Photo by Dan Cristian Pădureț on Unsplash
Throughout the study of economics, many variables change over time—think GDP, the number of workers, or the level of technology. The growth of these variables can be modeled either with discrete growth or continuous growth.
You’ll need to be familiar with both ideas and mathematical models of growth while constructing your own models and reading papers, so in this article we’ll explore these two different approaches to quantifying growth.
Discrete growth assumes that a variable grows at some step interval. Say you own a bond that pays out monthly – your investment grows at a set, discrete rate monthly. Continuous growth assumes that growth happens at every instance (we can use seconds as a metric). Since there’s additional growth on each instance of growth, continuous growth means compound interest or compound growth.
We can use this formula to describe discrete growth:
Xₜ = (1 + g)ᵗ X₀Discrete Growth Formula
Where g is the growth rate at each interval (say, the monthly payout on the bond) expressed as a decimal. t is the number of time periods, say t number of years. X₀ is the starting point or initial value of whatever we’re solving for, say revenue, population, capital, etc. Note that we can use just (1 + g)ᵗ X₀ to solve for things like future value:
FV = 100(1 + 0.1)¹⁰
Using the general discrete growth formula, let’s say we’re calculating the average growth rate of a given variable over time and we’re asking what constant growth rate would have gotten us from X₀ to X₁ over t periods.
We’ll simply solve for g:
Xₜ = (1 + g)ᵗ X₀(1 + g)ᵗ = Xₜ/X₀1 + g = (Xₜ/X₀)^1/tg = ((Xₜ/X₀)^1/t) – 1
So, if time periods is 5, final value is 150, and starting value is 100:
g = ((150/100)^.2) — 1g = 1.5^.2 – 1g = 1.0844717712 – 1g = .0844717712g = 8.44% per time period
Moving on to continuous growth, the general formula is:
Xₜ = eᵗᵍX₀Continuous Growth Formula
Where g is the growth rate per period, e is the constant Euler number (2.718…), and t is once again the number of time periods we’re calculating for. Let’s once again isolate g to pretend we’re solving for a growth rate:
Xₜ/X₀ = eᵗᵍln(eᵗᵍ) = ln(Xₜ/X₀)tg = ln(Xₜ) – ln(X₀)g = (ln(Xₜ) — ln(X₀))/t
So, those are the two formulas you need to know to get started with calculating discrete and continuous growth! Hope it helps.
Hope you found it helpful! Here’s another article detailing nominal and real GDP (here), and one more about how to calculate inflation using multiple GDP deflators (here). Also:
Read about the Income Approach to GDP here.Read more economics stories here.To learn more about the oil market, consider reading about PADD Districts, the Why WTI and Brent are Crude Oils, and Why There are Price Differences Among Crude Oils, and my Oil & Gas Terms Guide.