In: Economics
Suppose that the market for microprocessors is dominated by just two firms. All microprocessors produced are sold at the market-clearing price which depends on TOTAL industry output. The market demand was estimated to be P = 30 – QTOTAL, where QTOTAL is the combined output of two firms in million.
The only decision variable for each firm is how many
microprocessors to produce. Each firm must decide whether to build
a plant suited to produce high volume, low volume, or to produce no
microprocessors at all. Once the output decision is made, it is
final. Regardless of the volume of production, each microprocessor
costs a firm $7 to produce.
High and low volumes are 10 million and 5 million microprocessors,
respectively.
a. (6 pts) Suppose the game is played simultaneously. Present the game in the normal form, using the table below and expressing payoffs in TOTAL PROFIT amounts.
(Consider, for example, the case when Firm 1 produces 5 mln units and firm 2 produces 10 mln units. The total industry output is 15 million. According to the demand equation, the market price will be P = 30 – 15 = $15.
The profit margin is then (P–ATC) = 15 – 7 = $8 per microprocessor. Finally, the profits of the two firms are 8·5=$40mln and 8·10=$80mln, respectively… and so on.)
Firm 2 |
||||
Volume=10mln |
Volume=5mln |
Volume=0mln |
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Firm 1 |
Volume=10mln |
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Volume=5mln |
40, 80 |
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Volume=0mln |
b. (2.5 pts) Does any of the firms have a dominant strategy? If so, state what those are.
c. (2.5 pts) Does any of the firms have a secure strategy? If so, state what those are.
d. (3 pts) You are Firm 1, and you are concerned only with your own profit. According to the textbook and the lecture, which strategy should you play? Explain your choice.
Next, suppose one of the firms is able to commit to a certain plant size first and make it known to the other firm.
e. (4 pts) Present this game in the extensive form (a tree). Make sure all the nodes, branches, and payoffs are properly labeled!
f. (4 pts) Identify the most likely outcome of the game
presented in part e.
Does changing the rules of the game change your prediction about
the most likely outcome? Comment briefly.
a) After writing the payoffs as per the instructions given in part A we get the total profits as the payoffs. 10, 5 and 0 million productions are the strategies here. As per the diagram if we want to find the payoff for (10,10) we write total profit = 30-(10+10)- 7= $3. now we multiply 10*3 and 10*3 i.e the total profit and the amount produced in millions. After solving the game by nash equilibrium we get (10,5) and (5,10) as the solution of the game.
b) Firm 1 has strategy 5 and 10 as dominant over strategy 0. If we see the diagram we see that in strategy 10 of firm 1 ( 30>0,80>0,120>0) , 0 being the payoff for firm 1's strategy 0. For strategy 5 of firm 1 again ( 40> 0). For firm 2 also 10 and 5 strategies are dominant over strategy 0 because (30>0, 80>0, 130>0)
c) Secure strategy given the best payoff given the worst payoffs in absence of dominant strategies. If we ignore the dominat strategies and play using nash equilibrium strategy we get (10,5) and (5,10) as the secure strategies. If firm 1 chooses strategy 10, firm 2 will choose strategy 5. If firm 1 chooses 5 then firm will choose 10, if firm 1 chooses strategy 0 then firm 2 will choose strategy 10. Now from the point of view of firm 2, if firm 2 chooses strategy 10 firm 1 will choose 5, if firm 2 chooses strategy 5 firm 1 will choose strategy 10, if firm 2 chooses strategy 0 firm 1 will choose strategy 10.
d) If i am firm 1 and i only keep my profit in mind then i will choose (10,0) as it maximizes my payoff and firm 2's payoff is zero. Now if a plant is able to commit to a certain plant size and it is know to the other firm the other firm will play strategy (5,5) or(10,10) or (0,0) as both gets equal payoff. If the strategy would have been taken simultaneously as in case of nash the firms cannot trust each other then the equilibrium strategies will would have been (10,5) (5,10) no one would have any incentive to deviate and they never end up getting the ideal payoff i.e (5,5). But when the strategy is known to the other firm both of them both of them want to get the same payoff without obviously harming their own interest.