A potential dividend from workforce ageing in Australia.

Abstract

This study explores by simulation analysis the extent of a potential dividend from workforce ageing in Australia. The analysis suggests that the optimum workforce is older than the actual workforce today. Hence, workforce ageing implies that the actual age distribution of the workforce cart be expected to shift closer to the optimum age distribution, generating a dividend in terms of aggregate labour productivity. Simulations suggest that the dividend is likely to be non-trivial in magnitude, although the size of the effect depends very much on tire values of elasticities of substitution about which little is known. Nevertheless, plausible scenarios can be constructed in which the dividend is sufficient to offset significantly the costs of projected declines in the worker to population ratio.

1. Introduction

The costs of population ageing for national economic well being depend, essentially, on how ageing affects national output per capita via two variables: labour productivity and the employment to population ratio which Cutler et al. (1990) called the support ratio. The product of these two variables is simply national output per capita, expressed as follows:

Y/N [equivalent to] Y/L L/N (1)

where Y is national output, N is population and L is aggregate employment. The effect of population ageing on labour productivity (Y/L) is critical because labour productivity growth could potentially offset--indeed swamp --the economic burden that will occur through a falling support ratio. The magnitude and direction of the effect of population ageing on productivity growth remain, however, an elusive question--unresolved in theory and empirically. For a synthesis and critique of the literature, see Chapters 4 to 7 in Birdsall, Kelley and Sinding (2001) and Chapters 1 to 8 in Mason (2001).

Traditional growth accounting can be used to identify three sources of growth in labour productivity (Y/L) (see the Appendix). One factor--technical progress--is a source of long run growth in Y/L; and two other factors affect short run growth of Y/L: investment and the capital deepening effects of labour force growth (a negative effect if labour force growth is positive).

This paper focuses on a potential third source of short run growth in Y/L, which can occur through changes in the age distribution of the labour force. This channel was first explored by Prskawetz and Fent (2004) who provided a simulation analysis for the Austrian economy. The purpose here is to extend their analysis in a particular way and apply it to the Australian economy.

It is of interest to compare the potential magnitude of the effect on output per capital of a changing workforce age distribution with that of increases in labour force participation rates (LFPRs). The latter implies an increase in L/N and therefore an increase in output per capita, from (1). Much policy attention has been directed towards increasing LFPRs, particularly of older workers, in Australia and other ageing countries (see the next section). Hence, as a comparative exercise, simulations are also reported here of the effects of increases in LFPRs on output per capita in the absence of any age distribution effect. Similarly, the age distribution effect is simulated without assuming any increases in LFPRs.

The next section briefly reviews a range of policies being developed in OECD countries to boost the employment of older workers as a way of offsetting the falls in L/N that would occur at current labour force participation rates. Sections 3 explains the theory behind the potential dividend from workforce ageing. Section 4 reports the simulation results, including a comparison of the age distribution effect with the effect of higher LFPRs. Section 5 concludes the paper.

2. Policies to Boost the Support Ratio (L/N)

Population ageing eventually lowers the support ratio because declining fertility and declining mortality ultimately imply a lower ratio of workers to consumers. Assuming no changes in LFPRs and no change in unemployment, the support ratio for Australia would fall steadily to be 15 percent lower by the year 2050 than it is today, and a further 5 percent lower by the year 2100. This would imply commensurately lower output per capita and therefore lower living standards--that is, lower than they would have been with a constant support ratio.

In OECD countries, considerable policy attention is being directed towards increasing LFPRs of older workers and ensuring that there is demand for their labour. (1) This attention to both the supply side and the demand side is important. It would be wasteful if older workers seeking jobs were unable to find them because of discrimination--however subtle it may be--against older workers. Similarly ineffective would be a situation where the benefits of older workers were being increasingly recognised by employers but older workers showed no greater willingness to participate in the labour market.

On the supply side, policies are focusing on superannuation, pension policy and welfare reform; incentives for older workers to enter labour market training programs; and various forms of careers/employment guidance for older workers. Of these, the changes to superannuation and pension arrangements have received the greatest attention. In Australia, bodies that represent superannuation funds such as ASFA and IFSA (2) have called for changes to taxation arrangements affecting superannuation. These include: abolishing the superannuation contributions surcharge (3) that is imposed on the superannuation contributions of high income persons; removing the limits on tax-deductible superannuation contributions; removing the work test on contributions; increasing the age up to which working members of super funds can make personal superannuation contributions; lowering tax rates on superannuation contributions and superannuation fund earnings; and changing the tax rules on income streams by allowing variable income streams, thereby encouraging superannuation funds to invest away from interest-bearing assets to growth assets.

There are reasons, aside from policy drivers, to expect the participation rate of older workers to increase in the coming decades. Day and Dowrick (2004) provide evidence that the decline in fertility since the 1960s has been associated with a substantial increase in female LFPRs. They argue that this will continue--in particular with respect to older women, as the higher education attainments of young women today will result in much higher LFPRs of older women in the future. On that latter point, Dowrick and McDonald (2002) make the following interesting comparison: only 25 percent of the current cohort of 50 year old women completed the final years of high school education, whereas in coming decades nearly 80 percent of 50 year old women will have completed Year 12. They argue that the resulting boost in the LFPRs of older women in the future will be enhanced by the fact that the LFPRs of the current 25-34 year old women are much higher than in the past. The resulting familiarisation with the paid workforce, and the accumulation of human capital which this implies, will lead to a higher LFPR of this cohort as they enter their 50s and beyond compared with the current cohorts of women of those ages.

Policies are also being directed to the demand side of the labour market. An important determinant of labour demand is labour costs per unit of output, which depend on labour productivity relative to labour costs. For given labour costs, increased investment in human capital of workers increases their productivity and therefore increases labour demand. Hence policy attention is being directed largely toward boosting the human capital of older workers. Measures to do this include (see Encel 2003 for further details): career counselling; courses to upgrade skills and assistance with job applications; financial assistance with accredited training courses undertaken by job seekers; personal advisers to provide more intensive individual assistance to unemployed people with special difficulties; and training places in information technology for persons aged over 45.

Other policies to boost demand for older workers include the abolition in 1999 of compulsory retirement in the Australian public service and the introduction of anti-age discrimination legislation.

The rationale for this intensive policy focus on older workers derives from the fact that the duration of unemployment for workers over 45 years is twice the duration for younger workers, even though the rate of unemployment is lower for older workers. Long term unemployment is a more serious problem in social and economic terms than short term unemployment.

Later in the paper simulations are reported which show the potential for these policies to offset the effect of ageing on L/N and therefore on output per capita (see the discussion of Figure 7).

3. A New Dividend from Population Ageing

It is well-known that age-dependent differences in the productivity of workers imply changes in aggregate labour productivity in response to changes in the age distribution of the workforce. There is an additional factor related to the age distribution, however, that is much less known. Prskawetz and Fent (2004) show that the degree of substitutability between workers of different ages is another, potentially important, factor affecting aggregate labour productivity. The idea is that there is an optimum age mix of the workforce which depends on the degree of substitutability of workers and their relative productivity levels. An implication is that demographic change could move the actual age mix either closer to or further from the optimum mix and therefore affect aggregate labour productivity. The magnitude of this age distribution effect depends on how substitutable workers are by age for given relative productivity levels.

Macroeconomic models of population ageing have typically assumed that once workers of different ages are adjusted for their productivity differences they become identical inputs in economic terms. That is, they are infinitely substitutable. The seminal example is Cutler et al (1990); for an Australian application see Guest and McDonald (2001); and for New Zealand see Guest, Bryant and Scobie (2004). The assumption of infinite substitutability is clearly unrealistic, because it does not account for any degree of complementarity between workers of different ages. Examples of complementary age-dependent skills include the physical strength of young workers that complements the skills that older workers have in managing people, including mentoring younger workers, and in making decisions. Such complementarities are typically assumed away in calculating the index of aggregate employment used in modelling the macroeconomic impact of demographic change.

We develop a simple model to illustrate the role of worker age substitutability in determining the aggregate labour input for the economy. The exposition is diagrammatic here; see Appendix B for an algebraic treatment. Consider Figure 1 in which there are only two types of labour: young and old. A 45[degrees] line represents all of the possible combinations of the proportion, [infinity]Y, of young workers, and the proportion, [infinity]O of old workers. The actual distribution must be at some point on this 45[degrees] line, which we will call the age distribution line, with horizontal and vertical intercepts being equal 1. Assume also that these two types of workers operate with no capital; hence we have a labour only economy. (4)

[FIGURE 1 OMITTED]

We consider three cases. First, young and old workers are assumed to be equally productive and perfectly substitutable. In this case the isoquants, representing the locus of workforce distributions that yield equal output per worker, are also 45[degrees] lines and the maximum output per worker that this workforce can produce is represented by an isoquant that is also the 45[degrees] age distribution line. (5) Hence in Figure 1, the points 2005 and 2030 are meant to represent two hypothetical age distributions of the workforce in the year 2005 and 2030, respectively, that generate the same maximum output per worker. In other words, maximum output per worker is completely independent of the workforce age distribution. The workforce could become either older or younger without having any effect on output per worker.

In the second case (Figure 2) we assume that the old workers are more productive than young workers but that they are still perfectly substitutable. In this case the isoquant is drawn with an angle less than 45[degrees] to the horizontal axis as shown in Figure 2. (6) It is still linear, reflecting perfect substitutability between old and young workers. In this case, if the age distribution were to become older between 2005 and 2030, output per worker would increase as represented by an isoquant that is further out from the origin, as shown in Figure 2. In fact, the maximum output per worker would be achieved if all workers were old, in which case the vertical intercept of the isoquant would be at point (0,1) in the figures.

[FIGURE 2 OMITTED]

In the third case (Figure 3) we relax the assumption of perfect substitutability of workers by age. The isoquant is now convex, rather than linear. The maximum output per worker occurs at the point of tangency of the isoquant to the age distribution line. The point at which this occurs depends on the shape of the isoquant which in turn depends on the degree of substitutability between older and younger workers. We have no evidence on the magnitude of such a substitution parameter--there appears to be no literature on this. The approach in the simulations will be to adopt a scenario analysis in which a range of parameter values are assumed. For illustrative purposes, Figure 3 is drawn under the assumption that the tangency point occurs where the workforce is older than it is at points 2005 and 2030. in this hypothetical scenario, aggregate output per worker would increase in the future as the workforce age distribution becomes older.

[FIGURE 3 OMITTED]

Note that, in Figure 3, relative wages of young workers increase as the age distribution becomes older. This is because the ratio of young wages to old wages at any given point on the age distribution line is equal to the slope of the isoquant at that point. That in turn is derived from the profit maximizing condition that the marginal labour costs of producing an extra unit of output are equal for labour inputs of all ages:

[w.sub.Y]/[MPL.sub.Y]=[w.sub.O]/[MPL.sub.O] or, rearranging,

[w.sub.Y]/[w.sub.O]=[MPL.sub.Y]/[MPL.sub.O] (2)

where, respectively, [W.sub.y] and [W.sub.o] are the real wages of young and old workers, and [MPL.sub.y] and [MPL.sub.o] are the marginal products of labour of young and old workers. The ratio of the marginal products--the right hand side of (2)--is the slope of the isoquant. As we move upwards along the age distribution line the slopes of the isoquants increase where they cut the age distribution line. At the optimum distribution, the isoquant is tangential to the age distribution line which, because it is the 45[degrees] line, implies that the marginal products and therefore real wages are equal.

For the general case the optimum age distribution is given in Prskawetz and Fent (2004), citing Lain (1989), as (7)

[[pi].sub.j]/[[pi].sub.i]=[([[alpha].sub.j]/[[alpha].sub.i].sup.[sigma]] (3)

where the [pi]s are the shares of the labour force in age groups denoted by i and j, the [alpha]s are the productivity weights and o is the elasticity of substitution (ES) (8) between labour inputs i and j. If the productivity weights are equal then [[alpha].sub.i]=[[alpha].sub.j] and the optimum age shares are equal no matter what the value of the ES. If, on the other hand, older workers are more productive than younger workers, then their optimum labour force share is higher than for younger workers by a factor given by the ES. The lower the ES, the less variation in optimum labour shares for any given productivity weights. Intuitively, a low ES implies that different types of labour are highly complementary--they cannot be easily substituted. Therefore small adjustments to the type of labour employed cause relatively large changes in marginal products, and hence relatively small adjustments away from equality of labour shares are required to equate marginal products.

The central point that emerges from the above analysis is that the effect of changes in the age distribution of the workforce on average labour productivity depends on the degree of substitutability between workers of different ages.

The simulation analysis in this paper follows broadly the approach in Prskawetz and Fent (2004) and extends it by replacing their 'fuzzy CES' function (9) with a CRESH function (10), special cases of which include the CES, additive and Leonteif functions. The simulations here use Australian data, whereas Prskawetz and Fent (2004) used Austrian data. The results suggest that the age distribution effect could be as important for growth as policies that aim to increase the labour force participation rates (LFPR) of older workers. The effect could offset the effect on output per capita of projected declines in the employment to population ratio associated with population ageing. It would amount to a dividend from population ageing that has hitherto not been recognised.

See Appendix B for further technical analysis and description of the data.

4. Simulation Results

The aim of the simulations is to determine the effect on labour productivity (Y/L) and output per capita (Y/N) of the projected change in the age distribution of the labour force under alternative assumptions about the ES between labour of different ages. The direction and size of this effect depend on whether the age distribution is projected to move closer to the optimum distribution or further away from it, and on the elasticities of substitution.

Four cases are compared:

* An infinite ES between labour of all ages.

* CES=2.

* CES=1 (i.e. Cobb Douglas).

* CRESH (see Appendix B for the way this is parameterised).

Figure 4 illustrates the key result and Figure 5 illustrates the reason for that result. Aggregate labour productivity is relatively constant in the case of infinite elasticity because the lower proportion of younger workers can be replaced, with infinite ease, by older workers. Relaxing this assumption by assuming a finite elasticity leads to increases in aggregate labour productivity over time. The reason is that the age distribution is projected to become closer to the optimum distribution, as illustrated in Figure 5. The resulting effect on labour productivity is therefore described here as the age distribution effect. The magnitude of the age distribution effect depends on the differences between the optimum age distribution and the projected actual distribution. The optimum distribution itself depends on the ES; the higher the ES, the more hump-shaped is the optimum distribution and therefore the closer it is to the projected actual distributions. This is illustrated in Figure 5 by plotting the optimum distribution under two constant values of the ES: ES=1 and ES=2. Therefore the gain in labour productivity over time is greater for CES=1 than for CES=2 (see Figure 4).

[FIGURES 4-5 OMITTED]

As suggested in Section 1, one way to gauge the magnitude of the age distribution effect is to compare its effect on output per capita (Y/N) with that of increases in LFPRs. On current LFPRs, the support ratio would be expected to fall by about 15 per cent over the next 50 years on the basis of the ABS Series B projections used in these simulations. The effect on Y/N is illustrated in Figure 7 by the series 'CES=inf (base LFPR)', which shows that Y/N declines by 15 percent over the next 50 years. To offset this, there would have to be increase in Y/L and/or LFPRs. Figure 4 shows that an increase in Y/L of roughly 15 per cent would be achieved through the age distribution effect in the case of CES=1. The net effect on Y/N is illustrated in Figure 7, which shows that a CES of 1 prevents Y/N from falling below its present level over the next 100 years. Higher elasticities would generate smaller increases in Y/L and therefore a smaller offset of the effect of a declining support ratio on Y/N. See, for example, the series for CES=2 in Figure 7, in which case Y/N finishes up 10 per cent below its current level after about 50 years.

[FIGURE 7 OMITTED]

As suggested above, an increase in LFPRs, especially of older workers, could also prevent a falling support ratio from reducing output per person. It is therefore interesting to compare the effect on output per person of the age distribution effect with that of an increase in the LFPRs of older workers. This question is addressed by running a simulation in which the LFPR of 60-64 and 65-69 year olds both increase gradually over time until they equal the LFPR of 55-59 year olds in 2002. The simulated paths of the LFPR for these age groups are illustrated in Figure 6 for both males and females. The result is that by about 2040 the LFPR of all workers over 55 is the same. This is quite an extreme scenario. For example, it implies that by 2040 three out of four 69 year old males will be in the labour force. (11) This can be compared with the scenario simulated by Gruen and Garbutt (2003), in which the LFPR of all workers increases gradually over the next 20 years to reach the 80th percentile of the distribution of current participation rates across the OECD. Their scenario implies that the LFPR of 60-64 year olds will reach 60 and 40 percent for males and females, respectively, after 20 years. This compares with 70 and 50 percent, simulated here, which is therefore extremely optimistic. The simulated increases in LFPR assume an infinite ES (i.e. equation B6 in Appendix B), since this is the typical assumption implicitly adopted in modelling demographic change as in the Cruen and Garbutt (2003) simulations.

[FIGURE 6 OMITTED]

The effect on Y/N is illustrated in Figure 7 by the series CES=inf (higher LFPR). The gap between this series and the series with base LFPR, labelled CES=inf (higher LFPR) gives the LFPR effect on Y/N. The result is that Y/N falls by only 5 percent rather than 15 per cent. Hence the LFPR effect boosts Y/N in the long run by about 10 per cent. This is equivalent to the age distribution effect for an ES somewhere in between I and 2 (see Figure 7).

The important policy implication is that a positive age distribution effect would take some pressure off the need to raise LFPRs of older workers in order to prevent declining output per capita as a result of population ageing.

5. Conclusion

Higher labour productivity growth and/or higher LFPRs would be required to offset the costs of a falling worker to population ratio on output per capita. Neither of these remedies is easily achieved through government policies. For example, increasing the LFPRs of older workers requires difficult policy changes in the areas of taxation, welfare and retirement incomes.

The analysis in this paper, extending earlier work of Prskawetz and Fent (2004), identifies a potential dividend from the change in the age distribution of the workforce that could raise labour productivity without requiring any change in policy. The fundamental assumption behind this result is that workers of different ages are not perfectly substitutable. The degree of substitutability--or, inversely, complementarity--affects the optimum age mix of the workforce. Simulations presented here suggest that the age mix of the workforce is projected to move closer to the optimum mix, yielding a dividend in terms of higher labour productivity.

The simulations suggest that the magnitude of this dividend is sensitive to the elasticities of substitution between labour of different ages, about which little is known. The size of the dividend could be anywhere from trivial to substantial. However, with elasticities that seem plausible the dividend could be roughly equivalent to the gains that would occur through the largest conceivable increases in the LFPRs of older workers. Put another way, the gains could be sufficient to offset substantially the effect on output per capita of the decline in the worker to population ratio that will occur with no increase in LFPRs. This should not deter policy makers from removing distortions that act as disincentives to labour force participation. That would be good policy irrespective of population ageing. However, the need to do so in order to offset the effect of ageing on output per person could be less pressing than previously thought.

Sensitivity of the magnitude of the dividend to the values of the elasticity parameters clearly suggests that there is a need for empirical work to investigate the magnitudes of these elasticities. An initial step might be to undertake a microeconometric study of the relationship between firms' performances and the age distributions of their workforces, using cross-section or panel data, controlling for as many other factors as possible. The aim would be to determine whether some age distributions are more likely to be associated with higher firm performance than others--in other words, whether the age distribution of the workforce matters. The next step might be to attempt to estimate production functions where the labour variable is disaggregated by age group. For example, one could attempt to estimate the parameters of a CRESH function of capital and labour inputs by age. The author plans to attempt such empirical work.

Appendix A

This Appendix derives the four sources of growth in per capita output referred to in the introduction. See Mason (2001, p.224) for a similar derivation.

We define the following variables and parameters: Y is output, N is population, L is employment, A is an exogenous technology parameter, i is the output share of investment net of depreciation, [infinity] is the output elasticity of capital. Let [y.sub.N]=Y/N, [y.sub.L]=Y/L, [y.sub.K]=Y/K, k=K/L, and let '^' denote growth rate.

We start with the following decomposition of output per capita:

Y/N=(Y/L) (Y/L) and therefore

[[??].sub.N] =[[??].sub.L] + [??] - [??]. (A1)

Assume a Cobb-Douglas production function for simplicity without affecting the qualitative result:

Y/L = [A(K/L).sup.[alpha]] and therefore

[[??].sub.L] = [??] + [[alpha][??]. (A2)

By definition:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (A3)

Substituting (A3) into (A2):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (A4)

Equation (A4) shows that growth of labour productivity is a function of three variables: technical progress, investment, and the effect of labour force growth on capital deepening.

Substituting (A4) into (A1):

[y.sub.N] = [??] + [[alpha]iy.sub.K] - [alpha][??] + ([??] - [??]) (A5)

Equation (A5) shows that growth of output per capita is a function of the three variables on the right hand side of (A4) plus the rate of change in the employment to population ratio.

Appendix B

This Appendix provides further technical analysis of the model used for the simulations.

Following Prskawetz and Fent (2004), we assume that output is produced by labour only and that there is no technical progress. Hence output is simply an index of labour inputs. The fuzzy CES function in Prskawetz and Fent (2004) is replaced here by a CRESH function of labour inputs distinguished by age:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (B1)

where [[alpha].sub.i] is the productivity weight of labour of age i, k is the number of age groups, [L.sub.i] is the number of workers of age i, Y is the index of composite labour inputs and [[rho].sub.i] is a parameter that represents the flexibility, or versatility, of [L.sub.i], meaning the degree to which [L.sub.i] can substitute for any other input, [L.sub.j]. We assume here that all labour inputs are substitutes to some degree, which restricts [[rho].sub.i] such that -[infinity]<[[rho].sub.i] <1. The larger the absolute value of [rho] the more easily [L.sub.i] is substitutable for any other labour input with a given value of [[rho].sub.j]. This implies that two labour inputs with high absolute values of [[rho].sub.i] will be good substitutes and two inputs with low absolute values of [[rho].sub.i] will be poor substitutes.

The elasticity of substitution (12) (ES) between [L.sub.i] and [L.sub.j], which we will define as [[sigma].sub.j], is given by Hanock (1971, p.699):

[[sigma].sub.ij] = [a.sub.i] [a.sub.j]/[k.summation over (m = 1)] [s.sub.m][a.sub.m]

where [a.sub.i] = 1/1 - [[rho].sub.i] and [s.sub.m] is the factor share of [L.sub.i].

Restrictions exist on the range of values of the % that yield a unique solution for the CRESH function (Hanoch 1971). The binding restriction in the present application is that [[alpha].sub.i][[rho].sub.i] must be of the same sign for all i, assuming that all [a.sub.i]>0 which implies that all labour inputs are substitutes to some degree.

Given all [[alpha].sub.i]>0 by definition, we must have for all i, either 0<[[rho].sub.i] <1 ([a.sub.i]>1) or [[rho].sub.i] <0 (0<[a.sub.i]<1).

Returns to scale are assumed here to be constant, which implies f(Y)=Y. If [[rho].sub.i]=[rho] then (B1) becomes CES:

MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (B3)

As [rho] in (B3) approaches zero we have the Cobb Douglas function in which the ES=1:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (B4)

As [rho] in (B3) approaches [infinity], the ES approaches zero, in which case the function is of the Leonteif form:

Y = [min.sub.i=1,..,k] L/[a.sub.i] (B5)

where the [a.sub.i] are constants.

Finally, if [rho] in (B3) is equal to 1, we have the additive function for which the ES=[infinity]:

Y = [k.summation over (i = 1)] [[alpha].sub.i][L.sub.i] (B6)

In (B6) the index of labour inputs is a sum of labour inputs by age, weighted by age-dependent productivity levels. The implication of (B6) is that effective units of labour of different ages, [a.sub.i][L.sub.i], are infinitely substitutable. In other words, once workers of different ages are adjusted for their productivity differences they become identical inputs in economic terms.

Despite this shortcoming of (B6), it is the typical functional form that has been commonly adopted to define the labour index in macroeconomic models applied to modelling demographic change. The seminal example is Cutler et al (1990); for an Australian application see Guest and McDonald (2001); and for New Zealand see Guest, Bryant and Scobie (2004). The production functions in these models include capital. Here capital is excluded for simplicity.

Other Assumptions and Data

In applying the CRESH function we assume that the degree of flexibility of workers varies with their age. In particular, we assume that middle age workers are more flexible than either young workers or older workers. The degree of flexibility is a hump shape function of age, rising to middle age then falling to old age. The intuition for this is that middle age workers, defined here as workers in the 35 to 54 age group, will be more substitutable for young workers than will older workers; and will also be more substitutable for older workers than will younger workers.

Table B1 gives the matrix of values of the ES in (B2) that are used in the simulations. These values were chosen arbitrarily, subject to the restrictions on the parameters mentioned above and such that the resulting values for the ES are in the range of values for ES that are commonly used in applications of CES functions. The denominator in (B2) is equal to 2, the value based on the data applied in the simulations. The hump shape of [a.sub.i] is evident, as is the intuitively appealing result, described above, that workers closer in age to the versatile middle age groups have higher ES.

It would also have been desirable to parameterise the CRESH function such that workers in adjacent age groups are better substitutes than are workers further away in age. Unfortunately, the hump-shaped pattern violates this property. This is clear from Table B1. For example, the ES between 15-19 year olds and 20-24 year olds is lower than that between 15-19 year olds and 35-44 year olds. This is a shortcoming of the hump-shape profile of [a.sub.i] illustrated here and used in the simulations.

This shortcoming could have been overcome by assuming a declining pattern of [a.sub.i]. The advantage of such a pattern is that the ES between labour of a given age, i, and labour of any other age, j, becomes less the further away j is from i. That is, [[sigma].sub.ij] > [[sigma].sub.ik] for all k>j. (13) The problem with this pattern is the implicit assumption that the youngest workers are the most versatile and substitute more easily for other workers than do any other age group. This implies for example that 15-19 and 20-24 year olds substitute more easily than 20-24 and 25-34 year olds, and so on, such that the ES between adjacent groups declines monotonically as the ages of the adjacent groups increase. Other patterns of [a.sub.i] would yield alternative matrices of ES.

Finally, the productivity weights, [a.sub.i], are calibrated as follows. Males and females are distinguished in terms of their age-dependent productivity weights but their productivity adjusted labour units are assumed to be infinitely substitutable at any given age. Hence [([alpha]L).sub.i]=[([alpha]L).sub.males]+[([alpha]L).sub.i,females] and therefore [[alpha].sub.i] is a weighted average of the productivity weights for males and females: (14) [[alpha].sub.i] = [([alpha]L).sub.i,males] + [([alpha]L).sub.i,females]/[L.sub.i].

In the simulations, the productivity weights by sex were proxied by the age distribution of full-time average weekly earnings by sex at August 2003, obtained from ABS Catalogue 6310.0. The labour force (in persons, not hours worked) and the labour force participation rate by age and sex were obtained from ABS Cat 6291.055.001. The population projections by age and sex, 2002-2100, were obtained from ABS Cat 3222.0, Series B. Of the three series of population projections published by the ABS, Series B is based on assumptions for international migration, fertility and mortality which yield population levels that are in between those of Series A and C.

References

Birdsall, N., Kelley, A. and Sindig, W. (eds.) (2001), Economic Growth and Poverty in the Developing World, Oxford University Press.

Blackorby, C. and Russell, R. (1981), 'The Morishima Elasticity of Substitution; Symmetry, Constancy, Separability, and its Relationship to the Hicks and Allen Elasticities', Review of Economic Studies, vol 48, pp. 147-158.

Cutler, D.M., Poterba, J.M., Sheiner, L.M. and Summers, L.H. (1990), 'An Aging Society: Opportunity or Challenge?', Brookings Papers on Economic Activity, vol 1, pp.1-74.

Day, C. and Dowrick, S. (2004), 'Ageing Economics: Human Capital, Productivity and Fertility', Agenda, vol 11, pp. 3-20.

Dowrick, S. and McDonald, P. (2002), 'Comments on the Intergenerational Report, 2002-03', mimeo, Australian National University.

Encel, S. (2003). Age Can Work, Report to ACTU and Business Council of Australia.

Gruen, D. and Garbutt, M. (2003), 'The Output Implications of Higher Labour Force Participation', Treasury Working Paper, 2003-2. Available at: http://www.treasury.gov.au/.

Guest, R. and McDonald, I.M. (2001), 'Ageing, Immigration and Optimal National Saving in Australia', Economic Record, vol 77, pp. 117-134.

Guest, R. and McDonald, I. (2004), 'The Effect of World Fertility Scenarios on International Living Standards?', Economic Record, vol 48, pp.13-26.

Guest, R., Bryant, J. and Scobie, G. (2004, forthcoming), 'Population Ageing in New Zealand: Implications for Living Standards and the Optimal Rate of Saving', New Zealand Economic Papers.

Hanoch, G. (1971), 'CRESH Production Functions', Econometrica, vol 39, pp. 695-712.

Lam, D. (1989), 'Population Growth, Age Structure, and Age-Specific Productivity', Journal of Population Economics, vol. 2, pp. 189-210.

Mason, A. (2001), 'Population, Capital and Labor', in Mason, A.(ed.), Population Change and Economic Development in Asia: Challenges Met, Opportunities Seized, Stanford University Press, pp. 209-230.

Prskawetz, A. and Fent, T. (2004), Workforce Ageing and Economic Productivity: The Role of Supply and Demand for Labour: An Application to Austria, Vienna Institute of Demography, Austrian Academy of Sciences.

Ross Guest, Griffith University 

Endnotes

(1) The OECD has published a series of country studies called Ageing and Employment Policies, covering about 20 countries, in which it explains the policy initiatives that have been taken, and are planned to be taken, to boost the employment of older workers. The series is unfinished at the time of writing (Feb, 2005). See http://www.oecd.org/findDocument/0,2350,en_2649_34747_1_119699_1_1_37435,0 0.html

(2) ASFA is the Association of Superannuation Funds of Australia and IFSA is the Investment and Financial Services Association Limited. See for example IFSA (2002).

(3) The tax is being reduced from 15 per cent to 10 per cent over 3 years and applies to income over $114,981 in the 2003-04 year. A lower surcharge applies to income over $94,691 in the 2003-04 year.

(4) This simplifies the exposition. Capital could be introduced without invalidating the underlying principle.

(5) For this to be true, we must assume constant returns to scale and appropriately define the units of output.

(6) The dotted segments represent infeasible age distributions because the proportions sum to greater than 1.

(7) This derivation requires the assumption, adopted throughout the analysis, that workers are paid their marginal products. In that case the marginal rate of substitution is equal to the relative prices of labour i.e.

[w.sub.i]/[w.sub.j] = [[alpha].sub.i]/[[alpha].sub.j][([[pi.sub.j]/[[pi].sub.i]).sup.I/[alpha].

Equality of marginal products implies [w.sub.j]=[w.sub.i] and therefore

[[pi].sub.j]/[[pi].sub.i] = [([alpha].sub.j]/[[alpha].sub.i].sup.[alpha].

(8) The ES is a measure of the degree of curvature of the isoquant in Figure 3. See Appendix 13 for further discussion.

(9) CRESH stands for constant ratio of elasticity, homogeneous. The CRESH function was introduced by Hanoch (1971).

(10) CES stands for constant elasticity of substitution (see Appendix B for further discussion).

(11) The baseline assumption that the LFPR of workers 70 and over is zero is maintained. That is, the LFPR of workers over 65 reported by the ABS is assumed to apply to only to workers aged 65-69. This implies some downward bias in the employment of older workers in the simulations because the LFPR of workers 70 years and over is positive although smaller than that of 65-69 year olds.

(12) ES refers here to the Allen-Uzawa pairwise elasticity of substitution, which is the n-factor analogue of the two-factor Hicks ES of substitution. The former Allen-Uzawa elasticity is equal to the Miroshima elasticity in the case where the technology is CES and therefore all the as are equal (Blackorby and Russell, 1981).

(13) Ideally we would impose the less restrictive condition that [[sigma].sub.ij] > [[sigma].sub.km] for all i-j>k-m, but it is not possible to find a unique set of values of [[rho].sub.i] that satisfies this condition.

(14) It is of course possible to treat females as a separate labour group with age dependent ES, but this creates additional anomalies of the kind illustrated in Table B1. The assumption of infinite elasticities between males and females of any given age is preferred in order to avoid these anomalies.

Table B1: ES for CRESH function for hump-shaped pattern of [a.sub.i]

i          [a.sub.i]       [[sigma].sub.ij]   [[sigma].sub.ij]
                                 i=15              i=20-
                                 -19                 24

15-19         1.00
20-24         1.50               0.75
25-34         2.00               1.00               1.50
35-44         3.00               1.50               2.25
45-54         3.00               1.50               2.25
55-59         2.00               1.00               1.50
60-64         1.50               0.75               1.13
65-69         1.00               0.50               0.75

i       [[sigma].sub.ij]   [[sigma].sub.ij]   [[sigma].sub.ij]
             i=25-              i=35-              i=45-
               34                 44                 54

15-19
20-24
25-34
35-44         3.00
45-54         3.00               4.50
55-59         2.00               3.00               3.00
60-64         1.50               2.25               2.25
65-69         1.00               1.50               1.50

i       [[sigma].sub.ij]   [[sigma].sub.ij]
             i=55-              i=60-
               59                 64

15-19
20-24
25-34
35-44
45-54
55-59
60-64         1.50
65-69         1.00               0.75