  A recent economic literature deals with models of random growth in which the
size (say, log-wealth) of an individual economic agent is subject to
light-tailed random additive shocks over time. The distribution of these shocks
depends on a latent Markov modulator. It has been shown that if Markov
modulation is irreducible then the models give rise to a distribution of sizes
across agents whose upper tail resembles, in a particular sense, that of an
exponential distribution. We show that while this need not be the case under
reducible Markov modulation, the upper tail will nevertheless resemble that of
an Erlang distribution. A novel extension of the Rothblum index theorem from an
affine setting to a holomorphic one allows us to characterize the shape
parameter for the relevant Erlang distribution.
