Question

Consider two countries, A and B, whose respective industries produce goods qA and qB. Total world output of the good is given by Q=qA+qB. There is a world demand given by p=102-Q. Suppose that the cost function for country A is given by CA(qA)=7qA while the cost function in country B is given by CB(qB)=3qB. The production of the good generates greenhouse gas emissions which cause global climate change. Total world emissions are 0.5 per unit of good, such that total world emissions are 0.5Q. If the two countries' industries compete in a Cournot fashion, what will the total world emissions be?

Answer 1

We are given,

qA = quantity produced by country A, C(qA) = 7qA : Total cost of A

qB = quantity produced by country B , C(qB) = 3qB : Total cost of B

Total World Output: Q = qA + qB

World Price is given by: p = 102 – Q such that,

p = 102 – (qA + qB)            {Inverse demand function faced by both countries}

It is assumed that the two countries compete in a Cournot fashion, which means that both the countries decided upon their quantities independently but simultaneously, i.e. at the same time.

Step1: Derive the profit function for Country A

Profit A = Total Revenue of A – Total Cost of A

= TR(A) – TC(A)   

= p.qA - 7qA

= qA[102 – (qA + qB)] - 7qA

=102qA – qA2 – qAqB – 7qA

ΠA = 95qA– qA2 – qAqB

Step 2: Partially differentiate proft function for country A and equalize it to zero to find its best response function.

dΠA/dqA = 95 – 2qA – qB

Let dΠA/dqA = 0

95 – 2qA – qB = 0

95 – qB = 2qA

qA = 47.5 – 0.5qb {Best response function of A}

Step3: Derive the profit function for Country B

Profit B = Total Revenue of B – Total Cost of B

= TR(B) – TC(B)   

= p.qB - 3qB

= qB[102 – (qA + qB)] – 3qB

= 102qB – qAqB – qB2 – 3qB

ΠB = 99qB – qAqB – qB2

Step 4: Partially differentiate profit function for country B and equalize it to zero to find its best response function.

dΠB/dqB = 99 – qA – 2qB

Let dΠB/dqB = 0

99 – qA – 2qB = 0

99 – qA = 2qB

qB = 49.5 – 0.5qA           {Best response function of B}

Step 5: Putting the value of qB in the best response function of country A:

qA = 47.5 – 0.5qb

qA = 47.5 – 0.5 (49.5 – 0.5qA)

qA = 47.5 – 24.75 + 0.25qA

qA – 0.25qA = 22.75

0.75qA = 22.75

qA = 22.75 / 0.75

qA = 30.33

Step 5: Putting the value of qA in the best response function of country B:

qB = 49.5 – 0.5qA

qB = 49.5 – 0.5(30.33)

qB = 49.5 – 15.165

qB = 34.33

Thus world output Q = qA + qB

Q = 30.33 + 34.33

Q = 64.66

World Price, p = 102 – Q

p = 102 - 64.66

p = 37.34

Total world emission = 0.5Q

= 0.5 (64.66)

= 32.33