Maurice has the following utility function: U(X,Y)-20X+80Y - x2-2Y2 where X is h
ID: 1205512 • Letter: M
Question
Maurice has the following utility function: U(X,Y)-20X+80Y - x2-2Y2 where X is his consumption of CDs with a price of $1 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. Maurice's utility is maximized when he consumesCDs andmovie videos. (Enter your responses using integers.) es CDs and movie videos. (Enteryo r repo ser Maurice's lity s m ized when he ons sin g egers.Explanation / Answer
1. U = 20X + 80Y - X^2 - 2Y^2
dU/dX = 20 - 2x
Puting dU/dX = 0
dU/dX = 20 - 2x = 0
X = 20/2 = 10
dU/dY = 80 - 4Y
Puting dU/dy = 0
dU/dY = 20 - 2Y = 0
Y = 80/4 = 20
So consumes 10 CDs and 20 Movie Videos
2.
Maximize
U = 10X^2Y
Subject to Budget constraint
20X + 5Y = 1500
Setting up lagrange
L = 10X^2Y - (20X + 5Y - 1500)
dL/dX = 20XY - 20
dl/dY = 10X^2 - 5
dL/d = 20X + 5Y - 1500
Putting equal to zero
dL/dX = 20XY - 20 = 0
dl/dY = 10X^2 - 5 = 0
dL/d = 20X + 5Y - 1500 = 0
Solving 1 and 2
=20 XY/20 = 10X^2/5
= XY = 2X^2
= Y = 2X
Putting in eq 3
20X + 5(2X) - 1500 = 0
30X = 1500
X = 50
Y = 2*50 = 100
So, Consumes 50 Blue ray discs and 100 DVDs