This chapter involves the input--output model. Here, there are several goods under production, and some of each is needed to meet the production of the others, and there is also an external demand for each good. The model involves a matrix known as the technology matrix and a related matrix know as the Leontief matrix. It is shown how to solve such problems and it is explained that, in general (under very reasonable conditions), there always will be a solution. It is also shown how to approximate the solution using powers of the technology matrix.

This chapter studies the case of a small efficient firm in a perfectly competitive market. Breakeven and startup points are defined. Relationships between marginal cost, average cost and average variable cost at breakeven and startup points are investigated, and it is shown how to derive the supply set of such firms.

The chapter starts by discussing how we can determine the long-term qualitative behaviour of the solutions to second-order recurrence equations. In particular, in some cases, it can be seen that the solution is oscillatory. In the context of the multiplier-accelerator model, this corresponds to what are known as business cycles. The chapter concludes with an analysis of a dynamic macroeconomic model that is more realistic than the multiplier-accelerator one.

The chapter starts by formulating the standard problem in the theory of the firm: namely, to minimise combined capital and labour costs while producing a certain output. A general formulation of a constrained optimisation problem (with one constraint) is given and it is explained how to solve such problems by the method of Lagrange multipliers. This leads to a method to calculate the cost function of firms given their production functions and capital and labour unit costs. This enables us to derive the supply sets for efficient small firms with Cobb--Douglas production functions.

This chapter provides an introduction to mathematical modelling in economics through the study of supply and demand sets, equilibrium and the effect of the imposition of an excise tax.

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.

The cobweb model is introduced and analysed fully for the case of linear supply and demand, and conditions for stability are derived.

The inverse of a square matrix is defined and it is explained how the existence of an inverse is related to the question of whether systems of linear equations corresponding to the matrix have unique solutions. A method for determining when a matrix is invertible and, when it is, finding the inverse, is shown: this involves row operations. An explicit formula for the inverse in the 2 × 2 case is given. This is applied to IS--LM analysis, a macroeconomic application.

The profit-maximisation problem (for production of one good) is introduced as motivation for the development of general optimisation techniques. The general concept of a critical (or stationary) point is presented, together with the method for finding such points and classifying their nature in two different ways: by examining the sign of the derivative around the point and by using the second-derivative test. Optimisation on intervals and infinite intervals is then discussed (where the end-points must be taken into consideration). Additional economic and financial applications are given.

Matrices are introduced and it is explained how matrix addition, scalar multiplication and multiplication of two matrices works. As an example of matrix multiplication, it is demonstrated how investment portfolios can be modelled, and their returns in various states quantified by multiplication with a returns matrix. The concept of an arbitrage portfolio is explained.

This chapter introduces (with production function as an example) functions of more than one variable. Then we define partial and second partial derivatives and explain how to calculate them, and present the chain rule for partial differentiation.

Important special functions and their properties are described. In particular, the exponential function and its connection to continuous compounding is discussed, together with the logarithm and trigonometrical functions. It is explained how one can interpret not just integer, but rational and then irrational powers of positive real numbers. The derivatives of exponential, logarithmic and trigonometrical functions are studied.

The standard integrals of the previous chapter are of fairly limited use, so this chapter develops some much more widely applicable techniques. These are integration by substitution, integration by parts and integration by partial fractions.

The general matrix formulation of a system of linear equations is described. It is explained that a system may have no, one or infinitely many solutions. We begin to describe a general approach to solving such systems by performing row operations on the augmented matrix in order to reduce this to echelon form. This chapter gives examples in the case where the system has a unique solution. (The next chapter considers other cases.)

This is a discussion of sequences and first-order recurrence (or difference) equations and the behaviour of the solutions to such equations. It contains some economic applications.