What is Lagrange

The Lagrange method

Another approach used to calculate optimal bundles of consumer goods is Lagrange method. It is used for Determination of an optimum taking into account Constraints.
This method is briefly presented here for the sake of completeness, as the notation differs from the previous one. However, the results are identical to the procedure discussed above.

The goal is again to maximize the utility of a household. A Cobb-Douglas utility function will serve as an example.

Click here to expandExample with Cobb-Douglas utility function

$ \ m = 64 $,
$ \ p_1 = 2 $,
$ \ p_2 = 8 $
Utility function: $ \ u = (x_1 \ cdot x_2) ^ {0.5} $

Lagrange optimization under constraints

The utility function should take into account the Budget constraint as a secondary condition can be maximized. To do this, the Lagrange function be formulated. It results as:

Click here to expand

$ \ L (x_1, x_2, \ lambda) = objective function + \ lambda \ cdot (secondary condition) $

"$ \ \ lambda $" is that Lagrange multiplier. As we shall see, it disappears in the course of the calculation. Its determination is possible, but should not be of further interest to us here. This is part of an advanced course on microeconomics.

Before we set up the Lagrange function for our example, we just have to take a look at the Secondary condition throw. It has to be transformed so that there is a zero on one side of the equation. In our example, the budget constraint becomes $ \ 64 = 2x_1 + 8x_2 $ so $ \ 64-2x_1-8x_2 = 0 $.

If we now set up the complete function, we get:
$$ \ L (x_1, x_2, \ lambda) = (x_1 \ cdot x_2) ^ {0.5} + \ lambda \ cdot (64-2x_1-8x_2) $$ The next step is to derive all three variables $ \ x_1, x_2 $ and $ \ \ lambda $.
This results in three functions:
$$ \ {dL \ over dx_1} = 0.5 \ cdot x1 ^ {- 0.5} \ cdot x_2 ^ {0.5} - \ lambda \ cdot 2 = 0 $$ $$ \ {dL \ over dx_2 } = 0.5 \ cdot x1 ^ {0.5} \ cdot x_2 ^ {- 0.5} - \ lambda \ cdot 8 = 0 $$ $$ \ {dL \ over d \ lambda} = 64-2x_1- 8x_2 = 0 $$

It is important that the first two functions not only represent the derivation of the utility function, but also from the constraint $ \ - \ lambda \ cdot 2 $ (in general: $ \ - \ lambda p_1 $) or $ \ - \ lambda \ cdot 8 \ (- \ lambda p_2) $ must be added. The last derivation only gives the reformed budget constraint.
With the first two equations, $ \ - \ lambda \ cdot 2 $ or $ \ - \ lambda \ cdot 8 $ are brought to the other side in the next step. Then they are each divided by 2 ($ \ p_1 $) or 8 ($ \ p_2 $), so that only $ \ \ lambda $ is on one side of the equation.
Since both functions have $ \ \ lambda $ on one side, they can be equated. So we get:
$$ \ {0.5 \ times x_1 ^ {- 0.5} \ times x_2 ^ {0.5} \ over 2} = {0.5 \ times x_1 ^ {0.5} \ times x_2 ^ {- 0.5} \ over 8} $$ If this equation is multiplied, the result is: $ \ x_2 = {1 \ over 4} \ cdot x_1 $. This can be used again in the normal way in the budget restriction. Then the result can be determined. It reads here (16; 4).