12.1 Suppose the market for oil is characterized by the demand . Suppose there are two firms, Shell and Caltex. Suppose that both firms has a cost of 1 per unit of oil supplied. Assuming symmetry, solve for the following:

Let firm 1 be Caltex, firm 2 be shell:

12.1.1 Cartel solution. How much each of the firms is producing and what is the resulting price? What are the firms’ profits?

If these two firms forms a cartel, they would jointly act like a monopoly.

Profit maximization requires

Due to symmetry, each of the firms is producing

12.1.2 Competitive market solution.

Due to symmetry, each of the firms is producing

12.1.3 Cournot solution.

Firstly, treat Shell’s quantity as given:

(12.)

For Caltex to maximize its profit, the FOC requires:

(12.)

Secondly, treat Caltex’s quantity as given:

For Shell to maximize its profit, the FOC requires:

(12.)

The Cournot-Nash equilibrium is given as:

Market output , market price , and firms’ profits are:

12.1.4 Bertrand solution.

12.1.5 Stackelberg solution. Suppose Caltex moves first and chooses its quantity.

Sub Eq. (12.) into Eq. (12.):

FOC:

Sub into Eq. (12.) yields:

Hence the market output , market price

12.1.6 Now suppose Caltex moves first and chooses its price, then Shell observes Caltex’s price and decides about its own.

12.2 Suppose now Caltex has the cost of 3, and Shell has the cost of 1.

12.2.1 Cournot solution.

First, treat Shell’s quantity as given:

(12.)

For Caltex to maximize its profit, the FOC requires:

(12.)

Second, treat Caltex’s quantity as given:

For Shell to maximize its profit, the FOC requires:

(12.)

The Cournot-Nash equilibrium is given as:

Market output , market price , and firms’ profits are:

12.2.2 Stackelberg solution. Suppose Caltex moves first and chooses its quantity.

Sub Eq. (12.)into Eq. (12.):

FOC:

Sub into Eq. (12.) yields:

Hence the market output , market price