WATCA zero-tax threshold correction: divide by 2.333 not 1.333

watca_zero_tax_thresholds.ipynb

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    "# 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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       "<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",
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      "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                    $105,800  "
      ]
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   "source": [
    "import pandas as pd\n",
    "import numpy as np\n",
    "\n",
    "# WATCA parameters\n",
    "amounts = {\"Single\": 46_000, \"Head of household\": 64_400, \"Joint\": 92_000}\n",
    "phase_out_multiple = 1.75\n",
    "\n",
    "# 2026 projected standard deductions (TCJA extended under OBBBA)\n",
    "std_ded = {\"Single\": 16_100, \"Head of household\": 24_150, \"Joint\": 32_200}\n",
    "\n",
    "rows = []\n",
    "for fs in amounts:\n",
    "    amt = amounts[fs]\n",
    "    sd = std_ded[fs]\n",
    "    po_range = amt * 0.75\n",
    "    po_rate = amt / po_range  # 1.333\n",
    "    combined_rate = 1 + po_rate  # 2.333\n",
    "\n",
    "    # Incorrect formula (divides by phase-out rate only)\n",
    "    wrong = amt + sd / po_rate\n",
    "    # Correct formula (divides by combined rate)\n",
    "    correct = amt + sd / combined_rate\n",
    "\n",
    "    rows.append({\n",
    "        \"Filing status\": fs,\n",
    "        \"Exemption amount\": f\"${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)"
   ]
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   "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)."
   ]
  },
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       "<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",
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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                      $0  "
      ]
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   "source": [
    "def taxable_income(agi, exemption_amount, standard_deduction):\n",
    "    \"\"\"Compute taxable income under WATCA.\"\"\"\n",
    "    po_end = exemption_amount * phase_out_multiple\n",
    "    po_range = po_end - exemption_amount\n",
    "    fraction = np.clip((po_end - agi) / po_range, 0, 1)\n",
    "    exemption = exemption_amount * fraction\n",
    "    return max(0, agi - standard_deduction - exemption)\n",
    "\n",
    "\n",
    "rows = []\n",
    "for fs in amounts:\n",
    "    amt = amounts[fs]\n",
    "    sd = std_ded[fs]\n",
    "    po_rate = amt / (amt * 0.75)\n",
    "    wrong = amt + sd / po_rate\n",
    "    correct = amt + sd / (1 + po_rate)\n",
    "\n",
    "    rows.append({\n",
    "        \"Filing status\": fs,\n",
    "        \"AGI at wrong threshold\": f\"${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)"
   ]
  },
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   "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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