1: Suppose John had a utility function of U=X^2/3Y^1/3 . Derive Johns demand function from his utility function showing all the necessary steps.
i know that the MUx=MUy and first i derive the equation to
2/3X^-1/3 1/3Y^-2/3
then im stuck i need help with the simplification
see my answer to your earlier post.
To derive John's demand function from his utility function, we need to apply the concept of marginal utility and equate the marginal utilities of both goods, X and Y.
Let's start by taking the partial derivative of the utility function with respect to X, holding Y constant:
∂U/∂X = (2/3)X^(-1/3)Y^(1/3)
Next, we take the partial derivative of the utility function with respect to Y, holding X constant:
∂U/∂Y = (1/3)X^(2/3)Y^(-2/3)
Since John's demand for X and Y should equate their marginal utilities, we set the above two equations equal to each other:
(2/3)X^(-1/3)Y^(1/3) = (1/3)X^(2/3)Y^(-2/3)
Now let's simplify this equation. To do this, we can start by canceling out the common factors on both sides. In this case, we can cancel out (1/3), (X^(-1/3)), and (Y^(-2/3)):
2Y^(1/3) = X^(2/3)
Next, we isolate X by raising both sides of the equation to the power of 3/2:
(2Y^(1/3))^(3/2) = (X^(2/3))^(3/2)
Simplifying this further, we get:
2^(3/2) * (Y^(1/3))^(3/2) = X^2
Simplifying the exponents, we have:
2^(3/2) * Y^(3/2 * 1/3) = X^2
2^(3/2) * Y^(1/2) = X^2
Taking the square root of both sides of the equation:
X = (2^(3/2) * Y^(1/2))^0.5
Finally, we can simplify further:
X = (2^(3/2) * Y^(1/2))^0.5
X = (2^(3/2))^0.5 * (Y^(1/2))^0.5
X = 2^(3/4) * Y^(1/4)
So, the derived demand function for X from John's utility function is X = 2^(3/4) * Y^(1/4).