Debt and Incomplete Financial Markets:

A Case for Nominal GDP Targeting

Kevin D. Sheedy

London School of Economics

First draft: 7
th
February 2012
This version: 15
th
February 2013
Abstract
Financial markets are incomplete, thus for many agents borrowing is possible only by ac-
cepting a nancial contract that species a xed repayment. However, the future income that
will repay this debt is uncertain, so risk can be ineciently distributed. This paper argues that
a monetary policy of nominal GDP targeting can improve the functioning of incomplete nan-
cial markets when incomplete contracts are written in terms of money. By insulating agents
nominal incomes from aggregate real shocks, this policy eectively completes the market by
stabilizing the ratio of debt to income. The paper argues that the objective of nominal GDP
should receive substantial weight even in an environment with other frictions that have been
used to justify a policy of strict ination targeting.
JEL classifications: E21; E31; E44; E52.
Keywords: incomplete markets; heterogeneous agents; risk sharing; nominal GDP targeting.

I thank Carlos Carvalho, Wouter den Haan, Monique Ebell, Albert Marcet, and Matthias Paustian
for helpful comments. The paper has also beneted from the comments of seminar participants at Banque
de France,

Ecole Polytechnique, National Bank of Serbia, PUCRio, Sao Paulo School of Economics,
the Centre for Economic Performance annual conference, the Joint French Macro Workshop, LBS-CEPR
conference Developments in Macroeconomics and Finance, and the London Macroeconomics Workshop.

Department of Economics, London School of Economics and Political Science, Houghton Street, Lon-
don, WC2A 2AE, UK. Tel: +44 207 107 5022, Fax: +44 207 955 6592, Email: k.d.sheedy@lse.ac.uk,
Website: http://personal.lse.ac.uk/sheedy.
1 Introduction
Following the onset of the recent nancial crisis, ination targeting has increasingly found itself under
attack. The frequent criticism is not that it has failed to achieve what it purports to do to avoid
a repeat of the inationary 1970s or the deationary 1930s but that central banks have focused
too much on price stability and too little on nancial markets.
1
Such a view implicitly supposes
there is a tension between the goals of price stability and nancial stability when the economy is hit
by shocks. However, it is not clear why this should be so, there being no widely accepted argument
for why stabilizing prices in goods markets causes nancial markets to malfunction.
The canonical justication for ination targeting as optimal monetary policy rests on the presence
of pricing frictions in goods markets (see, for example, Woodford, 2003). With infrequent price
adjustment due to menu costs or other nominal rigidities, high or volatile ination leads to relative
price distortions that impair the ecient operation of markets, and which directly consumes time
and resources in the process of setting prices. While there is a consensus on the importance of
these frictions when analysing optimal monetary policy, it is increasingly argued that monetary
policy must also take account of nancial-market frictions such as collateral constraints or spreads
between internal and external nance.
2
These frictions can magnify the eects of both shocks and
monetary policy actions and make these eects more persistent. But the existence of a quantitatively
important credit channel does not in and of itself imply that optimal monetary policy is necessarily
so dierent from ination targeting unless new types of shocks are introduced (Faia and Monacelli,
2007, Carlstrom, Fuerst and Paustian, 2010, De Fiore and Tristani, 2012).
This paper studies a simple and compelling friction in nancial markets that immediately and
straightforwardly leads to a stark conict between the ecient operation of nancial markets and
price stability. The friction is a modest one: nancial markets are assumed to be incomplete. Those
who want to borrow can only do so through debt contracts that specify a xed repayment (eectively
issuing non-contingent bonds). The argument is that many agents, households in particular, will nd
it very dicult to issue liabilities with state-contingent repayments resembling equity or derivatives.
Implicitly, it is assumed to be too costly to write lengthy contracts that spell out in advance dierent
repayments conditional on each future state of the world.
The problem of non-contingent debt contracts for risk-averse households is that when borrowing
for long periods, there will be considerable uncertainty about the future income from which xed
debt repayments must be made. The issue is not only idiosyncratic uncertainty households do not
know the future course the economy will take, which will aect their labour income. Will there be a
1
White (2009b) and Christiano, Ilut, Motto and Rostagno (2010) argue that stable ination is no guarantee of
nancial stability, and may even create conditions for nancial instability. Christiano, Motto and Rostagno (2007)
suggest that credit growth ought to have a role as an independent target of monetary policy. Contrary to these
arguments, the conventional view that monetary policy should not react to asset prices is advocated in Bernanke and
Gertler (2001). Woodford (2011) makes the point that exible ination targeting can be adapted to accommodate
nancial stability concerns, and that it would be unwise to discard ination targetings role in providing a clear
nominal anchor.
2
Starting from Bernanke, Gertler and Gilchrist (1999), there is now a substantial body of work that integrates
credit frictions of the kind found in Carlstrom and Fuerst (1997) or Kiyotaki and Moore (1997) into monetary DSGE
models. Recent work in this area includes Christiano, Motto and Rostagno (2010).
1
productivity slowdown, a deep and long-lasting recession, or even a lost decade of poor economic
performance to come? Or will unforeseen technological developments or terms-of-trade movements
boost future incomes, and good economic management successfully steer the economy on a path
of steady growth? Borrowers do not know what aggregate shocks are to come, but must x their
contractual repayments prior to this information being revealed.
The simplicity of non-contingent debt contracts can be seen as coming at the price of bundling
together two fundamentally dierent transfers: a transfer of consumption from the future to the
present for borrowers, but also a transfer of aggregate risk to borrowers. The future consumption
of borrowers is paid for from the dierence between their uncertain future incomes and their xed
debt repayments. The more debt they have, the more their future income is eectively leveraged,
leading to greater consumption risk. The ip-side of borrowers leverage is that savers are able to
hold a risk-free asset, reducing their consumption risk.
To see the sense in which this bundling together of a transfer of risk and borrowing is inecient,
consider what would happen in complete nancial markets. Individuals would buy or sell state-
contingent bonds (Arrow-Debreu securities) that make payos conditional on particular states of
the world (or equivalently, write loan contracts with dierent repayments across all states of the
world). Risk-averse borrowers would want to sell relatively few bonds paying o in future states
of the world where GDP and thus incomes are low, and sell relatively more in good states of the
world. As a result, prices of contingent bonds paying o in bad states would be relatively expensive
and those paying o in good states relatively cheap. These price dierences would entice savers
to shift away from non-contingent bonds and take on more risk in their portfolios. Given that the
economy has no risk-free technology for transferring goods over time, and as aggregate risk cannot
be diversied away, the ecient outcome is for risk-averse individuals to share aggregate risk, and
complete markets allow this to be unbundled from decisions about how much to borrow or save.
The ecient nancial contract between risk-averse borrowers and savers in an economy subject
to aggregate income risk (abstracting from idiosyncratic risk) turns out to have a close resemblance
to an equity share in GDP. In other words, borrowers repayments should fall during recessions
and rise during booms. This means the ratio of debt liabilities to GDP should be more stable than
it would be in a world of incomplete nancial markets where debt liabilities are xed while GDP
uctuates.
With incomplete nancial markets, monetary policy has a role to play in mitigating ineciencies
because private debt contracts are typically denominated in terms of money. Hence, the real degree
of state-contingency in nancial contracts is endogenous to monetary policy. If incomplete markets
were the only source of ineciency in the economy then the optimal monetary policy would aim
to make nominally non-contingent debt contracts mimic through variation in the price level the
ecient nancial contract that would be chosen with complete nancial markets.
Given that the ecient nancial contract between borrowers and savers resembles an equity
share in GDP, it follows that a goal of monetary policy should be to stabilize the ratio of debt
liabilities to GDP. With non-contingent nominal debt liabilities, this can be achieved by having a
non-contingent level of nominal income, in other words, a monetary policy that targets nominal
2
GDP. The intuition is that while the central bank cannot eliminate uncertainty about future real
GDP, it can in principle make the level of future nominal GDP (and hence the nominal income of
an average person) perfectly predictable. Removing uncertainty about future nominal income thus
alleviates the problem of nominal debt repayments being non-contingent.
A policy of nominal GDP targeting generally deviates from ination targeting because any
uctuations in real GDP would lead to uctuations in ination of the same size and in the opposite
direction. Recessions would feature higher ination and booms would feature lower ination, or
even deation. These ination uctuations can be helpful because they induce variation in the real
value of nominally non-contingent debt, making it behave more like equity, which promotes ecient
risk sharing. A policy of strict ination targeting would convert nominally non-contingent debt into
real non-contingent debt, which would imply an uneven and generally inecient distribution of risk.
The ination uctuations that occur with nominal GDP targeting would entail relative-price
distortions if prices were sticky, so the benet of ecient risk sharing is most likely not achieved
without some cost. It is ultimately a quantitative question whether the ineciency caused by incom-
plete nancial markets is more important than the ineciency caused by relative-price distortions,
and thus whether nominal GDP targeting is preferable to ination targeting.
This paper presents a model that allows optimal monetary policy to be studied analytically in
an incomplete-markets economy with heterogeneous agents. The basic framework adopted is the
life-cycle theory of consumption, which provides the simplest account of household borrowing and
saving. The model contains overlapping generations of individuals: the young, the middle-aged, and
the old. Individuals are risk averse, having an Epstein-Zin-Weil utility function. Individuals receive
incomes equal to xed age-specic shares of GDP (labour supply is exogenous, but this simplifying
assumption can be relaxed). The age-prole of income is assumed to be hump shaped: the middle-
aged receive the most income; the young receive less income; while the old receive the least. Real
GDP is uncertain because of aggregate productivity shocks, but there are no idiosyncratic shocks.
Young individuals would like to borrow to smooth consumption, repaying when they are middle-
aged. The middle-aged would like to save, drawing on their savings when they are old. The economy
is assumed to have no investment or storage technology, and is closed to international trade. There
are no government bonds and no at money, and no taxes or scal transfers such as public pensions.
In this world, consumption smoothing is facilitated by the young borrowing from the middle-aged,
repaying when they themselves are middle-aged and their creditors are old. It is assumed the only
nancial contract available is a non-contingent nominal bond. The basic model contains no other
frictions, and initially assumes that prices and wages are fully exible.
The concept of a natural debt-to-GDP ratio provides a useful benchmark for monetary policy.
This is dened as the ratio of (state-contingent) debt liabilities to GDP that would prevail were
nancial markets complete, which is independent of monetary policy. The actual debt-to-GDP
ratio in an economy with incomplete markets would coincide with the natural debt-to-GDP ratio if
forecasts of future GDP were always correct ex post, but will in general uctuate around it when
the economy is hit by shocks. The natural debt-to-GDP ratio is thus analogous to concepts such as
the natural rate of unemployment and the natural rate of interest.
3
If all movements in real GDP growth rates are unpredictable then the natural debt-to-GDP ratio
turns out to be constant (or if utility functions are logarithmic, the ratio is constant irrespective
of the statistical properties of GDP growth). Even when the natural debt-to-GDP ratio is not
completely constant, plausible calibrations suggest it would have a low volatility relative to real
GDP itself.
Since the equilibrium of an economy with complete nancial markets would be Pareto ecient in
the absence of other frictions, the natural debt-to-GDP ratio also has desirable welfare properties.
A goal of monetary policy in an incomplete-markets economy is therefore to close the debt gap,
dened as the dierence between the actual and natural debt-to-GDP ratios. It is shown that doing
this eectively completes the market in the sense that the equilibrium with incomplete markets
would then coincide with the hypothetical complete-markets equilibrium. Monetary policy can aect
the actual debt-to-GDP ratio and thus the debt gap because that ratio is nominal debt liabilities
(which are non-contingent with incomplete markets) divided by nominal GDP, where the latter is
under the control of monetary policy.
When the natural debt-to-GDP ratio is constant, closing the debt gap can be achieved by
adopting a xed target for the level of nominal GDP. With this logic, the central bank uses nominal
GDP as an intermediate target that achieves its ultimate goal of closing the debt gap. This turns
out to be preferable to targeting the debt-to-GDP ratio directly because a monetary policy that
targets only a real nancial variable would leave the economy without a nominal anchor. Nominal
GDP targeting uniquely pins down the nominal value of incomes and thus provides the economy
with a well-dened nominal anchor.
It is important to note that in an incomplete-markets economy hit by shocks, whatever action a
central bank takes or fails to take will have distributional consequences. Ex post, there will always
be winners and losers. Creditors lose out when ination is unexpectedly high, while debtors suer
when ination is unexpectedly low. It might then be thought surprising that ination uctuations
would ever be desirable. However, the ination uctuations implied by a nominal GDP target
are not arbitrary uctuations they are perfectly correlated with the real GDP uctuations that
are the ultimate source of uncertainty in the economy, and which themselves have distributional
consequences when individuals are heterogeneous. For individuals to share risk, it must be possible
to make transfers ex post that act as insurance from an ex-ante perspective. The result of the
paper is that ex-ante ecient insurance requires ination uctuations that are negatively correlated
with real GDP (a countercylical price level) to generate the appropriate ex-post transfers between
debtors and creditors.
It might be objected that there are innitely many state-contingent consumption allocations
that would also satisfy the criterion of ex-ante eciency. However, only one of these the hypo-
thetical complete-markets equilibrium associated with the natural debt-to-GDP ratio could ever
be implemented through monetary policy. Thus for a policymaker solely interested in promoting ef-
ciency, there is a unique optimal policy that does not require any explicit distributional preferences
to be introduced.
The model also makes predictions for how dierent monetary policies will aect the volatility
4
of nancial-market variables such as credit and interest rates. It is shown that policies implying an
inecient distribution of risk, for example, ination targeting, are associated with greater volatility
in nancial markets when compared to the nominal GDP targeting policy that allows the economy
to mimic the hypothetical complete-markets equilibrium. Stabilizing ination implies that new
lending as fraction of GDP is excessively procyclical: credit expands too much during a boom and
falls too much during a recession. Similarly, ination targeting implies that real interest rates will
be excessively countercyclical, permitting real interest rates to fall too much during an expansion.
These ndings allow the tension between price stability and ecient risk sharing to be seen in more
familiar terms as a trade-o between price stability and nancial stability.
Determining which of these objectives is the more quantitatively important requires introducing
nominal rigidity into the model, allowing for there to be a cost associated with ination uctuations
due to relative-price distortions. Nominal rigidity is introduced with a simple model of predeter-
mined price-setting, but in a way that allows the welfare costs of ination to be calibrated to match
levels found in the existing literature. With both incomplete nancial markets and sticky prices,
optimal monetary policy is a convex combination of a nominal GDP target and a strict ination
target. After calibrating all the parameters of the model, the conclusion is that the nominal GDP
target should receive approximately 95% of the weight.
This paper is related to a number of areas of the literature on monetary policy and nancial
markets. First, there is the empirical work of Bach and Stephenson (1974), Cukierman, Lennan
and Papadia (1985), and more recently, Doepke and Schneider (2006), who document the eects of
ination in redistributing wealth between debtors and creditors. The novelty here is in studying the
implications for optimal monetary policy in an environment where ination uctuations with such
distributional eects may actually be desirable because nancial markets are incomplete.
The most closely related theoretical paper is Pescatori (2007), who studies optimal monetary
policy in an economy with rich and poor individuals, in the sense of there being an exogenously
specied distribution of assets among otherwise identical individuals. In that environment, both
ination and interest rate uctuations have redistributional eects on rich and poor individuals, and
the central bank optimally chooses the mix between them (there is a need to change interest rates
because prices are sticky, with deviations from the natural rate of interest leading to undesirable
uctuations in output). Another closely related paper is Lee (2010), who develops a model where
heterogeneous individuals choose less than complete consumption insurance because of the presence
of convex transaction costs in accessing nancial markets. Ination uctuations expose households
to idiosyncratic labour-income risk because households work in specic sectors of the economy, and
sectoral relative prices are distorted by ination when prices are sticky. This leads optimal monetary
policy to put more weight on stabilizing ination. Dierently from those papers, the argument here
is that ination uctuations can actually play a positive role in completing otherwise incomplete
nancial markets (and where debt arises endogenously owing to individual heterogeneity).
3
3
In other related work, Akyol (2004) analyses optimal monetary policy in an incomplete-markets economy where
individuals hold at money for self insurance against idiosyncratic shocks. Kryvtsov, Shukayev and Ueberfeldt (2011)
study an overlapping generations model with at money where monetary policy can improve upon the suboptimal
level of saving by varying the expected ination rate and thus the returns to holding money.
5
The idea that ination uctuations may have a positive role to play when nancial markets are
incomplete is now long-established in the literature on government debt (and has also been recently
applied by Allen, Carletti and Gale (2011) in the context of the real value of the liquidity available
to the banking system). Bohn (1988) developed the theory that nominal non-contingent government
debt can be desirable because when combined with a suitable monetary policy, ination will change
the real value of the debt in response to scal shocks that would otherwise require uctuations in
distortionary tax rates.
Quantitative analysis of optimal monetary policy of this kind was developed in Chari, Christiano
and Kehoe (1991) and expanded further in Chari and Kehoe (1999). One nding was that ination
needs to be extremely volatile to complete the market. As a result, Schmitt-Grohe and Uribe (2004)
and Siu (2004) argued that once some nominal rigidity is considered so that ination uctuations
have a cost, the optimal policy becomes very close to strict ination targeting. This paper shares
the focus of that literature on using ination uctuations to complete nancial markets, but comes
to a dierent conclusion regarding the magnitude of the required ination uctuations and whether
the cost of those uctuations outweighs the benets. First, the benets of completing the market
in this paper are linked to the degree of household risk aversion, which is in general unrelated to
the benets of avoiding uctuations in distortionary tax rates, and which proves to be large in the
calibrated model. Second, the earlier results assumed government debt with a very short maturity.
With longer maturity debt (household debt in this paper), the costs of the ination uctuations
needed to complete the market are much reduced.
4
This paper is also related to the literature on household debt. Iacoviello (2005) examines the
consequences of household borrowing constraints in a DSGE model, while Guerrieri and Lorenzoni
(2011) and Eggertsson and Krugman (2012) study how a tightening of borrowing constraints for
indebted households can push the economy into a liquidity trap. Dierently from those papers, the
focus here is on the implications of household debt for optimal monetary policy. Furthermore, the
nding here that the presence of household debt substantially changes optimal monetary policy does
not depend on there being borrowing constraints, or even the feedback eects from debt to aggregate
output stressed in those papers. C urdia and Woodford (2009) also study optimal monetary policy
in an economy with household borrowing and saving, but the focus there is on spreads between
interest rates for borrowers and savers, while their model assumes an insurance facility that rules
out the risk-sharing considerations studied here. Finally, the paper is related to the literature on
nominal GDP targeting (Meade, 1978, Bean, 1983, Hall and Mankiw, 1994) but proposes a dierent
argument in favour of that policy.
The plan of the paper is as follows. Section 2 sets out the basic model and derives the equilibrium
conditions. The main optimal monetary policy results are given in section 3. Section 4 introduces
some extensions of the basic model and studies the observable consequences of following a suboptimal
monetary policy. Section 5 introduces sticky prices and hence a trade-o between incomplete markets
and price stability. Finally, section 6 draws some conclusions.
4
This point is made by Lustig, Sleet and Yeltekin (2008) in the context of government debt.
6
2 A model of a pure credit economy
The population of an economy comprises overlapping generations of individuals. Time is discrete and
is indexed by t. A new generation of individuals is born in each time period and each individual lives
for three periods. During their three periods of life, individuals are referred to as the young (y), the
middle-aged (m), and the old (o), respectively. An individual derives utility from consumption
of a composite good at each point in his life. There is no intergenerational altruism. At time t,
per-person consumption of the young, middle-aged, and old is denoted by C
y,t
, C
m,t
, and C
o,t
.
Individuals have identical lifetime utility functions, which have the Epstein-Zin-Weil functional
form (Epstein and Zin, 1989, Weil, 1990). Future utility is discounted by subjective discount factor
(0 < < ), the intertemporal elasticity of substitution is (0 < < ), and is the coecient
of relative risk aversion (0 < < ). The utility U
t
of the generation born at time t is
U
t
=
V
1
1

y,t
1
1

, where V
y,t
=
_
C
1
1

y,t
+
_
E
t
_
V
1
m,t+1
1
1
_
1
1

_
1
1
1

,
V
m,t
=
_
C
1
1

m,t
+
_
E
t
_
V
1
o,t+1
1
1
_
1
1

_
1
1
1

, and V
o,t
= C
o,t
. [2.1]
The utility function is written in a recursive form with V
y,t
, V
m,t
, and V
o,t
denoting the continuation
values of the young, middle-aged, and old in terms of current consumption equivalents.
5
The number of young individuals born in any time period is exactly equal to the number of
old individuals alive in the previous period who now die. The economy thus has no population
growth and a balanced age structure. Assuming that the population of individuals currently alive
has measure one, each generation of individuals has measure one third. Aggregate consumption at
time t is denoted by C
t
:
C
t
=
1
3
C
y,t
+
1
3
C
m,t
+
1
3
C
o,t
. [2.2]
All individuals of the same age at the same time receive the same income, with Y
y,t
, Y
m,t
, and
Y
o,t
denoting the per-person incomes (in terms of the composite good) of the young, middle-aged,
and old, respectively, at time t. Age-specic incomes are assumed to be time-invariant multiples of
aggregate income Y
t
, with
y
,
m
, and
o
denoting the multiples for the young, middle-aged, and
old:
Y
y,t
=
y
Y
t
, Y
m,t
=
m
Y
t
, Y
o,t
=
o
Y
t
, where
y
,
m
,
o
(0, 3) and
1
3

y
+
1
3

m
+
1
3

o
= 1.
[2.3]
Real GDP is specied as an exogenous stochastic process. This assumption turns out not to aect
the main results of the paper, but is relaxed later.
6
The growth rate of real GDP between period
5
The functional form reduces to the special case of time-separable isoelastic utility when the coecient of relative
risk aversion is equal to the reciprocal of the intertemporal elasticity of substitution ( = 1/).
6
The introduction of an endogenous labour supply decision need not aect the results unless prices or wages are
sticky.
7
Figure 1: Age prole of non-nancial income
Age
Income/GDP per person
1
Young Middle-aged Old
1
1 + (1 +)
1
t 1 and t, denoted by g
t
(Y
t
Y
t1
)/Y
t1
, is given by
g
t
= g + x
t
, where Ex
t
= 0, Ex
2
t
= 1, and x
t
[x, x], [2.4]
with x
t
being an exogenous stationary stochastic process with bounded support. The growth rate
g
t
has mean g and standard deviation . Dening in terms of the parameters , , , g, and
(and the stochastic process of x
t
), the following parameter restriction is imposed:
0 < < 1, where E
_
(1 + g
t
)
1

11/
1
. [2.5]
The income multiples
y
,
m
, and
o
for each generation are parameterized to specify a hump-
shaped age prole of income in terms of and a single new parameter :

y
= 1 ,
m
= 1 + (1 +), and
o
= 1 . [2.6]
The income multiples are all well-dened and strictly positive for any 0 < < 1. The general
pattern is depicted in Figure 1. As 0, the economy approaches the special case where all
individuals alive at the same time receive the same income irrespective of age, while as 1, the
dierences in income between individuals of dierent ages are at their maximum with old individuals
receiving a zero income. Intermediate values of imply age proles that lie between these extremes,
thus the parameter can be interpreted as the gradient of the age prole of income over the life
cycle. The presence of the coecient in the specication [2.6] implies that the income gradient
from young to middle-aged is less than the gradient from middle-aged to old.
7
There is assumed to be no government spending and no international trade, and the composite
good is not storable, hence the goods-market clearing condition is
C
t
= Y
t
. [2.7]
The economy has a central bank that denes a reserve asset, referred to as money. Reserves
7
Introducing this feature implies that the steady state of the model will have some convenient properties. See
section 2.3 for details.
8
held between period t and t + 1 are remunerated at a nominal interest rate i
t
known in advance at
time t. The economy is cash-less in that money is not required for transactions, but money is used
by agents as a unit of account in writing nancial contracts and in pricing goods. One unit of the
composite good costs P
t
units of money at time t, and
t
(P
t
P
t1
)/P
t1
denotes the ination
rate between period t1 and t. Monetary policy is specied as a rule for setting the nominal interest
rate i
t
. Finally, in equilibrium, the central bank will maintain a supply of reserves equal to zero.
2.1 Incomplete nancial markets
Asset markets are assumed to be incomplete. No individual can sell state-contingent bonds (Arrow-
Debreu securities), and hence in equilibrium in this economy, no such securities will be available to
buy. The only asset that can be traded is a one-period, nominal, non-contingent bond. Individuals
can take positive or negative positions in this bond (save or borrow), and there is no limit on
borrowing other than being able to repay in all states of the world given non-negativity constraints
on consumption. With this restriction, no default will occur, and thus bonds are risk free in nominal
terms.
8
Bonds that have a nominal face value of 1 paying o at time t + 1 trade at price Q
t
in terms of
money at time t. These bonds are perfect substitutes for the reserve asset dened by the central
bank, so the absence of arbitrage opportunities requires that
Q
t
=
1
1 + i
t
. [2.8]
The central banks interest-rate policy thus sets the nominal price of the bonds.
Let B
y,t
and B
m,t
denote the net bond positions per person of the young and middle-aged at the
end of time t (positive denotes saving, negative denotes borrowing). The absence of intergenerational
altruism implies that the old will make no bequests (B
o,t
= 0) and the young will begin life with no
assets. The budget identities of the young, middle-aged, and old are respectively:
C
y,t
+
Q
t
P
t
B
y,t
= Y
y,t
, C
m,t
+
Q
t
P
t
B
m,t
= Y
m,t
+
1
P
t
B
y,t1
, and C
o,t
= Y
o,t
+
1
P
t
B
m,t1
. [2.9]
Maximizing the lifetime utility function [2.1] for each generation with respect to its bond holdings,
subject to the budget identities [2.9], implies the Euler equations:
E
t
_

_
P
t
P
t+1
_
_
_
V
m,t+1
E
t
_
V
1
m,t+1
1
1
_
_
_
1

_
C
m,t+1
C
y,t
_

_
= Q
t
= E
t
_

_
P
t
P
t+1
_
_
_
V
o,t+1
E
t
_
V
1
o,t+1
1
1
_
_
_
1

_
C
o,t+1
C
m,t
_

_
. [2.10]
8
With the utility function [2.1], marginal utility tends to innity as consumption tends to zero. Thus, individuals
would not choose borrowing that led to zero consumption in some positive-probability set of states of the world,
so this constraint will not bind. Furthermore, given that the stochastic process for GDP growth in [2.4] has nite
support, for any particular amount of borrowing, it would always be possible to set the standard deviation to be
suciently small to ensure that no default would occur.
9
With no issuance of government bonds, no bond purchases by the central bank (the supply
of reserves is maintained at zero), and no international borrowing and lending, the bond-market
clearing condition is
1
3
B
y,t
+
1
3
B
m,t
= 0. [2.11]
Equilibrium quantities in the bond market can be summarized by one variable: the gross amount
of bonds issued.
9
Let B
t
denote gross bond issuance per person, L
t
the implied real value of the
loans that are made per person, and D
t
the real value of debt liabilities per person that fall due at
time t. Assuming that (as will be conrmed later) the young will sell bonds and the middle-aged
will buy them, these variables are given by:
B
t

B
y,t
3
, L
t

Q
t
B
t
P
t
, and D
t

B
t1
P
t
. [2.12]
It is convenient to introduce variables for age-specic consumption, loans, and debt liabilities mea-
sured relative to GDP Y
t
. These are denoted with lower-case letters. The real return (ex post) r
t
between periods t 1 and t is dened as the percentage by which the real value of debt liabilities
is greater than the real amount of the corresponding loans. These denitions are listed below:
c
y,t

C
y,t
Y
t
, c
m,t

C
m,t
Y
t
, c
o,t

C
o,t
Y
t
, l
t

L
t
Y
t
, d
t

D
t
Y
t
, and r
t

D
t
L
t1
L
t1
. [2.13]
Using the denitions of the debt-to-GDP and loans-to-GDP ratios from [2.13] it follows that:
d
t
=
_
1 + r
t
1 + g
t
_
l
t1
. [2.14a]
The real interest rate
t
(ex-ante real return) between periods t and t+1 is dened as the conditional
expectation of the real return between those periods:
10

t
= E
t
r
t+1
. [2.14b]
Using the age-specic incomes [2.3] and the denitions in [2.12] and [2.13], the budget identities
in [2.9] for each generation can be written as:
c
y,t
= 1 + 3l
t
, c
m,t
= 1 + (1 +) 3d
t
3l
t
, and c
o,t
= 1 + 3d
t
. [2.14c]
Similarly, after using the denitions in [2.12] and [2.13], the Euler equations in [2.10] become:
E
t
_

_
(1 + r
t+1
)(1 + g
t+1
)

_
_
_
(1 + g
t+1
)v
m,t+1
E
t
_
(1 + g
t+1
)
1
v
1
m,t+1
1
1
_
_
_
1

_
c
m,t+1
c
y,t
_

_
= 1
= E
t
_

_
(1 + r
t+1
)(1 + g
t+1
)

_
_
_
(1 + g
t+1
)v
o,t+1
E
t
_
(1 + g
t+1
)
1
v
1
o,t+1
1
1
_
_
_
1

_
c
o,t+1
c
m,t
_

_
, [2.14d]
9
In equilibrium, the net bond positions of the household sector and the whole economy are of course both zero
under the assumptions made.
10
This real interest rate is important for saving and borrowing decisions, but there is no actual real risk-free asset
to invest in.
10
where v
m,t
V
m,t
/Y
t
and v
o,t
V
o,t
/Y
t
denote the continuation values of middle-aged and old
individuals relative to GDP. Using equation [2.1], these value functions satisfy:
v
m,t
=
_
c
1
1

m,t
+
_
E
t
_
(1 + g
t+1
)
1
v
1
o,t+1
1
1
_
1
1

_
1
1
1

, and v
o,t
= c
o,t
. [2.14e]
The ex-post Fisher equation for the real return on nominal bonds is obtained from the no-
arbitrage condition [2.8] and the denitions in [2.12]:
1 + r
t
=
1 + i
t1
1 +
t
. [2.15]
Finally, goods-market clearing [2.7] with the denition of aggregate consumption [2.2] requires:
1
3
c
y,t
+
1
3
c
m,t
+
1
3
c
o,t
= 1. [2.16]
Before examining the equilibrium of the economy under dierent monetary policies, it is helpful
to study as a benchmark the hypothetical world of complete nancial markets.
2.2 The complete nancial markets benchmark
Suppose it were possible for individuals to take short and long positions in a range of Arrow-Debreu
securities for each possible state of the world. Suppose markets are sequentially complete in that
securities are traded period-by-period for states of the world that will be realized one period in
the future, and that individuals only participate in nancial markets during their actual lifetimes
(instead of all trades taking place at the beginning of time).
11
Without loss of generality, assume the
payos of these securities are specied in terms of real consumption, and their prices are quoted in
real terms. Let K
t+1
denote the kernel of prices for securities with payos of one unit of consumption
at time t +1 in terms of consumption at time t. The prices are dened relative to the (conditional)
probabilities of each state of the world.
Let S
y,t+1
and S
m,t+1
denote the per-person net positions in the Arrow-Debreu securities at the
end of period t of the young and middle-aged respectively (with S
o,t+1
= 0 for the old, who hold
no assets at the end of period t). These variables give the real payos individuals will receive (or
make, if negative) at time t + 1. The price of taking net position S
t+1
at time t is E
t
K
t+1
S
t+1
(if
negative, this is the amount received from selling securities).
In what follows, the levels of consumption obtained with complete markets (and the corre-
sponding value functions) are denoted with an asterisk to distinguish them from the outcomes with
incomplete markets. The budget identities of the young, middle-aged, and old are:
C

y,t
+E
t
K
t+1
S
y,t+1
= Y
y,t
, C

m,t
+E
t
K
t+1
S
m,t+1
= Y
m,t
+S
y,t
, and C

o,t
= Y
o,t
+S
m,t
. [2.17]
Maximizing utility [2.1] for each generation with respect to holdings of Arrow-Debreu securities,
11
This distinction is relevant here. As will be seen, sequential completeness is the appropriate notion of complete
markets for studying the issues that arise in this paper.
11
subject to the budget identities [2.17], implies the Euler equations:

_
_
_
V

m,t+1
E
t
_
V

1
m,t+1
1
1
_
_
_
1

_
C

m,t+1
C

y,t
_

= K
t+1
=
_
_
_
V

o,t+1
E
t
_
V

1
o,t+1
1
1
_
_
_
1

_
C

o,t+1
C

m,t
_

, [2.18]
where these hold in all states of the world at time t +1. Market clearing for Arrow-Debreu securities
requires:
1
3
S
y,t
+
1
3
S
m,t
= 0. [2.19]
Let S
t+1
denote the gross quantities of Arrow-Debreu securities issued at the end of period t.
By analogy with the denitions of L
t
and D
t
in the case of incomplete markets (from [2.12]), let
L

t
denote the value of all securities sold, which represents the amount lent to borrowers, and let
D

t
be the state-contingent quantity of securities liable for repayment, the equivalent of borrowers
debt liabilities. Supposing, as will be conrmed, that securities would be issued by the young and
bought by the middle-aged, these variables are given by:
S
t+1

S
y,t+1
3
, L

t
E
t
K
t+1
S
t+1
, and D

t
S
t
. [2.20]
In what follows, let c

y,t
, c

m,t
, c

o,t
, l

t
, d

t
, and r

t
denote the complete-markets equivalents of the
variables dened in [2.13].
Starting from the denitions in [2.13] and [2.20], it can be seen that equation [2.14a] also holds for
the complete-markets variables d

t
, r

t
, and l

t1
. The real interest rate is dened as the expectation
of the real return, so equation [2.14b] also holds for

t
and r

t+1
. Using the age-specic income levels
from [2.3] and the denitions from [2.13] and [2.20], the budget identities [2.17] can be written as
in equation [2.14c] with l

t
and d

t
. The denition of the real return r

t
together with [2.20] implies
that E
t
[(1 +r

t+1
)K
t+1
] = 1. Using these denitions again, the Euler equations [2.18] imply that the
equations in [2.14d] hold for c

y,t
, c

m,t
, c

o,t
, v

m,t
, v

o,t
, and r

t
, with the value functions satisfying the
equivalent of [2.14e].
It is seen that all of equations [2.14a][2.14e] in the incomplete-markets model hold also under
complete markets. The distinctive feature of complete markets is that the Euler equations in [2.18]
also imply the following equation holds in all states of the world:
_
_
_
(1 + g
t+1
)v

m,t+1
E
t
_
(1 + g
t+1
)
1
v

1
m,t+1
1
1
_
_
_
1

_
c

m,t+1
c

y,t
_

=
_
_
_
(1 + g
t+1
)v

o,t+1
E
t
_
(1 + g
t+1
)
1
v

1
o,t+1
1
1
_
_
_
1

_
c

o,t+1
c

m,t
_

.
[2.21]
This condition reects the distribution of risk that is mutually agreeable among individuals who have
access to a complete set of nancial markets. The condition equates the growth rates of marginal
utilities of those individuals whose lives overlap (and their consumption growth rates in the case of
time-separable utility).
12
2.3 Equilibrium conditions
There are nine endogenous real variables: the age-specic consumption ratios c
y,t
, c
m,t
, and c
o,t
; the
value functions v
m,t
and v
o,t
; the loans- and debt-to-GDP ratios l
t
and d
t
; and the real interest rate

t
and the ex-post real return r
t
. Real GDP growth g
t
is exogenous and given by [2.4]. Common to
both incomplete and complete nancial markets are the ten equations in [2.14a][2.14e] and [2.16].
By Walras law, one of these equations is redundant, so the goods-market clearing condition [2.16]
(seen to be implied by [2.14c]) is dropped in what follows.
With incomplete markets, the equilibrium conditions [2.14a][2.14e] are augmented by the ex-
post Fisher equation [2.15], to which must be added a monetary policy rule since this equation refers
to the nominal interest rate. Thus, two equations are added, corresponding to the two extra nominal
variables, the ination rate
t
and the nominal interest rate i
t
. With complete markets, one extra
equation [2.21] is appended to the system [2.14a][2.14e]. There are no extra endogenous variables,
but condition [2.21] renders redundant one of the two equations in [2.14d]. Since none of the
equilibrium conditions includes nominal variables, the complete-markets equilibrium is independent
of monetary policy.
Finding the equilibrium with incomplete markets is complicated by the fact that the real payo
of the nominal bond is endogenous to monetary policy. However, owing to the substantial overlap
between the equilibrium conditions under incomplete and complete markets, there is a sense in
which there is only one degree of freedom for the equilibria in these two cases to dier, and thus
only one degree of freedom for monetary policy to aect the equilibrium with incomplete markets.
To make this analysis precise, dene
t
to be the realized debt-to-GDP ratio relative to its
expected value:

t

d
t
E
t1
d
t
, with
t
=
_
1 + r
t
1 + g
t
_
_
E
t1
_
1 + r
t
1 + g
t
_
. [2.22]
The second equation states that
t
is also the unexpected component of portfolio returns r
t
relative
to GDP growth g
t
, where that equation follows from [2.14a]. With complete markets, equations
[2.13] and [2.20] imply that

t
= (S
t
/(1 +g
t
))/E
t1
[S
t
/(1 +g
t
)]. Since the portfolio S
t
is a variable
determined by borrowers and savers choices,

t
is also determined. With incomplete markets, it
can be seen from equation [2.15] that
t
will depend on monetary policy. But once
t
is determined,
portfolio returns in all states of the world (relative to expectations) are known, which closes the
model.
If
t
has been determined then equation [2.22] implies that the debt-to-GDP ratio d
t
is a state
variable. In the model, the debt-to-GDP ratio is a sucient statistic for the wealth distribution
at the beginning of period t. It would therefore be expected that there is a unique equilibrium
conditional on having determined
t
. There are two caveats to this. First, since the model does not
feature a representative agent, there is the possibility of multiple equilibria if substitution eects
were too weak relative to income eects. Second, since the model features overlapping generations of
individuals, there is the possibility of multiple equilibria due to rational bubbles. Suitable parameter
restrictions will be imposed to rule out both of these types of multiplicity.
13
Given that d
t
is a state variable, the uniqueness of the equilibrium will depend on the system
of equations [2.14a][2.14e] having the saddlepath stability property together with a unique steady
state. This issue is investigated by examining the perfect-foresight paths implied by equations
[2.14a][2.14e]. Starting from time t
0
onwards, suppose there are no shocks to real GDP growth
( = 0 in [2.4]) so g
t
= g, and suppose there is no uncertainty about portfolio returns, hence
t
= 1.
With future expectations equal to the realized values of variables, equations [2.14b] and [2.14d]
reduce to:

t
= r
t+1
, and (1+r
t+1
)(1+g
t+1
)

_
c
m,t+1
c
y,t
_

= 1 = (1+r
t+1
)(1+g
t+1
)

_
c
o,t+1
c
m,t
_

. [2.23]
The perfect-foresight paths are determined by equations [2.14a], [2.14c], and [2.23] (with no uncer-
tainty, [2.14e] is redundant). The analysis proceeds by reducing this system to two equations in
two variables: one non-predetermined variable, the real interest rate
t
, and one state variable, the
debt-to-GDP ratio d
t
.
Proposition 1 The system of equations [2.14a], [2.14c], and [2.23] has the following properties:
(i) Any perfect foresight paths {
t
0
,
t
0
+1
,
t
0
+2
, . . .} and {d
t
0
, d
t
0
+1
, d
t
0
+2
, . . .} must satisfy a pair
of rst-order dierence equations F(
t
, d
t
,
t+1
, d
t+1
) = 0.
(ii) The system of equations has a steady state:

d =

3
,

l =

3
, c
y
= c
m
= c
o
= 1, and r = =
1 + g

1 =
(1 + g)
1

1, [2.24]
where [2.4] and [2.5] imply = (1 + g)
1
1

when = 0. The steady state is not dynamically

inecient ( > g) if satises 0 < < 1. Given 0 < < 1, this steady state is unique if and
only if:
(, ), where

1 +
< (, ) <
1
2
, lim
0
(, ) = 0, and
(, )

> 0. [2.25]
(iii) If the parameter restrictions [2.5] and (, ) are satised then in the neighbourhood of
the steady state there exists a stable manifold and an unstable manifold. The stable manifold
is an upward-sloping line in (d
t
,
t
) space, and the unstable manifold is either downward sloping
or steeper than the stable manifold.
Proof See appendix.
Focusing rst on the steady state, note that given the age prole of income in Figure 1 and a
preference for consumption smoothing, the young would like to borrow and the middle-aged would
like to save. In the absence of any uctuations in real GDP, and with the parameterization of the age
prole of income in [2.6], the model possesses a steady state where the age-prole of consumption is
at over the life-cycle. The parameterization [2.6] also has the convenient property that the value of
debt obligations at maturity relative to GDP is solely determined by the income age-prole gradient
parameter , while the formula for the equilibrium real interest rate is identical that found in an
14
Figure 2: Borrowing and saving patterns
Young Middle-aged Old
Repay Lend
Time
Young Middle-aged Old
Young Middle-aged Old
Generation t
Generation t + 1
Generation t + 2
t t + 1 t + 2 t + 3 t + 4
Repay Lend
economy with steady-state real GDP growth of g and a representative agent having discount factor
and elasticity of intertemporal substitution . Greater changes in individuals incomes over the
life-cycle imply that there will be more borrowing in equilibrium, while faster GDP growth or greater
impatience increase the real interest rate. With the parameter restriction 0 < < 1 from [2.5], the
steady state is not dynamically inecient (the real interest rate exceeds the growth rate g).
The equilibrium borrowing and saving patterns are depicted in Figure 2. The young borrow from
the middle-aged and repay once they, the young, are middle-aged and the formerly middle-aged are
old.
12
Lending to the young provides a way for the middle-aged to save. Note that all savings are
held in the form of inside nancial assets (private IOUs) created by those who want to borrow.
Under the models simplifying assumptions, there are no outside assets (for example, government
bonds or at money).
13
Given that 0 < < 1, Proposition 1 shows that the steady state [2.24] is unique if the elas-
ticity of intertemporal substitution is suciently large relative to the gradient of the age-prole of
income. These two conditions are sucient to rule out multiple equilibria, and will be assumed in
what follows. Out of steady state, the dynamics of the debt ratio and the real interest rate are de-
termined by the rst-order dierence equation from Proposition 1, which in principle can be solved
for (d
t+1
,
t+1
) given (d
t
,
t
). With a unique steady state, the model has the property of saddlepath
12
The model is designed to represent a pure credit economy where the IOUs of private agents are exchanged for
goods, and where IOUs can be created without the need for nancial intermediation. The role of money is conned
to that of a unit of account and a standard of deferred payments (what borrowers are promising to deliver when their
IOUs mature). The downplaying of moneys role as a medium of exchange is in line with Woodfords (2003) cashless
limit analysis where the focus is on the use of money as a unit of account in setting prices of goods.
13
The trade between the generations would not be feasible in an overlapping generations model with two-period
lives. In that environment, saving is only possible by acquiring a physically storable asset or holding an outside
nancial asset. In the three-period lives OLG model of Samuelson (1958), the age prole of income is monotonic, so
there is little scope for trade between the generations. As a result, the equilibrium involving only inside nancial
assets is dynamically inecient. This ineciency can be corrected by introducing an outside nancial asset. Here,
under the parameter restrictions, the steady-state real interest rate is above the economys growth rate, which is
equivalent to the absence of dynamic ineciency (Balasko and Shell, 1980). There are then no welfare gains from
introducing an outside asset.
15
stability: starting from a particular debt ratio d
t
0
at time t
0
, there is only one real interest rate
t
0
consistent with convergence to the steady state.
14
3 Monetary policy in a pure credit economy
This section analyses optimal monetary policy in an economy with incomplete markets subject to
exogenous shocks to real GDP growth. A benchmark for monetary policy analysis is the equilibrium
in the hypothetical case of complete nancial markets.
3.1 The natural debt-to-GDP ratio
In monetary economics, it is conventional to use the prex natural to describe what the equilibrium
would be in the absence of a particular friction, such as nominal rigidities or imperfect information.
For instance, there are the concepts of the natural rate of unemployment, the natural rate of interest,
and the natural level of output. Here, the friction is incomplete markets, not nominal rigidities,
but it makes sense to refer to the equilibrium debt-to-GDP ratio in the absence of this friction as
the natural debt-to-GDP ratio. Just like any other natural variable, the natural debt-to-GDP
ratio is independent of monetary policy, while shocks will generally perturb the actual equilibrium
debt-to-GDP ratio away from its natural level, to which it would otherwise converge. Furthermore,
the natural debt-to-GDP ratio has eciency properties that make it a desirable target for monetary
policy.
The natural debt-to-GDP need not be constant when the economy is hit by shocks (just as the
natural rate of unemployment may change over time), but there are two benchmark cases where it
is in fact constant even though shocks occur. These cases require restrictions either on the utility
function or on the stochastic process for GDP growth.
Proposition 2 Consider the equilibrium of the economy with complete nancial markets (the so-
lution of equations [2.14a][2.14e] and [2.21]). If either of the following conditions is met:
(i) the utility function is logarithmic ( = 1 and = 1 in [2.1]);
(ii) real GDP follows a random walk (the random variable x
t
in [2.4] is i.i.d.);
then the equilibrium is as follows, with a constant natural debt-to-GDP ratio:
d

t
=

3
, l

t
=

3
, c

y,t
= c

m,t
= c

o,t
= 1,

t
=
1 +E
t
g
t+1

1, and r

t
=
1 + g
t

1. [3.1]
The real interest rate is also constant (

t
= (1 + g)/ 1) when GDP growth is i.i.d.
Proof See appendix.
14
Proposition 1 establishes the saddlepath stability property locally for parameters for which there is a unique
steady state. Numerical analysis conrms the saddlepath stability property holds globally for these parameters. See
appendix for further details, including a discussion of why non-convergent paths cannot be equilibria.
16
Intuitively, the case of real GDP following a random walk can be understood as follows. If a
shock has the same eect on the level of GDP in the short run and the long run then it is feasible
in all current and future time periods for each generation alive to receive the same consumption
share of total output as before the shock. Since the utility function is homothetic, given relative
prices for consumption in dierent time periods, individuals would choose future consumption plans
proportional to their current consumption. If consumption shares are maintained then no change of
relative prices is required. In this case, all individuals would have the same proportional exposure
to consumption risk. Since each individual has a constant coecient of relative risk aversion, and
as this coecient is the same across all individuals, constant consumption shares are equivalent
to ecient risk sharing. For constant consumption shares to be consistent with individual budget
constraints it is necessary that debt repayments move one-for-one with changes in GDP. Thus, the
ecient nancial contract between borrowers and savers resembles an equity share in GDP, which
is equivalent to a constant natural debt-to-GDP ratio.
If the short-run and long-run eects of a shock to GDP dier then it is not feasible at all times
for generations to maintain unchanged consumption shares because generations do not perfectly
overlap. Relative prices of consumption at dierent times will have to change, which will generally
change individuals desired expenditure shares of lifetime income on consumption at dierent times.
However, with a logarithmic utility function, current consumption will be an unchanging share of
lifetime income, and so ecient risk sharing (given that all individuals have log utility) requires
stabilization of individuals consumption shares. This again requires debt repayments that move in
line with GDP.
The debt gap

d
t
is dened as the actual debt-to-GDP ratio (d
t
) relative to what the debt-to-
GDP ratio would be with complete nancial markets (d

t
):

d
t

d
t
d

t
. [3.2]
This concept is analogous to variables such as the output gap or interest-rate gap found in many
monetary models. The next section justies the claim that the goal of monetary policy should be
close the debt gap, that is, to aim for

d
t
= 1.
3.2 Pareto ecient allocations
Before considering what can be achieved by a central bank setting monetary policy, rst consider
the economy from the perspective of a social planner who has the power to mandate allocations
of consumption to specic individuals by directly making the appropriate transfers. The planner
maximizes a weighted sum of individual utilities subject to the economys resource constraint.
Starting at some time t
0
, the welfare function maximized by the planner is
W
t
0
= E
t
0
2
_
1
3

t=t
0
2

tt
0

t
U
t
_
, [3.3]
which includes the utility functions [2.1] of all individuals alive at some point from time t
0
onwards.
17
The Pareto weight assigned to the generation born at time t is denoted by
tt
0

t
/3, where the
variable
t
is scaled for convenience by the term
tt
0
(using from [2.5] as a discount factor),
and by the population share 1/3 of that generation when its members are alive. A Pareto-ecient
allocation is a maximum of [3.3] subject to the economys resource constraints for a particular
sequence of Pareto weights {
t
0
2
,
t
0
1
,
t
0
,
t
0
+1
, . . .}, where the weight
t
for individuals born
at time t may be a function of the state of the world at time t.
15
The Lagrangian for maximizing the social welfare function subject to the economys resource
constraint C
t
= Y
t
(with aggregate consumption C
t
as dened in [2.2]) is:
L
t
0
= E
t
0
2
_
1
3

t=t
0
2

tt
0

t
U
t
+

t=t
0

tt
0

t
_
Y
t

1
3
C
y,t

1
3
C
m,t

1
3
C
o,t
_
_
, [3.4]
where the Lagrangian multiplier on the time-t resource constraint is
tt
0

t
(the scaling by
tt
0
is for convenience). Using the utility function [2.1], the rst-order conditions for the consumption
levels C

y,t
, C

m,t
and C

o,t
that maximize the welfare function [3.3] are:

t
C

y,t
=

t
,
t1
_

_
_
V

m,t
E
t1
[V

1
m,t
]
1
1
_1

m,t
=

t
, and

t2
_

_
2
_
V

o,t
E
t1
[V

1
o,t
]
1
1
_1

_
V

m,t1
E
t2
[V

1
m,t1
]
1
1
_1

o,t
=

t
for all t t
0
. [3.5]
Since the rst-order conditions are homogeneous of degree zero in the Pareto weights
t
and the
Lagrangian multipliers
t
, one of the weights or one of the multipliers can be arbitrarily xed. The
normalization
t
0
Y
1
t
0
is chosen, which has the convenient implication that a 0.01 change in the
value of the welfare function is equivalent to an exogenous 1% change in real GDP in the initial
period.
16
Since the normalization uses output Y
t
0
at time t
0
, the Pareto weights
t
0
2
and
t
0
1
may be functions of the state of the world at time t
0
, but the ratio
t
0
1
/
t
0
2
must depend only
on variables known at time t
0
1. The welfare function [3.3] and rst-order conditions [3.5] can be
rewritten in terms of stationary variables as follows:
W
t
0
= E
t
0
2
_
1
3

t=t
0
2

tt
0

t
u
t
_
, with
t

t
Y
1
1

t
, u
t

U
t
Y
1
1

t
,
t

t
Y
t
, and
t
0
1.
[3.6]
15
This means that an individual comprises not just a specic person but also a specic history of shocks up to the
time of that persons birth. But the weight is not permitted to be a function of shocks realized after birth because
this would result in an essentially vacuous notion of ex-post eciency where every non-wasteful allocation of goods
could be described as ecient for some sequence of weights that vary during individuals lifetimes. See appendix for
further discussion.
16
Applying the envelope theorem to the Lagrangian [3.4] yields W
t
0
/Y
t
0
=
t
0
, and hence by setting
t
0
= Y
1
t
0
it follows that W
t
0
/ log Y
t
0
= 1.
18
Manipulating the rst-order conditions [3.5] and using the denitions in [2.13] and [3.6] leads to:

t
=

t
c

y,t
, and

t+1

t
= (1 + g
t+1
)
1
1

_
(1 + g
t+1
)v

m,t+1
E
t
[(1 + g
t+1
)
1
v

1
m,t+1
]
1
1
_1

_
c

m,t+1
c

y,t
_

= (1 + g
t+1
)
1
1

_
(1 + g
t+1
)v

o,t+1
E
t
[(1 + g
t+1
)
1
v

1
o,t+1
]
1
1
_1

_
c

o,t+1
c

m,t
_

for all t t
0
, [3.7]
where these equations hold in all states of the world. There is a well-dened steady state for all
of the transformed variables in [3.6]. Using Proposition 1 together with equations [2.1], [3.6], and
[3.7], it follows that = 1 and = 1. Given the parameter restriction [2.5], this shows the welfare
function is nite-valued for any real GDP growth stochastic process consistent with [2.4].
The equations in [3.7] imply that the risk-sharing condition [2.21] is a necessary condition for any
Pareto-ecient consumption allocation. This equation is an equilibrium condition with complete
nancial markets, so the complete-markets equilibrium will be Pareto ecient.
17
However, there are
many other Pareto-ecient allocations satisfying the resource constraint [2.16] and the risk-sharing
condition [2.21].
Now return to the analysis of monetary policy where the policymaker is a central bank with
a single instrument, the nominal interest rate i
t
. The central bank operates in an economy with
incomplete markets where the equilibrium conditions are [2.14a][2.14e] and [2.15]. The central
bank maximizes the welfare function [3.3] subject to the incomplete-markets equilibrium conditions
as implementability constraints (including [2.14c], which implies the resource constraint [2.16]).
The solution will depend on which Pareto weights
t
are used, which capture the distributional
preferences of the policymaker.
Two questions regarding eciency and distribution naturally arise when studying the central
banks constrained maximization problem. First, the extent to which the central bank will be able to
achieve a Pareto-ecient consumption allocation. Second, the considerations that should guide the
choice of the Pareto weights determining the policymakers distributional preferences. The second
question is less familiar in optimal monetary policy analysis because much existing work is based
on models with a representative agent. The approach adopted here is to assume the central bank
strives for Pareto eciency and will always sacrice distributional concerns to eciency (that is, it
has a lexicographic preference for eciency). The following result provides some guidance for such
a central bank.
Proposition 3 (i) A state-contingent consumption allocation {c

y,t
.c

m,t
, c

o,t
} is Pareto ecient
from t t
0
onwards if and only if it satises the resource constraint [2.16] for all t t
0
, the
risk-sharing condition [2.21] for all t t
0
, and is such that v

o,t
0
c

o,t
0
/v

m,t
0
c

m,t
0
depends
only on variables known at time t
0
1. The complete-markets equilibrium (with markets open
from at least time t
0
1 onwards) is Pareto ecient from t t
0
.
17
There are two caveats to this claim specic to overlapping generations models: the question of whether the utility
functions of the individuals considered by the social planner should be evaluated as expectations over shocks realized
prior to birth, and the possibility of dynamic ineciency. As discussed in appendix, while these issues are potentially
important, neither of them is relevant in this paper.
19
(ii) If a Pareto-ecient consumption allocation can be implemented through monetary policy from
time t
0
onwards then this allocation must be the complete-markets equilibrium (with markets
open from time t
0
1 onwards).
Proof See appendix.
The rst part conrms that the complete-markets equilibrium is one of the many Pareto-ecient
consumption allocations. More importantly, the second part states that the complete-markets equi-
librium is the only Pareto-ecient allocation that can be implemented in an incomplete-markets
economy by a central bank setting interest rates (rather than by a social planner who can make
direct transfers). The intuition is that the risk-sharing condition [2.21] is necessary for Pareto ef-
ciency, but this is also the only equation that diers between the equilibrium conditions of the
incomplete- and complete-markets economies. This result is useful because it provides a unique
answer to the question of the choice of Pareto weights for a central bank that always prioritizes
eciency over distributional concerns. This avoids the need to specify the political preferences of
the central bank when analysing optimal monetary policy in a non-representative-agent economy.
Therefore, in what follows, monetary policy is evaluated using the Pareto weights

t
consistent with
the complete-markets equilibrium.
3.3 Optimal monetary policy
Optimal monetary policy is dened as the constrained maximum of the welfare function [3.3] subject
to the equilibrium conditions [2.14a][2.14e] and [2.15] as constraints, and using Pareto weights

t
consistent with the complete-markets equilibrium. Monetary policy has a single instrument, and
this can be used to generate any state-contingent path for one nominal variable, for example, the
price level (accepting the equilibrium values of other nominal variables). For simplicity, monetary
policy is modelled as directly choosing this nominal variable, while the question of what interest-rate
policy would be needed to implement it is deferred for later analysis.
In characterizing the optimal policy it is helpful to introduce the denition of nominal GDP
M
t
P
t
Y
t
. Given the denitions of ination
t
and real GDP growth g
t
, the dynamics of nominal
GDP can be written as M
t
= (1 +
t
)(1 + g
t
)M
t1
. Using this equation together with [2.14a]
and [2.15], the following link between the unexpected components of the debt-to-GDP ratio d
t
and
nominal GDP is obtained:
d
t
E
t1
d
t
=
M
1
t
E
t1
M
1
t
. [3.8]
This equation indicates that stabilizing the ratio of debt liabilities to income is related to stabilizing
the nominal value of income. The intuition is that d
t
can be written as a ratio of nominal debt
liabilities to nominal income. Since nominal debt liabilities are not state contingent, any unpre-
dictable change in the ratio is driven by unpredictable changes in nominal GDP. This leads to the
main result of the paper.
20
Proposition 4 The complete-markets equilibrium can be implemented by monetary policy in the
incomplete-markets economy, closing the debt gap (

d
t
= 1) from [3.2]. This equilibrium is obtained
if and only if monetary policy determines a level of nominal GDP M

t
such that:
M

t
= d

t
1
X
t1
, [3.9]
where d

t
is the debt-to-GDP ratio in the complete-markets economy and X
t1
is any function of
variables known at time t 1.
Proof See appendix.
To understand the intuition for this result, consider an economy where shocks to GDP are
permanent or individuals have logarithmic utility functions. In those cases, Proposition 2 shows that
ecient risk sharing requires debt repayments that rise and fall exactly in proportion to income.
Decentralized implementation of this risk sharing entails individuals trading securities with state-
contingent payos, or equivalently, writing contracts that spell out a complete schedule of varying
repayments across dierent states of the world. Incomplete nancial markets preclude this, and the
assumption of the model is that individuals are restricted to the type of non-contingent nominal
debt contracts commonly observed. In this environment, ecient risk sharing will break down when
debtors are obliged to make xed repayments from future incomes that are uncertain.
In an economy that is hit by aggregate shocks, irrespective of what monetary policy is followed,
there will always be uncertainty about future real GDP. However, there is nothing in principle
to prevent monetary policy stabilizing the nominal value of GDP. In the absence of idiosyncratic
shocks, nominal GDP targeting would remove any uncertainty about nominal incomes, ensuring
that even non-contingent nominal debt repayments maintain a stable ratio to income in all states
of the world, and thus achieves ecient risk sharing.
3.4 Discussion
The importance of these arguments for nominal GDP targeting obviously depends on the plausi-
bility of the incomplete-markets assumption in the context of household borrowing and saving. It
seems reasonable to suppose that individuals will not nd it easy to borrow by issuing Arrow-Debreu
state-contingent bonds, but might there be other ways of reaching the same goal? Issuance of state-
contingent bonds is equivalent to households agreeing loan contracts with nancial intermediaries
that specify a complete menu of state-contingent repayments. But such contracts would be much
more time consuming to write, harder to understand, and more complicated to enforce than conven-
tional non-contingent loan contracts, as well as making monitoring and assessment of default risk a
more elaborate exercise.
18
Moreover, unlike rms, households cannot issue securities such as equity
that feature state-contingent payments but do not require a complete description of the schedule of
payments in advance.
19
18
For examples of theoretical work on endogenizing the incompleteness of markets through limited enforcement of
contracts or asymmetric information, see Kehoe and Levine (1993) and Cole and Kocherlakota (2001).
19
Consider an individual owner of a business that generates a stream of risky prots. If the rms only external
nance is non-contingent debt then the individual bears all the risk (except in the case of default). If the individual
21
Another possibility is that even if individuals are restricted to non-contingent borrowing, they can
hedge their exposure to future income risk by purchasing an asset with returns that are negatively
correlated with GDP. But there are several pitfalls to this. First, it may not be clear which asset
reliably has a negative correlation with GDP (even if GDP securities of the type proposed by
Shiller (1993) were available, borrowers would need a short position in these). Second, the required
gross positions for hedging may be very large. Third, an individual already intending to borrow
will need to borrow even more to buy the asset for hedging purposes, and the amount of borrowing
may be limited by an initial down-payment constraint and subsequent margin calls. In practice,
a typical borrower does not have a signicant portfolio of assets except for a house, and housing
returns most likely lack the negative correlation with GDP required for hedging the relevant risks.
In spite of these diculties, it might be argued the case for the incomplete markets assumption
is overstated because the possibilities of renegotiation, default, and bankruptcy introduce some
contingency into apparently non-contingent debt contracts. However, default and bankruptcy allow
for only a crude form on contingency in extreme circumstances, and these options are not without
their costs. Renegotiation is also not costless, and evidence from consumer mortgages in both the
recent U.S. housing bust and the Great Depression suggests that the extent of renegotiation may be
ineciently low (White, 2009a, Piskorski, Seru and Vig, 2010, Ghent, 2011). Furthermore, even ex-
post ecient renegotiation of a contract with no contingencies written in ex ante need not actually
provide for ecient sharing of risk from an ex-ante perspective.
It is also possible to assess the completeness of markets indirectly through tests of the ecient
risk-sharing condition, which is equivalent to correlation across consumption growth rates of indi-
viduals. These tests are the subject of a large literature (Cochrane, 1991, Nelson, 1994, Attanasio
and Davis, 1996, Hayashi, Altonji and Kotliko, 1996), which has generally rejected the hypothesis
of full risk sharing.
Finally, even if nancial markets are incomplete, the assumption that contracts are written in
terms of specically nominal non-contingent payments is important for the analysis. The evidence
presented in Doepke and Schneider (2006) indicates that household balance sheets contain signicant
quantities of nominal liabilities and assets (for assets, it is important to account for indirect exposure
via households ownership of rms and nancial intermediaries). Furthermore, as pointed out by
Shiller (1997), indexation of private debt contracts is extremely rare. This suggests the models
assumptions are not unrealistic.
The workings of nominal GDP targeting can also be seen from its implications for ination and
the real value of nominal liabilities. Indeed, nominal GDP targeting can be equivalently described
as a policy of inducing a perfect negative correlation between the price level and real GDP, and
ensuring these variables have the same volatility. When real GDP falls, ination increases, which
wanted to share risk with other investors then one possibility would be to replace the non-contingent debt with
state-contingent bonds where the payos on these bonds are positively related to the rms prots. However, what
is commonly observed is not issuance of state-contingent bonds but equity nancing. Issuing equity also allows for
risk sharing, but unlike state-contingent bonds does not need to spell out a schedule of payments in all states of the
world. There is no right to any specic payment in any specic state at any specic time, only the right of being
residual claimant. The lack of specic claims is balanced by control rights over the rm. However, there is no obvious
way to be residual claimant on or have control rights over a household.
22
reduces the real value of xed nominal liabilities in proportion to the fall in real income, and vice
versa when real GDP rises. Thus the extent to which nancial markets with non-contingent nominal
assets are suciently complete to allow for ecient risk sharing is endogenous to the monetary policy
regime: monetary policy can make the real value of xed nominal repayments contingent on the
realization of shocks. A strict policy of ination targeting would be inecient because it converts
non-contingent nominal liabilities into non-contingent real liabilities. This points to an inherent
tension between price stability and the ecient operation of nancial markets.
20
That optimal monetary policy in a non-representative-agent
21
model should feature ination uc-
tuations is perhaps surprising given the long tradition of regarding ination-induced unpredictability
in the real values of contractual payments as one of the most important of all inations costs. As
discussed in Clarida, Gal and Gertler (1999), there is a widely held view that the diculties this
induces in long-term nancial planning ought to be regarded as the most signicant cost of ina-
tion, above the relative price distortions, menu costs, and deviations from the Friedman rule that
have been stressed in representative-agent models. The view that unanticipated ination leads to
inecient or inequitable redistributions between debtors and creditors clearly presupposes a world
of incomplete markets, otherwise ination would not have these eects. How then to reconcile this
argument with the result that incompleteness of nancial markets suggests nominal GDP targeting
is desirable because it supports ecient risk sharing? (again, were markets complete, monetary
policy would be irrelevant to risk sharing because all opportunities would already be exploited)
While nominal GDP targeting does imply unpredictable ination uctuations, the resulting real
transfers between debtors and creditors are not an arbitrary redistribution they are perfectly cor-
related with the relevant fundamental shock: unpredictable movements in aggregate real incomes.
Since future consumption uncertainty is aected by income risk as well as risk from uctuations in
the real value of nominal contracts, it is not necessarily the case that long-term nancial planning
is compromised by ination uctuations that have known correlations with the economys funda-
mentals. An ecient distribution of risk requires just such uctuations because the provision of
insurance is impossible without the possibility of ex-post transfers that cannot be predicted ex ante.
Unpredictable movements in ination orthogonal to the economys fundamentals (such as would
occur in the presence of monetary-policy shocks) are inecient from a risk-sharing perspective, but
there is no contradiction with nominal GDP targeting because such movements would only occur if
policy failed to stabilize nominal GDP.
22
It might be objected that if debtors and creditors really wanted such contingent transfers then
they would write them into the contracts they agree, and it would be wrong for the central bank to try
to second-guess their intentions. But the absence of such contingencies from observed contracts may
simply reect market incompleteness rather than what would be rationally chosen in a frictionless
20
In a more general setting where the incompleteness of nancial markets is endogenized, ination uctuations
induced by nominal GDP targeting may play a role in minimizing the costs of contract renegotiation or default when
the economy is hit by an aggregate shock.
21
It is implicitly assumed dierent generations do not form the innitely lived dynasties suggested by Barro (1974).
22
The model could be applied to study the quantitative welfare costs of the arbitrary redistributions caused by
ination resulting from monetary-policy shocks. See section 5 for further details.
23
world. Reconciling the non-contingent nature of nancial contracts with complete markets is not
impossible, but it would require both substantial dierences in risk tolerance across individuals
and a high correlation of risk tolerance with whether an individual is a saver or a borrower. With
assumptions on preferences that make borrowers risk neutral or savers extremely risk averse, it
would not be ecient to share risk, even if no frictions prevented individuals writing contracts that
implement it.
There are a number of problems with this alternative interpretation of the observed prevalence of
non-contingent contracts. First, there is no compelling evidence to suggest that borrowers really are
risk neutral or savers are extremely risk averse relative to borrowers. Second, while there is evidence
suggesting considerable heterogeneity in individuals risk tolerance (Barsky, Juster, Kimball and
Shapiro, 1997, Cohen and Einav, 2007), most of this heterogeneity is not explained by observable
characteristics such as age and net worth (even though many characteristics such as these have
some correlation with risk tolerance). The dispersion in risk tolerance among individuals with
similar observed characteristics suggests there should be a wide range of types of nancial contract
with dierent degrees of contingency. Risk neutral borrowers would agree non-contingent contracts
with risk-averse savers, but contingent contracts would be oered to risk-averse borrowers.
Another problem with the complete markets but dierent risk preferences interpretation relates
to the behaviour of the price level over time. While nominal GDP has never been an explicit target
of monetary policy, nominal GDP targetings implication of a countercyclical price level has been
largely true in the U.S. during the post-war period (Cooley and Ohanian, 1991), albeit with a
correlation coecient much smaller than one in absolute value, and a lower volatility relative to
real GDP. Whether by accident or design, U.S. monetary policy has had to a partial extent the
features of nominal GDP targeting, resulting in the real values of xed nominal payments positively
co-moving with real GDP (but by less) on average. In a world of complete markets with extreme
dierences in risk tolerance between savers and borrowers, ecient contracts would undo the real
contingency of payments brought about by the countercyclicality of the price level, for example,
through indexation clauses. But as discussed in Shiller (1997), private nominal debt contracts have
survived in this environment without any noticeable shift towards indexation. Furthermore, both
the volatility of ination and correlation of the price level with real GDP have changed signicantly
over time (the high volatility 1970s versus the Great Moderation, and the countercyclicality of
the post-war price level versus its procyclicality during the inter-war period). The basic form of
non-contingent nominal contracts has remained constant in spite of this change.
23
Finally, while the policy recommendation of this paper goes against the long tradition of citing
the avoidance of redistribution between debtors and creditors as an argument for price stability,
it is worth noting that there is a similarly ancient tradition in monetary economics (which can be
traced back at least to Bailey, 1837) of arguing that money prices should co-move inversely with
productivity to promote fairness between debtors and creditors. The idea is that if money prices
fall when productivity rises, those savers who receive xed nominal incomes are able to share in
23
It could be argued that part of the reluctance to adopt indexation is a desire to avoid eliminating the risk-sharing
oered by nominal contracts when the price level is countercyclical.
24
the gains, while the rise in prices at a time of falling productivity helps to ameliorate the burden
of repayment for borrowers. This is equivalent to stabilizing the money value of incomes, in other
words, nominal GDP targeting. The intellectual history of this idea (the productivity norm) is
thoroughly surveyed in Selgin (1995). Like the older literature, this paper places distributional
questions at the heart of monetary policy analysis, but studies policy through the lens of mitigating
ineciencies in incomplete nancial markets, rather than with looser notions of fairness.
4 Equilibrium in a pure credit economy
In cases where Proposition 2 applies, [3.1] fully characterizes the equilibrium of the economy if the
optimal monetary policy of nominal GDP targeting from Proposition 4 is followed. The equilibrium
with optimal policy under conditions where Proposition 2 is not applicable, or where a non-optimal
monetary policy is followed, cannot generally be found analytically. In what follows, log-linearization
is used to nd an approximate solution to the equilibrium in these cases.
4.1 Log-linear approximation of the equilibrium
The log-linearization is performed around the non-stochastic steady state of the model ( = 0 in [2.4])
as characterized in Proposition 1 (which is valid for suciently small values of the standard deviation
of real GDP growth). Log deviations of variables from their steady-state values are denoted with
sans serif letters,
24
for example, d
t
log d
t
log

d, while for variables that do not necessarily have
a steady state,
25
the sans serif equivalent denotes simply the logarithm of the variable, for example,
Y
t
log Y
t
. In the following, terms that are second-order or higher in deviations from the steady
state are suppressed.
First consider the set of equations [2.14a][2.14e] common to the cases of complete and incomplete
nancial markets. The equation for debt dynamics [2.14a], the denition of the real interest rate
[2.14b], the budget identities [2.14c], and the Euler equations [2.14d] for each generation have the
following log-linear expressions:

t
= E
t
r
t+1
, d
t
= r
t
g
t
+ l
t1
, c
y,t
= l
t
, c
m,t
= d
t
l
t
, c
o,t
= d
t
, [4.1a]
c
y,t
= E
t
c
m,t+1

t
+E
t
g
t+1
, and c
m,t
= E
t
c
o,t+1

t
+E
t
g
t+1
, [4.1b]
observing that the value functions v
m,t
and v
o,t
and the coecient of relative risk aversion do not
appear in these equations.
Proposition 5 The log linear approximation of the solution of equations [4.1a][4.1b] is determined
only up to a martingale dierence stochastic process
t
(E
t1

t
= 0) such that
t
= d
t
E
t1
d
t
is
the unexpected component of the debt-to-GDP ratio dened in [2.22]. Given
t
, the debt-to-GDP
24
For all variables that are either interest rates or growth rates, the log deviation is of the gross rate, for example,
g
t
log(1 + g
t
) log(1 + g).
25
The level of GDP can be either stationary or non-stationary depending on the specication of the stochastic
process for g
t
.
25
ratio is given by
d
t
= d
t1
+(2f
t1
+E
t1
f
t
) +
t
, with f
t

_
1

=1

1
E
t
g
t+
. [4.2]
Given a debt ratio d
t
satisfying [4.2], the other endogenous variables must satisfy:
l
t
=
1
d
t

1
f
t
,
t
=
1

E
t
g
t+1
+

(d
t
+(f
t
+E
t
f
t+1
)) , [4.3a]
c
y,t
= (d
t
+f
t
) , c
m,t
= ((1 )d
t
f
t
) , c
o,t
= d
t
, and [4.3b]
r
t
= d
t
+
1
d
t1
+
1
f
t1
+ g
t
. [4.3c]
All coecients , , , (/), , and are functions only of and the ratio /, and all
are increasing in the ratio /. Formulas for the coecients are given in appendix. The coecients
satisfy 0 < < 1, 0 < < 1, || < 1, || < 1, and both and are positive and bounded.
Proof See appendix
The variable f
t
includes all that needs to be known about expectations of future real GDP growth
to determine equilibrium saving and borrowing behaviour given individuals desire for consumption
smoothing over time. An increase in f
t
leads to a reduction in lending l
t
and a higher real interest rate

t
. However, whether expectations of future growth have a positive or negative eect on f
t
depends
on the relative strengths of income and substitution eects. With strong intertemporal substitution
( > 1), expectations of future growth increase lending by the middle-aged to the young (the eects
of f
t
on the consumption of these two groups always have opposite signs because lending involves a
transfer of resources), while the eect is the opposite if intertemporal substitution is weak ( < 1).
Any unanticipated movements in the debt ratio d
t
constitute transfers from the middle-aged to
the old. These have the eect of pushing up real interest rates because aggregate desired saving falls
following this transfer, and higher real interest rates reduce borrowing by the young. Consistent
with this, consumption of the old is increasing in d
t
, while consumption of both the young and
the middle-aged is decreasing in d
t
(the coecient measures how the eects are spread between
the young and middle-aged in equilibrium). Note that the size of these eects is increasing in
the parameter , and as this parameter tends to zero, the economy behaves as if it contained a
representative agent.
26
With incomplete markets, the system of equations [4.1a][4.1b] is closed (that is,
t
is deter-
mined) by a description of monetary policy and equation [2.15], which has the following log-linear
form:
r
t
= i
t1

t
. [4.4]
With complete markets, the system [4.1a][4.1b] is closed by the risk-sharing equation [2.21]. This
26
With = 0, equations [4.3a] and [4.3b] imply
t
= (1/)E
t
g
t+1
and c
y,t
= c
m,t
= c
o,t
= 0. This means that
C
y,t
= C
m,t
= C
o,t
= C
t
= Y
t
and the representative-agent consumption Euler equation Y
t
= E
t
Y
t+1

t
holds.
Strictly speaking, this limiting case is not a representative-agent model, but because all individuals receive the same
incomes, there is limited scope for trade, so to a rst-order approximation, the economy behaves as if it contained a
representative agent.
26
can be log-linearized as follows:
1

_
(c

m,t+1
c

y,t
) (c

o,t+1
c

m,t
)
_
+
_

1

_
_
(v

m,t+1
E
t
v

m,t+1
) (v

o,t+1
E
t
v

o,t+1
)
_
= 0,
with v

m,t
=
1
1 +
c

m,t
+

1 +
E
t
[c

o,t+1
+ g
t+1
], and v

o,t
= c

o,t
, [4.5]
where the second line log linearizes the value functions appearing in [2.14e]. The following result
rst characterizes the complete-markets equilibrium, then states the equations for the gaps between
variables and their values in the hypothetical complete-markets equilibrium, and nally provides the
link between these gaps and the ination rate.
Proposition 6 The equilibrium with complete nancial markets is given by equations [4.2] and
[4.3a][4.3c] with

t
= d

t
E
t1
d

t
given by

t
=
_
2

1 +
( 1)

(1 +)
_
1
_

__
2 +

1 +
( 1)

_
(f
t
E
t1
f
t
)
+

1 +
( 1)

(E
t
f
t+1
E
t1
f
t+1
)
_
+
1

1 +
( 1)

(E
t
g
t+1
E
t1
g
t+1
)
_
. [4.6]
The debt gap

d
t
d
t
d

t
(from [3.2]) in the incomplete-markets economy must satisfy:
E
t

d
t+1
=

d
t
. [4.7a]
The debt gap is a sucient statistic for describing all deviations of the economy from the hypothetical
complete-markets equilibrium (for example, the real interest rate gap
t

t

t
):

l
t
=
1

d
t
,
t
=

d
t
, c
y,t
=

d
t
, c
m,t
= (1 )

d
t
, and c
o,t
=

d
t
. [4.7b]
The ination rate in the incomplete-markets economy satises:

t
= i
t1

d
t

d
t1
r

t
. [4.7c]
Proof See appendix
The proposition shows how the complete-markets debt-to-GDP ratio d

t
(the natural debt-to-
GDP ratio) can be characterized for general utility function parameters and a general stochastic
process for real GDP growth. In the absence of further shocks, the economy will approach d

t
in
the long run (the debt gap will shrink to zero according to equation [4.7a], noting that || < 1).
However, the debt gap is not automatically closed in the short run following shocks without a
monetary policy intervention. The behaviour of the debt-to-GDP ratio following a shock depends
on the behaviour of nominal GDP (see equation [3.8]):
d
t
E
t1
d
t
= (M
t
E
t1
M
t
). [4.8]
The class of policies that close the debt gap are characterized in Proposition 4. The simplest is a
target for nominal GDP (M
t
= P
t
+ Y
t
) that moves inversely with the natural debt-to-GDP ratio
27
d

t
, that is, M

t
= d

t
. This policy achieves

d
t
= 0, but requires uctuations in ination. The
equilibrium ination rate and nominal interest rate are:

t
= g
t
(d

t
d

t1
), and i
t
=

t
E
t
g
t+1
(E
t
d

t+1
d

t
). [4.9]
The optimal policy only allows nominal GDP to uctuate if the natural debt-to-GDP ratio is time
varying. With real GDP following a random walk or logarithmic utility, d

t
= 0, in which case
the target reduces to M

t
= 0 and the required ination uctuations simply mirror the uctuations
in real GDP growth in the opposite direction. In general, it is a quantitative question how much
optimal policy deviates from a completely stable level of nominal GDP.
4.2 Non-logarithmic utility and predictable variation in GDP growth
To study how much optimal policy deviates from a constant nominal GDP target when the utility
function is not logarithmic and real GDP does not follow a random walk, consider the following
stochastic process for real GDP growth:
g
t
=
t
+
t1
, with
t
i.i.d.(0,

). [4.10]
In this rst-order moving-average process, the parameter represents the dierence between the
long-run eect of a shock
t
on the level of GDP minus its short-run eect ( = 0 corresponds to
the case of a random walk where the long-run eect is identical to the short-run eect). If > 0
then the long-run eect on GDP is greater than the eect in the short run, and vice versa for < 0.
Substituting this stochastic process into [4.6] yields an expression for the innovation

t
= d

t
E
t1
d

t
to the natural debt-to-GDP ratio:

t
= (

1)(Y
t
E
t1
Y
t
), where

= 1+

(1)

_
2 +

1+
(1)

_
+
1

1
1+
(1)

_
2

1+
(1)

(1 +)
.
[4.11]
The coecient

determines how much the debt-to-GDP ratio should rise or fall following a shock
to the level of GDP: the debt ratio should positively co-move with GDP if

> 1, and negatively co-

move if

< 1. As can be seen from the expression for

, determining which case prevails requires

assumptions on the preference parameters and the GDP stochastic process. As an example, consider
the plausible case where intertemporal substitution is relatively low ( < 1) and risk aversion is at
least what would be implied by a time-separable utility function ( 1/). In this case, following
a negative shock to GDP, the debt-to-GDP ratio should rise if the shocks eects are smaller in the
long run than the short run, while the ratio should fall if the long-run eects are larger. Intuitively,
if the economy is expected to recover in the future, debt liabilities should fall by less than current
income does, while if GDP is expected to deteriorate further, the real value of debt liabilities should
fall by more than current income.
Proposition 7 If real GDP growth is described by the stochastic process [4.10] then optimal mon-
etary policy can be described as a constant target for weighted nominal GDP P
t
+

Y
t
= 0, where
28
the weight

on real output is given in equation [4.11].

Proof See appendix.
In the case where real GDP is described by stochastic process [4.10], Proposition 7 shows that
optimal monetary policy can equivalently be expressed in terms of a target for a stable level of
weighted nominal GDP, where is the weight on real GDP relative to the weight on the price level
(standard nominal GDP targeting is = 1). The optimal policy implies P
t
=

Y
t
, so

can
also be interpreted as the optimal countercyclicality of the price level. As an example, consider
the plausible case where the elasticity of intertemporal substitution is relative low ( < 1) and risk
aversion is relatively high ( 1/). It can be seen from [4.11] that when the long-run eect of a
shock to GDP is smaller than its initial eect ( < 0) then

< 1, so following a negative shock to

real GDP, the price level should rise by less than if the shock were permanent. Given parameters
and , the size of the deviation of

from 1 depends on the deviation of the parameter from

zero.
Figure 3: Optimal monetary policy when GDP shocks have dierent short-run and long-run
eects weight

assigned to real GDP relative to price level

Risk aversion ()
Short run > Long run ( < 0)
Intertemporal substitution ()
Short-run to long-run dierence (||)
Risk aversion ()
Short run < Long run ( > 0)
Intertemporal substitution ()
Short-run to long-run dierence (||)
0 0.5 1
0.5 1 1.5 2
2 4 6 8 10
0 0.5 1
0.5 1 1.5 2
2 4 6 8 10
0.75
1
1.25
0.75
1
1.25
0.75
1
1.25
0.75
1
1.25
0.75
1
1.25
0.75
1
1.25
Notes: Monetary policy is P
t
+

Y
t
= 0, where the formula for

is given in [4.11]. The graphs

show the eects of varying one parameter, holding other parameters constant at their baseline values,
given in Table 1 (with the baseline value of set to 0.5).
29
The quantitative deviation of the optimal monetary policy from pure nominal GDP targeting
thus depends rst on how much the stochastic process for real GDP diers from a random walk.
There is an extensive literature that attempts to determine whether shocks to GDP have largely
permanent or transitory eects, in other words, whether GDP is dierence stationary or trend
stationary (see, for example, Campbell and Mankiw, 1987, Durlauf, 1993, Murray and Nelson,
2000). This literature has not reached a consensus, but episodes such as the Great Depression and
the recent Great Recession point towards the existence of shocks where the economy has no strong
tendency to return to the trend line that was expected prior to the shock. For the stochastic process
[4.10], real GDP is described by a random walk when = 0, while the level of GDP is stationary
when = 1, and when 1 < < 0, a partial recovery is expected following a negative shock. The
evidence then suggests a relative low value of may be appropriate.
Even when the parameter signicantly diers from zero, how far optimal policy is from pure
nominal GDP targeting depends on preference parameters. A range of plausible values for these are
studied, as discussed later in section 5.6. For the coecient of relative risk aversion, values between
0.25 and 10 are considered, with 5 as the baseline estimate. For the elasticity of intertemporal
substitution, the range is 0.26 to 2, with 0.9 as the baseline. The values of and are chosen
to match the average real interest rate and debt-to-GDP ratio as described in section 5.6. The
implied values of

are shown in Figure 3. Apart from cases where risk aversion or intertemporal
substitution are extremely low, the value of

lies approximately between 0.75 and 1.25, even with

almost complete trend reversion in real GDP. Therefore, the quantitative deviation of optimal policy
from pure nominal GDP targeting due to trend reversion in real GDP appears to be small.
4.3 Implementation of optimal monetary policy
The analysis so far has assumed that the central bank can directly set the nominal price level or the
nominal value of income. The optimal monetary policy results have thus been stated as targeting
rules, rather than instrument rules. The following result shows how the nominal GDP target can be
implemented by a rule for adjusting the nominal interest rate in response to deviations of nominal
GDP from its target value. This is analogous to the Taylor rules that can be used to implement a
policy of ination targeting.
27
Proposition 8 Suppose the nominal interest rate is set according to the following rule:
i
t
=

t
(E
t
g
t+1
+E
t
d

t+1
d

t
) +(M
t
M

t
), [4.12]
where M

t
= d

t
is the target for nominal GDP. If > 0 then M
t
= M

t
and

d
t
= 0 is the unique
equilibrium in which nominal variables remain bounded. If = 0 then there are multiple equilibria
for the debt gap

d
t
, in all of which nominal variables remain bounded.
Proof See appendix.
27
The use of Taylor rules to determine ination and the price level is studied by Woodford (2003). The determinacy
properties of Taylor rules have been criticized by Cochrane (2011).
30
4.4 Consequences of directly targeting nancial variables
Finally, given that the optimality of targeting nominal GDP derives from its eect on the ratio of
debt liabilities to income, it might be argued that a more immediate way of implementing optimal
policy would be to target the debt-to-GDP ratio directly. While a targeting rule of d
t
= d

t
is feasible
(there is one instrument and one target to hit), this policy has the serious drawback that it fails to
provide a nominal anchor.
Proposition 9 Suppose monetary policy is adjusted to meet the target d
t
= d

t
. The equilibria of
the economy are:

d
t
= 0,
t
= e
t1
+

t1
r

t
, and i
t
=

t
+ e
t
, [4.13]
where e
t
is any arbitrary stochastic process observed at time t.
Proof See appendix.
While this targeting rule achieves Pareto eciency (because

d
t
= 0), it does not uniquely deter-
mine ination expectations because it specied solely in terms of a ratio. Nominal GDP targeting
both achieves eciency and provides a nominal anchor.
4.5 Consequences of ination targeting
The choice of monetary policy in an economy with incomplete nancial markets not only aects the
distribution of risk, but also has implications for the quantity and price of credit. In particular, the
model predicts that if the central bank reduces uctuations in the price level below those consistent
with an ecient distribution of risk then this increases the procyclicality of credit. The more the
price level is stabilized, the more lending rises and interest rates fall during an expansion. In other
words, more stable prices lead to larger uctuations in nancial variables.
Proposition 10 Suppose monetary policy implements the targeting rule
t
= 0 (strict ination
targeting). The unique equilibrium of the economy is then:

d
t
=

d
t1
(Y
t
E
t1
Y
t
) (d

t
E
t1
d

t
), [4.14]
with implied nominal interest rate i
t
=

t
+

d
t
. In the case where real GDP is described by the
stochastic process [4.10], a monetary policy target of P
t
+Y
t
= 0 implies the equilibrium has the
following features:

d
t
=

d
t1
(

)(Y
t
E
t1
Y
t
), and [4.15]

l
t
=

l
t1
+
1
(

)(Y
t
E
t1
Y
t
), and
t
=
t1
(

)(Y
t
E
t1
Y
t
),
where

is dened in [4.11].
Proof See appendix.
31
The proposition reveals that too much price stability relative to that consistent with an ecient
distribution of risk ( <

) implies the loan-to-GDP ratio rises by more than is ecient following

a positive shock to real GDP, and the equilibrium real interest rate falls more than is ecient (that
is, falls below the natural interest rate). Achieving greater stability in nancial markets is seen to
require some sacrice of price stability in goods markets.
The size of these eects for plausible parameter values is depicted in Figure 4 (the coecient of
relative risk aversion is set to 5, the elasticity of intertemporal substitution is set to 0.9, and the
parameters and are set to match the average real interest rate and debt-to-GDP ratio see
Table 1). With strict ination targeting, lending increases by approximately 1.5% following a 1%
rise in GDP, while for a temporary shock, the ecient outcome is for lending to rise by slightly less
than GDP. The eects on the real interest rate are smaller, but strict ination targeting leads to fall
by about 0.30.4% more than is ecient (in the case of a permanent shock, the ecient outcome is
for the real interest rate to remain unchanged).
Figure 4: Response of nancial variations to a positive shock to real GDP under dierent
assumptions on the completeness of markets and monetary policy
Dierence between long-run and short-run eects on GDP ()
Loans-to-GDP ratio (gross), percentage deviation
Representative agent
Complete markets
Ination targeting
Dierence between long-run and short-run eects on GDP ()
Real interest rate (expected), percentage deviation
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1
1 0.8 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1
1
0
1
0
0.2
0.4
0.6
Notes: Percentage deviations from steady state on impact following an unexpected 1% increase in
the level of real GDP. Representative agent is the limiting case of 0. Complete markets is
also the outcome when the optimal monetary policy is followed under incomplete markets. Ination
targeting is strict ination targeting under incomplete markets. The long-run eect of the shock on
GDP is larger than the short-run eect when > 0, and vice versa for < 0.
The intuition for these results can be understood by looking at the eects of a transfer between
savers and borrowers. Given the pattern of earnings over the life-cycle, savers are older than borrow-
32
ers. An unexpected change in ination is thus economically equivalent to a redistribution between
younger and older individuals. Overlapping generations models have been widely used to study the
eects of such intergenerational transfers in the context of public debt and pensions (Samuelson,
1958, Diamond, 1965, Feldstein, 1974). A redistribution from younger to older individuals reduces
desired saving and raises real interest rates (and reduces capital accumulation in a model with an
investment technology). A policy of nominal GDP targeting that implies an unexpected decrease
in ination when real GDP unexpectedly rises thus generates a transfer from debtors (younger in-
dividuals) to creditors (older individuals). A policy of strict ination targeting fails to generate this
transfer following the shock to real GDP. Since the eects of the transfer are to reduce desired saving
(and hence in equilibrium the amount of lending) and raise interest rates, strict ination targeting is
responsible for increasing lending too much in a boom and reducing the real interest rate too much.
These eects are also at work following a pure monetary policy shock, where an unexpected
loosening of policy increases lending and reduces the real interest rate.
Proposition 11 Suppose monetary policy is described by M
t
= M
t1
+
t
with an exogenous policy
shock
t
i.i.d.(0,

). The equilibrium of the economy is then:

d
t
=

d
t1

t
(d

t
E
t1
d

t
), and
t
=
t
g
t
, [4.16]
with nominal interest rate i
t
=

t
E
t
g
t+1
+

d
t
. A positive shock
t
reduces the real return r
t
, the
real interest rate
t
, and increases the loans-to-GDP ratio l
t
.
Proof See appendix.
4.6 The maturity of debt
The analysis so far has assumed borrowers have one loan contract over their period of borrowing with
a single monetary repayment at maturity. In this case, all ination cumulated over the borrowing
period that was not anticipated at the beginning of the contract reduces the real value of debt by
the same percentage amount. In general, with repayments over the term of the loan, or with a
sequence of loan contracts over the borrowing period, the eect of ination is smaller (except for
the case of a single jump in the price level before the rst repayment, unanticipated when the initial
loan contract was agreed).
The duration of a loan contract is dened as the average maturity of the repayments weighted
by their contribution to the present discounted value of the loan. Duration is the elasticity of the
value of the repayments with respect to a parallel shift in the term structure over the term of the
loan. Now consider the case where any ination that is unanticipated at the beginning of the loan
period is spread evenly over the term of the loan. This ination has a larger eect on the real value
of repayments made later in the term of the loan. To introduce this into the model where borrowing
takes place over one discrete time period, let denote the duration of debt relative to the period of
borrowing (0 < 1), and let i

t+1
denote the overall nominal interest rate between period t and
33
t + 1. Assume this eective nominal rate is given by:
1 + i

t+1
= (1 + i
t
)
_
1 +
t+1
1 +E
t

t+1
_
1
, [4.17]
which implies an ex-post real return of 1 + r

t+1
= (1 + i

t+1
)/(1 +
t+1
). The standard case where
the duration of debt is the same as the period of borrowing is obtained by setting = 1.
Proposition 12 If the eective nominal interest rate is given by [4.17] then all the results of
Proposition 5 and Proposition 6 continue to hold with equation [4.7c] replaced by:

t
+ (1 )E
t1

t
= i
t1

d
t

d
t1
r

t
, [4.18]
where i
t
= E
t
i

t+1
is the expected nominal rate over the term of the loan. Unexpected changes in the
debt-to-GDP ratio are associated with unexpected changes in weighted nominal GDP P
t
+

Y:

(d
t
E
t1
d
t
) =
_
{P
t
+

Y
t
} E
t1
{P
t
+

Y
t
}
_
, where

=
1
, [4.19]
which replaces equation [4.8]. Pareto eciency is achieved using a monetary policy target of P
t
+

Y
t
=

t
, and when the stochastic process for real GDP is [4.10], by using the target P
t
+

Y
t
= 0.
Proof See appendix.
The eect of shorter maturity debt ( < 1) is to increase the amount of ination required
to achieve the ecient real state-contingency of debt obligations (assuming that ination occurs
uniformly over the term of borrowing). To implement this, the weight assigned to real GDP in the
weighted nominal GDP target must be scaled by a factor of

> 1 (in addition to any scaling

needed because of dierences between the short-run and long-run eects of shocks).
5 Policy tradeos: Incomplete markets versus sticky prices
With fully exible prices, the ination uctuations resulting from the optimal monetary policy of
nominal GDP targeting are without cost, but the conventional argument for ination targeting is
that such ination uctuations lead to a misallocation of resources. This section adds sticky prices
to the model to analyse optimal monetary policy subject to both incomplete nancial markets and
nominal rigidities in goods markets. To do this, it is necessary to introduce dierentiated goods,
imperfect competition, and a market for labour that can be hired by dierent rms.
5.1 Dierentiated goods
Consumption in individuals lifetime utility function [2.1] now denotes consumption of a composite
good made up of a measure-one continuum of dierentiated goods. Young, middle-aged, and old
individuals share the same CES (Dixit-Stiglitz) consumption aggregator over these goods. The price
34
level P
t
is the minimum expenditure required per unit of the composite good:
P
t
= min
_
[0,1]
P
t
()C
i,t
()d s.t. C
i,t
= 1, where C
i,t

__
[0,1]
C
i,t
()
1

d
_
1
for i {y, m, o},
[5.1]
with C
i,t
() denoting consumption of good [0, 1] per individual of generation i at time t and
P
t
() the nominal price of this good. The parameter ( > 1) is the elasticity of substitution
between dierentiated goods. The price level and each individuals expenditure-minimizing demand
functions for the dierentiated goods are given by:
P
t
=
__
[0,1]
P
t
()
1
d
_ 1
1
, and C
i,t
() =
_
P
t
()
P
t
_

C
i,t
for all [0, 1] and i {y, m, o}.
[5.2]
5.2 Firms
There is a measure-one continuum of rms in the economy, each of which has a monopoly on the
production and sale of one of the dierentiated goods. Each rm is operated by a team of owner-
managers who each have an equal claim to the prots of the rm, but cannot trade their shares.
Firms simply maximize the prots paid out to their owner-managers.
28
Consider the rm that is the monopoly supplier of good . The rms output Y
t
() is subject to
the linear production function
Y
t
() = A
t
N
t
(), [5.3]
where N
t
() is the number of hours of labour hired by the rm, and A
t
is the exogenous level
of TFP common to all rms. The rm is a wage taker in the perfectly competitive market for
homogeneous labour, where the real wage in units of composite goods is w
t
. The real prots of rm
are J
t
() = P
t
()Y
t
()/P
t
w
t
N
t
(). Given the production function [5.3], the real marginal cost of
production common to all rms irrespective of their levels of output is k
t
= w
t
/A
t
.
Firm faces a demand function derived from summing up consumption of good over all gen-
erations (each of which has measure 1/3). Using each individuals demand function [5.2] for good
and the denition [2.2] of aggregate demand C
t
for the composite good, the total demand function
faced by rm is Y
t
() = (P
t
()/P
t
)

C
t
, and prots as a function of price P
t
() are as follows (with
the rm taking as given the general price level P
t
, real aggregate demand C
t
, and real marginal cost
k
t
):
J
t
() =
_
_
P
t
()
P
t
_
1
k
t
_
P
t
()
P
t
_

_
C
t
. [5.4]
At the beginning of time period t, a group of rms is randomly selected to have access to all
28
The participation of a specic team of managers is essential for production, and managers cannot commit to
provide labour input to rms owned by outsiders. In this situation, managers will not be able to sell shares in rms,
so the presence of rms does not aect the range of nancial assets that can be bought and sold.
35
information available during period t when setting prices. For a rm among this group, P
t
() is
chosen to maximize the expression for prots J
t
() in [5.4]. Since the prot function [5.4] is the
same across rms, all rms in this group will chose the same price, denoted by

P
t
. The remaining
group of rms must set a price in advance of period-t information being revealed, choosing P
t
()
to maximize expected prots E
t1
J
t
(). All rms in this group will choose the same price

P
t
that
satises the rst-order condition in expectation. The rst-order conditions for

P
t
and

P
t
are:

P
t
P
t
=
_

1
_
k
t
, and E
t1
_
_

P
t
P
t

_

1
_
k
t
__

P
t
P
t
_

C
t
_
= 0, [5.5]
where the term /( 1) represents each rms desired (gross) markup of price on marginal cost.
29
The proportion of rms setting a price using period t 1 information relative to those using period
t information is denoted by the parameter (0 < < ), and rms are randomly assigned to these
two groups.
5.3 Households
An individual born at time t has lifetime utility function [2.1], with the consumption levels C
y,t
, C
m,t
,
and C
o,t
now referring to consumption of the composite good [5.1]. Labour is supplied inelastically,
with the number of hours varying over the life cycle.
30
Young, middle-aged, and old individuals
respectively supply
y
,
m
, and
o
hours of homogeneous labour. Individuals also derive income
from their role as owner-managers of rms, and it is assumed that the amount of income from this
source also varies over the life cycle in the same manner as labour income. Specically, each young,
middle-aged, and old individual belongs respectively to the managerial teams of
y
,
m
, and
o
rms. The non-nancial real incomes of the generations alive at time t are:
31
Y
y,t
=
y
w
t
+
y
J
t
, Y
m,t
=
m
w
t
+
m
J
t
, and Y
o,t
=
o
w
t
+
o
J
t
, with J
t

_
[0,1]
J
t
()d. [5.6]
The coecients
y
,
m
, and
o
are parameterized in terms of and as in [2.6].
The assumptions on nancial markets are the same as those considered in section 2. In the
benchmark case of incomplete markets with a one-period, risk-free, nominal bond, the budget iden-
tities are as given in [2.9]; in the hypothetical case of complete markets, the budget identities are as
in [2.17], in both cases with consumption C
i,t
and income Y
i,t
reinterpreted according to equations
[5.1] and [5.6].
29
It is implicitly assumed that rms using the preset price will be willing to satisfy whatever level of demand is
forthcoming. Technically, this requires that

P
t
/P
t
k
t
holds in all states of the world, which will be true for shocks
within some bounds given the presence of a positive steady-state markup.
30
The case of endogenous labour supply is taken up in appendix, but it is possible to study the cost of relative
price distortions in a model with an exogenous aggregate labour supply.
31
Individuals receive xed fractions of total prots J
t
because all variation in prots between dierent rms is
owing to the random selection of which rms receive access to full information when setting their prices.
36
5.4 Equilibrium
The young, middle-aged, and old have per-person labour supplies H
y,t
=
y
, H
m,t
=
m
, and
H
o,t
=
o
. The aggregate supply of homogeneous labour is therefore H
t
= (1/3)H
y,t
+ (1/3)H
m,t
+
(1/3)H
o,t
, which is xed at H
t
= 1 given [2.3]. Given aggregate demand C
t
, market clearing
(1/3)C
y,t
() +(1/3)C
m,t
() +(1/3)C
o,t
() = Y
t
() for dierentiated good holds because rm meets
all forthcoming demand. The aggregate goods- and labour-market clearing conditions are:
C
t
= Y
t
, where Y
t

_
[0,1]
P
t
()
P
t
Y
t
()d, and
_
[0,1]
N
t
()d = 1, [5.7]
with Y
t
now being the real value of output summed over all rms, which must equal C
t
given [5.1].
Using the denition of prots J
t
() and equations [5.6] and [5.7], it follows that J
t
= Y
t
w
t
, and
hence Y
y,t
=
y
Y
t
, Y
m,t
=
m
, and Y
o,t
=
o
Y
t
, as in equation [2.3].
Given the aggregate goods-market clearing condition from [5.7], and the individual demand and
production functions in [5.2] and [5.3], satisfaction of the labour-market clearing equation in [5.7] is
equivalent to real GDP given by the aggregate production function:
Y
t
=
A
t

t
, with
t

_
_
[0,1]
_
P
t
()
P
t
_

d
_
1
, [5.8]
where the term
t
represents the eects of relative-price distortions on aggregate productivity.
Let p
t


P
t
/P
t
denote the relative price of goods sold by the fraction 1/(1 + ) of rms that
set a price using period t information, and p
t


P
t
/P
t
the relative price for the fraction /(1 + )
of rms using period t 1 information. The formula for the price index P
t
in [5.2] implies p
t
=
(1 ( p
1
t
1))
1
1
, while equation [5.5] is equivalent to p
t
= (/( 1))k
t
. Using these equations,
the rst-order condition [5.5], the aggregate goods-market clearing condition [5.7], the denitions of
real GDP growth g
t
and ination
t
, and E
t1

P
t
=

P
t
, it follows that:
1 +
t
1 +E
t1

t
=
p
1
t
E
t1
p
1
t
, and E
t1
__
p
t

_
1
_
p
1
t
1
__ 1
1
_
p

t
(1 + g
t
)
_
= 0. [5.9a]
Using equation [5.8], real GDP growth g
t
and relative-price distortions
t
are given by:
1 + g
t
= (1 + a
t
)

t1

t
, and
t
=
_
p

t
+
_
1
_
p
1
t
1
__

1
1 +
_
1
, [5.9b]
where a
t
(A
t
A
t1
)/A
t1
is TFP growth. The equilibrium of the model with incomplete mar-
kets (given exogenous TFP A
t
) is then the solution of equations [2.15][2.14e] and [5.9a][5.9b],
augmented with a monetary policy equation.
Consider rst the hypothetical case where all prices are exible and set using full information
( = 0), with the resulting equilibrium values being denoted with a . In this case, [5.9b] implies

t
= 1, so equilibrium real GDP growth with exible prices is g
t
= (A
t
A
t1
)/A
t1
, which is simply
equal to growth in exogenous TFP. This corresponds to the Pareto-ecient level of aggregate output

Y
t
= A
t
.
Returning to the analysis for a general value of , in a non-stochastic steady state, the unique
37
solution of equations [5.9a] and [5.9b] is

p = 1 and

= 1. Assuming the steady-state growth rate
of A
t
is zero, the steady state of the model is then as described in Proposition 1. Log-linearizing
equations [5.9a][5.9b] around the unique steady state yields:
g
t
= A
t
A
t1
,
t
= 0, and
t
E
t1

t
= p
t
. [5.10]
This means that real GDP growth is equal to the exogenous growth rate of TFP up to a rst-order
approximation.
5.5 Optimal monetary policy
Optimal monetary policy maximizes social welfare [3.3] using the Pareto weights derived from the
equilibrium with complete nancial markets and exible prices:
W
t
0
= E
t
0
2
_
1
3

t=t
0
2

tt
0

t
U
t
_
, [5.11]
where

t
is constructed using

Y
t
as real GDP and g
t
as real GDP growth.
32
With both incomplete
nancial markets and sticky goods prices, monetary policy has competing objectives to meet with
the nominal interest rate as the single policy instrument.
Proposition 13 The welfare function W
t
0
in [5.11] can be written as W
t
0
= E
t
0
2
L
t
0
+ terms
independent of monetary policy + third- and higher-order terms, where L
t
0
is the quadratic loss
function:
L
t
0
=
1
2

t=t
0

tt
0
E
t
0
_

d
2
t
+(
t
E
t1

t
)
2
_
, where [5.12a]
=

2
3
_
2

_
1 +
2
_
+
_

1

_
(1
2
)
_
1 +
(1 )
2
1 +
__
. [5.12b]
The coecient on the squared debt-to-GDP gap

d
t
= d
t
d

t
is strictly positive.
Proof See appendix
The quadratic loss function [5.12a] shows that just two variables capture all that needs to be
known about the economys deviation from Pareto eciency. First, the loss from imperfect risk-
sharing in incomplete nancial markets is proportional to the square of the gap

d
t
= d
t
d

t
between
the debt-to-GDP ratio and its value with complete markets. Second, the loss from misallocation of
resources owing to sticky prices is proportional to the square of the ination surprise
t
E
t1

t
.
Optimal monetary policy minimizes the quadratic loss function using the nominal interest rate
i
t
as the instrument, and subject to rst-order approximations of the constraints involving the
endogenous variables, the debt-to-GDP gap

d
t
, and ination
t
. The debt-to-GDP gap must satisfy
32
As discussed in section 3.2, the complete-markets weights are the only ones for which monetary policy can achieve
ecient risk-sharing. The use of exible-price output ensures the weights are independent of monetary policy, unlike
in general those derived using actual GDP.
38
equation [4.7a], while in the general case where debt has average maturity , ination must satisfy
equation [4.18]. The two constraints are:

d
t
= E
t

d
t+1
, and
t
+ (1 )E
t1

t
= i
t1

d
t

d
t1
r

t
, [5.13]
where r

t
an exogenous variable determined using [4.3c] with the real GDP growth rate from [5.10].
Proposition 14 The rst-order condition for minimizing the loss function [5.12a] subject to the
constraints in [5.13] is

d
t
E
t1

d
t
=
(1
2
)

(
t
E
t1

t
). [5.14]
The rst-order condition is satised if monetary policy achieves the following target:
P
t
+

Y
t
=

t
, with

=
_
1 +
(1
2
)

_
1
, [5.15]
and with

is as dened in [4.19], or if the stochastic process for productivity growth is given by

[4.10], the target is P
t
+

Y
t
= 0, with

is as dened in [4.11].
Proof See appendix.
The optimal monetary policy can be expressed as target for weighted nominal GDP (the weight
on real GDP is scaled by

and

even with fully exible prices). Compared to the case of

exible prices, the weight on real GDP relative to the price level must be scaled down by

< 1.
This pushes monetary policy in the direction of strict ination targeting, which corresponds to the
case where

= 0. The optimal monetary policy is essentially a comprise between the nominal
GDP target that would achieve ecient risk sharing, and the strict ination target that would avoid
relative-price distortions.
The value of

is larger when risk aversion is higher or when the life-cycle income gradient
is higher (both of which increase the term in [5.15]). Intuitively, these parameters increase the
importance of risk sharing. The value of

is lower when the price elasticity is larger, or is
higher so prices are stickiness. These parameters increase the importance of avoiding relative-price
distortions. A quantitative assessment of whether optimal monetary policy is closer to nominal
GDP targeting or strict ination targeting requires calibrating these parameters.
5.6 Calibration
Let T denote the length in years of one discrete time period. In the model, the length of an
individuals lifetime is 3T, while shocks to GDP occur every T years. In choosing T there is a
trade-o between a realistic representation of the length of an individuals lifetime (suggesting T
between 15 and 20, excluding childhood) and allowing for the relevant shocks to occur at a realistic
frequency. Given that the model is more likely relevant for permanent shocks to GDP rather than
for transient business-cycle episodes, T is set to 10 years, which still allows for a realistic horizon
over which individuals borrow and save (the term of borrowing and saving is for T years). Values
39
of T between 5 and 15 years are considered in the sensitivity analysis. The parameters of the model
, , , , , , and are then set to match features of U.S. data. The calibration is summarized
in Table 1 and justied below.
Table 1: Calibration of parameters
Parameter Value Target
Directly calibrated
Relative risk aversion () 5 Values well within range of estimates obtained in
Intertemporal substitution () 0.9 the literature see discussion in text
Price elasticity of demand () 3

Borrowing/saving period (T) 10
Indirectly calibrated
Discount factor () 0.59 Real interest rate of 7%; real GDP growth of 1.7%
*
Life-cycle income gradient () 0.66 Household gross debt-to-income ratio of 130%
*
Debt maturity () 0.5 Average duration (T
f
) of debt of 5 years

Price stickiness () 0.0044 Median duration (T

p
) of a price spell of 8 months

*
Source: Authors calculations using series from Federal Reserve Economic Data (http://research.
stlouisfed.org/fred2)

Source: Doepke and Schneider (2006)

Source: Nakamura and Steinsson (2008)

The parameter is related to the steady-state real interest rate and real GDP growth rate
(see Proposition 1). Let R and G denote the annual rates of interest and GDP growth, so that
1 + = e
RT
and 1 + g = e
GT
. Equation [2.24] implies that = e
(RG)T
. Given the focus on
household debt, it is natural to consider interest rates on the types of loans oered to households
in choosing R.
From 1972 through to 2011, there was an average annual nominal interest rate of 8.8% on 30-
year mortgages, 10% on 4-year auto loans, and 13.7% on two-year personal loans, while the average
annual change in the personal consumption expenditure (PCE) price index over the same time period
was 3.8%. The average credit-card interest rate between 1995 and 2011 was 14%. For comparison,
30-year Treasury bonds had an average yield of 7.7% over the periods 19772001 and 20062011.
The implied real interest rates are 4.2% on Treasury bonds, 5% on mortgages, 6.2% on auto loans,
9.9% on personal loans, and 12% on credit cards.
33
Given this wide range of interest rates, the
sensitivity analysis considers values of R from 4% up to 10%. The baseline real interest rate is set
to 7% as the midpoint of this range.
34
Over the period 19722011 used to calibrate the interest rate, the average annual growth rate
of real GDP per capita was 1.7%. Together with the baseline real interest rate of 7%, this implies
33
Average PCE ination over the periods 19772001 and 20062011 was 3.5%, and 2% over the period 19952011.
The real interest rate on government bonds is close to the conventional calibration of a 4% annual real interest rate
used in many real business cycle models.
34
This would imply a spread of 2.8% between the interest rates on loans to households and Treasury bonds. C urdia
and Woodford (2009) consider a spread of 2% between borrowing and saving rates.
40
that 0.59 using = e
(RG)T
.
In the model, the parameter sets the gradient of the age-prole of income (see Figure 1),
but also determines the steady-state debt-to-GDP ratio (see Proposition 1). Given the focus on
debt rather than on the specic reasons for household borrowing, is chosen to match observed
levels of household debt. Let D denote the measured ratio of gross household debt to annual
household income. This corresponds to what is dened as the loans-to-GDP ratio in the model (the
empirical debt ratio being based on the amount borrowed rather than the subsequent value of loans
at maturity), with an adjustment made for the fact that the level of GDP in the model is total
income over T years.
According to equation [2.24], the steady-state loans-to-GDP ratio is

l = /3, and thus D =
T/3, from which it follows that = 3D/T. Note that in the model, all GDP is consumed, so
for consistency between the data and the models prediction for the debt-to-GDP ratio, either the
numerator of the ratio should be total gross debt (not only household debt), or the denominator
should be disposable personal income or private consumption. Since the model is designed to
represent household borrowing, and because the implications of corporate and government debt
may be dierent, the latter approach is taken.
In the U.S., like a number of other countries, the ratio of household debt to income has grown
signicantly in recent decades. To focus on the implications of levels of debt recently experienced,
the model is calibrated to match average debt ratios during the ve years from 2006 to 2010. The
sensitivity analysis considers the full range of possible debt ratios from 0% to the models theoretical
maximum (approximately 196%, corresponding to = 1 with 0.59). Over 20062010, the
average ratio of gross household debt to disposable personal income was approximately 124%, while
the ratio of debt to consumption was approximately 135%. Taking the average of these numbers,
the target chosen is a model-consistent debt-to-income ratio of 130%, which implies 0.66.
35
There is an extensive literature estimating the elasticity of intertemporal substitution . Taking
the balance of evidence as pointing towards an elasticity less than one, but not substantially so,
the baseline value of is set to 0.9.
36
The sensitivity analysis explores a range of values between
0.26 (the lower bound (, ) consistent with the model having a unique steady state according to
Proposition 1 with 0.66 and 0.59) and 2.
37
35
This calibration implies the log dierence between the peak and initial income levels over the life-cycle is ap-
proximately 1.2 (see equation [2.6]). Empirical age-earnings proles are less steep than this, see for example Murphy
and Welch (1990), where the peak-initial log dierence of income is approximately 0.8. In the model, that would
be consistent with 0.42 and a debt-to-GDP ratio of approximately 83%, which is considered in the sensitivity
analysis. The model does not however capture all the reasons for household borrowing so it is to be expected that
observed debt levels are higher than can be explained by the age-prole of income.
36
Since g 0.19 with G equal to 1.7%, and given that = (1 + g)
1
1

in steady state, the baseline value of

implies 0.6.
37
There is limited consensus among the various studies in the literature. Early estimates suggested large elasticities,
such as those from the instrumental variables method applied by Hansen and Singleton (1982). That work suggested
an elasticity somewhere between 1 and 2 (this early literature has one parameter to capture both intertemporal
substitution and risk aversion). Those high estimates have been criticized for bias due to time aggregation by Hall
(1988), who nds elasticities as low as 0.1 and often insignicantly dierent from zero. Using cohort data, Attanasio
and Weber (1993) obtain values for the elasticity of intertemporal substitution in the range 0.70.8, while Beaudry
and van Wincoop (1996) nd an elasticity close to one using a panel of data from U.S. states. A recent study
41
In estimating the coecient of relative risk aversion , one possibility would be to choose values
consistent with household portfolios of risky and safe assets. But since Mehra and Prescott (1985)
it has been known that matching the equity risk premium may require a risk aversion coecient
above 30, while values in excess of 10 are considered by many to be highly implausible. Subsequent
analysis of the equity risk premium puzzle has attempted to build models consistent with the large
risk premium but with much more modest degrees of risk aversion.
38
Alternative approaches to estimating risk aversion have made use of laboratory experiments,
observed behaviour on game shows, and in a recent study, the choice of deductible for car insurance
policies (Cohen and Einav, 2007).
39
The survey evidence presented by Barsky, Juster, Kimball
and Shapiro (1997) potentially provides a way to measure risk aversion over stakes that are large
as a fraction of lifetime income and wealth.
40
The results suggests considerable risk aversion, but
most likely not in the high double-digit range for the majority of individuals. Overall, the weight
of evidence from the studies suggests a coecient of relative risk aversion above one, but not
signicantly more than 10. A conservative baseline value of 5 is adopted, and the sensitivity analysis
considers values from as low as 0.25 up to 10.
In the model, the parameter represents the elasticity of the real value of debt liabilities with
respect to the total amount of ination occurring over loan period that was not initially anticipated.
This follows from equation [4.17], which implies an ex-post real return of r

t+1
=
t
(
t+1
E
t

t+1
).
To calibrate , the strategy is to use data on the duration of household debt liabilities. The duration
T
f
of a sequence of loan repayments is dened as the average maturity of those payments weighted
by their contribution to the present discounted value of all repayments.
Doepke and Schneider (2006) present evidence on the duration of household nominal debt liabil-
ities. For the most recent year in their data (2004), the duration lies between 5 and 6 years, while
the duration has not been less than 4 years over the entire period covered by the study (19522004).
This suggests a baseline duration of T
f
5 years. The sensitivity analysis considers the eects of
having durations as short as one quarter, and longer durations up to the theoretical maximum of
10 years (given T = 10).
by Gruber (2006) makes use of variation in capital income tax rates across individuals and obtains an elasticity of
approximately 2. Following Weil (1989), it has also been argued that low values of the intertemporal elasticity lead
to a risk-free rate puzzle, and many papers in the nance literature assume elasticities larger than one (for example,
Bansal and Yaron, 2004, use 1.5). Finally, contrary to these larger estimates, the survey evidence of Barsky, Juster,
Kimball and Shapiro (1997) produces an estimate of 0.18.
38
For example, Bansal and Yaron (2004) assume a risk aversion coecient of 10, while Barro (2006) chooses a more
conservative value of 4.
39
Converting the estimates of absolute risk aversion into coecients of relative risk aversion (using average annual
after-tax income as a proxy for the relevant level of wealth) leads to a mean of 82 and a median of 0.4. The stakes
are relatively small and many individuals are not far from being risk neutral, though a minority are extremely risk
averse. As discussed in Cohen and Einav (2007), the estimated level of mean risk aversion is above that found in
other studies, which are generally consistent with single-digit coecients of relative risk aversion.
40
Respondents to the U.S. Health and Retirement Study survey are asked a series of questions about whether they
would be willing to leave a job bringing in a secure income for another job with a chance of either a 50% increase
in income or a 50% fall. By asking a series of questions that vary the probabilities of these outcomes, the answers
can in principle be used to elicit risk preferences. One nding is that approximately 65% of individuals answers fall
in a category for which the theoretically consistent coecient of relative risk aversion is at least 3.8. The arithmetic
mean coecient is approximately 12, while the harmonic mean is 4.
42
The denition of duration (in years) implies that it is equal to the percentage change in the
real value of a sequence of repayments following a parallel upward shift by 1% (at an annual rate)
of the nominal term structure. To relate this to the model, suppose that any ination occurring
between period t and t +1 is uniformly spread over that time period. Ination
t+1
E
t

t+1
that is
unexpected when contracts covering the period were written would therefore shift up the nominal
term structure by (
t+1
E
t

t+1
)/T (at an annual rate) once the shock triggering it becomes known.
Given that is the elasticity of the real value of debt liabilities with respect to total unexpected
ination over T years, this suggests setting = T
f
/T, and hence 0.5.
In the model, the extent of nominal rigidity is captured by the parameter . As was seen in
section 5.5, the only role of this parameter in determining optimal monetary policy is as part of
the coecient of the squared unexpected ination term in the loss function [5.12a]. The form of
nominal rigidity in the model is that some fraction of prices are predetermined before shocks to
GDP are realized. However, it is desirable to evaluate the welfare costs of ination using the more
conventional Calvo (1983) pricing model with staggered price adjustment taking place at a higher
frequency.
Woodford (2003) demonstrates that Calvo pricing implies that the welfare costs of ination
appear in the utility-based loss function as squared ination terms (additively separable from other
terms, as in [5.12a]). Supposing that individual price adjustment occurs at a constant rate within
each discrete time period, and with ination uniformly spread over each period (to be consistent
with the analysis of inations eects on the real value of debt liabilities), appendix shows that the
formula for the welfare costs of ination with Calvo pricing are bounded by:
L
,t
0


2
_
T
p
T
_
2

t=t
0

tt
0
E
t
0

2
t
, [5.16]
where T
p
is the expected duration of a price spell (in years). The term L
,t
denotes the welfare
costs of ination as a fraction of the initial T years steady-state real GDP, which is in same units
as the loss function [5.12a] given the normalization of the Pareto weights adopted in section 3.2,
hence L
t
0
and L
,t
0
are comparable.
The calibration strategy for the parameter is to set it so that the coecient of the ination
term in the loss function is the same as would be found in the Calvo model for parameters consistent
with the measured average duration of a price spell.
41
Comparison of [5.12a] and [5.16] suggests
setting = (T
p
/T)
2
to capture the welfare costs of ination.
42
There is now an extensive literature
41
This strategy is much simpler than the alternative of actually building Calvo price adjustment into the model,
which would entail working with a quarterly or monthly time period to capture high-frequency price adjustment. The
debt contracts in this alternative model would span many discrete time periods, vastly increasing the dimensionality
of the models state space. The simplication adopted does ignore the possibility of interactions between staggered
price adjustment and nominal debt contracts, though arguably there is no obvious reason to suggest such interactions
might be quantitatively important.
42
The only dierence between the utility-based loss functions of the two forms of nominal rigidity is that the
predetermining pricing assumption implies the term in ination is unanticipated ination squared, rather than all
ination squared. In the model, anticipated ination E
t1

t
is ination that is anticipated before nancial contracts
over the period between t 1 and t are written. Such ination has no bearing on the real value of debt liabilities
arising from these contracts. As can be seen from equation [5.14], optimal monetary policy is therefore completely
characterized by the behaviour of unanticipated ination
t
E
t1

t
, and so can be implemented by a target that
43
measuring the frequency of price adjustment across a representative sample of goods. Using the
dataset underlying the U.S. CPI index, Nakamura and Steinsson (2008) nd the median duration of
a price spell is 79 months, excluding sales but including product substitutions. Klenow and Malin
(2010) survey a wide range of studies reporting median durations in a range from 34 months to
one year. The baseline duration is taken to be 8 months (T
p
2/3), implying 0.0044. The
sensitivity analysis considers average durations from 3 to 12 months.
There are two main strategies for calibrating the price elasticity of demand . The direct ap-
proach draws on studies estimating consumer responses to price dierences within narrow consump-
tion categories. A price elasticity of approximately three is typical of estimates at the retail level
(see, for example, Nevo, 2001), while estimates of consumer substitution across broad consump-
tion categories suggest much lower price elasticities, typically lower than one (Blundell, Pashardes
and Weber, 1993). Indirect approaches estimate the price elasticity based on the implied markup
1/(1), or as part of the estimation of a DSGE model. Rotemberg and Woodford (1997) estimate
an elasticity of approximately 7.9 and point out this is consistent with the markups in the range
of 10%20%. Since it is the price elasticity of demand that directly matters for the welfare conse-
quences of ination rather than its implications for markups as such, the direct approach is preferred
here and the baseline value of is set to 3. A range of values from the theoretical minimum elasticity
of 1 up to 36 is considered in the sensitivity analysis, with the extremely large range chosen to allow
for possible real rigidities that raise the welfare cost of ination in exactly the same way as a higher
price elasticity.
43
The mapping between calibration targets and parameters is summarized below:
= e
(RG)T
, =
3D
T
, =
T
f
T
, and =
_
T
p
T
_
2
. [5.17]
5.7 Results
The consequences of sticky prices for optimal monetary policy can be seen from the

coecient
in equation [5.15], which represents the weight on the monetary policy optimal with fully exible
prices relative to the weight on strict ination targeting (as would be optimal were nancial markets
complete). The value of

under the baseline calibration is 0.95, indicating that the quantitatively
dominant concern is to allow ination uctuations to help complete nancial markets, rather than
avoid these to minimize relative-price distortions.
The extent to which this conclusion is sensitive to the calibration targets and the resulting
parameter values can be seen in Figure 5. The panels plot the value of

as each target is varied
is consistent with zero expected ination at the beginning of the time period.
43
The model does not include real rigidities, but these would increase the welfare cost of ination. For example, if
marginal cost is increasing in rm-level output then the multipying squared ination in the loss function needs to
be replaced by (1+elasticity of marginal cost w.r.t. rm-level output). Assuming a Cobb-Douglas production
function with a conventional labour elasticity of 2/3, the elasticity of marginal cost with respect to output is 1/2.
Taking the value of = 7.8 from Rotemberg and Woodford (1997), the term in the loss function should be set to
36 rather than 7.8 to capture this eect. The sensitivity analysis allows for this by considering a wider range of
values to mimic the eects of real rigidities of this size. Assuming large real rigidities is controversial: Bils, Klenow
and Malin (2012) present some critical evidence.
44
Figure 5: Optimal monetary policy with sticky prices weight (

) assigned to exible-price
optimal monetary policy target relative to strict ination targeting
Risk aversion () Intertemporal substitution ()
Annual interest rate, % Debt-to-GDP ratio, %
Borrowing/saving period, years (T) Duration of debt, years (T
f
)
Price elasticity of demand () Duration of price spell, years (T
p
)
0.4 0.6 0.8 1 10 20 30
2 4 6 8 10 5 10 15
0 50 100 150 4 6 8 10
0.5 1 1.5 2 2 4 6 8 10
0.25
0.5
0.75
1
0.25
0.5
0.75
1
0.25
0.5
0.75
1
0.25
0.5
0.75
1
0.25
0.5
0.75
1
0.25
0.5
0.75
1
0.25
0.5
0.75
1
0.25
0.5
0.75
1
Notes: The formula for the weight

is given in equation [5.15]. Strict ination targeting corresponds
to

= 0, while the optimal monetary policy with exible prices corresponds to

= 1. Each panel
varies one parameter or calibration target holding constant all others at the baseline values given in
Table 1.
over the plausible ranges identied earlier. It can be seen immediately that the calibration targets
for , the real interest rate (and hence ), T, and T
p
make little dierence to the results. The results
are most sensitive to the steady-state debt-to-GDP ratio, the coecient of relative risk aversion, the
duration of debt contracts, and the price elasticity of demand. However, within a very wide range of
plausible values of these calibration targets, the weight on nominal GDP targeting is never reduced
signicantly below 0.5.
45
6 Conclusions
This paper has shown how a monetary policy of nominal GDP targeting facilitates ecient risk
sharing in incomplete nancial markets where contracts are denominated in terms of money. In an
environment where risk derives from uncertainty about future real GDP, strict ination targeting
would lead to a very uneven distribution of risk, with leveraged borrowers consumption highly
exposed to any unexpected change in their incomes when monetary policy prevents any adjustment
of the real value of their liabilities. This concentration of risk implies that volumes of credit, long-
term real interest rates, and asset prices would be excessively volatile. Strict ination targeting does
provide savers with a risk-free real return, but fundamentally, the economy lacks any technology that
delivers risk-free real returns, so the safety of savers portfolios is simply the ip-side of borrowers
leverage and high levels of risk. Absent any changes in the physical investment technology available
to the economy, aggregate risk cannot be annihilated, only redistributed.
That leaves the question of whether the distribution of risk is ecient. The combination of
incomplete markets and strict ination targeting implies a particularly inecient distribution of
risk when individuals are risk averse. If complete nancial markets were available, borrowers would
issue state-contingent debt where the contractual repayment is lower in a recession and higher in
a boom. These securities would resemble equity shares in GDP, and they would have the eect
of reducing the leverage of borrowers and hence distributing risk more evenly. In the absence of
such nancial markets, in particular because of the inability of households to sell such securities,
a monetary policy of nominal GDP targeting can eectively complete the market even when only
non-contingent nominal debt is available. Nominal GDP targeting operates by stabilizing the debt-
to-GDP ratio. With nancial contracts specifying liabilities xed in terms of money, a policy that
stabilizes the monetary value of real incomes ensures that borrowers are not forced to bear too much
of the aggregate risk, converting nominal debt into real equity.
While the model is far too simple to apply to the recent nancial crises and deep recessions
experienced by a number of economies, one policy implication does resonate with the predicament of
several economies faced with high levels of debt combined with stagnant or falling GDPs. Nominal
GDP targeting is equivalent to a countercyclical price level, so the model suggests that higher
ination can be optimal in recessions. In other words, while each of the stagnation and ination
that make up the word stagation is bad in itself, if stagnation cannot immediately be remedied,
some ination might be a good idea to compensate for the ineciency of incomplete nancial
markets. And even if policymakers were reluctant to abandon ination targeting, the model does
suggest that they have the strongest incentives to avoid deation during recessions (a procyclical
price level). Deation would raise the real value of debt, which combined with falling real incomes
would be the very opposite of the risk sharing stressed in this paper, and even worse than an
unchanging ination rate.
It is important to stress that the policy implications of the model in recessions are matched by
equal and opposite prescriptions during an expansion. Thus, it is not just that optimal monetary
policy tolerates higher ination in a recession it also requires lower ination or even deation
46
during a period of high growth. Pursuing higher ination in recessions without following a symmetric
policy during an expansion is both inecient and jeopardizes an environment of low ination on
average. Therefore the model also argues that more should be done by central banks to take away
the punch bowl during a boom even were ination to be stable.
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A Appendices
Appendices are available in the expanded version of the paper:
http://personal.lse.ac.uk/sheedy/papers/DebtIncompleteMarketsExpandedVersion.pdf
51