= Solution
Let $m$ be a nonzero invariant measure. Balance at state $i\geq1$ gives
$$
m_{i-1}\lambda_{i-1}
=m_i\lambda_i(1+\rho_i),
$$
and hence, after choosing $m_0>0$,
$$
m_i=\frac{m_0\lambda_0}
{\lambda_i\prod_{j=1}^i(1+\rho_j)}.
$$
Balance at zero requires
$$
m_0\lambda_0=\sum_{i\geq1}m_i\lambda_i\rho_i
=m_0\lambda_0\sum_{i\geq1}
\frac{\rho_i}{\prod_{j=1}^i(1+\rho_j)}.
$$
The sum telescopes:
$$
\sum_{i=1}^n\frac{\rho_i}{\prod_{j=1}^i(1+\rho_j)}
=1-\frac1{\prod_{j=1}^n(1+\rho_j)}.
$$
Consequently a nonzero invariant measure exists exactly when
$$
\boxed{\prod_{j=1}^{\infty}(1+\rho_j)=\infty.}
$$
It can be normalized to an invariant probability distribution exactly when, in addition,
$$
\sum_{i=0}^{\infty}
\frac1{\lambda_i\prod_{j=1}^i(1+\rho_j)}<\infty,
$$
where the empty product for $i=0$ is one and an overall constant has been suppressed.
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