## Sunday, November 26, 2006

### If Prices Are Signals, Are Taxes Noise?

Economists often say that prices are signals. A high price signals to the market that something is relatively scarce. Consumers should minimize their use of it or producers should try to produce more of it. If that happens, the signal has done its job. But what if some "noise" distorts the signal? It won't do its job very well.

In analog and digital communications, signal-to-noise ratio, often written S/N or SNR, is a measure of signal strength relative to background noise. If it is too low, communication is difficult.

What if we called the price-to-tax ratio the signal-to-noise ratio of the economy? It would be a measure of how much a market has been distorted and how poorly the market will perform as a result. A high ratio means the market will work well. As it falls, the market performs more poorly.

Harvard Professor Greg Mankiw issued a challenge to economists to explain the damage done by taxes. It is something that we can show using supply and demand curves, calculating something called deadweight loss triangles. This loss actually grows at a faster rate than the tax grows (impyling that the tax does disproportionate damage). The loss stems from the fact that a tax raises the price, thereby reducing profits for producers since consumers buy less (who are also hurt, obviously paying more and getting less). Sellers and buyers each pay part of the tax, depending on the supply and demand conditions. But that kind of analysis does not work in newspaper columns. He wondered if there was some other way to explain this disproportionate damage without having to explain supply and demand curves and calculating the area of trianlges. Below is what I posted:

"Suppose that the income tax rate is 0%. Then the ratio of taxes paid to disposable income (tax/DI) is 0 since you pay no taxes and 0 divided by anything is zero.

If a tax of 10% is then enacted, the ratio becomes 10/90, since you keep 90% of what you make and pay 10%. This is about .111. So the tax went up 10 percentage points, while the tax/DI went up .111. So they rose about the same amount.

But as the tax rate goes up, very quickly the tax/DI starts rising alot more than the tax rate. If the tax rate goes from 10% to 20%, the tax/DI goes from .111 to .25. If the tax rate then goes from 20% to 30%, the tax/DI goes from .25 to .429. From 30% to 40%, tax/DI goes to .667.

So the tax/DI ratio keeps rising at a faster and faster rate than the increases in the tax rate. Think of this as a noise-to-signal ratio. It can be applied more directly to Mankiw's original question. The more we tax a product, the faster the noise-to-signal ratio increases.

Since economists always say that prices are signals that steer resources to their most efficient use, the greater this noise-to-signal ratio, the less efficient the market will become and this damage accelerates with each incremental increase in price."

What I have below is technical and probably only economists will be interested. But it shows that my noise-to-signal ratio tracks the rise in deadweight loss very well.

Suppose we have a demand line with a slope of -1. Intercept is 11. Supply starts at the origin and has a slope of 1. So equilibrium price is 5.5 and equilibrium quantity is 5.5. An excise tax of 1 raises price to 6 and creates a deadweight loss of .25. My noise-to-signal ratio would be .5 over 5 since the seller pays .5 of the tax and actually receives 5. That ratio is .1. If the tax is 2, the price is 6.5. DWL is 1 (so the tax doubled yet DWL quadrupled). My noise-to-signal ratio would be 1/4.5 = .222. So the tax doubled yet my noise-to-signal ratio more than doubled. In the numbers below, the first column is the tax level, the 2nd is the noise-to-signal ratio and the 3rd is the DWL

0-0.0-0
1-0.1-0.25
2-0.222-1
3-0.375-2.25
4-0.571-4
5-0.833-6.25
6-1.2-9
7-1.75-12.25
8-2.67-16
9-4.5-20.25
10-10-25

If you graph this, up to a tax of 8, the line is almost straight. Beyond that, DWL rises much more slowly as the tax rises. But for the most part, as my noise-to-signal ratio rises the DWL rises at a steady rate.

#### 1 comment:

champ said...

Very interesting information. I guess since I'm not quite an economist yet, I can only understand so much of your personal analysis at the bottom. It does make since, but I could go for a better understanding.

i like the s/n and price to tax amalgam(I hope I used that word right)