### Legallizing Pot Won't Raise Tax Revenues

**Matthew Martin**1/11/2013 03:51:00 PM

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Lately I've been hearing a lot of people who oppose marijuana, but think it would be worth legalizing on the grounds that we can tax it and generate more revenues. All that a specific tax on marijuana does is redistribute income from the producers and users of marijuana to the non-producers and non-users. Thus, unless the "legalize-and-tax" cohort is driven by hatred of marijuana users, the implicit assumption they rely on is that legalization is guaranteed to increase GDP--that is, that all production of marijuana would be above and beyond the amount of all other things we currently produce. However, this logic is wrong, wrong, wrong.

Perhaps the easiest way to explain whats wrong with this is to simply note that there is only a finite amount of labor supply available in the United States. Growing marijuana requires labor, which means that we will have to divert workers from other pursuits. Thus, at least part of the increase in marijuana output must be matched by a corresponding decrease in output elsewhere in the economy. Hence, the effect on GDP, and by extension on tax revenues, is ambiguous. Here is, basically, a slightly more mathematical way of saying that:

Lets group all of the things that the economy produces into two categories: one is marijuana, denoted $M_j$, and the other is consumption of all other goods and services, denoted $C$. We will suppose that people have preferences such that they always want positive amounts of both marijuana and consumption, with preferences represented by the utility function $$U=\alpha ln \left(C\right)+\beta ln\left(M_j\right),$$ where $\alpha , \beta \geq 0.$Now we will postulate a hypothetical production function in which marijuana are substitutes, but it takes twice as much inputs (capital and labor) to make marijuana as it does to make the typical consumption good. We will suppose that capital and labor supply are fixed, and normalize the maximum amount of consumption we can produce to one. The production function therefore looks something like this: $$C+2M_j\leq 1.$$ Now lets solve for equilibrium in two scenarios, the first where marijuana is illegal, the second where it is legal (assuming perfect enforcement of the law). We will denote GDP by $Y$.

Marijuana illegal: We have $M_j=0$, $Y=1$, $C=1$.

Marijuana legal: $M_j=\frac{1}{2} \frac{\beta}{\alpha+\beta}$, $C=\frac{\alpha}{\alpha+\beta}$, $Y=\frac{\alpha+\frac{\beta}{2}}{\alpha+\beta}$.

It should be pretty clear in this case that output $Y$ is actually higher in the case where marijuana is illegal whenever $\beta>0$. This means that legalize-and-tax logic is all wrong: no matter how much you tax marijuana and consumption you'll never be able to raise as much government revenue in the economy where marijuana is legal as you could in the economy where it is illegal. If you're concerned about the budget, in this hypothetical the fiscally responsible thing to do would be to ban all marijuana.

This is, of course, only a hypothetical. It is possible that production of marijuana is efficient enough that we would see no change, or maybe an increase in GDP, but I wouldn't bet on that. But my point is considerably more general than that--in fact there are plenty of cases where we could increase GDP by prohibiting certain goods. One example that comes to mind is meat, which takes at least ten times as much inputs as vegetables to produce (that is, it takes ten times as much grain to raise enough meat for us to eat as we would need if we just ate the grain directly). A general prohibition on meat would definitely and unambiguously raise GDP and tax revenues. But then, we like to eat meat, and there's nothing wrong with sacrificing GDP to satisfy our preferences.

Now, if you don't believe me that you can raise more revenue from consumption taxes with marijuana illegal than legal, here's an appendix with the math after the jump:

In the case where marijuana is legal, the household problem is to maximize:

\begin{align}

\stackrel{\max}{_{C,M_j}} \alpha ln \left(C\right)+\beta ln\left(M_j\right) \\

s.t.~\left(1+\tau_c\right) C + 2 \left(1+\tau_m\right) M_j \leq 1

\end{align}

where $\tau_c$ is the sales tax on consumption and $\tau_m$ is the specific tax on marijuana. The FOCs are

\begin{align}

\frac{\alpha}{C}&= \left(1+\tau_c\right)\lambda \\

\frac{\beta}{M_j}&= 2\left(1+\tau_m\right)\lambda \\

\left(1-\tau_c\right)C+ 2\left(1+\tau_m\right) M_j &=1

\end{align}

Solving this system gives us the solution:

\begin{align}

C&=\frac{1}{1+\tau_c}\frac{\alpha}{\alpha+\beta} \\

M_j &=\frac{1}{2}\frac{1}{1+\tau_m}\frac{\beta}{\alpha+\beta} \\

R &=\frac{\tau_c}{1+\tau_c}\frac{\alpha}{\alpha+\beta}+\frac{1}{2}\frac{\tau_m}{1+\tau_m}\frac{\beta}{\alpha+\beta}

\end{align}

where $R$ is government tax revenue. Note from the revenue equation that the maximum revenue possible is bounded by $$R_{bound}=\frac{\alpha}{\alpha+\beta}+\frac{1}{2}\frac{\beta}{\alpha+\beta}$$.

By contrast, consider the case where marijuana is illegal:

\begin{align}

\stackrel{\max}{_{C}} \alpha ln \left(C\right) \\

s.t.~\left(1+\tau_c\right) C + \leq 1

\end{align}

The solution here is trivial:

\begin{align}

M_j&=0 \\

C&=\frac{1}{1+\tau_c} \\

R&= \frac{\tau_c}{1+\tau_c}

\end{align}

Note that in this case the maximum tax revenue is bounded by $$R_{bound}=1$$

Thus, in this case, the fiscally responsible course would be to ban, not legalize marijuana.