Fixing Tax Cliffs with Control Theory

The problem illustrated here isn't really high taxation.

From a control-systems perspective, it is a badly designed transfer function.

Think of:

Gross income as the input
Disposable income as the output
Taxation and benefit withdrawal as the mechanism mapping one to the other

At certain thresholds, such as £100,000, the current system introduces a discontinuity: an extremely small increase in gross income can trigger the loss of a large amount of support.

In other words:

The input goes up, but the output goes down.

That is terrible system design.

The problem is not necessarily that the state wants higher earners to contribute more.

The problem is the way that contribution is implemented.

If we want a progressive system, then the proportion taken from the next pound earned can progressively increase with gross income.

But there should be no bands, cliffs or abrupt changes.

The rate itself should be a smooth function of nominal gross income.

The fundamental constraint

Let gross income be:

x

and disposable income after tax and withdrawn benefits be:

y=f(x)

The most basic design requirement should be:

\frac{dy}{dx}>0

Everywhere.

Or, equivalently:

x_2>x_1 \quad\Longrightarrow\quad f(x_2)>f(x_1)

In plain English:

Earning another pound should always leave you with more disposable income than you had before.

There should be no point at which earning more makes somebody poorer.

That does not mean everybody has to face the same marginal rate.

A progressive system can still progressively increase the proportion of each additional pound collected as income rises.

It simply means the gradient of disposable income must never become negative.

Don't create tax bands — define a function

Rather than saying:

income between £X and £Y is taxed at one rate;
income between £Y and £Z is taxed at another;
a benefit disappears when income exceeds some threshold;

define the marginal deduction rate directly as a function of nominal gross income:

m(x)

where:

m(x)=\text{proportion of the next £1 removed at gross income }x

Then disposable income satisfies:

\frac{dy}{dx}=1-m(x)

If we impose:

0\leq m(x)<1

then automatically:

\frac{dy}{dx}>0

and therefore disposable income always rises with gross income.

The tax system can still be progressive.

But it progresses continuously.

There is no point at which somebody crosses an arbitrary line and suddenly faces an entirely different rule.

An illustrative smooth tax function

There are many mathematical functions that could achieve this.

One convenient example is a logistic function:

m(x) = m_0 + \frac{m_1-m_0} {1+e^{-k(x-x_0)}}

where:

\begin{aligned} m_0 &= \text{lower marginal deduction rate} \\[6pt] m_1 &= \text{upper marginal deduction rate} \\[6pt] x_0 &= \text{centre of the transition} \\[6pt] k &= \text{steepness of the transition} \end{aligned}

For the worked example, consider the following smooth progressive function:

m(x) = 0.20 + \frac{0.3793666} {1+e^{-(x-100,000)/25,000}}

This is not intended as a proposed UK tax schedule.

The numbers are simply illustrative.

What matters is the structure.

There are no tax bands.

There isn't a 20% band, followed by a 40% band, followed by a 45% band.

Instead, nominal gross income $x$ is passed into one function:

x\longmapsto m(x)

and that function determines the marginal deduction rate applying to the next pound earned at that precise level of income.

What marginal rate does the function produce?

Evaluating the function at different nominal gross incomes gives:

Gross income $x$	Marginal deduction rate $m(x)$	Next £1 retained
£40,000	23.16%	76.84p
£50,000	24.52%	75.48p
£75,000	30.20%	69.80p
£90,000	35.22%	64.78p
£100,000	38.97%	61.03p
£110,000	42.71%	57.29p
£125,000	47.73%	52.27p
£150,000	53.41%	46.59p
£200,000	57.25%	42.75p

There is nothing special about £100,000.

There is nothing special about £125,000.

There is nothing special about £150,000.

They are simply different inputs to the same continuous function.

Someone earning £99,999 and someone earning £100,001 are essentially adjacent points on the same curve.

There is no threshold to cross.

From the marginal rate to total deductions

The marginal rate tells us what happens to the next pound earned.

Total deductions are obtained by accumulating those marginal deductions over the person's income.

Mathematically:

T(x) = \int_0^x m(s)\,ds

For our illustrative function, this becomes:

T(x) = 0.20x + 9,484.165 \left[ \ln\left(1+e^{(x-100,000)/25,000}\right) - \ln\left(1+e^{-4}\right) \right]

Disposable income is then simply:

y(x)=x-T(x)

So the entire system can be thought of as:

x \longrightarrow m(x) \longrightarrow T(x) \longrightarrow y(x)

Gross income goes in.

The function determines the marginal rate at that income.

Accumulating those marginal deductions determines total deductions.

And disposable income falls out.

Worked example: £50,000 income

Take someone earning:

x=£50,000

The marginal deduction rate at that precise income is:

m(50,000)\approx24.52\%

Their accumulated deductions are approximately:

T(50,000)=£11,032

Disposable income is therefore:

y(50,000) = 50,000-11,032

=\boxed{£38,968}

There is an important distinction here.

The 24.52% rate is not applied to the entire £50,000 salary.

It is the marginal rate applying to the next pound earned at that particular income.

Earlier pounds were subject to slightly lower marginal rates generated by exactly the same function.

Worked example: £100,000 income

Now take someone earning:

x=£100,000

The marginal deduction rate is:

m(100,000)\approx38.97\%

Total deductions are approximately:

T(100,000)=£26,402

Disposable income is therefore:

y(100,000) = 100,000-26,402

=\boxed{£73,598}

Again, there is no £100,000 tax band.

£100,000 is simply another input to the function:

m=m(100,000)

Worked example: £150,000 income

Now take:

x=£150,000

The marginal deduction rate at this income is approximately:

m(150,000)=53.41\%

Total deductions are approximately:

T(150,000)=£50,000

and disposable income is approximately:

y(150,000)=£100,000

The average deduction rate at this income is therefore approximately:

\frac{50,000}{150,000} = 33.33\%

This illustrates the important distinction between marginal and average rates.

At £150,000, the next pound faces a marginal deduction of around 53p, while the average deduction across the whole income is around 33p in the pound.

That happens because the marginal rate was progressively lower at lower nominal incomes.

Run the function across different incomes

We can now evaluate exactly the same function at any nominal gross income.

Gross income	Marginal rate	Total deductions	Average deduction rate	Disposable income	Next £1 retained
£40,000	23.16%	£8,651	21.63%	£31,349	76.84p
£50,000	24.52%	£11,032	22.06%	£38,968	75.48p
£75,000	30.20%	£17,799	23.73%	£57,201	69.80p
£90,000	35.22%	£22,693	25.21%	£67,307	64.78p
£100,000	38.97%	£26,402	26.40%	£73,598	61.03p
£110,000	42.71%	£30,487	27.72%	£79,513	57.29p
£125,000	47.73%	£37,283	29.83%	£87,717	52.27p
£150,000	53.41%	£50,000	33.33%	£100,000	46.59p
£200,000	57.25%	£77,937	38.97%	£122,063	42.75p

The important property is visible all the way through the table.

As gross income rises:

x\uparrow

disposable income also rises:

y\uparrow

The proportion of each additional pound retained gradually decreases as income increases, creating progressivity.

But at no point does earning an additional pound make somebody poorer.

Look closely around £100,000

This is where the contrast with the cliff-edge system becomes particularly clear.

Suppose we evaluate the same function at:

x=£99,999

then:

x=£100,000

and then:

x=£100,001

Nothing special happens.

The function doesn't even know that £100,000 happens to be a nice round number.

Disposable income moves approximately:

£73,597.60 \rightarrow £73,598.21 \rightarrow £73,598.82

So:

\Delta x=+£1

produces approximately:

\Delta y=+£0.61

around this point.

The marginal rate itself also changes by an infinitesimal amount as income changes.

There is no cliff.

There isn't even a kink.

The function is smooth.

Compare that with a cliff

Now compare that behaviour with the problem in the original example.

Imagine that a household receives £20,000 of support at:

x=£100,000

but loses that support once income exceeds £100,000.

Then an increase from:

£100,000 \rightarrow £100,001

might produce:

\Delta x=+£1

while simultaneously producing approximately:

\Delta Y=-£19,999

in total disposable resources.

That is the pathological behaviour.

The issue becomes much easier to see when the tax and benefit system is treated as an input-output relationship.

A positive movement in the input has generated an enormous negative movement in the output.

The effective marginal deduction rate around that transition is vastly greater than 100%.

What about childcare and other benefits?

Exactly the same principle should apply to benefit withdrawal.

Suppose childcare support is:

B(x)

Rather than saying:

Support exists below £100,000 and disappears above £100,000.

make support itself a smooth function of nominal gross income.

For example:

B(x) = \frac{B_{\max}} {1+e^{k_B(x-x_B)}}

where:

\begin{aligned} B_{\max} &= \text{maximum support} \\[6pt] x_B &= \text{centre of the withdrawal curve} \\[6pt] k_B &= \text{steepness of withdrawal} \end{aligned}

The support then declines progressively as nominal gross income increases.

The derivative:

\frac{dB}{dx}

tells us how much support is being withdrawn for the next pound earned.

Again, there is no threshold.

There is simply another continuous function of nominal gross income.

The combined system is what matters

This is probably the most important point.

Government may think of:

income tax;
National Insurance;
childcare support;
personal allowance;
housing support;
child benefit;
other income-dependent transfers;

as separate policies.

The person earning the money doesn't experience them separately.

They experience the combined transfer function.

Suppose:

T(x)

represents taxation and:

B_1(x),B_2(x),\ldots,B_n(x)

represent income-dependent transfers.

Then total disposable resources are:

Y(x) = x-T(x)+\sum_{i=1}^{n}B_i(x)

Differentiate:

\frac{dY}{dx} = 1 - \frac{dT}{dx} + \sum_{i=1}^{n}\frac{dB_i}{dx}

The hard system constraint should therefore be:

\boxed{ \frac{dY}{dx}>0 }

everywhere.

You don't merely test whether each individual policy looks sensible in isolation.

You test the combined system:

If this person's gross income increases slightly, what happens to their total disposable resources?

If the answer is that their disposable resources decrease, the system has failed.

Define the desired behaviour first

This is where I think control and systems engineering provides a much better way of thinking about taxation.

Rather than building a tax and benefits system as an ever-growing collection of rules such as:

IF income > X THEN remove Y

start by specifying the behaviour we actually want from the overall system.

For example:

\begin{aligned} Y(x) &\text{ is continuous} \\[6pt] \frac{dY}{dx} &>0 \\[6pt] \frac{d^2Y}{dx^2} &\text{ is bounded} \\[6pt] 0\leq m_{\mathrm{effective}}(x)&<1 \end{aligned}

In plain English:

Disposable income should be continuous and monotonically increasing, while changes in marginal rates should themselves be gradual rather than abrupt.

That last condition matters.

It is possible to eliminate outright cliffs while still creating sharp kinks.

A properly smooth system should ideally avoid both.

Government could then optimise the functions subject to whatever political objectives it chooses:

required tax revenue;
redistribution objectives;
work incentives;
childcare support;
fiscal sustainability;
desired progressivity;
minimum and maximum marginal rates.

Conceptually:

\text{Political objectives} \longrightarrow \text{Mathematical constraints} \longrightarrow m(x),B_i(x) \longrightarrow \text{System behaviour}

You specify the system-level behaviour first, then engineer the functions that produce it.

That is a much more coherent approach than continually adding thresholds, exemptions and special cases and hoping that the resulting system behaves sensibly.

This isn't difficult to administer

One objection might be that this sounds considerably more complicated than tax bands.

For somebody calculating their tax manually, perhaps.

For a computer, it is trivial.

The system receives nominal gross income:

x

and evaluates:

m(x)

along with the relevant transfer functions:

B_i(x)

and calculates total liability and disposable income.

It is just software evaluating a function.

Modern payroll systems already perform far more complicated calculations.

There is no computational reason why a digital tax system needs to approximate a continuous economic relationship using a collection of crude steps.

Bands and cliffs made considerably more sense when taxation had to be calculated using paper forms, printed tables and manual arithmetic.

That technological constraint no longer exists.

Tax the gradient, not the threshold

The principle can be reduced to something very simple:

Tax the gradient, not the threshold.

Don't ask:

Which tax band has this person entered?

Ask:

Given their nominal gross income, what should the marginal deduction rate be at this precise point?

Mathematically:

m=m(x)

Every additional pound simply moves the person a tiny distance further along the same continuous curve.

There is no crossing point.

There is no sudden reclassification.

There is no cliff.

If we define the overall marginal retention rate as:

r(x)=1-m_{\mathrm{effective}}(x)

then the basic constraint becomes:

0<r(x)\leq1

and therefore:

\Delta x>0 \quad\Longrightarrow\quad \Delta Y>0

Or, more simply:

An increase in gross income must never cause a reduction in disposable resources.

That doesn't determine what the tax rate should be.

It doesn't determine how progressive the tax system should be.

It doesn't determine the size of the state.

And it doesn't determine how much redistribution there should be.

Those are political choices.

What the mathematics does is impose a basic systems-design requirement:

Whatever political choices are made, the resulting transfer function should be continuous, smooth, monotonic and free from perverse cliff-edge incentives.

The politics determines the desired distribution.

The mathematics should ensure the system actually behaves properly.

Fixing Tax Cliffs with Control Theory

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