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March 18th, 2012, 11:28 AM   #1
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Some Thoughts on Elasticity

In another thread I tried using separation of variables to see what curves of constant elasticity would look like:
viewtopic.php?f=46&t=28699
This seems to have failed for the likely reason that the variables aren't separable but I can't yet articulate well enough why this is the case beyond saying the variables aren't independent since they are constrained to a curve.

To approach elasticity (specifically price elasticity of demand) in a more intuitive manner I think we should start by recognizing that it is often used in the context of revenue:
http://en.wikipedia.org/wiki/Total_revenue_test

In the above link Wikipedia's derivation relating a change in revenue to price elasticity is unintuitive. Since it has been some time since I learned of the concept of price elasticity of demand I can remember if a clear motivation was given for the definition.

However, after some thought it is clear that:

given the price elasticity is (for y equal to quantity and x equal to price) that:
(1)

from this we can write a difference expression for the total revnue:
(2)

expanding,
(3)

and then rearranging gives;
(4)

one obtains a linear relationship between a percentage change in X to a change in revenue.

So here is a good time to pause and reflect on what has been done. Elasticity was used to relate a change in revenue with respect to a percentage change in x (price). When delta X is equal to zero there is a surprisingly linear relationship between x an y:


Let us denote changes in the revenue with respect to a percentage change in x with the variable w




(for lack of a creative term), then relates to a change in w as follows:

(5)

Now the percentage change depends on some reference point. That is we measure the percent change in X with respect to some point x but if we wanted a more uniform dependent variable we can take logarithms of both sides. Using log base 1.1 it follows that points of 1 unit apart will represent a change of 10%. Using this particular logarithm we can write

(6)

which again for lack of creative terms will write as:

(7)

Now, whether these transformation are useful remains to be determined. I hope though it gives more intuitive ways to look at elasticity.
John Creighton is offline  
 
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