Can you check my answer
Consider an individual with a utility function of the form: U(x,y)=xy , and where m=1, px=2, and py=3 , where m is the individual's income or wealth, pi is the price of good i.
Calculate the individual's optimal consumption bundle, (x,y) = (Blank 1,Blank 2).
If your answer is a decimal, enter it to 1 decimal place (e.g. 5.5).
I got (x,y)=(.83,.06)
To check your answer, we can use the utility maximization principle. The individual's utility function is U(x, y) = xy, and the individual has an income of 1 (m = 1). The prices of good x and good y are given as px = 2 and py = 3, respectively.
To find the optimal consumption bundle, we need to maximize the individual's utility function subject to the budget constraint. The budget constraint can be expressed as: px * x + py * y = m.
Let's set up the problem mathematically:
Maximize U(x, y) = xy
Subject to: 2x + 3y = 1
To solve this problem, we can use the method of Lagrange multipliers. Define the Lagrangian function as:
L(x, y, λ) = xy + λ(1 - 2x - 3y)
Taking partial derivatives with respect to x, y, and λ, we get:
∂L/∂x = y - 2λ
∂L/∂y = x - 3λ
∂L/∂λ = 1 - 2x - 3y
Setting each partial derivative equal to zero, we have:
y - 2λ = 0 ...(1)
x - 3λ = 0 ...(2)
1 - 2x - 3y = 0 ...(3)
From equation (1), we find y = 2λ. Substituting this into equation (3), we get:
1 - 2x - 6λ = 0
Simplifying, we have -2x - 6λ = -1
From equation (2), we find x = 3λ. Substituting this into the equation above, we get:
-2(3λ) - 6λ = -1
-6λ - 6λ = -1
-12λ = -1
λ = 1/12
Substituting this value of λ back into equations (1) and (2), we find:
y = 2(1/12) = 1/6
x = 3(1/12) = 1/4
Therefore, the optimal consumption bundle is (x, y) = (1/4, 1/6).
Comparing this with your answer of (x, y) = (0.83, 0.06), it seems there was an error in your calculations. The correct answer is (1/4, 1/6), which can be rounded to one decimal place as (0.3, 0.2).
Please double-check your calculations to verify the correct answer.