Answer to Question #148246 in Microeconomics for Jyotiramay Rout

Question

If Z = f(x,y) = xy. Find the maximum value for f(x,y) if X and y are constrained to sum to 1 (That is, x+y = 1). Solve the problem in two ways : by substitution and by using the Lagrangian multiplier method.

Answer 1

Substitution method

sum of x and y is 1. i.e. x + y = 1, therefore this can be written as, y = (1 - x)

f(x,y) = x(1 - x)

f_x = 1 - 2x

Hence, x =1/2

Lagrangian multiplier method

We know that, x + y = 1. This can be written as 1 - x - y = 0

L= f(x,y) + λ(1 - x - y)

Taking the first order condition, we get;

∂L/∂x = y - λ

∂L/∂y = x - λ

∂L/∂λ = 1- x - y

y - λ = 0

y =λ

x - λ = 0

x = λ

Therefore, x = y

1 -x - y = 0

1 = x + y

Since y = x, then 1 =x + x

1 = 2x

Therefore, x = 1/2