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Summary

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.

We find all real-valued general solutions $f:S\rightarrow \mathbb{R}$ of the d’Alembert functional equation with involution

$$\begin{eqnarray}\displaystyle f(x+y)+f(x+\unicode[STIX]{x1D70E}y)=2f(x)f(y) & & \displaystyle \nonumber\end{eqnarray}$$

for all

$x,y\in S$, where

$S$ is a commutative semigroup and

$\unicode[STIX]{x1D70E}~:~S\rightarrow S$ is an involution. Also, we find the Lebesgue measurable solutions

$f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ of the above functional equation, where

$\unicode[STIX]{x1D70E}:\mathbb{R}^{n}\rightarrow \mathbb{R}^{n}$ is a Lebesgue measurable involution. As a direct consequence, we obtain the Lebesgue measurable solutions

$f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ of the classical d’Alembert functional equation

$$\begin{eqnarray}\displaystyle f(x+y)+f(x-y)=2f(x)f(y) & & \displaystyle \nonumber\end{eqnarray}$$

for all

$x,y\in \mathbb{R}^{n}$. We also exhibit the locally bounded solutions

$f:\mathbb{R}^{n}\rightarrow \mathbb{R}$ of the above equations.

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23 - Second-Order Recurrence Equations

Summary

ON THE REAL-VALUED GENERAL SOLUTIONS OF THE D’ALEMBERT EQUATION WITH INVOLUTION

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23 - Second-Order Recurrence Equations

Summary

ON THE REAL-VALUED GENERAL SOLUTIONS OF THE D’ALEMBERT EQUATION WITH INVOLUTION