Question

Natalie consumes only apples and tomatoes. Her utility function is U(x, y) = x 2y 8 , where x is the number of apples consumed and y is the number of tomatoes con-sumed. Natalie’s income is $320, and the prices of apples and tomatoes are $4 and $3, respectively. How many apples will she consume?

Answer 1

Answer:

16 apples.

Step-by-step explanation:

$U(x, y) = x^2y^8$

$MU_x =\frac{\partial U}{\partial x} = 2xy^8\\MU_y = \frac{\partial U}{\partial x} = 8x^2y^7$

Price of apples, Px=$4

Price of tomatoes,  Py=$3

Ratio of their Marginal Utilities

$\frac{MU_x}{MU_y} = \frac{Px}{Py}$

$\frac{y}{4x} = \frac{4}{3}$

$y=\frac{16x}{3}$

Since Natalie’s income is $320

320 = xPx+yPy

$320=4x + 3*\frac{16x}{3} \\320= 4x+16x = 20x\\x=\frac{320}{20} = 16\\Also\\y=\frac{16x}{3}=\frac{16X16}{3}=\ 85.33$

Since x=16, Natalie will consume 16 apples.