Question

1. Consideramarketwithtwoidenticalfirms.Themarketdemandis

?(?) = 26 − 2?,
where ? = ?1 + ?2 indicates total output and ? is the market price. Suppose that

both firms have the following identical total cost function:

??(??) = 2?? ,? = 1,2.

1. Computethedifferencebetweenaverageandmarginalcostsforbothfirms.

2. DerivetheCournotbest-responsefunctions.

3. Solve for the Cournot-Nash equilibrium. Calculate the quantity, price and profit for each firm.

4. Nowsupposethat,duetopoorinvestmentdecisions,thetotalcostfunctionof firm 1 changes to:

   ??(?1) = 2.5?1.

   The cost function of firm 2 is the same as before. Compute the Cournot-Nash equilibrium in this case and compare it with the equilibrium derived under “1c” above. What is the intuition behind your finding?

From the book Industrial Organization as I am currently studying for an exam

Answer 1

1) The market demand

P=26-2Q=26-2(q₁+q₂)

TC for firm 1=2q_1, MC=2 (dTC/dq₁), AC=2 (2q₁/q₁)

Similarly TC For firm 2=2q₂, MC=2, AC=2

Therefore the difference between average and marginal cost for both firms is 0

TR for firm 1=Pq₁ = q₁(26-2(q₁+q₂))

MR for firm 1=dTR/dq₁= 26-4q₁-2q₂

For profit maximization for firm 1:
MC=MR

1) 2=26-4q₁-2q₂

Thus best response for firm 1 when Firm 2 produces q₂

is q₁=(24-2q₂)/4= 6-0.5q₂

Similarly for firm 2:

TR for firm 2=Pq₂ = q₂(26-2(q₁+q₂))

MR for firm 1=dTR/dq₂= 26-2q₁-4q₂

For profit maximization for firm 2:
MC=MR

2) 2=26-2q₁-4q₂

Thus best response for firm 2 when Firm 1 produces q₁

is q₂=(24-2q₂)/4= 6-0.5q₁

For both 1) and 2) To be true, we solve them simaltaneously to get

48=6q₁+6q₂ (From adding both equations)

and q₁=q₂(From subtracting one equation from another)

Therefore q₁=q₂=48/12=4

This is the cournot Nash equilibrium where both q₁=q₂=4, their best response to each others quantity will also be 6-0.5q=6-2=4 thus it is a Nash equililbrium

Now the cost function for firm 2 is same as before so the profit maximization condition MC=MR will remain same as above which is

2) 2=26-2q₁-4q₂

However the cost function for firm 1 is TC=2.5q₁

MC=2.5

Therefore the profit maximizing condition for firm 1 is

2.5=26-4q₁-2q₂

For both to be true, we add and subtract them to find

47.5=6q₁+6q₂

and

0.5=2q₂-2q₁ or

q₂=q₁+0.25

So 47.5=12q₁+1.5

q₁=46/12=3.833 and q₂=5.33

This is the new Cournot Nash equilibrium and in this case the intuition is that the Firm 1 produces less at an equilibrium because it has a higher Marginal cost and since MR decreases as it increases its own quantity.the MC=MR condition will occur sooner for first 1 and thus it will produce less than firm 2, and since firm 1 now produces less, firm 2 will have a higher value of q₂ because q₁ will be less in the profit maximizing condition for Firm 2.

Hope it's clear and helps. Do ask for any clarifications required.