### Consumption taxes and the "double taxation" of income

**Matthew Martin**6/19/2015 01:17:00 PM

A short sentence in this piece about Rand Paul's tax plan set off a chain reaction in my brain that can only be resolved by blogging. Here it is:

"This is the traditional argument for flat taxes: they're a form of consumption tax, and economists think consumption taxes help growth by exempting savings and investment from taxation."Clicking through to their link brings us to a common talking point about consumption taxes:

"Economists tend to find that consumption taxes are better for the economy than income taxes, because income taxes discriminate against savers.I'm afraid Dylan Matthews has engaged in an unintended slight-of-hand here: you don't generally pay income taxes on investment income.

To see why, imagine you make $50,000 in wages and there's a flat 20 percent tax on all income. You'd pay $10,000 in taxes on your wages. That leaves you with $40,000.

Now you've got a decision to make: do you want to take $5,000 of the $40,000 you have left and invest it, or do you want to take that $5,000 and spend it on a really awesome television? If you invest it and make money off the stocks, then the thing you bought with your money—the profits those stocks made for you—will get taxed again. If you just buy the TV, the government doesn't tax you a second time."

^{1}We have a capital gains tax that is separate from the income tax.

So the real difference between income and consumption taxes is that consumption taxes let you invest first and then pay taxes on both the income and the interest earned on it, while income taxes require you to pay taxes first, reducing the amount you can invest, but then doesn't tax the interest income. The question is, which is more efficient?

The answer is that consumption taxes are weakly more efficient, but not for the reasons commonly argued.^{2} In fact, income taxes don't distort the *investment* decision any more than consumption taxes do. The only difference between the two is that the income tax distorts the labor supply decision more. For illustration, consider the basic DSGE model of an infinitely lived representative household.The household problem is to maximize \begin{align*}\max_{C_t,L_t,K_{t+1}}U=\sum_t \beta^t u\left(C_t,1-L_t\right)&\\ subject~to~K_{t+1}-\left(1-\delta\right)K_t+\left(1+s\right)C_t\leq\left(1-\tau\right)w_tL_t+r_tK_t& \end{align*} where [$]C_t[$],[$]L_t[$], and [$]K_t[$] are consumption, labor, and capital, respecively, in period [$]t[$], [$]w_t[$] and [$]r_t[$] are the wage rate and interest rate, [$]\beta,\delta \gt 0[$] are constants, and [$]u\left(\centerdot \right)[$] is a period utility function satisfying standard convexity and Inada assumptions. We'll let [$]u_{c,t}[$] and [$]-u_{l,t}[$] denote the derivatives of utility respect to consumption and labor respectively. Firms seek to maximize profits according to a linearly homogenous production function given by [$]Y=AL_t^\alpha K_t^{1-\alpha}[$] by solving the problem:[$$]\max_{L_t,K_t} AL_t^\alpha K_t^{1-\alpha}-w_tL_t-r_tK_t.[$$] And finally, the government taxes consumption and labor at rates [$]s[$] and [$]\tau[$] respectively to finance government spending [$]G_t[$] according to it's budget constraint given by [$$]sC_t+\tau w_t L_t=G_t.[$$]

The first order conditions for the household problem are \begin{align} Labor~Supply:&~\frac{u_{c,t}}{u_{l,t}}=\frac{\left(1+s\right)}{\left(1-\tau\right) w_t} \\ Euler:&~u_{c,t}=\beta E_t\left[u_{c,t+1}\left(1-\delta+r_{t+1}\right)\right]\\ Budget~Constraint:&~K_{t+1}-\left(1-\delta\right)K_t+\left(1+s\right)C_t=\left(1-\tau\right)w_tL_t+r_tK_t \end{align} Notice right away we can see that the first order condition for investment [$]K_{t+1}[$] actually doesn't have any tax rates in it at all. These taxes don't actually distort the investment decision at all, but rather distort the consumption and labor decisions—this shouldn't be very surprising since the two taxes are a consumption tax and income tax, after all. From the firm problem we have: \begin{align} w_t&=\alpha A\left(\frac{K_t}{L_t}\right)^{1-\alpha} \\ r_t&=\left(1-\alpha\right) A\left(\frac{L_t}{K_t}\right)^{\alpha} \end{align} and of course, [$$]s=\frac{G_t-\tau w_tL_t}{C_t}[$$] follows from the government's budget.

It turns out this model has a steady-state solution, so stripping out all the time subscripts and substituting out [$]r[$]: \begin{align} \frac{u_c}{u_l}&=\frac{\left(1+s\right)}{\left(1-\tau\right) w} \\ \left(1+s\right)C&=\left(1-\tau\right)wL+\frac{1-\beta}{\beta}K\\ w&=\alpha A\left(\frac{K}{L}\right)^{1-\alpha} \\ \frac{1-\beta+\delta\beta}{\beta}&=\left(1-\alpha\right) A\left(\frac{L}{K}\right)^{\alpha} \label{fixedK}\\ s&=\frac{G_t-\tau wL}{C} \end{align} From here it should be readily apparent why it is incorrect to say the income tax discourages investment. To see this, consider the case where labor supply is fixed. Then from equation [$]~\eqref{fixedK}[$] we have that [$]K[$] is fixed as well—that is, if the income tax does not cause labor supply to change, then it follows that it does not have any effect on investment either. And from there it follows that if labor supply is fixed, then households are totally indifferent between income and consumption taxes, and that consumption is identical either way. In other words, the income tax doesn't discourage investment, it discourages labor, and affects investment only indirectly by reducing labor supply and thus the amount workers have to invest.

But in the real world, labor supply is not fixed. And if you do the algebra from the system above you'll see that it turns out that when labor is not perfectly inelastic, the income tax causes a relatively larger distortion of the labor supply decision than consumption taxes do, meaning that the consumption tax is relatively more efficient. That is, for a given amount of government revenue, financing it with consumption taxes will result in higher consumption, higher labor supply, and higher utility.

Of course, this is an extremely silly way to analyze tax policy for the simple reason that the vast majority of government spending is actually just transfers from rich to poor. That is, transfers based on *income*. Switching from income to consumption taxes can therefore exacerbate the labor disincentive through the spending side of the ledger in even more detrimental ways.

1 There are two types of capital gains taxes in the US: for short-run investments sold (I think) within a year of being purchased, the capital gains are taxed as personal income under the income tax. For long-term investments they are taxed according to a separate flat tax rate and not considered income for the income tax.

2 A standard practice in economics is to call a policy associated with a *potential Pareto improvement* more efficient, regardless of whether the potential improvement actually exists within the real-world policy space. A potential Pareto improvement is a policy that makes at least one person better off but where all of the people harmed by it could theoretically be fully compensated with lump-sum transfers financed with the surplus from those made better off, such that every individual would either prefer or be indifferent to the policy change plus transfers. In practice, however, these theoretical transfers are never actually feasible because the government doesn't possess that kind of information, and can't costlessly administer that kind of bureaucracy. Thus, the relevance of economic "efficiency" is pretty dubious—it would be more informative to look at the welfare effects of the real-world policy, including the distributional effects. When income distribution is taken into account, income taxes are generally preferable to consumption taxes because income is more highly correlated with household types (although this is not unambiguous; the correct answer depends on the empirical incentive effects, nature of the household heterogeneity, and criteria by which welfare is measured.)