## Questions:

1. In duopoly market, with using Cournot-Nash equilibrium and diagram calculate:

- Nash equilibrium quantities.
- Equilibrium price.
- Profit to each firm.

2. Suppose that the firms cooperate to act as a monopoly and want to form OPEC cartel in the market, then:

- What do you think the objectives of this cartel?

- Calculate the following in monopoly market and compare them with previous results:

- Quantity of output.

- Profit to each firm.

3. Suppose that (SA) firm is expected to produce the same quantity in cartel contract. But (KPC) firm wanted to cheat, then:

- In your opinion, what are the main incentives that push the firm to deviate and cheat?

- Calculate the following for both firms:

- Quantity of output.

- Profit to each firm.

- Explain the previous results.

4. Summarize this case in the pay-off matrix and decide what the right behaviour each firm will take in the market (Nash equilibrium).

### Answers:

### Answer 1:

As per the Cournot equilibrium:

Response function from SA will be:

Profit = Total Revenue (TRsa) – Total Cost (TCsa)

= (55 – Qsa - Qkpc) x Qsa – 10 x Qsa

First Order Condition of profit maximisation:

dProfit/dQsa = 55 – 2 x Qsa – Qkp – 10 ………… (i)

Second Order Condition of profit maximisation:

dProfit/dQsa = 0

From (i) we get,

Qkpc = 45 – 2 x Qsa ………. (ii)

Response function of KPC:

Profit = Total Revenue (TRkpc) – Total Cost (TCkpc)

= (55 – Qsa – Qkpc) x Qkpc – 10 x Qkpc

First Order Condition of profit maximisation:

dProfit/dQkpc = 55 – 2 x Qkpc – Qsa ………… (iii)

Second Order Condition of profit maximisation:

dProfit/dQkpc = 0

From (iii) we get

Qkpc = (55 – Qsa)/2 …………. (iv)

From (iii) and (iv) we get,

45 – 2 x Qsa = (55 – Qsa)/2

90 – 4 x Qsa = 55 – Qsa

- 3 x Qsa = - 35

Qsa = 35/3

Therefore, Qsa = 11.66

Qkpc = (55 – 11.66)/2 ……… from (iv)

Therefore, Qkpc = 21.67

Price = (55 – 11.66 – 21.67)

Therefore, Price = 21.67

Profit SA = (21.67 x 11.66) – 10 x 11.66

Therefore, Profit SA = 136.07

Profit KPC = (21.67 x 11.66) – 10 x 11.66

Therefore, Profit SA = 136.07

From the above calculation, it can be seen that,

- Nash equilibrium quantities are – (11.66 & 21.67)
- Equilibrium price – (21.67)

- Profit to each firm – (136.07)

### Answer 2:

Being a monopoly cartel objective of the same will be to enhance the profit of the same (Waldman & Jensen, 2016). As per the profit maximisation policy of the monopoly, the price would be:

MR = MC (MR = Marginal Revenue & MC = Marginal Cost)

MR = dTR/Dq (TR = Total Revenue)

TR = P x Q

= (55 – Q) x Q

= 55Q – Q^{2}

MR = 55 – 2Q

MC = 10

Therefore,

55 – 2Q = 10

- 2Q = -45

Q = 45/2

Therefore, Q = 22.5

Price = 55 – 22.5

Therefore, Price = 22.5

Profit = TR – TC

= Q x P – Q x C

= (22.5 * 22.5) – (22.5 * 10)

= 281.253

Profit to each firm = 281.253/2 = 140.626

Therefore, Profit = 281.253

- quantity output = 22.5
- Price = 22.5

- Profit to each firm = 140.626

Considering the cartel scenario, it can be stated that, price has been enhanced whereas quantity output has been decreased leading to rise in the profit of both the firm.

### Answer 3:

If one firm produce at same quantity as previous and the other one cheats, then the cartel will fail (Klein & Schinkel, 2018). There is often a tendency from the business firms who enters into cartel to break the agreements in the last period. Thus, the cartel will last, until one firm wants to opt out and produce at the mutually beneficial output (Bloch, 2018).

If one firm produce at cartel decided level of output, then profit will be,

Profit = (55 – 22.5 – Qkpc) x Qkpc – 10 x Qkpc = 0

Therefore, (22.5 – Qkpc) x Qkpc = 10 x Qkpc

Therefore, 22.5 – Qkpc = 10

Therefore, Qkpc = 12.5

Price = 55- 22.5 – 12.5 = 20

Profit = (55 – 22.5 – 12.5) x 12.5 – 10 x 12.5

= (20 x 12.5) – 125

= 250 – 125

= 125

Total output by both the firm = 22.5 + 12.5 = 35

Price = 22.5 + 20 = 42.5

Profit in case of non-cooperation = 125

- Quantity output is 12.5
- Price will be 20

- Profit to each firm will be 125

### Answer 4:

A | B | ||

Collude | Compete | ||

A | Collude | 140.626, 140.626 | 140.626, 125 |

B | Compete | 125,140.626 | 136.07,136.07 |

As per the above table it can be seen, AA represents collusion between both the firms. AB represents first firm goes for collusion and the second one produce competitive output, vis-?-vis in case of the BA. BB represents competitive pricing between both the options. Thus, it can be seen that if the first firm goes for competitive pricing and second firm goes for collusion, then Nash equilibrium will be achieved (Telser, 2017).

## Reference:

Bloch, F. (2018). Coalitions and networks in oligopolies.

Klein, T., & Schinkel, M. P. (2018). Cartel Stability by a Margin.

Telser, L. G. (2017). Competition, collusion, and game theory. Routledge.

Waldman, D., & Jensen, E. (2016). Industrial organization: theory and practice. Routledge.