Question

1.    Given the utility function of the form: U (x, y) = 4x2 + 3xy + 6y2: maximize utility subject to the budget constraint: x + y = 56. Then find the utility maximizing level of output x and y?

Answer 1

$U(x,y) = 4x^2 +3xy+6y^2$

Budget line

$56 = x+y \\ P(x)=1 \\ P(y) = 1$

At optimum : $\frac{MU(x)}{P(x)}= \frac{MU(y)}{P(y)}$

$MU(x) = \frac{ΔU}{Δx} = 8x+3y \\ MU(y) = \frac{ΔU}{Δy} = 3x+12y \\ \frac{8x+3y}{1} = \frac{3x+12y}{1} \\ 8x+3y=3x+12y \\ 5x=9y \\ x= \frac{9}{5}y = 1.8 y$

Putting x=1.8y in budget line

$56 = 1.8y+y \\ 56 = 2.8y \\ y = \frac{56}{2.8} = 20 \\ x= 36 \\ U(x,y)_{max} = 4 \times 36^2 + 3 \times 36 \times 20 + 6 \times 20^2 \\ = 5184 + 2160 + 2400 = 9744$