19.2 UBI determination and formulas
UBI determination and formulas
Relevant parts of the constitution
From the constitution:
Constant Value Description \(R_{\mathrm{svc}}\) 0.30 The fraction of each tax-inclusive monetary payment allocated as consumption tax dedicated to funding public services. \(R_{\mathrm{ubi}}\) 0.20 The fraction of each tax-inclusive monetary payment allocated as consumption tax dedicated to funding the Universal Basic Income (UBI). This establishes the degree of redistribution from higher‑consumption to lower‑consumption citizens. \(f_{\mathrm{start}}\) 0.50 The base UBI weighting for children from birth. \(a_{\mathrm{ramp}}\) 13 The age at which the UBI weighting begins to increase linearly. \(a_{\mathrm{adult}}\) 18 The age at which a citizen reaches full UBI weighting (1.0).
- For any monetary payment received into an R‑account at tax-inclusive price p, the component of consumption tax allocated to UBI equals: \[ T_{\mathrm{ubi}} = T_{\mathrm{ext}} + p\,R_{\mathrm{ubi}}. \] Here \(T_{\mathrm{ext}}\) denotes the externality cost (e.g., pollution) associated with that supply.
- The total UBI pool generated by \(T_{\mathrm{ubi}}\) shall be distributed to all citizens. The individual payment \(U_{i}\) for a citizen \(i\) of age \(a_i\) is determined by the formula: \[ U_{i} = \frac{\sum T_{\mathrm{ubi}}}{\sum W(a_j)} \times W(a_i) \]
where \( W(a) \) is the UBI weighting factor by age \(a\):
\[ W(a) = \begin{cases} f_{\mathrm{start}} & 0 \le a \lt a_{\mathrm{ramp}} \\ f_{\mathrm{start}} + (1 - f_{\mathrm{start}}) \frac{a - a_{\mathrm{ramp}}}{a_{\mathrm{adult}} - a_{\mathrm{ramp}}} & a_{\mathrm{ramp}} \le a \lt a_{\mathrm{adult}} \\ 1.0 & a \ge a_{\mathrm{adult}} \end{cases} \]- For any monetary payment received into an R‑account at tax-inclusive price p, the component of consumption tax allocated to public services is: \[ T_{\mathrm{svc}} = p\,R_{\mathrm{svc}}. \]
Age dependent UBI
The "Teenage Raise" under the constants \( f_{\mathrm{start}} = 0.5, a_{\mathrm{ramp}} = 13, a_{\mathrm{adult}} = 18 \) given in the constitution:
- A 13-year-old gets 0.5 units.
- A 14-year-old gets 0.6 units.
- A 15-year-old gets 0.7 units.
- A 16-year-old gets 0.8 units.
- A 17-year-old gets 0.9 units.
- An 18-year-old gets 1.0 units.
The affect of the age dependent UBI is to help families. Note that the UBI depends on age. It doesn't depend on when a child leaves their parents. Dependency determines whether a guardian has control over a person's UBI, it doesn't affect how much UBI is given to a person. That only depends on age. That prevents traps such as entitlement to a person changing when they leave their parents, which for example might cause a child to be forced to stay in a violent home.
Consumption tax rates on tax-inclusive versus tax-exclusive prices
The consumption‑tax rates \(R_{\mathrm{svc}}\) and \(R_{\mathrm{ubi}}\) are on tax-inclusive prices. This can be a source a confusion, because often consumption tax rates are defined on tax-exclusive prices. For example, Australia has 10% GST on tax-exclusive prices, which corresponds to about 9.1% on tax-inclusive prices.
The inclusive and exclusive tax rates are related as follows:
\[ t_\text{exclusive} = \frac{t_\text{inclusive}}{1 - t_\text{inclusive}} \] \[ t_\text{inclusive} = \frac{t_\text{exclusive}}{1 + t_\text{exclusive}} \]Because both \(R_{\mathrm{svc}}\) and \(R_{\mathrm{ubi}}\) are defined on the same tax‑inclusive base, they add linearly. This would not hold if the rates were defined on a tax‑exclusive base. The total consumption tax is 0.2 + 0.3 = 0.5 (i.e. 50%) on tax-inclusive prices. This corresponds to 100% on tax-exclusive prices.
Note that inclusive rates can approach 100%, while exclusive rates blow up to infinity as the inclusive rate approaches 100%.
Definitions and basic quantities
The VoTa system allocates a uniform consumption tax rate \( R_{\mathrm{tot}} \) across two destinations: a universal basic income (UBI) component and a services component. The following derivations assume:
- Every person consumes all income, including their UBI. This is a fair assumption in the long term - the ultimate purpose of money is to spend it.
- Every person receives the same UBI (ignoring age adjustments).
- All tax rates are tax-inclusive.
Under these assumptions, each individual’s tax base is their total consumption \( C_i \), which equals their pre-tax income \( Y_i \) plus their UBI \( U \).
| Parameter | Description |
|---|---|
| \( R_{\mathrm{tot}} = R_{\mathrm{ubi}} + R_{\mathrm{svc}} \) | Total consumption tax rate, decomposed into the UBI component and the services component. |
| \( N \) | Total number of persons in the state. |
| \( Y_i \) | Pre-tax, pre-transfer income of person \( i \) (wages, salary, business income, etc.). |
| \( \overline{Y} = \frac{\sum Y_i}{N} \) | Mean pre-tax, pre-transfer income. |
| \( C_i = Y_i + U \) | Tax-inclusive consumption of person \( i \), assuming all income and UBI are consumed. |
| \( T_i = R_{\mathrm{tot}} C_i \) | Consumption tax paid by person \( i \). |
Determining the UBI
The UBI is funded by the UBI component of the consumption tax. Total UBI revenue is \( R_{\mathrm{ubi}} \sum C_i \). Dividing by the population \(N\) gives the UBI per person:
\[U = R_{\mathrm{ubi}} \frac{\sum C_i}{N}.\]
Using the tax‑inclusive identity \( C_i = Y_i + U \), we obtain
\[U = R_{\mathrm{ubi}} \frac{\sum (Y_i + U)}{N}= R_{\mathrm{ubi}}(\overline{Y} + U).\]
Solving for \(U\):
\[U = \overline{Y}\,\frac{R_{\mathrm{ubi}}}{1 - R_{\mathrm{ubi}}}.\]
This expression is the closed‑form sum of a geometric series representing the recursive self‑funding of UBI inside a tax‑inclusive system:
\[U = \overline{Y}\left(R_{\mathrm{ubi}} + R_{\mathrm{ubi}}^2 + R_{\mathrm{ubi}}^3 + \cdots\right).\]
Thus the UBI is mechanically determined by the UBI tax rate and mean income; it is not an independent policy choice.
Disposable income
In a pure consumption‑tax system, disposable income and tax‑exclusive consumption coincide. Disposable consumption for person \(i\) is defined as consumption minus tax:
\[ C_i^{\mathrm{disp}} = C_i - T_i. \]Since the total consumption‑tax rate is \(R_{\mathrm{tot}}\), the tax paid is \(T_i = R_{\mathrm{tot}} C_i\), giving
\[ C_i^{\mathrm{disp}} = (1 - R_{\mathrm{tot}})\, C_i. \]Using the tax‑inclusive identity \(C_i = Y_i + U\), disposable consumption decomposes into disposable UBI plus net‑of‑tax earned income:
\[ C_i^{\mathrm{disp}} = U^{\mathrm{disp}} + (1 - R_{\mathrm{tot}})\, Y_i, \]where the disposable UBI is
\[ U^{\mathrm{disp}} = (1 - R_{\mathrm{tot}})\, U. \]Thus the disposable income \(C_i^{\mathrm{disp}}\) is an affine (straight‑line) function of gross income \(Y_i\), with:
- The slope \((1 - R_{\mathrm{tot}})\), the net‑of‑tax rate.
- The intercept \(U^{\mathrm{disp}}\), the disposable UBI.
Effective marginal tax rate
Here, “effective” means that transfers are treated as part of the tax system. The effective marginal tax rate (EMTR) is the marginal tax taking transfers into account, and the effective tax rate (ETR) is the average tax taking transfers into account.
The Effective Marginal Tax Rate (EMTR) is defined as the derivative of tax paid with respect to income:
\[ \text{EMTR}_i = \frac{dT_i}{dY_i} = R_{\mathrm{tot}}. \]The derivative describes the rate applied to a small change in income. Actual tax amounts are calculated in integer currency units.
Because the tax is levied uniformly on total consumption, the EMTR is constant and equal to the total consumption‑tax rate.
A flat tax ensures that for every extra dollar earned, the same percentage is taken in tax. You always keep a consistent percentage of your additional earnings, which is a clear and consistent incentive to work more hours and earn more.
Effective tax rate
The Effective Tax Rate (ETR) is an average tax rate and is defined as total tax paid divided by gross income:
\[ \text{ETR}_i = \frac{Y_i - C_i^{\mathrm{disp}}}{Y_i} = R_{\mathrm{tot}} - \frac{U^{\mathrm{disp}}}{Y_i}. \]The ETR falls with income because the disposable UBI \(U^{\mathrm{disp}}\) reduces the average tax burden more for low‑income individuals than for high‑income individuals. In other words, the UBI is progressive in average-rate terms.
Allocation to services
The average per‑person allocation to services \(S\) is determined by the service share of the consumption tax. Total service revenue is \(R_{\mathrm{svc}} \sum C_i\), so dividing by the population \(N\) gives
\[ S = \frac{R_{\mathrm{svc}} \sum C_i}{N}. \]Using the tax‑inclusive identity \(C_i = Y_i + U\), this becomes
\[ S = R_{\mathrm{svc}}(\overline{Y} + U). \]Unlike the UBI, the purchasing power of \(S\) is not reduced by a factor of \((1 - R_{\mathrm{tot}})\). There is no distinction between a “nominal” and a “disposable” funding level for services. Service funding flows directly to operators and their suppliers via transfers into B‑accounts, which do not trigger consumption tax. As a result, \(S\) retains full purchasing power over intermediate inputs.
The relative strength of service funding compared to UBI can be expressed directly in terms of their constitutional shares. Each dollar of UBI, \(U\), is spent as taxed consumption and therefore has real purchasing power \(U(1 - R_{\mathrm{tot}})\). By contrast, each dollar of service funding, \(S\), is spent on intermediate inputs via B‑accounts and retains its full value. Hence, per unit of tax‑inclusive private consumption, the ratio of real redistribution through services versus UBI is
\[ \frac{\text{purchasing power for services}}{\text{purchasing power of UBI}} = \frac{R_{\mathrm{svc}}}{R_{\mathrm{ubi}}(1 - R_{\mathrm{tot}})}. \]For example, with \(R_{\mathrm{ubi}} = 0.2\) and \(R_{\mathrm{svc}} = 0.3\), and thus \(R_{\mathrm{tot}} = 0.5\), this ratio is \(\frac{0.3}{0.2 \cdot (1 - 0.5)} = 3\), meaning that, in real terms, the service allocation is three times stronger than the UBI redistribution.
Interpretation
These relationships highlight three structural features of VoTa:
- The EMTR is constant and equal to the total consumption tax rate.
- The ETR declines with income because the UBI is progressive in average-rate terms.
- The UBI is not an independent policy choice: it is determined by the UBI tax rate and mean income.
Equivalence of flat income tax and flat consumption tax
Under standard simplifying assumptions (no savings, no intermediate consumption tax, and competitive pricing), a flat income tax at rate is equivalent to a flat consumption tax at the same rate on tax-inclusive prices. Both systems ultimately tax the same base, factor income, and therefore produce identical net incomes and government revenue.
The following is an example where there is a supplier A of an intermediate product I and a seller B of a final product F. The only difference between the two columns is how the $4 final price is decomposed into intermediate prices, factor incomes, and tax. In both systems A and B end up with $1 each, and the tax office receives $2.
| Parameter | 50% flat income tax | 50% flat consumption tax on final goods |
|---|---|---|
| price of I (before any tax) | $2 | $1 |
| income tax on A | 50% x income = 50% x $2 = $1 | $0 |
| consumption tax on I | $0 | $0
(because only final goods are taxed) |
| net income to A = gross income - tax | $2 - $1 = $1 | $1 - $0 = $1 |
| price of F including any consumption tax | $4 | $4 |
| gross income to B = sales - costs | $4 - $2 = $2 | $4 - $1 = $3 |
| income tax to B | 50% x income = 50% x $2 = $1 | $0 |
| consumption tax on F | $0 | 50% x price = 50% x $4 = $2 |
| net income to B = gross income - tax | $2 - $1 = $1 | $3 - $2 = $1 |
This result follows from Atkinson, A. B., & Stiglitz, J. E. (1976). “The Design of Tax Structure: Direct versus Indirect Taxation.” Journal of Public Economics, 6(1–2), 55–75.