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# Marginal cost & differential calculus

## Video transcript

let's say I run some type of a factory and I've studied my operations and I'm able to figure out how my cost varies as a function of quantity over a week on a weekly period and so to visualize that let me draw it draw this cost function so this is my cost axis this right over here could be my quantity axis so that's quantity or Q my Q X let me just call that Q that's my Q axis and my function might look something like this my function might look something like this it seems reasonable to me even if I produce nothing I still have fixed costs I have to pay rent on the factory I have to probably pay people even if we produce nothing and so let's say that fixed costs in the week is one thousand one thousand dollars and then as my quantity increases so do my cost so if I produce one hundred units right over here then my cost goes up to 1300 1300 if I produce more than that you see my cost increase and they increase it of ever-faster rate now I go into a lot more depth on things like cost functions in the economics playlist but what I want to think about in the calculus context is what would the derivative of this represent what would the derivative of C with respect to Q which could be real so written as C prime of Q what does that represent well if we think about it visually we know that we can think about the derivative as the slope of the tangent line so for example that's the tangent line when Q is equal to 100 so the slope the slope of that tangent line is you could view as C prime or it is C prime of 100 but what is that slope telling us well the slope is the slope is the change this is a change in our cost divided by the change in our quantity and it's the slope of the tangent line we've this is what we first learned in calculus as we get to smaller and smaller and smaller changes in quantity we essentially take the limit as our change in quantity approaches 0 that's why we get that instantaneous chain so one way to think about it one way to think about it is this is the instantaneous this is the this is the rate right on the margin at which our cost is changing with respect to quantity so if I were to produce just another drop another atom of whatever I'm producing how at what rate is my cost going to increase and the reason why I'm saying it right on the margin as we see that it's not constant if our cost function were aligned it we would have a constant slope the tangent line would essentially be the cost function but do we see it changes right over here are the incremental atom to produce here cost less than the incremental atom right over here the slope has slope has gone up but it might make sense maybe I'm using some raw material out there in the world and as I use more and more of it it becomes more and more scarce and so the market price of it goes up and up and up but you might say well why why are you even you know why do I even care about the rate at which my costs are increasing my cut the rate at which my costs are increasing on the margin and which is why this is called marginal marginal cost well the reason why you care about is you might try you might be trying to figure out when do I stop producing let's say this is orange juice if I know that next gallon is going to cost me five dollars to produce and I can sell it for \$6 and I'm going to do it but if that next gallon if I'm up here I've already produced a lot and I'm taking all the oranges off the market and now I have to transport oranges from the other side of the planet or whatever it might be and now that incremental of GAAP incremental gallon of oranges or a gallon of orange juice cost me ten dollars to produce and I'm not going to be able to sell it for more than \$6 and doesn't make sense for me to produce it anymore so in a calculus context or as you say in an economics contest context if you can model your model your cost as a function of quantity the derivative of that is the marginal cost it's the rate at which costs are increasing for that incremental for that incremental unit and there's other there's other similar ideas if we had if we modeled our profit as a function of quantity if we took the derivative that would be our marginal profit for your modeled revenue that would be our marginal revenue how much is the function increasing on the margin as or how much is the function increase as we increase our input R as we increase our quantity on the margin
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