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So my book likes to assume that I already have a business degree while learning calculus so I need your help to clarify my book's questions.

It asks:

Price-demand equation. The marginal price for a weekly demand of x bottles of shampoo in a drugstore is given by: $p'(x)=\frac{-6000}{(3x+50)^2}$ Find the price-demand equation if the weekly demand is 150 when the price of a bottle of shampoo is 8. What is the weekly demand when the price is 6.50?

What exactly is being asked? I integrated the marginal price function to get a price-function (I suppose) but now what?

My assumed price-function: $p(x)=\frac{2000}{(3x+50)}$

Paze
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  • Thank you. I still don't understand what they are asking, though.

    The weekly demand is noted by x. The price of EACH bottle is 8...So that makes 150*8=1200 dollars as the price function... I mean it just seems they already put all the variables. I don't understand the question. Please elaborate on the question that I put.

     – Paze Nov 20 '13 at 00:20

1 Answers1

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The question in your textbook indeed doesn't make sense in the economics terms, but sort of makes sense in the context of it being a calculus textbook.

In economics and finance, "marginal" refers to the derivative.

For example, marginal cost function is defined as the amount by which the total cost changes when you produce an additional unit of the good. Cost function $C(Q)$ corresponds to the cost of producing $Q$ units. In the context of calculus, provided that the cost function is differentiable, marginal cost function is the derivative of the cost function $C(Q)$ with respect to $Q$, $\frac{dC}{dQ}$.

The reason I brought up the cost function is because it is sometimes called price function. However, while your calculation of the anti-derivative is correct, usually the total cost function is an increasing function of the number of units produced. So, I think that your textbook is mis-using the term cost function.

The price-demand equation maps the price per unit of the good to the number of consumers willing to buy the good at that price (we assume that each consumer buys a single unit of good). This is also known as the demand curve.

Now, I will guess what your calculus book question means (and this is only a guess). I think what your book means by a "price function" is (a function of) a "demand function", which maps the number of consumers willing to buy the good (in your case shampoo) to price of the good, with demand curves for most goods decreasing (I am very confident that in the great majority of markets shampoo has a decreasing demand curve). I guess (and that's just a guess) that the book's price function however maps the number of units of shampoo sold to the "producer's cost" of the shampoo. So, plugging in 150, we get the "price" of $\frac{2000}{3\times150+5}=4$. The price of the shampoo as sold to the store might be, as you had guessed in the comment, $P(x)=\frac{2000}{3\times150+5}+C$, which makes sense in the context of this being a calculus textbook whose goal it is to teach you the concept of integration constant and not necessarily the laws of supply and demand. But really, had we not known that this is a calculus textbook, we could put in any function $f(x,y)$ that satisfies $8=f(4,8)$ would be valid as a map from the "producer's cost" to "price in the store"...

I hope that I didn't confuse you further, and you seem to have figured this out on your own. I think your teacher/professor will accept your answer in the context of the calculus class.

M.B.M.
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  • I'm only stuck with the last part now. Can you help with that? What is the weekly demand when the price is 6.50? – Paze Nov 20 '13 at 00:32
  • I'm editing my answer now... yes, the calculus book question makes no sense (or, at the very least, is very confusing.) – M.B.M. Nov 20 '13 at 00:35
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    Well I think I got this figured out (it's still a strange way to present the question).

    I input $8=\frac{2000}{3\cdot x+50}+C$ and get C=4.

    So the equation is $P(x)=\frac{2000}{3x+50}+4$ and then for the last part, I should for some reason keep the SAME cost function as before and put 6.50 instead of 8.

     – Paze Nov 20 '13 at 00:38
  • Sorry, I got distracted. I'll finish the edit to my answer now. Yes, that may be one of the possibilities that the book meant. The other candidate might have been $P(x)=\frac{2000}{3x+50}\times C$, or really any $f(x,y)$ such that $8=f\left(\frac{2000}{3x+50},C\right)$. I hope that the rest of that calculus book is better. – M.B.M. Nov 20 '13 at 01:02
  • Finished edit... – M.B.M. Nov 20 '13 at 01:13