Daisy received a tape recorder as a birthday gift and is not able to return it. Her utility function is U(x, y, z) = x + z f(y), where z is the number of tapes she buys, y is the number of tape recorders she has, and x is the amount of money she has left to spend. f(y) = 0 if y

Question

Daisy received a tape recorder as a birthday gift and is not able to return it. Her utility function is U(x, y, z) = x + z f(y), where z is the number of tapes she buys, y is the number of tape recorders she has, and x is the amount of money she has left to spend. f(y) = 0 if y < 1 and f(y) = 8 if y is 1 or greater. The price of tapes is $1 and she can easily afford to buy dozens of tapes. How many tapes will she buy?

Quizplus · Accepted Answer

Since Daisy can afford to buy dozens of tapes, we know that the marginal utility of tapes is positive. To maximize her utility, she should keep buying tapes until the marginal utility of tapes equals the price of tapes. The marginal utility of tapes is the partial derivative of her utility function with respect to z: MUz = ∂U/∂z = 11ea6c55_6e0e_6b37_9875_89a57022877f_TB6162_11 8. Since the price of tapes is $1, she should keep buying tapes as long as the marginal utility of tapes is at least $1. In this case, her marginal utility of tapes is $8, so she will continue buying tapes until MUz = $1. Solving for z, we get: 11ea6c55_6e0e_6b37_9875_89a57022877f_TB6162_11 8 = $1 z = 1/8 She can only buy a whole number of tapes, so she will buy 16 tapes (since she cannot buy 17 tapes with the money she has left).

Daisy Received a Tape Recorder as a Birthday Gift and Is