Constraint 3 restricts investment to be no larger than the endowment plus initial borrowing from abroad. Constraint 4 is the feasibility constraint in period 1. The social optimum assigns x units of consumption to each impatient; this is financed by period 1 borrowing abroad, 6, and possibly by liquidating some portion of the long term asset. Constraint 5 is the feasibility constraint for period 2 and 6 is the external credit ceiling, both of which are self-explanatory there.
Constraint 7 is the incentive compatibility or truth-telling constraint for patient agents, derived under the assumption that the commercial bank can monitor each agent’s transactions with the domestic banking system but not her consumption or her world transactions. If a patient agent lies about her type she will be given x units of consumption in period 1; given the assumption just stated, the best she can do then is to exchange them at the world market for x units of period 2 consumption. On the other hand, she can obtain у of period 2 consumption from telling the truth; hence 7 ensures that patient depositors will not lie.
Finally, 8 contains the obvious non-negativity constraints.
The study of the social planning problem yields useful properties of the solution. It can be shown that I — 0 -that is, there is no liquidation in period 1 of the long term investment. This should be obvious, since the bank faces no aggregate uncertainty, and liquidating the long term asset in period 1 is costly. Given this fact, it can be shown that the value of the social problem is superior to the value of autarky. This also should be intuitive, since the bank pools resources to prevent the inefficient liquidation of the long term asset; in contrast, the long term asset must be liquidated with positive probability in autarky.
The other features of the solution follow from marginal optimality conditions. It is easy to see that the credit ceiling 6 will bind at the optimum. Given this result, the choices of x and у must satisfy the following equation, which can be thought of as a social transformation curve:
where 9 is a coefficient in the unit interval. If the coefficient of relative risk aversion is exactly 1 (the case of log utility) we have 9 = A. In that case per capita consumption is у = Rw and x = w, and each set of consumers receives the technological return corresponding to their period of consumption; patient consumers consume more because investments kept until period 2 are more productive. If cr > 1 we have 9 > A, so that patient consumers who get to consume in the “high productivity” period cross-subsidize impatient consumers who get to consume in the “low productivity” period.