Question

Georgina consumes only grapefruits and pineapples. Her utility function is U (x comma y )equals x to the power of 0.8 end exponent y to the power of 0.2 end exponent, where x is the number of grapefruits consumed and y is the number of pineapples consumed. Georgina’s income is $105, and the prices of grapefruits and pineapples are $4 and $3, respectively. How many grapefruits will she consume?

Answer 1

Answer:

21

Explanation:

Given that:

The utility function U(x, y) = $x^{0.8} y^{0.2}$

The budget line income is:

105=4x +3y

The equation MRTS is:

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

where;

$MU_x(x,y) = 0.8 \times x^{0.8-1}\times y^{0.2} \\ \\ \implies 0.8 \times x^{-0.2}\times y^{0.2}$

$MU_y(x,y) = 0.2 \times x^{0.8}\times y^{0.2-1} \\ \\ \implies 0.8 \times x^{0.8}\times y^{-0.8}$

and:

$P_y= 3$

$P_x = 4$

∴

Using the equation MRTS:

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

$\dfrac{ 0.8 \times x^{-0.2}\times y^{0.2} }{0.8 \times x^{0.8}\times y^{-0.8}} = \dfrac{4}{3}$

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

4x = 12y

x = 12y/4

x = 3y

Replacing the value of x into the budget line income, we have:

105 = 4x + 3y

105 = 4(3y) + 3y

105 = 12y + 3y

105 = 15y

y = 105/15

y = 7

Then, from x = 3y

x = 3(7)

x = 21

Thus, she will consume 21 gapefruits