Appendix 2
Further details on the simulations
In this appendix we provide some supplementary results from the simulations. We show the paths for consumption and wealth on a phase diagram, and paths for debt to GDP and the interest rate under alternative values of λ. The parameter λ measures the responsiveness of the interest rate to the level of debt to GDP, which in this model indicates the degree of capital immobility.
Appendix Figures 4 and 5 show time paths for debt and the interest rate under alternative values of the capital immobility parameter λ. This parameter links the pattern of population ageing with the return to saving, via Equation (7), which models the interest rate as a function of debt. In the case of a zero endogenous interest rate premium, λ=0, there is no link between ageing and the return to saving. The country borrows and lends at an unchanging interest rate. As a result there is perfect consumption smoothing and consumption drops immediately to its new steady state level. If, on the other hand, λ>0 and capital is not perfectly mobile, a fall in consumption (a rise in saving) lowers the return to saving which moderates the initial fall in consumption. The greater the value of λ, the greater the response of the interest rate to a given change in debt, which in turn moderates the change in debt. This illustrated in Appendix Figures 4 and 5. The larger the change in debt the larger the change in wealth (for a given capital stock). Hence greater values of λ imply smaller changes in wealth. This is apparent in the phase diagram in Appendix Figure 5.
The most striking feature of the simulations illustrated in Appendix Figures 4, 5 and 6 is the sensitivity of optimal paths in debt and in consumption-wealth to small variations in the path of the interest rate. The path of the interest rate is itself fairly insensitive to changes in the parameter for interest rate premium, λ, for values greater than 0.01, but is highly sensitive to movement away from the λ=0 case. This illustrates the knife-edge characteristic of the λ=0 case, in that minor variations from this hypothetical case can materially affect the path of the endogenous variables.
