If two out of three Cournot firms merge and the two resulting firms compete in Cournot fashion. Will the profits of the merged firm exceed the total pre-merger profits of the two merged firms? Assume that each firm still has to incur its own fixed cost, marginal cost is constant, and the demand is linear.
Two identical firms A and B compete in Cournot fashion. Which firm’s reaction function is shown below
What is the highest profit that a Stackelberg leader can make if it cannot credibly commit to quantity (and neither can the follower)
Cournot (#2-20pts, #3-5pts)
Setup: the demand is given by .There are two firms: A and B. The cost of each firm is given by. The firms compete in Cournot fashion by simultaneously choosing quantity
Write firm A’s residual demand
Some answers had the demand written as a function of both quantities – partial credit.
QB=6-0.5QA Write firm B’s reaction function
Technically reaction function has to be of the form QB=…QA... many answers just wrote the right side of the equation. I did not take points away for this
Find firm A’s quantity
Find firm B’s quantity
(+) Continuing with the previous setup. Can firm B increase profits through investment that would double its (fixed cost) but lower marginal cost by half?
Yes because the profit will increase to 20 from 8.
With the new cost the reaction function becomes . The quantities of B and A are respectively 6 and 3. The price becomes 9.
A common mistake here was to assume that both firms A and B will increase the fixed cost and have lower marginal cost. If the rest of the solution was correct it resulted in a minimal reduction.
Stackelberg (#4-20pts, #5-pts)
Setup: the demand is given by There are two firms. The cost of each firm is given by The firms compete in Stackelberg fashion by sequentially choosing quantity.
Find leader’s residual demand after the leader takes into account follower’s quantity
Present this game in extensive form and show how to find an equilibrium of this game by adding arrows
Wrong profits without calculation were not credited. I did my best to trace the mistakes.
(+)Continuing with the previous setup, can the leader prevent entry by producing 20 units? Show calculations.
No, the follower will make profit of 29.
Follower’s reaction function is , the leader demand after follower enters is then , leader’s MR is then . Setting it equal to the marginal cost leader will produce 9x2=18, the follower reacts with 9. The price is then P=19-0.5(9+18)=5.5.
Yes, or no without calculations were not credited but even a wrong answer received partial credit if the calculations were correct and the wrong answer resulted from an arithmetic mistake.
P= 2 Q= 5
Setup: the demand is given by . There are two firms. The cost of each firm is given by . The firms compete in Bertrand fashion by simultaneously choosing price
Find Bertrand equilibrium price and quantity
On the graph below, explain why your answer to the previous question is not an equilibrium if producers are capacity constrained. Assume each firm has capacity that equals half of the Bertrand quantity. (notice this question is not asking to prove that the equilibrium does not exist)
Game theory (#7-5pts, #8-15pts)
Find all Nash equibria of the following game or show that none exist
The demand is given by .There are two firms: A and B. The cost of each firm is given by. Setup the collusion game as a prisoner’s dilemma game. Assume that when both firms collude they produce monopoly output and split the profits; when both firms compete they revert to Cournot equilibrium; when one firm colludes and the other competes the colluding firm still produces half of monopoly output and the competing firm produces its best response quantity.