This Appendix derives the probability of default for the case in which the partial repayment function can be approximated by a linear curve. For simplicity of exposition we do it for the case in which employment nh is constant, and focuses on intermediation between foreign lenders and domestic banks.19
Suppose that in the relevant region domestic banks’ revenue is given by the linear approximation
with A, B > 0. The threshold value δ of the aggregate shock δ that makes banks indifferent between partial default and repayment is thus’
and the probability of bank default is
Using equation (14), we infer that, with g(δ)=l/2δ
Solving equation (A3) yields
that is, using equation (A2):
Using equations (A1) and (A3), we can infer that the slope of the borrower’s interest rate-bank cost of fund curve is
In general, raising the borrower’s interest rate has two opposing effects on the bank’s expected profits. First, the revenue goes up in states of nature where no default takes place, increasing expected profits. Second, the higher borrower’s interest rate increases the probability of default, thereby increasing the expected cost of financial intermedialion. In terms of equation (A5), the first effect is 1 - Pr(d/h) (per unit of loan), and the second is given by Ch/2Kh Bδm. Hence, in general, the borrower’s inieresi rale-bank cosi of credit curve is backward bending, with the backward-bending portion depicting an inefficient region. The analysis in the paper assumes that competition among banks implies that, if Pr(dlb) >0, the country operates on the upward-sloping portion of the curve. The denominator of equation (A5) is hence positive.
Using equations (A1) and (A2), we can also solve for the contractual interest rale facing domestic banks, r*h. in terms of Pr(d/b). Substituting the result in equation (A4) yields a quadratic equation for the probability of default:
Applying the implicit function theorem to this equation yields
which can be combined w’itb equation (20) to give equation (21).
Note also that, for δm given:
If creditors’ capacity to enforce partial repayment is small (that is, if K or Kh is small), or if the cost of financial intermediation is large enough, there is no internal solution—that is, the value of Pr that solves the quadratic equation given above is outside the [(),1] interval. Furthermore, for certain parameter values we may observe multiple equilibria, as is the case where there are two values of Pr. satisfying 0< Pr < 1. corresponding to low or high interest rate rates. Henceforth we assume that the model’s parameters are such that an internal equilibrium exists. Specifically, we assume that creditors’ bargaining power is large enough, and that the cost of financial intermediation is small enough to ensure that the probability of default is zero in the absence of aggregate volatility (δ = 0). and the probability of default is positive for large enough volatility. We also assume that, in the presence of multiple equilibria, the markcl chooses the equilibrium associated with the lower interest rate. This is also the equilibrium associated with the lower probability of defauli and the higher welfare level. It can be shown that these assumptions imply dial, in an internal equilibrium satisfying (0 < Pr < 1) equation (A6) is positive.
A similar analysis applies for the impact of higher volatility on the producer’s probability of default. The main difference between analyzing the partial effects ∂Pr(d/b) /∂δm and ∂Pr(d/b) /∂δm is that, as can be inferred from Proposition 2. higher volatility increases the cost of funds for domestic banks by
whereas higher volatility dues not affect the domestic banks’ expected cost oi’ funds on world capital markets (which is equal to the safe interest rale řbf). Adjusting for this effect, and assuming that the default repay mein Φ(.) is linear, it follows that
which can be rearranged to give
where Ω is defined by
Using equation (A6), this expression becomes
which can be substituted in equation (20) to give equation (21).
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Agenor, Pierre-Richard, and Joshua Aizenman, 1998, “Volatility and the Welfare Costs of Financial Market Integration” (unpublished: Washington: Economic Development Institute. World Bank. March).
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Pierre-Richard Agenor is a Senior Economist in the Research Department of the IMF. Joshua Aizenman is Champion Professor of International Economics at Dartmouth College. The authors would like to thank, without implication, Guillermo Calvo, Luis Catao, Robert Flood, Joseph Haimowitz, Martin Kaufman, Paulo Neuhaus, Maurice Obstfeld, Andrew Powell, Evan Tanner. Carlos Vegh, an anonymous referee, and participants at seminars at Dartmouth College, the University of Washington, and the WEA Conference held in Seattle (July 9-14) for helpful discussions and comments on an earlier version. Brooks Calvo provided excellent research assistance.
There appears to be some agreement that this shift in market sentiment was not entirely warranted by fundamentals. On the one hand, the real exchange rate had indeed appreciated substantially since the introduction of the Convertibility Plan in April 1991. and the current account deficit (as a share of output) was increasing. On the other hand, inflation was low and falling, output and exports were growing at a relatively high rate (with real GDP growing by more than 7 percent a year between 1991 and 1994), and ample liquid reserves appeared to be available to defend the fixed parity between the U.S. dollar and the peso.
In addition to these studies, Kaufman (1996) uses the Stiglitz-Weiss model of credit rationing (see Jaffee and Stiglitz, 1990) to argue ihai the credit crunch in Argentina resulted from an increase in the share of illiquid borrowers induced by the rise in interest rates, and increased incidence of adverse selection problems. Essentially, banks faced greater difficulties screening out between “safe” and “risky” borrowers because those borrowers mosi willing to pay a higher interest rate on loans were precisely those for which the potential risk of default had increased. Catao (1997, p. 6) estimates that problem loans had already exceeded 10 percent of the loan portfolio of all financial institutions by end-1994.
In contrast to some of the existing studies, our model is static and partial equilibrium in nature. In particular, we do not explicitly model consumption decisions or central bank regulations. Some of these features could be added at the cost of greater complexity, but without adding much insight.
Modeling domestic saving would be relevant under financial autarky, in which case all lending is supported by domestic saving. Adding a domestic saving schedule would not change the key results of this paper. In countries where the real interest rate that prevails in autarky is high relative to the international rate, the marginal source of funds is foreign saving. Henceforth we assume that funds are relatively scarce in the country under consideration, and that as a result the interest rate that prevails in autarky exceeds the world rate. See Agenor and Aizenman (1998) for a discussion of more general cases,
Although we use, throughout the paper, the expression “aggregate productivity shock” to refer to 5, a more general interpretation is to think of 8 as a composite (or reduced-form) shock to output. See the discussion in the concluding section.
The enforcement cost can be related, in particular, to the idea of costly state verification (see Townsend, 1979). That is, it costs C to verify the realization of e;, and to force the producer to repay accordingly. Although C is modeled as a fixed monitoring and enforcement cost per loan, the analysis can be extended to allow for a variable cost, proportional to the size of the loan, without changing the key results derived below. K could also be made endogenous. For earlier models of impeded creditworthiness with costly state verification in a related context, see Aizenman, Gavin, and Hausmann (1996), Bemanke and Gertler (1989), Boot, Thakor, and Udell (1991), Calvo and Kaminsky (1991), and Eaton (1986).
As is usual in the literature, repayment obligations are assumed not to be contingent on the state of nature. Loan agreements in practice typically specify a single contractual rate, which lenders are allowed to adjust only it’the borrower violates some specific terms of the contract.
Note that if the bargaining power of creditors is too low, so that β/K > 1, the probability of default is always positive and condition (4) is always violated.
Note that Φ(.) is a function of δ through the effect of that variable on yhas shown in equation (1).
That is, banks diversify away the i.i.d. risk.
Note that rL is taken as given by each producer in determining optimal employment. The reason is that we assume the existence of a large group of ex ante homogeneous producers; all of them are charged the same interest rate by lenders. As shown earlier, rt is determined in equilibrium by a break-even condition that internalizes all the information about the distribution of shocks—including idiosyncratic shocks.
An Appendix providing formal derivations of these results is available upon request. This proposition assumes that volatility is sufficiently small to warrant an internal solution, which is such that the country operates on the upward-sloping portion of the borrower’s interest rate-bank cost of funding curve. See the discussion in the Appendix, following equation (A5), for a formal statement of this condition.
A similar analysts would apply to labor, where the ultimate welfare effect associated with the fall in employment is given by the reduction in employment times the difference between the producers’ real wage and the supply price of labor.
Our discussion focuses on the welfare effects of volatility, comparing the welfare to the benchmark of integrated capital markets in the absence of volatility. To compare welfare with financial autarky, one should model the naiure of financial intermediation and domestic saving in autarky. See Agenor and Aizenman (1998) for such a comparison in a related framework.
The case of high values of C and Ch may be particularly relevant for Argentina. As discussed by Caiao (1997) and Powell, Broda, and Burdisso (1997), severe limitations to the seizure of collateral property still prevail in that country. Judicial aclions take lime, have uncertain outcomes, and are relatively costly—thereby affecting lending rates by raising the potential cost of default.
An example of supply complementarity is the case where producers use non-traded inputs provided by other producers. Default by some producers due to higher cost of credit will trigger default by other producers due to the increase in the price of needed inputs, implying that the increase in the cost of credit may trigger higher volatility of domestic productivity shocks of domestic producers. Similar examples of demand complementarities will arise if producers have some market power—as is the case in a world of monopolistic competition.
In Argentina, various other factors have also played a role in this process An increase in the perceived risk of confiscation of bank deposits—as occurred in December 1989, when the government, in an effort to reduce inflation, forced the conversion of time deposits and public sector debt into U.S. dollar-denominated government (BONEX) securities—and the fact that bank deposits were not insured certainly played a role in the bank run and the credit crunch that look place in earl) 1995 (see Catao, 1997). There may also have been increased doubts about the sus-lainability of full convertibility of current and capital account transactions, as well as perceived constraints on the lender-of-last-resort function of the central bank, under the currency board in place since the Convertibilitv Plan was introduced in early 1991.
Recall that in the expression determining the welfare effects of higher volatility, (equation (19)), changes in employment are of secondary importance by virtue of the envelope theorem.