# Term for reciprocal percentage

I find myself struggling to succinctly express the concept of gain or loss percentages that are equivalent in the sense that they are reciprocal. For instance, take the following sentence:

An investor with log-utility will experience the same magnitude of utility change for a gain or loss of a certain percentage of wealth.

What is intended is that the investor would lose as much utility from a 20% decrease in wealth as they would gain from a 25% increase (because (1+25%)=1/(1+-20%)). Obviously, I can say "change in log" but that makes the above sentence feel redundant. And I could also say "for percentage gains or losses that are reciprocal when added to unity", but that's pretty verbose. Is there a nice succinct term/way to express this idea of "same" reciprocal percentages?

The log-utility function is $$U(c)=\log(c)$$ where $$c$$ is the investor's wealth. We know that $$\frac{dU(c)}{dc}=\frac{1}{c}\,.$$ In other words, the absolute change $$dU$$ of the utility under an absolute change $$dc$$ of wealth is $$dU=\frac{dc}{c}\,.$$ The right hand side of this is obviously the relative change of wealth.
Regarding your numerical example. The relation $$1+25\%=\frac{1}{1-20\%}\,.$$ says nothing else than a $$25\%$$ increase of wealth followed by a $$20\%$$ loss brings us back to the initial wealth $$c_0$$: $$c_1=c_0(1+25\%)\,,\quad\quad c_2=c_1(1-20\%)=c_0(1+25\%)(1-20\%)=c_0\,.$$ I'd say that is is the succint way of expressing "same reciprocal percentage".
Closely related to this is the following asymmetry of the log-utility: The corresponding log-utility does not got back to the initial utility: $$U_1=U_0+25\%\,,\quad\quad U_2=U_1-20\%=U_0+5\%\,.$$ This happens only when an absolute gain is followed by an equal absolute loss. The log-utility function has an asymmetry which is used to model risk aversion.