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.

We start by introducing the key ingredients in macroeconomic modelling: investment, production, income and consumption, and explain the corresponding equilibrium conditions. Modelling these quantities in discrete time, we describe the multiplier-accelerator model, a classic model of macroeconomic dynamics, and an example of a second-order recurrence equation. We then embark on describing how to solve linear constant-coefficient second-order recurrence equations in general. The general solution is the sum of the solution of a corresponding homogeneous equation and a particular solution. 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.