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frez [133]
3 years ago
13

​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.
Business
1 answer:
ankoles [38]3 years ago
7 0

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.

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