In period 1, the commercial bank meets withdrawals by borrowing b on it remaining credit line abroad, and then drawing on the emergency credit from the Central Bank. Because the emergency credit is unlimited, the bank does not close. other
Withdrawing depositors then go to the Central Bank, which is committed to selling them dollars at a unity exchange rate. To honor its commitment, the Central Bank uses first the dollars obtained from the commercial bank, and then liquidates the domestic asset up to the limit l+\ this implies that, just as under a currency board, the maximum quantity of dollars that the Central Bank can sell in period 1 is b + rl+. After the Central Bank sells this quantity of dollars, it stops selling more 33; if this happens while there are still agents attempting to purchase dollars, we say that there is a “balance of payments crisis.”
Under these assumptions one can show that if all depositors act honestly the social optimum is implemented. Now, the same conditions under which there was a bank run in the previous regime imply that there is a balance of payments crisis with a lender of last resort. In particular, there is an equilibrium in which all depositors claim to be impatient and the Central Bank is unable to service all dollar demands in period 1 if and only if 15 holds.
A sketch of why 15 is a sufficient condition is as follows (necessity can be proven along the lines of the argument given in the appendix). If all depositors claim to be impatient in period 1, they withdraw their fimds and run to the Central Bank with x pesos. The Central Bank obtains b dollars from the bank (which uses the remainder of its available international credit), and rl+ from liquidation of the long term investment it now owns. Given 15, the Central Bank will close while there are still some agents trying to buy dollars.
The preceding result reveals that, if the Central Bank acts as a lender of last resort, the vulnerability to a crisis is the same as with a currency board. The difference is whether the crisis is expressed as a bank run or a balance of payments crisis} but this has no economic importance. The intuition is that, when 15 holds, implementing the socially optimal allocation under fixed rates requires that the financial system as a whole be internationally illiquid in the following sense: given the commitment to fixed rates, the total amount of demandable debt of the system, 5, is a short term dollar obligation which exceeds the quantity of dollars that the system can tap in period l.