Question

​U(X,Y)equals=20Xplus+80Yminus−Upper X squaredX2minus−2Upper Y squaredY2 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 ​$22. He plans to spend ​$6565 on both forms of entertainment. Determine the number of CDs and video rentals that will maximize​ Maurice's utility.

Answer 1

Answer:

The number of CDs = 111.36

The number of movie videos = 242.72

N/B: I choose not to round up the answers.

Explanation:

The method used is the Lagrangian method. Basically, the optimization problem we are trying to solve is  the utility function $u(x,y) = 20x+80y -x^2 -y^2$

subject to the constraint

$11x + 22y = 6565$.

So the optimization problem(Lagrangian) is

$\Delta = 20x + 80y -x^2 -y^2- \lambda(11x+22y-6565)$,

where $\lambda$ is a constant called the Lagrange multiplier.

To find the optimal consumption, we need to maximize the Lagrangian with respect to the variables $x,y,\lambda$. This we do by differentiating $\Delta$ with respect to each variable and then equate to 0.

$\Delta_x : 11\lambda = 20 - 2x ........................(1) \\\Delta_y: 11\lambda = 40 -y .........................(2) \\\Delta_\lambda = 11x + 22y = 6565............................(3)$

Equate (1) and (2), to get $y = 20+2x$ and substitute into (3) to get $x = 111.36$. Substituting $x = 111.36$ into $20+2x$ to get the corresponding value of $y$.