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)