# 4 Projecting interest rates and growth rates

## 4.1 Economic growth, the golden rule and dynamic efficiency

Consider first the closed economy version of the neoclassical growth model (Solow,1956; Swan,1956). The growth path of output is determined by the growth rates of the labour force (*n*) and labour-augmenting technical change (*μ*). Labour augmenting technical change increases the productivity of labour, while allowing factor shares to remain constant for a given capital-output ratio. Growth in *effective* labour is the sum of growth in actual labour and the rate at which it is augmented (*n* + *μ*). In steady-state, output per worker and real wages grow at *μ*. Furthermore, capital grows at the same rate as output and so the capital-output ratio is constant. With output and capital growing at the same rate as the effective labour force (i.e., *n* + *μ*), output per effective worker and capital per effective-worker (capital intensity) are constant.[7]

The level of per-worker consumption is maximised at a capital-to-effective labour ratio *k*_{gold} that is determined by *n* + *μ*. This is the so-called “golden rule” of capital accumulation in a closed economy without capital depreciation. At this point the marginal product of capital equals *n* + *μ*, and assuming perfect capital markets, this equals the real interest rate *r*.

Levels of capital intensity above that implied by the golden rule denote a region of dynamic inefficiency. Reducing capital intensity toward *k*_{gold} requires a fall in the (exogenous) saving ratio, which sees the marginal product of capital rise and *r *increase to *n* + *μ*.

If capital intensity is below the golden rule rate, the economy is in the dynamically efficient region. An increase in capital intensity requires capital accumulation via a higher saving ratio. As capital intensity increases toward *k*_{gold}, the marginal product of capital falls, as does *r*, toward *n* + *m*. Based on the above, it is common to see an economy described as dynamically efficient if the real interest rate is equal to, or greater than the real GDP growth rate.

In a closed economy operating in the dynamically efficient region, the decline in the marginal product of capital as adjustment to the golden rule occurs implies a fall in the rate of return on saving. In a small open economy with perfect capital mobility, world capital markets set the real interest rate. The return to saving is set by the exogenous world real interest rate.

Even if the real interest rate is, in the longer run, determined on world capital markets, it will not necessarily be appropriate to assume a constant real interest rate. The chosen profile of the real interest rate may need to reflect worldwide influences, including demographic change. For a small, open economy like New Zealand, the impact of domestic demographics and economic growth on interest rates is likely to be minor. Worldwide demographics are likely to be more relevant.

Population ageing is likely to see a slowdown in labour force growth, slower economic growth and falling returns to domestic capital, creating an incentive to invest offshore. Global capital markets may facilitate intergenerational capital shifts in response to differences in rates of population ageing across regions. In an examination of the various issues, Diamond (1999) concludes that on balance, slower projected economic growth may reduce the return on capital (bonds and equities), but the effect is probably considerably less than one-for-one (for a summary of these issues, see the Appendix in McCulloch and Frances, 2001).

## 4.2 Interest rate and growth rate assumptions used in practice

Long-term fiscal projections generally rely on an assumption of dynamic efficiency, so that *r* > *g. *For example, in their 50-year fiscal projections for major OECD countries, Chand and Jaeger (1996) project GDP growth on the basis of labour-augmenting technical change and labour inputs (where the latter is influenced by demographic change). They assume real interest rates of 3.5%, which are on average around 2 percentage points in excess of the real GDP growth rates.

Both Auerbach (1994, 1997) and the CBO investigate the effects of different interest rate assumptions on the fiscal gap. Auerbach (1994) finds that reducing the differential between *r* and *g* reduces the size of the fiscal gap over time horizons of around 30 and 70 years as the lower cost of debt servicing dominates the calculation. Recall from Section 3 that Auerbach assumes that primary deficits persist beyond 2070. In this case, projected primary deficits over the permanent time horizon tend to dominate the calculation. So, with a lower interest rate, primary deficits in the future, which are larger as a share of GDP, matter more. Over the permanent horizon the fiscal gap increases when the excess of *r* over *g *is reduced.

The CBO’s 10-year fiscal projections assume that the interest rate on government debt exceeds the growth rate of output. The differential is assumed to increase through time as rising fiscal deficits crowd out investment, interest rates rise and real economic growth slows. (The projections assume that when the government holds assets these pay the same average interest rate as government debt.)

The recent OECD study on the fiscal implications of population ageing utilised national projection models with an agreed set of macroeconomic and demographic assumptions (see OECD, 2001; Dang, Antolin and Oxley, 2001). OECD analysis indicates that a range of factors will affect future real interest rates, including growth across regions and saving and investment balances (see Turner, Giorno, De Serres, Vourc’h and Richardson, 1998). Given the high degree of uncertainty around the real interest variable, the cross-country OECD exercise proposed that countries use a (constant) real risk free interest rate of 4% over the period to 2050.[8]

#### Notes

- [7]For details see Wells (1995, Chapter 13) and Barro and Sala-i-Martin (1995, Chapter 1). The LTFM only projects the path of output using labour input projections and exogenous labour productivity growth (see Appendix). See Benge and Wells (2001) for an analysis in the open economy context. Stiroh (1998) provides a useful survey of the growth models used in the fiscal projections of four US government agencies.
- [8]In their reference scenario, Turner et. al. (1998) project a gradual rise in the world real interest rate from 5% to 5.7% by around 2030, followed by a decline to just below 5% by 2100. This rate is calculated as the weighted average of real interest rates (net of any sovereign risk premium) in each region where the weights reflect the share of each region in world output.