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Unformatted text preview: Demange (2012) and
Acemoglu, Ozdaglar, and TahbazSalehi (2013) generate greater contagion by making debts to ﬁnancial
institutions subordinate to other payment obligations. With priority given to outside payments, shocks
produce greater losses within the network. In practice, bank debt and bank deposits are owned by both
ﬁnancial institutions and nonﬁnancial ﬁrms and individuals, so characterizing seniority based on the
type of lender is problematic. 3.3 Contagion with truncated shocks In this section we shall show that the preceding results are not an artifact of the beta distribution:
similar bounds hold for a variety of shock distributions. Under the beta distribution the probability is
zero that a node loses all of its outside assets. One could easily imagine, however, that the probability
o
of this event is positive. This situation can be modeled as follows. Let Xi ≥ 0 be a primary shock 14 o
(potentially unbounded in size) and let Xi = ci ∧ Xi be the resulting loss to i’s outside assets. For
o
example Xi might represent a loss of income from an employment shock that completely wipes out i’s outside assets. Assume that the primary shocks have a joint distribution function of form
F o (xo , ..., xo ) =
1
n H o (xo /ci ),
i (29) 1≤i≤n where H o is a distribution function on the nonnegative real line. In other words we assume that the
shocks are i.i.d. and that a given shock xo aﬀects every dollar of outside assets ci equally. (We shall
i
consider a case of dependent shocks after the next result.)
In general a random variable with distribution function G and density g is said to have an increasing
failure rate (IFR) distribution if g (x)/(1 − G(x)) is an increasing function of x. Examples of IFR
distributions include all normal, exponential, and uniform distributions and, more generally, all logconcave distributions. Observe that truncating the shock can put mass at ci and thus assign positive
probability to a total loss of outside assets.
Theorem 2. Assume the primary shocks are i.i.d. and IFRdistributed, and that the net worth of every
node is initially nonnegative. Let D be a nonempty subset of nodes and let i ∈ D. Contagion from i to
/
D is impossible if
wj > wi βi (λi − 1)
j ∈D and it is weak if
wj > wi βi
j ∈D λi /λj . (30) j ∈D Corollary 3. Assume that all nodes hold the same amount of outside assets ci ≡ c. Under the assumptions of Theorem 2, contagion is weak from any node to any other set of nodes.
This is immediate upon rewriting (30) as
cj λ−1 /
j
j ∈D Proof of Theorem 2 . λ−1 ≥ βi ci .
j
j ∈D Through relabeling, we can assume that the source node for contagion is i = 1 and that the infected nodes are D = {2, 3, . . . , m}. By Proposition 1 we know that contagion is weak
from 1 to D if
wj ) ≤ P (X1 > w1 )P (X2 > w2 ) · · · P (Xm > wm ). P (X1 > w1 + (1/β1 )
2≤j ≤m 15 (31) o
Since X1 = c1 ∧ X1 , the lefthand side is zero when w1 + (1/β1 ) 2≤j ≤m w...
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This document was uploaded on 02/20/2014 for the course ECON 101 at Pontificia Universidad Católica de Chile.
 Spring '11
 Traferri

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