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.