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Menu Pricing of Large Language Models

Dirk Bergemann, Alessandro Bonatti, Alex Smolin

TL;DR

This work develops a framework for the optimal pricing and product design of LLMs in which a provider sells menus of token budgets to users who differ in their valuations across a continuum of tasks, and shows that competitive pressure reshapes both the intensive and extensive margins of compute provision.

Abstract

We develop a framework for the optimal pricing and product design of LLMs in which a provider sells menus of token budgets to users who differ in their valuations across a continuum of tasks. Under a homogeneous production technology, we show that users' high-dimensional type profiles are summarized by a scalar index, reducing the seller's problem to one-dimensional screening. The optimal mechanism takes the form of committed-spend contracts: buyers pay for a budget that they allocate across token classes priced at marginal cost. We extend the analysis to environments with multiple differentiated models and to competition between a proprietary leader and an open-source fringe, showing that competitive pressure reshapes both the intensive and extensive margins of compute provision. Each element of our theory (token-budget menus, maximum- and minimum-spend plans, multi-model versioning, and linear API pricing) has a direct counterpart in the observed pricing practices of providers such as Anthropic, OpenAI, and GitHub.

Menu Pricing of Large Language Models

TL;DR

This work develops a framework for the optimal pricing and product design of LLMs in which a provider sells menus of token budgets to users who differ in their valuations across a continuum of tasks, and shows that competitive pressure reshapes both the intensive and extensive margins of compute provision.

Abstract

We develop a framework for the optimal pricing and product design of LLMs in which a provider sells menus of token budgets to users who differ in their valuations across a continuum of tasks. Under a homogeneous production technology, we show that users' high-dimensional type profiles are summarized by a scalar index, reducing the seller's problem to one-dimensional screening. The optimal mechanism takes the form of committed-spend contracts: buyers pay for a budget that they allocate across token classes priced at marginal cost. We extend the analysis to environments with multiple differentiated models and to competition between a proprietary leader and an open-source fringe, showing that competitive pressure reshapes both the intensive and extensive margins of compute provision. Each element of our theory (token-budget menus, maximum- and minimum-spend plans, multi-model versioning, and linear API pricing) has a direct counterpart in the observed pricing practices of providers such as Anthropic, OpenAI, and GitHub.

Paper Structure

This paper contains 46 sections, 16 theorems, 136 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Literature
  3. Model
  4. Task Environment
  5. Notation
  6. Mechanism Design
  7. Mapping the Setting to Practice
  8. Gain function, $g$
  9. Efficient Solution
  10. Unconstrained problem
  11. Constrained problem
  12. Menus of Token Budgets
  13. Optimal Budget Mechanism
  14. Indirect Mechanisms
  15. Multiple Models and Competition
  16. ...and 31 more sections

Key Result

Proposition 1

Under the efficient allocation, all buyer types $w$ with the same aggregate type $\theta(w)$ consume the same number of fine-tuning tokens, consume the same total number of inference tokens in each class, and obtain the same total payoff. The number of inference tokens allocated to task $i$ is propo

Figures (3)

  • Figure 1: Leader-Fringe allocations across different regimes. Example with $\theta\sim U[0,1]$, $\sigma=1/2$, $\hat{\sigma}_L=1/4$, $c_F=1/10$, $c_L=1/8$.
  • Figure 2: Anthropic subscription tiers (January 2026). Model access is constant across paid tiers; differentiation occurs through usage allocations.
  • Figure 3: OpenAI ChatGPT subscription tiers (January 2026). Higher tiers grant access to more capable models and larger usage allocations.

Theorems & Definitions (26)

  • Proposition 1: Efficient Allocation
  • Corollary 1: Constrained Efficient Allocation
  • Proposition 2: Buyer Indirect Utility
  • Lemma 1: Cost Function
  • Proposition 3: Optimal Menu
  • Definition 1: Maximum-Spend Mechanism
  • Definition 2: Minimum-Spend Mechanism
  • Definition 3: Two-Part-Tariff Mechanism
  • Proposition 4: Indirect Implementation
  • Lemma 2: Buyer-Optimal Payoff
  • ...and 16 more