Answer:
To find the optimum combination of goods using the Lagrange method, we need to set up the Lagrangian function:
L(X,Y,λ) = √(X^2 + Y^2) + λ(200 - 3X - 4Y)
where λ is the Lagrange multiplier.
We then take partial derivatives of L with respect to X, Y, and λ, and set them equal to 0 to find the critical points:
∂L/∂X = X/√(X^2 + Y^2) - 3λ = 0
∂L/∂Y = Y/√(X^2 + Y^2) - 4λ = 0
∂L/∂λ = 200 - 3X - 4Y = 0
Solving these equations, we get:
X/√(X^2 + Y^2) = 3λ
Y/√(X^2 + Y^2) = 4λ
200 = 3X + 4Y
Squaring the first two equations and adding them together, we get:
X^2 + Y^2 = 25λ^2
Substituting into the third equation, we get:
200 = 3X + 4Y = 3√(X^2 + Y^2)λ + 4√(X^2 + Y^2)λ = 5√(X^2 + Y^2)λ
Substituting X^2 + Y^2 = 25λ^2, we get:
200 = 5(25λ^2)^(1/2)λ = 125λ
λ = 200/125 = 1.6
Substituting λ back into the first two equations, we get:
X/√(X^2 + Y^2) = 3(1.6) = 4.8
Y/√(X^2 + Y^2) = 4(1.6) = 6.4
Squaring both equations and adding them together, we get:
X^2 + Y^2 = 20^2
Finally, substituting this into the budget constraint, we get:
3X + 4Y = 200
Solving these two equations simultaneously, we get:
X = 60 and Y = 80
Therefore, the optimal combination of goods is to buy 60 units of X and 80 units of Y, which costs 3(60) + 4(80) = 420 Birr and gives a utility of U(60,80) = √(60^2 + 80^2) ≈ 100.
Step-by-step explanation: