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dP/P = E*dD/D

where 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):

ln|P| = E*ln|D| + constant

**P = D**^{E}+ constantSo, 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!