# My Labor Model

To figure out how people make labor decisions, the natural approach is to assume people maximize some utility function that takes consumption and labor as it's parameters.

Two additional intuitive simplifying assumption are that (1) consumption's and labor's effects on utility are additive separable and (2) that the elasticity of each of these components is fixed.

This all implies a utility function that looks like

$$ - k \cdot c^{-\alpha} / \alpha - L^\beta / \beta $$where $k$, $\alpha$, and $\beta$ are parameters; $L$ is labor, and $c$ is consumption given by

$$ c = w L - \tau(w L) $$

where $w$ is the person's wage and $\tau$ is the tax function.

Optimizing by taking the derivative and setting it to zero yields

$$ k w (1 - \tau'(w L)) {\left( w L - \tau(w L) \right)}^{-\alpha - 1} = L^{\beta - 1} $$If we let $t$ be the marginal tax rate and the "virtual income" (todo: cite) be $y$, this becomes

$$ {\left( k w (1 - t) \right)}^{-1/(\alpha+1)} \left( (1-t) w L + y \right) = L^{(1-b)/(\alpha+1)} $$While this doesn't have a nice closed-form solution, we can compute how $L$ should change when the other parameters change by computing using partial derivatives. For instance, let $Q$ be the percent of your post-tax income that isn't "virtual" - i.e. what you "actually earned":

$$ Q = \frac{(1-t)wL}{(1-t)wL + y} $$Then a 1% increase in $(1-t)w$ should cause a percent change in $L$ of

$$ \frac{1 - Q(\alpha + 1)}{\beta-1-Q} $$while a 1% increase in $y$ relative to post-tax income should cause a percent change in $L$ of

$$ - \frac{\alpha+1}{\beta-1-Q} $$Before we continue, I want to point out some things about these equations:

- $Q \geq 0$. In a flat tax system $Q=1$; in a progressive tax system $Q \lt 1$; and in a regressive tax system $Q \gt 1$.
- $\beta$ is almost certainly large than 2, which makes the denominator always positive.
- Any halfway reasonable person would agree $\alpha \gt -1$, which means $\alpha+1 \gt 0$.
- From the above, we can conclude (a) welfare always reduces $L$ and (b) higher marginal taxes reduce $L$ only if $\alpha/(\alpha+1) \lt 1-Q$ - that is, only $y$ is large enough relative to your post-tax income.
- In particular, under a flat tax, higher tax rates
*increase*labor. This may seem counterintuitive (and it contradicts most of the literature) but it actually matches the historical data: a flat tax of 80% is equivalent to wages being 4x lower, which is exactly what the US was like in the 1940s/50s, when we were actually working*more*hours.

Consider high earners. Let's assume $y \approx 0$ (i.e. $Q \approx 1$), the the wage-labor elasticity equation simplifies into

$$ \delta L = -\frac{\alpha}{\beta-1-Q} $$From the labor economics literature (TODO: link) high earners have a labor~wage elasticity of ~0.15. So, this implies that $\beta \approx \alpha/0.15 + 2$. Recall our estimate of $\alpha \approx 0.35$ (TODO: link) implying that $\beta \approx 4.3$. This in turn lets us compute the labor-welfare elasticity as 0.59.

Running with this, our equation becomes

Consider a single man making ~\$25k. For them, $Q \approx 0.5$. If we plug in these numbers and the $\alpha=0.35$ and $\beta \approx 4$ from above, we get a wage-labor elasticity of 0.12 and a wage-welfare elasticity of 0.48.

Note that this both the 0.12 and the 0.48 estimate are ~20% lower than the estimates for the high-earners.