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Economics: Marginal Cost & Revenue - Problem 2

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

The marginal revenue function is the derivative of the total revenue function, r(x). To find the marginal revenue, take the derivative of the revenue function to find r'(x). It gives the approximate cost of producing the next item (if x=5), r'(5) tells you the approximate cost of producing the 6th item). This value can be compared to the actual revenue of that item (using this example, the actual revenue r(6)).

I want to talk about marginal revenue. Let's take a look at an example. Suppose R(x) is the total revenue from selling x skateboards. The revenue function R(x) is 60x minus 0.01x² in dollars.

I'm going to do three things. A; find the marginal revenue function. B; find R'(500) and give the units. C; find the actual revenue from the sale of the 501st skateboard, and compare that with R'(500).

Remember that R'(500) is the marginal revenue at 500 units. It should give me an approximation of what it would cost to produce that 501st. Or rather what revenue I gain from selling that 501st skateboard. So this is the actual revenue that I'll get from the 501st skateboard. It will be interesting to compare the two.

So let me do my work over here. First, let's calculate marginal revenue. I have my revenue function written up here R(x) equals 60x minus 0.01x². Just remember that marginal revenue is the derivative of revenue. So R'(x). The derivative of 60x is just 60. The derivative of 0.01x² is 0.02x. That's it. That's your marginal revenue function.

Part b asks to find R'(500) and give units. R'(500) is 60 minus 0.02 times 500. Point 02 times 500 is 10, 60 minus 10 is 50. So this is just 50. Now what are the units? Well, keep in mind that R'(x) is just dr/dx; the derivative of revenue with respect to x.

The units of the top would be the units of revenue which are in dollars. The units of the bottom would be skateboards. So this would be dollars per skateboard. I'll write that over here; dollars per skateboard. So this 50 represents the amount of money I could get by selling that 501st skateboard. The extra amount of revenue.

Now let's compare that to the actual amount of revenue, that you would get from selling the 501st skateboard. I need to calculate R(501) minus R(500). So I'm going to calculate each of these individually, and then subtract in the end. Let me actually get rid of this, because I'm going to need this space.

So R(501). Using this function 60 times 501 minus 0.01 times 501². 60 times 501 is 60 times 500 which is 30,000 plus 60 times 1. So 30,060. Then I have 0.01 times 501². 501² is 251,001. 0.01 times that is 2510.01. You just move the decimal place two places to the left.

So I get 30,060 minus 2510.01. This difference ends up being 27,549.99. 27549.99, that number in itself isn't important, we're going to end up subtracting from that R(500) which we'll calculate now. Much easier calculation.

60 times 500 minus 0.01 times 500². 60 times 500 is 30,000. This is 250,000. 250,000 times 0.01 is 2,500. So this difference is 27,500. So these are the two numbers that I'm subtracting. I'm subtracting 27,500 from 27,549 and 99 cents. The difference is $49, and 99 cents.

Now compare this number to the 50 dollars I got as an estimate using the marginal revenue. Marginal revenue is a very good approximation of the actual revenue generated by selling that 501st skate board.

So this lengthy calculation I just want to point out that it's not really necessary. The only advantage to it is that this is the actual extra revenue from the sale of that 501st skateboard, and this is an approximation. But anyway it's a very good comparison.

Marginal revenue remember is just the derivative of the revenue function, and it can be used to get an approximate value with the extra revenue gained by selling one more skateboard.

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