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