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 combinations of X and Y, we need to find the bundle of X and Y that maximizes utility subject to the consumer's budget constraint.
First, let's set up the budget constraint:
PX * X + PY * Y = M
12 * X + 4 * Y = 240
Now, let's rearrange the equation to solve for X:
X = (240 - 4 * Y) / 12
Next, we substitute this expression for X into the utility function to get a function of Y only:
U = 5 * sqrt(X) * sqrt(Y)
U = 5 * sqrt((240 - 4 * Y) / 12) * sqrt(Y)
To maximize utility, we take the derivative of U with respect to Y and set it equal to zero:
dU/dY = 0
Now, let's calculate the derivative and set it equal to zero:
(5/2) * sqrt((240 - 4 * Y) / 12) / sqrt(Y) - (5/2) * sqrt((240 - 4 * Y) / 12) / (2 * sqrt(Y)) = 0
Simplifying this equation, we get:
(5/2) * sqrt((240 - 4 * Y) / 12) / sqrt(Y) - (5/4) * sqrt((240 - 4 * Y) / 12) / sqrt(Y) = 0
Combining like terms, we have:
(5/4) * sqrt((240 - 4 * Y) / 12) / sqrt(Y) = 0
Multiplying both sides by 4/5, we get:
sqrt((240 - 4 * Y) / 12) / sqrt(Y) = 0
Squaring both sides, we have:
(240 - 4 * Y) / 12 = 0
Simplifying, we get:
240 - 4 * Y = 0
Solving for Y, we find:
Y = 60
Now, we can substitute this value of Y back into the budget constraint to find the value of X:
12 * X + 4 * 60 = 240
12 * X + 240 = 240
12 * X = 0
X = 0
Therefore, the utility maximizing combination of X and Y is X = 0 and Y = 60.
To calculate the marginal rate of substitution of X for Y (MRSX,Y) at equilibrium, we need to take the derivative of the utility function with respect to X and divide it by the derivative of the utility function with respect to Y:
MRSX,Y = (∂U/∂X) / (∂U/∂Y)
For our utility function, the marginal utilities are given by:
∂U/∂X = 5 / (2 * sqrt(X))
∂U/∂Y = 5 / (2 * sqrt(Y))
Substituting the equilibrium values of X = 0 and Y = 60, we get:
∂U/∂X = 5 / (2 * sqrt(0))
∂U/∂Y = 5 / (2 * sqrt(60))
Since the square root of 0 is undefined, we cannot calculate the MRSX,Y at equilibrium.
Interpretation: The marginal rate of substitution (MRSX,Y) represents the amount of Y that a consumer is willing to give up in order to obtain one more unit of X while keeping utility constant. In this case, the MRSX,Y is undefined, meaning the consumer is not willing to give up any amount of Y to obtain more X. This can be interpreted as the consumer having a strong preference for Y over X and is not willing to trade any amount of Y for X at the given prices and income.