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w o r k i n g
p
a
p
e
r
0 2 0 4
Monetary Policy in a Financial Crisis
by Lawrence J. Christiano,
Christopher Gust, and Jorge Roldos
FEDERAL RESERVE BANK
OF CLEVELAND
Working papers
of the Federal Reserve Bank of Cleveland are preliminary materials
circulated to stimulate discussion and critical comment on research in progress. They may
not have been subject to the formal editorial review accorded official Federal Reserve Bank
of Cleveland publications. The views stated herein are those of the authors and are not
necessarily those of the Federal Reserve Bank of Cleveland, the Board of Governors of the
Federal Reserve System or of the International Monetary Fund.
Working papers are now available electronically through the Cleveland Fed’s site on the
World Wide Web: www.clev.frb.org.
Working Paper 02-04
June 2002
Monetary Policy in a Financial Crisis
by Lawrence J. Christiano, Christopher Gust, and Jorge Roldos
What are the economic effects of an interest rate cut when an economy is in the midst of a financial
crisis? Under what conditions will a cut stimulate output and employment, and raise welfare?
Under what conditions will a cut have the opposite effects? We answer these questions in a general
class of open economy models, where a financial crisis is modeled as a time when collateral
constraints are suddenly binding. We find that when there are frictions in adjusting the level of
output in the traded good sector and in adjusting the rate at which that output can be used in other
parts of the economy, then a cut in the interest rate is most likely to result in a welfare-reducing fall
in output and employment. When these frictions are absent, a cut in the interest rate improves asset
positions and promotes a welfare-increasing economic expansion.
JEL Classification: E5, F3, F4
Key Words: financial crisis, exchange rates, collateral constraint
Lawrence J. Christiano is at Northwestern University, NBER, and serves as a Research Associate
at the Federal Reserve Bank of Cleveland and the Federal Reserve Bank of Chicago. Christopher
Gust is at the Federal Reserve Board of Governors. Jorge Roldos is at the International Monetary
Fund.
The authors are grateful for advice and comments from Fabio Braggion, Peter Clark, Martin
Eichenbaum, Andy Neumeyer, Sergio Rebelo and Martin Uribe. The first author acknowledges
support from a grant to the National Bureau of Economic Research from the National Science
Foundation. The authors are grateful for the support of the International Monetary Fund in the
preparation of this manuscript.
1. Introduction
In recent years there has been considerable controversy over the appropriate monetary policy
in the aftermath of a financial crisis. Some argue that the central bank should raise domestic
interest rates to defend the currency and halt the flight of capital. Others argue that interest
rate reductions are called for. They note that a country that has just experienced a financial
crisis is typically sliding into a steep recession. They appeal to the widespread view that in
developed economies like the US, central banks typically respond to situations like this by
reducing interest rates. These authors urge the same medicine for emerging market economies
in the wake of a financial crisis. They argue that to raise interest rates at such a time is a
mistake, and is likely to make a bad situation even worse. One expositor of this view, Paul
Krugman (1999, pp.103-105), puts it this way:
“But when financial disaster struck Asia, the policies those countries followed in response
were almost exactly the reverse of what the United States does in the face of a slump.
Fiscal austerity was the order of the day; interest rates were increased, often to punitive
levels....Why did these extremely clever men advocate policies for emerging market
economies that would have been regarded as completely perverse if applied at home?”
We describe a framework that allows us to articulate the two views just described. The
framework has two building blocks. First, we assume that to carry out production, firms
require domestic working capital to hire labor and international working capital to purchase
an imported intermediate input. Second, we adopt the asset market frictions formalized in
the limited participation model as analyzed in Lucas (1990), Fuerst (1992) and Christiano and
Eichenbaum (1992, 1995). The limited participation assumption has the consequence that an
expansionary monetary action makes the domestic banking system relatively liquid and induces
firms to hire more labor. To the extent that the imported intermediate input complements labor,
the interest rate drop leads to the increased use of this factor too. This is in the spirit of the
traditional liquidity channel emphasized in the closed economy literature, which stresses the
positive effects of an interest rate cut on output. So, absent other considerations, the model
rationalizes the Krugman view outlined above.
Our model has an additional feature which may be particularly relevant during a crisis.
We suppose that a crisis is a time when international loans must be collateralized by physical
assets such as land and capital, and that this restriction is binding. To understand how collateral
affects the monetary transmission mechanism in our model, it is useful to consider a simplified
version of our collateral constraint expressed in units of the foreign currency:
Q
K ≥ R∗ z + B.
S
Here, B represents the stock of long-term external debt; z represents short-term external borrowing to finance a foreign intermediate input; R∗ represents the associated interest rate; K
represents domestic physical assets like land and capital; Q is the value (in domestic currency
units) of a unit of K; and S represents the nominal exchange rate. We suppose that under
normal conditions, the collateral constraint is not binding, while it suddenly binds with the
onset of a crisis. This may be because in normal times, output in addition to land and capital
1
is acceptable as collateral. Then, in a crisis the fraction of domestic assets accepted as collateral
by foreigners suddenly falls.1 In any case, in our analysis we model the imposition of a binding
collateral constraint as an exogenous, unforeseen event.2
We then compare the ensuing transition path of the economy under two scenarios. In the
benchmark scenario, the monetary authority does not adjust policy in response to the collateral
shock. In the alternative scenario, the monetary authority reduces the domestic rate of interest
relative to what it is in the benchmark scenario. We find that in the benchmark scenario, output
and employment are low during the transition to the new steady state. The shadow-cost of
international debt, B, is higher while the collateral constraint is binding, and the economy
responds by increasing the current account and paying down the debt. In the new steady state
the debt is reduced to the point where the collateral constraint is marginally nonbinding. That
is, the collateral constraint is satisfied as an equality, but with a zero multiplier.
Although the transition path after a collateral shock is of independent interest because it
captures key features of actual economies in the aftermath of a crisis, it is not the central focus
of our analysis. Our key objective is to understand the impact on the transition of a cut in
the interest rate. We study this by comparing the dynamic equilibrium of the economy under
the benchmark and alternative scenarios. We now briefly describe the results. In doing so,
we make use of the fact that R∗ and K are held fixed throughout the paper. We also find it
convenient in summarizing the results here to ignore the impact of the interest rate cut on B.3
Finally, in describing the intuition for the results we make use of our numerical finding that
whenever there is a monetary policy-induced cut in the interest rate, there is a depreciation of
the currency, i.e., a jump in S. Using these observations and the collateral constraint evaluated
at equality, it is easy to see why it is that for some versions of our model an interest rate cut
produces a contraction, and for others it produces an expansion.
The contraction outcome is perhaps the easiest to understand. When S jumps, the left side
of the collateral constraint falls. Supposing that Q does not jump very much, this means that
the right side must be reduced, i.e., z must fall. Our assumption that the imported intermediate
1
Our characterization of a crisis as a time when collateral constraints suddenly bind is not unprecendented.
For example, Caballero (2000, p. 5) states that a crisis is a time of “...sudden loss in the international appeal
of a country’s assets.” He also states that a (p.4) “crisis is a time when (a) a significant fraction of firms
or economic agents are in need of financing to either repay debt or implement new investments needed to
save high return projects — I will refer to these agents as ‘distressed firms’ — and (b) on net, the economy
as a whole needs substantial external resources but does not have enough assets and commitment to obtain
them.”
2
In some respects our framework resembles a reduced form representation of the environment considered
in Albuquerque and Hopenhayn (1997) and further developed in Cooley, Marimon and Quadrini (2001) and
Monge (2001). There, an investment project requires an initial fixed investment, followed by a sequence of
expenditures to make the investment project productive. The papers in this literature derive the optimal
dynamic contract between the entrepreneur and a bank, as well as a sequential decentralization. In the
latter, the initial fixed investment is financed by long term debt that resembles our B, and the sequence
of expenditures is financed by working capital loans with the entrepreneur being restricted by a collateral
constraint that resembles the one we adopt. This literature suggests a variety of factors that could cause
collateral constraints to suddenly become binding. For example, if there is a shock that causes the court
system to be overwhelmed by bankruptcy filings and other business in a recession, collateral constraints could
suddenly bind because lenders now understand that the default option is more attractive to the marginal
entrepreneur who wishes to borrow.
3
As noted earlier in the introduction, in the full analysis reported in the body of the paper, B is treated
as a variable that moves endogenously over time.
2
good is important in domestic employment and production, ensures that a recession follows. In
this outcome, the currency mismatch between assets and liabilities in the collateral constraint
plays the central role.
That an expansion outcome is possible is also easy to see. If the nominal interest rate
cut succeeds in reducing the real interest rate used to discount future flows, then asset prices,
Q, may in fact jump a substantial amount. Indeed, in closed economy settings when there
are no currency mismatches in balance sheets, it is often considered the ‘natural’ outcome
that a cut in the interest rate lifts asset prices and improves balance sheets. If the rise in
Q is sufficiently strong to offset the nominal depreciation, then the left side of the collateral
constraint is increased by the interest rate cut. In this case, there is room in the collateral
constraint for z to go up, and for domestic production to rise.
The above discussion suggests that the contraction outcome is most likely in economies
where an increase in z does not lead to a substantial increase in Q, the value of productive
capital and land. Two features promote this possibility in our model environment. The first
occurs if increases in z encounter strong decreasing returns in production, and complementary
factors of production cannot be brought in to offset this. The second occurs if there is little
substitutability between traded and nontraded goods in the production of final goods. By
inhibiting the ability of the economy to effectively exploit increases in z, these two features
reduce the likelihood that an increase in z is associated with a substantial rise in Q. We find
that when these frictions are not present, then an interest rate cut tends to be associated with
the expansion outcome.
The role of asset prices in propagating shocks is a topic that is of independent interest. The
existing literature focuses on the role of asset prices in magnifying and propagating the effects
of shocks.4 We obtain the magnification effect here too, in the version of the model that implies
the expansion outcome. In that model, the response of output and employment to an interest
rate cut is the same sign and stronger than what it is when the collateral constraint is ignored
altogether. Interestingly, in the version of the model that implies the contraction outcome, the
collateral constraint actually has the effect of changing the sign of the economy’s response.
The organization of the paper is as follows. The next section presents our general model.
Section three presents a version of the model simplified by the assumption that the stock of
external long-term debt is held constant. The advantage of this simplification is that the
model can be studied analytically. The insights that are obtained from this are useful for
understanding the more relevant version of the model, in which the long-term external debt is
determined endogenously. Numerical methods are used to study this version of the model in
the third section of the paper. The final section concludes.
2. The Model
We adopt a standard traded-good/non-traded good small open economy model. The model
has households, firms, a financial intermediary, and a domestic monetary authority.
4
For recent papers on closed economy models that emphasize the role of asset prices in magnifying and
propagating shocks, see Bernanke, Gertler and Gilchrist (1999) and Carlstrom and Fuerst (1997, 2000) and
the literature that they cite. For open economy models that assign an important role to asset prices, see
Mendoza and Smith (2000) and the literature they cite.
3
2.1. Households
There is a representative household, which derives utility from consumption, ct , and leisure as
follows:
∞
X
β t u(ct , Lt ),
(2.1)
t=0
where Lt denotes labor. We adopt the following specification of utility:
u(c, L) =
h
c−
i1−σ
ψ0
1+ψ
L
1+ψ
1−σ
(2.2)
.
The household begins the period with a stock of liquid assets, M̃t . Of this, it deposits Dt
with the financial intermediary, and the rest, M̃t −Dt , is allocated to consumption expenditures.
The cash constraint that the household faces on its consumption expenditures is:
Pt ct ≤ Wt Lt + M̃t − Dt ,
(2.3)
where Wt denotes the money wage rate and Pt denotes the price level.
The household also faces a flow budget constraint governing the evolution of its assets:
h
i
M̃t+1 = Rt (Dt + Xt ) + PtT πt + Wt Lt + M̃t − Dt − Pt ct .
(2.4)
Here, Rt denotes the gross domestic rate of interest, πt denotes lump-sum dividend payments
received from firms, and Xt is a liquidity injection from the monetary authority. Also, πt is
measured in units of traded goods, and PtT is the domestic currency price of traded goods.
The term on the right of the equality reflects the household’s sources of liquid assets at the
beginning of period t + 1 : interest earnings on deposits and on the liquidity injection, profits
and any cash that may be left unspent in the period t goods market.
The household maximizes (2.1) subject to (2.3)-(2.4), and the following timing constraint.
A given period’s deposit decision is made before that period’s liquidity injection is realized,
while all other decisions are made afterward. The Euler equation associated with the labor
decision is:
Wt
(2.5)
ψ0 Lψt =
.
Pt
We refer to this as the labor supply equation. The intertemporal Euler equation associated
with the deposit decision is:
Pt
(2.6)
uc,t = βRt uc,t+1
.
Pt+1
4
2.2. Firms
There are two types of representative, competitive firms. The first produces the final consumption good, c, purchased by households. Final goods production requires tradeable and
non-tradeable intermediate goods which are produced by the second type of representative
firm. We now discuss these two types of firms.
2.2.1. Final Good Firms
The production function of the final good firms is:
c=
(
h
T
(1 − γ) c
i η−1
η
h
N
+ γc
i η−1
η
)
η
η−1
, 1 ≥ η ≥ 0, 0 < γ < 1,
(2.7)
where cT and cN denote quantities of tradeable and non-tradeable intermediate inputs, respectively. One interpretation is that these firms are retailers that package traded and non-traded
intermediate goods into a final consumption good. Here, η denotes the elasticity of substitution
in production between the two intermediate inputs. For later purposes, it is useful to note that
as η → 0,
n
o
c = min (1 − γ) cT , γcN .
As noted in the introduction, this specification of the technology for producing final goods
increases the likelihood that the economy will contract when there is a cut in the domestic rate
of interest.
Let P T and P N denote the prices of traded and non-traded goods. Zero profits and efficiency
imply that the price of c, P, and these input prices have the following relationship:
p=
Ã
1
1−γ
!1−η
Ã
pN
+
γ
1
!1−η 1−η
, p=
P
.
PT
(2.8)
For η 6= 1, efficiency also dictates:
γ
p =
1−γ
N
Ã
(1 − γ) cT
γcN
!1
η
, pN =
PN
.
PT
(2.9)
When η = 0, this expression is replaced by (1 − γ) cT = γcN . The object, P, in the model corresponds to the model’s ‘consumer price index’, denominated in units of the domestic currency.
The object, p, is the consumer price index denominated in units of the traded good.
5
2.2.2. Intermediate Inputs
A single representative firm produces the traded and non-traded intermediate inputs. That
firm manages three types of debt, two of which are short-term. The firm borrows at the
beginning of the period to finance its wage bill and to purchase a foreign input, and repays
these loans at the end of the period. In addition, the firm holds the outstanding stock of
external (net) indebtedness, Bt . In terms of assets, the firm owns all the economy’s physical
capital. This specification of the firm allows us to abstract from problems associated with the
poor distribution of collateral among firms, that is emphasized by Caballero and Krishnamurthy
(2001).
The firm’s optimization problem is:
max
∞
X
β t Λt+1 πt ,
(2.10)
t=0
where
∗
∗
N
T
πt = pN
t yt + yt − wt Rt Lt − R zt − r Bt + (Bt+1 − Bt ),
(2.11)
denotes dividends, denominated in units of traded goods. Here, wt = Wt /PtT is the wage rate,
denominated in units of the traded good. Also, Bt is the stock of external debt at the beginning
of period t, denominated in units of the traded good; R∗ is the gross rate of interest (fixed in
units of the traded good) on loans for the purpose of purchasing zt ; and r∗ is the net rate of
interest (again, fixed in terms of the traded good) on the outstanding stock of external debt.
The price, Λt+1 , is taken parametrically by firms. In equilibrium, it is the multiplier on πt in
the (Lagrangian representation of the) household problem:5
uc,t+1 PtT
β
T
pt+1 Pt+1
1
uc,t+1 pTt
=
β,
T
pt+1 pt+1 1 + xt
Λt+1 =
where
pTt
(2.12)
PtT
=
.
Mt
Here, Mt is the aggregate stock of money at the beginning of period t, which evolves according
5
The intuition underlying (2.12) is straightforward. The object Λt+1 in (2.12), is the marginal utility of
one unit of dividends, denominated in traded goods, transferred by the firm to the household at the end of
period t. This corresponds to PtT πt units of domestic currency. The households can use this currency in
period t + 1 to purchase PtT πt /Pt+1 units of the consumption good. The value, in period t, of these units
T
of consumption goods is βuc,t+1 PtT πt /Pt+1 , or βuc,t+1 PtT πt /(pt+1 Pt+1
), where uc,t is the marginal utility of
consumption. This is the first expression in (2.12).
6
to:
Mt+1
= 1 + xt .
Mt
(2.13)
With one exception, we adopt the convention that a price expressed in lower case indicates the
price has been scaled by the price of traded goods. The exception, pTt , is the domestic currency
price of traded goods, scaled by the beginning of period stock of money. Alternatively, pTt is
the inverse of a measure of real balances.
The firm production functions are:
y
T
V
=
½
θ [µ1 V ]
³
= A KT
yN =
³
KN
ξ−1
ξ
´ν ³
+ (1 − θ) [µ2 z]
LT
´α ³
LN
´1−ν
´1−α
ξ−1
ξ
¾
ξ
ξ−1
,
(2.14)
,
,
where ξ is the elasticity of substitution between value-added in the traded good sector, Vt , and
the imported intermediate good, zt . In the production functions, K T and K N denote capital
in the traded and non-traded good sectors, respectively. They are owned by the representative
intermediate input firm. We keep the stock of capital fixed throughout the analysis. It does
not depreciate and there exists no technology for making it bigger.
Our specification of technology is designed to encompass a variety of cases. In one, there is
no substitutability between z and V in production, i.e., ξ = 0, so that
y T = min {µ1 V, µ2 z} .
(2.15)
An optimizing producer sets V = (1/µ1 )y T and z = (1/µ2 )y T , so that the share of value-added,
V, in total output, y T , is 1/µ1 and the share of imported intermediate inputs in total output is
1/µ2 . We impose that these shares sum to unity. In another specification, z is the only variable
factor of production and occurs when ξ = µ1 = µ2 = ν = 1 :
³
y T = AK T
´θ
z 1−θ .
(2.16)
Later, we shall see that (2.15) is associated with the expansion outcome, the outcome in which
a cut in the interest rate produces an increase in output and employment. This is because,
in (2.15) a fall in productivity can be avoided when z increases as long as V is increased
simultaneously. In the case of (2.16), there are no complementary factors that can be adjusted
to overcome the fall in productivity associated with an increase in z, when θ > 0. As explained
in the introduction, this production function is associated with the contraction outcome, that
is, one in which a cut in the interest rate produces a fall in output and employment.
We impose the following restriction on borrowing:
7
Bt+1
→ 0, as t → ∞.
(1 + r∗ )t
(2.17)
We suppose that international financial markets impose that this limit cannot be positive. That
it cannot be negative is an implication of firm optimality.
N
T
The firm’s problem at time t is to maximize (2.10) by choice of Bt+j+1 , yt+j
, yt+j
, zt+j , LTt+j
and LN
t+j , j = 0, 1, 2, ..., subject to the various constraints just described. In addition, the firm
takes all prices and rates of return as given and beyond its control. The firm also takes the
initial stock of debt, Bt , as given. This completes the description of the firm problem in the
pre-crisis version of the model, when collateral constraints are ignored.
The crisis brings on the imposition of the following collateral constraint:
τ N qtN K N + τ T qtT K T ≥ R∗ zt + (1 + r∗ )Bt + wt Rt Lt ,
(2.18)
i
where Lt ≡ LTt + LN
t . Here, q , i = N, T denote the value (in units of the traded good) of a
unit of capital in the non-traded and traded good sectors, respectively. Also, τ i denotes the
fraction of these stocks accepted as collateral by international creditors. The left side of (2.18)
is the total value of collateral, and the right side is the payout value of the firm’s debt. It is
the total amount that the firm would have to pay, to completely eliminate all its debt by the
end of period t. Before the crisis, firms ignore (2.18), and assign a zero probability that it will
be implemented. With the coming of the crisis, firms believe that (2.18) must be satisfied in
every period henceforth, and do not entertain the possibility that it will be removed.
The equilibrium value of the asset prices, qti , i = N, T, is the amount that a potential firm
would be willing to pay in period t, in units of the traded good, to acquire a unit of capital and
start production in period t. We let λt ≥ 0 denote the multiplier on the collateral constraint
(= 0 in the pre-crisis period) in firm problem. Then, qti is the derivative of the Lagrangian
representation of the firm’s problem with respect to Kti :
i
+ λt τ i qti +
qti = V MPk,t
∞
n
o
β X
i
i
+ λt+j τ i qt+j
β j−1 Λt+1+j V MPk,t+j
Λt+1 j=1
(2.19)
or,
qti
=
i
+ β ΛΛt+2
V MPk,t
qi
t+1 t+1
1 − λt τ i
, i = N, T.
(2.20)
i
denotes the period t value (in terms of traded goods) marginal product of capital
Here, V MPk,t
in sector i. With our assumptions on technology, these are:
N
= αpN
V MPk,t
t
ytN
,
KN
8
V
T
MPk,t
=
³
ytT
µ1 Vt
ν µK1 VTt
h
´ 1ξ
1−
θν µK1 VTt ,
(1+λt )R∗
µ2
i
ξ 6= 0
.
, ξ=0
When λt ≡ 0, (2.19) is just the standard asset pricing equation. It is the present discounted
value of the value of the marginal physical product of capital. When the collateral constraint is
binding, so that λt is positive, then qti is greater than this. This reflects that in this case capital
is not only useful in production, but also for relieving the collateral constraint. In our model
capital is never actually traded, since all firms are identical. However, if there were trade, then
the price of capital would be qti . If a firm were to default on its credit obligations, the notion
is that foreign creditors could compel the sale of its physical assets in a domestic market for
capital. The price, qti , is how much traded goods a domestic resident is willing to pay for a unit
of capital. Foreign creditors would receive those goods in the event of a default. We assume
that with these consequences for default, default never occurs in equilibrium.
We now derive the Euler equations of the firm. Differentiating the date 0 Lagrangian
representation of the firm problem with respect to Bt+1 :
1=β
Λt+2
(1 + r∗ )(1 + λt+1 ), t = 0, 1, 2, ... .
Λt+1
(2.21)
Following standard practice with small open economy models, we assume β(1 + r∗ ) = 1, so
that6
Λt+1 = Λt+2 (1 + λt+1 ), t = 0, 1, 2, ... .
(2.22)
A high value for λ, which occurs when the collateral constraint is binding, raises the effective
rate of interest on debt. The interpretation is that when λ is large, then the debt has an
additional cost, beyond the direct interest cost. This cost reflects that when the firm raises
Bt+1 in period t, it not only incurs an additional interest charge in period t + 1, but it is
also further tightens its collateral constraint in that period. This has a cost because, via the
collateral constraint, the extra debt inhibits the firm’s ability to acquire working capital in
period t + 1. Thus, when λ is high, there is an additional incentive for firms to reduce π and
‘save’ by paying down the external debt. Although the firm’s actual interest rate on external
debt taken on in period t is 1 + r∗ , it’s ‘effective’ interest rate is (1 + r∗ ) (1 + λt+1 ) .
The firm’s first order conditions for labor in the non-traded and traded sectors, and for z
are, when ξ 6= 0:
Ã
6
ytT
µ1 Vt
!1
(1 − α)pN
t
ξ
θ(1 − ν)
ytN
LN
t
= wt (1 + λt )Rt
(2.23)
µ1 Vt
= wt (1 + λt )Rt
LTt
(2.24)
See, for example, Obstfeld and Rogoff (1997).
9
Ã
ytT
µ2 zt
!1
ξ
(1 − θ) µ2 = (1 + λt )R∗
(2.25)
The presence of Rt on the right side of (2.23)-(2.24) reflects that to hire labor, firms must
borrow cash in advance in the domestic money market, at the gross interest rate, Rt . When
the collateral constraint is binding, then the effective interest rate is higher than Rt . The gross
interest rate on short term foreign loans, R∗ , appears on the right of (2.25) because firms must
borrow foreign funds in advance to acquire zt . Note that the effective foreign interest rate is
higher than the actual interest rate when the collateral constraint is binding.
When ξ = 0, then of course (2.23) still holds, but (2.24) and (2.25) are replaced by:
"
#
(1 + λt )R∗
µ1 Vt
(1 − ν) T 1 −
= wt (1 + λt )Rt
Lt
µ2
µ1 Vt = µ2 zt
(2.26)
(2.27)
Ignoring the term in square brackets in (2.26), this is just the marginal product of LT in
producing µ1 Vt . The term in square brackets reflects that expansions in y T also requires an
increase in z.
2.3. Financial Intermediary and Monetary Authority
The financial intermediary takes domestic currency deposits, Dt , from the household at the
beginning of period t. In addition, it receives the liquidity transfer, Xt = xt Mt , from the
monetary authority.7 It then lends all its domestic funds to firms who use it to finance their
employment working capital requirements, Wt Lt . Clearing in the money market requires Dt +
Xt = Wt Lt , or, after scaling by the aggregate money stock,
dt + xt = wt pTt Lt ,
(2.28)
where dt = Dt /Mt .
The monetary authority in our model simply injects funds into the financial intermediary.
Its period t decision is taken after the household has selected a value for Dt , and before all
other variables in the economy are determined. This is the standard assumption in the limited
participation literature. It is interpreted as reflecting a sluggishness in the response of household
portfolio decisions to changes in market variables. With this assumption, a value of xt that
deviates from what households expected at the time Dt was set produces an immediate reaction
7
In practice, injections of liquidity do not occur in the form of lump sum transfers, as they do in our
model. It is easy to show that our formulation is equivalent to an alternative, in which the injection occurs
as a result of an open market purchase of government bonds which are owned by the household, but held by
the financial intermediary. We do not adopt this interpretation in our formal model in order to conserve on
notation.
10
by firms and the financial intermediary but not, in the first instance, by households. The name,
‘limited participation’, derives from this feature, namely that not all agents react immediately
to (or, ‘participate in’) a monetary shock. As a result of this timing assumption, many models
exhibit the following behavior in equilibrium. An unexpectedly high value of xt swells the
supply of funds in the financial sector, since Dt on the left side of (2.28) cannot fall in response
to a positive xt shock. To get firms to absorb the increase in funds, a fall in the equilibrium
rate of interest is required. When that fall does occur, they borrow the increased funds and use
them to hire more labor and produce more output.
We abstract from all other aspects of government finance. The only policy variable of the
government is xt .
2.4. Equilibrium
We consider a perfect foresight, sequence-of-market equilibrium concept. In particular, it is a
sequence of prices and quantities having the properties: (i) for each date, the quantities solve
the household and firm problems, given the prices, and (ii) the labor, goods and domestic
money markets clear.
Clearing in the money market requires that (2.28) hold and that actual money balances,
Mt , equal desired money balances, M̃t . Combining this with the household’s cash constraint,
(2.3), we obtain the equilibrium cash constraint:
pTt pt ct = 1 + xt .
(2.29)
According to this, the total, end of period stock of money must equal the value of final output,
ct . Market clearing in the traded good sector requires:
ytT − R∗ zt − r∗ Bt − cTt = − (Bt+1 − Bt ) .
(2.30)
The left side of this expression is the current account of the balance of payments, i.e., total
production of traded goods, net of foreign interest payments, net of domestic consumption. The
right side of (2.30) is the change in net foreign assets. Equation (2.30) reflects our assumption
that external borrowing to finance the intermediate good, zt , is fully paid back at the end of
the period. That is, this borrowing resembles short-term trade credit. Note, however, that
this is not a binding constraint on the firm, since our setup permits the firm to finance these
repayments using long term debt. Market clearing in the non-traded good sector requires:
ytN = cN
t .
(2.31)
It is instructive to study this model’s implications for interest parity. Combining the house-
11
hold and firm intertemporal conditions, (2.6) and (2.21), with (2.12), we obtain
Rt+1 = (1 + r∗ )
T
Pt+1
(1 + λt+1 ) , t = 0, 1, 2, ...
PtT
(2.32)
T
On the right hand side, of this expression, (1 + r∗ )Pt+1
/PtT is the rate of interest on external
debt, expressed in domestic currency units. Expression (2.32) with λ = 0 is the usual interest
rate parity relation. When λ > 0, there is a collateral premium on the domestic rate of interest.
Expression (2.32) highlights our implicit assumption that foreign and domestic markets for
loanable funds are isolated, at least in times when the collateral constraint is binding. When
λ > 0, so that the domestic interest rate exceeds the foreign rate, lenders of foreign currency
would prefer to exchange their currency for domestic currency and lend in the domestic currency
market. Similarly, firms borrowing domestic funds for the purpose of paying their wage bill
would prefer to borrow in the foreign currency market and convert the proceeds into domestic
currency. That λ > 0 is possible in equilibrium reflects that we rule out this type of cross-border
borrowing and lending.8
As an empirical proposition, interest rate parity does poorly. In response to this, researchers
often introduce exogenously a term like our λ in (2.32). In conventional practice, λ is interpreted
as reflecting a risk premium. Our setup may provide an alternative interpretation.
Details about computing equilibrium for this model are reported in the appendix.
3. Qualitative Analysis of the Equilibrium
Our full model is not analytically tractable and so to understand its implications for the questions we ask requires numerical simulation. However, in the special case in which long-term
external debt is constant, it is possible to obtain analytic results, at least locally. This is the
case considered in this section. The next section considers the case where the debt is a choice
variable. We identify a set of sufficient conditions which guarantee that a cut in the domestic
rate of interest is contractionary. Under these assumptions, z is the only variable factor of
production in the production of traded goods and it is subject to diminishing returns; traded
and non-traded goods are not very substitutable in the production of final goods; and the
size of the external debt is small. The assumptions that the elasticity of substitution between
traded and non-traded goods is low and that the debt is low appears to be crucial to the result.
That is, it is possible to construct examples where a combination of the other assumptions does
not hold and where an interest rate cut still produces a recession. However, in the examples
considered below, a modest degree of substitution between traded and non-traded goods and a
modest amount of external debt always has the consequence that an interest rate cut produces
an expansion.
8
Our market-segmentation assumption may capture what actually happens in the aftermath of a financial
crisis. Domestic residents may be fearful of borrowing in foreign markets because of concerns about exchange
risk (hedging markets tend to become very illiquid at times like this). Similarly, foreign residents may not
want to lend in domestic markets. While our market-segmentation assumption may be plausible, the factors
that justify it are not present in our model.
12
In the first subsection below, we describe the nature of the monetary experiments analyzed
here. The second subsection identifies a particular version of our model for which we have
analytic results. That section also explains why our strategy of characterizing monetary policy
in terms of the interest rate simplifies the technical analysis of the model, while entailing no loss
of generality. The third subsection investigates the properties of that model, and of deviations
from that model.
3.1. The Nature of the Policy Experiment
In our analysis, we compare two equilibria, for t = 0, 1, 2, .... . In both, the collateral constraint
is binding in each date. In each case, we characterize monetary policy by the choice of the
nominal interest rate, Rt , in the domestic money market. In the baseline equilibrium, Rt is
held constant, Rt = Rs , in each period. Our restriction that the current account is always
zero guarantees that the relative prices and quantities in this equilibrium are time-invariant.
In the policy intervention equilibrium, the monetary authority unexpectedly implements a onetime drop in the interest rate in t = 0, i.e, R0 < Rs , Rt = Rs for t ≥ 1. This drop has a
non-neutral impact on allocations because of our assumption - taken from the literature on
the limited participation models of money - about the timing of actions by different agents
during the period. At the beginning of the period, the household makes a deposit decision.
Then, the monetary authority takes its action and after that all the other period t variables
are determined. We assume that at the beginning of period t = 0, when the household makes
its deposit decision, it expects Rt = Rs for t ≥ 0. At the beginning of period t = 1, 2, ... the
household expects Rt = Rs despite the fact that its expectation was violated in period t = 0.
Given the assumptions of our model, the relative prices and quantities in the baseline and
policy intervention equilibria are identical in t ≥ 1, but they differ in t = 0. Our analysis
focuses on this difference in period 0. In particular, we investigate what conditions guarantee
that output and employment in t = 0 for the policy intervention equilibrium are lower than they
are in the baseline equilibrium. Because they are time invariant, we refer to values of relative
prices and quantities in t ≥ 0 in the baseline equilibrium, and t > 0 in the policy intervention
equilibrium as their steady state values. Because of the simplicity of these equilibria, the analysis
has a static flavor. It only involves comparing the steady state relative prices and quantities
with the t = 0 values of the variables in the policy intervention equilibrium.
3.2. A Simplified Model
Throughout this section, we assume Bt+1 ≡ Bt . In addition, we assume that z is essential in
production of the traded good, and that labor cannot be adjusted in that sector. We capture
this with the specification, ξ = ν = µ1 = µ2 = 1, so that the traded goods production function
is given by (2.16). For simplicity, we also exclude the wage bill from the collateral constraint:
τ N q N K N + τ T q T K T ≥ R∗ z + (1 + r∗ )B.
(3.1)
With these simplifications, we can analyze the response of the variables at date 0 to the
13
t = 0 cut in the domestic rate of interest as the intersection of two curves — each one involving
the endogenous variables, pN and L, and the exogenous policy variable, R (when there is no
risk of confusion, we drop time subscripts). The first curve summarizes equilibrium in the labor
market, and so we refer to it as the LM (‘Labor Market’) curve. The other curve, because
it incorporates restrictions from the asset market, is called the AM (‘Asset Market’) curve.
We now discuss these in turn. The simplicity of the analysis reflects in part the fact that
we characterize policy in terms of the interest rate, rather than the money supply. The last
subsection below shows that this involves no loss of generality, since there is always a money
growth rate that can support any interest rate policy, as long as R > 1.
3.2.1. Labor Market
Equating the household and non-traded good firm Euler equations for labor, (2.5) and (2.23),
we obtain:9
³
´α
N
N
(1
−
p
K
α)
(3.2)
.
RLψ+α =
ψ0 p
In this expression, it is understood that p is the simple function of pN given in (2.8). As noted
above, we think of R as an exogenous variable. So, this expression characterizes the relationship
between L and pN imposed by equilibrium in the labor market. It is easy to see that this LM
equation is positively sloped when graphed with pN on the vertical axis and L on the horizontal.
A higher pN is consistent with a higher L because it shifts the labor demand curve to the right,
while leaving the location of labor supply unchanged.10 It is also easy to see that a fall in R
shifts the LM equation to the right. This reflects that a fall in R shifts labor demand to the
right and this results in an increase in equilibrium L for a fixed level of pN .
3.2.2. Asset Market
We now turn to the AM equation. This is constructed by combining the production functions in
both sectors, (2.14), the first order condition for the intermediate input, (2.25), the pN equation,
(2.9), and the collateral constraint, (3.1), under the assumption that it is binding. Substitute
the expression for asset prices, (2.19), into the collateral constraint, (3.1), evaluated with an
equality and assume that τ N = τ T = τ to obtain:
τ
[θy T + αpN y N + Ωpc] = R∗ z + (1 + r∗ )B,
1 − λτ
(3.3)
where Ω = psβcs (qsN K N + qsT K T ) is a constant. Absence of a time subscript indicates t = 0, and
the subscript, s, denotes steady state. Here, we have used the fact, Λ2 /Λ1 = pc/ps cs . The first
9
The absence of a multiplier in (3.2) reflects that we now drop the wage bill from the collateral constraint.
Following convention, we think of labor supply and demand as corresponding to the Euler equations,
(2.5) and (2.23). We think of these relationships in a diagram with W/P on the vertical axis and L on the
horizontal.
10
14
two terms in the left hand side of the collateral constraint are the value of the marginal product
of capital at t = 0 (V MPKi ), multiplied by the respective capital stocks. The third is the present
discounted value of future cash flows. Using the zero profit condition on final consumption good
firms, pc = cT + pN cN , we can write current spending in terms of non-tradeables as
"
(1 − γ)pN
pc = 1 +
γ
#η−1
pN cN .
(3.4)
Substituting this into (3.3), our expression for the collateral constraint reduces to:
"
N
(1 − γ)p
τ
θy T + α + Ω 1 +
1 − λτ
γ
= R∗ z + (1 + r∗ )B
#η−1
pN y N
(3.5)
Equilibrium in the goods market yields the following expression for pN :
N
p =
Ã
1−γ
γ
! 1−η Ã
η
T
c
cN
!1
η
=
Ã
1−γ
γ
! 1−η
η
A
³
K
T
´θ
1−θ
∗
∗
1
η
− R z − r B
.
α
(K N ) L1−α
z
(3.6)
Finally, take into account the first order condition for z:
³
(1 − θ)A K T
´θ
z −θ = (1 + λ)R∗ .
(3.7)
Equations (3.5), (3.6), and (3.7) represent three equations in the four unknowns, λ, z, pN and
L. The third defines λ as a function of z and the second defines z as a function of pN (it is singlevalued as long as λ ≥ 0) and L. So, the three equations can be used to define a relationship
between pN and L alone. This relationship is what we call the AM curve.
It is clear that the slope of the AM curve is essential in determining whether an interest
rate cut is expansionary or contractionary. For example, if it is downward sloped, then a shift
right in the LM curve induced by a cut in the interest rate drives L up and pN down. The
contractionary case results when the AM curve is positively sloped and cuts the LM curve from
below. In general, it is not possible to say what the slope of the AM curve is. We shall see in
the next subsection that for particular parameter configurations, it is possible to determine the
slope.
Finally, we find it useful to define the version of the AM curve that holds when the collateral
15
constraint is not binding.11 In this case, finite z requires θ > 0. When the collateral constraint
is not binding, we lose one equation, (3.5), and one variable, λ, from our system. As a result,
the AM curve is defined simply by (3.6) and (3.7) with λ = 0. It is trivial to see that in this
case, the AM curve is definitely downward sloped.
3.2.3. Equilibrium
As the previous discussion indicates, to construct the AM curve it is necessary to first compute
the values of the variables in the baseline equilibrium (i.e., the steady state values of the
variables). This is a straightforward exercise, which is discussed in the appendix. In the
numerical experiments reported in this paper, we always found that the steady state of the
model is unique.
In the remainder of this subsection we verify that for a given period 0 interest rate, R, the
values of pN , L defined by the intersection of the AM and LM curves correspond to a policy
intervention equilibrium. By this we mean that, given such values of pN and L, values for p, cN ,
cT , c, w, λ, z, y T , y N , qT , q N , pT , and x can be found which satisfy all the equilibrium conditions
for t = 0. Verifying that this is true for all but the last two variables is straightforward. For
example, p can be constructed from pN using (2.8), cN can be constructed from the non-traded
good production function, and so on.
We now briefly discuss the construction of pT and x. Divide the money market clearing
condition, (2.28), by the equilibrium cash constraint, (2.29), to obtain:
d+x
w L pN (1 − α) cN L 1 − α pN cN
=
=
=
1+x
p c
pRL
c
R
pc
1−α
·
¸,
=
³
´
(1−γ)pN η−1
R 1+
γ
after using (2.23) and (3.4). Since d is predetermined at its steady state value, this expression
can be used to deduce x. Obviously, there is always an x that satisfies this expression, for any
R > 1.12 Whether a cut in R requires that the monetary authority increases or decreases x
depends upon the response of pN . We can then determine pT from (2.29).
Finally, we use a standard argument to deduce the nominal exchange rate from pT . We
assume purchasing power parity in foreign and domestic traded goods. Then, taking the initial
stock of money and the foreign price level as predetermined, we can interpret variations in pT
as reflecting movements in the nominal exchange rate.
11
The AM curve in this case is a bit of a misnomer, since asset prices do not appear.
We only consider equilibria with R > 1. Accordingly, in our calculations we impose that the cash in
advance constraint is always binding.
12
16
3.3. Effects of an Interest Rate Cut
In this section, we examine the response of equilibrium outcomes at t = 0 to an interest rate
cut. Consider first the case when the collateral constraint is not binding. As noted above, in
this case the AM curve is downward sloping. From this we conclude:
Proposition 1 If the collateral constraint is not binding, then a cut in R produces a rise
in L, a fall in pN , and no change in z.
The monetary transmission mechanism underlying this result corresponds to the standard
mechanism emphasized in the literature on the limited participation model of money. A cut
in R reduces the cost of hiring labor, and so results in an expansion in employment and a
rise in the production of non-traded goods. The cut in the interest rate produces a fall in the
marginal cost of producing non-traded goods, relative to the marginal cost of producing traded
goods, and this results in the fall in pN . The central bank engineers the cut in R by producing
a suitable move in x.
We now turn to the case when the collateral constraint is binding in both the baseline and
policy intervention equilibria. We begin with the case, θ = 0, when z is the only factor of
production in the traded good sector. In this case, a cut in R is always expansionary. When
θ = 0, substitution of (3.6) and (3.7) into (3.5) results in the following analytic representation
of the AM curve:
"Ã
!
#Ã
!η−1
1−γ
λτ
Ω−1
1 − λτ
γ
(
Ã
!
)
∗
∗
³ ´1−η
[r + λ(1 + r )] B
λτ
N
(α + Ω) p
−
=
.
(pN )η y N
1 − λτ
(3.8)
³
´η
In addition, it is evident from (3.6) that when θ = 0, z is an increasing function of pN y N .
Finally, as long as A > R∗ , λ is a positive constant.
Note first that when B = 0, (3.8) pins down a unique value for pN , so that the AM equation
is horizontal. In this case, a cut in R produces a rise in L and no change in pN or z. The
intuition for this is simple, and can been seen by inspecting (3.5) and (3.6). Note that, when
B = θ = 0 two things happen. First, an equiproportional rise in z and y N produces no change
in pN . This is because with B = θ = 0 there are no diminishing returns as cT increases with
z. Second, for fixed pN , an equiproportional increases in y N and z produces equiproportional
increases in the left and right side of the collateral constraint. Under these circumstances,
the collateral constraint simply does not get in the way of the type of expansion in output
associated with an interest rate cut that occurs when the collateral constraint is nonbinding.
On the contrary, the collateral constraint amplifies the response of employment to an interest
rate shock by preventing the decline in pN that Proposition 1 says would occur in the absence
of that constraint.
When B > 0 then both proportionality results cited in the previous paragraph fail, and
the AM curve is no longer horizontal. For example, there are now diminishing returns in
transforming additional z into extra cT . With B > 0 the AM curve has a negative slope,
17
according to (3.8).1314 Loosely, a rise in B produces a clockwise rotation in the AM curve. As a
N
result,
³ a cut
´η in R generates a rise in L and a fall in p when B > 0. Equation (3.8) also shows
that pN y N rises with the cut in R for 0 ≤ η < 1. This implies that the cut in R generates
a rise in z. We summarize these findings in a proposition:
Proposition 2 (i) When θ = B = 0, A > R∗ , a cut in R produces a rise in L and z, and
no change in pN .
(ii) When θ = 0, B > 0 and A > R∗ , a cut in R produces a rise in L and z, and a fall
in pN .
We conclude from this discussion that when θ = 0, our simple environment cannot rationalize
the notion that an interest rate cut produces a recession.
We now turn to the case, θ > 0. Suppose first that η = 1. From (3.6), we see that z can
be expressed as a function of pN y N .15 According to (3.7) λ is a function of z, and, hence of
pN y N . Substituting these results into (3.5), we conclude that when θ > 0 and η = 1, the AM
curve pins down pN y N . In particular, the curve is downward-sloping. As a result, a cut in R
produces a rise in L and a fall in pN . Because pN y N remains unchanged, it follows that z does
not change. The AM curve and the LM curves before and after the cut in the interest rate are
displayed in Figure 1.16 We summarize this finding as follows:
Proposition 3 When θ > 0 and η = 1, then a cut in R produces a rise in L, a fall in pN ,
and no change in z.
We have not been able to obtain analytic results for 0 ≤ η < 1, when θ > 0. However, when
we linearize the AM curve about steady state we find, for η = 0:17
n
o
∗
∗
T
dpN
pN θy − (1 − λτ ) [r + λ(1 + r )] B (1 − α)
=
.
dL
L
τ λ(α + Ω)pN y N
Note that when B = 0, this expression is definitely positive. If, in addition, the slope is steeper
than the slope of the LM curve, we know that with a small cut in the interest rate, there is a
13
Equation (3.8) suggests the possibility that when η > 1 and large enough, then the AM curve may be
positively sloped with B > 0, perhaps even steeper than the LM curve. The latter case is the one that is
required for a cut in R to generate a recession. We have not considered this case because we view the case,
η > 1, as empirically implausible. Still, analysis of this case may yield insights into the nature of our model,
and we plan to do this in future drafts.
14
The slope of the AM curve is given by:
1−α
dpN
[γ + λ (1 + r∗ )] B
.
=−
N
N
∗
dL
η [γ + λ (1 + r )] B/p + (1 − η) y λτ (α + Ω) /(1 − λτ ) L
¢θ
¡
This requires that the function mapping z into A K T z 1−θ − R∗ z − r∗ B be invertible. It is invertible,
given that we restrict z to those values that satisfy (3.7) with λ ≥ 0.
16
The parameter values used in this figure are: β = 1/1.05, α = 0.25, θ = 0.6, xs = 0.06, ψ0 = 0.3,
N
K = K T = 1, A = 1.9, R∗ = 1 + r∗ = 1.05, τ = 0.01, B = 0, η = 0.9.
17
See the appendix for a derivation.
15
18
period 0 set of equilibrium allocations in which L and pN are both lower. In this case, z must
fall too. We have constructed numerical examples with B = η = 0, in which the AM curve
indeed does cut the LM curve from below and a cut in the interest rate does generate a drop in
L and z. In these examples, we verified numerically that there is a unique intersection to the
LM and AM curves. Figure 2 displays the AM and LM curves for one example with η = 0.18
So far, we have found the following. We have an example with diminishing returns in
the production of traded goods, zero elasticity of substitution between traded and non-traded
goods, and low external debt, in which a cut in R induces a fall in output. However, we find
that substantial deviation from any one of these assumptions reverses the result.
Consider, for example, the parameter, η. We found that when it was increased to about 0.2,
then an interest rate cut leads to an expansion in L. However, z still falls in this case. It falls
enough so that GDP falls too, when measured in base year prices. When η was increased to
0.3, then GDP actually rises.19
Consider the effect of raising B. The preceding discussion suggests the possibility that
increasing the debt could rotate the AM curve clockwise from a position with positive slope to
one with a negative slope. In numerical experiments we have found that this is indeed the case.
Figure 3 displays the results of one such experiment. It corresponds to the model economy
underlying Figure 2, except that B has been increased to 0.1, or 27 percent of GDP.20 We find
it intriguing that the addition of substantial amount of external debt can convert a situation
from one in which an interest rate cut results in a contraction, into one in which it results in
an expansion. The economic interpretation of this finding deserves further exploration.
We have also explored more basic perturbations on the production function, by changing
µ1 , µ2 , ν, ξ from their values of unity in the above examples. One consistent result we found is
that reductions in ν, which opens up a role for variable labor in the production of y T , moves the
system in the direction of the result that a drop in R produces an expansion in the economy.21
By reducing the costs associated with diminishing labor productivity of reallocating labor across
sectors, dropping ν seems to help support assets values and prevent a tightening of the collateral
constraint in the wake of a cut in R. This seems to operate in two ways. First, a reallocation
of resources away from the non-traded good sector and towards the traded good sector limits
the fall in pN after a cut in R. Other things the same, this supports asset values in that sector.
Second, the allocation of labor towards the traded good sector pushes up asset values there by
raising the productivity of capital in that sector. Although we did find values for µ1 , µ2 , ν, ξ
that imply a large reduction in output and employment after an interest rate cut, the reduction
18
The parameter values underlying this example are: β = 1/1.05, α = 0.25, θ = 0.6, xs = 0.06, ψ = 1,
ψ0 = 0.3, K N = K T = 1, A = 1.9, R∗ = 1 + r∗ = 1.05, τ = 0.01, γ = 0.5, B = 0. When, η = 0, we obtain
N T
the following steady state properties for this model: Ls = 0.604, pN
s cs /cs = 1.459, λ = 0.796. We defined
N N
T
∗
GDP as ps c + c + r B. In this example, we found that this quantity drops 6 percent with a 4 percentage
point cut in R.
19
When the example of the previous footnote was modified so that η = 0.2, 0.3 the percent change in base
year GDP induced by a 4 percentage point drop in R is −0.12 and 0.3 respectively.
20
We did an experiment using the parameter values from the previous footnote. We set η = 0 and B = 0.4.
In the steady state, this implies a debt to GDP ratio of 0.60, or 60 percent. We found that L and z rise 1.5
and 3.1 percent, respectively, with a 4 percentage point drop in R. With B = 0, L and z both drop by 7.9
and 23 percent, respectively with the same drop in R.
21
For example, ν was reduced from unity to 0.85, then a four percentage point cut in R produces an 0.04
percent jump in GDP and an 0.87 percent jump in total employment. Recall from a previous footnote that
when ν = 1, then there is a 7.94 percent drop in employment and a 6 percent drop in GDP.
19
in output and employment was converted into an expansion with the introduction of a modest
amount of substitutability between cN and cT and a modest amount of external debt.22
4. Quantitative Analysis
In this section we study versions of our model in which the external debt is endogenous. We saw
in the previous section how the implications of a model for the effects of a domestic interest rate
cut are sensitive to assumptions. To further clarify the nature of this sensitivity, this section
analyzes two versions of our model: one that rationalizes the view that an interest rate cut
reduces output and utility and another that rationalizes the opposite view.
The nature of the monetary experiment is similar to the one studied in the previous section.
There is a benchmark analysis, in which monetary policy is treated as constant and the economy
is confronted with a binding collateral constraint. When the debt is endogenous, the economy
responds to this situation by running a current account surplus until the debt is reduced to
the point where the collateral constraint is marginally non-binding. At this point, the current
account drops to zero and the economy is in a steady state. During the transition, output is
low because the binding collateral constraint inhibits borrowing. This scenario is depicted in
Figure 4.
We analyze the impact on the transition path of a cut in the nominal rate of interest
implemented by the monetary authority. As in the previous section, the policy intervention has
a non-neutral impact because it can affect the degree of liquidity in domestic financial markets.
This in turn reflects our specification that monetary actions occur at a point in time when the
household’s deposit decision is a predetermined variable.23 We adopt the following timing. The
first thing that happens in period 0 is that the collateral constraint is imposed, and is believed
to be in place forever. Second, households make their deposit decision under the assumption
that the economy is in the no policy reaction equilibrium. Third, the monetary action is taken.
The timing in this experiment is depicted in Figure 5. Although we have not verified this,
we suspect that the outcome of our analysis is not sensitive to the nature of policy in the no
policy reaction equilibrium. That is, the difference between what happens to the economy in
the benchmark scenario and the policy intervention scenario is qualitatively unaffected by the
nature of the benchmark scenario.
The impact on the transition path of the interest rate cut is very different for the two
economies that we consider. The two economies differ in the way they model production in
the traded good sector. In one, labor plays no role and output is the Cobb-Douglas function
of z and K T only given in (2.16). In the other, labor is used in the traded good sector. In this
case, the production function is given by the Leontief specification in (2.15), with value-added,
V, given by the specification in (2.14). In each model economy, the production function in the
non-traded sector corresponds to the specification in (2.14). Also, in each model production of
22
For example, with µ1 = 1, µ2 = 2.1, ξ = 0.7, ν = 0.85, B = 0, a four percentage point cut in R produces
a 15 percent drop in total employment and a 14 percent drop in real GDP. When B is then raised to 0.1
(so that the debt to GDP ratio in the steady state is 0.18) and η is increased to 0.3, then total employment
rises 2 percent and GDP rises by 1 percent, with a four percentage point interest rate cut.
23
It is only predetermined for one period. After that, the deposit decision is free to respond.
20
the consumption good involves zero elasticity of substitution between traded and non-traded
goods. Finally, preferences in the two economies are the same.
Consistent with the analysis of the previous section, we find that the model without labor in
the traded good production function has the implication that output contracts, foreign capital
inflows dry up and welfare falls with a cut in the domestic rate of interest. The other model
implies that an economic expansion follows a cut in the interest rate.
The following section discusses the parameter values used for the two models. The section
after that presents and discusses the numerical simulation results.
4.1. Parameter Values and Steady State
The parameter values for the two versions of our model are displayed in Table 1. Consider first
the parameter values for the version in which labor enters in the production of tradables (see
left side of Table 1). These were chosen to replicate several stylized features of recent crises
countries, in particular, Thailand and Korea.
The share of tradables in total production for Korea, assuming that tradables correspond
to the non-service sectors, was approximately one third before the crisis.24 Combining this
assumption with estimates of labor shares from Young (1995), we estimate shares of capital
for the tradable and nontradable sectors in Korea to be respectively 0.48 and 0.21.25 Uribe
(1995) and Rebelo and Vegh (1995) estimate the same shares to be 0.52 and 0.37. We adopt
values that are close to both these point estimates by specifying ν = 0.50 and α = 0.36. There
is conflicting evidence on the appropriate value of σ. For example, Reinhart and Vegh (1995)
estimate the elasticity of intertemporal substitution in consumption for Argentina to be equal
to 0.2. Higher values are used in macroeconomic studies. In our analysis, we set σ = 1. We take
the foreign interest rate to be equal to 6 percent and we assume and we assume that β = 1/1.06.
We also assume a money growth rate, x, of 6 percent to obtain a nominal domestic interest
rate of 12.3 percent, roughly in line with the experience of Korea and Thailand in the years
before the crises. We set ψ = 3, implying a labor supply elasticity of 1/3. This is in between the
elasticity used in standard business cycle models, and the elasticity often reported in empirical
analyses of labor supply.
The parameters, µ1 and µ2 , in the production technology were chosen to reproduce the ratio
of imported intermediate inputs in manufacturing to manufacturing value-added in Korea for
the year, 1995. In that year, this ratio is 0.40, or, z/V = 0.40.26 This, together with the facts,
24
According to Bank of Korea (1996), Table 20-1, pages 198-199, total Korean GDP in 1995 was 352
trillion won and value-added in the tradable sector (agriculture, mining and quarrying, and manufacturing)
was 118 trillion won.
25
According to Young (1995), Table VII, page 660, the share of labor in South Korean GDP in the period
1966-1990 was 0.703. The corresponding figure for manufacturing was 0.521. To obtain the share of labor
in the non-manufacturing sector, we solved 0.33 × 0.521 + 0.67 × x = 0.703, for x. The result is x = 0.793.
For the purpose of these estimates, we identify tradables with the manufacturing sector and non-tradables
with the rest.
26
This ratio was obtained as follows. Table 4 of Bank of Korea (1998), reports that the value of total
intermediate inputs was 69 percent of gross output in manufacturing in 1995. Thus, value added in manufacturing was 31 percent of gross output. Table 13 reports that the ratio of imported intermediate inputs to
total inputs in manufacturing was 18 percent in 1995, or 12.4 (= 0.18 × 0.69) percent of gross output. Our
result is obtained as the ratio, 12.4/31.0 = 0.40.
21
µ1 V = µ2 z and 1/µ1 + 1/µ2 = 1, implies µ1 = 1.40 and µ2 = 1.4/0.4.
We chose τ and the stock of debt in the initial steady state equilibrium so that the initial and
final debt to output ratio correspond roughly to the experience of Korea and Thailand. Korea’s
(Thailand’s) external debt started at 33% of GDP by end-1997 (60.3%) and is forecasted to be
at 26.8% of GDP (51% of GDP) and the end of the year 2000. Based on these observations,
we aimed to parameterize the models so that the model economy starts in the range, 30-60%,
and then drops by an amount in the range of 8 - 10 precentage points.
We base our calibration of the relative size of K T and K N just prior to the 1997 Asian
crisis on Korean data. To our knowledge, there do not exist direct, published estimates of
sectoral Korean capital for that year. Instead, we followed two strategies. The first is the one
in Fernandez de Cordoba and Kehoe (1998). Using the definition of the share of capital income
in value added, and assuming the rental rate of capital is the same in all sectors,
Kj =
sj K j
y , j = T, N.
s y
Here, sj is the share of capital income in value-added, y j , in sector j, j = T, N. These shares,
including the aggregate share, s, were taken from Young, as discussed above. We obtained the
aggregate capital to aggregate output ratio, K/y, from Summers and Heston (1991). Table
II, page 353, reports that this is 16, 659/12, 275 = 1.36 in 1985. We estimate output for the
tradable and nontradable sectors in 1995 using data from the Bank of Korea.27 Pursuing these
calculations, we find that K T is 259, 330 trillion won and K N is 220, 861 trillion won. To convert
into units of account in the model, we normalize K T = 1 and set K N = 0.85.
The second strategy for estimating the relative size of K T and K N in 1995 for Korea uses the
ratio of sectoral investment. We obtained annual sectoral investment for the years 1990-1995
from the OECD.28 Consistent with our definition of sectoral GDP, we compute investment in
tradables using the sum of investment in manufacturing, mining and agriculture. We compute
investment in nontradables as total gross fixed capital formation minus tradables investment.
The average of the ratio of investment in nontradables to investment in tradables over our
sample is 2.2. If we normalize K T = 1, this suggests K N = 2.2. The two estimation strategies
just described produce very different estimates of the relative size of K T and K N . Each rests
on a different set of assumptions that are, at best, crude approximations. The first strategy
assumes the rental rate of capital in the two sectors is the same. In our model, and most likely
in reality too, there is no such requirement since capital is fixed in place. The second strategy
assumes the capital stock is growing at the same rate in each sector and that depreciation is
the same.29 Again, this is at best an approximation.
We decided to go with the larger estimate of the relative size of K N . In particular, we
27
See the footnote prior to the previous one for details.
The data were obtained from the web at http://www.sourceoecd.org.
29
Implicitly, we assume the following capital accumulation technology for sector i : Ki,t+1 = (1 − δ)Ki,t +
Ii,t . If the growth rate of the capital stock in each sector is µ, then we have
28
[µ − (1 − δ)] Ki,t = Ii,t ,
so that the ratio of investment in two sectors equals the ratio of the capital stock across the same two sectors.
22
set K T = 1, K N = 2. We did this because there is some direct evidence to support this - the
investment data just described - and to promote the ability of the model to match the relatively
large size of the nontraded sector suggested by the work of Burstein, Eichenbaum and Rebelo
(2001). They argue that the ratio of the value of cN to the value of cT is in the range of 3 − 4
for emerging market economies.30
The parameter values we chose for the version of the model in which labor does not appear
in the traded good production function are reported in the right side of Table 1. A difference
from the parameters in the left is that the labor supply elasticity is higher for this version of
the model. We stress this version’s implication that a cut in R generates a fall in output, and
we presume that a lower labor supply elasticity would only have made this contraction worse.
The steady state properties of the no-collateral constraint version of the model are meant
to capture the pre-crisis situation, and these are reported in Table 2. The collateral constraint
is imposed in period 0, and the economy eventually converges to the new steady state, one in
which the collateral constraint is not binding. The properties of that steady state are reported
in Table 3.
To evaluate the plausibility of our model parameterization, it is useful to consider data
sectoral employment. Bank of Korea estimates suggest that the ratio of employment in nontradables to employment in tradables averaged 1.52 over the period, 1991-1995.31 We also
obtained sectoral employment from the OECD for 1997. Here, we found that the percent of
total employment in agriculture and manufacturing was 42.3 while the percent in services was
57.7. The ratio of the two suggests the ratio of employment in nontradables to tradables was
1.38. Both these figures suggest that our ‘labor in the traded good sector’ version of the model
overstates the amount of labor in the nontraded good sector. This result is largely driven by
the relatively high consumption of nontraded goods used in the calibration of the model.
In comparing tables 3 and 4, one other feature is worth noting. In the model in the left side
of these tables, the debt to GDP ratio falls about 10 percentage points of total output, which
corresponds roughly to Thailand’s experience (see above). In the model on the right, the debt
to GDP ratio falls about 26 percentage points of output, which is rather large. Overall, we view
the parameter values as forming a reasonable basis for carrying out the exercises that interest
us.
4.2. Baseline Scenario
We now consider the dynamic effects of the imposition of the collateral constraint, when monetary policy takes the form of a constant money growth rate throughout the transition to the
new steady state. Figures 6a-6b shows the variables of the model in equilibrium, as the economy transits from the high initial debt to the lower level of debt in the steady state where
the collateral constraint is marginally non-binding. The results in Figures 6a-6b pertain to
the economy in which labor is used in the traded good sector. Note that firms respond to the
30
We include what Burstein, Eichenbaum and Rebelo (2001) model as the ‘distribution sector’ in our
non-traded good sector.
31
This is an average over quarterly ratios. The denominator has employment in the tradable sector,
which we measure as the sum of employment in agriculture, mining and manufacturing. The numerator
has employment in the nontradable sector. We measure this as total employment minus employment in the
tradable sector. These Bank of Korea data were obtained from the IMF’s edss data base.
23
enormous 70 percentage point jump in the shadow cost of foreign borrowing by paying off the
external debt. The current account jumps from zero to 4 percent of aggregate output in the
period that the collateral constraint is imposed. The current account remains above 1 percent
of aggregate output for roughly 3 years. The imposition of the collateral constraint generates
a general cutback in borrowing for working capital purposes. But, this shows up primarily in
a reduction in domestic borrowing for labor. There is actually a small rise in borrowing to
finance imports of z, because the great emphasis placed on paying off the long-term international debt.32 Total employment drops roughly 10 percent, but this drop is experienced by the
nontraded sector while the traded sector experiences a rise in employment. Total output in the
traded good sector actually expands, but domestic consumption of both traded and nontraded
goods falls.33 The overall slowdown in economic activity contributes to a fall in asset values,
as the marginal physical product of assets decline. There is a small drop in the domestic rate
of interest, presumably as the demand for funds falls with the reduction in output.
Finally, the nominal exchange rate exhibits an immediate 58 percent depreciation followed
by a long period of appreciation as the exchange rate returns to its original position (see P T ).
Interestingly, the 58 percent depreciation has a relatively small impact on P, the consumer price
index. The inflation rate in period 0 is roughly 20 percent, which is only 14 percentage points
higher than what it would have been in the absence of the collateral shock. There are two
reasons for this. First, traded goods make up only about one-third of the consumer price index.
Second, the relative price of non-traded goods, pN , falls by about 38 percent in the period of
the collateral constraint, so that the domestic currency price of non-traded goods rises by only
about 20 percent.
Figures 7a-7b display the corresponding results for the model in which labor cannot be
used in the traded sector. With one exception, the effects are qualitatively similar to what we
saw in the previous figures. The exception is that now imports of the intermediate good fall
substantially. The main difference in the results has to do with magnitudes. The effects tend
to be larger because the debt in this version of the model is much higher in the initial steady
state than it is in the version just discussed. Thus, the shadow cost of borrowing jumps 648
percent in the period of the imposition of the collateral constraint. Associated with this there
is a major reduction in employment and imports of the intermediate good. Capital inflows (to
finance z) display the ‘sudden stop’ feature emphasized by Calvo (1998). Finally, the collateral
constraint triggers an immediate real and nominal depreciation that overshoots.
We take it that these characteristics of our models correspond reasonably well, at least qualitatively, with what actually happened after the 1997 financial crises in several Asian countries
(see Boorman, et. al. (2000).) On this basis, we feel justified in using these models to study
the effects of a cut in domestic interest rates in the wake of a financial crisis. We turn to this
now.
32
The numerical experiments we have run suggests that a fall in total borrowing is a robust response to
the imposition of collateral constraint. For some parameterizations, this implies a rise in z, and for others,
a fall.
33
These numbers are not reported in the figures.
24
4.3. The Effect of an Interest Rate Cut
We now suppose that in period 0, after the collateral constraint has been imposed, the monetary
authority temporarily deviates from its constant money growth path. It does so by doing
whatever is necessary with the money supply to obtain a given reduction in the period 0 rate
of interest. This policy action, which is unanticipated, is executed after the household has
made its deposit decision. Agents expect, correctly, that the monetary authority will revert
to its constant money growth path in t ≥ 1. This one-time change policy has no impact on
the ultimate steady state to which the economy is headed. It only affects the nature of the
transition path. We ask what it does to the economic variables along that path, and whether
things are made better or worse in a welfare sense.
Results regarding the contemporaneous impact of the policy intervention are reported in
Table 4.34 The cut in the interest rate is 9 percentage points in each model economy. It is
accomplished by a one-time change in money growth in period 0, after which the steady state
money growth rate of 6 percent is resumed. In explaining what happens in the two models,
we adopt the approach taken in the introduction, which centers the discussion on the collateral
constraint, (3.1), which we reproduce here:
τ qN K N + τ qT K T = R∗ z + (1 + r∗ )B.
Here, we have not included the wage bill, which we abstract from for purposes of this discussion
(though, not in the computational experiments). In both model economies, the cut in the
interest rate generates a nominal depreciation (see Table 4).35 Other things the same, this
makes the left side, the asset side, of the collateral constraint fall. To see this, recall that the
collateral constraint is measured in units of traded goods. So if only P T rose, and no other
price - when measured in domestic currency units - or quantity changed, the asset side would
fall, requiring a fall in z. There is another price effect that may have a similar impact on
the collateral constraint. In particular, the cut in the domestic rate of interest, by having a
relatively large impact on marginal costs in the non-traded good sector, may cause the relative
price of goods produced in that sector, pN , to fall. This has a further depressive effect on the
asset side of the collateral constraint, because pN is used to value the productivity of the assets
in the non-traded good sector.36 If this were the whole story, then the interest rate cut leaves us
with a mismatch between the asset and liability sides of the collateral constraint, which could
only be resolved by reducing capital inflows through a cut in z.
But, this is not the whole story. Inspection of the collateral constraint reveals another
option: one could in principle increase z. Of course, this has the wrong effect on the liability
side of the collateral constraint. However, this problem is somewhat alleviated if the external
34
We have not been able to establish formally the uniqueness properties of the baseline equilibrium and the
policy intervention equilibrium. To build confidence that we do have uniqueness, we searched numerically
for other equilibria and never found any.
35
This corresponds to the traditional view of monetary policy in open economies. See Mussa (2000) for
evidence from episodes of emerging market crises that seem to provide support for the traditional view.
36
Recall that pN enters in the value of the marginal product of capital in the non-traded good sector (see
(2.19).)
25
debt is very large.37 In this case, the percentage increase in the liability side of the collateral
constraint associated with a given rise in z is small. What about the impact of a rise in z
on the asset side of the collateral constraint? In general - and in our models specifically one expects the increased use of z to raise the value of the economy’s assets by raising their
marginal physical product. However, this channel is not strong enough if z is subject to strongly
diminishing returns. In this case, the rise in asset values associated with a rise in z is small
and likely to be dominated by the rise in liabilities.38
This is the situation in the model where labor does not enter the traded good sector. In
that model, the equilibrium response of z to a cut in R involves a reduction in z, not an
increase. This reduction in z sets into motion additional forces in our model which keep it
falling. In particular, the lack of substitutability between traded and non-traded goods in the
production of final consumption goods has the consequence that a fall in z reduces demand for
the non-traded good, so that employment there falls. This has the effect of further reducing
asset values, aggravating the assets and liability mismatch in the collateral constraint. The
effects on asset prices can be seen in Table 4, which shows that q T rises by a very small amount,
and is dominated in the collateral constraint by the fall in qN . Hence, the value of assets as a
whole falls.
The situation is different in the model where labor does play a role in the traded good
sector. Now the option of restoring equality to the collateral constraint by increasing z is a
greater possibility. This is because an infusion of labor into the traded good sector can work
against the diminishing returns associated with an increase in z. As a result, a rise in z could
in principle raise the asset side of the collateral constraint by more than the liability side. The
tables indicate that this is precisely what happens in the model in which labor enters the traded
good sector.
We calculated the present discounted value of utility from period 0 on, for our baseline
scenario and for the scenario in which the monetary authority responds by cutting the rate
of interest. We did this for each of our two models. Note that utility in the steady state to
which the economy converges after the collateral constraint is imposed is higher than utility
in the pre-crisis steady state. This reflects the wealth effects of the reduced level of debt in
the collateral-constrained steady state. In the case of the model with labor in the traded good
sector, the present discounted utility in the equilibrium with the interest rate cut is higher than
what it is when the monetary authority does not react. Utility falls with the interest rate cut
in the other model.
Figures 8a-8b display the impact of the interest rate cut on the whole dynamic path, for the
version of our model in which labor can be used in the traded goods sector. We see that the
cut in R lifts up asset values, relaxes the collateral constraint (note how λ falls), and stimulates
37
In this context, our analysis of the previous section is relevant. There we presented evidence that
suggests: (i) if a model is to rationalize the notion that an interest rate cut generates a recession, then the
AM curve must be positively sloped and cut the LM curve from below, and (ii) increasing the external debt
rotates the AM curve clockwise. Conditions (i) and (ii) suggest that if an economy with low debt produces
a recession with an interest rate cut, then the recession will be smaller or it may even turn into a boom for
a model in which the external debt is higher.
38
There is another channel that could in principle be operative. A cut in the domestic interest rate, by
increasing the supply of nontraded goods (recall, the interest rate cut reduces the marginal cost of those
goods), raises the demand for traded goods, to the extent that these complement with nontraded goods. If
so, then the shadow cost of the collateral constraint is likely to increase and this can have the effect of raising
asset prices. This effect does not appear to be strong in the particular numerical examples displayed here.
26
output and employment. Note that the economy takes advantage of the boom to raise the
current account even further, accelerating the transition to the new steady state. Interestingly,
the depreciation of the currency now generates an almost equal increase in domestic inflation.
The reason for this is that pN rises with this policy. Presumably, this reflects a wealth effect,
which leads to greater purchases of non-tradable goods and, hence, in their marginal cost. This
stands in interesting contrast with the results displayed in Figure 6b. There, the depreciation
of the currency is associated with a contraction, producing a negative wealth effect, which - by
damping pN - causes the resulting general inflation to be a lot smaller than the rise in the price
of traded goods.
Figures 9a-9b display the impact of the interest rate cut in the model in which labor does not
appear in the traded goods production function. The results here could not be more different
from what we saw in Figure 8a-8b. Now, the collateral constraint tightens, employment and
output drop, asset prices fall, there is a depreciation in the nominal exchange rate, together
with a much smaller rise in inflation, presumably due to the negative wealth effect associated
with the fall in R.
We now briefly discuss the welfare consequences of the cut in interest rates in this model.
In our model, absent the collateral constraint, the frictions associated with R > 1 imply that
reducing the rate of interest is always desirable. In our environment, the optimality of the
Friedman rule - and, hence, of cutting the interest rate - is not so obvious because of the
presence of the collateral constraint. Table 5 reports the utility levels associated with the
various equilibria. First, notice that utility in the pre-crisis steady state is always lower than
what it is in the new steady state. This simply reflects that in the latter, there is less external
debt and so foreigners are imposing fewer ‘taxes’ in the form of interest rate payments. Note
too, that utility in the transition to the new steady state is lower than utility in the old steady
state: although people are happier in the new steady state, if they were given the choice whether
to stay in the old equilibrium or transit to the new one, they would prefer the old equilibrium.
Finally, consider the welfare calculations associated with the central question that interests
us. We can evaluate the impact on utility of the cut in the interest rate by comparing the
discounted utility of the paths with and without the monetary policy interventions. Here, we
find that utility goes down in the model with no labor in the traded good sector. In this model,
the transition to the new steady state is made harder by the interest rate cut. In the other
model, the transition is made easier.
5. Conclusion
We analyzed a small open economy model in which firms require two types of working capital: domestic currency to hire domestic inputs and foreign currency to finance imports of an
intermediate input. We adopt a reduced form model of a financial crisis, and ask what is the
economic impact of a cut in the domestic rate of interest at such a time. We model a financial
crisis as a time when collateral constraints on borrowing are imposed and are binding. Our
notion of a ‘financial crisis’ corresponds to what some might think of as a ‘credit crunch’.
In our model, application of binding collateral constraints causes the economy to run a
current account surplus and bring its debt down to the steady state in which the collateral
constraint is marginally non-binding. During the transition, the collateral constraint limits the
27
amount of borrowing that firms can do, and so leads to a reduction in output and employment.
In addition, asset values fall with the slowdown in activity, and real and nominal exchange
rates depreciate and overshoot with the onset of the crisis. These features of the transition
dynamics in our model correspond - at least qualitatively - with what was observed in the
Asian crises that began in late 1997. We believe that this justifies taking our reduced form
model of a financial crisis seriously, as a laboratory for studying the economic effects of a cut
in the domestic interest rate in the aftermath of a financial crisis.
To understand our analysis of the effects of the interest rate cut, it is sufficient to keep
in mind firms’ collateral constraint: the requirement that the value of their assets be no less
than the value of their liabilities. We model the former as consisting of productive assets such
as land and capital in the domestic economy. Also, most of firms’ liabilities take the form of
international debt. Our framework captures the tensions emphasized in the literature that are
created by operation of this collateral constraint.
First, an interest rate cut engineered by the central bank produces a nominal exchange rate
depreciation in our model. Other things the same, this tightens the collateral constraint by
producing a fall in the value of the domestic assets of the firm, while not affecting the value of
international liabilities. This effect arises from the widely discussed mismatch in the currency
denomination of assets and liabilities. This effect could be compounded if in addition to a
nominal depreciation, there is also a real depreciation. Second, an interest rate cut can also
alleviate the collateral constraint by pushing up asset values.
We find that the first scenario - the one in which currency mismatch problems cause an
interest rate cut to produce a contraction - is more likely when there are limitations in how
flexibly the economy can exploit an increase in the quantity of the intermediate good. The
second scenario - the one in which an interest rate cut produces an expansion by inflating asset
values - is more likely when these limitations are not present. We conclude that resolving the
debate over the effects of an interest rate cut in the aftermath of a financial crisis requires
understanding how much short-run flexibility there is in the economy. We suspect that there is
relatively little such flexibility, at least in the short run, so that the contraction scenario may
be the most plausible one.
A. Appendix
In this technical appendix we discuss various issues raised in the text. The first subsection
discusses the computation of the steady state in the version of the model of section 3 in which
the collateral constraint is binding. The second subsection derives the linearization formulas
used in the local analysis in section 3. The third subsection discusses the solution of the version
of our model analyzed in section 4, in which the current account is not constrained to be zero.
28
A.1. Steady State in the Model of Section 3
For convenience, we repeat some of the equations of the model here:
RLψ+α =
³
ψ0 p
´α
´θ
= R∗ z + (1 + r∗ )B.
pN (1 − α) K N
(A.1)
.
The collateral constraint is:
³
τ N αpN K N
´α
L1−α
1 − λτ N − β
³
τ T θA K T
+
z 1−θ
1 − λτ T − β
(A.2)
The first order necessary condition for z in the traded good sector is:
³
(1 − θ) A K T
´θ
z 1−θ = (1 + λ)zR∗ .
(A.3)
The price equation is:
³
1
´θ
η
T
z 1−θ − R∗ z − r∗ B
1 − γ A K
N
p =
.
(K N )α L1−α
γ
(A.4)
When η = 0, we replace (A.4) with
³
KN
´α
³
L1−α = A K T
´θ
z 1−θ − R∗ z − r∗ B.
(A.5)
The unknowns are L, pN , z, λ.
We now discuss how to find the steady state when θ > 0 and η = 0. [the case, η > 0 will
be added later]. We use the equations, (A.1)-(A.3) and (A.5) to define a mapping from z to z 0 ,
whose fixed point corresponds to an equilibrium. Rewrite (A.2) as follows:
N
p =
³
1 − λτ N − β
´
"
θ
R∗ z + (1 + r∗ )B −
τ N α (K N )α L1−α
τ T θA(K T ) z 1−θ
1−λτ T −β
#
,
(A.6)
Combining (A.5) and (A.3), we obtain (the discussion below assumes B = 0, which will be
29
fixed later):
³
KN
´α
·
³
L1−α = z A K T
"
´θ
z −θ − R∗
1+λ ∗
= z
R − R∗
1−θ
λ+θ ∗
=
Rz
1−θ
or,
"
λ + θ R∗ z
L=
1 − θ (K N )α
#
#
¸
(A.7)
1
1−α
(A.8)
.
Note that (A.3) defines a mapping from z to λ. Taking this and (A.8) into account, (A.6) defines
a mapping from z to pN :
(A.9)
pN = f (z),
where
f (z) =
"
N
1 − λτ − β
1 − λτ T − β
#"
1−θ
λ+θ
# h³
´
i
1 − λτ T − β − τ T (1 + λ)θ/(1 − θ)
τNα
.
(A.10)
Here, it is understood that λ is the function of z implied by (A.3).
Solving (A.1) for L :
L=
³
´α 1
(1 − α) K N ψ+α
h
i
1
Rψ
+1
0
.
pN
Combining this with (A.7), we obtain:
"
∗
λ+θ R z
1 − θ (K N )α
#
1
1−α
³
(1 − α) K N
´α
(1 − α) K N
=
Rψ0
³
1
pN
´α
1
ψ+α
´
.
+1
Solve this for z :
³
N
1−θ K
z =
λ + θ R∗
= g(pN , λ),
30
Rψ0
³
³
1
pN
´α 1−α
ψ+α
´
+1
(A.11)
say. We can use this and f in (A.9) to define a mapping from z into itself:
z 0 = g(pN , λ) = g (f (z), λ(z)) = h(z),
say, where λ(z) summarizes (A.3). It is easy to see that h is an increasing function of z as long
as τ N = τ T .
To actually find the fixed point, if it exists, it is useful to be able to
³ restrict the´set of
candidate equilibrium values of z. We know that we must have λ ≥ 0, 1 − λτ N − β ≥ 0,
³
´
1 − λτ T − β ≥ 0, pN ≥ 0. The first of these implies an upper bound on z, and the others
imply a lower bound. (These conditions imply L ≥ 0, so we don’t list that separately.) From
(A.9) we see that pN ≥ 0 requires:
"
#
(1 − θ) (1 − β)
1−θ
θ
=
−θ
λ ≤ T 1 − β − τT
1−θ
τ
τT
This places a lower bound on z :
z≥
(
"
#) −1
(1 − θ) (1 − β)
R∗
−θ+1
θ
T
τT
(1 − θ)A (K )
θ
.
We want this lower bound (of course!) to be less than the upper bound on z implied by λ ≥ 0.
This places a restriction on the parameters:
θ≤
(1 − θ) (1 − β)
,
τT
τT ≤
(1 − θ) (1 − β)
.
θ
or,
This is a pretty tight upper bound on τ T .
The value of a unit of capital in the non-traded and traded good sectors is, respectively:
N
q =
³
αpN K N
´α−1
L1−α
1 − λτ N − β
31
T
, q =
³
θA K T
´θ−1
z 1−θ
1 − λτ T − β
.
A.2. Linearization of the Model in Section 3
We derive provide formulas for linearizing the model of section 3 about its steady state. Define
the percent deviation of a variable from its steady state value as x̂ = dx/x.
Linearizing the LM curve around the steady state we obtain:
R̂ + (ψ + α)L̂ = p̂N − p̂.
or
Ã
pγ
R̂t + (ψ + α)L̂t = 1 − N
p
!η−1
p̂N ,
t
where we have used the fact that
p̂ =
Ã
pγ
pN
!η−1
p̂N
Finally, rearrange to obtain
p̂N
t =
where
R̂t + (ψ + α)L̂t
1−
Ã
pγ
1− N
p
³
pγ
pN
´η−1
!η−1
(A.12)
,
≥ 0,
after evaluating p in terms of pN .
To linearize the AM curve, begin with the pN equation:
p̂N = −
1−α
1
L̂ + ĉT
η
η
From the resource constraint we obtain:
·
³
·
³
dcT = (1 − θ)A K T
so that
ĉT =
(1 − θ)A K
32
T
cT
´θ
´θ
¸
z −θ − R∗ dz,
z
−θ
−R
∗
¸
z ẑ,
where ẑ = dz/z. Then,
·
¸
³
´θ
1−α
1
z
(1 − θ)A K T z −θ − R∗ T ẑ,
p̂ = −
L̂ +
η
η
c
N
or
Ã
!
1−α
λR∗ z
p̂ = −
ẑ.
L̂ +
η
ηcT
N
We can now obtain ẑ as a function of p̂N as follows:
ẑ =
1−α
L̂ +
η
λR∗ z
ηcT
p̂N
i
cT h
N
=
(1
−
+
α)
L̂
η
p̂
.
λR∗ z
(A.13)
³
From the first order condition for labor in the traded goods sector, (1 − θ)A K T
(1 + λ)R∗ , we obtain
+ λ) =
−θẑ = (1 d
= λ̂
´θ
(1 + λt ) − (1 + λ)
λt − λ
=
(1 + λ)
1+λ
z −θ =
(A.14)
λ
.
1+λ
Totally differentiating the expression for the binding collateral constraint with respect to L,
λ, z, and pN , we obtain:
τ
R∗ zλλ̂
1 − λτ
"
#
µ
¶
N η−1
³
´α
(1
−
τ
γ)p
(1 − α)pN K N
α + Ω 1 +
+
L1−α L̂
1 − λτ
γ
µ
¶
³
´α
τ
+
{αpN K N L1−α
1 − λτ
"
#
"
#
N η−1
N η−1
³
´α
(1
−
(1
−
γ)p
γ)p
+ (η − 1)
}p̂N
+pN K N L1−α Ω 1 +
γ
γ
= R∗ z −
³
(1 − θ) τ θA K T
1 − λτ
´θ
z 1−θ
ẑ
33
or,
τ
R∗ zλλ̂
1 − λτ
"
#
µ
¶
N η−1
³
´α
(1
−
γ)p
τ
(1 − α)pN K N
α + Ω 1 +
+
L1−α L̂
1 − λτ
γ
(A.15)
"
#
¶
N η−1
³
´
(1
−
γ)p
τ
N
N α 1−α
1 + η
p̂N
+
Ω
+
p
K
L
α
1 − λτ
γ
µ
= R∗ z −
³
(1 − θ) τ θA K
1 − λτ
T
´θ
z
1−θ
ẑ
Let’s simplify the expression on (1 − α)L̂. According to the collateral constraint:
"
³
´θ
(1 − γ)pN
τ
[θA K T z 1−θ + α + Ω 1 +
1 − λτ
γ
#η−1
³
´α
pN K N
L1−α ]
so that,
"
(1 − γ)pN
τλ
α + Ω 1 +
1 − λτ
γ
τλ
θy T
1 − λτ
τλ
(1 − θ)y T − R∗ z −
θy T + λ(1 + r∗ )B
1 − λτ
1
− R∗ z + λ(1 + r∗ )B
y T − θy T
1 − λτ
θy T
+ λ(1 + r∗ )B
cT + r∗ B −
1 − λτ
θy T
+ [r∗ + λ(1 + r∗ )] B
cT −
1 − λτ
= λR∗ z + λ(1 + r∗ )B −
=
=
=
=
#η−1
pN y N
Now consider the term on p̂N .
"
(1 − γ)pN
τλ
ηcT −
α + Ω 1 + η
1 − λτ
γ
34
#η−1
pN y N
= R∗ z + (1 + r∗ )B
"
#η−1
pN y N
#
(1 − γ)pN
τλ
= cT −
α + Ω 1 +
1 − λτ
γ
+(η
− 1) cT
"
(1 − γ)pN
τλ
Ω
−
1 − λτ
γ
η−1
pN y N
"
(1 − γ)pN
θy T
τλ
=
− [r∗ + λ(1 + r∗ )] B + (η − 1) cT −
Ω
1 − λτ
1 − λτ
γ
#η−1
pN y N .
But,
"
(1 − γ)pN
τλ
Ω
c −
1 − λτ
γ
T
#η−1
pN y N
"
(1 − γ)pN
τλ
Ω
= y −R z−r B−
1 − λτ
γ
∗
T
∗
#η−1
= y T − (1 + λ)R∗ z + λ [R∗ z + (1 + r∗ ) B]
"
(1 − γ)pN
τλ
Ω
−λ (1 + r ) B − r B −
1 − λτ
γ
∗
∗
pN y N
#η−1
"
pN y N
³
´θ
(1 − γ)pN
τλ
[θA K T z 1−θ + α + Ω 1 +
= yT +
1 − λτ
γ
"
#η−1
#η−1
pN y N ]
(1 − γ)pN
τλ
Ω
−
pN y N − (1 + λ)R∗ z − [r∗ + λ (1 + r∗ )] B
1 − λτ
γ
³
´θ
τλ
= y T − (1 − θ)y T +
[θA K T z 1−θ + (α + Ω) pN y N ] − [r∗ + λ (1 + r∗ )] B
1 − λτ
τ
λ
[θy T + (α + Ω) pN y N ] − [r∗ + λ (1 + r∗ )] B
= θy T +
1 − λτ
τλ
θy T
+
(α + Ω)pN y N − [r∗ + λ (1 + r∗ )] B
=
1 − λτ
1 − λτ
So:
"
(1 − γ)pN
τλ
α + Ω 1 + η
1 − λτ
γ
#η−1
pN y N
"
(1 − γ)pN
τλ
θy T
= ηcT −
− [r∗ + λ(1 + r∗ )] B + (η − 1) cT −
Ω
1 − λτ
1 − λτ
γ
= ηcT − {
θy T
− [r∗ + λ(1 + r∗ )] B +
1 − λτ
35
#η−1
pN y N
Ã
!
θy T
τλ
+
(α + Ω)pN y N − [r∗ + λ (1 + r∗ )] B }
(η − 1)
1 − λτ
1 − λτ
(
Ã
!)
θy T
θy T
τλ
∗
∗
T
N N
= ηc −
− η [r + λ(1 + r )] B + (η − 1)
+
(α + Ω)p y
1 − λτ
1 − λτ
1 − λτ
Substituting the above expression into (A.15):
τ
R∗ zλλ̂
1 − "λτ
#
1 T
θy T
∗
∗
+ c −
+ [r + λ(1 + r )] B (1 − α)L̂
1 − λτ
λ
"
#
1
τλ
θy T
T
N N
∗
∗
− (η − 1)
(α + Ω)p y + η [r + λ(1 + r )] B p̂N
+ ηc − η
1 − λτ
1 − λτ
λ
= R∗ z −
³
(1 − θ) τ θA K T
1 − λτ
´θ
z 1−θ
ẑ
Substituting out for λ̂ in terms of ẑ from (A.14) and for ẑ in terms of L̂ and p̂N from (A.13),
we obtain:
i
τ
cT h
∗
N
(1
−
+
−
R zθ (1 + λ)
α)
L̂
η
p̂
1−
λR∗ z
λτ
"
#
T
1 T
θy
∗
∗
+ [r + λ(1 + r )] B (1 − α) L̂
+ c −
1 − λτ
λ
"
#
T
1
τ
λ
θy
+ ηcT − η
− (η − 1)
(α + Ω)pN y N + η [r∗ + λ(1 + r∗ )] B p̂N
1 − λτ
1 − λτ
λ
= R∗ z −
³
(1 − θ) τ θA K T
1 − λτ
´θ
cT h
i
z 1−θ
N
(1
−
+
α)
L̂
η
p̂
λR∗ z
Collecting terms and simplifying (using the first order necessary condition for z and the collateral constraint):
"
#
1 T
θy T
+ [r∗ + λ(1 + r∗ )] B (1 − α) L̂
c −
1 − λτ
λ
"
#
1
τλ
θy T
∗
∗
T
N N
+ ηc − η
− (η − 1)
(α + Ω)p y + η [r + λ(1 + r )] B p̂N
1 − λτ
1 − λτ
λ
½
µ
¶
¾
τ
τ
∗
∗
∗
= R z+
[θ(1 + λ)R z]
R zθ (1 + λ) −
1 − λτ
1 − λτ
36
×
=
i
cT h
N
(1
−
+
α)
L̂
η
p̂
λR∗ z
i
cT h
(1 − α) L̂ + η p̂N
λ
or
"
Ã
!
#
1 T
[r∗ + λ(1 + r∗ )] B
θy T
cT
+
(1 − α) L̂
−
c −
1 − λτ
λ
λ
λ
" Ã
!
#
1
η [r∗ + λ(1 + r∗ )] B N
τλ
θy T
cT
T
N N
+
− (η − 1)
(α + Ω)p y
−η +
ηc − η
p̂
1 − λτ
1 − λτ
λ
λ
λ
= 0
Multiply by λ, cancel terms, as appropriate, multiply by 1 − λτ, to get:
³
´
−ηθy T − (η − 1)τ λ(α + Ω)pN y N + (1 − λτ ) η [r∗ + λ(1 + r∗ )] B p̂N
=
n
o
θy T − (1 − λτ ) [r∗ + λ(1 + r∗ )] B (1 − α) L̂.
or,
N
n
o
− θy T − (1 − λτ ) [r∗ + λ(1 + r∗ )] B (1 − α)
p̂
= sAM ,
=
ηθy T + (η − 1)τ λ(α + Ω)pN y N − (1 − λτ ) η [r∗ + λ(1 + r∗ )] B
L̂
(A.16)
say, where sAM is the slope of the AM curve. From this expression we can see that for the
Cobb-Douglas case (η = 1), the slope is definitely negative:
p̂N
= −(1 − α) < 0
L̂
So, with perfect substitutability, the AM curve slopes downward, and a cut in R must produce
a fall in pN and a rise in L. With η = 0:
N
p̂
=
L̂
n
o
θy T − (1 − λτ ) [r∗ + λ(1 + r∗ )] B (1 − α)
τ λ(α + Ω)pN y N
.
Note that when B = 0, this expression is definitely positive. It cannot be signed when B > 0.
37
By substituting out for p̂N in (A.16) from the LM curve, (A.12), we find:
R̂
1−
³
pN
γp
´1−η
+ sLM L̂ = sAM L̂,
where
ψ+α
sLM =
1−
³
pN
γp
´1−η .
Then,
1
L̂
µ
=
³ N ´1−η ¶ .
R̂
AM
LM
(s
− s ) 1 − pγp
So, for a cut in R, R̂ < 0, to produce an equilibrium drop in employment, L̂ < 0, we need
sAM > sLM . Since the latter is positive, this requires that sAM be more than just positive. It
must be big enough. The condition is:
ηθy T
n
o
θy T − (1 − λτ ) [r∗ + λ(1 + r∗ )] B (1 − α)
+ (η − 1)τ λ(α +
Ω)pN y N
− (1 −
λτ ) η [r∗
+ λ(1 +
r∗ )] B
>
ψ+α
1−
³
pN
γp
´1−η .
A.3. Solving the Main Model
We stress the case where there is substitution between cN and cT , and between zt and Vt . The
case of no substitutability involves obvious modifications on the discussion below and so is not
included. A technical manuscript which covers this case in detail is available on request.
We begin with a discussion of the steady state in which all real variables and relative prices
with time subscripts are constant. We consider two steady states. The pre-crisis steady state
is one in which the initial level of debt, say B0 , violates (2.18). We define the post-crisis steady
state as one in which the initial level of debt, say B∞ , has the property that the collateral
constraint is satisfied as an equality. We then discuss the computation of the dynamic path
taking the economy from first steady state to the second. We do this under two scenarios. In
our baseline scenario government policy, defined here in terms of money growth, is constant.
In the alternative scenario the money growth rate is adjusted to hit a particular target interest
rate in period 0, while money growth is returned to the money growth rate is changed in the
period when the collateral constraint is imposed.
A.3.1. Steady State
To understand how these steady states are computed, it is useful to note that - subject to
feasibility - corresponding to any initial level of debt there is a unique steady state when the
38
collateral constraint, (2.18), is ignored. To find the post crisis steady state, we simply alter the
initial debt until the collateral constraint is satisfied as an equality. Then, B0 is selected as a
number bigger than B∞ , to be consistent with data as discussed in the text.
The following discussion explains how we find the steady state corresponding to an arbitrary
initial value of the debt.
The steady state interest rate, R, is determined from (2.6), (2.13), and the fact that pt =
Pt /Mt is constant in steady state:
1+x
.
R=
β
From here on, we treat R as a known quantity.
Rewriting the firm’s first order condition for z :
Ã
R∗
y = µ2 z
(1 − θ) µ2
T
!ξ
.
The resource constraint in the traded good sector is:
y T − cT − R∗ z = r∗ B.
Combining this with the previous expression,
µ
1
1−θ
¶ξ Ã
R∗
µ2
!ξ−1
− 1 R∗ z = cT + r∗ B.
(A.17)
Here is an algorithm for finding the steady state which involves a nonlinear search in the single
variable, cT .
Suppose cT is given. Then, z can be computed from (A.17). LT may then be obtained from:
z=
µ1 V 1
µ2 θ
Ã
∗
R
(1 − θ) µ2
!ξ−1
−
Ã
−ξ
! ξ−1
1−θ
θ
(A.18)
Given LT , LN may be obtained by combining the price equation:
γ
p =
1−γ
N
Ã
39
(1 − γ) cT
γcN
!1
η
(A.19)
and equality of V MPL ’s across the two sectors:
³
(1 − α)pN K N
´α ³
LN
´−α
1
= −
θ
Ã
1−θ
θ
!Ã
∗
R
(1 − θ) µ2
1
!1−ξ 1−ξ
³
θ(1 − ν)µ1 A K T
´ν ³
LT
´−ν
.
Substitute the former into the latter, to obtain:
γ
(1 − α)
1−γ
1
Ã
1−θ
= −
θ
θ
Ã
(1 − γ) cT
γ (K N )α (LN )1−α
!Ã
R∗
(1 − θ) µ2
or
³
LN
where
D=
½
1
θ
−
³
1−θ
θ
´³
!1
η
KN
1
!1−ξ 1−ξ
´−[ 1 (1−α)+α]
η
R∗
(1−θ)µ2
³
1
´1−ξ ¾ 1−ξ
γ
(1 − α) 1−γ
³
´α ³
LN
´−α
³
´ν ³
θ(1 − ν)µ1 A K T
LT
´−ν
,
= D,
(A.20)
³
θ(1 − ν)µ1 A K T
(1−γ)cT
γ(K N )α
´1
η
(K N )α
´ν ³
LT
´−ν
With LN , cT in hand, compute pN from the price equation, (A.19). Finally, assess whether
labor supply equals labor demand in the traded good sector:
Ã
1 1
1−θ
−
R θ
θ
!Ã
R∗
(1 − θ) µ2
1
!1−ξ 1−ξ
³
θ(1 − ν)µ1 A K T
´ν ³
LT
´−ν
³
= ψ0 LT + LN
´ψ
p.
(A.21)
Adjust c until (A.21) holds exactly. The five equations, (A.17)-(A.21), can be used in this
way to pin down the five variables, LN , LT , pN , cT , z.
The variables, p and c, may be obtained from (2.8) and (2.7). Then, obtain pT from
T
ppT c = 1 + x.
The wage rate comes from
w = ψ0 Lψ p, L = LT + LN
We now obtain the steady state value of d, the ratio of deposits to the beginning of period
40
money stock. Dividing (2.3) by P T :
pc = wL +
1
d
− T,
T
p
p
so that
d = pT [wL − pc] + 1.
We require 0 ≤ d ≤ 1.
Now we go for the asset values. From (2.20),
q i = V MPKi +
so that
qi =
qi
,
1 + r∗
1 + r∗
V MPKi , i = N, T,
∗
r
where
αy N
,
KN
"
#1
y T ξ θνµ1 V
=
.
µ1 V
KT
V MPKN = pN
V MPKT
The value of collateral is, in units of the traded good,
qN K N + qT K T
This completes the discussion of the steady state.
A.3.2. The Transition Path
We imagine that in date 0 the economy has an initial debt level of B0 . At this level of debt the
collateral constraint is binding. In the baseline equilibrium, money growth is kept constant.
That is, xt = x for t = 0, 1, 2, .... We compute the equilibrium path of the economy to the
new steady state where the debt level is B∞ . In the second equilibrium, x0 6= x, but xt = x for
t = 1, 2, ... . In this equilibrium, the monetary adjustment is unanticipated in the sense that
when households make their deposit decision in the beginning of period 0 they do so under the
assumption that they are in the baseline equilibrium. As noted above, they do not adjust this
41
decision when it turns out that x0 6= x.
We first consider the computation of the baseline equilibrium. We then discuss the computation of the equilibrium in which there is a monetary intervention. The basic strategy is
based on solving a system of non-linear equations in the Lagrange multipliers on the collateral
constraint. For a given set of Lagrange multipliers, we compute a sequence of candidate allocations and prices, imposing the following conditions: (i) quantities and prices eventually end up
in the new steady state; (ii) the initial level of debt is B0 ; and (iii) all equilibrium conditions
except the collateral constraint are imposed. We then evaluate the collateral constraint at each
date. We adjust the Lagrange multipliers until it is satisfied. If the multipliers turn out to
violate non-negativity, then we conclude there is no equilibrium.
Baseline Scenario At date 0, B0 is given. We want to compute an equilibrium set of
sequences,
T
N
N
T
qtT , qtN , cTt , cN
t , Lt , Lt , pt , pt , Rt , wt , zt , Bt+1 , λt , t = 0, 1, 2, 3, ...,
for a given sequence of xt ’s. These 13 sequences must satisfy 13 equilibrium conditions. These
are the two equations defining the q’s (2.20); the firm’s intertemporal Euler equation (2.22), and
its three intra-temporal Euler equations, (2.23), (2.24), (2.25); the marginal condition relating
pN to the consumption goods, (2.9); the resource constraint in the traded and non-traded good
sectors, (2.30) and (2.31); the collateral constraint, (2.18); the household’s intra- and intertemporal Euler equations, (2.5), (2.6); and finally, the cash in advance constraint, (2.29).39
We seek an equilibrium which converges asymptotically to the steady state where the debt
is B∞ and the collateral constraint is marginally non-binding. This means that all the above sequences converge to their values in the steady state equilibrium whose computation is discussed
in the previous section.
Here is our strategy for accomplishing this. We assume that the system arrives in a steady
state in period T + 2 (in practice, we found that T = 10 works well.) We specify exogenously
(below, we explain in detail how this is done), a sequence, λ ≡ (λ0 , λ1 , ..., λT +1 ), with λt = 0
for t ≥ T + 2. Also, ΛT +2 = Λs , where the subscript, ‘s’, means steady state. Similarly, all
the other 13 variables are assumed to be in the new steady state for t ≥ T + 2. The value of
T that we used in the calculations satisfies the property that the economy has for all practical
purposes achieved convergence to the new steady state before T + 2.
The idea is to vary λ until the collateral constraint,
τ N qtN K N + τ T qtT K T − R∗ zt − (1 + r∗ )Bt = 0
is satisfied for t = 0, 1, ..., T + 1. (This equation is satisfied by construction for t ≥ T + 2.) To
do this, we need to compute a mapping from λ to the qti ’s, the zt ’s and the Bt ’s.
39
Equilibrium also requires that the limiting condition, (2.17), be satisfied. We can verify that this is
satisfied ex post, when we have found a set of Lagrange multipliers which produce allocations where the
collateral constraint is satisfied in each period.
42
First, we set up a mapping:
³
´
³
´
T
N
N
T
T
Λt+1 , pTt → cN
t , ct , pt , wt , Rt , Lt , Lt , Λt , pt−1 ,
(A.22)
starting with t = T + 2 and ending with t = 1. We then handle t = 0 separately.
Dates, t ≥ 1 The object, Λt is obtained using (2.22):
Λt = Λt+1 (1 + λt ),
which is an equation that is available for t = 1, ... T + 2. Then, we make use of (2.12) to solve
for pTt−1 :
1
uc,t pTt−1
(A.23)
Λt =
β,
T
pt pt 1 + xt−1
which is available for t = 1, ..., T + 2. To solve this, we require the other variables first. We do
variables
by setting
this using our equilibrium conditions and the given Λt , pTt . We find these
³
´
N
T
up a one-dimensional search for Lt . So, suppose that in addition to Λt , λt , pt , we have LN
t .
³
N
Then, from our assumptions about technology, cN
t = K
T
following two equations can be solved for pN
t and ct :
pN
t
γ
=
1−γ
Ã
´α ³
LN
t
(1 − γ) cTt
γcN
t
´1−α
!1
N
. Given LN
t and ct , the
η
(A.24)
pt pTt ct = 1 + xt ,
where
p =
c =
Ã
(
h
1
1−γ
!1−η
T
(1 − γ) c
Ã
pN
+
γ
i η−1
η
h
1
!1−η 1−η
N
+ γc
i η−1
η
)
(A.25)
η
η−1
.
(A.26)
This can be treated as a one dimensional search problem in cTt alone.
We now have LN , pN , p, cN and cT in hand. The next step is to find z and LT . One equation
43
that is useful for this purpose is the first order condition for z :
Ã
Since
"
we have
µ V
yT
1
= θ
zµ2
µ2 z
yT
µ2 z
# ξ−1
!1
ξ
"
ξ
ξ−1
ξ
Ã
1
µ1 V
=
µ2 z
θ
#
1+λ ∗
µ2 =
R.
1−θ
+ 1 − θ
(1 + λ)R∗
(1 − θ) µ2
³
, V = A KT
!ξ−1
´ν ³
LT
ξ
! ξ−1
Ã
1−θ
−
θ
´1−ν
,
(A.27)
.
We require
ξ
µ2 − (1 − θ) 1−ξ R∗
ξ
> λ,
(1 − θ) 1−ξ R∗
which guarantees that, as long as 0 ≤ ξ ≤ 1, the object in braces is positive. This is necessary,
for z and V to be positive. This is a restriction we place on λ.
Equation (A.27) involves two unknowns, z and LT . We need another equation to pin these
two variables down. Before obtaining these, it is useful to work on the expression for the
marginal product of labor in the traded good sector:
V
MPLT
=
Ã
yT
µ1 V
!1
ξ
θ(1 − ν)µ1
Vt
,
LTt
or,
1
Ã
1−θ
V MPLT = −
θ
θ
!Ã
(1 + λ)R∗
(1 − θ) µ2
1
!1−ξ 1−ξ
³
θ(1 − ν)µ1 A K T
´ν ³
´−ν
³
´ν ³
LT
(A.28)
.
Equating the V MPL ’s in each sector:
Ã
1
1−θ
yN
(1 − α)pN N = −
L
θ
θ
!Ã
(1 + λ)R∗
(1 − θ) µ2
44
1
!1−ξ 1−ξ
θ(1 − ν)µ1 A K T
LT
´−ν
.
This can be solved for LT :
³
LT = LN
³
·
1
θ
´α
ν
Then, setting V = A K T
´ν ³
z=
−
LT
³
1−θ
θ
´1−ν
´³
´
(1+λ)R∗ 1−ξ
(1−θ)µ2
¸
³
1
ν
´ν
KT
.
(A.29)
, we obtain z from (A.27):
µ1 V
µ2 θ (1 − θ) µ2
uc,t
θ(1 − ν)µ1 A
(1 − α)pN (K N )α
"
#
1 (1 + λ)R∗ ξ−1
Compute
1
1−ξ
−
Ã
−ξ
! ξ−1
1−θ
θ
Ã
´1+ψ
ψ0 ³ T
= ct −
Lt + LN
t
1+ψ
!−σ
.
.
(A.30)
(A.31)
Now, it is possible to solve for pTt−1 using (A.23). But, we are not done yet, because we started
with a guess for LN
t .
From the labor supply equation, (2.5),
³
w = pψ0 LT + LN
´ψ
.
(A.32)
From (2.32) we obtain Rt :
βRt = (1 + xt−1 ) (1 + λt ) (pTt /pTt−1 ),
(A.33)
t = 1, 2, ... . Evaluating the product of R obtained from here and w obtained from (A.32), we
can evaluate (2.24):
N
T
N yt
(A.34)
;
Λ
)
≡
(1
−
− wt Rt .
f (LN
,
λ
,
p
α)p
t
t
t
t
t
LN
t
N
T
The idea is to adjust LN
t until f (Lt ; Λt , λt , pt ) = 0.
Date t = 0 We now have in hand,
³
´
T
N
N
T
T
cN
t , ct , pt , wt , Rt , Lt , Lt , Λt , pt−1 , for t = 1, ..., T + 2.
45
³
´
T
N
N
T
Next, we seek cN
for t = 0. Note that we do not have Λ0 , since (??) is
t , ct , pt , wt , Rt , Lt , Lt
not available for t = 0. This means that we cannot find LN
0 by setting f = 0, in (A.34) as we
do for t = 1, ..., T + 1. We replace equation (A.33) by
βR0 uc,1
uc,0
=
.
T
p0 p0
p1 pT1 (1 + x0 )
(A.35)
N
N
N
T
Then, we solve for LN
0 as follows. Fix L0 . Solve for p0 , p0 , c0 , c0 , c0 using the iterative
T
algorithm described around (A.24). Then, compute L0 using (A.29), z0 using (A.30), uc,0 using
(A.31), and w0 using (A.32). Solve for w0 R0 using
w0 R0 = (1 −
α)pN
0
Ã
KN
LN
0
!α
,
and compute R0 from w0 R0 /w0 . Finally, evaluate
g(LN
0 ) =
uc,0
uc,1 βR0
−
.
T
p0 p0
p1 pT1 1 + x0
N
T
Adjust LN
0 until g(L0 ) = 0. Another way to write the g function substitutes based on pp c =
1 + x,
uc,1 c1 βR0
uc,0 c0
−
g(LN
,
0 ) =
1 + x0 1 + x1 1 + x0
or,
g(LN
0 ) = uc,0 c0 − uc,1 c1
βR0
,
1 + x1
The next step is to evaluate the qti ’s and the Bt ’s. The qti ’s can be solved recursively from
(2.20). The Bt ’s can be obtained by simulating (2.30) forward, for the fixed B0 .
In practice, we found that the following parameterization of the λ’s works well. We let λt
for t = 0, 1, ..., N − 1, N < T be free parameters and we set λt for N < t < T + 2 by linear
interpolation and imposing λT +2 = 0. We chose the free λ’s to enforce exactly the collateral
constraints in periods t = 0, 1, 2, ... N − 1 and T. The adequacy of this computational strategy
can be evaluated ex post by evaluating the collateral constraints for dates with t 6= T and
t∈
/ {0, 1, ..., N }. We found that this procedure works well for T = 10 and N = 6.
Surprise Scenario We now suppose that d0 is set according to the equilibrium in the previous
subsection. In reflection of this, we drop the household’s dynamic first order condition, (A.35),
from consideration in period 0. The computational strategy for finding the equilibrium in this
scenario is essentially identical to what we described in the previous subsection, apart from the
obvious changes that must be made to for handling period 0.
46
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49
Table 1: Parameter Values, Two Models Used in Analysis
Labor in Traded Good Sector No Labor In Traded Good Sector
β
0.94
γ
0.27
β
0.94
γ
0.33
ψ
3.00
R
1.12
ψ
1
R
1.12
R∗
1.06
r∗
0.06
R∗
1.06
r∗
0.06
α
0.36
KN
2.00
α
0.36
KN
11.25
ν
0.50
KT
1.00
ν
1
KT
5.
µ1
1.4
µ2
3.50
µ1
1
µ2
1
τ
0.185
θ
NA
τ
0.11
θ
0.6
ψ0
4.59
σ
1
ψ0
0.0036
σ
2
A
0.87
ξ
0.
A
3.2
ξ
1
Table 2: Steady State for Two Models, Ignoring Collateral Constraint
Labor in Traded Good Sector
No Labor In Traded Good Sector
L
0.54 z
0.13 L
26.
z
6.8
LT
0.13 LN
0.40 LT
NA
LN
26
cT
0.27 cN
0.72 cT
9.69
cN
19.4
w
1.05 V
0.32 w
0.44
V
NA
pN cN
cT
2.79 y T
0.44
2.12
yT
18.1
pN
1.04 pT
1.05 pN
1.06
pT
0.035
qT
2.73 q N
2.37 q T
38.46 qN
11.63
B
0.70
0.66 B
20
B
pN cN +yT −R∗ z
pN cN
cT
2
B
pN cN +y T −R∗ z
0.66
Table 3: Steady State for Two Models, Respecting Collateral Constraint
Labor in Traded Good Sector
No Labor In Traded Good Sector
L
0.54 z
0.12 L
28.
z
6.85
LT
0.13 LN
0.41 LT
NA
LN
28
cT
0.27 cN
0.72 cT
10.01 cN
20.05
w
1.05 V
0.32 w
0.44
V
NA
pN cN
cT
2.85 y T
0.44
2.69
yT
18.1
pN
1.06 pT
1.03 pN
1.34
pT
0.029
qT
2.68 qN
2.43 qT
38.46 q N
15.21
B
0.59
0.55 B
14.62
B
pN cN +y T −R∗ z
pN cN
cT
3
B
pN cN +y T −R∗ z
0.40
Table 4: Contemporaneous Effect of Cut in R at Date 0 (Relative to constant x path)
Change in:
Labor in Traded Good Sector No Labor In Traded Good Sector
R
-0.09
-0.09
λ
-0.014
0.09
L
1.75
-0.43
z
1.12
-2.04
LT
2.24
NA
LN
1.53
-0.43
cT , cN
0.98
-0.27
W /P T
9.25
-4.53
W /P
12.85
4.13
Current Account
2.78
-3.89
pT
3.50
4.57
Real Exchange Rate
1.00
1.04
pN
0.31
-8.16
qT
0.62
0.10
qN
0.61
-0.40
V MPkT
2.00
-0.82
V MPkN
1.29
-8.41
P
3.70
0.26
Percent Change in:
4
Table 5: Welfare Analysis
Equilibrium
Present Discounted Value of Utility
Labor in
No Labor in
Traded Good Sector Traded Good Sector
Pre-crisis steady state
-30.64399
-3.39104
Collateral-constrained steady state -30.51503
-3.33955
Baseline transition path, t ≥ 0
-30.68844
-3.94647
Path, t ≥ 0, with interest rate cut
-30.68228
-3.95733
5
Figure 1: The Effect of an Interest Cut with η = 0.9 and No Debt
29
LM Curve (R high)
28.5
28
AM Curve
LM Curve (R low)
27.5
pn
27
26.5
26
25.5
25
24.5
24
0.013
0.0135
0.014
0.0145
L
0.015
0.0155
0.016
Figure 2: The Effect of an Interest Cut with η = 0 and No Debt
1.9
1.8
1.7
1.6
LM Curve (R low)
pn
1.5
1.4
1.3
1.2
LM Curve (R high)
1.1
AM Curve
1
0.65
0.7
0.75
L
0.8
Figure 3: The Effect of an Interest Cut with η = 0 and Modest Debt
0.36
LM Curve (R high)
AM Curve
0.35
LM Curve (R low)
0.34
pn
0.33
0.32
0.31
0.3
0.29
0.28
0.31
0.32
0.33
0.34
0.35
0.36
L
0.37
0.38
0.39
0.4
Figure 4:
Response of Debt and Output to Collateral Shock In
the Absence of Monetary Policy Response
Bt
Level of International Debt
in Old Steady State
High Shadow Value of
Debt Induces Firms to
Pay it Down
Financial
Crisis
Level of Debt in
New Steady State
t
Output
Output During
the Transition
t
Figure 5: Model Timing
Collateral Constraint
Imposed Unexpectedly
Monetary
Action
0
1
Household
Deposit Decision
Production,
Consumption Occur
2
t
Figure 6a: Transition to Lower Debt In Aftermath of Crisis, No Policy Response, Expansion Scenario Model
Debt (B )
Multiplier (λ)
t
0
0.6
-5
% dev from ss
0.8
0.4
0.2
0
0
5
10
15
-10
-15
-20
20
0
5
Current Account
% dev from ss
% of ss output
0.02
0.01
0
0
5
10
15
15
20
15
20
4
2
0
-2
20
0
5
N
L /L
5
0.5
0
0.45
-5
0.4
-10
0.35
0
10
T
Labor (L)
% dev from ss
20
6
0.03
-15
15
Imports (z)
0.04
-0.01
10
5
10
15
20
Notes: % dev from ss - Percent Deviation from Pre-Crisis Steady State
% of ss output - Percent of Pre-Crisis Gross Output
0.3
0
5
10
Figure 6b: Transition to Lower Debt In Aftermath of Crisis, No Policy Response, Expansion Scenario Model, (Cont'd)
Price of Capital in Nontraded Sector (qN)
0
5
-5
0
% dev from ss
% dev from ss
Price of Capital in Traded Sector (qT)
-10
-15
-20
-25
0
5
10
15
-5
-10
-15
-20
20
0
5
Price of Nontraded Good (pN)
20
60
0
% dev from ss
% dev from ss
15
Price of Traded Good (PT)
10
-10
-20
-30
-40
10
0
5
10
15
20
40
20
0
-20
0
5
10
15
20
15
20
Inflation Rate (P /P )
Domestic Interest Rate (R)
t
1.14
t-1
1.3
1.2
1.12
1.1
1
1.1
0
5
10
15
20
Note: % dev from ss - Percent Deviation from Pre-Crisis Steady State
0.9
0
5
10
Figure 7a: Transition to Lower Debt In Aftermath of Crisis, No Policy Response, Contraction Scenario Model
Debt (B )
Multiplier (λ)
t
8
0
% dev from ss
6
4
2
0
0
5
10
15
-10
-20
-30
20
0
5
Current Account
% dev from ss
% of ss output
15
20
-20
0.02
0.01
0
5
10
15
20
-40
-60
-80
-100
0
5
10
T
Labor (L)
Consumption of Traded Good (c )
20
% dev from ss
50
% dev from ss
20
0
0.03
0
-50
-100
15
Imports (z)
0.04
0
10
0
5
10
15
20
Notes: % dev from ss - Percent Deviation from Pre-Crisis Steady State
% of ss output - Percent of Pre-Crisis Gross Output
0
-20
-40
-60
-80
0
5
10
15
20
Figure 7b: Transition to Lower Debt In Aftermath of Crisis, No Policy Response, Contraction Scenario Model, (Cont'd)
Price of Capital in Nontraded Sector (qN)
20
40
0
20
% dev from ss
% dev from ss
Price of Capital in Traded Sector (qT)
-20
-40
-60
0
5
10
15
0
-20
-40
20
0
5
Price of Nontraded Good (pN)
20
600
20
% dev from ss
% dev from ss
15
Price of Traded Good (PT)
40
0
-20
-40
-60
10
0
5
10
15
20
400
200
0
-200
0
5
10
15
20
15
20
Inflation Rate (P /P )
Domestic Interest Rate (R)
t
2
t-1
4
1.8
3
1.6
2
1.4
1
1.2
1
0
5
10
15
20
Note: % dev from ss - Percent Deviation from Pre-Crisis Steady State
0
0
5
10
Figure 8a: Effect on Transition of Policy Cut in Interest Rate, Expansion Scenario Model
Debt (B )
Multiplier (λ)
t
% dev from baseline
0.1
-0.005
-0.01
-0.015
0
5
-4
10
15
-0.2
0
20
5
10
15
20
15
20
15
20
0
-0.5
0
5
-4
dev from baseline
4
0
15
0.5
2
0.5
10
1
Labor (L)
1
5
Imports (z)
0
1.5
0
1.5
5
0
-0.1
Current Account
x 10
10
-5
0
-0.3
20
% dev from baseline
dev from baseline
15
% dev from baseline
dev from baseline
0
x 10
10
T
N
L /L
2
0
-2
-4
5
10
15
20
0
5
Notes: % dev from baseline - Percent Deviation from Baseline Path of Constant Money Growth
% dev from baseline - Deviation from Baseline Path (Percentage Points)
10
Figure 8b: Effect on Transition of Policy Cut in Interest Rate, Expansion Scenario Model (Cont'd)
Price of Capital in Traded Sector (qT)
Price of Capital in Nontraded Sector (qN)
0.8
% dev from baseline
% dev from baseline
0.8
0.6
0.4
0.2
0
-0.2
0
5
10
15
0.6
0.4
0.2
0
-0.2
20
0
5
Price of Nontraded Good (pN)
% dev from baseline
% dev from baseline
0.4
0.2
0
5
10
15
4.5
4
3.5
3
20
0
5
10
15
20
15
20
Inflation Rate (P /P )
Domestic Interest Rate (R)
t
t-1
0.06
dev from baseline
0.1
dev from baseline
20
5
0.6
0.05
0
-0.05
-0.1
15
Price of Traded Good (PT)
0.8
0
10
0
5
10
15
20
0.04
0.02
0
-0.02
0
5
Notes: % dev from baseline - Percent Deviation from Baseline Path of Constant Money Growth
% dev from baseline - Deviation from Baseline Path (Percentage Points)
10
Figure 9a: Effect on Transition of Policy Cut in Interest Rate, Contraction Scenario Model
Debt (B )
Multiplier (λ)
t
0.3
% dev from baseline
dev from baseline
0.2
0.15
0.1
0.05
0
0
5
-4
15
0
-0.1
20
5
10
15
15
20
15
20
-2
-4
-6
20
0
5
10
T
Consumption of Traded Good (c )
0
% dev from baseline
0
% dev from baseline
10
Imports (z)
Labor (L)
-1
-2
-3
-4
5
0
-5
0
0
Current Account
x 10
0
-10
0.1
% dev from baseline
dev from baseline
5
10
0.2
0
5
10
15
20
-0.5
-1
-1.5
-2
-2.5
0
5
Notes: % dev from baseline - Percent Deviation from Baseline Path of Constant Money Growth
% dev from baseline - Deviation from Baseline Path (Percentage Points)
10
15
20
Figure 9b: Effect on Transition of Policy Cut in Interest Rate, Expansion Scenario Model (Cont'd)
Price of Capital in Traded Sector (qT)
Price of Capital in Nontraded Sector (qN)
0.2
% dev from baseline
% dev from baseline
0.2
0
-0.2
-0.4
-0.6
-0.8
0
5
10
15
0
-0.2
-0.4
-0.6
-0.8
20
0
5
Price of Nontraded Good (pN)
% dev from baseline
% dev from baseline
-5
0
5
10
15
4
2
0
-2
20
0
5
10
15
20
15
20
Inflation Rate (P /P )
Domestic Interest Rate (R)
t
t-1
0.015
dev from baseline
0.1
dev from baseline
20
6
0
0.05
0
-0.05
-0.1
15
Price of Traded Good (PT)
5
-10
10
0
5
10
15
20
0.01
0.005
0
-0.005
-0.01
0
5
Notes: % dev from baseline - Percent Deviation from Baseline Path of Constant Money Growth
% dev from baseline - Deviation from Baseline Path (Percentage Points)
10
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