# Utility Labor Model

## My Darlings

Most people probably won't enjoy reading this, but it took a bunch of work and I'm proud of it, so here we are.

## The Model

Suppose we model people as utility-maximizing agents whose utility function follows $$ - \left((1-t) \cdot w \cdot L \right)^{1-\epsilon} / (1-\epsilon) - f(L) $$

Where $t$ is a flat tax, $w$ is the wage, $L$ is the amount of labor, $w$ is an amount of money provided to everyone by the government, and $f$ is a function denoting how much people value their free time.

I'm fairly confident there is no closed-form solution for $L$. Even so, this model suggests some interesting ways to estimate $\epsilon$. Before we get into that, though, I want to address a potential issue: most countries don't use a flat tax and fixed refund.

When someone is deciding how long to work, most taxes and transfers can be envisioned as a fixed refund followed by a series of tax brackets.

For instance, suppose a country has two tax brackets: 10% for income between \$0 and \$100,000 and 20% for income above \$100,000.

It should be pretty clear that for most people with low-moderate incomes, this system operates as an effective 10% flat tax since they can't reasonably chose to make over the \$100,000 threshold. What's less obvious is that for people making significantly more than \$100,000, the system is equivalent to a 20% flat tax with a \$10,000 refund.

From this it is clear that the only people for which this system differs from a flat-tax-and-refund are the people near the border: those who can choose between making below \$100,000 and above \$100,000.

This is an important distinction and we'll come back to it. For now, I just want to note that most tax systems have fairly few brackets, which means most people face a system that is effectively a flat tax with a fixed refund. For this reason, this simple model is a non-terrible approximation of the tax code for analytical purposes.

## Comparing Distortions

Like I said, I don't think $L$ has a closed-form solution, so, instead, consider a simple scenario. Suppose at present $t=0$ and $w=0$. Now consider two policies:

- Raise $t$ from $0$ to $\delta$.
- Reduce $t$ from $0$ to $-\delta$.
- Raise $w$ from $0$ to $\delta \cdot w \cdot L_0$ where $L_0$ is the amount of labor worked today.
- Reduce $w$ from $0$ to $-\delta \cdot w \cdot L_0$.

Finally, let $\delta$ be small.

From this starting point, we can prove (mathematically) that (a) the welfare changes cause the same-sized labor distortions (b) the tax changes cause the same-sized labor distortions and (c) the welfare changes will cause a greater decrease in labor than the tax change. In particular, the change will be $\frac{\epsilon}{\epsilon-1}$ times larger.

One thing to note is that if $\epsilon = 1$, then this ratio rises to infinity. If $\epsilon \lt 1$, then the ratio goes *negative* to reflect the fact that this suggests higher flat taxes would cause people to work *more* (as discussed in the Historical Labor section).

This allows us to form a thought experiment: how much would *you* change your labor in each scenario? I'm particularly interested in pairing (1) with (4) and (2) with (3):

- How much more would you work with a -1% flat tax vs a 1%-of-present-income flat refund.
- How more more would you work with a 1% flat tax vs a 1%-of-present-income head tax?

Using my preferred estimate of $\epsilon \approx 1.35$, this suggests the ratio if $3.9$. That is, a guaranteed welfare of 1% of present income is as distorionary as a -3.9% flat tax.

I want to point out that this approach is fairly similar to FFF in that it assumes linear separability between two things and makes use of changes to these by outside forces to estimate elasticity.

As far as I can tell, this thought experiment is mostly useless as an intuition pump since most people probably can't predict how they'd change their labor in these two questions. However, I think it is *potentially* fruitful for two reasons. First, the method is quite sensitive. For instance, if the ratio comes out at 3 vs 4, that distinguishes between $\epsilon = 1.5$ and $\epsilon = 1.33$, which is to say that this method potentially has high ability to distinguish between small differences in $\epsilon$. Second, the IRS has really detailed income tax data (unfortunately, they charge thousands of dollars for anonymized versions of it) which I believe could be leveraged to conduct this kind of analysis.

## Bracket Discontinuities

As I alluded to above, the simplicity of a flat-tax-with-fixed-refund breaks down in the region between tax brackets. This actually provides a second way to use this model to estimate $\epsilon$.

Like I said before, we can simulate each individual bracket as a flat tax with a fixed refund. Now we can consider two tax brackets. I won't go into the math, but this model predicts there should be gaps in the income distribution.

In progressive tax regimes, there should be gaps just above the change in tax brackets where no one earns that much. Conversely, in regressive tax regimes, there should be gaps just before the change in tax brackets where no one earns that much.

The mathematical justification is ornate, but the intuition is straightforward: if your first 39 hours each week are taxed at 10% and your 40th hour will be taxed at 25%, it is unlikely utility-maximizing to work that last hour. After all, you're being paid 17% less after-tax for that hour than you were being paid for the first 39 hours.

The size of these gaps depends on $\epsilon$ and $L$, so (in theory) if we analyze these gaps, we can estimate $\epsilon$ and $f$.

In practice, this approach to estimating these parameters runs into severe difficulties. The first is that the vast majority of people don't have the kind of control over their work hours that would make these gaps emerge; instead, labor distortions usually appear as people dropping out of the labor force, switching to part-time work, or choosing lower-earning careers over time. The second is that we have no compelling reason to think $\epsilon$ and $f$ are the same for everyone, which means we shouldn't really expect clean gaps to emerge.