# Economics formula

This is an economics question, but if I can get the correct answer to this formula I can answer the question. Any help is appreciated, thanks!

If the demand $Q_x^d$ for a product given the price $P_x$ is

$$\ln Q_x^d = 10 – 5 \ln P_x$$

then product $x$ is:

1. Elastic
2. Inelastic.
3. Unitary elastic.
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A few things. First, welcome to math.SE! I tagged this as homework as that's how the question reads. HW questions are fine, but if it's not, feel free to retag. Second, the community here is more than willing to help you learn, but not to just give you the answers. If you provide some more context and offer an explication of your work this far, people we be much more willing to help. Thirdly, I left you '1n' in paraens in your equation because I'm not sure what you're referring to. Any additional information w.r.t to that part of equation will be helpful. –  Drew Christianson Sep 27 '11 at 21:27
Oh, and this is math.SE not econ.SE, so you may want to define elastic/inelastic in the question. –  Drew Christianson Sep 27 '11 at 21:28
Is "$1n$" supposed to be $\ln$ (natural log)? –  Zev Chonoles Sep 27 '11 at 21:33
Chris: your equations are unreadable and you appear to be merely requesting the answer in lieu of actual understanding, hence the downvotes. I suggest you clarify the question and add what your thoughts on it are so far. @Drew: Even if it's blatantly obvious, I don't think you're supposed to tag other people's questions as [homework] for them without permission (though other tags are fine), maybe I'm misremembering some point of etiquette though. Zev: Yeah, probably. –  anon Sep 27 '11 at 21:40
–  Charles Sep 27 '11 at 21:47

The price elasticity $E$ of a product which has price $P$ given a demand $Q$ is defined as the percentage change in demand for each percentage change in price, i.e.

$$E = \frac{d \ln Q}{d \ln P}\qquad (1)$$

or alternatively

$$E = \frac{P}{Q} \frac{dQ}{dP}\qquad (2)$$

Since in your case you have

$$\ln Q = 10 - 5\ln P$$

You should be able to differentiate $\ln Q$ with respect to $\ln P$ (equation (1)) to get the result you require.

Alternatively, you could express $Q$ in terms of $P$ by exponentiating both sides of the formula, giving

$$Q = e^{10 - 5\ln P} = \frac{e^{10}}{P^5}$$

and apply equation (2) to find $E$, which may be easier if you're not that experienced with differentiation.

Once you have calculated $E$, you can decide whether the good in question is elastic or inelastic by noting that elastic goods have $|E|>1$ and inelastic goods have $|E|<1$.

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... and presumably unitary elastic goods have $|E|=1$. –  Henry Sep 27 '11 at 23:43
(+1)... even though this is quite generous (considering how terrible the OP was, and that the other Chris has done zero to contribute to the solution of his own problem in the past two hours!) –  The Chaz 2.0 Sep 28 '11 at 0:05
I feel like it's better to show first-timers the benefit of the doubt (fix their questions, give them good answers etc) since they don't know the ropes yet, and there are many reasons that someone might not look at a question they've posted for hours or even days after the original posting. –  Chris Taylor Sep 28 '11 at 7:02