Тестовое задание
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 U(N, L) = N \times L, 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 N \times (1 + 10\%) = 1.1 \times N after one year, so later he can spend L = $1000 – 1.1 \times N dollars.
The total utility of spending N dollars now and 1000 - 1.1 \times N dollars later is:
U(N,\ 1000 - 1.1N) = N \cdot (1000 - 1.1N) = 1000N - 1.1N^2
The quadratic function has a maximum at:
N^* = \frac{1000}{2 \cdot 1.1} = \mathbf{454.55}
He pays back 454.55 \times 1.1 = 500 and spends L = 500 on vacations later.