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Some mathematical relations important to general counsel are curvilinear

Many relationships between metrics are not linear – there is not constant change in one in step with changes in another. In addition, to note a few, there are exponential functions, power-law functions, and U-shaped functions. As an example of a U-function, law departments with few lawyers per billion probably have high total legal spending as a percentage of revenue (the top left of the U) but that as departments bulk up internally their spending ratio declines (slides down the U). At some point, however, with more and more internal lawyers per unit of revenue, efficiency lags and the spending ratio starts to climb (back up to the top of the other side of the U shape). Whew! A graph is worth a thousand words.

As it turns out, I have often referred to linear functions, but never to those that are curvilinear (See my post of March 10, 2005: patent litigation costs may rise linearly, but certainly not exponentially; Dec. 22, 2005: M&A staffing does not change linearly; Nov. 20, 2006: linear decrease in spending ratios with increasing departmental size; Jan. 3, 2007: linear compared to exponential; Dec. 26, 2007: people demands rise faster than linearly as departments grow in size; Aug. 4, 2008: client satisfaction scores are nonlinear; Feb. 24, 2009: power law distributions on a log-log scale are linear; May 6, 2009: costs do not rise linearly as a case nears trial; June 14, 2009: staff does not grow linearly, but as a step function; Aug. 25, 2009 #4: legal departments as complex, nonlinear, adaptive systems; and Dec. 30, 2009: the longer the litigation, the higher the fees – but not a linear function.).

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