I am hoping for a simple intuition for why price stickiness induces indeterminacy of the interest rate, $i_t$, in the benchmark New Keynesian model. When prices are flexible, the natural rate is pinned down. However, it seems that price stickiness is the unique factor that breaks this. Is this correct?
I can do the pages of derivations and show to myself that $i_t$ is not uniquely determined without a policy rule, but in all that math I cannot see the economic intuition. I only see the math intuition of more variables than equations. I am missing the forest for the trees - I do not have an intuition for the core reason that introducing price stickiness makes $i_t$ indeterminate.
One thought I had is that since the firm problem becomes dynamic, and current price setting depends on future price setting that perhaps there is some sort of coordination indeterminacy? That doesn't seem correct to me though.