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MaxGhenis/watca_zero_tax_thresholds.ipynb

Created March 6, 2026 02:18
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WATCA zero-tax threshold correction: divide by 2.333 not 1.333
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watca_zero_tax_thresholds.ipynb
{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# WATCA zero-tax thresholds: corrected calculation\n",
"\n",
"The Working Americans' Tax Cut Act creates a cost-of-living exemption that phases out between the exemption amount and 175% of the exemption amount. This means the phase-out range is 75% of the exemption, giving an implicit phase-out rate of 1/0.75 = **133%** per dollar of AGI.\n",
"\n",
"But when computing the AGI at which taxable income first becomes positive, we need to account for **both** effects of each additional dollar of AGI above the phase-out start:\n",
"1. **+$1.00** directly to AGI (increases taxable income)\n",
"2. **-$1.333** reduction in exemption (also increases taxable income)\n",
"\n",
"Combined: each $1 of AGI increases taxable income by **$2.333**, not $1.333.\n",
"\n",
"So the correct formula is:\n",
"\n",
"$$\\text{zero\\_tax\\_threshold} = \\text{phaseout\\_start} + \\frac{\\text{standard\\_deduction}}{1 + 1.333} = \\text{phaseout\\_start} + \\frac{\\text{standard\\_deduction}}{2.333}$$"
]
},
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{
"data": {
"text/html": [
"<div>\n",
"<style scoped>\n",
" .dataframe tbody tr th:only-of-type {\n",
" vertical-align: middle;\n",
" }\n",
"\n",
" .dataframe tbody tr th {\n",
" vertical-align: top;\n",
" }\n",
"\n",
" .dataframe thead th {\n",
" text-align: right;\n",
" }\n",
"</style>\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>Filing status</th>\n",
" <th>Exemption amount</th>\n",
" <th>Standard deduction</th>\n",
" <th>Wrong threshold (/ 1.333)</th>\n",
" <th>Correct threshold (/ 2.333)</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>0</th>\n",
" <td>Single</td>\n",
" <td>$46,000</td>\n",
" <td>$16,100</td>\n",
" <td>$58,075</td>\n",
" <td>$52,900</td>\n",
" </tr>\n",
" <tr>\n",
" <th>1</th>\n",
" <td>Head of household</td>\n",
" <td>$64,400</td>\n",
" <td>$24,150</td>\n",
" <td>$82,512</td>\n",
" <td>$74,750</td>\n",
" </tr>\n",
" <tr>\n",
" <th>2</th>\n",
" <td>Joint</td>\n",
" <td>$92,000</td>\n",
" <td>$32,200</td>\n",
" <td>$116,150</td>\n",
" <td>$105,800</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
],
"text/plain": [
" Filing status Exemption amount Standard deduction \\\n",
"0 Single $46,000 $16,100 \n",
"1 Head of household $64,400 $24,150 \n",
"2 Joint $92,000 $32,200 \n",
"\n",
" Wrong threshold (/ 1.333) Correct threshold (/ 2.333) \n",
"0 $58,075 $52,900 \n",
"1 $82,512 $74,750 \n",
"2 $116,150 $105,800 "
]
},
"execution_count": 1,
"metadata": {},
"output_type": "execute_result"
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],
"source": [
"import pandas as pd\n",
"import numpy as np\n",
"\n",
"# WATCA parameters\n",
"amounts = {\"Single\": 46_000, \"Head of household\": 64_400, \"Joint\": 92_000}\n",
"phase_out_multiple = 1.75\n",
"\n",
"# 2026 projected standard deductions (TCJA extended under OBBBA)\n",
"std_ded = {\"Single\": 16_100, \"Head of household\": 24_150, \"Joint\": 32_200}\n",
"\n",
"rows = []\n",
"for fs in amounts:\n",
" amt = amounts[fs]\n",
" sd = std_ded[fs]\n",
" po_range = amt * 0.75\n",
" po_rate = amt / po_range # 1.333\n",
" combined_rate = 1 + po_rate # 2.333\n",
"\n",
" # Incorrect formula (divides by phase-out rate only)\n",
" wrong = amt + sd / po_rate\n",
" # Correct formula (divides by combined rate)\n",
" correct = amt + sd / combined_rate\n",
"\n",
" rows.append({\n",
" \"Filing status\": fs,\n",
" \"Exemption amount\": f\"${amt:,.0f}\",\n",
" \"Standard deduction\": f\"${sd:,.0f}\",\n",
" \"Wrong threshold (/ 1.333)\": f\"${wrong:,.0f}\",\n",
" \"Correct threshold (/ 2.333)\": f\"${correct:,.0f}\",\n",
" })\n",
"\n",
"pd.DataFrame(rows)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Verification\n",
"\n",
"At the correct threshold, taxable income should be exactly zero. At the wrong threshold, it's positive (meaning the filer already owes tax)."
]
},
{
"cell_type": "code",
"execution_count": 2,
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"shell.execute_reply": "2026-03-06T01:34:09.518130Z"
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{
"data": {
"text/html": [
"<div>\n",
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" }\n",
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"\n",
" .dataframe thead th {\n",
" text-align: right;\n",
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"</style>\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>Filing status</th>\n",
" <th>AGI at wrong threshold</th>\n",
" <th>TI at wrong threshold</th>\n",
" <th>AGI at correct threshold</th>\n",
" <th>TI at correct threshold</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>0</th>\n",
" <td>Single</td>\n",
" <td>$58,075</td>\n",
" <td>$12,075</td>\n",
" <td>$52,900</td>\n",
" <td>$0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>1</th>\n",
" <td>Head of household</td>\n",
" <td>$82,512</td>\n",
" <td>$18,112</td>\n",
" <td>$74,750</td>\n",
" <td>$0</td>\n",
" </tr>\n",
" <tr>\n",
" <th>2</th>\n",
" <td>Joint</td>\n",
" <td>$116,150</td>\n",
" <td>$24,150</td>\n",
" <td>$105,800</td>\n",
" <td>$0</td>\n",
" </tr>\n",
" </tbody>\n",
"</table>\n",
"</div>"
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"text/plain": [
" Filing status AGI at wrong threshold TI at wrong threshold \\\n",
"0 Single $58,075 $12,075 \n",
"1 Head of household $82,512 $18,112 \n",
"2 Joint $116,150 $24,150 \n",
"\n",
" AGI at correct threshold TI at correct threshold \n",
"0 $52,900 $0 \n",
"1 $74,750 $0 \n",
"2 $105,800 $0 "
]
},
"execution_count": 2,
"metadata": {},
"output_type": "execute_result"
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],
"source": [
"def taxable_income(agi, exemption_amount, standard_deduction):\n",
" \"\"\"Compute taxable income under WATCA.\"\"\"\n",
" po_end = exemption_amount * phase_out_multiple\n",
" po_range = po_end - exemption_amount\n",
" fraction = np.clip((po_end - agi) / po_range, 0, 1)\n",
" exemption = exemption_amount * fraction\n",
" return max(0, agi - standard_deduction - exemption)\n",
"\n",
"\n",
"rows = []\n",
"for fs in amounts:\n",
" amt = amounts[fs]\n",
" sd = std_ded[fs]\n",
" po_rate = amt / (amt * 0.75)\n",
" wrong = amt + sd / po_rate\n",
" correct = amt + sd / (1 + po_rate)\n",
"\n",
" rows.append({\n",
" \"Filing status\": fs,\n",
" \"AGI at wrong threshold\": f\"${wrong:,.0f}\",\n",
" \"TI at wrong threshold\": f\"${taxable_income(wrong, amt, sd):,.0f}\",\n",
" \"AGI at correct threshold\": f\"${correct:,.0f}\",\n",
" \"TI at correct threshold\": f\"${taxable_income(correct, amt, sd):,.0f}\",\n",
" })\n",
"\n",
"pd.DataFrame(rows)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The wrong thresholds produce **positive** taxable income (the filer already owes tax at that AGI), while the correct thresholds produce exactly $0.\n",
"\n",
"## Intuition\n",
"\n",
"The error comes from only considering the exemption shrinkage ($1.333/dollar) while forgetting that the additional AGI itself adds to taxable income ($1.00/dollar). The standard deduction buffer erodes at the **sum** of these rates: $2.333 per dollar, so it runs out ~43% sooner than the incorrect formula predicts."
]
}
],
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