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

21

Step-by-step 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:

`(MU_x)/(MU_y ) =( Px)/(Py)`

where;

`MU_x(x,y) = 0.8 * x^(0.8-1)* y^(0.2) \\ \\ \implies 0.8 * x^(-0.2)* y^(0.2)`

`MU_y(x,y) = 0.2 * x^(0.8)* y^(0.2-1) \\ \\ \implies 0.8 * x^(0.8)* y^(-0.8)`

and:

`P_y= 3`

`P_x = 4`

∴

Using the equation MRTS:

`(MU_x)/(MU_y ) =( Px)/(Py)`

`( 0.8 * x^(-0.2)* y^(0.2) )/(0.8 * x^(0.8)* y^(-0.8)) = (4)/(3)`

`(4y )/(x) = (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