Why cheaper technology could make pricing menus more complicated

A short list of basic, standard and premium plans looks like a natural way to sell a digital service. But what keeps that list short? A new theoretical paper argues that the answer may lie less in sellers having exhausted profitable ways to distinguish customers than in the costs of managing all those distinctions.

In Complex Menus in Discriminatory Pricing, Patrick Lahr, Axel Niemeyer and Marek Pycia show that, in a benchmark monopoly model with multiple product attributes, exact revenue maximization generically requires an infinite menu. Once costs of complexity enter the picture, smaller menus can become optimal. As those costs approach zero, however, optimal menu size grows without bound for generic buyer populations. The work is an arXiv preprint; its supplied landing page lists no journal publication or peer-review status.

Letting customers sort themselves

The analysis concerns screening: sellers design versions and prices so that customers reveal something about their willingness to pay through their choices. Everyone can face the same menu. The mechanism does not require a company to identify each shopper and quote a personalized price.

The authors begin with a seller offering products that vary along two dimensions. Think of a cloud service combining processing capacity and memory. Customers differ in how much they value each attribute. The seller knows the distribution of those preferences, but not an individual customer's type.

Buyers choose whichever option gives them the greatest value after payment, and can decline to buy. In the baseline, value increases linearly with each attribute and production is costless. This deliberately stripped-down setting isolates price discrimination from explanations based on complicated production costs or diminishing returns to quality.

Adding a cheaper, inferior version creates a familiar tension. It may attract customers who previously bought nothing, but it may also tempt premium customers to trade down. In the comparable one-dimensional linear model, one paid offering is enough to maximize revenue. With multiple attributes, the seller has more ways to balance those effects.

What an infinite menu means

The paper establishes its result through mathematical proofs, not an experiment or an estimated model of an industry. Its key construction adds a continuous family of slightly different downgraded products and makes small changes to the distribution of buyer preferences. These products separate customers along curved boundaries in preference space. A finite menu creates only straight boundary segments and cannot reproduce that separation exactly.

For any fixed ceiling on menu size, the authors show that distributions requiring more options are both dense and open in the distribution spaces they study: such distributions can be found arbitrarily close to any starting distribution, and the need to exceed that ceiling survives sufficiently small perturbations. Combining these results yields the generic requirement for infinite menus.

Here, “generic” is a topological term, not a measured share of businesses or a probability that a randomly selected market behaves this way. Nor does infinity mean an actual firm will list endlessly many plans. It describes an exact optimum in a model without complexity costs. The paper also notes that increasingly large finite menus can approximate optimal revenue; its main theorem does not quantify the practical revenue gain from another option.

Cheaper choice is not automatically better choice

The authors then introduce seller-side costs that increase with menu size, covering activities such as designing, administering and marketing versions. Under their assumptions, optimal menus are finite when these costs are positive. Reducing the cost scale never reduces the smallest optimal menu size, and generically sends it upward without bound as costs vanish.

Buyer-side costs produce a subtler result. The model lets customers decide whether the expected benefit of inspecting a menu justifies the effort; those who inspect learn their preferences and choose their best option. Vanishing cognitive costs again lead generically to unbounded menu growth. But the adjustment need not be monotonic: higher cognitive costs can sometimes make a larger menu optimal by inducing the seller to concentrate on high-value customers.

More variety is not necessarily a welfare improvement. The paper gives an example in which lower seller-side complexity costs make a basic version worthwhile while allowing a higher premium price. Consumer surplus and total surplus fall, even before counting menu-management costs. This is a possibility, not a universal prediction.

The scope matters. The genericity proof does not establish the same result when attribute valuations are required to be statistically independent. Fine differentiation must also be technologically feasible, and competition is not the main model. Cloud computing motivates the analysis rather than validating it empirically, while AI-driven cost reductions are a proposed application, not an estimated effect. The central insight is conditional but consequential: making choices cheaper to produce and navigate can strengthen the incentive to multiply them.