Editing Duopoly (section)

====Bertrand model in game theory====

The [[Bertrand competition]] was developed by a French mathematician called [[Joseph Louis François Bertrand]] after investigating the claims of the Cournot model in "Researches into the mathematical principles of the theory of wealth, 1838".<ref name=":0" /> According to the Cournot model, firms in a duopoly would be able to keep prices above marginal cost and hence be extremely profitable.<ref>{{Cite journal |last=Vives |first=Xavier |author-link=Xavier Vives |date=October 1984 |title=Duopoly information equilibrium: Cournot and bertrand |journal=[[Journal of Economic Theory]] |volume=34 |issue=1 |pages=71–94 |doi=10.1016/0022-0531(84)90162-5 |issn=0022-0531}}</ref>  Bertrand took issue with this. In this market structure, each firm could only choose whole amounts and each firm receives zero payoffs when the aggregate demand exceeds the size of the amount that they share with each other. The market demand function is <math>Q(P)=a-bP</math>. The Bertrand model has similar assumptions to the Cournot model:

* Two firms
* Homogeneous products
* Both firms know the market demand curve
* However, unlike the Cournot model, it assumes that firms have the same MC. It also assumes that the MC is constant.

The Bertrand model, in which, in a [[Game theory|game]] of two firms, competes in price instead of output. Each one of them will assume that the other will not change prices in response to its price cuts. When both firms use this logic, they will reach a [[Nash equilibrium]].

* Consider price competition among two firms ({{math|1=''i'' = 1, 2}}) selling homogeneous good
* Downward sloping market demand {{math|''D''(''p'')}}, with {{math|''{{prime|D}}''(''p'') < 0}}
* Constant, symmetric marginal cost {{math|1= ''c''{{sub|1}} = ''c''{{sub|2}} = ''c''}}
* Static game: firms set prices simultaneously
* Rationing rule of demand:

# lowest priced firm wins all demand at its price
# if prices are tied, each firm gets half of market demand at this price

* Firm {{mvar|i}}{{'}}s profits: <math display=inline>\Pi_i = (p_i-c)D_i(p_i, p_j)</math>

Let {{mvar|p{{sup|m}}}} be the monopoly price, <math display=inline>p^m = \operatorname{argmax}_p(p-c)D(p)</math>

* Firm {{mvar|i}}{{'}}s best response {{math|''R{{sub|i}}''(''p{{sub|j}}'')}} is:

<math display="block>R_i(p_j) = \begin{cases}
p^m, & \text{if } p_j > p^m \\
p_j - c, & \text{if } c < p_j \le p^m \\
c, & \text{if } p_j \le c
\end{cases}</math>

For rival prices above cost, each firm has incentive to undercut rival to get the whole demand. If rival prices below cost, firms make losses when it attracts demand; firm better off charging at cost level. Nash equilibrium is {{math|1= ''p''{{sub|1}} = ''p''{{sub|2}} = ''c''}}.