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) = 24 if y is 1 or greater. The price of tapes is $4 and she can easily afford to buy dozens of tapes. How many tapes will she buy?

Answer 1

Answer:

The answer is 9 tapes

Explanation:

U(x,y,z)=x+z^1/2*f(y)

Since Daisy has one tape recorder, f(y)=24. Utility function becomes

U(x,y,z)=x+24*z^1/2=x+24z^1/2

Marginal utility from x=MUX=dU/dx=1

Marginal utility from z=MUY=dU/dz=12/z^1/2

Utility maximization requires that

MUX/MUY=PX/PY

1/[12/z^1/2]=1/4

z^1/2=3

Taking square at both sides

(z^1/2)²=(3)²

z=9 (answer)