Suppose that there is limited commitment in the credit market, but lenders are uncertain about the value of collateral. Each consumer has a quantity of collateral H, but from the point of view of lender, there is a probability a that the collateral will be worth p in the future period, and probability 1 - a that the collateral will be worthless in the future period. Suppose that all consumers are identical.
(a) Determine the collateral constraint for the consumer, and show the consumer's lifetime budget constraint in a diagram.
(b) How will a decrease in a affect the consumer's consumption and savings in the current period, and consumption in the future period? Explain your results.