Equilibrium in a two commodity market

Let us consider a two-commodity market model in which the two commodities are related to each other.  Let us assume the functions for both commodities are linear.  The two commodities are complementary commodities say (cars (c ) and petrol (P).  The functions representing the commodities are as follows:

Qdc = 820 - 10 Pc - 4Pp                                  Qdp = 590 - 2Pc - 6Pp

Qsc = -120 + 6Pc                                                     Qsp = - 240 + 4Pp

At equilibrium,

1)       Qdc = Qsc                                          

820 - 10Pc - 4Pp = - 120 + 6Pc

       940 - 16Pc - 4Pp = 0

2)       Qdp = Qsp

590 - 2Pc - 6Pp = -240 + 4Pp

830 - 2Pc - 10Pp = 0

There are now therefore two equations:

940 - 16Pc - 4Pp = 0. .....................(i)

830 - 2Pc - 10Pp = 0  ......................(ii)

Multiply (ii) by 8 which gives (iii).  Subtract (i) from (iii) to eliminate Pc and solve for Pp.

6,640 - 16 Pc - 80Pp = 0..(iii)

- (940 - 16Pc -   4Pp = 0  ...................(i)

5,700             -  76Pp = 0

                          Pp = 75

Substituting Pp = 75 in (i) we obtain:

940 - 16Pc - 4(75) = 0

16pc = 640

Pc = 40

Substituting Pc = 40 and Pp = 75 into Qd or Qs for each market

1)       Qdc = 820 - 10 (40) - 4 (75)

= 820 - 400 - 300

Qdc = 120 = Qsc

2)       Qdp = 590 - 2 (40) - 6 ( 75)

= 590 - 80 - 450

Qdp = 60 = Qsp