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

The number of CDs = 111.36

The number of movie videos = 242.72

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

Step-by-step 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`.