Question

4. Ethel is trying to decide whether to have 0 cars, 1 car, or 2 cars. If x is the number of cars she has and y is the amount of money she has per year to spend on other stuff, Ethel’s utility function is U(x, y), where U(0, y) = y^1/2, U(1, y) = (15/14)y^1/2, and U(2, y) = (10/9)y^1/2. Suppose that it costs $2,000 a year to have 1 car and $4,000 a year to have 2 cars. Ethel finds that the right thing to do depends on her income.

a.         What is her willingness to pay for 1 car if her income is M?

b.         What is the lowest income at which she would have a car?

c.         What is the lowest income at which she would have 2 cars?

Please explain why (containing the solving process)

Thank you!

Answer 1

Utility when 0 car= M^1/2.

Utility when 1 car= 15/14(M-2000)^1/2.

Increase in utility is  15/14(M-2000)^1/2.- M^1/2. This is the willingness to pay for 1 car.

The lowest income at which she will buy 2 cars has to satisfy the inequality 15/14(M-2000)^1/2> M^1/2.So 225/196(M-2000)>M. So 29M/196>2000. So M>2000*196/29. SO M>13517.24. This is the minimum income required to buy 1 car.

For 2 cars, 100/81(M-4000)>M. So 19/81M>4000. So M>4000*81/19. So M>17052.63. This is the minimum income to buy 2 cars.