What is an equilibrium position
The Lagrange approach or the Lagrange method is a helpful instrument in microeconomics, but it is also used again and again in maths and physics. We explain in three simple steps how you can set up the Lagrange function with the help of the Lagrange multiplier and thus get to your goal quickly!
The easiest way to understand the Lagrange approach is to use our Video look at it! Here we explain the method to you using an example without you having to read our detailed article.
Would you like to start right away and apply your knowledge? Then take a look at our exercise!
The Lagrange function solves mathematical optimization problems with several variables as a system of equations. The objective function must include at least as many constraints as there are variables. In 1788, Joseph-Louis Lagrange found the Lagrange function, a method for solving a scalar function through the introduction of the Lagrange multiplier.
Setting up and solving the Lagrange function using an example
So that you understand the Lagrange approach one hundred percent, we will explain the whole thing to you using an example. Imagine your boss asks you the following task: You should determine the optimal distribution of temporary workers and permanent employees for a project. You have a prescribed budget for this.
So that you can optimally fill your project with temporary workers and permanent employees, you use the Lagrange method. You can use these when you want to maximize certain variables. In our example it is the permanent employees and temporary workers. At the same time, however, the Lagrange method has a secondary condition that restricts the variables. In our case it is the budget given for the project.
The Lagrange method in three steps
So, let's get started: To solve the task, you proceed in three steps:
- First you set up the Lagrange approach.
- In the second step, you have to derive for each variable so that you get multiple derivatives.
- Finally, you have a system of equations that you can solve with a few tricks.
Add lagrange multiplier lambda
In order to set up the Lagrange approach, you need an objective function that you want to optimize. In our case this is the maximized benefit - more on that in a moment. You also have to observe a secondary condition. In the example, the secondary condition is the budget for the project.
Another component is the Lagrange multiplier, which is represented by the Greek letter lambda. You have to multiply this by the secondary condition.
So let's do that directly for our example. If we employ someone, we have a use - after all, someone works for us. Therefore we set up a so-called utility function. Because we want to maximize utility, that is our objective function. Typically it looks like this: Our utility functionu depends on and .
stands for temporary workers and for the permanent employees. Because permanent employees tend to be more productive, we get more benefit from employing them. That is why the potency is at also a little higher than at .
Is this your first time hearing about utility functions? Then watch our video Utility function and indifference curves at.
We have a budget of € 2000 for our project. So this is our secondary condition. The temporary workers receive a wage of € 100, while the permanent employees are paid € 200.
Our secondary condition can therefore be set up very easily. We distribute the budget of 2000 € among a certain number of temporary and permanent employees. So it means:
In order to be able to operate immediately with the Lagrange multiplier, we resolve the secondary condition here to zero.
That shouldn't be too difficult. We just bring the right term with minus to the other side and then we’ve got it.
Since we now have our objective function u () and know the constraint, we can finally set up our Lagrange function:
L is the objective function combined with the Lagrange multiplier, as well as the constraints:
Derive the Lagrange function
In the second step we have to partially derive for all variables that occur in the Lagrange method. These are for the temporary workers, for permanent employees and the Lagrange multiplier lambda.
Let's trace our function off, this gives:
We always find the optimum where the slope is zero - like when you reach the summit while mountain climbing. Therefore we have to set the derivative equal to zero.
The partial derivative works according to the same principle .
If the derivation went too fast for you, watch the video Derive a power function in the area differential calculus I again. Then it should work with the left.
There remains the partial derivative according to lambda, i.e. the Lagrange multiplier. You can determine that directly without doing a lot of calculations. The trick here is that the lambda derivative is simply the constraint. So you can write that down right away.
From the partial derivatives we can then set up three equations. We need that in the next step and to be able to determine. You should always bring the lambda to one side so that you can simply shorten it out in the last step.
For the first partial derivative, we add 100 times lambda on both sides. 100 can also be shortened later, so just do it yourself and leave the 100 at the lambda. This is our first equation. We are now doing the same thing with the partial derivative and go completely analog to in front.
We can also rewrite the secondary condition in such a way that the budget of € 2000 is on one side.
You can already see that we can only find the lambda in the first two equations.
Solve the system of equations - reduce the Lagrange multiplier
So we now have a system of equations that consists of three equations.
Let's just look at the first and the second:
If we divide equation 1 by equation 2, then on the left is 100 times lambda divided by 200 times lambda. It works the same way on the right, so just write one below the other and don't forget the fraction line!
Now we can simplify that by shortening 100 lambda and 200 lambda on the left. That comes on the right with the negative power, always on the other side of the fraction line. The so moves down, that up.
To then finally we get the relationship of . This is our fourth equation.
As a final step, we only need the third and fourth equations. The we put in our budget condition and solve it on.
So it results:
From this we can calculate that equals 8.
We put that into the fourth equation a what we use for get equal to 6.
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