## 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

- Oligopolisitic Competition: http://people.stfx.ca/tleo/iolecture4.pdf
- Price Competition Under Product Differentiation http://www00.unibg.it/dati/corsi/8915/28786-horizontal_oligopoly_web.pdf
- Oligopoly Games under Asymmetric Costs http://www.princeton.edu/~sircar/Public/ARTICLES/static axis_games_final.pdf
- Bertrand competition http://web.cenet.org.cn/upfile/97289.pdf