Jimmy isn’t earning much money this year (only $20K) but he knows he’ll earn a lot of money next year ($70K). His utility function over consumption this year (c1) and consumption next year (c2) is given by u(c1, c2) = ln c1 + β ln c2, where β is a parameter indicating Jimmy’s patience. Jimmy has two ways of borrowing money: he can take out a loan at 50% interest, up to a maximum loan amount of $20K. (That is, if he borrowed the maximum amount of $20K, he would have to pay back $30K next year.) In addition, he can use a credit card that charges 100% interest (so for every dollar he borrows today, he has to pay back $2 in a year); there’s no limit to the amount he can charge on that card. If he saved money, he wouldn’t earn any interest on that savings.
(a) On a graph showing consumption this year vs. consumption next year, in thousands of dollars, plot Jimmy’s endowment point (label it W) and budget constraint.
(b) For what value(s) of β would Jimmy take out the maximum loan amount of $20K, but not charge anything to the credit card (i.e., set c1 = 40K)? Explain your reasoning.
(c) Suppose β = 0.2. What would Jimmy do? On the diagram from part (a), show his optimal consumption bundle (label it X∗ ), and the indifference curve passing through it.