Question

Using the Lagrangian method, solve the following problem. I.e., write down the Lagrangian, the FOCs, and find (x ∗ , y∗ ) and λ ∗ . Find f(x ∗ , y∗ ). Max L = lnx + lny subject to x + y = 1.

Answer 1

To solve this problem using the Lagrangian method, we set up the Lagrange equations and solve them to find the optimal solution. The Lagrangian is defined as L = ln(x) + ln(y), and the constraint equation is x + y = 1. By solving the Lagrange equations, we find that the optimal solution is (x*, y*) = (1/2, 1/2) and the corresponding value of the Lagrange multiplier is 1. Finally, substituting these values into the objective function, we find that f(x*, y*) = ln(1/4).

Step-by-step explanation:

To solve this problem using the Lagrangian method, we start by defining the Lagrangian as L = ln(x) + ln(y). The constraint equation is x + y = 1. Now, we set up the Lagrange equations:

1. ∂L/∂x - λ(∂f/∂x) = 0
2. ∂L/∂y - λ(∂f/∂y) = 0
3. x + y = 1

From the first equation, we have 1/x - λ/x = 0, which simplifies to 1 = λ. Similarly, from the second equation, we have 1/y - λ/y = 0, which again simplifies to 1 = λ. Substituting these values of λ into the third equation, we get x = y = 1/2. Therefore, the optimal solution is (x*, y*) = (1/2, 1/2) and λ* = 1.

Finally, we can find f(x*, y*) by substituting the values of (x*, y*) into the objective function. So, f(x*, y*) = ln(1/2) + ln(1/2) = ln(1/4).