Amoroso-Robinson relation

The Amoroso-Robinson relation (after Luigi Amoroso and Joan Robinson ) describes in microeconomics a relationship between the marginal revenue of a good and its price elasticity .

definition

Be given by the proceeds that a monopolist can obtain by selling units of a good i . It is the inverse demand function that indicates for a given quantity of goods which price ensures that consumers ask exactly that amount. It is the inverse function of the demand function . Let further be the price elasticity of the demand for good i at a price of goods equal to p. The following Amoroso-Robinson relation describes the relationship between marginal revenue and price elasticity from the point of view of a monopoly provider: ${\displaystyle E(x_{i})=p(x_{i})\cdot x_{i}}$ ${\displaystyle x_{i}}$ ${\displaystyle p(x_{i})}$ ${\displaystyle \epsilon _{i,p}\equiv D_{i}'(p)\cdot (p/x_{i})}$

${\displaystyle E'(x_{i})=p(x_{i})\cdot \left(1+{\frac {1}{\epsilon _{i,p}}}\right)}$

Because a profit-maximizing monopolist chooses the quantity in such a way that marginal revenue and marginal costs coincide, that is, the right-hand side of this equation in the profit maximum also coincides with marginal costs; this correspondence is also referred to in the literature as the Amoroso-Robinson relation. However, the above equation applies in general and regardless of whether the monopolist has chosen the profit-maximizing amount. ${\displaystyle E'(x_{i})=K'(x_{i})}$

Assuming a negative price elasticity, the Amoroso-Robinson relation can also be written in the form using the absolute amount . ${\displaystyle E'(x_{i})=p(x_{i})\cdot \left(1-1/\left|\epsilon _{i,p}\right|\right)}$

meaning

The Amoroso-Robinson relation can be seen as follows:

• The marginal revenue agrees with the price if the direct price elasticity of the demand as a result of perfect competition approaches infinity (horizontal price-sales function ), i.e. for . ${\displaystyle \epsilon _{i,p}\rightarrow \infty }$
• The marginal revenue is smaller than the price if the demand is not completely elastic (negatively inclined price-sales function).
• The marginal revenue is negative if the demand is inelastic.

In addition, the Amoroso-Robinson relationship is important for deriving the degree of monopoly (see Lerner index ).

Derivation

The starting point is the revenue function . Note that the price is not necessarily a constant, as would be the case in perfect competition, but can in turn depend on the output volume. The marginal revenue is accordingly . Factoring out then leads to: ${\displaystyle E(x_{i})=p(x_{i})\cdot x_{i}}$ ${\displaystyle E'(x_{i})=p(x_{i})+p'(x_{i})\cdot x_{i}}$ ${\displaystyle p(x_{i})}$

${\displaystyle E'(x_{i})=p(x_{i})\cdot \left(1+{\frac {p'(x_{i})\cdot x_{i}}{p(x_{i})}}\right)}$

The second term in the parenthetical expression is (in Leibnitz notation) or, taking into account the fact that the amount of the demand function forms . The price elasticity of demand is in turn given in this notation by . Finally, as already stated, and are inverse functions. ${\displaystyle (\mathrm {d} p/\mathrm {d} x_{i})\cdot (x_{i}/p)}$ ${\displaystyle (\mathrm {d} p/\mathrm {d} D_{i})\cdot (D_{i}/p)}$ ${\displaystyle \epsilon _{i,p}}$ ${\displaystyle (\mathrm {d} D_{i}/\mathrm {d} p)\cdot (p/D_{i})}$ ${\displaystyle p(x_{i})}$ ${\displaystyle D_{i}(p)}$

literature

• Friedrich Breyer: Microeconomics. An introduction. 5th edition. Springer, Heidelberg a. a. 2011, ISBN 978-3-642-22150-7 .
• Michael Heine and Hansjörg Herr: Economics. Paradigm-oriented introduction to micro- and macroeconomics. Oldenbourg, Munich 2013, ISBN 978-3-486-71523-1 .
• Susanne Wied-Nebbeling and Helmut Schott: Fundamentals of microeconomics. Springer, Heidelberg a. a. 2007, ISBN 978-3-540-73868-8 .

Individual evidence

1. ↑ See also the derivation, e.g. Breyer 2011, p. 72; Heine / Herr 2013, p. 111 f .; Wied-Nebbeling / Schott 2007, p. 216 ff.

Web links

• Amoroso-Robinson relation - definition in the Gabler Wirtschaftslexikon