title: Markup rule
text: # Markup rule
  
  A markup rule is the pricing practice of a producer with market power, where a firm charges a fixed mark-up over its marginal cost.
  
  ## Derivation of the markup rule
  
  Mathematically, the markup rule can be derived for a firm with price-setting power by maximizing the following expression for profit:
  
      $\pi =P(Q)\cdot Q-C(Q)$
      where
      Q = quantity sold,
      P(Q) = inverse demand function, and thereby the price at which Q can be sold given the existing demand
      C(Q) = total cost of producing Q.
      $\pi$ = economic profit
  
  Profit maximization means that the derivative of $\pi$ with respect to Q is set equal to 0:
  
      $P'(Q)\cdot Q+P-C'(Q)=0$
      where
      P'(Q) = the derivative of the inverse demand function.
      C'(Q) = marginal cost–the derivative of total cost with respect to output.
  
  This yields:
  
      $P'(Q)\cdot Q+P=C'(Q)$
  
  or "marginal revenue" = "marginal cost".
  
      $P\cdot (P'(Q)\cdot Q/P+1)=MC$
  
  By definition $P'(Q)\cdot Q/P$ is the reciprocal of the price elasticity of demand (or $1/\epsilon$). Hence
  
      $P\cdot (1+1/{\epsilon })=P\cdot \left({\frac {1+\epsilon }{\epsilon }}\right)=MC$
  
  Letting $\eta$ be the reciprocal of the price elasticity of demand,
  
      $P=\left({\frac {1}{1+\eta }}\right)\cdot MC$
  
  Thus a firm with market power chooses the output quantity at which the corresponding price satisfies this rule. Since for a price-setting firm $\eta <0$ this means that a firm with market power will charge a price above marginal cost and thus earn a monopoly rent. On the other hand, a competitive firm by definition faces a perfectly elastic demand; hence it has $\eta =0$ which means that it sets the quantity such that marginal cost equals the price.
  
  The rule also implies that, absent menu costs, a firm with market power will never choose a point on the inelastic portion of its demand curve (where $\epsilon \geq -1$ and $\eta \leq -1$). Intuitively, this is because starting from such a point, a reduction in quantity and the associated increase in price along the demand curve would yield both an increase in revenues (because demand is inelastic at the starting point) and a decrease in costs (because output has decreased); thus the original point was not profit-maximizing.
  
  