This is from an economics paper and I have a question about how I am specifying $z_{t+1}$ and $z^{+}$.
Suppose the entrepreneurs live for two periods only. They are only active in the day and are born with an investment opportunity that transforms $x_{t}$ units of current day output to $z_{t}f(x_{t})$ units of day output in date $t+1$.
Productivity evolves stochastically over time and follows a Markov process
\begin{equation*}
\begin{aligned}
Pr[z_{t+1}\leq z^{+}|\eta _{t} = \eta] = G(z^{+}|\eta)
\end{aligned}
\end{equation*}
where $\eta_{t} \in \left \{ b,g \right \}$ denote news received at date $t$ at the beginning of the night.
Define $z(\eta)=\int z^{+}dG(z^{+}|\eta)$ and assume that $G(z^{+}|g) \leq G(z^{+}|b)$ $\Rightarrow z(b) \leq z(g)$. So that good news stochastically dominates bad news.
Is this right way to define $z_{t+1}$? By $z^{+}$, I am also referring to $z_{t+1}$.