How to Work With Differences of Sums in Economics
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 let’s back up and look at a general economics example:
At time t = 1, we can note revenue across two products:
- P1 = 100
- P2 = 120
At time t = 2, this is the corresponding revenue:
- P1 = 150
- P2 = 160
To calculate the change in total revenue, we can just add the sum of the differences of each product. 150–100 = 50 and 160–120 = 40, so the overall difference is 50 + 40, or 90.
Let’s now think about multiplicative constants that multiply changing variables. Let’s say we have:
- aXₜ
Where a is a constant multiplier and Xₜ is the value of X at time period t. Given such, we can note that the change in aX, or ΔaXₜ, equals the value at time period 2 minus the current value, or ΔaXₜ = aX₂ — aX₁. In turn, we can pull out a to produce:
- a(X₂ — X₁) = aΔXₜ
Thus, we’ve shown that ΔaXₜ = aΔXₜ, or that the change in aXₜ is equivalent to the change in the base variable multiplied by the constant.
Now, let’s apply both those rules by asking what happens if we take the difference of a difference. I’ll lay this out for you:
Thus we can see that when looking at change in the difference between two variables, we can also find the change in each variable individually, and then sum them.
As stated, these seem simple (and they are), but they’re important cornerstones that are useful to internalize when going about more advanced calculations. Hope that helps!
Leave any questions in the comments! Check out my other articles on economics here (useful for studying, or just learning).
