We ask a lot of our federal income tax system. Its stated purpose is to fund the government, but we also expect it to do something about income inequality. The system redistributes income, implicitly or explicitly, through tax preferences and credits, tax-funded benefit programs, and a progressive rate scale.

Throughout the system, its basic revenue-collection function is intertwined with various redistribution features. Policy makers are thus unable to address revenue issues independent of fairness issues. Simpson and Bowles estimate that over $1 trillion of revenue is lost annually to tax preferences, and not all of them are progressive.

Fortunately, there is a simple way to generate a tax scale which separates redistribution from revenue collection. This article presents the technique, which is *policy neutral* with regard to redistribution, and also explores some of the political implications.

The *z-transform* is a technique commonly used in statistics to change the parameters of a probability distribution. The same technique can also be used to change the parameters of an income distribution. Consider the form of the z-statistic:

Revenue is collected by subtracting from the population’s mean income * μ*. Redistribution is accomplished by reducing the standard deviation

*. Ideally, policy makers would eliminate all tax preferences, and leave redistribution entirely to the formula. The diagram below illustrates how reducing the standard deviation “squeezes” more people into the middle-income range.*

**σ**For an example, consider the income distribution shown below. This example is based on CPS income survey data for 114 million households earning less than $200,000 per year. This population has a mean income of $57,500 with a standard deviation of $43,400. From these households, we will collect tax revenue of $430 billion. This has the effect of reducing the mean from $57,500 to $53,700, an average of $3,800 from each of the households, with the burden distributed according to z-score.

Having established our after-tax mean income level, we decide on $32,100 as the new standard deviation. Economists agree that, while high levels of income inequality can be damaging, the free enterprise system requires some degree of disparity. The z-score preserves *relative disparity* while reducing the absolute amounts. Someone earning at the 1.95 level, roughly $142,000 in this example, would still be at the 1.95 level after taxes. That is,

Where * x *is an individual income figure, and the prime symbol (’) denotes “after tax.”

We convert each income level to a z-score using the pretax mean and standard deviation, then alter these two parameters as above, and convert the z‑scores back to income levels using the new parameters:

Below is a graph of the income distribution, after tax.

Observe that the after tax distribution has a narrower range, and no one in the “under $5,000” category. Those 4,000 households each received a $12,000 tax credit. The top category, households approaching $200,000, each paid a tax of $40,100 or 20%. This 26% reduction in standard deviation is the least disruptive relative to our current tax scale, which favors taxpayers in the $50,000 to $100,000 range. A taxpayer earning $10 million would have a z‑score of 229 and pay a 26% tax.

Once the parameters are set, the z-transform can be used to generate tax tables, as shown below, or (better) a few lines on the tax form:

- Subtract $57,500 from your gross income. The result may be negative.
- Divide this number by $43,400. This is your z-score.
- Multiply your z-score by $32,100, and then add $53,700. This is your net income.
- Subtract your net income from your gross income. This is your tax due.

For example, someone earning the population mean of $57,500 would have a z-score of zero and pay the average tax of $3,800.

We chose the CPS dataset for this example because its $5,000 increments provide a better illustration than the broad income bands used by the IRS. Indeed, the prospect of a *continuously variable rate scale* is one of the z-transform’s advantages.

For a second example, we use the IRS data. This is the complete population of 140 million tax returns. It has a mean income of $55,700, similar to the CPS data, but a standard deviation of $252,500. The distribution is skewed by a small number of households with income above $5 million. This “long tail” of high incomes is a *power law* distribution, with exponent* α = 1.7*. For the sake of clarity, we do not show here the z-transform technique applied to log-scaled income data.

From a policy perspective, the high standard deviation limits our ability to raise taxes on this group without also impacting a large number of households in the middle bands. We suspect that natural skew in the data is exaggerated by the broad IRS reporting bands.

In this example, we will again reduce the (higher) standard deviation by 26%. We collect $870 billion by setting the mean after-tax income to $49,500. The structure of the IRS data obscures the shape of the distribution, so we present the results in tabular form.

Recognizing that our tax system is a vehicle for income redistribution as well as revenue collection, this approach separates the two functions. It uses a simple formula which allows policy makers to set goals for both functions explicitly and independently. It has the added benefit of being continuously variable over the range of incomes. This eliminates well-known problems associated with rate brackets.

We close our exposition of the z-transform approach with a few political results. An individual tax payment (or credit) is given by:

Where * Δµ *is the individual amount required to fund the federal government. It has no subscript because it is the same for every taxpayer. It is determined in advance, and it can be printed on the tax form:

In our example from the 2009 IRS data, * Δµ *is $6,200. This is what each taxpayer contributes toward funding the government. The rest

**, is paid (or received) as redistribution. Even taxpayers below the mean, and receiving a credit, will see the amount by which**

*r**reduces their credit. Redistribution will not excuse inefficiency.*

**Δµ**Total tax revenue is given by:

Everyone is better off when * µ *increases year over year, or

*decreases. Note that*

**Δµ***. Solving for*

**∑r = 0****where**

*x’*

*x**gives:*

**= 0**This is the national minimum income. In the CPS example above, it is roughly $12,000. Our current minimum income is roughly $45,000, although it is never stated as such. The z-transform method states it unambiguously, and rewards any incremental earnings. It avoids the patchwork of benefit programs identified by Alexander, and the “welfare cliff.”

The scale of redistribution is given by the change in standard deviation between the pre- and post-tax distributions. It is natural to state this is as a percentage change:

In both examples, above, we “reduce inequality by 26%.” This is how politicians can state their intention to be more or less progressive. Taxpayers with negative z-scores (those receiving tax credits) will tend to favor a higher * Δσ*, while those above zero will favor a lower one. These self-interested tendencies come into balance when the median income, which evenly splits the voting population, is also the mean.