#### An example of Pareto interpolation for Pennsylvania in 2015

Row # Income brackets Lower bound (si) Number of returns (Ni) Cumulative # of returns (Ni*) Adjusted gross income (Yi) Cumulative adjusted gross income (Yi*)
1 Under \$1 0 70,220 6,200,580 -\$4,832,017,000 \$405,142,214,000
2 \$1 to \$24,999 1 2,234,270 6,130,360 \$25,883,054,000 \$409,974,231,000
3 \$25,000 to \$49,999 25,000 1,428,170 3,896,090 \$51,938,139,000 \$384,091,177,000
4 \$50,000 to \$74,999 50,000 864,230 2,467,920 \$53,231,431,000 \$332,153,038,000
5 \$75,000 to \$99,999 75,000 573,080 1,603,690 \$49,640,844,000 \$278,921,607,000
6 \$100,000 to \$199,999 100,000 776,750 1,030,610 \$104,312,881,000 \$229,280,763,000
7 \$200,000 to \$499,999 200,000 207,390 253,860 \$59,138,493,000 \$124,967,882,000
8 \$500,000 to \$999,999 500,000 32,210 46,470 \$21,558,860,000 \$65,829,389,000
10 \$1,000,000 or more 1,000,000 14,260 14,260 \$44,270,529,000 \$44,270,529,000
Total 6,200,580 \$405,142,214,000
Row # (yi = Yi*/Ni*) Pareto coefficient (bi = yi/si) ai = (bi/(bi – 1) pi% = Ni*/N* ki = si * [pi power(1/ai)]
1
2
3 98,584 3.9434 1.340 0.573 16,498
4 134,588 2.6918 1.591 0.363 26,445
5 173,925 2.3190 1.758 0.236 32,979
6 222,471 2.2247 1.817 0.152 35,395
7 492,271 2.4614 1.684 0.037 28,397
8 1,416,600 2.8332 1.545 0.007 19,858
10 3,104,525 3.1045 1.475 0.002 15,289
Row # Min [Abs(pi – 10)] P90 = ki/[0.1 power 1/ai] Min [Abs(pi – 5)] P95 = ki/[0.05 power 1/ai] Min [Abs(pi – 1)] P99 = ki/[0.01 power 1/ai]
1
2
3 0.473 0.523 0.563
4 0.263 0.313 0.353
5 0.136 0.186 0.226
6 0.052 \$125,730 0.102 0.142
7 0.063 0.013 \$168,161 0.027
8 0.093 0.043 0.003 \$390,860
10 0.098 0.048 0.008

Note: N* (tax units for Pennsylvania in 2015) is 6,799,252.

Source: Authors’ analysis of state-level tax data from Sommeiller 2006 extended to 2015 using state-level data from the Internal Revenue Service SOI Tax Stats (various years) and Piketty and Saez 2016