There are 100 identical individuals on an island. At date 0, each individual has 1 dollar. 50 of them are early consumers and they would like to consume at date 1. The rest are late consumers who would like to consume at date 2. Individuals learn whether they are early or later consumers only at date 1. That is, at date 0, each individual only knows that there is 50% chance that she will be a late consumer. There are two investment projects in this economy. First project generates 1.4 dollars at date 2 for each dollar invested at date 0. The liquidation value of this project at date 1 is 1 dollar. In the case of liquidation, the project does not generate anything at date 2. The second project also requires 1 dollar investment at date 0 and it generates 1.5 dollars at date 2. This project has no liquidation value at date 1. In addition to these investment projects, each individual can also carry cash in his pocket from date 0 to date 1 or from date 1 to date 2.
a) Describe each individual’s optimal investment decision at date 0. What is the welfare (i.e., sum of payoffs of everyone) on the island?
b) A penniless banker just landed on the island. At date 0, he collects individuals’ money under the following contract: for each dollar collected he promises to pay 1.1 dollars if withdrawn at date 1 or “r” dollars if withdrawn at date 2. To be able to pay the required amount at date 1, the banker keeps 55 dollars (= 50 x 1.1) as cash at date 0 and she invests the remaining funds in one of the projects. What is the interest rate on this deposit contract (i.e., what is r)? Calculate banker’s profit. Assume that the banker has the bargaining power (i.e., if a depositor is indifferent between bringing his money to the banker or making the investment himself, he prefers bringing to the banker)
c) What is the welfare in the economy now.
d) At date 1, it turns out that the number of early consumers is 30 instead of 50. Moreover, suppose that you learned that you are a late consumer and you know that other late consumers are not withdrawing their money early. Would you withdraw your money at date 1? Show your calculations. Assume that at date 2, if there is more demand than the cash available in the bank, existing cash will be distributed equally to depositors.