00:01
Hello everyone, let us look into the solution.
00:04
So, the answer is c1 of q1 is equal to alpha q1 take and c2 of q2 we can take it as alpha q2 where alpha belongs to closed interval 1, 2 and the market demand here is p equal to 2 minus q where q equal to q1 plus q2.
00:28
Then from this we can have the profit.
00:31
Now from this we can have the profit of firm 1 equal to pq1 minus cq1 and we can have phi 1 equal to 2 minus q1 minus q2 multiplied by q1 minus alpha q1 and phi 1 equal to by substituting we can write it as 2 q1 minus q1 square minus q1 q2 minus alpha q1.
00:59
Hence, we can have d phi 1 by d q1 is equal to 2 minus 2 q1 minus q2 minus alpha equal to 0 that is we are differentiating with respect to q1.
01:12
So, from this we can have 2 q1 plus q2 equal to 2 minus alpha mark it as equation 2, equation 1.
01:21
Then similarly we can have the profit of firm 2 it is pq2 minus cq2 therefore, we can have phi 2 equal to 2 minus q1 minus q2 multiplied by q2 minus q2.
01:37
So, which is equal to 2 q2 minus q1 q2 minus q2 square minus q2.
01:45
Now we have to differentiate this with respect to q2 d phi 2 by d q2 equal to 2 minus q1 minus 2 q1 minus 1 equal to 0.
01:56
Hence, we can have q1 plus 2 q2 equal to 1 mark it as equation 2.
02:05
So, now we have to solve these two equations.
02:08
So, from the equation 2 we can have q1 equal to 1 minus 2 q2.
02:13
Now substitute the value in the first one we can have 2 q1 plus q2 equal to 2 minus alpha implies 2 times 1 minus 2 q2 plus q2 equal to 2 minus alpha.
02:30
So, which is 2 minus 4 q2 plus q2 equal to 2 minus alpha.
02:38
Hence, 2 minus 3 q2 equal to 2 minus alpha then bring this to the right side it is after simplification we make it as 3 q2 equal to alpha because 2 will get cancelled therefore, q2 equal to alpha by 3...