Discrete dividend budget equation ($C_t+S_t{\theta }_{t+1}=(S_t+d_t){\theta }_t$)

When a productive stock pays $d_t$ before trading at ex-dividend price $S_t$, an investor entering with ${\theta }_t$ shares receives $d_t{\theta }_t$ and sells or buys shares to leave with ${\theta }_{t+1}$. The displayed budget constraint accounts for the available consumption and the cost of the new holding. Here ${\theta }_t$ is known before date $t$, while $C_t$ and ${\theta }_{t+1}$ may depend on the information revealed at that date.