From Population Growth to TFP Growth

Layer 1: Overview

This paper asks how the well-documented slowdown in labor-force growth affects aggregate total factor productivity (TFP) growth, a question that prior work on business dynamism had left unanswered. The authors build a general-equilibrium business-dynamics model that embeds two engines of productivity growth: innovation by young entrants (a step-size improvement over the leading-productivity frontier, in the spirit of Romer 1990 and Aghion-Howitt 1992) and steady productivity growth by mature leading businesses. Population (labor-force) growth determines the demographic composition of the business stock, because the number of firms must grow in proportion to the labor force along any balanced growth path (BGP). A slower labor force therefore shifts the firm distribution toward older incumbents.

The paper’s central theoretical result is a “sufficient statistic” for whether slower population growth reduces TFP growth: the employment-size growth rate of surviving old businesses, which converges to the ratio gS/gX (the productivity growth of leading businesses divided by average economy-wide productivity growth). If gS/gX < 1 — i.e., old firms’ productivity grows more slowly than the economy average — then a lower labor-force growth rate raises the share of old firms and drags down aggregate productivity growth. Both the sign and the magnitude of the effect are characterized in closed form.

The model is calibrated to U.S. and Japanese establishment data (Business Dynamics Statistics; Economic Census and Establishment/Enterprise Census), targeting the life-cycle profiles of exit rates, average employment size by age, and the employment growth rate of surviving businesses, with the reference period 1980–1999. The U.S. labor-force growth rate used in calibration is 1.67 percent per year (average 1980–1999); Japan’s is 0.72 percent. A key calibrated quantity is gS: 1.060 for the U.S. and 1.030 for Japan, reflecting the faster decline in the size of surviving old establishments in Japan relative to the U.S. The benchmark model adds entry congestion (parameter ϕ = 0.55, taken from Karahan, Pugsley and Sahin 2024) and spillovers from young to old firms’ productivity growth (γ = 0.342, estimated from BDS data using venture capital investment as an IV).

Main quantitative findings across BGPs: In the U.S., the projected decline in labor-force growth from approximately 2.59 percent (1970–1980) to 0.26 percent (2050–2060) implies a long-run reduction in TFP growth of approximately 0.3 percentage points. In Japan, the decline from approximately 1.86 percent (1950–1960) to −0.97 percent (2050–2060) — a drop of more than 3 percentage points — implies a long-run reduction in TFP growth of approximately 0.6 percentage points. These effects are substantially attenuated when congestion and spillovers are removed: the U.S. effect falls from 0.30 to 0.19 percentage points and the Japan effect falls from 0.63 to 0.41 percentage points in the simplest model, so roughly 65 percent of the benchmark effect is attributable to the core mechanism alone.

For the transition analysis, the model accounts for approximately 49.7 percent of the observed U.S. TFP growth slowdown between 1980–1999 and 2000–2019 (an observed decline of 0.184 percentage points, model-explained 0.091 percentage points). In Japan, the model explains approximately 24.2 percent of a larger observed slowdown of 0.451 percentage points (model: 0.109 pp). A critical feature of the dynamics is that TFP growth responds sluggishly to population growth changes. Two transitional counterbalancing forces explain this: (1) a “level-vs-growth” effect — on impact, a higher share of older (larger and more productive) firms temporarily raises productivity growth in levels even while it lowers the growth rate in the long run; and (2) a “labor-reallocation” effect — fewer entrants means less labor in the innovation sector and more in production, temporarily raising the production-sector labor share and boosting measured TFP growth. Both effects fade as the economy converges to the new BGP.

Looking forward, the expected further decline in TFP growth from population aging is -0.05 to -0.06 percentage points for the U.S. between 2020 and 2100 (benchmark, without incorporating forecasts), and -0.14 to -0.17 percentage points for Japan over the same horizon. When BLS/CAO forecasts for labor-force growth through 2060 are incorporated, these magnitudes rise to -0.07 to -0.08 pp (U.S.) and -0.24 to -0.34 pp (Japan) between 2020 and 2100. Cross-sectional IV regressions using lagged state birth rates as instruments confirm that a 1-percentage-point change in labor-force growth maps to approximately a 0.1 to 0.2 percentage-point change in labor productivity growth across U.S. states, consistent with model predictions. Local projections using U.S. state data 1977–2019 show that the dynamic pattern in data (initial positive then negative response of productivity growth to a labor-force shock) mirrors the model’s transitional dynamics closely.

Layer 2: Deep Dive

What is the paper’s core theoretical result, and what is the ‘sufficient statistic’?

The main result (Lemma 4) states that if the employment-size growth rate of surviving old businesses is negative — equivalently, if gS/gX < 1 — then an increase in the labor-force growth rate raises average productivity growth, and vice versa. The ‘sufficient statistic’ is gS/gX, the ratio of old-firm productivity growth to economy-wide average productivity growth. This ratio asymptotically equals the employment growth rate of surviving old firms in a BGP (Lemma 3). Lemma 5 further shows that the magnitude of the effect is increasing in how fast old firms’ size shrinks, i.e., larger when gS/gX is further below 1. This means the calibration of the life-cycle profile of surviving business growth is the decisive input for the quantitative results.

What are the two growth engines in the model and how do they interact?

The first engine is innovation by new entrants: innovators choose a step size g relative to the average leading-firm productivity frontier χ, paying convex research costs. The free-entry condition ties the step size to structural parameters (research cost slope and entry cost), making g* constant in equilibrium. The second engine is the exogenous (in the benchmark) or endogenous (in extensions) productivity growth of leading businesses at rate gS per period. Both engines operate simultaneously: gX is determined by a weighted average of these two sources, where the weight on the old-firm engine equals their share in the firm distribution. Population growth affects this weight by determining the number of new entrants relative to incumbents.

What identification strategy is used in the empirical validation and what are the threats?

Two empirical strategies are used. First, local projections (Jordà 2005) using U.S. state-level data 1977–2019 regress the change in labor productivity growth over horizons i = 0 to 8 years on the change in labor-force growth, controlling for seven lags of each variable and a quadratic time polynomial. This establishes that the dynamic pattern in the data mirrors the model-predicted non-monotonic response (initial positive effect, then negative and significant effects at 2–5 years). Second, cross-sectional IV regressions for U.S. states average 2004–2024 data and use the lagged state birth rate (pushed back 20 years) as an instrument for labor-force growth, with controls for initial GDP per capita and state population. The main threat is reverse causality: workers may relocate to states with higher expected productivity growth. The authors note the IV addresses this by using birth rates from 20 years prior. A further threat acknowledged is knowledge spillovers across states, which would bias the local-projection coefficient downward.

What does the paper say about the role of entry congestion and innovation spillovers?

Entry congestion modifies the free-entry condition to make entry costs rise with the ratio of entrants to population (with elasticity ϕ = 0.55). This means that when population growth slows and fewer entrants arrive, entry costs fall, which discourages innovation intensity (lower g*), adding a second channel through which slower population growth lowers TFP growth. Innovation spillovers allow the productivity growth of leading businesses (gS) to respond positively to lagged aggregate productivity growth (with elasticity γ = 0.342, estimated via IV). When population growth slows and productivity growth falls, spillovers to incumbents also fall, amplifying the total effect. Together, these features explain roughly 35 percent of the benchmark effect beyond what the core mechanism delivers alone: the U.S. effect rises from 0.19 pp (no congestion, no spillovers) to 0.30 pp in the benchmark.

What are the robustness checks on the BGP results?

Five alternative productivity processes are considered. Case 1 is a standard two-state AR(1), Case 2 allows transition probabilities to depend on age, Case 3 uses deterministic productivity growth by type (high and low) with age-dependent transitions, Case 4 is the benchmark (asymmetric absorbing high-productivity state with tenure-dependent productivity history), and Case 5 cuts the productivity jump θ in half. All five deliver similar qualitative results, with the long-run U.S. effect ranging from -0.15 to -0.22 percentage points compared to -0.19 in the benchmark. The AR(1) specification (Case 1) yields the smallest effect because it misses the growth of young and old businesses in the data. Endogenous exit is examined in a separate extension: the exit rate declines further when population growth falls (amplifying the old-firm share effect), but this is nearly exactly offset by higher innovation incentives from longer business horizons, resulting in very small net change. Endogenous innovation by leading businesses is also explored and found to amplify the result at low population growth rates (making the effect nonlinear and potentially larger in future decades), but its impact at observed historical ranges is modest.

How do the transitional dynamics differ from the BGP comparison, and why?

The BGP comparison provides the long-run effect of a permanently different population growth rate on TFP growth. The transition shows that convergence to this new BGP is very slow — taking more than 20 years to reach the new steady-state share of young businesses after a step decline in population growth. This slowness is driven by two counterbalancing forces. The level-vs-growth effect: on impact, a lower entry rate raises the share of larger, more productive older firms, which temporarily boosts the level of productivity growth even as the long-run growth rate falls (because young firms have lower productivity levels despite faster productivity growth). The labor-reallocation effect: fewer entrants mean less labor in the innovation sector, reallocating workers to production, which temporarily raises the production-employment share and therefore measured TFP growth. As a result, the model accounts for 49.7 percent of the U.S. TFP growth slowdown between 1980–1999 and 2000–2019, not the full long-run 0.30 pp effect. The sensitivity analysis shows that lower sS, lower β, or higher gS all speed up convergence.

How does this paper relate to Karahan, Pugsley and Sahin (2024) and Hopenhayn, Neira and Singhania (2022)?

Both prior papers show that slower labor-force growth reduces business dynamism by generating a startup deficit and shifting the firm age distribution toward older incumbents. They share the basic Hopenhayn (1992) firm-dynamics structure with this paper. The key distinction is that those papers focus on entry rates, exit rates, employment concentration, and labor market dynamics as outcomes, whereas Inokuma and Sanchez focus on TFP growth. As a validation exercise, this paper shows its model also reproduces the decline in U.S. business dynamism (entry rate, exit rate, share of young establishments) when fed the trend in labor-force growth.

How does this paper relate to Peters and Walsh (2022)?

Peters and Walsh (2022) also studies population growth and productivity. Their framework builds on Klette and Kortum (2004) and emphasizes scale effects, variety expansion, market concentration, and markups, abstracting from firm life-cycle dynamics. This paper instead builds on Hopenhayn (1992) and focuses on how innovation intensity varies with firm age. The two mechanisms are complementary: the life-cycle mechanism in this paper would add 56 percent to the productivity growth decline found in Peters and Walsh (Peters and Walsh find approximately 0.23 pp per 1 pp decline in population growth, almost all from varieties; Inokuma and Sanchez find 0.13 pp per 1 pp for the U.S., so the combined effect would be roughly 0.36 pp).

What heterogeneity is documented in the paper?

The most important heterogeneity is between the U.S. and Japan. Japan’s establishments exhibit a much flatter size profile by age (the ratio of employment in establishments 29+ years to age-1 establishments is 1.5 in Japan versus 3.5 in the U.S.) and a sharper decline in the size of surviving old establishments, yielding a calibrated gS of 1.030 for Japan versus 1.060 for the U.S. This implies a larger sufficient statistic |1 - gS/gX| for Japan and therefore a larger elasticity of TFP growth to population growth: 0.6 pp effect for Japan versus 0.3 pp for the U.S. over their respective projected population growth declines. Within the model, the two types of firms (laggard and leading) have different survival rates (sS > sU), different productivity levels (leading firms are roughly 200 vs 10 employees on average), and different exit dynamics (laggards face much higher exit rates, especially when young).

What are the policy implications and their scope conditions?

The paper does not focus on policy prescriptions, but the implied lesson is that policies affecting the entry rate of new firms — or the productivity life-cycle of mature incumbents — are the primary levers for mitigating the TFP drag from aging populations. Because the effect operates through firm-age composition, any policy that encourages new business formation (lowering entry costs, relaxing congestion) would partially offset the demographic headwind. The scope conditions are important: the main result holds under a perfectly elastic supply of new businesses, constant entrant innovation intensity, and exogenous survival/productivity profiles. Congestion and spillovers amplify the mechanism. When exit is endogenous, competing forces nearly cancel, so the result is robust. The direction of the effect depends critically on gS < gX (i.e., old firms’ productivity growing more slowly than average), which is empirically verified for both the U.S. and Japan. If the sufficient statistic were positive (gS > gX), slower population growth would raise TFP growth.

What does the paper say about scale effects and how they interact with the life-cycle mechanism?

In a CES variety model (as in Peters and Walsh 2022), gTFP = g_tilde_X + (1/(sigma-1)) * gN, adding a direct scale effect where slower population growth reduces the number of varieties and TFP directly. Calibrating sigma = 4 (consistent with Jones 2022), this implies a 0.33 pp TFP decline per 1 pp population growth decline from the variety channel. The life-cycle mechanism in this paper adds 0.13 pp for the U.S. and 0.22 pp for Japan per 1 pp decline. Thus the two mechanisms together would imply a 0.46 to 0.55 pp decline per 1 pp of population growth slowdown — 30 to 60 percent larger than the variety channel alone.

What is the ’level-vs-growth’ effect and how does it arise?

When population growth slows suddenly, the entry rate falls and fewer young firms enter. This means the firm pool immediately becomes more skewed toward older, larger, more productive incumbents. On impact, this raises the average level of productivity in the economy (because old firms have higher levels, even if slower growth rates). This temporarily boosts the growth rate of average productivity in the short run, even though in the long run the effect is to lower TFP growth (because old firms’ productivity growth rate gS is below gX). This transient positive effect on TFP growth counterbalances and delays the long-run decline, contributing to the sluggish response.

What role does the discount factor and household preferences play in the results?

The household problem involves standard intertemporal optimization with risk aversion ε = 2 and discount factor β = 0.96. These parameters enter the speed of convergence in the transition: lower β increases the speed of convergence (sensitivity analysis shows β has an elasticity of -4.212 for convergence speed). Along the BGP, household preferences determine the interest rate through the Euler equation and affect the capital share α-tilde, which varies across BGPs. The paper notes that d(alpha-tilde)/d(gM) is likely negative, meaning that lower population growth also reduces the capital share, amplifying the effect on TFP growth, though extreme parameter values could reverse this.

What are the data sources and what moments are targeted in calibration?

For the U.S.: establishment-level data from the Business Dynamics Statistics (BDS), spanning 1978 onwards; labor force data from BLS Current Population Survey (1949–2019) and Lebergott (1966) for 1900–1948; TFP from Penn World Table 10.0; venture capital investment from PwC/CB Insights MoneyTree. For Japan: establishment data from the Establishment and Enterprise Census (1981–2006) and Economic Census (2009–2021); labor force from Statistics Bureau of Japan; TFP from PWT 10.0. Calibration targets 32 moments for the U.S. (31 life-cycle bars plus average productivity growth) and 20 for Japan. The targeted moments are the exit rate by establishment age (with equal weighting), the average employment size profile by age, and the growth rate of surviving establishments by age. Ten parameters are jointly estimated to minimize the distance between model-implied and data moments.

Key Concepts

Sufficient statistic (gS/gX): The employment-size growth rate of surviving old businesses, which asymptotically equals the ratio of old-firm productivity growth (gS) to economy-wide average productivity growth (gX). This single ratio determines both the sign (if less than 1, slower population growth reduces TFP growth) and the magnitude (the faster gS/gX falls below 1, the larger the effect) of population growth’s impact on productivity growth along balanced growth paths.

Leading versus laggard businesses: The paper’s two-type firm classification. Laggard businesses start with productivity θ·χ·g (below the frontier), grow at a flat rate, and face high exit rates; they can transition to the leading group with age-dependent probability λ_a. Leading businesses begin at or above the frontier (productivity χ·g at entry), grow at constant rate gS per period, and face lower exit rates. The share of leading versus laggard firms — and the speed at which laggards transition — determines the life-cycle productivity profile that is central to the sufficient statistic.

Level-vs-growth effect: A transitional counterbalancing force: when population growth slows, fewer young (small, low-productivity-level) firms enter, immediately raising the average level of productivity in the firm pool and temporarily boosting measured productivity growth, even though the long-run effect is negative. The short-run level gain outweighs the long-run growth-rate loss, delaying the TFP growth decline.

Labor-reallocation effect: A second transitional counterbalancing force: lower entry rates reduce the number of workers employed in innovation (research and development) activities, reallocating them to goods production. This increase in the production-sector labor share temporarily raises measured TFP growth. Like the level-vs-growth effect, it fades as the economy converges to the new balanced growth path.

Entry congestion: An extension to the free-entry condition in which the per-entrant cost rises with the ratio of the entry rate to population growth (with elasticity ϕ = 0.55). When population growth slows, congestion costs fall, reducing the incentive to invest in high-step-size innovation, thus providing a second channel through which slower population growth reduces TFP growth beyond the core composition channel.

Innovation spillovers: A mechanism by which the productivity growth of already-leading businesses (gS) responds positively to lagged aggregate productivity growth gX (with estimated elasticity γ = 0.342). This link means that when population growth slows and gX falls, mature firms also grow more slowly, amplifying the initial effect. Calibrated using OLS and IV (venture capital investment as instrument) regressions of old-establishment productivity growth on aggregate past productivity growth.

Balanced growth path (BGP) comparison: The primary analytical exercise: comparing steady-state TFP growth rates across economies that differ only in their constant labor-force growth rate. This isolates the long-run equilibrium effect, abstracting from the transitional dynamics that counteract the decline in the short run. The BGP effect is larger than what is observed during any historical transition window because of the slow convergence.