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
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) twosector growth model and made the investment goods twosector 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 pricetaking agent. Hayashi (1982) showed that linearly homogenous convex adjustment costs were required for the specification of a Ramsey model with pricetaking 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
3
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 infinitehorizon Ramsey model must impose a noPonzigame 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 SalehiMartin (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 ( )
( ) r s t
t t t s
t
b q k w k e ds
∞
− −
≤ +
_{∫}
where b is the stock of debt, qkis the value of the capital stock , 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.
( )
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 longrun 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 steadystate level of debt (b) is assumed to satisfy
the solvency constraint (see, e.g., SchmidtGrohé 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 steadystate borrowing constraints, where ρ is the rate of time preference, r is the world interest rate, and b k, , andc are steadystate values of debt, capital, and consumption:
0
( ) ( )
.
( ) ( )
. ( )
( )
. ( ) ( )
. 0
w k c b
i r b k
r w k c r
ii c r b k
r
w k c
iii r b b k
r
iv b aqk b ak a
ρ
ρ
ρ ρ
−
= ⇒ = +
−
= ⇒ = +
−
= ⇒ = +
≤ ⇒ ≤ ≤
%
1 ≤
Given the steadystate capital stock (k ), the first closure assumption determines the steadystate level of debt as a function of the initial stock of debt ( ). The second assumption determines steadystate debt by pinning down steadystate consumption via the exogenous world interest rate, while the third assumption pins down steadystate debt by equating the exogenous rate of time preference with the endogenous debtdetermined interest rate. If and the collateral constraint binds, consumption will equal the real wage in equilibrium.
0
b
1
a=
9
binding rather than on the effect of fractional changes in the collateral requirementwhen the constraint is binding. Barro, Mankiw and SalaiMartin (1995) and Lane (2001) modify the CohenSachs 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 nonbinding 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
2. The Model
The open economy model with convex investment costs is wellknown and is exposited by a number of graduatelevel textbooks, such as Blanchard and Fischer (1989, 5869) and
Turnovsky (1997, 5777). 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:
0 0 ,
0
max ( ) ,
( , ) ( )
t c i
t t t t t t
t t t
u c e dt subject to k b and
b c i k rb f k
k i k
ρ
ψ δ
∞ −
= + + −
= −
∫
& &
(1)
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:
(2)
[
]
( )( ) ( , ) r s t
t t s s s s
t
q k f k k ψ i k e ds
∞
− −
′
=
_{∫}
−Consumption is constant and is equal to permanent income:
(3)
[
]
( ) ( )0 0 0
0 0
( ) ( , ) r s t ( _{o} ) ( )_{t} r s t c rb r f k ψ i k e ds r q k b r w k e d
∞ ∞
− − − −
= − +
∫
− = − +∫
sIn this paper the agent has more limited access to the world capital market. This limited access takes the form of a borrowing constraint:
t t
b ≤aq k_{t}
t
(4) 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 ( ). Sections 5 and 6 will generalize the model to allow a to take on fractional values. The assumption 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
1
a=
1
a=
t t
b ≤q k is then added to the maximization problem given by (1). The present value Hamiltonian associated with the maximization problem) is:
[
]
( ) ( ) ( , ) * ( ) * ( )
t
eρH =u c +λ f k −rb c− −ψ i k +q i−δk +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 firstorder necessary conditions are:
( )
u c′ =λ (6)
( , )
i i k q
ψ = (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
( ) _{k}( , ) ( )
q_{&}= −f k′ +ψ i k + +r δ q (9)
The state equation for capital is derived from the first order condition ψ_{i}( , )i k =q(equation 7): ( , )
k&=i q k −δk (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 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
1
a=
1
a= is governed by the same speed of adjustment as the canonical model:
1
0
( ) t
t
k = +k k −k eµ where
2 1,2
( )
2 2 _{ii}
r r f k
µ
ψ
′′ ⎛ ⎞
= _{⎜ ⎟} −
⎝ ⎠
m (11)
are the characteristic roots of equations (9) and (10). The adjustment dynamics are the same for the canonical and borrowingconstrained 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 borrowingconstrained economy is equal to the real wage:
( , ) ( )
( ) '( ) ( )
b c i k rqk f k qk qk
c f k f k k w k
ψ
= + + − = +
⇓
= − =
& _{&} &
The borrowing constraint will bind in equilibrium if γ ρ= − >r 0 where γ is the shadow value of the borrowing constraint in steadystate 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 thatr≤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_{t} =q k_{t} _{t}, the capital stock is financed 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 nonbinding (b_{t} <q k_{t} _{t}) under several conditions. First, if the initial capital stock ( ) is sufficiently large, the agent can borrow against it to perfectly smooth consumption, assuming that
0
k
r
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 en t it.
igenvalues, which correspond to capital and the price of capital, are given by equatio
ough to allow the agent to temporarily smooth consumption by borrowing agains 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 e
n (11). Debt accumulation and consumption dynamics are governed by the third and fourth eigenvalues,
3 0, 4 r
µ = µ =
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:
3 4
1 2
1 2
t t
t
rt
c c A e A e
c A A e
µ µ
= + +
= + +
where and are constants that are determined by the model. If the borrowing constraint is never binding, adjustment is saddlepath so that
1
A A2
3
1
t t
c = +c A eµ
11
See Turnovsky (1997), pp. 9498 for a discussion and several mathematical examples of this kind of temporary dynamics.
12
The linearized system (with ρ =r) can be written as:
( ) 0 0
1 0 0 0
1 ( ) 1
0 0 0 0
ii ii
q r f k q q
k k
f k r b b
b
c c c
ψ ψ
−
⎛ ⎞ ⎛ − ′′ ⎞⎛ ⎞
⎜ ⎟ ⎜ ⎟⎜ _{−} ⎟
⎜ ⎟_{=}⎜ ⎟⎜ ⎟
⎜ ⎟ ⎜ − ′ ⎟⎜ _{−} ⎟
⎜ ⎟ ⎜ _{⎟⎜}⎜ ⎟_{⎟}
⎜ ⎟ _{⎝} _{⎠⎝} _{−}
⎠ ⎝ ⎠
& & & &
which, with ρ =r implies that A_{1}=0. Therefore, during the period of a temporarily nonbinding budget constraint, consumption evolves according to:
2
rt t
c = +c A e
Since the initial steadystate (borrowingconstrained) consumption is given by c =w( _{0}
2 0 ( )0
A = −c w k .
) k , then
This implies that during the temporary period in which b<qk, consumption will exceed l stock:
(12)
perio o nt
t
c
wage income ( )w k_{t} due to the agent’s ability to borrow against the initial capita
0 0 0
( ) ( ( )) rt
t
c =w k + c −w k e
implying that the growth rate of consumption during the d before the borr wing constrai becomes binding is:
0 0 ( )0
c −w k
Throughout the transi ( ) 1
t
rt t
c r
w k c
e− =
+
&
(13)
tion, the growth rate of consumption is continuous and, with ρ=r, is given from equation (8) by:
t
t
c c
γ σ =
&
.
During the period after the constraint becomes binding, consumption equals the real e at which the borrowing constraint becomes binding )
T , then
wage. Letting T denote the tim (c_{T} =w k(
( )
T T T T T
T T T T
c _{=} k w k′ k _{=}_{α} k ( )
c w k k k
& &
)
where the latter equality assumes a neoclassical production function y=kα. Evaluating equations (11) and (12) at time T and equating the growth rate of consumption from equation
T gives three equations (13) with the growth rate of the real wage from equation (14) at time
that determine k_{T}, c_{0}, and Tas functions of the initial capital stock k_{0}:
1
0
( ) T
T
k = +k k −k eµ (15)
[
]
0 ( )0 ( ) ( )0
rT T
c =w k + w k −w k e− (16)
0 1
( ) ( )
( )
T T
T
w k w k k k
r
k w k
αµ − = −
T
(17)
where µ_{1}is the negative eigenvalue f
equation (17) determines , the size of the capital stock when the borrowing constraint becomes binding. Given , equation (15) determines T, the time at which the constra becomes binding. Given and T, equation (16) then determines the initial level of consumption ( ). In particular, it can be shown that
rom equation (11). Given the initial capital stock (k_{0}),
T
k
T
k int
T
k
0
c
0
0, 0, 0
T c
k T > >
= ∂ =
∂ _{>} ∂
0 0 0
k k < k <
∂ ∂ ∂ (18)
The ambiguous effect of an incre
borrowing constraint becomes binding is due to two offsetting factors. A higher initial capital n 15) while the increase in lowers T (also equation 15). Similarly, with an increased and T the wealth effect associated with a larger initial capital stock (via a higher wage) may be offset by
ase in the initial capital stock on the time at which the
stock raises the time T capital stock (equation 17).13 This increase in k_{T}raises T (equatio
0
k k_{T}
13
a transition path of greater duration that tends to lower the initial level of consumption c_{0} (equation 16).
4. Closed Capital Account
f this model was analyzed by Abel and Blanchard (1983), but they id not characterize the shortrun dynamics or adjustment speed of the economy
A closed economy version o
d . This section
does that to provide a comparison to the economy with the capital account opened. In a closed economy the net stock of debt is zero so that
( ) ( , )
f k = +c ψ i k (19)
Making use of the linear homogeneity of the investment function, the growth rate of the capital stock is:
( ) _{k}
f k c k
k k
i
ψ _{δ}
ψ
The present value Hamiltonian is − −
= −
& (20)
( ) ( )
t k
i
f k c k
eρ u c λ ψ δk
ψ
⎡ − − ⎤
= + _{⎢} − _{⎥}
⎣ ⎦
H (21)
By making use of the first order condition ( ( )u c′ =λ ψ_{i} ) and the costate equation
'( )
( )
k
i
f k ψ
λ _{= −}⎡ − _{−} _{ρ δ}_{+} ⎤
⎢ ⎥
&
(22)
the growth rate of consumption can be expressed as (see Appendix 1):
λ _{⎣} ψ _{⎦}
'( ) k ( ) i ii
f k
c&= −ψ_{σ}−_{ψ}ρ δ ψ+ (23)
i
i
c ψ
Linearizing equations (20) and (23) around the steady state (with ψ_{i} =1 andψ_{k} =0) gives
0 ( )
1 k
ρ
⎢ ⎥
⎣ ⎦ _{⎢}_{−} _{⎥}
⎣ ⎦
&
ii
f k c
c σ ψ ⎡ _{′′} ⎛ _{+} ⎞⎤
⎡ ⎤_{=}_{⎢} _{⎜} _{⎟}_{⎥}
⎝ ⎠
⎢ ⎥& (24)
with corresponding eigenvalues given by:
2 1,2
( )
2 2
f k
ρ ρ
µ _{σ}
ii
c ψ
′′ ⎛ ⎞
= ± _{⎜ ⎟} −
⎝ ⎠ (25)
Comparing equations (11) and (25), the adjustm
capital account is closed because of the concave utility function (whose curvature is measured
+
ent speed of the economy is slower when the
byσ c). With an open capital account, the adjustment speed is only limited by the convex investment costs and becomes infinite as adjustment costs become linear (ψ_{ii} =0).
Figure 1 shows the consumption paths that correspond to the closed economy, the borrowingconstrained economy (including the transitionally unconstrained one), and the
canoni in
ially Pledgeable Capital Income
be generalized to include partially pledgeable come:
≤ (
cal open economy with perfect consumption smoothing. Steadystate consumption the borrowingconstrained economy is less than steadystate consumption in the closed economy by the amount of net capital income, which is used to make interest payments on the debt.
5. Part
The analysis of the previous sections can in
0 1
t t t
When a=0 the economy is closed to capital flows and when a=1 capital income is fully pledgeable. In the intermediate cases, the agent can borrow against a fraction of future capital income. This could be the result of limited capability of foreign lenders to punish the agent in the case of default (i.e., the CohenSachs argument), or it could arise from moral hazard and legal restrictions that prevent creditors from seizing more than a proportion of the agent’s collateral, as emphasized by Holmstrom and Tirole (1998).
The optimization problem for the representative agent can be expressed as the following presentvalue Hamiltonian:
[
]
( ) ( , ) ( ) * ( )
t
e Hρ =u c −λ c+ψ i k +rb− f k +q i−δk +γ * (aqk−b) (26) Firstorder necessary conditions are:
'( ) u c =λ
( , ) *
i i k q where q q
ψ = = λ
The costate equations are:
c c r
λ λ σ& = _{&} = − −ρ γ (27)
[
]
'( ) _{k}( , ) (1 ) )
q&= −f k +ψ k i + + + −r δ a γ q (28)
oustime analogue of the assetpricing equation used by Cohen and Sachs to characterize the borro
Equation (28) is the continu
wing constraint with a linear production function
( ( )f k = Ak).14 Cohen and Sachs do not provide a solution for a more general production
model parameters. Noting that along a constrained growth path where _{b}=_{h k}* , q 0 14
Cohen and Sachs write the borrowing constraint as _{b}≤_{h k}* where the paper solves fo as a function of
, r _{h}*
=
& and c c_{&} =κ (where κ is a constant) and normalizing ψ_{k} =0, then *
[(1 )( ) ]
h =aq=aA −a ρ− +r σκ +r
The credit constraint is increasing in total factor productivity A and the pledgeable income ratio a, and decreasing in both the rate of time preference and the interest rate (given the assumption
δ + .
r ρ > ) geneity
. These are the results of Cohen and Sachs (see Table 1 and equation A3.13 of their paper). The endo of in their model reflects both the pledgeable income ratio and the price of capital (q) that corresponds to alternative steadystate growth rates.
function, noting that “once the linear production technology is abandoned, an analytical solution appears out of reach.” In this paper’s continuous time setting, the more general solution can derived for a neoclassical production function ( ) (f k f k′( )≥0, f′′( )k <0).15
In the general case of partially pledgeable capital inc
ome (b=aqk), the adjustment speed of the economy around a steady state is the following (see Appendix 1 for the
derivation):
2 1
2
"( )
2 2 _{(1} _{)}
ii
f k
a c
ρ ρ
µ
σ ψ
⎛ ⎞
= − _{⎜ ⎟} −
⎝ ⎠ _{−} _{+} (29)
As intuition would suggest, funding investment with a larger proportion of foreign loans raises
ent depends on the quadrat
the speed of capital accumulation, since foreign funding is supplied perfectly elastically. The dependence of the rate of growth on the loantovalue ratio has also been noted by Cohen and Sachs (1986) and Aghion, Banerjee, and Piketty (1999) in AK models, as well as by Aghion, Howitt, and MayerFoulkes (2005) in a Schumpeterian growth model.
A novel result shown by equation (29) is that the speed of adjustm
ic of the agent’s net worthtovalue constraint (1−a). The effect of an increase inthe loantovalue ratio (a) on the adjustment speed is greate autarchy. Increasing the loantovalue ratio has both the direct effect of relaxing the borrowing constraint (b=aqk) as well as
st at
the indirect effect of raising the valuation of the capital stock (
(
∂ ∂b q)(
∂ ∂q a)
. This second effect can be viewed as a multiplier that depends on the elasticity of installation costs(q=ψ_{i}( , )i k ) with respect to the borrowing constraint, so that the combined effect is:
15
1
db a q
qk⎡ ∂ ⎤
= +
⎣ ⎦. (30)
da ⎢ q a∂ ⎥
The model with partial collateralization of the capital stock encompasses four special cases. The standard Ramsey growth model (a=0,ψ_{ii} =0) and the AbelBlanchard (1983) model (a=0,ψ_{ii} >0) are limiting cases with a closed capital account (a=0). With an open account, the limiting cases
perfect
include both the canonical open economy Ramsey model with consumption smoothing (a=1,ψ_{ii} >0) and the model of instantaneous adjustment of the capital stock (a=1,ψ_{ii} =0).
Opening the economy raises the adjustm eed of the economy. In the presence of nonconvexities, th in
6. Nonconvexities
ent sp
e difference adjustment speeds can make a critical difference in the adjustment dynamics. This section assumes that the production function is characterized by nonconvexities arising from an external effect arising from public knowledge (k). This knowledge is a byproduct of capital accumulation and is taken as given by the agent as in Romer (1986). Greater knowledge results in a more durable capital stock so that the
production function can be written as ( , k)F k = f k( )−δ(k)k, δ′(k)≤0. The depreciation function (k)δ is convexconcave, so that the rate of improvement in the durability of the capital stock initially increases before decreasing. This allows the possibility of three
16
analytical re of this section would be the same with that assumption, but the expression for the eigenval 16
It is more common to introduce the nonconvexity in the production function , as in Romer (1986). The
sults ues
(equation 31) would not be as clean.
( , k)
equilibria (k k k, ,ˆ ), as shown in Figure 2, where the vertical distance between thef k′( ) (k)
and δ curves equals ρat the equilibria.
At k and k ,f′′( )k −δ′(k)<0 so that the equilibria are saddlepoints. At kˆ, ( ) (k) 0
f′′k −δ > c ian is
positive, indicatin
′ ian is convex and the determinant of the Ja ob g that the equilibrium is unstable:
so that the Hamilton
2
'(k)k '(k)k f ( )k
ρ δ ρ δ
1,2
2
'(k)
2 2 _{(1} _{)}
ii
a c
δ µ
σ ψ
′′
− −
= _{⎜} _{⎟} −
⎝ ⎠ _{−} _{+}
m (31)
The equilibrium at will be a node if both eigenvalues are positive and a focus if both are
own in equation (31), a (the ratio of pledgeable income to debt) is a bifurcating variabl
−
⎛ ⎞
ˆ
k
complex.17 As sh
e. Partial opening of the economy to borrowing (a= → >0 a 0) will increase the size of the second term under the radical. If the equilibrium in autarchy (a=0) is a node, then for some positive value of the loantovalue ration (a≤1) there will be a bifurcation in the dynamics for ψ_{ii}small enough, at which value th aracteristic roots will switch from b positive to com lex.
Figure 3 graph
e ch eing
p
s the adjustment paths of consumption and the capital stock. Point is the uns
1
e
table equilibrium. When the capital account is closed (a=0), I assume that the dynamics are nodal, so that an initial endowment of capital k> ll lead to an accumulation of capital along the dashed path leading to point e_{2}, while an initial endowment of capital
ˆ
kwi
will produce a reduction in the capital st the dashed path leading to . Figure
ˆ
k <k ock along e_{3}
17
3 also graphs the adjustment path associated with focal dynamics for a large enough loanto ratio. Any initial endowment k≥k
% will allow the agent to accumulate capita n the
solid adjustment path leading toe_{4}. Thus, an increased loantovalue ratio, by increasing the speed of adjustment, allows the agent to attain the larger longrun capital stock from a lower initial capital stock than with the nodal dynamics.
An important aspect of the adjustment process involves the role of debt. It is common to describe nodal dynamics as being determined by “history” (i.e., the
value l o
initial endowment of capital) while focal dynamics are also determined by expectations in the spiral region. In 18 Figure 3 the region to the right of the spiral (k >k%) is characterized by saddlepath dynamics for any initial endowment of capital. In the area of the spiral (k≤ ≤k k%
% ), there are multiple
consumption choices (and corresponding inve nt choices) that are consistent with saddlepath adjustment and the transversality condition at infini
In most models there is nothing in the agent’s optimizing behavior that pins down the adjustment path in this region. Hence, the idea that “expectations” ar
stme
ty.19
e critical to the dynamic behavior of the econom
example,
y. With a collateral constraint, however, the initial stock of bonds (b_{0}) is an additional predetermined state variable. With an initial collateral constraint given by
0 0 0
b ≤aq k , the price of capital will be uniquely determined if the constraint is binding. A jump from a lower level of consumption in the spiral region to a higher level of consumption, for
would reduce investment and lower q, thereby violating the collateral constraint.
18
See Krugman (1991) and Matsuyama (1991) for this interpretation. Skiba (1978) is the original reference on the dynamics associated with the Ramsey model with a convexconcave production function.
19
For example, high consumption and low investment take place along the top of the spiral, while low
With a binding borrowing constraint, therefore, the potential indeterminacy of adjustment p in the presence of increasing returns is eliminated.
aths
gman (1979) have used a twoperiod
Fisheri unt
noncon a traditional
20
Hirshleifer (1958), McKinnon (1973), and Kru
an investment analysis to illustrate the dependence of multiple equilibria on the disco rate in the presence of increasing returns to scale. Similarly, the presence of a node or focus at
k% also depends on the discount rate via its impact on the speed of adjustment. To illustrate the importance of the speed of adjustment in the presence of vexities, assume (as in Hirshleifer, McKinnon, and Krugman) that there is
production technology that employs a constant returns production technology over the relevant range (shown by the dotted curve going through point e_{0} in Figure 3). The economy is
initially in longrun equilibrium at point e _{0} where f k′( )= +ρ δ. There is also a second technology characterized by increasing re rns to s om a declining depreciati rate in aggregate capital as outlined above. With a closed capital account and an initial capita stock k_{0}, adopting the new technology will cause an initial upward jump in consumption from point to the dashed adjustment path and a decline in savings as the economy’s capital stock declin to
ising fr on
l tu cale ar
0
e
es k. With an open capital account and a sufficiently large fraction of pledgeable capital income (so that a is large enough to produce a bifurcation in the dynamics with a larg enough spiral) consumption will initially fall and the economy will move along the solid path toward the longrun equilibrium e_{4}.
e
20
The agent may wish to reduce the capital stock to the long run equilibrium . Unlike the closed economy, in which the interest rate endogenously falls with the decline in the capital stock along the dashed adjustment path, the borrowing constrained open economy can sell capital on the world market for bonds that pay the world interest rate. As discussed in Section 2, the dynamics of adjustment with a declining capital stock will be the same as in the canonical open economy Ramsey model with adjustment costs.
3
The increased speed of adjustment requires a higher savings rate and an initial
reduction in consumption. An increase in the domestic savings rate raises investment and the price of capital (q=ψ_{i}( , )i k ), thereby relaxing the borrowing constraint b=aqk and
increasing access to foreign capital.21 Unlike the twoperiod Fisherian analysis, the concavity of the utility function and the convexity of the investment function also matter for adjustment in the presence of scale economies because they affect the speed of adjustment.
Aghion, Banerjee, and Piketty (1999) and Aghion, Bacchetta, and Banerjee (2004) have also constructed models in which changes in the loantovalue ratio can alter the adjustment dynamics. However, their models employ either linear (Ak) or Leontief production functions and rely on cobweb dynamics created by price movements arising from the interaction between savers and entrepreneurs.22 In addition, they characterize unstable equilibria as indicative of negative consequences of capital market liberalization in developing countries.23
In this paper’s model, in contrast, nonconvexities in production discourage the representative agent from undertaking investment when borrowing is prohibited ( ). A capital market liberalization that relaxes the collateral constraint creates focal (spiral) dynamics that, rather than creating instability, offer the possibility of successfully reaching the higher equilibrium capital stock. The higher convergence speed associated with a larger loantovalue ratio reduces the present value of losses that accompany production in the convex region of the production frontier ( ).
0
a=
0
k&=
21
We can think of a parametric reduction in σ in equation (31) as capturing an increased propensity to save. 22
See also Matsuyama (2008) for similar results involving the allocation of credit between alternative projects in an overlapping generations model. Aoki, Benigno, and Kiyotaki (2006) also generate this type of result by employing dual domestic and foreign borrowing constraints in a model with entrepreneurs and workers. 23
The dynamics of consumption and investment in Figure 3 are not inconsistent with the results of several papers. Jappelli and Pagano (1994) find that liquidity constraints can
increase the growth rate by raising saving and channeling these funds to firms. This paper’s explanation is consistent with Jappelli and Pagano (1994), but offers the additional insight that greater saving raises the collateral value of capital, thereby permitting greater borrowing. Prasad, Rajan, and Subramian (2006) hypothesize that fastgrowing lower income countries cannot easily access foreign capital (low a) so that their current account deficits are smaller than in more developed countries (high a) with similar growth rates. This explanation is in agreement with this paper’s model. In the model of Aghion, Comin, and Howitt (2006), financial restrictions caused by agency problems are alleviated by increased domestic saving, thus raising foreign investment and growth and thereby creating a complementarity between domestic and foreign saving. This complementarity channel is distinct from the one in this paper, but to the extent that the borrowing constraint in this paper is meant to capture
unmodeled agency problems (as in Matsuyama 2008), then the link between increased saving and the collateral value of the capital stock is not inconsistent with their model.
7. Conclusion
This paper has taken the canonical small open economy Ramsey model with adjustment costs for investment and added a borrowing constraint in the tradition of Cohen and Sachs (1986) and Kiyotaki and Moore (1997). The paper shows that the borrowing constraint, which can be expressed as a collateral requirement on the value of the capital stock, provides an attractive alternative to several standard borrowing constraints that have been used in the literature.24
24
The model encompasses the standard closed economy Ramsey model, the AbelBlanchard closedeconomy Ramsey model with convex investment costs, and the canonical small open economy model with convex investment costs as special cases.
The model can be extended in several directions. A number of papers (e.g., Chari, Kehoe, and McGrattan 2005 and Mendoza 2008) refer to collateral constraints that depend on the value of the capital stock ( ) as endogenous constraints, in contrast to fixed limits ( e ), which are referred to as exogenous constraints. A useful extension to this model would be the introduction of a costofcollateralization function that would make the loantovalue (a) a choice variable and thus provide the collateral constraint with an additional degree of endogenization. Following the general theme of Barro, Mankiw and SalaiMartin (1995), another extension of the model could specify two types of capital—one with a
borrowing constraint and one without a constraint. Finally, following Brock (2009) the model can be extended to include fiscal variables, where the representative agent and government operate with distinct borrowing constraints.
References
Abel, A. and O. Blanchard. 1983. An intertemporal model of saving and investment. Econometrica 51, 67592.
Adolfson, M., S. Laséen, J. Lindé, and V. Mattias. 2007. Bayesian estimation of an open economy DSGE model with incomplete passthrough. Journal of International Economics 72, 481511.
Aghion, P., P. Bacchetta, and A. Banerjee. 2004. Financial development and the instability of open economies. Journal of Monetary Economics 51, 10771106.
Aghion, P., A. Banerjee, and T. Piketty. 1999. Dualism and macroeconomic volatility. Quarterly Journal of Economics 114, 135997.
Aghion, P., D. Comin, and P. Howitt. 2006. When does domestic saving matter for economic growth? NBER Working Paper 12275.
Aghion, P., P. Howitt, and D. MayerFoulkes. 2005. The effect of financial development on convergence: theory and evidence. Quarterly Journal of Economics 120, 173222.
Aiyagari, S. and M. Gertler. 1999. Overreaction of asset prices in general equilibrium. Review of Economic Dynamics 2, 335.
Aoki, K., G. Benigno, and N. Kiyotaki. 2006. Adjusting to capital account liberalization. Working Paper, Princeton University.
Barro, R., N.G. Mankiw, and X. SalaiMartin. 1995. Capital mobility in neoclassical models of growth. American Economic Review 85, 103115.
Barro, R. and X. SalaiMartin. 2004. Economic Growth. Cambridge, MA: MIT Press (second edition).
Blanchard, O. 1983. Debt and the current account deficit in Brazil. In P. Armella, R. Dornbusch, and M. Obstfeld, eds., Financial policies and the world capital market: the problem of Latin American countries. Chicago: University of Chicago Press.
Blanchard, O. and C. Kahn. 1980. The solution of linear difference models under rational expectations. Econometrica 48, 13051311.
Blanchard, O. and S. Fischer. 1989. Lectures on Macroeconomics. Cambridge, MA: MIT Press.
Brock, P. 1988. Investment, the current account, and the relative price of nontraded goods in a small open economy. Journal of International Economics 24, 23553.
Brock, P. 2009. Capital account liberalization in an open economy Ramsey model. Economics Department, University of Washington.
Bruno, M. 1976. The twosector open economy and the real exchange rate. American Economic Review 66, 56677.
Buiter, W. 1984. Saddlepoint problems in continuous time rational expectations models: a general method and some macroeconomic examples. Econometrica 52, 66580.
Caballero, R. and A. Krishnamurthy. 2001. International and domestic collateral constraints in a model of emerging market crises. Journal of Monetary Economics 48, 51348.
Chari, V., P. Kehoe, and E. McGrattan. 2005. Sudden stops and output drops. American Economic Review papers and proceedings 95, 38187.
Christiano, L., M. Eichenbaum, and C. Evans. 2005. Nominal rigidities and the dynamic effects of a shock to monetary policy. Journal of Political Economy 113, 145.
Cohen, D. and J. Sachs. 1986. Growth and external debt under risk of debt repudiation. European Economic Review 30, 52960.
Devereux, M. and D. Poon. 2004. A simple model of optimal monetary policy with financial constraints. University of British Columbia, Department of Economics.
Evans, D. and B. Jovanovic. 1989. An estimated model of entrepreneurial choice under liquidity constraints. Journal of Political Economy 97, 80827.
Fischer, S. and J. Frenkel. 1972. Investment, the twosector model and trade in debt and capital goods. Journal of International Economics 2, 21133.
Frenkel, J. and C. Rodriguez. 1975. Portfolio equilibrium and the balance of payments: a monetary approach. American Economic Review 65, 67488.
Giavazzi, F. and C. Wyplosz. 1984. The real exchange rate, the current account, and the speed of adjustment. In J. Bilson and R. Marston, eds., Exchange Rate Theory and Practice.
Chicago: University of Chicago Press.
Hartl, R., P. Kort, G. Feichtinger, and F. Wirl. 2004. Multiple equilibria and thresholds due to relative investment costs. Journal of Optimization Theory and Applications 123, 4982. Hayashi, F. 1982. Tobin’s marginal q and average q: a neoclassical interpretation. Econometrica 50, 21324.
Hirshleifer, J. 1958. On the theory of optimal investment decision. Journal of Political Economy 66, 329352.
Holmstrom, B. and J. Tirole. 1998. The private and public provision of liquidity. Journal of Political Economy.
Jappelli, T. and M. Pagano. 1994. Saving, growth, and liquidity constraints. Quarterly Journal of Economics 109, 83109.
Kiyotaki, N. and J. Moore. 1997. Credit cycles. Journal of Political Economy 105, 21148. Krugman, P. 1979. Interest rate ceilings, efficiency, and growth: a theoretical analysis. MIT. Mimeograph.
Krugman, P. 1991. History versus expectations. Quarterly Journal of Economics 106, 65167. Lane, P. 2001. International trade and economic convergence: the credit channel. Oxford Economic Paper 53, 22140.
Lipton, D. and J. Sachs. 1983. Accumulation and growth in a twocountry model: a simulation approach. Journal of international economics 15, 13539.
Lucas, R. and E. Prescott. 1971. Investment under uncertainty. Econometrica 39, 65981. Matsuyama, K. 1987. Current account dynamics in a finite horizon model. Journal of International Economics 23, 299313.
Matsuyama, K. 1991. Increasing returns, industrialization, and indeterminacy of equilibrium. Quarterly Journal of Economics 106, 61750.
Matsuyama, K. 2008. Aggregate implications of credit market imperfections. NBER Macroeconomics Annual 2007. Cambridge: MA.
McKinnon, R. 1973. Money and Capital in Economic Development. Washington, D.C.: Brookings Press.
Obstfeld, M. 1981. Macroeconomic Policy, ExchangeRate Dynamics, and Optimal Asset Accumulation. The Journal of Political Economy 89, 11421161.
Obstfeld, M. 1989. Fiscal deficits and relative prices in a growing world economy. Journal of Monetary Economics 23, 46184.
Obstfeld, M. and K. Rogoff. 1995. The intertemporal approach to the current account. In G. Grossman and K. Rogoff, Handbook of International Economics, volume 3. New York: NorthHolland.
Prasad, E., R. Rajan, and A. Subramanian. Foreign capital and economic growth. 2007. NBER working paper 13619.
Romer, P. 1986. Increasing returns and longrun growth. Journal of Political Economy 94, 10021037.
Sachs, Jeffrey. 1981. The current account and macroeconomic adjustment in the 1970s. Brookings Papers on Economic Activity, 20168.
SchmittGrohé, S. and M. Uribe. 2003. Closing small open economy models. Journal of international economics 61, 16385.
Skiba, A. 1978. Optimal growth with a convexconcave production function. Econometrica 46, 527540.
Svensson, Lars and Assaf Razin. 1983. The terms of trade and the current account: the HarbergerLaursenMetzler effect. Journal of Political Economy 91, 97125.
Tirole, J. 2006. The theory of public finance. Princeton: Princeton University Press. Turnovsky, S. 1997. International macroeconomic dynamics. Cambridge, MA: MIT Press. Turnovsky, S. and P. Sen. 1991. Fiscal policy, capital accumulation, and debt in an open economy. Oxford Economic Papers 43, 124.
Uzawa, H. 1961. On a twosector model of economic growth, I. Review of Economic Studies 29, 407.
Appendix 1: Convergence Speed with Partially Pledgeable Capital Income
The representative agent faces the following utility maximization problem:
0
( ) ,
t
u c e max
c i
ρ ∞
−
∫
subject to initial state variables k b_{0}, _{0} and
( , ) ( )
b& = +c ψ i k +rb− f k (A1)
k&= +i δk (A2)
0
b≤aqk ≤ <a 1 (A3)
where ( , )ψ i k is linearly homogeneous and qk is the value of installed capital.
The optimization problem for the representative agent can be expressed as the following currentvalue Hamiltonian:
[
]
*( ) ( , ) ( ) ( ) * ( )
t
e Hρ =u c −λ c+ψ i k +rb− f k +q i−δk +γ aqk−b . (A4) Firstorder necessary conditions are:
'( ) u c =λ
*
( , )
i i k q where q q
ψ = = λ .
The costate equations are: c
r c
λ σ ρ γ
λ = = − −
& _{&}
(A5)
[
]
'( ) _{k}( , ) (1 ) )
q&= −f k +ψ k i + + + −r δ a γ q . (A6)
With a binding collateral constraint (b=aqk):
( , ) ( )
b&=aqk&+aqk& = +c ψ i k +raqk− f k so that
[
]
( ) ( ) _{k} ( ) (1 ) ( , ) ( )
Noting that aqi+aψ_{k}k ≡a(ψ_{i}i+ψ_{k}k)=aψ( , )i k by the linear homogeneity of the investment function,
[
]
( ) ( ) (1 ) ( , )
c= f k −af k k′ − −a ψ i k −a qkγ (A7)
and
( ) ( ) (1 ) (1 )
(1 )
k
i
f k af k k c a a qk a k
k a k γ ψ _{δ} ψ ′ − − + − − − = − & − 1 (A8)
for 0≤ <a . Using equation (A6), the capital accumulation constraint can be written as:
( ) ( )
(1 )
k i
i
f k c k ar k
k a ak ψ δ ψ ψ ω − − − + = − +
& . (A9)
where ω ≡ ∂ ∂ =q k ψ_{ii}(∂ ∂ +i k) ψ_{ik} .The second term in the denominator equals zero around a steady state since by linear homogeneity of the installation cost function the price of capital is scale invariant: ω ψ δ ψ= _{ii} + _{ik} =0. In addition, around a steady state 2 2
0
k q k
ω
∂ ∂ ≡ ∂ ∂ = for a quadratic installation cost function (see appendix 2), so that
( ) ( )
(1 )
k i
i
f k c k a r k
k k a ψ δ ψ _{δ} ψ − − − + = − −
& (A10)
and
( ) ( )
(1 )
k
i
f k a r
k
k a
ψ δ ψ_{i} _{δ}
ψ
′ − − +
∂ _{=}
∂ −
&
− (A11)
around a steady state.
The constrained optimization problem can be written as the following Hamiltonian:
( ) ( )
( )
(1 )
t k i
i
f k c k a r k
e H u c k
a ρ _{λ} ψ δ ψ _{δ} ψ ⎡ − − − + ⎤ = + _{⎢} − −
⎣ ⎦⎥ . (A12)
The firstorder necessary condition is: '( ) (1 ) i
u c =λ −aψ (A13)
( ) ( ) (1 )
k i
i
f k a r
a ψ δ ψ λ _{ρ δ} λ ψ ′ − − + = − + + − &
. (A14)
From the economy’s resource constraint (A1),∂ ∂ = − −c i (1 a)ψ_{i}, so that
2
( )
(1 ) ( )
(1 )
( )(1 ) (1 )
ii i
i ii
i i
u c
a u c
a c
u c a c a
ψ ψ ψ ψ c λ σ λ ψ λ ψ λ ⎡ _{′′} ′ ⎤ − _{⎢} − _{⎥} − ⎛ ⎞ ⎣ _{⎦ = ⇒} _{+} _{= −} ⎜ ⎟ ′ − _{⎝} − _{⎠} & &
& & (A15)
"( ) where '( ) cu c u c σ ≡ − .
Combining this with the costate equation (A14) gives the equation: c&
2 ( ) ( ) ( ) (1 ) (1 ) k i i ii i
f k a r
a c c a ψ δ ψ _{ρ δ} ψ ψ σ ψ ′ − − + + + − = + −
& . (A16)
Noting again from equation (A6) that q_{&}= −f k'( )+ψ_{k}( , )k i + + + −
[
r δ (1 a) )γ]
q and c c γ σ= &from equation (A5) (with ρ=r), around a steady state the c& equation is25:
2 ( ) ( ) (1 ) k i ii i
f k c
a c c c a ψ _{σ} _{ρ δ} ψ ψ σ ψ ′ − _{+} _{+} _{+} = + − &
& (A17)
so that, by combining terms,
( ) ( ) (1 ) (1 ) k i ii i i f k c a c a ψ ρ δ ψ ψ σ ψ ψ ′ − + + = − + −
& . (A18)
25
The derivation also makes use of the property of the installation cost function that 2 2 0
q k q k
Linearizing around the steady state and making use of the steadystate properties of the installation cost function, the dynamic adjustment equations around a steady state ( ,c k ) are given by:26
0 "( ) (1 )
(1 ) 1
1
ii
f k a
c _{c} _{a} c c
k k k
a
ψ σ
ρ
⎛ ⎡ ⎤⎞
− +
⎜ ⎢ ⎥⎟ _{−}
⎛ ⎞_{=}_{⎜} _{⎣} − _{⎦} ⎛ ⎞
⎜ ⎟ _{⎜} ⎟_{⎟}⎜ ⎟
−
⎝ ⎠
⎝ ⎠ _{−}
⎜ ⎟
−
⎝ ⎠
&
& . (A19)
The eigenvalues of the linearized system are:
2 1,2
2
"( )
0 1
2 2 _{(1} _{)}
ii
f k
a a
c
ρ ρ
µ
σ ψ
⎛ ⎞
= ± _{⎜ ⎟} − ≤ <
⎝ ⎠ _{−} _{+} . (A20)
The special case of a closed capital account can be determined by setting a=0. The solution converges to the case with fully pledgeable capital as a→1.
26
Appendix 2. The Investment Function
The investment function ( , )ψ i k is linearly homogeneous: ψ_{i}i+ψ_{k}k =ψ( , )i k , where i= +k& δk.27
( , ) ( , )
0, 0, (0) 0
general specification steady state values k
i k i k k k k
k ψ φ ψ δ δ φ φ φ − ⎛ ⎞ = + _{⎜ ⎟} = ⎝ ⎠ ′′
≥ > =
/
&
2 2
2 3
2
( , ) 1 ( , ) 1
1
( , ) ( , ) (0) 0
( , ) ( , ) 0
( , ) ( , ) (0) 0
( , ) ( , ) i i ii ii k k kk kk ik ik k
i k k k
k k
i k k k k
k k k i
i k k k
k k
k i k i
i k k k
k k k k k
k i
i k k k
k k ψ φ ψ δ ψ φ ψ δ φ ψ φ ψ δ δ ψ φ φ ψ δ φ ψ φ ψ δ ⎛ ⎞ ′ = + _{⎜ ⎟} = ⎝ ⎠ ⎛ ⎞ ′′ ′′ = _{⎜ ⎟} = ⎝ ⎠ ⎛ ⎞ ′ = − _{⎜ ⎟} = ⎝ ⎠ ⎛ ⎞ ⎛ ⎞ ′ ′′ ′′
= _{⎜ ⎟} + _{⎜ ⎟} = >
⎝ ⎠ ⎝ ⎠ ⎛ ⎞ ′′ = − ⎜ ⎟ ⎝ ⎠ & & & & & & >  1 2 2 2  1 (0) 0
( , ) ( , ) 0
( , ) ( , ) 0
(0) ii ik q ik kk q k q
k k k k
k
k k k k
q k k δ φ ψ δ δ ψ δ ψ δ δ ψ δ δ φ = = ′′ = − <
∂ _{=} _{+} _{=} ∂ + = ∂ _{=} _{′′′} ⎛ ⎞ ⎜ ⎟ ∂ ⎝ ⎠ 27
An alternative form of the investment function is ( , )i k i 1 k
k ψ = ⎡_{⎢} +ϕ⎛ ⎞_{⎜ ⎟}⎤_{⎥}
⎢ ⎝ ⎠⎥
⎣ ⎦
&
. To ensure the convexity of installation
costs this specification requires that2 k i k
k k k
ϕ′⎛ ⎞_{⎜ ⎟}+ ϕ′′⎛ ⎞_{⎜ ⎟}
⎝ ⎠ ⎝ ⎠
& &
0
> , which is less intuitive than the assumption
0
k k φ′′⎛ ⎞_{⎜ ⎟}>
⎝ ⎠
&
0 c&=
c
k r
c_{ρ}_{=}
closed
c

0
k k
1.
F igure G row th w ith C ollateralized B orro w ing
b qk
c _{=}
0, 0 k&= b=
0
c&=
⎫⎪
⎬
⎪⎭
[ '( ) ] rb%= f k −δ k
1
e
2
e
3
e
0 0 0
b q k
c _{<}
T
k _{k} k
0
ρ
'( )
f k
(k)
δ
'( ) (k)
f k −δ
 ˆ
k
ρ
ρ
ρ 2
0 c&=
c
k
1
e

0
k k
0 c&=
⎫
⎪⎪
⎬
⎪
⎪⎭
rb%=ark
2
e
0 c&=
0 ( 0) k&= b=
4
e
k
% k%
3
e e0•

k _{k}ˆ 