Question

there are 2 goods x & y has utility function v=x²+y³ two commodities 2 and 6 respectively & his/her budget is 40 what is the quantities of x & y what will be the maximize utility

Answer 1

Solution:

U = X² + Y³

Price of X = 2

Price of Y = 6

Budget constraint:

I = PxX + PyY

40 = 2X + 6Y

Utility is maximized at the point where: $\frac{MUx}{Px} = \frac{MUy}{Py}$

MUx = $\frac{\partial U} {\partial x} = 2X$

MUy = $\frac{\partial U} {\partial y} = 3y^{2}$

$\frac{MUx}{Px} = \frac{MUy}{Py}$

$\frac{2X}{2} = \frac{3y^{2} }{6 }$

$X = \frac{y^{2}}{2}$

Substitute in the budget constraint to get Y:

40 = 2X + 6Y

40 = ($\frac{y^{2}}{2}$) + 6Y

40 = Y² + 6Y

Y = 4

Substitute to get X:

X = $\frac{y^{2}}{2} = \frac{4^{2}}{2} = \frac{16}{2} = 8$

X = 8

U (x,y) = (8, 4)

The quantities of X and Y that will maximize quantity = 8 and 4