My Math Forum Some Thoughts on Elasticity

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 March 18th, 2012, 11:28 AM #1 Member   Joined: Sep 2010 Posts: 63 Thanks: 0 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) $E=\frac{% \Delta y}{% \Delta x}$ from this we can write a difference expression for the total revnue: (2) $R*(1+% \Delta R)= (x+ % \Delta x)(y+% \Delta y)=(x+ %\Delta x)(y+\frac{1}{E} %\Delta x)$ expanding, (3) $\Delta R= (xy-R) + (\frac {x}{E}+y)%\Delta x+\frac{1}{E} \left (%\Delta x \right)^2= (\frac {x}{E}+y)%\Delta x+\frac{1}{E} \left (%\Delta x \right)^2$ and then rearranging gives; (4) $\frac{\Delta R}{% \Delta x}= (\frac {x}{E}+y)+\frac{1}{E} \left (%\Delta x \right)$ 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: $\left( \frac{\Delta R}{% \Delta x} \right )_o= (\frac {x}{E}+y)$ Let us denote changes in the revenue with respect to a percentage change in x with the variable w $w=\frac{\Delta R}{% \Delta x}$ $w_o=(\frac {x}{E}+y)$ (for lack of a creative term), then $\Delta x$ relates to a change in w as follows: (5) $w-w_o=\frac{1}{E} \left (%\Delta x \right)$ 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) $log_{1.1}\left ( w-w_o \right )=log_{1.1} \left ( \frac{1}{E} \right) + log_{1.1}\left (%\Delta x \right)$ which again for lack of creative terms will write as: (7) $z=z_o+\Delta z$ Now, whether these transformation are useful remains to be determined. I hope though it gives more intuitive ways to look at elasticity.

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