We continue discussion of row operations to solve linear systems. In particular, we see how to characterise when a system has no solutions (is inconsistent) and, if consistent, we show how the method can be used to find all (possibly infinitely many) solutions, and to express these in vector notation. Here, the notion of the rank of the system, which determines the number of free parameters in the general solutions, is shown to be important. Continuing the earlier discussion of portfolios, we explain how the existence of an arbitrage portfolio is determined by the existence or otherwise of state prices.

We start the chapter with a mathematical model of how consumers might anticipate market trends and what effect this will have on the evolution of prices. This leads us to second-order differential equations. We then embark on describing how to solve linear constant-coefficient second-order differential equations. The general solution is the sum of the solution of a corresponding homogeneous equation and a particular solution. In an analogous way to the way in which second-order recurrence equations are solved, there is a general method for determining the solution of the homogeneous equation, involving the solution of a corresponding quadratic equation known as the auxiliary equation. We explain how to find particular solutions and how to use initial conditions. We also discuss the behaviour of the solutions obtained.

The derivative is introduced as an instantaneous rate of change and it is shown how this can be determined from first principles. Techniques (sum, product, quotient and composite function rules) are then explained and the connection with small changes is illustrated. Economic interpretations via marginals are given.

Elasticity of demand is introduced and it is shown how this characterises how revenue will change upon a price increase (the two distinct possibilities being represented by elastic and inelastic demand). Profit maximisation is considered in general, and it is shown that, when maximising profit, marginal revenue and marginal cost must be equal. Two very different cases are then studied: that in which the firm is a monopoly and that of perfect competition.

Optimisation of two-variable functions is motivated via the example of a firm producing two goods, where the concepts of complementary and substitute goods are discussed. The general idea of a critical point is discussed and it is explained that there can be different types of critical point: maxima, minima and saddle points. It is explained how the second partial derivatives may be used to determine what type of critical point one has. This is shown explicitly for the case in which the two-variable function is quadratic, and then stated in general. Examples involving profit-maximisation are given.

The determinant of a 2 × 2 and a 3 × 3 matrix are defined explicitly, and a more general way of (defining and) calculating determinants of larger matrices is described, involving the use of row operations to transform a matrix to upper-triangular form. It is then explained that a non-zero determinant is equivalent to invertibility. Cramer's rule is presented and a general method (based on the co-factor matrix) is given for inverting 3 × 3 matrices, an alternative to the row operations procedure described in .

Continuing from the previous chapter, this chapter explores the powerful applications of diagonalisation. We demonstrate first how it can be used to determine the powers of a matrix, which can then be applied to solve a coupled system of recurrence equations. An alternative approach to solving such systems also uses diagonalisation, but uses it to effect a change of variable so that the corresponding system in the new variables is much simpler to solve (and can then be used to revert to the solution in the original variables). We show how an analogous approach can be used to solve coupled systems of differential equations. This closing chapter provides an interesting link between the calculus and linear algebra aspects of the course.

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.