Question

Suppose the utility function is given by 20 In x1 + x2 and the budget constraint by
P1x1 + 12 = m.
3.1 Write the Lagrangian, find the first-order conditions for utility maximization, and
solve these first-order conditions to find the demand functions for good x, and
X2, as a function of the price pı and the income m. Then, to be concrete, suppose
P1 = 4, p2 = 1, and m = 40. Find the optimal consumption bundle. How much
money does the consumer spend on good 1?​

Answer 1

Answer:

The Lagrange Multiplier Method

Sometimes we need to to maximize (minimize) a function that is subject to some sort of

constraint. For example

Maximize z = f(x, y)

subject to the constraint x + y ≤ 100

For this kind of problem there is a technique, or trick, developed for this kind of problem

known as the Lagrange Multiplier method. This method involves adding an extra variable to

the problem called the lagrange multiplier, or λ.

We then set up the problem as follows:

1. Create a new equation form the original information

L = f(x, y) + λ(100 − x − y)

or

L = f(x, y) + λ [Zero]

2. Then follow the same steps as used in a regular maximization problem

∂L

∂x = fx − λ = 0

∂L

∂y = fy − λ = 0

∂L

∂λ = 100 − x − y = 0

3. In most cases the λ will drop out with substitution. Solving these 3 equations will give

you the constrained maximum solution

Explanation: