A consumer consuming two commodities X and Y has the utility function
U(X,Y) XY 2X . The prices of the two commodities are 4 birr and 2 birr respectively. The
consumer has a total income of 60 birr to be spent on the two goods.
a) Find the utility maximizing quantities of good X and Y.
b) Find the
MRS X ,Y
at equilibrium
To find the utility maximizing quantities of goods X and Y, we need to find the point where the consumer's budget constraint is tangent to the indifference curve representing the highest possible utility.
The consumer's budget constraint can be expressed as:
4X + 2Y = 60
We can rewrite this equation as:
2X + Y = 30
Now, let's find the marginal utility of X and Y:
MUx = ∂U/∂X = Y + 2
MUy = ∂U/∂Y = X
If the consumer is maximizing utility, MUx/MUy = Px/Py, where Px and Py are the prices of goods X and Y respectively.
In this case, (Y + 2)/X = 4/2 = 2
Solving this equation, we get Y + 2 = 2X ----(1)
Now, let's substitute equation (1) into the budget constraint equation:
2X + (2X - 2) = 30
4X = 32
X = 8
Substituting X = 8 into equation (1):
Y + 2 = 2(8)
Y + 2 = 16
Y = 14
Therefore, the utility maximizing quantities of goods X and Y are X = 8 and Y = 14.
To find the marginal rate of substitution (MRS) of X and Y at equilibrium, we need to find the slope of the indifference curve at the utility maximizing point.
MRS X,Y = (∂U/∂X)/(∂U/∂Y) = (Y + 2)/X
Substituting X = 8 and Y = 14:
MRS X,Y = (14 + 2)/8 = 16/8 = 2
Therefore, the MRS X,Y at equilibrium is 2.