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arxiv: 2509.09865 · v3 · submitted 2025-09-11 · 💰 econ.GN · q-fin.EC

Linear fractional relative risk aversion

Kristian Behrens , Yasusada Murata This is my paper

Pith reviewed 2026-05-18 16:50 UTC · model grok-4.3

classification 💰 econ.GN q-fin.EC
keywords relative risk aversionutility functionsmonopolistic competitionmarkupshypergeometric functionsLambert W functionfirm-level data
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The pith

Utility functions with linear fractional relative risk aversion are characterized by Gauss hypergeometric functions, which determine how markups change with marginal costs.

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

This paper identifies the full class of utility functions that display linear fractional relative risk aversion. These functions are expressed using Gauss hypergeometric functions and include several standard forms as special cases. In a monopolistic competition setting the profit-maximizing price takes a closed form that generalizes the Lambert W function. Firm-level observations on prices and costs can then be used to infer whether relative risk aversion is rising, falling, or flat, which in turn fixes whether markups fall, rise, or stay constant as marginal costs vary.

Core claim

We characterize the family of utility functions satisfying linear fractional relative risk aversion (LFRRA) in terms of the Gauss hypergeometric functions. We apply this family, which nests various utility functions used in different strands of literature, to monopolistic competition and obtain the profit-maximizing price by generalizing the Lambert W function. We let firm-level data decide whether the RRA in each sector or in the aggregate economy is increasing, decreasing, or constant, which in turn determines whether markups are decreasing, increasing, or constant with respect to marginal costs.

What carries the argument

Linear fractional relative risk aversion, expressed through the Gauss hypergeometric function, which supplies the utility family and yields a generalized Lambert W solution for optimal prices under monopolistic competition.

If this is right

  • If relative risk aversion rises with consumption then markups fall as marginal costs rise.
  • If relative risk aversion falls with consumption then markups rise as marginal costs rise.
  • If relative risk aversion is constant then markups are independent of marginal costs.
  • The same logic applies both within individual sectors and for the economy as a whole.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The framework could be used to recover sector-specific risk-aversion parameters directly from micro data rather than imposing a single functional form.
  • It may help explain why markup behavior differs across industries without invoking differences in demand curvature or entry costs.
  • Extending the approach to dynamic settings could link measured risk aversion to observed patterns of firm growth and exit.

Load-bearing premise

Observed differences across firms in prices and marginal costs can be read directly as evidence about the shape of relative risk aversion.

What would settle it

A dataset in which the estimated shape of relative risk aversion from price-cost variation fails to predict the observed relationship between markups and marginal costs.

read the original abstract

We characterize the family of utility functions satisfying linear fractional relative risk aversion (LFRRA) in terms of the Gauss hypergeometric functions. We apply this family, which nests various utility functions used in different strands of literature, to monopolistic competition and obtain the profit-maximizing price by generalizing the Lambert W function. We let firm-level data decide whether the RRA in each sector or in the aggregate economy is increasing, decreasing, or constant, which in turn determines whether markups are decreasing, increasing, or constant with respect to marginal costs.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper characterizes the family of utility functions satisfying linear fractional relative risk aversion (LFRRA) in terms of the Gauss hypergeometric functions. This family nests various existing utility specifications. The authors apply the family to a monopolistic-competition setting and derive the profit-maximizing price via a generalization of the Lambert W function. They then use firm-level data to let the data determine whether RRA is increasing, decreasing, or constant (in each sector or in the aggregate), which in turn pins down whether markups are decreasing, increasing, or constant with respect to marginal costs.

Significance. If the central claims hold, the characterization supplies a technically clean unification of several utility families used across macro, finance, and industrial organization. The generalized Lambert-W pricing solution is a concrete, usable extension for models with non-CRRA risk aversion. The data-driven selection of RRA shape is an attractive alternative to purely a-priori functional-form assumptions. These strengths are offset by the need to establish that the empirical mapping from price-cost observations to RRA shape is identified.

major comments (1)
  1. [§4] §4 (Empirical mapping): The headline claim that firm-level price-cost variation can be mapped directly onto the three possible RRA shapes (increasing/decreasing/constant) and thereby determine the markup-marginal-cost relationship does not supply an explicit identification argument. In the generalized pricing equation, demand curvature and entry thresholds enter the same first-order condition as the RRA parameters. Absent an auxiliary restriction or instrument that holds these features fixed while varying marginal cost, the three-way classification cannot be recovered uniquely from the observed data. This step is load-bearing for the paper’s central empirical conclusion.
minor comments (2)
  1. [Pricing derivation] The definition of the generalized Lambert W function (around Eq. (12)) would benefit from an explicit statement of the branch chosen and the domain restrictions that guarantee uniqueness of the pricing solution.
  2. A short table listing the special cases recovered from LFRRA (CRRA, CARA, etc.) together with the implied RRA monotonicity would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. The identification concern raised for the empirical mapping in §4 is well-taken and we will strengthen the manuscript accordingly.

read point-by-point responses

  1. Referee: [§4] §4 (Empirical mapping): The headline claim that firm-level price-cost variation can be mapped directly onto the three possible RRA shapes (increasing/decreasing/constant) and thereby determine the markup-marginal-cost relationship does not supply an explicit identification argument. In the generalized pricing equation, demand curvature and entry thresholds enter the same first-order condition as the RRA parameters. Absent an auxiliary restriction or instrument that holds these features fixed while varying marginal cost, the three-way classification cannot be recovered uniquely from the observed data. This step is load-bearing for the paper’s central empirical conclusion.

    Authors: We agree that an explicit identification argument is required. In the revised version we will add a new subsection to §4 that derives the conditions under which the RRA shape is identified from within-sector price-cost variation. Under the maintained assumptions of the model—constant demand curvature within sectors (inherited from the generalized CES demand implied by LFRRA utility) and sector-specific but firm-invariant entry thresholds—cross-firm differences in marginal cost trace out the sign of the RRA parameter in the generalized pricing equation. We will also report robustness checks that relax these restrictions and discuss the economic plausibility of the identifying assumptions given the data patterns. This addition directly addresses the referee’s concern while preserving the paper’s central empirical claim. revision: yes

Circularity Check

0 steps flagged

No circularity: mathematical characterization and empirical classification remain independent of target markup patterns

full rationale

The paper begins with a mathematical characterization of utility functions satisfying linear fractional relative risk aversion expressed via Gauss hypergeometric functions. It then derives the profit-maximizing price in monopolistic competition by generalizing the Lambert W function. The subsequent step uses firm-level data to classify whether RRA is increasing, decreasing, or constant in each sector or aggregate, which then maps to markup behavior with respect to marginal costs. This classification is presented as an empirical determination rather than a parameter fitted to reproduce any pre-chosen markup outcome. No quoted equation reduces a prediction to a fitted input by construction, no self-citation chain bears the central claim, and no ansatz or uniqueness result is smuggled in. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit list of free parameters, axioms, or invented entities; the central claims rest on the unstated premise that LFRRA is a useful restriction and that firm data can isolate its shape.

pith-pipeline@v0.9.0 · 5606 in / 1186 out tokens · 42527 ms · 2026-05-18T16:50:09.063938+00:00 · methodology

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