Question

Question 2: Simultaneous quantity choiceTwo firms F1 and F2 produce a homogeneous product and compete on the same market. The market price is described by the inverse demand curveP= 11−2Q, where Q is total industry output andPis the market price. To keep things simple, suppose that each firm can produce either 1 or 2 units (these are the only possible choices of production).Further suppose that both firms have a constant marginal cost equal to 2, so that the total cost of firmi= 1,2 producing quantity qi∈{1,2}is given byC(qi) = 2qi. Further suppose that firms’ production choices are simultaneous.

1. Since each firm can choose one of two possible output levels, there are four possible combinations of output choices: (q1,q2) = (1,1), (q1,q2) = (2,1),(q1,q2) = (1,2), and (q1,q2) = (2,2). Find the market price P under each of these possible output choices.

Now consider exactly the same firms, inverse demand function, cost function, and quantity choices as above. In contrast to before, however, suppose that firm 1 chooses its quantity q1 first, then firm 2 observes firm 1’s quantity choice and chooses its quantity q2.

1. Draw the game tree for this game (using the payoffs you found above).

2. What is Firm 1’s strategy set? What is Firm 2’s strategy set? (For F2, bear in mind that a strategy is a

complete contingent plan for how to play the game!)

3. Using backward induction, predict the outcome of this game. Discuss your

answer. Does moving first benefit Firm 1?

Answer 1

Given inverse demand function: P= 11-2Q
Q=q1+q2
MC=2
The market price under the 4 possible alternatives can be calculated as follows
1 ( q1,q2)= (1,1)
Substituting in the inverse demand function and as well as substituting for Q we get
1. 11-2(1+1)= 7
2. ( q1,q2)= (2,1)
We get 11-2( 2+1)= 5
3. ( q1,q2)= (1,2)
We get 11-2(1+2)=5
4. ( q1,q2)= (2,2)
We get 11-2(2+2)= 3