A.H. comments on What is wrong with this “utility switch button problem” approach?

[A.H.]

For $h_a$, this is the policy that is optimal when $q=0$ which has $u\left(h_a\right)=a\left(h_a\right)$. Then $\frac{du}{dE\left(K\right)}<0$.

Please could you explain how you get $\frac{du}{dE\left(K\right)}<0$ when $u=a=E\left(A\right(1-K\left)\right)$?

Possibly a dumb question but I don’t have a good intuition for what it means to differentiate an expected value with respect to an expected value.

I can see that this is the case when $A$ is positive (as expected for a utility function) and uncorrelated with $K$, but is is true in general? Even when $A$ is strongly correlated (or anti-correlated) with $K$? They would presumably be correlated in some way since they both depend on the policy pursued.

Also, how would this work for a utility function A which is negative? In theory we should be able to apply an affine shift to a utility function so that it has a negative expected value. If A was uncorrelated with K but has a negative expected value then $\frac{du}{dE\left(K\right)}>0$ , right? Can we do a similar affine shift to B to ensure that there won’t necessarily be a value of q where $\frac{du}{dE\left(K\right)}=0$?