⁶³The Hadamard operators $⊙$ and $⊘$ represent element-wise multiplication and division, respectively. As a numeric detail, $^j Pt$ is reset to 1 when $j φt = 0$ , which would otherwise cause it to be undefined. When no households perceive growth rate $Φj$ , the average perception of productivity does not exist for this state and is quantitatively irrelevant, but must exist for (37) to not fail in the next period.