Exponential Functions and Models

Exponential change multiplies by a fixed factor each period, where linear change adds a fixed amount.

The model

y = a · bˣ, with a the initial amount and b the factor applied each period.

Reading b

A 6% rise gives b = 1.06 and a 6% fall gives b = 0.94. The factor includes the original amount.

Compounding more often

With n compoundings a year, A = P(1 + r/n)^(nt). More frequent compounding earns more, with rapidly shrinking gains.

The continuous limit

Compounding without pause gives A = Pe^(rt), where e is about 2.718. It is the ceiling the others approach.

The shape

Growth starts slowly and becomes very steep. Decay falls fast then flattens, approaching zero without reaching it.

Growth that feeds on itself

y = abˣ multiplies by a constant factor each step rather than adding a constant amount. The change depends on how much is already present, which is what distinguishes it from linear growth.

Reading the growth factor

b greater than 1 grows, between 0 and 1 decays. A percentage change of r gives b = 1 + r, so a 5% annual rise is a factor of 1.05. Converting a percentage into a factor is the key modelling step.

Why e appears

Continuous compounding leads to the limit of (1 + 1/n)ⁿ, which is e. It is the natural base for any process where change happens continuously rather than in discrete steps.

Exponential models expire

Populations run out of resources and epidemics run out of hosts, so real curves flatten. An exponential model describes an early phase, and knowing where it stops applying is part of using it.