Solving the Linear Bertrand Model using Mathematica

General information about the model:

We assume constant marginal costs due to the assumption of a linear cost function. This implies that C'1 = c1. A simple, linear Bertrand demand function for firm 1 can be written on the following form:
In[]:=
Q1=A-bP1+kP2
Out[]=
A-bP1+kP2

Solving the model:

We can now derive firm 1's profit function and solve his profit-maximization problem by using partial derivatives. We will then obtain firm 1's best response function for every price that his competitor, firm 2, sets (firm 2's pricing strategy).
In[]:=
π1=(P1-c1)Q1
Out[]=
(-c1+P1)(A-bP1+kP2)
In[]:=
D[π1,P1]
Out[]=
A-bP1-b(-c1+P1)+kP2
In[]:=
Simplify[A-bP1-b(-c1+P1)+kP2]
Out[]=
A+b(c1-2P1)+kP2
In[]:=
Solve[D[π1,P1]==0,P1]
Out[]=
P1
A+bc1+kP2
2b

In[]:=
R1=P1==
A+bc1+kP2
2b
;
Firm 2 is a naive competitor. This means that he always wants to undercut his competitor, firm 1, on price. We therefore assume firm 2 will react in a way where he sets his price as a percentage, m, of his competitor, where m < 1. Firm 2's best response function will can therefore mathematically be expressed as:
In[]:=
R2=P2==mP1;
Solving this system of equations will yield the market equilibrium prices from each firm:
In[]:=
Eq=Solve[{R1,R2},{P1,P2}]
Out[]=
P1-
-A-bc1
2b-km
,P2
(A+bc1)m
2b-km

In[]:=
Simplify[Eq]
Out[]=
P1
A+bc1
2b-km
,P2
(A+bc1)m
2b-km
