Difference Equations Amounts And Rates

In difference equation models, the difference in the quantity of interest can result from a net gain or loss in absolute terms, an amount, or relative to the present value of the quantity. In practice amounts in and amounts out add up to a net amount in or out and rates in and rates out likewise net out to a single rate in or out. This is the logic that let's us write a general difference equation as

(1)\begin{equation} P_{n+1} = a P_{n} + b \end{equation}

where **a** is the net rate and **b** is the net amount.

If we start with that distinction between amounts and rates, there are four basic types of difference equations, two simple and two mixed:

Type | Description | Example | Equation |

Simple | amount in and amount out | College enrollment, new recruits, graduates | $Enrollment_{i} = Enrollment_{i-1} + Recruits - Graduates$ |

Simple | rate in and rate out | Population with a birth rate and a death rate | $Population_{i} = Population_{i-1} + Population_{i-1} \times BirthRate - Population_{i-1} \times DeatRate$ |

Mixed | amount in and rate out | Reservoir, rainfall, leakage | $Reservoir_{i} = Reservoir_{i-1} + RainFall - Reservoir_{i-1} \times LeakageRate$ |

Mixed | rate in and amount out | Endowment, interest, expenditure | $Endowment_{i} = Endowment_{i-1} + Endowment_{i-1} \times InterestRate - AnnualEndowmentExpenditure$ |