Optimal Monetary Policy with Heterogeneous Agents
Galo Nuño
Carlos Thomas
Banco de España
Banco de España
First version: March 2016
This version: May 2017
Abstract
Incomplete markets models with heterogeneous agents are increasingly used for policy
analysis. We propose a novel methodology for solving fully dynamic optimal policy problems
in models of this kind, both under discretion and commitment, that employs optimization
techniques in function spaces. We illustrate our methodology by studying optimal monetary policy in an incomplete-markets model with non-contingent nominal assets and costly
in‡ation. Under discretion, an in‡ationary bias arises from the central bank’s attempt to
redistribute wealth towards debtor households, which have a higher marginal utility of net
wealth. Under commitment, this in‡ationary force is counteracted over time by the incentive
to prevent expectations of future in‡ation from being priced into new bond issuances; under
certain conditions, long run in‡ation is zero as both e¤ects cancel out asymptotically. We
…nd numerically that the optimal commitment features …rst-order initial in‡ation followed
by a gradual decline towards its (near zero) long-run value. Welfare losses from discretionary
policy are …rst-order in magnitude, a¤ecting both debtors and creditors.
Keywords: optimal monetary policy, commitment and discretion, incomplete markets,
Gateaux derivative, nominal debt, in‡ation, redistributive e¤ects, continuous time
JEL codes: E5, E62,F34.
The views expressed in this manuscript are those of the authors and do not necessarily represent the views of
Banco de España or the Eurosystem. The authors are very grateful to Antonio Antunes, Adrien Auclert, Pierpaolo
Benigno, Saki Bigio, Marco del Negro, Emmanuel Farhi, Jesús Fernández-Villaverde, Luca Fornaro, Jesper Lindé,
Alberto Martin, Alisdair Mckay, Kurt Mitman, Ben Moll, Omar Rachedi, Frank Smets, Pedro Teles, Fabian Winkler,
conference participants at the ECB-NY Fed Global Research Forum on International Macroeconomics and Finance,
T2M Conference in Lisbon, and seminar participants at the University of Nottingham for helpful comments and
suggestions. We are also grateful to María Malmierca for excellent research assistance. This paper supersedes a
previous versions entitled “Optimal Monetary Policy in a Heterogeneous Monetary Union.” All remaining errors
are ours.
1
1
Introduction
Ever since the seminal work of Bewley (1983), Hugget (1993) and Aiyagari (1994), incomplete
markets models with uninsurable idiosyncratic risk have become a workhorse for policy analysis in
macro models with heterogeneous agents.1 Among the di¤erent areas spawned by this literature,
the analysis of the dynamic aggregate e¤ects of …scal and monetary policy has begun to receive
considerable attention in recent years.2
As is well known, one di¢ culty when working with incomplete markets models is that the state
of the economy at each point in time includes the cross-household wealth distribution, which is an
in…nite-dimensional object.3 The development of numerical methods for computing equilibrium in
these models has made it possible to study the e¤ects of aggregate shocks and of particular policy
rules. However, the in…nite-dimensional nature of the endogenously-evolving wealth distribution
has made it di¢ cult to make progress in the analysis of optimal …scal or monetary policy problems
in this class of models.
In this paper, we propose a novel methodology for solving fully dynamic optimal policy problems in incomplete-markets models with uninsurable idiosyncratic risk, both under discretion and
commitment. We exploit the fact that the continuous-time dynamics of the cross-sectional distribution are characterized by a partial di¤erential equation known as the Kolmogorov forward (KF)
or Fokker-Planck equation, and therefore the problem can be solved by using calculus techniques
in in…nite-dimensional Hilbert spaces. To this end, we employ a generalized version of the classical
di¤erential known as Gateaux di¤erential.
We illustrate our methodology by analyzing optimal monetary policy in an incomplete-markets
economy. Our framework is close to Huggett’s (1993) standard formulation. As in the latter,
households trade non-contingent claims, subject to an exogenous borrowing limit, in order to
smooth consumption in the face of idiosyncratic income shocks. We depart from Huggett’s real
framework by considering nominal non-contingent bonds with an arbitrarily long maturity, which
allows monetary policy to have an e¤ect on equilibrium allocations. In particular, our model
features a classic Fisherian channel (Fisher, 1933), by which unanticipated in‡ation redistributes
wealth from lending to borrowing households.4 In order to have a meaningful trade-o¤ in the choice
of the in‡ation path, we also assume that in‡ation is costly, which can be rationalized on the basis
of price adjustment costs; in addition, expected future in‡ation raises the nominal cost of new debt
1
For a survey of this literature, see e.g. Heathcote, Storesletten & Violante (2009).
See our discussion of the related literature below.
3
See e.g. Ríos-Rull (1995).
4
See Doepke and Schneider (2006a) for an in‡uential study documenting net nominal asset positions across
US household groups and estimating the potential for in‡ation-led redistribution. See Auclert (2016) for a recent
analysis of the Fisherian redistributive channel in a more general incomplete-markets model that allows for additional
redistributive mechanisms.
2
2
issuances through in‡ation premia. Finally, we depart from the standard closed-economy setup
by considering a small open economy, with the aforementioned (domestic currency-denominated)
bonds being also held by risk-neutral foreign investors. This, aside from making the framework
somewhat more tractable,5 also makes the policy analysis richer, by making the redistributive
Fisherian channel operate not only between domestic lenders and borrowers, but also between the
latter and foreign bond holders.6
On the analytical front, we show that discretionary optimal policy features an ’in‡ationary bias’,
whereby the central bank tries to use in‡ation so as to redistribute wealth and hence consumption.
In particular, we show that at each point in time optimal discretionary in‡ation increases with the
average cross-household net liability position weighted by each household’s marginal utility of net
wealth. This re‡ects the two redistributive motives mentioned before. On the one hand, in‡ation
redistributes from foreign investors to domestic borrowers (cross-border redistribution). On the
other hand, and somewhat more subtly, under market incompleteness and standard concave preferences for consumption, borrowing households have a higher marginal utility of net wealth than
lending ones. As a result, they receive a higher e¤ective weight in the optimal in‡ation decision,
giving the central bank an incentive to redistribute wealth from creditor to debtor households
(domestic redistribution).
Under commitment, the same redistributive motives to in‡ate exist, but they are counteracted
by an opposing force: the central bank internalizes how investors’expectations of future in‡ation
a¤ect their pricing of the long-term nominal bonds from the time the optimal commitment plan
is formulated (’time zero’) onwards. At time zero, in‡ation is close to that under discretion, as no
prior commitments about in‡ation exist. But from then on, the fact that the price of newly issued
bonds incorporates promises about the future in‡ation path gives the central bank an incentive
to commit to reducing in‡ation over time. Importantly, we show that under certain conditions on
preferences and parameter values, the steady state in‡ation rate under the optimal commitment
is zero;7 that is, in the long run the redistributive motive to in‡ate exactly cancels out with the
incentive to reduce in‡ation expectations and nominal yields for an economy that is a net debtor.
We then solve numerically for the full transition path under commitment and discretion. We
5
We restrict our attention to equilibria in which the domestic economy remains a net debtor vis-à-vis the rest of
the World, such that domestic bonds are always in positive net supply. As a result, the usual bond market clearing
condition in closed-economy models is replaced by a no arbitrage condition for foreign investors that e¤ectively prices
the nominal bond. This allows us to reduce the number of constraints in the policy-maker’s problem featuring the
in…nite-dimensional wealth distribution.
6
As explained by Doepke and Schneider (2006a), large net holdings of nominal (domestic currency-denominated)
assets by foreign investors increase the potential for a large in‡ation-induced wealth transfer from foreigners to
domestic borrowers.
7
In particular, assuming separable preferences, then in the limiting case in which the central bank’s discount
rate is arbitrarily close to that of foreign investors, optimal steady-state in‡ation under commitment is arbitrarily
close to zero.
3
calibrate our model to match a number of features of a prototypical European small open economy,
such as the size of gross household debt or their net international position.8 We …nd that optimal
time-zero in‡ation, which as mentioned before is very similar under commitment and discretion, is
…rst-order in magnitude. We also show that both the cross-border and the domestic redistributive
motives are quantitatively relevant for initial in‡ation. Under discretion, in‡ation remains high due
to the in‡ationary bias discussed before. Under commitment, by contrast, in‡ation falls gradually
towards its long-run level (essentially zero, under our calibration), re‡ecting the central bank’s
e¤orts to prevent expectations of future in‡ation from being priced into new bond issuances. In
summary, under commitment the central bank front-loads in‡ation so as to transitorily redistribute
existing wealth from lenders to borrowing households, but commits to gradually undo such initial
in‡ation.
In welfare terms, the discretionary policy implies sizable (…rst-order) losses relative to the
optimal commitment. Such losses are su¤ered by creditor households, but also by debtor ones.
The reason is that, under discretion, expectations of permanent future positive in‡ation are fully
priced into current nominal yields. This impairs the very redistributive e¤ects of in‡ation that
the central bank is trying to bring about, and leaves only the direct welfare costs of permanent
in‡ation, which are born by creditor and debtor households alike.
Finally, we compute the optimal monetary policy response to an aggregate shock to the World
interest rate. In particular, we compare impulse responses under a policy of zero in‡ation with
those under the optimal commitment plan ‘from a timeless perspective.’ In both cases, the rise
in interest rates reduces aggregate consumption. In the case of commitment in‡ation rises slightly
on impact, as the central bank tries to partially counteract the negative e¤ect of the shock on
household consumption. However, the in‡ation reaction is an order of magnitude smaller than
that of the shock itself. Intuitively, the value of sticking to past commitments to keep in‡ation
near zero weighs more in the central bank’s decision than the value of using in‡ation transitorily
so as to stabilize consumption in response to an unforeseen event.
Overall, our …ndings shed some light on current policy and academic debates regarding the
appropriate conduct of monetary policy once household heterogeneity is taken into account. In
particular, our results suggest that an optimal plan that includes a commitment to price stability
in the medium/long-run may also justify a relatively large (…rst-order) positive initial in‡ation
rate, with a view to shifting resources to households that have a relatively high marginal utility of
net wealth.
Related literature. Our main contribution is methodological. To the best of our knowledge,
ours is the …rst paper to solve for a fully dynamic optimal policy problem, both under commitment
8
These targets are used to inform the calibration of the gap between the central bank’s and foreign investors’
discount rates, which as explained before is a key determinant of long-run in‡ation under commitment.
4
and discretion, in a general equilibrium model with uninsurable idiosyncratic risk in which the
cross-sectional net wealth distribution (an in…nite-dimensional, endogenously evolving object) is
a state in the planner’s optimization problem. Di¤erent papers have analyzed Ramsey problems
in similar setups. Dyrda and Pedroni (2014) study the optimal dynamic Ramsey taxation in an
Aiyagari economy. They assume that the paths for the optimal taxes follow splines with nodes set
at a few exogenously selected periods, and perform a numerical search of the optimal node values.
Acikgoz (2014), instead, follows the work of Davila et al. (2012) in employing calculus of variations
to characterize the optimal Ramsey taxation in a similar setting. However, after having shown that
the optimal long-run solution is independent of the initial conditions, he analyzes quantitatively
the steady state but does not solve the full dynamic optimal path.9 Other papers, such as Gottardi,
Kajii, and Nakajima (2011), Itskhoki and Moll (2015), Bilbiie and Ragot (2017) or Challe (2017),
analyze optimal Ramsey policies in incomplete-market models in which the policy-maker does
not need to keep track of the wealth distribution.10 In contrast to these papers, we introduce a
methodology for computing the full dynamics under commitment in a general incomplete-markets
setting. Regarding discretion, we are not aware of any previous paper that has quantitatively
analyzed the Markov Perfect Equilibrium (MPE) in models with uninsurable idiosyncratic risk.
A recent contribution by Bhandari et al. (2017), released after the …rst draft of this paper was
circulated, has also addressed the issue of optimal control with commitment based on an alternative
methodology. They employ standard calculus techniques to obtain the …rst-order conditions of
the Ramsey problem in a problem with a …nite number of agents. The computation is then
performed by simulating a large number of agents and applying perturbation techniques to improve
the computational tractability. Our paper, by constrast, employs in…nite-dimensional calculus to
obtain the …rst order conditions, so that the the solution reduces to an extension of a standard
heterogeneous-agents model in which there is a direct mapping from the equilibrium objects to the
optimal policies –in‡ation in our case.
The use of in…nite-dimensional calculus in problems with non-degenerate distributions is employed in Lucas and Moll (2014) and Nuño and Moll (2015) to …nd the …rst-best and the constrainede¢ cient allocation in heterogeneous-agents models. In these papers a social planner directly decides
on individual policies in order to control a distribution of agents subject to idiosyncratic shocks.
Here, by contrast, we show how these techniques may be extended to game-theoretical settings
9
Werning (2007) studies optimal …scal policy in a heterogeneous-agents economy in which agent types are
permanently …xed. Park (2014) extends this approach to a setting of complete markets with limited commitment
in which agent types are stochastically evolving. Both papers provide a theoretical characterization of the optimal
policies based on the primal approach introduced by Lucas and Stokey (1983). Aditionally, Park (2014) analyzes
numerically the steady state but not the transitional dynamics, due to the complexity of solving the latter problem
with that methodology.
10
This is due either to particular assumptions that facilitate aggregation or to the fact that the equilibrium net
wealth distribution is degenerate at zero.
5
involving several agents who are moreover forward-looking. Under commitment, as is well known,
this requires the policy-maker to internalize how her promised future decisions a¤ect private agents’
expectations; the problem is then augmented by introducing costates that re‡ect the value of deviating from the promises made at time zero.11 If commitment is not possible, the value of these
costates is zero at all times.12
Our techniques can also be applied in discrete-time settings, as we describe in the online
appendix. Notwithstanding, our baseline analysis assumes continuous time because it helps to
improve the e¢ ciency of the numerical solution, as discussed in Achdou, Lasry, Lions and Moll
(2015) or Nuño and Thomas (2015). This is due to two properties of continuous-time models.
First, the HJB equation is a deterministic partial di¤erential equation which can be solved using
e¢ cient …nite-di¤erence methods. Second, the dynamics of the distribution can be computed
relatively quickly as they amount to calculating a matrix adjoint: the KF operator is the adjoint
of the in…nitesimal generator of the underlying stochastic process. This computational speed is
essential as the computation of the optimal policies requires several iterations along the complete
time-path of the distribution.13
Aside from the methodological contribution, our paper relates to several strands of the literature. As explained before, our analysis assigns an important role to the Fisherian redistributive channel of monetary policy, a long-standing topic that has experienced a revival in recent
years. Doepke and Schneider (2006a) document net nominal asset positions across US sectors and
household groups and estimate empirically the redistributive e¤ects of di¤erent in‡ation scenarios.
Adam and Zhu (2014) perform a similar analysis for Euro Area countries, adding the cross-country
redistributive dimension to the picture.
A recent literature addresses the Fisherian and other channels of monetary policy transmission
in the context of general equilibrium models with incomplete markets and household heterogeneity.
In terms of modelling, our paper is closest to Auclert (2016), Kaplan, Moll and Violante (2016),
Gornemann, Kuester and Nakajima (2012), McKay, Nakamura and Steinsson (2015) or Luetticke
(2015), who also employ di¤erent versions of the incomplete-markets, uninsurable idiosyncratic
risk framework.14 Other contributions, such as Doepke and Schneider (2006b), Meh, Ríos-Rull
and Terajima (2010), Sheedy (2014), Challe et al. (2015) or Sterk and Tenreyro (2015), analyze
the redistributive e¤ects of monetary policy in environments where heterogeneity is kept …nite11
In the commitment case, we construct a Lagrangian in a suitable function space and obtain the corresponding
…rst-order conditions. The resulting optimal policy is time inconsistent (re‡ecting the e¤ect of investors’in‡ation
expectations on bond pricing), depending only on time and the initial wealth distribution.
12
Under discretion, we work with a generalization of the Bellman principle of optimality and the Riesz representation theorem to obtain the time-consistent optimal policies depending on the distribution at any moment in
time.
13
In a home PC, the Ramsey problem presented here can be solved in less than …ve minutes.
14
For work studying the e¤ects of di¤erent aggregate shocks in related environments, see e.g. Guerrieri and
Lorenzoni (2016), Ravn and Sterk (2013), and Bayer et al. (2015).
6
dimensional. We contribute to this literature by analyzing fully dynamic optimal monetary policy,
both under commitment and discretion, in a standard incomplete markets model with uninsurable
idiosyncratic risk.
Although this paper focuses on monetary policy, the techniques developed here lend themselves
naturally to the analysis of other policy problems, e.g. optimal …scal policy, in this class of
models. Recent work analyzing …scal policy issues in incomplete-markets, heterogeneous-agent
models includes Heathcote (2005), Oh and Reis (2012), Kaplan and Violante (2014) and McKay
and Reis (2016).
Finally, our paper is related to the literature on mean-…eld games in Mathematics. The name,
introduced by Lasry and Lions (2006a,b), is borrowed from the mean-…eld approximation in statistical physics, in which the e¤ect on any given individual of all the other individuals is approximated
by a single averaged e¤ect. In particular, our paper is related to Bensoussan, Chau and Yam (2015),
who analyze a model of a major player and a distribution of atomistic agents that shares some
similarities with the Ramsey problem discussed here.15
2
Model
We extend the basic Huggett framework to an open-economy setting with nominal, non-contingent,
long-term debt contracts and disutility costs of in‡ation. Let ( ; F; fFt g ; P) be a …ltered probability space. Time is continuous: t 2 [0; 1). The domestic economy is composed of a measure-one
continuum of households that are heterogeneous in their net …nancial wealth. There is a single,
freely traded consumption good, the World price of which is normalized to 1. The domestic price
(equivalently, the nominal exchange rate) at time t is denoted by Pt and evolves according to
dPt =
where
tion).
2.1
2.1.1
t
t Pt dt;
(1)
is the domestic in‡ation rate (equivalently, the rate of nominal exchange rate deprecia-
Households
Output and net assets
Household k 2 [0; 1] is endowed with an income ykt units of the good at time t, where ykt follows
a two-state Poisson process: ykt 2 fy1 ; y2 g ; with y1 < y2 . The process jumps from state 1 to state
15
Other papers analyzing mean-…eld games with a large non-atomistic player are Huang (2010), Nguyen and
Huang (2012a,b) and Nourian and Caines (2013). A survey of mean-…eld games can be found in Bensoussan, Frehse
and Yam (2013).
7
2 with intensity 1 and vice versa with intensity 2 .
Households trade a nominal, non-contingent, long-term, domestic-currency-denominated bond
with one another and with foreign investors. Let Akt denote the net holdings of such bond by
household k at time t; assuming that each bond has a nominal value of one unit of domestic
currency, Akt also represents the total nominal (face) value of net assets. For households with a
negative net position, ( ) Akt represents the total nominal (face) value of outstanding net liabilities
(‘debt’for short). We assume that outstanding bonds are amortized at rate > 0 per unit of time.16
The nominal value of the household’s net asset position thus evolves as follows,
dAkt = (Anew
kt
Akt ) dt;
is the ‡ow of new assets purchased at time t. The nominal market price of bonds at
where Anew
kt
time t is Qt . Let ckt denote the household’s consumption. The budget constraint of household k
is then
Qt Anew
ckt ) + Akt :
kt = Pt (ykt
Combining the last two equations, we obtain the following dynamics for net nominal wealth,
dAkt =
We de…ne real net wealth as akt
equations (1) and (2),
dakt =
Qt
Akt dt +
Pt (ykt ckt )
dt:
Qt
(2)
Akt =Pt . Its dynamics are obtained by applying Itô’s lemma to
t
Qt
akt +
ykt
ckt
Qt
dt:
(3)
We assume that each household faces the following exogenous borrowing limit,
akt
(4)
:
where
0.
For future reference, we de…ne the nominal bond yield rt implicit in a nominal bond price Qt as
the discount rate for which the discounted future promised cash ‡ows equal the bond price. The
R1
discounted future promised payments are 0 e (rt + )s ds = = (rt + ). Therefore, the nominal
bond yield is
rt =
16
Qt
:
This tractable form of long-term bonds was …rst introduced by Leland and Toft (1986).
8
(5)
2.1.2
Preferences
Household have preferences over paths for consumption ckt and domestic in‡ation
at rate > 0,
Z 1
Uk0 E0
e t u(ckt ; t )dt ;
t
discounted
(6)
0
with uc > 0, u > 0, ucc < 0 and u < 0.17 From now onwards we drop subscripts k for ease
of exposition. The household chooses consumption at each point in time in order to maximize its
welfare. The value function of the household at time t can be expressed as
v(t; a; y) =
max
fcs gs2[t;1)
Et
Z
1
e
(s t)
u(cs ;
s )ds
(7)
;
t
subject to the law of motion of net wealth (3) and the borrowing limit (4). We use the short-hand
notation vi (t; a) v(t; a; yi ) for the value function when household income is low (i = 1) and high
(i = 2): The Hamilton-Jacobi-Bellman (HJB) equation corresponding to the problem above is
vi (t; a) =
@vi
@vi
+ max u(c; (t)) + si (t; a; c)
c
@t
@a
+
i
[vj (t; a)
vi (t; a)] ;
(8)
for i; j = 1; 2; and j 6= i; where si (t; a; c) is the drift function, given by
si (t; a; c) =
(t) a +
Q (t)
yi c
; i = 1; 2:
Q (t)
(9)
The …rst order condition for consumption is
uc (ci (t; a) ; (t)) =
1 @vi (t; a)
:
Q (t) @a
(10)
Therefore, household consumption increases with nominal bond prices and falls with the slope of
the value function. Intuitively, a higher bond price (equivalently, a lower yield) gives the household
an incentive to save less and consume more. A steeper value function, on the contrary, makes it
more attractive to save so as to increase net asset holdings.
2.2
Foreign investors
Households trade bonds with competitive risk-neutral foreign investors that can invest elsewhere
at the risk-free real rate r. As explained before, domestic bonds are amortized at rate . Foreign
17
The general speci…cation of disutility costs of in‡ation nests the case of costly price adjustments à la Rotemberg.
See Section 4.1 for further discussion.
9
investors also discount future future nominal payo¤s with the accumulated domestic in‡ation (i.e.
exchange rate depreciation) between the time of the bond purchase and the time such payo¤s
accrue. Therefore, the nominal price of the bond at time t is given by
Q(t) =
Z
1
e
(r+ )(s t)
t
Rs
t
u du
(11)
ds:
Taking the derivative with respect to time, we obtain
_
Q(t) (r + + (t)) = + Q(t);
(12)
_
where Q(t)
dQ=dt: The partial di¤erential equation (12) provides the risk-neutral pricing of the
nominal bond. The boundary condition is
lim Q(t) =
t!1
where
2.3
r + + (1)
(13)
;
(1) is the in‡ation level in the steady state, which we assume exits.
Central Bank
There is a central bank that chooses monetary policy. We assume that there are no monetary
frictions so that the only role of money is that of a unit of account. The monetary authority
chooses the in‡ation rate t .18 In Section 3, we will study in detail the optimal in‡ationary policy
of the central bank.
2.4
Competitive equilibrium
The state of the economy at time t is the joint density of net wealth and output, f (t; a; yi ) fi (t; a),
i = 1; 2. The dynamics of this density are given by the Kolmogorov Forward (KF) equation,
@fi (t; a)
=
@t
@
[si (t; a) fi (t; a)]
@a
i fi (t; a)
+
j fj (t; a);
(14)
a 2 [ ; 1); i; j = 1; 2; j 6= i: The density satis…es the normalization
2 Z
X
1
fi (t; a) da = 1:
(15)
i=1
18
This could be done, for example, by setting the nominal interest rate on a lending (or deposit) short-term
nominal facility with foreign investors.
10
We de…ne a competitive equilibrium in this economy.
De…nition 1 (Competitive equilibrium) Given a sequence of in‡ation rates (t) and an initial wealth-output density f (0; a; y), a competitive equilibrium is composed of a household value
function v(t; a; y), a consumption policy c(t; a; y); a bond price function Q (t) and a density f (t; a; y)
such that:
1. Given ; the price of bonds in (12) is Q.
2. Given Q and , v is the solution of the households’ problem (8) and c is the optimal consumption policy:
3. Given Q; ; and c, f is the solution of the KF equation (14).
Notice that, given ; the problem of foreign investors can be solved independently of that of
the household, which in turn only depends on and Q but not on the aggregate distribution.
We can compute some aggregate variables of interest. The aggregate real net …nancial wealth
in the economy is
2 Z 1
X
at
afi (t; a) da:
(16)
i=1
P2 R 0
( a) fi (t; a) da. Aggregate
We may similarly de…ne gross real household debt as bt
i=1
consumption is
2 Z 1
X
ct
ci (a; t) fi (t; a) da;
i=1
where ci (a; t)
c(t; a; yi ), i = 1; 2, and aggregate output is
2 Z
X
yt
1
yi fi (t; a) da:
i=1
These quantities are linked by the current account identity,
2 Z
X
dat
=
dt
i=1
2 Z
X
=
i=1
=
Qt
1
1
X
@fi (t; a)
a
da =
@t
i=1
2
a
Z
@
(si fi ) da
i fi (t; a) +
@a
2
2 Z 1
X
X
1
asi fi j +
si fi da
@
(si fi ) da =
@a
t
at +
yt
a
i=1
ct
Qt
1
j fj (t; a)
da
i=1
(17)
;
11
where we have used (14) in the second equality, and we have applied the boundary conditions
s1 (t; ) f1 (t; ) + s2 (t; ) f2 (t; ) = 0 in the last equality.19
Finally, we make the following assumption.
Assumption 1 The value of parameters is such that in equilibrium the economy is always a net
debtor against the rest of the World: at 0 for all t:
This condition is imposed for tractability. We have restricted households to save only in bonds
issued by other households, and this would not be possible if the country was a net creditor visà-vis the rest of the World. In addition to this, we have assumed that the bonds issued by the
households are priced by foreign investors, which requires that there should be a positive net supply
of bonds to the rest of the World to be priced. In any case, this assumption is consistent with the
experience of the small open economies that we target for calibration purposes, as we explain in
Section 4.
3
Optimal monetary policy
We now turn to the design of the optimal monetary policy. Following standard practice, we assume
that the central bank is utilitarian, i.e. it gives the same Pareto weight to each household. In order
to illustrate the role of commitment vs. discretion in our framework, we will consider both the case
in which the central bank can credibly commit to a future in‡ation path (the Ramsey problem),
and the time-consistent case in which the central bank decides optimal current in‡ation given the
current state of the economy (the Markov Perfect equilibrium).
3.1
Central bank preferences
The central bank is assumed to be benevolent and hence maximizes economy-wide aggregate
welfare,
Z 1X
2
CB
U0 =
vi (0; a) fi (0; a)da:
(18)
i=1
It will turn out to be useful to express the above welfare criterion as follows.
Lemma 1 The welfare criterion (18) can alternatively be expressed as
U0CB
Z
0
19
1
e
s
Z
1
X2
i=1
u (ci (s; a) ; (s)) fi (s; a)da ds:
(19)
This condition is related to the fact that the KF operator is the adjoint of the in…nitesimal generator of the
stochastic process (3). See Appendix A for more information. See also Appendix B.6 in Achdou et al. (2015).
12
3.2
Discretion (Markov Perfect Equilibrium)
Consider …rst the case in which the central bank cannot commit to any future policy. The in‡ation
rate then depends only on the current value of the aggregate state variable, the net wealth
MP E
distribution ffi (t; a)gi=1;2 f (t; a); that is, (t)
[f (t; a)] : This is a Markovian problem
in a space of distributions. The value functional of the central bank is given by
W M P E [f (t; )] =
f
max
s gs2[t;1)
Z
1
e
(s t)
t
" 2 Z
X
1
u (cis (a) ;
s ) fi (s; a)da
i=1
#
ds;
(20)
subject to the law of motion of the distribution (14). Notice that the optimal value W M P E and
the optimal policy M P E are not ordinary functions, but functionals, as they map the in…nitedimensional state variable f (t; a) into R.
Let f0 ( ) ffi (0; a)gi=1;2 denote the initial distribution. We can de…ne the equilibrium in this
case.
De…nition 2 (Markov Perfect Equilibrium) Given an initial distribution f0 ; a symmetric
Markov Perfect Equilibrium is composed of a sequence of in‡ation rates (t) ; a household value
function v(t; a; y), a consumption policy c(t; a; y); a bond price function Q (t) and a distribution
f (t; a; y) such that:
1. Given ; then v; c; Q and f are a competitive equilibrium.
2. Given c; Q and f ,
is the solution to the central bank problem (20).
The fact that v; c; Q and f are part of a competitive equilibrium needs to be imposed in the
de…nition of Markov Perfect Equilibrium, as it is not implicit in the central bank’s problem (20).
Using dynamic programming arguments, the problem (20) can be expressed recursively as
W
MP E
[f (t; )] =
f
max
s gs2[t;
)
Z
e
(s t)
t
Z
1
X2
i=1
u (cis ;
s ) fi (s; a)da
ds+e
(
t)
W M P E [f ( ; )] ;
(21)
for any > t and subject to the law of motion of the distribution (14).
The following proposition characterizes the solution to the central bank’s problem under discretion.
Proposition 1 (Optimal in‡ation - MPE) In addition to equations (14), (12), (8) and (10),
if a solution to the MPE problem (20) exists, the in‡ation rate function (t) must satisfy
2 Z
X
i=1
1
a
@vi
@a
u (ci (t; a) ; (t)) fi (t; a) da = 0:
13
(22)
In addition, the value functional must satisfy
W
MP E
[f (t; )] =
2 Z
X
1
vi (t; a) fi (t; a)da;
(23)
i=1
The proof is in the Appendix. The idea is to derive a Hamilton-Jacobi-Bellman (HJB) equation
for the policy-maker in a suitable function space. To this end we need to employ the concept of
Gateaux derivative, which extends the concept of derivative from Rn to in…nite-dimensional spaces,
and the Riesz Representation Theorem, which allows decomposing the Gateux derivatives of the
central bank value functional W M P E as an aggregation of individual values vi (t; a) across agents.20
The optimal in‡ation is then obtained by taking the standard derivative at any point in time of
this HJB equation. The use of the Gateaux derivative can be seen as a formalization of variational
arguments.
Equation (22) captures the basic static trade-o¤ that the central bank faces when choosing
in‡ation under discretion. The central bank balances the marginal utility cost of higher in‡ation
across the economy (u ) against the marginal welfare e¤ects due to the impact of in‡ation on the
i
). For borrowing households (a < 0), the latter
real value of households’nominal net positions (a @v
@a
e¤ect is positive as in‡ation erodes the real value of their debt burden, whereas the opposite is
true for creditor ones (a > 0). Moreover, provided that the value function is concave in net wealth
@ 2 vi
< 0; i = 1; 2; ; and given Assumption 1 (the country as a whole is a net debtor), the central
@a2
bank has a double motive to use in‡ation for redistributive purposes.21 On the one hand, it will try
to redistribute wealth from foreign investors to domestic borrowers (cross-border redistribution).
On the other hand, and somewhat more subtly, since borrowing households have a higher marginal
utility of net wealth than creditor ones, the central bank will be led to redistribute from the latter
to the former, as such course of action is understood to raise welfare in the domestic economy as
a whole (domestic redistribution).
3.3
Commitment
Assume now that the central bank can credibly commit at time zero to an in‡ation path f (t)g1
t=0 .
The optimal in‡ation path is now a function of the initial distribution f0 (a) and of time: (t)
20
In fact, one of the advantages of the example of an small open economy is that the social value of an agent
coincides with its private value. In the case of closed economy version of the model this would not be the case,
making the computations more complex, but still tractable.
21
The concavity of the value function is guaranteed for the separable utility function presented in Assumption 2
below.
14
R
[t; f0 (a)] : The value functional of the central bank is now given by
R
W [f0 ( )] =
f
s ;Qs ;v(s;
max
);c(s; );f (s; )gs2[t;1)
Z
1
e
s
0
Z
1
X2
i=1
u (cis ;
s ) fi (s; a)da
ds;
(24)
subject to the law of motion of the distribution (14), the bond pricing equation (12), and household’s HJB equation (8) and optimal consumption choice (10). The optimal value W R and the
optimal policy R are again functionals, as in the discretionary case, only now they map the initial
distribution f0 ( ) into R, as opposed to the distribution at each point in time. Notice that the
central bank maximizes welfare taking into account not only the state dynamics (14), but also the
households’HJB equation (8) and the investors’bond pricing condition (12). That is, the central
bank understands how it can steer households’and foreign investors’expectations by committing
to an in‡ation path. This is unlike in the discretionary case, where the central bank takes the
expectations of other agents as given.
De…nition 3 (Ramsey problem) Given an initial distribution f0 ; a Ramsey problem is composed of a sequence of in‡ation rates (t) ; a household value function v(t; a; y), a consumption
policy c(t; a; y); a bond price function Q (t) and a distribution f (t; a; y) such that they solve the
central bank problem (24).
If v; f; c and Q are a solution to the problem (24), given , they constitute a competitive
equilibrium, as they satisfy equations (14), (12), (8) and (10). Therefore the Ramsey problem
could be rede…ned as that of …nding the such that v; f; c and Q are a competitive equilibrium
and the central bank’s welfare criterion is maximized.
The Ramsey problem is an optimal control problem in a suitable function space. The following
proposition characterizes the solution to the central bank’s problem under commitment.
Proposition 2 (Optimal in‡ation - Ramsey) In addition to equations (14), (12), (8) and
(10), if a solution to the Ramsey problem (24) exists, the in‡ation path (t) must satisfy
2 Z
X
1
a
i=1
where
@vit
@a
u (ci (t; a) ; (t)) fi (t; a) da
(t) Q (t) = 0;
(25)
(t) is a costate with law of motion
d (t)
=(
dt
and initial condition
r
(t)
) (t) +
2 Z
X
i=1
(0) = 0.
15
1
@vit a + yi ci (t; a)
fi (t; a) da
@a
Q (t)2
(26)
The proof can also be found in the Appendix. The idea in this case is to construct a Lagragian
in a suitable function space and to obtain the …rst-order conditions by taking Gateaux derivatives.
In the appendix we show that the Lagrange multiplier associated with the KF equation (14), which
represents the social value of an individual household, coincides with the private value vit (a). In
addition, the Lagrange multipliers associated with the households’HJB and …rst-order conditions
(8) and (10), are zero. Therefore, the only nontrivial Lagrange multiplier is the one associated
with the bond pricing equation (12), denoted by (t) in Proposition 2.
The equation determining optimal in‡ation under commitment (25), is identical to that in the
discretionary case (22), except for the presence of the costate (t). Intuitively, (t) captures the
value to the central bank of promises about time-t in‡ation made to foreign investors at time 0.
Such value is zero only at the time of announcing the Ramsey plan (t = 0), because the central
bank is not bound by previous commitments, but it will generally be di¤erent from zero at any
time t > 0. By contrast, in the MPE case no promises are made at any point in time, hence the
absence of such costate. Therefore, the static trade-o¤ between the welfare cost of in‡ation and
the welfare gains from in‡ating away net liabilities, explained above in the context of the MPE
solution, is now modi…ed by the central bank’s need to respect past promises to investors about
current in‡ation. If (t) < 0, then the central bank’s incentive to create in‡ation at time t > 0
so as to redistribute wealth will be tempered by the fact that it internalizes how expectations of
higher in‡ation a¤ect investors’bond pricing prior to time t.
Notice that the Ramsey problem is not time-consistent, due precisely to the presence of the
(forward-looking) bond pricing condition in that problem.22 If at some future time t~ > 0 the central
bank decided to re-optimize given the current state f t~; a; y ; the new path for optimal in‡ation
R
R
~ (t)
t; f t~;
would not need to coincide with the original path (t)
[t; f (0; )], as
the value of the costate at that point would be ~ t~ = 0 (corresponding to a new commitment
formulated at time t), whereas under the original commitment it is
t~ 6= 0.23
Importantly, none of the techniques introduced above are restricted to continuous-time problems. In fact, the equivalent discrete-time model can also be solved using the same techniques
at the cost of slightly more complicated notation. Appendix E shows how our methodology can
be used to solve for optimal policy, both under discretion and commitment, in the discrete-time
version of our model.
22
As is well known, the MPE solution is time consistent, as it only depends on the current state.
We also note that in the Ramsey equilibrium the Lagrange multipliers associated to households’HJB equation
(8) and optimal consumption decision (10) are zero in all states (see the Appendix). That is, households’forwardlooking optimizing behavior does not represent a source of time-inconsistency, as the monetary authority would
choose at all times the same individual consumption and saving policies as the households themselves.
23
16
3.4
Some analytical results
In order to provide some additional analytical insights on optimal policy, we make the following
assumption on preferences.
Assumption 2 Consider the class of separable utility functions
u (c; ) = uc (c)
u ( ):
The consumption utility function uc is bounded, concave and continuous with ucc > 0; uccc < 0 for
c > 0. The in‡ation disutility function u satis…es u > 0 for > 0; u < 0 for < 0; u > 0
for all , and u (0) = u (0) = 0.
We …rst obtain the following result.
Lemma 2 Let preferences satisfy Assumption 2. The optimal value function is concave.
The following result establishes the existence of a positive in‡ationary bias under discretionary
optimal monetary policy.
Proposition 3 (In‡ation bias under discretion) Let preferences satisfy Assumption 2. Optimal in‡ation under discretion is then positive at all times: (t) > 0 for all t 0:
The proof can be found in Appendix A. To gain intuition, we can use the above separable
preferences in order to express the optimal in‡ation decision under discretion (equation 22) as
u ( (t)) =
2 Z
X
1
( a)
i=1
@vi
fi (t; a) da:
@a
(27)
That is, under discretion in‡ation increases with the average net liabilities weighted by each household’s marginal utility of wealth, @vi =@a. Notice …rst that, from Assumption 1, the country as
P2 R 1
a whole is a net debtor:
( a) fi (t; a) da = ( ) at
0. This, combined with the strict
i=1
concavity of the value function (such that debtors e¤ectively receive more weight than creditors),
makes the right-hand side of (27) strictly positive. Since u ( ) > 0 only for > 0, it follows
that in‡ation must be positive. Notice that, even if the economy as a whole is neither a creditor
or a debtor (at = 0), the concavity of the value function implies that, as long as there is wealth
dispersion, the central bank will have a reason to in‡ate.
The result in Proposition 3 is reminiscent of the classical in‡ationary bias of discretionary
monetary policy originally emphasized by Kydland and Prescott (1977) and Barro and Gordon
(1983). In those papers, the source of the in‡ation bias is a persistent attempt by the monetary
17
authority to raise output above its natural level. Here, by contrast, it arises from the welfare gains
that can be achieved for the country as a whole by redistributing wealth towards debtors.
We now turn to the commitment case. Under the above separable preferences, from equation
(25) optimal in‡ation under commitment satis…es
u ( (t)) =
2 Z
X
1
( a)
i=1
@vi
fi (t; a) da + (t) Q (t) :
@a
(28)
In this case, the in‡ationary pressure coming from the redistributive incentives is counterbalanced
by the value of time-0 promises about time-t in‡ation, as captured by the costate (t). Thus, a
negative value of such costate leads the central bank to choose a lower in‡ation rate than the one
it would set ceteris paribus under discretion.
Unfortunately, we cannot solve analytically for the optimal path of in‡ation. However, we
are able to establish the following important result regarding the long-run level of in‡ation under
commitment.
Proposition 4 (Optimal long-run in‡ation under commitment) Let preferences satisfy Assumption 2. In the limit as ! r, the optimal steady-state in‡ation rate under commitment tends
to zero: lim (1) = 0.
!r
Provided households’ (and the benevolent central bank’s) discount factor is arbitrarily close
to that of foreign investors, then optimal long-run in‡ation under commitment will be arbitrarily
close to zero. The intuition is the following. The in‡ation path under commitment converges over
time to a level that optimally balances the marginal welfare costs and bene…ts of trend in‡ation.
On the one hand, the welfare costs include the direct utility costs, but also the increase in nominal
bond yields that comes about with higher expected in‡ation; indeed, from the de…nition of the
yield (5) and the expression for the long-run nominal bond price (13), the long-run nominal bond
yield is given by the following long-run Fisher equation,
r (1) =
= r + (1) ;
Q (1)
(29)
such that nominal yields increase one-for-one with (expected) in‡ation in the long run. On the
other hand, the welfare bene…ts of in‡ation are given by its redistributive e¤ect (for given nominal
yields). As ! r, these e¤ects tend to exactly cancel out precisely at zero in‡ation.
Proposition 4 is reminiscent of a well-known result from the New Keynesian literature, namely
that optimal long-run in‡ation in the standard New Keynesian framework is exactly zero (see e.g.
Benigno and Woodford, 2005). In that framework, the optimality of zero long-run in‡ation arises
18
from the fact that, at that level, the welfare gains from trying to exploit the short-run outputin‡ation trade-o¤ (i.e. raising output towards its socially e¢ cient level) exactly cancel out with the
welfare losses from permanently worsening that trade-o¤ (through higher in‡ation expectations).
Key to that result is the fact that, in that model, price-setters and the (benevolent) central bank
have the same (steady-state) discount factor. Here, the optimality of zero long-run in‡ation re‡ects
instead the fact that, at zero trend in‡ation, the welfare gains from trying to redistribute wealth
from creditors to debtors becomes arbitrarily close to the welfare losses from lower nominal bond
prices when the discount rate of the investors pricing such bonds is arbitrarily close to that of the
central bank.
Assumption 1 restricts us to have > r, as otherwise households would we able to accumulate
enough wealth so that the country would stop being a net debtor to the rest of the World. However,
Proposition 4 provides a useful benchmark to understand the long-run properties of optimal policy
in our model when is very close to r. This will indeed be the case in our subsequent numerical
analysis.
4
Numerical analysis
In the previous section we have characterized the optimal monetary policy in our model. In this
section we solve numerically for the dynamic equilibrium under optimal policy, using numerical
methods to solve continuous-time models with heterogeneous agents, as in Achdou et al. (2015)
or Nuño and Moll (2015). Before analyzing the dynamic path of this economy under the optimal
policy, we …rst analyze the steady state towards which such path converges asymptotically. The
numerical algorithms that we use are described in Appendices B (steady-state) and C (transitional
dynamics).
4.1
Calibration
The calibration is intended to be mainly illustrative, given the model’s simplicity and parsimoniousness. We calibrate the model to replicate some relevant features of a prototypical European
small open economy.24 Let the time unit be one year. For the calibration, we consider that the
24
We will focus for illustration on the UK, Sweden, and the Baltic countries (Estonia, Latvia, Lithuania). We
choose these countries because they (separately) feature desirable properties for the purpose at hand. On the one
hand, UK and Sweden are two prominent examples of relatively small open economies that retain an independent
monetary policy, like the economy in our framework. This is unlike the Baltic states, who recently joined the euro.
However, historically the latter states have been relatively large debtors against the rest of the World, which make
them square better with our theoretical restriction that the economy remains a net debtor at all times (UK and
Sweden have also remained net debtors in basically each quarter for the last 20 years, but on average their net
balance has been much closer to zero).
19
economy rests at the steady state implied by a zero in‡ation policy.25 When integrating across
households, we therefore use the stationary wealth distribution associated to such steady state.26
We assume the following speci…cation for preferences,
u (c; ) = log (c)
2
2
:
(30)
As discussed in Appendix D, our quadratic speci…cation for the in‡ation utility cost, 2 2 , can be
micro-founded by modelling …rms explicitly and allowing them to set prices subject to standard
quadratic price adjustment costs à la Rotemberg (1982). We set the scale parameter such that
the slope of the in‡ation equation in a Rotemberg pricing setup replicates that in a Calvo pricing
setup for reasonable calibrations of price adjustment frequencies and demand curve elasticities.27
We jointly set households’discount rate and borrowing limit such that the steady-state net
international investment position (NIIP) over GDP (a=y) and gross household debt to GDP (b=y)
replicate those in our target economies.28
We target an average bond duration of 4.5 years, as in Auclert (2016). In our model, the
Macaulay bond duration equals 1= ( + r). We set the world real interest rate r to 3 percent. Our
duration target then implies an amortization rate of = 0:19.
The idiosyncratic income process parameters are calibrated as follows. We follow Huggett
(1993) in interpreting states 1 and 2 as ’unemployment’ and ’employment’, respectively. The
transition rates between unemployment and employment ( 1 ; 2 ) are chosen such that (i) the
unemployment rate 2 = ( 1 + 2 ) is 10 percent and (ii) the job …nding rate is 0:1 at monthly
25
This squares reasonably well with the experience of our target economies, which have displayed low and stable
in‡ation for most of the recent past.
26
The wealth dimension is discretized by using 1000 equally-spaced grid points from a = to a = 10. The upper
bound is needed only for operational purposes but is fully innocuous, because the stationary distribution places
essentially zero mass for wealth levels above a = 8.
27
The slope of the continuous-time New Keynesian Phillips curve in the Calvo model can be shown to be given
by ( + ), where is the price adjustment rate (the proof is available upon request). As shown in Appendix D,
in the Rotemberg model the slope is given by " 1 , where " is the elasticity of …rms’demand curves and is the
scale parameter in the quadratic price adjustment cost function in that model. It follows that, for the slope to be
the same in both models, we need
" 1
=
:
( + )
Setting " to 11 (such that the gross markup "= (" 1) equals 1.10) and to 4=3 (such that price last on average
for 3 quarters), and given our calibration for , we obtain = 5:5.
28
According to Eurostat, the NIIP/GDP ratio averaged minus 48.6% across the Baltic states in 2016:Q1, and
only minus 3.8% across UK-Sweden. We thus target a NIIP/GDP ratio of minus 25%, which is about the midpoint
of both values. Regarding gross household debt, we use BIS data on ’total credit to households’, which averaged
85.9% of GDP across Sweden-UK in 2015:Q4 (data for the Baltic countries are not available). We thus target a
90% household debt to GDP ratio.
20
frequency or 1 = 0:72 at annual frequency.29 These numbers describe the ‘European’labor market
calibration in Blanchard and Galí (2010). We normalize average income y = 1 +2 2 y1 + 1 +1 2 y2 to
1. We also set y1 equal to 71 percent of y2 , as in Hall and Milgrom (2008). Both targets allow us
to solve for y1 and y2 . Table 1 summarizes our baseline calibration. Figure 1 displays the value
functions vi (a; 1)
vi (a) and the consumption policies ci (a), for i = 1; 2 in the zero-in‡ation
30
steady state.
Table 1. Baseline calibration
Parameter
Value
r
0.03
5.5
0.19
0.72
0.08
0.73
1.03
1
2
y1
y2
0.0302
-3.6
4.2
Description
Source/Target
world real interest rate
standard
scale in‡ation disutility
slope NKPC in Calvo model
bond amortization rate
Macaulay duration = 4.5 years
transition rate unemployment-to-employment
monthly job …nding rate of 0.1
transition rate employment-to-unemployment
unemployment rate 10 percent
income in unemployment state
Hall & Milgrom (2008)
income in employment state
E
( (y) = 1
subjective discount rate
NIIP -25% of GDP
HH debt/GDP ratio 90%
borrowing limit
Steady state under optimal policy
We start our numerical analysis of optimal policy by computing the steady state equilibrium to
which each monetary regime (commitment and discretion) converges. Table 2 displays a number
of steady-state objects. Under commitment, the optimal long-run in‡ation is close to zero (-0.05
percent), consistently with Proposition 4 and the fact and r are very closed to each other in our
calibration.31 As a result, long-run gross household debt and net total assets (as % of GDP) are
very similar to those under zero in‡ation. From now on, we use x x (1) to denote the steady
state value of any variable x. As shown in the previous section, the long-run nominal yield is
r = r + , where the World real interest rate r equals 3 percent in our calibration.
29
Analogously to Blanchard and Galí (2010; see their footnote 20), we compute the equivalent annual rate
1
=
12
X
m i 1
1 )
(1
1
as
m
1 ;
i=1
m
1
where
is the monthly job …nding rate.
30
Importantly, while the …gure displays the steady-state value functions, it should be noted that their concavity
is preserved in the time-varying value functions implied by the optimal policy paths.
31
As explained in section 3, in our baseline calibration we have r = 0:03 and = 0:0302.
21
(a) Value function, v(a)
(b) Consumption, c(a)
4
1.3
3
1.2
z = z2
2
1.1
1
1
0
0.9
-1
0.8
-2
0.7
-3
-4
z = z1
-3
-2
-1
0
1
2
3
4
-3
Assets, a
-2
-1
0
1
2
3
4
Assets, a
Figure 1: Steady state with zero in‡ation.
Table 2. Steady-state values under optimal policy
In‡ation,
Nominal yield, r
Net assets, a
Gross assets (creditors)
Gross debt (debtors), b
Current acc. de…cit, c y
units
Ramsey
MPE
%
0:05
2:95
24:1
65:6
89:8
0:63
1:68
4:68
0:6
80:0
80:6
0:01
%
% GDP
% GDP
% GDP
% GDP
Under discretion, by contrast, long run in‡ation is 1.68 percent, which re‡ects the in‡ationary
bias discussed in the previous section. The presence of an in‡ationary bias makes nominal interest
rates higher through the Fisher equation (29). The economy’s aggregate net liabilities fall substantially relative to the commitment case (0:6% vs 24:1%), mostly re‡ecting larger asset accumulation
by creditor households.
4.3
Optimal transitional dynamics
As explained in Section 3, the optimal policy paths depend on the initial (time-0) net wealth
distribution across households, ffi (0; a)gi=1;2 , which is an (in…nite-dimensional) primitive in our
model.32 In the interest of isolating the e¤ect of the policy regime (commitment vs discretion) on
32
As explained in section 3.1, in our numerical exercises we assume that the income distribution starts at its
ergodic limit: fy (yi ) = j6=i = ( 1 + 2 ) ; i = 1; 2. Also, in all our subsequent exercises we assume that the time-0
net wealth distribution conditional on being in state 1 (unemployment) is identical to that conditional on state 2
(employment): fajy (0; a j y2 ) = fajy (0; a j y1 )
f0 (a). Therefore, the initial joint density is simply f (0; a; yi ) =
22
the equilibrium allocations, we choose a common initial distribution in both cases. For the purpose
of illustration, we consider the stationary distribution under zero in‡ation as the initial distribution.
In section 4.5 we will analyze the robustness of our results to a wide range of alternative initial
distributions.
Consider …rst the case under commitment (Ramsey policy). The optimal paths are shown by
the green solid lines in Figure 2.33 Under our assumed functional form for preferences in (30), we
have from equation (28) that initial optimal in‡ation is given by
(0) =
2 Z
1X
1
( a)
i=1
@vi (0; a)
fi (0; a) da;
@a
where we have used the fact that (0) = 0, as there are no pre-commitments at time zero.
Therefore, the time-0 in‡ation rate, of about 4:6 percent, re‡ects exclusively the redistributive
motive (both cross-border and domestic) discussed in Section 3. This domestic redistribution can
be clearly seen in panels (h) and (i) of Figure 2: the transitory in‡ation created under commitment
gradually reduces both the assets of creditor households and the liabilities of debtor ones. The
cross-border redistribution is apparent from panel (g): the country as a whole temporarily reduces
its net liabilities vis-à-vis the rest of the World.
As time goes by, optimal in‡ation under commitment gradually declines towards its (near) zero
long-run level. The intuition is straightforward. At the time of formulating its commitment, the
central bank exploits the existence of a stock of nominal bonds issued in the past. This means
that the in‡ation created by the central bank has no e¤ect on the prices at which those bonds
were issued. However, the price of nominal bonds issued from time 0 onwards does incorporate
the expected future in‡ation path. Under commitment, the central bank internalizes the fact that
higher future in‡ation reduces nominal bond prices, i.e. it raises nominal bond yields, which hurts
net bond issuers. This e¤ect becomes stronger and stronger over time, as the fraction of total
nominal bonds that were issued before the time-0 commitment becomes smaller and smaller. This
gives the central bank the right incentive to gradually reduce in‡ation over time. Formally, in the
equation that determines optimal in‡ation at t 0,
(t) =
2 Z
1X
i=1
1
( a)
@vi
1
fi (t; a) da +
(t) Q (t) ;
@a
(31)
the (absolute) value of the costate (t), which captures the e¤ect of time-t in‡ation on the price
of bonds issued during the period [0; t), becomes larger and larger over time. As shown in panels
f0 (a) j6=i = ( 1 + 2 ), i = 1; 2.
33
We have simulated 800 years of data at monthly frequency.
23
(a) In.ation, :
(b) MV-weighted net debt
6
4
-2
2
Ramsey
0
%
2
%
%
0
MPE
4
-2
(c) Costate, 7
6
-4
7(t)Q(t)
7(t)
0
10
20
0
30
0
10
20
years
(d) Nominal bond price, Q
(e) Nominal bond yield, r
90
8
85
6
80
4
-6
30
years
0
10
20
30
years
(f) Real yield, r ! :
3
2.5
%
%
2
1.5
75
0
10
20
0
30
10
20
(g) Net assets, a7
(h) Gross assets (creditors)
-20
-22
10
20
years
30
10
20
30
(i) Gross assets (debtors)
68
-82
66
-84
64
60
0
years
62
0
0.5
30
years
% GDP
% GDP
0
years
-18
-24
1
% GDP
70
2
-86
-88
0
10
years
20
30
-90
0
10
years
20
Figure 2: Dynamics under optimal monetary policy and zero in‡ation.
24
30
(c) and (b) of Figure 2, the increase in j (t) Q (t)j dominates that of the marginal-value-weighted
P R1
i
( a) @v
average net liabilities, i
f (t; a) da, which from equation (31) produces the gradual
@a i
fall in in‡ation.34 Importantly, the fact that investors anticipate the relatively short-lived nature
of the initial in‡ation explains why nominal yields (panel e) increase much less than instantaneous
in‡ation itself. This allows the ex-post real yield rt
t (panel f) to fall sharply at time zero, thus
giving rise to the aforementioned redistribution.
In summary, under the optimal commitment the central bank front-loads in‡ation in order
to redistribute net wealth towards domestic borrowers but also commits to gradually reducing
in‡ation in order to prevent in‡ation expectations from permanently raising nominal yields.
Under discretion (dashed blue lines in Figure 2), time-zero in‡ation is 4:3 percent, close to the
value under commitment.35 In contrast to the commitment case, however, from time zero onwards
optimal discretionary in‡ation remains relatively high, declining very slowly to its asymptotic value
of 1:68 percent. The reason is the in‡ationary bias that stems from the central bank’s attempt
to redistribute wealth to borrowing households. This in‡ationary bias is not counteracted by any
concern about the e¤ect of in‡ation expectations on nominal bond yields; that is, the costate (t)
in equation (31) is zero at all times under discretion. This in‡ationary bias produces permanently
lower nominal bond prices (higher nominal yields) than under commitment. Contrary also to the
Ramsey equilibrium, the discretionary policy largely fails to deliver the very redistribution it tries
to achieve. The reason is that investors anticipate high future in‡ation and price the new bonds
accordingly. The resulting jump in nominal yields (panel e) undoes most of the instantaneous
in‡ation, such that the ex-post real yield (panel f) barely falls.
4.4
Welfare analysis
We now turn to the welfare analysis of alternative policy regimes. Aggregate welfare is de…ned as
Z
1
2
X
i=1
vi (0; a) fi (t; a)da =
Z
0
1
e
t
Z
1
2
X
u (ci (t; a) ; (t)) fi (t; a)dadt
W [c] ;
i=1
Table 3 displays the welfare losses of suboptimal policies vis-à-vis the Ramsey optimal equilibrium.
We express welfare losses as a permanent consumption equivalent, i.e. the number
(in %)
34
Panels (b) and (c) in Figure 2 display the two terms on the right-hand side of (31), i.e. the marginal-valueweighted average net liabilities and (t) Q (t) both rescaled by the in‡ation disutility parameter . Therefore, the
sum of both terms equals optimal in‡ation under commitment.
35
Since (0) = 0, and given a common initial wealth distribution, time-0 in‡ation under commitment and
discretion di¤er only insofar as time-0 value functions in both regimes do. Numerically, the latter functions are
similar enough that (0) is very similar in both regimes.
25
that satis…es in each case W R cR = W [(1 + ) c], where R denotes the Ramsey equilibrium.36
The table also displays the welfare losses incurred respectively by creditors and debtors.37 The
welfare losses from discretionary policy versus commitment are of …rst order: 0.31% of permanent
consumption. This welfare loss is su¤ered not only by creditors (0.23%), but also by debtors
(0.08%), despite the fact that the discretionary policy is aimed precisely at redistributing wealth
towards debtors. As explained in the previous subsection, under the discretionary policy the
increase in nominal yields undoes most of the impact of in‡ation on ex post real yields and hence
on net asset accumulation. As a result, discretionary policy largely fails at producing the very
redistribution towards debtor households that it intends to achieve in the …rst place, while leaving
both creditor and debtor households to bear the direct welfare costs of permanent positive in‡ation.
Table 3. Welfare losses relative to the optimal commitment
Economy-wide Creditors Debtors
Discretion
Zero in‡ation
0.31
0.05
0.23
-0.17
0.08
0.22
Note: welfare losses are expressed as a % of permanent consumption
We also compute the welfare losses from a policy of zero in‡ation, (t) = 0 for all t
0.
As the table shows, the latter policy approximates the welfare outcome under commitment very
closely, for two reasons. First, the welfare gains losses su¤ered by debtor households due to
the absence of initial transitory in‡ation are largely compensated by the corresponding gains for
creditor households. Second, zero in‡ation avoids too the welfare costs from the in‡ationary bias.
4.5
Robustness
Steady state in‡ation. In Proposition 4, we established that the Ramsey optimal long-run in‡ation
rate converges to zero as the central bank’s discount rate converges to that of foreign investors, r.
In our baseline calibration, both discount rates are indeed very close to each other, implying that
Ramsey optimal long-run in‡ation is essentially zero. We now evaluate the sensitivity of Ramsey
36
Under our assumed separable preferences with log consumption utility, it is possible to show that
W R cR
W [c]
1.
37
That is, we report a>0 and a<0 , where
=
exp
a>0
0
W R;a>0
a<0
W M P E;a>0
1;
de…ned analogously, R and where for each policy regime we have de…ned W a>0
0 P2
a<0
a>0
and a>0 do not exactly add up
i=1 vi (0; a) fi (t; a)da, W
i=1 vi (0; a) fi (t; a)da. Notice that
a>0
, as the expontential function is not a linear operator. However, is su¢ ciently small that
+ a>0 .
Rwith
1 P2
to
= exp
26
optimal steady state in‡ation to the di¤erence between both discount rates. From equation (31),
Ramsey optimal steady state in‡ation is
=
2 Z
1X
i=1
1
( a)
@vi
1
fi (a) da +
Q;
@a
(32)
where the …rst term on the right hand side captures the redistributive motive to in‡ate in the
long run, and the second one re‡ects the e¤ect of central bank’s commitments about long-run
in‡ation. Figure 3 displays
(left axis), as well as its two determinants (right axis) on the
right-hand side of equation (32). Optimal in‡ation decreases approximately linearly with the gap
r. As the latter increases, two counteracting e¤ects take place. On the one hand, it can be
shown that as the households become more impatient relative to foreign investors, the net asset
distribution shifts towards the left, i.e. more and more households become net borrowers and come
close to the borrowing limit, where the marginal utility of wealth is highest.38 As shown in the
…gure, this increases the central bank’s incentive to in‡ate for the purpose of redistributing wealth
towards debtors. On the other hand, the more impatient households become relative to foreign
investors, the more the central bank internalizes in present-discounted value terms the welfare
consequences of creating expectations of higher in‡ation in the long run. This provides the central
bank an incentive to committing to lower long run in‡ation. As shown by Figure 3, this second
’commitment’e¤ect dominates the ’redistributive’e¤ect, such that in net terms optimal long-run
in‡ation becomes more negative as the discount rate gap widens.
Initial in‡ation. As explained before, time-0 optimal in‡ation and its subsequent path depend
on the initial net wealth distribution across households, which is an in…nite-dimensional object.
In our baseline numerical analysis, we set it equal to the stationary distribution in the case of zero
in‡ation. We now investigate how initial in‡ation depends on such initial distribution. To make
the analysis operational, we restrict our attention to the class of Normal distributions truncated
at the borrowing limit . That is,
f (0; a) =
(
(a; ; ) = [1
0;
( ; ; )] ; a
a<
;
(33)
where ( ; ; ) and ( ; ; ) are the Normal pdf and cdf, respectively.39 The parameters and
allow us to control both (i) the initial net foreign asset position and (ii) the domestic dispersion
in household wealth, and hence to isolate the e¤ect of each factor on the optimal in‡ation path.
38
The evolution of the long-run wealth distribution as
r increases is available upon request.
As explained in Section 5.2, in all our simulations we assume that the initial net asset distribution conditional
on being in a boom or in a recession is the same: fajy (0; a j y2 ) = fajyP
(0; a j y1 )
f0 (a). This implies that the
marginal asset density coincides with its conditional density: f (0; a) = i=1;2 fajy (0; a j yi ) fy (yi ) = f0 (a).
39
27
100
0
Redistributive component (percent), lhs
50
-0.2
Steady-state in.ation, : (percent), rhs
0
-0.4
-50
-0.6
Commitment component (percent), lhs
-100
0
0.05
0.1
0.15
0.2
r7 ! ; (percent)
Figure 3: Sensitivity analysis to changes in
28
r:
-0.8
0.25
Notice also that optimal long-run in‡ation rates do not depend on f (0; a) and are therefore exactly
the same as in our baseline numerical analysis regardless of and .40 This allows us to focus here
on in‡ation at time 0, while noting that the transition paths towards the respective long-run levels
are isomorphic to those displayed in Figure 2.41 Moreover, we report results for the commitment
case, both for brevity and because results for discretion are very similar.42
Figure 4 displays optimal initial in‡ation rates for alternative initial net wealth distributions.
In the …rst row of panels, we show the e¤ect of increasing wealth dispersion while restricting
the country to have a zero net position vis-à-vis the rest of the World, i.e. we increase and
simultaneously adjust to ensure that a (0) = 0.43 In the extreme case of a (quasi) degenerate
initial distribution at zero net assets (solid blue line in the upper left panel), the central bank has
no incentive to create in‡ation, and thus optimal initial in‡ation is zero. As the degree of initial
wealth dispersion increases, so does optimal initial in‡ation.
The bottom row of panels in Figure 4 isolates instead the e¤ect of increasing the liabilities
with the rest of the World, while assuming at the same time ' 0, i.e. eliminating any wealth
dispersion.44 As shown by the lower right panel, optimal in‡ation increases fairly quickly with the
external indebtedness; for instance, an external debt-to-GDP ratio of 50 percent justi…es an initial
in‡ation of over 6 percent.
We can …nally use Figure 4 to shed some light on the contribution of each redistributive motive
(cross-border and domestic) to the initial optimal in‡ation rate, (0) = 4:6%, found in our baseline
analysis. We may do so in two di¤erent ways. First, we note that the initial wealth distribution
used in our baseline analysis implies a consolidated net foreign asset position of a (0) = 25%
of GDP (y = 1). Using as initial condition a degenerate distribution at exactly that level (i.e.
= 0:205 and ' 0) delivers (0) = 3:1% (see panel d). Therefore, the pure cross-border
redistributive motive explains a signi…cant part (about two thirds) but not all of the optimal time-0
in‡ation under the Ramsey policy. Alternatively, we may note that our baseline initial distribution
has a standard deviation of 1:95. We then …nd the ( ; ) pair such that the (truncated) normal
distribution has the same standard deviation and is centered at a (0) = 0 (thus switching o¤
the cross-border redistributive motive); this requires = 2:1, which delivers (0) = 1:5% (panel
40
As shown in Table 2, long-run in‡ation is 0:05% under commitment, and 1:68% under discretion.
The full dynamic optimal paths under any of the alternative calibrations considered in this section are available
upon request.
42
As explained before, time-0 in‡ation in both policy regimes di¤er only insofar as the respective time-0 value
functions do, but numerically we found the latter to be always very similar to each other. Results for the discretion
case are available upon request.
43
We verify that for all the calibrations considered here, the path of at after time 0 satis…es Assumption 1. In
particular, the redistributive e¤ect from foreign lenders to the domestic economy due to the initial positive increase
in in‡ation is more than compensated by the increase in debtors’consumption.
44
That is, we approximate ’Dirac delta’distributions centered at di¤erent values of . Since such distributions
are not a¤ected by the truncation at a = , we have a (0) = , i.e. the net foreign asset position coincides with .
41
29
(a) Initial density, f(0)
1.2
(b) Initial in.ation, :(0)
1.5
< = 0.01
1
< = 0.4
0.8
1
0.6
0.4
0.2
0
0.5
< = 0.8
< = 1.2
< = 2.1
-3
-2
-1
0
1
2
0
3
0
0.5
1
1.5
2
<
Assets, a
(c) Initial density, f(0)
(d) Initial in.ation, :(0)
20
7 = -1.5
1
15
0.8
7 = -1
0.6
10
0.4
7 = -0.5
5
0.2
7=0
0
-3
-2
-1
0
1
2
0
-1.5
3
-1
-0.5
0
7
Assets, a
Figure 4: Ramsey optimal initial in‡ation for di¤erent initial net asset distributions.
b). We thus …nd again that the pure domestic redistributive motive explains about a third of
the baseline optimal initial in‡ation. We conclude that both the cross-border and the domestic
redistributive motives are quantitatively important for explaining the optimal in‡ation chosen by
the monetary authority.
4.6
Aggregate shocks
So far we have restricted our analysis to the transitional dynamics, given the economy’s initial
state, while abstracting from aggregate shocks. We now extend our analysis to allow for aggregate
disturbances. For the purpose of illustration, we consider a one-time, unanticipated, temporary
change in the World real interest rate. In particular, we allow the World real interest rate r to vary
over time and simulate a one-o¤, unanticipated increase at time 0 followed by a gradual return to
its baseline value of 3%. The dynamics of rt following the shock are given by
drt =
r
(r
30
rt ) dt;
(a) Inter. real rate, r7
(b) Nominal bond yield, r
(c) Real yield, r ! :
1
0.5
0
0
5
10
15
:=0
0.2
Ramsey
0.1
0
20
0
5
years
(d) In.ation, :
5
10
5
15
15
20
(f) Consumption, c7
0.5
-0.5
-1
-1.5
20
10
years
0
5
10
15
0
-0.5
-1
-1.5
20
0
5
10
15
20
years
years
years
(g) Net assets, a7
(h) Gross assets (creditors)
(i) Gross assets (debtors)
8
6
4
2
5
10
15
20
1.5
Dev. from ss (%)
2
Dev. from ss (%)
10
0
0
0
Dev. from ss (%)
Dev. from ss (p.p.)
Dev. from ss (p.p.)
0.02
0
0.1
20
(e) Bond price, Q
0.04
Dev. from ss (%)
15
0
0.06
0
10
0.2
years
0.08
0
Dev. from ss (p.p.)
Dev. from ss (p.p.)
Dev. from ss (p.p.)
1.5
1.5
1
0.5
0
0
5
years
10
years
15
20
1
0.5
0
0
5
10
15
20
years
Figure 5: Impact of an international interest rate shock under commitment (from a timeless
perspective).
with r = 0:03 as in Table 1 and r = 0:5: We consider a 1 percent increase in rt : Notice that,
up to a …rst order approximation, this is equivalent to solving the model considering an aggregate
stochastic process drt = r (r rt ) dt + dZt with = 0:01 and Zt being a Brownian motion. In
fact the impulse responses reported in Figure 5 coincide with the ones obtained by considering
aggregate ‡uctuations and solving the model by …rst-order perturbation around the deterministic
steady state, as in the method of Ahn et al. (2017).
The dashed red lines in Figure 5 display the responses to the shock under a strict zero in‡ation
policy, t = 0 for all t. The shock raises nominal (and real) bond yields, which leads households
to reduce their consumption on impact. The reduction in consumption induces an increase in the
amount of gross assets in the case of creditors and a reduction in gross debts in the case of debtors.
This allows consumption to slowly recover and to reach levels slightly above the steady state after
roughly 5 years from the arrival of the shock.
The solid lines in Figure 5 display the economy’s response under the optimal commitment
31
policy. An issue that arises here is how long after ‘time zero’ (the implementation date of the
Ramsey optimal commitment) the aggregate shock is assumed to take place. Since we do not want
to take a stand on this dimension, we consider the limiting case in which the Ramsey optimal
commitment has been going on for a su¢ ciently long time that the economy rests at its stationary
equilibrium by the time the shock arrives. This can be viewed as an example of optimal policy ’from
a timeless perspective’, in the sense of Woodford (2003). In practical terms, it requires solving the
optimal commitment problem analyzed in Section 3.3 with two modi…cations: (i) the initial wealth
distribution is the stationary distribution implied by the optimal commitment itself, and (ii) the
initial condition (0) = 0 (absence of precommitments) is replaced by (0) = (1), where the
latter object is the stationary value of the costate in the commitment case. Both modi…cations
guarantee that the central bank behaves as if it had been following the time-0 optimal commitment
for an arbitrarily long time.
In the case of commitment in‡ation rises slightly on impact, as the central bank tries to partially
counteract the negative e¤ect of the shock on household consumption. However, the in‡ation
reaction is an order of magnitude smaller than that of the shock itself. Intuitively, the value
of sticking to past commitments to keep in‡ation near zero weighs more in the central bank’s
decision than the value of using in‡ation transitorily so as to stabilize consumption in response to
an unforeseen event. Therefore, we conclude that sticking to a zero in‡ation policy would produce
outcomes rather similar to pursuing the Ramsey optimal in‡ation path.
5
Conclusion
We have analyzed optimal monetary policy, under commitment and discretion, in a continuoustime, small-open-economy version of the standard incomplete-markets model extended to allow
for nominal noncontingent claims and costly in‡ation. Our analysis sheds light on a recent policy
and academic debate on the consequences that wealth heterogeneity across households should have
for the appropriate conduct of monetary policy. Our main contribution is methodological: to the
best of our knowledge, our paper is the …rst to solve for a fully dynamic optimal policy problem,
both under commitment and discretion, in a standard incomplete-markets model with uninsurable
idiosyncratic risk. While models of this kind have been established as a workhorse for policy
analysis in macro models with heterogeneous agents, the fact that in such models the in…nitedimensional, endogenously-evolving wealth distribution is a state in the policy-maker’s problem
has made it di¢ cult to make progress in the analysis of fully optimal policy problems. Our
analysis proposes a novel methodology for dealing with this kind of problems in a continuous-time
environment.
We show analytically that, whether under discretion or commitment, the central bank has an
32
incentive to create in‡ation in order to redistribute wealth from lending to borrowing households,
because the latter have a higher marginal utility of net wealth under incomplete markets. It also
aims at redistributing wealth away from foreign investors, to the extent that these are net creditors
vis-à-vis the domestic economy as a whole. Under commitment, however, these redistributive
motives to in‡ate are counteracted by the central bank’s understanding of how expectations of
future in‡ation a¤ect current nominal bond prices. We show moreover that, in the limiting case
in which the central bank’s discount factor is arbitrarily close to that of foreign investors, the
long-run in‡ation rate under commitment is also arbitrarily close to zero.
We calibrate the model to replicate relevant features of a subset of prominent European small
open economies, including their net foreign asset positions and gross household debt ratios. We
show that the optimal policy under commitment features …rst-order positive initial in‡ation, followed by a gradual decline towards its (near zero) long-run level. That is, the central bank frontloads in‡ation so as to transitorily redistribute existing wealth both within the country and away
from international investors, while committing to gradually abandon such redistributive stance. By
contrast, discretionary monetary policy keeps in‡ation permanently high; such a policy is shown
to reduce welfare substantially, both for creditor and for debtor households, as both groups su¤er
the consequences of the redistribution-led in‡ationary bias.
Our analysis thus suggest that, in an economy with heterogenous net nominal positions across
households, in‡ationary redistribution should only be used temporarily, avoiding any temptation
to prolong positive in‡ation rates over time.
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Appendix
Mathematical preliminaries
First we need to introduce some mathematical concepts. An operator T is a mapping from one
vector space to another. Given the stochastic process at in (3), de…ne an operator A
Av
(t;a)
s1 (t; a) @v1@a
+
1
[v2 (t; a)
v1 (t; a)]
(t;a)
s2 (t; a) @v2@a
2
[v1 (t; a)
v2 (t; a)]
+
;
(34)
so that the HJB equation (8) can be expressed as
v=
@v
+ max fu (c; ) + Avg ;
c
@t
v1 (t;a)
u(c1 ; ) 45
where v
and u (c; )
:
v2 (t;a)
u(c2 ; )
Let
[ ; 1) be the valid domain. The space of Lebesgue-integrable functions L2 ( ) with
the inner product
Z
2 Z
X
hv; f i =
vi fi da =
v T f da; 8v; f 2 L2 ( ) ;
i=1
is a Hilbert space.46 Notice that we could have alternatively worked in
= R as the density
f (t; a; y) = 0 for a < :
Given an operator A, its adjoint is an operator A such that hf; Avi = hA f; vi : In the case
of the operator de…ned by (34) its adjoint is the operator
Af
@(s1 f1 )
@a
@(s2 f2 )
@a
1 f1
+
2 j2
i fi
+
1 j1
;
with boundary conditions
si (t; ) fi (t; ) = lim si (t; a) fi (t; a) = 0; i = 1; 2;
a!1
45
46
The in…nitesimal generator of the process is thus @v
@t + Av:
See Luenberger (1969) or Brezis (2011) for references.
38
(35)
such that the KF equation (14) results in
@f
= A f;
@t
for f
f1 (t;a)
f2 (t;a)
(36)
: We can see that A and A are adjoints as
hAv; f i
=
=
Z
T
(Av) f da =
2 Z
X
si
i=1
2
X
vi si fi j1 +
2 Z
X
vi
i=1
Zi=1
=
v T A f da = hv; A f i :
@vi
+
@a
i
[vj
@
(si fi )
@a
vi ] fi da
i fi
+
j jj
da
We introduce the concept of Gateaux and Frechet derivatives in L2 ( ) ; where
generalizations of the standard concept of derivative to in…nite-dimensional spaces.47
Rn as
De…nition 4 (Gateaux derivative) Let W [f ] be a functional and let h be arbitrary in L2 ( ) :
If the limit
W [f + h] W [f ]
(37)
W [f ; h] = lim
!0
exists, it is called the Gateaux derivative of W at f with increment h: If the limit (37) exists for
each h 2 L2 ( ) ; the functional W is said to be Gateaux di¤erentiable at f:
If the limit exists, it can be expressed as W [f ; h] =
concept is that of the Fréchet derivative.
d
d
W [f + h] j
=0 :
A more restricted
De…nition 5 (Fréchet derivative) Let h be arbitrary in L2 ( ) : If for …xed f 2 L2 ( ) there
exists W [f ; h] which is linear and continuous with respect to h such that
lim
khkL2 (
) !0
jW [f + h]
W [f ]
khkL2 ( )
W [f ; h]j
= 0;
then W is said to be Fréchet di¤erentiable at f and [W f ; h] is the Fréchet derivative of W at f
with increment h:
The following proposition links both concepts.
Theorem 1 If the Fréchet derivative of W exists at f , then the Gateaux derivative exists at f and
they are equal.
47
See Luenberger (1969), Gelfand and Fomin (1991) or Sagan (1992).
39
Proof. See Luenberger (1969, p. 173).
The familiar technique of maximizing a function of a single variable by ordinary calculus can be
extended in in…nite dimensional spaces to a similar technique based on more general derivatives.
We use the term extremum to refer to a maximum or a minimum over any set. A a function
f 2 L2 ( ) is a maximum of W [f ]if for all functions h; kh f kL2 ( ) < " then W [f ] W [h]. The
following theorem generalizes the Fundamental Theorem of Calculus.
Theorem 2 Let W have a Gateaux derivative, a necessary condition for W to have an extremum
at f is that W [f ; h] = 0 for all h 2 L2 ( ) :
Proof. Luenberger (1969, p. 173), Gelfand and Fomin (1991, pp. 13-14) or Sagan (1992, p. 34).
In the case of constrained optimization in an in…nite-dimensional Hilbert space, we have the
following Theorem.
Theorem 3 (Lagrange multipliers) Let H be a mapping from L2 ( ) into Rp : If W has a
continuous Fréchet derivative, a necessary condition for W to have an extremum at f under the
constraint H [f ] = 0 at the function f is that there exists a function 2 L2 ( ) such that the
Lagrangian functional
L [f ] = W [f ] + h ; H [f ]i
(38)
is stationary in f; that is., L [f ; h] = 0:
Proof. Luenberger (1969, p. 243).
Finally, according to De…nition 5 above, if the Fréchet derivative W [f ] of W [f ] exists then it
is linear and continuous. We may apply the Riesz representation theorem to express it as an inner
product
Theorem 4 (Riesz representation theorem) Let W [f ; h] : L2 ( ) ! R be a linear continuous functional. Then there exists a unique function w [f ] = Wf [f ] 2 L2 ( ) such that
W [f ; h] =
W
;h
f
=
2 Z
X
wi [f ] (a) hi (a) da:
i=1
Proof. See Brezis (2011, pp. 97-98).
Proof of Proposition 1. Solution to the MPE
The idea of the proof is to employ dynamic programming in order to transform the problem of the
central bank in a family -indexed by time- of static calculus of variations problems. Then we solve
each of these problems using di¤erentiation techniques in in…nite-dimensional Hilbert spaces.
40
Step 1: Dynamic programming Given an initial condition f (t; ) we have an optimal control
path f (s)gs2[t;1) : The …rst step is to apply the Bellman’s Principle of Optimality
W [f (t; )] =
=
f
f
max
1
s gs=0
max
1
s gs=0
Z
1
e
(s t)
t
Z
t0
e
(s t)
t
2 Z
X
i=1
2 Z
X
u (c(s; a);
u (c(s; a);
(39)
s ) fi (s; a)dads
s ) fi (s; a)dads
+e
(t0 t)
W [f (t0 ; )] ;
i=1
where W [f ] is the central bank’s optimal value functional. For compactness, we will henceforth
use the inner product notation
2 Z
X
u (ct ;
t ) fi (t; a)da
i=1
= hu; f i :
Given the Riesz
theorem (Theorem 4), the Fréchet derivative W [f (t; ); h] can
D representation
E
be expressed as Wf ; h : If we compute the derivative with respect to time t of W [f (t; )] we
may combine this with the chain rule to obtain
@
W [f (t; )] =
@t
W @f
;
f @t
=
w;
@f
@t
;
(40)
where w (t; a) = Wf : [0; 1)
! R2 : Notice that, as there is no aggregate uncertainty the
dynamics of the distribution only depend on time. As it will be clear below w(t; a) is the central
bank’s value at time t of a household with net wealth a.
Taking derivatives with respect to time in equation (39) and the limit as t0 ! t:
W [f (t; )] = max hu; f i + wt ;
t
@f
@t
;
(41)
@
where we used (40) to substitute for @t
W [f (t; )].
Equation (41) is the Hamilton-Jacobi-Bellman (HJB) equation of the problem (39) in the
in…nite-dimensional space L2 ( ). The last term of the right-hand side equation is
w;
@f
@t
= hw; A f i = hAw; f i ;
(42)
where we have applied the KF equation (36) and the fact that A is the adjoint operator of A.
Using (42), we may express the functional Hamilton-Jacobi-Bellman (41) equation as
W [f ] = max fhu; f i + hAwt ; f i g :
t
41
(43)
Step 2: Optimal in‡ation The …rst order condition with respect to in‡ation in (43) is
@
fhu; f i + hAwt ; f i g
@
2 Z
X
@si @wi
fi da =
u +
u fi
@ @a
i=1
(44)
0=
=
2 Z
X
i=1
afi
@wi
@a
da;
where in the second equality we have used the de…nition of A (equation 34) and in the third
equality we have used the de…nition of s (equation 9).
Therefore the optimal in‡ation should satisfy
2 Z
X
afi
i=1
@wi
@a
(45)
u fi da = 0:
Step 3: Central Bank’s value function In order to …nd the value of w(t; ), we compute the
Gateaux derivative of the HJB equation (43):
W [f (t; ); h] =
d
hu; f + hi + hAw [f + h] ; f + hi j
d
=0
:
(46)
The right-hand side results in
d
hu + Aw [f + h] ; f + hi j
d
=0
d
w [f + h]j
d
= hu; hi +
=0
;A f
+ hw [f ] ; A hi ;
where we have applied the envelope condition for [f ] and the fact that A is the adjoint operator
of A (equation 42).
D 2
E
W
d
; h is the second Gateaux derivative of W [f ] : It is related
The term d w [f + h]j =0 =
f2
to the partial derivative of w with respect to time:
@w
@ W
=
[f (t; )] =
@t
@t f
f
W
f
;
@f
@t
2
=
W @f
;
f 2 @t
;
(47)
where we have applied the chain rule (40). Hence applying the associative and commutative
42
properties of the inner product
d
w [f + h]j
d
=0
"
2 Z
X
2
2 Z
X
#
2
W
W
@f
@fi
=
da
;h ;
hj (a0 ) da0
2
2
f
@t
f
@t
i=1
j=1
" 2 Z
#
2 Z
2 Z
2
2
X
X
X
W @fi
W @f
0
0
=
da
h
(a
)
da
=
hj (a0 ) da0
;
j
2
2
f
@t
f
@t
i=1
j=1
j=1
;A f
=
2
=
W @f
;
f 2 @t
;h
=
@w
;h
@t
;
where in the …rst equality we have used the KF equation (14) and in the fourth equality we have
2
used the relationship between @w
and fW2 (equation 47). Equation (46) hence becomes
@t
hw; hi = hu; hi +
@w
;h
@t
+ hw; A hi =
u+
@w
Aw; h
@t
;
where we have used equation (42) (with h instead of f ).
Grouping all the terms at the right-hand side we obtain:48
0=
2 Z
X
i=1
@w
u (c; ) +
+ Aw
@t
T
w
h(a)da:
Equation (48) should be satis…ed for any h 2 L2 ( ), which requires that u (c; )+ @w
+Aw
@t
for all a 2 ; y 2 fy1 ; y2 g ; or equivalently
wi (t; a) = u(ci ; ) +
@wi
@wi
+ si (t; a)
+
@t
@a
i
(wk (t; a)
wi (t; a)) ; i = 1; 2; k 6= i:
(48)
w = 0;
(49)
Equation (49) is identical to the individual HJB equation (8) with w ( ) replacing v ( ). The
boundary conditions are also the same (state constraints on the domain ) and therefore their
solutions should coincide: w(t; a; y) = v(t; a; y); that is, the central bank’s social value w ( ) equals
the private value v ( ).
48
The application of equation (42) requires that the boundary condition si (t; ) hi (t; )
=
lima!1 si (t; a) hi (t; a) = 0; i = 1; 2 is sati…ed, which is guaranteed by the fact that hi (t; ) = 0 in the
cases in which fi (t; ) takes a …xed value due to the boundary (35).
43
Proof of Proposition 2. Solution to the Ramsey problem
Step 1: Statement of the problem The problem of the central bank is given by
W [f0 ( )] =
s ;Qs ;v(s;
f
max
);c(s; );f (s; )g1
s=0
2 Z
X
i=1
1
Z
s
e
0
u (cs ;
s ) fi (s; a)da
ds;
subject to the law of motion of the distribution (14), the bond pricing equation (12) and the individual HJB equation (8). This is a problem of constrained optimization in an in…nite-dimensional
Hilbert space that includes also time, whic we denote as ^ = [0; 1)
. We de…ne L2 ^
as
(;)
the space of functions f that verify
Z
e
t
^
Z
2
jf j =
1
0
Z
t
e
2
jf j dtda =
Z
1
e
t
0
kf k2 dt < 1:
We need …rst to prove that this space, which di¤ers from L2 ^ is also a Hilbert space. This is
done in the following lemma, the proof is in the Online Appendix.
Lemma 3 The space L2 ^
with the inner product
(;)
(f; g) =
Z
t
e
fg =
^
Z
1
e
t
0
hf; gi dt = e
t
f; g
^
is a Hilbert space.
Step 2: The Lagragian The Lagrangian is de…ned in L2 ^
as
(;)
L [ ; Q; f; v; c]
Z
0
+
1
Z
e
1
t
hu; f i dt +
t
e
Z
1
e
0
(t) Q (r +
t
(t; a) ; A f
@f
@t
dt
Q_ dt
+ )
0
+
Z
1
e
t
(t; a) ; u + Av +
e
t
(t; a) ; uc
0
+
Z
1
0
1 @v
Q @a
@v
@t
v
dt
dt
where e t (t; a), e t (t; a), e t (t; a) 2 L2 ^ and e t (t) 2 L2 [0; 1) are the Lagrange
multipliers associated to equations (14), (10), (8) and (12), respectively. The Lagragian can be
44
expressed as
Z
L =
0
+
1
Z
t
e
1
e
u+
t
@
+A
@t
+ h (0; ) ; f (0; )i
T
T !1
Q (r +
@
;v
@t
h ; ui + A
0
+ lim e
+
T
lim e
Q_ ; f
+ )
+
1 @v
Q @a
; uc
1
e
t
0
;
@f
@t
2 Z
X
dt =
1
i=1 0
2 Z
X
=
Z
t
e
i
1
i 0
t
fi e
=
da +
i=1
+
2 Z
X
1
0
i=1
2 Z
X
Z
(0; a) da
i
t
e
= h (0; ) ; f (0; )i
lim
T !1
@ i
@t
fi
1
0
i=1
fi (0; a)
1
t
e
0
2
X
i=1
P2
vi si i j1 dt;
vi si i j1 and integrated by
i=1
@fi
dadt
@t
i=1
2 Z
X
Z
h (0; ) ; v (0; )i +
where we have applied h ; A f i = hA ; f i ; h ; Avi = hA ; vi +
parts
Z
dt
(T; ) ; f (T; )
T !1
(T; ) ; v (T; )
dt
i
T
lim e
T !1
Z
fi
2 Z
X
@
e
@t
e
T
t
dadt
i
fi (T; a)
i
(T; a) da
i=1
dadt
+
(T; ) ; f (T; )
Z
1
@
@t
t
e
0
;f
dt;
and
Z
0
1
e
t
@v
;
@t
v dt =
2 Z
X
1
0
=
i=1
2 Z
X
i=1
=
lim
T !1
Z
i=1
2
XZ 1
i=1
=
lim e
T !1
t
i ei
2 Z
X
0
T
t
e
v
1
0
@vi
@t
i
da
vi dadt
2 Z
X
0
i=1
T
e
Z
vi (T; a)
i
1
Z
vi
(T; a) da
@
e
@t
2 Z
X
t
i
+
i
vi (0; a)
i
dadt
(0; a) da
i=1
e
t
vi
@ i
@t
(T; ) ; v (T; )
45
dadt
h (0; ) ; v (0; )i +
Z
0
1
e
t
@
;v
@t
dt;
Step 3: Necessary conditions In order to …nd the extrema, we need to take the Gateaux
derivatives with respect to the controls f , ; Q; v and c. The Gateaux derivative with respect to
f is
d
L [ ; Q; f + h; v; c] j
d
=0
= h (0; ) ; h (0; )i
lim e
T !1
Z 1
@
e t u+
+A
@t
0
t
which should equal zero for any function e
of f (0; ) is given. We obtain
h 2 L2 ^
T
(T; ) ; h (T; )
;h
dt;
such that h (0; ) = 0; as the initial value
@
+A ;
(50)
@t
and taking this into account and considering functions such that limT !1 h (T; ) 6= 0; we obtain
the transversality condition limT !1 e T (T; a) = 0: Equation (50) equals the individual HJB
equation (8). The boundary conditions are also the same (state constraints on the domain )
and therefore their solutions should coincide: (t; a; y) = v(t; a; y); that is, the Lagrange multiplier
(t; a; y) equals the private value v ( ).
In the case of c (t; a) ; the Gateaux derivative is
=u+
d
L [ ; Q; f; v; c + h (t; a)] j
d
=0
=
Z
0
+
1
Z
t
e
1
uc
t
e
1@
Q @a
; uc
0
h; f
1 @v
Q @a
dt
h
+ h ; ucc hi
dt;
@
@
(A ) = Q1 @a
: The Gateaux derivative should be zero for any function e t h 2 L2 ^ .
where @a
Due to the …rst order conditions (10) and to the fact that ( ) = v ( ) this expression reduces to
Z
1
e
t
0
h (t; a) ; ucc (t; a) h (t; a)i dt = 0;
so that (t; a) = 0 for all (t; a) 2 ^ ; that is, the …rst order condition (10) is not binding as its
associated Lagrange multiplier is zero.
In the case of v (t; a) ; the Gateaux derivative is
d
L [ ; Q; f; v + h (t; a) ; c] j
d
=0
=
Z
1
e
t
A
0
+ lim e
T
T !1
46
@
;h
@t
(T; ) ; h (T; )
dt
h (0; ) ; h (0; )i +
2
X
i=1
hi si i j1 ;
where we have already taken into account the fact that (t; a) = 0: Proceeding as in the case of
f; we conclude that the condition that the derivative should be zero at the extremum yields a
Kolmogorov forward equation
@
=A ;
(51)
@t
with boundary conditions
si (t; )
lim e
i
T
(t; ) =
lim si (t; a)
a!1
(T; ) = 0;
T !1
i
(t; a) = 0; i = 1; 2;
(0; ) = 0:
This is a KF equation in ; with an initial density of (0; ) = 0: Therefore, the distribution at
any point in time should be zero ( ; ) = 0:
Both the Lagrange multiplier of the HJB equation and that of the …rst-order condition are
zero, re‡ecting the fact that the HJB condition is not binding, that is, that the monetary authority
would choose the same consumption as the households.
The Gateaux derivative in the case of (t) :
d
L [ + h (t) ; Q; f; v; c] j
d
=0
=
Z
1
t
e
u
0
where we have already taken into account the fact that
(t; a) = v (t; a) ; the optimality condition results in
(t) Q (t) =
2 Z
X
a
i=1
@vi
@a
u
a
@
@a
hdt;
+ Q; f
(t; a) = 0. Taking into account that
fi (t; a) da:
Finally, in the case of Q (t) the Gateaux derivative is
d
L [ ; Q + h (t) ; f; v; c] j
d
=0
=
Z
1
e
h @
a
Q2 @a
t
0
(y
h
c) h @
+ h (r +
Q2 @a
+ )
Integrating by parts
Z
0
1
e
t
D
_ f
h;
E
dt =
=
=
Z
1
e
h_ h1; f i dt =
0
Z 1
1
t
e
h0 +
e t (_
Z 10
(0) h (0) +
e t h( _
t
0
47
Z
1
e
t
_
hdt
0
) hdt
) h; f i dt:
i
_h ; f
dt
Therefore, the optimality condition in this case is
Z
0
1
e
t
Q2
a
@
@a
(y
Q2
c) @
+ (r +
@a
so that, using similar arguments as in the case of
Q2
a
@v
@a
(y
Q2
c) @v
;f
@a
+
) + _;f
hdt + (0) h (0) = 0;
above we can show that
+ (r +
+
(0) = 0 and
) + _ = 0; t > 0:
If we have applied the fact that ( ) = v ( ) this results in
d
=(
dt
1 X
) + 2
Q (t) i=1
2
r
48
Z
@vit
[ a + (y
@a
c)] fi (t; a) da:
Online appendix (not for publication)
A. Additional proofs
Proof of Lemma 1
Given the welfare criterion de…ned as in (18), we have
U0CB
=
=
=
Z
Z
Z
0
=
Z
1
1
2
X
vi (0; a)fi (0; a)da =
i=1
" 2 Z
2
X
X 1
i=1
2
1X
j=1
t
e
j=1
1
0
2
X
t
e
i=1
Z
Z
1
1
Z
1
t
e
0
u(c; )
"
Z
1
2
X
E0
Z
1
e
t
0
i=1
u(ct ;
#
t )dtja (0)
= a; y (0) = yi fi (0; a)da
u(c; )f (t; a
~; y~j ; a; yi )dtd~
a fi (0; a)da
2 Z
X
1
#
f (t; a
~; y~j ; a; yi )fi (0; a)da d~
adt
i=1
u(c; )fj (t; a
~)d~
adt;
where f (t; a
~; y~j ; a; y) is the transition probability from a (0) = a; y (0) = yi to a (t) = a
~; y (t) = y~j
and
2 Z 1
X
f (t; a
~; y~j ; a; yi )fi (0; a)da;
fj (t; a
~) =
j=1
is the Chapman–Kolmogorov equation.
Proof of Lemma 2
In order to prove the concavity of the value function we express the model in discrete time for an
arbitrarily small t: The Bellman equation of a household is
vt t (a; y) =
max
0
a 2 (a;y)
+e
t
uc
2
X
i=1
h
Q (t)
t
1+
(t)
Q (t)
t a+
y t
Q (t)
a0
u ( (t))
t
vt+t t (a0 ; yi ) P (y 0 = yi jy) ;
y t
Q(t)
i
where (a; y) = 0; 1 + Q(t)
(t)
t a+
; and P (y 0 = yi jy) are the transition probabilities of a two-state Markov chain. The Markov transition probabilities are given by 1 t and
2 t:
We verify that this problem satis…es the conditions of Theorem 9.8 of Stokey, Lucas and
Prescott (1989): (i)
is a convex subset of R; (ii) the Markov chain has a …nite number
49
of values; (iii) the correspondence (a; y) is nonempty, compact-valued and continuous; (iv)
t
the function uc is bounded, concave and continuous and e
2 (0; 1); and (v) the set Ay =
f(a; a0 ) such that a0 2 (a; y)g is convex. We may conclude that vt t (a; y) is concave for any
t > 0: Finally, for any a1 ; a2 2
!) vt t (a2 ; y) ;
vt t (!a1 + (1
!) a2 ; y)
!vt t (a1 ; y) + (1
lim vt t (!a1 + (1
!) a2 ; y)
lim !vt t (a1 ; y) + (1
v (t; !a1 + (1
!) a2 ; y)
!v (t; a1 ; y) + (1
t!0
!) vt t (a2 ; y) ;
t!0
!) vt (t; a2 ; y) ;
so that v (t; a; y) is concave.
Proof of Lemma 3
We need to show that L2 ^
is complete, that is, that given a Cauchy sequence ffn g with
(;)
limit f : limn!1 fn = f then f 2 L2 ^
kfn
(;)
: If ffn g is a Cauchy sequence then
fm k( ; ) ! 0; as n; m ! 1;
or
kfn
fm k2( ; )
=
Z
e
^
t
jfn
D
fm j = e
2
as n; m ! 1: This implies that e
2
t
2
t
(fn
fm ) ; e
2
t
(fn
fm )
E
^
= e
t
2
(fn
fm )
^
! 0;
fn is a Cauchy sequence in L2 ^ . As L2 ^ is a complete
space, then there is a function f^ 2 L2 ^
such that
lim e
2
n!1
t
fn = f^
(52)
under the norm k k2^ : If we de…ne f = e 2 t f^ then
lim fn = f
n!1
under the norm k k( ; ) , that is, for any " > 0 there is an integer N such that
kfn
2
f k2( ; ) = e
2
t
2
(fn
f)
50
^
= e
2
t
fn
f^
2
^
< ";
where the last inequality is due to (52). It only remains to prove that f 2 L2 ^
kf k2( ; )
=
Z
e
t
^
2
jf j =
Z
:
(;)
2
^
f^ < 1;
as f^ 2 L2 ^ :
Proof of Proposition 3: In‡ation bias in MPE
As the value function is concave in a; then it satis…es that
@vi (t; a
~)
@a
@vi (t; 0)
@a
@vi (t; a
^)
; for all a
~ 2 (0; 1); a
^ 2 ( ; 0); t
@a
0; i = 1; 2:
(53)
In addition, the condition that the country is a net debtor (at < 0) implies
2 Z
X
2 Z
X
0
( a) fi (t; a)da
i=1
i=1
1
0
(a) fi (t; a)da; 8t
(54)
0:
Therefore
2 Z
X
i=1
0
1
@vi (t; a)
da
afi
@a
2 Z
2 Z
@vi (t; 0) X
@vi (t; 0) X 1
afi da
@a
@a
i=1 0
i=1
Z
2
X 0
@vi (t; a)
( a) fi (t; a)
da;
@a
i=1
0
( a) fi (t; a)da
(55)
(56)
where we have applied (53) in the …rst and last stepts and (54) in the intermediate one. The
optimal in‡ation in the MPE case (22) with separable utility u = uc u is
2 Z
X
i=1
1
@vi
afi
@a
u fi da =
2 Z
X
1
i=1
afi
@vi
da + u = 0:
@a
Combining this expression with (55) we obtain
u =
2 Z
X
1
( a) fi
i=1
@vi
da > 0:
@a
Finally, taking into account the fact that u > 0 only for
51
> 0 we have that
(t) > 0.
Proposition 4: optimal long-run in‡ation under commitment in the limit as r !
In the steady state, equations (26) and (28) in the main text become
(
2 Z
1 X
) + 2
Q i=1
r
Q=u ( )+
1
@vi
[ a + (yi
@a
2 Z
X
1
a
i=1
respectively. Consider now the limiting case
equations then become
ci )] fi (a) da = 0;
@vi
fi (a) da;
@a
! r , and guess that
2 Z
1 X 1 @vi
Q =
[ a + (yi
Q i=1
@a
2 Z 1
X
@vi
fi (a) da;
Q =
a
@a
i=1
! 0. The above two
ci )] fi (a) da;
as u (0) = 0 under our assumed preferences in Section 3.4. Combining both equations, and using
the fact that in the zero in‡ation steady state the bond price equals Q = +r ; we obtain
2 Z
X
1
i=1
@vi
@a
ra +
yi
ci
Q
(57)
fi (a) da = 0:
In the zero in‡ation steady state, the value function v satis…es the HJB equation
vi (a) = uc (ci (a)) + ra +
yi
ci (a)
Q
@vi
+
@a
i
[vj (a)
vi (a)] ; i = 1; 2; j 6= i;
(58)
where we have used u (0) = 0 under our assumed preferences. We also have the …rst-order
condition
@vi
@vi
ucc (ci (a)) = Q
) ci (a) = ucc; 1 Q
:
@a
@a
We guess and verify a solution of the form vi (a) = i a + #i ; so that ucc (ci ) = Q i . Using our guess
in (58), and grouping terms that depend on and those that do not, we have that
i
= r
i
+
#i = uc uc;
c
i
1
(
j
(59)
i) ;
(Q i ) +
yi
1
uc;
c
Q
52
(Q i )
i
+
i
(#j
#i ) ;
(60)
for i; j = 1; 2 and j 6= i. In the limit as r ! , equation (59) results in j = i
; so that
consumption is the same in both states. The value of the slope can be computed from the
boundary conditions.49 We can solve for f#i gi=1;2 from equations (60),
1
#i = uc uc;
c
1
yi
ucc; 1 (Q )
+
Q
(
=
in (57), we obtain
(Q ) +
for i; j = 1; 2 and j 6= i. Substituting
2 Z
X
@vi
@a
1
ra +
yi
ci
fi (a) da = 0:
Q
i=1
(yj yi )
;
i + j + )Q
i
(61)
Equation (61) is exactly the zero-in‡ation steady-state limit of equation (17) in the main text
(once we use the de…nitions of a, y and c), and is therefore satis…ed in equilibrium. We have thus
veri…ed our guess that ! 0:
B. Computational method: the stationary case
B.1 Exogenous monetary policy
We describe the numerical algorithm used to jointly solve for the equilibrium value function,
v (a; y), and bond price, Q; given an exogenous in‡ation rate . The algorithm proceeds in 3 steps.
We describe each step in turn. We assume that there is an upper bound arbitrarily large { such
that f (t; a; y) = 0 for all a > {: In steady state this can be proved in general following the same
reasoning as in Proposition 2 of Achdou et al. (2015). Alternatively, we may assume that there is
a maximum constraint in asset holding such that a {; and that this constraint is so large that
it does not a¤ect to the results: In any case, let [ ; {] be the valid domain.
Step 1: Solution to the Hamilton-Jacobi-Bellman equation Given ; the bond pricing
equation (12) is trivially solved in this case:
Q=
49
r+
+
:
The condition that the drift should be positive at the borrowing constraint, si ( )
s1 ( ) = r +
and
=
y1
ucc; 1 (Q )
= 0;
Q
ucc (r Q + y1 )
:
Q
In the case of state i = 2, this guarantees s2 ( ) > 0.
53
(62)
0, i = 1; 2, implies that
The HJB equation is solved using an upwind …nite di¤erence scheme similar to Achdou et al.
(2015). It approximates the value function v(a) on a …nite grid with step a : a 2 fa1 ; :::; aW g,
where aj = aj 1 + a = a1 + (j 1) a for 2 j J. The bounds are a1 = and aI = {, such
that a = ({
) = (W 1). We use the notation vi;j
vi (aj ); i = 1; 2, and similarly for the
policy function ci;j .
Notice …rst that the HJB equation involves …rst derivatives of the value function, vi0 (a) and
vi00 (a). At each point of the grid, the …rst derivative can be approximated with a forward (F ) or a
backward (B) approximation,
vi0 (aj )
@F vi;j
vi0 (aj )
@B vi;j
vi;j+1
vi;j
a
vi;j
a
vi;j
1
;
(63)
:
(64)
In an upwind scheme, the choice of forward or backward derivative depends on the sign of the drift
function for the state variable, given by
si (a)
for
a
a+
Q
(yi
ci (a))
;
Q
(65)
0, where
v 0 (a)
ci (a) = i
Q
1=
(66)
:
Let superscript n denote the iteration counter. The HJB equation is approximated by the following
upwind scheme,
n+1
vi;j
n
vi;j
+
n+1
vi;j
(cni;j )1
=
1
2
2
n+1 n
n+1 n
+@F vi;j
si;j;F 1sni;j;F >0 +@B vi;j
si;j;B 1sni;j;B <0 +
i
v n+1
i;j
n+1
vi;j
;
for i = 1; 2; j = 1; :::; J, where 1 ( ) is the indicator function and
sni;;jF =
sni;j;B =
yi
a+
Q
yi
a+
Q
h
h
Q
n
@F vi;j
Q
Q
n
@B vi;j
Q
i1=
i1=
;
(67)
:
(68)
Therefore, when the drift is positive (sni;;jF > 0) we employ a forward approximation of the derivn+1
n+1
ative, @F vi;j
; when it is negative (sni;j;B < 0) we employ a backward approximation, @B vi;j
. The
term
n+1
n
vi;j
vi;j
n+1
n
! 0 as vi;j
! vi;j
: Moving all terms involving v n+1 to the left hand side and the
54
rest to the right hand side, we obtain
n+1
vi;j
n
vi;j
n+1
+ vi;j
=
(cni;j )1
1
2
2
n+1
+ vi;j
1
n
i;j
n+1
+ vi;j
n
i;j
n+1
+ vi;j+1
n
i;j
+
n+1
i v i;j ;
(69)
where
sni;j;B 1sni;j;B <0
n
i;j
a
;
sni;j;F 1sni;j;F >0
n
i;j
a
sni;j;F 1sni;j;F >0
n
i;j
a
+
sni;j;B 1sni;j;B <0
;
for i = 1; 2; j = 1; :::; J. Notice that the state constraints
In equation (69), the optimal consumption is set to
cni;j
=
i;
a
n
@vi;j
Q
a
0 mean that sni;1;B = sni;J;F = 0:
1=
(70)
:
where
n
n
n
n
@vi;j
= @F vi;j
1sni;j;F >0 + @B vi;j
1sni;j;B <0 + @vi;j
1sni;F
n
= Q(cni;j )
In the above expression, @vi;j
sni = 0 :
cni;j =
Equation (69) is a system of 2
1
:
0 1s n
i;B 0
where cni;j is the consumption level such that s (ai )
aj Q + y i :
Q
J linear equations which can be written in matrix notation as:
vn+1
vn + vn+1 = un + An vn+1
55
where the matrix An and the vectors v n+1 and un are de…ned by
2
n
A =
n
1;1
6
6 n
6 1;2
6
6 0
6
6 ..
6 .
6
6
6 0
6
6 0
6
6
6 2
6 .
6 ..
4
0
n
1;1
0
0
0
1
0
n
1;2
n
1;2
0
0
0
1
n
1;3
n
1;3
n
1;3
0
..
.
0
..
.
..
.
0
0
0
...
0
..
.
..
.
n
1;J 1
...
0
0
...
0
..
.
n
1;J 1
n
1;J
0
...
n
1;J 1
n
1;J
0
...
0
2
6
6
6
6
6
6
6
un = 6
6
6
6
6
6
4
1
(cn
1;1 )
1
1
(cn
1;2 )
1
1
(cn
1;J )
1
1
(cn
2;1 )
1
1
(cn
2;J )
1
0
..
.
0
0
0
n
2;1
n
2;1
..
2
2
2
2
2
2
..
.
.
..
.
..
.
.
..
.
n
2;J
2
..
.
..
1
..
.
0
2
0
2
2
2
The system in turn can be written as
3
3
2
n+1
v1;1
n+1
v1;2
n+1
v1;3
..
.
7
6
7
6
7
6
7
6
7
6
7
6
7
6
7
6
7
n+1
7 ; vn+1 = 6
6 v1;J
1
7
6
7
6 v n+1
7
6 1;J
1 7
6 n+1
7
6 v2;1
0 7
6 .
6 .
.. 7
7
4 .
. 5
n+1
n
v2;J
2;J
0
..
.
..
.
0
3
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
7
5
(71)
7
7
7
7
7
7
7
7:
7
7
7
7
7
5
Bn vn+1 = dn
(72)
where ; Bn = 1 + I An and dn = un + 1 vn .
0 J
The algorithm to solve the HJB equation runs as follows. Begin with an initial guess fvi;j
gj=1 ;
i = 1; 2. Set n = 0: Then:
n
n J
1. Compute f@F vi;j
; @B vi;j
gj=1 ; i = 1; 2 using (63)-(64).
2. Compute fcni;j gJj=1 ; i = 1; 2 using (66) as well as fsni;j;F ; sni;j;B gJj=1 ; i = 1; 2 using (67) and (68).
n+1 J
3. Find fvi;j
gj=1 ; i = 1; 2 solving the linear system of equations (72).
n+1
n+1
4. If fvi;j
g is close enough to fvi;j
g, stop. If not set n := n + 1 and proceed to 1.
Most computer software packages, such as Matlab, include e¢ cient routines to handle sparse
matrices such as An :
56
Step 2: Solution to the Kolmogorov Forward equation The stationary distribution of
debt-to-GDP ratio, f (a), satis…es the Kolmogorov Forward equation:
d
[si (a) fi (a)]
Z da
1
1 =
f (a)da:
0 =
i fi (a)
+
i f i (a);
i = 1; 2:
(74)
We also solve this equation using an …nite di¤erence scheme. We use the notation fi;j
The system can be now expressed as
0=
fi;j si;j;F 1sni;j;F >0
fi;j 1 si;j
1;F 1sn
i;j
1;F >0
(73)
fi;j+1 si;j+1;B 1sni;j+1;B <0
a
fi;j si;;jB 1sni;;jB <0
a
fi (aj ):
i fi;j +
or equivalently
fi;j
1 i;j 1
then (75) is also a system of 2
+ fi;j+1
i;j+1
+ fi;j
i;j
+
i f i;j
= 0;
(75)
J linear equations which can be written in matrix notation as:
AT f = 0;
(76)
where AT is the transpose of A = limn!1 An . Notice that An is the approximation to the operator
A and AT is the approximation of the adjoint operator A : In order to impose the normalization
constraint (74) we replace one of the entries of the zero vector in equation (76) by a positive
constant.50 We solve the system (76) and obtain a solution ^
f . Then we renormalize as
fi;j = P
J
j=1
f^i;j
f^1;j + f^2;j
:
a
Complete algorithm The algorithm proceeds as follows.
Step 1: Individual economy problem. Given ; compute the bond price Q using (62) and
solve the HJB equation to obtain an estimate of the value function v and of the matrix A.
Step 2: Aggregate distribution. Given A …nd the aggregate distribution f .
B.2 Optimal monetary policy - MPE
In this case we need to …nd the value of in‡ation that satis…es condition (22). The algorith proceeds
as follows. We consider an initial guess of in‡ation, (1) = 0. Set m := 1: Then:
50
In particular, we have replaced the entry 2 of the zero vector in (76) by 0:1.
57
i f i;j ;
Step 1: Individual economy problem problem. Given (m) ; compute the bond price Q(m)
using (62) and solve the HJB equation to obtain an estimate of the value function v(m) and
of the matrix A(m) .
Step 2: Aggregate distribution. Given A(m) …nd the aggregate distribution f (m) .
Step 3: Optimal in‡ation. Given f (m) and v(m) , iterate steps 1-2 until
2 X
J 1
X
(m)
(m)
vi;j+1
(m)
aj fi;j
vi;j
1
2
i=1 j=2
satis…es51
(m)
(m)
+
= 0:
B.3 Optimal monetary policy - Ramsey
Here we need to …nd the value of the in‡ation and of the costate that satisfy conditions (26) and
(25) in steady-state. The algorith proceeds as follows. We consider an initial guess of in‡ation,
(1)
= 0. Set m := 1: Then:
Step 1: Individual economy problem problem. Given (m) ; compute the bond price Q(m)
using (62) and solve the HJB equation to obtain an estimate of the value function v(m) and
of the matrix A(m) .
Step 2: Aggregate distribution. Given A(m) …nd the aggregate distribution f (m) .
Step 3: Costate. Given f (m) ; v(m) ;compute the costate
(m)
=
1
Q(m)
2
(m)
2 X
J 1
vi;j+1
X
(m)
4
aj fi;j
i=1 j=2
Step 4: Optimal in‡ation. Given f (m) ; v(m) and
r
(m)
(m)
+
1
2
(Q(m) )
2
2 X
J 1
X
4
(m)
(m)
using condition (25) as
3
(m)
vi;j
1
2
(m) 5
+
:
, iterate steps 1-3 until
(m)
(m)
a j + yi
i=1 j=2
(m)
ci;j
(m)
fi;j
satis…es
(m)
vi;j+1
vi;j
2
1
3
5:
C. Computational method: the dynamic case
C.1 Exogenous monetary policy
We describe now the numerical algorithm to analyze the transitional dynamics, similar to the one
described in Achdou et al. (2015). With an exogenous monetary policy it just amounts to solve the
51
This can be done using Matlab’s fzero function.
58
dynamic HJB equation (8) and then the dynamic KFE equation (14). De…ne T as the time interval
considered, which should be large enough to ensure a converge to the stationary distribution and
discretize it in N intervals of lenght
T
t= :
N
The initial distribution f (0; a; y) = f0 (a; y) and the path of in‡ation f t gTt=0 are given. We
proceed in three steps.
Step 0: The asymptotic steady-state The asymptotic steady-state distribution of the model
can be computed following the steps described in Section A. Given N ; the result is a stationary
destribution fN , a matrix AN and a bond price QN de…ned at the asymptotic time T = N t:
Step 1: Solution to the Bond Pricing Equation The dynamic bond princing equation (12)
can be approximated backwards as
(r +
n
+ ) Qn = +
Qn+1
Qn
t
; () Qn =
Qn+1 +
1 + t (r +
t
; n=N
n+ )
1; ::; 0;
(77)
where QN is the asymptotic bond price from Step 0.
Step 2: Solution to the Hamilton-Jacobi-Bellman equation The dynamic HJB equation
(8) can approximated using an upwind approximation as
(vn+1
vn = un + An vn +
vn )
t
;
where An is constructing backwards in time using a procedure similar to the one described in Step
n+1
1 of Section B. By de…ning Bn = 1t + I An and dn = un + V t ; we have
vn = (Bn )
1
dn :
(78)
Step 3: Solution to the Kolmogorov Forward equation Let An de…ned in (71) be the
approximation to the operator A. Using a …nite di¤erence scheme similar to the one employed in
the Step 2 of Section A, we obtain:
fn+1
fn
t
= AT
n fn+1 ; () fn+1 = I
tAT
n
1
fn ; n = 1; ::; N
where f0 is the discretized approximation to the initial distribution f0 (b):
59
(79)
Complete algorithm The algorith proceeds as follows:
Step 0: Asymptotic steady-state. Given
AN , bond price QN :
Step 1: Bond pricing. Given f
N 1
n gn=0 ;
N;
compute the stationary destribution fN , matrix
1
compute the bond price path fQn gN
n=0 using (77).
1
N 1
Step 2: Individual economy problem. Given f n gN
n=0 and fQn gn=0 solve the HJB equation
1
(78) backwards to obtain an estimate of the value function fvn gN
n=0 , and of the matrix
1
fAn gN
n=0 .
1
(k)
Step 3: Aggregate distribution. Given fAn gN
n=0 …nd the aggregate distribution forward f
using (79).
C.2 Optimal monetary policy - MPE
In this case we need to …nd the value of in‡ation that satis…es condition (22)
Step 0: Asymptotic steady-state. Compute the stationary destribution fN , matrix AN , bond
(0)
1
price QN and in‡ation rate N : Set (0) f n gN
n=0 = N and k := 1:
Step 1: Bond pricing. Given
(k 1)
; compute the bond price path Q(k)
(k)
1
fQn gN
n=0 using (77).
Step 2: Individual economy problem. Given (k 1) and Q(k) solve the HJB equation (78)
(k)
1
backwards to obtain an estimate of the value function v(k)
fvn gN
n=0 and of the matrix
(k)
1
A(k) fAn gN
n=0 .
Step 3: Aggregate distribution. Given A(k) …nd the aggregate distribution forward f (k) using
(79).
Step 4: Optimal in‡ation. Given f (k) and v(k) , iterate steps 1-3 until
(k)
n
2 X
J 1
X
(k)
(k)
aj fn;i;j
vn;i;j
2
This is done by iterating
(k)
n
with constant
=
(k 1)
n
= 0:05:
60
satis…es
(k)
vn;i;j+1
i=1 j=2
(k)
(k)
n ;
1
+
(k)
n
= 0:
C.3 Optimal monetary policy - Ramsey
In this case we need to …nd the value of the in‡ation and of the costate that satisfy conditions
(26) and (25)
Step 0: Asymptotic steady-state. Compute the stationary destribution fN , matrix AN , bond
(0)
1
price QN and in‡ation rate N : Set (0) f n gN
n=0 = N and k := 1:
Step 1: Bond pricing. Given
(k 1)
; compute the bond price path Q(k)
(k)
1
fQn gN
n=0 using (77).
Step 2: Individual economy problem. Given (k 1) and Q(k) solve the HJB equation (78)
(k)
1
backwards to obtain an estimate of the value function v(k)
fvn gN
n=0 and of the matrix
(k)
1
A(k) fAn gN
n=0 .
Step 3: Aggregate distribution. Given A(k) …nd the aggregate distribution forward f (k) using
(79).
Step 4: Costate. Given f (k) and v(k) , compute the costate
(k)
n+1
=
(k)
n
1+
t
+
(k)
Qn
with
(k)
0
2
(k)
f
(k) N 1
n gn=0
using (26):
(k)
vn;i;j
(k)
t
r
2
2 X
J 1
X
4
(k+1)
(k)
cn;i;j fn;i;j
aj + yi
(k)
vn;i;j+1
2
i=1 j=2
1
3
5;
= 0:
Step 5: Optimal in‡ation. Given f (k) , v(k) and
(k)
n
2 X
J 1
X
vn;i;j+1
vn;i;j
2
i=1 j=2
iterate steps 1-4 until
(k)
satis…es
(k)
(k)
(k)
aj fn;i;j
(k)
1
+
(k)
n
Q(k)
n
(k)
n
= 0:
This is done by iterating
(k)
n
=
(k 1)
n
(k)
n :
D. An economy with costly price adjustment
In this appendix, we lay out a model economy with the following characteristics: (i) …rms are
explicitly modelled, (ii) a subset of them are price-setters but incur a convex cost for changing
their nominal price, and (iii) the social welfare function and the equilibrium conditions constraining
the central bank’s problem are the same as in the model economy in the main text.
61
Final good producer
In the model laid out in the main text, we assumed that output of the consumption good was
exogenous. Consider now an alternative setup in which the consumption good is produced by a
representative, perfectly competitive …nal good producer with the following Dixit-Stiglitz technology,
Z 1
"=(" 1)
(" 1)="
yt =
yjt
dj
;
(80)
0
where fyjt g is a continuum of intermediate goods and " > 1. Let Pjt denote the nominal price of
R1
Pjt yjt dj, subject
intermediate good j 2 [0; 1]. The …rm chooses fyjt g to maximize pro…ts, Pt yt
0
to (80). The …rst order conditions are
yjt =
Pjt
Pt
"
(81)
yt ;
for each j 2 [0; 1]. Assuming free entry, the zero pro…t condition and equations (81) imply
R1
Pt = ( 0 Pjt1 " dj)1=(1 ") .
Intermediate goods producers
Each intermediate good j is produced by a monopolistically competitive intermediate-good producer, which we will refer to as ’…rm j’henceforth for brevity. Firm j operates a linear production
technology,
yjt = njt ;
(82)
where njt is labor input. At each point in time, …rms can change the price of their product but
face quadratic price adjustment cost as in Rotemberg (1982). Letting P_jt
dPjt =dt denote the
change in the …rm’s price, price adjustment costs in units of the …nal good are given by
t
P_jt
Pjt
!
2
P_jt
Pjt
!2
C~t ;
(83)
where C~t is aggregate consumption. Let jt
P_jt =Pjt denote the rate of increase in the …rm’s
price. The instantaneous pro…t function in units of the …nal good is given by
jt
Pjt
yjt wt njt
Pt
Pjt
Pjt
=
wt
Pt
Pt
=
62
t
(
jt )
"
yt
t
(
jt ) ;
(84)
where wt is the perfectly competitive real wage and in the second equality we have used (81) and
(82).52 Without loss of generality, …rms are assumed to be risk neutral and have the same discount
factor as households, . Then …rm j’s objective function is
Z
E0
1
t
e
jt dt;
0
with jt given by (84). The state variable speci…c to …rm j, Pjt , evolves according to dPjt =
jt Pjt dt. The aggregate state relevant to the …rm’s decisions is simply time: t. Then …rm j’s value
function V (Pjt ; t) must satisfy the following Hamilton-Jacobi-Bellman (HJB) equation,
V (Pj ; t) = max
j
(
Pj
Pt
wt
)
@V
@V
(Pj ; t) +
(Pj ; t) :
t ( j ) + j Pj
@Pj
@t
"
Pj
Pt
yt
The …rst order and envelope conditions of this problem are (we omit the arguments of V to ease
the notation),
@V
~
;
(85)
jt Ct = Pj
@Pj
@V
= "wt
@Pj
("
Pj
1)
Pt
"
Pj
Pt
yt
+
Pj
j
@V
@ 2V
+ Pj
@Pj
@Pj2
:
In what follows, we will consider a symmetric equilibrium in which all …rms choose the same price:
Pj = P; j = for all j. After some algebra, it can be shown that the above conditions imply the
following pricing Euler equation,53
"
dC~ (t) 1
dt C~ (t)
#
(t) =
"
1
"
"
1
w (t)
1
d (t)
1
+
:
dt
C~t
(86)
Equation (86) determines the market clearing wage w (t).
Households
The preferences of household k 2 [0; 1] are given by
E0
Z
1
e
t
log (~
ckt ) dt;
0
52
In the proofs of Propositions 1 and 2 w have been used to denote the social value function. There is no possibility
of confusion.
53
The proof is available upon request.
63
where c~kt is household consumption of the …nal good. We now de…ne the following object,
ckt
c~kt
c~kt +
C~t
Z
1
t
(
jt ) dj;
0
R
i.e. household k 0 s consumption plus a fraction of total price adjustment costs (
t ( ) dj) equal to
that household’s share of total consumption (~
ckt =C~t ). Using the de…nition of t (eq. 83) and the
symmetry across …rms in equilibrium (P_jt =Pjt = t ; 8j), we can write
ckt = c~kt + c~kt
2
2
t
= c~kt 1 +
2
t
2
(87)
:
Therefore, household k’s instantaneous utility can be expressed as
log(~
ckt ) = log (ckt )
log 1 +
2
2
t
2
= log (ckt )
2
2
t
+o
2
2
t
!
;
(88)
where o(kxk2 ) denotes terms of order second and higher in x. Expression (88) is the same as
the utility function in the main text (eq. 30), up to a …rst order approximation of log(1 + x)
around x = 0, where x 2 2 represents the percentage of aggregate spending that is lost to price
adjustment. For our baseline calibration ( = 5:5), the latter object is relatively small even for
relatively high in‡ation rates, and therefore so is the approximation error in computing the utility
losses from price adjustment. Therefore, the utility function used in the main text provides a fairly
accurate approximation of the welfare losses caused by in‡ation in the economy with costly price
adjustment described here.
Households can be in one of two idiosyncratic states. Those in state i = 1 do not work. Those
in state i = 2 work and provide z units of labor inelastically. As in the main text, the instantaneous
transition rates between both states are given by 1 and 2 , and the share of households in each
state is assumed to have reached its ergodic distribution; therefore, the fraction of working and
non-working households is 1 = ( 1 + 2 ) and 2 = ( 1 + 2 ), respectively. Hours per worker z are
such that total labor supply 1 +1 2 z is normalized to 1.
An exogenous government insurance scheme imposes a (total) lump-sum transfer t from working to non-working households. All households receive, in a lump-sum manner, an equal share of
R1
aggregate …rm pro…ts gross of price adjustment costs, which we denote by ^ t Pt 1 0 Pjt yjt dj
R1
wt 0 njt dj. Therefore, disposable income (gross of price adjustment costs) for non-working and
64
working households are given respectively by
t
I1t
I2t
2= ( 1
+
2)
+ ^ t;
t
wt z
1= (
1+
2)
+ ^ t:
We assume that the transfer t is such that gross disposable income for households in state i equals
a constant level yi , i = 1; 2, with y1 < y2 . As in our baseline model, both income levels satisfy the
normalization
2
1
y1 +
y2 = 1:
1+ 2
1+ 2
Also, later we show that in equilibrium gross income equals one: ^ t + wt 1 +1 2 z = 1. It is then
easy to verify that implementing the gross disposable income allocation Iit = yi , i = 1; 2, requires
2
^ t . Finally, total price adjustment costs are assumed to be
a transfer equal to t = 1 +2 2 y1
1+ 2
distributed in proportion to each household’s share of total consumption, i.e. household k incurs
ykt 2 fy1 ; y2 g denote
adjustment costs in the amount (~
ckt =C~t )( 2 2t C~t ) = c~kt 2 2t . Letting Ikt
household k’s gross disposable income, the law of motion of that household’s real net wealth is
thus given by
dakt =
=
Qt
t
akt +
Qt
t
akt +
Ikt
c~kt
c~kt
Qt
ykt
ckt
Qt
dt;
t =2
dt
(89)
where in the second equality we have used (87). Equation (89) is exactly the same as its counterpart
in the main text, equation (3). Since household’s welfare criterion is also the same, it follows that
so is the corresponding maximization problem.
Aggregation and market clearing
In the symmetric equilibrium, each …rm’s labor demand is njt = yjt = yt . Since labor supply
1
z = 1 equals one, labor market clearing requires
1+ 2
Z
1
njt dj = yt = 1:
0
Therefore, in equilibrium aggregate output is equal to one. Firms’pro…ts gross of price adjustment
costs equal
Z 1
Z 1
Pjt
^t =
yjt dj wt
njt dj = yt wt ;
0 Pt
0
65
such that gross income equals ^ t + wt = yt = 1.
Central bank and monetary policy
We have shown that households’welfare criterion and maximization problem are as in our baseline
model. Thus the dynamics of the net wealth distribution continue to be given by equation (14).
Foreign investors can be modelled exactly as in Section 2.2. Therefore, the central bank’s optimal
policy problems, both under commitment and discretion, are exactly as in our baseline model.
E. The methodology in discrete time
The aim of this Appendix is to illustrate how the methodology can be extended to discrete-time
models. We assume again that ( ; F; fFt g ; P) is a …ltered probability space but time is discrete:
t 2 N.
E.1 Model
Households
Output and net assets. The domestic price at time t, Pt , evolves according to
Pt = (1 +
t ) Pt 1 ;
(90)
where t is the domestic in‡ation rate.
Household k 2 [0; 1] is endowed with an income ykt per period, where ykt follows a two-state
Markov chain: ykt 2 fy1 ; y2 g ; with y1 < y2 . The transition matrix is
P=
"
p11 p12
p21 p22
#
:
Outstanding bonds are amortized at rate > 0 per period. The nominal value of the household’s
net asset position Akt evolves as follows,
Akt+1 = Anew
kt + (1
) Akt ;
where Anew
is the ‡ow of new issuances. The nominal market price of bonds at time t is Qt and
kt
ckt is the household’s consumption. The budget constraint of household k is
Qt Anew
kt = Pt (ykt
66
ckt ) + Akt :
The dynamics for net nominal wealth are
Akt+1 = (1 + rt ) Akt +
where rt Qt
is the nominal bond yield.
The dynamics of the real net wealth as akt
akt+1 =
1
1+
Pt (ykt ckt )
:
Qt
(91)
Akt =Pt are
(1 + rt ) akt +
t
ykt
ckt
= st (akt ; ykt ) :
Qt
(92)
From now onwards we drop subscripts k for ease of exposition. For any Borel subset A~ of
we de…ne the transition function associated to the stochastic process at as
i
h
~ yt+1 = yj jat = a; yt = yi ); i; j = 1; 2:
~ yj = P(at+1 2 A;
Ht (a; yi ) ; A;
This transition function equals
h
i
~ yj
Ht (a; yi ) ; A;
= pij 1A~ (st;i (a)) ;
where 1A~ ( ) is the indicator function of subset A~ and st;i (a) st (a; yi ) :
Preferences. Household have preferences over paths for consumption ckt and domestic in‡ation t discounted at rate > 0,
U0
E0
"
1
X
t=0
t
u(ct ;
t)
#
(93)
:
We use the short-hand notation vi (t; a) v(t; a; yi ) for the value function when household income
is low (i = 1) and high (i = 2): The Bellman equation results in
vi (t; a) = max u(ct ;
ct
t)
+ (T vi ) (t + 1; a); i = 1; 2;
(94)
where operator T is the Markov operator associated with (92), de…ned as54
(T vi ) (t + 1; a) = Et [v(t + 1; at+1 ; yt+1 )jat = a; yt = yi ]
2 Z
2
X
X
0
0
=
vj (t + 1; a ) Ht [(a; yi ) ; (da ; yj )] =
pij vj (t + 1; st;i (a)):
j=1
(95)
j=1
54
Notice that we consider the complete space R as the borrowing limit a¤ects the dynamics through the admissible
consumption paths.
67
The …rst order conditions of the individual problem are
uc (ci ) +
T
@vi
@a
(t + 1; a)
@st;i (a)
= uc (ci )
@ci
T
@vi
@a
(t + 1; a)
(1 +
t ) Qt
=0
(96)
Foreign investors
The nominal price of the bond at time t is given by
+ (1
) Qt+1
:
(1 + t ) (1 + r)
Qt =
Distribution dynamics
The state of the economy at time t is the joint density of net wealth and output, f (t; a; yi ) fi (t; a),
i = 1; 2: The dynamics of this density are given by the Chapman–Kolmogorov (CK) equation,
fi (t; a) = (T fi ) (t
where the adjoint operator Tt
(T fi ) (t
1; a) =
2 Z
X
Ht
(97)
1; a)
is de…ned as
1
1
0
0
[(a ; yj ) ; (a; yi )] fj (t
0
1; a )da =
j=1
2
X
j=1
pji
fj (t
1; st 11;j (a))
;
dst 1;j =da
where st;i1 (a) is the inverse function of st;i (a) : if a0 = st;i (a) then a = st;i1 (a0 ) :
The proof of the CK equation is as follows. Let
a; yt = yi ) =
P(at
be the joint probability of at
Z
a
fi (t; a0 ) da0 ;
1
a and yt = yi : It is equal to
2
X
j=1
pji
Z
st
1
1;j (a)
fj (t
1; a0 ) da0 ;
1
and taking derivatives with respect to a:
fi (t; a) =
2
X
j=1
pji fj t
1; st 11;j
2
dst 11;j (a) X
fj (t 1; st 11;j (a))
(a)
=
pji
;
da
dst 1;j =da
j=1
where we have applied the inverse function theorem.
68
(98)
The proof that T and T are adjoint operators is straightforward.55 If we de…ne T v(t; ) =
[T v1 (t; ); T v2 (t; )]T and T f (t; ) = [T f1 (t; ); T f2 (t; )]T the inner product results in
hT v(t + 1; ); f (t; )i =
=
2 Z
X
(T vi ) (t + 1; a)fi (t; a)da =
j=1
i=1
i=1
2 Z
X
2
2 Z X
X
i=1
2
X
pij vj (t + 1; st;j (a))fi (t; a)da
j=1
pij fi (t; a)vj (t + 1; st;j (a))da:
By changing variables a0 = st;i (a) :
hT v(t + 1; ); f (t; )i =
=
2 Z X
2
X
j=1
i=1
2 Z
X
j=1
=
pij fi (t; st;i1 (a0 ))vj (t + 1; a0 )
2 Z
X
j=1
"
2
X
i=1
da0
dst;i =da
#
fi (t; st;i1 (a0 ))
pij
vj (t; a0 )da0
dst;i =da
(Tt fj ) (t; a0 )vj (t; a0 )da0 = hv(t + 1; ); Tt f (t; )i :
E. 2 Optimal monetary policy
Central bank preferences
The central maximizes economy-wide aggregate welfare,
U0CB
=
1
X
t
t=0
Z
1
X2
i=1
u (ci (t; a) ; (t)) fi (t; a)da :
(99)
Markov Perfect Equilibrium
Step 1: Dynamic programming Given an initial condition f (t; ) we have an optimal control
path f (s)g1
s=t : The …rst step is to contruct the Bellman equation
W [f (t; )] = max
= max
Z X
2
Z
i=1
2
X
u (ci (t; a) ; ) fi (t; a)da + W [f (t + 1; )]
u (ci (t; a) ; ) fi (t; a)da + W [T f (t; )] ;
i=1
where we have applied the CK equation (97):We de…ne f (t; ) = [f1 (t; ); f2 (t; )]T
55
Alternatively, a general proof can be found in theorem 8.3 in Stockey and Lucas (1989).
69
(100)
Step 2: Optimal in‡ation The …rst order condition with respect to in‡ation in (100) is
2 Z
X
u (ci (t; a); ) fi (t; a)da +
i=1
@
@
W [T f (t; )] = 0:
In order to compute the …nal term we apply the chain rule and the Riesz representation theorem
as we did in the continuous-time case:
@
W [T f (t; )] =
@
W [f (t + 1; )] @T f (t; )
;
f
@
=
2 Z
X
wi (t + 1; a)
i=1
where h ; i is the inner product in the Hilbert space L2 (R) and wi (t + 1; ) =
L2 ( ) :
As T and T are adjoint operators
2 Z
X
wi (t + 1; a)
i=1
@T f
da =
@
w(t + 1; );
@T f
@
=
In order to compute
@T wi (t + 1; a)
@
=
@
@
=
@T wi
@
"
2
X
j=1
pij
[fi (t + 1; )] 2
@T wt+1
; f (t; )
@
we need to take into account the de…nition of T in equation (95):
#
pij wj (t + 1; st;i (a)) =
j=1
2
X
W
f
@
hw(t + 1; ); T f (t; )i
@
@
hT w(t + 1; ); f (t; )i =
@
2 Z
X
@T wi (t + 1; a)
=
fi da:
@
i=1
=
@T fi
da;
@
2
X
pij
j=1
@wj
st;i (a)
(t + 1; st;i (a))
=
@a
(1 + t )2
@wj
@st;i (a)
(t + 1; st;j (a))
@a
@
Et at+1 @w
(t + 1; at+1 ; yt+1 )jat = a; yt = yi
@a
(1 +
t)
2
so that the optimal in‡ation should satisfy
2 Z
X
i=1
E at+1 @w
(t + 1; at+1 ; yt+1 )jat = a; yt = yi
@a
(1 +
2
t)
70
!
u (ci (t; a); ) fi (t; a) da = 0:
(101)
;
Step 3: Central Bank’s HJB In order to …nd the value of w(t; ), we compute the Gateaux
derivative of the Bellman equation (100) with respetc to f .
Z X
2
d
d
d
W [f + h]j =0 =
u (ct ; ) (fi + hi ) da
+
W [T (f + h)]j
d
d
d
i=1
=0
Z
Z
Z
2
2
2
X
X
X
wi (t; a) hi (a) da =
u (ci ; ) hi (a)da +
wi (t + 1; a) T hi (t; a) da;
i=1
i=1
=0
;
i=1
and applying again the fact that T and T are adjoint operators:
2 Z
X
i=1
wi (t; a) hi (a) da =
2 Z
X
u (ci ; ) hi (a)da +
i=1
2 Z
X
i=1
T wi (t; a) hi (a) da:
This equality should be satis…ed for any h 2 L2 (R) so that we have
wi (t; a) = u (ci ; ) + Et [w(t + 1; at+1 ; yt+1 )jat = a; yt = yi ] ; i = 1; 2:
As this equation is the same as the individual Bellman equation (94). The boundary conditions
are also the same (state constraints on the domain ) and therefore its solution should be the
same: wi (t; a) = vi (t; a):
Proposition 5 (Optimal in‡ation - MPE, discrete time) If a solution to the MPE problem
(100) exists, the in‡ation rate function t must satisfy
2 Z
X
i=1
"
@v
(t + 1; at+1 ; yt+1 )jat = a; yt = yi
Et at+1 @a
(1 +
t)
2
#
u (ci (t; a); ) fi (t; a) da = 0:
Notice that this proposition is the the equivalent of Proposition 1 in discrete time.
71
Ramsey problem
Lagragian In this case the Lagragian can be written as
L [ ; Q; f; v; c] =
1
X
t
t=0
1
X
hut ; ft i +
+
t
1
X
+
+
t
t=0
1
X
+ T vt+1
t ; ut
t
ft
ft
1
+ (1
) Qt+1
(1 + t ) (1 + r)
t=0
1
X
t; T
t=0
Qt
t
t
t ; uc;t
(1 +
t=0
vt
t ) Qt
@vt+1
@a
T
;
where t t (a), t t (a), t t (a) e t t are Lagrange multipliers.
The problem of the central bank in this case is
max
s ;Qs ;vs (
f
);cs ( );fs ( )g1
s=0
L [ ; Q; f; v; c] :
(102)
We can apply the fact that T and T are adjoint operators to express
t
t
t
t ; uc;t
(1 +
t ; ut
ft
ft
=
t+1
+ T vt+1
vt
=
t
t; T
T
t ) Qt
@vt+1
@a
t
=
T
t+1 ; ft
h t ; ut
vt i +
h t ; uc;t i
(1 +
t
t+1
h t ; ft i ;
t+1
hT
t ) Qt
t ; vt+1 i ;
T
t;
@vt+1
@a
:
Necessary conditions In order to …nd the maximum, we need to take the Gateaux derivative
with respect to the controls f , ; Q; v and c.
The Gateaux derivative with respect to ft ( ) in the direction h is
t
hut ; ht i +
t+1
T
t
t+1 ; ht
h t ; ht i = 0:
Expression (103) should equal zero for any function hi (t; a) 2 L2 ([0; 1)
i
(t; a) = u(ct;i ;
t)
+ (T
which is a Bellman equation equal to (94) and thus
72
i
i ) (t; a) ;
(t; a) = vi (t; a) :
(103)
R) ; i = 1; 2 :
In the case of ct (a) ; the Gateaux derivative is
t+1
t
huct ht ; ft i
t
+
(1 +
h t ; ucc;t ht i +
ht T
t ) Qt
t+1
(1 +
t)
t+1
@ +1
; ft
@a
t
+
h t ; uct ht i
(1 +
t ) Qt
t ; ht T
@vt+1
@a
2
2
T
t ; ht
Q2t
@ vt+1
@a2
;
2
vt+1
@
t+1
where we have applied the fact that @c
T @v@a
= (1+ 1t )Qt T @ @a
2 : This expression should be zero
2
for any function hi (t; a) 2 L ([0; 1) R) ; i = 1; 2: Notice that
t;
1
uct
(1 +
T
t ) Qt
@vt+1
@a
ht
=0
due to the …rst order condition of the individual problem (96). Analogously,
1
ft ; uct
as
(1 +
T
t ) Qt
@
t+1
@a
ht
=0
= v: Therefore the optimality condition with respect to c results in
t
ucc;t +
(1 +
t)
2
Q2t
T
@ 2 vt
@a2
=0
As the instantaneous utility function is assumed to be strictly concave, ucc;t < 0 and the individual
2
value function v is also strictly concave @@av2t < 0 for all t and a then
ucc;t +
2
t)
(1 +
T
Q2t
@ 2 vt
@a2
<0
and the equality is only satis…ed if i (t; ) = 0; i = 1; 2:
In the case of vt (a) ; the Gateaux derivative is
t
t
h t ; ht i +
hT
t 1 ; ht i ;
where we have taken into account the fact that i (t; ) = 0: The Gateaux derivative should be
zero for any function hi (t; a) 2 L2 ([0; 1) R) ; i = 1; 2 so that we obtain a CK equation that
describes the propagation of the “promises”to the individual households:
t
where
1
=T
t 1;
= 0 as there are no precommitments. Hence
73
i (t;
) = 0; i = 1; 2::
In the case of Qt ; we compute the standard (…nite-dimensional) derivative:
@
T vt+1 ; ft + t
@Qt
(yt ct )
T vt+1 ; ft +
Q2t
t+1
Q2t
a
(1
t 1
t
1
t
)
= 0;
(1 + t 1 ) (1 + r)
(1
)
= 0;
1
(1 + t 1 ) (1 + r)
t 1
t
and thus
t
=
t 1
(1 +
2 Z
X
)
+ 2
( a + yi
Qt i=1
1 ) (1 + r)
(1
t
ci (t; a)) T
@vi
@a
(t + 1; a) fi (t; a) da:
The lack of any precommitment to bondholders implies 1 = 0:
Finally, we compute the standard derivative with respect to t :
t
hu t ; ft i
hu t ; ft i +
(1 +
t)
2
T
+ (1
) Qt+1
@
T vt+1 ; ft + t t
= 0;
2
@ t
(1 + t ) (1 + r)
@vt+1
Qt+1
at+1
= 0;
; ft + t
@a
(1 + t )2 (1 + r)
t+1
and hence
t Qt+1
= (1 + r)
2 Z
X
i=1
T
a
@vi
@a
(t + 1; a)
u (t; a) fi (t; a) da:
The solution to the Ramsey problem in discrete time is given by the following proposition
Proposition 6 (Optimal in‡ation - Ramsey discrete time) If a solution to the Ramsey problem (102) exists, the in‡ation path (t) must satisfy
t Qt+1
= (1 + r)
2 Z
X
i=1
where
t
T
a
@vi
@a
(t + 1; a)
u (t; a) fi (t; a) da;
(104)
ci (t; a)) T
@vi
@a
(105)
(t) is a costate with law of motion
=
t 1
(1 +
2 Z
X
)
+ 2
( a + yi
Qt i=1
1 ) (1 + r)
(1
t
and initial condition
1
(t + 1; a) fi (t; a) da:
= 0.
Notice that this proposition is the the equivalent of Proposition 2 in discrete time.
74