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COLLATERAL CONSTRAINTS AND MACROECONOMIC ADJUSTMENT IN AN OPEN ECONOMY

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This paper analyzes a small open economy Ramsey growth model with convex investment costs and a collateral constraint on borrowing. Optimal control methods are used to characterize the dynamics of investment, consumption, and debt. The analysis demonstrates that the economy's adjustment speed depends on the fraction of the capital stock that can be used as collateral. In the presence of non-convexities, a higher loan-to-value of the capital stock may produce a bifurcation in the dynamics by increasing the economy's adjustment speed. In contrast to the canonical small open economy model with convex investment costs, domestic and foreign savings are growth-rate complements due to the interaction between domestic savings, the price of capital, and the borrowing constraint. The standard closed economy Ramsey model, the Cohen-Sachs debt repudiation model, and the canonical small open economy model with adjustment costs are shown to be special cases of the analysis.
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January 2009








COLLATERAL CONSTRAINTS AND MACROECONOMIC ADJUSTMENT
IN AN OPEN ECONOMY






Philip L. Brock

Department of Economics
University of Washington
Seattle, Washington 98195
plbrock@u.washington.edu






Abstract

This paper analyzes a small open economy Ramsey growth model with convex investment
costs and a collateral constraint on borrowing. Optimal control methods are used to
characterize the dynamics of investment, consumption, and debt. The analysis demonstrates
that the economy’s adjustment speed depends on the fraction of the capital stock that can be
used as collateral. In the presence of non-convexities, a higher loan-to-value of the capital
stock may produce a bifurcation in the dynamics by increasing the economy’s adjustment
speed. In contrast to the canonical small open economy model with convex investment costs,
domestic and foreign savings are growth-rate complements due to the interaction between
domestic savings, the price of capital, and the borrowing constraint. The standard closed
economy Ramsey model, the Cohen-Sachs debt repudiation model, and the canonical small
open economy model with adjustment costs are shown to be special cases of the analysis.


1
I. Introduction

During the late 1970s and early 1980s economists began a concerted effort to develop
optimizing intertemporal models of open economies1. As part of this push various attempts
were made to open the Ramsey growth model to capital flows. Simply opening the standard
Ramsey model produces instantaneous adjustment of the capital stock, since domestic saving is
no longer a constraint on capital formation. In order to slow down the adjustment of the capital
stock two principal approaches were taken. The first opened the Uzawa (1961, 1964) two-
sector growth model and made the investment goods sector nontradable.2 The second
approach incorporated convex investment costs into the basic Ramsey model so that the cost of
investment increased with the rate of investment. Both approaches slowed the speed of capital
stock adjustment and created a relative price of investment goods to consumption goods that
could be interpreted as a real exchange rate.
The adoption of the second approach was initially slowed by the difficulty of
integrating the microeconomics literature on firm adjustment costs, such as Lucas and Prescott
(1973), into a Ramsey model with a representative price-taking agent. Hayashi (1982) showed
that linearly homogenous convex adjustment costs were required for the specification of a
Ramsey model with price-taking firms. 3 Hayashi’s paper proved to be instrumental in the
development of the open economy Ramsey model. In short order papers by Blanchard (1983),
Lipton and Sachs (1983), and Giavazzi and Wyplosz (1984) incorporated linearly

1 See, e.g., Sachs 1981 and Svensson and Razin 1983 for early models.
2 Papers laying the groundwork for this approach include Fischer and Frenkel (1972) and Bruno (1976). See
Murphy (1986), Brock (1988), and Obstfeld (1989) for early examples of explicitly intertemporal optimizing open
economy models with nontraded investment
3Without a linearly homogeneous installation cost function, firms are price makers that generate monopoly rents.
Hayashi (1982) characterized the problem as one of determining the assumptions necessary to make the marginal
“q” valuation of the capital stock the same as the Tobin’s “q” average value of the capital stock.


2
homogeneous investment costs into open economy versions of the Ramsey model, and many
other papers soon followed.4
By the 1990s the open economy Ramsey model with convex investment costs had been
incorporated into graduate textbooks and became one of the canonical models of international
macroeconomics.5 Nevertheless, there has remained the question of how to close this model
with a borrowing constraint. Any infinite-horizon Ramsey model must impose a no-Ponzi-
game condition on the representative agent, which in the presence of complete markets is
equivalent to a solvency condition requiring that the value of debt cannot exceed the present
value of the economy’s net capital income plus wage income.6 The solvency condition
generates an upper bound on debt and a lower bound of zero on consumption. For standard
utility functions that imply positive consumption (e.g., ones that satisfy the Inada conditions),
assumptions in addition to the solvency condition must be incorporated to solve for the
model’s steady state equilibrium.
As is well known, if both the rate of time preference ( ρ ) and the world interest rate (r)
are taken as constants, then the existence of a steady state for the small open economy model
requires, in addition to the solvency condition, that ρ = r . This in turn gives rise to a zero root
in the dynamics, so that the steady state depends on initial debt and consumption equals

4 See, e.g., Matsuyama (1987), Brock (1988), and Turnovsky and Sen (1991).
5 See, e.g., Blanchard and Fischer (1989), Obstfeld and Rogoff (1995), Turnovsky (1997), and Barro and Saleh-i-
Martin (2004). Convex investment costs also have an increasingly important role in dynamic stochastic general
equilibrium models such as Christiano, Eichenbaum and Evans (2005) and Adolfson et. al (2007).
6 See Levine and Zame (1996). This condition can be expressed algebraically as

− ( − )
b q k +
(
w k ) r s t
e
ds


t
t
t
s
t
where b is the stock of debt, qk is the value of the capital stock , (
w k) is the real wage, and r is the world interest
rate. With convex, linearly homogenous investment costs, the value of the capital stock equals the present
discounted value of future rentals on capital net of adjustment costs.


3
permanent income.7 For many purposes modelers may wish to have steady states that are
invariant to initial levels of debt. One alternative way to close the model is to pin down long-
run consumption ( c ) by assuming that the rate of time preference is increasing in consumption
(e.g. Obstfeld 1981). A second alternative is to specify the borrowing rate as an increasing
function of the amount borrowed where the steady-state level of debt ( b ) is assumed to satisfy
the solvency constraint (see, e.g., Schmidt-Grohé and Uribe 2003). The assumption I make in
this paper is that debt must be collateralized by a fraction a of the value of the capital stock.
Current and future labor income cannot be pledged against repayment of a loan.8
Evans and Jovanovic (1989) were among the first to use a collateral requirement on
capital in an optimizing model.9 Cohen and Sachs (1986) were the first to introduce a
collateral constraint in an open economy Ramsey model with convex investment costs, but
their model proved to be analytically tractable only for linear production functions. Mendoza
(2008) has recently used a collateral constraint, but focuses on whether or not the constraint is

7 Three papers in the early 1980s that developed techniques which permitted the analytical characterization of
open economy models with zero roots are Blanchard and Kahn (1980), Buiter (1984), and Giavazzi and Wyplosz
(1985).
8 The four alternative closure assumptions generate the following alternative steady-state borrowing constraints,
where ρ is the rate of time preference, r is the world interest rate, and b , k , and c are steady-state values of
debt, capital, and consumption:
(
w k ) − c (b )
0
.
i ρ = r

b = k +
r
(
w k ) − c (r)
.
ii ρ (c) = r

b = k +
r

(
w k ) − c
ii .
i ρ = r(b%)

b (ρ ) = k +
r
.
iv b aqk

b ak
0 ≤ a ≤ 1
Given the steady-state capital stock ( k ), the first closure assumption determines the steady-state level of debt as a
function of the initial stock of debt ( b ). The second assumption determines steady-state debt by pinning down
0
steady-state consumption via the exogenous world interest rate, while the third assumption pins down steady-state
debt by equating the exogenous rate of time preference with the endogenous debt-determined interest rate.
If a = 1 and the collateral constraint binds, consumption will equal the real wage in equilibrium.
9 The paper by Aiyagari and Gertler (1999) is also an early closed-economy model that employs a collateral
constraint of the type used in this paper.


4
binding rather than on the effect of fractional changes in the collateral requirement when the
constraint is binding. Barro, Mankiw and Sala-i-Martin (1995) and Lane (2001) modify the
Cohen-Sachs model by replacing convex investment costs with an additional factor of
production that requires nontraded investment. Caballero and Krishnamurthy (2001),
Devereux and Poon (2004), Chari, Kehoe, and McGrattan (2005), and Braggion, Christiano,
and Roldós (2007) also employ collateral constraints, but like Mendoza (2008) restrict their
analyses to the change in model dynamics when there is a switch from a binding to a non-
binding constraint.
Several papers motivate the collateral constraint by the ability of foreign debtors to
seize the capital stock in the case of default (see, e.g., Barro et. al. 1995, Lane 2001, and Chari
et. al. 2005). In this paper, like that of Cohen and Sachs, international lenders cannot seize the
capital stock due to adjustment costs that fix capital in the short run, so the borrowing
constraint refers to the amount of present and future capital income that can be pledged against
the debt. This limit on pledgeable income could reflect the ability of foreign creditors to
impose a penalty equal to a fraction of the economy’s output in the case of default, as in Cohen
and Sachs, or it could reflect the inability of agents to pledge wage income because of the
inalienability of human capital as in Kiyotaki and Moore (1997).10 Following Matsuyama
(2008) and Mendoza (2008), I assume that the collateral requirement captures the outcome of
unmodeled agency problems or legal restrictions that prevent the representative agent from
pledging more than a fraction of capital income when incurring debt.



10 There are, of course, other reasons related to adverse selection and moral hazard. Similarly, Holmstrom and
Tirole (1998) posit a private benefit that cannot be pledge by an entrepreneur to creditors.


5

2. The Model

The open economy model with convex investment costs is well-known and is exposited by a
number of graduate-level textbooks, such as Blanchard and Fischer (1989, 58-69) and
Turnovsky (1997, 57-77). The canonical form of the model is the following, where f (k) is a
neoclassical production function, ψ (i, k) represents convex investment costs, b is debt, r is the
world interest rate, ρ is the rate of time preference, and u(c) is a concave utility function:


−ρ
max u(c)
t
e
dt
subject to k , b and

0
0
c,i
0
b& = c +ψ (i , k ) + rb f (k )
(1)
t
t
t
t
t
t
k& = i − δ k
t
t
t

With the rate of time preference set equal to the interest rate, the model is separable in
production and consumption decisions. With a linearly homogenous investment costs, the
value of the capital stock (qk) is equal to the sum of the discounted value of rental income from
capital net of investment costs:




q k =
f k k −ψ i k
e
ds

t
t
[
s
s
s
s ]
(
)
( )
( ,
)
r s t
(2)
t

Consumption is constant and is equal to permanent income:








c = −rb + r
f (k) −ψ (i, k)
r s t
e
ds = r(q k b ) + r
(
w k ) r s t
e
d


s
0
[
] ( )
(
)
(3)
o 0
0
t
0
0




Consumption is constant because the agent can use the world capital market to create an
annuity from current debt and future income.


6

In this paper the agent has more limited access to the world capital market. This
limited access takes the form of a borrowing constraint:
b aq k (4)
t
t
t
The agent can pledge only a fraction a of the capital stock as collateral and no part of wage
income. The capital stock cannot literally serve as collateral since adjustment costs prevent its
direct pledging to creditors. Since the value of the capital stock equals the present value of
capital income net of investment costs, the better interpretation of the borrowing constraint is
that at most a fraction a of present and future net capital income can be pledged to lenders.
This section looks at the effect of the constraint when all net capital income can be
pledged ( a = 1 ). Sections 5 and 6 will generalize the model to allow a to take on fractional
values. The assumption a = 1 is the assumption made by Kiyotaki and Moore (1997). It
signifies that capital income can be pledged to lenders, but that borrowers cannot pledge their
wage income. The borrowing constraint b q k is then added to the maximization problem
t
t
t
given by (1). The present value Hamiltonian associated with the maximization problem) is:
ρt
e H = u(c) + λ [ f (k) − rb c −ψ (i, k)] + q *(i − δ k) + γ *(qk b) (5)
where −λ is the shadow value of debt, q * is the shadow value of installed capital, and γ * is
the shadow value attached to the borrowing constraint. The first-order necessary conditions
are:
u (
c) = λ (6)
ψ (i,k) = q (7)
i
where q = q * λ is the market value of capital. The Euler equation for consumption is:


7
c
c& =
(r + γ − ρ)
σ

(8)
where σ = − cu (
′′ c) u (′c) and γ = γ * λ .
The evolution of the price of capital is given by
q& = − f (′k) +ψ (i, k) + (r + δ )q (9)
k
The state equation for capital is derived from the first order condition ψ (i, k ) = q (equation 7):
i
k& = i(q, k) − δ k (10)
These latter two equations are identical to the canonical model. The consumption equation
differs from the canonical open economy Ramsey model by the term γ in the consumption
Euler equation. Thus, when a = 1 the borrowing constraint of this model alters the Euler
equation while leaving the dynamics of capital stock adjustment unchanged from the canonical
model. Capital accumulation with the borrowing constraint a = 1 is governed by the same
speed of adjustment as the canonical model:
2
µ
r
r
f ′ (
k)
1
k = k + (k k )
t
e where µ
=

t
0
m
(11)
1,2
⎜ ⎟
2
⎝ 2 ⎠
ψii
are the characteristic roots of equations (9) and (10). The adjustment dynamics are the same
for the canonical and borrowing-constrained economy because with a = 1 borrowing for
investment is completely collateralized by future capital income.
With a binding constraint (b=qk) the current account equation (1) implies that
consumption in the borrowing-constrained economy is equal to the real wage:
b& = c +ψ (i, k) + rqk f (k ) = q&k + qk&

(12)
c = f (k) − f '(k)k =
(
w k )


8
The borrowing constraint will bind in equilibrium if γ = ρ − r > 0 where γ is the
shadow value of the borrowing constraint in steady-state equilibrium. As distinct from the
canonical model, where ρ = r is required for a stationary steady state, the borrowing
constrained model will have a stationary steady state provided that r ≤ ρ . In essence, when the
autarchic interest rate ( ρ ) exceeds the world interest rate (r), imposing a borrowing constraint
is an alternative assumption to imposing equality between the world interest rate and the rate of
time preference in the steady state, including the assumptions that the interest rate is an
increasing function of the stock of debt or that the rate of time preference is a decreasing
function of consumption.
In summary, with a binding collateral constraint b = q k , the capital stock is financed
t
t
t
by borrowing from foreigners. Agents cannot borrow against future wage income so that
consumption is equal to the real wage rather than to a constant value as in the canonical model.

3. Nonbinding borrowing constraint

The borrowing constraint may be non-binding ( b < q k ) under several conditions. First, if the
t
t
t
initial capital stock ( k ) is sufficiently large, the agent can borrow against it to perfectly
0
smooth consumption, assuming that ρ = r . Second, if the initial capital stock is greater than
the long-run capital stock, labor income will decline over time. In this case the agent will
accumulate foreign assets to smooth consumption so that the borrowing constraint will not
bind. Once again, a stationary steady state will exist provided that ρ = r . In these two cases,
consumption will equal permanent income and the model’s dynamics will be the same as those
of the canonical model.


9
In addition, there is a third possibility. The initial capital stock may not be large
enough to allow the agent to borrow against it to perfectly smooth consumption, but it may be
large enough to allow the agent to temporarily smooth consumption by borrowing against it.
In this case, the dynamics are described by a temporary period in which consumption exceeds
wage income before the borrowing constraint becomes binding and consumption equals wage
income.11
The canonical model is described by four dynamic equations in k, q, b, and c.12 The
first two eigenvalues, which correspond to capital and the price of capital, are given by
equation (11). Debt accumulation and consumption dynamics are governed by the third and
fourth eigenvalues,
µ = 0, µ = r
3
4
During the temporary period in which the borrowing constraint is not binding, the
trajectory of consumption will be governed by the third and fourth eigenvalues as follows:
µ
µ
3t
4t
c = c + A e
+ A e
t
1
2

rt
= c + A + A e
1
2
where A and A are constants that are determined by the model. If the borrowing constraint
1
2
is never binding, adjustment is saddlepath so that
µ3t
c = c + A e

t
1

11 See Turnovsky (1997), pp. 94-98 for a discussion and several mathematical examples of this kind of temporary
dynamics.
12 The linearized system (with ρ = r ) can be written as:
q&⎞ ⎛ r
f (
′′ k) 0 0⎞⎛ q q
⎜ ⎟ ⎜
⎟⎜

k&
1 ψ
0
0
0
k
⎜ ⎟
k
ii

⎟⎜

=
⎜ ⎟ ⎜

&
− ′
⎟⎜

b
1 ψ
f (k)
r
1

ii
b b
⎜ ⎟ ⎜
⎟⎜

⎜ ⎟


c& ⎠ ⎝ 0
0
0
0 ⎠⎝c c


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