Amoroso–Robinson relation

The Amoroso -Robinson Relation ( after Luigi Amoroso and Joan Robinson) describes in microeconomics a certain transformation of the marginal revenue function, which reveals a relationship between the price of a good, the marginal revenue generated by his sale and the direct price elasticity of demand for this commodity.

Definition

Be given by the revenue that can be generated through the sale of units of a commodity i. It is at the so -called price-demand function that specifies for a given quantity of goods on the price that ensures that consumers also ask exactly this amount. With her is the inverse function of ( inverse ) demand function. Be on the price elasticity of demand for good i in a goods price of p. Then the Amoroso -Robinson relation holds

Importance

The Amoroso -Robinson relation can refer to the following:

• The marginal revenue is consistent with the price, if the direct price elasticity of demand - as a result of perfect competition - (absolute) is infinite (horizontal price-demand function). ( More precisely: The agreement applies in the limit for. )
• The marginal revenue is less than the price if the demand is not perfectly elastic ( negatively sloped price-demand function).
• The marginal revenue is negative if the direct price elasticity of demand (absolute value) is less than 1. No ( the profits of interested ) provider offers therefore to the inelastic range.

In addition, the Amoroso -Robinson relation is important for the derivation of the monopoly degree.

Derivation

The starting point is the revenue function. Note that the price is not necessary here is a constant as in perfect competition necessarily be the case, but, in turn, may depend on the amount of output. The marginal revenue is accordingly. This means that one

The second term in the bracket expression is ( in Leibnitz notation) or, taking into account the fact that the amount is in accordance with the demand function. The price elasticity of demand is again given precisely in this spelling by. As already pointed out, finally, and are inverse functions. Consequently, also holds that

Where the elasticity of demand under the law of demand is negative, and therefore

- The desired relation.