Appraisal and vulnerability in 3 spoonfuls: change the denominator, change the map

Most countries tax immovable property, and most of them argue about it badly. The argument usually skips the part that decides the answer: before any map is coloured, someone has to choose what is added up, what it is divided by, over which territory it is aggregated, and which cases are left out. Change any of those and the map can change while the underlying data stay identical. This post works through that problem with Chilean data, because Chile happens to publish the pieces needed to do it honestly: a national cadastre of every taxable property, and an official index that ranks small civic territories by socio-territorial vulnerability. The mechanics, though, are not Chilean. Any jurisdiction that assesses property for tax and then maps the result against a deprivation measure faces exactly the same four choices. The question fits in one small fraction: Adding up the assessed value inside a territory answers how much administrative value was allocated there. Dividing that same total by households, by residents or by square metres answers different questions. None of them is «the correct one» by nature; the error appears when one is presented under another's name. The arithmetic is usually innocent. The narrative is not always. Reading contract I cross two Chilean administrative registers: the real-estate cadastre of the Servicio de Impuestos Internos (SII) -Chile's tax authority, roughly the counterpart of the IRS or HMRC- and the Índice Global de Vulnerabilidad Socioterritorial (IGVUST), a socio-territorial vulnerability index published by the Ministry of Social Development and Family. The unit of analysis is the neighbourhood unit, not the parcel, the household or the person. A word on that unit, because it has no clean equivalent elsewhere and it drives half of what follows. A Chilean unidad vecinal (UV) is a civic territory drawn for neighbourhood organisation and local participation - closer to a British ward or an American neighbourhood association boundary than to a census tract. Crucially, it was never designed to tile the country. Large stretches of rural Chile belong to no UV at all. A census geography would cover everything by construction; this one does not, and pretending otherwise is the first way to get the map wrong. | Concept | What it means here | What it does not mean | |---|---|---| | SII cadastre | Administrative register of real estate and its characteristics. | A population census or a register of residents. | | Parcel | Cadastral unit identified by municipality, block and parcel number. | A dwelling, a household, an owner or a person. | | Fiscal appraisal | Administrative valuation used as the base of the property tax. | Sale price, income, or the wealth of whoever lives there. | | IGVUST | A ranking of neighbourhood units by socio-territorial vulnerability. | An individual diagnosis or a causal mechanism. | | RSH | Chile's Registro Social de Hogares, the means-testing registry that supplies the households and people used as denominators. In this processing it sums 15,978,644 people, close to 85% of the country's population. | A complete census, or a universe with even coverage across municipalities. | | Neighbourhood unit (UV) | Territory defined for neighbourhood organisation and participation. | An exhaustive mesh covering all of Chile parcel by parcel. | | Denominator | The magnitude the allocated appraisal is divided by. | Small print added afterwards: it defines the question. | The RSH matters because it is the source of the households and people I use as denominators. Nationally it is broad -around 85% of the population- but it does not cover every municipality equally. In municipalities with low relative RSH enrolment, as can happen in the wealthiest districts of Santiago, an indicator «per RSH household» can inflate because the denominator is narrow, not because there is more appraised value. That is a coverage artefact, not a finding. Fiscal appraisal is also not market price: the SII builds it from the characteristics of the property and its homogeneous valuation zone, not from an observed transaction. It can be a useful territorial signal as long as it keeps its surname, fiscal - Rosen's (1974) hedonic price theory explains why surroundings weigh on the valuation of a differentiated good such as housing, but this post observes neither transactions nor household wealth. One note on notation for readers used to the short scale: Chilean Spanish uses billón for 10¹². Throughout this English version I write trillion for that same quantity. Chilean pesos traded around 980 CLP per US dollar in July 2026, so the national total below -587.4 trillion CLP- is on the order of US$600 billion. Data cut: 19 July 2026. Editorial date: 26 July 2026. All relationships are descriptive and depend on the UV mesh used. The question I aggregated the fiscal appraisal of SII parcels to UV scale and compared it against the IGVUST national vulnerability ordering, holding the numerator fixed and changing the lens: | Measure | Question it answers | |---|---| | Total appraisal | How much fiscal appraisal was allocated to this UV? | | Appraisal per RSH household | How much allocated appraisal corresponds to each registered household in the UV? | | Appraisal per RSH person | How much corresponds to each registered person? | | Appraisal per m² of parcel area | How much corresponds to each square metre of allocated parcel surface? | The short answer: per household and per person there is almost no relationship with vulnerability. Per square metre a strong national relationship appears, but it dissolves when you look only at predominantly urban UVs. This is not a glamorous paradox between wealth and vulnerability. It is more sober, and therefore more useful: the denominator, the universe and the scale are part of the result. A warning before continuing: none of this says how much the people living there earn, what their house is worth on the market, or who owns it. Turning a territorial association into a statement about persons is precisely the ecological inference Robinson (1950) warned about more than seventy years ago. Before the spoonfuls: why we tax what cannot move The property tax has a less exotic logic than its public reputation suggests. Property is a visible base, immobile, and tied to the territory where services are delivered. In Chile, property tax revenue is municipal: part stays in the municipality of origin and part feeds the Fondo Común Municipal, an equalisation fund that redistributes resources for street lighting, green areas, infrastructure and social programmes (SII, Impuesto Territorial). Without a cadastre, appraisals and parcel-level location, that architecture simply does not work. None of this is a Chilean peculiarity. Recurrent taxes on immovable property hold a relevant place in local finance across many countries. The comparison assembled by the World Bank shows collections close to 2%-3% of GDP in the United States, Canada and the United Kingdom, and significant shares of local revenue. These are not copies of the Chilean system, but they share the same intuition: part of the value that accumulates in a territory helps finance that territory (World Bank, 2020). The comparative evidence does not say that any property tax is fair by definition. It says something more uncomfortable: design rules. The OECD and the IMF highlight its immobile base, its revenue potential and its link to local services, but they recommend up-to-date valuations, moderate rates, and targeted or deferred relief for owners with low liquidity. The IMF itself uses the British Council Tax as an example of how overly compressed bands can produce a regressive outcome (OECD, 2022; IMF, 2024). The instrument does not arrive progressive from the factory. There is, of course, a flashier route: delete a line from the bill and rebuild the cost on another spreadsheet. The sum can balance; the distribution need not. When an exemption stops looking at income or value and the compensation reproduces prior revenue, the tax does not disappear: it changes pocket, fund, or postcode. The accounting stays calm. The territory may not. Seen that way, the cadastre stops being a collection of tax rolls and becomes what it actually is: the infrastructure that lets you measure the base, divide it, and argue about who benefits from each rule. Now, the three spoonfuls. Spoonful 1: building the numerator without closing the leak The original extract holds 10,343,893 records. A record is not the same as a unique parcel; administrative databases have echoes too. After deduplicating the cadastral key -municipality, block and parcel number- 9,401,277 parcels remain. The goal is to distribute their appraised value across 6,891 UVs before testing any denominator. Parcels and UVs are both polygons, but their boundaries do not coincide. A parcel can fall entirely inside one UV, straddle several, or touch none. I use areal apportionment by intersection area, a form of areal interpolation (Goodchild, Anselin & Deichmann, 1993): where $$a_p$$ is the parcel's fiscal appraisal and $$A_u$$ the total allocated to the neighbourhood unit. If the parcel sits entirely inside one UV, it contributes everything; if it straddles the boundary in half, it contributes half. Two methodological decisions matter. I use geometric area to apportion. Among unique parcels, 10.7% have no usable declared surface and 2.8% have no geometry at all. Declared surface can serve for auditing or sensitivity checks, but it is not enough to place a parcel in space when there is no polygon. So I do not impute UV allocation from reported square metres alone. If a fallback is implemented later, it has to enter the analytical pipeline first, with an explicit location rule, and only then the post and the viewer. A pretty map is not a licence to invent geometry. I do not renormalise. If $$\sum_u f_{p,u} Before the IGVUST: the size of the observed appraisal Before crossing anythi