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Economics: Elasticity Differential Equation Question 
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#1
Aug2313, 12:00 AM

P: 30

Hi. Let me preface this by saying that I know nothing about economics. However, I learned a little bit about the concept of price elasticity of demand (that for something really elastic, if price goes up a little, demand will go down a lot), and I came across an equation relating price, demand, and elasticity. It was this:
dP/P = E*dD/Dwhere P is price, E is elasticity, and D is demand, with P and D > 0. In solving this, I get the following (unless I completely forgot how Calculus II works): lnP = E*lnD + constant So, I'm assuming that with an elastic good, if price goes up, demand will go down (I believe that is the Law of Demand). But this means that E has to be negative, does it not? Based on that equation, the only way I can see Price going up and Demand going down is if E is negative. And if E = 0, then a good that is completely inelastic will have a constant price and a constant demand? (P = 1 + constant)? But then if E = 1, then it's P = D + constant, so what does an elasticity with 1 mean? And if E > 0, then if price goes up, demand goes up? What kind of good would be like that? "Status" goods? Anyway, would someone who is familiar with economics correct me or verify and/or elaborate on what I seem to get here? Thanks! 


#2
Aug2313, 02:49 AM

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$$\varepsilon = \frac{dQ/Q}{dP/P}$$ where Q is the quantity demanded and P is the price. 


#3
Aug2713, 01:46 PM

P: 30

First of all, thanks for your response!
which is, ##ce^{lnD^E}## = ##cD^E##, correct? Of course, as you said, it should have been ##Q = cP^E## (why do they use Q for demand? Odd choice. Oh, wait, nevermind. dD/D is kind of awkward I guess, and Lord help you if you use D as a differential operator.) In any case, if E is not a constant, you wouldn't still use this form of an equation? Say, E(Q,P,R...) = some quotient of partial derivatives, I guess? So, ##Q = cP^0 => Q = c##, a constant. Demand doesn't change. Now, if E = 1, you'd have Q = cP, which is just a straight line of slope c, right? (but then you'd have demand going up as price goes up). But what does that mean, if anything at all? As far as the critical points here, I understand when E < 0. That's pretty much any normal good where demand goes down as price goes up. As for E = 0, again, that makes sense. It's a constant, so demand doesn't change. But I'm not clear on any significance for E = 1, and also, when would revenue be maximized? I suppose with a revenue function I could find that just using the maxima finding technique from calculus, but I'm not sure from this equation. Thanks again for all your work here. 


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Aug2813, 12:48 AM

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Economics: Elasticity Differential Equation Question



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