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Federal Reserve Bank of Chicago

Quantitative Easing in Joseph’s Egypt
with Keynesian Producers
Jeffrey R. Campbell

November 2014
WP 2014-15

Quantitative Easing in Joseph’s Egypt with
Keynesian Producers
Jeffrey R. Campbell∗
November 5, 2014

Abstract
This paper considers monetary and fiscal policy when tangible assets can
be accumulated after shocks that increase desired savings, like Joseph’s biblical prophecy of seven fat years followed by seven lean years. The model’s
flexible-price allocation mimics Joseph’s saving to smooth consumption. With
nominal rigidities, monetary policy that eliminates liquidity traps leaves the
economy vulnerable to confidence recessions with low consumption and investment. Josephean Quantitative Easing, a fiscal policy that purchases either
obligations collateralized by tangible assets or the assets themselves, eliminates both liquidity traps and confidence recessions by putting a floor under
future consumption. This requires no commitment to a time-inconsistent
plan.

∗

Federal Reserve Bank of Chicago and CentER, Tilburg University
E-mail: jcampbell@frbchi.org.

I am very grateful to Gadi Barlevy, Bob Barsky, Marco Bassetto, Charlie Evans, and Simon
Gilchrist for many discussions on this topic; to Refet Gürkaynak for his insightful conference discussion; and to seminar participants at UPF-CREI for their responses to a very preliminary version
of this paper. The views expressed are those of the author. They do not necessarily represent the
views of the Federal Reserve Bank of Chicago, the Federal Reserve System, or its Board of Governors.
JEL Codes: E12, E63.
Keywords: Zero Lower Bound, Liquidity Trap, Confidence Recession, Storage, Equilibrium Multiplicity, Competitive Devaluation.

“Accordingly, let Pharaoh find a man of discernment and wisdom, and set
him over the land of Egypt. And let Pharaoh take steps to appoint
overseers over the land and organize the land of Egypt in the seven years of
plenty. Let all the food of these good years that are coming be gathered,
and let the grain be collected under Pharaoh’s authority as food to be
stored in the cities. Let that food be a reserve for the land for the seven
years of famine which will come upon the land of Egypt, so that the land
may not perish in the famine.”
Genesis 41:33-36 in Berlin and Brettler (2004)

1

Introduction

This paper considers monetary and fiscal policy following unexpected increases of
desired savings when tangible assets can be accumulated. Such increases of desired savings are the standard driving force in the modern theory of liquidity traps
(Krugman, 1998; Eggertsson and Woodford, 2003), in which the zero lower bound
makes monetary policy inappropriately tight and thereby sends the economy into
recession. The story of Joseph prophetically forecasting seven fat years followed by
seven lean years shares liquidity trap models’ shock to desired savings, but the Bible
records a superior macroeconomic outcome based on the accumulation of grain in
the prosperous years which was drawn down during the following famine. Joseph
achieved this using two advantages not available to liquidity-trap models’ policy
makers. First, monetary frictions like sticky prices and the zero lower bound on
interest rates presumably did not limit his policy options. Second, Joseph possessed a storage technology. In contrast, prominent liquidity trap models feature
a fallacy of composition: Tangible assets cannot be accumulated, but individuals
believe they can intertemporally substitute consumption using bonds. Since those
models’ liquidity traps arise when desired savings at potential output exceeds the
supply of bonds, one might reasonably speculate that the storage technology was
the key to Joseph’s success. That is, adding storage or another form of capital
accumulation to the standard new Keynesian model can remove both the fallacy
of composition and the threat of liquidity traps. Krugman (1998) addresses this
possibility by arguing that adjustment costs make avoiding a liquidity trap using
capital accumulation infeasible. Accordingly, Christiano, Eichenbaum, and Rebelo
1

(2011) incorporate investment adjustment costs into their model of liquidity traps
with capital accumulation.
In this paper, I add a storage technology without adjustment costs to a New
Keynesian model with nominal rigidities and then characterize the monetary and
fiscal policies that can be used to avoid a recession and successfully accumulate
assets during fat years for later consumption. Since the Joseph story is familiar
from both the Bible and Broadway, I use it throughout the paper as a mnemonic
device. The model’s shock to desired savings qualitatively resembles the productivity sequence in the Joseph story, but it has one year of plenty and an infinite
horizon of famine instead of the biblically specified seven years for each phase. This
Egyptian scenario is the supply-side analogue of the preference shock employed by
Eggertsson and Woodford (2003) and Christiano, Eichenbaum, and Rebelo (2011).
Because production requires elastically-supplied labor, the economy can fall into a
recession. Although the original Joseph story featured literal storage prominently,
here it represents wealth accumulation in general. This can be achieved by accumulating inventories, running a current account surplus, or investing in productive
capital.
Without the possibility of storage, monetary policy can avoid a liquidity trap if
inflation expectations are high enough for some positive nominal rate of interest to
clear the bond market when output equals its flexible-price level. When storage is
possible, such inflation expectations are necessary but not sufficient for this. Even
when monetary policy is consistent with the flexible-price allocation, the economy
has a continuum of distinct equilibrium allocations. A deflationary coordination
game underlies this multiplicity: If firms with flexible prices expect deflation, they
choose low prices. This lowers both real aggregate consumption and marginal cost
and thereby confirms their expectations. When there is no storage, households’
optimal bond purchases resolve this indeterminacy: the Euler equation determines
the level of current consumption given a fixed real interest rate and the rational
anticipation that consumption will equal its flexible-price level when the shock to
desired savings has passed. With storage, future consumption is a free variable. This
allows the static coordination game’s multiplicity to manifest itself in a dynamic
setting. Because storage bounds the real interest rate from below, it cannot be
lowered any further to lift the economy out of such a recession.
This multiplicity implies that avoiding a recession in the economy’s initial fat
2

year might require a policy to raise current savings and thereby increase future
consumption. Both Eggertsson and Woodford (2003) and Werning (2012) advocate
lifting expectations of future consumption by committing to low future interest
rates which lead consumption to overshoot its long-run level. A policy maker that
exchanges nominal bonds for loans backed by storage achieves the same goal without
requiring commitment. The natural non-negativity constraint on storage prevents
households from offsetting the resulting real wealth accumulation. After the shock
to desired savings has passed, unwinding the monetary authority’s position increases
consumption; and the rational expectation of this keeps consumption and output
at their flexible-price levels when the propensity to save is still high. A substantial
fraction of assets currently on the Federal Reserve’s balance sheet are securities
backed by claims to accumulated capital, namely housing; and the Bank of Japan
and European Central Bank are both purchasing assets backed by loans to the
private sector. Therefore, it seems reasonable to label the balance-sheet expansion in
the model a form of quantitative easing. Instead of increasing “aggregate demand,”
it removes recessionary outcomes by shrinking the equilibrium set. Although the
liabilities on Pharoh’s balance sheet offsetting Joseph’s accumulation of grain went
unrecorded, one might reasonably consider such an accumulation of real assets by
the sovereign to be a prototype for this form of quantitative easing. Hence, I label it
Josephean. Such Josephean quantitative easing (JQE) is fiscal policy because it has
obvious implications for the size of the government’s consolidated balance sheet.
Unlike the closed economies of many liquidity trap models, small open economies
can intertemporally substitute consumption by running a current account surplus,
investing the proceeds abroad, and repatriating them in the future. Krugman (1998)
dismissed the possibility that such trade-facilitated intertemporal substitution could
lift an economy out of a liquidity trap based on an analysis that takes the shortcut
(his word) “that one can ignore the effect of the current account on the future investment income of the country.”1 This paper shows that accounting for the country’s
future investment income is crucial for designing appropriate policy in a liquidity
trap. In an international context, JQE resembles a monetary authority accumulating foreign reserves to implement an export-promoting competitive devaluation of
its currency. However, JQE does not operate through the real exchange rate. Instead, it (possibly) improves outcomes by putting a floor on expectations of future
1

See (Krugman, 1998, Page 164).

3

domestic consumption, just as it does in a closed economy. The foreign country
experiences a current account deficit that reverses itself when the possibility of a
liquidity trap has passed in the home country. Such unstable international capital
flows are not an undesirable side effect of JQE; they are its goal.
Although early models of liquidity traps featured a fallacy of composition, recently Correia, Farhi, Nicolini, and Teles (2013) and Christiano, Eichenbaum, and
Rebelo (2011) have examined them in models with capital accumulation. Indeed,
this paper’s model is nearly a special case of that in Correia, Farhi, Nicolini, and
Teles (2013). Those authors characterize the Pigouvian taxes that allow a competitive equilibrium to coincide with the optimal allocation. This paper complements
theirs by showing how policy can make the flexible-price allocation the unique equilibrium using JQE. (Since firms’ markups might be part of a preexisting scheme that
grants monopoly rights to induce innovation, I consider only the modest goal of implementing the flexible-price allocation instead of the more ambitious aspiration of
achieving a completely distortion-free allocation.)
Christiano, Eichenbaum, and Rebelo (2011) quantitatively examined the government spending multiplier under the common assumptions that an interest-rate
rule satisfying the Taylor principle governs the nominal interest rate (subject to the
zero-lower-bound) and that equilibrium sequences converge to the unique steady
state with active monetary policy (Benhabib, Schmitt-Grohé, and Uribe, 2001).
In the present model, such an imposition of local determinacy indeed eliminates
equilibrium multiplicity. When the interest-rate rule and its inflation target are
appropriately chosen, the unique equilibrium implements the flexible-price allocation. However, this criterion eliminates a continuum of other less desirable equilibria
merely because they induce the monetary authority to drive the economy into the
zero lower bound permanently. Aruoba, Cuba-Borda, and Schorfheide (2014) document that such a equilibrium replicates the Japanese experience since 1995 well.
Furthermore, neither market-clearing nor individual optimality requires inflation to
equal the monetary authority’s target in the long run (Cochrane, 2011). Therefore
there are neither theoretical nor empirical grounds for removing such outcomes from
consideration ex-ante. I show that an appropriate choice of JQE can remove them
from the equilibrium set ex-post.
Previous work on fiscal policy in liquidity traps has focused on purchasing public goods that impact neither the marginal utility of private consumption nor pro4

duction possibilities. Christiano, Eichenbaum, and Rebelo (2011) argue that the
multiplier effects of such government spending are large in liquidity traps caused by
shocks to desired savings, because the usual crowding out of investment by debtfinanced government spending disappears when the nominal interest rate is stuck
at the zero lower bound. This paper shows that debt-financed government spending that decreases the marginal utility of future private consumption can guide the
economy to the flexible-price outcome. That is, fear of falling into a liquidity trap
following an increase in desired savings need not justify expanded purchases of public goods. Instead, it calls for the policy authority to save on behalf of households
and thereby eliminate recessions driven by households’ failure to coordinate on an
equilibrium with both high consumption and high investment.
The shocks to desired savings in liquidity-trap models are usually interpreted as
stand-ins for the balance-sheet repair that follows financial-market turmoil. Eggertsson and Krugman (2012) expand on this by explicitly modeling the financial turmoil
as a “Fisher-Minsky-Koo” moment, in which a contraction of consumer credit and
debt deflation reduce aggregate demand. Fornaro (2013) shows that consumer debt
forgiveness then can be Pareto improving: Borrowers’ consumption increases while
savers’ consumption remains the same. I anticipate that adding capital accumulation to that environment can determine the potential of unconventional policy to
mitigate such a liquidity trap (holding fixed the dysfunctional consumer-credit market) by encouraging real wealth accumulation. However, that extension lies beyond
this paper’s scope.
The remainder of this paper proceeds as follows. The next section contains the
model’s primitive assumptions, and Section 3 presents its flexible-price allocation.
Section 4 adds nominal rigidities and characterizes the resulting recessionary equilibria. These can usefully be divided into two classes, liquidity traps and confidence
recessions. Section 5 shows how JQE can destroy these equilibria, and it places
this paper’s results in the context of previous theoretical characterizations of QE.
Section 6 develops the interpretation of the model as a small open economy that
stores consumption by trading the aggregate good with a large foreign sector. Section 7 offers concluding remarks on the relevance of JQE for current policy at the
Bank of Japan and the European Central Bank. The model of the text embodies a
linear storage technology. An appendix demonstrates that this paper’s key results,
the existence of confidence recessions and JQE’s ability to eliminate both them and
5

liquidity traps, are robust to allowing the marginal cost of storage to increase with
its quantity.

2

Primitive Assumptions

The model features three key features of New Keynesian economies, monopolistic
competition so that goods’ prices are set by specific agents, nominal rigidities which
generate a Phillips curve trading off inflation and output, and a market for nominal
bonds with an interest rate set by a policy authority subject to the zero lower bound.
Additionally, the policy authority can issue nominal bonds and invest the proceeds
within the storage technology. This access to a technology for intertemporal transformation of goods does not distinguish the policy authority from the economy’s
households. Henceforth I anthropomorphize this authority and name him “Joseph”.
The presentation of the model’s primitive assumptions follows the conventional
preferences-technology-trading opportunities road map. A single representative
household populates the model economy. Its preferences over streams of consumption goods and time spent at work are
U ({Ct }, {Nt }) =

∞
X

β t (ln Ct − θNt ) , with

t=0

Z

1

Ct (j)

Ct =

ε−1
ε

ε
 ε−1

dj

.

0

Here Nt is time spent at work, Ct (j) is the consumption of good j (with j ∈
[0, 1]) in year t, and ε > 1 is the elasticity of substitution between any two of
the differentiated goods. It is well-known that quasi-linear preferences like these
feature an infinite Frisch elasticity of labor supply. I adopt them here for algebraic
convenience. Since there is no uncertainty in this economy and I restrict attention
to deterministic equilibria, risk-aversion plays no role in this analysis. I (implicitly)
set the elasticity of intertemporal substitution to one only to avoid unnecessary
parameter proliferation.
Without storage, the natural non-negativity constraint on time at work would
be irrelevant because the marginal rate of substitution between consumption and
leisure grows without bound as consumption goes to zero. Since storage creates the

6

possibility of consumption without work, I make this constraint explicit with
Nt ≥ 0.

(1)

Replacing (1) with positive lower bound on hours worked, which is perhaps more
realistic, would leave this paper’s results unchanged.
The technology for producing each of the differentiated goods is the same: one
unit of labor yields At units of the good in question. To make a liquidity trap
possible, I assume that A0 = AH and At = AL < AH for all t ≥ 1. This is the
Egyptian scenario mentioned above.
The economy’s other technology is that for storage. To have S units of the
aggregate good available next year, one must invest S/(1 − δ) units of the aggregate
good today. Here, δ is the depreciation rate on storage. So that this technology
cannot be used to transfer resources from the future into the present, I require
St ≥ 0.

(2)

Although it is natural to assume that δ ≥ 0, the analysis below only requires
β(1 − δ) < 1.

(3)

This more general bound on δ will be helpful when interpreting “storage” as investment abroad with a positive real return.
Trade occurs in a labor market, product markets, and financial markets. The
labor market is perfectly competitive with nominal wage Wt . Product markets conform to the familiar monopolistic competition framework. Each product’s monopolist chooses its nominal price taking as given all other products’ prices, aggregate
income, and the household’s demand system for all of the differentiated products.
The functions Pt (·) and Yt (·) give all of the monopolists’ nominal prices in year t
and their corresponding quantities sold.
The model’s nominal rigidity resembles Fisher’s (1977) model of overlapping
labor contracts. Each year, half of the economy’s producers set their nominal prices
for the current and next years. By eliminating intertemporal trade-offs in price
setting inherent in the more commonly employed Calvo (1983) specification, I focus
the analysis on intertemporal substitution and the obstacles to its efficient execution.
7

Joseph sets the interest rate for nominal bonds subject to the zero lower bound.
Before setting the interest rates for bonds purchased in t that mature in t + 1, he
observes storage brought into the year St , the nominal wage Wt , producers’ price
choices and real outputs Pt (·) and Yt (·), and households’ consumption Ct . Joseph
collects this information, the rationally anticipated path for At , and the complete
histories of consumption, storage, and nominal wages and prices through year t − 1
into the information set Ωt and inputs it into the interest rate rule it = ρ(Ωt ). Joseph
selects this rule at the beginning of time and thereafter follows its prescriptions absolutely. Wicksellian models (like this one) place the zero lower bound in the monetary
policy rule as a stand-in for the analogous no-arbitrage condition that would come
out of an explicit specification for money demand. Accordingly, I henceforth require
ρ(Ωt ) ≥ 0 for all possible Ωt .
To undertake JQE, Joseph issues nominal bonds, uses the proceeds to acquire the
aggregate good, and directly invests the goods acquired in the storage technology.
It is the restriction to investing in assets that directly contribute to real national
wealth that distinguishes JQE from general quantitative easing, not the direct use
of the storage technology per se. (The appendix extends the model to have Joseph
invest in privately-issued assets backed by stored goods. This modification changes
no result.) Let Qt+1 denote the amount of the aggregate good available in t + 1
from Joseph’s storage investments during t, and use Bt+1 to represent the nominal
redemption value of the bonds Joseph issued in t to finance that storage. In contrast
with the Pigouvian policy maker in Correia, Farhi, Nicolini, and Teles (2013), Joseph
has access to no other tax instruments. Therefore, given Q0 = B0 = 0, the sequences
Qt and Bt must satisfy the feasibility constraint

Qt+1 = (1 − δ)

3



Bt+1
− Bt /Pt + Qt .
1 + it

(4)

The Flexible-Price Allocation

The equilibrium allocation when producers face no nominal rigidities serves as a
baseline for the subsequent analysis. Whether the household or the government
undertakes storage is a matter of indifference when prices are flexible, so I assume
for this section that Qt+1 = Bt+1 = 0 for all t ≥ 0.
Begin the construction of a flexible-price equilibrium with the household’s pur8

chases of differentiated goods, and let Yt denote the quantity of the aggregate good
created.
ε
Z 1
 ε−1
ε−1
Yt ≡
Yt (j) ε dj
0

The household’s optimal allocation of nominal consumption expenditures across
differentiated goods has the familiar form:

Yt (j) = Yt

Pt (j)
Pt

−ε

with Pt the aggregate price index
Z
Pt ≡

1

Pt (j)1−ε

1
 1−ε

dj.

0

By construction, Pt Yt is the household’s total nominal expenditure on goods.
Given the household’s initial holdings of nominal bonds and the aggregate good
from storage; utility maximization requires choosing sequences of aggregate consumption, hours worked, the values of all assets subject to the budget constraint
P t Ct +

St+1
Bt+1
+ Pt
≤ Wt Nt + Bt + Pt St + Dt ,
1 + it
1−δ

(5)

and the non-negativity constraints in (1) and (2). Here, Dt is the dividend earned
from the household’s ownership of the producers,
Dt = Pt (Ct + St+1 /(1 − δ) − St ) − Wt Nt ;
and it is the interest rate given by ρ(·).2
In an equilibrium, the sequences for Ct , Nt , Bt+1 , and St+1 solve the household’s
utility maximization problem given the sequences for Dt , Wt , Pt , and it . If we denote the Lagrange multipliers on the year t budget constraint and the non-negativity
constraints on storage and labor with β t λt /Pt , β t λt νt , and β t λt υt ; the utility maximization problem yields familiar conditions for optimal labor supply, optimal bond
purchases, and optimal storage.
2

Producers are entirely equity financed. Since they face unlimited liability, Dt may be negative.

9

θCt =

Wt
+ υt
Pt

Pt C t
Pt+1 Ct+1
Ct
+ νt
1 = β(1 − δ)
Ct+1
1 = β(1 + it )

(6)
(7)
(8)

These, together with the transversality condition
lim β t

t→∞

St+1 + Bt+1 /Pt+1
=0
Ct+1

are necessary and sufficient for the household’s utility maximization.
In the flexible-price baseline, producers always set the optimal monopoly price,
so


ε
Wt
Pt =
.
(9)
ε − 1 At
The only remaining requirements for a flexible-price equilibrium are bond market
clearing, Bt+1 = 0 for all t ≥ 0, and the aggregate resource constraint
At Nt = Ct + St+1 /(1 − δ) − St .
From Walras’s law, this guarantees labor-market clearing.
There are many flexible-price equilibria, but they all share a single allocation of
consumption, storage, and hours worked. Since I repeatedly reference this allocation’s values below, I denote the associated values of Ct , St , and Nt with C̃t , S̃t ,
and Ñt . The tilde should bring flexibility to mind. To reduce the number of cases
under review, I henceforth suppose that the economy starts with no consumption
available from storage: S0 = 0.

3.1

Mild Famines

When AL /AH is sufficiently close to one, the flexible-price allocation does not use
storage. In this sense, the foreseen famine is “mild.” To begin this case’s equilibrium
analysis, suppose that indeed S̃t = 0 for all t ≥ 1. The optimal price-setting
condition in (9) determines Wt /Pt . Substituting this into (6) and imposing the
10

resource constraint gives

C̃t =

ε−1
ε



At
, and
θ

(10)

Ñt = C̃t /At .

(11)

Setting S̃t+1 = 0 is consistent with (8) for t ≥ 1, because C̃t is constant from t = 1
onwards and β(1 − δ) < 1. For the household also to choose S̃1 = 0, we requre that
C0 /C1 is not too large.
1
βAH
.
<
L
A
1−δ

(12)

The inequality in (12) puts a lower bound on AL which defines a “mild” famine.
When it holds good, S̃t = 0 and expressions for C̃t and Ñt in (10) and (11) together
give the unique flexible-price equilibrium allocation.
Completing the construction of an equilibrium requires finding an interest rate
rule for Joseph and sequences of nominal prices and wages that are consistent with
this allocation. For the rule, consider
(

? −1 C̃t+1

it = max 0, π β

C̃t

 π φ
t
?
π

)
−1 .

(13)

This is a censored inflation targeting rule with a time-varying intercept. In it, π ? is
the target inflation rate, π0 ≡ P0 π ? , and πt ≡ Pt /Pt−1 for t ≥ 1.3 When πt = π ? , the
underlying non-censored rule tracks the nominal interest rate consistent with the
flexible-price allocation. This is the “natural” interest rate. Otherwise φ regulates
the response of it to deviations from the inflation target. If φ > 1, the rule satisfies
the “Taylor principle.”
To construct an equilibrium using (13), set π ? ≥ βAH /AL and Pt = π ? t . With
these values, (13) gives
At+1
it = π ? β −1
− 1.
At
This satisfies (7) for all t ≥ 0; so this sequence of allocations, prices, and interest
3
Both with flexible prices and sticky price plans, there is no loss to Joseph’s economy from
setting π ? 6= 1 as long as all equilibria considered are deterministic.

11

rates forms a flexible-price equilibrium.
Since inflation never deviates from its target, any value of φ is consistent with
achieving πt = π ? always. However, this result only holds if π ? ≥ βAH /AL , which
in turn guarantees that the equilibrium avoids the zero lower bound. If instead
π ? < βAH /AL , then (13) sets i0 to zero if π0 = π ? . With this interest rate, clearing
the nominal bond market (given Ct = C̃t always) requires π1 to exceed π ? . For
reasons familiar from Benhabib, Schmitt-Grohé, and Uribe (2001), the evolution of
subsequent inflation depends on φ. If φ < 1, then inflation temporarily overshoots
π ? . If instead φ = 1, then inflation remains permanently at π1 . Finally, with φ > 1
the Taylor principle induces Joseph to raise the nominal interest rate more than
one-for-one with inflation. Bond-market clearing then requires inflation to rise even
further, leading to an explosive inflation sequence. Regardless of its implications for
inflation, the interest rate rule has no influence on flexible price allocations.

3.2

Severe Famines

Now, suppose that (12) does not hold, so AL is low enough to induce storage. As
in the case of a mild famine, υ0 = 0 and Equation (10) determines C̃0 .
To determine C̃t for t ≥ 1, hypothesize that S̃1 > 0 and S̃t = 0 for t ≥ 2. The
first assumption and (8) give us
C̃1 = β(1 − δ)C̃0 .

(14)

Since the economy faces a severe famine, this exceeds the value of C1 consistent
with setting N1 > 0. So that this value of C̃1 is also consistent with the household
setting S2 to zero, assume
AH
(15)
1 ≥ β 2 (1 − δ)2 L .
A
That is, the rate of return from saving across two years is not too large. Although
none of the results below depend on this particular limit on the duration of storage
and its attendant vacation, I henceforth assume that (15) holds good to keep the
analysis simple. With this, S̃t = 0 and (10) characterizes C̃t for all t ≥ 2.
Given the sequences for C̃t and S̃t in hand, the budget constraints determine Ñt
for t = 0 and t ≥ 2. The consumption sequence and (7) determine equilibrium real
interest rates. To decompose these into nominal interest rates and inflation; use the
12

interest rate rule in (13). If the given value of π ? > β C̃0 /C̃1 , then i0 > 0 and πt = π ?
for all t. Otherwise, i0 = 0 and πt > π ? always.
Figure 1 summarizes this section’s results with (qualitative) plots of the flexibleprice allocation and its associated real interest rate over time. In each panel, the
blue line with circles corresponds to the case of a mild famine, while the orange line
with squares gives analogous values for a severe famine. The upper-right panel plots
productivity for the two cases, which share a common value for AH . The upper-left
panel gives consumption, which begins at C̃0 in both cases. With the mild famine,
it falls to C̃1 and stays there forever. With a severe famine, the household carries
wealth into year 1, so C̃1 > C̃2 in this case. Regardless, consumption reaches its
long-run value in year 2. The lower-left panel gives the associated gross real interest
rates, which do not depend on the particular interest rate rule employed by Joseph.
With a mild famine, this equals AL /(βAH ). Making the foreseen famine worse by
reducing AL reduces this until it reaches 1 − δ. For even lower values of AL , the
household’s use of the storage technology keeps this “natural” interest rate from
falling. With a mild famine, the real interest rate reaches its long-run value, 1/β,
in year 1. In the case of a severe famine, the real interest rate remains below 1/β
in year 1 because consumption is still higher than its long-run value.4 Finally, the
lower-right panel gives hours worked in the two cases. Under a mild famine, the
ratio of consumption to wages is constant. Since these preferences satisfy standard
balanced-growth restrictions, this means that wage changes’ income and substitution
effects exactly offset to leave hours worked constant. The case of a severe famine
shows the Lucas and Rapping (1969) theory of intertemporal substitution and labor
supply in action. Temporarily high real wages in year 0 induce the household to
expand labor supply, accumulate savings, and raise consumption in future years.
The future consumption boom lowers hours worked for the one year that it lasts.

4

Equilibria with Nominal Rigidities

This section shows how nominal rigidities can interfere with implementing the
flexible-price allocation, which requires reexamining producers’ optimal pricing decisions and appropriately redefining equilibrium. For this section, I continue to hold
4

As drawn, (1 + i1 )/π2 in the case of a severe famine is less than (1 + i0 )/π1 with a mild famine.
This is possible, but not necessary.

13

Figure 1: The Flexible-Price Allocation and Real Interest Rate
Productivity, At

Consumption, Ct

AH

C̃0

Mild AL
β(1 − δ)C̃0
Severe AL

0

1

2

3

0

Real Interest Rate, (1 + it )/πt+1

3

(1 +

β) ε−1
θε

AL
βAH

ε−1
θε

1−δ
Severe

2

Hours Worked, Nt

1
β

Mild

1

AL
βAH

Year
0

1

Mild Famine:

2
βAH
AL

Year

0
3

<

0

1
1−δ

1

Severe Famine:

14

2
βAH
AL

3

≥

1
1−δ

Bt+1 = Qt+1 = 0, so the resulting equilibria are without JQE.
Denote the price chosen by a firm in year t − j that will apply in year t with Ptj ;
so Pt0 is the price chosen by producers with a current price choice, and Pt1 is the
price for t chosen by producers that set their year t price in t − 1.
Since there is no uncertainty, the optimal price choices are
Pt0
Pt1




ε
Wt
=
∀t ≥ 0 and
ε − 1 At


ε
Wt
=
∀t ≥ 1.
ε − 1 At

(16)
(17)

The right-hand sides of (16) and (17) are identical, but they apply to different years.
The preset price, P01 , is one of the economy’s initial conditions; which I normalize
to one. With these firm-level prices, the aggregate price index is

Pt =

1 0 1−ε 1 1 1−ε
P
+ Pt
2 t
2

1
 1−ε

(18)

The definition of an equilibrium with nominal rigidities simply adds the prices
from pre-existing plans to the economy’s initial conditions; replaces (9) with (16),
(17) and (18); and accounts for the effects of pricing distortions on aggregate output
in the aggregate resource constraint.

1
2



Pt0
Pt

At Nt
−ε
 1 −ε = Ct + St+1 /(1 − δ) − St .
P
+ 21 Ptt

(19)

Here, total output is written as a linear function of hours worked, with productivity
dependent on differentiated goods’ producers’ relative prices. This is at its maximum
when all goods’ nominal prices equal each other.5

4.1

The Fundamental Multiplicity

Generically, this economy has multiple equilibrium allocations. When π1 indexes the
equilibrium set, Joseph can guide the economy to a desired outcome by appropriately managing inflation expectations. (The tools of an inflation-targeting regime
(Bernanke and Mishkin, 1997) could be useful for this task.) In other cases, π1
5

ε

To show this, use (18), the fact that x ε−1 is convex if ε > 0, and Jensen’s inequality.

15

is constant across equilibria and instead they differ in the expected (and realized)
value of C1 . (This paper shows how JQE can manage these expectations of future
consumption.) Both cases can be understood as specific instances of a fundamental
multiplicity that arises in the model when the only dynamic considerations come
from firms’ preset prices. That is, the economy has neither a bond market nor a
storage technology.
In year 0, half of the firms have nominal prices fixed at P01 = 1, while the other
half can choose their nominal prices. Additionally, the household provides labor
and spends all of its income on consumption. Given P01 = 1, equilibrium in this
alternative economy requires C0 , N0 , W0 , P0 , and P00 to satisfy the optimal labor
supply condition in (6) with υ0 set to zero, the optimal pricing condition in (17),
the price aggregation rule in (18), and the resource constraint in (19) with St and
St+1 set to zero.
Four conditions restrict five unknowns. To show mechanically that this under
determination indeed results in equilibrium multiplicity, select any
1

C0 > 2 1−ε C̃0 .

(20)

The optimal labor supply condition then determines the real wage W0 /P0 . Together,
the conditions for optimal flexible prices and price aggregation imply that the price
level solves
1

1−ε ! 1−ε
C0
1 1
+
P0
.
(21)
P0 =
2 2
C̃0
This can be interpreted as the economy’s Phillips curve, positively connecting the
consumption gap, C0 /C̃0 , with inflation, π0 /π ? ≡ P0 . The assumed lower bound
for C0 guarantees that the unique solution for P0 is real. Because firms’ choices of
current prices do not constrain their choices of future prices, it does not have the
familiar dependence on expected future inflation.
Figure 2 presents this Phillips curve graphically. It begins arbitrarily close to
1
1
the point (2 1−ε , 0), crosses through the 45◦ line at (1, 1), and asymptotes to 2 ε−1 as
C0 /C̃0 goes to infinity. Any point on this Phillips curve is consistent with equilibrium: Given C0 and P0 from such a point, W0 and P00 can be obtained immediately
from (6) and (17). The resource constraint then determines N0 .
Equilibrium multiplicity in the full model can be better understood in light of
16

Figure 2: The Phillips Curve
P0 ≡ π0 /π ?
1

2 ε−1

1

C0 /C̃0
1

1

2 1−ε

the fundamental multiplicity by considering the Euler equation for optimal nominal
bond purchases, (7). Given i0 , π1 , and C1 , this determines C0 and thereby selects one
of many points on the Phillips curve of Figure 2. In the “standard” analysis of the
liquidity traps with discretionary monetary policy, i0 = 0 and C1 is assumed to equal
C̃1 . Then, inflation expectations determine current macroeconomic performance, as
in Krugman (1998). For the present model, Subsection 4.3 covers this case in detail
by assuming that the famine is mild. With storage, C1 becomes endogenous; even
if we assume that it equals its flexible-price value given S1 . In this case – which is
covered in Subsection 4.4 given a severe famine – the economy can have multiple
equilibria with constant (across equilibria) values of i0 and π1 .
Economically, the fundamental equilibrium multiplicity reflects a coordination
failure (Cooper and John, 1988). Producers with flexible prices must coordinate
on an expectation of real marginal cost (W0 /(A0 P0 )). Increasing this expectation
raises their prices and lowers the economy’s average markup over marginal cost,
thereby boosting economic activity. This raises marginal cost through (6), so firms’
expectations of higher marginal cost are fulfilled.6 The lower bound on C0 in (20)
reflects a limit to the amount of damage nominal rigidity can do to this economy:
6

A host of macroeconomic models feature equilibrium multiplicity. Among those, the one
most closely related to this fundamental multiplicity is that of Shleifer (1986). That model also
omits external effects of production and derives equilibrium multiplicity from a static coordination
failure. However, its coordination failure concerns technological development, which is arguably
more relevant for medium-run fluctuations than are this model’s short-run pricing decisions.

17

Equilibrium consumption cannot fall below that which would occur if P01 /P00 were
driven to infinity so that half of the economy’s goods are effectively not available.

4.2

The Flexible-Price Allocation Replicator

An equilibrium can exhibit the familiar Keynesian connection between disinflation
and output in the initial year, because adjustments of P00 influence consumption
and marginal cost and thereby change the average markup of producers with fixed
nominal prices. However, as Cochrane (2013) noted, this is not a necessary feature
of equilibrium in a new Keynesian economy. When Joseph follows an interest rate
rule like (13), then there always exists an equilibrium that implements the flexibleprice allocation. As in the examples considered by Cochrane in the standard threeequation model, this requires inflation to overshoot π ? when this target is too low.
I call this equilibrium the flexible-price allocation replicator. Begin its construction by selecting π ? ≥ β; assigning the interest rate rule in (13) to Joseph with
the given value of π ? ; and setting Ct , Nt , and St+1 to C̃t+1 , Ñt+1 , and S̃t+1 respectively. Select P00 = 1 and π1 to satisfy (7) given i0 , C0 , and C1 . If π ? ≥ β C̃0 /C̃1 ,
then π1 ≤ π ? . Otherwise, π1 > π ? . In either case, inflation progresses thereafter
according to πt /π ? = (πt−1 /π ? )φ .
This equilibrium construction restates a basic piece of intuition from Krugman
(1998) and Eggertsson and Woodford (2003): It is possible to implement the flexibleprice allocation if short-run inflation expectations are high enough. To the extent
that central banks facing actual liquidity traps have set inflation targets that are
both credible and inappropriately low, the flexible-price replicator deliniteates the
problem of escaping a liquidity trap rather than solving it.

4.3

Mild Famines

Although liquidity traps driven by low inflation expectations and a mild anticipated
famine are not this paper’s focus, they provide a necessary baseline for the results
when the famine is severe. Its characterization begins with the following intuitive
proposition, which allows us to restrict the analysis of this subsection to equilibria
with S1 = 0.
Proposition 1. There exists no equilibrium in which both S1 > 0 and π1 < 1/(1−δ).

18

Proof. Suppose otherwise. Since S1 > 0, we can rearrange (7) and (8) to get
1 + i0
= (1 − δ).
π1
Therefore
1 + i0 = π1 (1 − δ) < 1
This violates the zero-lower bound on interest rates.
Intuitively, a liquidity trap caused by a mild famine decreases C0 and so reduces the
household’s incentive to save. Even with consumption at its higher flexible-price
level, this incentive is insufficient to induce positive storage, so S1 = 0.
To construct a liquidity trap equilibrium that resembles the “discretionary” equilibrium of Eggertsson and Woodford (2003), presume that consumption, storage, and
1
hours worked equal C̃t , S̃t , and Ñt for t ≥ 1. Next, fix a value for π ? ∈ [β, β2 ε−1 ).
(The upper bound on π ? helps ensure that the lower bound in (20) does not constrain C0 in a recession with a high inflation target that is not credible. See Footnote
7 below for details.) With this, select π1 ∈ [β, βAH /AL ). Given this value for π1 ,
C0 can be determined from
C0 =

π1 C̃1
n
 
o
L
β max 1, π ? β −1 AAH P0 (C0 )φ

(22)

In (22), P0 (C0 ) is the Phillips curve from Section 4.1. Since its left-hand side strictly
increases with C0 while its right-hand side weakly decreases with C0 , (22) uniquely
determines C0 . The assumed upper bound on π1 guarantees that C0 < C̃0 . To
ensure that C0 also exceeds its lower bound in (20), assume that7
1
AL
> 2 1−ε .
H
A

7

(23)

For this demonstration, suppose first that π ? ≤ βAH /AL . In this case, C0 = π1 C̃1 /β; so
(23) and the presumption that π1 > β immediately imply (20). If instead π ? > βAH /AL , use
P0 (C0 ) ≤ 1, to show that 1 + i0 ≤ π ? β −1 AL /AH . Therefore, we have
1
C0
π1 C̃1
π 1 AH
AL
π1
β
=
≥
= ? ≥ ? > 2 1−ε
?
−1
L
βπ β A AH
π
π
C̃0
β(1 + i0 )C̃0

Here, the final inequality comes from the assumed upper bound on π ? .

19

Since S0 = 0; N0 , W0 , P00 , and P0 can be obtained from C0 following the logic of
Section 4.1.
The determination of πt for t ≥ 2, combines the interest rate rule with the
first-order condition that characterizes the household’s demand for bonds to get
πt+1


 φ 
? −1 πt
= β max 1, π β
π?

(24)

for all t ≥ 1. This difference equation and the presumed value of π1 yield the desired
inflation sequence. I summarize this equilibrium construction with a
Proposition 2. Suppose that
• βAH /AL < 1/(1 − δ),
1

• π ? ∈ [β, β2 ε−1 ), and
• (23) holds ;
and select π1 ∈ [β, βAH /AL ). Then there exists a equilibrium in which πt equals the
given value of π1 for t = 1, St+1 = 0 for all t ≥ 0, C0 < C̃0 and Ct = C̃t for all
t ≥ 1.
Proof. The only requirement that the proposed equilibrium does not satisfy by construction is (8). To verify that the value of ν0 required to satisfy this (given C0 and
C1 ) is not negative, use (22) and the proposition’s first stated assumption to get
π1 (1 − δ)
βAH /AL (1 − δ)
βC0
(1 − δ) =
≤
<1
C1
1 + i0
1 + i0
So ν0 = 1 − (1 − δ)βC0 /C1 > 0.
In the equilibria of Proposition 2, short-run inflation expectations that are rational but too low cause a real recession. Whether or not i0 = 0 as in other models’
liquidity traps depends on π ? and φ. If π ? ≤ βAH /AL , then i0 must equal zero. If
instead π ? > βAH /AL , then the initial deflation can force i0 to hit the zero lower
bound if φ is large enough. These equilibria can be unambiguously labelled liquidity
traps. However, i0 can exceed zero if both π ? > βAH /AL and φ is small.
With the caveat that i0 might be positive, I will hereafter refer to the equilibria
of Proposition 2 as liquidity traps. Clearly, their multiplicity (indexed by π1 ) arises
20

from the fundamental multiplicity described above. Their traditional interpretation
labels C0 aggregate demand. In this story, monetary policy that is made too tight by
the zero lower bound and inappropriately low inflation expectations lowers aggregate
demand through (7), and this brings about an accompanying deflation. Indeed, the
equilibrium construction does lead from the determination of C0 in the bond market
to the value of P0 required to support that outcome. In this sense, the equilibrium
of Proposition 2 conforms to the familiar pattern of other new Keynesian models of
liquidity traps.
Implicitly, Proposition 2 embodies the now conventional policy prescription for
avoiding a liquidity trap: Somehow convince the public that π1 will exceed the inverse of the natural rate of interest (βAH /AL ). This would allow Joseph to set
nominal bonds’ real return to the value required by the flexible price allocation with
a positive nominal interest rate. If we set π ? > βAH /AL , φ > 1 and presume that
πt = π ? in the long run (for large t), then the inflation target’s assumed long-run
credibility and the Taylor principle mathematically guarantee that π1 = π ? . In light
of Cochrane’s (2011) extensive critique of this scheme’s economic foundations (or
lack thereof), I choose not to adopt it as a useful resolution of equilibrium indeterminacy. Nevertheless, it is worth understanding how such an “active” monetary
policy selects from the equilibrium set if only because it is commonly embodied in
applied work.

4.4

Severe Famines

Although storage occurs in the flexible-price allocation when the expected famine is
severe (βAH /AL > 1/(1 − δ)), it need not do so when there are nominal rigidities.
Intuitively, the possibility of storage is irrelevant when bonds offer a higher real rate
of return. To see this more formally, note that Proposition 1’s preconditions did
not exclude the case of a severe famine. Therefore, it guarantees that S1 = 0 in all
equilibria with π1 < 1/(1 − δ); even if S̃1 > 0. The following corollary guarantees
that such liquidity traps exist.
Corollary 2.1. Suppose that
• 1/(1 − δ) < βAH /AL ,
1

• π ? ∈ [β, β2 ε−1 ), and
21

• (23) holds;
and select π1 ∈ [β, 1/(1 − δ)), then there exists an equilibrium with the consumption,
hours worked, and storage sequences from the equilibrium of Proposition 2 with the
same value of π1 .
Proof. The construction preceding Proposition 2 goes through without modification
if 1/(1 − δ) < βAH /AL . To verify that the value of ν0 required to satisfy (8) is not
negative, use (22) and the upper bound for π1 to get
π1 (1 − δ)
1
βC0
(1 − δ) =
<
≤1
C1
1 + i0
1 + i0
So again, ν0 = 1 − (1 − δ)βC0 /C1 > 0.
Clearly, achieving the flexible-price allocation requires π1 ≥ 1/(1 − δ), but such
fortunate inflation open up the possibility of falling into a confidence recession in
which C0 < C̃0 , N0 < Ñ0 , and S1 < S̃1 even though (1 + i0 )/π1 = (1 − δ). To
construct one, select
 L

A ε−1
, C̃1 .
C1 ∈
(25)
θ ε
The lowest value for C1 in this range is that from the flexible-price allocation without
storage. Proceeding, suppose that (1 + i0 )/π1 = (1 − δ). (Below, this will be
confirmed for i0 given by (13) and some π1 ≤ π ? .) With this supposition and the
given value of C1 , (7) determines
C0 = β −1 (1 − δ)−1 C1 .
The assumption in (23) guarantees that the lowest possible value of C0 exceeds
its lower bound from (20) because β(1 − δ) < 1. Applying Section 4.1’s analysis
then determines values for W0 , P00 , and P0 consistent with (6), (17), and (18).
The optimal labor supply condition requires that N1 = 0 whenever C1 exceeds its
lower bound in (25), so the resulting requirement that year 0 production equals
C0 + S1 = C0 + C1 /(1 − δ) determines N0 .
The equilibrium construction continues by feeding π0 ≡ P0 π ? into the interest
rate rule (13) to yield i0 . With this in hand, the unique value of π1 consistent
with both bond-market clearing and the household’s non-negative choice for S1 is
22

Figure 3: The Intertemporal-Substitution (IS) Curve
(1 + i0 )/π1

1−δ
C0
L

β −1 (1 − δ)−1 Aθ

ε−1
ε

C̃0

(1 + i0 )/(1 − δ) ≥ 1/(1 − δ). With this, (13) and (24) yield it and πt for t ≥ 2.
The upper bound on the return from storage in (15) allows us to set the remaining
values of Ct , St , and Nt to C̃t , S̃t , and Ñt respectively. Again, I summarize this
equilibrium with a
Proposition 3. Suppose that βAH /AL < 1/(1 − δ), and select C1 from the interval
in (25). Then there exists an equilibrium with the given value of C1 and C0 < C̃0 .
In this equilibrium, (1 + i0 )/π1 = (1 − δ). Furthermore, C0 and N0 are strictly
increasing with the chosen value for C1 .
In these confidence recessions, households’ choices of storage are strategic complements: One household’s optimal choice of S1 increases with all other households’
choices. This complementarity arises from the fundamental multiplicity combined
with the endogeneity of C1 .8
Figure 3 summarizes these results with an intertemporal-substitution (IS) curve,
which gives the combinations of real interest rates and consumption consistent with
the equilibria of Corollary 2.1 and Proposition 3. An empty orange circle denotes
8

One might hypothesize that the equilibrium multiplicity demonstrated by Proposition 3 arises
from the anticipation of different paths for inflation and nominal interest rates. To show that this is
incorrect, set φ to zero. In this special case of extremely passive interest-rate policy, the equilibria
of Proposition 3 share common inflation and interest-rate sequences: πt = π ? and it always equals
the “natural” interest rate in the intercept of (13). This justifies the claim at the start of Section
4.1 that confidence recessions can share interest-rate and inflation sequences.

23

the limiting equilibrium as C0 is driven to its lower bound in (20), while a solid
blue circle marks the equilibrium that implements the flexible-price allocation.9 If
bonds’ real return exceeds 1 − δ, then S1 = 0. Over this range, the IS curve inherits
its shape from the Euler equation (7). These equilibria are the model’s liquidity
traps from Corollary 2.1. Corollary 2.1 says that Joseph could choose any of these
equilibria if he had complete control over π1 .
If instead bonds’ real return equals (1 − δ), Proposition 3 tells us that any
C0 ∈ [β −1 (1 − δ)−1

AL ε − 1
, C̃0 )
θ ε

is consistent with equilibrium. Therefore, the IS curve becomes horizontal. All
points on this horizontal segment to the left of the blue dot represent confidence
recessions. In all but one of these, the transitional equilibrium point denoted by a
blue circle filled with orange where the IS curve’s horizontal segment begins, S1 > 0.
The transitional equilibrium is both a liquidity trap and a confidence recession. In it,
the non-negativity constraint on storage does not bind. Nevertheless, the household
chooses S1 = 0.
The IS curve’s horizontal segment suggests that Joseph might not be able to
avoid a confidence recession even if monetary policy could somehow determine (1 +
i0 )/π1 . If both π ? > 1/(1 − δ) and φ > 0, then the only equilibrium with π1 = π ?
implements the flexible-price allocation. In this special case, Joseph can indeed
guide the economy to the flexible-price allocation if he (somewhat magically) could
set π1 = π ? . Given any interest rate rule, properly-implemented JQE can accomplish
this goal

5

Josephean Quantitative Easing

Incorporating JQE into the analysis requires only relaxing the assumption that
B1 = Q1 = 0 and modifying the resource constraint to account for Joseph’s storage.

1
2



At Nt

 1 −ε = Ct + (St+1 + Qt+1 )/(1 − δ) − St − Qt
−ε
Pt0
1 Pt
+ 2 Pt
Pt

9

Because Proposition 3’s interval for C1 is open to the right, it does not include this flexible-price
equilibrium. However, this equilibrium is indeed the limit as C1 → C̃1 .

24

I maintain the assumption that Joseph sets Bt = Qt = 0 for t ≥ 2 to mimic the
flexible-price allocation’s absence of storage after the famine’s first year.
Given total storage, its decomposition between St+1 and Qt+1 is of no consequence to the household. Nevertheless, Joseph might prefer public storage because
setting B1 > 0 and Q1 to the resulting real goods accumulated can impact the equilibrium set through two channels. First, the accumulation of a primary surplus and
its offsetting liabilities allows the fiscal theory of the price level to determine P1 . In
turn, this requires Joseph’s real cost of funds (the real return on nominal bonds) to
equal the real return on his storage investments. In the liquidity traps proven to
exist by Corollary 2.1, the real return on nominal bonds exceeds the cost of storage.
Therefore, these are inconsistent with even a small amount of JQE. I summarize
this first channel in the following
Proposition 4. If B1 > 0, then in any equilibrium,
1 + i0
= (1 − δ).
π1

(26)

Furthermore, there exists no equilibrium with consumption and prices equal to those
from an equilibrium proven to exist by Corollary 2.1.
Proof. To prove that Equation (26) must hold in an equilibrium with B1 > 0, use
(4) for year 0, Q1 = (1 − δ)B1 /((1 + i0 )P0 ), to eliminate Q1 from the same equation
for year 1, B1 /P1 = Q1 . Remove B1 from the resulting equation and rearrange. For
the second assertion, note that in the referenced equilibria we have
1−β

C0
C0 1 + i 0
(1 − δ) = ν0 > 0 = 1 − β
,
C1
C 1 π1

which contradicts Equation (26).
In theory, even a small amount of JQE can substitute for inflation-expectations
management by other (unmodeled) means, such as the communications protocols
of an inflation-targeting regime. In practice, its efficacy at this task depends on
whether or not households expect (4) to hold in year 1. If Joseph could recover any
capital loss incurred from deflation by taxing households, then deflation might occur
in equilibrium. In that case, nominal bonds’ real rate of return would exceed the
real rate of return on storage, so Proposition 4’s conclusions would not hold. For
25

this reason, JQE might best be delegated to a monetary authority without access
to a reliable stream of tax revenues, such as the ECB.
With this potentially important caveat in place, we can proceed to consider the
second channel for JQE to influence the equilibrium set: The primary surplus Q1
places a floor on C1 . This in turn bounds C0 from below and thereby eliminates
confidence recessions with consumption beneath the bound. Unsurprisingly, this
channel’s efficacy depends on the magnitude of B1 , not just its sign.
To develop this in more detail, define
C 0 ≡ (1 − δ)−1 β −1

AL ε − 1
and P 0 ≡ P0 (C 0 ).
θ ε

These are the initial consumption and price level in the worst equilibrium of Proposition 3; which is the transitional equilibrium in Figure 3’s IS curve. (The underlines
indicate that these are lower bounds.) With this notation, we can state
Proposition 5. Define B 1 ≡ βC 0 P 0 max{1, π ? (1−δ)P φ0 }. For each B1 ∈ [B 1 , π ? C̃1 ],
there exists a threshold C̄0 (B1 ) for C0 such that
1. there is no equilibrium with C0 < C̄0 (B1 );
2. any equilibrium of Proposition 3 with C0 ≥ C̄0 (B1 ) has a corresponding equilibrium with the given value of B1 and the same sequences for Ct and Nt ;
3. C̄0 (B̄1 ) = C 0 ;
4. C̄0 (B1 ) is strictly increasing in B1 ; and
5. C̄0 (π ? C̃1 ) = C̃0 .
Proof. From Proposition 4, we know that any equilibrium with B1 > 0 has P0 >
P 0 , so the amount of the aggregate good that Joseph can take into year 1 with
the funds raised by issuing nominal bonds with redemption value B1 must exceed
(1 − δ)B1 /(P 0 max{1, π ? (1 − δ)P φ0 }). The definition of B 1 ensures that this lower
bound equals β(1 − δ)C 0 , which in turn is below any equilibrium value of C1
consistent with B1 > B 1 .
Define
Υ(C, B) ≡ β(1 − δ)C −

(1 − δ)B
,
P0 (C) max{1, (1 − δ)π ? P0 (C)φ }
26

and with this define C̄0 (B1 ) implicitly from Υ(C̄0 (B1 ), B1 ) = 0. That is, Q1 = C1
if C0 = C̄0 (B1 ). We know that Υ(C 0 , B1 ) ≤ 0, because issuing B1 ≥ B 1 bonds
facing the same price level and nominal interest rate as the worst equilibrium of
Proposition 3 and investing the proceeds yields Q1 ≥ β(1 − δ)C 0 . Alternatively
Υ(C̃0 , B) ≥ 0, because the assumed upper bound on B1 keeps Q1 ≤ β(1 − δ)C̃0
when the price level and interest rate are those from the flexible-price allocation.
Furthermore, Υ(C, B) is strictly increasing with C. Therefore, there exists exactly
one value of C̄0 (B1 ) that satisfies its definition. With this in hand the proposition’s
third and fifth conclusions can be directly verified by substitution into Υ(C, B), and
the fourth conclusion follows from noting that increasing B strictly decreases this
function.
Proposition 4 immediately implies that no equilibrium exists with C0 < C 0 .
The first conclusion’s demonstration only requires us to demonstrate the same if
C0 ∈ [C 0 , C̄0 (B1 )). For this, assume the opposite.
• Since P0 (C0 ) is strictly increasing, we know that P0 < P0 (C̄0 ) and 1 + i0 ≤
max{1, (1 − δ)π ? P0 (C̄0 (B1 ))φ }. Therefore, Q1 > β(1 − δ)C̄0 (B1 ) > β(1 − δ)C0 .
From Proposition 4, (7), and (8), we know that C1 = β(1 − δ)C0 , so Q1 > C1 .
• The upper bound on the return to storage in (15) can be rewritten as 1 ≥
β(1 − δ)C̃1 /C̃2 . Since in the hypothesized equilibrium C1 < C̃1 and because
(6) requires that C2 ≥ C̃2 in any equilibrium; we know that 1 > β(1−δ)C1 /C2 .
From (8) and the complementary slackness condition, we therefore can conclude that S2 = 0.
Because Q1 > C1 and Q2 = 0 by assumption, the resource constraint requires
S2 > 0. Therefore, these two conclusions of assuming that C0 < C̄0 (B1 ) contradict
each other.
All that remains to be demonstrated is the Proposition’s second conclusion.
Begin this by adopting the original equilibrium’s sequences for Ct , Nt , Wt , Pt0 ,
1
Pt+1
, Pt , Dt , and it . Then, set Q1 = (1 − δ)B1 /(P0 (1 + i0 )). Since C0 ≥ C̄0 (B1 ),
Q1 ≤ β(1 − δ)C̄0 (B1 ) ≤ C1 . Therefore, we can set S1 = C1 − Q1 without violating
the non-negativity constraint on storage. To complete the candidate equilibrium,
set St = Bt = Qt = 0 for all t ≥ 2. The sequences for Ct , Nt , Bt+1 , and St+1 solve
the household’s utility maximization problem given the sequences for Dt , Wt , Pt ,
and it ; because the household is indifferent between directly accumulating C1 and
27

indirectly doing so by purchasing bonds with the same rate of return. Firms’ original
pricing decisions remain optimal; and Bt+1 , Qt+1 , and it satisfy (4). Therefore, the
candidate is indeed an equilibrium.
To summarize, the two channels for JQE allow Joseph to destroy all liquidity
traps and confidence recessions by setting B1 = π ? C̃1 . A slightly smaller balancesheet expansion eliminates some confidence recessions, but leaves those with C0
slightly below C̃0 in place. Finally, a very small balance sheet expansion eliminates
liquidity traps by equating nominal bonds’ real return with that of storage, but it
leaves room for households to coordinate on a confidence recession with too little
saving. Regardless, JQE requires no commitment to time-inconsistent interest-rate
or balance-sheet policies.
Ricardian equivalence with policy commitment provides the point of departure
for most theoretical discussions of QE. For example, Eggertsson and Woodford
(2003) “argue that the possibility of expanding the monetary base through central bank purchases of a variety of types of assets does little if anything to expand
the set of feasible paths for inflation and real activity that are consistent with equilibrium under some (fully credible) policy commitment.”10 Nothing in this paper
contradicts this assertion. Here, JQE potentially improves economic outcomes by
shrinking the set of feasible paths that are consistent with equilibrium.
Previous models of liquidity traps featuring policy-relevant QE have either featured frictions that impede private borrowing and lending (Cúrdia and Woodford,
2011; Gertler and Karadi, 2011), financial markets segmented by asset maturity
(Chen, Cúrdia, and Ferrero, 2012), or limited commitment that can be overcome
somewhat by manipulating the maturity structure of the monetary authority’s balance sheet (Bhattari, Eggertsson, and Gafarov, 2014). In all of those approaches,
QE can potentially improve a given equlibrium outcome. In contrast, JQE has no
impact on an equilibrium if it would have occurred anyways. Instead, it guides
households’ expectations towards the flexible-price allocation. The quality of that
guidance depends on how close B1 is to π ? C̃1 .
With segmented financial markets, the monetary authority can influence assets’
relative prices by changing their relative supplies. Accordingly, empirical investigations of QE have concentrated on measuring its impact on asset prices. The
10

See Page 143 of Eggertsson and Woodford (2003).

28

present economy is Ricardian, but it would be incorrect to conclude from that fact
alone that JQE does not influence asset prices. When it is successful, in the sense
that it eliminates a confidence recession that would have otherwise occurred, longdated interest rates fall because both C0 and C1 rise while all Ct for t ≥ 2 remain
unchanged. Although Eggertsson and Woodford (2003) emphasize that forwardguidance can expand current economic activity by reducing long-dated real interest
rates, the analogous reduction in this model is a consequence of such an economic
expansion; not its cause.

6

An Open Economy Interpretation

Although I have developed the analysis of JQE in a closed economy, the most
straightforward interpretation of the model’s linear storage technology is as a representation of using international trade to achieve intertemporal substitution. For
this, suppose that the economy is small relative to a large foreign sector. The aggregate good can be shipped either to or from the foreign sector at the iceberg
transportation cost τ . The real rate of return available in the foreign sector is rf .
Then, if we define δ with
1 − δ = (1 − τ )2 (1 + rf );
we can interpret storage as shipping aggregate goods abroad, selling them, investing
the proceeds in foreign bonds, and repatriating the proceeds in the next year by
shipping the aggregate good back home. Two notable differences exist between this
open economy extension and the original closed economy. First, the restriction that
St ≥ 0 should be interpreted as a limit on uncollateralized international borrowing.
Second, the possibility of importing the good eliminates all equilibria in which P0 <
(1 − τ ). Thus, greater openness to trade improves the worst possible equilibrium
but does not necessarily eliminate the possibility of falling into a liquidity trap.
With this caveat, any equilibrium of the closed economy has a doppelganger in the
small-open-economy extension.11
11

If we suppose that the foreign sector uses a currency subject to no inflation with a price-level
of 1, then we can introduce a market for the exchange of home and foreign currencies. If the
aggregate good is shipped in either direction, then the price of foreign currency in units of home
currency is et = Pt /(1 − τ ). Therefore, international trade never occurs unless purchasing-power

29

In the open economy, JQE mimics the monetary mechanics of a sterilized competitive devaluation (swap interest-bearing domestic liabilities for foreign assets), and
the resulting current account surplus matches the conventional Mundell-Fleming
model’s predictions. This paper is not the first to notice the strong resemblance between sterilized interventions and quantitative easing. For example, Rajan (2014) labels such interventions (tongue in cheek) as “Quantitative External Easing” (QEE).
He reports
Indeed, some advanced economy central bankers have privately expressed
their worry to me that QE “works” primarily by altering exchange rates,
which makes it different from QEE only in degree rather than in kind.12
It is inconceivable that these anonymous central bankers had JQE in mind when
confiding with Rajan, but from this model’s perspective QE and QEE are indeed
cut from the same cloth. Nevertheless, changes to the real exchange rate play
no role in JQE’s effectiveness. Instead, it works by coordinating home-country
households’ savings decisions and thereby enabling them to substitute consumption
intertemporally using international trade. As noted in the introduction, the foreign
sector’s initial current-account deficit and its eventual reversal are not side effects
of JQE. Together, they are its goal.
If the foreign sector itself also faces a Keynesian shortfall in aggregate demand,
then JQE can easily turn into a beggar-thy-neighbor affair. However, without
foreign-sector inefficiencies it results in a Pareto-efficient allocation of world resources. This suggests that the international monetary policy cooperation advocated by Rajan (2014) can indeed improve worldwide macroeconomic performance
when these two possibilities can be distinguished. Further investigation of this point
is certainly worthwhile, but it requires an explicit model of the foreign sector that
lies beyond this paper’s scope.
parity holds up to the constant iceberg transportation cost. If there is no international trade, then
equilibrium only requires et ∈ [Pt (1 − τ ), Pt /(1 − τ )]. One might conclude that JQE depreciates
the currency if it eliminates an equilibrium with S1 = 0 and et < Pt /(1 − τ ) that otherwise would
have occurred. However, such a depreciation is not logically necessary and so cannot be said to
cause the home country’s initial current-account surplus.
12
Page 6 of Rajan (2014)

30

7

Conclusion

A shock to the demand for real assets leads households to accumulate goods in
storage for consumption, just as in the biblical Joseph story, when prices are flexible.
Price stickiness can disrupt this outcome and send the economy into a recession even
when nominal bonds’ real return is consistent with the flexible-price allocation. In
this sense, conventional interest-rate policy and forward guidance that manipulates
inflation expectations cannot necessarily guide the economy to its potential. QE that
purchases real assets, JQE, can fill this gap. JQE puts a floor on future national
wealth and consumption, and the expectation of high future consumption raises
current consumption and output. Ironically, the full solution to the “paradox of
thrift” coordinates an increase in savings.
Although the Federal Reserve has begun to end its balance-sheet expansion, QE’s
relevance for global monetary policy has not abated. Under the heading of “Quantitative and Qualitative Easing,” the Bank of Japan currently purchases about U80
Trillion per month of securities. Although most these are Japanese sovereigns, the
remainder are a wide variety of private assets. The analysis of JQE suggests that tilting this portfolio choice towards private assets could improve Japanese economic performance by increasing the economy’s real wealth accumulation. To date, the ECB’s
quantitative easing has more closely resembled JQE. Its Targeted Long-Term Refinancing Operations directly extend credit to banks that themselves expand credit
to private non-financial borrowers, and it just began direct purchases of securities
backed by Euro-area non-financial private assets.13 This paper provides a possible
justification for the ECB’s quantitative easing: Increasing European households’
real wealth boosts their expectations of future consumption and thereby improves
current economic performance.

13

See the Introductory Statement to Mario Draghi’s press conference on 4 September 2014.

31

Appendix: Increasing Marginal Costs of Storage
This appendix replaces the linear storage technology employed in the text with a
concave technology that is represented by a convex cost function. So that the profits
associated with the resulting scarce storage opportunities are properly accounted,
I take the storage technology out of the households’ hands and add banks to the
model. There is a unit mass of banks, each of which can produce S units of the
aggregate good next year by investing Ξ(S) units of the aggregate good in the storage
technology. This input-requirement/cost function is twice differentiable everywhere,
and satisfies Ξ(0) = 0, Ξ0 (0) > 0, and Ξ00 (S) > 0. As did the households’ investments
in the model’s text, banks’ investments must satisfy St+1 ≥ 0. This technology’s
analogue to (3) is
β
< 1.
(A1)
0
Ξ (0)
This ensures that storage is not worthwhile when consumption is constant. Finally,
the aggregate resource constraint with flexible prices and this technogy is
At Nt = Ct + Ξ(St+1 ) − St .
Banks finance their inputs by issuing nominal bonds. In the next year, they use
the proceeds from selling storage technology’s output to retire them. Any remaining
proceeds are returned to the representative household as dividends. Just like those
of the economy’s firms, these dividends can be negative because banks face unlimited
liability. Banks in year t choose St+1 to maximize real dividends in period t + 1,
St+1 − Ξ(St+1 )(1 + it )/πt+1 . If we use ωt to denote the nonnegativity constraint’s
Lagrange multiplier, then the first-order necessary condition for this problem is
1 + ωt = Ξ0 (St+1 )(1 + it )/πt+1 .

(A2)

If ωt > 0, then the cost of storage investment exceeds its benefit, so St+1 = 0.

I

The Flexible-Price Allocation

With the text’s linear storage technology, famines were classified into severe and
moderate depending on whether or not the flexible price allocation set S̃1 > 0.
With the more general convex cost of storage, it is useful to divide famines into
32

three categories; severe, intermediate, and moderate. In a severe famine, S̃1 > 0
and Ñ1 = 0. That is the household saves in order to take a vacation during the
famine’s first year. In an intermediate famine, S̃1 > 0 but Ñ1 > 0. The household
uses storage to reallocate hours worked from year 1 to year 0 and thereby save on
its utility cost, but the consumption profile is the same as that in a mild famine,
when S1 = 0 and the storage technology is irrelevant.
I.1

Mild Famines

To define a mild famine, replace (12) with
βAH
≤ Ξ0 (0).
L
A

(A3)

When this holds, the flexible-price allocation is exactly the same as that in the text.
I.2

Severe Famines

To define a severe famine, first denote the flexible-price allocation’s consumption for
years t ≥ 1 with a mild anticipated famine using
C? ≡

AL ε − 1
.
θ ε

In a severe famine, the marginal cost of increasing storage at C ? is less than its
benefit when C1 = C ? . That is
βAH
> Ξ0 (C ? )
AL

(A4)

(As in the text, C̃0 depends only on AH , ε, and θ.) This guarantees that S̃1 ≥ C̃1 .
Suppose for the moment that S̃1 = C̃1 as in the text. With this, combining banks’
profit maximization condition with (7) gives
β C̃0 = C̃1 Ξ0 (C̃1 ).

(A5)

This implicitly defines C̃1 . With this in hand, replacing (15) with
β C̃1
< Ξ0 (0)
C̃2
33

(A6)

guarantees that indeed S2 = 0.14 Aside from this modification to C̃1 and the attendant change to Ñ0 , the flexible-price allocation with a severe famine is the same as
that in the text.
I.3

Intermediate Famines

With the linear technology of the text, Ξ0 (0) = Ξ0 (C ? ), so either (A3) or (A4) must
hold. The assumption that Ξ00 (S) > 0 creates a third case.
Ξ0 (0) <

βAH
≤ Ξ0 (C ? )
L
A

(A7)

This says that the marginal benefit of storage when C0 = C̃0 and C1 = C ? exceeds its
marginal cost when there is no storage but is less than its marginal cost when storage
equals or exceeds C ? . In this case, the flexible-price allocation’s consumption profile
equals that from a mild famine. To retrieve S̃1 , use banks’ profit maximization
condition.
β C̃0
= Ξ0 (S̃1 )
(A8)
C̃1
With S̃1 in hand, the resource constraint immediately yields Ñ0 and Ñ1 . All of the
allocation’s other quantities equal those from a mild famine.

II

Equilibria with Nominal Rigidities

Much of the text’s analysis of equilibria with nominal rigidities applies to the model
with Ξ00 (S) > 0 with little or no modification. Section 4.1’s characterization of
the Phillips curve has nothing to do with the storage technology, and adapting the
flexible-price allocation replicator of Section 4.2 to this economy is a simple exercise.
As in the text, adding nominal rigidities introduces a possible production inefficiency
into the resource constraint.
14

Together, (A4) and (A6) require that
β 2 AH /AL ∈ (βΞ0 (C ? ), Ξ0 (0)Ξ0 (C ? )) .

This interval is non-empty because (A1) guarantees that β < Ξ0 (0) and (6) and the convexity of
Ξ(·) together ensure that Ξ(C̃1 ) ≥ Ξ(C ? ). Therefore, the assumptions in (A4) and (A6) can be
simultaneously satisfied.

34

1
2

II.1



At Nt

 1 −ε = Ct + Ξ(St+1 ) − St .
−ε
Pt0
1 Pt
+ 2 Pt
Pt

(A9)

Mild Famines

With this change, the analogues to Propositions 1 and 2 are
Proposition A1. There exists no equilibrium in which both S1 > 0 and π1 < Ξ0 (0).
Proof. Suppose otherwise. Since S1 > 0, we can rearrange (A2) to get
1 + i0 =

π1
0
Ξ (S1 )

<

π1
<1
Ξ0 (0)

This violates the zero-lower bound on interest rates.
Proposition A2. Suppose that
• βAH /AL < Ξ0 (0),
1

• π ? ∈ [β, β2 ε−1 ), and
• (23) holds ;
and select π1 ∈ [β, βAH /AL ). Then there exists a equilibrium in which πt equals the
given value of π1 for t = 1, St+1 = 0 for all t ≥ 0, C0 < C̃0 and Ct = C̃t for all
t ≥ 1.
Proof. The proposed equilibrium is that constructed in the text, and the only requirement that the proposed equilibrium does not satisfy by construction is (A2).
To verify that the value of ω0 required to satisfy this is not negative, use the upper
bound for π1 and the Proposition’s first stated assumption to get
1 + i0 0
1 + i0 0
Ξ (0) > 1.
Ξ (0) >
π1
βAH /AL
So ω0 =

1+i0 0
Ξ (0)
π1

− 1 > 0.

The analogue to Corollary 2.1 applies to both intermediate and severe famines
Corollary A2.1. Suppose that
35

• Ξ0 (0) < βAH /AL ,
1

• π ? ∈ [β, β2 ε−1 ), and
• (23) holds;
and select π1 ∈ [β, Ξ0 (0)), then there exists an equilibrium with the consumption,
hours worked, and storage sequences from the equilibrium of Proposition A 2 with
the same value of π1 .
Proof. The proposed equilibrium is from construction preceding Proposition 2, and
this goes through without modification if Ξ0 (0) < βAH /AL . To verify that the value
of ω0 required to satisfy (A2) is not negative, use the upper bound for π1 to get
1 + i0 0
Ξ (0) > 1 + i0 ≥ 1.
π1
So again, ω0 =
II.2

1+i0 0
Ξ (0)
π1

− 1 > 0.

Intermediate Famines

With an intermediate famine, a type of recessionary equilibrium arises that does not
appear in the model of the text, a storage recession. To construct one, set C1 = C̃1
and select
C̃1 Ξ0 (0)
, C̃0 ].
(A10)
C0 ∈ [
β
The lower end of this interval is the largest C0 from an equilibrium of Corollary
1
A2.1, and this exceeds 2 1−ε C̃0 . Therefore, we can apply Section 4.1 to find values
for P00 , P0 , and W0 consistent with any such C0 , (6), (17), and (18). Continuing,
combine (7) with (A2) to yield
βC0
= Ξ0 (S1 )
C̃1
This implicitly defines S1 as an increasing function of C0 . This can be no greater
than S̃1 because C0 ≤ C̃0 . To complete the equilibrium allocation’s construction,
set Ct = C̃t , St = 0, and Nt = Ñt for t ≥ 2.
To get this equilibrium allocation’s accompanying nominal interest rates and
prices, plug π0 ≡ π ? P0 into (13) to get i0 ; and use this and the real interest rate
36

(1 + i0 )/π1

1/Ξ0 (0)
1/Ξ0 (S̃1 )
C0
β −1 Ξ0 (0)C̃1

C̃0

Figure A1: The IS Curve with an Intermediate Famine
implied by S1 to set π1 .
π1 = (1 + i0 )Ξ0 (S1 )

(A11)

Equations (13) and (24) then give it and πt for t ≥ 2. I summarize this with a
Proposition A3. Suppose that βAH /AL > Ξ0 (0), and select C0 from the interval
in (A10). Then there exists an equilibrium with the given value of C0 and C1 = C̃1 .
Furthermore, S1 is strictly increasing with the chosen value for C0 .
Economically, a storage recession occurs when a high real interest rate (supported
by a lack of real investment) resolves the fundamental multiplicity with a low value
of C0 .
Figure A1 plots the IS curve for the case with an intermediate famine. As in
Figure 3, the empty orange circle indicates the limit of the liquidity trap equilibria
1
as C0 is driven to its lower bound of 2 1−ε C̃0 , and the solid blue circle denotes
the equilibrium that implements the flexible-price allocation. For real interest rates
above 1/Ξ0 (0), the economy is in a liquidity trap, and below that but above 1/Ξ0 (S̃1 )
it is in a storage recession. The blue circle filled with orange is the transitional
equilibrium that falls into both of these categories.
II.3

Severe Famines

Since both Corollary 2.1 and Proposition 3 apply when the anticipated famine is
severe, either a liquidity trap or a storage recession is possible in this case. To
37

(1 + i0 )/π1

1/Ξ0 (0)
1/Ξ0 (C ? )
C0
β −1 (1 − δ)−1 C ?

C̃0

Figure A2: The IS Curve with a Severe Famine
construct a confidence recession, begin by selecting
h

C1 ∈ C ? , C̃1 .

(A12)

Then set S1 = C1 and C0 = β −1 C1 Ξ0 (C1 ). Given C0 , retrieve P0 from the Phillips
curve and then get W0 and P00 from (6) and (17). The resource constraint in (A9)
then determines N0 , and N1 = 0. The interest-rate rule in (13) gives i0 ; and
setting π1 = (1 + i0 )Ξ0 (C1 ) ensures that both (7) and (A2) are satisfied. For t ≥ 2;
setting St = 0, Ct = C̃t ; Nt = Ñt , πt using (24), and it using (13) completes the
equilibrium construction. The upper bound on the return to storage in (A6) ensures
that this corner solution for S2 maximizes bank profits given C0 and C1 . With this
construction in hand, I can state the following analogue to Proposition 3.
Proposition A4. Suppose that Ξ0 (C ? ) < βAH /AL , and select C1 from the interval
in (A12). Then there exists an equilibrium with the given value of C1 and C0 < C̃0 .
In this equilibrium, C0 and N0 are strictly increasing with the chosen value for C1 .
Figure A2 plots the IS curve for this model that is the direct analogue of that
in Figure 3. As in the earlier IS curves, the empty orange circle indicates the
limit attaned from driving C0 to its lower bound, and the solid blue dot marks the
equilibrium that implements the flexible-price allocation. Blue circles are at the two
transitional equilibria. The left equilibrium is both a liquidity trap and a storage
recession, while the right one is both a storage recession and a confidence recession.
38

The IS curve is differentiable at the first one but not at the second. Unlike the
one in Figure 3, the IS curve has no horizontal segment. The discussion in the
text emphasized the fact that Joseph might not be able to guide the economy to
the flexible-price allocation even if he (somewhat magically) could set (1 + i0 )/π1
directly. Here with Ξ00 (·) > 0, such extremely-effective guidance of real interest
rates does indeed destroy all recessionary equilibria. Of course, the sensitivity of
economic outcomes with respect to π1 depends inversely on Ξ00 (·). If this is very
small, then the IS curve is nearly horizontal and very small changes in π1 can have
large impacts on C0 . To the extent that actual central bankers can only influence
π1 imprecisely and indirectly, the ability of JQE to eliminate recessionary outcomes
remains of interest.

III

Josephean Quantitative Easing

In the text, Joseph invested directly in the storage technology. Here, banks make this
investment decision; so the specification of JQE must be suitably modified. Joseph
issues nominal bonds with nominal redemption value B1 and uses the proceeds to
make loans to banks at the same nominal interest rate. Banks invest the proceeds of
their borrowing, both from Joseph and from the private sector, and repay the loans
in the next period with the proceeds of the storage technology. Since Ξ00 (·) > 0,
the banks will have profits following a positive investment in storage. These are
returned to the representative household as dividends.
Of course, neither the household nor the banks care about the fraction of a given
investment in storage that Joseph intermediates. However, it is relevant for the
equilibrium analysis because Joseph’s intermediation choice places a lower bound
on total borrowing. To denote that it is a lower bound, I use B t to indicate the
year t redemption value of bonds issued by Joseph in year t − 1. I continue to use
Bt to represent the household’s total bond holdings, so the bonds directly issued by
banks are worth Bt − B t ≥ 0 on redemption.
The text separated the influence of JQE into two channels, which corresponded to
the elimination of liquidity traps by setting B 1 > 0 and the elimination of confidence
recessions by setting B 1 = π ? C̃1 . The presence of storage recessions (which requires
Ξ00 (·) > 0), makes these two channels less distinct. Therefore, I place the results
analogous to Propositions 4 and 5 within the following single proposition.

39

Proposition A5. Suppose that Ξ0 (0) < βAH /AL . For each

i
B 1 ∈ 0, Ξ−1 (S̃1 ) max{1, π ? /Ξ0 (S̃1 )} ,
there exists a threshold C̄0 (B 1 ) for C0 such that
1. there is no equilibrium with C0 < C̄0 (B 1 );
2. any equilibrium of Proposition A3 or A4 with C0 ≥ C̄0 (B 1 ) has a corresponding
equilibrium with some B1 ≥ B 1 and the same sequences for Ct and Nt ;
3. C̄0 (B 1 ) is strictly increasing in B 1 ; and
4. C̄0 (Ξ−1 (S̃1 ) max{1, π ? /Ξ0 (S̃1 )}) = C̃0 .
Proof. Define
B ? ≡ P0 (β −1 C ? Ξ0 (C ? ))Ξ−1 (C ? ) max{1, π ? P0 (β −1 C ? Ξ0 (C ? ))φ /Ξ0 (S̃1 )}
If the anticipated famine is intermediate, then B ? equals the upper bound of the
admissible interval for B 1 . If instead it is severe, then it lies in this interval’s interior.
To define C̄0 (B 1 ), first define








B
−1
−Ξ Ξ
max 1,π ? P0 (C)φ /Ξ0 (C ? )}

 {

Υ(C, B) ≡

βC
B
0
−1

 θ(C) − Ξ Ξ
max{1,π ? P0 (C)φ /Ξ0 (S̃1 )}
βC
C?

0

if B ≤ B ?
otherwise;

where θ(C) definitionally satisfies θ(C)Ξ0 (θ(C)) = C. If B < B ? , then
Υ(β −1 C ? Ξ0 (0), B) < 0 and Υ(β −1 C ? Ξ0 (C ? , B) ≥ 0.
If instead B ≥ B ? , then
Υ(β −1 C ? Ξ0 (C ? ), B) ≤ 0 and Υ(C̃0 , B) ≥ 0.
In either case, Υ(C, B) is strictly increasing with C, so we can define C̄0 (B 1 ) with
Υ(C̄0 (B 1 ), B 1 ) = 0.
Since Υ(C, B) is strictly decreasing with B, the proposition’s third assertion
is easily proven; and its fourth assertion can be easily verified by noting that the
40

proposed threshold satisfies the definition given here. To demonstrate the first
assertion, suppose that it is not true. That is, there exists an equilibrium with C0 <
C̄0 (B 1 ). This requires P0 < P0 (C̄0 (B 1 )) and 1+i0 ≤ max{1, π ? P0 (C̄0 (B 1 ))φ /Ξ0 (S̃1 )}.
Both of these
 changes raise the real purchasing power of Joseph’s bond issuance, so
S1 > Ξ−1 B1 / max{1, π ? P0 (C̄0 (B 1 ))φ /Ξ0 (S̃1 )} . Therefore,


 β C̄ (B )
0
1
Ξ0 (S1 ) > Ξ0 Ξ−1 B1 / max{1, π ? P0 (C̄0 (B 1 ))φ /Ξ0 (S̃1 )} =
C̃1
However, equilibrium and S1 > 0 require that Ξ0 (S1 ) = βC0 /C1 . Therefore, we can
conclude that C0 > C̄0 (B 1 ), a contradiction.
The final assertion to be proven is the second. This proceeds by construction,
exactly paralleling the analogous demonstration for Proposition 5.
Note that the portion of this proof that demonstrates JQE’s effectiveness in
destroying recessionary equilibria is somewhat different from the analogous demonstration from the text. Since Ξ00 (·) > 0, the proof proceeds by showing that JQE
bounds the marginal cost of storage from below. If I allowed Ξ00 (S) to equal zero
while still maintaining the convexity of Ξ(·), then this proof strategy would not be
available. In this alternative, adapting the proof of the text (which bounds C1 from
below) is straightforward.

41

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43

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Following the Financial Crisis
Sumit Agarwal, Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet,
and Douglas D. Evanoff

WP-11-03

Federal Reserve Policies and Financial Market Conditions During the Crisis
Scott A. Brave and Hesna Genay

WP-11-04

The Financial Labor Supply Accelerator
Jeffrey R. Campbell and Zvi Hercowitz

WP-11-05

Survival and long-run dynamics with heterogeneous beliefs under recursive preferences
Jaroslav Borovička

WP-11-06

A Leverage-based Model of Speculative Bubbles (Revised)
Gadi Barlevy

WP-11-07

Estimation of Panel Data Regression Models with Two-Sided Censoring or Truncation
Sule Alan, Bo E. Honoré, Luojia Hu, and Søren Leth–Petersen

WP-11-08

Fertility Transitions Along the Extensive and Intensive Margins
Daniel Aaronson, Fabian Lange, and Bhashkar Mazumder

WP-11-09

Black-White Differences in Intergenerational Economic Mobility in the US
Bhashkar Mazumder

WP-11-10

Can Standard Preferences Explain the Prices of Out-of-the-Money S&P 500 Put Options?
Luca Benzoni, Pierre Collin-Dufresne, and Robert S. Goldstein

WP-11-11

Business Networks, Production Chains, and Productivity:
A Theory of Input-Output Architecture
Ezra Oberfield

WP-11-12

Equilibrium Bank Runs Revisited
Ed Nosal

WP-11-13

Are Covered Bonds a Substitute for Mortgage-Backed Securities?
Santiago Carbó-Valverde, Richard J. Rosen, and Francisco Rodríguez-Fernández

WP-11-14

The Cost of Banking Panics in an Age before “Too Big to Fail”
Benjamin Chabot

WP-11-15

1

Working Paper Series (continued)
Import Protection, Business Cycles, and Exchange Rates:
Evidence from the Great Recession
Chad P. Bown and Meredith A. Crowley

WP-11-16

Examining Macroeconomic Models through the Lens of Asset Pricing
Jaroslav Borovička and Lars Peter Hansen

WP-12-01

The Chicago Fed DSGE Model
Scott A. Brave, Jeffrey R. Campbell, Jonas D.M. Fisher, and Alejandro Justiniano

WP-12-02

Macroeconomic Effects of Federal Reserve Forward Guidance
Jeffrey R. Campbell, Charles L. Evans, Jonas D.M. Fisher, and Alejandro Justiniano

WP-12-03

Modeling Credit Contagion via the Updating of Fragile Beliefs
Luca Benzoni, Pierre Collin-Dufresne, Robert S. Goldstein, and Jean Helwege

WP-12-04

Signaling Effects of Monetary Policy
Leonardo Melosi

WP-12-05

Empirical Research on Sovereign Debt and Default
Michael Tomz and Mark L. J. Wright

WP-12-06

Credit Risk and Disaster Risk
François Gourio

WP-12-07

From the Horse’s Mouth: How do Investor Expectations of Risk and Return
Vary with Economic Conditions?
Gene Amromin and Steven A. Sharpe

WP-12-08

Using Vehicle Taxes To Reduce Carbon Dioxide Emissions Rates of
New Passenger Vehicles: Evidence from France, Germany, and Sweden
Thomas Klier and Joshua Linn

WP-12-09

Spending Responses to State Sales Tax Holidays
Sumit Agarwal and Leslie McGranahan

WP-12-10

Micro Data and Macro Technology
Ezra Oberfield and Devesh Raval

WP-12-11

The Effect of Disability Insurance Receipt on Labor Supply: A Dynamic Analysis
Eric French and Jae Song

WP-12-12

Medicaid Insurance in Old Age
Mariacristina De Nardi, Eric French, and John Bailey Jones

WP-12-13

Fetal Origins and Parental Responses
Douglas Almond and Bhashkar Mazumder

WP-12-14

2

Working Paper Series (continued)
Repos, Fire Sales, and Bankruptcy Policy
Gaetano Antinolfi, Francesca Carapella, Charles Kahn, Antoine Martin,
David Mills, and Ed Nosal

WP-12-15

Speculative Runs on Interest Rate Pegs
The Frictionless Case
Marco Bassetto and Christopher Phelan

WP-12-16

Institutions, the Cost of Capital, and Long-Run Economic Growth:
Evidence from the 19th Century Capital Market
Ron Alquist and Ben Chabot

WP-12-17

Emerging Economies, Trade Policy, and Macroeconomic Shocks
Chad P. Bown and Meredith A. Crowley

WP-12-18

The Urban Density Premium across Establishments
R. Jason Faberman and Matthew Freedman

WP-13-01

Why Do Borrowers Make Mortgage Refinancing Mistakes?
Sumit Agarwal, Richard J. Rosen, and Vincent Yao

WP-13-02

Bank Panics, Government Guarantees, and the Long-Run Size of the Financial Sector:
Evidence from Free-Banking America
Benjamin Chabot and Charles C. Moul

WP-13-03

Fiscal Consequences of Paying Interest on Reserves
Marco Bassetto and Todd Messer

WP-13-04

Properties of the Vacancy Statistic in the Discrete Circle Covering Problem
Gadi Barlevy and H. N. Nagaraja

WP-13-05

Credit Crunches and Credit Allocation in a Model of Entrepreneurship
Marco Bassetto, Marco Cagetti, and Mariacristina De Nardi

WP-13-06

Financial Incentives and Educational Investment:
The Impact of Performance-Based Scholarships on Student Time Use
Lisa Barrow and Cecilia Elena Rouse

WP-13-07

The Global Welfare Impact of China: Trade Integration and Technological Change
Julian di Giovanni, Andrei A. Levchenko, and Jing Zhang

WP-13-08

Structural Change in an Open Economy
Timothy Uy, Kei-Mu Yi, and Jing Zhang

WP-13-09

The Global Labor Market Impact of Emerging Giants: a Quantitative Assessment
Andrei A. Levchenko and Jing Zhang

WP-13-10

3

Working Paper Series (continued)
Size-Dependent Regulations, Firm Size Distribution, and Reallocation
François Gourio and Nicolas Roys

WP-13-11

Modeling the Evolution of Expectations and Uncertainty in General Equilibrium
Francesco Bianchi and Leonardo Melosi

WP-13-12

Rushing into American Dream? House Prices, Timing of Homeownership,
and Adjustment of Consumer Credit
Sumit Agarwal, Luojia Hu, and Xing Huang

WP-13-13

The Earned Income Tax Credit and Food Consumption Patterns
Leslie McGranahan and Diane W. Schanzenbach

WP-13-14

Agglomeration in the European automobile supplier industry
Thomas Klier and Dan McMillen

WP-13-15

Human Capital and Long-Run Labor Income Risk
Luca Benzoni and Olena Chyruk

WP-13-16

The Effects of the Saving and Banking Glut on the U.S. Economy
Alejandro Justiniano, Giorgio E. Primiceri, and Andrea Tambalotti

WP-13-17

A Portfolio-Balance Approach to the Nominal Term Structure
Thomas B. King

WP-13-18

Gross Migration, Housing and Urban Population Dynamics
Morris A. Davis, Jonas D.M. Fisher, and Marcelo Veracierto

WP-13-19

Very Simple Markov-Perfect Industry Dynamics
Jaap H. Abbring, Jeffrey R. Campbell, Jan Tilly, and Nan Yang

WP-13-20

Bubbles and Leverage: A Simple and Unified Approach
Robert Barsky and Theodore Bogusz

WP-13-21

The scarcity value of Treasury collateral:
Repo market effects of security-specific supply and demand factors
Stefania D'Amico, Roger Fan, and Yuriy Kitsul
Gambling for Dollars: Strategic Hedge Fund Manager Investment
Dan Bernhardt and Ed Nosal
Cash-in-the-Market Pricing in a Model with Money and
Over-the-Counter Financial Markets
Fabrizio Mattesini and Ed Nosal
An Interview with Neil Wallace
David Altig and Ed Nosal

WP-13-22

WP-13-23

WP-13-24

WP-13-25

4

Working Paper Series (continued)
Firm Dynamics and the Minimum Wage: A Putty-Clay Approach
Daniel Aaronson, Eric French, and Isaac Sorkin
Policy Intervention in Debt Renegotiation:
Evidence from the Home Affordable Modification Program
Sumit Agarwal, Gene Amromin, Itzhak Ben-David, Souphala Chomsisengphet,
Tomasz Piskorski, and Amit Seru

WP-13-26

WP-13-27

The Effects of the Massachusetts Health Reform on Financial Distress
Bhashkar Mazumder and Sarah Miller

WP-14-01

Can Intangible Capital Explain Cyclical Movements in the Labor Wedge?
François Gourio and Leena Rudanko

WP-14-02

Early Public Banks
William Roberds and François R. Velde

WP-14-03

Mandatory Disclosure and Financial Contagion
Fernando Alvarez and Gadi Barlevy

WP-14-04

The Stock of External Sovereign Debt: Can We Take the Data at ‘Face Value’?
Daniel A. Dias, Christine Richmond, and Mark L. J. Wright

WP-14-05

Interpreting the Pari Passu Clause in Sovereign Bond Contracts:
It’s All Hebrew (and Aramaic) to Me
Mark L. J. Wright

WP-14-06

AIG in Hindsight
Robert McDonald and Anna Paulson

WP-14-07

On the Structural Interpretation of the Smets-Wouters “Risk Premium” Shock
Jonas D.M. Fisher

WP-14-08

Human Capital Risk, Contract Enforcement, and the Macroeconomy
Tom Krebs, Moritz Kuhn, and Mark L. J. Wright

WP-14-09

Adverse Selection, Risk Sharing and Business Cycles
Marcelo Veracierto

WP-14-10

Core and ‘Crust’: Consumer Prices and the Term Structure of Interest Rates
Andrea Ajello, Luca Benzoni, and Olena Chyruk

WP-14-11

The Evolution of Comparative Advantage: Measurement and Implications
Andrei A. Levchenko and Jing Zhang

WP-14-12

5

Working Paper Series (continued)
Saving Europe?: The Unpleasant Arithmetic of Fiscal Austerity in Integrated Economies
Enrique G. Mendoza, Linda L. Tesar, and Jing Zhang

WP-14-13

Liquidity Traps and Monetary Policy: Managing a Credit Crunch
Francisco Buera and Juan Pablo Nicolini

WP-14-14

Quantitative Easing in Joseph’s Egypt with Keynesian Producers
Jeffrey R. Campbell

WP-14-15

6