1.Given utility function U= PxX+PyY where PX = 12 Birr, Birr, PY = 4 Birr and the income of the consumer is, M= 240 Birr.
A. Find the utility maximizing combinations of X and Y.
B. Calculate marginal rate of substitution of X for Y (MRSX,Y) at equilibrium and interpret your result.
Solutions:
A. Utility function is given:
U= PxX+PyY
Putting the values of Px and Py in the utility function:
U= 12X + 4Y
Differentiate with respect to X:
"\\frac{dU}{dX}= MUx = 12"
Differentiate with respect to Y:
"\\frac{dU}{dY}=MUy=4"
At the equilibrium level:
"\\frac{MUx}{MUy}=\\frac{Px}{Py}"
"\\frac{12}{4}=\\frac{12}{4}"
"3=3"
Thus, consumer is in equilibrium condition
Budget function:
M = PxX + PyY
240 = 12X + 4Y ......Divide by 4
60 = 3X + Y
Thus, the equilibrium utility maximizing combinations of X and Y is :
3X + Y = 60
B. At the equilibrium level:
MRSxy = "\\frac{Px}{Py}"
="\\frac{12}{4}"
= 3
Thus, the marginal rate of substitution of X and Y (MRSxy) is 3.
Therefore, it means that the consumer is willing to give up 3 units of X to obtain an additional unit of Y at the same utility level.
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