Amoroso–Robinson Relation¶

Relate marginal revenue to a selling price and the signed own-price elasticity of demand along a differentiable price–quantity schedule.

Version: v1 · 2026-10-03 · History
Domain-specific #: 12980
Domain group: Social Sciences
Origin domain: Economics & Finance
Subdomains: Microeconomics, Industrial Organization → Economics & Finance
Aliases: Amoroso Robinson Formula

Core Idea¶

The Amoroso–Robinson relation is a calculus identity connecting the incremental revenue from selling another unit with the current selling price and the local own-price elasticity of demand. Suppose a seller faces a differentiable inverse demand schedule $p(q)$, with positive price $p$ and positive quantity $q$. Revenue is $R(q)=p(q)q$, so marginal revenue is $MR=dR/dq=p+q p'(q)$. If locally invertible demand has signed elasticity $\varepsilon=(dq/dp)(p/q)=p/[q p'(q)]$, substitution gives $MR=p(1+1/\varepsilon)$. For ordinary downward-sloping demand $p'<0$, $\varepsilon<0$; with positive magnitude $\eta=|\varepsilon|$, the same statement is $MR=p(1-1/\eta)$.^[1]^[2]

The identity does not assume that the seller maximizes profit or even has a specified cost function. It decomposes one extra unit's revenue into its price $p$ and the loss $q p'(q)$ from lowering a common price on units already sold. Hence marginal revenue is positive in the elastic region $\eta>1$, zero at unit elasticity, and negative in the inelastic region $0<\eta<1$. The derivative formulation needs the local price–quantity schedule; at a kink, zero quantity, or zero price, the stated elasticity form may be undefined even though a one-sided revenue calculation is possible.^[1]^[2]

The familiar markup result is a conditional consequence, not the identity itself. At a differentiable interior profit maximum, $MR=MC$. If marginal cost is strictly positive, then $MR>0$ and the optimum is in an elastic region. When $MC=0$, unit elasticity can be an interior revenue maximum. Rearranging the interior first-order condition yields $(p-MC)/p=-1/\varepsilon=1/\eta$, the Lerner equality under those assumptions. It is not a universal equation for every seller or every observed price–cost margin.^[1]^[2]

Structural Signature¶

Sig role-phrases:

Price–quantity demand schedule: the locally differentiable inverse demand $p(q)$ whose slope records the change in common selling price required to alter quantity.
Revenue product: $R(q)=p(q)q$, the joint dependence on per-unit receipts and units sold.
Signed own-price elasticity: $\varepsilon=(dq/dp)(p/q)<0$ on a locally invertible downward-sloping schedule; the absolute-value convention must be converted with a minus sign.
Marginal revenue identity: $MR=p+qp'=p(1+1/\varepsilon)$, exposing additional-unit receipts and repricing loss in one expression.
Optional optimizing context: a cost schedule and interior $MR=MC$ permit a markup implication, but neither is needed for the identity.

These roles jointly distinguish this relation from a generic derivative of revenue. The elasticity form normalizes the local slope: two markets with different price or quantity units can exhibit the same proportional responsiveness while having different dollar-valued marginal revenues. The expression identifies a local comparison, not a global optimum or an empirical demand estimate.

What It Is Not¶

The relation is not the Lerner Index. The latter is the ratio $(p-MC)/p$; only at an appropriate interior optimum does the relation plus $MR=MC$ turn that ratio into $-1/\varepsilon$. It is not a standalone theory of monopoly power: the identity also applies to a nonoptimizing seller and to a differentiated firm's residual demand. Nor does it state that every profit-maximizing price lies in the strictly elastic region; zero marginal cost permits unit elasticity, and a corner or nondifferentiable optimum can evade the interior first-order inference.^[1]^[2]

It is also not the derivative $dR/dp=q(1+\varepsilon)$, which asks how revenue changes with price. Amoroso–Robinson as used here asks how revenue changes with quantity, $dR/dq$. Horizontal demand gives $MR=p$ as the slope tends to zero and $\varepsilon\to-\infty$; treating infinite elasticity as an ordinary finite reciprocal or using the elasticity expression at $q=0$ suppresses its limiting character.

Scope of Application¶

The relation applies wherever a common-price seller can represent revenue locally by $p(q)q$ with a differentiable, locally invertible demand schedule. It is a mathematical relationship along the schedule, not a claim that an estimated elasticity stays constant as output changes. Linear demand and constant-elasticity demand instantiate the same roles with different shapes; the former has elasticity that varies by point, while the latter fixes it by model assumption. Both appear in original instructor treatments of monopoly pricing.^[1]^[2]

The result can also be read against a firm's residual demand under product differentiation, provided that firm's price–quantity response, not the whole market's elasticity, supplies $\varepsilon$. In multi-price discrimination, dynamic pricing, bundles, or multiple products, the simple scalar formula may need a segment-specific or multivariate derivative and cannot be pasted onto aggregate sales uncritically. Where no smooth inverse demand exists, $p+qp'$ remains the starting decomposition only if the derivative exists; one-sided, discrete, or set-valued analysis is a different treatment.

Clarity¶

The relation separates two effects often collapsed in the phrase “one more sale.” The new unit yields current price $p$, but selling it along a downward-sloping common-price demand curve requires a lower price for previous units. The sign of $MR$ depends on which effect dominates, encoded by elasticity. The number one in $1+1/\varepsilon$ is not an arbitrary empirical threshold: it arises because a one-percent price reduction that expands quantity by exactly one percent leaves first-order revenue unchanged.^[1]^[2]

This also clarifies when the Lerner reading is available. Observing or estimating elasticity alone gives a local marginal-revenue relation. Inferring a profit-maximizing markup additionally requires an interior optimum and the relevant marginal cost. Conflating those steps makes a definition look like a behavioral prediction and makes the model's assumptions disappear.

Manages Complexity¶

The identity compresses a potentially unfamiliar slope $p'(q)$ into a scale-free elasticity plus the observed price. Once $p$ and the correctly signed local elasticity are known, the sign and magnitude of marginal revenue follow without writing the entire demand function. At a positive-cost interior optimum it also supports a compact markup condition. But this compression has a cost: local derivative estimates may be noisy; different prices, units or customer segments can change the relevant elasticity. The identity cannot repair an invalid empirical demand model.^[2]

The linear case illustrates what is being compressed. With $p=a-bq$, revenue is $aq-bq^2$, so $MR=a-2bq$: the marginal-revenue slope doubles the inverse-demand slope because an output increase both sells the marginal unit and discounts prior units. The elasticity form states the same result at each point even though that elasticity is not constant.^[2]

Abstract Reasoning¶

Start with the revenue product rather than a quoted markup formula. Differentiate $R=p(q)q$ with respect to $q$, giving $MR=p+qp'$. Define elasticity with its direction and sign: $\varepsilon=(dq/dp)(p/q)$, so local invertibility gives $1/\varepsilon=(q/p)(dp/dq)$. Substitute to obtain the named relation. Only then, if the question is about an interior profit optimum, impose $MR=MC$ and inspect the sign of $MC$ before declaring an elastic-region result.^[1]^[2]

If $\eta=|\varepsilon|$ is the reported statistic, rewrite $MR=p(1-1/\eta)$; using $1+1/\eta$ would reverse the economics. If a horizontal price-taker demand curve is approached, the second term tends to zero, so $MR\to p$. If demand is kinked, ask for appropriate one-sided changes rather than claiming the smooth identity at the kink.

Knowledge Transfer¶

The portable mathematical skeleton is the product rule plus a dimensionless derivative: derivative of “unit receipt times quantity” equals current unit receipt plus the effect of quantity on unit receipt. That skeleton can be recognized elsewhere, but the Amoroso–Robinson identity is tied to economic price, sold quantity, revenue and own-price demand elasticity. It does not become a general prime merely because the calculus travels.

Within economics the mapping travels between both functional forms and pricing arrangements. In a single linear market, $\varepsilon$ changes with $q$ and the relation reconstructs $a-2bq$ point by point. In isoelastic demand, $\varepsilon$ is constant and the relation makes marginal revenue a constant fraction of price. Under third-degree price differentiation, the same relation is applied separately to each segment's own-price demand, with common marginal cost linking the two interior choices. The cost/optimization extension is shared only when valid segment-level interior $MR=MC$ conditions are added.^[1]^[2]

Examples¶

Linear inverse demand. Let $p(q)=a-bq$ with $a,b>0$, in the positive-price interval. Then $R=aq-bq^2$, $MR=a-2bq$, and $\varepsilon=-p/(bq)$. Therefore $p(1+1/\varepsilon)=p-bq=a-2bq$. With constant marginal cost $c$ satisfying $0<c<a$, the interior solution is $q^*=(a-c)/(2b)$ and $p^*=(a+c)/2$; its elasticity magnitude is $(a+c)/(a-c)>1$. If $c=0$, the revenue maximum instead has $\eta=1$. This is algebra on Varian's linear-demand model, not a measured market.^[2]

Mapped back: The demand schedule is $a-bq$; the revenue product is $aq-bq^2$; signed elasticity is $-p/(bq)$; the identity gives $a-2bq$; and the optional cost condition locates an elastic interior optimum only when $c>0$.

Constant-elasticity residual demand. Let $q=A p^{-\eta}$ with $A>0$ and $\eta>1$. The inverse schedule $p(q)=(A/q)^{1/\eta}$ has signed elasticity $-\eta$ everywhere in its interior. Thus $MR=p(1-1/\eta)$. For a seller with constant positive marginal cost $c$, an interior optimum satisfies $p^*=c\eta/(\eta-1)$. The example shows why the same identity yields a fixed markup factor here but a changing elasticity along linear demand; it does not claim that every real firm has constant-elasticity demand.^[1]^[2]

Mapped back: The schedule is $(A/q)^{1/\eta}$; revenue is that price times $q$; signed elasticity is $-\eta$; the identity produces a constant fraction of price as $MR$; and the positive-cost interior condition, if imposed, gives the stated price.

Separately priced submarkets. Wiese's two-market exercise gives $p_1=12-4q_1$ and $p_2=8-\tfrac12q_2$, with common marginal cost $4$. Applying the relation within each segment gives $MR_1=12-8q_1$ and $MR_2=8-q_2$. At a differentiable interior optimum both equal the shared marginal cost: $q_1=1$, $p_1=8$, $\varepsilon_1=-2$; $q_2=4$, $p_2=6$, $\varepsilon_2=-3$. The more elastic segment receives the lower price here, but that comparison depends on separate pricing being feasible and the stated cost structure. It is not a claim that every market can be segmented.^[1]

Mapped back: Each submarket has its own $p_i(q_i)$ schedule and $p_iq_i$ revenue product; signed own-price elasticities are $-2$ and $-3$ at the solved outputs; each marginal-revenue identity converts its segment elasticity into $MR_i=4$; common marginal cost is the additional optimizing condition, not part of either identity.

Structural Tensions¶

Additional-unit receipts versus repricing losses. Along downward-sloping common-price demand, adding output earns $p$ on the new unit but lowers receipts on the $q$ units already sold by the first-order amount $qp'<0$. More output and preservation of the old unit price cannot both be had on that schedule. Diagnostic: At the contemplated output, is $|\varepsilon|$ above, equal to, or below one, making net marginal revenue respectively positive, zero, or negative?^[1]

Local derivative fidelity versus stable estimation. The literal identity uses the derivative at the contemplated output. Estimating demand from a wider price range can stabilize a noisy slope, yet averaging across segments or smoothing a kink can miss the local response that determines marginal revenue; narrowing the window increases sensitivity to sampling error. Diagnostic: Is the price–quantity schedule locally smooth and identified at this decision point, or is the reported elasticity a broad-range approximation that changes across the interval? This is an inference tradeoff, not a failure of the algebra.

Structural–Framed Character¶

This identity sits toward the structural end within a constitutively economic frame. Its evaluative weight is low: the equality judges neither fairness nor desirability of a price. Its dependence on human practice is substantial but bounded: price, sales, and demand are market constructions, while the product-rule inference is mathematical once those quantities are specified. The named form originated in economics instruction, not an institutional rule or legal entitlement; institutional origin is therefore low. The vocabulary of elasticity and marginal revenue travels among monopoly, differentiated-firm and segmentation analyses, but the full named identity does not travel as a domain-neutral rule about arbitrary products. In another field one would recognize the product-rule skeleton, not import Amoroso–Robinson as an economic law unless its price–quantity roles were genuinely present. These five criteria support a domain-specific entry, not a new prime.^[1]^[2]

Structural Core vs. Domain Accent¶

The structural core is a two-factor product differentiated against one factor, with the response term normalized by elasticity. The domain accent is indispensable to this named identity: quantity sold, common unit price, sales revenue and own-price demand elasticity give the reciprocal term its sign and economic meaning. Prime Price Elasticity is a proposed strict presupposition because the identity needs its local response measure; generic prime Relation does not distinguish this economic calculus identity enough to serve as a useful strict parent. The product-rule skeleton is already ordinary mathematics, not an independently admitted new prime. A broader “marginal product–elasticity identity” would require a separate prime-admission review rather than being smuggled in by renaming this entry.

This entry presupposes Price Elasticity.

The proposed typed DAG edge is a strict composition-presupposition to live Price Elasticity: the elasticity can be studied without this relation, whereas the relation's defining expression cannot omit the elasticity. Live Elasticity is a broader ancestor of that component. Live Demand supplies the economic schedule, and live Lerner index is a conditional derived neighbor after interior $MR=MC$, not an alias or genus. No canonical edge has been changed here.

Relationships to Other Abstractions¶

Current abstraction Amoroso–Robinson Relation Domain-specific

Parents (1) — more general patterns this builds on

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

Show alternative path (1)

Neighborhood in Abstraction Space¶

Amoroso–Robinson Relation sits in a moderately populated region (48^th 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

Not to Be Confused With¶

Do not equate the identity with a monopoly equilibrium, the Lerner index, a generic revenue derivative, or the price derivative $dR/dp=q(1+\varepsilon)$. The formula uses signed own-price demand elasticity; with the positive magnitude convention, the sign switches. Nor is “a monopolist always chooses an elastic point” a stand-alone theorem without positive marginal cost and interior differentiability. Finally, a horizontal demand curve gives $MR=p$ in the limiting price-taking case; writing $\varepsilon=-\infty$ is shorthand for a limit, not a finite observed elasticity.^[1]^[2]

References¶

[1] Harald Wiese, Monopoly and Monopsony, Leipzig University original lecture slides, especially slide 15/53 (named Amoroso–Robinson derivation), slide 12/53 (isoelastic positive-cost exercise) and slides 35–37 and 40/53 (segmented prices and original two-linear-submarket exercise). registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m

[2] Hal R. Varian, Intermediate Microeconomics textbook outline, author-hosted original teaching outline, Market Demand p. 64 and Monopoly pp. 92–93 (PDF pp. 64, 92–93). Historical naming priority is not inferred from these notes. registry ↩a ↩b ↩c ↩d ↩e ↩f ↩g ↩h ↩i ↩j ↩k ↩l ↩m ↩n