3. Given utility function U=
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
To find the utility-maximizing combination of X and Y, we need to equate the marginal utility per Birr spent on X and Y.
First, we need to calculate the marginal utility for X and Y using the given utility function:
MUx = dU/dX = 10X^(-1/2)
MUy = dU/dY = 2Y^(-1/2)
Next, we equate the marginal utilities:
MUx/Px = MUy/Py
(10X^(-1/2))/12 = (2Y^(-1/2))/4
Simplifying the equation, we have:
10X^(-1/2) = Y^(-1/2)
Now we can solve for Y in terms of X:
Y^(-1/2) = 10X^(-1/2)
Y = 100X
Since X + Y = M (the total income), we can substitute the value of Y into the equation:
X + 100X = 240
101X = 240
X = 240/101 ≈ 2.3762 Birr
Now we can calculate the value of Y:
Y = 100X
Y ≈ 237.62 Birr
Therefore, the utility-maximizing combination of X and Y is approximately X = 2.3762 Birr and Y = 237.62 Birr.
To calculate the marginal rate of substitution (MRSX,Y) at equilibrium, we need to differentiate the utility function with respect to X and Y:
dU/dX = 10X^(-1/2)
dU/dY = 2Y^(-1/2)
Now, we can calculate the MRSX,Y by taking the derivative of the utility function:
MRSX,Y = (dU/dX)/(dU/dY)
MRSX,Y = (10X^(-1/2))/(2Y^(-1/2))
Substituting the values of X and Y at equilibrium:
MRSX,Y = (10(2.3762)^(-1/2))/(2(237.62)^(-1/2))
MRSX,Y ≈ 5
The interpretation of the MRSX,Y is that the consumer is willing to give up 5 units of good Y to obtain one additional unit of good X, while keeping utility constant.