Left-digit bias may hide the welfare cost of sales taxes

When demand falls abruptly at a whole-dollar price, consumers may not be misreading the price. They may simply stop considering the product. A new theoretical and empirical preprint explores how that distinction changes the interpretation of sales-tax experiments—and the surplus economists calculate when a tax eliminates a purchase.

A small price increase across a whole-dollar boundary can produce a disproportionately large drop in demand, a pattern known as left-digit bias. One explanation is that consumers misperceive prices just below round numbers. Kaushil Patel’s preprint, Left-Digit Bias: Theory and Empirical Evidence from a Tax Salience Experiment, proposes a different mechanism: consumers screen products using price thresholds before deciding whether buying is worthwhile.

Posted as arXiv:2611.05342v1 on November 3, 2026, the paper combines a mathematical model with a reanalysis of an existing sales-tax experiment. Its central argument is that weak purchasing responses to taxes do not, by themselves, establish that taxation destroys little economic surplus.

A threshold is not necessarily a valuation

In the model, consumers first consider a product only if its posted price falls below an attention threshold. Among products they consider, they buy when their willingness to pay exceeds the cost. Someone with a whole-dollar threshold can therefore stop considering a product even though they value it above that threshold.

The model allows different valuations and thresholds across consumers. It assumes valuations have no point masses and restricts finite attention thresholds to positive integers. Under those assumptions, concentrations of binding thresholds generate abrupt demand drops at whole-dollar prices.

For these consumers, the threshold applies to the posted price, while the purchase decision incorporates the full tax-inclusive cost. A tax can initially leave their maximum acceptable posted price unchanged because their valuation remains above the taxed cost at the threshold. Larger taxes can eventually push that cutoff below the threshold, a movement the paper calls “debunching.” Weak tax responses can thus arise without consumers overlooking the tax.

What the experiment contributes

Patel uses public replication data from a 2018 experiment by Dmitry Taubinsky and Alex Rees-Jones. It involved 4,223 US consumers reporting maximum acceptable posted prices for 20 household products. Participants were randomly assigned to face no tax, their local sales tax, or three times that rate in an initial module. A second module asked everyone for untaxed cutoffs.

After comprehension and cutoff restrictions, Patel’s estimation sample contains 58,478 consumer-product observations from 2,991 consumers. The analysis classifies exact untaxed reports at integers from 2 through 14 as binding attention thresholds. This is an empirical assumption, not an independently observed mental process. Such reports account for about 13% of eligible observations in the no-tax arm, not 13% of individual consumers.

The estimated order-adjusted log cutoff response, divided by the tax rate, is approximately 0.269 for binding-threshold observations under local taxes and 0.383 under triple taxes. At small tax rates, this measure approximates the behavioral response relative to full optimization, but it is not identical to that response. The increase between tax arms is not statistically significant.

Estimated gross debunching is 41.5% under local taxes and 52.2% under triple taxes. That difference is also statistically inconclusive. These estimates require reporting assumptions and adjustments using the no-tax group, because reported cutoffs change substantially between modules even without taxation.

A joint fit, with important qualifications

The paper finds that one distribution of willingness to pay relative to attention thresholds can reproduce four estimated moments: response and debunching in each taxed arm. Under a common-distribution assumption across arms and local tax rates, between 43.8% and 52.2% of binding-threshold types have valuations no more than 10% above their thresholds. These are identification bounds conditional on the estimated moments and model restrictions, not confidence intervals.

The evidence does not uniformly support the mechanism. In the triple-tax arm, reports show significant excess movement from one integer to a lower integer. The model cannot explain that relocation as changes in true cutoffs. Reporting variation also prevents a clean test of its predicted lower bound on movers’ taxed cutoffs. Matching four aggregate moments therefore does not establish that the model explains every observed pattern.

Why the mechanism changes welfare calculations

The welfare comparison turns on what an untaxed cutoff reveals. Under the interpretation used by Taubinsky and Rees-Jones, it represents willingness to pay. Under Patel’s binding-threshold interpretation, it lies below willingness to pay. The same purchase eliminated by taxation consequently represents more lost surplus in the latter model.

This result assumes perfectly elastic supply, a fixed producer price equal to marginal social cost, and tax revenue treated as an efficient transfer. The quantitative welfare exercise adds a common valuation-to-threshold distribution across thresholds and covers only binding-threshold observations.

The implication is not that taxes necessarily have large welfare costs. It is that purchasing behavior alone may leave a decisive question unresolved: does a consumer’s cutoff reveal what the product is worth, or where consideration stops?