ISI question

[komal]

Two firms 1 and 2 sell a single, homogeneous, infinitely divisible good in a market. Firm 1 has 40 units to sell and firm 2 has 80 units to sell. Neither firm can produce any more units. There is a demand curve: p = a - q , where q is the total amount placed by the firms in the market. So if  qi is the amount placed by firm ith firm, q = q1 + q2 and p is the price that emerges. a is positive and a measure of market size. It is known that a is either 100 or 200. The value of a is observed by both firms. After they observe the value of a, each firm decides whether or not to destroy a part of its output. This decision is made simultaneously and independently by the firms. Each firm faces a constant per unit cost of destruction equal to 10. Whatever number of units is left over after destruction is sold by the firm in the market.

Show that a firm’s choice about the amount it wishes to destroy is independent of
the amount chosen by the other firm.

[s]

Hi,

when a=100,
find out reaction fn of 2 firms
and i got q1 = 110/3 = q2 and q= 220/3...
firm 1 sells 110/3 units and destroys 40 - 110/3 units....firm 2 sells 110/3 units and destroys 80 - 110/3

when a=200,
q1=q2=70,
so firm 1 sells 40 has nothing to destroy, firm 2 sells 70 destoys 10 units..

so firm 2 always destroys firm1 destroys only when a=100

[Amit Goyal]

Let q(i) denotes the amount of output sold by firm i.
Firm 1's objective:
Maximize, with respect to q(1),
(a - q(1) - q(2))q(1) - 10(40 - q(1))
subject to 0 ≤ q(1) ≤ 40.
Differentiating the objective we get,
(a - 2q(1) - q(2)) + 10
Given q(2), if (a - 2q(1) - q(2)) + 10 ≥ 0 at q(1) = 40, then the best response of firm 1 is 40.
if (a - 2q(1) - q(2)) + 10 ≤ 0 at q(1) = 0 then the best response of firm 1 is 0.
if (a - 2q(1) - q(2)) + 10 = 0 at some 0 ≤ q(1) ≤ 40 then the best response of firm 1 is q(1) = (a + 10 - q(2))/2.
To summarize, best response correspondence of firm 1 is:
q(1) (q(2)) = 40 if  (a - 70 ≥ q(2))
                = 0 if (a + 10 ≤ q(2))
                = (a + 10 - q(2))/2 if (a - 70 ≤ q(2) ≤ a + 10)

Firm 2's objective:
Maximize, with respect to q(2),
(a - q(1) - q(2))q(2) - 10(80 - q(2))
subject to 0 ≤ q(2) ≤ 80.

Similar to above, best response correspondence of firm 2 is:
q(2) (q(1)) = 80 if  (a - 150 ≥ q(1))
                = 0 if (a + 10 ≤ q(1))
                = (a + 10 - q(1))/2 if (a - 150 ≤ q(1) ≤ a + 10)

Put a = 200, the Nash equilibrium is: (q(1),q(2)) = (40, 80)
Put a = 100, the Nash equilibrium is: (q(1),q(2)) = (110/3, 110/3)

[komal]

Thanks for replying..
sir, a firm's response function(q1) is dependent on another firm's quantity q2, does not this mean that one firm's decision of destroying depends on no. of units destroyed by another?
question is asking to show that firms decision would be independent of what other firm has destroyed...
I m confused at this point.

[Amit Goyal]

If you put a = 200 in the best response correspondence, you will get answer to your question.

[Amit Goyal]

The best response is not independent when a = 100, it is independent when a = 200.

[Rayee]

there is also a sub-part to this question,
"Show also that the amount destroyed by firm 2 is always positive, while firm 1 destroys a part of its output if and only if a = 100". In this line of argument if a=200, then firm 2 does not destroy anything. how do i show the above mentioned condition then?

[Amit Goyal]