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Unformatted text preview: alls with bankruptcy costs In the absence of bankruptcy costs, when a node fails its remaining assets are simply divided among
its creditors. To capture costs of bankruptcy that go beyond the immediate reduction in payments, we
introduce a multiplier γ ≥ 0 and suppose that upon a node’s failure its assets are further reduced by γ pi − (ci +
¯ pj aji − xi ) , (60) j =i up to a maximum reduction at which the assets are entirely wiped out. This approach is analytically
tractable and captures the fact that large shortfalls are considerably more costly than small shortfalls,
where the ﬁrm nearly escapes bankruptcy. The term in square brackets is the diﬀerence between node
i’s obligations pi and its remaining assets. This diﬀerence measures the severity of the failure, and the
¯
factor γ multiplies the severity to generate the knockon eﬀect of bankruptcy above and beyond the
immediate cost to node i’s creditors. We can think of the expression in (60) as an amount of value
destroyed or paid out to a ﬁctitious bankruptcy node upon the failure of node i. 23 The resulting condition for a payment vector replaces (3) with pi = pi ∧ (1 + γ )(ci +
¯ pj aji − xi ) − γ pi .
¯
j =i (61) + Written in terms of shortfalls si = pi − pi , this becomes
¯
sj aji − wi + xi ]+ ∧ pi .
¯ si = (1 + γ )[ (62) j =i Here we see explicitly how the bankruptcy cost factor γ magniﬁes the shortfalls.
Let Φs denote the mapping from the vector s on the right side of (62) to the vector s on the left
γ
side, and let Φp similarly denote the mapping of p deﬁned by (61). We have the following:
γ
Proposition 3. (a) For any γ ≥ 0, the mapping Φs is monotone increasing, bounded, and continuous
γ
n
on R+ , and the mapping Φp is monotone decreasing, bounded, and continuous. It follows that Φs has a
γ
γ ¯
least ﬁxed point s and Φp has a greatest ﬁxed point p = p − s. (b) For any s, Φs (s) is increasing in γ ,
γ
γ
and for any p, Φp (p) is decreasing in γ . Consequently, the set of default nodes under the minimal s and
γ
maximal p is increasing in γ . The ﬁxed point s (and p) is unique if (1 + γ )A has spectral radius less
than 1.
Proof of Proposition 3. Part (a) follows from the argument in Theorem 1 of Eisenberg and Noe (2001).
For part (b), write vi = ci + (pA)i − xi and observe that
Φp (p)i =
γ vi − γ (¯i − vi ), vi < pi ;
p
¯
pi
¯
otherwise, from which the monotonicity in γ follows. The maximal ﬁxedpoint is the limit of iterations of Φp
γ
starting from p by the argument in Section 3 of Eisenberg and Noe (2001). If γ1 ≤ γ2 , then the iterates
¯
of Φp1 are greater than those of Φp2 , so their maximal ﬁxedpoints are ordered the same way. But then
γ
γ
the set of nodes i for which pi < pi at the maximal ﬁxedpoint for γ1 must be contained within that for
¯
γ2 . The same argument works for Φs . Uniqueness follows as in the case without bankruptcy costs.
γ
This result conﬁrms that bankru...
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 Spring '11
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