Ramsey-optimal policy with
aggregate shocksóto the best of our knowledge, for the Örst time in an incomplete-markets economy
with uninsurable risk and limited participation.11
Households participate infrequently in Önancial markets. When they do, they can freely adjust
their portfolio and receive dividends from Örms. When they do not, they can only use money to
smooth consumption. Denote by - the probability to keep participating in period t + 1, conditional
upon participating at t (hence, the probability to switch to not participating is 1 " -). Likewise, call
/ the probability to keep non-participating in period t + 1, conditional upon not participating at t
(hence, the probability to become a participant is 1 " /). The fraction of participating households is
n = (1 " /) = (2 " - " /), and the fraction 1 " n = (1 " -) = (2 " - " /) does not participate.
Furthermore, households belong to a family whose head maximizes the intertemporal welfare of
family members using a utilitarian welfare criterion (all households are equally weighted), but faces
some limits to the amount of risk sharing that it can do. Households can be thought of as being in
two states or "islands". All households who are participating in Önancial markets are on the same
island, called P . All households who are not participating in Önancial markets are on the same
island, called N . The family head can transfer all resources across households within the island, but
cannot transfer some resources between islands.
10
See also Curdia and Woodford (2009) for another application of the "infrequent participation" structure in a
di§erent context (with savers and borrowers) with sticky prices.
11
Bilbiie (2008) analyzes optimal monetary policy in a sticky-price model when some agents are (employed) handto-mouth, and there is no self-insurance other than labor. Curdia and Woodford (2009) also look at optimal policy in
their model, around a steady-state with perfect insurance.
7
Households in the participating island work at real wage wt , whereas households in the nonparticipating island work to get a Öxed home-production amount 4 (which is also their Öxed labor
supply). Thus, we assume that Önancial risks (participating or not) and labor market risks (employed
or not) are perfectly correlated, to simplify the exposition. This isolates the channel that we want
to emphasize: self-insurance through money in face of unemployment risk.
The timing is the following. At the beginning of the period, the family head pools resources within
the island. The aggregate shocks are revealed and the family head determines the consumption/saving
choice for each household in each island. Then, households learn their next-period participation status
and have to move to the corresponding island accordingly, taking only money with them. The key
assumption is that the family head can not make transfers to households after the idiosyncratic shock
is revealed and it will take this as a constraint for the consumption/saving choice.
The áows across islands are as follows. The total measure of households leaving the N island
each period is the number of households who participate next period: (1 " n) (1 " /). The measure
of households staying on the island is thus (1 " n) /. In addition, a measure (1 " -) n leaves the P
island for the N island at the end of each period.
Total welfare maximization implies that the family head pools resources at the beginning of
the period in a given island and implements symmetric consumption/saving choices for all houseP
holds in that island. Denote as bPt+1 and Mt+1
, the per-capita period t bonds and money balances
respectively, in the P island, after the consumption-saving choice. The real money balances are
P
mPt+1 = Mt+1
=Pt ; where Pt is the price level. The end-of-period per capita real values (after the consumption/saving choice but before agents move across islands) are ~bP and m
~ P :Denote as mN , the
t+1
t+1
t
per capita beginning-of-period capital money in the N island (where the only asset is money). The
end-of-period values (before agents move across islands) are m
~N
t+1 . We have the following relations,
after simpliÖcation (as bonds do not leave the P island, we have bP = ~bP ):
t+1
t+1
mPt+1 = -m
~ Pt+1 + (1 " -) m
~N
t+1
mN
~ Pt+1 + /m
~N
t+1 = (1 " /) m
t+1 :
The program of the family head is (with 9 t = (Pt " Pt!1 )=Pt!1 denoting the net ináation rate)
"
! P "1+' #
%
&
! P P N
"
! P"
! N"
lt
4 1+'
W bt ; mt ; mt ; Xt =
max
n u ct " '
+ (1 " n) u ct " '
~P
1+'
1+'
fcP
t ;bt+1 ;
N P
m
~P
~N
t+1 ;m
t+1 ;ct ;lt g
"
!
+#EW bPt+1 ; mPt+1 ; mN
t+1 ; Xt+1
8
subject to
cPt + ~bPt+1 + m
~ Pt+1 = wt ltP " = Pt
1 + it!1 P
mPt
1
bt +
+ dt ;
1 + 9t
1 + 9t n
N
mt
N
N
m
~N
t+1 + ct = 4 " = t +
1 + 9t
(1)
+
m
~ Pt+1 ; m
~N
t+1 $ 0
(2)
(3)
and the laws of motion for money áows outlined above, relating mjt+1 to m
~ jt+1 . Equation (1) is the
per capita budget constraint in the P island: P -households (who own all the Örms) receive dividends
dt =n, and the real return on money and bond holdings. With these resources they consume and
save in money in bonds, and pay taxes/receive transfers = Pt (lump-sum taxes include any new money
created or destroyed). Equation (2) is the budget constraint in the N island. Finally (3) are positive
constraints on money holdings and are akin to credit constraints in the heterogeneous-agent literature.
The variable Xt in the value function refers to all relevant period t information necessary to form
rational expectations. Using the Örst-order and envelope conditions, we have:
! "
1 + it 0 ! P "
u0 cPt $ #E
u ct+1 or ~bPt+1 = 0
1 + 9 t+1
! "
'
!
"
!
"(
u0 cPt $ #E -u0 cPt+1 + (1 " -) u0 cN
t+1
1
or m
~ Pt+1 = 0
1 + 9 t+1
!
"
'
!
"
!
"(
1
u0 c N
$ #E (1 " /) u0 cPt+1 + /u0 cN
or m
~N
t
t+1
t+1 = 0
1 + 9 t+1
! "
! "'
wt u0 cPt = ' ltP
(4)
(5)
(6)
(7)
The Örst Euler equation corresponds to the choice of bonds: there is no self-insurance motive,
for they cannot be carried to the N island: the equation is the same as with a representative agent.
The money choice of P -island agents is governed by (5), which takes into account that money can be
used when moving to the N island. The third equation (6) determines the money choice of agents
in the N island, and the last equation labor supply.
The important implication of this market structure is that the Euler equations (5) and (6) have
the same form as in a fully-áedged incomplete-markets model ‡ la Bewely-Huggett-Aiyagari. In
particular, the probability 1 " - measures the uninsurable risk to switch to "low income" (unem-
ployment) next period, risk for which money is the only means to self-insure. This is why money is
held in equilibrium for self-insurance purposes, despite being a dominated asset.
Production and Price Setting. The Önal good is) produced by a Örm
using intermediate goods
"
+ "!1
R1
1! 1"
as inputs. The Önal sector production function is Yt = 0 (yt (z))
dz
, where yt is the amount
of type z intermediate good used in production. Denote as Pt (z) the price of intermediate goods
z. Demand for an individual product is Yt (z) = (Pt (z) =Pt )!" Yt with the welfare-based price index
9
Pt =
)R
1
0
Pt (z)1!" dz
1
+ 1!"
. Each individual good is produced by a monopolistic competitive Örm,
indexed by z, using a technology given by: Yt (z) = At lt (z). Cost minimization taking the wage as
given, implies that real marginal cost is Wt = (At Pt ) : The problem of producer z is to maximize the
present value of future proÖts, discounted using the stochastic discount factor of their shareholders,
the participants.
When price adjustment is frictionless, prices of all Örms are equal to a constant markup over
nominal marginal costóreal marginal cost is constant Wt = (At Pt ) = (" " 1) =": We assume that
Örms are subject to nominal rigidities: Örms incur a quadratic adjustment cost ‡ la Rotemberg
(1982) for changing
their prices,
)
+ cost which is homogenous across Örms. ProÖts of each Örm are thus
given by dt = 1 "
wt
At
" /2 9 2t Yt , anticipating that the equilibrium is symmetric. Maximization of
their present discounted value gives rise to the nonlinear forward-looking "New Keynesian Phillips
curve" whose derivation is described in detail in the Appendixówhere we replaced the labor supply
! "' ! P "$
schedule wt = ' ltP
ct :
!
! "' ! P "$
%, P -$
&
ct
ct
Yt+1
" ' ltP
9 t (1 + 9 t ) = #Et
9 t+1 (1 + 9 t+1 ) +
+."1 ;
(8)
Yt
D
At
cPt+1
where . % 1 " (" " 1) (1 + E) =" captures the steady-state distortion and E is a corrective sales
subsidy; in particular, when the subsidy is equal to the desired net markup E = (" " 1)!1 ; there is
no steady-state distortion associated to monopolistic competition and elastic labor, . = 0. These
considerations will be useful when studying Ramsey policy below.
Money Creation and Budget of the State. New money is created through "helicopter
12
CB
drops" and we consider uniform taxation = Pt = = N
t = = t . . Denote by mt+1 the (real value of) new
tot
money created in period t, and by Mt+1
the total nominal quantity of money in circulation at the
tot
end of each period. In nominal terms, Mt+1
= Mttot + Pt mCB
t+1 , and in real terms
mtot
t+1
mtot
t
=
+ mCB
t+1
1 + 9t
(9)
Hence, the total period t net taxes/transfers are = t = "mCB
t+1 :
Market clearing and equilibrium Since there is no public debt, the period t market for bonds
is nbpt+1 = 0. The money market clears mtot
~N
~ Pt+1 and so does the labor market
t+1 = (1 " n) m
t+1 + nm
12
Di§erent ways of money creation, i.e. open mrket operations have some starkly di§erent positive and normative
implications that are the subject of a companion paper (Bilbiie and Ragot, 2016). In that paper, we also leave open
the possibility of exogenous redistribution by choosing type-speciÖc transfers:! P
t =
!
N
n ! t; ! t
=
1!!
1!n ! t
where ! is the
share of total taxes paid by all type-N agents. When ! = n we have the case of entirely lump-sum transfers. At
the other extreme, it can be easily shown that there exist processes ! jt that redistribute money so as to replicate
Woodfordís cashless limit. Finally, there exists a value of ! that restores neutrality and Wallaceís 1981 logicói.e.
"keeping Öscal policy constant" in the sense of Önding an (exogenous) redistribution that un-does the endogenous
redistribution triggered by a monetary policy shock in our framework.
10
lt = nltP . Denoting by ct total consumption and Yt = At lt market-produced output (or earned total
income), we have that the goods market will also clear, by Walrasí Law:
)
D 2+
ct % ncPt + (1 " n) cN
=
1
"
9 Yt + (1 " n) 4:
t
2 t
(10)
Note for further use that there is a resource cost of changing prices (ináation), which is isomorphic
to the welfare cost of relative price dispersion in a Calvo-type model, see e.g. Woodford (2003). In
Appendix A we provide the summary of model equations and the equilibrium deÖnition.
Steady state. The analysis of the modelís steady state (deÖned as an allocation where real
variables are constant and nominal variables grow at a constant rate 9) provides a series of Örst
insights into its monetary structure. The Euler equation for bonds implies that their real return is
always equal to the inverse of the discount factor
1+i
= # !1 :
1+9
DeÖning the new variable consumption inequality as qt % cPt =cN
t , the self-insurance Euler equation
delivers:
P
q%
c
=
cN
1+4
5
"-
1"-
! %1
> 1:
Letting the steady-state share of unemployment beneÖts (home production) in average consumption be 4 c % 4=c, and the share of N householdsí consumption in total be h:
cN
1
h%
=
;
c
1 + n (q " 1)
we Önd (as long as it is positive) the steady-state money demand share, or inverse consumption
velocity of money:
mtot
h " 4c
=
;
c
2"/"Subject to a caveat of existence of a monetary equilibrium discussed in detail in Appendix B,
H%
steady-state money demand is equal to the share of (non-home-produced) consumption of N, divided
by a parameter capturing the degree of overall churning, the sum of the transition probabilities from
one state to another. Under the restriction - + / > 1 (that we return to below), this parameter
is between 0 and 1. For a given level of home production, this expression implicitly deÖnes upper
bounds on the degree of market incompleteness (as described by - and /) so that steady-state money
demand is positive.13 Conversely, for given - and / there exists a threshold 4 beyond which P agents
choose not to hold money: the outside option is too good and there is no need to self-insure.
!1
!1
The formal restriction is, for the case of zero steady-state ináation and treating n as a parameter: $ < 1"n '(!1 !1
<
c
q
(1!')(
!1
1
1
c
& : In terms of the original parameter ( we have 1!)
1!* >
4 + (1!*)'(1!( c ) " 2 :
13
11
2.1
Simple Monetary NK Model with Heterogeneous Agents
It is instructive to pause and compare the household side of our model with that of the seminal
HANK papers reviewed in the introduction. This helps understand how ours is a simpliÖed version
of that frameworkówhat mechanisms it still captures, and what it misses in order to gain tractability.
Take Örst our concept of liquidity, which is di§erent from Kaplan, Moll and Violante (2014)ówhere
bonds are liquid and equity and housing illiquid. In assuming that bonds and equity are illiquid while
money is liquid, we follow the deÖnition of the monetary theory that we reviewed;14 a similar notion
of liquidity is also used by the quantitative HANK model of Bayer, Luetticke, Pham-Dao, and Tjaden
(2015). Second, our constrained unit-MPC household are "wealthy hand-to-mouth", similarly to
KVMótheir wealth is just located on the P island, where they have a positive probability of going
(back). Third, unlike in KVM, our constrained households are unemployed. An earlier literature
already clariÖed the amplifying, Keynesian e§ect on monetary transmission of hand-to-mouth who
are employed and thus have endogenous income (see Bilbiie, 2004; 2008 and the discussion in the Introduction). We abstract from that well-understood general equilibrium channel to isolate and better
understand another, that we emphasize belowóendogenous movements in liquidity; this is also consistent with the uninsurable risk being related to unemployment.15 Lastly, the assumptions we used
to reduce heterogeneity and history-dependence have a close counterpart in the sticky-price
literature that is probably clear to readers well-seasoned in NK models: our participation/insurance
scheme is conceptually similar to the Calvo model of price stickiness. Whereas KVMís portfolio decision based on a quadratic transaction cost for illiquid assets is conceptually close to the Rotemberg
model of price stickinessóyet since past state variables enter this decision, complex distributional
dynamics occur that our simpliÖcation abstracts from.
Thus, our model cannot Öt the detailed distribution of asset holdings and wealth, nor reproduce
movements in portfolio shares, nor realistic idiosyncratic income processesóit does not capture the
rich household wealth dynamics of fully-áedged Bewley-Aiyagari-Huggett models; in particular, it
does not capture tails of the distributionóhouseholds who have a long stream of good (or bad)
luck. But it does captures market incompleteness through one parameter through which, as we shall
see, "history matters"óeven though for just one period. The simpliÖcations "buy" us a closedform solution, analytical determinacy conditions and, perhaps most importantly, computing optimal
policy. In that sense, we see our framework as complementary to fully-speciÖed HANK models, and
14
Tongue in cheek, one may label this a MONK model, as in "monetarist New Keynesian". See Weill (2007),
Rocheteau and Weill (2011), Kiyotaki and Moore (2012) and Cui and Sadde (2016) for recent sophisticatd reÖnements
(as well as reviews) of the concept of liquidity in recent monetary theory, including a di§erent view of liquidity based
on asset resalability (while our is on limited participation).
15
This is thus closer to Ravn and Sterk (2013), Challe et al (2015), Gornemann et al (2013), Bayer et al (2015)
and McKay, Nakamura and Steinsson (2015). Except for the last, all papers also consider endogenous unemployment
riskówhich we abstract from.
12
as an extra step in the direction of a synthesis.
3
Loglinearized Monetary HANK: Liquidity and Inequality
We are ready to analyze the monetary policy implications of our model by considering a local solution
around a steady state with zero ináation 9 = 0 (a summary of all loglinearized equilibrium conditions
around an arbitrary ináation rate is in Appendix B). Denote log-deviations of any variable by a hat,
unless speciÖed otherwise. Aggregate labor supply is proportional to participantsí labor supply
^lt = ^lP ; so the linearized labor supply equation is
t
'^lt = w^t " *^
cPt
(11)
)
+
^
c^t = (1 " (1 " n) 4 c ) lt + a
^t
(12)
c^t = np^
cPt + (1 " n) h^
cN
t
(13)
The economy resource constraint is:
with aggregate consumption (denoting p % cP =c = qh):
The Euler equation of participants and the self-insurance equation are given by, respectively
c^Pt = Et c^Pt+1 " * !1 (it " Et 9
^ t+1 )
!1
c^Pt = -#Et c^Pt+1 + (1 " -#) Et c^N
Et 9
^ t+1
t+1 + *
(14)
(15)
!
"
CB
Let x^t % mCB
=mtot be the deviation of new money issued by the central bank today,
t+1 " m
as a fraction of steady-state total money. The equation governing money growth is hence
x^t = m
^ tot
^ tot
^t
t+1 " m
t +9
(16)
The linearized budget constraint of non-participants is:
c^N
t =
" H
1 " - H ! tot
m
^t "9
^ t + x^t :
1"nh
h
(17)
New money x^t reaches N agents within the period (because money is issued through helicopter drops)
and the Pigou e§ect reduces the value of their outstanding real balances. Finally, the price-setting
equation is the loglinearized version of (8):
9
^ t = #Et 9
^ t+1 +
with
%
"!1
/
(w^t " a
^t ) :
ranging from 0 (Öxed prices) to 1 (áexible prices).
13
(18)
A local rational expectations equilibrium consists of a vector of processes c^N
^Pt ; c^t ; ^lt ; w^t ; ^{t ; 9
^ t ; x^t ;
t ;c
m
^ tot
t+1 that satisfy the equations (11) to (18). To close the model, we need to specify how monetary
policy is conducted. In this section, we focus on the positive question: what are the e§ects and transmission mechanism of monetary policy, and how does a central bank ensure equilibrium determinacy
in an incomplete-markets economy. We have two options for specifying monetary policy, as is well
understood since the celebrated analysis of Sargent and Wallace (1975) or from the more modern
textbook treatments of Woodford (2003) or Gali (2008). Under a money growth rule, money creation
x^t is exogenousówe study this in detail elsewhere. Here, we focus on interest-rate rules, whereby
money creation x^t is endogenous and equilibrium determination requires adding an extra equation
(a Taylor-type interest rate rule), the most basic form of which is:
^{t = N9 t " "t ;
(19)
where "t is an iid monetary policy "shock" capturing discretionary changes in interest rates. In the
last section, we derive optimal policy and compare it with simple rules.
3.1
Aggregate Demand, Liquidity, and Inequality
Combining (15) and (14) we obtain a core equation of our model, which captures the link between
interest rates (the price of liquidity) and inequality in consumption deÖned as q^t % c^Pt " c^N
t :
Et q^t+1 =
Et c^Pt+1
"
Et c^N
t+1
* !1
=
^{t
1 " -#
(20)
This illustrates the link between monetary policy and inequality in our model: more liquidity (lower
interest rates) leads to less inequality: as the opportunity cost of holding liquidity falls, P agents
hold more of itñleading to higher consumption for N (and lower for P) agents tomorrow: hence,
lower future inequality. This e§ect is stronger, the more intertemporal substitution there is (higher
* !1 ) and the higher is -. This equilibrium outcome of our model is consistent with the empirical
Öndings documenting a positive correlation between expansionary, ináationary monetary policy and
redistribution (less inequality); see for example Doepke and Schneider (2006), Adam and Zhu (2014),
and Coibion et al (2013).
Monetary-NK IS curve. Aggregate demand in our economy is made of demand of the two
types, participants and nonparticipants. Demand of participants is determined by an Euler equation,
but in contrast to the standard RA model (and to models with hand-to-mouth agents) this Euler
equation includes an insurance/precautionary saving motive (15). That equation thus links the
two components of aggregate demand: participantsí (employed agents) and non-participants (unemployed). Demand of non-participants is determined by the previous accumulation of money balances,
and by the money transfer received, as in (17). Ináation has an impact on both agentsí demand:
14
realized ináation reduces the real value of money balances (and hence, the income and consumption)
of N agents; while expected future ináation ináuences the insurance decision of P agents.
Combining (17), and (16) we obtain an equation linking aggregate demand of N agents to money
transfers and ináation:
(1 " n) h^
cN
= H (- " n) x^t + H (1 " -) m
^ tot
t
t+1
! tot
"
= H (1 " n) m
^ tot
^t "9
^t
t+1 " H (- " n) m
(21)
The key measure of market incompleteness in our model is:
- " n = (1 " n) (- + / " 1) > 0;
which captures the direct e§ect on demand of non-participants of an increase in liquidity xt . Indeed,
- " n captures the idea that the conditional probability to remain P is higher than the unconditional
probability of becoming P , i.e. the share of P in total population. The parameter thus measures the
incumbentsí advantage, the "memory" of the process, or the trialsí not being independent: - > 1 " /
implies that it is more likely for an employed to keep their job than for an unemployed to become
employed, which is a natural restriction. In equilibrium, - " n is hence the elasticity (integrated
across all N agents) of N agentsí consumption to a monetary transfer (for given future real money
balances). The same parameter captures also the elasticity of Nís aggregate demand to ináation, for
given real money balancesóthat is, the Pigou e§ect discussed previously.
The aggregate IS curve of our economy puts together this, (21), with the Euler equation of
participants (14) and the deÖnition of aggregate consumption (13):
c^t = Et c^t+1 " np* !1 (^{t " Et 9
^ t+1 ) + H (1 " -) Et 9
^ t+1
(22)
+H (- " n) x^t " H (1 " n) Et x^t+1
Through what we could call the monetary-New Keynesian IS curve (22), aggregate demand
depends on money (liquidity), interest, and prices (ináation)óhat tip to Patinkin (1956). There
are three main di§erences with respect to the aggregate IS curve of a standard representative-agent
economy, corresponding to these three components.
First, money (liquidity) creation a§ects aggregate demand directly, through its impact on aggregate demand of N agents discussed in detail above. This e§ect is proportional to - " n > 0; which
captures market incompleteness in our model as explained above.16
16
A particular exogenous redistribution through transwers n! P = $! is one instance of a Wallace (1981)-type
"constant Öscal policy" that will undo the e§ect of monetary injections; we study this issue in more detail in our
companion paper.
15
Second, the interest-elasticity of aggregate demand is lower than in a representative-agent economy: np* !1 < * !1 and decreasing with the share of constrained. This is the opposite with respect
to a modelósuch as Bilbiie, 204, 2008óin which nonparticipants are employed, and hence have endogenous labor income. In that model, the interest elasticity of aggregate demand is increasing with
the share of hand-to-mouth nonparticipants: in response to a cut in interest rates, demand expands,
labor demand shifts, and the wage increases; income of the constrained increases, leading to a further ampliÖcation on demand.17 We abstract from these (by now well understood) considerations
and focus on the role of money for self-insurance against unemployment risk, thus assuming that all
nonparticipants are unemployed.18
Lastly, expected ináation matters for aggregate demand over and above its e§ect through
the ex-ante real interest rate.
Higher expected ináation creates more demand today (through
H (1 " -) Et 9
^ t+1 ) by intertemporal substitution, because it diminishes the real value of liquidity
tomorrow. This expected ináation channel is "as if" N agents were at the zero lower bound perma-
nently. Combining the last two e§ects (at given nominal interest), it is clear that expected ináation
has asymmetric e§ects on aggregate demand.
Money demand and LM curve.
Since our model embeds a microfoundation for money
demand, it is instructive to analyze it in detail to draw a parallel with alternative, more familiar
models. Money (liquidity) demand is determined by (5), wherein we deÖne the marginal utility of P
' !
"
! P "(
0
agents to hold money for insurance purposes, call it vt = (1 " -) # u0 cN
= (1 + 9 t+1 )
t+1 " u ct+1
or, using (4):
vt
it
=
P
1 + it
(ct )
This implicitly deÖnes money demand in our model; it has the same familiar form as in money-inu0
utility or cash-in-advance frameworks, except that the value of money here is linked to self-insurance
and liquidity. Indeed, from the loglinear approximation of this and of (5) around zero steady-state
ináation, we get that
v^t = "* (1 " #) c^Pt + #it
17
See Bilbiie (2004, 2008) for a full analysis of a cashless model with employed nonparticipants (including the case
where at high values of the share of non-participants, the interest elasticity of aggregate demand changes sign). See
Gali, Lopez-Salido and Valles (2004, 2007) for related models with hand-to-mouth agents focusing on di§erent issues.
Eggertsson and Krugman (2012) use a similar aggregate demand structure to analyze deleveraging and liquidity traps.
Werning (2016) also emphasizes the possibility of demand ampliÖcation when the income of constrained agents is
correlated to aggregate incomeóas it is endogenously (through labor income) in the cited papers by Bilbiie and Gali
et al.
18
This is more in line with Ravn and Sterk (2013) and McKay, Nakamura and Steinsson (2015). See also Bilbiie
(2016) for further discussion of the di§erence between the two aggregate demand models and their di§erent implications
for forward guidance. See Ravn and Sterk (2016) for another, comlpementary channel that delivers ampliÖcation: the
endogenous response of vacancies and separation rates, and hence of employment.
16
where the loglinearized marginal utility of money (which is decreasing with money mt+1 , insofar as
utility is concave) can be written, using the deÖnition of inequality, as
v^t = " (1 " #) *Et c^N
^ t+1 + (1 " -) #*Et qt+1 :
t+1 " (1 " #) Et 9
There are three components to the beneÖt of money v^t . First, it decreases when Et c^N
t+1 increases,
as money is used to self-insure. Second, it decreases with expected ináation (the return on money
decreases). Third, it increases when expected inequality increases, because of the insurance motive;
with # close to 1, this last e§ect is the largest.
The LM curve of our model is Önally (using again the self-insurance equation):
,
(1 " -) H tot
H (1 " -)
!1 -#
!1
m
^
= c^t " np*
^{t " np * "
Et 9
^ t+1
(1 " n) h t+1
1 " -#
(1 " n) h
H
"H (- " n) x^t " np Et x^t+1
h
(23)
Money demand depends positively on total consumption (income e§ect), and negatively on interest
rates (the opportunity cost to hold money). The e§ect of expected ináation is ambiguous: positive
when * is high (income e§ect dominates), or negative when * is low (substitution e§ect dominates).
Lastly, current and expected money creation a§ect money demand because of redistributive e§ects:
they change the relative wealth of P and N.
Aggregate supply and Phillips curve. Aggregate supply is described by a Phillips curve
obtained by using labor supply (11), the economy resource constraint (12), and the deÖnition of
aggregate consumption (13) to eliminate the real wage in (18):
9
^ t = #Et 9
^ t+1 +
'
(
(0
'np + *) c^Pt + '
0 (1 " n) h^
cN
^t ;
t " (1 + ') a
(24)
where we use the extra notation '
0 = ' (1 " (1 " n) 4 c )!1 : This aggregate supply schedule links
ináation to the quantities demanded by both agents.19
The IS curve (22), together with the LM equation (23), the Phillips curve (24) and a speciÖcation
of monetary policy (including (16) and (19)) fully describe our aggregate model. The model is
thus reminiscent of both "old" monetarist and Keynesian models, yet it relies on a speciÖcation
with incomplete markets and limited participation and captures some key features of more recent
"HANK" models.
19
Both agents consume the output, so movements along the labor supply curve concern them both: thus, the real
wage depends on total consumption with elasticity '
% . However, since only participants work, they are the only ones
subject to the income e§ect: thus, the real wage depends only on consumption of participants with elasticity equal to
the income e§ect +. This generates an asymmetry in the ináationary e§ects of consumption of the two agents.
17
4
Understanding Monetary Transmission: Liquidity, Inequality, and Determinacy
In this section, we revisit some classic topics on the positive side of New Keynesian models, and show
how they are fundamentally transformed in an economy with incomplete markets. First, we analyze
the e§ect and transmission of interest rate changes, focusing on the endogenous response of liquidity.
Then, we look at determinacy properties of interest rate rules: from how standard prescriptions
fail to how they can be restored, and how it all depends on market incompleteness, inequality and
liquidity.
4.1
Liquidity E§ect: Sticky Prices or Incomplete Markets?
To understand the transmission of monetary policy shocks (discretionary interest rate changes),
consider the closed-form solution of our model for the case of one-time, iid shocks. We assume for
this subsection that the equilibrium is unique, i.e. that N in (19) is "high enough" in a sense that we
make precise subsequently.
The equilibrium responses are intuitive. First, the nominal interest rate does not move: since
the shock is iid and there is no income e§ect on P agents, their consumption and nominal interest
d^
cP
d^{t
stay unchanged d"tt = 0; d"
= 0. It follows that ináation responds proportionally
t
d^
4t
d"t
= N!1 . This
ináation hurts the liquidity constrained since it reduces the value of their money balances via the
Pigou e§ect. Despite this, the equilibrium e§ect is positive:
d^
cN
t
d"t
= [(1 " n) hN '
0 ]!1 .20 This holds
because equilibrium real balances increase: the endogenous transfer that governs the demand for
'
(
xt
liquidity is H d^
= ( '
0 )!1 + H (1 " -) (N (1 " n))!1 .
d"t
Demand for liquidity increases through two channels: on the one hand, sticky prices and elastic
labor ( '
0 )!1 ; on the other, incomplete markets H (1 " -). The stickier are prices, the larger the
demand e§ect and thus the equilibrium expansion in income of P agents. The more elastic is labor, the
larger the response of hours worked by P agents, and the higher their income. Both of these features
imply a higher income e§ect on "precautionary saving", aka self-insurance demand for liquidity. The
second channel is immediate: the higher the risk of moving to the N state, the higher the self-insurance
motive. These e§ects combine to deliver the "liquidity e§ect" (used here to denote the increase in
real money balances following an interest rate cut) emphasized in the following Proposition.
Proposition 1 Liquidity E§ect. The equilibrium e§ect of an interest rate cut on money demand
20
Aggregate demand increases in this simple case only because of an "indirect e§ect" (in the language of Kaplan
et al, 2015) that is due to the endogenous liquidity expansion: the "direct e§ect" is nil because interest rates do not
move.
18
is
dm
^ tot
( '
0 )!1 " H (- " n)
t+1
H
=
:
d"t
N (1 " n)
A liquidity e§ect (
dm
^ tot
t+1
d"t
> 0) emerges when prices are sticky enough and labor is elastic enough.21
The second term (real balances decreasing proportionally to (- " n)) captures the negative substi-
tution e§ect on precautionary saving: since there is ináation, the return on holding liquidity for
self-insurance goes down, and so does its demand. A corollary is that with áexible prices (or inelastic
labor), money demand decreases proportionally to market incompleteness: there is no income e§ect
on precautionary saving, only the negative substitution e§ect. This is an artefact of helicopter drop
money creation, and changes when implementing monetary policy through open market operations
(see Alvarez and Lippi, 2014, for the áexible-price case and our companion paper Bilbiie and Ragot,
2016 for the case of sticky prices).
4.2
How (Badly) the Taylor Principle Fails with Incomplete Markets
Our model has striking implications for the determinacy properties of interest rate rules: it is no
longer su¢cient that the policy rule be "active", in Leeperís (1991) terminology.22 In particular,
market incompleteness requires much larger response of interest rates to ináation than the "Taylor
principle" in order to ensure equilibrium determinacy. For reasonable degrees of market incompleteness and with Taylor rule parameters in the range of those estimated by empirical studies, the
equilibrium is indeterminate.
How does a self-fulÖlling, sunspot ináationary equilibrium occur with a standard Taylor rule with
incomplete markets.23 Suppose there is a sunspot shock to ináationary expectations, and the Taylor
rule simply neutralizes ináation (thus delivering ^{t " Et 9
^ t+1 = 0 in (22))ówhich, in a representativeagent model, ensures determinacy. In our model, it does not: aggregate demand increases (hence,
through the Phillips curve, ináation becomes self-fulÖlling) and the reason is two-fold. First, demand
increases because a sunspot to expected ináation tells P agents to hold less liquidity for self-insurance.
21
The liquidity e§ect is most often referred to as the reverse experiment: the fall in interest rates triggered by an
increase in exogenous money supply. We study that experiment in the companion paper, which analyzes this economy
under money supply rules. The result that sticky-price models deliver a liquidity e§ect is emphasized by Christiano,
Eichenbaum, and Evans (2005). The same authors compared sticky-price and limited-participation modelsí ability to
deliver a liquidity e§ect in previous work, see Christiano (1991), Christiano and Eichenbaum (1992, 1995); see also
Fuerst (1992).
22
Our model has a unique local equilibrium under money growth rules, just like old Keynesian models studied by
Sargent and Wallace and New Keynesian models with money in the utility function studied by Woodford (2003) or
Gali (2008).
23
A Taylor rule that responds to expected ináation would lead to an uninteresting type of indeterminacy: equation
(20) would contain only expectations and no actual variable, thus implying automatically sunspot equilibria.
19
More importantly, endogenous liquidity xt increases: there is an income e§ect on precautionary
saving/demand for liquidity that dominates the substitution e§ect described earlier. This boosts
the consumption of N agents today. This second mechanism survives even under Öxed prices, as we
show below. Thus, indeterminacy in our model is of a fundamentally di§erent nature than the one
of the standard, representative-agent NK modelñeven though the two mechanisms complement each
other.24
Our novel mechanism is serious at even moderate level of market incompleteness. The determinacy
conditions are emphasized in the following Proposition, proved in Appendix B.3:
Proposition 2 The ModiÖed Taylor Principle with Incomplete Markets. The model has a
locally unique RE equilibrium if and only if the Taylor coe¢cient satisÖes
N > max (1; N% ) ,
when N% %
2(1+5))
'!1 $ !1 +1+)
'!1 $
(1!n)h
!1
2 1!)* !(1+)
' $)
> 0: Otherwise, when N% < 0 there exists no N that delivers a unique
RE equilibrium.
The requirement for determinacy is stronger than the Taylor Principle N% > 1 as long as labor is
elastic enough.25 The threshold is also increasing with price stickiness (
!1
), and it is larger than
1 even in the extreme case of áexible prices, insofar as labor is elastic enough (namely, '
0 !1 * >
(-# " np) = (1 " -#)). In the other limit with Öxed prices, the threshold tends to inÖnity and no
Taylor rule delivers determinacy; we discuss this in detail below, as it is key for understanding the
novel indeterminacy channel occurring under incomplete markets.
The more striking implication of our model is the dependence of the threshold response N% upon
incomplete-market parameters - and /: for reasonable values of market incompleteness, the threshold
becomes very large. For illustration, Figure 1 plots the threshold as a function of the probability
of staying participant-employed -, for the domain - > n; for two di§erent values of the share of
participants n : n = 0:5 (solid blue line) and n = 0:8 with dashed red line. The parameter values
(used in the more rigorous calibration provided in Table 1) are standard.26
24
It is also diferent from the indeterminacy occurring in models with employed "hand-to-mouth" agents such as Gali
et al (2004) and Bilbiie (2004, 2008), both of whom rely upon endogenous labor by hand-to-mouth.
25
The necessary for determinacy to obtain at all (," > 0, i.e. + '
% !1 < 2 (1!n)h
1!)' " 1) is satisÖed for reasonable
parameter values. Consider the case of inelastic labor, no discounting & = 1 and log utility, it can be easily seen
that the condition is always satisÖed since it implies $ " n > 0 > n " 1. By continuity, the condition is satisÖed also
with discounting and elastic labor. The condition is in fact somewhat moot: in a version of the model where labor is
demand-determined (with ct in the Phillips curve), it boils down to 2 (1 " n) h > 1 " $& which is satisÖed for resonable
discounting.
26
Unlike in Table 1, here we use ' = 1; under a Taylor rule, higher elasticity will preclude determinacy but somewhat
articiliallyósee previous footnote.
20
150
φ∗
100
50
0
0.5
0.6
0.7
0.8
0.9
a
Figure 1: Threshold N% as a function of - for given n
(0:5 blue solid, 0:8 red dashed).
The vertical scale is correct: under this otherwise standard calibration, the Taylor coe¢cients
necessary for determinacy are in double-digit (if not triple-digit) territory. It is clear that the Taylor principle is very far from being su¢cient for ensuring determinacy in our incomplete-markets
economy, even at moderate levels of market incompleteness (for instance for low n = 0:5 and high
- = 0:9; the threshold is still around 5). Whereas for high n; the determinacy region starts squeezing
and the threshold N becomes very large. The threshold is increasing in n; at given -; and decreasing
in -; at given n: In other words, it is a decreasing function of - " n; the parameter capturing market
incompleteness and governing the e§ect of liquidity on aggregate demand.27 The lower this para-
meter (the closer conditional and unconditional probability), the larger the endogenous movements
in liquidity necessary to ináuence aggregate demand, and the higher the response needed to obtain
determinacy.
Liquidity, Inequality, and Indeterminacy with Fixed Prices
To understand and emphasize the novel channel causing indeterminacy under incomplete markets, it
is useful to isolate it from the standard indeterminacy due equilibrium ináation variations that is at
the heart of representative-agent NK models. We just saw that the two interact in nontrivial ways,
but the mechanism emphasized here is more general than our model and worth isolating.
27
Take log case for simplicity, replace h; Expression
1!n
1!)'
)
!1
!1
1 + n '1!)
+!1
is decreasing in n and increasing in $.
The former is immediate and holds generallyÑ the latter by continuity starting from & = 1 and holds for & close to 1.
It follows that the threshold is increasing with n and decreasing with $: Evidently, the threshold ," is higher than 1
!
"
if $ " n is high enough, namely (1!n)h
% !1 + (1 + &) !1 + 1 :
1!)' < 1 + '
21
To do so, consider for now the case of Öxed prices, i.e. horizontal aggregate supply:
= 0: An
interest-rate peg leads to indeterminacy even in this case, and this indeterminacy is two-fold in a
sense that will become clear below. Before delving into this, recall the implications of an interest rate
peg in standard RA models. In the cashless model analyzed in Woodford (2003), with Öxed prices
nothing happens: indeterminacy can only occur through self-fulÖlling ináation. In models with cash
(either through CIA or MIU), there is one source of indeterminacy that is well-understood since the
seminal paper of Sargent and Wallace: the level of liquidity today xt (which, with Öxed prices, fully
determines aggregate demand and income today) is indeterminate; the same holds true also in more
modern treatments, see e.g. Gali (2008). Things are very di§erent in our model.
The following two equations in inequality and liquidity fully characterize our model under Öxed
prices:28
Et qt+1 =
* !1 %
i
1 " -# t
H
H
qt = -#Et qt+1 " (- + / " 1) x^t + Et x^t+1
h
h
It is immediate to see that the eigenvalues of this system are 0 and - + / " 1; in particular, they
are both lower than 1: the indeterminacy occurring under an interest rate peg i%t = 0 is thus two-
fold. First, inequality today qt is arbitrary since the Örst equation only determines Et qt+1 . Call this
sunspot qtS ; bearing in mind that the allocation of consumption between the two agents today is not
pinned down. Worse still: given this sunspot, consider the second equation governing demand for
liquidity:
Et x^t+1 = (- + / " 1) x^t +
h S
q
H t
Its root - + / " 1 is stable (- + / < 2), but it should not: liquidity x is a jump variable and its
demand determined in forward-looking fashion. The result is an inÖnity of paths of liquidity that
can support any consumption allocation: consumption of both agents today can be anything.
The foregoing analysis illustrates that endogenous movements in liquidity, demanded for selfinsurance purposes, can lead to equilibrium indeterminacy even when prices are Öxed. This new
mechanism compounds with the standard NK indeterminacy to seriously challenge conventional
policy prescriptions at even moderate levels of market incompleteness. Next we ask: what can the
central bank do about it.29
28
The second equation is obtained by rewriting the self-insurance Euler equation using the N agentí budget constraint
and the equilibrium under Öxed prices 0 t = 0; and replacing the deÖnition of inequality.
29
Evidently, in the model with horizontal aggregate supply the central bank can restore determinacy by adopting
an interest rate rule that responds to the level of liquidity; we treat this in the more general case below.
22
4.3
Inequality, Liquidity, and the Taylor Principle
The image depicted in the previous section is pessimistic: a central bank operating in an economy
with incomplete markets loses any control of the economy because of this powerful ampliÖcation of
endogenous liquidity demand, generated by heterogeneity and the need to self-insure. Let us thus
turn to the optimistic side: the central bank can break this vicious spiral by responding to either
its consequence (expected inequality) or directly its source (liquidity)óthus exploiting the close link
between liquidity and inequality captured by (20).
Consider a rule responding to (future) inequality:
it = N4 9 t + Nq Et q^t+1 :
(25)
Take the same thought experiment as before, with a sunspot to ináation Et 9 t+1 and a policy rule
that neutralizes its e§ect on the ex ante real interest rate. The key element of our reasoning is the
endogenous reaction of liquidity x when the central bank responds to future inequality: liquidity
goes down! That is because a sunspot shock to ináation Et 9 t+1 increases inequality, as P hold less
money for tomorrow. This triggers higher interest rates, which reduce money demand todayóthus
invalidating the sunspot. A virtuous spiral is put in place that can be illustrated by replacing (20)
in our inequality-augmented Taylor rule:
it =
N4
9t
1 " Nq [* (1 " -#)]!1
Responding to expected inequality by Nq magniÖes implicitly the response to ináation hyperbolically
(meant in a literal sense):
The full necessary and su¢cient determinacy conditions for (25) are provided in Proposition 7 in
Appendix B.3. Here, we focus on a slightly di§erent question: what response to inequality restores
the Taylor Principle as a su¢cient condition for ensuring a unique equilibrium. In other words, if we
were to tell the central bank to adopt an inequality objective in order to ensure determinacy without
having to revisit its standard ináation objective, how should that be designed?
Corollary 3 If the central bank follows the rule (25), the Taylor Principle N4 > 1 is a su¢cient
requirement for determinacy (in the case N% > 0) if the inequality response satisÖes
)
+
!
"
!1
% !1
~
* (1 " -#) 1 " (N )
< Nq < * (1 " -#) 1 + N
~ > 0 is given in the Appendix.
where N
The response to inequality should be "close to" (1 " -#) * in a sense made speciÖc by the Corol-
lary. Since 1 " -# is in fact (1 " -) #q $ , this implies that the central bank, in designing its inequality
response, needs to take into account unemployment risk, and long-run inequality. The determinacy
23
result provided here has a close connection to welfare: as we shall see next, optimal policy in our
model seeks to reduce inequality, and a rule of the form (25) is also a reasonable approximation to
optimal policy.
An interest-rate rule with liquidity feedback
Endogenous changes in liquidity are key to sustain self-fulÖlling áuctuationsóso perhaps determinacy
can be restored by responding directly to them:
it = Nx xt :
(26)
Adopting this rule with Nx > 0 is akin to specifying a supply schedule for liquidity. Since xt stands
also for the growth rate of nominal money (insofar as we loglinearize around a steady-state where this
growth rate is zero), this is also related to a money growth rule with a long tradition in monetary
theory. In particular, Canzoneri, Henderson and Rogo§ (1983) and McCallum (1986) studied a
money supply rule whereby the central bank responds to current deviations of nominal interest from
its target.30 Our rule is the reverse of theirs, in that it treats nominal interest as the instrument
and the money growth rate as the indicator variable. Its determinacy properties are sharpóthey are
akin to the Taylor principle (and summarized in the following Proposition).
Proposition 4 A Taylor Principle for liquidity-based interest rate rules. The model has
a locally unique RE equilibrium if and only if nominal interest rates increase more than one-to-one
with liquidity
Nx > 1.
The proof for the case of áexible prices is presented in Appendix B.3. We already covered the
intuition above: increasing nominal interest when liquidity increases breaks the vicious spiral that
leads to self-fulÖlling áuctuations head-on. Yet the same property that is a virtue for restoring
determinacy becomes a curse from a welfare stand-point: (26) represses movements in liquidity
precisely when they are necessary. Shocks that increase inequality also increase liquidity demand
for insurance purposes. A central bank would optimally accommodate this demand by liquidity
injections, but rule (26) prevents it from doing so. This brings us to the last topic of our study,
optimal monetary policy in this economy.
5
Optimal Monetary Policy: Inequality and Ináation
Optimal monetary policy needs to Önd the balance between two main distortions in this economy.
The Örst is inequality: a scope for providing insurance, speciÖc to an incomplete-markets, limited30
This rule ensures determinacy in a Sargent-Wallace-type model, and saves wage setters the cost of indexing to
nominal interest in Canzoneri, Henderson and Rogo§ís model.
24
participation setup like ours. The second is costly price adjustment: the standard distortion that
operates in a representative-agent NK model. This section analyzes how this trade-o§ is resolved
in our model. We solve the full Ramsey-optimal policy, provide a second-order approximation ‡
la Woodford (2003) that is useful to understand the policy trade-o§s, and analyze optimal policy
quantitatively. The general theme is that inequality implies large optimal deviations from price
stability.
5.1
Ramsey-Optimal Policy
Following a long tradition started by Lucas and Stokey (1983), we assume that the central bank
acts as a Ramsey planner who maximizes aggregate welfare. In our economy, this entails calculating
welfare of the two agents and weighting them by their shares in population. The constraints of
the planner are the rearranged private equilibrium conditions: self-insurance (5), Phillips curve
(8), and economy resource constraint (10).31 We denote the system of these three constraints by
!
"
P
6t cPt ; cN
t ; lt ; 9 t : As it is by now well understood, the optimal policy problem of the central bank
P
can be written as choosing the allocation fcPt ; cN
t ; lt ; 9 t g to maximize the following Lagrangian:
max
N P
fcP
t ;ct ;lt ;4 t g
E0
1
X
#t
t=0
( "
!
"
P
n u ct
)
! P "1+' #
%
&
! N"
lt
4 1+'
"'
+ (1 " n) u ct " '
+ ! t 6t
1+'
1+'
(27)
where ! t is the vector of three costate, Lagrange multipliers, one for each constraint in 6t . The
Örst-order conditions of this problem are outlined in Appendix C.2, as is the proof of the following
Proposition.
Proposition 5 The optimal long-run ináation rate is such that # " 1 ) 9 % ) 0
As in other NK models incorporating di§erent theories of money demand (e.g. Khan et al, 2003;
Schmitt-Grohe and Uribe, 2004, 2007) the long-run ináation rate ranges from the Friedman rule
under áexible prices and optimal subsidy, to zero ináation under sticky prices and inelastic labor. A
low ináation rate allows agents to self-insure, but generates price adjustment costs. An ináation rate
close to 0 minimizes price adjustment costs, but decrease the ability of agents to self-insure, as the
return on money decreases.
But unlike in other NK models, including those incorporating money demand, the central bank
also uses ináation optimally over the cycle in our economyóto provide insurance and decrease
31
Since the nominal interest rate only enters the Euler equation for bonds, the problem can be regarded as one where
the planner chooses the allocation directly; once consumption of participants and ináation are known, the optimal
interest rate is determined by the Euler equation. By similar reasoning, once consumption of non-participants is
determined, along with ináation, the quantity of real money balances is fully determined too. These simpliÖcations
only apply when money is issued via helicopter drops; our companion paper compares this with open market operations
implementation; see Appendix C.2.
25
inequality. We Örst illustrate formally the trade-o§ faced by a central bank by deriving a secondorder approximation to the aggregate utility function, which contains an "inequality" motive, and
then explore the quantitative signiÖcance of this novel trade-o§.
5.2
A second-order approximation to welfare
To understand the relevant policy trade-o§s, we derive a second-order approximation ‡ la Woodford
(2003, Ch. 6) to the aggregate welfare function, around a steady-state with imperfect insurance
(p > 1 > h), an optimal subsidy inducing marginal-cost pricing in steady state (. = 0 in (8)), and
arbitrary steady-state ináation.32 In the Appendix C.3 we prove the following.
Proposition 6 Solving the welfare maximization problem is equivalent to solving
4
%
&5
1
1 X t
1 " * ! N "2
2
2
2
N
min
E0
# 94 9
~ t + 9c c^t + 9q q^t " 9 c^t +
c^t
N
2
fcP
t ;ct ;qt ;4 t g 2
t=0
where the optimal relative weights are
94 =
D
1+'
/ 2 ; 9c = * " 1 +
1 " 29
1 " (1 " n) 4 c
9 = 2 (1 " n) h (q $ " 1) ; 9q = (* " 1) np (1 " n) h:
The Proposition transparently illustrates the novel trade-o§ implied by our framework between
inequality and stabilization (of ináation and aggregate demand).33 The last two terms pertain to
inequality. The last one represents the Örst-order welfare beneÖt of increasing N agentsí welfare
by increasing their consumption levelóintuitively, its weight 9 is proportional to the steady-state
inequality distortion captured by (q $ " 1) ; since the Örst-order beneÖt exists only insofar as the
steady state is distorted to start with. This is analogous to the linear beneÖt of increasing output
above the natural rate when the steady-state is Örst-order distorted in the standard New Keynesian
model, see the next footnote. The distortion vanishes when the steady-state is egalitarian (p = h,
perhaps through a steady-state insurance scheme, if enough Öscal lump-sum instruments are available
to undertake such policy) or, trivially, when n = 1 (the standard cashless representative-agent NK
model). Replacing the equilibrium q reveals that 9 is proportional to # !1 " 1 + 9: the distortion
also becomes arbitrarily small when the steady state tends toward the Friedman rule, 9 ! 1 " # !1 .
32
Bilbiie (2008) also derives a quadratic loss function in a cashless economy with hand-to-mouth agents. Curdia and
Woodford (2009) and Nistico (2015) also do this for cashless models with infrequent access to credit markets; unlike
us, they focus on an e¢cient equilibrium with insurance when calculating optimal policy.
33
Note that 0
~ t = 0 t + 0 is ináation level, so the function is written so that target ináation is zero absent aggregate
shocks, 0
~ t = 0. The optimal target is the optimal long-run ináation found in the Ramsey problem above, the equivalent
of which is here the steady state of the solution of the relevant linear-quadratic problem.
26
The other new term is 9q q^t2 , which captures the welfare cost of the volatility of inequality (naturally,
this drops out if all agents are identical, n = 0 or n = 1).
Because of the linear term in the loss function, second-order terms in the private constraints
matter for welfare34 : as long as there is steady-state inequality, ináation and aggregate demand
volatility will matter for welfare beyond their direct e§ects through 94 and 9c . The reason is by
now intuitively clear: when the steady-state has q > 1; increasing consumption of non-participants
provides a Örst-order welfare beneÖt; the only way to achieve this beneÖt, absent Öscal instruments,
is monetary. As we show next, a quantitative analysis of optimal policy in a calibrated version of
our model suggests that ináation volatility is desirable in this framework. Pursuing price stability
instead, even around an optimally chosen ináation target, has large welfare costs.
5.3
Optimal deviations from price stability and inequality: a quantitative evaluation
To analyze optimal policy in our model we calibrate the model at quarterly frequency. We follow,
for common parameters pertaining to preferences and the supply side, the classic papers in optimal
policy in NK models, Khan, King and Wollman (2003) and Schmitt-GrohÈ and Uribe (2007): the
inverse elasticity of labor supply is ' = :25, and * = 1. The elasticity of substitution between goods
is " = 6 and we introduce the steady-state subsidy E = 1=(""1) to avoid steady-state distortions due
to monopolistic competitionóthus isolating our novel channel as a motivation for deviations from
price stability. Both cited papers use di§erent models of staggered pricing, and assume that prices
stays unchanged on average for 5 periods; this implies a Phillips curve slope (our ) of around 0:05.
Given our ", the price adjustment cost parameter that delivers the same
is D = 100. The discount
factor is # = 0:98, as in other studies with heterogeneous agents (Eggertsson and Krugman, 2012;
Curdia and Woodford, 2009); we consider larger values for robustness below. We use the same labor
productivity process as Khan et al, with autocorrelation 0:95 and standard deviation of 1%.
Three parameters pertain to market incompleteness and money demand: the probabilities to
keep participating (-) and non-participating (/), and home production when non-participating (or
unemployment beneÖts) 4. Since we perfectly correlated Önancial market and labor market participation to obtain our tractable model, two calibrations are possible: one that targets Önancial market
participation and money demand, and the other labor market variables. We use the former as a
benchmark and report the latter for robustness.
34
This is analogous to the linear beneÖt of increasing output above the natural rate when the steady-state is
Örst-order distorted in the standard New Keynesian model. See Woodford (2003; Ch. 6), Benigno and Woodford
(2005, 2012) and Schmitt-Grohe and Uribe (2007) for an analysis of this when the distortion pertains to monopolistic
distortion, i.e. + > 0; including explanations of the second-order corrections that are necessary to correctly evaluate
! N "2
welfare. The quadratic term 1!,
c^t
represents merely a second-order correction.
2
27
We target three data features in our benchmark calibration. First, the number of participants n:
in the US economy roughly half of the population participates in Önancial markets, either directly or
indirectly (Bricker et al. 2014), and this is stable over time. We thus take n = :5, which implies the
restriction - = /. Second, the velocity of money (roughly speaking, H!1 in our notation): considering
a broad money aggregate, the quarterly velocity (GDP=M 2) is around 2 over the period 1982 " 2007
(chosen to avoid the zero lower bound period). Third, consumption inequality q between participating
and non-participating agents captures the lack of insurance due to market incompleteness. Since
agents participate infrequently in Önancial markets (Vissing-Jorgensen, 2002) and one cannot keep
track of their participation status, it is hard to Önd an exact empirical counterpart to q. We take as
a proxy the fall of nondurable consumption when becoming unemployed, which is estimated between
10% and 20% (see e.g. Chodorow-Reich and Karabarbounis 2014), and target the conservative value
of 10% for this object (q !1 " 1 in our model). These three targets jointly imply - = / = 0:9, and
4 = 0:783. Table 1 presents our parameters and the implied Ramsey steady-state values for our
target variablesówhich are determined by the exact Ramsey equilibrium conditions outlined in the
Appendix.
Optimal long-run deviations from price stability
The optimal asymptotic (steady-state) ináation rate is 9 = ":79%. As expected, this is higher than
the ináation implies by the Friedman Rule (which is "2%), because prices are sticky, just like in
standard monetary models with sticky prices, e.g. Khan et al (2003). More equilibrium deáation
occurs if prices are more áexible, labor is more elastic, and - is higher. The Örst two elements are
standard (the former was Örst noticed by Chari Christiano Kehoe, 1997; see also Schmitt-Grohe and
Uribe, 2004). The last part has a standard interpretation too: at given n; higher - implies more
elastic money demand. As we will show below, less elastic money demand (lower - " n), implies less
optimal deáationóas in Khan, King and Wollman, although for a di§erent theory of money demand.
Preferences
Production and price setting
Heterogeneity
#
*
'
"
D
/a
Ea
-
/
4
:98
1
:25
6
100
:95
:01
:9
:9
:78
(cN "cP )=cP
9
cp
cN
l
mtot
q
":79%
:98
:87
1:07
:46
1:12
Model outcome
GDP/M2
n
2
50%
"11%
Table 1: Baseline calibration
28
What is the welfare cost of ináation? This is a classic question in monetary economics, going back
at least to Baileyís 1956 calculation.35 The welfare cost of (steady-state) ináation in our framework
can be easily calculated, in the Lucas (1987) traditionówe review this brieáy in the Appendix. We
Önd that moving from a steady-state annualized ináation rate of 2% (:5% quarterly) to the optimal
rate of "3:2% (":79% quarterly) is equivalent to a permanent increase in consumption of :61%óthis
is in line with numbers found in the literature, e.g. Lucas (2000) and Imorohoglu (1992), although
slightly larger.
Our modelís implications for optimal policy in the long run are thus rather standard. But in the
short run, things are di§erent and this is intimately related to inequality in our model: with long-run
inequality (q = 1:12), the steady state is distorted and this has implications for short-run optimal
policy.
Optimal short-run deviations from price stability
The inequality channel requires the central bank to accommodate some ináation volatility, as doing
otherwise leads to large welfare losses. This is true in our economy even when the source of business
cycles is a shock that, in standard NK model with money demand but no inequality, generates no
such trade-o§: a plain vanilla labor productivity shock.
Recall what happens in the baseline NK model in response to this shock: not much. A welfaremaximizing central bank keeps prices unchanged and ináation at zero, as this shock creates no
trade-o§: the central bank can close the output gap costlessly, a well-known result labeled "divine
coincidence" by Blanchard and Gali (2007). This result changes but only slightly when the steady
state is distorted (. > 0 in our notation), as analyzed in detail by Benigno and Woodford (2005):
productivity shocks then have a "cost-push" dimension, creating a trade-o§. Quantitatively, however,
this is mootósubject to one caveat mention in the next footnote. The same is true in models
incorporating a variety of other frictionsóin particular, in models with monetary frictions such as
Khan, King and Wollman (2003) and Schmitt-Grohe and Uribe (2004, 2007): price stability is a
robust policy prescription. Even though these models do imply ináation volatility under the optimal
Ramsey policy, the welfare cost of eliminating such volatility is generically negligible.36
35
A large literature analyzed this question usig a variety of frameworks. To cite juste some prominent examples,
Lucas (2000) found that reducing ináation from 10 to 0 percent annually results in a 1 percent increase in consumption.
Analyzing a monetary framework closer in spirit to the one our model embeds (based on the Bewley model), Imorohoglu
(1992) showed that welfare e§ects of ináation are larger in incomplete-markets economies. See Doepke and Schneider
(2006), Erosa and Ventura (2002) and Ragot (2014) for reviews.
36
See for instance Table 2 in Schmitt-GrohÈ and Uribe (2007); See also Bilbiie, Fujiwara and Ghironi (2014) for
a result on optimal short-run price stability in a model with entry and variety, and a review of the literature using
other distortions. As Benigno nd Woodford (2005) show analytically, this result changes ñ price stability ceases to be
optimalóif, on top of + > 0; the share of government spending in steady-state output is also non-zero.
29
This is no longer the case in our model: optimal Ramsey policy requires volatile ináation, and this
volatility matters for welfare. To see the Örst part of this argument, consider the impulse responses
to a productivity shock presented in Figure 2, for three economies. With black solid line, we have
our economy under optimal policyóobtained by solving (27). With blue dashed line, we have our
monetary economy under what we label "Strict ináation targeting" (SIT): the central bank perfectly
stabilizes ináation around the Ramsey-optimal steady state ináation (this is implemented by a Taylor
rule with large N4 and the optimal 9 % target). Finally, we show with red circle line optimal policy
in a standard cashless equilibrium37 , a comparison with which illustrates the extent of risk-sharing
provided by money in our model. All variables are in percentage deviation from steady-state, except
the ináation and interest rates, which are in deviation from steady state.
Figure 2: Responses to a labor productivity shock under optimal Ramsey policy in our model (solid black),
strict ináation targeting in our model (blue dash), and optimal policy in cashless model (red circles).
The responses of the cashless model are standard: ináation does not move, and output is equal
to its natural rate. Since labor productivity a§ects only P agents, their consumption increases (and
37
Since in the non-monetary equilibrium the steady-state ináation rate is 0, we recalibrated it to have the same
steady state allocation. In particular, we reduce output by -2 0 2 and introduce a transfer between N and P households,
such that the steady-state consumption and labor supply are the same in the monetary and non-monetary equilibrium,
and only the steady-state ináation is di§erent.
30
so does inequality) and the nominal interest rate goes down.
In our monetary economy, the planner reduces inequality by providing insurance: compared to
the red circle line, the black solid line shows that consumption of N agents increases, and inequality
decreases. The planner issues money and interest rates fall; the result is ináation (due to the demand
e§ect on Örms), which erodes N agentsís purchasing power (money balances) via the Pigou e§ect.
Consider now the allocation when this ináation is absent (blue dashed line): more money is
issued, and the real value of balances is much higher: thus, consumption of N agents responds by
more, and is more volatile. Since consumption of P agents is largely unchanged, the same is true for
inequality. We will now show that this extra volatility is costly in terms of aggregate welfare.38
Table 3 reports the standard deviations of the main variables for the (Ramsey-)optimal and
SIT allocation. The volatility of ináation is comparable to that obtained by Khan et al (2003).
Because of limited risk-sharing, N agentsís consumption volatility is higher than P agentsí. More
importantly, N agentsís consumption volatility is higher under strict ináation targeting than under
optimal policyñas a result, the volatility of our inequality measure is twice as large. This di§erence
in volatilities translates into a large welfare cost of price stability (around the optimal asymptotic
ináation rate): agents need to be compensated by :08% of consumption every period in order to live
in an economy with stable prices, rather than in one with optimal policy and ináation volatility.39
This welfare cost is high, much higher than those usually found in the literature reviewed above,
even though our optimal ináation volatility is of comparable size; it is of the same magnitude as the
total welfare cost of uncertainty in incomplete-markets models (see Krusell and Smith, 1998, and
Lucas, 2003). The reason is by now clear: our long-run equilibrium is one with inequality (imperfect
insurance), which for a planner is a distortion. In standard NK models with distorted steady state,
this is not enough to generate signiÖcant costs of price stability. Here, it is, because volatility has Örstorder e§ect through the level of N agentsí consumption. In terms of our second-order approximation,
this e§ect makes it "as if" the weight on ináation volatility in true Ramsey loss function were smaller
38
It is by now well known, starting with the ináuential paper of King and Wolman (1999), that welfare calculations
depend crucially upon the initial values of the Lagrange multipliersóthat can be set to 0, or to their Ramsey steadystate values.Under the former choice, policy is not timeless-optimal: initial period t0 ináation has no consequence
for prior expectations, thus policy chosen in any later period is not a continuation of t0 policy. In the second case,
policy is timeless-optimal in King and Wolman (1999) and Khan et al.(2003)ís sense (Woodford, 2003, uses a di§erent
deÖnition). The numers we report are for the former, t0 -optimal case; in the timeless-optimal case, the welfare losses
are very close to zero in all cases; se also Bilbiie, Fujiwara and Ghironi (2014) for further discussion in a di§erent
context.
39
The welfare losses are similar for the two types of agents: thus, our simpliÖed heterogeneity misses some of the
distributional e§ects emphasized by Krusell and Smith (1998)óin their framework and the subsequent literature (see
Lucas 2003 for a review) the welfare beneÖts of eliminating uncertainty are asymmetric among the poor, the rich, and
the middle class. One would expect that such an asymmetry ocurs in a richer model of the wealth distribution also
for the welfare costs that we calculate.
31
than 94 :
Standard deviation (%)
Welfare (%)
c^
N
c^
q^
9
?W
Ramsey
2:6
3:3
:6
:05
SIT
2:6
3:7
1:2
0
"
Ramsey
2:5
3:2
:6
:05
SIT
2:5
3:8
1:3
0
Economies
P
Baseline
No SS inequality (# ! 1)
:08
"
:00
Table 2: Standard deviations and welfare losses (percent)
That small ináation volatility translates into high welfare gains in our model is not so much due
to volatility itself, as it is due to inequality. To illustrate this, consider an economy where the steadystate inequality distortion vanishes, q ! 1, which amounts to taking # ! 1 and re-calibrating 4 to
get the same steady state money velocity; evidently, the optimal long-run ináation rate converges to
0. As the bottom panel of Table 2 illustrates, the volatility of ináation under Ramsey policy in this
economy is unchanged. Nevertheless, this volatility no longer means welfare: without an inequality
trade-o§, the central bank can safely and costlessly pursue price stability (just as in Khan et al, 2003
and Schmitt- Grohe and Uribe, 2007).
How do simple interest rate rules perform in this framework? To answer this, we optimized over
simple rules of the form studied in the previous section, following the method outlined by SchmittGrohe and Uribe (2007). An inequality-augmented rule with Nq = :1 reproduces the same allocation
as strict ináation targeting, but with a much smaller ináation response (N4 = 3), which is desirable
for determinacy.40
Robustness
As a Örst robustness check we report the same outcomes for economies with more áexible prices
(D = 50) and less elastic labor (' = 1).41 The upper panel of Table 3 contains the results.
40
We also optimized over liquidity rules and found that they generate much lower welfareóthe intuition being that
they preclude liquidity provision, which is what is needed to implement Ramsey policy.
41
For each economy, in order to perform meaningful welfare comparisons we calibrate the discount factor & and
home production 5, to start from the same steady state: this gives :973 and :79 for the Örst and :982 and :765 for the
second calibration (results are similar when we keep these parameters unchanged).
32
Standard deviation (%)
SS
P
c^
c^
q^
9
?W
"
Economies
9
D = 50
"1:54
Ramsey
2:7
3:2
:5
:08
SIT
2:6
3:8
1:2
0
":6
Ramsey
2:1
2:7
:6
:04
SIT
2:0
3:0
:9
0
":36
Ramsey
3:0
5:1
2:1
:02
SIT
3:1
5:5
2:4
0
'=1
Labor market calibration
(- = :95; / = :5; 4 = :5)
(%)
Welfare (%)
N
:13
"
:03
"
:06
Table 3: Robustness analysis
Both the ináation volatility its welfare beneÖt increase as prices become more áexible and labor
supply more elastic. The reason is that with more áexible prices (lower D), the cost of using ináation
is lower: in the limit, as prices become áexible, ináation essentially becomes a lump-sum taxñ an
insight originally due to Chari, Christiano, and Kehoe (1997) and also discussed by Schmitt-Grohe
and Uribe (2004).
The second alternative calibration we consider is based on labor market risk. Instead of matching
Önancial market variables (n; H; and q) as in our previous calibration, we draw on the labor market
literature, in particular Shimer (2005), to Önd parameter values for -; / and 4. At quarterly frequency,
the job loss probability is 5% and the average job Önding probability 50% for the post-war periodó
these two numbers imply - = :95; / = :5 and thus n = :94; the gross replacement ratio is set to
4=w = 50% (see also Challe and Ragot, 2014). The lower panel of Table 3 contains the results,
assuming that all other parameters are as in the baseline; apart from the reported numbers, it is
worth mentioning that the quarterly velocity of money is somewhat higher (2:33), and the fall in
consumption when becoming unemployed is now 24%óin the upper range of empirical estimates
discussed above.
The optimal steady-state ináation rate is "0:36%: there is less deáation than in the baseline
calibration, for there is less money in circulation. This is similar to the optimal deáation rate obtained
by Khan et al for their calibration with low money demand elasticity (obtained by estimating money
demand over a shorter sample); indeed, since - is very close to n our calibration also implies low
money demand elasticity. The similarities go further: as in that model, optimal policy also implies
lower ináation volatility under this calibration; but the parallel stops here, for this smaller volatility
is still associated with a large welfare cost in our model. Agents are willing to sacriÖce :06% of
consumption every period in order to live in an economy with optimally volatile ináation, rather
than in an economy with stable prices. Our result thus survives even in this economy with very low
unemployment risk.
All our previous calibrations assumed that there is an optimal subsidy that undoes the steadystate monopolistic distortion, . = 0; this allows isolating the novel channel that operates in our
33
framework. We now report one last set of robustness checks, assuming that there is no such subsidy
E = 0:
Economies
9 SS (%)
sd(9)(%)
?W
0
Baseline
-1.54
.06
.48
D = 50
-2.6
.1
.55
'=1
-0.1
.04
.15
#!1
-.7
.05
0
-.6
.02
.33
Labor Market
Table 4: A distorted steady state, E = 0
As Table 4 shows, the welfare losses are now much larger. The two long-run distortions complement each other and generate signiÖcant losses from price stability. The notable exception is the
case when there is no steady-state inequality: the welfare loss is, again, zero; as our second-order
approximation showed, the linear term in the loss function disappears in this case. This result is
related to Benigno and Woodford, who showed that a distorted steady state only implies signiÖcant
deviations from price stability when the steady-state government spending share is non-zero. Our
framework thus identiÖes another channelówhich, when shut o§, makes price stability again optimal
even when the monopolistic distortion is large. But when our channel is at work, i.e. when long-run
inequality matters (q > 1), the optimal deviations from price stability can be very large indeed if
supply-side distortions are also an issue (. > 0).
6
Conclusions
In monetary policy analysis, a new synthesis looms: the integration of sticky-price, New Keynesian
models and models of heterogeneous agents, incomplete markets and limited participation. This very
active research area (that we reviewed in the Introduction) started in the early 2000s and is going
at full speed. We hope to have contributed to these convergence e§orts an analytically tractable
framework that allows a fully-áedged NK-style monetary policy analysisóincluding determinacy
of interest rate rules and optimal monetary policyóin a model that captures key mechanisms of
heterogeneous-agent incomplete-markets models, and includes a deep reason for money as a selfinsurance device.
In our model, inequality matters for monetary policy, and vice versa. The key to this link is
the interplay between liquidity and aggregate demand. Aggregate demand depends on: money, or
liquidity, which relaxes the constraint of non-participating agents; interest, because of intertemporal
substitution by participating agents; and prices, or ináation, because a Pigou e§ect operates for nonparticipating agents, and (expected) ináation is the relevant return for holding liquidity. Liquidity
34
and inequality are intimately related in our model: todayís liquidity (lower interest rates) implies
tomorrowís lower inequality.
The dependence of aggregate demand on endogenous liquidity has stark implications for the
ability of central banks to control the economy by setting interest rates. The "Taylor principle", the
standard requirement ensuring this control in the baseline NK model, fails badly even at moderate
levels of market incompleteness: the response of interest rates to ináation needed to restore control of
the economy is in double-digit territory for standard calibrations. This is due to a vicious spiral that
makes expectations-driven, sunspot shocks self-fulÖlling through endogenous liquidity adjustment.
This mechanism operates even under Öxed prices, and is thus independent of the usual NK intuition
for indeterminacyóalthough it interacts with it in subtle ways. The good news is that the central
bank can break this spiral and restore control of the economy by mildly responding, on top of
a standard, Taylor-style response to ináation, to a measure of future inequalityóthus exploiting
the intimate link between liquidity and inequality in this model. It can achieve this purpose also by
responding to liquidity directly, but this policy rule is welfare-dominated precisely because it prevents
provision when it is needed.
Inequality is also key for optimal monetary policy: in an economy with a long-run equilibrium
characterized by consumption inequality, deviations from price stability are optimal. This holds,
Örst, in the long run: the optimal ináation target should be between zero and Friedman rule; this is
no surpriseóit is true in most monetary models. But in our framework, unlike in others, it is also
true in the short run. Optimal policy implies ináation volatility in response to (productivity) shocks
that otherwise create no trade-o§.
What is more, this volatility matters for welfare. A policy of stabilizing prices (albeit around the
optimal ináation target) incurs a large welfare loss. This happens because, when there is long-run
inequality, short-run volatility has a Örst-order e§ect on constrained agents: optimal policy requires
giving less weight to ináation stabilizationówhich de facto implies giving more weight to constrained
agents. A simple rule responding to inequality is as good as price stability, but without relying on
unreasonably high responses for determinacy.
While we view our study as a step in the direction of the new synthesis that we mention at
the outset of these concluding comments, we think such e§orts should continue, for much remains
to be done. Our tractable framework allows the calculation of determinacy conditions and optimal
policy, but it inherently misses several other, surely important links between inequality and monetary
policy, as we acknowledged in Section 2.1 above. Incorporating such channels as those present in the
"HANK" models reviewed above (more realistic wealth distributions; endogenous portfolio shares;
nominal debt; endogenous unemployment risk; etc.) is paramount in order to attain a thorough
understanding of how monetary policy works and how it should be conducted in a world where
inequality matters.
35
7
References
Adao, B., I. Correia, and P. Teles, 2003. "Gaps and Triangles," Review of Economic Studies, 70(4), 699-713
Adam, K. and J. Zhu, 2014 "Price Level Changes and the Redistribution of Nominal Wealth Across the
Euro Area" Journal of the European Economic Association
Adam, K. and P. Tzamourani, 2016, ìDistributional Consequences of Asset Price Ináation in the Euro
Areaî, European Economic Review, 89, 172ñ192.
Algan, Y. and X. Ragot. 2010., "Monetary Policy with Heterogeneous Agents and Borrowing Constraints." Review of Economic Dynamics, 13, 295-316.
Algan, Y., E. Challe and X. Ragot, 2010. "Incomplete Markets and the Output-Ináation Trade-o§",
Economic Theory, 46(1), 2011, 55-84
Alvarez F., A. Atkeson and C. Edmond. 2009. "Sluggish Responses of Prices and Ináation to Monetary
Shocks in an Inventory Model of Money Demand." The Quarterly Journal of Economics, MIT Press, vol.
124(3), pages 911-967, August.
Alvarez, F., A. Atkeson and P. J. Kehoe. 2002. "Money, Interest Rates, and Exchange Rates with
Endogenously Segmented Markets," Journal of Political Economy, University of Chicago Press, vol. 110(1),
pages 73-112, February.
Alvarez, F. and A. Atkeson, 1997. "Money and exchange rates in the Grossman-Weiss-Rotemberg
model." Journal of Monetary Economics 40(3), pages 619-640, December.
Alvarez, F. and F. Lippi. 2009. ìFinancial Innovation and the Transactions Demand for Cash.î Econometrica 77 (2): 363ñ402.
Alvarez, F. and F. Lippi. 2014. "The demand of liquid assets with uncertain lumpy expenditures."
Journal of Monetary Economics, forthcoming.
Alvarez, F. and F. Lippi. 2014. "Persistent Liquidity E§ects and Long-Run Money Demand." American
Economic Journal: Macroeconomics 6(2), 71-107, April.
Ascari, G., A. Colciago and L. Rossi, 2016 "Determinacy analysis in high order dynamic systems: The
case of nominal rigidities and limited asset market participation", Mimeo Oxford
Auclert, A. 2016, "Monetary Policy and the Redistribution Channel" Mimeo Stanford
Benigno, P., and M. Woodford. 2005. ìInáation Stabilization and Welfare: The Case of a Distorted
Steady State.î Journal of the European Economic Association, 3(6): 1185ñ1236.
Benigno, P., and M. Woodford. 2012. ìLinear-Quadratic Approximation of Optimal Policy Problems.î
Journal of Economic Theory, 147(1): 1ñ42.
Bernanke, B. S. 2007, "The Level and Distribution of Economic Well-Being," speech delivered at the
Greater Omaha Chamber of Commerce, Omaha, Nebraska
Bernanke, B. S. 2015, "Monetary Policy and Inequality", Brookings.edu
36
Bewley, T. F. 1977. "The Permanent Income Hypothesis: A Theoretical Formulation". Journal of
Economic Theory 16: 252ñ92
Bewley, T. F. 1983 "A Di¢culty with the Optimum Quantity of Money" Econometrica, 1983, 51(5),
1485-504
Bilbiie F. O. 2004. "Limited Asset Market Participation, Monetary Policy, and (Inverted) Keynesian
Logic", Working Paper Nu¢eld College, Oxford University
Bilbiie F. O. 2008. "Limited Asset Market Participation, Monetary Policy, and (Inverted) Aggregate
Demand Logic." Journal of Economic Theory 140, 162-196.
Bilbiie, F.O., 2016, "Optimal Forward Guidance", CEPR Discussion Paper.
Bilbiie, F. O., I. Fujiwara, and F. Ghironi, 2014, "Optimal Monetary Policy with Endogenous Entry and
Product Variety." Journal of Monetary Economics, 64, 1-20
Bilbiie, F.O., and R. Straub, 2012, "Changes in the Output Euler Equation and Asset Markets Participation" Journal of Economic Dynamics and Control 36(11), 1659-1672
Bilbiie F. O. and R. Straub, 2013, "Asset Market Participation, Monetary Policy Rules and the Great
Ináation", Review of Economics and Statistics, 95(2), pp 377-392
Bilbiie F. O., T. Monacelli and R. Perotti 2013, ìPublic Debt and Redistribution with Borrowing Constraints", The Economic Journal 2013, F64-F98, 02
Bricker, J., L. Dettling, A. Henriques, J. Hsu, K. Moore, J. Sabelhaus, J. Thompson, and R. Windle.
2014. ìChanges in U.S. Family Finances from 2010 to 2013: Evidence from the Survey of Consumer
Finances,î Federal Reserve Bulletin, 100(4).
Cao, S, C. A. Meh, J.V. R¨os-Rull and Y. Terajima. 2012. "Ináation, Demand for Liquidity and
Welfare", Working Paper.
Campbell, J. Y. and Mankiw, N. G. 1989. îConsumption, Income, and Interest Rates: Reinterpreting
the Time Series Evidence,î in Blanchard, O. and Fisher, S., eds, NBER Macroeconomics Annual 185-216,
MIT Press.
Canzoneri, M., D. Henderson, and K. Rogo§. 1983. ìThe Information Content of the Interest Rate and
Optimal Monetary Policy.î Quarterly Journal of Economics 98: 545-566.
Challe, E. and X. Ragot, 2014 ìPrecautionary Saving Over the Business Cycle,î The Economic Journal.
Challe, E; Ragot, X. Matheron, J. and J. Rubio-Ramirez, (2016), "Precautionary saving and aggregate
demand", Quantitative Economics, Forthcoming
Chari,V.V., L.J. Christiano, and P.J. Kehoe, 1991, Optimal Öscal and monetary policy: some recent
results, Journal of Money, Credit, and Banking 23 519ñ539.
Christiano, L. J., 1991 "Modelling the Liquidity E§ect of a Monetary Shock" Quarterly Review Minneapolis Fed
Christiano, L. J. and M. Eichenbaum 1992. "Liquidity e§ects and the monetary transmission mechanism." American Economic Review PP
37
Christiano, L. J. and M. Eichenbaum 1995. "Liquidity e§ects, monetary policy and the business cycle."
Journal of Money, Credit and Banking 27(4,1): 1113-1136.
Christiano, L. J., M. Eichenbaum and C. Evans, 2005, "Nominal Rigidities and the Dynamic E§ects of
a Shock to Monetary Policy," Journal of Political Economy, 113(1): 1-45
Chodorow-Reich, G. and L. Karabarbounis 2014, ìThe Cyclicality of the Opportunity Cost of Employment,î Mimeo Harvard.
Clarida, R., J. Gali and M. Gertler, 1999, "The Science of Monetary Policy: A New Keynesian Perspective," Journal of Economic Literature, 37 1661-1707
Cloyne, J., C. Ferreira, and P. Surico 2015 "Monetary Policy when Households have Debt: New Evidence
on the Transmission Mechanism" CEPR Discussion Paper
CúurÈ, B. 2012, ìWhat can monetary policy do about inequality?î Speech at Extreme Poverty and
Human Rights, Fourth World Committee, European Parliament, Brussels
Coibion, O., Y. Gorodnichenko, L. Kueng and J. Silvia. 2012. "Innocent Bystanders? Monetary Policy
and Inequality in the U.S", IMF Working Paper 12/199.
Correia, I. and P. Teles 1999, "The optimal ináation tax", Review of Economic Dynamics.
Cui, W. and S. Radde, 2016, "Search-Based Endogenous Asset Liquidity and the Macroeconomy",
Mimeo UCL and ECB
Curdia, V. and M. Woodford, 2009, "Credit Frictions and Optimal Monetary Policy", Forthcoming,
Journal of Monetary Economics.
Den Haan,W., Rendahl, P. and M. Riegler 2016, "Unemployment (Fears) and Deáationary Spirals",
working paper.
Deutsche Bundesbank Monthly Report (2016), ìDistributional E§ects of Monetary Policyî, September
2016.
Doepke, M. and M. Schneider 2006, "Ináation and the Redistribution of Nominal Wealth", Journal of
Political Economy 114 (6), 1069-97
Draghi, M., 2016 ìStability, Equity and Monetary Policyî 2nd DIW Europe Lecture, DIW Berlin
Eggertsson G. and P. Krugman 2012, "Debt, Deleveraging, and the Liquidity Trap: A Fisher-MinskyKoo Approach, Quarterly Journal of Economics, 127(3): 1469-1513.
Erosa, A., Ventura,G. 2002. "On Ináation as a Regressive Consumption Tax." Journal of Monetary
Economics 49, 761-795.
Feldman, M and Gilles, C. 1985. "An Expository note on Individual risk without aggregate uncertainty."
Journal of Economic Theory 35, 26-32.
Felippa, C. A. and K. C. Park, 2004. "Synthesis tools for structural dynamics and partitioned analysis
of coupled systems." In A. Ibrahimbegovic and B. Brank (Eds.), Multi-Physics and Multi-Scale Computer
Models in Nonlinear Analysis and Optimal Design of Engineering Structures under Extreme Conditions.
Friedman, M. 1969, The Optimum Quantity of Money, Macmillan
38
Fuerst, T. 1992. "Liquidity, loanable funds and economic activity." Journal of Monetary Economy
29(1),3-24.
Furceri, D., P. Loungani,and A. Zdzienicka, 2016 "The E§ects of Monetary Policy Shocks on Inequality"
International Monetary Fund Working Paper
Gali, J. 2008, "Monetary Policy, Ináation, and the Business Cycle: An Introduction to the New Keynesian Framework and Its Applications", Princeton University Press
Gali, J., Lopez-Salido D. and J. Valles. 2004. "Rule-of-Thumb Consumers and the Design of Interest
Rate Rules." Journal of Money, Credit, and Banking 36(4), 739-764.
GalÌ, J., D. LÛpez-Salido, and J. VallÈs, 2007. "Understanding the E§ects of Government Spending on
Consumption". Journal of the European Economic Association, March, vol. 5 (1), 227-270.
Green, E. 1994. "Individual-level randomness in a nonatomic population", working paper.
Gottlieb, C. 2015, "On the Distributive E§ects of Ináation" Mimeo EUI and Oxford
Gornemann, N., K. Kuester, and M. Nakajima, 2012 ìMonetary Policy with Heterogeneous Agents,î
Working Papers 12-21, Federal Reserve Bank of Philadelphia
Grossman S.J and Weiss L. 1983. "A transactions-based model of the monetary transmission mechanism." American Economic Review. 73 (1983). 871-880.
Iacoviello, M., 2005 House Prices, Borrowing Constraints, and Monetary Policy in the Business Cycle,
American Economic Review
Imrohoroglu, A., 1992. The welfare cost of in·ation under imperfect insurance. Journal of Economic
Dynamics and Control 16, 79-91.
Ireland, P. 1996. "The role of countercyclical monetary policy." Journal of Political Economy 104(4),
704-723.
Jappelli, T and Pistaferri, L, 2014, "Fiscal Policy and MPC Heterogeneity", American Economic Journal:
Macroeconomics, 6(4), 107-36
Kaplan, G. and G. Violante, 2014 ìA Model of the Consumption Response to Fiscal Stimulus Payments,î
Econometrica
Kaplan, G., G. Violante, and J. Weidner, 2014 ìThe Wealthy Hand-to-Mouth,î Brookings Papers on
Economic Activity.
Kaplan, G., B. Moll and G. Violante 2015, "Monetary Policy According to HANK", mimeo Princeton
and NYU
Kehoe, T., D. Levine, and M. Woodford, The optimum quantity of money revisited,î in Partha Dasgupta,
Douglas Gale, Oliver Hart, and Eric Maskin, editors, The Economic Analysis of Markets and Games: Essays
in Honor of Frank Hahn, MIT Press, 1992, 501ñ26.
Khan, A., R. G. King, and A. L. Wolman 2003. ìOptimal Monetary Policy.î Review of Economic
Studies, 70(4), 825-860.
39
Khan, A. and J. Thomas 2011. "Ináation and Interest Rates with Endogenous Market Segmentation",
working paper.
King, R. G., and A. L. Wolman, 1999 ìWhat Should the Monetary Authority Do When Prices are
Sticky?,î in John B. Taylor, editor, Monetary Policy Rules, The University of Chicago Press, 1999, 349-398.
Kiyotaki, N. and J. Moore, 1997 "Credit Cycles," Journal of Political Economy, 105 (1997), 211-248
Kiyotaki, N and M. Gertler, 2010. "Financial Intermediation and Credit Policy in Business Cycle Analysis," in Handbook of Monetary Economics, Volume 3A, eds B. Friedman and M. Woodford, Amsterdam:
Elsevier
Kiyotaki, N. and J. Moore, 2012, "Liquidity, Business Cycles, and Monetary Policy," mimeo Princeton
Krusell, P. and A. Smith. 1998. "Income and wealth heterogeneity in the macroeconomy." Journal of
Political Economy 106(5), 867-896.
Leeper, E. M., 1991, "Equilibria under active and passive monetary and Öscal policies", Journal of
Monetary Economics, 129-147
Lippi, F., S. Ragni and N. Trachter 2015. "Optimal monetary policy with heterogeneous money holdings." Journal of Economic Theory 339-368.
Lucas, R. E. Jr. 1987. ìModels of Business Cycles.î Oxford: Basil Blackwell.
Lucas R. E. Jr., 1990 "Liquidity and Interest Rates." Journal of Economic Theory 50, 237 - 264.
Lucas, R.E. Jr., 2000. "Ináation and Welfare". Econometrica 68, 247-274
Lucas, R. E., Jr. 2003 "Macroeconomic Priorities". American Economic Review 93 (1): 1ñ14.
Lucas, R. E., Jr., and N. L. Stokey, 1983, ìOptimal Fiscal and Monetary Policy in an Economy without
Capital,î Journal of Monetary Economics 12, 55-93.
McCallum, B. T. "Some Issues Concerning Interest Rate Pegging, Price Level Determinacy, and the
Real Bills Doctrine," Journal of Monetary Economics, 17(1), 135-160
McKay A. and R. Reis, "The Role of Automatic Stabilizers in the U.S. Business Cycle" , Econometrica,
Vol. 84, No. 1, January 2016.
McKay A., Nakamura E. and J. Steinsson 2016, "The power of forward guidance revisited", American
Economic Review.
Mersch, Y., 2014, "Monetary policy and economic inequality", Keynote Speech at Corporate Credit
Conference, Zurich
Misra, K. and P. Surico, 2014, "Consumption, Income Changes, and Heterogeneity: Evidence from Two
Fiscal Stimulus Programs"American Economic Journal: Macroeconomics, 6(4),
Nistico, S., 2015, "Optimal monetary policy and Önancial stability in a non-Ricardian economy." Forthcoming, Journal of the European Economic Association
Patinkin, D. 1956, "Money, Interest and Prices: An Integration of Monetary and Value Theory", Second
edition (1965), MIT Press
Pigou, A.C., 1943. "The Classical Stationary State". Economic Journal 53 (212): 343ñ351
40
Ragot, X. 2014. "The case for a Önancial approach to money demand." Journal of Monetary Economics,
in press.
Ravn, M. and V. Sterk, 2013 ìJob Uncertainty and Deep Recessions,î forthcoming, Journal of Monetary
Economics.
Ravn, M. and V. Sterk, 2016 ìMacroeconomic Fluctuations with HANK and SAM: an Analytical Approach,î Mimeo, University College London.
Rocheteau, G., and P. O. Weill 2011: ìLiquidity in Frictional Asset Markets,î Journal of Money, Credit
and Banking, 43, 261ñ282.
Rocheteau, G., P. O. Weill and T. Wong, 2015, "A Tractable Model of Monetary Exchange with Ex-Post
Heterogeneity" Mimeo UC Irvine, UCLA and Bank of Canada
Rocheteau, G., P. O. Weill and T. Wong, 2016, "Working through the Distribution: Money in the Short
and Long Run" Mimeo UC Irvine, UCLA and Bank of Canada
Rotemberg, J. J. 1982, ìMonopolistic Price Adjustment and Aggregate Outputî. Review of Economic
Studies, 49(4), 517-531
Samuelson, P. 1958. "An Exact Consumption-Loan Model of Interest with or without the Social Contrivance of Money." Journal of Political Economy, 66, 467.
Scheinkman, J.A. and Weiss, L. (1986), ëBorrowing constraints and aggregate economic activityí, Econometrica, 54(1), 23-45.
Schmitt-Grohe, S. and M. Uribe, 2004, "Optimal Fiscal and Monetary Policy under Sticky Prices."
Journal of Economic Theory, February 2004, 114(2): 198-230
Schmitt-Grohe, S. and M. Uribe, 2007, "Optimal Simple and Implementable Monetary and Fiscal Rules."
Journal of Monetary Economics 54, 6
Sheedy, K. D. 2014, "Debt and Incomplete Financial Markets: A Case for Nominal GDP Targeting,"
Brookings Papers on Economic Activity, 301-373
Shimer, R. 2005. "The cyclical behaviour of equilibrium employment and vacancies." American Economic Review 95(1), pp. 25ñ49.
Sterk, V. and S. Tenreyro, 2016, "The Transmission of Monetary Policy through Redistributions and
Durable Purchases" Mimeo UCL and LSE
Surico, P. and R. Trezzi, 2016, "Consumer Spending and Fiscal Consolidation: Evidence from a Housing
Tax Experiment", Mimeo LBS and FRB
Vissing-Jorgensen, A. 2002. "Towards an explanation of the households portfolio choice heterogeneity:
Non Önancial income and participation cost structure" NBER Working Paper 8884.
Vissing-Jorgensen, A. 2002. "Limited Asset Market Participation and the Elasticity of Intertemporal
Substitution" Journal of Political Economy.
Wallace, N. 1981. A Modigliani-Miller Theorem for Open-Market Operations, The American Economic
Review, Vol. 71, No. 3 (Jun., 1981), pp. 267-274
41
Wallace, N. 2014. "Optimal money-creation in pure-currency economies: a conjecture." Quarterly Journal of Economics 129(1), 259-274.
Weill, P. O. 2007 ìLeaning Against the Wind,î Review of Economic Studies, 74(4), 1329ñ1354.
Werning, I., 2016, "Incomplete markets and aggregate demand", mimeo
Woodford, M. 1990. "Public Debt as Private Liquidity" American Economic Review 80(2), pp. 382-88
Woodford, M., 2003, Interest and prices: foundations of a theory of monetary policy, Princeton University
Press
Woodford, M. 2012. "Methods of Policy Accommodation at the Interest-Rate Lower Boundî, Jackson
Hole Symposium.
Yellen, J. L. 2014, "Perspectives on Inequality and Opportunity from the Survey of Consumer Finances,"
speech delivered at the Conference on Economic Opportunity and Inequality, Federal Reserve Bank of
Boston, Boston, Mass.
42
A
Model Summary
The equations describing our model in the general case are:
! "
1 + it 0 ! P "
u0 cPt = #E
u ct+1
1 + 9 t+1
! "
'
!
"
!
"(
u0 cPt = #E -u0 cPt+1 + (1 " -) u0 cN
t+1
1
1 + 9 t+1
!
"
'
!
"
!
"(
1
u0 c N
$ #E (1 " /) u0 cPt+1 + /u0 cN
t
t+1
1 + 9 t+1
1
1"- N
cPt + m
~ Pt+1 = wt ltP " = Pt + dt +
m
~ Pt +
m
~
n
1 + 9t
1 + 9t t
1"/ P
/
N
N
m
~N
m
~t +
m
~N
t+1 + ct = 4 " = t +
1 + 9t
1 + 9t t
%, P -$
&
%
&
ct
Yt+1
""1
" wt
9 t (1 + 9 t ) = #Et
9 t+1 (1 + 9 t+1 ) +
" (1 + E)
Yt
D
" " 1 At
cPt+1
Yt = nAt ltP
! "' ! P "$
wt = ' ltP
ct
,
wt D 2
dt = 1 "
" 9 Yt
At 2 t
nm
~ Pt + (1 " n) m
~N
t
nm
~ Pt+1 + (1 " n) m
~N
=
+ mCB
t+1
t+1
1 + 9t
= t = "mCB
t+1
!
1"!
= Pt = = t ; = N
=t
t =
n
1"n
m
~ Pt+1 ; m
~N
t+1 $ 0
where mCB
t+1 is a period t monetary shock (new money created) and At exogenous labor productivity.
The economy resource constraint follows by Walrasí law:
)
D 2+
ncPt + (1 " n) cN
=
1
"
9 Yt + (1 " n) 4
t
2 t
and can replace (for instance) the P agentsí budget constraint in the system above.
CB
P
P
N
An equilibrium of the economy is a sequence fcPt ; cN
~ Pt+1 ; m
~N
t , ct 9 t , mt+1 ; m
t+1 ; wt ; lt ; = t ; = t ; = t ; dt ; Yt g
satisfying the previous conditions. Assuming that nominal bonds are in zero net supply, we guess-
an-verify the structure of the equilibrium with m
~N
t = 0; i:e: non-participating households never hold
money at the end of the period. The conditions for households in the N "island not to hold money,
which we check holds in the equilibrium we consider, is:
! "
'
!
"
!
"(
u0 c N
> #E (1 " /) u0 cPt+1 + /u0 cN
t
t+1
43
1
:
1 + 9 t+1
Consider the conditions for a monetary equilibrium to exist.42
In a monetary steady-state,
N "agents do not hold money when 1 + 9 > #, while P -agents save in money comparing the gain
to self insure and the opportunity cost (deáation). Thus, we have cP > cN and q > 1. Since
it is costly for P agents to save (the return on money is lower than the discount factor), they
rationally choose not to perfectly self-insure. Using this inequality in the condition (6), we have
! " '
! "
! "( 5
u0 cN > (1 " /) u0 cP + /u0 cN 1+4
: N agents do not hold money at the end of each period,
and m
~ N = 0 In steady state, positive money demand requires the restriction that outside option not
be too good:
"
4 c < 04 = h = 1 + n
,
1 + 9 " -#
# (1 " -)
- %1
"1
!#!1
(28)
Under a Taylor rule, the steady-state ináation rate 9 is determined by the central bankís target
and the above condition is parametric. Under Ramsey policy, 9 is endogenous (it depends, among
other things, on 4) and the above condition deÖnes a threshold implicitly.
B
Derivations and Proofs
B.1
New Keynesian Phillips curve
The intermediate goods producers solve
"
#
,
-2
1
X
D
P
(z)
t
maxE0
Qs0;t (1 + E) Pt (z)Yt (z) " Wt lt (z) "
" 1 Pt Yt ;
Pt (z)
2
P
(z)
t!1
t=0
!
"$
where Qs0;t % # ts P0 cP0 =Pt cPt
is the marginal rate of intertemporal substitution of participants
between times 0 and t; and E is a sales subsidy. Firms face demand for their products from two
sources: consumers, and Örms themselves (in order to pay for the adjustment cost); the demand
function for the output of Örms z is Yt (z) = (Pt (z)=Pt )!" Yt : Substituting this into the proÖt function,
42
Monetary variables are generally not uniquely determined. There always exists an equilibrium of our model where
money has no value. If agents anticipate that money is not traded in the future, they do not accept money today and
the price of money is 0. The reason for the existence of a non-monetary equilibrium is the same as in the monetary
overlapping-generations model of Samuelson (1958). In such a cashless equilibrium, the consumption of N "agents is
cN = 5 t in each period. The consumption of P -agents is easily determined; this is akin to the standard cashless New
Keynesian model studied in Woodford (2003).
44
the Örst order condition is, after simplifying:
,
-!" "
,
-!1 #
Pt (z)
Wt Pt (z)
0 = Q0;t
Yt (1 + E) (1 " ") + "
Pt
Pt
Pt
,
Pt (z)
1
" Q0;t DPt Yt
"1
+
Pt!1 (z)
Pt!1 (z)
4
%
,
&5
Pt+1 (z)
Pt+1 (z)
+ Et Q0;t+1 DPt+1 Yt+1
"1
Pt (z)
Pt (z)2
(29)
In a symmetric equilibrium all producers make identical choices (including Pt (z) = Pt ); deÖning net
!
"$
ináation 9 t % (Pt =Pt!1 ) " 1, and noticing that Q0;t+1 = Q0;t # cPt =cPt+1 (1 + 9 t+1 )!1 ; equation (29)
becomes:
%,
&
-$
cPt
Yt+1
9 t (1 + 9 t ) = #Et
9 t+1 (1 + 9 t+1 ) +
Yt
cPt+1
%
&
""1
" wt
+
" (1 + E)
D
" " 1 At
(30)
Loglinearization of this gives rise to the equation used in text.
B.2
Loglinearized equilibrium conditions
Table 1 outlines the equilibrium conditions loglinearized around an arbitrary steady state, denoting
t
t
9
^ t = ln 1+4
and ^{t = ln 1+i
.
1+4
1+i
Table 1: Summary of loglinearized equilibrium conditions
c^Pt = Et c^Pt+1 " * !1 (^{t " Et 9
^ t+1 )
!
"
A5
A5
P
P
!1
Self-insurance/money demand c^t = 1+4 Et c^t+1 + 1 " 1+4 Et c^N
Et 9
^ t+1
t+1 + *
1!B D
Budget constraint N
c^N
^ tot
^ ) " Dh 1!!
=^ + Ghc ^4 t
t = n(1+4) h (m
t "9
1!n t
)t
+
^
Ec res cons (replace Pís BC)
c^t = (1 " (1 " n) 4 c ) a
^t + ^lt " /4(1+4)
^
, 2 9
t + (1 " n) 4 c 4 t
1! 4
Euler P
2
Aggregate labor
Labor supply
Aggregate C
Transfer
Money growth
^lt = ^lP
t
P
'^l = w^t " *^
cP
t
c^Pt
=
1
c^
np t
"^
= t = x^t
"
t
(1!n)h N
c^t
np
x^t = m
^ tot
^ tot
^t
t+1 " m
t +9
1+4 "!1
Phillips curve
9
^ t = #Et 9
^ t+1 + 1+24
(w^t " a
^t )
/
In an equilibrium with áexible prices (denoted with a star), the log deviation of the real wage
is proportional to labor productivity w^t% = a
^t : Combining equations (11), (12) and (13) we obtain:
%
(0
'np + *) c^Pt % + '
0 (1 " n) h^
cN
= (1 + ') a
^t ;
t
and the áexible-price equilibrium can be easily solved by combining (31), (16), (17) and (15)
45
(31)
B.3
Proofs of Indeterminacy Propositions
The model under the simplest interest-rate rule is disconnected in the following sense. The deÖnition
of money growth and the budget constraint of N, combined, give a di§erence equation in money
balances that is stable ( A!n
< 1), given paths for c^N
^t :
t and 9
1!n
m
^ tot
t+1 =
" h
- " n ! tot
m
^t "9
^ t + c^N
1"n
H t
(32)
The following system, written in canonical form and obtained by combining the other equilibrium
conditions including the simple Taylor rule, then determines c^Pt ; c^N
^ t independently of money
t and 9
balances.
Et c^N
t+1
Et c^Pt+1
-#N
-#
= (1 "
+ (1 + F " (1 "
" # +
* !1 9
^ t + * !1
(33)
"t
1 " -#
1 " -#
!
"
= (1 " -#) H^
cN
^Pt + N " # !1 * !1 9
^ t " * !1 "t
t + (1 + F " (1 " -#) H) c
-#) H^
cN
t
,
-#) H) c^Pt
!1
* !1 Et 9
^ t+1 = " (1 " -#) H^
cN
^Pt + # !1 * !1 9
^t;
t " (F " (1 " -#) H) c
with the shorthand notation that will prove useful below F = # !1 (1 + * !1 '
0 ) ; H = # !1 * !1 '
0 (1!n)h
:
1!A5
The solution with iid shock follow immediately since model is purely forward-looking, as long as
equilibrium is unique (which we study below):
^{t = 0; 9
^ t = N!1 "t :
Euler equation of P: c^Pt = 0 so Phillips curve gives
!1
'
0 (1 " n) h^
cN
t = N "t
Back to determinacy, we ignore shocks. )
The transition
+ matrix of the canonical system is
A5H
!1
(1 " -#) H 1 + F " (1 " -#) H " # + 1!A5
(1 " -#) H
1 + F " (1 " -#) H
" (1 " -#) H " (F " (1 " -#) H)
with characteristic polynomial:
N " # !1
,
# !1
!
"
!
"
J % (X) = X 3 " F + 1 + # !1 X 2 + # !1 + N (F " H) X + NH
For both Propositions, the polynomial needs three roots outside unit circle (and no root inside).
We transform this into a Hurwitz polynomial that is easier to analyze (we use results in Ascari,
Colciago and Rossi, 2016; and Felippa and Park); use the change of variable
X=
1+z
;
1"z
and consider the new polynomial in z
!
"
!
"
J (z) = 2 + F + 2# !1 + NF " 2NH z 3 +z 2 (4 + F " NF + 4NH)+ 2 " 2# !1 " F " NF " 2NH z+F (N " 1)
46
Conditions that all roots of J (z) be positive (i.e. all roots of J* outside unit circle) are the same
as those needed for all roots of J ("z) be negative; the latter are readily available in engineering as
the Rooth-Hurwitz criterionósee Felippa and Park (page 20-22; eq. 4.18), and Ascari et al (2016)
for an economics application. We distinguish two cases.
Case I N > 1 Conditions are
!
"
2 " 2# !1 " F " NF " 2NH < 0
!
"
2 + F + 2# !1 + NF " 2NH < 0
!
"
!
"
2 " 2# !1 " F " NF " 2NH (4 + F " NF + 4NH) < 2 + F + 2# !1 + NF " 2NH F (N " 1)
Case II N < 1; change polynomial to normalize constant to positive
!
"
J2 (:) = "J ("z) = 2 + F + 2# !1 + NF " 2NH z 3
!
"
" (4 + F " NF + 4NH) z 2 + 2 " 2# !1 " F " NF " 2NH z + F (1 " N)
Conditions:
!
2 " 2# !1 " F " NF " 2NH
"
> 0
2 + F + 2# !1 + NF " 2NH > 0
!
"
!
"
" (4 + F (1 " N) + 4NH) 2 " 2# !1 " F " NF " 2NH > F (1 " N) 2 + F + 2# !1 + NF " 2NH
Proof of Proposition 2: ModiÖed Taylor Principle
We look at the case where N = N4 > 0: Rule out Case 2, because second condition cannot hold (rhs
negative).
Take Case I, Örst requirement is the Taylor principle N > 1: Second requirement always holds
(rhs is negative).
Take last condition,
Case I.1 2H > F
*'
0 !1 < 2
implies
%
N>N %
(1 " n) h
"1
1 " -#
2 (1 + #) '
0 !1 *
!1
+1+'
0 !1 *
2 (1!n)h
" (1 + '
0 !1 *)
1!A5
and the third requirement always holds then (4H > 2H > F ): N (4H " F ) > " (4 + F ) :
The threshold for * '
0 !1 can be simpliÖed as follows. Take the no-discounting limit # ! 1; the
1!n
right-hand side tends to 2 1!A
" 1 > 1: Therefore, a su¢cient condition is * '
0 !1 < 1 < 2 (1!n)h
"1
1!A5
and we restrict attention to values ' > (1 " (1 " n) 4 c ) *; which is a realistic restriction.
Case I.2 2H < F; last condition never fulÖlled, so this case cannot occur. This proves the
proposition.
47
The Inequality-Augmented Taylor Principle
)
+!1
Hq
Recall the notation N = N4 1 " $(1!A5)
; where N4 > 0: Thus N > 0 is equivalent to Nq <
* (1 " -#)
Proposition 7 The Inequality-Augmented Taylor Principle. If the central bank follows the
rule (25), the model has a locally unique RE equilibrium if and only if the Taylor coe¢cients satisfy:
Case 1: if N > 0 and N% > 0 : same as proposition 2.
N > max (1; N% )
Case 2.i N < 0 and N% > 0
~
"N > N
Case 2.ii N < 0 and N% < 0
~ < "N < "N% ;
N
~ > 0 is deÖned below.
where N
)
The proof for case 1 is immediate, as it is just a rescaling by 1 "
Hq
$(1!A5)
+!1
> 0. The more
challenging case is Case 2: an overall negative response to ináation can now ensure determinacy.
CASE 2 Take N < 1; Örst condition is
N<
2 " 2# !1 " F
<0
F + 2H
which can only hold when N < 0 i.e. Nq > * (1 " -#)). So Örst we have
!
"
2 # !1 " 1 + F
N4
>
Hq
F + 2H
"1
(34)
$(1!A5)
Further
N (2H " F ) < 2 + F + 2# !1
Case 2i when 2H > F this is automatically satisÖed. Third condition
!
"
!
"
" (4 + F (1 " N) + 4NH) 2 " 2# !1 " F " NF " 2NH > F (1 " N) 2 + F + 2# !1 + NF " 2NH
! !
"
"
(1 + NH) 2 # !1 " 1 + F + N (F + 2H) > F (1 " N) (1 " NH)
!
"
("NH)2 " # !1 + F + F H !1 ("NH) + # !1 " 1 > 0
48
Su¢cient condition43
~ % 1 " -#
"N > N
(1 " n) h
,
1 + *'
0
!1
+
!1
*'
0
!1
,
"
1 " -# !
1+#
1 + *'
0 !1
(1 " n) h
--
;
which can be shown by direct substitution to be strictly larger than the threshold 34; thus delivering
the condition in the proposition.
Case 2ii When 2H < F same conditions from last requirement, plus,
~ < "N < "N%
N
) Corrolary
+ 1 in text is obtained as follows. Consider Case 1 Örst, the requirement is N4 >
Hq
1 " $(1!A5) max (1; N% ) .
N4 > 1 is su¢cient if the threshold RHS is less )than 1, which
+ gives the Örst inequality.
Hq
~ if RHS is less than 1, which gives
Take now Case 2i, N4 > 1 is su¢cient for N4 > $(1!A5) " 1 N
second inequality.
Liquidity augmented rule
Take the special case of áexible prices, D = 0. Under the liquidity rule, the model can be written as,
using the shorthand notation
(1!n)h
(1!A5)
np
= Z; (1!n)D
=D
"
1 " - ! tot
m
^t "9
^t
1"n
"
1 " - ! tot
= (* " Nx (1 " Z) D) c^Pt " (1 " Z) Nx
m
^t "9
^t
1"n
!
"
"
n
= "D^
cPt % +
m
^ tot
^t
t "9
1"n
Et c^Pt+1 = "* !1 ZDNx c^Pt " * !1 ZNx
Et 9
^ t+1
m
^ tot
t+1
Noticing that only m
^ tot
^ t and redeÖning it as a new variable we have the characteristic polynomial
t "9
of the new 2x2 system:
,
,
,
1"1"1"2
!1
!1
J (X) = X + Nx Z * D +
" 1 " (Nx " 1)
X " ZNx
+* D ;
1"n
1"n
1"n
which needs two roots outside the unit circle because both variables are controls. The conditions for
both roots to be outside the unit circle are as follows. The Örst condition is the "Taylor principle"
emphasized in our proposition, coming from:
J (1) = " (Nx " 1)
1"< 0 ! Nx > 1
1"n
43
The necessary and su¢cient conditions are of course ", 2
!
q
2
' !1 +F H !1 +F + (' !1 +F H !1 +F ) !4(' !1 !1)
;1
2H
0;
!
q
2
' !1 +F H !1 +F ! (' !1 +F H !1 +F ) !4(' !1 !1)
[
2H
But for & !1 close to 1 the Örst interval is almost empty and the lower threshold is close to (but lower than) the
su¢cient one given in text.
49
The second condition 1 > 0 > "ZNx
J ("1) < 0; implying in addition:
! 1!A
1!n
Nx >
"
+ * !1 D is always satisÖed; the third requirement is
1!A
1!n
$ !1 np+D(1!A)
2 (1!A5) . " 1!A
1!n
h
2"
;
where we used that the denominator is positive (which we show below).
This requirement is weaker than the Taylor principle if the threshold is lower than 1, i.e. (1 " -#) Dh <
* !1 np+H (1 " -). We now prove this in a limit case and then appeal to a continuity argument (taking
* !1 for simplicity). Take the limit of no discounting, # ! 1; we then have h ! 1 and H !
(1!n)(1!G c )
:
1!A
The condition then becomes n > 0 which always holds. To see that the denominator is positive, in
the same special case we obtain 2n > (n " -) (1 " 4 c ) ; which always holds as the RHS is negative.
C
Optimal PolicyñDerivations and Proofs
C.1
Friedman Rule with Flexible Prices
First we show that, as in other monetary economies, the price level is indeterminate at the Friedman
!
"! P
"
rule. For i = 0, the steady state implies 1+9 = #; cP = cN = c and mCB = 1 " # !1 nm
~ + (1 " n) m
~N :
! "' ! P "$
The real allocation is determined by 1 = ' lP
c
; c = nlp + (1 " n) 4 and
,
,
- ,
,
1
1
1
1
P
c = 4+
1 " / " (1 " (1 " #) N)
"1 n m
~ +
/ " # " (1 " (1 " #) N)
" 1 (1 " n) m
~N
#
#
#
#
There is indeterminacy, even though the real variables c and lp are uniquely determined as the steadystate Örst best values: monetary variables m
~ P and m
~ N must only satisfy one equation, so the real
quantity of money is indeterminate.
Second, we show (in the nonlinear model) convergence to the Örst-best allocation (when D = 0)
if 2 " - " / > # !1 " 1; the steady-state allocation converges to the Örst best when i "! 0+ . In this
case, 1 + 9 "! # + . For 0 < k < 1, deÖne ^lt (k) as the unique solution to the equation
(n + (1 " n) k)
,
At
'
- %1 )
+! '
^lP (k) % = At ^ltp (k) + (1 " n) 4 t
t
As the left hand side is decreasing and the right hand side increasing in ^ltp , there always exists a
positive solution to the previous equation, whatever At ; 4 t > 0. DeÖne c^Pt (k) as
c^Pt
nAt ltp + (1 " n) 4 t
(k) %
n + (1 " n) k
p
For any k < 1, we show that the can reach allocations where cPt = c^Pt (k), cN
t = kct (k) and
lt = ^ltP (k). When k equals 1, the allocation is exactly the Örst-best allocation. When k approaches
50
1, the allocation can be made arbitrarily close to the Örst-best allocation and the nominal interest
rate it tends toward 0+ . Take now the model equations from Appendix A, for the case of áexible
prices D = 0 and using the money market equilibrium to substitute for m
~ Pt+1 . We proceed by guess
and verify. At any period, the variables mCB
and m
~ Pt are predetermined. As a consequence, assume
t
that the period t money creation mCB
t+1 is determined by the following law :
!
"
1 u0 c^Pt!1
1"/
CB
P
mt+1 = k^
ct " 4 "
m
~ Pt
P
!$
# u0 (^
+
(1
"
-)
k
ct )
It is easy to show that the allocation cPt = c^Pt ; cN
cPt ; it = - + (1 " -) k !$ ; = t = "mCB
t = k^
t+1 and
! "
'
( u0 c^Pt
!$
!
"
1 + 9 t = # - + (1 " -) k
u0 c^Pt!1
is an equilibrium of the model, because it satisÖes all equations. The equilibrium is locally unique,
which can be shown by standard perturbation methods (we prove local determinacy in a more general
case above).
C.2
Optimal Ramsey Policy
The constraints of the Ramsey planner are (these are the model equations, with relevant substitutions
and using the economy resource constraint instead of the P budget constraint):
"
! P "
! N "#
0
0
!
"
-u
c
+
(1
"
-)
u
ct+1
t+1
u0 cPt = #Et
1 + 9 t+1
)
+
D 2
P
ncPt + (1 " n) cN
t = 1 " 9 t nAt lt + (1 " n) 4
2
"
#
! "' ! P "$
%, P -$
&
P
ct
At+1 lt+1
ct
""1
" ' ltP
9 t (1 + 9 t ) = #Et
9 t+1 (1 + 9 t+1 ) +
" (1 + E)
D
""1
At
cPt+1
At ltP
,
1"!
mtot
1 1 " - tot
t
N
tot
ct = 4 t +
mt+1 "
+
m
1"n
1 + 9t
1 + 9t 1 " n t
! "
1 + it 0 ! P "
u0 cPt = #Et
u ct+1
1 + 9 t+1
As long as money is created through helicopter-drop, within-period transfers, only the Örst three
equations above are constraints for the Ramsey planner.44 Indeed, once cP and 9 are known, i follows
from the Euler equation for bonds (which hence will not bind as a constraint). Similarly, once the
allocation of consumption of N and ináation have been chosen, the quantity of money delivering it
can be recovered through the following equation:
tot
cN
t = 4 t + mt+1 "
44
- " ! mtot
t
;
1 " n 1 + 9t
Whereas with open-market operations, all of the above equations are constraints; we analyze this case and provide
a welfare comparison in the companion paper.
51
where, implicitly, we concentrate only on equilibria where money is used.
The central bank chooses cP ; cN ; lP ; 9 to maximize the objective deÖned in text, subject to the
above system of 3 constraints which we denoted in text by 6t and write here explicitly for reference:
"
! P "1+' #
%
&
1
X
!
"
! N"
lt
4 1+'
t
P
max E0
# fn u ct " '
+ (1 " n) u ct " '
(35)
N P
1
+
'
1
+
'
fcP
t ;ct ;lt ;4 t g
t=0
h
! P "!$
! P "!$
! N "!$ i
+! 1t (1 + 9 t+1 ) ct
" #- ct+1
" # (1 " -) ct+1
(36)
h
)
+
i
D 2
P
+! 2t ncPt + (1 " n) cN
t " 1 " 9 t nAt lt " (1 " n) 4
2
"
!#
! "' ! P "$
%, P -$
&
ct
ct
Yt+1
" ' ltP
+! 3t 9 t (1 + 9 t ) " #Et
9 t+1 (1 + 9 t+1 ) "
+."1 g
Yt
D
At
cPt+1
The solution is a system of 4 Örst order conditions and 3 constraints, for 4 variables and 3 co-states
(the Lagrange multipliers on the constraints). The Örst-order conditions of the Ramsey problem are
for each variable respectively
! "!$
! "!$!1
! "!$!1
0 = n cPt
" *! 1t (1 + 9 t+1 ) cPt
+ *! 1t!1 - cPt
! "' ! P "$!1
' ltP
ct
"
+ ! 2t n " *! 3t
D
At
! N "!$
! "!$!1
0 = (1 " n) ct
+ *! 1t!1 (1 " -) cN
+ ! 2t (1 " n)
t
! "'!1 ! P "$
)
! P "'
ct
D 2+
" ' ltP
0 = "n' lt
" ! 2t 1 " 9 t nAt " ! 3t '
2
D
At
!
"
!$
0 = ! 1t!1 # !1 cPt!1
+ ! 2t D9 t nAt ltP + (! 3t " ! 3t!1 ) (1 + 29 t )
plus the three constraints with complementary slackness.
A steady-state of the Ramsey problem is deÖned by
! "!$
1 " - ! N "!$
! 1 = 0 or cP
= 1+4
c
"5
)
D 2+ P
! 2 = 0 or ncPt + (1 " n) cN
=
1
"
9 nl + (1 " n) 4
t
2
' ! P "' ! P "$
(
"
! 3 = 0 or 9 (1 + 9) =
' l
c
" (1 " .)
D (1 " #)
! "!$
! "!$!1
! "' ! P "$!1
"
0 = n cP
" *! 1 (1 + 9 " -) cP
+ ! 2 n " *! 3 ' lP
c
D
! "!$
! "!$!1
0 = (1 " n) cN
+ *! 1 (1 " -) cN
+ ! 2 (1 " n)
)
+
! P "'
D
" ! "'!1 ! P "$
0 = "n' l
" ! 2 1 " 9 2 n " ! 3 ' ' lP
c
2
D
! "!$
0 = ! 1 # !1 cP
+ ! 2 D9nlP
52
The Proof of Proposition XX is now immediate. With áexible prices D = 0 and optimal subsidy,
the only solution to the above system of equations is perfect insurance through the Friedman Rule:
1+9
= 1 ! cP = cN = c
#
With sticky prices and inelastic labor ' ! 1, the intratemporal optimality condition disappears
from the set of constraints, labor is Öxed, and it can be easily shown that ináation tends to zero
(9 = 0 solves the above system).
Computing the welfare cost To calculate the welfare cost of ináation, we proceed in the standard
way pioneered by Lucas (1987). Denote with an upper-script SS the allocation for the ináation rate
9 SS and no shock. We denote the welfare of an economy where ináation is, say 9 % , as V % : We then
compute the proportional decrease in consumption for all households in the economy with ináation
rate 9 % to equalize the two welfare measures. Formally we compute ?W to have
( "
)
! P SS "1+' #
%
1
1+' &
X
!!
"
"
!!
"
"
l
4
SS
E0 V % = E0
# t n u 1 " ?W cPt SS " ' t
+ (1 " n) u 1 " ?W cN
"'
t
1
+
'
1+'
t=0
C.3
Welfare function and second-order approximation
The second-order approximation technique used is described in detail in Woodford (2003, Chapter
6), Benigno and Woodford, and Bilbiie (2008). A second-order approximation of P agentsí utility
delivers
UtP
,
1 " * ! P "2
1 + ' )^P +2
P P
P
^
"U =
+
c^t
+ UL l
lt +
lt
2
2
,
,
) +2 -! P "1!$ P 1 " * ! P "2 lP
1
+
'
P
^lP
= c
c^t +
c^t " P ^lt +
t
2
c
2
P
UcP cP
,
c^Pt
where second equality used SS under subsidy w = 1; ULP = "UCP
For N agents
UtN
Aggregating:
!
Ut " U = c
"
P !$
!
+n c
!
= c
"
N 1!$
ncP c^Pt " nlP ^ltP +
"
P 1!$
Take the linear term Örst
!
"U
N
c
,
,
cN
cP
,
1 " * ! N "2
+
c^t
2
c^N
t
-!$
(1 " n) cN c^N
t
1 " * ! P "2 lP 1 + ' )^P +2
c^t " P
lt
2
c
2
"
P !$
c^
ct " l^lt +
,
cN
cP
-!$
53
-
!
-
!
! "1!$ 1 " * ! N "2
+ (1 " n) cN
c^t
2
" 1 (1 " n) cN c^N
t
!
Economy resource constraint to second order is (denote ?t = 1 " /2 9 2t ):
)
+
^ t where
c^t = (1 " (1 " n) 4 c ) at + ^lt + ?
^t = "
?
D9
1
D
92
/ 2 9t "
1 " 29
2 1 " /2 9 2 t
Note that under zero ináation the linear term disappears. The squared term captures the welfare
cost of ináation.
Linear term becomes hence
where we recall q $ = 1 +
!
c
"
P !$
(1+4)5 !1 !1
;
1!A
)
^ t + at + (q $ " 1) (1 " n) h^
c ?
cN
t
+
At Friedman rule this is unity, and the linear term drops out.
Else, it is larger than 1 and linear term has positive coe¢cient ñ increasing consumption of N close
the inequality gap providing a Örst-order beneÖt.
Quadratic term is (ignore price dispersion because in quadratic terms it becomes third or fourth
order):
!
cP
"!$ ,
) ! "
! N "2 +
c
1+'
P 2
$
2
(1 " *) np c^t + (1 " n) hq c^t
"
c^
2
1 " (1 " n) 4 c t
Thus the loss function becomes, rearranging and ignoring terms independent of policy
0
1
^ t " (q $ " 1) (1 " n) h^
"?
cN
! P "!$
t
A
! N "2 + 1+'
L = c
c @ $!1 ) ! P "2
1
$
2
+ 2 1!(1!n)G
c
^
+ 2 np c^t + (1 " n) hq c^t
c t
0
)
+ 1
!
"
^ t " (q $ " 1) (1 " n) h c^N + 1!$ c^N 2
"?
! P "!$
t
t
2
)
+ A
= c
c @ $!1
$!1
1
+ 2 np (1 " n) h^
qt2 + 1+'
+
c^2t
2 1!(1!n)G c
2
Adding and subtracting the steady-state ináation constant and ignoring all terms independent of
policy, we obtain the loss function in text.
54