Suppose a business produces a product using two materials, and there is a function to model the amount of product created based on quantities of raw materials. The business obviously also has a budget. How can the business maximize the production of the product while staying within budget?

Enter
Lagrange Multipliers! We’ve done optimization in calculus before, but Lagrange
Multipliers use optimization *with constraints*. In this example, the
constraint is a monetary one — budget.

Using two functions: the production function:

– f(x,y), where f is a quantity of product produced using x quantities of a certain raw material X and y quantities of another raw material Y and the constraint function:

– g(x,y)
= p_{1}x + p_{2}y ≤ c where p_{1} is the
price of raw material X, p_{2} is the price of y raw material
Y, and c is the budget

The Lagrange Multiplier λ is the scalar at which:

grad f = λ grad g

“grad f”
is the *gradient vector* of a function, a vector that points in the
direction of greatest increase for f(x,y). Using this, we can find not only λ,
the Lagrange Multiplier, but we can also find the values of x and y at which f
is maximized. In economic terms, we can find the maximum amount of product we
can make staying within budget.

The Lagrange Multiplier is more than just a tool that allows us to find these values. It has significance of its own. λ allows us to find out how to price our product, because it is equal to the change in production units per 1 unit increase in budget.

Optimization with constraints has applications far beyond economics. It is a useful tool in any sort of budget on a system, whether due to money, energy, utility, population, or any other metric!