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Maurice has the following utility function: U (X Y) =20X+80Y-X^2-2Y^2, where X is his consumption of CDs with a price of ​$11 and Y is his consumption of movie​ videos, with a rental price of ​$2. He plans to spend ​$50 on both forms of entertainment. Determine the number of CDs and video rentals that will maximize​ Maurice's utility.

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Answer:

1 CD and 19 movie videos

Explanation:

This is a quadratic programming problem. Given the utility function, product price and budget constraint. the following relation between X and Y is:

[tex]Y =\frac{50-11X}{2}[/tex]

When that is inserted in the utility function, the function is:

[tex]U(X) = -X^{2} -420X+200-\frac{121X^{2}-1100X+2500 }{2}[/tex]

In order to find the maximization parameter X, the first derivative of the function is needed (leveled with zero), and it is:

[tex]-123X + 130 = 0[/tex]

The value for X is 1,06 which can be rounded to 1. From the first relation, we see that Y is 19.