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^1/2 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^1/2 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? A)18 B)14 C)16 D)20 E)We need to know the price of tape recorders to solve this problem.

Quizplus · Accepted Answer

To maximize utility, Daisy should use all of her money. Therefore, we can set x = 0 in the utility function. The only variable left to determine is z (the number of tapes she buys). We can rewrite the utility function as: 
U(y, z) = 10z + U1(y) 
where U1(y) = f(y)/2 if y < 1 and U1(y) = 4 + f(y)/2 if y >= 1. 
To maximize U(y,z), we can take the derivative with respect to z: 
dU(y, z)/dz = 10 
This derivative is positive, which means that U(y, z) is increasing in z. Therefore, Daisy should buy as many tapes as possible. Since she can easily afford to buy dozens of tapes, we should calculate how many dozens of tapes she can buy without exceeding her budget. 
Her budget is $10 (because she can spend all of her money on tapes). Therefore, she can buy up to 10 tapes. This is less than one dozen, so the answer is 10.

Daisy Received a Tape Recorder as a Birthday Gift and Is