Finally, since 4 and 5 hold with equality and liquidation is zero, the optimal investment strategy is given by
Next we discuss how this allocation can be implemented in a decentralized fashion.
Demand deposits and bank runs
The previous subsection identified the social optimum as the best allocation that, given the environment, the bank can achieve in principle. The bank must, in addition, find a mechanism to implement that allocation. One natural way, which will be the focus of this paper, is via demand deposits.
Demand deposits are contracts that stipulate that each agent must surrender her endowment and her capacity to borrow abroad to the bank in period 0. The bank invests к in the long term technology and borrows d in period 0 and b in period 1 . In return, the agent is promised the option to withdraw, at her discretion, either x units of consumption in period 1 or у in period 2.
We shall impose two additional assumptions on the problem. First, the bank must respect a sequential service constraint which requires, loosely speaking, that the commercial bank attend to the requests of depositors on a first come-first served basis. The existence of sequential service constraints can be justified by more primitive features of the environment, as suggested by Wallace (1988).
Second, for our benchmark case we will assume that the bank is committed to repay any foreign debt under all circumstances. This is not necessarily a realistic assumption, but it is the easiest to handle: it allows us to abstract, until the next section, from the possibility of foreign creditor panics. To ensure that the foreign debt is always repaid, the bank is committed to limit any possible period 1 liquidation of the long term investment to:
As a result of these assumptions, the timing of events is as follows. In period 1 depositors arrive to the bank in random order. Upon arrival, each agent reports may withdraw x if the bank is still open. The commercial bank services withdrawal requests sequentially, first by borrowing abroad (up to / — d), then by liquidating the long term investment up to the maximum if withdrawal requests exceed the maximum liquidation value of the bank, given by (/ — d) + r/+, the bank closes and disappears. Finally, if the bank did not close in period 1, in period 2 the bank liquidates all of its remaining investments, repays its external debt, and pays у dollars plus any profits to agents that did not retire their deposits in period 1.
Given the demand deposit system just described, depositors face a strategic decision about when to withdraw their funds; in other words, they are players engaged in an (anonymous) game. Hence the outcomes of a demand deposit system are given by the equilibria of such game; an equilibrium is a description of the strategies of each depositor and aggregate outcomes such that the aggregate outcomes are implied by the depositors’ strategies and each depositor strategy is optimal for her given the aggregate outcomes.