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 Economics Economics Forum - Financial Mathematics, Econometrics, Operations Research, Mathematical Finance, Computational Finance

 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) 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. Tags elasticity, thoughts Thread Tools Show Printable Version Email this Page Display Modes Linear Mode Switch to Hybrid Mode Switch to Threaded Mode Similar Threads Thread Thread Starter Forum Replies Last Post shunya Algebra 1 January 24th, 2014 02:53 PM Hedge Number Theory 20 July 8th, 2013 02:01 AM CherryPi Calculus 2 April 18th, 2012 04:19 AM billymac00 Number Theory 2 June 29th, 2010 09:24 PM natus zeri Number Theory 1 October 22nd, 2007 03:07 AM

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