1. The present version extends the model in Guest & McDonald (2005, forthcoming) by allowing for traded and non-traded goods and hence allowing for a flexible real exchange rate; and by finding optimal paths that start and finish in a steady state (the prior model did not specify an initial or final steady state). In the present model, prospective demographic change is treated as a shock to which the economy adjusts by following a path to a new steady state. For an application of the present model to a world economy consisting of nine regions, see Guest & McDonald (2004) Guest, R. and McDonald, I. M. 2004. The effect of world fertility scenarios on international living standards?. The Economic Record, 48(1): 13–26. [CSA] [Google Scholar].

2. The hypothesis of conditional convergence, which controls for various characteristics of economies, has received stronger empirical support. However, even the testing of this weaker hypothesis faces some serious econometric problems. See Durlauf & Quah in Taylor & Woodford (1999) Taylor, J. B. and Woodford, M., eds. 1999. Handbook of Macroeconomics, Volume 1A, Amsterdam: Elsevier.  [Google Scholar] for a comprehensive discussion of the theory and empirical tests of convergence hypotheses.

3. One form of comparison utility is the ‘outward-looking’ model that we adopt here. The other form is the ‘inward-looking’ model in which consumers compare their consumption with their own past consumption rather than the consumption of others. Both forms of comparison utility generate a type of habit formation in consumption which implies the sort of persistence in consumption that we observe in the data. This is the main motivation for adopting comparison utility in this paper, and from that point of view it does not matter whether we adopt the outward-looking model or the inward-looking model.

4. Elmendorf & Sheiner refer to a capital intensity effect that we call a labour productivity effect because it is perhaps more descriptive for this model (see discussion below).

5. We are referring throughout this discussion to optimal living standards – that is to living standards assuming the optimal response to population ageing is followed.

6. The state of economic development of a country is represented in the model by the country's steady state capital to output ratio. As explained in the Appendix, the developed countries, being Japan, Australia, New Zealand, Singapore, South and North Korea, are assumed to have a value of the steady state capital to output ratio equal to 2.0, while the value for the other countries is equal to 1.5.

7. We follow the assumption in Obstfeld & Rogoff (1996) Obstfeld, M. and Rogoff, K. 1996. Foundations of International Macroeconomics, MIT Press.  [Google Scholar] that only tradable goods can be transformed into capital. They describe this assumption as ‘inessential but helpful’ Obstfeld & Rogoff (1996, p. 204). In our version of their model this assumption allows an analytic solution for l _N and l _T ; otherwise numerical methods would be required.

8. The dependency ratio = (N−L)/L = N/L − 1, where N is the total population and L is the working age population. The constant term (−1) can be omitted in defining μ _t .

9. The intuition for the difference between r and r ^C is as follows. P ^C is a monotonic increasing function of p; and e = P ^N /P ^T . Hence if P falls over time (i.e. P ^C _t / P ^C _{t + 1} is rising), then T goods are becoming more expensive relative to N goods. Therefore, a dollar of expenditure buys fewer traded goods relative to units of the consumption index, than before. Hence, the own interest rate on T goods has to rise to equal a given own interest rate on the consumption index.

10. is the ‘world interest rate’ that adjusts to ensure that world saving equals world investment. The interest rate, r, for each region represents the equilibrium interest rate, which is both the return on saving and the cost of capital (less depreciation). That is, capital importing regions face a higher equilibrium than do capital exporting regions.

11. A steady state rate of labour productivity growth of 1.5 percent implies a rate of total factor productivity growth of approximately 1 percent given the elasticity in the production function.