Thursday, September 6, 2012

Question from Mark

Mark asked:

First off, thanks for your videos. They help.My question:Assuming I have 0 means lottery X, (-6 with 1/2 probability and 6 with 1/2 probability) and utility function u(w)=w for w<=10 and u(w)=1/2w+5 for w>=10
Can I apply the Arrow-Pratt approximation of pi(w;X)=1/2 (sigma)^2A(w)?
My hunch is no since A(w)=0 in either case of u(w)...i think?? My question is what does A(w)=0 mean? and why does Arrow-Pratt not work here?

My response - The Arrow-Pratt Measure applies to utility functions that are twice differentiable.  In the example above the utility function is piece-wise linear, with a kink at 10.   It is not differentiable at the kink, so it is outside the class of functions for which the measure is intended.  Alternatively, if you prefer, when w is not 10, the individual is risk neutral (for small gambles).  When w is 10, the individual is infinitely risk averse.


  1. That makes sense. I keep coming across the statement that this approximation is only good "for small risk." What does that mean?

  2. In the context of Mark's example, you can contrast having w for certain with a lottery that gives w + epsilon with probability 1/2 and w - epsilon with probability 1/2, where epsilon is a small positive number. Note that the expected value of this lottery is w. By keeping epsilon small there is small risk and the expectation remains the same. You need both of those to make the comparison.