In this article we’ll explore a few different rules about the differences of sums and differences of differences as applied in economics.
These may seem simple and reasonably obvious upon a close examination, but they’re useful across the board in equations and computation and, once internalized, will serve you well in analyzing economic relationships.
If you’re curious about growth rates as well, check out this read.
Let’s say you have two variables, X and Y. These variables change over time and we’ll use “t” to denote the time period. X₁ is the value of X at time period at t = 1, Y₂ is Y at time period t = 2, and so on.
The sum of X and Y at any time can be written as:
- Xₜ + Yₜ
For example, at time period 5, the sum of X and Y is:
- X₅ + Y₅
The change in the sum of X and Y, which we can denote through delta of Xₜ + Yₜ, can be written as:
- Δ(Xₜ + Yₜ) = (X₂+ Y₂) - (X₁ + Y₁)
Let’s simplify this:
- Δ(Xₜ + Yₜ) = X₂+ Y₂ - X₁ - Y₁
- Δ(Xₜ + Yₜ) = X₂ - X₁ + Y₂ - Y₁
Using our same delta notation:
- Δ(Xₜ + Yₜ) = ΔXₜ + ΔYₜ
Put simply, the difference of a sum (change in X + Y) equals the sum of the differences (the change in X and Y individually). This seems upon enough upon closer look, but…
