Pricing of Competing Products

BI Solutions

December 2015


Executive Summary: Objectives and Background


  • Objectives:

    • analyze sales /price data for two competitive products A and B in light of competitive pricing theory/game theoretic approach
    • explain product B vs. product A price equilibrium
    • explain product B vs. product A price parallelism phenomenon
  • Data: 2014 yearly (one data point) and quarterly (four data points) sales/price data for competing products A and B

  • Background:

    • Product B dominates market regardless of the fact that product A is significantly cheaper than products B (on average, by $183 per unit). In 2014 products A market share remains practically unchanged
    • As soon as product B price is changed, the competitor’s product A price is changed more-or-less simultaneously and in more-or-less similar proportion (hypothesis of parallel pricing)
  • Study restrictions:

    • Since only four quarterly data points are available and each product price are practically unchanged during that period, dynamic price competition models could not be applied
    • Only Bertrand price  competition model and diverse its generalizations can be applied


Executive Summary: Results


  • Stable 2014 product B dominance can be due to one (or several ones) of the following reasons:
    • product A is not a perfect substitute for product B (products are not homogeneous), and customer perception of product B is preferable
    • product A capacity constraints do not allow additional production that can satisfy increased product A customer demand
    • Marginal cost of these two products are different (asymmetric marginal costs), and product B marginal cost is smaller than product A (product B has superior production technology) 
    • Reality of product A – product B sales and price relationship reflect the hypothesis of a simultaneous combination of these three assumptions. In this case product B can dominate the market and product B price can be significantly higher than product A price 
    • In other words, game-theoretical price equilibrium is in agreement with observed product A – product B  price “equilibrium” (prices are significantly different, and the price of a product with smaller marginal cost and larger capacity is higher)
    • Additional product A price reduction will not lead to significant increase in its market share
  • Parallel (product B vs. product A) Pricing is normal for duopoly if products are not a perfect substitute (Bertrand model with differentiated products)



Product B dominates market, and 2014 price – sales relationship for competing products are stable



yearly and quarterly data



Optimal Product B Net Price is $493 per Unit




further net price decrease



Let's Back Up For A Moment



Violated Assumptions of Bertrand Model (one shot pricing game): perfect substitute, equality of marginal cost, and absence of capacity constraints



  • Questions:

    • What would  be the best price strategy and price equilibrium for competing products A and B?
  • Assumptions of Bertrand model:

    • Only two firms in the market
    • Products are homogeneous (products are perfect substitutes)
    • Firms set prices simultaneously
    • Each firm has the same constant marginal cost c (Constant Returns to Scale)
    • Consumers always buy from the firm that offers the lowest price
    • No capacity constraints (each firm can meet all of the demand of the market)
  • Bertrand model results:

    • Nash equilibrium (best strategy of each firm):
      • Both firms will set up the same product price and the price equal c  (pure strategy):
                    p1 = p2 = c
    • Since  Product A – Product B price reality different from this equilibrium, some Bertrand model assumptions are severely violated   
  • Violated assumptions:

      • products are perfect substitutes (Reality: products are not perfect substitutes)
      • each firm has the same constant marginal cost (Reality: marginal costs for these products are asymmetric)
      • no capacity constraints (Reality: there are capacity constraints, and for Product B capacity constraints are less restrictive)



note1 and note2



Bertrand model with capacity constraints


  • Undercutting rival’s price to attract more buyers is not very useful if the firm cannot produce more output to meet the increased demand

    • Bertrand model: firms can produce as much as they want at the same unit cost
    • real world: firms have limited production capacity
  • Additional assumptions:

    • Each firm has a capacity constraint of k1 and k2, therefore if a demand for homogeneous product is greater than they can produce, they are not able to meet it (a firm cannot sell more than its capacity)
    • total industry capacity (k1 k2) is “small” relative to market demand
      • If (k1 k2) is large relative to market demand, the capacity is not fully utilized, and Nash equilibrium will be the same as in the classical Bertrand model
  • Nash Equilibrium:
    • p1 = p2 = P , where P is the price at which market demand is exactly equal  (k1 k2). In other words,  both firms price at the point where there is no unused capacity
  • Conclusion:

  • The more severe the capacity constraints the less competitive the market is and the higher the extent of market power
  • Since the Product A – Product B price “equilibrium” (prices are significantly different) is deviated from in this theoretical equilibrium (prices are equal), several Bertrand model assumptions are severely violated simultaneously


Bertrand model with asymmetric marginal cost and imperfect product substitutes


  • Assumption relaxation:

    • Firms have different constant marginal cost (firm 1 marginal cost is c1, and firm 2 marginal cost is c2), c2 > c1
  • Additional assumption:

    • Assume that ∆ is the smallest monetary unit so that costs and prices have to be a multiple of ∆, i.e. ci , pi = λ∆ with λ being an integer, i= 1, 2
  • Nash Equilibrium:

    • two equilibria exist (c2 and p2 for product A, and c1 and p1 for product B):
      • p1 = c2,  p2 = c2 ∆
      • p1 = c2 − ∆ ,  p2 = c2  (observed reality)
    • if one firm has a superior production technology (Product B)- marginal cost is significantly lower than the other one, the Product B manufacturer can charge the highest price that is a little bit lower than the marginal cost of the other firm and practically take all the business (“limit pricing”)
  • General case of asymmetric marginal cost: 

    • Equilibrium: any price from [c1, c2]
  • Conclusion:

    • Theoretical price equilibrium is in agreement with observed Product A– Product B  price “equilibrium” (prices are significantly different). It means that asymmetric marginal costs play a great role in Product A – Product B market stability.
    • Similar results exists for  price competition with imperfect product substitutes


Parallel Pricing: Product B vs. Product A


  • Background hypothesis:

    • As soon as Product B price is changed, the competitor’s product Product  A price is changed more-or-less simultaneously and in more-or-less similar proportion
    • This is the hypothesis of parallel pricing/Price leadership

  • Definition of Price Leadership/Parallel Pricing (in oligopolistic market):

    • Direct competitors change prices more-or-less simultaneously and in similar proportion
    • Dominant competitor changes the price first and publishes its price ahead of other firms in  the market, and the other firms match the announced price (price leadership)

  • Bertrand model with differentiated products induces parallelism:

    • Parallelism is not a sufficient indicator of a collusive conduct
    • Competition itself elicits parallel pricing



Literature