An idea for estimating payroll taxes by narrow geographic areas

(Note: The post below later became this working paper on estimating federal taxes by geography, and this report summarizing the results.)

Let’s say we want to know how much payroll taxes are paid by residents of some U.S. county. Seems easy. Just go to the Bureau of Economic Analysis (BEA) website, and look it up the Regional Economic Accounts, right?

Unfortunately, that won’t do. BEA accounts give total payroll taxes remitted by employers in each county, and that counts both residents and nonresidents who commute in from other areas. Turns out official figures for counties are hard to come by. Same for cities, congressional districts or ZIP codes.

So that’s our problem. Here’s a rough-and-ready solution another economist and I worked out recently. It uses a simple fact about the U.S. income distribution—that it approximately follows what statisticians call an exponential distribution—to generate decent estimates of payroll taxes for counties, cities and other narrow geographic areas.

Some Basic Facts
Here’s how payroll taxes work. Earnings are subject to 15.3 percent payroll taxes. Of that, 2.9 percent is for Medicare, and 12.4 percent is for Social Security. But the Social Security portion applies only to about the first $90,000 of income—$87,900 in 2004.

This "capped" structure becomes a huge pain in the neck when estimating payroll taxes by geography. Some areas are richer than others, which means they have more earners above the cap. So to estimate payroll taxes by area, you’ve got to have knowledge of the full income distribution for each area. And that’s not easy to find.

So here’s what we do. First, go to the BEA’s website and gather two pieces of data for each county you want to examine: county population, and the earnings of county residents. Note that to find resident earnings, you’ll have to manually sum "earnings by place of work" and "adjustment for residence" from published BEA tables.

Take those earnings and divide by population. That’s the average earnings for the county. Let’s call this .

Some Math You’ll Need
Most U.S. incomes can be approximately described by what’s called an exponential probability density function. Here’s what it looks like:

(1) .

In equation (1), is the number for mean county earnings we calculated above. The next step is to turn equation (1) into a cumulative probability distribution that lets us plug in values for some income level, and find the percentage of people earning less than that amount.

Let’s call this new function F(x). Here’s what it looks like:

(2) .

Equations (1) and (2) look complex, but there’s an elegant feature of them. They’ve got just one parameter: . And we already know that from above. That means we can use them to estimate the income distributions of counties by knowing only one fact about each one—mean county income.

Here’s how it works.

Step One: How Many People Earn Above and Below the Payroll Cap?
Using equation (2), we calculate the percentage of residents earning above and below the payroll tax cap. The 2004 cap was $87,900. Plug this in for x in equation (2), with equal to average county income. That gives us the percentage of people earning below the cap. Then subtract that figure from one. That’s the percentage of people who earn above the cap. Multiplying each of these by county population, and set the results aside. We’ll need those in a minute.

Step Two: How Much Was Earned Above the Cap?
Taxable earnings for those above the cap are simple to calculate. Take the number of people earning above the cap from step one, and multiply by the 2004 cap of $87,900. That’s the total taxable income of folks earning above the cap. Sit that number aside until the end.

Step Three: How Much Was Earned Below the Cap?
Finding taxable earnings for people below the cap is harder. Every penny of earnings of that group are subject to payroll taxes. That means we have to derive the full earnings of folks below the cap analytically from the exponential distribution above.

The simplest way to do that is to calculate the probability-weighted average earnings of everybody below the cap. Then we’ll just multiply that by the number of people below the cap.

To do this, we put on our math hat and write the following integral for the probability-weighted average earnings of people below the cap,

(3) ,

where k is the 2004 payroll cap of $87,900. Think of equation (3) like this. If we take every possible earnings level from zero to k, and multiply each by the probability that somebody will earn that amount, we’ll end up with a probability-weighted average wage for everybody together. That’s what equation (3) gives you.

So let’s solve it. Thinking hard back to high-school calculus, we integrate this by parts. Here’s what we get:

(4) .

With a little more algebra, this monster simplifies to,

(5) ,

where W is the probability-weighted average earnings of folks below the cap. That’s the number we need.

Step Four: Put It All Together
Now we’re home free. All that’s left is to take the W we calculated in equation (5), and multiply it by the population earning below the payroll tax cap, which we calculated back in step one. That gives us the total income earned by everybody below the cap. Then we add that to the taxable income of everybody above the payroll cap, which we found in step two.

Putting those two together and multiplying by the 12.4 percent payroll tax rate, we’ve got total Social Security payroll taxes for the area. And we’re done.

Table 2 shows some estimates using this method for the five sample counties from earlier. These estimates can then be used to allocate nationwide payroll tax aggregates to those counties.

Table 2. Here's what some results of this method look like for a few sample counties in 2004.

County

State

Population

Mean Resident Earnings

Percentage Residents Below Cap

Probability-Weighted Mean Earnings Below Cap

Social Security Payroll Tax Per Capita

Medicare Payroll Tax Per Capita

Effective Payroll Tax Rate

Los Angeles

California

9,917,331

$26,224

96.5%

$22,228

$3,041

$761

14.5%

King (Seattle)

Washington

1,777,746

$39,818

89.0%

$25,773

$4,043

$1,155

13.1%

St. Louis

Missouri

1,007,723

$33,334

92.8%

$24,656

$3,619

$967

13.8%

Denver

Colorado

555,991

$38,571

89.8%

$25,621

$3,968

$1,119

13.2%

Washington

D.C.

554,239

$48,065

83.9%

$26,228

$4,480

$1,394

12.2%

As far as I can tell, the above works as a rough estimate for cities and counties. But be careful drilling down to really small areas. The math probably falls apart if you plug in Hyannis, Nebraska, population 287. Otherwise, enjoy.

Further Reading
Banerjeea, Anand et al. 2006. "A Study of the Personal Income Distribution in Australia." Physica A 370: 54–9.

Borges, Ernesto P. 2003. "Empirical Nonextensive Laws for the County Distribution of Total Personal Income and Gross Domestic Product." Physica A 334: 255-66.

Bureau of Economic Analysis. 2005. "Local Area Personal Income." (Available at http://bea.gov/regional/pdf/overview/Regional_LAPI.pdf.) Washington, D.C.: U.S. Commerce Department.

Dragulescu, A. and V.M. Yakovenko. 2000. "Evidence for the Exponential Distribution of Income in the U.S.A." The European Physical Journal B 20: 585-9.

National Institute of Standards and Technology. 2007. Engineering Statistics Handbook. (Available at http://www.itl.nist.gov/div898/handbook/). Washington, D.C.: U.S. Commerce Department, Chapter 1.3.6.6.7.

Silva, A. C., and V. M. Yakovenko. 2005. "Temporal Evolution of the ‘Thermal’ and ‘Superthermal’ Income Classes in the U.S.A. During 1983-2001." Europhysics Letters 69: 304-10.

Posted by Andrew on Monday February 12, 2007 | Feedback?



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