Mr. Mouse has no money for his vacation today, but after a year he will have $1000 for his vacation. He can borrow some money at an annual interest rate of 10% to go on vacation now, pay back the loan after one year and use the rest of the money for the vacation after one year as well. His total utility function is , where N is the amount he spends on the vacation now and L is the amount he spends on the vacation in one year. How much should he spend now to maximize his total utility?
Borrowing and spending N dollars now means paying back after one year, so later he can spend L = $1000 – 1.1 N dollars.
The total utility of spending dollars now and dollars later is:
The quadratic function has a maximum at:
He pays back and spends on vacations later.