A generalization of Ramsey rule on discount rate with regime switching

I generalize the following rule of Ramsey (1928) on the discount rate with regime switching: the discount rate is the sum of the rate of pure time preference and the product of the consumption elasticity of marginal utility and the consumption growth rate. The Ramsey rule can be extended to regime-dependent interest-rate formulas for discounting future regime changes. Notwithstanding debate about empirically plausible values of the rate of pure time preference, I theoretically show that the effect of pure time preference is overwhelmingly dominated by the effect of the regime switching parameter. This is closely associated with consumption smoothing consequences across regimes.