Using Constrained Optimization and Lagrange Multipliers to Maximize Production Subject to a Budget

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) = p1x + p2y ≤ c where p1 is the price of raw material X, p2 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!

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