The Hyperbolic Sales Tax

Calculating the Fair Tax

In common usage sales tax, tax, is calculated as a fraction, crate (pronounced 'see rate' for common rate), of the established retail price, price. The calculated amount is added, at the time of sale, to the price to generate a total amount, cash, to be paid by the customer. We'll be using those bold names to represent numbers in the calculations to come.

tax = crate × price
cash = price + tax
cash = price + crate × price
cash = price ×（1 + crate）

Retailers mark up prices above what they pay for stock. Some use a crate but the rates can seem pretty high when divulged to customers. By changing the base to which the rate is applied so that the markup rate is a fraction of the cost to the customer a more palatable number becomes publishable. We'll call it a jrate (pronounced 'jay rate', see note 1). Because we're talking about applying markup logic to a tax we'll just call the markup amount a tax right now. The formulas are:

cash = price + tax
tax = jrate × cash

These equations require solving to make cash and tax depend only on price and jrate. Using SAM² we get the formulas:

cash = jrate × cash + price
cash ×（1－jrate）= price

cash = price ⁄（1－jrate）
tax = price ⁄（1－jrate）－price

The next to last form is used by the merchant to place price tags on shelved merchandise and is quite different from what is done at a cash register when common sales tax is added. The Fair Tax (Note 3), as presently proposed, moves the jrate style of calculation to the cash register and would be applied before or after state and local taxes are applied. In either case reprogramming of cash registers will be the order of the day.

It is reasonable to ask what value of crate corresponds to a jrate determined by tax law yet to come. The algebra is ninth grade level but it does require some attention to detail. Start by computing the actual tax for both methods of calculation and setting them equal to each other and then apply some more SAM.

price × crate = price ⁄（1－jrate）－price
crate = 1 ⁄（1－jrate）－1
crate = 1 ⁄（1－jrate）－（1－jrate）⁄（1－jrate）
crate =（1－1 + jrate）⁄（1－jrate）

crate = jrate ⁄（1－jrate）

We can also represent the reverse conversion from crate to jrate. The algebra is left for the reader.

jrate = crate ⁄（1 + crate）

Some numeric values for pairs of crate and jrate are easily done in your head. For instance for jrate = 1⁄2（50%）（1－jrate）is also 1⁄2 and the result is 1⁄2 divided by 1⁄2 which is unity（100%）. Similarly for jrate = 1⁄4（25%）the corresponding crate is（1⁄4）divided by（3⁄4）which is 1⁄3 or 33%. You can try some others like jrate = 23% with a pocket calculator. Note that for jrate = 1（100%）there is no answer. If the tax is 100% of the cash transferred there will never be anything left for the merchant and no sale will occur.

Note 1: The use of "j" needs a reason. I attended a four year high school in Philadelphia which was for college bound boys. In 1948 the school was more than half populated by young men whose parents worshiped in synagogues and typically ran small retail establishments on street corners before such stores were driven out of business by the likes of Penn Fruit. It was those merchants who would buy stock for $.50 and sell it for $1.00 while describing the process as a Jewish 50% markup. In freshman algebra class it was a joke because it was so clearly wrong. But today it's no joke. In any case that's where the j comes from even though in today's world it might be considered less than politically correct.

Note 2: The acronym SAM stands for either simple or suitable algebraic manipulation. In this paper it's pretty much the simple version.

Note 3: I see absolutely no reason for use of jrates in the field of taxation. It has to be the lower numbers that convince protagonists that they should essentially lie to the public while proselytizing for their idea. The US of A does have a truth in lending law that makes things like that illegal when selling automobiles but it apparently doesn't apply to politicians. It embarrasses me because I do believe that a consumption tax like the Fair Tax is a good thing for the country and all of its people. The Fair Tax Act calls jrates an "inclusive" calculation "the same way income taxes are calculated" which makes some sense if taxed income is all spent on what would become subject to the Fair Tax. For a price of $100 one would have to earn $100 plus $100 times his marginal tax rate to pay for it. But not all income will be spent on goods subject to the Fair Tax. Surely investment in corporate stock equity would not be a sales-taxable event. Rent? Interest on a consumer loan? Baby sitting? Purchase of a house? With much of disposable income going to pay for items that would not be taxed the comparison loses its meaning. The eventual rate will be determined by the requirements of government budgets yet to be. As a crate it's easy to understand. As a jrate it isn't and would become a real pain if the associated crate needs to be above unity.

For those who are beyond 9th grade algebra the crate vs jrate curve is more than the single arc shown in the graph. There are negative values for both parameters that can be plotted, The result of that is two curves that asymptotically approach the lines jrate=1 and crate= -1. The transformation x = jrate - 1 and y = crate + 1 converts the relationship to a simple xy = -1 where I have dropped the multiplication sign for "real" mathematicians. xy = 1 is well known as the 45 degree hyperbola in quadrants 1 and 3 for which the coordinate axes are the asymptotes, With -1 on the right side it's the same thing in quadrants 2 and 4. Thus the title of this page. I wonder if the term will be considered good or bad hyperbole in the advertising required to convince ordinary citizens that the Fair Tax is a good idea. A suitable logo might use the two knappes of a right angled cone shown with an intersecting plane one unit from the axis.

Calculating the Fair Tax

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