Population and incomes
The groups of the population, what each one receives from the budget and pays into it, what it thinks of the government, and the levers that decide who pays and who receives. This page covers equations E27, E28 and E29.
In plain words
The game divides the population of each country by age, activity, wealth and area, urban or rural. Each budget line passes through the income of certain groups. Pensions go to retirees, income tax is mostly paid by the better-off, and VAT weighs more heavily, relative to income, on the least well-off (E27). Each group judges the government on the country's economy, like everyone else, but also on what has become of its own income since the government took office, and the country's popularity is the average of its voters (E28). Finally, two levers change the way a line is shared between rich and poor, without changing its amount (E29).
In the game
The “Society” tab shows seven groups of the country, low-income households, the middle class, well-off households, retirees, the unemployed, young people and rural dwellers, with the income in dollars and the popularity of each one. The overview shows the country's Gini coefficient. Before ordering a measure, the player sees what it would change in the income and the opinion of each group. They also set who pays and who receives, through the income tax schedule, bracket by bracket, through the VAT rates specific to food, energy and medicines, through each of the other benefits, and through a minimum pension that they read in dollars per month, and parliament votes on all of this like the rest of the budget (E25). A group that the government makes poorer can take to the streets (E33). If a game setting switches the groups off, popularity goes back to that of E12.
The basics
A quintile is a fifth of the population, once the population is ranked from the lowest income to the highest. The first quintile thus brings together the least well-off 20%, and the fifth the best-off 20%. In France, the first quintile holds 7.7% of household income, and the fifth 39.7%.
The game combines four divisions. Age has four bands, from under 15 to 65 and over, and activity has five, child, employed, unemployed, inactive and retired. Wealth follows the five quintiles, and the area is urban or rural. Each person thus falls into a single cell of this combination, for example “employed, 25 to 64, third quintile, urban”, and a table gives, for each country, the share of the population in each cell. Since the sources publish each division separately, with only a few combinations of two, the game brings them together with a standard method that respects everything the data say and adds nothing else. When unemployment rises, the table moves workers into unemployment, without changing their age or their quintile, and every 1st of January, it follows the ageing of the population where the data give its age pyramid.
A household's disposable income is what it has left to live on. It comes first from the market, through wages and income from capital, then the government adds pensions and other benefits to it and takes taxes away from it, so that each budget line affects some groups more than others. Who really pays a tax is something that studies measure, and the game takes their figures, by country group. In advanced countries, the best-off quintile pays more than half of income tax. In emerging and low-income countries, pensions go mostly to the better-off, because only declared jobs give a right to them.
The Gini coefficient measures income inequality. It is 0 when everyone has as much, and 100 when a single person has everything. Since the game computes it on the five quintiles, it stays a little lower than that of surveys, which see each household, and France thus starts at 29.4, against 31.8 for the World Bank in the same year.
Show how the game builds the groups, and what the screen reads from themThe table of each country, what it respects, how it follows unemployment, income in dollars, the Lorenz curve and the Gini coefficient.
The table
Each dimension divides the whole population into categories: age (under 15, from 15 to 24, from 25 to 64, and 65 and over), activity (child, employed, unemployed, inactive, retired), wealth (the five income quintiles), area (urban, rural). Each person is in a single category of each dimension, and so in a single cell of their cross-tabulation: “employed, 25 to 64, third quintile, urban”. A country's table gives the share of its population in each cell. Multiplied by its population, it gives the number of people.
What the table respects
- The shares of each dimension that the sources give: the share of people aged 65 and over, that of the unemployed…
- The two-way cross-tabulations that a source gives: activity by age, activity by quintile.
- The combinations that are impossible by definition, which hold nobody. A child (under 15) is neither employed, nor unemployed, nor inactive, nor retired in the sense of these statistics, which only count people from age 15. An adult is not a “child”.
Today, for 186 countries (World Bank, United Nations, ILO), the table holds: the shares of age, activity and area, one fifth of the population in each quintile, and the cross-tabulation of age and activity, which a rule builds. People aged 15 to 24 work and are unemployed at their own rates (ILO), and those aged 25 and over at the rate that the country's unemployment leaves once the young are counted. People aged 65 and over outside the labour force are the “retired”, and those aged 15-64 outside the labour force the “inactive”. For the 62 high-income countries, the table also crosses activity and wealth. Employees, the unemployed and retirees are spread across the quintiles as in the OECD's distributional national accounts (average of six countries). In France, half of the unemployed are in the poorest quintile. For 160 countries, the table finally carries the share of disposable income held by each quintile, which E27 reads.
How it is built
By iterative proportional fitting (Deming and Stephan 1940), the method used to build synthetic populations:
- each possible cell starts from the product of the shares of its categories, which is independence; an impossible cell is zero;
- for each target in turn (a dimension, a cross-tabulation), the cells of each category of the target are multiplied by what this category lacks to reach its share:
where b is the category (or the combination of categories) of the target to which cell c belongs, and tb its share;
- the process starts again, as long as a share deviates from its target by more than the tolerance
(10−10), at most 500 times.
When the targets agree, the result is the table closest to independence that respects them all. Cells that no cross-tabulation links stay independent of one another, insofar as the impossible combinations allow.
| Quantity | Value | Status | Source |
|---|---|---|---|
| maximum passes | 500 | beyond that, targets that do not converge contradict each other; measured: 11 passes at most on an invented world where three cross-tabulations form a loop | |
| tolerance | 10−10 | the gap left between a share and its target | |
| cell budget | 20,000 per country | the product of the categories of all the dimensions; beyond that, the game refuses to load the table |
Why. The sources publish each dimension separately, and a few two-way cross-tabulations, but almost never the cross-tabulation of all of them. Fitting is the most neutral way to bring them together. It adds no information that the data do not have, and it respects everything they say.
What it simplifies. A cross-tabulation that no source gives is taken to be independent. Today, area is crossed with nothing, and outside high-income countries, neither is wealth. There, retirees and the unemployed are spread across the quintiles like the working population, and towns like the countryside, which is wrong. It is written here for that reason, and it is what a new source comes to correct, dimension by dimension. The distribution by quintile of high-income countries is the average of six countries (Czechia, France, Israel, Italy, Slovenia, Sweden), applied to all of them. The “retired” are a definition (65 and over, outside the labour force), not a count of pensions. A 62-year-old retiree is counted as inactive. Without rates specific to young people (a few countries), they take those of the 15-64 age group. The population ages from year to year where the data give it an age pyramid (E58). It does not grow richer or move during a game, and activity follows unemployment (below). Sources that contradict each other leave a gap, which the game reports when it loads the data.
The table follows unemployment
The table is fitted at the start to the rates of the sources, then on every 1st of January to the age pyramid (E58). The model's unemployment moves every day (E6). On the day it is read, the table is that of the last 1st of January, with the unemployed shifted to the day's unemployment. In each combination of the other dimensions (an age, a quintile, an area), the labour force stays the same size, and its unemployment rate is the starting one multiplied by the same ratio,
where ub0 is the unemployment rate of combination b in the starting table, u0 that of the whole table and ut the model's unemployment (E6).
Why. This keeps the structure that the sources describe. Young people, twice as often unemployed as others in France, stay so, and take on twice as many points when unemployment rises. This is the order of magnitude of the sensitivity of youth unemployment to the business cycle.
What it simplifies. The ratio is the same everywhere. Youth unemployment is no more sensitive to the business cycle, in proportion, than that of others. Nobody changes quintile on losing their job. The person stays in the starting quintile.
What the screen reads from incomes
These quantities do nothing to the simulation. They are read, on the 1st of each month, from what E27 has computed, and the screen shows them. They do not enter the hash of the game state.
The day's shares. The share of household disposable income held by quintile q is
with the shares xc of the day's table and the incomes rc of E27. At the start, it is the survey's share. It then moves with the budget lines and unemployment. Without the groups (the groups.enabled setting switched off), the screen keeps the survey's shares.
In dollars. The disposable income of a person, in dollars per year, is their income in points of output per head, times the day's nominal output per head:
where GDP0$ is GDP in dollars in the starting year and YNt the day's nominal GDP, 1 at the start. The income of a group is read in the same way on its own cells only. The income of a person in quintile q is Sq / pq · R, where pq is the share of the population in the quintile, one fifth.
The Lorenz curve and the Gini coefficient. The curve joins (0, 0) to the points (Pq, Lq), where Pq and Lq are the share of the population and the share of income of the q least well-off quintiles, up to (1, 1). The Gini coefficient is twice the area between the diagonal and the curve, each step counted as a trapezium:
It is 0 when each quintile holds one fifth of income.
Out of work for each person in work. The number of people who are not in work (children, the unemployed, the inactive, retirees) for each person who is: (1 − e) / e, where e is the share of the population in work in the day's table.
Why. A person's income reads better in money than as a change. This is what the studio asked for. The Gini is the best-known measure of inequality, and, read on the day's shares, it responds to the policy pursued. One more point of GDP of social benefits brings France's Gini down from 29.4 to 28.8 in two months. The ratio between people out of work and those who have a job says how many people each job supports. It rises with unemployment.
What it simplifies. The Gini is computed between quintiles, as if everyone in a quintile had the same. It is lower than that of the surveys, which see each household (29.4 for France at the start, on the shares of the 2023 survey, against 31.8 the same year at the World Bank), and it does not exceed 80, the value it would reach if the richest quintile held everything. The dollars are those of the starting year, at that year's exchange rate, grown by the country's growth and inflation. The model has no exchange rate, and high inflation inflates the dollars in it like the country's currency. The population is that of the start, grown in line with the country's age pyramid (E56).
E27. The income of the groups
In plain words
Household disposable income is split into flows, work, capital, pensions, other benefits and each tax. The other benefits are themselves five flows, unemployment insurance, family benefits, housing benefits, sickness and disability, and minimum income benefits, each with those it reaches. Each flow is first shared out among the five quintiles, according to the data, then, within each quintile, among those it reaches. Wages thus go to those who have a job, pensions to retirees and unemployment insurance to the unemployed, whereas VAT is paid by all adults. Within the same cell of the table, everyone receives the same. One more point of GDP on a line therefore changes the income of each group according to the share of this line that comes to it.
In France, the least well-off quintile pays 10.5% of VAT, and the best-off 34.3%. France's GDP is 3,366 billion dollars for 68.7 million inhabitants, or about 49,000 dollars per person, so one more point of GDP of VAT takes 490 dollars per inhabitant on average. A person in the first quintile pays 258 dollars of it a year, and a person in the fifth 840. Relative to income, however, it is the first quintile that pays the most, because its income is much smaller.
Since everyone in a cell receives the same, a retiree pays as much income tax and VAT as an employee in the same quintile. The studio knows that this rule probably overstates the share of pensions in retirees' net income, and therefore the effect of a cut in pensions on their income.
In the formula:
| Symbol | What it is | In the example |
|---|---|---|
| \(r_c\) | the income per person of cell \(c\), in points of output | |
| \(M_f\) | the amount of flow \(f\), positive if it is received, negative if it is paid | one more point of VAT |
| \(s_{f,q}\) | the share of the flow that comes to quintile \(q\) | 10.5% for the first, 34.3% for the last |
| \(h_{f,a}\) | what a person with activity \(a\) receives, compared with the others in the quintile; zero for those the flow does not reach | 1 for each adult |
| \(x_c\) | the share of the population in cell \(c\) | |
| \(Y\) | real output |
Show the full formula and its sourcesThe flows one by one, the calibration at the start on each country's survey, the sources by country group, and what the equation leaves out.
Household disposable income is that of E14, taken flow by flow: what the market pays, for labour and for capital; plus pensions and the other benefits; minus each tax. Each flow f is split between the quintiles, then, within a quintile, between the activities it reaches. The income per person of cell c (quintile q, activity a) is, in points of output,
where Mf is the amount of the flow (positive when received, negative when paid), sf,q the share of the flow that goes to quintile q, hf,a what a person of activity a gets compared with the others in the quintile (zero for those the flow does not reach), and xc' the shares of the table. The real income of a person in the cell is rc times real output Y. A flow whose share falls on a quintile where it reaches nobody is lost.
| Flow | Amount | Split between the quintiles | Reaches, within the quintile |
|---|---|---|---|
| labour | fixed, in points of output (calibrated at the start) | calibrated at the start | those in work |
| capital | same | same | adults |
pensions (pensions) | the budget line (E26) | data | retirees |
other benefits (other_social), in five flows (E53) | the share of each benefit in the line; for unemployment benefit, what the number of unemployed makes of it (E54) | unemployment, as the insurance pays it; family, the same amount per child; housing, the same amount per person in the two least well-off quintiles; sickness and disability, the same amount per inactive person; exclusion, data | the unemployed; children; everyone; the inactive; adults |
income tax (income_tax) | the line | data | those in work, retirees |
social contributions (social_contributions) | the line | data | those in work |
VAT (vat), excise duties (excises) | the line | the disposable income of each quintile times the rate it pays; for VAT, then shifted by its rates by family of goods (E38) | adults |
customs duties (customs) | the line | the disposable income of each quintile | adults |
corporate tax, property taxes (corporate_tax, property_tax) | the line | like capital income | adults |
The calibration, at the start. The disposable income of households is D = 100 − A − Σ taxes + pensions + other benefits, in points of output (A is the state's other revenue, which no household pays). Each quintile holds the share of it given by its country's survey (World Bank), or, without a survey, the median of the countries in its group. With the benefits and taxes split as above, the market income of each quintile is what remains: its disposable income, minus what it receives, plus what it pays. This market income is split between labour and capital according to the labour share that the accounts give for this quintile. Measured: none of the 182 calibrated countries leaves a quintile without market income.
| Quantity | Value | Status | Source |
|---|---|---|---|
| share of disposable income of each quintile | per country | World Bank, Poverty and Inequality Platform, through the World Development Indicators | |
| shares of the flows by quintile, VAT and excise rates, labour share | per group of countries | data, derived | OECD (distributional national accounts; OECD/KIPF 2014), World Bank (ASPIRE; Poverty and Shared Prosperity 2022) |
Why. This is what gives each budget line its winners and its losers. A point of pensions goes to retirees. Half of a point of income tax is paid by the richest quintile in advanced countries. A point of VAT weighs more, relative to income, on the poorest (12.2% of its income against 7.7% for the richest, OECD/KIPF 2014). In emerging and low-income countries, pensions go first to the better-off (half to the richest quintile in middle-income countries, three quarters in low-income countries, ASPIRE), so cutting them there does not hit the same people. The income of each quintile at the start is the one that the country's survey measures.
What it simplifies. Within a cell, everyone gets the same, whatever their age or their area. A retiree pays as much income tax and VAT as an employee of the same quintile, which probably exaggerates the share of pensions in the net income of retirees, and therefore the effect of a cut in pensions on their income. Four of the five benefits go to those whom a rule designates, not to those whom a source measures (E53). The shares are those of a group of countries, not of the country: the average of 14 advanced OECD countries, two countries for emerging ones (Mexico, Costa Rica), and, for low-income countries, several flows with no source of their own (the labour share, contributions, corporate tax) that take the values of emerging countries. Market income grows like output, and the income of people who work grows like output per person in work. The calibration says nothing about interest on the debt, like E14.
E28. The opinion of the groups
In plain words
Everyone first judges the country's economy, as E12 says, then looks at what has become of their own income since the government took office, compared with that of all households. One point of income gained or lost through a decision of the state is worth 0.2 points of popularity, whereas one point gained or lost through the economic cycle is worth a quarter as much, 0.05 points. What moves without anyone deciding it, such as pensions that follow the number of retirees each year, or unemployment benefit that follows the number of unemployed, counts like the economic cycle. The unemployed stand 2.5 points below the others, and the others a little above, all the more so as the unemployed are numerous among voters. The country's popularity is the average of its voters, each one weighing according to their chances of voting.
The state takes 9.4% of their income from retirees, and 1% from the income of all households. A retiree therefore loses 8.4 points more than the average, which is worth 0.2 × 8.4 = 1.68 points of popularity less. Since a retiree is not unemployed, they also stand 2.5 × 0.028 = 0.07 points above, because the unemployed make up 2.8% of French voters. In total, retirees sit 1.61 points below the country's target.
The studio has each group's opinion start from that of the whole country, instead of building it from the group's own experience alone, because studies find that the country's economy accounts for most of each person's reaction.
Not everyone votes as much. Where voting is voluntary, the game takes the turnout of each activity and each quintile from Finnish election studies, according to which an unemployed person votes about three quarters as often as the average, and a person in the best-off quintile one and a fifth times as often. Where voting is compulsory and the law is enforced, as in Brazil, everyone weighs the same.
In the formula:
| Symbol | What it is | In the example |
|---|---|---|
| \(P^{*}_c\) | the popularity target of cell \(c\) | 1.61 points below \(P^{*}\) |
| \(P^{*}\) | the target of E12, that of a voter who judged only the country's economy | |
| \(p_c\), \(\bar p\) | what the state has changed in the income of the cell, and in that of all households, in % | −9.4; −1 |
| \(m_c\), \(\bar m\) | what the market has changed in them | 0; 0 |
| \(b_P\), \(b_M\) | what a point of income changed by the state is worth, and one changed by the market | 0.2; 0.05 |
| \(d_U\) | the gap between the unemployed and the others | 2.5 points |
| \(w_U\), \(w_c\) | the weight of the unemployed among voters, and that of the cell | 0.028 |
Show the full formula and its sourcesWho votes and how much, compulsory voting, every parameter with its source, what it gives for France, and what the equation leaves out.
The popularity of E12 judges the country's economy, and that is first of all how each person judges it. A person also weighs their own income since the government took office, and weighs what the state has done to it more than what the market has done to it. Losing a job costs the government a few more points. The popularity target of cell c is
and the country's popularity is the average over its voters:
During a campaign, the player's promises add to each target what voters grant them in advance (E32).
P* is the target of E12. mc is the change, in %, in the real income of a person in the cell since the government took office, due to the market: today's output and unemployment, with the budget lines that the government found when it arrived. pc is what today's lines add to it. Three of these lines move without anyone deciding it, and those the government found then move with them, under the rules it found. Pensions follow the number of retirees (E59), long-term care follows the very old (E66), and unemployment benefit the number of unemployed (E54). What moves in this way belongs to the market, not to the government. p̄ and m̄ are the same changes for the income of all households together. wc is the weight of the cell among voters, wU that of the unemployed. Each cell, each group and the country move towards their target at the pace of popularity (E12), and stay between 0 and 100.
Incomes are read on the first of each month, with the month's decisions in force, and on the day a government takes office. It is judged from the economy it inherits. At the start of the game, the government in office is judged from the first day, as in E12.
Who votes, and how much. The weight of a cell is its share of the population, times the share of its age band that has the right to vote, times, where voting is not compulsory, the weight of a person of its categories among those who vote:
where v is the country's voting age, Amin and Amax the ages of the band (ages are taken to be evenly spread within it: 0.7 of those aged 15 to 24 vote where the voting age is 18, and 0.9 where it is 16), and τd,k the turnout of a person of category k of dimension d, compared with the average. Where voting is compulsory, and the law enforced, all the τ equal 1. Whoever has the right to vote votes.
| Quantity | Advanced countries | Emerging | Low-income | Status | Source |
|---|---|---|---|---|---|
| voting age | 18, except the countries named | same | same | CIA World Factbook (2026); V-Dem v16: 16 in Austria, Malta, Brazil, Argentina, Ecuador, Cuba, Nicaragua; 17 in Greece, Indonesia; 20 in Taiwan, Bahrain, Cameroon; 21 in Singapore, Kuwait, Lebanon, Oman… | |
| compulsory and enforced voting | — | Brazil, Bolivia, Ecuador, Peru, Uruguay | — | Factbook, Kostelka, Singh and Blais (2022), V-Dem; also Australia, Singapore (advanced), North Korea, Samoa | |
| turnout by age (voluntary voting) | carried by activity | 15-24: 0.84; 65 and over: 1.17 | 15-24: 0.875; 65 and over: 1.17 | CSES module 5, seven middle-income countries (OECD, Society at a Glance 2024, figure 8.11); Afrobarometer, round 9 (18-35, and 56 and over) | |
| turnout by activity (voluntary voting) | employed 1.061; unemployed 0.753; inactive 0.796; retired 0.997 | — | — | Statistics Finland, 2023 parliamentary elections, electoral register | |
| turnout by quintile (voluntary voting) | 0.824; 0.910; 0.994; 1.072; 1.200 | — | — | same, by quintile of disposable income |
Compulsory voting. Enforced compulsory voting adds 14.5 to 18.5 points of turnout. Compulsory voting with no enforced sanction adds 7.5 to 10 (Kostelka, Singh and Blais 2022, over all democratic elections from 1945 to 2017). When turnout falls, the gaps between young and old, rich and poor widen (Persson, Solevid and Öhrvall 2013: a Swedish election rerun in 2011, where turnout fell from 80.6 to 44.1%). Conversely, the effect of education on turnout shrinks under weakly enforced compulsory voting, and even more under strictly enforced voting (Dassonneville, Hooghe and Miller 2016). The model keeps the two clear-cut cases. Where the law is enforced, nobody abstains more than anyone else. Elsewhere, each person's abstention follows the sources. A country where voting is compulsory in law but the sanction is not applied (Belgium since 2003 according to V-Dem, Luxembourg, Argentina, Greece, Mexico, Thailand…) is counted as a voluntary-voting country.
| Parameter | Value | Status | Source |
|---|---|---|---|
own_income_policy bP | 0.2 points per point of a person's own income changed by the state | Healy, Persson and Snowberg 2017, table 5: 0.20 to 0.21, Swedish tax register matched to the 2010 election survey; one point of probability of voting for the government is counted as one point of popularity | |
own_income_market bM | 0.05 points per point of a person's own income changed by the market | the same table: 0.044 to 0.059 for the election year, 0.069 to 0.11 for recession years | |
unemployed_gap dU | 2.5 points | Upward and Wright 2024, table 1: losing a job removes 2.1 to 3.1 points of support for the party in power, up to more than six years later |
Why. The literature does not allow the opinion of each group to arise from its own experience alone. The country's economy makes up most of each person's reaction (Kinder and Kiewiet 1979; Di Tella, MacCulloch and Oswald 2001: 86% of what a point of unemployment costs falls on those who keep their job). The literature does, on the other hand, measure the effect of a person's own income, from registers, and finds it four times stronger when it comes from a decision of the state than from the business cycle. Voters reward a tax cut and hardly punish a recession that the government did not cause (Healy et al. 2017). Hence the two coefficients, and the gap measured since the government took office. In this study, a change in income in 2006-2007 still counts in the 2010 vote. The unemployment of some does not make the country more discontented than E12 already says. The unemployed sit below the others, and the others a little above, in proportion to the weight of the unemployed among voters. Each voter counts for one vote, whereas E12 counts each euro of income. A tax paid by few people, even rich ones, costs less popularity than a VAT paid by everyone.
The results. Measured on France, in a calm world, a cut in pensions of one point of GDP takes 9.4% of their income from retirees and, four months later, 1.2 points of popularity among them, against 0.4 among low-income households, the middle class or well-off households, and 0.4 for the country. At the start, the unemployed stand 2.5 points below the others, and the country does not move.
What is read without going through income. Two things enter the “political” share of what a group holds the state responsible for, without any income moving. One is a raised retirement age, for those who were approaching it (E65), and the other is the help for very old people that the state pays, for those aged 65 and over (E67). Protest movements (E33) read them in the same way.
What it simplifies. The coefficients come from a Swedish study and a British study, applied to all countries. One point of individual probability of voting is counted as one point of popularity. Nothing distinguishes what each group thinks of inflation or unemployment. The literature says that low-income households fear unemployment more and retirees fear inflation more, without giving a size in points of popularity. No loss weighs more than a gain. Nothing punishes a cut in pensions among retirees in its own right. The studies that look for such a penalty do not find it (Bansak, Bechtel and Margalit 2021; Ciobanu). Turnout by activity and by quintile comes only from Finland, and the weights of several dimensions are multiplied as if the dimensions were independent. Outside advanced countries, only age counts. Compulsory voting is all or nothing. Enforced, it erases every difference. Without a sanction, it changes nothing, even though it still adds 7.5 to 10 points of turnout. Under compulsory voting, those for whom voting is optional (16- and 17-year-olds and people over 70 in Brazil) vote like the others. The level of each country's turnout is not read. The weights are relative, and the gaps in a voluntary-voting country are those of Finland, whether its turnout is 50 or 80%. Reading each country's turnout would require V-Dem, whose licence (CC BY-SA 4.0) would extend to the data drawn from it. Abstention does not respond to discontent. A median fiscal consolidation adds 0.6 points of abstention (Hübscher, Sattler and Wagner 2023), which the model does not count. No group leans towards a party. The parties' votes (E20) do not read the groups.
E29. Who pays, who receives
In plain words
The player decides how much each budget line weighs, and two levers also decide how two of these lines are shared among the quintiles. Each lever moves a share of the line from the distribution given by the data towards another one. Progressivity makes income tax heavier for the better-off, or brings it closer to a tax proportional to income. The minimum pension pays a share of pensions, in equal parts, to the retirees of the two least well-off quintiles. People respond to these levers by declaring less of an income that is taxed more, and by not always claiming a means-tested minimum to which they are entitled. The player does not hold progressivity itself, since they set the income tax schedule bracket by bracket, and this lever serves non-player governments. Who pays VAT depends on its rates by family of goods, and who receives the other benefits depends on the share of each of the five.
The player raises France's minimum pension to 20%. Since 38% of what a means-tested minimum would pay is never claimed, the share of pensions that really goes through the means test is only 0.2 × (1 − 0.38) / (1 − 0.38 × 0.2) = 0.134, so the quintiles' shares go 13.4% of the way towards the targeted distribution. Pensions then cost 1 − 0.38 × 0.2 = 0.924 times what they cost before, because the minimum that nobody claims is not paid. In the game, the player reads and sets this minimum in dollars per month, which is what a low-income retiree who claims it receives, and the game converts it into this share of pensions.
In the formula:
| Symbol | What it is | In the example |
|---|---|---|
| \(v\), \(w\) | the value of the lever in %, and the same as a fraction | 20; 0.2 |
| \(s_q\), \(s'_q\) | the share of the line that goes to quintile \(q\), before and after | |
| \(t_q\) | the targeted distribution, the same amount for each retiree in the first two quintiles | |
| \(f_q\) | the quintile's share of income before taxes, towards which a proportional tax moves | |
| \(r\) | the share of the minimum pension that nobody claims | 0.38 |
| \(w'\) | the share of the line that really goes through the means test | 0.134 |
| \(k\) | what the line raises or costs, compared with before, once people have responded | 0.924 |
In parliament, the parties judge these levers on the slider of the economic left and right, where progressivity and the minimum pension go towards “more state”. A non-player government that comes to power sets the two levers according to whether it is further to the left or to the right than the one at the start (E24).
Show the full formula and its sourcesThe shares of each lever, how people respond, how far a lever goes, its vote, every parameter with its source, what it gives for France, and what the equation leaves out.
The player decides how much each budget line weighs (E26). Two levers also decide how two lines are split between the quintiles. VAT is set family of goods by family of goods, and its rates cost or raise what they shift (E38); the other benefits are set one by one. Each lever moves the shares sq that the data give the line (E27), by a share w = v/100 where v is its value, towards another distribution, which the lever names. People respond to it, and the line raises or costs a little less than before.
The player does not set progressivity. They set the income tax schedule, bracket by bracket (E52). The lever stays in the model for non-player governments, and applies to the shares that the schedule gives.
| Lever | Line | Values | Towards what |
|---|---|---|---|
income_tax_progressivity | income tax | −100 to +100% | below zero, a tax proportional to income before taxes; above zero, as much further away from it |
minimum_pension | pensions | 0 to 100%, read in dollars per month | the same amount for each retiree in the two least well-off quintiles, means-tested |
The shares
Progressivity moves the shares away from those of a proportional tax, or brings them closer to them:
where fq is the share of quintile q in household income before taxes, at the start: market income, pensions and the other benefits (E27). At w = −1, s′ = f, and everyone pays the same share of their income. Above zero, no quintile pays less than nothing.
The minimum pension moves towards a target distribution t:
where nq is the number of people the line reaches in quintile q (the retirees), each counted by what they receive from it (h of E27). Each one receives the same amount. A means-tested minimum is not claimed by everyone entitled to it. A share r is never paid out, and of what is paid out, the share that goes through the means test is
with r = minimum_pension_non_take_up.
How people respond
The line raises or costs k times what it would without a response:
- income tax: income that is taxed more is declared less. Each quintile, with income before taxes Yq and average rate aq = sqM/Yq (M the tax at the start), changes the income it declares by ΔYq = −ε Yq (a′q − aq)/(1 − aq), ε =
taxable_income_elasticity. This missing income is taxed at the new rate:
More progressive, the tax raises a little less. Closer to a proportional tax, it raises a little more;
- the minimum pension: k = 1 − r w, since the minimum that nobody claims is not spent.
When the lever goes from v to v′, the line goes from L to L · k(v′)/k(v). This is what the state collects or pays out. The player sees it before giving the order, in the deficit and in the amount for the year. Going back to the previous value gives back the previous line.
How far
A lever goes no further than what it still changes. Where the data make the richest quintile pay all of the income tax (low-income countries, where only the richest decile pays direct taxes, E27), it cannot be more progressive. More generally, progressivity stops at the value where the last quintile below the proportional tax no longer pays anything, if only one quintile is then left above it. The minimum pension stops at zero where the line reaches nobody among the least well-off. Where the groups do not judge (the setting, or a country whose data do not describe its people), no lever moves.
What the player reads: the minimum pension in dollars
The model keeps the minimum pension as a share w of pensions. The player, for their part, reads and sets it in dollars per month, as the amount that a retiree in the two least well-off quintiles receives when they claim it:
- P is the pensions line paid out today, as a share of output; P0 is what it would be if everyone claimed what they are owed, and the share is taken on it;
- y is output per head, in today's dollars per year;
- n is the share of the population that the minimum targets, the retirees of the two least well-off quintiles, counted with the day's unemployment.
All those entitled to it, paid for twelve months, take exactly the share set aside: 12 a n × the population = w P0 × output.
No equation changes, since it is the same value, stated differently. The requested amount is brought back to the share w = 12 a n / (P0 y) at the time of the order, and it is the share that is voted on, applied and saved. Where the groups do not judge, the lever reads as before, in % of the line.
What it simplifies.
- The amount follows the pensions budget. With the share fixed, the minimum rises with output and prices, and rises or falls with the pensions line, so it is uprated on its own. A minimum frozen in dollars, which inflation erodes and which has to be uprated by a law, is not represented.
- It is an amount paid out, not a floor. Each low-income retiree receives a on top of what the remaining share of pensions leaves them; a real minimum tops up the pension to a threshold, and gives nothing to those already above it.
- Dollars, not the country's currency, because the game counts everything in dollars.
The vote, and non-player governments
In the vote (E25, proximity voting), the value of the lever, the requested value and the ideal of each party are counted in the points of output that the lever moves from one quintile to others, starting from the distribution in the data, |m| = ½ M Σq |s′q − sq|, with the sign of the lever. The country starts from zero, the gap between countries is the most the lever can move, and the response of people counts in the balance that the parties worry about. The slider and the pole of each lever are written in the game's data. Progressivity and the minimum pension move towards “more state” on economic_left_right, so the left votes for them and the right opposes them.
A non-player government that comes to power sets the two levers according to the gap between its party and the one at the start. This is its camp's distribution, a behaviour of E24, which pure rationality switches off.
| Parameter | Value | Status | Source |
|---|---|---|---|
taxable_income_elasticity ε | 0.25 | Saez, Slemrod and Giertz 2012: the best estimates range from 0.12 to 0.40; part of it is evasion, part is less work, all counted here as lost revenue | |
minimum_pension_non_take_up r | 0.38 | DREES 2022: at the end of 2016, half of the single people eligible for the minimum pension do not receive it; in amount, 790 million euros not paid out for 1,300 million paid, 790 / 2,090 = 0.38 |
| Quantity | Value | Status | Source |
|---|---|---|---|
| least well-off quintiles | 2: the least well-off 40% | the boundary of means-tested benefits is not a quintile; two quintiles keep the lever readable | |
| bounds of the levers | −100 to +100%, 0 to 100% | each end has a meaning: the proportional tax, all pensions paid as the minimum |
Why. Each budget line has its winners and its losers (E27). Changing who pays without changing how much is the heart of social policy (the progressivity of taxes, the minimum pension), and it lets the player arbitrate between groups. The ends of each lever are definitions, not calibrated quantities. The responses that matter most are in the model, with measured sizes. A progressive tax makes people declare less, and the minimum is not always claimed, which makes it cheaper for the budget and less generous for those it targets.
The results. Measured on the first day, at that instant, before the economy responds: the income of each group, the gap between its opinion and the country's target, the primary balance, and the vote.
| France | Low-income | Middle class | Well-off | Retirees | Opinion: low-income, well-off, all | Balance | Vote (seats) |
|---|---|---|---|---|---|---|---|
| progressivity +50% | +2.7% | +1.0% | −2.4% | +1.0% | +0.61; −0.66; +0.11 | −0.02 points | 54 for, 33 against |
| minimum pension 20% | +3.5% | −4.1% | −1.7% | −8.7% | +0.89; −0.80; −0.30 | +1.13 points | 67 for, 33 against |
The minimum pension saves what nobody claims, nearly two fifths of what it promises. In Ethiopia, income tax, paid by the richest quintile alone, cannot be more progressive. The lever stops at zero. Without a player, over ten years of elections without shocks, 109 countries out of 240 have a government that moves these levers, 49 further to the left than the one at the start, 60 further to the right. Progressivity there ranges from −16 to +14%.
What it simplifies. Responses are instant. Declared income and claims for aid follow the lever on the same day. The income that the better-off stop declaring is lost revenue, not lost output. Evasion and reduced work are not told apart, and growth does not move. Average rates stand in for marginal rates, which the table does not have. Non-take-up comes from France, for the minimum pension of single people, and applies everywhere. It depends neither on the amount of the minimum nor on its complexity. The “least well-off” are two quintiles, with no threshold and no phase-out. The parties judge these levers only on the economic left and right.