# Proof of Proposition

In document Persistent gaps and default traps (Page 40-44)

The proof establishes that the strategies are optimal given beliefs and other player’s strategy and beliefs are consistent with observed choices. Step 1 begins by assuming the the optimality of the borrower’s repayment choice in period 1, and establishes the optimality of subsequent choices. Step 2 confirms the optimality of period-1 repayment decision rule.

Step 1. Assume that the borrower’s decision rule at t = 1 is to repay if and

only if 1 exceeds some arbitrary thresholde1, that is, if 1 ≥ e1. At t = 2,

contingent on history h, with repayment obligation Dh2, the borrowers’s net payoff to repayment is ˜Y2−D2h, while sanctions and partial recovery of debt

following default leave it with (1−s) ˜Y2−cD2h. Clearly, in period 2 repayment

is rational if and only if ˜Y2 ≥[(1−c)/s]Dh2 ≡Y2∗h.

If the lender believes, as assumed above, that the borrower repays iff

1 ≥e1, it updates its prior beliefs Φ(1) as follows. Default signals that the

persistent shock was drawn from the lower tail of the distribution, truncated at e1 so that the posterior density function is given by

γd(1|e1) =

( φ(1)

Φ(e1) for 1 < e1

0 for 1 ≥e1

γr(1|e1) =

(

0 for 1 < e1

φ(1)

1−Φ(e1) for 1 ≥e1

Let Γh(1|e1) denote the associated cumulative distribution functions. Ob-

serve that Γd(1|e1) = Φ(1)/Φ(e1) is decreasing in e1, while Γr(1|e1) =

Φ(1)/(1−Φ(e1)) is increasing in e1. This difference will matter for our

results.

The lender’s strategy at t = 0 is to set a price that allows it to break even given the probability of default Φ(e1). Its strategy at t = 1 de-

pends on the expected default probabilities, which in turn depend on beliefs about future output consistent with posterior distributions over the persis- tent shock. Given distributions Gh( ˜Y2|e1) (over period-2 output) consistent

have πh

2(e1) =Gh((1−sc)D2h|e1). Substituting in equation (3), which specifies

the investment requirement in period 1, the bond issue Dh2 must satisfy [1−(1−c)πh2(D2h)]D2h =RfI1. (12)

Some useful properties follow directly from the Bayesian updating rule.

Lemma 1 The default premium is positive.

To see why note that, given persistence,Gr(·), the distribution of period-2

output conditional on repayment, dominatesGd(·) in the first-order stochas-

tic sense.27 This implies πr

2(D2) < πd2(D2) for any given D2. From (12) it

follows that Dd

2 > Dr2. As default probabilities are increasing in the amount

borrowed, we must have πr2(Dr2)< πd2(Dd2). Finally, using equations (5) and (6), it follows that bond prices are lower contingent on default (pd

1 < pr1), or

equivalently the default premium idir is positive. Lemma 2 Dd

2 is decreasing in e1 while D2r is increasing in e1.

Note thatDh

2 depends one1 as this conditions Γh and, through that,Gh.

Observe that Γd(·|e01)≤ Γd(·|e1) for e01 > e1 and so also Gd(·|e01) ≤Gd(·|e1).

This implies πd

2 is decreasing in e1 and consequently Dd2 is decreasing too.

In contrast, for e01 > e1 the distribution Γr(·|e01) ≥ Γr(·|e1), and so also

Gr(·|e01)≥Gr(·|e1): this implies that πr2 and D2r are increasing in e1.

Step 2. We now establish the existence of ane∗1consistent with Step 1, and the optimality of the borrower’s repayment decision rule in period 1. Consider any arbitrary threshold e1 such that the borrower defaults in period 1 if

1 < e1. The continuation payoff following action h for realization 1 is

V2h(1, e1) =

Z

max[ ˜Y2−Dh2,(1−s) ˜Y2−cD2h]dF|1( ˜Y2). (13)

Note that Vh

2 depends on 1 (as borrowers condition the distribution

F|1( ˜Y2)) of future income on the known realization of the persistent shock)

and on the threshold for default e1 (as this affects D2h). When choosing h

in period 1, the borrower takes into account the immediate payoff and the discounted value of the continuation payoff, Vh

2 . Thus repayment has payoff

V1r(1, e1) = ( ˜Y1−D1) +βV2r(1, e1), (14)

27A distributionA(x) is said to dominate distributionB(x) in the first-order stochastic

while default has payoff

V1d(1, e1) = ( ˜Y1−cD1) +βV2d(1, e1). (15)

Define g(1, e1) = V1r−V1d = β[V2r(1, e1)−V2d(1, e1)]−(1−c)D1(e1).

Note that β[Vr

2(1, e1)−V2d(1, e1)] represents the gains from repayment in

terms of future financial savings (given a positive default premium). On the other hand (1−c)D1(e1) represents the gains from default in terms of current

savings. To prove the optimality of the borrower’s strategy we show that (i)

g(1, e1) is increasing in its first argument,1; (ii) there exists ane∗1 such that

g(e∗1, e∗1) = 0. Together these imply that g(1, e∗1) ≥ 0 for 1 ≥ e∗1, so that

β[Vr

2(1, e1)−V2d(1, e1) ≥ (1−c)D1(e1). That is, it is rational to repay iff

1 ≥e∗1.

(i) As (1−c)D1(e1) does not vary with 1, it is sufficient to show that

β[V2r(1, e1) −V2d(1, e1)] is increasing in 1. Consider the following

partition of the support of ˜Y2, conditional on 1. Define EL = {Y˜2 :

˜

Y2 < Y2∗r} as the set of realizations of future output for which the

borrower will default in period 2 regardless of previous history; for

EH ={Y˜2 : ˜Y2 ≥Y2∗d}, the borrower repays regardless of default history,

and EM = {Y˜2 : Y2∗r ≤ Y˜2 < Y2∗d}, the realization in which prior

repayment induces future repayment and prior default induces future default. Evaluating β[Vr

2 −V2d] in each element of this partition, we

obtain β    Z EL c[D2d−Dr2]dF + Z EM [sY˜2+cDd2 −D r 2]dF + Z EH [D2d−D2r]dF   

F|1(·) is increasing in 1. Further, each of the integrands in the above

expression is positive and increasing. Since the default premium is positive with fixed borrowing needs we have thatD2d−Dr2 >0. Finally, notice that ˜Y2−D2r > (1−s) ˜Y2−cDr2 > (1−s) ˜Y2 −cDd2, hence the

integrand in the middle region is also positive. This proves thatg(1, e1)

in increasing in 1.

(ii) We prove the existence of an e∗1 such that g(e∗1, e∗1) = 0 in three steps. (ii.a) The immediate gain from default, (1−c)D1(e1), is increasing ine1. It

to zero) and from above by ((1−c)/c)RfI0 (when default is almost

sure event). This follows from equation (8). (ii.b) The future gain from repayment, β[Vr

2(1, e1)−V2d(1, e1)] is decreas-

ing in e1. First, observe that, from by definition (13), V2h(1, e1) is

decreasing in Dh2. Next, by Lemma 2, D2d is decreasing in e1 while Dr2

is increasing in e1. Combining these two, we have V2r decreasing in e1

and Vd

2 increasing ine1, so β[V2r−V2d] is decreasing in e1.

(ii.c) Since the functions (1−c)D1(e1) and β[V2r(1, e1)−V2d(1, e1)] are con-

tinuous, a value e∗1 exists provided only that that β is not too low relative to other parameters.28

### Proof of Proposition 2

For any givene1 an increase inρincreases the informational value of default.

To see why note that the lender’s distributionGd( ˜Y2d;ρ), written as a function

of ρ satisfies the following property: Gd( ˜Y2d;ρ) ≥ Gd( ˜Y2d;ρ0) for ρ > ρ0. In

words, observed default in period 1 leads to greater pessimism about future returns to bondholders for ρ0 > ρ. This implies a higher πd

2, so required D2d

is increasing in ρ. On the other hand, Gr( ˜Y2r;ρ) ≤ Gr( ˜Y2r;ρ0) for ρ > ρ0, so

that πr

2 and Dr2 are decreasing in ρ: Observed repayment suggests a more

optimistic outlook for future repayments. Thus, for given e1, a higher value

of ρ is associated with a higherβ[V2r(1, e1)−V2d(1, e1)]. Finally, remember

that by definition (13)Vh

2 (1, e1) is decreasing inDh2, and that from equation

(3), Dh

2 is decreasing in ph1. All these facts together imply that an increase

in ρ generates an increase in the default premium as stated.

From the above argument an increase inρimplies that at the equilibrium

e∗1 the gain from repayment exceeds the gain from default. Given that the gain from default, (1−c)D1(e1), is increasing in e1, in order to restore equi-

librium, the equilibrium threshold e∗1 must rise. The probability of default in period 1, given by Φ(e∗1) rises as well. By the break-even condition, this implies that sovereign spreads in period 0 must rise.

28For an ‘interior’ solution to exist, informational content from default should be suf-

ficiently valuable – this is the case when future investment needs I1 are large relative to I0. Our simulations, not reported here, suggest that equilbria can be found for a plausible

### Appendix 2: Data and Sources

In document Persistent gaps and default traps (Page 40-44)