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Advocard [28]
3 years ago
8

George has $49 which he decides to spend on x and y. commodity x costs $5 per unit and commodity y costs $11 per unit. he has th

e utility function u(x, y) = 3x 2 + 6y 2 and he can purchase fractional units of x and y. george will choose
Mathematics
1 answer:
FromTheMoon [43]3 years ago
7 0

We are given the equations:

<span>5 x + 11 y = 49                    --> eqtn 1</span>

<span>u = 3 x^2 + 6 y^2               --> eqtn 2</span>

 

Rewrite eqtn 1 explicit to y:

11 y = 49 – 5 x

<span>y = (49 – 5x) / 11               --> eqtn 3</span>

 

Substitute eqtn 3 to eqtn 2:

u = 3 x^2 + 6 [(49 – 5x) / 11]^2

u = 3 x^2 + 6 [(2401 – 490 x + 25 x^2) / 121]

u = 3 x^2 + 14406/121 – 2940x/121 + 150x^2/121

u = 4.24 x^2 – 24.3 x + 119.06

Derive then set du/dx = 0 to get the maxima:

du/dx = 8.48 x – 24.3 = 0

solving for x:

8.48 x = 24.3

x = 2.87

 

so y is:

y = (49 – 5x) / 11 = (49 – 5*2.87) / 11

y = 3.15

 

Answer:

<span>George will choose some of each commodity but more y than x.</span>

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