# Double Tax Rates, Quadruple the Economic Harm

At last Wednesday’s hearing on tax reform, three witnesses–Rosanne Altshuler, Larry Lindsey, and I–invoked a famous rule of thumb about taxes. We each told the Senate Budget Committee that high tax rates are disproportionately harmful for the economy and that:

If you double tax rates, you quadruple the resulting economic harm.

If a 10% tax rate on some activity does a certain amount of economic damage, for example, then it’s a reasonable guess that doubling the tax rate to 20% would multiply that damage by a factor of four.

It was nice to hear such agreement among the panelists, but judging by the senators’ reaction, this idea is not intuitive. So let me try to explain with a simple example.

Suppose there are five people who might buy a pizza. The first person values a pizza at \$14.50, the second at \$13.50, the third at \$12.50, the fourth at \$11.50, and the fifth at \$10.50.

If pizzas cost \$10, all five people will buy one. The first person gets a net benefit of \$4.50, since the pizza was worth \$14.50 to her, but she paid only \$10. The second person gets a benefit of \$3.50, and so on. Add it all up, and the benefit of the pizza market is \$12.50 (= \$4.50 + \$3.50 + \$2.50 + \$1.50 + \$0.50).

Now suppose that the government levies a 10% tax on pizzas; that lifts the price to \$11. Now only the four consumers who place the highest value on pizzas will buy them; Mr. \$10.50 won’t buy. The four remaining consumers now benefit by \$3.50 + \$2.50 + \$1.50 + \$0.50 = \$8.00 from buying pizza. The government collects \$4.00 in revenue, so the total economic benefit of the pizza market is \$12.00, \$0.50 less than before. That 50-cent loss falls on the hungry guy who no longer buys a pizza. The \$1 loss for each of the four buyers isn’t lost to the economy; instead, it transfers to the government.

Now suppose, instead, that the tax is 20%; pizzas now cost \$12 each, and only three consumers will buy. Their total benefit is \$4.50 (= \$2.50 + \$1.50 + \$0.50). The government collects \$6.00 in revenue, so the total economic benefit is \$10.50. That’s \$2.00 less than without a tax.

So there you have it. When you double the tax from 10% to 20%, you quadruple the economic harm from \$0.50 to \$2.00.

Why does this happen? Because doubling the tax doubles the number of consumers who drop out (from 1 to 2) and doubles the average economic value of the pizza sales that never happen (from \$0.50 to \$1.00). Two times two is four, so the overall effect is to quadruple the economic harm.

Put another way, the value of the second lost pizza (\$1.50) is three times larger than the value of the first one (\$0.50). So the economic harm of the 20% tax is four times the harm of the 10% tax.

This is a big deal when you design a tax system for the entire economy. To avoid needless economic harm, you should aim for low tax rates and the broadest possible tax base. If you need to raise \$6.00 from our mythical food economy, for example, it would be far better to levy a 5% tax on pizzas, tacos, and hamburgers, than a 20% tax on pizzas alone.

P.S. Did I cook the pizza example to get the increase to be exactly a factor of four? Of course. In the real world, the actual multiple will vary. If the fifth person valued the pizza at only \$10.25, for example, the loss from the 10% tax would have been \$0.25, and the loss from the 20% tax would have been seven times larger at \$1.75. Conversely, if the fourth person valued the pizza at only \$11.00, the loss from the 20% tax would have been \$1.50, only three times larger than the \$0.50 loss from the 10% tax. The double/quadruple rule of thumb assumes an even spread of consumers and their values — a reasonable starting assumption until you have more information.

## 11 thoughts on “Double Tax Rates, Quadruple the Economic Harm”

I think it should be reiterated that this only applies to taxes that target a certain activity, and that it requires a demand curve with a slope of 1 for the quadrupling of harm to apply.

When talking about broad-based taxes like an income tax, the imaginary perfectly rational man would first cut spending on those activities that give him the least marginal utility.

It’s also worth mentioning that taxes aren’t a black hole, so in this example the harm for pizza producers and consumers is quadruple the amount of the tax, but the tax revenue should be generating some utility, possibly enough to offset the harm in the pizza market.

2. pedromammoth says:

What about substitution? Does the \$10 pizza guy not eat, or does he increase value in some other product that he would have forgone if the pizza were within his budget?

1. Yes indeed. Since he passes on the pizza, he can afford to buy something else instead (likely some other food). But he won’t be as happy – by an amount equal to \$0.50.

3. Jane Quatam says:

I’m not certain about the intended lesson from this posting. Was it intended to trivialize pizza, economics or the average intellectual abilities of your readers.

I love these abstract economic models that only possess a single data point and live inside a massively manipulated universe. We have a buyer who abstractly values a pizza with no underlying reasoning as to why the pizza assumes different values – and yet apparently there is only one type of pizza in this hypothetical universe and no one in this universe knows how to make their own pizza.

Then with merely a bit of psycho economic mumbo jumbo we manipulate all the residents within the abstract/false universe and accept the results of a flawed model as producing a universal truth.

Truly magical thinking, it makes it easier to understand how the Federal Reserve comes up with their own deeply flawed plans cloaked in monetary semantics to dazzle the unwary and destroy the unwise.

These are almost as much fun as the weekly puzzler on Car Talk, I hope you can continue to entertain us with more of these remarkable hypothetical universes.

Can we have one that incorporates the price of a bowl of rice, a barrel of oil, and a trillion dollars in back door bailouts to fraudulent corporations, for bonus points include hyperinflation of commodities and rising political instabilities in oil producing nations that supply petroleum products used in agriculture and transport of food.

4. To amplify on what Jane Quatam wrote, it is completely logical-sounding and quantitative-seeming (but misguided) analogies like this that help one understand two things. 🙂 One is why leaders in the heterodox economics field have called for an investigation into why there has been no accountability in the mainstream economics profession even years after the failure to predict the most recent economic meltdown (while still shutting out alternative viewpoints). The other is why the World Economics Association (WEA) was just launched with a call by 141 economists to promote a pluralism of approaches to economic analysis. The USA was at its peak socioeconomically in some ways when it has a 91% marginal progressive tax rate under Franklin Roosevelt, with a growing middle class and lots of social optimism. The last thirty years of supply-side low-tax policy (coupled with high military spending) has led to stagnant wages and rising uncertainty for most people in the USA. Something is deeply wrong here. Part of it is the assumption that rich people spend most of their money on stuff like pizza instead of keep it stashed in government bonds.

To understand why, let’s do the numbers of the pizza a little different. Consider five people who are all hungry. One is a billionaire, because he owns state-granted intellectual monopolies (he lobbied for) on pizza-making. Another is a millionaire who works for the first, enforcing the artificial scarcity by calling in the police on anyone who makes pizza without a license. The third makes US\$7 an hour at the pizza place and is glad to get it, given the alternative. The two others are destitute, without a dime in their pockets; one has untreated communicable tuberculosis the millionaire is about to unknowingly catch, the other is so stressed out he does weird things in public. Now, it does not really matter to the billionaire what a piece of pizza costs — the billionaire actually loses money when he reaches to the floor to pick up a dropped \$1000 bill, given the value of his time. The millionaire might be a bit price sensitive, but convenience will be a bigger factor. Price matters a lot to the other three, but none can afford the product, even the working guy in the pizza parlor who lives off rice and beans and also has untreated hepatitis (no health care, but he can’t afford to tell his boss) that he spreads unfortunately to the billionaire. So, with not enough pizza selling, that one blue-collar working guy is about to lose his job, too. The two rich guys have to spend extra money in other places to wall off their lives so they don’t have to see this other suffering of 3/5 of the human race, but this gets spun as just the cost of being upscale.

But, let’s say there was again a marginal 91% tax on high incomes, as under Roosevelt, as well as even a substantial wealth tax, with the tax revenue distributed as a “basic income” as well as invested in public infrastructure and basic R&D and basic health care for all. Suddenly, everyone can afford pizza. The pizzas are much healthier because of the basic R&D, freely shared, so everyone who eats one lives longer, reducing health care costs. The pizza workers and customers are not so disease-ridden. The pizza parlor is even hiring, because there is *demand* for the product, and they can borrow money based on demand projections, and when they do finally automate completely, they know there will still be demand. And the rich guys are not as rich, but they live in a happier and healthier society, and they don’t have to wall themselves off from it, and they can be sure their grandchildren will always have a basic income, even if they are not so good with earning money. And, with more economic activity driven by intrinsic motivation instead of extrinsic motivation (search on “RSA Animate – Drive: The surprising truth about what motivates us”), the variety and quality of goods and services increases.

Eventually, with robotics, AI, 3D printers, open source software, better design, and voluntary social networks, we may transition towards a mix of a gift economy, planned economy, subsistence economy, and basic income economy, where rationing of things like pizza ceases to be a big issue. People can take their new-found free time and put it into raising happier kids and being more involved in their communities.

Fiat dollars are, a kind of generalized ration unit; as Iain Banks says, “Money is a sign of poverty”. But as long as we have monetary transactions, we have to ask what values and assumptions governs the movement of money in our society? It is an aspect of 21st century government to reflect our collective values and use taxes, subsidies, and regulations to deal with things like risks, externalities, and general social problems, and to prevent individuals and the businesses they own from privatizing profits but socializing expenses.

Still, I’d agree that if the only reason to tax is to spend much of it on a big war machine, then that is very problematical, but sadly that is what we have in the USA. Similarly, spending on school-like prisons is problematical. And much of the US medical establishment, much of which is tax supported, is also problematical in its own way, focusing on magic bullets but not the basics of nutrition, exercise, and healthful living. The big issue is not the tax rate, it is how the taxes are spent.