If you are familiar with Lagrange multiplier you are lucky! You can make money by doing this concept and applying that since usually Econ people using this concept. You can get idea of it from the wiki page but since Im in Civil Eng. I will try to explain it in my way.
If you had some structural courses you may heard about virtual work. So in a very simple case lets say what will happen to my structure's forces (for example if we are interested to forces) here to my single bar if I just change the displacement one unit. Economist usually using this concept and they call it shadow price and in many cases it meas if I change the limitation or constraint I have over my resources just one unit, what will happen to my cost or benefit!
Something that we need to be careful about is Lagrangian multiplier's sign. The sign of the multipliers can be positive, negative or in some case we may not have any sign (Lambda = 0) (I used Lambda because usually it used for Lagrange multipliers).
Usually you are trying to maximzie or minimize a objective function, I call it here: f(x,y)
'f' is my objective function and 'x' and 'y' are variables. I show the constraint function here by: g(x,y). The constraint function as its name shows is limited or constrained by something.
Sometimes for example when we want to go shopping we have limited amount of money and then we are constraint by the amount of money we want to spend, or in many problems we are limited by time, especially when we have exams!! ;)
g(x,y)<b
('b' here is our constraint.)
other situation is when we are limited at the other end, for example you should be 16 at least to get the driver license, or another example is, you want to do business, you bought something for lets say for 5 box, and you want to make at least 10% benefit, so your price should be greater than $5.5, and if you can sell your thing for more than that you are lucky!
g(x,y)>b
it is worth to mention that here I just have two variables but you may have thousands of them. The concept is the same.
You may have one or more than one constraint function and also you may have more than one objective function that called multi-objective function which we will talk about that later on.
You can get information about your function by the sign of the Lambda.
Lambda is how the objective function changes if I change the limitation (constraint) one unit
Lambda = df/db
In general there are two cases:
1) we are maximizing:
1-1) g(x,y)<b
Lambda is positive
1-2) g(x,y)>b
Lambda is negative
2) we are minimizing:
2-1) g(x,y)<b
Lambda is negative
2-2) g(x,y)>b
Lambda is positive
so for example when we are maximizing and g(x,y)<b, we are trying to reach the max but the constraint limits us and going uphill our slope will be positive.
If you want to visualize the case 1-2, its like the case 1-1, we want to go uphill again but its like we are going backwards, so if you walk backward and look to your front your slope is negative but the truth is you are going backward toward the summit.
Cases, 2-1 and 2-2 are the same, for example for the case 2-1, we want to minimize and if you walk downhill, when you look at your feet, your slope is negative. The backward walk is again is the same for the case 2-2.
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