Question

car ​as the following utility​ function: ​U(X,Y)equals10Upper X squaredY where X is her consumption of​ Blu-ray disks with a price of ​$40 and Y is her consumption of DVDs with a price of ​$10. She has ​$1800 with which to spend on the two goods. Determine the number of​ Blu-ray disks and DVDs that will maximize Cary​'s utility. Cary maximizes utility when she consumes nothing ​Blu-ray disks and nothing DVDs. ​(Enter your responses using integers.​)

Answer 1

She consumes 30 blue ray disk and 60 DVD respectively.

Explanation:

Consider the provided information.

`U(x,y)=10x^2y`

Calculate marginal utility of good

Differentiate with respect of x and y.

`MU_x=20xy`

`MU_y=10x^2`

Her consumption of​ Blu-ray disks (X) with a price of ​$40 and Y is her consumption of DVDs with a price of ​$10.

The ratio of marginal utility of good x to y is equal to the price ratio:

`(MU_x)/(MU_y)=(P_x)/(P_y)`

Substitute the respective values.

`(20xy)/(10x^2)=(40)/(10)`

`(2y)/(x)=4`

`2y=4x`

`y=2x`

She has ​$1800 with which to spend on the two goods.

Her consumption of​ Blu-ray disks (X) with a price of ​$40 and Y is her consumption of DVDs with a price of ​$10. She has ​$1800 with which to spend on the two goods.

Therefore,
`40x+10y=1800`

`4x+y=180`

Substitute the value of y in above equation.

`4x+2x=180`

`6x=180`

`x=30`

Thus,
`y=60`

She consumes 30 blue ray disk and 60 DVD respectively.