Two firms produce a homogeneous product for a certain market. Firm 1 produces Q₁ and Firm 2 produces Q₂. The two firms are price takers, and the market price is P = 15. The total costs of producing Q₁ and Q₂, respectively, are

C₁ = .1 Q₁²+ 5 Q₁ - .1 Q₁ Q_2. (total costs borne by Firm 1)

C₂ = .2 Q₂² + Q₂ + .1 Q₂ Q_1. (total costs borne by Firm 2)

a).Which firm is a source of positive externalities, and which is a source of negative externalities, if any? (Careful!) Explain briefly.

b.)Suppose that each firm chooses its own output so as to maximize its own profits, taking the output of the other firm as given. (No remedial action is taken against the externalities.) Calculate the levels of output that each firm will choose.

c.)How would you characterize (conceptually label) the solution you have obtained in part b? Explain briefly.

d.)Suppose that the two firms merge (thus internalizing the externalities), and that their objective is to maximize joint profits. Calculate the new levels of Q₁ and Q₂ that would be chosen.

e.)Compare the values of Q₁ and Q₂ in parts b and d, using as intuition your answer in part a. If there are no additional distortions in the market system, will total social welfare be higher under the assumptions of part b or those of part d? Explain.

a) Firm 1 is the source of negative externality for Firm 2

Firm 2 is the source of positive externality for Firm 1

b) Q1 = 60

Q2 = 20

c) Positive externality increases the production of firm 1 and negative externality suffered by firm 2 decreases its production.

d) Q1 = 50

Q2 = 35

e) Social welfare is higher when firm jointly produce i.e. in part d

Step-by-step explanation

a)

Negtiave externality is when the action of one firm decreases the welfare of other firm which does not directly involved in the externality making activity. Positive externality is when the action of one firm adds or increases the welfare of other.

Firm 1 is the source of negative externality for Firm 2 as it's quantity produced increases the cost of production of firm 2.

Firm 2 is the source of positive externality for Firm 1 as it's quantity produced decreases the cost of production of firm 1.

 

b)

Each firm will produces at a point and maximize ts profits where,

Price = Marginal cost as the firms are price takers and produce homogeneous goods and service.

MC 1 = d(C)/dQ₁ = 0.2Q₁ + 5 - 0.1Q₂

MC 2 = d(C)/dQ₂ = 0.4Q₂ + 1 + 0.1Q₁

 

Equating P=MC for firm 1;

15 = 0.2Q₁ + 5 - 0.1Q₂

10 = 0.2Q₁ - 0.1Q₂ ...... (1)

 

Equating P=MC for firm 2;

15 = 0.4Q₂ + 1 + 0.1Q₁

14 = 0.4Q₂ + 0.1Q₁ ...... (2)

 

Multiplying equation 1 with 4, we get,

40 = 0.8Q₁ - 0.4Q₂ ...... (3)

Now, we can solve equation 3 and 2 by substitution method as we have two equations with two variables and two unknowns.

By adding two equations, 3 and 2, the term 0.4Q₂ will cancel out and we'll left with,

54 = 0.9Q₁

Q₁* = 60

Q₂* = [14 - 0.1(60)]/0.4 = 20

 

c)

As firm 1 receives positive externality, the output produced is higher than firm 2 who suffer from negative externality given by firm 1.

Thus, positive externality increases the production of firm 1 and negative externality suffered by firm 2 decreases its production.

 

d)

The joint production by both the firm will remove the externality as the firm will now produce jointly and their joint cost function would be,

C1+C2 = 0.1Q₁² + 5Q₁ + 0.2Q₂² + Q₂

Note that the term involved externality i.e. 0.1Q₁Q₂ cancel out with each other.

 

Now, the joint profit function would be,

Joint profits = ?π? = Total revenue - total cost

= PQ - {0.1Q₁² + 5Q₁ + 0.2Q₂² + Q₂}

= 15(Q₁+Q₂) - {0.1Q₁² + 5Q₁ + 0.2Q₂² + Q₂}

 

The optimal production of firm 1 would be,

d(π)/dQ₁ = 15 - {0.2Q₁ + 5}

15 = 0.2Q₁ + 5

10 = 0.2Q₁

Q'₁ = 10/0.2 = 50

 

The optimal production of firm 2 would be,

d(π)/dQ₂ = 15 - { 0.4Q₂ + 1}

15 = 0.4Q₂ + 1

14 = 0.4Q₂

Q'₂ = 14/0.4 = 35

 

e)

The firm 2 produces more when the externality is removed as the negative externality reduces the firm's output and firm 1 relatively produces less when the firms jointly produce. This is because firm 1 was enjoying the positive externality by firm 1 and now the externality is gone.

The total output in the market when the firm produces individually was, Q*1+Q*2 = 60+20 = 80

The joint output when the firm internalize the externality is 50+35 = 85.

Thus, social welfare is higher when firm jointly produce i.e. in part d.