## wp1867 ## Source details **Canonical URL:** [wp1867](https://www.imf.org/-/media/files/publications/wp/2018/wp1867.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2018/wp1867.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2018/wp1867.pdf.json) --- ### Key empirical findings and quantitative facts - Model implies a trend decline in a key aggregate share of "3.5 percentage points from 1990 to 2015." - Model predicts growth rate of labor productivity (output per hour worked) declined by about "0.5 percentage points from 1990 to 2011." - Observed average productivity decline over the same period is "about 1 percentage point" (Fernald, 2015). - Changes in labor quality (age and cohort composition effects on human capital) are important since the 1990s; entry and later exit of the baby boomer generation raise then lower growth rates of human capital and productivity. ### Mechanisms emphasized by the model - Demographics as primary driver: - Trend decline mainly caused by changes in workforce composition as workers live longer and labor force participation declines with age. - Distributional channel: - Age-composition interacts with age-productivity curve, creating a distributional channel from demographics to output. - Labor-quality channel: - Variation in aggregate labor and capital explains much change, but changes in labor quality matter since the 1990s. - Closed-economy abstraction with robustness arguments for open-economy relevance: - Most advanced economies experienced similar demographic changes. - Observed international capital flows were opposite to those implied by U.S. aging. ### Model structure — households, firms, government, and policy - Households: - Overlapping generations with maximum lifespan T (age range 0 to T−1). - Cohort mass n_{s,t}; population normalized to n_{0,t} = 1 absent exogenous incoming changes. - Mortality: n_{s+1,t+1} = (1−γ_{s,t}) n_{s,t}; γ_{T−1,t} = 1. - Household problem maximizes V_{s,t} with preference shock ln χ_t = (1−ρ_χ) ln χ + ρ_χ ln χ_{t−1} + σ_χ ε_{χ,t}. - Unintentional bequests redistributed as annuities (Yaari (1965), Blanchard (1985)). - Budget constraint includes quadratic capital adjustment costs parameterized by φ_k and aggregate adjustment shock ln κ_t = (1−ρ_κ) ln κ + ρ_κ ln κ_{t−1} + σ_κ ε_{κ,t}. - Euler equation for bonds: E_t λ_{s+1,t+1} / λ_{s,t} = 1/β E_t Π_{t+1} R_t, with Π_t = p_t / p_{t−1}. - Firms: - Continuum of intermediate goods producers: y_t(i) = μ_{t}^{1−α} (k_{t−1}(i))^{α} (Z_t `_{t}(i))^{1−α}, with Z_t / Z_{t−1} = z. - Aggregate TFP μ_t: ln μ_t = (1−ρ_μ) ln μ + ρ_μ ln μ_{t−1} + σ_μ ε_{μ,t}. - Rotemberg quadratic price adjustment costs with parameter φ_p; elasticity of substitution ξ_t follows ln ξ_t = (1−ρ_ξ) ln ξ + ρ_ξ ln ξ_{t−1} + σ_ξ ε_{ξ,t}. - Rental rate r_t = α mc_t y_t / k_{t−1}; wage w_t = (1−α) mc_t y_t / `_{t}. - Price resetting yields a forward-looking New Keynesian Phillips curve with slope ς_p. - Monetary policy and ZLB: - Two regimes for nominal interest rate R_t: - Taylor-rule regime: R_t / R = (R_{t−1} / R)^{φ_r} (Π_t / Π^*)^{(1−φ_r) φ_π} (y_t / y^F_t)^{(1−φ_r) φ_y} ( (y_t / y_{t−1}) / (y^F_t / y^F_{t−1}) )^{φ_g} exp(σ_R ε_{R,t} ). - Zero lower bound regime: log(R_t) = 0. - Entry into the ZLB can occur if Taylor rule calls for negative nominal rates or via central bank manipulation of expected ZLB duration; estimation uses NY Fed survey data to discipline expected ZLB duration during "2009 to 2015." - Government and social security: - Pay-as-you-go social security funded by labor income tax τ_t. - Benefits above eligibility age T^* depend on expected remaining life, accumulated pre-tax lifetime earnings W_{s,t}, and replacement parameter ω: ξ_{s,t} = ω W_{s,t} / (T^* − 1) for s ≥ T^*; ξ_{s,t} = 0 for s < T^*. - Accumulated gross lifetime earnings W_{s,t} recursive definition provided. - Government budget constraint: ∑_s n_{s,t} ξ_{s,t} = ∑_s n_{s,t} z_s w_t `_{s,t} τ_t; τ_t adjusts to equalize outlays and revenues. - Lump-sum tax funds g_t with ln g_t = (1−ρ_g) ln g + ρ_g ln g_{t−1} + σ_g ε_{g,t}. ### Approximation to representative-agent framework and time-varying parameters - Representative-agent approximation used to capture demographic effects via five time-varying exogenous parameters: - θ_t = ∑_s n_{s,t} z_s (productivity shock capturing workforce size and composition). - A_t converts aggregate hours into efficiency units of labor: A_t = [∑_s n_{s,t} (ẑ_s)^{1+1/φ} (v_s λ_s)^{−1/φ}] / [∑_s n_{s,t} (ẑ_s)^{1/φ} (v_s λ_s)^{−1/φ}], with ẑ_s = z_s / θ_t. - φ_t maps to marginal utility: φ_t c_t^{−σ} = λ_t and φ_t = [ ∑_s n_{s,t} (λ_s)^{1/σ} ]^{σ}. - v_t — aggregate marginal disutility of labor: v_t = [ ∑_s n_{s,t} (ẑ_s)^{1/φ+1} (v_s λ_s)^{−1/φ} ]^{−φ}. - s_t — adjustment to discount factor due to changing longevity. - Representative-agent preferences: U(c_t, `_{t}) = φ_t c_t^{1−σ} / (1−σ) − v_t `_{t}^{1+φ} / (1+φ). - Aggregate production: y_t = θ_t^{1−α} k_t^{α} `_{t}^{1−α} and `_{t} = A_t h_t. - Aggregation logic: near-aggregation (Krusell and Smith (1998)) justifies approximation; wealth redistribution has limited aggregate impact. - All demographic changes and time paths of θ_t, A_t, φ_t, v_t, s_t are assumed perfectly foreseen; linearization around steady state associated with parameters at each point in time. ### Equilibrium conditions preserved in the approximation - Representative household optimality: - λ_t = φ_t χ_t c_t^{−σ}. - λ_t = v_t χ_t (A_t h_t)^{φ} w_t (1−τ_t). - Euler equations: - Bonds: E_t λ_{t+1} / λ_t = 1/β E_t Π_{t+1} R_t. - Capital: E_t λ_{t+1} / λ_t = E_t [ 1/β [ 1 + φ_k κ_t (k_t / k_{t−1} κ_{t−1}^{−1}) ] [ φ_k/2 (k_{t+1}^2 / k_t^2 κ_t^{−1}) + 1−δ + r_{t+1} ]^{−1} ]. - Goods market clearing: - y_t = c_t + k_t + g_t − (1−δ) k_{t−1} + φ_k/2 (k_t / k_{t−1} − 1)^2 k_{t−1} + φ_p/2 [ Π_t / Π_{t−1} − 1 ]^2 y_t. ### Assigned and calibrated parameters (selected) - Model frequency: quarterly. - Lifespan: individuals begin at 16 and live up to age 95 (80 more years). - Social security replacement ratio λ: 46.7%. - Retirement benefits from age 65 (T^* = 49). - Age-productivity: peak increase in wages of about 134% at age 45; comparable Guvenen et al. (2015) find 127% for mean worker. - Post-age-65 productivity: calibrated to decay by 20% a year after age 65. - Mortality calibration implies life expectancy for 16 year olds increases from about 77 years in 1950 to about 85 years in 2020. - Capital depreciation δ: 10.6% a year. - Quarterly discount factor β: 0.99875. - Capital share α: 1/3. - Intertemporal elasticity σ: 1. - Inverse Frisch elasticity φ: 2. - Implied capital-output ratio in 2000: about 2.7. - Price-markup parameter ξ steady state: 8, implying steady-state markup ξ/(ξ−1) = 14%. - Quadratic cost of capital adjustment φ_k: 40. - Annual inflation target Π*: 2.2%. - Taylor-rule parameters (Smets and Wouters (2007) posterior mode): φ_r = 0.81, φ_π = 2.03, φ_y = 0.08, φ_g = 0.22. ### Bayesian estimation, shocks, and variance decomposition (selected) - Estimation frequency: quarterly; observables 1984Q1–2015Q1 include log R_t, log Π_t, log(y_t / y_{t-1}), log(c_t / c_{t-1}), log(i_t / i_{t-1}), T_t. - ZLB handled via regime-switching; expected ZLB durations T_t between 2009 and 2015 from Blue Chip and NY Fed surveys. - MCMC: two independent chains of 150,000 draws. - Posterior findings: - Slope of Phillips curve ϕ_p: posterior centered around 0.01, translating to quarterly Calvo probability of price adjustment ≈ 10%. - Modal estimate of trend growth implies annual growth rate ≈ 1.35% (about 0.4 percentage points less than Smets and Wouters (2007)). - Forecast error variance decompositions: - One-year horizon for output: monetary policy shocks, exogenous government spending shocks, and investment shocks together ≈ 50% of forecast error variance. - Infinite horizon: about 70% of forecast error variance caused by markup shocks. - Fed Funds rate: about one-third of forecast error variance driven by preference shocks. - Monetary policy shocks: ≈ 5% of forecast error variance for output, consumption, and investment at one-year horizon; < 1% at infinite horizon; ≈ 23% of Fed Funds rate variance at short horizons. - Investment shocks: explain 27% of forecast error variance of output and 50% of investment at one-year horizon; 18% and 30% for output and investment at infinite horizon. - Role of demographics in estimation: - Holding demographics constant from 1984 increases estimated annual trend growth by 0.4%. - Excluding demographics increases inferred importance of technology shocks in explaining output and consumption. ### Effect of demographics — quantitative outcomes and channels - Demographics-only simulation initialized 1984 (all other shocks off) produces: - Fed Funds rate: declines by about 2 percentage points between 1990 and 2015. - Real interest rate: expected to fall by almost 1 percentage point over the period. - Employment: declines by about 2.5% between 1990 and 2015. - Output growth: slowdown relative to 1984–2007 trend. - Labor force participation and composition: - Model predicts labor force participation rate declines at similar pace to observed and is predicted to fall by a further 4 percentage points from 2020 to 2040. - Decline driven by compositional shifts toward lower-participation workers. - Capital accumulation and interest rates: - Increased longevity raises aggregate savings, increases capital-output ratio, reduces marginal product of capital. - Aging baby-boomer cohorts generate an increase then decrease in real interest rate around secular decline; oscillation driven by cohort lifecycle. - Decline in real interest rate reinforced and stays low beyond 2030. - Output and productivity decomposition: - Demographics are a substantial drag on total output growth throughout forecast to 2070. - Total output growth declines peak-to-trough by about 1.5 percentage points. - From 1980 to 2015, demographic trends cause output growth to decline by about 1.25 percentage points. - Channels: (i) aggregate hours supplied, (ii) physical capital changes via saving/retirement, (iii) average labor quality (composition × age-productivity). - Change in labor supply has large negative effect on productivity growth but positive effect on total growth when baby boomers enter around 1960. - Change in average labor quality peaks around 1990, adding roughly 0.3 percentage points to total growth and productivity growth; becomes a drag on productivity growth starting 2000 and reduces productivity growth until 2030. ### ZLB interaction, forward guidance, and counterfactuals - Nonlinear interaction with ZLB amplifies shocks: - Large negative investment shock in 2008 steady-state can bind ZLB for about two years, with output falling an additional 2 percentage points on impact. - Counterfactual holding demographics fixed at 1984: - Fed Funds rate: would have remained above zero after Great Recession, lowest at 0.5% in annual terms in 2012Q4. - Ten-year Treasury yields: lowest point corresponds to yields around 1.6%. - Employment gap: would have been about 2 percentage points smaller in 2009. - Forward guidance effects: - Without Fed forward guidance extension, inflation would have been lower by about 1 to 2 percentage points between 2009 and 2015. - Employment would have been lower by an additional 0.7 percentage points in 2012Q2; cumulatively over 2009–2015 employment would have fallen by 7.6%. - Strongest forward guidance announcements estimated between 2011Q3 and 2013Q3, with forward guidance component of total ZLB duration between 8 and 9 quarters. - Output since Great Recession: - Data gap between output and trend widens to 12% by 2015. - Demographics alone account for one-third — 4 percentage points — of the gap in 2015. - Absent expansionary forward guidance, gap would have been larger by, at most, 2 percentage points (primarily 2011–2013). - Shock contributions: - Investment shocks and markup shocks primarily explain why output is low relative to trend. - Government spending shocks partially offset declines between mid-2009 and 2012. - Preference shocks mainly responsible for ZLB binding; about one-third of Fed Funds rate forecast error variance explained by preference shocks. ### Robustness checks and alternative calibrations (selected) - Borrowing constraints: - Restricting borrowing early in life increases savings, raises capital-output ratio, lowers real interest rate relative to baseline; fluctuation magnitudes similar. - Time-varying age-productivity profiles: - Flattening age-productivity profiles reduces amplitude of labor productivity growth changes; overall aggregate pattern similar. - Female labor force participation and multiple skill types: - Calibrating to female series or introducing two skill types leaves baseline predictions largely unaffected from 1985 on. ### Conclusions and policy-relevant takeaways - Demographic trends (declining mortality and changing age composition) can generate low-frequency movements in output growth, productivity, real interest rate, and employment-population ratio consistent with observed trends. - Aging population alone accounts for about one-third of the decline in output per capita relative to pre-crisis trend by 2015. - Had population not aged, the ZLB would not have bound between 2009 and 2015. - Absent Fed forward guidance, the ZLB would have caused output to fall by an additional 2 percentage points between 2011 and 2013. - Investment and markup shocks were important contributors to the post-crisis output decline; government spending partially offset declines 2009–2011; preference shocks are highly contractionary when expected ZLB duration is fixed. - Results highlight importance of demographic trends for macro outcomes and suggest further research on demographics × housing market interactions and combining long-run trends with ZLB considerations in empirical macro frameworks. *Source: Excerpt from IMF Working Paper wp1867 (PDF).* ### 3.5 percentage points from 1990 to 2015. In the model, the trend decline is mainly caused by ### wp1867 - 3.5 percentage points from 1990 to 2015. In the model, the trend decline is mainly caused by ### Key empirical findings and quantitative facts - The model implies a trend decline in a key aggregate share of "3.5 percentage points from 1990 to 2015." - The model predicts that the growth rate of labor productivity (output per hour worked) has declined by about "0.5 percentage points from 1990 to 2011." - This modeled productivity decline is compared with an observed average decline of "about 1 percentage point" over the same period (Fernald, 2015). - The model highlights that changes in labor quality have been an important factor since the 1990s; the entry and later exit of the baby boomer generation raise then lower growth rates of human capital and productivity. ### Mechanisms emphasized by the model - Demographics drive trends in key macro variables: the trend decline is mainly caused by changes in the composition of the workforce as workers live for longer and the labor force participation rate declines with age. - Distributional channel: the age-composition of the workforce interacts with the age-productivity curve, creating a distributional channel from demographics to output. - Labor-quality channel: variation in aggregate labor and capital explains much change, but changes in labor quality (age and cohort composition effects on human capital) are important since the 1990s. - International factors are abstracted away (model is closed-economy). The paper argues the demographic link to falling real interest rates is robust to open-economy considerations for two reasons given: - Most advanced economies experienced similar demographic changes. - Observed international capital flows were opposite to those implied by U.S. aging. ### Model structure — households, firms, government, and policy - Households: - Overlapping generations with maximum lifespan T (age range 0 to T−1). - Cohort mass n_s_t; population size normalized to n_0_t = 1 in absence of exogenous incoming population changes. - Mortality: n_{s+1,t+1} = (1−γ_{s,t}) n_{s,t}. Permanent decreases in γ_{s,t} imply increases in longevity; γ_{T−1,t} = 1 at maximum lifespan T. - Household problem maximizes V_{s,t} = max_{c_{s,t}, `_{s,t}, a_{s,t}} { χ_t u(c_{s,t},`_{s,t}) + β(1−γ_{s,t}) E_t V_{s+1,t+1} } with preference shock ln χ_t = (1−ρ_χ) ln χ + ρ_χ ln χ_{t−1} + σ_χ ε_{χ,t}. - Unintentional bequests redistributed as annuities (Yaari (1965), Blanchard (1985)), eliminating mortality risk. - Budget constraint includes quadratic capital adjustment costs parameterized by φ_k and subject to aggregate adjustment shock ln κ_t = (1−ρ_κ) ln κ + ρ_κ ln κ_{t−1} + σ_κ ε_{κ,t}. - Euler equation for bonds: E_t λ_{s+1,t+1} / λ_{s,t} = 1/β E_t Π_{t+1} R_t, where Π_t = p_t / p_{t−1}. - Firms: - Continuum of intermediate goods producers with production y_t(i) = μ_{t}^{1−α} (k_{t−1}(i))^{α} (Z_t `_{t}(i))^{1−α}, with Z_t / Z_{t−1} = z governing trend growth. - Aggregate TFP μ_t follows ln μ_t = (1−ρ_μ) ln μ + ρ_μ ln μ_{t−1} + σ_μ ε_{μ,t}. - Price adjustment costs are Rotemberg quadratic with parameter φ_p; elasticity of substitution ξ_t follows ln ξ_t = (1−ρ_ξ) ln ξ + ρ_ξ ln ξ_{t−1} + σ_ξ ε_{ξ,t}. - Rental rate r_t = α mc_t y_t / k_{t−1}, wage w_t = (1−α) mc_t y_t / `_{t}. - First-order condition for price resetting (Rotemberg) leads, when log-linearized, to a forward-looking New Keynesian Phillips curve with slope denoted ς_p. - Monetary policy and ZLB: - Two regimes for nominal interest rate R_t: - Taylor-rule regime: R_t / R = (R_{t−1} / R)^{φ_r} (Π_t / Π^*)^{(1−φ_r) φ_π} (y_t / y^F_t)^{(1−φ_r) φ_y} ( (y_t / y_{t−1}) / (y^F_t / y^F_{t−1}) )^{φ_g} exp(σ_R ε_{R,t} ). - Zero lower bound regime: log(R_t) = 0. - Entry into the ZLB can occur if the Taylor rule calls for negative nominal rates or if the central bank manipulates expectations about ZLB duration (Fed can announce extensions). Estimation uses New York Federal Reserve survey data to discipline expected duration of the ZLB during "2009 to 2015." - Government and social security: - Pay-as-you-go social security funded by labor income tax τ_t. - Benefits above eligibility age T^* depend on expected remaining life, accumulated pre-tax lifetime earnings W_s_t, and replacement parameter ω: ξ_{s,t} = ω W_s_t / (T^* − 1) for s ≥ T^*; ξ_{s,t} = 0 for s < T^*. - Accumulated gross lifetime earnings W_s_t defined recursively: W_s_t = w_t z_s `_{s,t} + W_{s−1,t−1} if s < T^*; W_s_t = W_{s−1,t−1} if s ≥ T^*. - Government budget constraint: ∑_s n_s_t ξ_{s,t} = ∑_s n_s_t z_s w_t `_{s,t} τ_t. Tax rate τ_t adjusts to equalize social security outlays and tax revenues. - Lump-sum tax funds government expenditures g_t, with ln g_t = (1−ρ_g) ln g + ρ_g ln g_{t−1} + σ_g ε_{g,t}. ### Approximation to representative-agent framework and time-varying parameters - To handle computational burdens (wealth distribution), the model is approximated by a representative-agent framework with exogenous time-varying parameters that capture demographic effects. - Five time-varying, exogenous parameters through which demographics affect the aggregate economy: - θ_t — productivity shock capturing changes in workforce size and composition: θ_t = ∑_s n_s_t z_s. - A_t — converts aggregate hours into efficiency units of labor: A_t = [∑_s n_s_t (ẑ_s)^{1+1/φ} (v_s λ_s)^{−1/φ}] / [∑_s n_s_t (ẑ_s)^{1/φ} (v_s λ_s)^{−1/φ}], with ẑ_s = z_s / θ_t. - φ_t — multiplicative parameter on marginal utility of consumption mapping to population size and utility curvature: φ_t c_t^{−σ} = λ_t and φ_t = [ ∑_s n_s_t (λ_s)^{1/σ} ]^{σ}. - v_t — aggregate marginal disutility of labor: v_t = [ ∑_s n_s_t (ẑ_s)^{1/φ+1} (v_s λ_s)^{−1/φ} ]^{−φ}. - s_t — an adjustment to the discount factor caused by changing longevity (appears in representative-agent discounting). - Aggregation logic: near-aggregation arguments (Krusell and Smith (1998)) justify that aggregate dynamics are well-approximated because individual policy functions are sufficiently linear; redistribution of wealth across individuals has limited impact on aggregate dynamics. - Representative-agent preferences: U(c_t, `_{t}) = φ_t c_t^{1−σ} / (1−σ) − v_t `_{t}^{1+φ} / (1+φ), with aggregate production y_t = θ_t^{1−α} k_t^{α} `_{t}^{1−α} and `_{t} = A_t h_t. ### Equilibrium conditions preserved in the approximation - Representative household optimality: - λ_t = φ_t χ_t c_t^{−σ}. - λ_t = v_t χ_t (A_t h_t)^{φ} w_t (1−τ_t). - Euler equations: - Bonds: E_t λ_{t+1} / λ_t = 1/β E_t Π_{t+1} R_t. - Capital: E_t λ_{t+1} / λ_t = E_t [ 1/β [ 1 + φ_k κ_t (k_t / k_{t−1} κ_{t−1}^{−1}) ] [ φ_k/2 (k_{t+1}^2 / k_t^2 κ_t^{−1}) + 1−δ + r_{t+1} ]^{−1} ]. - Goods market clearing: - y_t = c_t + k_t + g_t − (1−δ) k_{t−1} + φ_k/2 (k_t / k_{t−1} − 1)^2 k_{t−1} + φ_p/2 [ Π_t / Π_{t−1} − 1 ]^2 y_t. - All demographic changes and corresponding time paths of θ_t, A_t, φ_t, v_t, s_t are assumed perfectly foreseen; linearization is performed around the steady state associated with parameters at each point in time. ### Estimation strategy and scope - Model set at quarterly frequency. - Estimation involves calibrating lifecycle parameters, specifying transitory and permanent demographic changes that drive trends, and estimating transitory shocks that govern the business cycle around demographic trends. - The model jointly accounts for demographics, the zero lower bound, and the business cycle, emphasizing their nonlinear interactions. *Source: Excerpt from IMF Working Paper wp1867 (PDF).* ### 4.1 Assigned and Calibrated Parameters ### wp1867 - 4.1 Assigned and Calibrated Parameters ### Lifecycle Parameters - Model frequency: quarterly. - Lifespan: individuals begin at 16 years of age and live for at most 80 more years, up to age 95. - Full retirement: imposed only in the last period of life. - Disutility of labor v_s: calibrated with a scaled cumulative density function of a normal distribution so that v_s increases in s; parameters chosen to broadly match labor force participation rates by age observed in 2000. - Social security replacement ratio λ: set to 46.7%. - Retirement benefits received from age 65 on (T* = 49). - Age-productivity parameters z_s: calibrated to age-experience earnings profile constructed from deflated full-time, full-year worker data (decennial Census 1960–2000; annual ACS 2001–2007). - Pooled education groups: high school dropouts, high school graduates, some college, completed college or higher. - Estimated earnings profile: peak increase in wages of about 134% at age 45; comparable estimate Guvenen et al. (2015) find 127% for the mean worker. - Post-age-65 productivity: calibrated to decay by 20% a year after age 65. - Lifecycle asset and participation patterns implied: - Labor force participation: rises when young, flattens during prime working life, declines rapidly around retirement. - Asset profile: hump-shaped, peaks around 60 years old; individuals borrow when young and begin accumulating assets around age 35. ### Mortality Profiles - Mortality probabilities γ^s_t: calibrated to actuarial probabilities reported by the Social Security Administration (cohort-specific survival rates). - Calibration covers cohorts born 1900 onwards; extrapolated values used for cohorts born before 1900 but alive in 1940. - Changes to actuarial probabilities: assumed exogenous and perfectly foreseen. - Resulting life expectancy for 16 year olds under calibration: increases from about 77 years in 1950 to about 85 years in 2020. ### Incoming Cohort Size Shocks - Anticipated shocks to incoming cohort sizes chosen so model change in observed cohort share equals change in data cohort share. - Changes to incoming population beyond 2015: assumed to decay to zero; population distribution converges to steady-state implied by mortality profile constant from 2070. - Median age of population (above 16) implied by calibration: declines from around the 1960s to around the 1980s to about 37 years of age, then steadily increases as baby-boomers age and longevity rises. ### Preference and Nominal Parameters - Capital depreciation δ: 10.6% a year. - Quarterly discount factor β: 0.99875. - Capital share α: 1/3. - Intertemporal elasticity of substitution σ: 1. - Inverse Frisch elasticity φ: 2. - Implied capital-output ratio in 2000: about 2.7 (matches BLS-MFP). - Price-markup parameter ξ steady state: 8, implying steady-state markup ξ/(ξ−1) = 14%. - Quadratic cost of capital adjustment φ_k: 40. - Quadratic cost of price adjustment φ_p and slope of Phillips curve ϕ_p: to be estimated. - Annual inflation target Π*: 2.2%. - Taylor rule parameters (Smets and Wouters (2007) posterior mode): φ_r = 0.81, φ_π = 2.03, φ_y = 0.08, φ_g = 0.22. ### Bayesian Estimation — Solution Method - Estimation targets: slope of the Philips curve, trend growth, and model shocks driving business cycle fluctuations around demographic trend. - Incorporates anticipated demographic changes as time-varying structural parameters and handles the ZLB via regime-switching. - Linearized rational-expectations form: A x_t = C + B x_{t-1} + D E_t x_{t+1} + F ε_t. - Time-varying linear representation when agents expect parameters to converge: x_t = J_t + Q_t x_{t-1} + G_t ε_t. - Solution requires knowledge of full anticipated path of structural matrices A_t, B_t, C_t, D_t, F_t. ### Bayesian Estimation — The Zero Lower Bound (ZLB) - ZLB implemented via regime-switching algorithm with two regimes: ZLB regime and Taylor-rule policy regime. - Agents have rational expectations over regime application each quarter; algorithm iterates on expected ZLB periods until consistent with agents’ expectations. - Backward iteration starting from system at expected exit from ZLB regime used to obtain time-varying representation consistent with expected ZLB duration. - Expected ZLB duration need not equal realized duration (calendar-based forward guidance can extend expected duration). ### Bayesian Estimation — Quarterly Data - Observables used (1984Q1–2015Q1): {log R_t, log Π_t, log(y_t / y_{t-1}), log(c_t / c_{t-1}), log(i_t / i_{t-1}), T_t}. - Growth rates of output per capita, consumption per capita, investment per capita, GDP deflator, Fed Funds rate. - Nominal interest rate not observable when ZLB binds between 2009Q1 and 2015Q1; handled with time-varying observation equation. - Sequence of expected ZLB durations T_t between 2009 and 2015 taken from Blue Chip Financial Forecasts (2009–2010) and NY Fed Survey of Primary Dealers (2011–2015). ### Bayesian Estimation — Parameter Estimates and Shock Roles - Priors: diffuse, common to Bayesian NK literature. - Prior for slope of Phillips curve wide: 10th percentile implies quarterly Calvo reset probability = 13%; 90th percentile implies 34%. - Prior for trend growth z has 10th/90th percentiles implying annual trend growth of about 1% and 3%, respectively. - MCMC: two independent chains of 150,000 draws. - Posterior findings: - Slope of Phillips curve ϕ_p: posterior centered tightly around 0.01, translating to a quarterly Calvo probability of price adjustment ≈ 10%. - Modal estimate of trend growth: implies annual growth rate ≈ 1.35% (about 0.4 percentage points less than Smets and Wouters (2007)). - Forecast error variance decompositions (Table 2 highlights): - One-year horizon for output: monetary policy shocks, exogenous government spending shocks, and investment shocks together ≈ 50% of forecast error variance. - Infinite horizon: about 70% of forecast error variance caused by markup shocks. - Fed Funds rate: about one-third of forecast error variance driven by preference shocks. - Monetary policy shocks: ≈ 5% of forecast error variance for output, consumption, and investment at one-year horizon; < 1% at infinite horizon; ≈ 23% of Fed Funds rate variance at short horizons. - Investment shocks: explain 27% of forecast error variance of output and 50% of investment at one-year horizon; 18% and 30% for output and investment at infinite horizon. - Role of demographics in estimation: - Holding demographics constant from 1984 increases estimated annual trend growth by 0.4%. - Excluding demographics increases inferred importance of technology shocks in explaining output and consumption, suggesting demographic trends manifest as highly persistent productivity-like changes. ### Key Quantitative Findings on Demographics and the Business Cycle - Demographic trends alone are responsible for about one-third of the decline in log output relative to pre-crisis trend (absent all other shocks). - Demographic changes contributed to the ZLB binding between 2009 and 2015; with demographics held constant from 1984, the Fed Funds rate would not have hit the ZLB. - Nonlinearities: interaction of demographic changes and the ZLB generates strong nonlinear effects. - Removing the forward guidance response of the Fed to a binding ZLB: output would have fallen by an additional 2% relative to trend, primarily between 2011 and mid-2013. *Source: wp1867 - 4.1 Assigned and Calibrated Parameters (IMF Working Paper text).* ### 5.1 Effect of Demographics ### 5.1 Effect of Demographics ### Effect of Demographics Alone - Simulation initialized in 1984 with only expected demographic trends (all other shocks off) produces: - Fed Funds rate: declines by about 2 percentage points between 1990 and 2015. - Real interest rate: expected to fall by almost 1 percentage point over the period. - Employment: demographic trends alone cause a decline in employment of about 2.5% between 1990 and 2015. - Output: demographic changes cause a slowdown in output growth relative to log output’s 1984 to 2007 trend. - Labor force participation and composition: - Model captures aggregate employment-population ratio dynamics well under lifecycle calibration. - Labor force participation rate in the model declines at a pace roughly as fast as observed and is predicted to continue to fall by a further 4 percentage points from 2020 to 2040. - Decline driven by compositional changes toward workers with lower participation rates. - Capital accumulation and interest rates: - As longevity increases, aggregate savings increase to fund longer retirements, raising the capital-output ratio and reducing the marginal product of capital. - Aging baby-boomer cohorts generate an increase and then a decrease in the real interest rate around the secular decline implied by increasing longevity; oscillation driven by workforce size and composition as baby boomers enter, age, save, and then withdraw from the labor market. - Decline in real interest rate reinforced and stays low beyond 2030. - Output growth and productivity decomposition: - Demographics are a substantial drag on total output growth; contribution to overall growth from demographics remains a drag throughout the forecast horizon to 2070. - Total output growth declines peak-to-trough by about 1.5 percentage points. - From 1980 to 2015, demographic trends cause output growth to decline by about 1.25 percentage points. - Three channels through which demographics alter output growth: (i) hours supplied (aggregate labor), (ii) changes in physical capital via saving/retirement, (iii) changes in average labor quality (composition interacting with age-productivity profile). - Decomposition findings: - Accelerating capital accumulation increases growth of labor productivity and total output up to 1995, after which growth rate starts to decline. - Change in labor supply has a large negative effect on productivity growth but a positive effect on total growth when baby boomers enter the labor force around 1960. - Change in average labor quality peaks around 1990, adding roughly 0.3 percentage points to total growth and productivity growth. - Contribution of labor quality becomes a drag on productivity growth starting in 2000 and is forecast to reduce productivity growth until 2030. ### Decline in Real Interest Rate and the ZLB - Nonlinear interaction between declining real interest rate and the ZLB can amplify shock effects: - In an experiment comparing 1990 vs 2008 demographic steady-states, a large negative investment shock: - When initialized at the 2008 steady-state, the shock is large enough to cause the ZLB to bind for about two years, with output falling by an additional 2 percentage points on impact. - Decline in the real interest rate can be quantitatively important in the presence of the ZLB. ### Holding Demographic Trends Constant from 1984 (Counterfactual) - Counterfactual simulation holding demographic profile fixed at 1984 and using estimated structural shocks: - Fed Funds rate: would have remained above zero after the Great Recession, falling to, at its lowest point, 0.5% in annual terms in 2012Q4. - Ten-year Treasury yields: this lowest point period corresponds to the time of the lowest 10-year Treasury yields, which were around 1.6%. - Employment gap: holding demographics fixed at 1984, the employment gap would have been closed just prior to the recession, and in 2009 would have been about 2 percentage points smaller. - Output: counterfactual response shows output growth would have been lower due to the 1984 workforce composition skewed towards younger workers, favorable to productivity growth between 1984 and the mid-2000s. ### Effect of Forward Guidance (Summary of Related Results) - Counterfactual with Fed acting passively (no forward guidance extension of expected ZLB duration beyond shocks): - Inflation: would have been lower by about 1 to 2 percentage points between 2009 and 2015. - Employment: would have been lower by a further 0.7 percentage points in 2012Q2; cumulatively, over 2009 to 2015, employment would have fallen by 7.6%. - Measured forward guidance stimulus (survey-implied ZLB durations minus shock-implied durations): - Some degree of forward guidance stimulus exists every quarter between 2009 and 2015. - Strongest forward guidance announcements occur between 2011Q3 and 2013Q3, when the forward guidance component of total duration is estimated to be between 8 and 9 quarters. - This period corresponds to low long-rate yields and explicit calendar-based Fed targets (e.g., “to mid-2013” guidance). ### Output Since the Great Recession (Decomposition and Shock Contributions) - Output gap relative to long-run linear trend: - Data gap between output and trend widens between 2008 and 2015 to 12% by 2015. - Demographics alone account for a third — 4 percentage points — of the gap in 2015. - Absent expansionary forward guidance, the gap would have been larger by, at most, 2 percentage points, primarily between 2011 and 2013. - Shock-by-shock contributions (method: set each shock to zero, recompute with endogenous Fed Funds response, difference from observed series): - Investment shocks and markup shocks primarily explain why output is low relative to trend. - Government spending shocks kept output from falling even further between mid-2009 and 2012. - Discount factor (preference) shocks have a small effect on output when monetary policy can offset them, but are primarily responsible for the ZLB binding; about a third of the forecast error variance of the Fed Funds rate is explained by preference shocks (Table 2), while a much smaller fraction of the forecast error variance of output, consumption, and investment is explained by preference shocks. - Fixed-ZLB-duration exercise (holding observed ZLB durations between 2009 and 2015 constant) magnifies marginal impacts: - Government spending shocks have larger marginal impact because they stimulate inflation and lower real interest rates when nominal rates are fixed. - Markup shocks raise inflation and lower real interest rates, producing positive output effects when interest rate is fixed. - Preference shocks become substantially contractionary when the interest rate is fixed and cannot accommodate them. ### Robustness and Alternative Calibrations - Borrowing constraints: - Introducing borrowing constraints (restricting borrowing early in life) increases savings, pushes up capital-output ratio, and lowers the real interest rate relative to baseline. - Magnitude of fluctuations of the real interest rate, participation rate, and output growth remain very similar to baseline. - Time-varying productivity profile: - Adopting time-varying (flattening) age-productivity profiles alters accumulation of human capital via growth and level effects. - Overall pattern of aggregate labor productivity remains similar; amplitude of change in labor productivity growth is smaller, with demographics contributing most to labor productivity growth in 1980 rather than 1990. - Female labor force participation and multiple skill types: - Calibrating age-productivity and labor disutility profiles to match female age-earnings and participation series from the 1940s–1990s leaves baseline predictions largely unaffected from 1985 on. - Introducing two worker types (low vs high skilled) does not change directions of aggregate predictions; important demographic dynamics captured primarily through changes in population size. ### Conclusions (Key Takeaways) - Demographic trends (declining mortality and changing age composition) can generate trends matching low-frequency movements in output growth, productivity, the real interest rate, and the employment-population ratio. - The aging population alone is responsible for a sizeable proportion of the decline in output per capita relative to its pre-crisis trend by 2015 — about one-third. - Had the population not aged, the ZLB would not have been a binding constraint between 2009 and 2015. - Absent forward guidance by the Fed, the ZLB would have caused output to fall by an additional 2 percentage points between 2011 and 2013. - Investment and markup shocks played important roles in causing the post-crisis output decline; government spending partially offset declines between 2009 and 2011; preference shocks are highly contractionary only when the expected ZLB duration is held fixed. - Results highlight the importance of demographic trends for macroeconomic outcomes and suggest further research on interactions between demographics and the housing market and on combining long-run trends with ZLB considerations in empirical macro frameworks. *Source: "5.1 Effect of Demographics" (wp1867) — extracted from the supplied IMF chapter content.* ### References ### References ### Cited Works - Aaronson, Stephanie, Thomaz Cajner, Bruce Fallick, Felix Galbis-Reig, Christopher Smith, and William Wascher, “Labor Force Participation: Recent Developments and Future Prospects,” Brookings Papers on Economic Activity, 2014, Fall, 197–255. - Aksoy, Yunus, Henrique S. Basso, Tobias Grasl, and Ron P. Smith, “Demographic Structure and Macroeconomic Trends,” 2015. - An, Sungbae and Frank Schorfheide, “Bayesian Analysis of DSGE Models,” Econometric Reviews, 2007, 26(2-4), 113–172. - Antolin-Diaz, Juan, Thomas Drechsel, and Ivan Petrella, “Following the Trend: Tracking GDP when Long-Run Growth is Uncertain,” 2014. - Aruoba, S Boragan, Pablo Cuba-Borda, and Frank Schorfheide, “Macroeconomic Dynamics Near the ZLB: A Tale of Two Countries,” Review of Economic Studies, 2017. - Attanasio, Orazio, Sagiri Kitao, and Giovanni L. Violante, “Global Demographic Trends and Social Security Reform,” Journal of Monetary Economics, 2007, 54, 144–198. - Auerbach, A. J. and L. J. Kotlikoff, Dynamic Fiscal Policy, Cambridge University Press, 1987. - Backus, David, Thomas Cooley, and Espen Henriksen, “Demography and Low-Frequency Capital Flows,” Journal of International Economics, 2014, 92, S94–S102. - Bernanke, Ben S., “Why Are Interest Rates So Low?,” 2015. - Binder, M. and H.M. Pesaran, “Multivariate Rational Expectations Models and Macroeconomic Modelling: A Review and Some New Results,” Cambridge Working Papers in Economics 9415, Faculty of Economics, University of Cambridge 1995. - Blanchard, Olivier J, “Debt, Deficits, and Finite Horizons,” Journal of Political Economy, April 1985, 93(2), 223–47. - Bloom, David E., David Canning, and Gunther Fink, “Implications of Population Aging for Economic Growth,” 2011. - Bloom, David E., and Jaypee Sevilla, “Economic Growth and the Demographic Transition,” Working Paper 8685, National Bureau of Economic Research December 2001. - Canova, Fabio, Filippo Ferroni, and Christian Matthes, “Approximating Time Varying Structural Models with Time Invariant Structures,” 2015. - Carvalho, Carlos, Andrea Ferrero, and Fernanda Nechio, “Demographics and Real Interest Rates: Inspecting the Mechanism,” 2015. - Casanova, Maria, “Revisiting the Hump-Shaped Wage Profile,” 2013. - Christiano, Lawrence J., Martin S. Eichenbaum, and Mathias Trabandt, “Understanding the Great Recession,” American Economic Journal: Macroeconomics, 2015, 7(1), 110–167. - Christiano, Lawrence, Martin Eichenbaum, and Sergio Rebelo, “When Is the Government Spending Multiplier Large?,” Journal of Political Economy, 2011, 119(1), 78 – 121. - Constantinides, George M., “Intertemporal Asset Pricing with Heterogeneous Consumers and Without Demand Aggregation,” The Journal of Business, 1982, 55(2), 253–267. - Eggertsson, Gauti B., Neil R. Mehrotra, and Jacob A. Robbins, “A Quantitative Model of Secular Stagnation,” 2017. Brown University. - Elsby, Michael W. L. and Matthew D. Shapiro, “Why Does Trend Growth Affect Equilibrium Employment: A New Explanation of an Old Puzzle,” American Economic Review, 2012, 102(4), 1378–1413. - Fernald, John, “Productivity and Potential Output Before, During and After the Great Recession,” Technical Report 18, Federal Reserve Bank of San Francisco 2012. - Fernald, John, “Productivity and Potential Output before, during, and after the Great Recession,” NBER Macroeconomics Annual, 2015, 29(1), 1–51. - Fernández-Villaverde, Jesús, Grey Gordon, Pablo A. Guerrón-Quintana, and Juan Rubio-Ramírez, “How Structural Are Structural Parameters?,” NBER Macroeconomics Annual, June 2007, 22, 83–132. - Feyrer, James, “Demographics and Productivity,” The Review of Economics and Statistics, February 2007, 89(1), 100–109. - Fujita, S and I Fujiwara, “Aging and Deflation: Japanese Experience,” 2014. - Gagnon, Etienne, Benjamin K. Johannsen, and David Lopez-Salido, “Understanding the New Normal: The Role of Demographics,” Working Paper, Board of Governors of the Federal Reserve System 2016. - Gordon, Robert, The Rise and Fall of Economic Growth: The U.S. Standard of Living Since the Civil War, Princeton University Press, 2016. - Guerrieri, Luca and Matteo Iacoviello, “Occbin: A Toolkit to Solve Models with Occasionally Binding Constraints Easily,” Journal of Monetary Economics, March 2015, 70, 22–38. - Guvenen, Fatih, Fatih Karahan, Serdar Ozkan, and Jae Song, “What Do Data on Millions of U.S. Workers Reveal about Life-Cycle Earnings Risk?,” Working Paper 20913, National Bureau of Economic Research January 2015. - Hamilton, James D., Ethan S. Harris, Jan Hatzius, and Kenneth D. West, “The Equilibrium Real Funds Rate: Past, Present and Future,” Working Paper 21476, National Bureau of Economic Research August 2015. - Hansen, Gary D. and Selahattin Imrohoroglu, “Consumption Over the Life Cycle: The Role of Annuities,” Review of Economic Dynamics, 2008, 11(3), 566–583. - Jaimovich, Nir and Henry Siu, “The Trend is the Cycle: Job Polarization and Jobless Recoveries,” 2012. - Jones, Callum, “Unanticipated Shocks and Forward Guidance at the Zero Lower Bound,” 2017. New York University. - Jones, Callum, Virgiliu Midrigan, and Thomas Philippon, “Household Leverage and the Recession,” 2017. - Kong, Y.C., B. Ravikumar, and G. Vandenbroucke, “Explaining Cross-Cohort Differences in Life-Cycle Earnings,” 2016. - Krusell, Per and Anthony A Smith Jr, “Income and Wealth Heterogeneity in the Macroeconomy,” Journal of political Economy, 1998, 106(5), 867–896. - Kulish, Mariano and Adrian Pagan, “Estimation and Solution of Models with Expectations and Structural Changes,” Journal of Applied Econometrics, 2016. - Kulish, Mariano, Christopher Kent, and Kathryn Smith, “Aging, Retirement, and Savings: A General Equilibrium Analysis,” The BE Journal of Macroeconomics, 2010, 10(1). - Kulish, Mariano, James Morley, and Tim Robinson, “Estimating DSGE Models with Zero Interest Rate Policy,” Journal of Monetary Economics, 2017, 88, 35–49. - Maestas, Nicole, Kathleen J. Mullen, and David Powell, “The Effect of Population Aging on Economic Growth, the Labor Force and Productivity,” Working Paper 22452, National Bureau of Economic Research July 2016. - Maliar, Lilia and Serguei Maliar, “The Representative Consumer in the Neoclassical Growth Model with Idiosyncratic Shocks,” Review of Economic Dynamics, 2003, 6, 362–380. - Reichling, Felix and Charles Whalen, “Review of Estimates of the Frisch Elasticity of Labor Supply,” Working Paper 13, Congressional Budget Office 2012. - Rios-Rull, Jose-Victor, Frank Schorfheide, Cristina Fuentes-Albero, Maxym Kryshko, and Raul Santaeulalia-Llopis, “Methods Versus Substance: Measuring the Effects of Technology Shocks,” 2012, 59(8), 826–846. - Rogoff, Kenneth, “Debt Supercycle, Not Secular Stagnation,” 2015. - Sims, Christopher, “Solving Linear Rational Expectations Models,” Computational Economics, 2002, 20, 1–20. - Summers, Lawrence H., “U.S. Economic Prospects: Secular Stagnation, Hysteresis, and the Zero Lower Bound,” Business Economics, 2014, 49(2), 65–73. - Swanson, Eric and John Williams, “Measuring the Effect of the Zero Lower Bound on Medium- and Long-term Interest Rates,” American Economic Review, 2014, 104(10), 3154–3185. - Wong, Arlene, “Population Aging and the Transmission of Monetary Policy to Consumption,” 2015. - Wu, Jing Cynthia and Fan Dora Xia, “Measuring the Macroeconomic Impact of Monetary Policy at the Zero Lower Bound,” 2016. manuscript. - Yaari, Menahem, “Uncertain Lifetime, Life Insurance, and the Theory of the Consumer,” Review of Economic Studies, 1965, 32, 137–150. ### Table 1: Estimated Parameters - Parameter listings (Prior / Posterior reported columns as in source): -  p U0.10.0 0.2 0.010.01 0.01 0.02 - 400×(z−1) N2.01.1 3.0 1.341.35 1.24 1.46 - ρ χ B0.50.3 0.7 0.940.94 0.93 0.95 - ρ μ B0.50.3 0.7 0.760.73 0.55 0.83 - ρ θ B0.50.3 0.7 0.960.96 0.95 0.97 - ρ g B0.50.3 0.7 0.950.95 0.94 0.97 - ρ κ B0.50.3 0.7 0.940.94 0.93 0.95 - 100×σ χ IG1.20.5 3.7 2.152.18 1.95 2.47 - 100×σ μ IG1.20.5 3.7 0.480.52 0.33 0.73 - 100×σ θ IG1.20.5 3.7 3.513.56 3.20 4.03 - 100×σ i IG1.20.5 3.7 0.160.16 0.15 0.18 - 100×σ g IG1.20.5 3.7 1.051.06 0.98 1.15 - 100×σ κ IG1.20.5 3.7 1.001.04 0.86 1.25 ### Table 2: Variance Decomposition Due to Shocks, % - A. Conditional, 4 Quarter Ahead - Fed Funds Rate: 28.5 7.8 5.8 22.9 8.3 26.8 - Inflation: 21.9 2.4 2.4 8.9 6.4 18.4 - Wages: 6.8 2.6 74.8 10.5 0.6 4.8 - Output: 0.8 0.3 47.7 4.3 19.6 27.4 - Consumption: 28.0 0.7 26.7 6.8 2.6 11.2 - Investment: 7.2 0.1 38.8 2.3 0.8 0.8 - B. Unconditional - Fed Funds Rate: 34.6 1.8 2.8 5.6 9.7 19.6 - Inflation: 20.5 1.1 55.7 4.2 5.8 12.7 - Wages: 1.7 0.4 93.0 2.0 0.2 2.7 - Output: 4.2 0.0 70.9 0.7 6.2 18.0 - Consumption: 7.0 0.1 70.0 1.0 13.6 8.4 - Investment: 8.6 0.0 59.1 0.5 1.2 30.5 ### Figures and Notes - Figure 1: Macroeconomic Trends - Figure 2: Individual Profiles - Notes: The data for the asset profile is taken from the Survey of Consumer Finances for 2013, HP-filtered, and normalized to the values of assets when 60. The productivity profile is computed from pooled Census and American Community Survey datasets. All details are in the Appendix. - Figure 3: Path of Variables, Demographics Only - Figure 4: Model Trends - Notes: The data for the capital-output ratio is extracted from the BLS’s Multifactor Productivity program. All details are in the Appendix. - Figure 5: Impulse Response to Large Investment Shock - Figure 6: Path of Variables, Demographics Fixed from 1984 - Figure 7: Path of Variables, No Forward Guidance - Figure 8: Decomposition of ZLB Durations, Quarters - Figure 9: Output Relative to Trend - Figure 10: Output Relative to Trend, Contribution of Shocks - Figure 11: Output Relative to Trend, Contribution of Shocks Fixing ZLB Durations *Source: wp1867 - References (wp1867 - References).* --- _Source: https://www.imf.org/-/media/files/publications/wp/2018/wp1867.pdf_