00:01
So here we're given a rather complicated cost function.
00:03
We're told that total cost is equal to 1 over 3 q cubed minus 5 q squared plus 20 cube plus 50.
00:15
And we're also told that the price is equal to 4.
00:20
So now we need to think about maximizing profit.
00:23
So we need to set up a profit function.
00:27
A, so profit is equal to price times, times quantity minus cost, which is equal to 4 q, because each thing sells, right, for 4, minus 1 over 3, q cubed minus 5 q squared, sorry, plus 20 q plus 50, right? i think i got all those signs right i did now i want to differentiate profit with respect to quantity to find the critical values so this gives me four minus q squared plus 10 q minus 20 is equal to zero this tells me that we've got a quadratic on our hands q squared minus 10 q try to bring everything over to the other side plus six is equal to zero.
01:30
Oh, sorry, i made a small mistake here.
01:33
This is actually minus 20, right? screwed that up.
01:38
Um, minus 20, right? the derivative of one over three q is q squared.
01:48
The derivative of five q squared is 10 q.
01:51
The derivative of 20 q is 20.
01:53
And then we have plus 50.
01:55
So this gives me the quadratic equation, q squared minus 10, q plus 16 is equal to zero.
02:04
If i bring everything back to the other side, this factors into q minus two, q minus eight equals to zero.
02:13
So we have two possible solutions, q equals two, q equals eight.
02:18
We need to check both of them.
02:19
This is a savage problem, right? so the profit when we have q is equal to two gives us, well, we have to plug back into the profit function and q equals 8 is going to give us.
02:35
So both of these need to be plugged back into the original profit function.
02:39
I'm just going to do that with a calculator.
02:41
So having plugged both these into my profit function and compute it, the profit that's resulting, we have a cubic profit function, so it has multiple roots.
02:51
It's not a quadratic.
02:52
The resulting profit would be q equals 8.
02:55
This is the best place to choose, right? and so this is going to lead to long run shutdown.
03:04
It will not lead to short run shutdown because this is better than producing nothing.
03:22
Because if we have q equals nothing, we get minus 50, right? that's the fixed cost, minus 50.
03:29
So this makes sense for this firm to stay in the market for the short run, but not the long run...