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The exact revenue-maximizing price for logit demand is found with a closed-form Lambert W expression and lies below the inflection point.

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2026-05-21 17:06 UTC pith:N6YE52ER

load-bearing objection The paper gives a clean closed-form Lambert W expression for the revenue-max price under standard logit demand, and the derivation holds up without approximation. the 1 major comments →

arxiv 2512.02247 v1 pith:N6YE52ER submitted 2025-12-01 math.OC

An exact pricing algorithm for revenue maximization under the logit demand function

Moddassir Khan Nayeem , Omar Abbaas , Suzan Alaswad , Sinan Salman This is my paper
classification math.OC
keywords logit demand functionrevenue maximizationoptimal pricingLambert W functionclosed-form solutioninflection point pricingdemand modelingpricing algorithm
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows how to calculate the precise price that maximizes revenue when customer demand follows the logit model, which describes how purchase probability changes with price in an S-shaped way. Most prior work either uses repeated numerical searches or simply prices at the inflection point where demand is most sensitive to price changes. The authors rearrange the revenue equation to isolate price and solve it directly with the Lambert W function, giving an analytical answer instead of an approximation. They prove and demonstrate numerically that this optimal price is always lower than the inflection point price under typical conditions. This results in both higher revenue for the seller and lower prices for consumers compared to the common heuristic.

Core claim

The revenue-maximizing price under the logit demand function is derived analytically by setting the derivative of revenue to zero and solving the resulting transcendental equation using the Lambert W function, yielding a closed-form expression. This optimal price is shown to be consistently lower than the price at the inflection point of the demand curve, leading to an average 20% reduction in price and 15% increase in revenue in numerical tests.

What carries the argument

The closed-form solution for optimal price obtained by expressing the first-order condition for revenue maximization in a form solvable by the Lambert W function.

Load-bearing premise

The revenue function under the logit demand model can be rearranged into an equation solvable by the Lambert W function without additional constraints or numerical approximation.

What would settle it

A calculation showing that the revenue at the proposed closed-form price is not greater than the revenue at the inflection point price for some valid parameter values.

Watch this falsifier — get emailed when new claim-graph text bears on it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes an exact closed-form algorithm for the revenue-maximizing price under the logit demand function q(p) = exp(α − βp)/(1 + exp(α − βp)). It derives the solution p* = [1 + W_0(exp(α − 1))]/β via the first-order condition of the revenue function R(p) = p · q(p), shows that this price lies below the inflection-point price α/β under standard parameter values, and reports numerical experiments indicating an average 20% price reduction together with a 15% revenue gain.

Significance. If the derivation holds, the closed-form expression supplies a simple, non-iterative method for optimal pricing that can be implemented directly in revenue-management software. The explicit comparison to inflection-point pricing and the accompanying numerical illustration of revenue improvement constitute a practical contribution to the logit-demand literature in operations research.

major comments (1)
  1. [Numerical experiments] Numerical experiments section: the reported averages (20% price reduction, 15% revenue gain) are presented without stating the sampling ranges or distributions for the parameters α and β, the number of Monte-Carlo replications, or the precise definition of the optimality gap. These details are required to assess whether the quantitative claims are robust or sensitive to particular parameter regimes.
minor comments (2)
  1. [Derivation] The manuscript should include a short paragraph confirming that the second-derivative test or boundary analysis (R → 0 as p → 0 and p → ∞) establishes that the critical point is a global maximum.
  2. [Main result] Consider adding a brief remark on the domain of α for which exp(α − 1) lies in the range where the principal branch W_0 yields a real, positive solution.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive comment on the numerical experiments and for the overall positive evaluation of the manuscript. We address the point below and will revise accordingly.

read point-by-point responses

  1. Referee: [Numerical experiments] Numerical experiments section: the reported averages (20% price reduction, 15% revenue gain) are presented without stating the sampling ranges or distributions for the parameters α and β, the number of Monte-Carlo replications, or the precise definition of the optimality gap. These details are required to assess whether the quantitative claims are robust or sensitive to particular parameter regimes.

    Authors: We agree that these implementation details are necessary to allow readers to evaluate the robustness of the reported averages. In the revised manuscript we will augment the Numerical experiments section with the sampling ranges and distributions used for α and β, the number of Monte-Carlo replications, and an explicit definition of the optimality gap (percentage revenue difference between the closed-form optimum and the inflection-point price). revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper's central derivation starts from the standard logit demand q(p) = exp(α − βp)/(1 + exp(α − βp)) and revenue R(p) = p · q(p), sets the first-order condition to zero, and rearranges algebraically into an equation solved by the principal branch of the Lambert W function. This step uses only elementary calculus and the known properties of W; no parameters are fitted to the target revenue quantity, no self-citation supplies a uniqueness theorem or ansatz, and the result is externally verifiable by direct substitution back into the FOC. Boundary behavior and uniqueness of the critical point are shown independently. The claim that the optimum lies below the inflection point follows directly from comparing the closed-form expressions and does not reduce to any input by construction.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the logit demand model being the correct functional form and on the revenue equation being solvable via Lambert W once parameters are given; no new entities are introduced.

free parameters (1)
  • logit demand parameters
    Scale and location parameters of the logit function are required to evaluate the closed-form price; these are treated as known inputs but would normally be estimated from data.
axioms (2)
  • domain assumption Consumer demand follows the standard logit functional form
    Invoked throughout the abstract as the basis for the pricing problem.
  • standard math The Lambert W function can be applied directly to the revenue equation
    Used to obtain the closed-form solution.

pith-pipeline@v0.9.0 · 5772 in / 1431 out tokens · 138199 ms · 2026-05-21T17:06:49.916579+00:00 · methodology

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read the original abstract

Determining the optimal selling price is a challenge in revenue management, especially in markets characterized by nonlinear and price-sensitive demand. While traditional models, such as linear, power, and exponential demand functions, offer analytical convenience, they often fail to capture realistic purchase dynamics, leading to suboptimal pricing. The logit demand function addresses these limitations through its bounded, S-shaped curve, offering a more realistic representation of consumer behavior. Despite its advantages, most existing literature relies on heuristic approaches, such as pricing at the inflection point, which prioritizes maximum price sensitivity but does not guarantee maximum revenue. This study proposes a novel, exact pricing algorithm that analytically derives the revenue-maximizing price under the logit demand function using the Lambert W function. By providing a closed-form solution, the approach eliminates reliance on heuristic iterative methods and corrects the common practice of considering the inflection point price as market price. In fact, we demonstrate that the optimal price is consistently lower than the inflection-point price under reasonable assumptions, leading to lower prices for consumers and higher revenue for sellers. Numerical experiments illustrate the proposed algorithm and examine the changes in the optimality gap as demand function parameters vary. Results indicate that the optimal price is consistently lower than the inflection-point price, with an average 20% price reduction accompanied by a 15% increase in revenue.

discussion (0)

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    R. M. Corless, G. H. Gonnet, D. E. Hare, D. J. Jeffrey, and D. E. Knuth, "On the Lambert W function," Advances in Computational mathematics, vol. 5, pp. 329-359, 1996

    work page 1996