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A Tractable Model of Monetary Exchange
with Ex-Post Heterogeneity
WP 17-06
Guillaume Rocheteau
University of California, Irvine
Pierre-Olivier Weill
University of California, Los Angeles
Russell Tsz-Nga Wong
Federal Reserve Bank of Richmond
A Tractable Model of Monetary Exchange with
Ex-Post Heterogeneity
Guillaume Rocheteau
Pierre-Olivier Weill
University of California, Irvine
University of California, Los Angeles
Tsz-Nga Wong
Federal Reserve Bank of Richmond
First version: August 2012
This version: November 2016
Working Paper No. 17-06
Abstract
We construct a continuous-time, New-Monetarist economy with general preferences that
displays an endogenous, non-degenerate distribution of money holdings. Properties of equilibria
are obtained analytically and equilibria are solved in closed form in a variety of cases. We study
policy as incentive-compatible transfers …nanced with money creation. Lump-sum transfers are
welfare-enhancing when labor productivity is low, but regressive transfers achieve higher welfare
when labor productivity is high. We introduce illiquid government bonds and draw implications
for the existence of liquidity-trap equilibria and policy mix in terms of "helicopter drops" and
open-market operations.
JEL Classi…cation: E40, E50
Keywords: money, in‡ation, risk sharing, liquidity traps.
We thank Zach Bethune and Tai-Wei Hu for insightful discussions of our paper and participants at the 2012 and
2014 Summer Workshop on Money, Banking, Payments, and Finance at the Federal Reserve Bank of Chicago, at the
Search-and-Matching Workshop at UC Riverside, at the 2015 annual Search-and-Matching workshop at the Federal
Reserve Bank of Philadelphia, at the 2014 SED annual meeting, at the 2014 SAET annual meeting, at the 2015
Midwest Macro Meeting, at the 2015 Minnesota Macro Theory Workshop, at the 2015 Shanghai Macro Workshop,
at the 2015 Econometric Society World Congress, at the 2015 NBER/CEME meeting and seminar participants
at Academia Sinica (Taipei), the Federal Reserve Bank of Richmond, the Hong Kong University of Science and
Technology, the EIEF institute in Rome, the University of California at Irvine, Indiana Univesity, Purdue University,
Bank of Canada and Singapore Management University for useful discussions and comments. The views expressed herein
are those of the authors and do not necessarily represent the views of the Federal Reserve Bank
of Richmond or the Federal Reserve System.
1
Introduction
We analyze a continuous-time New Monetarist economy, based on the competitive version of the
Lagos and Wright (2005) model developed by Rocheteau and Wright (2005), LRW thereafter.1
As in LRW, agents use a medium of exchange to …nance random consumption opportunities, and
make endogenous labor supply decisions. In contrast to LRW, but similar to Lucas (1980) and
Bewley (1980, 1983), preferences are general and allow for wealth e¤ects, leading to a continuous
distribution of money holdings. Our …rst contribution is methodological: we show that the model
remains tractable despite the unharnessed ex-post heterogeneity in money holdings. We characterize
the properties of equilibria, including policy functions, value functions, and distributions, and we
solve the model in closed form in a variety of cases. Our second contribution is to study policy
in the form of incentive-compatible transfer schemes …nanced with money creation. We show
analytically that these schemes can be designed to raise welfare, and in some cases approach the
…rst best, by trading o¤ the need to promote self-insurance and the need to provide risk sharing,
thereby proving the Wallace (2014) conjecture. Our third contribution consists in adding illiquid
government bonds as a policy instrument. We draw implications for the existence of liquidity traps
and welfare-enhancing policies combining "helicopter drops" of money and open-market operations.
While we study a version of our model with money and bonds at the end of the paper, we
focus most of our attention to a pure currency economy as it is the most transparent benchmark
in which to study the monetary policy trade-o¤ between self-insurance and risk-sharing. In our
model, ex-ante identical households, who enjoy consumption and leisure ‡ows, have the possibility
to trade continuously in competitive spot markets.2 At some random times households receive
idiosyncratic preference shocks that generate utility for lumps of consumption. These represent
large shocks that cannot be paid for by a contemporaneous income ‡ow, such as health shocks,
large housing repairs, durable goods expenditures, and so on. Following Kocherlakota (1998), lack
1
The market structure with sequential competitive markets is analogous to the one in Rocheteau and Wright
(2005) but it could be readily reinterpreted as one where households meet sellers bilaterally and at random and terms
of trades are determined by bargaining. See our companion paper, Rocheteau, Weill, and Wong (2015), for such a
reinterpretation.
2
The assumption of continuous time has several advantages. First, the distribution of real balances obeys a
smooth, in…nitesimal law of motion known as a Kolmogorov forward equation. As a result, the distribution admits
a smooth density, without spikes, except maybe at one boundary. Second, continuous time provides a sharp representation of the mismatch between ‡ow endowments and lumpy spending that generates a role for liquidity. Third,
under continuous time ex-post heterogeneity is generic even under the commonly-used quasi-linear preferences. Last,
our methodology can be used to write Lagos-Wright economies in continuous time, which can be useful to integrate
them with other continuous-time models, such as models of unemployment, models of price distribution, or menu
cost models.
1
of enforcement and anonymity prevent households from borrowing to …nance these spending shocks,
thereby creating a role for liquidity. Because shocks are independent across households, the model
generates heterogeneous individual histories and hence, possibly, heterogeneous holdings of money.
We provide a detailed characterization of the household’s consumption and saving problem
under a minimal set of assumptions on preferences. We show that households have a target for
their real balances, which depends on their rate of time preference, the in‡ation rate, and the
frequency of consumption opportunities. They approach this target gradually over time by saving
a decreasing fraction of their labor income ‡ow. When they are hit by a preference shock for
lumpy consumption, households deplete their money holdings in full, if their wealth is below a
threshold, or partially otherwise. Given the households’optimal consumption-saving behavior, we
can characterize the stationary distribution of real money holdings in the population, and we solve
for the value of money, thereby establishing the existence of an equilibrium. Under zero money
growth ("laissez-faire"), the steady-state monetary equilibrium is unique, and it approaches the
…rst best when households are patient. We study in detail the special case where households have
linear preferences over consumption and labor ‡ows. This version showcases the tractability of the
model as the equilibrium can be characterized in closed form, and it admits as a limit, when labor
productivity grows large, the New-Monetarist model of LRW.
We study monetary policy in the form of stationary transfer schemes …nanced by money creation. Incentive compatibility restricts transfers to be non-negative (because of lack of enforcement)
and non-decreasing (because agents can hide their money) in an agent’s real balances. We emphasize labor supply e¤ects through which money creation a¤ects output and welfare. We isolate a
single parameter, labor productivity, that determines the speed at which households insure themselves against preference shocks, and that parametrizes the policy trade-o¤ between providing risk
sharing and promoting self-insurance. We …nd that, if we restrict transfers to be lump-sum, e.g.,
as in Kehoe, Levine, and Woodford (1992), then the optimal in‡ation rate is positive when labor
productivity is low. For high labor productivity, however, positive in‡ation is suboptimal if the
government is restricted to lump sum transfers. In that case, we prove that positive in‡ation raises
welfare when engineered by a more general transfer scheme, in accordance with the Wallace (2014)
conjecture. This scheme prescribes ‡at transfers for low levels of wealth, so as to provide risk
sharing, and transfers that increase linearly with real balances for high levels of wealth in order to
neutralize the disincentive e¤ects of the in‡ation tax. We show that such an in‡ationary transfer
2
scheme provides insurance but also raises aggregate output and welfare. We go beyond this conjecture and characterize transfer schemes that generate allocations arbitrarily close to the …rst best.
For high labor productivity the optimal transfer as a function of real balances corresponds to a
step function. For low labor productivity, the optimal transfer is lump sum.
In the pure currency economy the policymaker has a single instrument, …at money, to address
the trade-o¤ between risk sharing and self insurance, hence the need of nonlinear transfers. We
extend our pure currency by adding another asset, illiquid nominal government bonds, that can
bear interest and hence promote self insurance in a similar, but perhaps more realistic way, than
non-linear transfers. In equilibrium the poorest households hold money only, which creates an
endogenous segmentation of asset markets – only households with su¢ cient wealth participate in
the bonds market. We show an equivalence between liquidity-trap equilibria where the nominal
interest rate is zero and equilibria of the pure-currency economy where a fraction of households do
not deplete their real balances following a preference shock. We use this equivalence to draw some
implications for the existence of liquidity-trap equilibria and for policy. Liquidity-trap equilibria
occur where labor productivity is low, idiosyncratic risk is high, and bonds are scarce. Households
have a high precautionary demand for assets because the pace of wealth accumulation is low. If
the bond supply is low, the bond yield is driven to zero, so that households are indi¤erent between
holding money and bonds. A combination of higher anticipated in‡ation through higher money
growth and a lower bond-money ratio through open-market operations is welfare improving.
We also provide examples where interest-bearing illiquid bonds are essential in that they can
raise welfare relative to the pure currency economy with lump-sum transfers. The optimal supply
of bonds is chosen such that the nominal interest rate is positive and it is combined with lump
sum transfers …nanced with money creation. As labor productivity falls it is optimal to raise both
in‡ation and the nominal interest rate.
Literature
Our model can be viewed as a continuous-time, competitive version of Lagos and Wright (2005)
with general preferences. Despite the presence of uninsurable idiosyncratic risk, the Lagos-Wright
model delivers equilibria with degenerate distributions of money holdings which can be solved
in closed form and can easily be integrated with the standard representative-agent model used
in macroeconomics.3 Yet, this gain in tractability comes at a cost: in the absence of ex-post
3
For a recent review of the literature, see Lagos, Rocheteau, and Wright (2016).
3
heterogeneity, monetary policy is exclusively about enhancing the rate of return of currency, thereby
making the Friedman rule omnipotent. Formulating tractable search-theoretic monetary models
without restrictions on money holdings, and with non-degenerate distributions, has been considered
challenging due to the interaction between bargaining and ex-post heterogeneity. Examples of such
models include Camera and Corbae (1999), Zhu (2005), Molico (2006), and Chiu and Molico (2010,
2011), all in discrete time. While our preference shocks for lumpy consumption are reminiscent to
random matching shocks in search models, we avoid the intricacies due to bargaining by assuming
competitive prices.4 Green and Zhou (1998, 2002) and Zhou (1999) assume price posting, undirected
search, and indivisible goods, which leads to a continuum of steady states. In contrast, the laissezfaire monetary equilibrium of our model is unique. Menzio, Shi, and Sun (2013) assume directed
search and free-entry of …rms and characterize the monetary steady state under general preferences
and a constant money supply. They only brie‡y discuss how one would solve their model with
money growth, and they do not analyze the resulting policy trade-o¤, which is the main focus of
our analysis. See also Sun and Zhou (2016) with an exogenous upper-bound on real balances.
Our approach is also closely related to incomplete market models where households self-insure
against idiosyncratic income risk by accumulating assets: …at money in Bewley (1980, 1983) and
Lucas (1980), physical capital in Aiyagari (1994), and private IOUs in Huggett (1993).5 We contribute to this literature by analyzing a tractable continuous-time model with a type of idiosyncratic
risk that is reminiscent to the one in random matching models and with non-trivial labor supply
decisions.6 While incomplete markets are most often solved by way of numerical methods, a few papers have developed analytically tractable frameworks. In particular Scheinkman and Weiss (1986);
Algan, Challe, and Ragot (2011); and Lippi, Ragni, and Trachter (2015) study Bewley economies
with quasi-linear preferences, with a special attention to logarithmic preferences for consumption.
Our model di¤ers in a number of ways allowing for a comprehensive study of the welfare and output
e¤ects of in‡ation and the policy trade-o¤ of monetary policy.7 Amongst Bewley models who work
4
Rocheteau, Weill, and Wong (2015) study a discrete-time version of the model with search and bargaining and
alternating market structures. The model remains tractable and can be used to study transitional dynamics following
one-time money injections.
5
See Ljungqvist and Sargent (2004, chapters 16-17) and Heathcote, Storesletten, and Violante (2009) for surveys.
6
In contrast to the seminal paper by Lucas (1980), we do not assume a cash-in-advance constraint (i.e., his
condition (1.4)) since agents can …nance their ‡ow consumption with their current labor income, and we rule out
credit arrangements from …rst principles. The nature of the idiosyncratic risk is also di¤erent: it takes the form of
random arrivals of opportunities to consume lumpy amounts of consumption, which is analogous to the idiosyncratic
liquidity risk in continuous-time random matching models. Also in contrast to Lucas (1980), we allow for endogenous
labor supply we study money growth under various schemes and we can compute classes of equilibria in closed form.
7
We characterize our model under general concave preferences and consider quasi-linear preferences only as a
special case. Even for this special case, our model di¤ers in important ways from the Scheinkman-Weiss model: risk
4
with numerical methods, Imrohoroglu (1992) and Dressler (2011) have studied the welfare cost of
in‡ation.
Our work also contributes to a recent literature developing continuous time methods to analyze
general equilibrium models with incomplete markets. Recently, Achdou, Han, Lasry, Lions, and
Moll (2015) have proposed numerical tools based on mean-…eld-games techniques to study a wide
class of heterogeneous-agent models in continuous time, with Huggett (1993) as their baseline. Our
idiosyncratic lumpy consumption opportunities are similar to the uncertain lumpy expenditures in
the Baumol-Tobin model of Alvarez and Lippi (2013). Our model of Section 5 with money and
bonds is closely related with the following di¤erences: we assume no cost to liquidate assets, and
we do not take the consumption path (both in terms of ‡ows and jump sizes) as exogenous; neither
do we assume that labor income is exogenous.
2
The environment
Time, t 2 R+ , is continuous and goes on forever. The economy is populated with a unit measure
of in…nitely-lived households who discount the future at rate r > 0. There is a single perishable
consumption good produced according to a linear technology that transforms h units of labor
into h units of output. Households have a …nite endowment of labor per unit of time, h < 1.
Alternatively, one can normalize labor endowment to one and interpret h as labor productivity.
Households value consumption, c, and leisure ‡ows, `, according to an increasing and concave
instantaneous utility function, u(c; `). We assume that both consumption and leisure are normal
goods, that u(c; `) is bounded above, i.e. supc
0 u(c; h)
kuk < 1, and bounded below so that
we can normalize u(0; 0) = 0. In addition to consuming and producing in ‡ows, households receive
preference shocks that generate lumps of utility for the consumption of discrete quantities of the
good. Lumpy consumption opportunities represent large shocks (e.g., replacement of durables,
health events and expenditures due to changes in family composition) that require immediate
spending.8 These shocks occur at Poisson arrival times, fTn g1
n=1 , with intensity
. The utility
of consuming y units of goods at time Tn is given by an increasing, concave, and bounded utility
function, U (y), and we normalize U (0) = 0.9 Taken together, the lifetime expected utility of a
is idiosyncratic and arises from lumpy consumption opportunities, instead of an aggregate risk on agents’ability to
work, and we impose a bound, h, on ‡ow labor supply that plays a key role for our normative analysis.
8
One could also interpret the preference shocks as random consumption opportunities in a decentralized goods
market with search-and-matching frictions. For such an interpretation, see Rocheteau, Weill, and Wong (2015).
9
If we think of the shock as the replacement of durables, then U (y) = #(y)=(r + ) is the discounted sum of
the utility ‡ows, #(y), provided by a durable good, where is the Poisson arrival rate at which a particular durable
5
household can be written as:
"Z
E
+1
e
rt
u ct ; h
ht dt +
0
1
X
e
rTn
#
U (yTn ) ;
n=1
(1)
given some adapted and left-continuous processes for ct , ht , and yt . We impose the following
additional regularity conditions on households’utility functions. First, U (y) is strictly increasing,
strictly concave, and twice continuously di¤erentiable; it also satis…es the Inada condition U 0 (0) =
+1. Second, u(c; `) can have either one of the following two speci…cations:
1. Smooth-Inada (SI) preferences: u(c; `) is strictly concave, and twice continuously di¤erentiable, and it satis…es Inada conditions with respect to both arguments, i.e., uc (0; `) = 1
and uc (1; `) = 0 for all ` > 0, u` (c; 0) = 1 for all c > 0;
2. Linear preferences: u(c; `) = minfc; cg + `, for some c
0.
The …rst speci…cation facilitates the analysis because it implies smooth policy functions for
households and strictly positive consumption and labor ‡ows. The second speci…cation is useful
to establish close contacts with the literature as it corresponds to the quasi-linear preferences
commonly used in monetary theory since Lagos and Wright (2005) to eliminate wealth e¤ects and
obtain equilibria with degenerate distributions of money balances.10 In our model, distributions
are non-degenerate even under quasi-linear preferences, because the feasibility constraint on labor,
h
h, can be binding for some agents in equilibrium.
In order to make money essential we assume that households cannot commit and there is
no monitoring technology (Kocherlakota, 1998). As a result households cannot borrow to …nance
lumpy consumption since otherwise they would default on their debt. The only asset in the economy
is …at money: a perfectly recognizable, durable and intrinsically worthless object. The supply of
money, denoted Mt , grows at a constant rate,
0, through lump-sum transfers to households.
(We consider alternative transfer schemes in Section 4.2.) Trades of money and goods take place
in spot competitive markets. The price of money in terms of goods is denoted
t.
For the purpose of studying policy and welfare, our …rst-best benchmark is the full-insurance
allocation. Under SI preference, it is the time invariant allocation (cF I ; hF I ; y F I ) solving
uc cF I ; h
hF I = u` cF I ; h
hF I = U 0 y F I :
(2)
expires, and y is the quality of the durable.
10
Lagos and Wright (2005) assume quasi-linear preferences of the form u(c) + `. See also Scheinkman and Weiss
(1986) for similar preferences. The fully linear speci…cation comes from Lagos and Rocheteau (2005). One can achieve
the same amount of tractability with the larger class of preferences studied in Wong (2016), including constant return
to scale, constant elasticity of substitution, and CARA.
6
Households equalize the marginal utilities of ‡ow consumption, of leisure, and of lumpy consumption. Under linear preferences, ‡ow labor and consumption can be at corners, so that
hF I = y F I = minf y ? ; hg;
(3)
and cF I = 0, where y ? is the quantity that equalizes the marginal utility of lumpy consumption and
the marginal disutility of work, U 0 (y ? ) = 1. If labor endowments are su¢ ciently large, then the
…rst-best allocation is such that households consume y ? whenever they receive a preference shock
and they supply y ? of their labor endowment. If the endowments are small, h < y ? , then y ? is
not feasible, so households supply their whole labor endowment, h, and share the output equally
among the
3
households with a desire to consume.
Stationary monetary equilibrium
In this section we study stationary monetary equilibria where aggregate real balances,
t Mt ,
are
constant. It implies the rate of return on money, _ t = t , is constant and equal to the negative of the
in‡ation rate, i.e.,
t
=
t.
0e
In order to determine the equilibrium time-zero value of money,
0,
we proceed as follows. First, we obtain from the government budget constraint that the real value
of the lump sum transfer received by households is
_ =
t Mt
t Mt
=
0 M0 .
In Section 3.1 we
take this transfer as given and solve the household’s consumption-saving problem and determine
its choice of real balances. In Section 3.2, we characterize the distribution of real money balances
as a function of the lump-sum transfer,
the existence of
3.1
0 M0 .
In Section 3.3, we use market clearing to establish
0.
The household’s problem
We analyze the household’s problem given any constant in‡ation rate,
lump sum transfer
=
0 M0 .
0, and given any real
Let W (z) denote the maximum attainable lifetime utility of a
household holding z units of real balances. In our supplementary appendix we establish that W (z)
is a solution to the Bellman equation:
W (z) = sup
Z
1
e
(r+ )t
u ct ; h
ht +
0
7
U (yt ) + W (zt
yt )
dt;
(4)
with respect to left-continuous plans for fct ; ht ; yt g, a piecewise continuously di¤erentiable plan for
zt , and subject to:
z0 = z
0
yt
z_t = ht
The e¤ective discount factor, e
factor, e
rt ,
(r+ )t ,
(5)
zt
(6)
ct
zt + :
(7)
in the household’s objective, (4), is equal to the time discount
multiplied by the probability that no preference shock occurs during the time interval
[0; t), i.e., Pr (T1
t) = e
t.
This e¤ective discount factor multiplies the household’s expected
period utility at time t, conditional on T1
of consumption and leisure, u ct ; h
t. The …rst term of the period utility is the utility ‡ow
ht . The second term is the expected utility associated with
a preference shock at time t, an event occurring with Poisson intensity . This expected utility is
the sum of U (yt ) from consuming a lump of yt units of consumption good and the continuation
utility W (zt
yt ) from keeping zt
yt real balances.
Equation (5) is the initial condition for real balances, and (6) is a feasibility constraint stating
that real balances must remain positive before and after a preference shock. In particular, the
constraint that yt
zt follows from the absence of enforcement and monitoring technologies that
prevent households from issuing debt. Finally, (7) is the law of motion for real balances. The rate
of change in real balances is equal to the household’s output ‡ow net of consumption, ht
the negative ‡ow return on currency,
Theorem 1 For given
ct , plus
z, and a ‡ow lump-sum transfer of real balances,
.
, Equation (4) has a unique bounded solution, W (z). It is strictly increas-
ing, strictly concave and continuously di¤ erentiable over [0; 1). It is twice continuously di¤ erentiable over (0; 1), except perhaps under linear preference, when this property may fail for at most
two points. Moreover,
W 0 (0)
r+
h
kuk
kU k
+
r
r
;
lim W 00 (z) =
z!0
1; and lim W 0 (z) = 0:
z!1
Finally, W solves the Hamilton-Jacobi-Bellman (HJB) equation:
rW (z) = max u(c; h
h) +
[U (y) + W (z
with respect to (c; h; y) and subject to c
0, 0
h
8
y)
W (z)] + W 0 (z) (h
h and 0
y
z.
c
z+
) ;
(8)
Establishing that the value function is well behaved is important because it will later allow us
to apply standard theorems in order to establish the existence of a unique stationary distribution
of real balances, and to show that the mean of the distribution,
M , is continuous in
, which
facilitates the proof of existence of an equilibrium. A perhaps surprising result is that W 0 (0) < 1
even though U 0 (0) = 1. Intuitively, a household with depleted money balances, z = 0, has a …nite
marginal utility for real balances because it has some positive time to accumulate real balances
before the next opportunity for lumpy consumption, E [T1 ] = 1= > 0.11
The HJB equation, (8), has a standard interpretation as an asset-pricing condition. If we think
of W (z) as the price of an asset, the opportunity cost of holding that asset is rW (z). The asset
yields a utility ‡ow, u(c; `), and a capital gain, U (y) + W (z
shock with Poisson arrival rate
y)
W (z), in the event of a preference
. Finally, the value of the asset changes over time due to the
accumulation of real balances, represented by the last term on the right side of (8), W 0 (zt )z_t .
Optimal lumpy consumption. From (8) a household chooses its optimal lumpy consumption
in order to solve:
V (z) = max fU (y) + W (z
0 y z
y)g :
(9)
In words, a household chooses its level of consumption in order to maximize the sum of its current
utility, U (y), and its continuation utility with z
y real balances, W (z
y). Because U 0 (0) = 1
but W 0 (0) < 1, a household always …nds it optimal to choose strictly positive lumpy consumption,
y(z) > 0, for all z > 0. Hence, the …rst-order condition of (9) is
U 0 (y)
W 0 (z
y);
(10)
with an equality if y < z. The following proposition provides a detailed characterization of the
solution to (10).
Proposition 1 (Optimal Lumpy Consumption) The unique solution to (10), y(z), admits the
following properties:
1. y(z) is continuous and strictly positive for any z > 0.
11
The main technical challenge in Theorem 1 is to establish that W (z) admits continuous derivatives of su¢ ciently
high order. One approach would have been to …nd an explicit solution of the HJB equation and apply a su¢ ciency
argument to show that this solution is equal to the value function of the household. Unfortunately, there are no
explicit solutions in general, so we go the other way. That is, we show that the value function of the household
necessarily solves a generalized HJB equation, using arguments from the theory of viscosity solutions (see, e.g.,
Bardi and Capuzzo-Dolcetta, 1997). Based on this generalized HJB, we are able to establish the desired smoothness
properties of the value function.
9
2. Both y(z) and z
y(z) are increasing and satisfy limz!1 y(z) = limz!1 z
3. y(z) = z if and only if z
y(z) = 1.
z1 , where z1 > 0 solves U 0 (z1 ) = W 0 (0).
Finally, V (z), is strictly increasing, strictly concave, and continuously di¤ erentiable over (0; 1)
with V 0 (z) = U 0 [y(z)].
Proposition 1 shows that, as long as real balances are below some threshold z1 , the household
…nds it optimal to deplete its real balances in full upon receiving a preference shock. This follows
because the utility derived from spending a small amount of real balances, U 0 (0) = 1, is larger
than the bene…t from holding onto it, W 0 (0) < 1. This result— the fact that liquidity constraints
bind over a nonempty interval of the support of the wealth distribution— is in contrast with the
standard incomplete-market model in continuous time where liquidity constraints never bind in the
interior of the state space (Achdou, Han, Lasry, Lions, and Moll, 2015), and it will play a key role
for the tractability of our model.
By induction we can construct a sequence of thresholds for real balances, fzn g+1
n=1 , such that:
z 2 [0; z1 ) =) z
y(z) = 0
z 2 [zn ; zn+1 ) =) z
y(z) 2 [zn
1 ; zn ) ,
8n
1:
If a household’s real balances belong to the interval [zn ; zn+1 ), the post-trade real balances of the
household following a preference shock, z
y(z), belong to the adjacent interval, [zn
1 ; zn ).
Hence,
the household is insured against n consecutive preference shocks, i.e., it would take n shocks to
deplete the real balances of the household. The properties of lumpy consumption, y(z), and posttrade real balances, z
Optimal saving.
y(z), are illustrated in Figure 1.
Next, we characterize a household’s optimal saving behavior. We …rst de…ne
the saving correspondence:
s(z)
h
c
z+
: (h; c) solves (8) :
Proposition 2 (Optimal Saving Correspondence) The saving correspondence, s(z), is upper
hemi-continuous, convex-valued, decreasing, strictly positive near z = 0, and admits a unique z ? 2
(0; 1) such that 0 2 s(z ? ).
1. SI preferences. The saving correspondence is singled-valued, strictly decreasing, and continuously di¤ erentiable over (0; 1).
10
z − y(z)
y(z)
45o
o
45
z1
Full depletion
z1
z2
Partial depletion
Figure 1: Left panel: Lumpy consumption. Right panel: Post-trade real balances.
2. Linear preferences. The saving correspondence is equal to:
8
8
h
z+
<
< >
[ c
z + ;h
z + ] if W 0 (z)
= 1:
s(z) =
:
:
c
z+
<
(11)
The …rst part of Proposition 2 highlights three general properties of households’saving behavior.
The …rst states that households save less when they hold larger real balances. The second property
of s(z) is that it is strictly positive near zero. The third property is that households have a target,
z ? < 1, for their real balances.
The second part of Proposition 2 provides a tighter characterization of s(z) under our two
preference speci…cations. Under SI preferences, the HJB equation, (8), de…nes a strictly concave
optimization problem leading to a smooth and strictly decreasing saving correspondence. Indeed,
the …rst-order conditions for consumption and leisure are
uc (c; `) = u` (c; `) = W 0 (z):
(12)
Given that ‡ow consumption and leisure are assumed to be normal goods, it follows that c and `
increase with z and so decrease with the marginal value of real balances. Under linear preferences
(8) de…nes a linear optimization problem delivering a bang-bang solution for s. Households work
maximally, h = h, and consume nothing, c = 0, when real balances are low enough and W 0 (z) > 1.
They stop working, h = 0, while consuming maximally, c = c, when real balances are large enough
and W 0 (z) < 1.
11
Next, we study the time path of a household’s real balances conditional on not receiving a
preference shock, namely, the solution to the initial value problem
z_t = s (zt ) with z0 = 0:
(13)
Under linear preferences this problem is well de…ned, in the sense that s(z) is single-valued, for all z
except when W 0 (z) = 1 because there are multiple optimal values for s. In this case, we choose the
s that is closest to zero. As a result, if z ? is such that W 0 (z ? ) = 1, this ensures that real balances
remain constant and equal to their stationary point.12 Given the unique solution to (13), we can
de…ne the time to reach z from z0 = 0:
T (z)
inff t
0 : zt
zj z0 = 0g:
(14)
Proposition 3 (Optimal Path of Real Balances) The initial value problem (13) has a unique
solution. This solution is strictly increasing for t 2 [0; T (z ? )), where zT (z ? ) = z ? , and it is constant
and equal to z ? for all t
T (z ? ). Under SI preferences, T (z ? ) = 1. Under linear preferences,
T (z ? ) < 1 if and only if 0 2 ( c
z? + ; h
z? +
).
Proposition 3 shows that a household accumulates real money balances until it reaches its target
z ? . Under SI preferences real balances reach their target asymptotically at an exponential speed
dictated by js0 (z ? )j. Under linear preferences s(z) may fail to be continuously di¤erentiable at z ?
and, as a result, the target may be reached in …nite time. For instance, in the laissez-faire economy
where
=
= 0, s(z) jumps downward at the target z ? , i.e., s(z) = h > 0 for all z < z ? , while
s(z ? ) = 0. Clearly, this implies that the target is reached in …nite time T (z ? ) = z ? =h < +1. In
Figure 2 we illustrate the path for real balances and the spending behavior of a household subject
to random preference shocks.
3.2
The stationary distribution of real balances
We now show that the household’s policy functions, y(z) and s(z), induce a unique stationary
distribution of real balances over the support [0; z ? ]. To this end, we de…ne the minimal time that
it takes for a household with z real balances at the time of a preference shock to accumulate strictly
more than z 0 real balances following that shock:
(z; z 0 )
0
)
max T (z+
12
T [z
y(z)] ; 0 ;
(15)
Under SI preferences a technical di¢ culty arises because s(z) is not continuously di¤erentiable at z = 0, and
hence the standard existence and uniqueness theorems for ODEs do not apply. One can nevertheless construct the
unique solution of (13) by starting the ODE at some z > 0 and "run it backward" until it reaches zero.
12
zt
«
y ( zT1 )
s(
T1
)
zt
T3
T2
Figure 2: Optimal path of real balances
for z; z 0 2 [0; z ? ]. Notice that
(z; z ? ) = 1 since the household never accumulates more than the
target. Let F (z) denote the cumulative distribution function of a candidate stationary equilibrium.
For given
it must solve the …xed-point equation:
Z 1
Z 1
Z
1 F (z 0 ) =
e u
Ifu (z;z 0 )g dF (z) du =
0
0
1
e
(z;z 0 )
dF (z);
(16)
0
where the second equality is obtained by changing the order of integration. The right side of (16)
calculates the measure of households with real balances strictly greater than z 0 . First, it partitions
the population into cohorts indexed by the date of their last preference shock. There is a density
measure, e
u,
of households who had their last preference shocks u periods ago. Second, in each
cohort there is a fraction dF (z) of households who held z real balances immediately before the shock.
Those households consumed y(z), which left them with z
y(z) real balances. If u
(z; z 0 ), then
su¢ cient time has elapsed since the preference shock for their current holdings to be strictly greater
than z 0 . A key observation is that the …xed-point problem in (16) is equivalent to the problem of
…nding a stationary distribution for the discrete-time Markov process with transition probability
function:
Q(z; [0; z 0 ])
1
e
(z;z 0 )
:
(17)
The function Q is the transition probability of the discrete-time Markov process that samples the
real balances of a given household at the arrival times fTn g1
n=1 of its preference shocks. This
observation allows us to apply standard results for the existence and uniqueness of stationary
13
distributions of discrete-time Markov processes.13 We obtain:
Proposition 4 (Stationary Distribution of Real Balances) For given
the …xed point prob-
lem, (16), admits a unique solution, F (z). This solution is continuous in the lump-sum transfer
parameter,
, in the sense of weak-convergence.
In addition to obtaining existence and uniqueness of a stationary distribution, Proposition
4 shows that F is continuous in
because all policy functions are appropriately continuous in
that parameter. This continuity property is helpful to establish the existence of a steady-state
equilibrium as it ensures that the market-clearing condition is continuous in the price of money, .
3.3
The real value of money
Equating the aggregate supply of real balances,
0 M0 ,
with the aggregate demand of real balances
as measured by the mean of the distribution F , we obtain the market-clearing condition
Z 1
M
=
z dF (z j 0 M0 );
0 0
(18)
0
where the right side makes it explicit that the stationary distribution depends on
lump-sum transfer,
=
0 M0 ,
0
via the the
since the household’s path for real balances, zt , depends on
.
From (18) money is neutral at a steady state in the sense that aggregate real balances are determined
independently of M0 .14 As is standard, however, a change in the money growth rate will have real
e¤ects by a¤ecting the rate of return on household savings. We now de…ne an equilibrium:
De…nition 1 A stationary monetary equilibrium is composed of: a value function, W (z j
0 M0 ),
and associated policy functions, that solve the household’s optimization problem (4); a distribution
of real balances, F (z j
0 M0 ),
that solves the condition for stationarity (16); a price,
0
> 0, that
solves the market-clearing condition (18).
In the de…nition, our notations emphasize that the value function and the distribution of real
balance depend on
via the real lump sum transfer,
M.
In order to establish the existence of an equilibrium we study (18) at its boundaries. As
0
approaches zero, the left side of (18) goes to zero, but the right side remains strictly positive because,
13
In Section 3.4, we use this characterization to provide a closed-form solution of the distribution in the case of
equilibria with full depletion of real balances (z ? < z1 ). In the general case, numerical calculations are needed and
we use an equivalent characterization in terms of Kolmogorov forward equations, as described in our supplementary
numerical appendix.
14
In Rocheteau, Weill, and Wong (2015) we show in a discrete-time version of our model that one-time money
injections are not neutral in the short run.
14
from Proposition 2, households accumulate strictly positive real balances even when they receive
no lump-sum transfer,
= 0. As
0
tends to in…nity, the left side of (18) becomes larger than
the right side because the lump sum transfer becomes so large that households only consume and
stop working. Finally, Proposition 4 established that the stationary distribution, F , is continuous
in
=
0 M0 .
Hence, we can apply the intermediate value theorem and obtain:
Proposition 5 (Existence and Uniqueness) For all
equilibrium. Moreover, the laissez-faire equilibrium,
=
0 there exists a stationary monetary
= 0, is unique.
From Proposition 5 a monetary equilibrium exists for all in‡ation rates. Indeed, we show in
Proposition 2 that, as a result of the Inada condition on U (y), s(z) is always strictly positive near
z = 0. In the laissez-faire where
=
= 0 the equilibrium has a simple recursive structure
allowing uniqueness to be proved. From Theorem 1 the value and policy functions are uniquely
determined independently of F . From Proposition 4, F is uniquely determined given the policy
functions.
3.4
Equilibria with full depletion of real balances
In this section we study the class of equilibria with full depletion, in which households …nd it
optimal to spend all their money holdings whenever a preference shock occurs, i.e., y(z) = z for all
z 2 [0; z ? ]. In this case our model is very tractable and it lends itself to a tight characterization
of decision rules and distributions. We also show that full depletion occurs under appropriate
parameter restrictions. These results will be used in the following sections.
The optimal path for real balances under full depletion. The ODE for the optimal path
of real balances, (7), can be rewritten as:
z_t = h( t )
where
t
c( t )
zt + ;
(19)
W 0 (zt ) is the marginal value of real balances, while h( t ) and c( t ) are the solutions to
max
u(c; h
h) + (h
c
z+
) :
(20)
c 0;h h
To solve for
t
we apply the envelope condition to di¤erentiate the HJB (8) with respect to z along
the optimal path of money holdings. This leads to the ODE:
r
t
=
U 0 (zt )
t
15
t
+ _ t;
(21)
where we use V 0 (zt ) = U 0 [y(zt )] = U 0 (zt ) from Proposition 1. The pair, (zt ;
t ),
solves a system of
two ODEs, (19) and (21).15 We can show that the stationary point of this system is a saddle point
and the optimal solution to the household’s problem is the associated saddle path.
The stationary distribution of real balances under full depletion. Under full depletion,
y(z) = z, the time it takes for a household to accumulate z 0 real balances following a preference
shock is
0 ). Hence, from (17), the transition probability function,
(z; z 0 ) = T (z+
Q(z; [0; z 0 ]) = 1
0 )
T (z+
e
;
(22)
does not depend on z. In words, the probability that a household holds less than z 0 is independent
of its real balances just before its last lumpy consumption opportunity, z. This result is intuitive
since households "re-start from zero" after a lumpy consumption opportunity. It follows that the
stationary probabilities coincide with the transition probabilities, i.e.,
F (z 0 ) = Q z; [0; z 0 ] :
(23)
Finally, the equilibrium equation for the price level, (18), simpli…es as well:
Z 1
Z 1
Z z?
zdF (z j 0 M0 ) =
[1 F (z j 0 M0 )] dz =
e T (z j
0 M0 =
0
0
0 M0 )
dz;
(24)
0
where our notation highlights that the time to accumulate real balances, T , is a function of the
real lump sum transfer,
=
0 M0 .
Verifying full depletion. From the …rst-order condition, (10), households …nd it optimal to
deplete their money holdings in full when a lumpy consumption opportunity occurs; y(z) = z for
all z 2 [0; z ? ], if and only if
U 0 (z ? )
W 0 (0) =
0.
In order to verify this condition one must solve for the equilibrium price,
real transfer,
=
0 M0 .
(25)
0,
and the associated
We turn to this task in the following proposition.
Proposition 6 (Su¢ cient Conditions for Full Depletion) Under either SI or linear preferences, there exists a threshold for the in‡ation rate,
F,
such that, for all
F,
all stationary
monetary equilibria feature full depletion.
A similar system of ODEs holds under partial depletion, where U 0 (zt ) is replaced by U 0 [y(zt )]. Hence, in order
to solve for this system, one also needs to solve for the unknown function y(z). In Appendix B, we provide a numerical
solution to this problem.
15
16
Under linear preferences, given any
such that, for all h
0, there exists a threshold for labor productivity, hF ,
hF , there exists a unique stationary monetary equilibrium, and this equilibrium
must feature full depletion.
Proposition 6 identi…es two conditions on exogenous parameters ensuring full depletion. If
in‡ation is large enough, then money holdings become "hot potatoes": they depreciate quickly so
that households always …nd it optimal to spend all their money when given the opportunity. Under
linear preferences, if labor productivity is large enough, then households spend all of their money
holdings when a preference shock hits because they anticipate that they can rebuild their money
inventories quickly.
3.5
The LRW economy
We now show that the model under linear preferences, u(c; `) = minfc; cg + `, can easily be solved
with pencil and paper, and closed-form solutions, including the non-degenerate distribution of real
balances, can be obtained for a broad set of parameter values.16 When labor productivity grows
very large, h ! 1, equilibrium converges to the LRW equilibrium with linear value functions and
a degenerate distribution of money holdings.
We focus on laissez-faire equilibria (
= 0) with similar spending patterns as in LRW, i.e.,
households deplete their real balances in full when they receive a preference shock. From (20)
households choose z_ = h
h to maximize z_ (
The solution is such that zt = ht for all t
1) where
solves the envelope condition (21).
T (z ? ) = z ? =h, where t is the length of time since the
last preference shock, and z ? is the stationary solution to (21). The marginal value of money at
the target is
= 1 because a household who keeps its real balances constant must be indi¤erent
between not working and working at a disutility cost of one in order to accumulate one unit of real
balances worth . From (21):
U 0 (z ? ) = 1 +
r
:
(26)
The marginal utility of lumpy consumption is equal to the marginal disutility of labor augmented
by a wedge, r= , due to discounting. If households are more impatient, or if preference shocks are
less frequent, households reduce their targeted real balances.
16
While we focus on linear preferences, we could achieve the same amount of tractability with the larger class of
preferences studied in Wong (2016), including, for example, constant return to scale, constant elasticity of substitution,
and CARA.
17
From (22)-(23) the steady-state distribution of real balances is a truncated exponential distribution,
F (z) = 1
e
z
h
Ifz<z ? g for all z 2 R+ :
(27)
Note that it has a mass point at the targeted real balances, 1 F (z ? ) = e
z ? =h ,
which is increasing
with h. From market clearing, (24), aggregate real balances are:
0 M0 =
h
1
e
Aggregate real balances are smaller than the target,
z?
h
:
(28)
M < z ? , and they are increasing with the
household’s labor endowment.
We now check the condition for full depletion of money balances, (25). Integrating (21) over
[t; T (z ? )] and using the change of variable z = ht, we obtain a closed-form expression for
function of z 2 [0; z ? ],
(z) = 1 +
Z
z?
r+
e ( h )(x
z)
U 0 (z ? )
U 0 (x)
z
h
dx:
as a
(29)
The marginal value of real balances is equal to the marginal disutility of labor, one, plus a discounted
sum of the di¤erences between the marginal utility of lumpy consumption on the path going from z
to z ? , U 0 (zt ), and at the target, U 0 (z ? ). It is easy to check that
0
(z) < 0, i.e., the value function is
strictly concave, and as z approaches z ? the marginal value of real balances approaches one. From
(29) the condition for full depletion, (25), can be expressed as
r
Z
T (z ? )
e
(r+ )
U 0 [z( )]
U 0 (z ? ) d :
(30)
0
The right side of (30) is monotone decreasing in h (since T (z ? ) = z ? =h) and it approaches 0 as h
tends to +1. So, we con…rm Proposition 6 by showing that the equilibrium features full depletion
if and only if h is above some threshold. Alternatively, (30) holds if households are su¢ ciently
impatient because the cost of holding money outweighs the insurance bene…ts from hoarding real
balances.
The following proposition establishes that as h tends to in…nity the equilibrium approaches an
equilibrium with degenerate distribution and linear value function, analogous to the one in LRW.
Proposition 7 (Convergence to LRW) As h ! 1 the measure of households holding z ? tends
to one, the value of money approaches z ? =M0 , and W (z) converges to z
18
z? +
[U (z ? )
z ? ] =r.
4
Policy
A central objective of the literature on pure currency economies is to characterize welfare-enhancing
policies taking into account the frictions in the environment (e.g., lack of monitoring and lack of
enforcement) that make money essential. The policymaker seeks to resolve the fundamental tradeo¤ between risk sharing and self insurance using as a policy tool a transfer scheme …nanced with
money creation. Transfers must be consistent with the inability of the government to monitor agents
and to enforce trades, i.e., they are not contingent on private histories and they are non-negative.
Finally, transfers must be non-decreasing since households can hide their assets when claiming
the transfer. This problem is challenging because it requires to solve households’ optimization
problem under possibly non-linear transfers and to determine how di¤erent transfer schemes a¤ect
the distribution of real balance and the value of money.
The literature has followed di¤erent paths to address these challenges. Part of the literature
abstracts from risk sharing by studying models with degenerate distributions so that policy has
a single objective, to promote self insurance. If lump-sum taxes are available, it su¢ ces to pay
interest on currency …nanced with taxes. In the absence of lump-sum taxation (because of lack of
enforcement) this requires non-linear transfers that reward the accumulation of large real balances
(e.g., Hu, Kennan, and Wallace, 2009; Andolfatto, 2010). Another branch of the literature studies
pure currency economies with heterogenous agents so that policy faces a trade-o¤, i.e., promoting
risk sharing may reduce incentives to self-insure, but then it is restricted to lump sum transfer
schemes (e.g., Kehoe, Levine, and Woodford, 1992; Molico, 2006). Results take the form of examples
where in‡ation is bene…cial, but there are also many cases where in‡ation by means of lump-sum
transfer is not optimal. Finally, Wallace (2014) conjectured that, once general transfers are allowed,
some in‡ation is always optimal. That is, one can always …nd transfers …nanced by money creation
that dominate laissez faire. In this section we address the optimal design of monetary policy in the
context of the economy described in Section 3.5 in three steps.
In the …rst step, we follow Kehoe, Levine, and Woodford (1992, Section 6) and study money
growth through lump-sum transfers.17 We go beyond their analysis because our economy features
17
Kehoe, Levine, and Woodford (1992) consider a discrete-time version of the Scheinkman and Weiss (1986) economy with two types of agents, buyers and sellers, that alternate through time, which leads to aggregate uncertainty.
They focus on two-state Markov equilibria where sellers hold all the currency at the end of each period and transfers
of money are lump sum. Section 6, which is the closest to what we do, specializes on logarithmic preferences. In
contrast, we have idiosyncratic shocks, no heterogeneity except the one coming from money holdings, and a rich
distribution of money holdings with full continuous support.
19
a distribution of money holdings with a continuous support (instead of two mass points) and
endogenous labor supply. Dealing with such heterogeneity is analytically challenging because one
needs to determine how money transfers and the in‡ation tax a¤ect the labor supplies of all agents
in the distribution and aggregate real balances. In this context, our key contribution is to show
that optimal in‡ation is decreasing in a fundamental parameter of the economy, labor productivity,
h, and it is zero if labor productivity is large enough.
In the second step, we depart from lump-sum transfers in instances where such transfers generate
no welfare gain. We prove the Wallace conjecture by designing an incentive-compatible transfer
scheme that combines a lump-sum component for risk sharing and a regressive component to
promote self insurance. This scheme generates welfare gains for the whole class of equilibria studied
in Section 3.5.
Wallace (2014) acknowledged that his "conjecture is weak in that it says only that some intervention is bene…cial". He asked: "Can we hope for stronger conclusions— perhaps, a characterization
of when an improvement comes from a small progressive scheme and when it comes from a small
regressive scheme? I think not". The third step addresses this question. Namely, we characterize policies that not only generate welfare gains but implement allocations close to the …rst best.
Our answer to Wallace’s question is that these near-optimal schemes are progressive for low labor
productivity, and regressive for high labor productivity.
4.1
Money growth through lump-sum transfers
We investigate the e¤ects of anticipated in‡ation implemented with lump-sum transfers on output
and welfare. Analyzing the case of money growth is challenging analytically because the transfer,
=
0 M0 ,
that a¤ects individual problems and the distribution of real balances is endogenous
and depends on the mean of the distribution. We will see that despite this di¢ culty a large class
of equilibria can be obtained in closed form.
In order to study the trade-o¤ between risk sharing and self insurance, it is instructive to focus
on equilibria with full depletion, y(z ? ) = z ? . In the presence of money growth,
> 0, the target for
real balances can take two expressions depending on whether the feasibility constraint, h(z ? )
h,
is slack or binding:
z?
min fzs ; zb g ;
20
(31)
where
zs
1
U0
1+
r+
;
and
zb
h
+
0 M0 :
(32)
The quantity zs can be interpreted as the ideal target that households aim for: it equalizes the
marginal utility of lumpy consumption, U 0 (z), and the cost of holding real balances, 1 + (r + )= .
It is feasible to reach only if h +
=h+
M
zs . The quantity zb is the highest level of real
balances feasible to accumulate, given households’…nite labor endowment, h, the in‡ation tax on
real balances,
z, and the lump-sum transfer,
. Thus zb is a constrained target. From (31) the
e¤ ective target, z ? , is the minimum between these two quantities.
From (19) the trajectory for individual real balances is zt = zb (1
e
t ).
Given that the time
since the last preference shock is exponentially distributed, the distribution of real balances is
F (z) = 1
z ? . If zb
and F (z) = 1 for all z
zb
z
zb
for all z < z ? ;
(33)
zs then households reach z ? = minfzb ; zs g only asymptotically,
and the distribution of real balances has no mass point. In contrast, if zb > zs then households
reach z ? in …nite time and the distribution has a mass point at z = z ? . Substituting the closed-form
expressions for T (z j
M) =
log (1
z=zb ) = and zb into the market-clearing condition (24), we
…nd after a few lines of algebra that aggregate real balances solve:
(
zs
0 M0
=
1
1 min 1;
+
h= + 0 M0
h= + 0 M0
The left side is strictly increasing in
+
and the right side is decreasing in
)
:
(34)
. Hence, (34) has
a unique solution, and there is a unique candidate equilibrium with full depletion. Finally, the
condition for full depletion of money balances is given by (30) where r is replaced with r +
T (z j
0 M0 )
=
log (1
and
z=zb ) = .
We now de…ne aggregate output and households’ex-ante welfare by:
Z
H( ; h)
h(z; ; h)dF (z; ; h)
Z
W( ; h)
h(z; ; h) + U (z) dF (z; ; h):
The pointwise limits for those quantities when labor productivity goes to in…nity are denoted by
H1 ( )
limh!1 H( ; h) and W 1 ( )
limh!1 W( ; h). The following proposition shows that
the e¤ects of money growth on H and W are qualitatively di¤erent depending on the size of h.
Proposition 8 (Output and welfare e¤ ects of in‡ation) In the quasi-linear economy:
21
1. Large labor productivity. Both H1 ( ) and W 1 ( ) are decreasing with .
2. Low labor productivity. If U (z)= [zU 0 (z)] is bounded above near zero, then there exists
some minimum in‡ation rate,
lim
!0 H(
) = lim
!1 H(
, and a continuous function H : [0; 1) ! R+ with limits
) = 0, such that, for all
and h 2 0; H( ) , there exists
an equilibrium with binding labor, h(z ? ) = h, and full depletion. In this equilibrium H( ; h)
attains its …rst-best level, h, and W( ; h) increases with .
3. Large in‡ation. As
! 1, H( ; h) ! 0 and W( ; h) ! 0.
The size of the labor productivity, h, determines the speed at which households can reach their
targeted real balances, and the extent of ex-post heterogeneity across households that prevails in
equilibrium. As a result, h proves to be a key parameter to determine the extent to which lump-sum
transfers of money provide risk-sharing and deter self-insurance and, ultimately, how they a¤ect
households’ex-ante welfare.
With large labor productivity, h ! 1, there is no role for risk-sharing as all households reach
their target almost instantly. However, money growth implemented with lump-sum transfers reduces the rate of return of money, which adversely a¤ects the incentives to self insure, as measured
by z ? . Hence, aggregate output, which is approximately
[U (z ? )
z ? , and social welfare, approximately,
z ? ], are decreasing with the in‡ation rate. These are the standard comparative statics
in models with degenerate distributions (e.g., Lagos and Wright, 2005).
With low labor productivity, risk-sharing considerations dominate because even though
re-
duces z ? , it takes a long time for households to reach z ? . Indeed, in the laissez-faire equilibrium
the time that it takes, in the absence of any shock, to reach the target, T (z ? ) = z ? =h, can be
arbitrarily large when h is small. Consider the regime where the equilibrium features both full
depletion, y(z ? ) = z ? , and binding labor, h(z ? ) = h. From Part 2 of Proposition 8 this regime
occurs when the in‡ation rate is not too low and the labor endowment not too high. Because
households cannot reach their ideal target, zs , they all supply h irrespective of their wealth, and
thus aggregate output is constant and equal to h. This output level is also the full-insurance one,
hF I = h. Indeed, the condition for the binding labor constraint is h= + h=
implies h <
zs < y ? , which
y ? , and from (3) hF I = h. In addition, aggregate real balances are equal to the
…rst-best level of consumption,
0 M0
= h= . Hence, risk-sharing is the only consideration for
policy as the only source of ine¢ ciency arises from the non-degenerate distribution of real balances.
22
Wealthy households who hold more real balances than the socially desirable level of consumption,
z > h= , pay a tax equal to (z
h= ) while poor households who hold fewer real balances than
the socially-desirable level of consumption, z < h= , receive a subsidy equal to (h=
z). Hence,
moderate in‡ation moves individual consumption levels toward the …rst best, thereby smoothing
consumption across households and raising their ex-ante welfare.
A calibrated example In the following we complement Proposition 8 and use a calibrated
example to study the relationship between the optimal in‡ation rate and h.18 We normalize a unit
of time to a year and we set r = 4%. The in‡ation rate is
a =(1
for the utility of lumpy consumption, U (y) = y 1
= 2%. We adopt a CRRA speci…cation
a). Provided that h
level of lumpy consumption is 1. The remaining parameters, a,
the …rst-best
, and h, are calibrated to the
distribution of the balances of transaction accounts in the 2013 SCF.19 These calibration targets
are matched with a = :31,
= 3:21 and h = 6:26.
In Figure 3 we distinguish four regimes: full versus partial depletion of real balances (y(z ? ) = z ?
versus y(z ? ) < z ? ) and slack versus binding labor constraint (h(z ? ) < h versus h(z ? ) = h). For
su¢ ciently high
and su¢ ciently low h, the equilibrium features full depletion and binding labor;
this corresponds to the area marked III in the …gure. The lower bond for in‡ation and the upper
boundary of this area correspond respectively to
below
and log H( ) in Proposition 8. As
is reduced
the equilibrium features partial depletion of real balances (areas I and II). Finally, provided
that h is su¢ ciently large, the equilibrium features both full depletion and slack labor (area IV). The
equilibria we characterize in closed form correspond to III and IV. Note that for all h
(i.e., log h
= 3:21
1:14) the …rst-best level of output is hF I = h, achieved in areas II and III.
Proposition 8 establishes that for high h in‡ation is detrimental to society’s welfare whereas
for low h positive in‡ation implemented with lump-sum transfers raises welfare relative to the
laissez-faire. In Figure 3 we illustrate these results by plotting with a black, thick curve the
welfare-maximizing in‡ation rate as a function of the labor endowment. As h increases the optimal
in‡ation rate decreases, and for a su¢ ciently high value of h the laissez-faire equilibrium dominates
18
Arguably, pure currency economies are not directly comparable to actual economies with multiple assets. The
purpose of our calibration exercise is to provide a numerical example illustrating some properties of our model with
parameter values that are plausible. See Appendix B for details on how to solve numerically the system of delay
di¤erential equations.
19
We adopt the following three targets: the ratio of the balances of the 80th-percentile household to the average
balances, F 1 (:8) = M , the ratio of the average balances to the average income, M=H, and the semi-elasticity of
money demand,
@ log M=(100 @ ). In the 2013 SCF, F 1 (:8) = M = 1:23 and M=H = :39. Aruoba, Waller
and Wright (2011) estimate that = :06. Transaction accounts in SCF include checking, savings, money market,
and call accounts, but they do not include currency.
23
Optimal
inflation rate
Figure 3: Region I: Slack labor & Partial depletion; Region II: Binding labor & Partial depletion;
Region III: Binding labor & Full depletion; Region IV: Slack labor & Full depletion.
any equilibrium with positive in‡ation. Moreover, when equilibria with full depletion and binding
labor exist (area III) then the optimal in‡ation rate is the highest one that is consistent with such
an equilibrium. A higher in‡ation rate would relax the labor constraint (area IV) and would reduce
output below its e¢ cient level, h. For values of h that are large enough such that region III does
not exist, then the optimal in‡ation rate corresponds to an equilibrium with slack labor and partial
depletion (area I). In area IV with slack labor and full depletion a reduction of the in‡ation rate is
always welfare improving.
4.2
Beyond lump-sum transfers
We have shown in Proposition 8 and Figure 3 that when h is su¢ ciently large, in‡ation implemented
through lump-sum transfers is welfare-worsening as the social cost of lowering z ? outweighs the risksharing bene…ts associated with lump-sum transfers. In contrast, Wallace (2014) conjectures that
money creation is almost always optimal in pure-currency economies, as long as one can depart from
lump-sum transfers. In accordance with this conjecture, we establish in the following that in‡ation
is optimal once one allows for more general, incentive-compatible, transfer schemes. This conjecture
is hard to verify for economies with a rich heterogeneity as ours, as one needs to determine how
transfers a¤ect individual labor supplies throughout the endogenous distribution and their overall
welfare e¤ect.
24
Suppose new money, M_ = M , is injected through the following transfer scheme:
8
z z?
<
0
? ?
(z) =
zz
1 if z 2 (z ; z0 ] ;
:
z
z > z0?
(35)
where z ? solves U 0 (z ? ) = 1 + (r + )= . The real transfer, (z), is non-negative because in pure
currency economies with no enforcement taxation is not feasible. The transfer is non-decreasing
so that households have no incentive to hide some of their money balances. Hence,
0. Moreover, we assume that
z
( z?
(z) is continuous,
z
= ( z0?
?
0 ) = (z0
0 and
0
z ? ) and
1
=
?
?
0 ) z0 = (z0
z ? ). From the government budget constraint, the sum of the transfers to
R
households net of the in‡ation tax must be zero, [ (z)
z] dF (z) = 0, where the distribution
F is now indexed by the transfer scheme. Hence,
and
z
1
0. So the …rst tier is a lump-sum
transfer, the second is a linear regressive transfer, and the third tier is neutral.
τ
z
z−
τ
1
(z)
τ0
z π*
z 0*
Figure 4: A socially bene…cial in‡ationary scheme
Our proposed scheme, illustrated in Figure 4, takes into account the trade-o¤ between selfinsurance and risk sharing in economies with non-degenerate distributions. It has a lump-sum
component,
0,
that improves risk sharing by transferring wealth from the richest households to the
poorest ones. The threshold for real balances below which households receive
0
is z ? , the target for
real balances in an economy with lump-sum transfers only. It has a regressive component,
zz
1,
that speci…es a transfer increasing with real balances for all households holding z between z ? and
z0? . The purpose of this component is to mitigate the disincentive e¤ect of in‡ation on households’
willingness to accumulate real balances. As a result, households accumulate the same amount they
25
would in the laissez-faire equilibrium, z0? .
In the following proposition we denote by h0F the threshold for labor endowments above which
there is full depletion in the laissez-faire equilibrium ( = 0); in other words (30) holds.
Proposition 9 (The Wallace conjecture) Suppose that h
h0F , the equilibrium features full
depletion. There exists an incentive-feasible, in‡ationary transfer scheme, (z) given by (35) with
> 0, such that:
(i) Society’s welfare is higher under (z) relative to the laissez-faire.
(ii) Aggregate real balances and output are higher under (z) relative to the laissez-faire.
(iii) The target for real balances is z0? , the same as under laissez-faire.
In order to prove that the transfer scheme is socially bene…cial we show that it not only redistributes wealth, but also raises aggregate real balances. In order to make the second claim we
establish that it takes longer to accumulate z0? under the transfer scheme, , than under laissez
faire. Relative to laissez faire, households accumulate real balances at a faster pace when they are
poor, because (z)
z > 0, and at a slower pace when they are rich, because (z)
z < 0. Even
though the sum of the net transfers across households is zero, only a fraction of the households
become su¢ ciently rich to be net contributors to the scheme before they are hit by a new preference shock. As a result, the burden on the rich households outweighs the subsidies they received
while being poor, and hence they reach their desired real balances later relative to the laissez faire.
It follows that there is a larger fraction of households who are producing making aggregate real
balances larger under the in‡ationary scheme. In summary, the transfer scheme, , raises society’s
welfare by redistributing a higher stock of real balances from rich to poor households without giving
incentives to households to lower their targeted real balances.
4.3
Near-e¢ cient policies
We now characterize policies that implement allocations close to the …rst best, and we study
how such policies vary with labor productivity. We start with the case where labor productivity
is low and we assume that the utility for lumpy consumption is linear with a satiation point,
U (y) = A minfy; yg. Similar preferences have been used in Kehoe, Levine, and Woodford (1992,
Section 5) and Green and Zhou (2005, Section 6).
Proposition 10 (Implementation of the First Best).
h < y [ (A
1)
r] = ( A
Assume U (y) = A minfy; yg.
If
r), then there is a monetary equilibrium under a lump-sum transfer
26
scheme, (z) = h=
for all z with
2
h=( y
h); (A
1)
r , that exactly implements a
…rst best.
If h <
y, then a …rst best allocation is one where there is full employment, h = h. We
implement this outcome with positive money growth through lump sum transfers. We construct
an equilibrium in which in‡ation is su¢ ciently high so that households have to work full time just
to maintain their targeted real balances at a level less than y. Because the marginal utility of
lumpy consumption is constant and equal to A for all z 2 [0; y], there is no welfare loss associated
with households’ex-post heterogeneity. Provided that the rate of time preference is not too large,
there is a range of positive in‡ation rates that implement a …rst-best allocation. The in‡ation rate
cannot be too low since otherwise households might …nd it optimal not to work when they reach
the target for real balances. For instance, if
= 0 households reach the target in a …nite amount
of time, and hence a fraction of them do not supply any labor. The in‡ation rate cannot too high
or households will not …nd it optimal to accumulating real balances.
For high labor productivity a …rst-best allocation is such that y = y ? where U 0 (y ? ) = 1. We
implement allocations that are close to the …rst best with a transfer scheme that is a step function,
(z j ) =
where z ? ( ) = y ?
=h, for
0
z
if
z < z?( )
;
z
z?( )
(36)
= 2 [ y ? + U (y ? )] =jU 00 (y ? )j.20 If households hold real balances
above z ? , then they receive a proportional transfer that exactly compensates for the in‡ation tax,
i.e., their asset position is protected against in‡ation. However, if they hold less than z ? , then they
no longer receive the proportional transfer. When z ? is close to y ? , this scheme rewards households
who hold real balances close to the …rst best and to punish those who hold too little real balances.
For this scheme to be nearly e¢ cient, there must be few households along the equilibrium path
who hold less than z ? , which requires h to be large. We formalize this notion in the following
proposition by measuring the welfare loss relative to the …rst best in consumption-equivalent units,
i.e., we compute the fraction
of the …rst-best consumption that an household would be willing to
pay in order to move from the equilibrium to the …rst best.
It should be noted that the transfer scheme (36) generates a surplus corresponding to the
in‡ation tax levied on households with z < z ? : after making lump sum transfers, the government
20
This transfer scheme is a generalization of the one studied in Bajaj, Hu, Rocheteau and Silva (2017) in the
context of an economy with a degenerate distribution of money holdings.
27
has money left in hand that it uses to purchase consumption goods. We assume that this surplus
is thrown away, which creates a welfare loss. Our e¢ ciency result below holds in spite of this loss.
Proposition 11 (Near-E¢ cient Schemes with High Labor Productivity.) Assume that
ry ? + [U (y ? )
y ? ] > 0. Then, there is some
such that, given any
> , as long as h is large
enough, there exists a monetary equilibrium that features full depletion of real balances with scheme
(z j ). Moreover, the welfare loss in terms of …rst-best consumption is:
1
=
h
1
y?
Z
y?
0
f[U (y ? )
y?]
[U (z)
z]g dz + o
1
h
:
(37)
The proof of the Proposition is challenging because of the discontinuity in the transfer scheme.
This discontinuity implies that the value function has a concave kink at z ? . As a result, the
standard optimality veri…cation argument, which requires smoothness, must be extended to handle
our particular case. The condition
ry ? +
[U (y ? )
y ? ] > 0 is necessary for households to have
incentives to hold real balances close to the …rst best, y ? . It states that the opportunity cost of
holding real balances, ry ? , is less than the expected surplus from holding such real balances in the
event of a preference shock for lumpy consumption,
[U (y ? )
y ? ]. It is the typical condition for
the implementation of the …rst best in monetary models with degenerate distribution (see, e.g., Hu,
Kennan, and Wallace, 2009).
The Proposition shows that the transfer scheme (z j ) leads to nearly e¢ cient allocation as
long as h is large enough. Moreover, it provides an intuitive estimate of the welfare loss. Namely,
to a …rst-order approximation in 1=h, the welfare loss is equal to the average surplus that is lost
by households with real balances strictly below the target, z ? . Notice that the estimate of
does
not account for the welfare loss incurred by households at the target. This is because the target z ?
is close to the …rst-best output, y ? , and the …rst-best output already maximizes utilitarian welfare.
Hence, the welfare loss incurred by households at z ? is of second order.
Finally, we note that the transfer scheme in Proposition 11 can be replicated without transfers
but with indivisible bonds. Bonds are of the pure-discount variety that pay z ? in terms of goods
to their bearer at maturity with Poisson arrival rate . They are perfectly recognizable and can
be traded at no cost. Their real price is q = z ? =(% + ), where where % is the real rate of return
on bonds. Under the conditions stated in Proposition 11 the supply of bonds, B, can be chosen
to implement an equilibrium with q = z ? and % = 0, i.e., bonds are traded at their face value.
Precisely, B must be equal to the measure of households at their targeted wealth, B = 1
28
F (z ? ).
In equilibrium, households accumulate real balance until they reach their targeted wealth, at which
point they switch to holding bonds. When they receive a preference shock, they fully deplete their
wealth, spending real balances if z < z ? , and bonds if z = z ? .
In the following section we elaborate on the idea that government bonds can serve as a policy
instrument to provide incentives to self insure, just like regressive transfers do.
5
Beyond pure currency: Money and illiquid bonds
We now consider a second instrument, beside …at money, to tackle the policy trade-o¤ between risk
sharing and self insurance. We introduce illiquid nominal government bonds that can bear interest.
With this extension, monetary policy can be implemented through both "helicopter drops" and
open market operations by purchasing or selling government bonds in exchange for money. We
will show the equivalence between equilibria of the economy with money and bonds that feature a
zero nominal interest rate –liquid traps –and equilibria of the pure currency economy that feature
partial depletion. We will use this equivalence and the analytical results from Section 4 to obtain
insights for the existence of liquidity trap equilibria and for policy. We will also provide examples
where interest-bearing bonds raise welfare relative to the pure currency economy with lump-sum
transfers, and we will compute the optimal policy for di¤erent productivity levels.
In addition to …at money, suppose there is a supply, Bt , of short-term, pure-discount, nominal
bonds that pay one unit of money at the time of maturity that occurs at Poisson arrival rate
> 0.
Bonds that expire are replaced with newly-issued bonds. The supply of bonds is growing at the
same rate
as the money supply to keep the ratio, Bt =Mt , constant and equal to B0 =M0 . The
price of money and bonds (in terms of goods) are denoted by
equilibria in which qt Bt and
t Mt
are constant, i.e., _ t =
t
t
and qt . We focus on stationary
= q_t =qt =
.
In a stationary equilibrium, the expected real rate of return on bonds, denoted by %, is constant
and solves
%qt =
(
t
0
qt ) + q_t ) % =
q0
1
:
(38)
The rate of return of bonds has two components. First, with intensity , this bond matures into
one unit of money, generating a capital gain of
0 =q0
1. Second, the value of the bond, which is
a claim on a unit of money, depreciates at the rate of in‡ation, . If bonds can serve as means of
payment, then money and bonds are perfect substitutes,
rate of return, % =
0
= q0 , and so they generate the same
. In what follows we assume that bonds are not as liquid as money: in
29
the event of a preference shock only …at money can be used to …nance lumpy consumption, e.g.,
because it is the only asset that can be authenticated instantly. However, households are free to
trade bonds in between lumpy consumption opportunities.21 We will see later that the illiquidity
of bonds can also be justi…ed on normative grounds.
In Supplementary Appendix I, we extend the analysis of the households’problem of Section 3
to the present environment. As in Theorem 1, we study the maximum attainable lifetime utility
of a household with ! units of wealth, W (!). We show that it is strictly increasing and strictly
concave, with W 0 (0) < +1 and W 0 (+1) = 0, that it is continuously di¤erentiable over [0; 1),
and that it solves the Hamilton-Jacobi-Bellman equation:
rW (!) = max u(c; h
h) +
c;h;y;z
subject to c
0, 0
h
h, 0
constraint of the government,
y
z
=
[U (y) + W (!
!, and !_ = h
0)
B0 (q0
y)
c + %(!
+ (
0 M0
W (!)] + W 0 (!)!_ ;
z)
z+
(39)
where, by the budget
+ q0 B0 ).22 In any equilibrium, the
nominal interest rate on government bonds is bounded below by zero, % +
0. If the inequality
is strict, then y = z, households do not hold more money than what they intend to spend in case
of a preference shock.
The …rst-order condition with respect to y is
U 0 (y)
W 0 (!
y)
(% + ) W 0 (!);
(40)
with an equality if y < !. The left side is the same as in the pure-currency economy: it is the
expected net utility of consuming a marginal unit of good at the time of a preference shock. The
right side is the expected opportunity cost of holding real balances until the next preference shock.
Following the same reasoning as in Proposition 1, y(!) is strictly positive and increasing with
wealth. Because W 0 (0) < +1, the poorest households hold only money.
From the HJB equation, (39), the marginal value of wealth,
(r + )
t
=
U 0 [y(! t )]
t
t
= W 0 (! t ), solves:
+ _ t:
(41)
Interestingly, (41) is identical to its version in the pure currency economy. Intuitively, the choice
of real balances is interior for all levels of wealth and, as a result, the marginal value of wealth
21
It takes an in…nitesimal amount of time to authenticate bonds, but that delay is large enough to miss an
opportunity to consume. The idea that assets are not acceptable because they lack recognizability has been formalized
in Lester, Postlewaite, and Wright (2012), Li, Rocheteau, and Weill (2012), and Hu (2013). Alternative explanations
for the coexistence of money and interest-bearing bonds include Zhu and Wallace (2007) and Lagos (2013).
22
In the household’s budget constraint we assumed that the portfolio of bonds is fully diversi…ed so that the return,
%(! z), is deterministic.
30
coincides with the marginal value of real balances. If y(!)
! does not bind, then the Envelope
Theorem applied to (39) gives
(r
%) (!) =
[ (!
(!)] + _ :
y)
(42)
The left side is the opportunity cost of wealth measured by the di¤erence between the rate of time
preference and the rate of return on bonds. The …rst term on the right side is the change in the
marginal value of wealth following an opportunity for lumpy consumption. The targeted wealth,
! ? , corresponds to the stationary solution to (42), _ t = 0. Together with (40) it implies
1+
r
%
W 0 (! ? )
W 0 [! ?
y(! ? )] :
(43)
The strict concavity of W (!) implies r > %. Even though bonds are illiquid, in the sense that
they cannot be used to …nance lumpy consumption, they do provide insurance services by allowing
households to replenish their holdings of liquid assets after a preference shock. By market clearing,
the richest households must hold some bonds, so that (43) holds at equality.
From the policy functions one can construct
(!; ! 0 ), the minimal time that it takes for a
household with wealth ! at the time of a preference shock to accumulate strictly more than ! 0 . A
stationary distribution of wealth, F (!), is a solution to
Z 1
Z
Z 1
0
u
Ifu (!;!0 )g dF (!) du =
1 F (! ) =
e
0
0
1
e
(!;! 0 )
dF (!):
(44)
0
By market clearing, bonds have to be held, which implies that the richest households do not
deplete their wealth in full when a preference shock occurs. It is a key di¤erence between equilibria
of the economy with illiquid bonds and equilibria of the pure-currency economy: the former must
feature partial depletion of wealth. Using that from (38) q0 =
0 =(% +
+ ); the market-clearing
conditions for real balances and bonds are:
0 B0
%+
+
0 M0
=
=
Z
1
Z0 1
[!
z(!)] dF (! )
(45)
z(!)dF (!);
(46)
0
where the right sides of (45) and (46) depend on the lump-sum transfer,
, and the real return on
bonds, %. A stationary monetary equilibrium is composed of a value function, W (!), a distribution
of wealth, F (!), a price of money,
0
> 0, and a real interest rate, %, solving (39), (44), (45) and
(46). The following proposition follows directly from market clearing (bonds have to be held) and
the fact that W 0 (0) < +1 (the poorest households want to spend all their wealth).
31
Proposition 12 (Properties of Equilibrium) Any equilibrium features:
(i) Endogenous segmentation: There is a threshold for wealth, !2 (0; ! ? ), below which households hold all their wealth in the form of money, i.e., bonds are held by households with wealth above
!.
(ii) Liquidity premium on bonds: The real interest rate on illiquid bonds is less then the discount
rate.
Our model generates a form of segmentation that is similar to the one in Grossman and Weiss
(1983) and Alvarez, Atkeson, and Kehoe (2002). Only a fraction of households hold and trade
bonds. In contrast to Alvarez, Atkeson, and Kehoe (2002), this segmentation does not require a
…xed cost in participating in the bond market – there is an opportunity cost, of course, since by
participating in the bond market a households holds a less liquid asset. The second result that
bonds command a liquidity premium equal to r
% > 0, is in contrast with LRW where the rate of
return of illiquid bonds is r.23 Indeed, in LRW, illiquid bonds have no insurance value because the
household can reach its targeted real balances instantly. It is through this liquidity premium that
changes in the supply of bonds can a¤ect the real interest rates, %.
We now focus on a subset of equilibria called a liquidity traps where the nominal interest rate
on bonds reaches its lower bound, % +
= 0. In such equilibria money and bonds are perfect
substitutes as savings vehicles.
Proposition 13 (Liquidity-Trap Equilibria: An Equivalence Result.) Consider a purecurrency economy with money growth rate, , and initial money supply, M0 = 1. Denote, fy 1 (z); h1 (z); c1 (z);
F 1 (z)g, a steady-state equilibrium with partial depletion, and let
R1
y 1 (z) dF 1 (z )
0 Rz
> 0:
1 1
1
0 y (z)dF (z )
Then, for all B0
(47)
, there is an equilibrium of the economy with initial money supply, M0 = 1,
and initial bonds supply, B0 , fy 2 (!); h2 (!), c2 (!); z 2 (!);
2
; %2 ; F 2 (!)g, with %2 =
F 1 (!), y 2 (!) = y 1 (!), h2 (!) = h1 (!), c2 (!) = c1 (!), z 2 (!) 2 y 1 (!); ! , and
2
0
=
, F 2 (!) =
1
0 = (1
+ B0 ).
Conversely, any liquidity-trap equilibrium of two-asset economy corresponds to an equilibrium of
the pure-currency economy that features partial depletion.
23
The idea that government bonds can pay a liquidity premium even if they are not used as medium of exchange
provided that they allow agents to reallocate liquidity in the presense of idiosyncratic preference shocks can be found
in Kocherlakota (2003). See also Berentsen, Camera, and Waller (2007), Li and Li (2013), Lagos and Zhang (2015),
and Geromichalos and Herrenbrueck (2016).
32
1
,
Any equilibrium of the pure-currency economy with partial depletion is equivalent in terms of
allocations, aggregate wealth, and welfare to a liquidity-trap equilibrium of the economy with money
and bonds. In order to get some intuition for this equivalence result, note that in pure-currency
economies households accumulate money balances for two motives: y(z) for a transaction motive
and z
y(z) for a precautionary motive. In equilibria that feature partial depletion the second
motive is active, i.e. z y(z) > 0 for some z in the support of the real balance distribution. Suppose
we introduce a small supply of illiquid nominal bonds in such an economy. If the nominal interest
rate on nominal bonds is strictly positive, then households want to ful…ll their precautionary motive
with bonds only. However, if the supply of bond is small enough, i.e. B0 =M0
, the bond market
would not clear. Hence, in equilibrium, the nominal interest rate must fall to zero.
We now use Proposition 13 and results from Section 3.5 to establish conditions for the existence
of liquidity-trap equilibria.
Corollary 2 (Existence of Liquidity-Trap Equilibria.) Consider the economy with linear
preferences and constant money supply. There exists a liquidity-trap equilibrium if and only if
h < h0F , where h0F solves
r
=
Z
z ? =h0F
e
(r+ )
U 0 [z( )]
U 0 (z ? ) d ;
0
and B0 =M0
.
Liquidity-trap equilibria exist when labor productivity is low and bonds are scarce. Indeed,
when h is low, households have a high precautionary demand for assets because the pace of wealth
accumulation is low. If the bond supply is low, the bond yield is driven to zero, so that households
are indi¤erent between holding money and bonds. One can show after some algebra that h0F is
increasing in
, which means liquidity traps occur when the idiosyncratic risk, measured by
, is
high. In contrast, liquidity-trap equilibria do not exist for any bond-money ratio in times of high
productivity and low idiosyncratic uncertainty, in which case % 2 (
; r).24
24
From a theoretical standpoint, this result is consistent with the absence of liquidity-trap equilibria (away from
the Friedman rule) in the LRW model with illiquid bonds. Williamson (2012), Andolfatto and Williamson (2015), and
Rocheteau, Wright, and Xiao (2015) obtain liquidity-trap equilibria in models with degenerate distributions where
bonds are partially acceptable as means of payment and markets are segmented or liquidity is reallocated through
intermediaries. Related to what we do, Guerrieri and Lorenzoni (2015) study liquidity traps in an heterogenousagents, incomplete-market model and characterize transitional dynamics following a one-time aggregate shock. A
treatment of liquidity traps in New-Keynesian models has been provided by Eggertsson and Woodford (2003). Those
models di¤er in fundamental ways from ours: they assume complete …nancial markets, they introduce money in the
utility function, and in the absence of nominal rigidities a liquidity trap corresponds to the Friedman rule.
33
We can now use results from Section 4 to obtain policy recommendations. The condition under
which a liquidity trap exists, namely, a low h, is the same condition for which risk-sharing considerations matter the most and in‡ation is bene…cial. So even though open-market operations,
interpreted as changes in the ratio B0 =M0 , are ine¤ective in liquidity traps, anticipated in‡ation
through a higher growth rate of governments’ liabilities, can raise welfare. According to the example in Figure 3, for very low values of h, welfare can be improved by raising in‡ation to a level
that induces full depletion of real balances; meanwhile, to prevent the nominal interest rate from
increasing, B0 =M0 has to be driven to zero eventually. So while an open-market operation alone is
ine¤ective, a combination of open-market operations and "helicopter drops" is useful.
For intermediate values of h, in‡ation is still bene…cial but the bond-money ratio can stay
positive. In order to illustrate this point we compute the combination of
and i that generates the
highest welfare for the parameter values in Section 4.1. The optimal policy,
implies
0 M0 =q0 B0
= 5:24,
= 1% and i = 4:75%,
= 0:0019, which is about 0:04% of aggregate output. In the absence
of bonds, the optimal policy under lump-sum transfers is
= 0. So the presence of interest-bearing
illiquid bonds is socially bene…cial and it is accompanied by some positive in‡ation rate. If labor
productivity is reduced by half, the optimal policy is
= 2% and i = 5:75%. It is optimal to
increase the rate of growth of nominal assets, which allows to …nance larger transfers and interest
payments on nominal bonds. The money-to-bond ratio is reduced to
0 M0 =q0 B0
= 2:91, but
increases to 0:0043, which is about 0:15% of aggregate output.
6
Other applications
In the following we illustrate additional insights and other tractable cases of our pure currency
economy. We …rst provide an example with quadratic preferences, allowing us to characterize in
closed form the transitional dynamics following a one-time money injection. Second, we assume
general preferences over c and h but linear and stochastic preferences over lumpy consumption, in
order to discuss the e¤ects of in‡ation on households’spending behavior.
6.1
Money in the short run
Suppose now that preferences are quadratic: U (y) = Ay
y 2 =2 and u(c; h h) = "c c2 =2 h2 =2.25
From (12), and assuming interiority, the optimal choices of consumption and labor in a steady
25
Notice that these preferences do not satisfy the Inada conditions imposed earlier. But previous results are not
needed as we are able to solve the equilibrium in closed form.
34
state are are ct = "
t
and ht =
t.
Under full depletion of real balances the stationary solution
?
to the system of ODEs, (19)-(21), is
= "=2 and z ? = A
(1 + r= ) "=2. We assume that
A > (1 + r= ) "=2 to guarantee z ? > 0. Along the saddle path trajectory
(z) =
where
=
r+
q
(r + )2 + 8
2
(z
z?) +
?
;
(48)
=2 < 0. It follows that the household’s policy functions are:
c(z) =
h(z) =
z?)
"
(z
2
" + (z
2
z?)
(49)
:
(50)
As households get richer their marginal value of wealth decreases, their consumption ‡ow increases,
and their supply of labor decreases. The condition for full depletion is A
c(z) is interior for all z if c(0)
z? >
z ? =2 +
?
and
0. It can be shown that the set of parameter values for which these
restrictions hold is non-empty.
The saddle path of (19)-(21) is such that zt = z ? 1
e
t
where t is the length of the time
interval since the last preference shock. Given that t is exponentially distributed the distribution
of real balances is:
z? z
z?
F (z) = 1
for all z
z?:
(51)
In contrast to the model of Section 3.5 the distribution of real balances has no mass point at z ? as
households reach their target asymptotically. Market clearing gives
Z z?
z?:
M=
[1 F (z)] dz =
(52)
0
As before aggregate real balances depend on all preference parameters (r; ",A) but not on M : money
is neutral in the long run.
We now turn to the transitional dynamics following a one-time increase in the money supply,
from M to M , where
> 1. In general, one should take into account that the rate of return
of money, _ = , might vary along the transitional path. Here, however, we guess and verify the
existence of an equilibrium where the value of money adjusts instantly to its new steady-state
value,
= . Along the equilibrium path aggregate real balances, Z =
M , are constant. To
check that our proposed equilibrium is indeed an equilibrium we show that the goods market
clears at any point in time. From (49) and (50) it is easy to check that aggregate consumption
R
R
is C
c(z)dFt (z) +
zdFt (z) = ["
(Z z ? )] =2 + Z while aggregate output is H =
35
R
h(z)dFt (z) = [" + (Z
z ? )] =2. From (52) it follows that C + Z = H, i.e., the goods market
clears. The predictions of the model for aggregate quantities are consistent with the quantity
theory: the price level moves in proportion to the money supply and real quantities are una¤ected.
So, from an aggregate viewpoint, money is neutral in the short run.26
However, money a¤ects the distribution of real balances and consumption levels across house-
holds, which is relevant for welfare under strictly concave preferences. We compute society’s welfare
R
at the time of the money injection as W (z)dF0 (z) where
F0 (z) = F [ z
(
1)Z] :
(53)
According to (53) the measure of households who hold less than z real balances immediately after
the money injection is equal to the measure of households who were holding less than z
just before the shock: they received a lump-sum transfer of size (
scaled down by a factor
1
(
1)Z
1)Z and their real wealth is
due to the increase in the price level. The value function, W (z), being
strictly concave ( (z) is a decreasing function of z), the reduction in the spread of the distribution
leads to an increase in welfare.
6.2
In‡ation and velocity
Suppose now that U (y) = Ay where A is an i.i.d. draw from some distribution
(A).27 We will
use this version of the model to capture the common wisdom according to which households spend
their real balances faster on less valuable commodities as in‡ation increases, thereby generating a
misallocation of resources.28
We conjecture that W (z) is linear with slope . The HJB equation, (8), becomes:
Z
rW (z) = max u(c; h h) +
V (z) + (h c
z+ )
c;h
where V (z)
R
(54)
V (z; A)d (A) with
V (z; A)
max fAy + W (z
0 y z
y)g = max (A
0 y z
26
) y + W (z):
(55)
This result is certainly not general, but it is a useful benchmark suggesting that the e¤ects of a one-time money
injection on aggregate real quantities will crucially depend on preferences that determine the relationship between
labor supply decisions and wealth. In Rocheteau, Weill, and Wong (2015) we study transitional dynamic following onetime money injections in a discrete-time version of our model with search and bargaining and quasi-linear preferences.
We show that the money injection a¤ects the rate of return of money, aggregate real balances, and output levels.
27
For a signi…cant extension of our model with linear utility for lumpy consumption, see Herrenbrueck (2014). The
model is extended to account for quantitative easing and the liquidity channel of monetary policy.
28
This wisdom has proved di¢ cult to formalize in models with degenerate distributions. See Lagos and Rocheteau
(2005); Ennis (2009); Liu, Wang, and Wright (2011), and Nosal (2011) for several attempts to generate the ‘hot
potato’e¤ect in this class of models.
36
From (55) the household spends all his real balances whenever A > . Di¤erentiating (54) and
RA
using that V 0 (z)
=
(A
)d (A), solves:
"Z
#
Z
A
A
(r + )
=
(A
[1
) d (A) =
(A)] dA:
(56)
Equation (56) has the interpretation of an optimal stopping rule. According to the left side of
(56), by spending its real balances the household saves the opportunity cost of holding money, as
measured by r + . According to the middle term in (56), if the household does not spend its real
balances, then it must wait for the next preference shock with A
Poisson arrival rate
[1
real balances is E [ A
. Such a shock occurs with
( )], in which case the expected surplus from spending one unit of
RA
(A
) d (A)= [1
( )]. Finally, the right side of (56) is
jA
]=
obtained by integration by parts. It is straightforward to check that there is a unique,
?
, solution
to (56), and this solution is independent of the household’s real balances as initially guessed. As
in‡ation increases
?
decreases and, in accordance with the "hot potato" e¤ect, households spend
their money holdings on goods for which they have a lower marginal utility of consumption. Given
?
(12) describes the ‡ow of consumption, c? , and hours, h? .
The real balances of a household who depleted its money holdings t periods ago are zt =
(h?
c? +
M ) (1
shock with A
?
e
t )=
. The probability that a household does not receive a preference
[1
over a time interval of length t is e
F (z) = 1
h?
[1
c? + ( M z)
h? c? + M
(
?
)]t .
( )]
for all z
Consequently,
h?
c? +
M
:
(57)
By market clearing, (18),
h?
[1
M=
c?
:
( ? )]
(58)
Aggregate real balances fall with in‡ation: because households save less, h? c? is lower, and because
they spend their real balances more rapidly,
[1
(
?
)] increases. The velocity of money, denoted
V, is de…ned as nominal aggregate output divided by the stock of money. From (58),
V
h?
=
M
[1
1
(
c?
h?
?
)]
:
(59)
The velocity of money increases with in‡ation for two reasons: households spend their real balances
more often following preference shocks, 1
decreases. A monetary equilibrium exists if h?
(
?
) increases, and the saving rate, (h?
c? )=h? ,
c? > 0, which holds if the in‡ation rate is not too
large and the preference shocks are su¢ ciently frequent.
37
Finally, if preferences over ‡ow consumption and leisure are also linear, then all households
RA
supply h provided that <
(A)] dA r. So in‡ation has no e¤ect on aggregate output.
1 [1
Welfare at a steady-state monetary equilibrium is
W=
It is increasing with
?
Z Z
AzdF (z)d (A)
h=h
"R A
Ad (A)
1
( ?)
?
#
1 :
and hence decreasing with . As in‡ation increases output is consumed by
households with lower marginal utilities, which reduces social welfare.
7
Conclusion
We constructed a continuous-time, pure-currency economy in which households are subject to idiosyncratic preference shocks for lumpy consumption. We o¤ered a complete characterization of
steady-state equilibria for general preferences. We provided closed-form solutions for a class of
equilibria where households fully deplete their money holdings periodically and for special classes
of preferences. We studied both analytically and numerically a version of our economy with quasilinear preferences resembling the New-Monetarist framework of Lagos and Wright (2005) and Rocheteau and Wright (2005). The equilibrium of this economy features a non-degenerate distribution
of real balances and a trade-o¤ for policy between self-insurance and risk sharing parameterized by
labor productivity. We derived a number of analytical results on this policy trade-o¤.
We studied incentive-compatible transfers …nanced with money creation and designed such
transfers to raise welfare by optimally trading o¤ risk-sharing and self-insurance. We showed that
the shape of the optimal transfers depends on labor productivity. We extended our analysis to
monetary policy implemented with both money growth and illiquid nominal bonds. We used our
results for pure currency economies to establish that liquidity traps occur when labor productivity
is low and idiosyncratic risk is high. Money growth through "helicopter drops" accompanied by
open-market operations to reduce the bond-money ratio are welfare enhancing.
Our model can easily be extended in several directions. It can incorporate search and bargaining
in order to feature a non degenerate distribution of prices, as shown in Rocheteau, Weill, and Wong
(2015). We adopted a discrete-time version of this model to study transitional dynamics following
monetary shocks. The model remains highly tractable and delivers new insights for the short-run
e¤ects of money. One can incorporate idiosyncratic employment risk, e.g., by adding a frictional
labor market, and private assets, such as capital and claims on …rms’pro…ts. The model will have
38
implications for how the distribution of liquidity a¤ects …rms’ entry, employment, and interest
rates.
39
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44
Appendix A: Proofs of Propositions
PROOF OF THEOREM 1. The Theorem summarizes a series of results from the Supplementary Appendix, proved in a more general case in which there are two assets, money and an illiquid
bond. Lemma I.3 shows that the Bellman equation has a unique bounded solution, and that this
solution is concave, continuous, and increasing. Lemma I.5 shows that the value function is strictly
increasing. Proposition I.7 shows that the value function is a viscosity solution of the HJB equation. Proposition I.13 uses this and other results to show that the value function is continuously
di¤erentiable. Proposition I.15 shows that the derivative of the value function is strictly decreasing.
This implies that the value function is strictly concave, that it is twice continuously di¤erentiable
almost everywhere, and that it is a classical solution of the HJB equation, i.e., that it satis…es (8).
Proposition I.17 shows that the value function is twice continuously di¤erentiable in a neighborhood of any z > 0, except perhaps if saving rate is zero, and under linear preferences if W 0 (z) = 1.
Since there is a unique level of real balances such that the saving rate is zero (see Proposition
2), and since W 0 (z) is strictly decreasing, this means that the value function is twice continuously
di¤erentiable except for two levels of real balances. Under SI preferences, Proposition III.4 shows
that the value function is twice continuously di¤erentiable even when the saving rate is zero. Under
linear preferences, Lemma V.6 shows that, in equilibrium, the value function is twice continuously
di¤erentiable over the support of the distribution of real balances.
To derive the bound on W 0 (0), consider some small " > 0. By working full time, ht = h and
consuming nothing, the household can reach " at time T" solving zT" = ", where z_t = h +
zt .
Solving this ODE explicitly gives:
T" =
1
log 1
"
h+
=
"
h+
+ o("):
Since utility ‡ows are bounded below by zero, we must have that W (0)
e
(r+ )T" W ("),
which
implies in turn that:
0
W (")
W (0)
1
e
(r+ )T"
W ("):
Dividing both side by " and taking the limit " ! 0, we obtain that:
W 0 (0)
r+
h+
W (0):
By taking the sup norm on both sides of (4) we obtain that rkW k
kuk + kU k, and the result
follows. Finally, the result that limz!1 W 0 (z) = 0 follows from the fact that W (z) is concave and
45
bounded, while the result that limz!0 W 00 (z) = 1 is proved in Corollary I.20 of the Supplementary
Appendix.
PROOF OF PROPOSITION 1.
The results follow directly because U (y) and W (z) are
both strictly concave and continuously di¤erentiable, because U 0 (0) = 1 while W 0 (0) < 1, and
because U 0 (1) = W 0 (1) = 0.
PROOF OF PROPOSITION 2. Note that the saving rate correspondence can be written as
s(z) = h
c
z + , where (h; c) 2 X( ) and X( ) = arg max u(c; h
with respect to c
[0; c] [0; h] if
0, 0
h
h) + (h
h. With linear preferences, X( ) = (c; 0) if
= 1 and X( ) = (0; h) if
c
z+
) ,
< 1, X( ) =
> 1. With SI preferences, one can easily check that X( )
is singled-valued and continuous, that the optimal consumption choice, c( ), is strictly decreasing,
and that the optimal labor choice, h( ), is increasing (see Lemma I.11 in the Supplementary
Appendix for details). Combined with the fact, established in Theorem 1, that W 0 (z) is strictly
decreasing and continuous, all the statements of the Lemma follow except for s(z) > 0.
To establish that s(z) > 0 near zero, recall that the value function is twice di¤erentiable almost
everywhere. Consider any z > 0 such that W 00 (z) exists. Then, we can apply the envelope condition
to the right side of the HJB equation (see Theorem 1 in Milgrom and Segal, 2002). We obtain that:
(r +
+ )W 0 (z) = V 0 (z) + W 00 (z)s(z):
From Proposition 1 we have that V 0 (z) = U 0 [y(z)]. Since y(z)
(60)
z and limz!0 U 0 (z) = 1, it follows
that limz!0 V 0 (z) = 1. From W 0 (0) < 1 and (60), limz!0 W 00 (z)s(z) =
1. Since W 00 (z)
0, it
then follows that s(z) > 0 for some z close enough to zero. Since s(z) is decreasing, it follows that
s(z) > 0 for all z close enough to zero.
PROOF OF PROPOSITION 3.
The proof is based on results from two sections of the
Supplementary Appendix: Section III.3, which studies the initial value problem in the case of SI
preferences, and Section IV.2, which explicitly solves for the solution to this problem in the case of
linear preferences.
PROOF OF PROPOSITION 4. See Section V.1 of the Supplementary Appendix for the
detailed application of Theorem 12.12 and 12.13 in Stokey, Lucas, and Prescott (1989).
Lemma 1 At the target level of real balances, z ? :
W 0 (z ? ) =
r+
46
+
V 0 (z ? ):
First, we note that the stationary distribution cannot
PROOF OF PROPOSITION 5.
be concentrated at z = 0, since Q(z; f0g) = 0 for all z. Hence, when
= 0, the left-hand side of
(18) is zero and so is less than the right-hand side, which is strictly positive. When
! 1, we
have from the upper bound of Theorem 1 that W 0 (z) ! 0 for all z 2 [0; 1). This implies that
labor supply is zero and consumption is strictly positive for all z 2 [0; z ? ], hence the saving rate
is s(z) <
z+
. Plugging s (z ? ) = 0, it follows that z
z? <
=
for all real balances z in
the support of the stationary distribution, [0; z ? ], implying that the right-hand side of (18 ) is less
than the left-hand side. Finally, note that (18) is continuous in
stationary distribution is continuous in
because, by Proposition 4, the
in the sense of weak convergence. The result then follows
by an application of the intermediate value theorem.
Note that, at z ? , there exists some optimal consumption and
PROOF OF LEMMA 1.
labor choices, (c? ; h? ) such that h?
(r + )W (z)
c?
u(c? ; h
(r + )W (z ? ) = u(c? ; h
z? +
= 0. Hence, from the HJB:
h? ) + V (z) + W 0 (z) (h?
c?
z+
)
h? ) + V (z ? );
where the inequality in the …rst equation follows because we evaluate the right side of the HJB at
a point that may not achieve the maximum. Taking the di¤erence between these two equations,
and recalling that h(z ? )
c(z ? )
(r + ) [W (z)
z? +
= 0, we obtain that:
W (z ? )]
The result follows by dividing both sides by z
[V (z)
V (z ? )]
W 0 (z) (z
z?)
z ? , for z > z ? and then for z < z ? , and taking the
limit as z ! z ? , keeping in mind that V (z) is di¤erentiable at z ? and that W (z) is continuously
di¤erentiable.
Lemma 2 Under either SI or linear preferences, for z 2 [0; z ? ]:
Z z?
0
W (z) =
V 0 (x)dG(x j z); where G(x j z) 1
r+ +
z
PROOF OF LEMMA 2.
e
(r+ + )[T (x+ ) T (z)]
First, recall from Theorem 1 that the value function is twice
continuously di¤erentiable over (0; 1), except perhaps under linear preferences, when this property
may not hold for at most two points. Hence, we can take derivatives on the right side of the HJB
equation along the path of real balances zt , except perhaps at two points. Applying the envelope
condition, we obtain that:
(r +
+ )W 0 (zt ) = V 0 (zt ) + W 00 (zt ) z_t ;
47
if zt < z ? , except perhaps at two points. At z = z ? , Lemma 1 shows that
+ )W 0 (z ? ) = V 0 (z ? ):
(r +
In all cases we can integrate this formula forward over the time interval [t; T (z ? )] and we obtain
that:
0
W (zt ) =
Z
T (z ? )
V 0 (zs )e
(r+ + )(s t)
ds + e
(r+ + )[T (z ? ) t]
t
V 0 (z ? )
:
r+ +
(61)
Consider the integral on the right side of (61). The inverse of zt when restricted to the time interval
[0; T (z ? )] is the strictly increasing function T (x), the time to reach the real balances x starting
from time zero. Let M (x)
1
e
(r+ + )[T (x) t]
and note that M
z(s) = 1
e
(r+ + )(s t) .
With these notations, the …rst integral can be written:
Z T (z ? )
Z T (z ? )
V 0 zs d [M zs ]
V 0 (zs ) e (r+ + )(s t) ds =
r
+
+
t
Z
Z t
=
V 0 (x) dM (x) =
V 0 (x) dG(x j z):
r+ +
r+ +
x2[z;z ? ]
x2[z;z ? )
where the second equality follows by an application of the change of variable formula for LebesgueStieltjes integral (see Carter and van Brunt, 2000, Theorem 6.2.1), and the second line follows
because G(x j z) = M (x) for all x 2 [z; z ? ). The result follows by noting that the second integral
can be written:
V 0 (z ? )
r+ +
G(z ? j z)
G(z ? j z) .
PROOF OF PROPOSITION 6.
that, at the target z ? , h?
c?
To establish the …rst point of the Proposition, we note
z? +
= 0, where (c? ; h? ) are optimal consumption and labor
R z?
?
?
choices when z = z ? . Since, in equilibrium, =
c? > 0.
0 zdF (z) < z , we obtain that h
This implies that the marginal value of real balances satis…es W 0 (z ? )
is independent of the rate of in‡ation, . With linear preferences,
solves h( )
= 1. With SI preferences,
c( ) = 0. Next, we use Lemma 1:
(r +
Since W 0 (z ? )
> 0, where the constant
+ )W 0 (z ? ) = U 0 [y(z ? )] :
, this implies that lim
Theorem 1 that W 0 (0)
(r + )=h
!1 y(z
?)
= 0. Finally, since we have established in
(kuk + kU k)=r, we obtain that W 0 (0) < U 0 [y(z ? )] for
large enough. Therefore, the solution of the optimal lumpy consumption problem is y(z ? ) = z ? ,
i.e., there is full depletion. We conclude that lim
!1 z
?
= lim
!1 y(z
?)
= 0.
The second part of the Proposition, which deals with linear preference, requires some notations
and results from Section 3.5. The proof can be found at the beginning of the proof of Proposition
8, in the paragraph "(i) Large labor endowments".
48
PROOF OF PROPOSITION 7. From (27) limh!1 F (z) = 0 for all z < z ? and F (z) = 1
for all z
z ? . From (28) limh!1 M = z ? . Finally, from (29) we compute the value function in
closed form:
W (z) = z + W (z ? )
(
W (z ? ) =
r
z?
U (z ? )
z?
r+
r+
From (62) limh!1 W (z) = z + W (z ? )
Z
Z
z?
z
z?
0
h
h
1
e
1
e
(r+ )(u z)
h
(r+ )u
h
i
i
U 0 (u)
U 0 (u)
U 0 (z ? ) du 8z < z ? (62)
)
U 0 (z ? ) du :
z ? and from (63) limh!1 W (z ? ) =
(63)
[U (z ? )
z ? ] =r.
PROOF OF PROPOSITION 8. Part (i): Large labor endowment. Fix some
We …rst note that y(z ? )
z?
zs , hence equilibrium aggregate demand is bounded by
0.
zs
independently of h. Equilibrium aggregate supply can be written:
F (z ? )h + 1
F (z ? ) h? :
To remain bounded as h ! 1, it must be the case that limh!1 F (z ? ) = 0. This also implies that,
for h large enough, there is an atom at z ? , so that W 0 (z ? ) = 1 and z ? = zs . Because F converges
to a Dirac distribution concentrated at z ? = zs , we have that limh!1 M = zs .
Next we argue that, as h is large enough, y(z ? ) = y(zs ) = zs , i.e., all equilibria must feature
full depletion. For this we use the expression for W 0 (z) derived in Lemma 2:
Z z?
Z z?
0
0
U [y(z)] dG(z j 0)
maxfU 0 z); W 0 (0) dG(z j 0)
W (0) =
r+ +
r+ +
0
0
"
#
Z
dG(z
j
0)
G(zs j 0)
max U 0 (z); W 0 (0)
+ 1 G(zs j 0) maxfU 0 (zs ); W 0 (0)g ;
r+ +
j
0)
G(z
s
z2[0;zs )
as long as h is large enough. To obtain the inequality of the …rst line, we have used that U 0 [y(z)] =
U 0 (z) if there is full depletion, while U 0 [y(z)] = W 0 [z
y(z)]
W 0 (0) if there is partial depletion.
To obtain the second line, we have used that z ? = zs as long as h is large enough. Substituting the
expression for T (z j
M ) into the de…nition of G(z j 0), we obtain that:
8
1+ r+
<
z
1
1
if z < zs
zb
G(z j 0) =
: 1
if z = zs :
Given that zb goes to in…nity as h goes to in…nity, one sees that G(z j 0) converges weakly to a
Dirac distribution concentrated at zs . We also have:
1+
G0 (z j 0)
G(zs j 0)
=
1+
1
zb
r+
1
1
1
z
zb
zs
zb
49
1+
1+
r+
1+
1
1
zs
zb
1
zb
1+ 1+
!
1
;
zs
as h goes to in…nity (since it implies zb ! 1). Thus, the conditional probability distribution,
G(z j 0)=G(zs j 0), has a density that can be bounded uniformly in h. Finally, our bound for W 0 (0)
in Theorem 1 can be written, in the case of linear preferences, as
r+
h
W 0 (0)
kU k
h+c
+
r
r
!1+
r
;
as h ! 1. Taken together, these observations imply that:
Z
dG(z j 0)
max U (z); W (0)
G(zs j 0)
z2[0;zs )
0
1
zs
0
Z
zs
0
n
max U 0 (z); 1 +
r
o
dz + "
for some " > 0 as long as h is large enough (note that the integral on the right side is well de…ned
Rz
since U (z) = 0 U 0 (z) dx). Together with the fact that G(zs j 0) ! 0 as h ! 1, we obtain that:
G(zs j 0)
Z
z2[0;zs )
as h ! 1. Hence, for any " > 0, W 0 (0)
enough. Picking " <
dG(z j 0)
! 0 and 1
G(zs j 0)
max U 0 (z); W 0 (0)
r+
r+ +
r+ +
G(zs j 0) ! 1
maxfU 0 (zs ); W 0 (0)g + " as long as h is large
U 0 (zs ), we obtain that W 0 (0) < maxfW 0 (0); U 0 (zs )g, which implies that
W 0 (0) < U 0 (zs ), for h large enough, i.e., there is full depletion.
Because H =
M under full depletion, and because the distribution of real balances converges
towards a Dirac distribution concentrated at zs , we obtain that limh!1 H = H 1 ( ) = zs , which
R
is decreasing in . Aggregate welfare can be written as
U (z) dF (z) H, the average utility
enjoyed from lumpy consumption net of the average disutility of supplying labor. As h ! 1, F
converges weakly to a Dirac distribution concentrated at zs , and H converges to
that welfare converges to W 1 =
[U (zs )
zs . It follows
zs ], which is decreasing with .
Part (ii): Small labor endowment. We have shown that there exists a unique candidate
equilibrium with full depletion. In this candidate equilibrium, the condition for binding labor is
that zs
zb or, using the de…nition of zs :
U 0 (zb )
Recall that zb =
h
+
h
1+
r+
:
is an increasing function of h. Since marginal utility is decreasing, the
condition for binding labor can be written:
h 2 0; H( ) where H( ) =
One immediately sees that lim
!0 H(
) = lim
+
!1 H(
50
U0
) = 0.
1
1+
r+
:
Next, we turn to the su¢ cient condition for full depletion. Using Lemma 2 we have, in the
candidate equilibrium with full depletion:
0
W (0) =
r+
+
Z
zb
0
U (z)dG(z) where G(z) = 1
0
1
z
zb
1+ r+
:
Substituting the expression for G(z) in the integral, we obtain:
0
W (0) =
1
zb
Z
zb
U (z) 1
0
where the inequality follows by using (1
U (0) = 0. Full depletion obtains if W 0 (0)
z=zb )
r+
z
zb
0
dz
r+
U (zb )
;
zb
1, integrating, and keeping in mind that
U 0 (zb ). Using the above upper bound for W 0 (0), we
obtain that a su¢ cient condition for full depletion is:
U (zb )
:
zb U 0 (zb )
Note that zb
(0; (U 0 )
1
(U 0 )
1
(1), that the function z 7! [U (z)
U (0)] = [zU 0 (z)] is continuous over
(1)] and, by our maintained assumption in the Lemma, bounded near zero. Hence, it
is bounded over the closed interval [0; (U 0 )
1
(1)]. Therefore, the condition for full depletion is
satis…ed if:
sup
z2[0;(U 0 )
1
U (z)
:
0 (z)
zU
(1)]
Output e¤ect of in‡ation. In the regime with binding labor, h(z) = h for all z 2supp(F ).
h
i
^ and for all 2 [ ; ], H = h.
Hence, for all h 2 0; h
Welfare e¤ect of in‡ation. From (34) in the regime with binding labor, M = h= . Hence,
an increase in the money growth rate through lump-sum transfers is a mean-preserving reduction
in the distribution of real balances. In this regime social welfare is measured by
Z
Z
W = [ h(z) + U (z)] dF (z) = h +
U (z)dF (z):
Given the strict concavity of U (y) money growth leads to an increase in welfare.
! 1, z ? ! 0, M ! 0, H ! 0, and W ! 0.
Part (iii): Large in‡ation. From (31), as
PROOF OF PROPOSITION 9.
The proof is structured as follows. Given a policy, ( ; ), we conjecture that households behave
as follows: y(z) = z for all z 2 [0; z0? ]; h(z) = h for all z < z0? , and h(z0? ) = 0. We also assume that
parameters are such that h + (z)
z > 0 for all z 2 [0; z0? ). Given this conjecture we will show
51
that: (i) Aggregate real balances under
(ii) Welfare under
are larger than under laissez faire (
0
=
1
=
z
= 0).
is larger than under laissez-faire. The second part of the proof will consist in
checking that: (iii) For
small enough, there is a transfer scheme, , of the form described in (35),
that balances the government budget; (iv) Households’conjectured behavior is optimal.
Guessing that the equilibrium features full depletion, and keeping in mind that (z0? ) = z0? by
construction, the government budget constraint under the transfer scheme, , is:
Z
Z T (z ? ; )
0
f [z(t)]
z(t)g e t dt = 0;
[ (z)
z] dF (z) =
(64)
0
where T (z0? ; ) is the time to accumulate z0? under the transfer scheme
and z(t) is the solution to
z for all z < z0? :
z_ = h + (z)
(65)
= 0 if z = z0? :
R
We denote Z
[1 F (z)] dz the aggregate real balances under the transfer scheme, , and Z0
R
[1 F0 (z)] dz the aggregate real balances under laissez faire. Moreover, denote T
T (z0? ; ) and
T0 = T (z0? ; 0) under laissez-faire.
RESULT #1: T > T0 and Z > Z0 :
PROOF: By construction the transfer scheme in (35) is such that there is a level of real balances,
zt^, with t^ 2 (0; T ), below which the net transfer to the household is positive, since
0
> 0, and
above which the net transfer is negative, since from (64) the sum of those transfers must be 0:
(zt )
zt > 0 for all t 2 0; t^
(zt )
zt < 0 for all t 2 t^; T
:
Dividing the government budget constraint by e t^, (64) becomes:
Z t^
Z T
e
e t
dt
+
[ (zt )
zt ]
[ (zt )
zt ]
^
t
^
e
e
t
0
Given t^, e
t=
e
t^
is decreasing in t, e
t=
e
t^ >
t
t^
dt = 0:
1 for all t < t^ and e
(66)
t=
e
t^ <
1 for all
t > t^. It follows that
Z
Z
t^
0
T
t^
[ (zt )
[ (zt )
e
zt ]
e
e
zt ]
e
t
t^
dt >
Z
t^
[ (zt )
zt ] dt
(67)
0
t
dt <
^
t
Z
t^
T
[ (zt )
From (66) and the two inequalities, (67)-(68),
Z T
Z T
e t
[ (zt )
zt ]
dt
=
0
>
[ (zt )
e t^
0
0
52
zt ] dt:
(68)
zt ] dt:
(69)
From (65) and (69),
Z
T
[ (zt )
zt ] dt =
0
Z
T
h dt = z0?
z_t
0
hT < 0;
where we used that z0 = 0 and zT = z0? . So T > T0 = z0? =h. As a result the measure of households
holding their targeted real balances is
F (z0? ) = e
1
T
z0? =h
<e
F0 (z0? ):
=1
The law of motion for aggregate real balances is Z_ = F (z0? )h
Z . At a steady state, Z =
F (z0? )h= , which is larger than Z0 = F0 (z0? )h= under laissez faire.
Social welfare is measured by the sum of utilities across households:
W =
Z
Z
[ h(z) + U (z)] dF (z) =
[ z + U (z)] dF (z);
(70)
R
R
where the second equality is obtained by market clearing, h(z)dF (z) =
y(z)dF (z) =
R
Rz 0
zdF (z). Using that U (z) z = 0 [U (x) 1] dx + U (0) (70) can be rewritten as
Z Z
W =
U 0 (x)
1 If0
x zg dxdF
(z) + U (0):
(71)
Changing the order of integration,
Z Z
0
U (x)
1 If0
x zg dxdF
(z) =
=
Z Z
Z
[1
x zg dF
If0
(z) U 0 (x)
F (z)] U 0 (x)
1 dx
1 dx:
(72)
Plugging (72) into (71):
W =
Z
F (x)] U 0 (x)
[1
RESULT #2: Social welfare under
PROOF: The welfare gain under
W
W0 =
=
Z
Z
[1
(73)
is higher than welfare at the laissez-faire.
relative to laissez faire is:
F (x)] U 0 (x)
[F0 (x)
1 dx + U (0):
1 dx
F (x)] U 0 (x)
Z
[1
F0 (x)] U 0 (x)
1 dx:
(74)
Given our conjecture that the equilibrium features full depletion we have:
F (z ;t ) = F0 (z0;t ) = 1
53
e
1 dx
t
;
where z
and z0;t denote the real balances of a household who received its last preference shock t
;t
periods ago under the transfer scheme
and under laissez-faire, respectively. Integrating the law
of motion of real balances, (65):
z0;t = ht for all t < z0? =h
Z t
z ;t = ht +
[ (z ;x )
z
;x ] dx
for all t
0
T :
By de…nition of the transfer scheme,
and, from (64),
z
;t~ =
RT
[ (z
0
;x )
(zt )
zt > 0 for all t 2 0; t^
(zt )
zt < 0 for all t 2 t^; T
z
< 0. It follows that there is a t~ 2 t^; T0 such that z0;t~ =
;x ] dx
zs . For all t 2 0; t~ , z0;t < z ;t . For all t 2 t~; T
;
and z0;t > z ;t . Equivalently, F (z) < F0 (z)
for all z < zs and F (z) > F0 (z) for all z > zs . From (74):
(Z
Z z?
zs
0
[F0 (x) F (x)] U 0 (x) 1 dx +
[F0 (x)
W
W0 =
0
F (x)] U (x)
)
1 dx : (75)
z~
0
By the de…nition of zs and the fact that U 0 (z) is decreasing:
[F0 (x)
F (x)] U 0 (x)
1
> [F0 (x)
F (x)] U 0 (~
x)
1
for all x 2 (0; zs )
[F0 (x)
F (x)] U 0 (x)
1
> [F0 (x)
F (x)] U 0 (~
x)
1
for all x 2 (zs ; z0? ) :
Plugging these two inequalities into (75):
W
0
W0
U (zs )
1
Z
z0?
[F0 (x)
F (x)] dx:
(76)
0
We proved that
Hence,
R z?
0
0
Z
[F0 (x)
zdF (z) =
Z
[1
Z
F (z)] dz
zdF0 (z) =
Z
[1
F0 (z)] dz:
U 0 (z0? ) = 1 + r= . Hence, U 0 (^
z)
F (x)] dx > 0. Moreover, U 0 (zs )
1 > 0. It
follows from (76) that W > W0 .
The transfer scheme, , is fully characterized by
1
= ( z?
?
?
0 ) z0 = (z0
lump-sum component,
RESULT #3: For
and
0
since
z
= ( z0?
z ? ). We now establish that for a given in‡ation rate,
0,
?
0 ) = (z0
z ? ) and
, there exists a
that balances the government budget.
su¢ ciently small, there is a
holds and z_t > 0 for all t 2 [0; T (z0? )).
54
0
2 (0; z ? ) such that
R
[ (z)
z] dF (z) = 0
PROOF: The government budget constraint, (64), can be re-expressed as
Z
( 0)
T (z0? ;
0)
z(t)g e t dt = 0;
f [z(t)]
0
By direct integration of the ODE for real balances, (65), one obtains that both z(t) and T (z0? ;
are continuous functions of
0.
is also continuous in
0
0.
If
Since (z) is, by construction, continuous in z, we obtain that ( 0 )
= 0, then from (35):
(
z
(z)
z=
z?
z ? z ? (z
0
Hence,
z0? )
z z?
:
z 2 (z ? ; z0? ]
if
zt < 0 for all t 2 (0; T (z0? )) and (0) < 0. If
(zt )
0
= z ? , then from (35):
z?
z z?
if
z
z 2 (z ? ; z0? ]
(z) =
z(t) > 0 for all t < T (z ? ) and
Consequently, [z(t)]
0)
(zt )
( z ? ) > 0. By the Intermediate Value Theorem there is
0
T ? . Hence,
zt = 0 for all t
2 (0; z ? ) such that
( 0 ) = 0.
Finally, for the transfer scheme to be feasible, it must be that z_ > 0 for all z < z0? . This requires
h+
z ? > 0, since net transfers achieve their minimum at z = z ? . This condition will be
0
satis…ed for
su¢ ciently small.
Finally, we need to check that household’s conjectured behavior is optimal: households …nd it
optimal to supply h units of labor until they reach z0? and to deplete their money holdings in full
when a preference shock occurs. The ODE for the marginal value of money is:
(r + )
RESULT #4: For
t<T ,
T
= 1, and
and
t
0
t
=
U 0 (zt )
t
+
t
0
(zt ) + _ t :
(77)
su¢ ciently small the solution to (77) is such that:
t
> 1 for all
U 0 (z0? ) for all t 2 [0; T ].
PROOF: Integrating (77), we obtain that the marginal value of money solves:
t
=1+
Z
T (z ? )
e
(r+ + )(s t)
U 0 (zs )
U 0 (z ? ) ds + e
(r+ + )[T (z ? ) t]
t
for all t
T ? , and
t
=1+
Z
T
e
(r+ +
z )(s
t)
t
for all t
T (z ? ), where we used that
T
U 0 (zs )
U 0 (z0? ) +
= 1 and T (z ? ) =
t 2 (T ? ; T ), zt < z0? and hence U 0 (zt ) > U 0 (z0? ). Given that
55
z
1
ln 1
z
>
T (z ? )
1 ;
ds;
z ? =(h +
(78)
(79)
0)
. For all
it follows from (79) that
t
> 1 for all t 2 (T ? ; T ). Similarly, for all t < T ? , zt < z ? and hence U 0 (zt ) > U 0 (z ? ). Given that
> 1, it follows from (78) that
T (z ? )
t
T (z ? ).
> 1 for all t
For the second part of the Lemma we note that, when
for all t 2 [0; T (z0? )]. Since
=
0
is continuous with respect to (t; ;
= 0, we obtain by uniform continuity that
Note that, by Result #3, the
hence it goes to zero as
=
0
= 0, we have that
0)
< U 0 (z0? ) for ( ;
t
t
< U 0 (z0? )
and since T (z0? ) is …nite at
0)
su¢ ciently small.
balancing the government budget constraint is less than z ? ,
goes to zero. Hence, when
is su¢ ciently small, the solution to the
ODE (77) satis…es all the properties of Result #4. This allows us to construct a candidate value
function for all z 2 [0; z0? ]. Namely, we let (z)
T (z) , where
Z z
(x) dx where rW (0)
W (z) W (0) +
t
is the solution to the ODE (77):
(0) h +
0
:
0
z0? :
Next, we construct a candidate value function for z
RESULT #5: For
and
0
su¢ ciently small, there exists a continuously di¤ erentiable and
bounded function, W (z), and two absolutely continuous functions, V (z) and (z), such that: For
z
z0? , W (z), V (z) and
(z) are the functions constructed following Result #4 ; For z
Z z
W (z) = W (z0 ) +
(x) dx
z0? :
(80)
z0
V (z) =
max U (y) + W (z
y2[0;z]
0
y)
c 0 (z) almost everywhere
(r + ) (z) = V (z)
(z) 2 [0; 1]:
(81)
(82)
(83)
Proof. We construct a solution to the problem (80)-(83) as follows. Suppose that we have
constructed a solution over some interval [z0? ; Z], where Z
U 0 (z0? ) = 1 +
r
sup
z0? . We …rst observe that:
(x) = sup
x2[0;z0? ]
(x);
(84)
x2[0;Z]
where the …rst equality and the …rst inequality follow from our construction of W (z) and (z) over
[0; z0? ], and the last equality follows because (z)
1 for z 2 [z0? ; Z]. We now show how to extend
this solution over the interval [Z; Z + z0? ]. First, we let:
V~ (z)
max U (y) + W (z
y2[z Z;z]
y);
which is well de…ned for all z 2 [Z; Z + z0? ], given that we have constructed W (z) for all z
since z
y
(85)
Z and
Z by the choice of our constraint set. Note that, in principle, the function V~ (z) di¤ers
56
from V (z) because it imposes the constraint that y
z
Z. Our goal is to show that, nevertheless,
V~ (z) = V (z). Precisely, if one extends (z) over [Z; Z + z0? ] using (82), and de…ne W (z) using (80),
then the household never …nds it optimal to choose y < z0? , implying that the additional constraint
we imposed to de…ne V~ (z) is not binding.
We …rst establish that V~ (z) is absolutely continuous and V~ 0 (z)
U 0 (z0? ). Consider …rst z 2
[Z; Z +z0? =2]. Given (84), it follows that the solution to (85) must be greater than z0? . By implication
since z
Z
z0? =2, the solution y to (85) must be greater than z
and after making the change of variable x = z
V~ (z)
Z + z0? =2. Given this observation
y, we obtain that
max
x2[0;Z z0? =2]
U (z
x) + W (x):
The objective is continuously di¤erentiable with respect to z, and its partial derivative is U 0 (z x)
U 0 (z0? =2) given that z
Z +z0? =2 and x
Z z0? =2. Proceeding to the interval z 2 [Z +z0? =2; Z +z0? ],
we make the change of variable x = z y in (85) and obtain that V~ (z) = maxx2[0Z] U (z x)+W (x).
Again, the objective is continuously di¤erentiable with a partial derivative with respect to z equal to
U 0 (z
x)
Z + z0? =2 and x
U 0 (z0? =2), since z
Z. Hence, in both cases, given that the objective
has a bounded partial derivative with respect to z, we can apply Theorem 2 in Milgrom and Segal
(2002): V~ (z) is absolutely continuous and the envelope condition holds, i.e., V~ 0 (z) = U 0 [y(z)]
whenever this derivative exists. By condition (84), it follows that y(z)
z0? , hence V~ 0 (z)
U 0 (z0? ),
as claimed.
Next, we construct a solution over [Z; Z + z0? ]. Given that the function V~ (z) constructed above
is absolutely continuous, we can integrate the ODE (82) with V~ 0 (z) and we obtain a candidate
solution:
~ (z) = (Z)e
Given that (Z)
1
1 and V~ 0 (x)
r+
c
(z Z)
+
c
Z
z
V~ 0 (x)e
r+
c
(z x)
dx
Z
U 0 (z0? ) = 1 + r= , one sees after direct integration that ~ (z)
U 0 (z0? ) for all z 2 [Z; Z + z0? ]. Now let
~ (z) = W (Z) +
W
Z
z
~ (x) dx:
Z
~ (z), (z) by ~ (z), and V (z) by V~ (z) over the interval
We now show that, if we extend W (z) by W
[Z; Z + z0? ], we obtain a solution of the problem (80)-(82) over [Z; Z + z0? ]: indeed, we have just
~ 0 (z)
shown that ~ (z) = W
U 0 (z0? ) for all z 2 [Z; Z + z0? ], implying that the constraint y
we imposed in the de…nition of V~ (z) is not binding. That is:
V (z) = max U (y) + W (z
y2[0;z]
y) =
max
y2[z Z;Z]
57
U (y) + W (z
y) = V~ (z):
z
Z
Hence, we have extended the solution from [z0? ; Z] to [Z; Z + z0? ]. Notice that the argument does
not depend on Z: we can start with Z = z0? , and repeat this extension until we obtain a solution
de…ned over [z0? ; 1).
Finally, we show that W (z) is bounded. By construction we have:
Z z
r+
r+
?
V 0 (x)e c (x
(z) = (z0? )e c (z z0 ) +
c z0?
Z z
W (z) = W (z0? ) +
(y) dy:
z0? )
dx
z0?
Plugging the …rst equation into the second, keeping in mind that (z0? ) = 1, and changing the order
of integration we obtain:
c
r+
c
W (z0? ) +
r+
c
?
W (z0 ) +
r+
W (z) = W (z0? ) +
1
+
+
e
r+
c
r+
r+
V (z)
+
r+
Z
[W (z) + kU k
r+
c
e
(z x)
z
z0?
V (z0? )]
[V (z)
where the …rst inequality follows because 1
inequality because W (z)
(z z0? )
h
V 0 (x) 1
e
r+
c
(z x))
i
dy
W (z0? )] ;
1 for all x 2 [z0? ; z], and the second
W (z) + kU k. Rearranging and simplifying we obtain that
W (z)
W (z0 ) +
c + kU k
;
r
establishing the claim.
RESULT #6: For
su¢ ciently small and
0
chosen, as in RESULT #3, to balance the govern-
ment budget constraint, the households conjectured behavior is optimal.
Proof. Consider the candidate value function constructed in Result #4 and #5. By construction, W (z) is continuously di¤erentiable and it solves the HJB equation:
(r+ )W (z) =
max
minfc; cg + h
h+
c 0;0 h h;0 y z
[U (y) + W (z
y)] + W 0 (z) [h
c + (z)
z] :
Then, the optimality veri…cation argument of Section VII in the supplementary appendix establishes
that W (z) is equal to the maximum attainable utility of a households, and that the associated
decision rules are optimal.
PROOF OF PROPOSITION 10.
We construct an equilibrium featuring full depletion
and such that z ? < y. The ODE for the marginal value of real balances, (21), becomes
(r + )
t
= (A
t)
+ _ t ; 8t 2 [0; T (z ? )] ;
58
(86)
with _ T (z ? ) = 0. With Poisson arrival rate,
generates a marginal surplus equal to A
, the household spends all its real balances, which
. The solution is
for all t 2 R+ . It is straightforward to check that A >
full depletion is optimal. From U 0 (y ) = A
zs
y. The condition
h=( y
0
t
=E e
(r+ )T1 A
= A=(r + + ),
= A=(r + + ), which guarantees that
1+(r + )= , so the unconstrained target is such that
h) implies zb = h(1= + 1= )
y. So z ? = zb = h(1= + 1= )
and the equilibrium features full employment, h = h. Finally, h <
y [ (A
1)
r] = ( A
r)
implies h < y, which implies that the …rst best is such that h = h.
PROOF OF PROPOSITION 11.
We start with a formal de…nition of . With linear
preferences, equilibrium welfare is:
W =
Z
minfc; c(z)g + h
h(z) + U [y(z)] dF (z):
First-best welfare is, for large enough h
W? = h +
[U (y ? )
y?] :
The welfare loss relative to the …rst-best allocation is then de…ned as the
h+
?
fU [y (1
)]
?
y g=
Z
minfc; c(z)g + h
2 [0; 1] solving:
h(z) + U [y(z)] dF (z):
We also provide an explicit formula for the threshold in‡ation
appearing in the Proposition.
We start by …xing some " > 0 small enough such that
[U (y ? )
U (")]
(r + )y ? > 0
The existence of such " is guaranteed by the maintained assumption that U (y ? )
(87)
(r + )y ? > 0.
Given such ", we choose
(r + )y ?
:
"
(88)
With these preliminaries in mind, we turn to the proof of the Proposition.
RESULT #1: Su¢ cient optimality conditions for the household’s problem.
We guess that, under the conditions stated in the Proposition, the household optimal policy is
For z < z ? : ‡ow consume c = 0, ‡ow work h = h, and lumpy consume y = z.
For z = z ? : ‡ow consume c = 0, ‡ow work h = 0, and lumpy consume y = z.
59
For z > z ? : ‡ow consume c = c, ‡ow work h = 0 and lumpy consume some y(z) > z ? .
The corresponding equations for the value function is:
for z < z ? ;
for z = z ? ;
(r + )W (z) =
[U (z) + W (0)] + W 0 (z) h
(r + )W (z ? ) = h +
for z > z ? ;
(89)
[U (z ? ) + W (0)]
(r + )W (z) = h + c +
Below we solve for W (z) explicitly for z
z
(90)
max [U (y) + W (z
y2[0;z]
y)]
W 0 (z)c:
(91)
z ? , and we provide an implicit construction of W (z) for
z > z ? . A non-standard feature of W (z), which is a consequence of the discontinuity of the transfer
scheme, is that it is not di¤erentiable at z ? , with a concave kink. This means in particular that
the standard optimality veri…cation argument, which assumes continuous di¤erentiability, does not
apply directly here. In Supplementary Appendix VI we extend that argument to our case and we
prove su¢ ciency. That is, the stated optimal consumption-saving policy is optimal if the function
W (z) is bounded, continuously di¤erentiable for z 6= z ? , and satis…es two Hamilton-Jacobi-Bellman
equations. First, for z 6= z ? :
(r + )W (z) = max u(c; h
with respect to c
h) +
c
z + (z)] ;
0, h 2 [0; h] and y 2 [0; z]. Second, for z = z ? ,
(r + )W (z) = max u(c; h
with respect to c
y)] + W 0 (z) [h
[U (y) + W (z
h) +
[U (y) + W (z
0, h 2 [0; h], y 2 [0; z] and subject to h
Given that u(c; h
c) = minfc; cg + h
c
y)] ;
z + (z) = 0.
h and given the transfer scheme (z j ) , we obtain
that the stated consumption-saving policy is optimal if there exists a bounded function W (z),
continuously di¤erentiable for z 6= z ? , satisfying (89)-(91) as well as:
W 0 (z) < U 0 (z ? ) for z < z ?
(92)
W 0 (z) > 1 for z < z ?
(93)
W 0 (z) < 1 for z > z ? :
(94)
Condition (94) is su¢ cient for full depletion, that is z = arg maxy2[0;z] U (y) + W (z
RESULT #2: A closed-form expression for W (z) for z
y).
z?.
We use (89)-(90) to construct a guess for the function W (z). First, taking limit in (89) as
z " z ? and comparing with (90), we obtain that, for W (z) to be continuous, it must be that
60
W 0 (z ? ) = h=(h
z ? ). Second, taking derivative in (89), we obtain that:
(r +
where we used our notation (z)
+ ) (z) = U 0 (z) +
0
(z) h
z ;
W 0 (z). After integration, using the terminal condition derived
above, (z ? ) = h=(h
z ? , we obtain:
" Z ?
z
h
(z) =
U 0 (y)
h
z h z
r+
h
h
y
z
Hence:
W (z) = W (0) +
Z
dy +
h
h
r+
z?
z
#
:
(95)
z
(y) dy;
0
where W (0) is obtained by equating the limit at z " z ? with (90)
Z z?
?
(z) dz:
rW (0) = h + U (z ) (r + )
0
Conversely, one easily shows that the function W (z) thus constructed satis…es (89) and (90). One
can also check that this function is convex near z ? with a slope strictly larger than one in a leftneighborhood of z ? .
RESULT #3: Veri…cation of the full time work condition, (93), for all h large enough.
We …rst derive a uniform lower bound over the interval z 2 ["; 1] and shows that it is greater
than one for large enough h. Clearly, in (95), the multiplicative term and the second term in the
square bracket are both minimized at z = " over z 2 ["; 1]. The integral term is clearly positive.
Therefore,
(z)
h
h
"
h
h
y?
"
r+
1
[ "
h
1
> 1+ [ "
h
= 1+
therefore, condition (88) ensures that, as long as
for all h large enough. The condition
>
(r + ) (y ?
1
h
")] + o
1
h
(r + )y ? ] + o
:
> , this expression is strictly greater than one
ensures that (z ? ) is su¢ ciently greater than one –
this is the sense in which the incentives to escape in‡ation and reach the target must be su¢ ciently
large.
Next, we derive a uniform lower bound for (z) over the interval [0; "] and we show that this
lower bound is strictly greater than one. The multiplicative term in (95) is greater than one. The
derivative of the integral between the square brackets is:
r+
Z z?
Z z?
d
h
y
r+
0
0
U (y)
dy =
U (z) +
U 0 (y)
dz z
h
z
h
z z
r+
U 0 (") +
U (y ? ) < 0
h
"
61
r+
h
h
y
z
dy
as long as h is large enough. The second term in the square bracket of (95) is increasing in z and
decreasing in z ? . Therefore, as long as h is large enough, we obtain that:
(z)
h
Z
y?
r+
h
0
U (y)
"
h
h
y
"
y?
h
dy +
r+
:
h
By an application of the Dominated Convergence Theorem, one easily sees that the integral on
the right-hand side converges towards U (y ? )
can be written 1
U (") as h ! 1. The second-term on the right-side
(r + )y ? =h + o(1=h). Taken together, we obtain
(z)
1+
1
h
[U (y ? )
U (")]
(r + )y ?
+o
1
h
:
Therefore, condition (87) ensures that the right-side will be strictly greater than one for all h large
enough.
RESULT #4: Veri…cation of the full depletion condition, (92).
z?
We obtain a uniform upper bound for (z) as follows. For z
is less than h=(h
y ? ). In the square bracket both (h
y)=(h
y ? we have that h=(h
z ? )=(h
z) and (h
z)
z) are
less than one. Therefore:
(z)
h
h
y?
h
i
1
U (y ? ) + 1 = 1 + [ y ? + U (y ? )] + o
h
h
On the other hand, with z ? = y ?
=h, U 0 (z ? ) = U 0 (y ? )
1
h
:
U 00 (y ? ) =h + o(1=h) = 1 +
jU 00 (y ? )j =h + o(1=h), since U 00 (z) < 0 and U 0 (y ? ) = 1 by assumption. Our choice of
that
ensures
> [ y ? + U (y ? )] =jU 00 (y ? )j, and so (94) holds for all h is large enough.
RESULT #5: Veri…cation of the full consumption condition, (94), for all h large enough. Given
that (92) and (93) hold, we xfollow the same steps as in the proof of Result #5, Proposition 9 to
z ? , and show that (z) 2 [0; 1],
construct the value function W (z) for z
PROOF OF PROPOSITION 13 . Let us …rst recall the problem of the household in the
pure-currency economy under a lump-sum transfer scheme,
rW (z) = max u(c; h
with respect to (c; h; y), subject to c
h) +
0, 0
[U (y) + W (z
h
h, 0
1:
y)
y
W (z)] + W 0 (z)z_ ;
z, and z_ = h
c+
(96)
1
z. Policy
functions are denoted y 1 (z), h1 (z), and c1 (z). The distribution of real balances across households
is F 1 (z). We compare equilibria of the pure currency economy to equilibria of an economy with
money and bonds such that % =
, i.e., money and bonds have the same rate of return. The
62
household problem, (39), becomes
rW (!) = max u(c; h
c;h;y
subject to c
0, 0
h
h, 0
y
h) +
[U (y) + W (!
!, and !_ = h
c
2
which is formally equivalent to (96) provided that
W (!)] + W 0 (!)!_ ;
y)
2.
!+
1.
=
(97)
The household problem (97)
If this condition holds, y 2 (!) = y 1 (!),
h2 (!) = h1 (!), c2 (!) = c1 (!); and the distributions of wealth across the two economies are the
same, F 2 (!) = F 1 (!). In order to check that
2
=
1
2
0
B0 +
we use the budget constraint of the
government:
2
where we used that q0 =
2
0,
=
( + )q0
=
2
0 B0
+
2
0
2
0 M0
;
B0 is the initial supply of bonds, and M0 = 1 is the initial supply of
money. By market clearing,
2
0 B0
+
2
0 M0
Z
=
1
Z0 1
=
!dF 2 (! )
zdF 1 (z )
0
1
0 M0
=
1
0;
=
which implies
2
0
and
2
=
1.
=
1
0
1 + B0
;
(98)
From (45), the clearing of the bonds market requires that there is a z 2 (!) 2 y 2 (!); !
such that:
2
0 B0
=
Z
1
z 2 (!) dF 2 (! ):
!
0
Such a function exists provided that
Z 1
Z
2
2
2
B
!
y
(!)
dF
(!
)
=
0 0
0
Substituting
2
0
1
z
y 1 (z) dF 1 (z ):
0
by its expression given by (98), we rewritte this inequality as:
R1
y 1 (z) dF 1 (z )
0 Rz
B0
:
1 1
1
0 y (z)dF (z )
The proof of the converse, namely, any liquidity-trap equilibrium of two-asset economy corresponds
to an equilibrium of the pure-currency economy that features partial depletion, is analogous and is
therefore omitted.
63
Appendix B: Numerical methods
Overview. In this section we provide a step-by-step numerical method to compute the stationary
equilibrium with standard packages, for example Matlab. A detailed discussion of the numerical
method is provided in the supplementary Appendix. To solve the system we need to start from
some initial values close to the solution. Step 1 suggests an e¢ cient method to compute initial
values of
and
0
: the solution to an economy with zero in‡ation and full depletion, which is close
to the equilibrium if the money growth rate is not very large but h is not very low. Given
0
and
,
Step 2 (or 2’under linear preferences) computes the system of delay di¤erential equations (DDE),
which summarizes the household’s optimal actions. Step 3 and 4 (or 4’under linear preferences)
computes the Kolmogorov forward equation (KFE), which solves the stationary distribution. Step
5 solves
0
and
as …xed points.29
Step 1a. Fix y (z) = z and
= 0. Solve the following values for initiation:
h(
?
) = c(
z? =
p =
where
)
r+
1
U0
?
h0 (
c0 (
)
?
?
)
?
) = c(
?
) = 0,
;
;
is the negative eigenvalue of the Jacobian given by
"
r+
4 U 00 (z ? ) 0 ?
=
1
h ( ) c0 (
2
(r + )2
Under linear preferences, we have h (
r+
?
?
1=2
?
)
#
1 :
= 1, z ? = (U 0 )
1
r+
and p =
U 00 (z ? ).
Step 1b. Use ode45 routine of Matlab to integrate the following ODE of
(z) backward from
z = z ? to z = 0:
0
where the initial values are given by
Step 1c. Having obtained
(z) =
(r + )
h( )
(z ? ) =
?
and
U 0 (z)
;
c( )
0
(z ? ) = p.
(z), use ode45 routine to integrate the following ODE of f (z)
29
The common approach in the literature is to use an "upwind" …nite-di¤erences algorithm to iterate a system
of PDEs composed of HJB and KFE. Instead, the equilibrium of our model de…ned as a system of DDEs can be
solved e¢ ciently with built-in Matlab routines with good control of error. For example, a laptop equipped with Intel
i5 2.30GHz CPU and 8GM RAM takes 15 seconds to compute the calibrated model in Section 4.3, with an error
tolerance of 10 6 . A …nite-di¤erences algorithm takes 27 times longer to converge, with an error tolerance of 5 10 3 .
64
forward from z = 0 to z = z ? :
+
f 0 (z) =
0
(z) [h0 ( ) c0 ( )]
f (z)
h( ) c( )
where the initial value is given by f (0) = 1. If s z ?
> 0 (for example under the slack labor
F z?
equilibrium of LRW models) then we construct the probability mass 1
by the following
KFE boundary condition
F z? =
1
s z? f z?
:
It obtains f (z).
Step 1d. The initial values of
0
and
are set to
0
=
(0) and
Step 2a. Jump to Step 2’a if under linear preferences. Given
R z?
=
0
0
and
zf (z) dz=
R z?
0
f (z) dz.
(from Step 1 if it is the
…rst time to run the iteration), use ddesd routine to integrate the following DDE system of z ( )
and
( )
z0 ( ) =
0
( ) = z 0 ( ) U 00
h
U0
h( ) c( )
z+
;
(r + + ) +
( )
i 1
1
[ ( )]
+ I [ ( ) < 0] z0 [
where the initial values are given by z (
0)
= 0 and
value if U 0 (0) = 1). Stop integrating whenever h (
stopping
and z as
?
(
0;
) and z ? (
0;
(
)
0)
c(
1
( )]
;
= U 0 (0) (or some arbitrary large
)
). It obtains z ( ) and
z( ) +
= 0. Denote the
( ).
Step 2b. De…ne
y (z)
zd
s (z)
1
U0
1
z
z?
y (z ? ) ;
h
z
1
(z)
c
(z) ;
z
1
(z)
(z
):
Jump to Step 3.
Step 2’a. Given
0
and
(from Step 1 if it is the …rst time to run the iteration), use ddesd
routine to integrate the following DDE of y (z)
(
1 if z (U 0 ) 1 ( 0 ) ;
0
h
y =
(z
1 + U 00 (y) (r+ + h)U 0 (y)
y)+
U 0 [y(z y)]
i
1
if z > (U 0 )
1
(
where the initial value is given by y (0) = 0. Stop integrating at either z = (h +
1+
r+
. It obtains y (z). Denote the stopping z as z ? (
zd
s (z)
0;
z?
y (z ? ) ;
h
z+ :
65
). De…ne
0) ;
) = or U 0 [y (z)] =
Step 2’b. Use ode45 routine to integrate the following ODE of
0
with the initial condition
(r +
(z) =
(0) =
0.
U 0 [y (z)]
;
)
+ ) (z)
h
(z
It obtains
Step 3. De…ne ' (z) as the solution to '
(z) forward from z = 0 to z ? :
?
(z). De…ne
(
0;
)=
(z ? ).
y (') = z. Consider the region [zd ; z ? ], where the
density function f (z) is simply an ODE solution of the following KFE:
+ s0 (z)
f (z) ; for all z 2 (zd ; z ? ) ;
s (z)
f 0 (z) =
which has closed-form solution
f (z) =
1
s (z)
s (zd )
; for all z 2 (zd ; z ? )
under linear preferences (and we need to construct the probability mass 1
F z?
by the same
KFE boundary condition in the Step 1c). Otherwise, use ode45 routine to solve f (t) forward from
z = zd to z = z ? with initial value f (zd ) = 1. It obtains f (z) for all z 2 [zd ; z ? ].
Step 4. Jump to Step 4’ if under linear preferences. Construct the "history" of
t 2 [ (z ?
zd ) ; 0], by setting
(t) = f (zd
to integrate the following DDE of
0
s0 (zd t)
s (zd t)
(t) =
where the initial value
t), where f (z) is given by Step 3. Use ddesd routine
(t) forward from t = 0 to t = zd :
s [' (zd t)]
[zd
s (zd t)2
(t)
' (zd
t)] ; for all t 2 (0; zd )
(0) is given by
(0) = 1
Having obtained
(t) for all
(t), set f (z) =
(zd
s z?
f z? :
s (zd )
z) for all z 2 [0; zd ). Jump to Step 5.
Step 4’. Under linear preferences, use ode45 routine to integrate the following DDE of
forward from t = 0 to t = zd :
0
(t) =
where the initial value
+
s (zd
t)
(0) is given by
(0) = 1
Having obtained
s [' (zd t)]
; for all t 2 [0; zd ] ;
s (zd t)2
s (zd )1
(t)
(t), set f (z) =
(zd
"
s z?
s (zd )
#
:
z) for all z 2 [0; zd ).
66
(t)
Step 5. De…ne a function
(
0;
) : R2+ ! R2 , where the …rst and second coordinates are
given by
) = (r + + ) ? + U 0 [y (z ? )] ;
R z?
zf (z) dz
(2)
( 0; ) =
;
R0 z ?
0 f (z) dz
(1)
?
where
solve
?
0
(
0;
, z ? , y and f are constructed given
?
and
such that
(
?
0;
?)
0
and
from previous steps. Use fsolve routine to
= 0.
Step 6. Finally, the stationary equilibrium is given by the marginal value function W 0 (z) =
z
1 (z;
?
0;
to z_ = s (z;
?)
and the density function f (z;
?
0;
? ),
?
0;
? ).
Agents accumulate real balances according
and the lumpy consumption is given by y (z;
?
0;
? ).
The above numerical
algorithm works whether the equilibrium features periodic full money depletion or periodic partial
money depletion.
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@FedFRASER