Ive been stuck on this forever now if someone could please walk me through it i was appreciate it:
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 don't know what hes asking or how to simplify it do i just solve for X and Y? help please!
I remember how to do this as "one-to-one" mapping problem, which can be solved graphically.
Let me try to solve with calculas and algebra.
CAVEAT EMPTOR - let the buyer beware.
First its MUx/Px=MUy/Py or MUx/MUy = Px/Py.
MUx,MUy are the first derivitive of U.
MUx = (2/3)X^-1/3 Y^1/3
MUy = (1/3)Y^-2/3 X^2/3
Put MUx over MUy (i.e. MUx/MUy) then collapse like terms. That is:
MUx/MUy = 2Y/X = Px/PY
Thus: Px = 2PyY/X
Let Z = total (fixed income). The guy spends all his income on X and Y. So,
Z=PxX + PyY so PyY = Z-PxX substitute this into the equation above. Thus:
Px = 2(Z-PxX)/X = 2Z/X - 2Px soooooo
Px = (2/3)Z/X a nice little demand function
QED
Thank you soo much im a little confused with the simplification so MUx/MUy simplifies to 2y over x ? and for some reason i thought that it would have to end in a demand function of X = something
Sure! I can help you with that. To derive John's demand function from his utility function, you need to find the conditions under which John maximizes his utility subject to his budget constraint.
Step 1: Write down the utility maximization problem:
Maximize U = X^(2/3) * Y^(1/3) subject to the budget constraint P_X * X + P_Y * Y = M, where P_X is the price of good X, P_Y is the price of good Y, and M is John's income.
Step 2: Set up the Lagrangian function:
Construct the Lagrangian function by including the budget constraint as a constraint in the utility maximization problem.
L = X^(2/3) * Y^(1/3) + λ(M - P_X * X - P_Y * Y), where λ is the Lagrange multiplier.
Step 3: Take partial derivatives:
Take the first partial derivative of the Lagrangian function with respect to X, Y, and λ, and set them equal to zero:
∂L/∂X = (2/3) * X^(-1/3) * Y^(1/3) - λ * P_X = 0
∂L/∂Y = (1/3) * Y^(-2/3) * X^(2/3) - λ * P_Y = 0
∂L/∂λ = M - P_X * X - P_Y * Y = 0
Step 4: Solve the partial derivative equations:
Solve these equations simultaneously to find the values of X and Y that maximize the utility function. Since you mentioned that the marginal utility of X is equal to the marginal utility of Y, you can use this condition as an additional equation to solve for X and Y.
By setting the two partial derivatives equal to each other, we have:
(2/3) * X^(-1/3) * Y^(1/3) / (1/3) * Y^(-2/3) * X^(2/3) = P_X / P_Y
Simplifying this equation yields:
2 * Y / X = P_X / P_Y
This equation represents John's demand function. It shows the relationship between the prices of the goods (P_X and P_Y) and the quantities demanded (X and Y) that maximize John's utility, given his income (M).
To further simplify or solve for X and Y, you would need to know either the prices of the goods or John's income (M). If any of these values are given, you can substitute them into the demand function to obtain specific values for X and Y.
I hope this explanation helps you understand the steps to derive John's demand function from his utility function. If you have any further questions, please let me know!