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“This lecture. We re gonna talk about a business concept called marginal value and this this concept applies to a wide variety of business concepts we can talk about marginal marginal cost marginal utility marginal profit to make the discussion kind of more concrete i m gonna talk specifically about marginal revenue. But just in the back of your mind keep in mind that this implies to a whole host of different topics as well so when we talk about marginal revenue. What we mean is how much revenue.

We would generate from selling one more of an item right so i have an example here. I have this very basic revenue function r. Is the revenue x. Is the number of units sold.

And you ve probably seen some some similar to this before what this is telling me just a first glance is that the company is selling this product for 1250. Each. So for example. The revenue generated by selling 10 items would be 1250 times 10 or 125.

Right. Then the revenue generated from selling for. Example 11 items would be 1250. Times.

11 which comes out to 100. 3750. Soon as what i did here. I picked two consecutive numbers right.

I pick 10 units versus. 11 units. Because this is gonna let me see what the marginal revenue for 10 units is the marginal revenue is the difference between two successive x values so the marginal revenue here is going to be 13750. 125 which is 1250.

So what does this tell me all right what this tells me in business terms is the margin is if i sell 11. Units i m going to make 1250 more than i would if i sold 10 years that s that s the marginal revenue concept is how much you would make from selling one more of an item and now notice here right this 1250 matches this 1250 up here. And that s not a coincidence when when the function that you re dealing with is a linear function like the revenue example. I picked here then the marginal revenue is going to be equal to the sub or it s going to be in business terms.

.

It s going to be the cost per unit. How much you re selling the item for now. That s not going to be the case for every function alright and we ll see on the next couple of slides. That if the revenue function is more complicated then the situation is a little more complicated and we re gonna have to get more creative alright.

So let s take a look at a more complicated concept. I have a new revenue. Function. Here now the revenue is equal to 100 x minus.

05. X. Squared. And i ve graphed.

That here. So. You can see visually. What s going on this is what you hopefully expected to happen right this revenue function here is a quadratic equation.

And the a term is negative. So you expect the graph to be a parabola opening down. Which is what we have here right now in business terms. You might think.

This is kind of on how can the revenue drop off as i sell more units shouldn t the revenue will always increase well think about a situation. Where you re you re putting product out there. And you re selling these products. And eventually you get to a to a point that the marketing people call market penetrance saturation right that s the point.

Where everybody in your market. Who wants your product has your product right there there are practical purposes. There are no new customers there s nobody out there looking to buy whatever. It is you re selling and at this point.

.

If you want to some more of your product. One way you can make this happen is by reducing your price because if you reduce your price. Then people who may not have been interested in buying it at the original price well they they might think that your new price is reasonable alright. But you know when you lower your price sometimes that that s you lower your price you increase your sales volume.

But sometimes that s still not enough to cause your your revenue to continue to rise and you may experience this kind of situation where your revenue drops off alright. So i want to think about what s going on with a marginal revenue in this situation and to do that what i ve done here on this next slide. I just blew up the part of the graph between 94 and 100 so so we ve got kind of a bigger graph here that i can draw on and what i want to do is i want to think about the marginal revenue for 97 units. In other words.

I m asking myself. If i if i sell 98 units how much more am i gonna make how much more can i make from selling 98 versus 97. Alright so you re on the on the pre on the first slide to find that marginal revenue. What i did is i calculated r of the first value 97 r of the next value and then i subtracted alright.

So visually what i m talking about is this i ve got uh. I ve got 97 that s this point on the graph and i ve got 98. That s this point on the graph. So if i subtract these two y values.

I get this alright and i could do that i could get out my calculator and i could i could figure out what r of 97 r. 9 adar that will be fine. But that s gonna get a little tedious especially if i m cutting do this from a lot of different numbers so here s what i m gonna do i m gonna draw the tangent line here at that point that i m interested in all right. And i want you to look at the value of the tangent line at 98.

And i want you to notice here that this distance the distance between my tangents value and 97. Is really close to the value. 2. The distance between the functions value in 97.

So what this tells me is ma. The tangent line. If i use the tangent line instead that s given me a actually a really a really good approximation of the marginal value right and now i want you to get to think back to that first slide. If we re looking at a linear function like this tangent line.

.

So i m is linear. Then the marginal value is just the mines slope all right so this is where i m gonna i m going to have to pull out a little calculus all. Right i. Have this.

Function. R. Of. X.

Equals. 100. X minus. 05.

X. Squared. And this is our revenue. And i m gonna find the derivative of this function right remember again referencing our calculus.

Here right the derivative is the slope of the tangent. Line. So this is going to be 100 minus. 2 times.

05. Is 1. So. This is gonna be just 100 minus x.

All right. So. This this r. Prime.

.

This derivative. Is the slope of the tangent line. And the slope of the tangent line is going to give me my approximate marginal revenue. All right so i want you to just kind of step back here look at the big picture think about we did all right.

I m trying to find a function that tells me the marginal revenue for this revenue model that i have here and to do that i m going to take the derivative. Because the derivative is the slope of the tangent line and the marginal revenue front line is just the slope so they and that s what i have here. Our prime is the slope of that tangent line. So what does this tell me let s work with this here.

A little bit. Let s say. I looked at our prime of life time in 97. Right our prime of 97.

This is 100 minus 97. That s three dollars all right so what does this tell me this tells me that selling 97 unit actually be selling 98 units is gonna net me three more dollars in revenue than i would have gotten from selling 97. All right let s look at another example. Let s look at our prime of 100.

The marginal revenue for selling 100 units right. That s 100 minus 100. That s zero all right so again. What does this tell me this tells me that the marginal revenue from selling 101 units versus a hundred the chain.

I m going to make zero dollars more and remember because if you think about the graph. It s not really zero did this is giving us an approximation. But this is an important value because this tells us where the function turns around this tells us where the revenue starts to drop off all right star starts to decrease. So what did we do here.

We developed a good method for getting an approximate value of the marginal revenue and i want to remind you what i said at the beginning right this applies to almost any business concept like you use the same concept to find marginal cost i could use it to find marginal utility marginal profit all of those business concepts if i can if i could find a function like my r of here that describes them i can find the corresponding marginal values. But ” ..

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