Mukesh spends all his $100 weekly income on two goods, X and Y .His utility function is given by U(X , Y ) = XY . If PX = 4 andPY = 10, how much of each good should he buy?
Question
Mukesh spends all his $100 weekly income on two goods, X and Y .His utility function is given by U(X , Y ) = XY . If PX = 4 andPY = 10, how much of each good should he buy?
Solution
To solve this problem, we need to use the concept of utility maximization subject to a budget constraint.
Step 1: Set up the budget constraint Mukesh's budget constraint is 4 and $10 respectively. So, 4X + 10Y = 100.
Step 2: Set up the utility function Mukesh's utility function is U(X, Y) = XY.
Step 3: Use the method of Lagrange multipliers to maximize the utility function subject to the budget constraint. The Lagrangian is L = XY - λ(4X + 10Y - 100), where λ is the Lagrange multiplier.
Step 4: Take the partial derivatives of the Lagrangian with respect to X, Y, and λ, and set them equal to zero to find the optimal values of X, Y, and λ. ∂L/∂X = Y - 4λ = 0 ∂L/∂Y = X - 10λ = 0 ∂L/∂λ = 4X + 10Y - 100 = 0
Step 5: Solve the system of equations From ∂L/∂X = 0, we get Y = 4λ Substitute Y into ∂L/∂Y = 0, we get X = 10λ Substitute X and Y into ∂L/∂λ = 0, we get 410λ + 104λ = 100, which simplifies to λ = 1
Step 6: Substitute λ = 1 back into the equations for X and Y to find the optimal quantities of X and Y. X = 101 = 10 Y = 41 = 4
So, Mukesh should buy 10 units of good X and 4 units of good Y to maximize his utility given his budget constraint.
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