2.1 The model

Firms in the economy employ labor inputs, L _i, where i = 1,…, M is the age of the labor inputs (workers). The aggregate quantum of labor, L _t, employed in a firm by the M workers is determined according to the CES labor index:

(1)$${\rm} L_t = \left[ {\sum\limits_{i = 1}^M \alpha_iL_{i,t}^\rho} \right]^{1/\rho}, $$

where α _i are the weighting parameters and ρ is the parameter governing the degree of substitution between pairs of L _i,t. (See section 3 for a comment on the assumption that ρ is age-invariant.) The firm determines the aggregate labor input L _t by minimizing total costs given an aggregate production technology. Our concern here, however, is only with the choice of the labor inputs L _i, having determined the optimal aggregate labor input L _t. The firm chooses L _i by minimizing a cost function: $C = \sum\nolimits_{i = 1}^M {w_iL_i}$. From the standard first-order condition for optimal L _i:

(2)$$\displaystyle{{w_{i,t}} \over {w_{\,j,t}}} = \displaystyle{{\partial L_t/\partial L_{i,t}} \over {\partial L_t/\partial L_{\,j,t}}} = \displaystyle{{\alpha _i} \over {\alpha _j}}\left( {\displaystyle{{L_{i,t}} \over {L_{\,j,t}}}} \right)^{\rho -1},\quad i\ne j,$$

where w _i,t is the wage of workers of age i in year t. Since L _t is the index value of all workers who are combining with the workers of age i in year t, then L _i,t/L _t is a measure of the workforce share of L _i,t in year t. For the analysis here, it is important to note that the marginal contribution of L _i,t to the aggregate index L _t depends not only on the size of L _i,t but also on the size of co-worker cohorts. To see this, note that the marginal contribution ∂L _t/∂L _i,t is given by the differentiation of (1):

(3)$$\displaystyle{{\partial L_t} \over {\partial L_{i,t}}} = \alpha _i\left( {\displaystyle{{L_{i,t}} \over {L_t}}} \right)^{\rho -1},\quad i = 1, \ldots, M.$$

This helps explain the result in (2) that the relative wage of workers depends on their relative labor size. Further we can show how the relative wage of a worker of age i responds to a change in its own size and the size of workers of another age, all else constant. The elasticity of the wage ratio, w _i/w _j, with respect to the labor input, L _i is given by:

(4)$$E\left[ {\displaystyle{{w_{i,t}} \over {w_{\,j,t}}},L_{i,t}} \right] = -1 + \rho \lt 0;\quad E\left[ {\displaystyle{{w_{\,j,t}} \over {w_{i,t}}},L_{i,t}} \right] = 1-\rho \gt 0.$$

By (4) increasing L _i will always decrease the relative wages of L _i. Moreover, a larger L _i will always increase the relative wages of the other labor groups L _j. The latter result implies that the effect of cohort size on the relative wages of that cohort depends on the absolute size of co-worker cohorts.

It is also noted that the relative size of a birth cohort depends not only on the sizes of past birth cohorts but on the sizes of future birth cohorts which cannot be known in advance of their birth. Moreover, immigration may change the future sizes of all birth cohorts as they progress through their working ages. In other words, history is not sufficient to determine the relative wage of a cohort according to its size.

The discussion in relation to (2) and (4) has implications for the size of the Easterlin effect, which is represented here as the effect of the relative size of a labor cohort on the discounted lifetime wages of that cohort. The discounted lifetime wages of a cohort entering the labor force at year t is given by

(5)$$W_n = \sum\limits_i^{} {w_{i,t-1 + i}{\lpar {1 + r} \rpar }^{1-i}}, $$

where r is a discount rate. As an example, suppose that a worker is aged i = 1 in the year t = 2005, then the worker's wage at that time will be w _1,2005. In order to calculate the wage level, w _i,t, we specify the aggregate wage bill for the economy:

(6)$$W_tL_t = \sum\limits_i {w_{i,t}L_{i,t}}, $$

where W _t is the wage per unit of the aggregate labor index, L _t, at time t, which is set exogenously at W _t = 1 for all t (this normalization is discussed in section 2.3). Dividing (6) by w _1,t and using (2) gives:

(7)$$w_{1,t} = \displaystyle{{W_tL_t} \over {\left( {L_{1,t} + \sum\nolimits_i {(w_{i,t}/w_{1,t})L_{i,t}}} \right)}},$$

which allows all w _i,t to be calculated by substituting for w ₁ in (2):

(8)$$w_{i,t}{\rm}={\rm } w_{1,t}\displaystyle{{\alpha _i} \over {\alpha _1}}\left( {\displaystyle{{L_i} \over {L_1}}} \right)^{\rho -1}{\rm,} \;i \gt 1.$$

Using this method, we calculate the discounted lifetime wages, W _n, for a worker entering the labor force for years between 2014 and 2100. The aim is to show the effect on W _n of alternative migration and fertility projections.

2.2 Data, calibration, and demographic projections

The number of age groups is M = 11, consisting of 10 five-year age groups, 15–64, and a group aged 65 and over. The values of L _i/L _t for the base year are given in Table 1. The parameters α _i are the labor input weights and are calibrated to the data variables w _i and L _i given the value of ρ which governs the relative degree of substitutability of workers of a given age with other workers. This calibration uses data on relative wages by age, as follows. Re-arranging (2):

(9)$$\displaystyle{{\alpha _i} \over {\alpha _j}} = \displaystyle{{w_{i,t}{\lpar {L_{\,j,t}/L_t} \rpar }^{\rho -1}{\rm}} \over {w_{\,j,t}{\lpar {L_{i,t}/L_t} \rpar }^{\rho -1}}}.$$

Table 1. Productivity weighting parameter values (α _i) by age (i) and rate of labor substitution (ρ) for baseline projection

To determine the values of w _i, we choose the average values of w _i for full-time employees (persons) for the 10-year period 2002–2011 from ABS (2012). For ρ we apply two cases: ρ = 0.5, implying a constant elasticity of substitution by the age of 2.0; (ii) ρ = 0 implying an elasticity of 1.0. The ratios α _i/α _j are then determined given the scaling restriction $\sum\nolimits_{i = 1}^M {\alpha _i = 1}$ which allows α ₁ to be calculated and hence all α _i to be obtained. The choices of ρ and α _i are interdependent in order to ensure that the calibrations are consistent with the known data for w _i,t and L _i,t [Temple (Reference Temple2012)]; hence, a change in one of these parameters implies a change in the other. Table 1 gives the values of α _i for alternative assumptions about ρ and the values of L _i/L and w _i for the base year. Given the values for L _i, α _i, and ρ, (1) is solved for the value of the index, L _t. The optimal values of w _i are then determined for years t = 2,…h from (7) and (8). In calculating discounted lifetime wages, we adopt two reference values: 0% and 5%, which represent a typical range of long run real interest rates for developed countries [Yi and Zhang (Reference Yi and Zhang2016)].

Projections of future values for L _i,t and, hence, L _t are generated by applying age-specific hours worked per person to the projections of the future numbers in the corresponding age group I (assumptions for labor force participation are described below):

(10)$$L_{i,t} = H_{i,t}N_{i,t},$$

where H _i,t denotes the hours worked per person in age group i at time t (i.e., the product of the employment to population ratio for age group i at time t and hours worked per employed person for in age group i at time t) and N _i,t is the projected population in age at time t. The future population numbers are projected using the standard cohort component method [Siegel and Swanson (Reference Siegel and Swanson2004)]. A baseline population projection was prepared using the following set of assumptions:

• • Fertility. All age-specific fertility rates remain constant at 2013 levels. The corresponding total fertility rate (TFR) is 1.89 births per woman [ABS (2014)]. Australia's TFR is higher than that for all more developed countries and its mean age at birth is older [UNPD (2017a)].

• • Mortality. All age–sex specific mortality rates remain constant at 2013 levels. The corresponding life expectancies at birth are 80.3 years for males and for 84.4 years for females [ABS (2015b)]. For both sexes, life expectancy at birth for Australia is among the highest in the world [UNPD (2017a)].

• • Labor force participation. This is calculated as hours worked per person. For both sexes for each age group, between 15–19 and 65+ hours worked per person continues at the average values over the 2010–2014 period [ABS (2015c)]. Average hours worked per person rise steeply between the 15–19 and 25–29 age groups, are broadly similar between the 25–29 and 50–54 age groups, and decrease steeply with increasing age above 55.

• • Net migration. Annual net migration is 200,000 per annum and the percentage age distributions for net migration are based on the average patterns over the period from 2004 to 2014 [ABS (2016)]. Sixty-two percent of net migration is between the ages of 15 and 34. Australia's rate of net migration is one of the highest in the world, and the mean age of its immigrants is somewhat younger than for other more developed countries for which data are available [United Nations (UNDP) (2017), Eurostat (2019)].

The baseline projection is compared to four variant projections:

1. (i) Low fertility, which is the lowest TFR over the past 20 years i.e., 2001 (Figure 2).

2. (ii) High fertility, which is the highest TFR over the past 20 years i.e., 2008 (Figure 2).

3. (iii) Low immigration, defined as 100,000 per annum net immigration, same percentage age–sex shares as the baseline migration.

4. (iv) High immigration, defined as 300,000 per annum net immigration, same percentage age–sex shares as the baseline migration.

Figure 2. Past and assumed future total fertility rates: Australia 1980–2025.

2.3 Cohort size and lifetime wages

First, we discuss the effects of cohort size on age-specific relative wages (2) for the baseline demographic scenario. We focus on the differences between the birth cohorts “1995–2000” (which spans the youngest working age group i.e., 15–19 in 2015), “2000–05” (which does so in 2020), “2010–15” (in 2025), “2020–25” (2040), and “2030–35” (2050). Figure 1 shows that the initial sizes of the 1995–2000 and 2000–05 birth cohorts are relatively small, and Figure 2 shows the dip in the TFRs over these periods (Figure 2). The TFR recovered after 2000–05 and is assumed to remain at 1.89, which is approximately the average of the past 20 years. The 2010–15 cohort is projected to be a significantly larger cohort, reflecting the significantly larger numbers of births between 2010 and 2015 (Figure 1). Subsequent cohorts are projected to be larger still in terms of absolute numbers (but not necessarily as a percentage of the working age population). Figure 3 shows the average working lifetime labour force shares $(l_n = \sum\nolimits_{i = 1}^M {L_{i,n-1 + i}/L_t} )/M$. The figure shows that l _2000–05 is slightly lower than l _1995–2005 and significantly lower than l _2010–15 which in turn is somewhat higher than l _2020–25 and l _2030–35.

Figure 3. Average lifetime share of labor force by cohort (n), baseline demographic scenario.

Turning to the implications of these relative lifetime labour force shares, l _n, for cohort lifetime wages, W _n, an important qualification is the normalisation W _t = 1 in (6). This assumption implies zero productivity-driven aggregate wage growth which would otherwise clearly affect cohort lifetime income and therefore lifetime income inequality. For example, if productivity-driven aggregate wage growth is 1%, cohorts 20 years apart would differ in their lifetime wages by 22% in the absence of any Easterlin effect and all else equal. Given that our focus is on the Easterlin effect and the extent to which it is affected by fertility and migration, aggregate productivity-driven wage growth is assumed to be zero. Given this assumption, the implications of l _n for W _n are shown in Figures 4 and 5 for the baseline demographic projection. More detailed results are discussed in the Appendix which gives tables showing the wages at each age for each cohort under alternative assumptions about the substitution parameter, ρ, and the discount rate, r. Figure 4 shows W _n for a zero discount rate and Figure 5 for a discount rate of 5%. Both figures show W _n for a relatively low (ρ = 0) and high (ρ = 0.5) substitution elasticity. It is important to note from (3) that under perfect labour substitution, ρ = 1, the age-specific wages, w _i, would not change from one cohort to another and nor would the discounted lifetime wage—hence the series in Figures 4 and 5 would be horizontal lines.

Figure 4. Lifetime wage, W[n]. Effect of substitution parameter. Baseline demographic scenario, r = 0.

Figure 5. Lifetime wage, W[n]. Effect of substitution parameter. Baseline demographic scenario, r = 5%.

For the undiscounted case (Figure 4), W _n for n = 1995–2000 is very marginally higher than W _n for n = 2000–05 (by 0.1% for ρ = 0.5 and 0.3% for ρ = 0). The strong similarity of the results for these cohorts reflects the stability in the number of births over the periods when they were (initally) formed (Figure 1). W _n is 1.5% greater for n = 2000–05 than for n = 2010–15 for the higher substitution case (ρ = 0.5) and 3.2% greater for the low substitution case (ρ = 0). Hence, the lower substitution case produces a greater effect on cohort lifetime wages. The differences between the results for n = 2000–05 and n = 2010–15 illustrate the Easterlin effect of imperfect labour substitution on lifetime wages; and the less the degree of substitution, the greater the effect. For the discounted case where r = 5% (Figure 5), W _n for n = 1995–2000 is virtually identical to n = 2000–05 [just 0.02% greater for the higher substitution case (ρ = 0.5) and 0.05% greater for the low substitution case (ρ = 0)]. In turn, n = 2000–05 is 1.6% greater than for n = 2010–15 for the higher substitution case (ρ = 0.5) and 3.2% greater for the low substitution case (ρ = 0). The effect of discounting is therefore small—the effect on W _n on the difference between n = 2000–05 and n = 2010–15 is 0.1 percentage points greater under both elasticity parameters. The effects of l _n on W _n for the other cohorts in Figures 4 and 5 are entirely consistent with the cases discussed for n = 2000–05 and 2010–15. From Figure 3, l _2020–25 and l _2020–25 are somewhat higher than l _1995–2000 and l _2000–05 and the values of W _n are commensurately lower than W _1995–2000 and W _2000–05 (Figures 4 and 5).

In the next section, we discuss the main contribution of this paper, arising from (4), which is to show that the Easterlin effects of l _n on W _n illustrated in Figures 4 and 5 depend on demographic projections for immigration and fertility.

2.4 Migration, fertility, and lifetime wages

2.4.1 Low fertility

Under the low fertility projection, the TFR is constant at 1.73 for all years beyond mid-2015, compared with 1.89 in the baseline projection. The low fertility assumption impacts on the numbers of entrants to the labour force post 2030, and therefore the 1995–2000 and 2000–05 cohorts will be working for most of their working lives with cohorts born during the low fertility period. The effect is that the 1995–2000 and 2000–05 cohorts have smaller successor co-worker cohorts for most of their working lives, and this reduces their marginal contribution to the labour index according to (3) and reduces their wages according to (2), (4) and (5). Hence, the Easterlin small cohort effect is mitigated by the smaller sizes of successor co-worker cohorts. This is illustrated in Figure 6 which shows the average lifetime labour force shares, l _n, for the five demographic projections. In the low fertility projection, the values for l _n for 1995–2000 and 2000–05 are greater than under the baseline projection (by 1.2% and 1.7%, respectively), reflecting the smaller co-worker cohorts under low fertility. The effect is to reduce the lifetime wages, W _n, for the 1995–2000 and 2000–05 cohorts by magnitudes that depend on the degree of labour substitution and the discount rate (Figures 7–10, Table A2). The reductions are smaller for higher labour substitution (ρ = 0.5) and higher discount rate (r = 5%) (Figure 8). The lower substitution case (ρ = 0) doubles the sizes of the effect on W _n for 1995–2000 and 2000–05, as does eliminating the discount rate. The largest effects are 1.8% for the 1995–2000 cohort and 2.5% for the 2000–05 cohort where ρ = 0 and r = 0 (Figure 9). Despite the similarity of the sizes of the two cohorts, the effects of the change to fertility on W _n for the 2000–05 cohort are greater than those on 1995–2000, due to the 5-year longer time the latter spends co-working with the cohorts whose size is reduced by the change in fertility (Figures 7–10). That the effects are proportionately greater with a higher discount rate is linked to the 15-year time lag between the change in fertility and its effect on numbers in the labour force.

Figure 6. Average lifetime share of labor force by cohort (n), various demographic projections.

Figure 7. Lifetime wage, W[n]. Effect of demographic projection. ρ = 0.5, r = 0.

Figure 8. Lifetime wage, W[n]. Effect of demographic projection. ρ = 0.5, r = 5%.

Figure 9. Lifetime wage, W[n]. Effect of demographic projection. ρ = 0, r = 0.

Figure 10. Lifetime wage, W[n]. Effect of demographic projection. ρ = 0, r = 5%.

Whereas under the baseline scenario 2020–25 and 2030–35 are “large cohorts”, under the low fertility scenario their initial sizes are reduced to such an extent that their average lifetime labour force shares are similar to those for the 1995–2000 and 2000–05 cohorts (Figure 6). The effects of low fertility on the share labour force (L _i,t/L_t) are greatest 5–15 years after the entry of these cohorts to the labour force, when their co-worker cohorts are mostly older cohorts whose sizes have not been reduced by low fertility and their labour force participation rate is high. Further in their future, they are working with younger cohorts which, like them, are reduced in size by low fertility, which is why above the age of 45 their L _i,t/L_t becomes higher under low fertility than under baseline (Tables A1 and A2). The similarity of their average lifetime share of labour force under the low fertility and baseline scenarios thus is due to a canceling out of trends with different directions at different ages.

The above analysis shows the effect of low fertility on the absolute levels of W _n. We can also compare the 1995–2000 and 2000–05 cohorts and the 2010–15 and subsequent cohorts under low fertility, and compare this with the gap under the baseline projection. This intergenerational comparison is a comparison of the Easterlin effect under baseline and low fertility. The same process that reduces the size of co-worker cohorts and therefore reduces W _n for the 1995–2000 and 2000–05 cohorts, described above, also reduces the intergenerational gap between these and subsequent cohorts in terms of W _n. For baseline demographic scenarios, W _n for the 1995–2000 and 2000–05 cohorts are, respectively, 3.5% and 3.2% above that for the 2010–15 cohort for the low substitution zero discount case (Figure 9). Under low fertility, these differences are reduced to 1.4% and 0.5%, respectively. Hence, in the low fertility case, the small size advantages of being in the 1995–2000 and 2000–05 cohorts are reduced considerably relative to 2010–15.

The advantage of the 1995–2000 and 2000–05 cohorts relative to the 2020–25 and 2030–35 cohorts is almost entirely eliminated in the low fertility case. There are slight differences in these magnitudes between substitution and discount assumptions (Figures 7–10). The increases to the value of W _n under the low fertility scenario for the 2020–25 and 2030–35 cohorts are greater under r = 5% than under r = 0%. This is because for these cohorts, the effect of low fertility on w _i,t is positive for the lower values of t (and i) when their co-workers are mostly drawn from older cohorts whose sizes are unchanged by low fertility and negative for the higher values of t when their co-workers are mostly from younger cohorts whose sizes also are reduced by low fertility. Indeed, for the higher values of t (and i), w _i,t for the 2020–25 and 2030–35 cohorts are lower under low fertility than under baseline, because of the smaller sizes of their successor cohorts.