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Working Paper Series
State Dependent Monetary Policy
WP 13-17
Francesco Lippi
EIEF and University of Sassari
Stefania Ragni
University of Sassari
Nicholas Trachter
Federal Reserve Bank of Richmond
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
State dependent monetary policy ∗
Francesco Lippi
Stefania Ragni
EIEF and University of Sassari
University of Sassari
Nicholas Trachter
Federal Reserve Bank of Richmond
Working Paper No. 13-17
October 23, 2013
Abstract
We study the optimal anticipated monetary policy in a flexible-price economy featuring heterogenous agents and incomplete markets, which give rise to a business cycle.
In this setting money policy has distributional effects that depend on the state of the
cycle. We parsimoniously characterize the dynamics of the economy and study the
optimal regulation of the money supply as a function of the state. The optimal policy
prescribes monetary expansions in recessions, when insurance is most needed by cashpoor unproductive agents. To minimize the inflationary effect of these expansions the
policy prescribes monetary contractions in good times. Although the optimal money
growth rate varies greatly through the business cycle, this policy “echoes” Friedman’s
principle in the sense that the expected real return of money approaches the rate of
time preference.
JEL classification: E50.
Keywords: Incomplete markets, Heterogenous agents, Liquidity, Precautionary savings,
Friedman rule, distributional effects
∗
We thank Fernando Alvarez, Aleks Berentsen, Francisco Buera, Isabel Correia, Piero Gottardi, Christian
Hellwig, Hugo Hopenhayn, Patrick Kehoe, Ricardo Lagos, David Levine, Fabrizio Mattesini, Guido Menzio,
John Moore, Ezra Oberfield, Nicola Pavoni, Facundo Piguillem, Tom Sargent, Rob Shimer, Alp Simsek,
Balasz Szentes, Aleh Tsyvinski, Harald Uhlig, Randall Wright, and Bill Zame for useful discussions. We
are grateful to seminar participants at the 1st Rome Junior Conference on Macroeconomics, 9th Hydra
Conference, Banque de France, EIEF’s summer macro lunch seminars and Macro Reading Group, Bank of
Italy, Bank of Portugal, FRB-Chicago Summer Workshop on Money, Banking, Payments and Finance (2012),
FRB-Boston, FRB-Richmond, FRB-St. Louis, Georgetown,Milano Bicocca, NYU, Penn State, Toulouse
School of Economics, UCLA, Universitat Autonoma Barcelona, Universitat Pompeu Fabra, Universidad
Torcuato Di Tella, Universidad de San Andres, Tinbergen Institute and University of Edinburgh. The
views expressed in this article are those of the authors and do not necessarily represent the views of the
Federal Reserve Bank of Richmond or the Federal Reserve System.Authors’ email addresses: flippi@uniss.it;
s.ragni@ba.iac.cnr.it; nicholas.trachter@rich.frb.org.
1
Introduction
We study optimal monetary policy in a competitive flexible-price economy where infinity-lived
agents are subject to idiosyncratic productivity shocks and money is valued in equilibrium
due to anonymity. The state of this economy is described by the wealth distribution, which
evolves through time following the history of shocks and determines the value of money and
aggregate output. Our objective is to characterize how to optimally regulate the money
supply as a function of the state of the economy, what we call an optimal state-dependent
monetary policy. A key feature of the setup is that monetary policy affects (and is affected
by) the wealth distribution, as in Wallace (1984) or Berentsen et al. (2005) and many other
monetary models whose first principles are explicitly spelled out. Although the propagation
of such redistributive effects of monetary policy is often “muted” by means of appropriate
assumptions for the sake of tractability, as in Lucas (1990), Shi (1997), or Lagos and Wright
(2005), in this paper we use an analytically tractable setup that allows us to study the role of
systematic monetary policy taking fully into account the dynamics of the wealth distribution.
The key assumption to get tractability, following Scheinkman and Weiss (1986), is that we
consider the simplest economy with time varying wealth distribution, namely one with two
types of agents.1 We see this as a convenient starting point to study the interactions between
the dynamics of the wealth distribution and monetary policy.
We think the question is interesting because it is novel in the theory and because the analysis provides a framework to interpret the large monetary expansions, sometimes observed
during deep recessions, with a mechanism that is completely different from the canonical one
relying on sticky prices.2 The properties of an optimal monetary policy in models where
incomplete markets and heterogenous agents allow for a potential redistributive role of monetary policy were first studied by Levine (1991). They have since been explored in a variety
of contexts by Kehoe et al. (1990), Shi (1999), Bhattacharya et al. (2005), Molico (2006),
Algan et al. (2011), and Tenreyro and Sterk (2013), for example. A common feature of
these models is a tension between the benefits of a deflationary policy, one that produces an
efficient return on money as under the Friedman’s rule, and the benefits of an expansionary
policy, which provides partial insurance to cash-poor agents. A novelty of this paper is that
while previous models focused on a constant rule, i.e., seeking the optimal constant rate of
monetary expansions, we consider a state-dependent monetary policy, in which the rate of
monetary expansion depends on the state of the economy.
1
This assumption is often used in macro theory for tractability, see e.g., Grossman and Weiss (1983),
Alvarez and Lippi (2013), or Golosov and Sargent (2013).
2
The assumption of flexible prices is useful to emphasize the workings of the redistributive role of monetary
policy, and to distinguish it from the better understood mechanism that arises with sticky prices. Both
channels are likely relevant in practice.
1
Our model extends Scheinkman and Weiss’s (1986) analysis, which assumes a constant
money supply, by letting the government control the money supply through lump-sum transfers.3 We provide a characterization of the price of money and of aggregate production in
terms of the policy rule in a competitive equilibrium. We adopt an ex-ante welfare criterion
and characterize an efficient monetary policy by solving a Ramsey problem.
Our results cast some light on the interactions between the dynamics of the assets distribution and the optimal anticipated policy. The main policy choice is a tradeoff between
the provision of insurance (through monetary expansions) and ensuring an efficient return on
savings (through contractions). We show that in our model the importance of the insurance
motive varies with the state of the economy, so that a state-dependent monetary policy allows
for a significant improvement compared with a constant policy. An expansionary monetary
policy turns out to be efficient in recessions, when poor and unproductive economic agents
benefit from some wealth redistribution. Surprisingly, in spite of the occasional large monetary expansions in states where the insurance motive is large, the optimal policy neutralizes
the inflationary effect of these expansions by contracting the money supply in states where
the insurance motive is small. In this way the state-dependent nature of the optimal policy
allows for the provision of insurance when mostly needed without severely distorting the return of the asset. Thus, although our setup creates a potentially beneficial role for monetary
expansions, the optimal rule prescribes an almost complete undoing of the inflationary effects
of those expansions, and it implements a policy that brings the expected return on money as
close as possible to the constant rate prescribed by the Friedman rule.
Our research question is related to the one analyzed by Molico (2006), who considers
a search model of money with a nondegenerate distribution of cash holdings, showing that
mild monetary expansions can be beneficial. In his model randomly matched agents may
exchange goods for money. The price paid by the buyer results from bargaining and depends
on the amount of money held by each agent upon entering the pairwise meeting. Therefore,
the distribution of money is nondegenerate and monetary injections, via lump-sum transfers,
can improve the terms of trade of poor buyers. Related results in the context of search
models of money and mechanism design are obtained by Berentsen et al. (2005), Green and
Zhou (2005), Deviatov and Wallace (2012), Wallace (2013). Our model departs from Molico
in a few important ways. First, we restrict attention to an economy with only two types of
agents, so that the wealth distribution is analytically tractable, while Molico does not restrict
the number of types and as a consequence his model must be solved numerically. Second,
and perhaps more importantly, we allow the planner to tie its policy to the distribution
3
As in Levine (1991) we assume that the government does not know which agent is productive, so that
the transfers are equal across agents. See Kehoe et al. (1990) for a thorough discussion of this assumption
and in particular Levine (1991) for a derivation of the equal-treatment restriction from first principles.
2
of cash holdings, while Molico considers a constant policy. A key feature of our model
is that business cycles, and the magnitude of fluctuations, depend on the tightness of the
borrowing constraint. Because the borrowing constraint is tighter in downturns the return
on money is high in recessions and low during booms. As a result inflation (the inverse of the
return on money) is positively correlated with aggregate activity, thus generating a “Phillips
curve’.’ This result relates to Guerrieri and Lorenzoni (2009) and Guerrieri and Lorenzoni
(2011), which explore the effects of borrowing constraints on business cycles in a model with
liquid assets. Similar to their papers, as the borrowing constraint becomes tighter, economic
fluctuations become more severe in our model. The relative simplicity of our setup allows
us to investigate the optimal provision of liquidity. Our analysis is also related to Algan
et al. (2011), that characterizes the output-inflation tradeoff in a flexible price economy with
incomplete markets and persistent wealth inequality among agents. While the setups are
similar, the focus is different as monetary policy is treated as an exogenous parameter in
these models.
The paper is organized as follows. Section 2 presents the set up of the model. Section 3
sets the ground for the analysis of monetary equilibria. Section 4 characterizes the value
of money in equilibrium under a state-dependent rule and compares it with the one with a
constant money rule. Section 5 defines an ex ante welfare criterion and studies the best state
dependent rule for the supply of liquidity. Section 6 concludes.
2
The model
This section describes the model economy: agents’ preferences, production possibilities, and
markets. Two useful benchmarks are presented: the (efficient) allocation with complete
markets and the optimal monetary policy with no uncertainty. To finalize, we also argue that
value functions and allocations are homogeneous in the exogenous parameters of interest; this
is useful as it allows us to reduce the dimensionality of the problem by doing an appropriate
normalization.
We consider two types of infinitely lived agents (with a large mass of agents of each type),
indexed by i = 1, 2, and assume that at each point in time only one type of agent can produce.
We further restrict attention to the case where agents of the same type play the same action
at every point in time so that we can discuss the model in terms of two representative agents,
one of each type. Because there are two agents in this economy, we can solve the model
by looking at the problem from the perspective of agent one (i.e., i = 1). Let It = {0, 1}
denote the productive type of the agent at time t. When It = 1 the agent can produce and
transforms labor into consumption one for one; we label this agent as productive. When
3
It = 0 the agent cannot produce; in this case we label the agent as unproductive. The
productivity of labor is state dependent: The duration of productivity spells is random,
exponentially distributed, with mean duration 1/λ > 0. Money is distributed at each time
t between the two agents so that m1t + m2t = Mt . The growth rate of the money supply
Rt
at time t is µt ; then, the money supply follows Mt = M0 e 0 µj dj , with M0 given. As in
Scheinkman and Weiss (1986) and Levine (1991), we let the individual state of an agent be
private information, precluding agents from issuing private debt.4 A key assumption is that
agents face a borrowing constraint restricting their unique savings instrument, money, to be
non-negative. Because of the assumption of anonymity, fiscal policy has limited powers in
this setup.5
Let ρ > 0 denote the time discount rate. Each agent chooses consumption ct , labor supply
`t , and depletion of money balances ṁ1t in order to maximize her expected discounted utility,
Z
max
{ct ,`t ,ṁ1t }∞
t=0
∞
e
E0
−ρt
(c̄ ln ct − It `t ) dt
(1)
0
subject to the constraints
ṁ1t ≤ (`t + τt − ct ) /q̃t
ṁ1t ≤ (τt − ct ) /q̃t
and `t = 0
m1t ≥ 0 , `t ≥ 0 , ct ≥ 0 , m10 , I0 given ,
if It = 1
(2)
if It = 0
(3)
(4)
where c̄ > 0 is a parameter governing the marginal utility of consumption, q̃t denotes the price
of money, i.e., the inverse of the consumption price level, τt denotes a government lump-sum
transfer to each agent, and where expectations are taken with respect to the productivity
process defining It and Mt conditional on time t = 0.
A monetary policy with µt > 0 is called expansionary, a policy with µt < 0 is called
contractionary. It is immediate that when the money supply is constant for all t (i.e., µt =
0 ∀ t) and c̄ = 1, the economy is the one analyzed by Scheinkman and Weiss (1986). The
monetary policy µt determines the transfers to the agents τt through the government budget
constraint,
q̃t µt Mt = 2τt .
The government transfer scheme implies that in the case of a contractionary policy, agents
4
Having a large mass of agents of each type is important for the argument as it implies that a single agent
cannot infer the productive state of a different agent given his own state.
5
In the online Appendix we discuss what allocations can be achieved using tax policy under various
assumptions about government powers (commitment vs. no commitment), types of available taxes (lump-sum
vs. distortionary), and government knowledge about the state (agent’s type observable vs. not observable).
4
must use their money holdings to pay taxes (i.e. τt < 0). The “tax solvency” constraint,
m1t ≥ 0, imposes this restriction. Notice that in the continuous time characterization of the
model the tax solvency constraint coincides with the borrowing constraint.
Note that the government cannot differentiate transfers across agent types. This follows
from the assumption that the identity of the productive type is not known to the government.
Levine (1991) shows in a similar setup that, because of anonymity, the best mechanism is
linear and resembles monetary policy.
Next we state two important remarks. The first one characterizes a symmetric efficient
allocation with complete markets (the proof is standard so we omit it):
Remark 1 Assume complete markets and an ex-ante equal probability of each productive
state. The symmetric efficient allocation prescribes the same constant level of consumption,
ct = c̄ for all t.
Thus, without borrowing constraints, the efficient allocation solves a static problem, and it
encodes full insurance: Agents consume a constant amount c̄ (since ex-ante agents are equal)
and the aggregate output is constant.
The second remark characterizes the optimal monetary policy in the case of no uncertainty. This helps highlighting the essential role of uncertainty in our problem. In particular,
consider the case where each agent oscillates deterministically between productive and unproductive cycles of length T . Without loss of generality, for the characterization of the
stationary equilibria, let us assume that the economy starts in period t = 0 with the agent
being productive and holding no money, so that m10 = 0. We have
Remark 2 Consider a deterministic production cycle of length T . The symmetric efficient allocation, ct = c̄ for all t is attained by deflating at the rate of time preference
µt = −ρ for all t.
A proof is available in Appendix B.1. This remark, together with the efficient allocation
described in Remark 1, shows that without uncertainty this economy replicates the result of
the optimality of the “Firedman rule” in Townsend (1980) and Bewley (1980).
To conclude, notice that, by inspection of the agent problem presented in equations (1)
Rt
to (4) and the evolution of money supply (Mt = M0 e 0 µj dj ), it is easily seen that the problem
is homogeneous on {λ, ρ, µt }: Allocations (the flows) are homogeneous of degree 0 while
prices and values (the stocks) are homogeneous of degree minus 1. This result is a natural
consequence of the Poisson rate of changing states λ, the discount rate ρ, and the monetary
expansion rate µt all being measured with respect to calendar time. This result is useful as
it shows that, after normalizing by λ, the model has only three parameters: the normalized
5
discount rate, ρ/λ, the normalized money growth rate, µt /λ, and c̄. This is useful as once
we treat µt as a policy instrument, the model has only two exogenous parameters: c̄ and the
normalized discount rate ρ/λ.
3
Characterization of monetary equilibrium
We look for an equilibrium where the price of money depends on the whole history of shocks,
as encoded in the current values of the money supply, the distribution of money holdings,
and the current state of productivity; that is, we let q̃t = q̃(Mt , m1t , It ). With a slight abuse
of notation this implies ct = c(Mt , m1t , It ), `t = `(Mt , m1t , It ), and ṁ1t = ṁ1 (Mt , m1t , It ).
As usual the nominal variables are homogenous of degree one in the level of money. With
this in mind, we simplify the state space by letting q̂(zt , It ) = Mt q(Mt , m1t , It ) denote the
m1
price of money when the agent holds zt ≡ Mtt share of total money balances-a measure of
the wealth distribution-where zt ∈ [0, 1]. Likewise the consumption and labor supply rules
are homogeneous of degree zero in the level of the money supply, cp (zt ) = c(Mt , m1t , 1),
cu (zt ) = c(Mt , m1t , 0), and `p (zt ) = `(Mt , m1t , 1).
Let xt denote the wealth share in the hands of the unproductive agent. Note that this
variable will record discrete jumps every time the identity of the productive type changes:
Whenever the identity of the productive type switches, the state x jumps. We allow the
planner to choose a Markovian monetary policy µt that is a continuous function of the
state x ∈ [0, 1]. As a result, notice that xt summarizes the whole history of the economy and,
without loss of generality, µt = µ(xt ). Given the symmetry of the problem we let q(xt ) denote
the price of money in terms of consumption units, which occurs when the unproductive type
assets are xt . That is, if agent 1 holds wealth zt , q(xt ) = q̂(zt , 0) if she is unproductive and
q(xt ) = q̂(1 − zt , 1) when she is productive. Next we define a monetary equilibrium.
Definition 1 Given the continuous policy rule µ(xt ), the initial level of money supply M0 ,
the initial productivity status I0 , the an initial distribution of money holdings x0 , a monetary
equilibrium is a price function q̃t = M1t q(xt ), with q : [0, 1] → R+ and a stochastic process
xt with values in [0, 1] such that, for all t, consumers maximize expected discounted utility
(equation (1)) subject to (2), (3) and (4), and the market clearing constraint cp (1 − xt ) +
cu (xt ) = `(1 − xt ) and the government budget constraint are satisfied.
From now on we omit the time index t to simplify the notation. A straightforward result
is that permanent deflations cannot be implemented in equilibrium.6 We state this result in
the next proposition.
6
This result relates to Bewley (1983), who showed that in a neoclassical growth model with incomplete
markets there is no monetary equilibrium if the interest rate is lower than the discount rate.
6
Proposition 1 There is no monetary equilibrium where µ(x) < 0 for all x. Moreover, all
monetary equilibria must satisfy µ(0) ≥ 0.
See Appendix B.2 for a proof. The economics of this result is simple. As the length of
the unproductive spell cannot be bounded above, there is a nonzero probability that a poor
unproductive agent fails to cover her tax obligations. The only way she can fulfill her tax
obligations is by keeping half of the money stock and not trading for goods. Because of no
trade, money has no value (i.e., q(x) = 0 for all x), there is no monetary equilibrium, and the
allocation is autarkic. Moreover for any rule, including those that may allow for monetary
contractions, the money growth rate cannot be negative at x = 0. This is immediate since
when x = 0 unproductive agents hold no assets and are unable to cover their tax obligations.
Solving the model requires characterizing the marginal value of money given by the Lagrange multipliers for ṁ in the problem defined in (1). Let p̃(M, m1 ) and ũ(M, m1 ) denote
the un-discounted multipliers associated with the constraints in equations (2) and (3), respectively, so that, e.g., ũ(M, m1 ) measures the marginal value of money for agent 1 when the
money supply is M , her wealth share is m1 /M , and she is unproductive. Likewise, p̃(M, m1 )
measures the marginal value for agent 1 when her wealth share is m1 /M and she is productive. Using the homogeneity in the level of money M , we can write ũ(M, m1 ) = u(z)/M and
p̃(M, m1 ) = p(z)/M .
Combining the first order conditions with respect to ` and cu gives
p(z) = q(x) ,
c̄
cp (z)
=
p(z)
q(x)
, and
c̄
cu (z)
=
u(z)
,
q(x)
(5)
where z is the share of money in the hands of the agent and x is the share of money in the
hands of the unproductive agent. These conditions equate marginal costs and benefits of an
additional unit of money. The first two equations apply when the agent is productive (i.e.
I = 1). The first one states that the marginal benefit of an a additional unit of money,
p(z), equals the cost of obtaining that unit, i.e. the disutility of work to produce and sell a
consumption amount q(x). The second equation states that the marginal cost of the foregone
unit of money, which is p(z), equals the marginal benefit, which is given by the product of
the price q(x) (consumption per unit of money) times the marginal utility of consumption
c̄/cp (z). Notice that combining these two equations implies that cp (z) = c̄ for all z. Finally,
the third equation applies when the agent is unproductive (i.e., I = 0) and states that the
marginal cost of the forgone unit of money u(z), equals the benefit that is given by the
additional units of consumption that can be bought with it: the product of the price q(x)
times the marginal utility of consumption c̄/cu (z).
7
The following functions define the evolution of money holdings,
ż u (z) = hu (z)
and
ż p (z) = hp (z) ,
(6)
where hu (z) is the change in the share of money holdings of an unproductive agent holding
a share z, and hp (z) is the analogue for a productive agent. It is immediate that hu (z) +
hp (1 − z) = 0. Consider the law of motion for z by type 1 when unproductive,
u
h (z) = µ(x)
1
−z
2
cu (z)
−
= µ(x)
q(x)
1
−z
2
−
c̄
,
u(z)
(7)
where we used the budget constraint of the unproductive agent, equation (3), the government
budget constraint, and the first order condition in equation (5).
For any z ∈ (0, 1) the marginal value of money for productive and unproductive agents,
p(z), and u(z), solve a system of differential equations, which is the continuous time counterpart of the discrete time Euler equations,
(ρ + µ(x)) p(z) = p0 (z) hp (z) + λ(u(z) − p(z)) ,
(8)
(ρ + µ(x)) u(z) = u0 (z) hu (z) + λ(p(z) − u(z)) .
(9)
The derivation is standard so we omit it. To provide some intuition consider the first equation:
When the agent is productive and holds a share of money z, the value flow, discounted by
the nominal rate (ρ + µ(x)), is equal to the change in the marginal value due to the evolution
of her money holdings, p0 (z)hp (z), and to the expectations of the change in value in case the
state switches and the agent becomes unproductive: λ(u(z) − p(z)).
To complete the description of the equilibrium we provide the boundary condition for
the marginal value of money p(z) and u(z). The boundary occurs when the unproductive
agent has no money. In this case an unproductive agent spends the whole money transfer to
finance her consumption, so that hu (0) = hp (1) = 0.7 The budget constraint gives that the
consumption of an unproductive agent with no money is limz→0 cu (z) = τ (0) = q(0)µ(0)/2.
Using equation (5),
2 c̄
,
(10)
lim u(z) =
z→0
µ(0)
with limz→0 u(z) = +∞ if limx→0 µ(x) = 0, and where the limit obtains because of Inada
conditions. This is an important result in our analysis. An expansionary policy provides an
upper bound to the marginal utility of money because the agent enjoys a positive consumption
even with no wealth. If there is no money growth when the unproductive agent is poor (i.e.,
7
We provide a formal proof of this statement in Section 4.
8
when µ(x) → 0 as x → 0), the agent is not able to consume in poverty and therefore Inada
conditions imply that her marginal utility diverges. Also, evaluating equation (8) at z = 1
and x = 0 gives
ρ µ(0)
u(1) = 1 + +
p(1) .
(11)
λ
λ
Notice that µ(0), the money growth rate when the unproductive agent has zero wealth,
appears in both boundaries. An implication is that the choice of the money growth rate
simultaneously affects the insurance needs of the unproductive agents and the production
incentives of the productive agents.
Using the first order conditions in equation (5) yields an expression for the consumption
of an unproductive agent,
p(1 − z)
.
(12)
cu (z) = c̄
u(z)
This is an important object because unproductive agents need to spend money to consume
and therefore monetary policy will affect their consumption behavior. Looking at the consumption level of the unproductive agent when she has no money gives an idea of the insurance
role of monetary expansions,
µ(0)
lim cu (z) =
p(1) ,
(13)
z→0
2
with limz→0 cu (z) = 0 if limx→0 µ(x) = 0, where the limit obtains from the agent’s budget
constraint. This is important for the welfare analysis that will follow because it shows that
monetary transfers provide the unproductive agent with a lower bound to her consumption
level. Without transfers, an agent with no money cannot consume.
Another interesting object is the expected real return on money or, equivalently, the
expected real interest rate. This object is useful in understanding the workings of the model
and illustrates how the market incompleteness affects the economy by generating a risk
premium. Let r(x) denote the expected (net) return on money,
˙
q̃(x)
xt = x
r(x) ≡ E
q̃(x)
,
which is just the expected growth rate of the price of money conditional on the unproductive
agent money holdings being x. Using that q̃(x) = p(1 − x)/M , the expected return can be
written as (see Appendix A)
p(x)
r(x) = ρ + λ
p(1 − x)
cu (1 − x) − c̄
cu (1 − x)
.
(14)
In a complete markets setting, such as the one described in Remark 2, consumption is constant
9
(at level c̄) and the expected real return equals the time discount, ρ. With incomplete
markets the return on money depends on the history of the shocks, as summarized by the
wealth distribution x. Recall, by equation (5), that the price of money is determined by
the productive agent q(x) = p(1 − x). Equation (14) shows that the expected real rate is
proportional to the change in the price of money associated to a switch of the state, i.e.,
p(x)/p(1 − x), and is proportional to the productive agent’s expected consumption growth.8
Finally, noting that 1/q̃(x) denotes the price of consumption in units of money, we can
compute the expected inflation rate
π(x) = −
r(x)
.
1 + r(x)
(15)
After determining the equilibrium functions u(z), p(z), cu (z) we will use these formulas to
argue that the model displays a “Phillips curve,” a positive correlation between the inflation
rate and the output level.
4
The value of money in equilibrium
The previous analysis showed that allocations in a monetary equilibrium are fully characterized by the Lagrange multipliers, u(z) and p(z), that solve the system of Euler equations
and associated boundary conditions. This section characterizes the properties of these multipliers, measuring the “value of money” to the productive and unproductive agents, under
two rules for the money supply. The first rule is state independent, i.e., equal to a constant
non-negative money growth value µ ∈ [0, +∞). The second rule assumes that the money
growth rate is a continuous function of the wealth share of the unproductive agent: µ(x).
Using equations (8)-(9) we can define the following system:
(ρ + λ + µ(1 − z))p(z) − λu(z)
,
hp (z)
(ρ + λ + µ(z))u(z) − λp(z)
u0 (z) =
.
hu (z)
p0 (z) =
(16)
(17)
The first equation describes the marginal value of money for a productive agent holding z,
so that x = 1 − z, i.e., monetary policy is a function of the wealth of the unproductive
agent. The second equation describes the marginal value for an unproductive agent holding
z, so that x = z. The solution of this system, together with the boundary condition, fully
8
Consumption remains constant with probability 1 − λ per unit of time, or it changes from c̄ to cu (1 − x)
with rate λ.
10
characterizes the value of money in equilibrium.
Next we state a result that is key in characterizing the problem.
Proposition 2 Assume µ(x) is continuous in [0, 1] and that the forcing terms on the right
hand side of (16)-(17) have no singularities in (0, 1). Then, for any c̄ > 0, it holds that
hu (z) < 0 for all z ∈ (0, 1) and limz→0 hu (z) = 0.
See Appendix B for a proof.9 The economic content of the proposition is that unproductive
agents deplete their share of money holdings as long as they remain unproductive. Proposition 2 allows us to characterize some interesting features of the Lagrange multipliers, the
marginal value of money p(z) and u(z), by representing their evolution in the corresponding
phase diagram. To this end we define the sets Lp and Lu with elements (U, P ) ∈ (R+ )2 ,
which evolve as z varies in [0, 1]: U is the unproductive agent marginal utility u(z) and P is
given by
Lp :
P =
λ
u(z), and Lu :
ρ + λ + µ(1 − z)
P =
ρ + λ + µ(z)
u(z),
λ
according to z that moves in its domain. These geometrical loci are very useful for investigating some features of the Lagrange multipliers. Indeed, by construction, p0 (z)hp (z) and
u0 (z)hu (z) are zero on Lp and Lu , respectively; thus, according to the sign of terms hp (z)
and hu (z), both Lp and Lu determine different regions in the phase plane (U, P ) where the
functions u(z), p(z) change behavior according to the sign of their derivative. Two examples
are shown in the top panels of Figure 1 where the arrows describe the increasing/decreasing
behavior for u(z) and p(z). The direction of the arrows is determined by taking into account
both the sign of terms p0 (z)hp (z), u0 (z)hu (z) and the evolution of money holdings established
in Proposition 2. The phase diagrams describe the model dynamics: The first case, plotted
in the upper-left panel, describes a constant money rule; the second case, describing a statedependent rule, is plotted in the upper-right panel. The bold dot in each plot corresponds to
p(1), which lies on Lp , the 45-degree
the boundary condition in equation (11): u(1) = ρ+λ+µ(0)
λ
line P = U is denoted by dotted line. The dashed curve is a possible path for the solution.
Next we exploit the phase diagram to investigate some features of the solution under the
different type of policies. In particular, our analysis allows us to provide global results for
the constant policy. However, when the policy is allowed to vary with the state, the same
results can be obtained only locally when x is sufficiently small (i.e., when the productive
agent holds most of the money).
9
This result arises from the requirement of continuity and uniform Lipschitz condition in the space (0, 1)
for the forcing terms of equations (16)-(17).
11
Figure 1: The marginal value of money
Phase diagrams
Constant policy
Lu
State dependent policy
p(z) I region
Lu
p(z)
II region
Lp
Lp
p(1)
p(1)
III region
u(z)
u(1)
u(1)
u(z)
The shadow value of money: Lagrange multipliers
Constant policy
State dependent policy
45
50
40
45
M ar ginal value of money
M ar ginal value of money
p(z )
u(z )
40
35
30
p(z)
u(z)
25
20
15
35
30
25
20
15
10
10
5
0
5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Share of money in the hands of the agent: z
Share of money in the hands of the agent: z
12
0.9
1
When µ is constant, so that µ(x) = µ ≥ 0 for every value of the aggregate state x, our
results extend those provided in Scheinkman and Weiss (1986), which only considered the
case where µ = 0. The phase diagram determines three regions where the functions u(z) and
p(z) change behavior according to their derivative sign. It is evident that the only positive
solutions of the problem must stay in the third region, where both functions are decreasing
on the entire (0,1) interval. More precisely, their path develops in the area that is dotted
in the figure. The next proposition characterizes the marginal value of money when µ is
constant.
Proposition 3 Under a constant policy µ(x) = µ ≥ 0 we have that for any c̄ > 0:
(i) p(z) < u(z),
and
(ii) p0 (z) < 0, u0 (z) < 0
for all
z ∈ (0, 1) .
See Appendix B.4 for a proof. The first part of the proposition establishes that when the
monetary policy is constant, the value of money for an unproductive agent is higher than the
value of money for a productive agent, u(z) > p(z), at all levels of money holdings z ∈ (0, 1).
This property, first highlighted by Scheinkman and Weiss (1986), seems intuitive: Because the
only difference between productive and unproductive agents holding z is that the productive
agent can work, an unproductive agent values more an extra unit of money; as a result, her
Lagrange multiplier is higher. Second, the proposition also states that the functions u(z)
and p(z) are decreasing in z at all levels of money holdings z ∈ (0, 1). The bottom-left panel
in Figure 1 displays these properties and the upper-left panel presents the phase diagram
explaining their origin: Since region III is the only admissible region for a solution to satisfy
the boundary condition, the curves satisfy the properties listed in Proposition 3.
When the monetary policy depends on the state x, some interesting new features arise.
Inspection of the upper-right panel of Figure 1 shows that the equilibrium Lagrange multipliers must reach the boundary condition, which occurs when x = 0 and the productive agent
money holdings are z = 1 (i.e., the bold dot in the figure), in the region that is under Lp and
Lu (i.e., the dotted region). As a result, when the productive agent money holdings z are
large enough, the solution path develops in the area where both p(z) and u(z) are decreasing
functions, as was the case for the constant policy. But crucially this result holds only locally,
and in general the functions p(z) and u(z) that solve the problem can take many shapes.
The next proposition states this result.
Proposition 4 Suppose that µ(x) is continuous and that the following assumptions hold:
(A1 ) the policy µ(x) satisfies
λ
ρ+λ+µ(0)
<
ρ+λ+µ(1)
;
λ
13
(A2 ) the policy µ(x) is strictly decreasing at x = 0, i.e. µ0 (x) < 0 for all 0 ≤ x < υ, where
υ is a suitable positive number.
Then, there is a neighborhood for the productive agent money holdings at z = 1, denoted by
the interval Iz , where for any c̄ > 0 the multipliers satisfy
(i) p(z) < u(z),
and
(ii) p0 (z) < 0, u0 (z) < 0
for all
z ∈ Iz .
See Appendix B.5 for a proof. The reason for the local similarity of the constant and the
state-dependent problem is the following. Start by noting that when the productive agent
money holdings are z = 1, the unproductive agent money holdings are x = 0; in this case,
by Proposition 1, µ(0) ≥ 0. It follows that by the continuity of µ(x), for any z ∈ Iz it is the
case that µ(1 − z) ≥ 0. This non-negativity constraint on the money growth rate, together
with assumptions A1 and A2 , produces a “local phase diagram,” and implied behavior of the
multipliers, that is similar to the behavior under a constant policy. As shown in the upperright panel of Figure 1, under the state dependent policy µ(x) we have that u(z) > p(z) and
that both curves are decreasing only when the unproductive agent money holdings are low
(i.e., x close to zero) and the productive agent money holdings are high (i.e., z close to one).
Assumptions A1 and A2 are important as they bound the local behavior of the dynamical
system. Assumption A1 guarantees that in the neighborhood of z = 1 the locus Lp lies below
the locus Lu , as depicted in the upper-right panel of Figure 1.10 Assumption A2 guarantees
that in the neighborhood of z = 1, the locus Lp is decreasing. While assumptions A1 and A2
are useful to analytically characterize the behavior of the economy under a state-dependent
policy, they are not imposed in the numerical solution of the optimal policy that is developed
below. It is interesting, however, that in all the cases that we analyzed they were always
satisfied at an optimum.
Notice that when z 6∈ Iz , money growth µ(1 − z) can be negative so that u(z) can be
below p(z) and one (or both) of them can be increasing. To understand this result it is
useful to contrast the state dependent case with the constant case. In the constant case the
only difference between productive and unproductive agents is the production opportunity
since policy is constant; as shown in Proposition 3 this immediately implies that u(z) >
p(z) for all z. But when the monetary rule varies with the state x, a comparative static
across productivity states, i.e. comparison of u(z) vs. p(z), also involves a different path for
the monetary rule. To make this point clear we make explicit the dependence of the marginal
value of money on the money rule µ. That is, let {p(z; µ(x)), u(z; µ(x))} denote the marginal
λ
To see this notice that this assumption can be extended to the inequality ρ+λ+µ(1−z)
≤
Lp lies below Lu . A sufficient condition for satisfying this assumption is µ(1) ≥ −ρ.
10
14
ρ+λ+µ(z)
λ
so that
value of money for an agent with money holdings z and where the money rule is µ. Consider
a productive agent holding money z. Under a constant policy her current value is p(z; µ) and,
if the state switches, her value will be u(z; µ). Under the state dependent rule her current
value is p(z; µ(1 − z)) and, if there is a state switch, her value would become u(z; µ(z)). This
shows that when the policy is state dependent the value of money across agent types differs
not only because of differences in production opportunities, but also because the money rule
depends on the wealth distribution. It follows that the restriction p(z) < u(z) needs not to
hold over the whole state space under a state-dependent rule. This feature, as we discuss
next, allows for a substantial welfare improvement.
To understand why relaxing the properties (i) and (ii) of Proposition 3 is important for
our problem, notice that the complete markets allocation features a constant consumption for
all agents, at c̄ (see Remark 1). If a money rule existed to implement a first best allocation,
it can be seen using equation (12) that the rule would imply
u(z) = p(1 − z)
for all z ∈ [0, 1]
(18)
or, in words, that the functions u(z), p(z) are symmetric around z = 1/2. Obviously this
feature cannot be achieved by a constant rule, since in that case the functions u(z), p(z)
are decreasing. A state-dependent policy has a chance of getting closer to this symmetric
benchmark, as shown by the lower-right panel of Figure 1. However, the next proposition
shows that the complete markets allocation cannot be sustained in general (see Appendix B.6
for the proof).
Proposition 5 Let ρ/λ > ρ̄ > 0. There is no monetary rule µ(x) that supports the complete
markets allocation as a monetary equilibrium.
5
The optimal supply of liquidity
In this section we define a welfare criterion and explore the optimal supply of liquidity. Let
v(x) denote the discounted present value of the sum of utilities of both types of agents,
where agents are given the same Pareto weight. This is a function of the money share of
unproductive agents, x. The continuous time Bellman equation is
cu (x)
ρ v(x) = c̄ ln c (x) + ln c̄ − 1 −
c̄
u
+ vx (x) hu (x) + λ ( v(1 − x) − v(x) ) .
(19)
The flow value ρ v(x) is given by the sum of the period utility for both agents plus the
expected change in the value function. The latter occurs because of the evolution of assets
15
(the change in x) as well as of the possibility that the identity of the productive agent will
change. Notice that in this case the state, i.e., the wealth of the unproductive agent, switches
from x to 1 − x.
We consider the problem from an ex-ante perspective, i.e., assuming that at the beginning
of time nature assigns the initial productive states and the planner can choose the initial
wealth distribution and a policy rule for money growth. We assume that the planner can
commit to the optimal policy. Note that because individual types are not observable, and
given the symmetry of the environment (and identical Pareto weights), the planner will give
the same amount of liquidity to every agent and therefore at the beginning of time x = 12 .
As a result, the planner chooses the function µ in order to maximize v(1/2).
To evaluate the policy it is useful to define a certainty equivalent compensating variation.
Let α denote the consumption equivalent cost of market incompleteness associated with a
given policy. That is, α solves the following equation
2c̄ ln (c̄(1 − α)) − 2c̄ = ρ v (1/2)
(20)
so that α measures the fraction of the consumption under complete markets that agents would
be willing to forgo to eliminate the volatility of consumption due to market incompleteness
for a given policy rule µ.11
Having established that the complete markets allocation cannot be achieved (Proposition 5), we look for the optimal policy µ̂(x) by searching numerically for the policy that
maximizes the ex-ante expected welfare v(1/2). Since the system of differential equations
that characterizes the equilibrium allocations cannot be solved in closed form, the numerical
approach represents the only means to develop a quantitative analysis. We restrict our attention to monetary policies that are piecewise linear continuous. For simplicity the results
discussed in this section are obtained by considering a policy function with five nodes.12 It
is also worth mentioning that neither assumption A1 or A2 are imposed in the estimation
routine. To aid in the understanding of the policy we also compute an alternative policy,
µ̄, which maximizes v(1/2) under the restriction that the policy has to be constant. In all
computations reported below, we set the scaling factor c̄ = 1. Figure 2 plots both policies
obtained for the baseline parametrization of the model where the normalized discount rate
ρ/λ equals 1/2. Given this parameter choice, setting the discount rate to the standard value
of 0.05 implies that λ equals 1/10, so that the average length of a productive state is 10 years.
11
Recall that under complete markets cu (z) = cp (z) = c̄ for all z, and lp (z) = 2c̄ for all z.
Alternatives with more nodes have been explored with little differences in the results. For instance, under
our benchmark calibration with ρ/λ = 1/2, the optimal policy µ̂(x) shares the same qualitative features under
both five and nine nodes. Moreover, the welfare gains of allowing for nine nodes are small.
12
16
Qualitatively similar results are obtained for other parameterizations for ρ/λ (see below).
Figure 2: Optimal policy (ρ/λ = 1/2)
0.5
State dependent policy ()
0.4
Constant ̄
0.3
0.2
()
0.1
0
-0.1
−
-0.2
-0.3
-0.4
-0.5
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Share of money in the hands of the unproductive agent:
The constant policy µ̄ consists of an expansion of the monetary base of 0.1 percent; µ̄ is
positive because the insurance motives outweigh the inflation costs. As shown in Figure 2,
the optimal policy µ̂(x) is very different: It expands the money supply when x is low and
it contracts the money supply when x is high; this happens because the optimal policy,
through its state-dependent nature, is able to decouple the insurance motives and inflation
costs. The welfare cost of market incompleteness α, as defined in equation (20), under the
constant policy is 31.7 percent while under the optimal policy is 3.3 percent, which shows
that the optimal policy increases welfare substantially.
We now describe the features of the optimal policy and interpret them in terms of the
trade-off between insurance motives and the cost of inflation. When the insurance motives
are very strong, which happens when x ≈ 0, the policy prescribes the largest expansion of
the money supply. As the insurance motives decrease, which happens as we take x away
from zero, the optimal money growth rate falls. Eventually, at x ≈ 0.2, the cost of inflation
(through the production incentives) outweighs the insurance role of money and, as a result,
the policy turns to prescribing monetary contractions. When x lies in an intermediate region
the policy prescribes the largest monetary contractions, much below the Friedman rule (−ρ).
These extreme contractions are a reflection of the large expansions that occur when x is low:
Because expansions damage the return of money, large (future) monetary contractions undo
the detrimental effects of the expansions on production incentives (as agents are forward
17
looking). Intuitively, the expansions and contractions compensate each other, striking a
balance between the insurance provision versus the production incentives.
As x increases toward 1 the level of the monetary contractions decreases, reaching a
minimum as x approaches 1. This is a reflection of the provision of insurance that might
be needed if a switch of the state occurred: When x is high, the unproductive agent is
rich, and her insurance needs are small. But the productive agent is poor and, if a state
switch occurs, she will be unproductive and with a high insurance need. Because of the
forward looking feature of monetary policy, the optimal policy incorporates this insurance
need, and even when x is high, the degree of monetary contractions decreases; this explains
the v-shape form of the policy. Nevertheless, because a productive agent can acquire money
by working, her insurance motives when poor are lower than those of a poor unproductive
agent. This explains why the policy is asymmetric, with larger monetary expansions when
the unproductive agent is poor.
Figure 3: Consumption and the return on money under optimal policy rule (ρ/λ = 1/2)
Consumption: cu (z)
Expected return: r(x)
3
0.3
State dependent policy ()
2.5
State dependent policy ()
Constant ̄
Constant ̄
0.2
2
0.1
()
()
1.5
0
-0.1
1
-0.2
0.5
-0.3
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0
1
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
Share of money in the hands of the unproductive agent:
Share of money in the hands of the agent:
Figure 3 illustrates the profiles of the consumption function and the expected return
on money under the optimal policy µ̂(x) and constant policy µ̄. The left panel of the figure
shows that the consumption of the unproductive type under the optimal policy is (i) smoother
than under the constant policy, and (ii) closer to the consumption implied by the complete
markets allocation (i.e. cu (z) = c̄ = 1). The smoother consumption profiles of the statedependent rule also yields a flatter profile for the expected return on money r(x), as shown in
the right panel of the figure. This flatter profile reflects the fact that the consumption of an
18
unproductive agent is less extreme under µ̂(x): The smaller expected changes in consumption
(hence marginal utilities) associated to a state switch dampen the risk premia and lead to
a smoother expected return of the asset. Moreover, because the consumption under the
optimal policy is closer to the consumption under complete markets, the expected rate of
return is close to the rate of discount ρ, which is the expected rate of return with a complete
markets allocation, as shown by equation (14).
Notice that, because the consumption of the unproductive agent cu (x) is increasing in
x, aggregate production, cp (1 − x) + cu (x) = 1 + cu (x), increases with x. Inspection of the
right panel of Figure 3 shows that the real return on money is high when x is low. To
understand this remember from the discussion of equation (14) that the return on money is
proportional to the consumption growth of the productive agent. At x = 0 the productive
agent is consuming c = 1, and expects, in case of a state switch, to consume cu (1) > 1, which
explains the high expected return at x = 0. An analogous logic explains the low returns
when production is high (when x is high). Recall from equation (15) that the expected
inflation rate π(x) is inversely related to r(x). This implies a positive correlation between
the aggregate production and inflation, i.e., the model features a “Phillips curve.”
Recall, as discussed at the end of Section 4, that replicating the first best allocation would
require that u(z) = p(1 − z). This means that, for any wealth level z, the marginal value of
money would have to be equal for both agent’s types; in other words, the marginal value of
money has to be symmetric around z = 1/2. This symmetry requirement cannot be handled
by a constant policy and, as showed in Proposition 5, no state-dependent rule can perfectly
achieve this outcome. However when µ is allowed to depend on x, the functions u(z) and
p(z) are able, at least partially, to produce the required symmetry. This can be seen in the
lower-right panel of Figure 1 where we plot u(z) and p(z) under the optimal money rule
µ̂(x). Moreover, the optimal policy µ̂(x) satisfies assumptions A1 and A2 so that u(z) and
p(z) satisfy Proposition 4.
Notice that the state-dependent pattern of the policy rule can be equivalently interpreted
in terms of the business cycle. As aggregate production is increasing in x, then the optimal
policy µ̂(x) is such that monetary expansions happen when aggregate production is low and
monetary contractions occur when it is high. In other words, the policy is expansionary
during recessions and contractionary during expansions.
To conclude this section we explore the effect of changing the average length of the
productive cycle, i.e., 1/λ, on the optimal policy. We do so in Figure 4 where we present
the policy under three different cases. The first case uses our baseline parameter value for
the probability of a state switch, λ = 1/10, so that the average duration of a productive
cycle is 10 years; the second case uses a low probability of a state switch, λ = 1/20, and
19
Figure 4: Optimal policy
1.5
1
basel i ne
l ong cycl es
short cycl es
µ (x)
0.5
0
−0.5
−1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
S hare of money i n the hands of the unproducti ve agent: x
Note on parameters: c̄ = 1. Baseline uses ρ/λ = 0.5, long cycles uses ρ/λ = 1, short
cycles uses ρ/λ = 0.1.
therefore the economy in this case exhibits long productive cycles (on average, 20 years),
while the third one uses a high probability of a state switch, λ = 1/2, so that the economy
displays short cycles (on average, two years). As the figure shows, the qualitative features
of the policy are similar for the three parameter configurations: high monetary injections
when insurance motives are the highest (when x is low), and high monetary contractions
when they are the lowest (when x is close to 1/2). Even though similar in qualitative terms,
they do differ in the level of monetary injections. Monetary expansions and contractions
are more extreme the shortest the production cycle. Our intuition is as follows. As the
production cycle shortens, the value of money increases so that the production distortion
that follows a monetary expansion is milder, and therefore, when x is small, the policy can
provide insurance through liquidity expansions at a higher rate. Moreover, as previously
discussed, to handle production incentives, the policy prescribes monetary contractions to
counterbalance the monetary expansions, which also naturally increase as the cycle becomes
shorter. Finally, as previously discussed, the welfare cost α is 3.3 percent under the baseline
configuration, while it increases to 11.7 percent when the cycle is long, and it falls to 0.3
percent when it is short. The improvement in welfare as the cycle shortens follows because
the insurance motives decrease as the average length of an unproductive spell falls, which,
in turn, decreases the degree of the trade-off between insurance motives and production
20
incentives. Notice that this analysis also applies if we evaluate changes in the discount rate
ρ. This follows as, for a given c̄, the unique parameter of the economy is ρ/λ, so that increases
in the length of productive cycles are analogous to decreases in the rate of time discount.
6
Concluding remarks
In a variety of monetary models policy has persistent effects on the distribution of wealth
(money or other assets). These distributional effects are often muted or kept to a minimum
in theoretical models to preserve tractability. This paper began to explore the way these
distributional concerns affect monetary policy in a monetary model that allows for a full
dynamic analysis of the (two-way) interactions between monetary policy and the wealth
distribution.
Two forces shape the design of monetary policy in this setup. As in many other monetary
models, efficiency requires distortions are minimized when the return on money equals the
rate of time discount (i.e., Friedman’s rule). However, with uncertainty and incomplete
markets an expansionary policy can be desirable due to insurance needs. A trade-off arises
as expanding the liquidity base dampens the return on the asset, therefore reducing the
production incentives. The regulation of liquidity strikes a balance between these two forces.
As we discussed in the paper, insurance motives and production incentives depend on the
wealth distribution that evolves through the business cycle. The novelty of our paper is that
we acknowledge this dependence and we explore how the state-dependent policy balances the
costs of anticipated inflation with the needs for insurance along the business cycle. This policy
allows for a dramatic improvement in welfare compared with a policy that does not respond to
the state. The optimal policy expands the supply of liquidity when the unproductive agents
are poor (when the insurance needs are large), and it contracts the liquidity base otherwise
to maximize production incentives. The principle underlying this prescription is due to the
state-dependent redistributive role of monetary policy, and it differs from the one arising in
sticky-price models. Because aggregate production is low when the unproductive group is
poor and high when they are rich, the best policy can be interpreted as counter-cyclical.
Interestingly, in spite of the policy being far away from contracting the money supply at the
rate of preference, the optimal policy “echoes” Friedman’s rule as the expected real return
of money approaches the rate of preference.
While the specific predictions of our model are likely to depend on specific modeling assumptions, the paper highlights the potential relevance of a transmission channel of monetary
policy that seems reasonable and little explored. Several interesting extensions are left for
future work. An important, and classical, assumption for our results is that the planner has
21
the ability to levy lump-sum taxes, and that agents are not allowed to renege from their obligations. However, as pointed out by Andolfatto (2013), if agents are allowed to voluntarily be
subject to taxation, the degree of monetary contractions has to be limited by the voluntary
nature of participation. Therefore, it is possible that the incentive-feasible allocation cannot
support the optimal policy we constructed in this paper, as this one requires large monetary
contractions. An interesting open question is to use the mechanism design toolkit to find the
optimal state-dependent policy implementable under voluntary participation. Another interesting extension is to reformulate our model, following Atkeson and Lucas (1992), considering
an endowment economy where agents are subject to preference shocks. If only two types of
agents are considered, as we did in this paper, the two setups share several similarities and
the solution developed for this paper can also be used in this other context. This would be
useful to check the robustness of the results. Finally, an extension that seems worth exploring
is to extend the model to allow for a larger set of agent types. Although this will come at
the cost of losing tractability, as the state space grows with the number of agent types, this
model, and the optimal policy, could probably be solved numerically. We conjecture that the
main insights of our paper would remain: monetary expansions when unproductive agents
are poor counterbalanced by monetary contractions in states where insurance motives are
small.
References
Algan, Y., Challe, E. and Ragot, X. (2011). Incomplete markets and the outputinflation
tradeoff, Economic Theory 46(1): 55–84.
Alvarez, F. and Lippi, F. (2013). Persistent liquidity effect and long run money demand,
American Economic Journal: Macroeconomics forthcoming.
Andolfatto, D. (2013).
60(4): 383–390.
Incentive-feasible deflation, Journal of Monetary Economics
Ascher, U., Mattheij, R. and Russell, R. (1988). Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, Prentice Hall Series in Computational Mathematics.
Atkeson, A. and Lucas, Robert E, J. (1992). On efficient distribution with private information, Review of Economic Studies 59(3): 427–53.
22
Berentsen, A., Camera, G. and Waller, C. (2005). The distribution of money balances and
the nonneutrality of money, International Economic Review 46(2): 465–487.
Bewley, T. (1980). The optimum quantity of money, in J. Kareken and N. Wallace (eds),
Models of monetary economies, Federal Reserve Bank of Minneapolis, pp. 169–210.
Bewley, T. (1983).
51(5): 1485–504.
A difficulty with the optimum quantity of money, Econometrica
Bhattacharya, J., Haslag, J. H. and Martin, A. (2005). Heterogeneity, redistribution, and
the friedman rule, International Economic Review 46(2): 437–454.
Deviatov, A. and Wallace, N. (2012). Optimal inflation in a model of inside money, mimeo,
Dept. of Economics, Penn State.
Golosov, M. and Sargent, T. J. (2013). Taxation, debt redistribution, Mimeo, NYU.
Green, E. and Zhou, R. (2005). Money as a mechanism in a bewley economy, International
Economic Review 46(2): 351–371.
Grossman, S. and Weiss, L. (1983). A transactions-based model of the monetary transmission
mechanism, American Economic Review 73(5): 871–80.
Guerrieri, V. and Lorenzoni, G. (2009). Liquidity and trading dynamics, Econometrica
77(6): 1751–1790.
Guerrieri, V. and Lorenzoni, G. (2011). Credit crises, precautionary savings, and the liquidity
trap, NBER Working Papers 17583, National Bureau of Economic Research, Inc.
Kehoe, T. J., Levine, D. K. and Woodford, M. (1990). The optimum quantity of money
revisited, Working Papers 404, Federal Reserve Bank of Minneapolis.
Lagos, R. and Wright, R. (2005). A unified framework for monetary theory and policy
analysis, Journal of Political Economy 113(3): 463–484.
Levine, D. K. (1991). Asset trading mechanisms and expansionary policy, Journal of Economic Theory 54(1): 148–164.
Lucas, R. J. (1990). Liquidity and interest rates, Journal of Economic Theory 50(2): 237–264.
Molico, M. (2006). The distribution of money and prices in search equilibrium, International
Economic Review 47(3): 701–722.
23
Scheinkman, J. A. and Weiss, L. (1986). Borrowing constraints and aggregate economic
activity, Econometrica 54(1): 23–45.
Shi, S. (1997). A divisible search model of fiat money, Econometrica 65(1): 75–102.
Shi, S. (1999). Money, capital, and redistributive effects of monetary policies, Journal of
Economic Dynamics and Control 23(4): 565–590.
Tenreyro, S. and Sterk, V. (2013). Durable goods and the transmission mechanism of monetary policy: an alternative framework, Mimeo, London School of Economics.
Townsend, R. M. (1980). Models of money with spatially separated agents, in J. Kareken and
N. Wallace (eds), Models of monetary economies, Federal Reserve Bank of Minneapolis,
pp. 265–304.
Wallace, N. (1984). Some of the choices for monetary policy, Quarterly Review 8(1).
Wallace, N. (2013). The optimality of inflation in ‘pure currency’ economies: a conjecture,
Quarterly Journal of Economics forthcoming.
Appendix
A
The return on money and inflation
Define the stochastic expected net return on money for a small time interval ∆ as
q̃t+∆
qt+∆
r(x)∆ = E
− 1 xt = x = E
− ∆µt − 1 xt = x ,
q̃t
qt
(21)
where q̃t = q(xt )/Mt . Without loss of generality consider the case where at time t agent 1
is productive with money holdings given by zt1 , and let zt2 denote the money holdings of the
unproductive agent at time t. Then, using that q(x) = p(z), we have
r(xt )∆ =
1
2
(1 − λ∆)p(zt+∆
) + λ∆p(zt+∆
)
− 1 − ∆µt ,
1
p(zt )
i
where we used equation (5). A first order Taylor expansion of p(zt+∆
) gives
p0 (zt1 )hp (zt1 )∆
r(xt )∆ =
+ λ∆
p(zt1 )
p(zt2 ) + p0 (zt2 )hp (zt2 )∆ p(zt1 ) + p0 (zt1 )hp (zt1 )∆
−
p(zt1 )
p(zt1 )
− ∆µt .
(22)
24
p0 (zt1 )hp (zt1 )
p(zt1 )
p(zt2 )
p(zt1 )
− 1 − µt .
Then, taking the limit as ∆ → 0 gives that r(xt ) =
+λ
2
1
p(z )
u(z )
We use (8) to get r(xt ) = ρ + λ p(zt1 ) − p(zt1 ) or, using that in this proof we assumed
t
t
that agent 2 was unproductive at time t, i.e. xt = zt2 ,
r(xt ) = ρ + λ
p(x)
u(1 − x)
−
p(1 − x) p(1 − x)
,
which yields the expression in the text. Define expected inflation as π(xt ) = E
h
1/q̃t+∆
1/q̃t
− 1 xt = x
r(xt )
it is easily seen that π(xt ) = − 1+r(x
.
t)
B
B.1
Proofs
Proof of Remark 2
Let i = 1 be the index for the unproductive agent, and consider her decision problem. The
t
money supply growth is µ = Ṁ
and let ũt denote the Lagrange multiplier of the money flow
Mt
constraint in equation (3). The first order condition with respect to ct gives: ut = qt /ct ,
where we used the homogeneity of degree -1 in the aggregate money supply Mt for both ũt
t
= qq̇tt − ċctt − µ where the last equality
and q̃t . The Euler equation for ṁ1t gives ρ = uu̇tt − Ṁ
Mt
uses ut = qt /ct . Notice that this is solved by ct = c̄, ċt = q̇t = 0, and µ = −ρ.
The constant level of q is pinned down by imposing that total nominal assets at the beginning of a cycle, which in the stationary equilibrium are held by the unproductive agent,
are equal the total nominal consumption and tax expenditures over the cycle of length T .
Without loss of generality let’s consider the first cycle, starting at time 0, where the unproductive agent holds all the money supply; m0 = M0 . Using that Mt = M0 eµt , µ = −ρ, and
q̃t = M1t q, we have
Z
0
T
c̄
dt +
q̃t
Z
T
ρMt dt = M0 ,
which gives
0
1 − e−ρT ∼
T
q = c̄
.
= c̄
−ρT
ρe
1 − ρT
where the approximation is accurate for small T . It is immediate to verify that this allocation
also solves the Euler equation of the productive agent and that her money holdings are never
negative.
B.2
Proof of Proposition 1
We first present a proof of the first part of the statement. That is, that there is no monetary
equilibrium where µ(x) < 0 ∀ x. We then present a proof of the second part of Proposition 1.
25
i
A contractionary policy µ(x) < 0 ∀ x requires agents to pay lump sum taxes (τ (x) <
0 ∀ x). Consider the case where agent 1 has fraction of money balances zt , and the current
state of the economy is It = 0, which means that agent 1 is unproductive. If zt is low enough,
given that λ > 0 and finite, the agent will fail to comply with the monetary authorities with
non-zero probability. On the other hand, consider the case where zt = 1. In this case the agent
is able to comply with her tax obligations with certainty, as she can make her consumption
profile to be arbitrarily low. This implies that there exists a threshold ζ ∈ (0, 1) such that
for z i ≥ ζ the agent is able to cover her lifetime tax needs with probability one. Note that
the threshold must be independent of the current state It as with positive probability the
states are reversed. In the next claim we characterize this threshold.
Claim 1 If µt < 0 ∀ t, for any state of the world It , there is a unique threshold: ζ = 1/2,
and a unique ergodic set where zt = 21 ∀ t, that ensures tax solvency.
Proof. We will first prove by contradiction that ζ 6∈ [0, 1/2). Then we will show that
ζ = 1/2 is enough to cover the lifetime tax obligations. Suppose that ζ < 1/2. Without
loss of generality assume that zt ∈ (ζ, 1/2) and agent 1 is unproductive. Conditional on no
reversal of the state, it follows that zt+∆ < zt for any ∆ > 0. Then for a given ∆ ∈ R+ ,
Pr [zt+∆ < ζ] > 0 and therefore the agent will fail to comply with her tax obligations with
positive probability. Then, ζ 6∈ [0, 1/2). Consider now the case where zt = ζ = 1/2. As the
agent can decide not to trade she can always keep her share of outstanding money balances
z above 1/2 and therefore for any µ ∈ (0, 1) she will be able to cover her tax needs. That
z = 1/2 is the ergodic set is trivial. If z < 1/2 there is a positive probability that an agent
fails to pay for her lifetime taxes. An unproductive agent with money holdings z > 1/2 is
willing to buy goods (and the productive one with z < 1/2 willing to take the money) until
z reaches 1/2.
Intuitively, given the uncertain duration of the productivity spell, the only value of money
holdings that ensures compliance with tax obligations for both types of agents is z = 1/2.
At this point, for any history of shocks, the identical lump-sum (negative) transfers reduce
the money holdings of both agents proportionally, leaving the wealth distribution unaffected.
This leads us to
Remark 3 Let µ(x) < 0 ∀ x: In the ergodic set there is no stationary monetary equilibrium
and consumption allocations are autarkic.
The proof of Remark 3 follows from noting that Claim 1 implies no trade in the ergodic
set. Productive agents have an unsatisfied demand for money and unproductive ones have
an unsatisfied demand for consumption goods.
26
Now we turn to prove that µ(0) ≥ 0. Consider the law of motion for the share of money
u (z)
held by type 1 when unproductive, hu (z) = µ(x) 12 − z − cq(x)
. Because cu (z) ≥ 0 and
q(x) > 0 (i.e. prices are positive in a monetary equilibrium) we can bound above the law of
motion hu (z): hu (z) ≤ µ(x) 21 − z . Notice that because the agent is unproductive z = x.
We apply this result into the previous equation and evaluate it at x = 0 to get hu (0) ≤ µ(0)
,
2
from where it can be seen that hu (0) < 0 if µ(0) < 0 which is inconsistent with the borrowing
and tax-solvency constraints. Therefore, µ(0) ≥ 0.
B.3
Proof of Proposition 2
c̄
The boundary condition (10) yields limz→0 hu (z) = 0. Moreover, we have hu (1/2) = − u(1/2)
<
0. Under the continuity assumption on µ(x), hu (z) is a continuous function; then the inequality can be extended over a suitable neighborhood of the asset z = 12 . More precisely,
it is possible to consider an interval ( 21 − δ, 12 + δ) ⊂ (0, 1) where hu (z) < 0. The further
assumption related to the lack of singularities yields that the function hu (z) cannot nullify
within the domain and its sign has to be uniform. It follows that the interval ( 21 − δ, 12 + δ)
overlaps the whole integration interval and the previous inequality holds for all z ∈ (0, 1). In
this way, the result is completely obtained.
The assumptions in the proposition mean that the solution (u(z), p(z)) evolves for z ∈
(0, 1) in a region of R2+ where the forcing term has no singularity and is continuous, with the
classical uniform Lipschitz condition locally satisfied. We recall that this kind of smoothness is a basic assumption which is needed for investigating the solution of the differential
problem. As discussed in Ascher et al. (1988), smoothness together with the requirement of
solvability of the algebraic system related to the boundary conditions are the crucial issues
for establishing the existence of a solution.
B.4
Proof of Proposition 3
The proof follows from the inspection of the phase diagram, which is plotted in upper-left
Figure 1. We assume the policy µ(x) = µ ≥ 0 is constant and exploit the geometric loci in
(R+ )2 defined as
ρ+λ+µ
, and Lu = (U, P ) ∈ (R ) | P =
U .
λ
(23)
λ
We first note that ρ+λ+µ
< 1 < ρ+λ+µ
; this implies that the locus Lp lies under Lu and
λ
the dotted line P = U is between them, as it is shown in the figure. Then, according to
equations (16)-(17), the terms p0 (z)hp (z) and u0 (z)hu (z) are zero on Lp and Lu , respectively.
Lp = (U, P ) ∈ (R+ )2 | P =
λ
U
ρ+λ+µ
27
+ 2
Thus, the uniform signs of hu (z) < 0 and hp (z) > 0 allow us to gain an insight about the
increasing/decreasing pattern of the Lagrange multipliers p(z) and u(z), as it is shown by
the arrows in the figure. It can be seen that region III is the only admissible area for a
p(1), which is on Lp . Indeed the
solution to satisfy the boundary condition u(1) = ρ+λ+µ(0)
λ
only positive solutions must stay in that region and their path develops in the dotted area.
Therefore, both p(z) and u(z) are decreasing functions: it follows that p0 (z) < 0 and u0 (z)
λ
everywhere. Moreover, in the same region, we have p(z) ≤ ρ+λ+µ
u(z) < u(z) for all z.
B.5
Proof of Proposition 4
The results in Proposition 4 can be stated by investigating the upper-right plot in Figure 1,
where some qualitative features of the solution are represented by exploiting the paths describing two specific different geometric loci in (R+ )2 . We define the following sets
Lp = {(U, P ) ∈ (R+ )2 | ∃ z ∈ [0, 1] so that U o
= u(z) solves equations (16)-(17) at the state z
λ
and P = ρ+λ+µ(1−z) U ,
and
Lu = {(U, P ) ∈ (R+ )2 | ∃ z ∈ [0, 1] so that U
o = u(z) solves equations (16)-(17) at the state z
ρ+λ+µ(z)
and P =
U .
λ
They can be considered as a generalization of the geometric loci already defined by (23) in
Appendix B.4, where the monetary policy is assumed to be state independent. As a difference
with respect to (23), here we consider an explicit dependence
on the state z for both U and P
λ
+ 2
u(z) ∈ Lp
in the plane (R ) : the sets are described by the points u(z), Γp (z) := ρ+λ+µ(1−z)
λ
u(z) ∈ Lu , as z varies in the domain [0, 1]. Again the terms
and u(z), Γu (z) := ρ+λ+µ(z)
0
p
0
u
p (z)h (z) and u (z)h (z) in equations (16)-(17) are zero on Lp and Lu , respectively. Then,
according to the sign of terms hp (z) and hu (z), both Lp and Lu determine different regions
in the plane (U, P ) where the functions u(z), p(z) change behavior according to the sign of
their derivative.
The following features are crucial to draw the loci Lp , Lu , and to characterize the solution
of the differential problem assuming A1 and A2
(P1 ) The loci may intersect in an even number of points, since condition Γp (z) = Γu (z) is
equivalent to have ρ2 + 2ρλ + (ρ + λ)(µ(z) + µ(1 − z)) + µ(z)µ(1 − z) = 0, which is a
symmetric formula with respect to z = 1/2.
(P2 ) Condition
λ
ρ+λ+µ(0)
<1≤
ρ+λ+µ(1)
λ
in assumption A1 can be extended in order to have
28
λ
the inequality ρ+λ+µ(1−z)
≤ ρ+λ+µ(z)
, satisfied in a suitable neighborhood of z = 1. It
λ
follows that, in the same neighborhood, the locus Lp is under Lu .
(P3 ) The boundary condition u(1) = ρ+λ+µ(0)
p(1) lies on Lp (denoted by a bold dot in the
λ
figures). It has to be reached by the solution in correspondence with z converging to 1.
(P4 ) Lp and Lu determine different regions in the phase plane (U, P ) where both functions
u(z), p(z) change behavior according to their derivative sign. Since Lp lies under Lu ,
as shown in the upper-right panel of Figure 1, the local behavior of u(z) and p(z) in
a neighborhood of z = 1 can be determined as follows. Let the arrows describe the
increasing/decreasing patterns for u(z) and p(z), accounting for the signs of hu (z) < 0
and hp (z) > 0 established by Proposition 2. It is evident that u(z) is decreasing when
the money holdings of the productive agent is near z = 1, since the solution must reach
the boundary condition (the bold dot on Lp ) according to P3 .
(P5 ) The increasing/decreasing shape
for Lp can be established
by noticing that, for any
u(z)µ0 (1−z)
λ
0
0
z ∈ [0, 1], Γp (z) = ρ+λ+µ(1−z) u0 (z)(ρ+λ+µ(1−z)) + 1 u (z). We remark that µ(1 − z) ≥ 0
and µ0 (1 − z) < 0 near z = 1. The first condition is due to the continuity for µ(x) and
µ(0) ≥ 0; the second relationship arises from assumption A2 . In this respect, since u(z)
is decreasing when z moves near 1 (see P4 ), then u0 (z) < 0. It follows that Γ0p (z) < 0
and the locus Lp decreases in a neighborhood of z = 1.
The previous features can be exploited in order to draw the picture in Figure 1, whose
inspection lets the proof of Proposition 4 be completed. It is evident that the only solutions
that are positive must reach that boundary condition in the dotted region, that is under Lp
and Lu (the dashed curve represents a possible path for the solution). Hence there exists a
threshold z̄ ∈ [0, 1] such that for all z ∈ Iz = [z̄, 1] the solution develops in the region where
both p(z) and u(z) are decreasing functions, that is equivalent to have p0 (z) < 0 and u0 (z) < 0.
p(1), the
In the same region we have p(z) < u(z). Indeed, since µ(0) ≥ 0 and u(1) = ρ+λ+µ(0)
λ
point (u(1), p(1)) ∈ Lp lies under the 45 degree line and p(1) < u(1). Since the function µ is
continuous then the inequality can be extended in Iz . This completes the proof.
B.6
Proof of Proposition 5
We prove that the complete markets allocation cannot be sustained when λρ > ρ̄ (i.e. when
the expected duration of the cycle is sufficiently long). To this end it suffices to show that
consumption is not equal to c̄ at all values of the state x ∈ [0, 1]. In particular let us
consider x = 0. The consumption equation (13) and the continuity of the consumption
function give that that cu (0) = µ(0)p(1)/2. Recall that µ(0) is finite (since µ(x) is continuous
29
over a compact set). The boundary condition in equation (11) can be rewritten as p(1) =
1
u
µ(0) u(1). Since u(1) is finite in any monetary equilibrium, it follows that c (0) can be
ρ
λ
+1+
λ
made arbitrarily small by increasing λρ , i.e. that limρ/λ→∞ cu (0) = 0. Therefore there exists
a threshold ρ̄ > 0 such that for all ρ/λ ∈ (ρ̄, ∞) we have that cu (0) < c̄, or that the complete
markets allocation cannot be implemented.
30
@FedFRASER