  We consider stochastic matrix models for population driven by random
environments which form a Markov chain. The top Lyapunov exponent $a$, which
describes the long-term growth rate, depends smoothly on the demographic
parameters (represented as matrix entries) and on the parameters that define
the stochastic matrix of the driving Markov chain. The derivatives of $a$ --
the "stochastic elasticities" -- with respect to changes in the demographic
parameters were derived by \cite{tuljapurkar1990pdv}. These results are here
extended to a formula for the derivatives with respect to changes in the Markov
chain driving the environments. We supplement these formulas with rigorous
bounds on computational estimation errors, and with rigorous derivations of
both the new and the old formulas.
