Inefficient Sovereign Defaults
Abstract
Many argue that financial crisis resolution should become more efficient and
opponents mainly emphasize the detrimental effects on incentives ex-ante of greater
efficiency ex-post. However, renegotiation theory suggests that debtors should be able
to “buy their way back to efficiency” ex-post and for this privilege creditors can
potentially charge them so much that ex-ante incentives are not distorted. Thus,
making crisis resolution efficient could be “reform without losers” (Lau et al., 2000).
We argue, however, that creditors may not be able to help themselves ex-post without
also helping the debtor: even if the debtor defaults it is still a high return country so
efficiency warrants that it receives a large capital inflow. Under a weak assumption
this raises its welfare. Thus, contracts which cannot be renegotiated and cause a sure
efficiency loss in case of default can still be efficient ex-ante.
1. Introduction
Financial crises in recent decades were associated with substantial output losses
(Sturzenegger and Zettelmeyer, 2006; Paoli et al., 2006). However, the efficient response
ex-post is to separate efficiency and distribution as much as possible. In particular, rather
than disturb the debtor economy by cutting trade or financial relations, or engage in
costly bargaining, efficient rescheduling would maximize the continuation value of the
defaulting economy and let creditors collect their claims over time with minimal
disturbance.
There are, roughly, two sides in this debate. One side wants precisely to minimize the
efficiency costs of crises once they occur and argues that the recovery of the deadweight
losses can be used to make both creditors and debtors better off. For example, rapid
rescheduling, new loans or debt reduction can ensure continued gains to financial trade,
improve the country’s incentives to invest or reform, and reduce negotiation costs and
uncertainty (Krugman 1988a,b; Krueger, 2002; Roubini and Setser, 2003). In contrast,
the other side sees output losses as a necessary evil when agency problems allow
countries to affect their likelihood of default and sovereign immunity makes contracts
imperfectly enforceable. If crises did not lead to output losses lenders would not trust
borrowers in the first place (Dooley and Verma, 2001; Atkeson, 1991; Shleifer, 2003;
Pitchford and Wright, 2007, Bulow, 2002).
2
However, it is unclear if this is really a zero-sum debate. In particular, inefficient crisis
resolution means that one side suffers losses not gained by the other side. Therefore, by a
Coasean argument renegotiation should allow debtors to “buy their way back to
efficiency” by giving creditors a share of the social gain. Indeed, if creditors can collect
the entire efficiency gain the debtor’s ex-ante incentives and its credit ceiling will be
unchanged (Kletzer and Wright, 2000). Thus, in the terminology used by Lau et al.
(2000) to describe China’s dual-track economic transition, there should be scope for
“reform with losers”. The “penalties” paid by the debtor might take the form of lump sum
payments, a rise in future interest rates, lower terms of trade or returns to domestic
factors employed abroad, or higher returns (e.g. less taxation) to foreign factors employed
at home. Alternatively, the debtor could supply political or military assistance to creditor
countries whose governments can then transfer the gain to private creditors (Cole and
Kehoe, 1996).
But if national bankruptcies can be resolved more efficiently while preserving ex-ante
incentives why are they not? This paper suggests an answer based on two general
propositions. First, any inefficiencies ex-post will provide incentives for the parties to
renegotiate. Second, in practice even colluding creditors cannot appropriate the whole
efficiency gain because restoring efficiency means restoring efficiency in the debtor’s
economy and under plausible conditions this will raise the debtor’s welfare.
Consequently, if creditors can help themselves ex-post they will also be helping the
debtor. In turn the debtor the will have a lower credit ceiling ex-ante and prefer a
renegotiation-proof contract. Thus, the paper supports the “necessary evil” side of the
3
rescheduling debate but strengthens the argument by explaining why a continuation
agreement cannot be designed so harshly from the debtor’s perspective that renegotiation
leaves it only indifferent.
Necessary ex-post inefficiencies can be modeled in several ways, for example using
information asymmetries and wars of attrition (Bulow and Klemperer, 1997; Alesina and
Drazen, 1991), costly and protracted bargaining (Pitchford and Wright, 2007)1, or a
setting where the gain to addressing the inefficiencies is rising over time for one or more
agents with veto-power (Velasco, 1998; Labán and Sturzenegger, 1994, 1998). In these
approaches, however, creditors are fighting over a fixed pie of repayments so the models
are silent on the changes in capital flows which are a defining feature of crises.
Therefore instead we model inefficiencies as the result of a common pool problem among
collecting creditors. The common pool regime arising after default leads to overextraction
of the debtor’s output and therefore a drop or reversal of the previous capital inflow.
Implicitly we thus assume that creditors in the common pool regime cannot coordinate on
efficient rescheduling.2
Beyond accounting for capital flow reversals the common pool approach is also
consistent with the creditor “panics” which are a stylized fact of debt crises in recent
decades (Morris and Shin, 2004; Radelet and Sachs, 1998; Fischer, 1999; Calvo and
1
Asymmetric information has also been used to explain the inefficiencies of real conflicts (wars), see
Fearon (1995).
2
Apart from giving commitment not to renegotiate lending with many dispersed creditors can lower risk
premia in a stochastic world since each lender can diversify. On the other hand poor incentives to acquire
information may raise premia. The only difference between common and private access we need for the
paper is in terms of renegotiation costs.
4
Mendoza, 1996; Diamond and Dybvig, 1983) and we can rationalize these panics from an
ex-ante perspective. Furthermore, based on their data set Tomz and Wright (2005) argue
that defaults are best viewed as a temporary suspension of sovereign capital flows
followed by resumption of lending on terms similar to those of pre-existing loans. This is
also consistent with the common pool approach.
2. A common pool model of crisis resolution
We first model the effects of default and then, in the next subsection, consider how
default outcomes affect the optimal contract given the inability of creditors in the
common pool regime to renegotiate.
2.1. Debt collections after default
To model the post-default regime we slightly adapt the dynamic common pool model
used in Tornell and Velasco (1992) to explain capital flight from developing countries.3
However, unlike in their paper and most others which analyze the effects of common
access to a resource here the first-best policy may not be private access. Instead, common
pool access ex-post default is part of the optimal debt contract because it commits lenders
not to renegotiate.
3
Dutta and Rowat (2006) present a more general commons model with a capital market.
5
There is a fixed number of international creditors n ≥ 2 receiving the competitive interest
rate r However, if the country fails to meet its financial commitments creditors collect
the debtor’s output under common pool conditions. We assume that the country’s entire
capital stock is borrowed. I later discuss this assumption in detail. The lifetime utility of a
creditor is
∞
Uj =∫
σ
σ −1
0
c j (t ) (σ −1) / σ e −δt dt , j = 1...n
(1)
with c(t ), δ ∈ [0,1] and σ > 0 denoting consumption in period t , the common discount
rate, and the common intertemporal elasticity of substitution. For σ = 1 the flow utility
is ln c(t ) . The capital stock of the debtor country follows
kɺ(t ) = y (k (t )) − d j (t ) − ∑ d i (t ),
(2)
k (0) > 0
(3)
i≠ j
where y (k ) is a standard neoclassical production function with diminishing returns to
capital due to a fixed non-tradable input such as labor. It also satisfies the Inada
conditions lim y ' (k ) = ∞ and lim y ' (k ) = 0 kɺ(t ), k (t ), d j (t ) and
k →0
k →∞
∑ d (t )
i
are the change
i≠ j
6
in the capital stock, the level of the capital stock, the real debt payment claimed by
creditor j , and the total real payments claimed by the other creditors at time t .
Finally, creditors can deposit funds abroad at the international interest rate r . The foreign
savings asset f has entirely private access and follows
fɺ j (t ) = rf j (t ) + d j (t ) − c j (t )
(4)
where we assume a zero starting stock for simplicity. However, the creditors can
potentially go short in the foreign asset. We show now, like Tornell and Velasco, that
despite the costless extraction, the common pool regime, and the positive return to the
international asset there remains an interior equilibrium where creditors do leave some
capital in the country.
The creditors play a differential game against each other since the claimant decisions of
each affects the decision problems of the rest. We restrict attention to symmetric closedloop Markov equilibria (which imply Markov perfection). Thus, players condition
behavior only on current asset stocks and the current behavior of other players.
Furthermore, we make the make the plausible assumption that d ' (k ) > 0 .
Apart from tractability these assumptions seem appropriate for the financial crises of
recent decades where both apparently self-fulfilling crises (e.g. Asia, 1997-8; Mexico,
1994) and fundamentals driven crises (Argentina, 2001) were associated with large
7
creditor “panics” once begun. These panics seemed to reflect high levels of dispersion
and anonymity among creditors eroding their capability or incentives to coordinate. On
the supply side, on the other hand, it was difficult to limit the payments creditors could
collect. Therefore collections occurred in unregulated conditions resembling a common
pool situation. Recent proposals to reform the international financial system by using
IMF sanctioned moratoriums, majority clauses, or even a sovereign bankruptcy court,
reflect precisely such concerns. However, we argue below that these measures may fail to
improve efficiency ex-ante.
The control problem of creditor j can be described by the current value Hamiltonian4
H=
σ
σ −1
c j (t ) (σ −1) / σ + λ (t ){y (k (t )) − d j (t ) − (n − 1)d (k (t ))}+ ψ (t ){rf (t ) + d j (t ) − c j (t )}
(5)
where d (k (t )) is the identical debt collections of the other n − 1 players. The first order
conditions for {c j (t ), d j (t )} and the state variables {k (t ), f j (t )} , and the possible
equilibria,
are described in the appendix. However, a key property of any interior
solution is that a creditor collects from the debtor until
4
In an interior equilibrium the capital stock of the debtor does not fall to zero so we do not have to consider
a non-negativity constraint on capital. The capital stock only falls to zero if there is a “grab race”
equilibrium. We discuss this case later.
8
y ' (k ) − (n − 1)d ' (k ) = r
(6)
where we have assumed that the left hand side is concave in the capital stock. (6) shows
two key results. First, there is excessive debt collection because the creditor income
maximizing capital stock k m satisfies
y ' (k m ) = r
(7)
and therefore
ke < km
(8)
where the superscript ‘e’ denotes an equilibrium outcome. Second, debt collections are
strategic complements: if other creditors respond more strongly to every capital stock
selected by a creditor, so that d ' (k ) rises, then the creditor prefers to lower k . However,
the Inada condition on the production function implies that in an interior equilibrium the
capital stock of the debtor remains larger than zero. In fact the interior equilibrium can be
relatively “friendly” because the creditors can potentially coordinate on non-aggressive
reaction functions d ' (k ) . In this case the perceived private return is close to the income
maximizing level and a net capital inflow may prevail even after default. The common
pool situation is therefore no prisoner’s dilemma but a coordination game.5 However,
5
Dutta and Sundaram (1993) derive conditions for when a tragedy of the commons does or does not have
to occur under common pool access. Copeland and Taylor (2004) present a model where regulatory
institutions form endogenously under international trade as a function of the costs and benefits of policing
9
comparing (6) and (7) common pool access always reduces the capital stock below the
income maximizing level. Also, creditors can potentially coordinate at the other extreme,
on the non-interior grab race equilibrium where each creditor tries to cease the entire
capital stock immediately. Since this equilibrium will be worse for the debtor, however
(see below), we can focus without loss on the interior equilibrium.
2.2. Ex-ante versus ex-post efficiency
This section considers lending with two types of creditors: a single creditor (or
equivalently a group of creditors which can coordinate to maximize their joint incomes
after default) and a group of non-coordinating creditors which will collect the debtor’s
capital stock under common pool conditions, as described above, if there is default.
Suppose first the case of a single creditor. With a single creditor it is possible to
renegotiate the contract ex-post and avoid the common pool inefficiency. The
renegotiation leads to k m . Consequently, creditors will leave the debtor with a larger
capital stock than under the common pool regime. However, except under extreme and
we believe unrealistic conditions this rise in the capital stock will leave the debtor better
off as well. A simple way to model this is to assume that the true production function
is (1 + a ) y (k ) , a > 0 , but the debtor can always consume ay (k ) or invest it for returns
creditors cannot access (for example, in non-tradables production, the informal sector, or
use of a resource. Thus common access, private access, or regulated common access can emerge. Future
research could construct a similar model for international lending. For example bank lending in the 1970s
and 1980s worked different institutionally from bond lending in the 1990s.
10
third country assets), so the function y (k ) is now the output accessible to the creditor(s).
a could reflect moral hazard or adverse selection problems giving rise to information
rents or sovereign immunity. The debtor is clearly better off when left with the stream
{ y (k m )} than with the stream { y (k e )} after default so common pool access ex-post will
raise the credit ceiling ex-ante. This raises ex-ante efficiency for a credit constrained
country.
More formally suppose that the debtor, risk neutral for simplicity, can appropriate
(1 + r )k (0)∆t
(9)
by failing to repay the creditor(s) for ∆t units of time before the creditor(s) begin(s) the
punishment. Postponing the punishment by ∆t is necessary because the model is in
continuous time and creditors can freely collect capital after default. The debtor discounts
the future at rate δ . Now a loan of k (0) can be enforced with a single creditor if,
assuming only financial means to punish are available,
(1 + r )k (0)∆t ≤ e
−δ∆t
∞

∞
 ∫ [(1 + a ) y (k (0)) − (1 + r )k (0)]e −δs dt − ∫ ay (k m (t ))e −δs dt 


t =0

 t =0
(10)
and with common access in the punishment phase due to lack of coordination and
appropriate contract design if
11
∞

∞
(1 + r )k (0)∆t ≤ e −δ∆t  ∫ [(1 + a ) y (k (0)) − (1 + r )k (0)]e −δt dt − ∫ ay (k e (t ))e −δt dt 
t =0

 t =0
(11)
and clearly the right hand side is larger in the second equation for any k (0) since
ke < km .
Due to the competitive lending market the debtor receives all gains to trade no matter
what the post default regime. However, post-default a single or coordinated creditor can
extract more from the country on the margin than can each creditor in an uncoordinated
group in a common pool regime. However, the higher and more efficient capital stock
inevitably helps the debtor. Any new capital stock less than each creditor’s income
maximizing level is not a Nash equilibrium.
2.2.3. Lump-sum fines removing the debtor’s surplus
Kletzer and Wright (2000) present a renegotiation-proof international lending model in
which they show that a contract exists such that, following default, lending remains
efficient subject to rationality and no-default constraints but all gains to trade now accrue
to creditors, that is, the continuation value of the debtor is zero and at the corner of the
Pareto frontier of feasible continuation values. This is particularly significant because the
default punishment replicates the punishment from future exclusion from the credit
market without being inefficient off the equilibrium path (hence the equilibrium is
renegotiation-proof). The punishment is implemented by the debtor making a lump sum
12
payment after default equal to its continuation value from the relationship, after which the
relationship resumes. This raises the question if a similar contract will work here:
suppose that after default, with either one or many creditors, the debtor must pay its
continuation value ay (k i ) / δ , i = e, m . Then its post-default payoff would be zero in
present-value terms and the feasible loan k (0) would rise. More importantly, the debtor
would be indifferent between borrowing from a single creditor or a group of
uncoordinated creditors since the punishment and in turn the credit ceiling is the same.
Additionally, since lenders prefer a single lender to punish off the equilibrium path
presumably lending should not involve common pool conditions after default but an
orderly ‘fine’ payment followed by resumed lending. The threat to not resume lending
before the fine is paid would be credible at every instant if the creditor expects the fine to
be paid at the next instant and the debtor finds it rational to pay the fine since otherwise
lending will not resume (as an analogy, in a two-player war of attrition (continue, quit) at
every instant is a perfect equilibrium).
Could such a ‘fine’ clause in the (implicit) contract make the common pool regime
unnecessary? First, in the model currently the country has no production without foreign
capital so its income would be zero and paying ay (k i ) / δ would then be infeasible. This
liquidity problem does not arise in Kletzer and Wright (2000) due to their endowment
output assumption. Realistically, here as well, the debtor should earn income without
foreign capital. However, suppose (1) that with the maximum fine the country can
finance, it still cannot finance the present value of continued lending with a single
creditor ay (k m ) / δ . After all, this value of access to foreign capital now and in all future
could be substantial and the real counterpart of the fine is net exports, the value of which
may be limited. Additionally, (2) even if the debtor can pay off ay (k m ) / δ with its future
13
net export stream, it may not be able to do so right away or even with a continuous
payment stream: if net exports nx accrue every instant, then the debtor can pay off its
agency rent continuously if nx ≥ ay (k m ) and otherwise it cannot. If it can not, the most
~
efficient capital level k post-default which leaves the debtor no surplus solves
~
~
nx = ay (k ) and such k < k m may be implemented by an appropriate choice of n , the
number of creditors who would engage in common pool debt collection after default.
Thus, the common pool regime can remain relevant by raising the debt ceiling ex-ante.
Additionally, while there is continuing debate, the evidence that creditors confiscate net
export earnings for debt repayment is limited (Mitchener and Weidenmier 2005,
Sandleris 2006, Martinez and Sandleris 2004).
2.3. Is the common pool always better?
In the perfect information model considered so far dispersed creditors and complete
commitment not to renegotiate the contract after default is always better. In practice,
however, involuntary or strategic defaults do happen - due perhaps to asymmetric
information (Atkeson, 1991), non-representative governments, or maturity mismatches in
a stochastic model. Because defaults do happen “on reality’s equilibrium path” ex-ante
efficiency and ex-post efficiency must be traded off. An easy way to model this is to put a
cap on the collection function d (k ) creditors can employ after default. The optimal cap
balances increased lending ex-ante when the cap is higher with a rising common pool
inefficiency cost ex-post, and the cap can accommodate both extremes. To implement
such a cap the market can use the contract specification or policy makers can use the
design of the international financial system (Pitchford and Wright, 2007).
14
3. Conclusion
Providing incentives for countries to repay national debts requires a punishment for
default. However, this alone does not mean that defaults should cause inefficiency
because efficient resolution coupled with a redistribution of the gains to continued trade
away from the debtor can, in principle, replicate the penalty from the inefficient
resolution. However, ex-post efficiency warrants leaving the debtor with a high capital
stock and this is likely to raise its welfare. Therefore, ex-post creditors cannot benefit
themselves without helping the debtor as well. In turn, to raise its credit ceiling ex-ante a
debtor may prefer creditor commitment not to renegotiate. One commitment device is to
use a dispersed anonymous creditor base with common pool access to debtor funds after
default. This can explain the frequency of creditor “panics” and output losses following
financial crises and predicts a trade-off between ex-post and ex-ante efficiency in
practice.
15
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20
Appendix
Tornell and Velasco (1992) consider the same model with a linear production technology,
y = ak , and solve for three equilibria where creditors collect a constant share of the
capital stock d = βk . One equilibrium is interior and the others are at the maximum and
minimum collection rates allowed in their paper. We now describe the solution to the
model in the main paper.
Interior equilibrium
The necessary first order conditions for the controls {c j (t ), d j (t )} and state variables
{k (t ), f j (t )} in (5) in an interior equilibrium are
c −j 1 / σ = ψ
(a1)
λ =ψ
(a2)
λ (δ + (n − 1)d ' (k (t )) − y ' (k ) ) = λɺ
(a3)
21
ψ (δ − r ) = ψɺ
(a4)
and the transversality conditions
lim λ (t )k (t )e −δt = 0 , limψ (t ) f j (t )e −δt = 0
t →∞
t →∞
(a5)
and (2) – (4). Since the Hamiltonian maximized with respect to the controls max H is
c j ,d j
concave in the state variables {k (t ), f j (t )} , as long as the constraint set is convex the first
order conditions are sufficient for a maximum. The maximized Hamiltonian is concave
since the first and second principal minors of the associated Hessian are negative and
zero, respectively, and the Hessian therefore negative semi-definite. The stock growth
constraints define a convex set if y (k (t )) − (n − 1)d (k (t )) is convex in k (t ) , which is true
since we assume in the text that y (k (t )) − (n − 1)d (k (t )) is concave.
(a1) and (a2) state that the value of consuming a unit of output today must be equal to the
value of increasing either of the two asset stocks by one unit and (a3) and (a4) define the
optimal growth rate of the identical shadow values of the two capital stocks. For the debt
claim the return for each agent is only y ' (k ) − (n − 1)d ' (k ) < y ' (k ) : due to the common
22
resource regime capital on the margin is perceived to yield the social return net of the rise
in collection by other creditors it leads to.
Combining (a1)-(a4) implies
cɺ j
cj
= −σ (δ + (n − 1)d (k (t )) − y ' (k ) ) = −σ (δ − r )
so that
c j (t ) = c j (0)eσ ( r −δ )t
(a6)
and from (2)
t
k (t ) = k (0) + ∫ y (k ( s )) − nd (k ( s ))ds
(a7)
0
and from (4)
t
f j (t ) = f j (0)e rt +
∫ (d (k ( s)) − c
j
( s ))e r (t − s ) ds
s =0
23
t
= f j ( 0) e +
rt
∫ d (k (s))e
s =0
r (t − s)
ds −
c j (0)(eσ ( r −δ ) t − e rt )
σ (r − δ ) − r
(a8)
using (a6). Now using the transversality condition for the international asset in (a5) to
find c j (0) we get
t

c j (0)(eσ ( r −δ )t − e rt ) 

limψ (t ) f j (t )e −δt = limψ (0)e (δ − r )t e −δt  f j (0)e rt + ∫ d (k ( s ))e r (t − s ) ds −


t →∞
t →∞
σ
(
r
δ
)
r
−
−
s =0


t

c j (0)(e (σ ( r −δ ) − r ) t − 1) 
− rs

=0
= limψ (0) f j (0) + ∫ d (k ( s ))e ds −


t →∞
−
−
σ
r
δ
r
(
)
s =0


(a9)
which says that the present discounted value of assets net of consumption converges to
zero. From (1) and (a6) bounded lifetime utility requires
σ (r − δ ) − r < 0
(a10)
which we will assume. Therefore in (a9)
24
c j ( 0) e ( σ ( r − δ ) − r ) t
goes to zero and therefore
t
c j (0) = (r − σ (r − δ )( f j (0) + lim ∫ d (k ( s ))e − rs ds )
t →∞
s =0
= ( r − σ ( r − δ ) k ( 0)
(a11)
where the last equation follows by arbitrage: all assets earn a private return r so the value
of what is collected equals the value of the initial asset stock k (0) .
While the capital stock of the debtor is inefficiently low in equilibrium (see (6)) there
does not have to be a net capital outflow: even ensuring
y ' (k ) − (n − 1)d ' (k ) = r
can require a capital inflow. Agents then take a short position in the international asset.
Grab race equilibrium
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If the equilibrium is not interior it is either a grab race or creditors cannot supply enough
capital because they cannot go short in the foreign asset. We first consider the grab race
equilibrium. Here each creditor wants to appropriate the entire capital stock immediately
because others are expected to do so. This lowers the creditor payoff for a given capital
stock since in (a11) each gets only k (0) / n and the capital stock is instantly depleted.
Equilibrium when creditors cannot go short in the foreign asset
If creditors cannot take a short position in the foreign asset and this constraint is binding
– the creditors may be hedge funds who wish to limit their exposure – then
y ' (k ) − (n − 1)c' (k ) > r
Since agents would like to take a short position in the foreign asset but cannot the
economy resembles a closed economy and c j (t ) = d j (t ) so (5) simplifies to
H=
σ
σ −1
c j (t ) (σ −1) / σ + λ (t ){y (k (t )) − c j (t ) − (n − 1)c(k (t ))}
(5’)
with first order conditions
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c −j 1 / σ = λ
(a1’)
λ (δ + (n − 1)c' (k (t )) − y ' (k ) ) = λɺ
(a3’)
and the transversality condition.
lim λ (t )k (t )e −δt = 0
t →∞
Then again
cɺ j
cj
= −σ (δ + (n − 1)c' (k (t )) − y ' (k ) )
and
c j (t ) = c j (0)eσ ( rˆ −δ )t ,
rˆ = y ' (k ) − (n − 1)d ' (k )
(a6’)
and capital accumulates according to
27
t
k (t ) = k (0) + ∫ y (k ( s )) − nc(k ( s ))ds
(2’)
0
And from the transversality condition
lim λ (t )k (t )e
−δt
t →∞
= lim λ (0)e
t →∞
(δ − rˆ ) t
e
−δt
t


 k (0) + ∫ y (k ( s )) − nc(k ( s ))ds 


0


t

c j (0)(e (σ ( rˆ −δ ) − rˆ ) t − 1) 

= λ (0) lim k (0) + ∫ y (k ( s )) − (n − 1)c(k ( s ))e − rˆs ds −

t →∞ 
ˆ
ˆ
−
−
σ
r
δ
r
(
)
0


(a9’)
and similarly to before to ensure bounded utility we assume
σ (rˆ − δ ) − rˆ < 0
(a10’)
which means that in (a9’)
c j (0)e (σ ( rˆ −δ ) − rˆ ) t
28
converges to zero and so
t



c j (0) = (rˆ − σ (rˆ − δ ) lim k (0) + ∫ y (k ( s )) − (n − 1)c(k ( s ))e − rˆs ds 
t →∞
0


t


= (rˆ − σ (rˆ − δ ) lim k (0) + ∫ y (k ( s ))ds − (n − 1)c j (0)e (σ ( rˆ −δ ) t − rˆ ) s ds 
t →∞
0


t
c j ( 0)



= (rˆ − σ (rˆ − δ ) lim k (0) + ∫ y (k ( s ))ds − (n − 1)

t →∞
ˆ
ˆ
−
−
(
r
σ
(
r
δ
)
0


(using symmetry in consumption)
so
t

1
c j (0) = (rˆ − σ (rˆ − δ )  k (0) + lim ∫ y (k ( s ))ds 
t →∞
n
0

(a11’)
29
which can be used in (a6’) to get consumption as a function of time. Substitution into (2’)
then gives a single first-order differential equation in the capital stock which can be
solved in principle after substituting specific functional forms for y (k ) and using
c j (t ) = d j (t ) for all creditors.
30