The Lagrange function is used to solve optimization problems in the field of economics. It is named after the Italian-French mathematician and astronomer, Joseph Louis Lagrange. Lagrange’s method of multipliers is used to derive the local maxima and minima in a function subject to equality constraints.
The existence of constraints in optimization problems affects the optimal value of the function since it implies a reduction of the feasible solutions space. Lagrange’s method multiplier is precisely used to measure how the correspondent restriction affects the optimal value of the objective function. In other words, they measure the degree of responsiveness of the problem’s optimal value due to changes in the constraints.
Mathematically, it is equal to the objective function’s first partial derivative regarding its constraint, and multiplying this last one by a lambda scalar (λ), which is an additional variable that helps to sort out the equation.
Let’s consider the following optimization problem:
The objective is to maximize (or minimize) a function with several variables
subject to the equality constraints
And we introduce the Lagrangian function,
Constrained optimisation plays a central role in Economics. For example, the consumer’s choice problem can be represented by either fixing a budget and obtaining its maximum utility (primal demand), or by setting a level of utility we want to achieve and minimise cost (dual demand). Other examples include profit maximization, along with various macroeconomic applications. Similarly to the previous example, the firm’s input decision has a dual nature. This is, it should find the optimal level of output given the resources available (minimising cost) or, having a budget constraint, solve the inputs allocation that results in the highest output (maximising production).