Question

If duopoly behaviour is one that is described by Cournot, the market demand curve is given by the equation q = 200 − 4p and both the firms have zero costs, find the quantity supplied by each firm in equilibrium and the equilibrium market price.

Answer 1

Market demand curve

Q = 200 − 4p

When the demand curve is a straight line and total cost is zero, the duopolist finds it most profitable to supply half of the maximum demand of a good.

At P = Rs 0, market demand is

Q = 200 − 4 (0)

= 200 units

If firm B does not produce anything, then the market demand faced by firm A is 200 units.

∴ The supply of firm `A = 1/2xx200` units

In the next round, the portion of market demand faced by firm B is `200 -200/2=200-100=100 `units

∴ Firm B would supply `1/2xx(200-200/2)` = 50 units

Thus, firm B has changed its supply from zero to 50 units. To this firm A would react accordingly and the demand faced by firm A will be `200-1/2xx(200-200/2)`

= 200 − 50

= 150 units

∴ Firm A would supply `= 150/2=75` units

The quantity supplied by firm A and firm B is represented in the table below.

Round	Firm	Quantity Supplied
1	B	0
2	A	`1/2[200-1/2xx200=100]`
3	B	`1/2[200-1/2xx200]=200/2+200/4`
4	A	`1/2[200-1/2(200-1/2xx200)]=200/2-200/4+200/8`
5	B	`1/2{200-1/2[200-1/2(200-1/2xx200)]}=200/2-200/4+200/8-200/16`

Therefore, the equilibrium output supplied by firm A

`= 200/2- 200/4+200/8-200/16+200/32-200/64+200/128+200/256+..............=200/3 `units.

Similarly, the equilibrium output supplied by firm B = `200/3` units.

Market Supply = Supply by firm A + Supply by firm B

`=200/3+200/3`

Equilibrium output or Market Supply = Q = `=400/3` units  ....... (1)

For equilibrium price

Q = 200 − 4p

4p = 200 − Q

`p=50-Q/4`

`p=50-1/4(400/3)`    .....(From(1)]

`p=50-100/3`

`p=(150-100)/3`

p = Rs `50/3`

Therefore, the equilibrium output (total) is `400/3` units and equilibrium price is Rs `50/3`.