Answer to Question #214385 in Microeconomics for Zebib Girmay

Question #214385

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


1
Expert's answer
2021-07-08T14:34:58-0400

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


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