How do you find the optimal condition given u (x1,x2)= sqrt(x1) + sqrt(x2)
Income is 12, p1 is 2, and p2 is 1
To find the optimal condition given the utility function u(x1, x2) = sqrt(x1) + sqrt(x2), we need to maximize this utility function subject to certain constraints. In this case, the constraints are income (I), the price of good 1 (p1), and the price of good 2 (p2).
Given that the income (I) is 12 and the prices are p1 = 2 and p2 = 1, the budget constraint can be written as:
p1 * x1 + p2 * x2 = I
Substituting the given values, we get:
2x1 + x2 = 12
To maximize the utility function, we can use the method of Lagrange multipliers. Define the Lagrangian function (L):
L(x1, x2, λ) = sqrt(x1) + sqrt(x2) - λ(2x1 + x2 - 12)
To find the optimal condition, we need to solve the following equations simultaneously:
∂L/∂x1 = 0
∂L/∂x2 = 0
∂L/∂λ = 0
Differentiating L with respect to x1, x2, and λ, we get:
∂L/∂x1 = (1/2) * (1/sqrt(x1)) - 2λ
∂L/∂x2 = (1/2) * (1/sqrt(x2)) - λ
∂L/∂λ = -2x1 - x2 + 12
Setting the derivatives equal to zero and solving the resulting equations, we can find the values of x1, x2, and λ that satisfy the conditions for maximum utility.
After finding the values of x1 and x2, you can plug them back into the budget constraint (2x1 + x2 = 12) to ensure that the optimal condition also fulfills the constraint.
To find the optimal condition in this case, we need to maximize the utility function u(x1, x2) = sqrt(x1) + sqrt(x2) subject to the constraint that the total income spent on goods 1 and 2 cannot exceed the total income available.
Let x1 be the quantity of good 1 consumed, and x2 be the quantity of good 2 consumed. The total amount spent on goods 1 and 2 is given by p1*x1 + p2*x2, where p1 is the price of good 1 and p2 is the price of good 2.
Given that the income is 12, p1 is 2, and p2 is 1, we can set up the following optimization problem:
Maximize: u(x1, x2) = sqrt(x1) + sqrt(x2)
Subject to: p1*x1 + p2*x2 ≤ income
Step 1: Express the inequality constraint as an equation:
2*x1 + x2 ≤ 12
Step 2: Solve the inequality constraint for one of the variables:
x2 ≤ 12 - 2*x1
Step 3: Substitute the inequality constraint into the utility function:
u(x1, x2) = sqrt(x1) + sqrt(12 - 2*x1)
Step 4: Differentiate the utility function with respect to x1 to find the critical points:
du/dx1 = 0
Step 5: Solve the equation du/dx1 = 0 to find the critical points.
The derivative of sqrt(x1) + sqrt(12 - 2*x1) with respect to x1 is:
1/2*sqrt(x1) - (2/sqrt(12 - 2*x1))
Setting this derivative equal to zero and solving for x1:
1/2*sqrt(x1) - (2/sqrt(12 - 2*x1)) = 0
Multiply through by 2*sqrt(x1) to eliminate the square roots:
sqrt(x1)*sqrt(12 - 2*x1) - 4 = 0
Square both sides to eliminate the square roots:
x1*(12 - 2*x1) - 16 = 0
12*x1 - 2*x1^2 - 16 = 0
2*x1^2 - 12*x1 + 16 = 0
Apply the quadratic formula:
x1 = (-(-12) ± sqrt((-12)^2 - 4*(2)*(16))) / (2*(2))
x1 = (12 ± sqrt(144 - 128)) / 4
x1 = (12 ± sqrt(16)) / 4
x1 = (12 ± 4) / 4
Finding the values of x1:
x1 = (12 + 4) / 4 = 4
x1 = (12 - 4) / 4 = 2
Step 6: Substitute these critical points back into the inequality constraint to find the corresponding values of x2.
For x1 = 4:
x2 ≤ 12 - 2*x1
x2 ≤ 12 - 8
x2 ≤ 4
For x1 = 2:
x2 ≤ 12 - 2*x1
x2 ≤ 12 - 4
x2 ≤ 8
Step 7: Evaluate the utility function at each critical point by substituting the corresponding values of x1 and x2:
For x1 = 4, x2 = 4:
u(4, 4) = sqrt(4) + sqrt(4)
u(4, 4) = 2 + 2
u(4, 4) = 4
For x1 = 2, x2 = 8:
u(2, 8) = sqrt(2) + sqrt(8)
u(2, 8) = √2 + 2√2
u(2, 8) = 3√2
Step 8: Compare the utility values and determine the optimal consumption.
The utility is higher at u(4, 4) = 4 compared to u(2, 8) = 3√2. Therefore, the optimal consumption is x1 = 4 and x2 = 4.