title: Differentiated Bertrand competition
text: # Differentiated Bertrand competition
  
  As a solution to the Bertrand paradox in economics, it has been suggested that each firm produces a somewhat differentiated product, and consequently faces a demand curve that is downward-sloping for all levels of the firm's price.
  
  An increase in a competitor's price is represented as an increase (for example, an upward shift) of the firm's demand curve.
  
  As a result, when a competitor raises price, generally a firm can also raise its own price and increase its profits.
  
  ## Calculating the differentiated Bertrand model
  
  * $q_1$ = firm 1's demand, $*q_1≥0$
  * $q_2$ = firm 2's demand, $*q_1≥0$
  * $A_1$ = Constant in equation for firm 1's demand
  * $A_2$ = Constant in equation for firm 2's demand
  * $a_1$ = slope coefficient for firm 1's price
  * $a_2$ = slope coefficient for firm 2's price
  * $p_1$ = firm 1's price level pr unit
  * $p_2$ = firm 2's price level pr unit
  * $b_1$ = slope coefficient for how much firm 2's price affects firm 1's demand
  * $b_2$ = slope coefficient for how much firm 1's price affects firm 2's demand
  * $q_1=A_1-a_1*p_1+b_1*p_2$
  * $q_2=A_2-a_2*p_2+b_2*p_1$
  
  The above figure presents the best response functions of the firms, which are complements to each other.
  
  ## Uses
  
  Merger simulation models ordinarily assume differentiated Bertrand competition within a market that includes the merging firms.
  
  