### Laffer Curve from first principles

It seems easy enough to me to create the Laffer Curve from first principles that I just went ahead in Excel and did it.
We can assume that as price rises, the Demand for a good drops. Conversely, the Supply increases. In the simplest case, each of these relationships is just a straight line.

In economics literature, the Supply and Demand curves are often shown with Price on the y-axis and the Supply and Demand on the x-axis. The point where they cross is the equilibrium price. For example, at a price of \$50, buyers will demand 50 units and suppliers will provide 50 units.

The Laffer Curve represents the amount of revenue the government might receive by imposing a sales tax. If the tax rate is 0, then the government will receive no money from the sales. If the tax rate is enormous, there will be no buyers, so the government will also receive no money. Somewhere in the middle, there should be a revenue-maximizing tax rate.

We can construct the Laffer Curve from the demand curve and a little bit of arithmetic. If the government imposes a \$1 sales tax and raises the price to \$51, then the sellers behavior should not change, since they are still getting their \$50 per sale.

However, the demand from the buyers should drop a little. In this straight line case, the demand will drop to 49 units, with the government getting \$1 per sale, or \$49 total.

If the tax is raised to \$2, demand drops to 48 units, with the government getting \$2 per sale, or \$96 total. As the tax rises further, the revenue-per-sale continue to rise, bu the total number of sales continues to fall. At some point, a revenue-maximizing tax level is reached and then, eventually, the market is killed off entirely.

This is the version of the diagram I have often seen in economics presentations, including at the Laffer Center’s website.

If the current tax rate is above the revenue-maximizing rate, then there is an ‘eat your cake and have it too’ situation where the government can reduce tax rates, but increase its own revenue by allowing the market to get larger.

However, real supply and demand curves are probably not straight lines. For the oil industry, I often see supply curves that look like the red line below. There are a few very low-cost producers (the Middle East), a large number of medium-cost producers (conventional sources outside the Middle East), and then a number of increasingly high-cost unconventional sources (tar sands, the Arctic, etc).

Demand is usually less elastic. There is a narrow bell curve of demand-vs-price, so a small change in price has little effect on demand. (Often, I have seen demand as just a downward-sloping straight line cutting through the supply curve).

In this case, repeating the arithmetic to find government revenue (that is, revenue-per-sale increasing with higher tax rates, while demand for units drops) gives the relationship shown below.

In principle, this is qualitatively the same as the simplified Laffer Curve above: Zero at low and high tax rates, and high in the middle.

But the thing I notice first about this new curve is the asymmetry. If the current tax rate is well above the revenue-maximizing rate, then the industry will die off very quickly.

Whatever your personal politics, you can probably find something here to support your current beliefs: Either high taxes kill markets and reducing taxes can increase market size and tax revenue. Or, current taxes levels in most markets are probably not too high, or else the market would not exist at all. Take your pick. I just thought it was interesting to see what the Laffer Curve might look like for a real product, not just the simplified form I have often seen before.