How to calculate change in deadweight loss when the tax rate changes

In my microeconomics course this week, I’m studying about markets and economic welfare. One thing I was learning about is deadweight loss.

Deadweight loss is basically the loss of value when a tax is introduced. In calculating deadweight loss, economists have to make a few assumptions.

First, they have to assume that the price before the tax is introduced is the equilibrium price. In other words, the price a good sells at is both the price consumers are willing to pay for that quantity of the good and the price producers are willing to sell for that quantity of good.

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The theory of supply and demand is that the higher the price of a good, the more of that good producers will want to sell and the fewer of that good consumers will want to buy, and vice versa.

Let’s look at it in a chart:

This graph shows how the theoretical model predicts that as a price increases, supply rises and demand falls, and as prices decreases, supply falls and demand rises.

The equilibrium price is when the purchase price for a quantity of a good intersects with the selling price for that quantity. Anything above that price and below the demand curve is consumer surplus. Anything below that price and above the supply curve is producer surplus.

The second assumption economists make related to deadweight loss is that introducing a tax reduces quantity sold and demanded. Their model predicts that the tax will increase the price sold, which reduces the quantity demanded. And if the producer is producing less quantity, the price they want to sell it would be less, too.

That loss of quantity demanded and quantity supplied results in a loss of both consumer and producer surplus. That loss of surplus is called deadweight loss.

I’m not convinced that introducing taxes absolutely decreases demand, let alone by a significant amount, but that’s not the point of this post. Let’s just assume that this theory is right.

I’m also not going to tell you how to calculate deadweight loss. You can find that in all sorts of places. My point here is to help you calculate the increase in deadweight loss when the tax rate increases.

If a tax increases, then—theoretically—the quantity demanded and supplied should shift left. That would expand the size of the deadweight loss.

And that’s where this formula comes in:

T2 = T1 × n ⇒ D2 = D1 × n2

I developed this formula when I was practicing calculating the deadweight loss on various tax rates and noticed a pattern in the results.

In my formula, T2 is the new tax and D2 is the new deadweight loss. So basically, if you multiply your current tax rate by a certain number to get the new tax rate, you multiply the current deadweight loss by the square of that number to get the new deadweight loss.

For example, let’s say the current tax rate (T1) is 3%, your deadweight loss (D1) is $9, and the new tax rate will be double that of the current rate (n). Here’s how the formula would look:

T2 = T1 × n D2 = D1 × n2

T2 = 3 × 2 ⇒ D2 = $9 × 22

T2 = 3 × 2 ⇒ D2 = $9 × 4

T2 = 6 ⇒ D2 = $36

And if you’re tripling the tax rate?

T2 = 3 × 3 ⇒ D2 = $9 × 32

T2 = 3 × 3 ⇒ D2 = $9 × 9

T2 = 9 ⇒ D2 = $81

What if you don’t know how much you’re multiplying it by? What if you know only how much you’ll be adding to the current tax rate?

Simply divide the new tax rate by the current tax rate to find n. In our example above, n = 6 ÷ 3, or n = 2.

If your current tax rate is 5% and it’s going up to 7%, then n = 7 ÷ 5 or
n = 1.4. If it’s going from 23% to 24%, it’s n = 24 ÷ 23 or n = 1.04. And so on.

And that’s my formula for determining the increase in deadweight loss when there’s an increase in the tax rate.

I couldn’t find any other formulas out there, so if I’m the first to create it, then I’m claiming ownership and calling it the Siever formula.

If someone else has already created it, well, then you can name it after her.

If you find my formula useful, let me know in the comments.

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By Kim Siever

Kim Siever is an independent queer journalist based in Lethbridge, Alberta. He writes daily news articles, focusing on politics and labour.

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