Changing the discount rate by adjusting the pure rate of time preference

The Ramsey (1928) equation decomposes the real discount rate into the pure rate of time preference plus a term that accounts for the changing marginal utility of consumption. Discussions about the appropriate discount rate to apply in Cost Benefit Analysis sometimes refer to variations induced by alternative values of the pure rate of time preference as if the two vary on a one-to-one basis. But the optimal consumption path, which determines the marginal product of capital and hence the discount rate, depends on the rate of time preference. Hence the discount rate depends on time preference through the marginal utility term. We derive an analytical expression of this relationship and show that the derivative of the discount rate with respect to time preference only equals unity in the steady state and converges from below. We estimate the derivative using US data from 1930 to 2015. Based on a semi-parametric regression model with time-varying coefficients we find it is about 0.9, but we cannot rule out 1.0 being included in the 95% confidence interval. The implied pure rate of time preference after 1980 is about 1.6 percent.