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Derivatives of the Stochastic Growth Rate

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.