Population Aging and Income Inequality in a Semi-Endogenous Growth Model

The model is a continuous-time Blanchard–Yaari perpetual-youth OLG economy grafted onto a simple semi-endogenous growth structure with external increasing returns to aggregate capital. On the production side, a unit continuum of identical firms uses the aggregate technology Y = A·K^α·K̄^γ·N^(1−α), with 0 < α < 1, γ > 0 and α + γ < 1; because the number of firms is normalised to one, K̄ = K, so the social production function Y = A·K^(α+γ)·N^(1−α) “exhibits increasing returns to scale, but the marginal product of capital is diminishing under the assumption of α + γ < 1.” Factor markets are competitive. On the household side, new households are born at size B_t growing at constant rate b, and each household dies with Poisson intensity m, so households are “heterogeneous in terms of their age and wealth holding.” The authors choose the OLG structure to “inspect the distribution effect of demographic change in a tractable manner,” and they allow a negative population growth rate (b < 0) subject to b + m > 0 so total population stays positive.

Population aging is represented by the average age of households, 1/(b+m), which rises when either the birth rate or the mortality rate falls. The stationary tail distribution of age solves a Kolmogorov forward equation and takes the form G(v) = e^(−(b+m)v), so the age density is (b+m)·e^(−(b+m)v) and the average age is exactly 1/(b+m). The paper states the resulting definition directly: “when the birth rate b or the mortality rate m decreases, the average age of households increases. Namely, 1/(b+m) represents the index of population aging in the steady state of population dynamics.” The authors motivate this with data on Japan, where the share aged 65 and over rose from 10% in 1980 to 29.2% in 2020 and is predicted to reach 35% in 2040, and where the total fertility rate fell below 2.0 by 1975 and reached 1.32 in 2020, with the total population declining since 2008.

In the steady-state growth equilibrium, the cumulative distribution of effective wealth, asset holdings and income is Pareto, with shape parameter ξ = (b+m)/(r − ρ − g_y) and support equal to human wealth.* The authors define a household’s “effective wealth” following Moll et al. (2021) as the sum of its growth-adjusted financial assets and human wealth. On the steady-state growth path this effective wealth grows at rate r − ρ − g_y, where r is the net return on capital, ρ the discount rate and g_y the growth rate of per capita income. Solving the Kolmogorov forward equation for the stationary complementary CDF gives a power law, and this is stated as Proposition 2. Because the distributions of financial assets and of income are affine transformations of effective wealth, they “have the same profile as that of the effective wealth.”

The measure is the tail index 1/ξ, the reciprocal of the Pareto shape parameter, and the paper decomposes it as the product of the growth rate of individual wealth and the level of population aging. Written out, 1/ξ = (r − ρ − g_y)/(b+m), i.e. “growth rate of individual wealth × level of population aging.” Substituting the steady-state growth rate gives 1/ξ = [(1−α−γ)(r−ρ) − γb] / [(1−α−γ)(b+m)]. “A larger 1/ξ means that the stationary distribution function has a fatter and longer tail so that inequality of income and wealth distribution rises. Thus, a higher r means a higher degree of inequality. Moreover, given r, a lower b or a lower m increases inequality.”

Because aging works through two channels with opposite signs, and the paper states the sign of the total derivative is “analytically indeterminate.” The indirect channel: “population aging increases the population share of older agents who accumulate larger levels of wealth than younger households. Hence, population aging accelerates the accumulation of aggregate capital, which lowers the steady-state rate of return on capital.” A lower r* shrinks the numerator of 1/ξ and so lowers inequality. The direct channel: a lower b or m raises 1/(b+m), the aging index, which raises inequality. This is stated as Proposition 3: “Population aging lowers the rate of return on capital in the steady-state growth equilibrium. The long-run impact of population aging on inequality depends on the relative strength of its indirect, negative effect on the rate of return on capital and its positive, direct effect on the index of inequality.” The steady-state r* is pinned down by an equation whose left-hand side represents the demand side of capital and right-hand side the supply side, which has a unique solution given ρ + g_y > 0 and m > g_y; a rise in b or m shifts the supply-side graph downward and raises r*.

Under the baseline calibration, longer life expectancy lowers inequality while a lower birth rate raises it — the two components of aging pull in opposite directions. The baseline sets the capital income share α = 0.35, the external effect γ = 0.3, the time discount rate ρ = 0.02 and the depreciation rate δ = 0.075; the authors describe α, ρ and δ as “conventional,” and select γ so that the steady-state growth rate of per capita income is g_y = γb/(1−α−γ) = 0.0174 at b = 0.02. Holding b at 0.02 and varying m from 0 to 0.05, “1/ξ monotonically increases with m,” so a fall in m — longer life expectancy — lowers inequality. Holding m at 0.02 and varying b from −0.01 to 0.05, “1/ξ monotonically decreases with b,” so a fall in b raises inequality. The paper is careful to describe these as holding “under plausible parameter values” rather than universally.

Because in a semi-endogenous growth model the birth rate feeds into the long-run growth rate of per capita income, while the mortality rate does not. As the paper puts it, “a change in the population growth rate affects the steady-state growth rate of income, but changes in the mortality rate do not affect the long-run growth rate of per capita income.” The mechanism for the birth rate is spelled out: “a decrease in b lowers the steady-state level of the rate of return on capital, r*. At the same time, a lower b reduces the growth rate of per capita income; hence, the detrended rate of return, r* − g_y may not decrease significantly or may even increase. Therefore, the degree of inequality may increase with a drop in the birth rate.” The asymmetry is visible in the algebra: the last term in the mortality-rate derivative, γb/(b+m), is positive when the birth rate is positive, whereas the corresponding term in the birth-rate derivative, −γm/(1−α−γ), “is strictly negative.”

Adding exogenous technical progress at rate x lets per capita income grow even with a shrinking population, and the paper finds that a permanent drop in x raises inequality, with the effect larger the more aged the population. With A_t growing at rate x, the steady-state per capita growth rate becomes g_y = (x + γb)/(1−α−γ), so “even if b < 0, the per capita income can grow if x > −γb.” Writing z = r − g_y, the inequality index becomes 1/ξ = (z* − ρ)/(b+m), and a higher x lowers z*, hence lowers 1/ξ, with the derivative scaled by 1/(b+m). This is Proposition 4: “If the exogenous productivity growth rate increases, then the gap between the rate of return on capital and the growth rate of per capita income is lowered, which reduces inequality.” The authors then draw a conditional inference rather than a result: since empirical work such as Daniele et al. (2020) and Maestas et al. (2022) suggests aging depresses productivity growth, “we may conjecture that a lower x may be associated with a larger 1/(b+m), which gives rise to a higher inequality in the steady-state growth equilibrium.”

Allowing households to retire from the labour force at Poisson intensity ς, the paper finds that a lower retirement probability decreases inequality in the stationary equilibrium. Labour supplied by a household born at s is e^(−ς(t−s)), so aggregate labour supply is (b+m)/(b+m+ς) times the population; ς = 0 recovers the baseline, and “a smaller ς means an increase in the labor participation rate.” The authors motivate the exercise by noting that “many people in aging economies tend to postpone their retirement,” and confirm that “a lower probability of retirement decreases inequality in stationary equilibrium.”

With Greenwood–Hercowitz–Huffman preferences, a more elastic labour supply raises long-run inequality. GHH preferences are used because they make labour supply independent of the wealth effect, “which substantially simplifies the analytical discussion”; the authors also cite Ascari and Rankin (2007) to note that under standard preferences with leisure as a normal good, “the labor supply of old agents with large wealth could be negative,” a deficiency GHH avoids. Individual labour supply is n = w^γ, so γ is the labour supply elasticity. A higher γ shifts the capital-supply schedule down and raises r*; the intuition offered is that the GHH term acts “the same role as subsistence consumption in the Stone-Geary utility function,” and this additional consumption “reduces capital accumulation, which yields a higher rate of return on capital in the steady state than the model with a fixed labor supply.” This is Proposition 6. The authors then add a conjecture, clearly flagged as such: since older workers “tend to be sensitive to labor-leisure choices,” aging “may increase the average elasticity of the labor supply function,” which “may enhance inequality in income and wealth distribution.”

The paper places itself in the class of dynamic models with random shocks generating power-law stationary distributions, and identifies its gap as the absence of population aging from all of them. It cites Benhabib et al. (2011, 2016) on OLG models with bequest motives and idiosyncratic income shocks producing a double Pareto wealth distribution; Jones (2014), who obtains a Pareto stationary income distribution in a birth-and-death model motivated by top income inequality in the sense of Piketty (2014); Nirei and Aoki (2016) on idiosyncratic investment shocks; and Moll et al. (2022) on idiosyncratic preference shocks in a study of automation. The paper states flatly: “None of the studies mentioned thus far consider the effect of population aging on income and wealth destitutions [distributions].”

Hiraguchi (2019) is identified as the closest analytically, and the paper lists three specific departures. First, “we employ a semi-endogenous growth model in which the population growth rate affects the steady-state growth rate of income,” whereas Hiraguchi treats a neoclassical growth model with an exogenously specified steady-state growth rate — this is precisely the feature that generates the asymmetry between the birth-rate and mortality-rate results. Second, Hiraguchi focuses on the relationship between the income growth rate, the return on capital and inequality, “whereas our concern is the distributional effect of demographic change.” Third, “we characterize the stationary distributions of income and wealth in a more general and detailed manner.”

The conclusion names three: the simplicity of the growth mechanism, the absence of endogenous demographics, and the absence of policy. “This study uses a simple semi-endogenous growth model in which external increasing returns sustain the persistent growth of per capita income. In this setting, the long-run growth rate of per capita income is proportional to the rate of change in population specified exogenously.” The authors suggest that “introducing the R&D activities of firms and the endogenous population change would enrich the analytical outcomes of this study.” They also note “we have not discussed the policy issues in our study,” and flag “examination of the distributional effects of income tax and intergenerational transfer programs in our model” as deserving further study. Note also that all results characterise stationary distributions in the steady-state growth equilibrium; the paper does not analyse transition paths.