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Question 1: Rebel without a cause Two drivers speed head-on toward each other and a collision is bound to occur unless one of them deviates at the last minute. If both deviate, everything is okay (they both win 1). If one deviates and the other does not, then it is a great success for the driver with iron nerves (he wins 2) and a great disgrace for the deviating driver (he loses 1). If both drivers have iron nerves, disaster strikes (both lose 2).

Question 2: Simultaneous quantity choice Two firms F1 and F2 produce a homogeneous product and compete on the same market. The market price is described by the inverse demand curve P = 11 − 2Q, where Q is total industry output and P is 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 firm i = 1, 2 producing quantity qi ∈ {1, 2} is given by C(qi) = 2qi . Further suppose that firms’ production choices are simultaneous.

question to be answered:

Question 3: Sequential quantity choice 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

Calculating the payoffs of this choice of units available to Firm 1 and Firm 2,

The profits earned from the choice of the units of goods produced will be the payoffs of both the firms,

Now, profits are calculated as TR - TC.

Market inverse demand funtion given as P = 11 - 2Q

TR = P.Q = (11 - 2Q).Q = 11Q - 2Q²

TC = 2Q

Profits = 9Q - 2Q²

Profits for both the firms when Q = 1, Profit = 7

Profits when Q = 2, Profit = 10

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(a) From the above payoffs, the game in the form of tree will be as

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(b)  

Both the firms would want to maximise their profits given all the variables.

So, Firm 1's strategy set will be to choose, 2 units of the quantity to produce.

And Firm 2 would also want to produce 2 units. Hence, strategy set will be to choose 2 units.

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(c) Since, both the firms would wants to maximise their profits, they both would want to produce 2 units of the good.

The nash equilibria for the following game would be ( 2 units, 2 units) where the outcomes of both the firms are maximum.

But when firm 1 gets to choose first, it would want to maximise its outcome and would try to make its market share higher than the other firm. This will be possible when Firm 2 would choose to produce 1 unit. Thus, to maximise its profit Firm 1 would choose 2 units and want Firm2 to Choose 1 unit and expect the outcome to be (10, 7).

But when, firm 1 would choose 2units and maximise its profits, firm 2 would want to compete neck to neck with firm 1 and would also try to maximise the profits. And hence choose 2 units only making the equilibrium output to again turn out to be (2 units, 2 units).

Thus, sequential choice and benefit of first choice to firm 1 does not change the outcome.

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