Example: Suppose a consumerโs utility function is given by U ( X ,Y ) = ๐ฟ^๐ ๐^๐ . Find MRSX,Y
To find the marginal rate of substitution (MRS) between X and Y, we need to calculate the derivative of the utility function with respect to X divided by the derivative of the utility function with respect to Y:
MRSX,Y = ๐ฟU/๐ฟX / ๐ฟU/๐ฟY
First, let's find ๐ฟU/๐ฟX:
๐ฟU/๐ฟX = 4X^3 ๐^2
Next, let's find ๐ฟU/๐ฟY:
๐ฟU/๐ฟY = 2X^4 ๐
Now we can find MRSX,Y:
MRSX,Y = (4X^3 ๐^2) / (2X^4 ๐)
Simplifying the expression, we get:
MRSX,Y = 2X^3 / Y
To find the Marginal Rate of Substitution (MRS) between X and Y in the utility function U(X,Y) = X^4 * Y^2, we need to take the partial derivatives of U(X,Y) with respect to X and Y.
The MRSX,Y is defined as the negative ratio of the derivative of U with respect to X over the derivative of U with respect to Y:
MRSX,Y = -dU/dX / dU/dY
Step 1: Finding the derivative of U with respect to X (dU/dX):
To find dU/dX, we differentiate U(X,Y) with respect to X while treating Y as a constant. Differentiating X^4, we get 4*X^(4-1) = 4*X^3. Since we are treating Y as a constant, the derivative of Y^2 with respect to X is 0. Therefore, dU/dX = 4*X^3.
Step 2: Finding the derivative of U with respect to Y (dU/dY):
To find dU/dY, we differentiate U(X,Y) with respect to Y while treating X as a constant. Differentiating Y^2, we get 2*Y^(2-1) = 2*Y^1 = 2*Y. In this case, Y is not a constant, so we also need to include the derivative of the inside function X^4 with respect to Y. Since X^4 is a constant with respect to Y, the derivative is 0. Therefore, dU/dY = 2*Y.
Step 3: Calculating the MRSX,Y:
Now we can calculate the MRSX,Y using the derivatives we found:
MRSX,Y = -dU/dX / dU/dY
= -(4*X^3) / (2*Y)
= -2*X^3 / Y
So, the Marginal Rate of Substitution (MRS) between X and Y in the utility function U(X,Y) = X^4 * Y^2 is -2*X^3 / Y.