Expand Universal Credit documentation page (#745) (#1700)

* Expand Universal Credit docs page (#745)

The existing page covered six legacy benefits, gave the Welfare Reform Act
2012 link, and showed a chart of selected element rates. Round it out with:

- a "How PolicyEngine computes Universal Credit" section explaining the
  uc_maximum_amount -> means test -> benefit cap pipeline and the
  would_claim_uc take-up input,
- a References section pointing at the primary statute (WRA 2012), the
  operational regs (SI 2013/376), the annual DWP rate uprating publication,
  and the UC official statistics page that policyengine-uk-data calibrates
  against,
- fix the rates dataframe so it builds in one pass instead of pd.concat in
  a loop (which raised FutureWarning) and corrects the "Amount(£m)" axis
  label to "Amount (£/month)",
- drop an empty trailing code cell.

* Fix taper-rate reference in UC docs

uc_earnings_taper_rate does not exist; the UC taper is the
gov.dwp.universal_credit.means_test.reduction_rate parameter (55%).

Co-Authored-By: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

---------

Co-authored-by: Claude Opus 4.8 (1M context) <noreply@anthropic.com>

Author: vahid-ahmadi

changelog.d/745.md

@@ -0,0 +1 @@
+- Expand the Universal Credit documentation page: add a "How PolicyEngine computes Universal Credit" section walking through `uc_maximum_amount` -> means test -> benefit cap, add a references block (Welfare Reform Act 2012, SI 2013/376, DWP rate uprating, official statistics), tidy the elements rates dataframe to drop the pd.concat-in-loop pattern, fix the £/month axis label, and remove an empty trailing cell.

docs/book/programs/gov/dwp/universal-credit.ipynb

@@ -22,7 +22,7 @@
     "* Housing Benefit for working-age claimants: Helps people with the cost of their rent if they are on a \n",
     "  low income.\n",
     "\n",
-    "Universal Credit parameters can be found in `policyengine_uk/parameters/gov/dwp/universal_credit` and logic in `policyengine_uk/variables/dwp/universal_credit.py`.\n",
+    "Universal Credit parameters live in `policyengine_uk/parameters/gov/dwp/universal_credit/` and the per-element formulas in `policyengine_uk/variables/gov/dwp/universal_credit/`.\n",
     "\n",
     "## Legislation\n",
     "\n",
@@ -41,6 +41,11 @@
     "The table below shows some of the rates covered by PolicyEngine-UK. "
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": "## How PolicyEngine computes Universal Credit\n\nPolicyEngine computes Universal Credit at the benefit-unit level in three stages:\n\n1. **Maximum entitlement** (`uc_maximum_amount`): sum the elements the benunit qualifies for — `uc_standard_allowance`, `uc_child_element`, `uc_disability_elements`, `uc_carer_element`, `uc_housing_costs_element`, and `uc_childcare_element`.\n2. **Means test**: the maximum entitlement is reduced by the benunit's countable earnings above its work allowance (`uc_work_allowance`) at the published taper rate (the `gov.dwp.universal_credit.means_test.reduction_rate` parameter, currently 55%), plus its unearned income. Asset rules apply on top of this — capital above the lower threshold deems a tariff income, and capital above the upper threshold disqualifies the benunit entirely.\n3. **Benefit cap**: the post-means-test award (`universal_credit_pre_benefit_cap`) is finally reduced by `benefit_cap_reduction` to produce `universal_credit`. The benefit cap only applies to benunits without a benefit-cap exemption (working enough hours, having a qualifying disability benefit, etc.).\n\nThe take-up step is handled by the input variable `would_claim_uc`, which is populated stochastically when the dataset is built so that PolicyEngine's caseload aggregates match published DWP claimant numbers rather than the full eligible population.\n\nParameters live in `policyengine_uk/parameters/gov/dwp/universal_credit/` and the per-element formulas in `policyengine_uk/variables/gov/dwp/universal_credit/`."
+  },
   {
    "cell_type": "code",
    "execution_count": 15,
@@ -202,34 +207,20 @@
     "    disabled_amount,\n",
     "    disabled_child_amount,\n",
     "    higher_amount,\n",
-    "]  # [...]\n",
-    "\n",
-    "dates = [\n",
-    "    \"2016-01-01\",\n",
-    "    \"2017-01-01\",\n",
-    "    \"2018-01-01\",\n",
-    "    \"2019-01-01\",\n",
-    "    \"2020-01-01\",\n",
-    "    \"2021-01-01\",\n",
-    "    \"2022-01-01\",\n",
-    "    \"2023-01-01\",\n",
-    "    \"2024-01-01\",\n",
     "]\n",
     "names = [\"Carer\", \"Child\", \"Disabled\", \"Disabled child\", \"First child\"]\n",
     "\n",
-    "import pandas as pd\n",
-    "\n",
-    "df = pd.DataFrame()\n",
+    "dates = [f\"{year}-01-01\" for year in range(2016, 2025)]\n",
     "\n",
-    "for date in dates:\n",
-    "    for element, name in zip(elements, names):\n",
-    "        # Append to a dataframe: row = date, column = element, value = amount\n",
-    "        new_row = {\"date\": date, \"element\": name, \"amount\": element(date)}\n",
-    "        # Append row to the dataframe\n",
-    "        df = pd.concat([df, pd.DataFrame([new_row])])\n",
+    "import pandas as pd\n",
     "\n",
+    "rows = [\n",
+    "    {\"date\": date, \"element\": name, \"amount\": element(date)}\n",
+    "    for date in dates\n",
+    "    for element, name in zip(elements, names)\n",
+    "]\n",
+    "df = pd.DataFrame(rows)\n",
     "\n",
-    "# merge element cells\n",
     "pivot_df = df.pivot(index=\"date\", columns=\"element\", values=\"amount\")\n",
     "pivot_df.fillna(\"\")"
    ]
@@ -1349,7 +1340,7 @@
     "    yaxis_range=[0, 500],\n",
     "    yaxis_tickformat=\",.0f\",\n",
     "    yaxis_tickprefix=\"£\",\n",
-    "    yaxis_title=\"Amount(£m)\",\n",
+    "    yaxis_title=\"Amount (£/month)\",\n",
     "    xaxis_title=\"Year\",\n",
     "    legend_title=\"Element\",\n",
     ")\n",
@@ -1359,11 +1350,16 @@
    ]
   },
   {
-   "cell_type": "code",
-   "execution_count": null,
+   "cell_type": "markdown",
    "metadata": {},
-   "outputs": [],
-   "source": []
+   "source": [
+    "## References\n",
+    "\n",
+    "- [The Welfare Reform Act 2012](https://www.legislation.gov.uk/ukpga/2012/5/contents) — the primary statute creating Universal Credit.\n",
+    "- [The Universal Credit Regulations 2013 (SI 2013/376)](https://www.legislation.gov.uk/uksi/2013/376/contents) — the operational regulations.\n",
+    "- DWP, [Benefit and pension rates](https://www.gov.uk/government/publications/benefit-and-pension-rates-2025-to-2026) — annual rate uprating.\n",
+    "- DWP, [Universal Credit official statistics](https://www.gov.uk/government/collections/universal-credit-statistics) — caseload and expenditure outturns used by `policyengine-uk-data` for calibration."
+   ]
   }
  ],
  "metadata": {
@@ -1388,4 +1384,4 @@
  },
  "nbformat": 4,
  "nbformat_minor": 1
-}
+}