## wpiea2019198-print-pdf - Section 4 demonstrates the steady states of the young and old societies. Then, in Section 5 we

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---

### Sections covered and flow
- Section 4: Demonstrates steady states of young and old societies and transition exercise.
- Section 5: Presents main impulse-response functions and discusses transmission mechanism of monetary policy.
- Section 6: Studies welfare consequences of aging and optimal monetary policy rules.
- Section 7: Provides empirical support for theoretical findings and summarizes main policy conclusions.

### Model architecture — key components
- Demographics and overlapping generations:
  - Two cohorts: young (working) and old (retired).
  - Net fertility rate denoted n; retirement probability for the young ωY; survival probability for retired 1−ωO.
  - Population identities: Nt = NOt + NYt (Equation (1)); ratios s t = NOt/NYt (Equation (4)); sY t = NYt/Nt = 1/(1+st) (Equation (5)).
  - Cohort growth rates: 1+gN,Y t = 1−ωY t−1 + n t (Equation (6)); 1+gN,O t = (1−ωO t−1) + ωY t−1 st−1 (Equation (7)); 1+gN t = (1+gN,Y t) (1+st)/(1+st−1) (Equation (8)).
- Households:
  - Retired cohort maximizes Bellman equation (9) subject to budget constraint (10); retired Euler equation (11); retired individual consumption (12) with IncO a,t(i) = TRYO a,t(i) ΩO t (13) and ΩO t defined in (14); MPCO t via (15); aggregate retired consumption: CO t = MPCO t IncO t + (1+rt−1) MPCO t (ωY t−1 BY t−1 + BO t−1) (16).
  - Young cohort Bellman equations (18)-(19) with budget constraint (20); two Euler equations (21)-(22); labor supply condition (23); young individual consumption CY b,t(i) = MPCY t IncY b,t(i) + MPCY t (1+rt−1) BY b−1,t−1(i) (24) with IncY b,t(i) defined in (25) and MPCY t in (26); aggregate young budget constraint (33).
  - State-contingent bonds allow full insurance against survival outcomes for the young.
- Firms:
  - Monopolistic competition with Calvo price stickiness: fraction 1−ωP can reset prices each period.
  - Production: Yt(j) = e−εP t At Kt−1(j)α Lt(j)1−α (35); demand Yt(j) = (P∗ t(j)/Pt)−φP Yt (36).
  - Investment adjustment costs and capital accumulation (37).
  - Marginal cost relations (38)-(40); Tobin-Q and no-arbitrage conditions (41)-(43).
  - Optimal price p∗ t(j) = φP/(φP−1) Z1 t / Z2 t (44) with Z1 t (45) and Z2 t (46). Price index condition (47).
  - Labor union optimization with Calvo stickiness: optimal wage v∗ t(j) = φV/(φV−1) W1 t / W2 t (50) and wage index condition (53). Wage inflation relation (54).
  - Profits: Profit t = Yt − wt Lt − Invt (55). Profits and union income accrue to the young.
- Fiscal policy:
  - Limited role: PAYG pensions and lump-sum taxes on the young; no government debt.
  - Tax rule: Taxt = TRt + Govt (56).
  - Individual pension at retirement: TRYO 0,t(i) = ν wt−1 Lb−1,t−1(i) (57).
  - Aggregate pension rules: TRYO t = ν ωY t−1 wt−1 Lt−1 (58).
  - Total pension expenditure recursive form: TRt = TRYO t + (1−ωO t−1) TRt−1 (60).
- Monetary policy:
  - Baseline policy rule with inflation reaction and interest rate smoothing: 1+it = (1+it−1) ρi ((1+ r̃ t)(1+πt)φπ)1−ρi eεi t (61).
  - Central bank’s long-term real rate follows: 1+ r̃ t = (1+ r̃ t−1) ρ r̃ (1+ r n t)1−ρ r̃ (62).
  - Fisher identity: 1+it = Et (1+ r t)(1+π t+1) (63).
- Market clearing:
  - No net financial assets across economy: 0 = BY t + BO t (64).
  - Goods market: Yt = CY t + CO t + Invt + Govt (65).

### Parametrization — selected parameters (Table 1)
- Demography:
  - Probability of retirement ωY = 0.0050 (Quarterly frequency)
  - Probability of death ωO = 0.0200 (Quarterly frequency)
  - Fertility rate n = 0.0100 (Quarterly frequency)
- Households:
  - Discount rate (old) βO = 0.9990 (Quarterly frequency)
  - Discount rate (young) βY = 0.9950 (Quarterly frequency)
  - Weight of consumption utility σ = 0.6000 (Only for youngs)
  - Inverse of intertemporal elasticity γ = 2.0000 (Same for both cohort)
- Firms:
  - Price elasticity φP = 6.0000
  - Wage elasticity φW = 6.0000
  - Capital share α = 0.3300
  - Capital depreciation δ = 0.0250 (Quarterly frequency)
  - Investment adjustment cost φInv = 2.0000
  - Share of price setters (Calvo) ωP = 0.8000 (Quarterly frequency)
  - Price indexation γP = 0.5000
  - Share of union wage setters (Calvo) ωV = 0.8000 (Quarterly frequency)
  - Nominal wage indexation γV = 0.5000
- Monetary policy:
  - Interest rate smoothing ρi = 0.7500 (Quarterly frequency)
  - Reaction to inflation φπ = 1.5000
- Fiscal policy:
  - Replacement rate ν = 1.0746 (Calibrated from TRY = 0.1)
  - Gov. cons. Gov = 0.2777 (Calibrated from GovY = 0.2)

### Permanent demographic shift — steady-state comparison (Table 2)
- Exercise: Old-age dependency ratio increases from 0.2000 to 0.8000 over a 50-year horizon.
- Selected steady-state comparisons (Young society → Graying society):
  - Old-age dependency ratios: 0.2000 → 0.8000
  - Share of the youngs Y: 0.8333 → 0.5556
  - Consumption to GDP: Total CY: 0.5622 → 0.6013
  - Consumption to GDP: Young CY Y: 0.4635 → 0.1677
  - Consumption to GDP: Old CO Y: 0.0986 → 0.4336
  - Bonds to GDP: Old BO Y: 0.2579 → 8.4030
  - Interest rate elasticity: Young εY: 0.6250 → 0.6250
  - Interest rate elasticity: Old εO: 0.5000 → 0.5000
  - Frisch elasticity η: 0.5313 → 0.1922
  - GDP Y: 1.3883 → 1.2098
  - Investment to GDP Inv Y: 0.2378 → 0.1691
  - Capital to GDP K Y: 7.9677 → 7.4986
  - Profit to GDP Profit Y: 0.2969 → 0.3656
  - Real interest rate r: 0.0046 → 0.0065
  - Transfer to GDP TR Y: 0.1000 → 0.4000
  - Public consumption to GDP Gov Y: 0.2000 → 0.2295

### Mechanisms and steady-state implications (Section 4)
- Retirement and survival probabilities imply retired accumulate more savings as lifetime increases (decreasing ωO).
- PAYG pension system requires higher pension benefits in the graying society, financed by taxes on the young — increasing young indebtedness.
- Behavioral assumption: older households are more patient (higher discount factor βO), amplifying young indebtedness.
- Supply-side adjustments with aging:
  - Lower fertility and a shrinking labor force lead firms to lower investment and reduce prices (disinflation/deflation) to offset profit loss.
  - Firms hire relatively more labor in the medium term; labor shortages increase real wage volatility as Frisch elasticity decreases.
  - Central bank initially decreases the interest rate to offset downward inflation pressure; in the long run normalizes policy rate as inflation returns to steady state.
  - Lower physical capital stock raises marginal product of capital and leads to a slightly higher equilibrium real interest rate in the new steady state.

### 5.1 Monetary Policy Shock — mechanism and dynamics
- Mechanism:
  - Central bank increases nominal interest rate relative to systematic Taylor-rule; price stickiness implies real interest rate change affecting real economy.
- Tighter monetary policy:
  - Forces firms to postpone investment.
  - Leads households to accumulate more savings or decrease their credit stock.
  - Shrinks aggregate demand, inducing share 1−ω_P of the firms to decrease their prices to reduce profit loss.
- Dynamics:
  - Due to price stickiness and price indexation, nominal adjustment is gradual; disinflation takes more than 4 quarters.
  - Central bank observes shrinking aggregate demand, decreases the nominal interest rate back to steady-state level, and the economy stabilizes at its initial steady state.

### 5.1 Effects of aging on the shock transmission
- Cohort heterogeneity:
  - In an old society, retired agents hold more savings, while workers have more debt over longer lifetimes.
  - In a young society, monetary restriction incentivizes postponing consumption and increasing savings.
  - In an old society, an interest rate hike can be interpreted by the old as additional income, so they increase consumption by a larger amount, while the young face higher credit costs and decrease consumption.
- Aggregate consequences:
  - Because aging changes cohort sizes, aggregate consumption and output gap fall less in grey societies.
  - Aging reallocates asset positions among generations, making monetary policy less effective — defined in the chapter as a larger drop in the inflation rate to the same monetary tightening — and making aggregate demand less elastic to interest rate changes.
- Empirical consistency: finding consistent with Wong (2018), who shows that the old react less to expansionary monetary policy shocks based on micro-level cohort-specific US data.
- Simulation notes:
  - Disinflation horizon: "more than 4 quarters".
  - Cohort-size assumption: young societies old-age dependency ratio = 20%; old societies labeled Aging = t80%.
  - Figures label "ar" as annualized rate.

### 6.3 Optimal Monetary Policy — welfare comparisons and optimized rules
- Comparative welfare losses under unchanged (non-optimal) monetary rules:
  - OADR scenarios s=0.20, s=0.40, s=0.60, s=0.80.
  - Initial losses normalized to s=0.20: Initial losses = 1.000, 0.959, 0.947, 1.154 for s=0.20, s=0.40, s=0.60, s=0.80, respectively.
  - Volatility and parameter outcomes (normalized by initial loss values):
    - Inflation volatility: 0.111, 0.126, 0.147, 0.191 for s=0.20, s=0.40, s=0.60, s=0.80.
    - Output gap volatility: 0.887, 0.831, 0.797, 0.959 for s=0.20, s=0.40, s=0.60, s=0.80.
    - Interest rate volatility: 0.002, 0.002, 0.003, 0.003 for s=0.20, s=0.40, s=0.60, s=0.80.
    - Frisch elasticity: 0.531, 0.441, 0.325, 0.192 for s=0.20, s=0.40, s=0.60, s=0.80.
- Optimal monetary policy regimes analyzed: Pure IT, Pure IT & Forward (Fwd), Flexible IT, Flexible IT & Fwd, Price level targeting, Nominal GDP targeting.
- Representative optimized outcomes (selected values from Table 4):
  - Pure IT (optimized): Optimized loss = 0.924, 0.885, 0.832, 0.760; φπ_t = 3.718, 4.217, 5.032, 6.751 for s=0.20, s=0.40, s=0.60, s=0.80.
  - Pure IT & Fwd (optimized): Optimized loss = 0.872, 0.837, 0.646, 0.646; φE_t π_{t+1} = 6.353, 6.880, 7.835, 9.541.
  - Flexible IT (optimized): Optimized loss = 0.478, 0.495, 0.528, 0.584; φπ_t = 5.756, 6.105, 6.966, 8.744; φŶ_t = 3.428, 3.391, 3.209, 2.646.
  - Flexible IT & Fwd (optimized): Optimized loss = 0.492, 0.513, 0.550, 0.612; φE_t π_{t+1} = 6.102, 6.537, 7.468, 9.026; φŶ_t = 3.164, 3.088, 2.820, 2.122.
  - Price level targeting (optimized): Optimized loss = 0.913, 1.093, 1.032, 0.898; φ̂P_t = 0.017, 0.603, 1.250, 3.130.
  - Nominal GDP targeting (optimized): Optimized loss = 0.499, 0.498, 0.499, 0.481; φ̂P_t^{Y_t} = 5.920, 6.063, 6.457, 7.862.
- Interpretation of optimized rules:
  - As s increases, optimal φπ generally rises; optimal φŶ typically falls (central bank places more weight on stabilizing inflation relative to output).
  - Flexible IT minimizes loss at lower OADRs; nominal GDP targeting yields consistently low optimized losses across OADRs and can be most effective at extreme OADR in some calibrations.

### Empirical evidence on aging, inflation level, and inflation volatility
- Historical relationship (1981–2017, developed economies U.S., U.K., Germany, France, Japan, Portugal):
  - Periods of disinflation correspond with periods of increases in the old-age dependency ratio.
- FE regression on inflation volatility (unbalanced panel, OECD, 1993–2017):
  - Model: σ(cpi_core)_{it} = α_i + δ_t + β OADR_{it} + γ X_{it} + u_{it}.
  - Data processing: non-food non-energy CPI detrended with HP-filter λ = 150000; yearly average standard deviation from trend used as time-variant volatility.
  - Table 5 FE regression coefficients:
    - σ(cpi_core)_{i,t−1} coefficient: 0.087∗ (0.043) in FE (1); 0.083. (0.043) in FE (2).
    - OADR_{it} coefficients: 0.021∗ (0.011) in FE (1); 0.020∗ (0.010) in FE (2); 0.019∗ (0.009) in FE (3); 0.018∗ (0.008) in FE (4).
    - σ(Y)_{it} coefficients: 0.015∗ (0.008) in FE (3); 0.017∗ (0.008) in FE (4).
  - Sample and fit statistics:
    - Country FE: Yes; Year FE: Yes.
    - N.obs = 793 for FE (1) and FE (2); 759 for FE (3) and FE (4).
    - R^2 = 0.250, 0.254, 0.307, 0.318 for FE (1)–FE (4).
    - Significance notation: ∗∗∗ is p<0.1%, ∗∗ is p<1%, ∗ is p<5%, . is p<10%.
  - Quantitative interpretation:
    - Median volatility in sample: 0.19.
    - FE regressions imply a 10 percentage point increase in the old-age dependency ratio increases inflation volatility by around 0.18–0.20.

### Mechanisms: why aging alters monetary transmission and welfare
- Channels identified:
  - Longer lifespans and longer planning horizons change saving behavior: young borrow more; retired accumulate more savings.
  - Interest rate changes have asymmetric effects: higher interest rates increase the cost for indebted young and raise income for savers (old).
  - Differences in consumption-savings preferences: the old are more patient than the young, so consumption responses differ.
  - Labor market effects: labor force shrinkage tightens the labor market; real wages react more strongly.
- Consequences:
  - Monetary policy transmission becomes less effective in older societies.
  - The same-size shock causes higher inflation volatility in an older society.
  - Aging increases social welfare loss via higher inflation and higher inflation volatility.

### Policy implications and recommendations
- For inflation-targeting central banks:
  - Increase the nominal interest-rate response to inflation shocks as OADR rises: central banks with IT regimes should more strongly react to nominal variables (increase φπ).
  - Mildly reduce the policy response to the output gap (lower φŶ) as societies age, focusing relatively more on inflation stabilization.
- Consideration of alternative regimes:
  - Nominal GDP targeting can be favorable and may become the most effective regime when OADR is extremely high.
  - Price-level targeting tends to be hawkish and often yields higher welfare losses than nominal GDP targeting in the analyzed settings.
- Robustness checks:
  - With a higher output-gap weight (1.0) nominal GDP targeting becomes optimal only at a higher OADR; with higher interest-rate smoothing weight (0.5) nominal GDP targeting can become most effective at lower OADR levels (alternative simulations reported in Appendix Tables A1 and A2).

### Contribution and research agenda
- Novel contributions claimed:
  - First paper to estimate the impact of aging on the volatility of inflation.
  - First to explore the impact of aging on short-run cyclical behavior and monetary policy transmission in an aged society using a multi-period DGE model with overlapping generations.
  - First to examine optimal monetary policy strategies in the presence of aging.
- Call for further research:
  - Authors urge more research to better understand implications of aging for central banks, central bankers, and the elderly.

*Source: wpiea2019198-print-pdf - Section 4 demonstrates the steady states of the young and old societies. Then, in Section 5 we*

### Section 4 demonstrates the steady states of the young and old societies. Then, in Section 5 we

### wpiea2019198-print-pdf - Section 4 demonstrates the steady states of the young and old societies. Then, in Section 5 we

### Sections covered and flow
- Section 4: Demonstrates steady states of young and old societies and transition exercise.
- Section 5: Presents main impulse-response functions and discusses transmission mechanism of monetary policy.
- Section 6: Studies welfare consequences of aging and optimal monetary policy rules.
- Section 7: Provides empirical support for theoretical findings and summarizes main policy conclusions.

### Model architecture — key components
- Demographics and overlapping generations:
  - Two cohorts: young (working) and old (retired).
  - Net fertility rate denoted n; retirement probability for the young ωY; survival probability for retired 1−ωO.
  - Population identities: Nt = NOt + NYt (Equation (1)); ratios s t = NOt/NYt (Equation (4)); sY t = NYt/Nt = 1/(1+st) (Equation (5)).
  - Cohort growth rates: 1+gN,Y t = 1−ωY t−1 + n t (Equation (6)); 1+gN,O t = (1−ωO t−1) + ωY t−1 st−1 (Equation (7)); 1+gN t = (1+gN,Y t) (1+st)/(1+st−1) (Equation (8)).
- Households:
  - Retired cohort maximizes Bellman equation (9) subject to budget constraint (10).
  - Retired Euler equation (11); retired individual consumption given by (12) with IncO a,t(i) = TRYO a,t(i) ΩO t (13) and ΩO t defined in (14). MPCO t defined via (15).
  - Aggregate retired consumption: CO t = MPCO t IncO t + (1+rt−1) MPCO t (ωY t−1 BY t−1 + BO t−1) (16).
  - Young cohort Bellman equations (18)-(19) with budget constraint (20). Two Euler equations (21)-(22). Labor supply condition (23).
  - Young individual consumption: CY b,t(i) = MPCY t IncY b,t(i) + MPCY t (1+rt−1) BY b−1,t−1(i) (24) with IncY b,t(i) defined in (25) and MPCY t in (26). Aggregate young budget constraint (33).
  - State-contingent bonds allow full insurance against survival outcomes for the young.
- Firms:
  - Monopolistic competition with Calvo price stickiness: fraction 1−ωP can reset prices each period.
  - Production: Yt(j) = e−εP t At Kt−1(j)α Lt(j)1−α (35); demand Yt(j) = (P∗ t(j)/Pt)−φP Yt (36).
  - Investment adjustment costs and capital accumulation (37).
  - Marginal cost relations (38)-(40); Tobin-Q and no-arbitrage conditions (41)-(43).
  - Optimal price p∗ t(j) = φP/(φP−1) Z1 t / Z2 t (44) with Z1 t (45) and Z2 t (46). Price index condition (47).
  - Labor union optimization analogous with Calvo stickiness: optimal wage v∗ t(j) = φV/(φV−1) W1 t / W2 t (50) and wage index condition (53). Wage inflation relation (54).
  - Profits: Profit t = Yt − wt Lt − Invt (55). Profits and union income accrue to the young.
- Fiscal policy:
  - Limited role: PAYG pensions and lump-sum taxes on the young; no government debt.
  - Tax rule: Taxt = TRt + Govt (56).
  - Individual pension at retirement: TRYO 0,t(i) = ν wt−1 Lb−1,t−1(i) (57).
  - Aggregate pension rules: TRYO t = ν ωY t−1 wt−1 Lt−1 (58).
  - Total pension expenditure recursive form: TRt = TRYO t + (1−ωO t−1) TRt−1 (60).
- Monetary policy:
  - Baseline policy rule with inflation reaction and interest rate smoothing:
    - 1+it = (1+it−1) ρi ((1+ r̃ t)(1+πt)φπ)1−ρi eεi t (61).
  - Central bank’s long-term real rate follows:
    - 1+ r̃ t = (1+ r̃ t−1) ρ r̃ (1+ r n t)1−ρ r̃ (62).
  - Fisher identity: 1+it = Et (1+ r t)(1+π t+1) (63).
- Market clearing:
  - No net financial assets across economy: 0 = BY t + BO t (64).
  - Goods market: Yt = CY t + CO t + Invt + Govt (65).

### Parametrization — selected parameters (Table 1)
- Demography:
  - Probability of retirement ωY = 0.0050 (Quarterly frequency)
  - Probability of death ωO = 0.0200 (Quarterly frequency)
  - Fertility rate n = 0.0100 (Quarterly frequency)
- Households:
  - Discount rate (old) βO = 0.9990 (Quarterly frequency)
  - Discount rate (young) βY = 0.9950 (Quarterly frequency)
  - Weight of consumption utility σ = 0.6000 (Only for youngs)
  - Inverse of intertemporal elasticity γ = 2.0000 (Same for both cohort)
- Firms:
  - Price elasticity φP = 6.0000
  - Wage elasticity φW = 6.0000
  - Capital share α = 0.3300
  - Capital depreciation δ = 0.0250 (Quarterly frequency)
  - Investment adjustment cost φInv = 2.0000
  - Share of price setters (Calvo) ωP = 0.8000 (Quarterly frequency)
  - Price indexation γP = 0.5000
  - Share of union wage setters (Calvo) ωV = 0.8000 (Quarterly frequency)
  - Nominal wage indexation γV = 0.5000
- Monetary policy:
  - Interest rate smoothing ρi = 0.7500 (Quarterly frequency)
  - Reaction to inflation φπ = 1.5000
- Fiscal policy:
  - Replacement rate ν = 1.0746 (Calibrated from TRY = 0.1)
  - Gov. cons. Gov = 0.2777 (Calibrated from GovY = 0.2)

### Permanent demographic shift — steady-state comparison (Table 2)
- Exercise: Old-age dependency ratio increases from 0.2000 to 0.8000 over a 50-year horizon.
- Selected steady-state comparisons (Young society → Graying society):
  - Old-age dependency ratios: 0.2000 → 0.8000
  - Share of the youngs Y: 0.8333 → 0.5556
  - Consumption to GDP: Total CY: 0.5622 → 0.6013
  - Consumption to GDP: Young CY Y: 0.4635 → 0.1677
  - Consumption to GDP: Old CO Y: 0.0986 → 0.4336
  - Bonds to GDP: Old BO Y: 0.2579 → 8.4030
  - Interest rate elasticity: Young εY: 0.6250 → 0.6250
  - Interest rate elasticity: Old εO: 0.5000 → 0.5000
  - Frisch elasticity η: 0.5313 → 0.1922
  - GDP Y: 1.3883 → 1.2098
  - Investment to GDP Inv Y: 0.2378 → 0.1691
  - Capital to GDP K Y: 7.9677 → 7.4986
  - Profit to GDP Profit Y: 0.2969 → 0.3656
  - Real interest rate r: 0.0046 → 0.0065
  - Transfer to GDP TR Y: 0.1000 → 0.4000
  - Public consumption to GDP Gov Y: 0.2000 → 0.2295

### Mechanisms and steady-state implications highlighted in Section 4
- Retirement and survival probabilities imply retired accumulate more savings as lifetime increases (decreasing ωO).
- PAYG pension system requires higher pension benefits in the graying society, financed by taxes on the young — increasing young indebtedness.
- Behavioral assumption: older households are more patient (higher discount factor βO), amplifying young indebtedness.
- Supply-side adjustments with aging:
  - Lower fertility and a shrinking labor force lead firms to lower investment and reduce prices (disinflation/deflation) to offset profit loss.
  - Firms hire relatively more labor in the medium term; labor shortages increase real wage volatility as Frisch elasticity decreases.
  - Central bank initially decreases the interest rate to offset downward inflation pressure; in the long run normalizes policy rate as inflation returns to steady state.
  - Lower physical capital stock raises marginal product of capital and leads to a slightly higher equilibrium real interest rate in the new steady state.

### Cyclical behavior and monetary policy transmission (Section 5 preview)
- The paper compares short-run cyclical reactions of the macroeconomy to:
  - A monetary policy shock.
  - A government expenditure shock.
  - A supply shock.
- Only one-off, temporary shocks are examined; all impulse response functions converge back to the demographic-specific steady state.

*Source: wpiea2019198-print-pdf - Section 4 demonstrates the steady states of the young and old societies. Then, in Section 5 we — IMF working paper content provided*

### 5.1  Monetary Policy Shock

### 5.1  Monetary Policy Shock

### Mechanism and dynamics
- When the central bank deviates from its systematic Taylor-rule by increasing the nominal interest rate, price stickiness implies a change in the real interest rate that affects the real economy.
- Tighter monetary policy:
  - Forces firms to postpone investment.
  - Leads households to accumulate more savings or decrease their credit stock.
  - Shrinks aggregate demand, inducing share 1−ω_P of the firms to decrease their prices to avoid (reduce) the loss in profits.
- Due to price stickiness and price indexation, nominal adjustment is gradual, and disinflation takes more than 4 quarters.
- The central bank observes shrinking aggregate demand, decreases the nominal interest rate back to its steady-state level, and the economy stabilizes at its initial steady state.

### Effects of aging on the shock transmission
- Aging generates significant differences in the output gap reaction (the deviation of GDP from its flexible-price equilibrium level).
- Heterogeneous cohort responses:
  - In an old society, retired agents hold more savings, while workers have more debt over their longer life-time.
  - In a young society, monetary restriction incentivizes postponing consumption and increasing savings.
  - In an old society, the interest rate hike can be interpreted by the old as additional income, so they increase consumption by a larger amount, while the young face higher credit costs and decrease consumption.
- Aggregate consequences:
  - Because aging changes cohort sizes, aggregate consumption and output gap fall less in grey societies.
  - Aging reallocates asset positions among generations, making monetary policy less effective — defined in the chapter as a larger drop in the inflation rate to the same monetary tightening — and making aggregate demand less elastic to interest rate changes.
- Empirical consistency: finding consistent with Wong (2018), who shows that the old react less to expansionary monetary policy shocks based on micro-level cohort-specific US data.

### Simulation setup and parameter notes
- Disinflation horizon: "more than 4 quarters" (as reported).
- Cohort-size assumption in simulations:
  - In young societies (on Figures 2-4 labeled without Aging) the old-age dependency ratio is 20%,  in old societies (on Figures 2-4 labeledAging) t80%.
- On the same figures "ar" is an abbreviation for annualized rate.

### Key quantitative and qualitative observations (from impulse responses)
- Interest rate: initial tightening by the central bank (impulse depicted in Figure 2).
- Inflation: falls gradually (annualized, QoQ), disinflation greater and more persistent when monetary policy is characterized as less effective in older societies.
- Output gap, investment, young consumption, retired consumption: cohort-specific impulse responses differ in magnitude and sign between young and old societies — specifically:
  - Young consumption falls more in old societies following a monetary tightening (due to higher credit costs).
  - Retired consumption can increase in old societies when retirees perceive the rate hike as additional income.
- Net result: in older societies monetary tightening produces a smaller decline in aggregate consumption and output gap but a larger decline in inflation for a given tightening, reflecting reduced transmission of monetary policy to demand.

_ Source: wpiea2019198-print-pdf - 5.1  Monetary Policy Shock _

### 6.3  Optimal Monetary Policy

### 6.3 Optimal Monetary Policy

### Comparative welfare losses under unchanged (non-optimal) monetary rules
- Old-age dependency ratio (OADR) scenarios considered: s=0.20, s=0.40, s=0.60, s=0.80.
- Initial losses normalized to s=0.20: Initial losses = 1.000, 0.959, 0.947, 1.154 for s=0.20, s=0.40, s=0.60, s=0.80, respectively.
- Volatility and parameter outcomes (normalized by initial loss values):
  - Inflation volatility: 0.111, 0.126, 0.147, 0.191 for s=0.20, s=0.40, s=0.60, s=0.80.
  - Output gap volatility: 0.887, 0.831, 0.797, 0.959 for s=0.20, s=0.40, s=0.60, s=0.80.
  - Interest rate volatility: 0.002, 0.002, 0.003, 0.003 for s=0.20, s=0.40, s=0.60, s=0.80.
  - Frisch elasticity: 0.531, 0.441, 0.325, 0.192 for s=0.20, s=0.40, s=0.60, s=0.80.
- Stylized result: As OADR increases, inflation volatility and interest-rate volatility increase; in very high OADR cases output gap volatility can dominate because of rising volatility in labor-market variables.

### Optimal monetary policy reaction functions (six regimes) — main findings
- Monetary policy regimes analyzed: Pure IT, Pure IT & Forward (Fwd), Flexible IT, Flexible IT & Fwd, Price level targeting, Nominal GDP targeting.
- Key qualitative findings:
  - In a younger society (low OADR), a flexible inflation-targeting (Flex. IT) regime with a strong reaction to inflation and a somewhat weaker reaction to the output gap minimizes welfare loss.
  - In an older society (higher OADR), the central bank should strongly react to inflation (higher φπ) and mildly react to the output gap (lower φŶ) compared to younger societies.
  - Nominal GDP targeting becomes increasingly favorable with aging and can be the most effective rule in a very old society (extremely high OADR).
  - Price level targeting generally generates higher welfare losses than nominal GDP targeting and is relatively hawkish (central bank tolerates larger output sacrifice to stabilize price level).
- Representative quantitative outcomes from optimized calibrations (selected values from Table 4):
  - Initial rule: Initial loss = 1.000, 0.959, 0.947, 1.154 for s=0.20, s=0.40, s=0.60, s=0.80; φπ_t (initial) = 1.500 for all s.
  - Pure IT (optimized): Optimized loss = 0.924, 0.885, 0.832, 0.760 for s=0.20, s=0.40, s=0.60, s=0.80; φπ_t = 3.718, 4.217, 5.032, 6.751.
  - Pure IT & Fwd (optimized): Optimized loss = 0.872, 0.837, 0.646, 0.646; φE_t π_{t+1} = 6.353, 6.880, 7.835, 9.541.
  - Flexible IT (optimized): Optimized loss = 0.478, 0.495, 0.528, 0.584; φπ_t = 5.756, 6.105, 6.966, 8.744; φŶ_t = 3.428, 3.391, 3.209, 2.646.
  - Flexible IT & Fwd (optimized): Optimized loss = 0.492, 0.513, 0.550, 0.612; φE_t π_{t+1} = 6.102, 6.537, 7.468, 9.026; φŶ_t = 3.164, 3.088, 2.820, 2.122.
  - Price level targeting (optimized): Optimized loss = 0.913, 1.093, 1.032, 0.898; φ̂P_t = 0.017, 0.603, 1.250, 3.130.
  - Nominal GDP targeting (optimized): Optimized loss = 0.499, 0.498, 0.499, 0.481; φ̂P_t^{Y_t} = 5.920, 6.063, 6.457, 7.862.
- Interpretation:
  - Flexible IT minimizes loss at lower OADRs (s=0.20) with relatively large φπ and substantial φŶ.
  - As s increases, optimal φπ generally rises; optimal φŶ typically falls (central bank places more weight on stabilizing inflation relative to output).
  - Nominal GDP targeting yields consistently low optimized losses across OADRs and becomes the most effective at extreme OADR in some calibrations.

### Empirical evidence on aging, inflation level, and inflation volatility
- Historical relationship (1981–2017, developed economies U.S., U.K., Germany, France, Japan, Portugal):
  - Periods of disinflation correspond with periods of increases in the old-age dependency ratio (Figure 5 summary).
  - This empirical pattern aligns with the model’s steady-state implication that increasing OADR exerts downward pressure on inflation.
- FE regression on inflation volatility (unbalanced panel, OECD, 1993–2017):
  - Model estimated: σ(cpi_core)_{it} = α_i + δ_t + β OADR_{it} + γ X_{it} + u_{it}.
  - Data processing: non-food non-energy CPI detrended with HP-filter λ = 150000; yearly average standard deviation from trend used as time-variant volatility.
  - Table 5 FE regression coefficients (OADR effect and controls):
    - σ(cpi_core)_{i,t−1} coefficient: 0.087∗ (0.043) in FE (1); 0.083. (0.043) in FE (2).
    - OADR_{it} coefficients: 0.021∗ (0.011) in FE (1); 0.020∗ (0.010) in FE (2); 0.019∗ (0.009) in FE (3); 0.018∗ (0.008) in FE (4).
    - σ(Y)_{it} coefficients: 0.015∗ (0.008) in FE (3); 0.017∗ (0.008) in FE (4).
  - Sample and fit statistics:
    - Country FE: Yes; Year FE: Yes.
    - N.obs = 793 for FE (1) and FE (2); 759 for FE (3) and FE (4).
    - R^2 = 0.250, 0.254, 0.307, 0.318 for FE (1)–FE (4).
    - Significance notation: ∗∗∗ is p<0.1%, ∗∗ is p<1%, ∗ is p<5%, . is p<10%.
  - Quantitative interpretation highlighted in text:
    - Median volatility in sample: 0.19.
    - FE regressions imply a 10 percentage point increase in the old-age dependency ratio increases inflation volatility by around 0.18–0.20.

### Mechanisms: why aging alters monetary transmission and welfare
- Channels identified:
  - Longer lifespans and longer planning horizons change saving behavior: young borrow more; retired accumulate more savings.
  - Interest rate changes have asymmetric effects: higher interest rates increase the cost for indebted young and raise income for savers (old).
  - Differences in consumption-savings preferences: the old are more patient than the young, so consumption responses differ.
  - Labor market effects: labor force shrinkage tightens the labor market; real wages react more strongly.
- Consequences:
  - Monetary policy transmission becomes less effective in older societies.
  - The same-size shock causes higher inflation volatility in an older society.
  - Aging increases social welfare loss via higher inflation and higher inflation volatility.

### Policy implications and recommendations
- For inflation-targeting central banks:
  - Increase the nominal interest-rate response to inflation shocks as OADR rises: central banks with IT regimes should more strongly react to nominal variables (increase φπ).
  - Mildly reduce the policy response to the output gap (lower φŶ) as societies age, focusing relatively more on inflation stabilization.
- Consideration of alternative regimes:
  - Nominal GDP targeting can be favorable and may become the most effective regime when OADR is extremely high.
  - Price-level targeting tends to be hawkish and often yields higher welfare losses than nominal GDP targeting in the analyzed settings.
- Robustness checks (Appendix summary):
  - With a higher output-gap weight (1.0) nominal GDP targeting becomes optimal only at a higher OADR; with higher interest-rate smoothing weight (0.5) nominal GDP targeting can become most effective at lower OADR levels (alternative simulations reported in Appendix Tables A1 and A2).

### Contribution and research agenda
- Novel contributions claimed:
  - First paper to estimate the impact of aging on the volatility of inflation.
  - First to explore the impact of aging on short-run cyclical behavior and monetary policy transmission in an aged society using a multi-period DGE model with overlapping generations.
  - First to examine optimal monetary policy strategies in the presence of aging.
- Call for further research:
  - Authors urge more research to better understand implications of aging for central banks, central bankers, and the elderly.

*Source: wpiea2019198-print-pdf - 6.3 Optimal Monetary Policy*

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*Source: wpiea2019198-print-pdf - References*

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_Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019198-print-pdf.pdf_