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
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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