# Why does the Laffer curve for consumption have no peak?

According to Trabant and Uhlig (2009), the Laffer curve for consumption has no peak.

As far as I can make out, the argument is based on the fact that money extracted by taxing consumption ends up back with consumers and then gets taxed again. However I don't understand why the standard idea that taxes deter activity doesn't create a peak.

This is not a general result. As the auhtors describe in page 10 of their paper, the result obtains only when government lump-sum transfers $s$ do not vary with tax revenues, and instead government spending $g$ does. Then consumption taxes do not affect labor supply, for any level of the tax rate. The relation becomes monotonic, and the Laffer curve with respect to the consumption tax rate becomes also monotonic. This is something that the authors are not very clear in the main text, but it can be detected in the various proofs at the end: they do not argue that the Laffer curve with respect to all three tax rates examined (consumption, labor, capital), becomes monotonic. Only its dimension with respect to the consumption tax.

And more over, on page 9, when government spending is fixed and it is lump-sum transfers that vary with tax revenues (their Proposition 2) they are not clear on what happens, with respect to consumption tax rate. They do say that here the usual Laffer curve emerges, but they comment on a scenario with $\tau^c$ fixed.

Mathematically, the result hinges on the different feasibility constraint that applies in the two cases (eq. $13$ for the fixed government spending case, eq. $14$ for the fixed-transfers case).

In the model, the difference between government spending and transfers, is that the first provides directly utility, while the second increases disposable income in a lump-sum fashion.

Consumption taxes affect utility: a higher consumption tax rate $\tau^c$ will decrease consumption and hence utility of the same income level. When this is offset by higher utility-generating government spending, while transfers remain unaffected, the incentives related to work/income generation ("activity") remain unaffected, pre-tax income remains the same, tax revenues increase: no extremum point in the Laffer curve. Essentially, this affect how much utility the individuals obtain from private consumption, and how much from government spending.

If alongside the increase in the consumption tax rate, we increase transfers keeping government spending fixed, this creates a substitution effect, a disincentive for labor income: it is better to sit back and collect, rather than work and collect. The individual can increase his utility indirectly, by decreasing the disutility from work. This creates conflicting forces: eventually, for a sufficiently high consumption tax rate (and a correspondingly higher level of transfers) he will decrease his labor supply and hence his labor income more than the increased transfers, thus lowering the consumption tax-base.

The above are of course static effects, "inside" each time period, since the Laffer curve is a static concept.

I think it is derived mathematically in the paper, but here's my take.

Preferences are defined by $log(c)-n$, while the budget constraint is $(1+\tau^c)c=(1+\tau^n)wn+s$. If $w$ is held constant and $s$ (transfers) equal tax receipts in equilibrium, then labor is equal to tax wedge $n=\varsigma=(1-r^n)/(1+r^c)$ and $c=wn$.

Using the equations defined in the paper, the Laffer curvers are

$$L(x)= (\tau_c+\tau_n)\frac{1-\tau^n}{1+\tau^c}w$$ where $x$ can be $\tau_c$ or $\tau_n$. Further simplifications in the algebra are made, but the essential point is that as tax revenues approach 1, transfers also approach 1 while labor supply approaches zero. As transfers are treated as income before taxes, the family will try to consume this transfer but will have to pay near 100% of it in form of tax consumption, rendering them useless.

Keep in mind that the $x$ axis shows consumption tax, whilst the $y$ axis represents tax revenues. As the tax rises, revenue flattens, but that doesn't mean that consumption is not deterred; in fact it is approaching zero, as labor supply is approaching zero.