Amoroso–Robinson Relation¶
Relate marginal revenue to a selling price and the signed own-price elasticity of demand along a differentiable price–quantity schedule.
Core Idea¶
For a differentiable inverse demand schedule \(p(q)\) at positive price and quantity, the Amoroso–Robinson relation expresses marginal revenue as \(MR=p(1+1/\varepsilon)\), where \(\varepsilon=(dq/dp)(p/q)<0\) is signed own-price demand elasticity. It follows from \(R=p(q)q\) and \(MR=p+qp'(q)\): selling another unit earns its price but lowering a common price reduces revenue on units already sold. With positive elasticity magnitude \(\eta=|\varepsilon|\), the same formula is \(MR=p(1-1/\eta)\). Thus \(MR\) is positive above unit elasticity, zero at unit elasticity and negative below it.[ref-eef541ca0172][ref-4acf5ab19c2b]
The relation requires neither monopoly nor profit optimization. At a differentiable interior profit maximum, adding \(MR=MC\) derives \((p-MC)/p=-1/\varepsilon\), the Lerner equality. If \(MC>0\), that optimum is strictly elastic; if \(MC=0\), unit elasticity is possible. At a corner or kink the smooth first-order inference need not hold.[ref-eef541ca0172][ref-4acf5ab19c2b]
Scope of Application¶
Varian's linear model \(p=a-bq\) gives \(MR=a-2bq\) and a different local elasticity at each output. Wiese's constant-elasticity model \(q=A p^{-\eta}\) with \(\eta>1\) gives \(MR=p(1-1/\eta)\) throughout its interior. In a distinct two-submarket case, Wiese specifies \(p_1=12-4q_1\), \(p_2=8-\tfrac12q_2\) and common marginal cost $4$: applying the relation separately yields \(q_1=1,p_1=8,\varepsilon_1=-2\) and \(q_2=4,p_2=6,\varepsilon_2=-3\) at the interior optimum. In each, the mapped roles are the inverse demand schedule, revenue \(pq\), signed elasticity, and resulting marginal-revenue derivative; marginal cost is an optional extra used only to locate an optimum. A segmented seller needs each submarket's own-price elasticity, not an arbitrary aggregate market elasticity.[ref-eef541ca0172][ref-4acf5ab19c2b]
Clarity¶
The formula makes the conflict between new-unit receipts and repricing loss visible. It is not the Lerner Index itself: that price–cost ratio needs a cost estimate and acquires a reciprocal-elasticity interpretation only under interior \(MR=MC\). It is also not \(dR/dp=q(1+\varepsilon)\), a different derivative. A horizontal demand curve yields \(MR=p\) as an infinite-elasticity limit, not as a finite reciprocal; at zero quantity the elasticity expression is undefined.[ref-eef541ca0172][ref-4acf5ab19c2b]
Manages Complexity¶
The identity replaces a demand slope with a scale-free elasticity and current price, allowing a quick local marginal-revenue check without solving the entire demand curve. This compression is useful but inherits the local derivative's limits: a noisy estimate, kink, or changing demand segment can make a broad average elasticity misleading at the contemplated output. The mathematical equality itself does not validate the demand model.[^ref-4acf5ab19c2b]
Abstract Reasoning¶
Differentiate \(R(q)=p(q)q\) first, obtaining \(p+qp'\). For locally invertible demand, \(1/\varepsilon=(q/p)p'\), so substitution yields \(MR=p(1+1/\varepsilon)\). Declare whether elasticity is signed or an absolute magnitude before using the formula. Only when optimization is separately assumed should \(MR=MC\) be imposed and the sign of \(MC\) checked. The proposed strict DAG relationship is a presupposition to live Price Elasticity, not to the conditional derived neighbor Lerner index.[ref-eef541ca0172][ref-4acf5ab19c2b]
Knowledge Transfer¶
Linear and isoelastic demand differ in how elasticity changes, while segmented pricing applies the same relation separately to each demand schedule. The price–revenue–elasticity roles map literally in all three. The underlying product-rule technique travels farther, but the named Amoroso–Robinson identity remains framed by economic price, sales revenue and own-price demand response. The generic mathematics is already represented by existing structural abstractions; it does not automatically make this named economic relation a prime.[ref-eef541ca0172][ref-4acf5ab19c2b]
[^ref-eef541ca0172]: Harald Wiese, Monopoly and Monopsony, Leipzig University original lecture slides, slides 12, 15, 35–37 and 40/53. [^ref-4acf5ab19c2b]: Hal R. Varian, Intermediate Microeconomics textbook outline, author-hosted original teaching outline, Market Demand p. 64 and Monopoly pp. 92–93.
Relationships to Other Abstractions¶
Current abstraction Amoroso–Robinson Relation Domain-specific
Parents (1) — more general patterns this builds on
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Amoroso–Robinson Relation presupposes Price Elasticity Prime
Its reciprocal-elasticity term requires a local own-price demand elasticity.
Hierarchy paths (2) — routes to 2 parentless roots
- Amoroso–Robinson Relation → Price Elasticity → Elasticity
- Amoroso–Robinson Relation → Price Elasticity → Marginal Analysis → Optimization
Neighborhood in Abstraction Space¶
Amoroso–Robinson Relation sits in a moderately populated region (48th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.
Family — Market Structure & Competitive Dynamics (31 abstractions)
Nearest neighbors
- Total revenue test — 0.87
- Break-even Point — 0.87
- Supply — 0.86
- Engel curve — 0.86
- Marginal Profit — 0.86
Computed from structural-signature embeddings · 2026-10-08