Lagrange method

Introduction

The Lagrange method in economics is used to solve utility maximisation problems where choice is constrained (by either income or a desired level of utility).  Using this method, you don’t need to know anything about your variables, and what they mean.  Consider consumer theory.  The optimum bundle of goods is found where the budget constraint line meets the indifference curve.  This can be solved graphically (by actually drawing the budget constraint and the indifference curve, and determining where they meet), or it can be solved mathematically using the Lagrange method.

The utility function is a way of converting a decision between two bundles of goods into a number that can then be used to rank various decisions.  Understanding a utility function allows a consumer to make the optimum choice between two bundles[1]. The indifference curve is a line showing all the bundles of goods a consumer finds equally desirable (that is, they all offer the same amount of utility). The slope of the indifference curve represents the consumer’s marginal rate of substitution – the maximum amount of one good the consumer will sacrifice to obtain one (more) unit of the other good. The marginal rate of substitution tends to change as the amounts of each good the consumer has changes. A consumer has an infinite number of indifference curves.

The budget constraint is an equation (or a line on a graph) that shows all possible combinations of goods that a consumer could purchase if all of their available money was spent on those goods.  The slope of the budget constraint reflects the relative prices of each good (the marginal rate of transformation).

The point where the indifference curve meets the budget constraint shows the optimum combination of goods – it shows the best possible bundle the consumer can buy, given the consumer’s budget constraint.  The Lagrange method is the way that this point can be found.

Marshallian demand functions

Marshallian demand functions assume a particular utility function, and solve for a budget constraint.  Consider a utility function:
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So the problem becomes:

This now is all the information required to solve the utility maximisation problem. The units don’t matter – this equation could refer to anything, and it would still be solved in the same way.

First order conditions

From here, Error: http://www.alisterair.com/wp-content/plugins/wpmathpub/phpmathpublisher/img/ must have write access Read the official wpmathpub plugin FAQ for more details.  The value of λ won’t change the equation overall, as it’s being multiplied by zero (income less expenditure, and expenditure equals income).  But this equation allows the utility maximisation problem to be solved[2].

The next step is to find the partial derivatives with respect to q₁, q₂ and λ, and to set these equal to zero.  These are the first order conditions.

FOOTNOTES
1. A consumer is very unlikely to be able to state their utility function in the way it’s used above.  We use utility functions to get a sense of what’s happening with consumer choices, but it’s a model – it won’t always accurately describe an individual consumer.↑
2. Unless you’re studying honours-level economics or greater, or mathematics, don’t worry too much how this works.  You’re unlikely to be asked to explain the proof of this.↑