Suppose that the total utility function of a consumer is given by TU(x,y) = 3x2 y and the prices of X and Y are 1 Birr and 2 Birr per unit, respectively. If the income of the consumer is 600 Birr and if he spends all of his income on the consumption of commodities of X and Y, find the optimum amount of X and Y that the consumer will consume at equilibrium and find MRTSx,y.
Solution:
Derive the budget constraint:
I = PxX + PyY
600 = X + 2Y
The utility maximizing rule is where ("\\frac{MUx}{MUy}) = (\\frac{Px}{Py})":
TU(x,y) = 3x2y
MUx = "\\frac{\\partial U} {\\partial x} = 6xy"
MUy = "\\frac{\\partial U} {\\partial y} = 3x^{2}"
"\\frac{Px}{Py} = \\frac{1}{2}"
"\\frac{6xy}{3x^{2} } = \\frac{1}{2}"
"\\frac{2y}{x } = \\frac{1}{2}"
Y = "\\frac{x}{4}"
Substitute in the budget constraint:
600 = X + 2Y
600 = X + 2("\\frac{x}{4}")
Multiply both sides by 4:
2400 = 4X + 2X
2400 = 6X
X = 400
Y = "\\frac{x}{4}" = "\\frac{400}{4}" = 100
TU(x,y) = (400,100)
The optimum amount of X and Y that the consumer will consume at equilibrium = 400 and 100
MRTSxy = "\\frac{MUx}{MUy}"
MUx = 6xy
MUy = 3x2
MRTSxy = "\\frac{6xy}{3x^{2} }" = "\\frac{2y}{x }" = 2(100)/400 = "\\frac{1}{2 }"
MRTSxy = "\\frac{1}{2 }"
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