An example of Pareto interpolation for Pennsylvania in 2011

Row # Income brackets Lower bound (si) Number of returns (Ni) Cumulative # of returns (Ni*) Adjusted gross income (Yi) Cumulative adjusted gross income (Yi*)
1 No income <= 0 82,325 6,183,225 -4,608,529 348,612,835
2 1–<25,000 1 2,419,804 6,100,900 28,102,112 353,221,364
3 25,000–< 50,000 25,000 1,458,749 3,681,096 52,856,101 325,119,252
4 50,000–< 75,000 50,000 859,952 2,222,347 52,954,678 272,263,151
5 75,000–< 100,000 75,000 543,875 1,362,395 47,004,707 219,308,473
6 100,000–< 200,000 100,000 633,858 818,520 84,200,638 172,303,766
7 200,000–< 500,000 200,000 151,006 184,662 43,064,934 88,103,128
8 500,000–<1,000,000 500,000 23,476 33,656 15,763,810 45,038,194
10 1,000,000 or more 1,000,000 10,180 10,180 29,274,384 29,274,384
11 Total 6,183,225 348,612,836
Row # (yi = Yi* / Ni*) Pareto Coefficient (bi= yi / si) ai = (bi / (bi-1) pi % = Ni* / N* ki = si * [pi power(1/ai)]
1 56,380
2 57,897 91.77
3 88,321 3.53 1.39 55.37 16,363
4 122,512 2.45 1.69 33.43 26,139
5 160,973 2.15 1.87 20.49 32,166
6 210,506 2.11 1.90 12.31 33,301
7 477,105 2.39 1.72 2.78 24,952
8 1,338,192 2.68 1.60 0.51 18,242
10 2,875,676 2.88 1.53 0.15 14,586
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 2.31 2.22 0.49
2 81.77 86.77 90.77
3 45.37 50.37 54.37
4 23.43 28.43 32.43
5 10.49 15.49 19.49
6 2.31 \$111,535 7.31 11.31
7 7.22 2.22 \$142,150 1.78
8 9.49 4.49 0.49 \$326,426
10 9.85 4.85 0.85

Note: Money amounts are in thousands of dollars. N* or tax units for Pennsylvania in 2011 is 6,648,369.

Source: Authors' analysis of state-level tax data from Sommeiller (2006) extended to 2011 using state-level data from the Internal Revenue Service SOI Tax Stats (various years), and Piketty and Saez (2012)