Taxes
The fifteen taxes that the player sets by their rate, and what each one raises. This page covers equation E35.
In plain words
The player does not decide how much a tax raises. They set its rate, such as the standard VAT rate or the corporate tax rate, and for the four taxes that have no single rate, they set the level of their schedule, which is 100% at the start. Revenue then follows from the rate. At the starting rate, the tax raises exactly what the data say, but when the rate rises, those who pay react by declaring less income, buying fewer taxed products, importing less or paying a smaller share of what they owe. Revenue therefore rises more slowly than the rate, and beyond a certain rate, it falls.
In the game
The “Decisions” tab shows each tax with its rate, and before ordering a change, the player sees what it does to the budget balance over one year, in dollars. In a democracy, parliament votes on each change of rate (E25). In France, at the start, the standard VAT rate, at 20%, raises 6.27 points of GDP on everything that does not have its own rate. Raised to 22%, it raises 6.50, and not the 6.90 points that the rule of three promised. Past 27.5%, each extra point makes revenue fall. Non-player governments do not touch their taxes, except for the tariff under a populist leader (E24).
The basics
A tax has a rate and a base. The rate is what the law takes, for example 20% of the price before tax for VAT, and the base is what the tax applies to, here household purchases. Revenue is roughly the rate multiplied by the base.
The gap between the statutory rate and the revenue is often wide, because some products pay a reduced rate, some incomes are exempt and part of the tax owed is never paid. The game does not detail these gaps. Instead, for each country, it measures what one point of rate raises at the start, that is, the starting revenue divided by the starting rate.
The base is not fixed, though. When the rate rises, taxpayers declare less income and companies declare their profits elsewhere, while taxed products sell less. Studies measure this reaction tax by tax, and the game takes their form and their figure. Raising a rate therefore raises less than the rule of three says, and for some taxes there is even a rate beyond which revenue falls, because the base shrinks faster than the rate rises.
The fifteen taxes fall into the seven revenue lines of the budget, and each tax takes from its line the people who pay it (E27) as well as the slider on which the parties vote on it (E25).
| Line | Taxes the player sets | What responds to the rate |
|---|---|---|
| income tax | the top marginal rate, which the whole schedule follows | declared income, according to what the taxpayer keeps |
| social contributions | the employee rate, the employer rate, the level of payroll taxes | the declared wage, and the wage bill, which bears part of what the employer pays |
| VAT | the standard rate, on everything that does not have its own rate, and the rates on food, on household energy and water, and on medicines | the share of the VAT owed that is actually paid, depending on the standard rate only |
| excise duties | the level, all products together | purchases, according to the price including taxes |
| corporate tax | the statutory rate | the profits declared in the country |
| property taxes | the level of taxes on land and buildings, the top rate of the wealth tax, the top rate of inheritance tax, the level of transfer taxes | declared wealth and inheritances; neither land and buildings nor transfers respond |
| customs duties | the average duty | imports, according to their price including duties |
A country that does not have a tax at the start sees it at zero. If it creates the tax, one point of it raises in that country what it raises in the countries of its group that already have it, scaled to the size of its economy.
E35. Taxes by their rate
In plain words
What a tax raises equals what one point of rate raised at the start, multiplied by the rate, then by what remains of the base at this rate, divided by what remained of it at the starting rate. At the starting rate, this last ratio equals 1, and the tax raises what the data say. The share of the base that remains depends on the tax. For VAT, it is the share of the tax owed that is actually paid, and it falls by 2.7 points each time the standard rate rises by one point. The studio chose this form because it is the only response to the standard VAT rate that studies have measured, and this is why the rates specific to food, energy and medicines do not change this share. What they shift is who pays VAT, since the least well-off spend a larger share of their budget on these goods.
In France, the standard VAT rate, at 20%, raises 6.27 points of GDP, so one point of rate raises 6.27 / 20 = 0.3135 points. At the start, 94.4% of the VAT owed is paid. At 22%, this share falls to 94.4 − 2.7 × 2 = 89%, and the base keeps only 89 / 94.4 = 94.3% of what it was. The standard rate then raises 0.3135 × 22 × 0.943 = 6.50 points, and not 0.3135 × 22 = 6.90.
In the formula:
| Symbol | What it is | In the example |
|---|---|---|
| \(R_t(\tau)\) | what tax \(t\) raises at rate \(\tau\), in points of GDP | 6.50 |
| \(\tau\), \(\tau_{t,0}\) | the requested rate, and the starting rate | 22%; 20% |
| \(R_{t,0}\) | what the tax raised at the start | 6.27 |
| \(c_t\) | what one point of rate raises, \(R_{t,0} / \tau_{t,0}\) | 0.3135 |
| \(g_t(\tau)\) | the share of the base that remains at rate \(\tau\) | 0.89 against 0.944 at the start |
| \(q_0\), \(k\) | for VAT, the share paid at the start, and what it loses per point of rate | 0.944; 2.7 |
Each tax has its own form. For income tax, the share that remains follows what the taxpayer keeps, 1 − τ, with an elasticity of 0.25, which means that declared income falls by 0.25% when what the taxpayer keeps falls by 1%. The rate that raises the most is then 1 / (1 + 0.25) = 80%. For corporate tax, each point of rate loses a fixed share of declared profits, and twice as much outside advanced countries. Customs duties, excise duties and employer contributions respond to the price including taxes, and the wealth tax to the return that remains to wealth once the tax is paid, while taxes on land and buildings and transfer taxes do not respond. All these responses apply from the day the new rate takes effect, and with their long-term strength. Studies do distinguish the short term from the long term, and fuel, for example, responds about 2.5 times more in the long term than in the short term. Even so, the studio has not added a delay, because it would be one more parameter.
Show the full formula and its sourcesThe fifteen taxes and the form of their base, the country that creates a tax, the vote on a rate, every parameter with its source, what it gives for France, and what the equation leaves out.
The player sets fifteen taxes by their statutory rate (the four VAT rates of E38, the corporate tax rate) or, where no rate sums up the tax, by the level of its schedule, 100% at the start. What a tax raises follows from its rate, and its base responds to it. A higher rate raises less than a rule of three would promise, and, beyond a certain rate, less in total.
Rt,0 and τt,0 are what the tax raises at the start (points of GDP) and its starting rate (%), taken from the data. ct is what one point of rate raises, and it holds everything that separates the statutory rate from the revenue (the base, exemptions, reduced rates, fraud). gt is the share of the base that remains at rate τ, which the study of each tax gives. At the starting rate, the tax raises what the data say, so the starting point does not move.
Tax (name) | What the rate says | OECD revenue items | Line | Base g(τ) |
|---|---|---|---|---|
income_tax_rate | top marginal rate of income tax; the whole schedule follows it, and it follows the average marginal rate of the schedule when a bracket moves (E52) | 1100, 1300 | income_tax | (1 − τ)ε |
employee_contribution_rate | employee contribution rate | 2100, 2300, 2400 | social_contributions | (1 − τ)ε |
employer_contribution_rate | employer contribution rate | 2200 | social_contributions | (1 + τ)−ρ |
payroll_tax_level | level of payroll taxes | 3000 | social_contributions | (1 + ts τ)−ρ |
vat_rate | standard rate of VAT, or of the general sales tax, which is the rate of everything that does not have its own rate | 5111, 5112, for the share of the rest (E38) | vat | q0 − k (τ − τ0) |
vat_food_rate, vat_energy_rate, vat_medicines_rate | VAT rate on food, on household energy and water, on medicines (E38) | 5111, 5112, for the share of the family | vat | does not respond |
excise_level | level of excise duties, all goods together | 5121 | excises | (1 + θ τ)−η |
corporate_tax_rate | statutory corporate tax rate, national and local | 1200 | corporate_tax | exp(−s τ) |
property_tax_level | level of property taxes | 4100 | property_tax | does not respond |
wealth_tax_rate | top rate of wealth tax | 4200 | property_tax | ((1 − τ) Rb − 1)ew |
inheritance_tax_rate | top rate of inheritance and gift taxes | 4300 | property_tax | (1 − τ)eh |
transfer_tax_level | level of transfer taxes and taxes on transactions | 4400 | property_tax | does not respond |
customs_rate | average applied customs duty, weighted by trade | 5123 | customs | (1 + τ)−σ |
Here τ is a fraction (a rate of 20% is 0.2; a level of 100% is 1). Each tax belongs to a line of E26, from which it inherits who pays it (E27) and the slider on which the parties vote on it (E25); the VAT rates by family have their own stake, and shift who pays VAT (E38).
Payroll taxes are paid on the wage bill, like employer contributions, and wages bear the same share ρ of them. Their level τ multiplies their starting rate on the wage bill, ts = Rs,0 / W, where the wage bill W, in points of GDP, is the base of employer contributions, that is, what one point of their rate raises times a hundred (the base of employee contributions where employers pay none; with neither, the base does not respond).
A country that does not have the tax at the start has a rate of zero, and a level starts at zero where the data count no revenue. The yield of one point is then borrowed. It is the median of c relative to a base, among the countries of its income group that have the tax, times the country's base. That base is household consumption for VAT, imports for customs and, for the taxes whose base the data do not give, the country's fiscal capacity, that is, what its taxes raise together. A country that levies few taxes would raise little from a new tax.
When the base responds. On the day the decision takes effect, with no delay, like the responses of E29.
With E29. The progressivity of the tax changes who pays it, and the taxpayers' response (E29) multiplies the line. A changed rate moves the line in proportion to what its taxes raise. The two combine. What people have done in response to the distribution stays in place when the rate changes.
At the vote (E25), a tax is weighed on the slider of its line, in points of GDP, but through its rate. What counts is what it would weigh at the requested rate on the base that the rate in force leaves,
that is, what it raises today, in proportion to the rate. A rate rise remains a rise beyond the rate that raises the most. The effect on the balance is what the measure really raises, and the cost of the change is counted in points of rate.
| Parameter | Value | Status | Source |
|---|---|---|---|
taxable_income_elasticity ε | 0.25 | Saez, Slemrod and Giertz 2012: 0.12 to 0.40; the parameter of E29, applied to the whole schedule and to employee contributions, which Lehmann, Marical and Rioux (2013) do not measure separately | |
corporate_base_semi_elasticity s | 0.98 / 1.8 / 1.8 | Beer, de Mooij and Liu 2020, table 2, 402 estimates from 37 studies; outside the OECD, a base about twice as sensitive (Crivelli, de Mooij and Keen 2016, table 3, 173 countries) | |
payroll_wage_incidence ρ | 0.66 | Melguizo and González-Páramo 2013, meta-analysis of 52 studies: employees bear two thirds of what employers pay, and the wage bill falls by as much; for employer contributions and payroll taxes | |
vat_compliance_slope k | 2.7 | Agha and Haughton 1996, 17 OECD countries: one point of standard rate means 2.7 points less compliance; small sample (Keen and Smith 2007), link confirmed on 35 countries (Pappadà and Zylberberg 2017) | |
excise_price_elasticity η | 0.4 | within the published ranges: tobacco, about 0.4 in rich countries (IARC 2011); alcohol 0.30 to 0.55, fuel 0.31 in the long run, corrected for publication bias (Nelson 2013; Havranek, Irsova and Janda 2012) | |
excise_price_wedge θ | 1 | excise duties are worth roughly the pre-tax price of tobacco and fuel; no open series by country | |
import_demand_elasticity σ | 5.03 | Head and Mayer 2014, table 5: the trade elasticity identified by tariffs; import demand product by product is much less elastic (Kee, Nicita and Olarreaga 2008: 1.67); a choice of the studio | |
wealth_return_elasticity ew | 0.77 | Jakobsen, Jakobsen, Kleven and Zucman 2020, table III: the long-run elasticity of taxable wealth to its net return, moderately wealthy; 1.15 for the very wealthy | |
wealth_gross_return Rb | 1.05 | the same study: the 5% gross return of its baseline simulation | |
inheritance_base_elasticity eh | 0.16 | Kopczuk and Slemrod 2001, United States 1916-1996, read in a review; Glogowsky 2021 finds less than 0.1 in the short run |
| Quantity | Value | Status | Source |
|---|---|---|---|
| starting VAT compliance q0 | 1 − the VAT gap; outside the Union, the Union median, 0.918 | study, choice | European Commission, VAT gap in the EU, 2025 |
| bounds of the rates | income and inheritance 0 to 90%; contributions 0 to 50%; VAT 0 to 40%; corporate tax 0 to 60%; wealth 0 to 10%; customs 0 to 100%; levels 0 to 300%; a country that starts above can stay there | what is observed, widened |
Why. A government does not decide that VAT will raise 7.4 points of GDP. It sets the rate, and the revenue follows, less than proportionally. Each form is the one the literature uses for the tax: the elasticity of taxable income to what the taxpayer keeps (Diamond and Saez 2011); the loss of declared profits per point of rate, a semi-elasticity; the tax-inclusive price for imported or taxed goods, and for wages, which bear part of employer contributions; compliance for VAT, the only response to the standard rate that is measured. The rate at which each tax raises the most falls where its formula puts it. It is 1/(1 + ε) = 80% for the income schedule, 1/s for corporate tax (about 100% in the OECD, 56% elsewhere), 1/(σ − 1) = 25% for customs, (Rb − 1)/(Rb(1 + ew)) = 2.7% for wealth and 1/(1 + eh) = 86% for inheritance. Employer contributions and excise duties, whose response is below one, have none.
The results. Measured on the first day, for France:
| Tax | Starting rate | Revenue, points of GDP | One tenth higher | Rate that raises the most | Revenue then |
|---|---|---|---|---|---|
income_tax_rate | 55.37 | 9.29 | 9.88 | 80.0 | 10.98 |
vat_rate | 20.00 | 6.27 | 6.50 | 27.5 | 6.77 |
vat_food_rate | 5.50 | 0.29 | 0.32 | the bound, 40 | 2.12 |
corporate_tax_rate | 25.82 | 2.25 | 2.42 | the bound, 60 | 3.75 |
wealth_tax_rate | 1.50 | 0.09 | 0.10 | 2.7 | 0.12 |
inheritance_tax_rate | 60.00 | 0.73 | 0.79 | 86.2 | 0.89 |
customs_rate | 1.33 | 0.11 | 0.12 | 24.8 | 0.74 |
The standard rate of French VAT raises 6.27 points at 20%. At 22%, it raises 6.50 and not 6.90, because compliance falls, and beyond 27.5% the revenue falls. In the United States, which has no VAT, the sales tax (4%) would raise up to 4.7 points at 19% on what is neither food, nor energy, nor medicines.
What it simplifies.
- The base responds only to its own rate. It does not follow consumption, imports or wages, which the model does not have. A recession lowers revenue along with output (E8); it does not lower the share of consumption in GDP.
- The response is the long-run one, and it comes on the same day. Studies distinguish the short and the long run (fuel responds about 2.5 times more in the long run). A delay would be one more parameter.
- One rate per tax, except VAT. The income schedule moves as a block (who pays remains the business of E29). VAT has four rates (E38). Excise duties are a level common to all goods.
- Part of the response is avoidance, not lost activity (corporate tax, wealth). The model counts it as lost revenue, with no effect on output.
- Outside the OECD, many starting rates come from Wikipedia or are estimated. No open series gives contribution rates outside the OECD. There, the starting rate is the median statutory rate of the countries that have one, and c comes from the country's revenue. A small formal economy raises little per point of contribution, but its real statutory rate can differ from the median by several points.