A more general formulation of the Lagrange multiplier method is given: that in which there are many variables and possibly more than one constraint. The general theory of the consumer is presented, the problem being to maximise utility subject to a budget constraint. Applying the Lagrange method to this problem, it is shown that the tangency conditions encountered inreappear as a consequence; and also that, when optimising, the marginal rate of substitution is equal to the price ratio. The general solution of the problem reveals how to express the demand quantities in terms of the budget and the prices, giving what are known as the Marshallian demand functions. The corresponding (maximum) value of the utility function (depending on the budget and the prices) is known as the indirect utility and it is explained that the partial derivative of this with respect to the budget (known as the marginal utility of income) is equal to the value of the Lagrange multiplier.

This chapter introduces the important idea of a vector through the example of bundles of goods. The dot product of two vectors is defined and it is shown how a budget constraint can be expressed in terms of dot product. It is explained how, in order to rank bundles according to a particular consumer's preference, we can use a utility function. Indifference curves are defined as the contours of the utility function. Linear and convex combinations and the concept of a convex set are explained. The utility maximisation problem -- to maximise utility subject to a budget constraint -- is explored and the relevance of convexity is emphasised.