00:01
Hello students, we are given a question here that the demand for a profit maximizing monopolist is q plus 2p is equal to 40.
00:10
And the average cost is given by ac is equal to 20 times q to the power minus 1 plus 4.
00:16
Now we are asked some question like first is at what output will monopolist maximize profit? okay, students? so first of all, what we need to understand here, we can say that what we are given, the function is given as.
00:30
Q plus 2p is equal to 40.
00:33
So we can just write that the p is equal to here, obviously, 2p is equal to 40 minus q and p can be written as 40 divided by 2.
00:43
It means 20 minus half of q.
00:46
So we are supposed to know that here, what we can understand? total revenue is total revenue can be written as a price times quantity.
00:57
So it means p times q so tr is equal to 20 minus half of q times q okay students which comes out as equals to total revenue is equals to 20 q minus half of q square okay students now here we can say that when we differentiate it up so what we will get the marginal revenue because the differentiation of total revenue gives us the function of marginal revenue which is 20 minus half of twice of q it means simply q okay students these are the functions as a total revenue and marginal revenue now as we are given further here that the ac is equals to average cost function is given as 20 times q to the power minus 1 plus 4 okay students so when we find the total cost so what we need to do here we can just say that we need to multiply it by q obviously because q times of it should be q times of ac now tc is equals to q times of 20 times of q to the power minus 1 plus 4 it means total cost gives us as comes out as the 20 plus 4 times of q okay students now when we differentiate it in order to find the marginal cost so when we differentiate total cost we get a function of marginal cost which is now 4 okay students so basically these are some functions now we will start with our first part what we are asked that at what output will monopolist maximize profit okay students so we are supposed to know that for profit maximization for profit maximizing okay students profit maximizing marginal revenue have to be equal to marginal cost okay so basically marginal revenue we have calculated as 20 minus q so 20 minus q is equal to marginal cost is 4 so basically q is equals to 20 minus 4 comes out as equals to here we can say that it is q star is equal to 16 units okay students now here we can say that when we calculate a p so basically when we put a q star is equals to 16 in this equation what we can get a p is equal to 20 minus half of 16.
03:32
Okay, students, so we can even mention below that the p is equal to 20 minus p star is equals to 20 minus half of 16.
03:40
So p star comes out as equals to 20 minus 8 in place 12.
03:45
Okay, students.
03:46
Now, second part, what we are asked? we are asked that calculate the price elasticity of demand when profits are maximized.
03:58
Okay, students.
03:59
So basically we are supposed to know that the price elasticity of demand, e can be written as dq by d .p, okay, students, times p divided by q.
04:11
So e is equal to dq by dp...