Hence, v ( 6 , 000 ) &lt; v ( 4 , 000 ) + v ( 2 , 000 ) and v ( − 6 , 000 ) &gt; v ( − 4 , 000 ) + v ( − 2 , 000 ). These preferences are in accord with the hypothesis that the value function is concave for gains and convex for losses.What this means and by how we can know if these align with convex and concave functions?

Question

Hence,   v  (  6  ,  000  )  &amp;lt;  v  (  4  ,  000  )  +  v  (  2  ,  000  ) and   v  (  −  6  ,  000  )  &amp;gt;  v  (  −  4  ,  000  )  +  v  (  −  2  ,  000  ). These preferences are in accord with the hypothesis that the value function is concave for gains and convex for losses.What this means and by how we can know if these align with convex and concave functions?

Feriheashrz · Accepted Answer

The easiest way to think about examples is the following:   f  (  x  )  =      x    a   is (strictly) convex for   a  &amp;gt;  1 and (strictly) concave for   a  &amp;lt;  1. Now try   a  =  .5 for concavity,   a  =  2 for convexity.The proposition given is just saying   (  x  +  y      )      .5    &amp;lt;  (  x      )      .5    +  (  y      )      .5   for positive   x  ,  y. Take squares and prove it on your own. The sign in inequality will reverse when   x  ,  y are negative.