A classic problem in economics is how to maximize utility despite limited resources, such as time and money. The Lagrangian method uses a technique from calculus to mathematically measure how consumers can achieve maximum satisfaction and businesses can maximize profit (or minimize costs) within given limits.
The Lagrangian method (also known as Lagrange multipliers) is named for Joseph Louis Lagrange (1736-1813), an Italian-born mathematician. His Lagrange multipliers have applications in a variety of fields, including physics, astronomy and economics.
A key theory in Neoclassical economics, the basis of most mainstream economic thought, is that consumers and businesses are rational actors who strive to maximize their utility. On the consumer side, this means obtaining the highest level of satisfaction from goods and services that a given consumer values highly. For businesses, maximum utility means maximizing profit.
Economists recognize that individuals and firms have unlimited wants, but only finite resources for satisfying those wants. Consumers have limited income for buying the goods and services they desire, and firms have only limited land, labor and capital for producing their products. These limited resources, then, present constraints.
The challenge, then, is how to achieve maximum satisfaction or profit within given constraints. Another challenge for firms is that of minimizing production costs while still meeting expected levels of output. The Lagrangian method provides a way to quantitatively resolve these issues, which some economists refer to as matters of constrained optimization.
The Lagrangian method applies differential calculus, involving the calculation of partial derivatives, to issues of constrained optimization. The owner of a business, for example, can use this technique to maximize profit or minimize costs given that the business has only a certain amount of money to invest.
A hypothetical consumer, who, for example, derives utility from collecting books and CDs, could use this method to determine how to obtain the optimal number of books and CDs, given that he or she has only $100 of disposable income to spend.
The Lagrange multiplier, represented in the equation by the lowercase Greek letter lambda (?), represents the rate of change in utility relative to the change in the budget constraint. In economics, this is known as the marginal value or marginal utility, the increase in utility gained from an increase in the budget constraint.
Based on the results of a Lagrangian analysis, an individual or firm has an empirical basis for making decisions on continued utility maximization within changes to the external constraints. A price increase for a favorite good, for example, may lead a consumer to purchase a lower quantity of that item or to work more hours to earn more income to afford the higher price.