3. Suppose the market for Widgets is dominated by two firms, firm A and firm B T
ID: 1149256 • Letter: 3
Question
3. Suppose the market for Widgets is dominated by two firms, firm A and firm B The market price of Widgets depends on the combined quantity of Widgets produces by firms A and B, qA + qB: P = 100-.5(Ta + qe). Assume further that the cost of making each Widget, c 5 is the same for both firms. Note that cach firm's profits are given by Pq cq where i-A,B indexes each firm. Suppose that due to the production process, each firm can choose to produce just four quantities of Widgets: 10, 20, 40 or 80 First, construct a 4x4 table showing the market price of Widgets when each firm produces one of its four possible output amounts of widgits, 10, 20, 40 or 80 (you will find 16 market prices). a. b. Using this price information the fact that c 5, construct a 4x4 table showing the profits earned by each firm from al possible quantity choice combinations made by the two firms Using the table, you constructed in part b, find the pure strategy Nash equilibrium ia) of this quantity choice game Suppose the cost to firm A of producing a widget drops to c-0, while the cost to firm B remains fixed a c-5 per Widget. Repeat parts a-c. c. d.Explanation / Answer
A)
Market price depends on the combined output:
P = 100 - 0.5(qa + qb)
Putting the values of qa and qb , we get different prices.
The first row shows various output levels of firm A and the first column shows various output levels of firm B. The values inside the table are various price levels that are arrived by putting quantity levels of both the firms in the demand equation:
B)
Profit = TR - TC
= P.Q - c.Q
For (10,10) output level, the corresponding price is 90.
Thus profit for Firm A =
= 90*10 - 5*10 = 850
Same for firm B.
Similarly, profit levels for both the firms for all output levels can be derived.
C)
When A produces 10 units, B's best response would be to produce 80
When A produces 20 units, B's best response would be to produce 80
When A produces 40 units, B's best response would be to produce 80
When A produces 80 units, B's best response would be to produce 40
Similarly, we can the best responses of A for various levels of B's output
When B produces 10 units, A's best response would be to produce 80
When B produces 20 units, A's best response would be to produce 80
When B produces 40 units, A's best response would be to produce 80
When B produces 80 units, A's best response would be to produce 40
Thus (40,80) and (80,40) are the pure strategy Nash Equilibrium.
D)
The price matrix will remain the same as it is not dependent on the cost
Profit Matrix
Nash Equilibrium
B's responses are the same as before.
A's responses are also same for 10, 20 and 40 units of B's output.
However, when B produces 80 units, A can produce 40 or 80 units.
Hence there is no pure Nash equilibrium as A can choose either 40 or 80 units.
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