## 25445. Farmer reserves a special thanks for Costas Azariadis, who suggested that he work on this problem more

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---

### Introduction
- New Keynesian classification:
  - Monetary policy active if the central bank raises the interest rate more than one-for-one in response to inflation; passive if less than one-for-one.
  - Fiscal policy active if the fiscal authority borrows to finance an arbitrary path of expenditure and taxes; passive if it adjusts expenditures and the tax rate to ensure fiscal solvency for all possible paths of the real interest rate.
- NK model uniqueness requirement:
  - Either monetary policy active and fiscal policy passive, or monetary policy passive and fiscal policy active.
  - Passive monetary policy with active fiscal policy → fiscal theory of the price level (FTPL).
- Post-2008 observation and FTPL debate:
  - Central banks maintained passive interest rate policies with a constant nominal interest rate peg for more than a decade; standard theory predicts indeterminate price level under a peg.
  - FTPL advocates claim price level is nevertheless uniquely determined when interest rate is pegged via reinterpretation of government debt accumulation as a debt valuation equation.
- Paper’s contribution:
  - Studies FTPL implications in Overlapping Generations (OLG) model rather than representative agent (RA) model.
  - Provides algorithm to construct steady-state equilibria for a class of T-period OLG models and studies local properties of dynamic equilibria around steady states.
  - Shows OLG models can possess indeterminate steady-state equilibria where money has value; provides a calibrated sixty-two generation example matching U.S. endowment profile.
- Key calibrated example results (sixty-two generation):
  - When monetary policy is passive and fiscal policy is active, the steady state displays two degrees of indeterminacy: both initial price level and real interest rate are not uniquely determined.
  - If monetary and fiscal policy are both active, degree of indeterminacy reduces from two to one: a linear combination of the real interest rate and the price level is determinate, but price level remains indeterminate.
  - Fiscal policy that does not respond to endogenous fluctuations in debt can, within parameter limits, be pursued without leading to exploding debt.

### Relationship to Previous Literature
- Core mechanism:
  - Local dynamics of prices near any steady-state in OLG models may be indeterminate of arbitrary degree (Kehoe and Levine, 1985).
- Differences from prior work:
  - Single good per period and a single type of person living T-periods; government issues interest-bearing nominal liability rather than fixed stock of money.
  - First long-lived example calibrated to U.S. micro data showing indeterminate steady-state when both monetary and fiscal policy are active.
  - Analysis restricted to linearized system around monetary steady state (contrast with prior work relying on global/non-linear dynamics).
- Relation to other strands:
  - Political-economy OLG literature often ignores indeterminacy or rules it out by calibration.
  - Distinct from bubbly/non-fundamental equilibria literature with credit constraints; here model is complete-markets OLG without credit constraints and contains ‘bubbly equilibria’.
  - Indeterminacy here does not rely on dynamic inefficiency, risk premia or liquidity effects; exists in dynamically efficient equilibria with no frictions other than lifecycle births/deaths.
  - Poses challenge to methods assuming unique transition paths (e.g., Eggertsson et al. (2019) fifty-six generation model with sticky prices).

### The Fiscal Theory of the Price Level (FTPL) — Mechanism and OLG Failure
- Government debt and definitions:
  - Nominal discount bond price Q_t, gross nominal interest rate i_t ≡ 1/Q_t − 1, gross inflation Π_{t+1} ≡ p_{t+1}/p_t.
  - Real debt b_t ≡ B_{t−1}/p_t; real primary deficit a^d_t ≡ g_t − τ_t.
  - Gross real return R_{t+1} ≡ (1 + i_t)/Π_{t+1} (Fisher parity).
- Real-form government budget equation:
  - b_{t+1} = R_{t+1} (b_t + d_t). (Equation (2))
- Debt valuation (forward iteration):
  - B_0 / p_1 = − Σ_{t=1}^∞ Q^1_t d_t + lim_{T→∞} Q^1_T b_T. (Equation (3))
- FTPL claim:
  - Treat Equation (3) as debt valuation equation that determines initial price level p_1 as function of fiscal policy path {−d_t}; under passive monetary policy FTPL implies unique p_1.
- OLG counterpoint:
  - In calibrated example, equilibrium real interest rates are not uniquely pinned down; Equation (3) may hold for more than one p_1.
  - Concludes FTPL logic does not extend to OLG model generally; indeterminacy can be more prevalent.

### Equilibria in the T-Generation OLG Model — Structure and Characterization
- Model features:
  - Government issues nominal liability B_t to finance real primary deficit d_t and pay interest at constant rate ī (bar i).
  - Inflation via Fisher parity: p_{t+1}/p_t = (1 + ī)/R_{t+1}, t = 1,...,∞.
  - Fiscal policy not responsive to outstanding debt → active fiscal policy; constant interest rate → passive monetary policy.
- Equilibrium equation sets:
  - Generic equations: market clearing in period T−1 and later (when agents are generic).
  - Non-generic equations: market clearing in first T−2 periods due to presence of T−1 non-generic generations born before period 1 with varying horizons and initial dollar assets/liabilities.
- Generic consumer problem and definitions:
  - Generation t maximizes lifetime utility U_t(c^t_t,...,c^t_{t+T−1}) subject to Σ_{k=t}^{t+T−1} Q^k_t (c^t_k − ẇ_{k−t+1}) ≤ 0.
  - Defines T excess demand functions ˜x^t_k(R_{t+1},...,R_{t+T−1}) ≡ ĉ^t_k − ẇ_{k−t+1}, and T−1 savings functions s^t_k computed recursively:
    - s^t_t = −˜x^t_t;
    - s^t_{k+1} = R_{k+1} s^t_k − ˜x^t_{k+1}, k = t,...,t+T−3.
  - Savings functions depend on R_{t+1},...,R_{t+T−1}.
- Non-generic consumers:
  - Indexed by birth date 1−j, j = 1,...,T−1; initial real wealth λ_{1−j} B_0 / p_1 ≡ ν_{1−j}.
  - Excess-demand functions depend on initial wealth distribution L ≡ {λ_0, λ_{−1},..., λ_{2−T}} (shares sum to 1 given B_0 ≠ 0).
- Generic equilibrium asset market clearing:
  - f(R_{t−T+3},...,R_{t+T−1}) = R_{t+1}^{−1} b_{t+1}. (Equation (6))
  - Combined with government accumulation b_{t+1} = R_{t+1} (b_t + d_t). (Equation (2))
- Non-generic equilibrium example and general count:
  - For T=4, G_4(Z^4_0) = 0 produces three restrictions on six elements of Z^4_0 = [R_5,R_4,R_3,R_2,b_2,b_1]'.
  - General T: unknown vector Z^T_0 has 3T−6 elements; function G_T(Z^T_0) has 2T−5 rows, leaving T−1 non-predetermined elements of X_{T−2}.

### Determinacy Properties of Equilibria — Local Dynamics and Blanchard-Kahn Conditions
- Linearization around steady state { R̄, b̄ }:
  - Deviations ˜R_t ≡ R_t − R̄ and ˜b_t ≡ b_t − b̄.
  - State vector X_t ≡ [R_{t+T−1}, R_{t+T−2},...,R_{t−T+4}, b_t]'.
  - Linear difference equation: J_1 ˜X_t = J_2 ˜X_{t−1}, t = T−1,... (Equation (8)), with J ≡ J_1^{−1} J_2.
- Proposition 1 (Blanchard-Kahn specialized):
  - Let K denote number of eigenvalues of J with modulus greater than 1.
  - If K > T−1: no bounded sequences satisfy equilibrium near X̄ → equilibrium does not exist.
  - If K = T−1: unique bounded sequence exists and converges to steady state (R̄, b̄) → steady state determinate.
  - If K ∈ {0,...,T−2}: there is a T−1−K dimensional subspace of initial conditions satisfying equilibrium equations; all sequences converge asymptotically to steady state → steady state indeterminate with degree T−1−K.
- Implication for FTPL:
  - In model with active fiscal and passive monetary policy (fiat money), FTPL predicts unique initial price level; calibrated example shows money can have value while FTPL fails — both initial price level and initial real interest rate can be indeterminate.

### A Sixty-Two Generation Example — Calibration and Key Features
- Construction:
  - Sixty-two generation model where each generation begins economic life at age 18 and a period corresponds to one year.
  - Preferences: u(c^t_t,...,c^t_{t+61}) = Σ_{i=1}^{62} β^{i−1} ([c^t_{t+i−1}]^α − 1)/α.
  - Model inspired by Kehoe and Levine (1983) three-generation example with hump-shaped income profile and high risk aversion.
- Key calibration inputs (Appendix A):
  - Representative generation: enters labour force at age 18, retires at age 66, lives to age 79.
  - Working-age income profile: Guvenen et al. (2015) data used for ages 25 to 60; ages 18–24 and 61–66 extrapolated by log-linear interpolation.
  - Retirement income: U.S. Social Security Administration data used for retirement portion.
  - Supplemental Security Income (SSI) in the U.S. estimated at 0.137% of GDP.
  - Baseline model parameters:
    - budget deficit d_t = 0
    - annual discount rate β = 0.953
    - elasticity of substitution α = 0.17 (corresponding to α = −5 and Arrow-Pratt risk aversion ρ = 6)
- Findings in sixty-two generation calibration:
  - Existence of a steady-state equilibrium where money has positive value.
  - Under passive monetary policy and active fiscal policy, steady state displays two degrees of indeterminacy (both initial price level and initial real interest rate indeterminate).
  - There exist indeterminate equilibria that are dynamically efficient (interest rate ≥ growth rate).

### Appendix A — Steady states, dynamics, and empirical relevance
- Steady states and properties:
  - Four steady-state equilibria found (labelled A, B, C, D).
  - Nature of equilibria:
    - Steady-States A, C, D: three equilibria in which steady-state government debt equals zero (autarkic steady-state equilibria).
    - Steady-State B: the golden-rule steady-state equilibrium; government debt positive at the golden-rule steady state.
  - Dynamic efficiency:
    - Steady-States B, C, D: associated with a non-negative interest rate and are dynamically efficient.
    - Steady-State A: associated with a negative interest rate of −47.5% and is dynamically inefficient.
  - Selected numeric steady-state entries preserved exactly:
    - Steady-State A: real interest factor value 0.525060611.
    - Steady-State C: real interest factor value 1.022060611.
    - Steady-State D: real interest factor value 1.13061610.
    - Steady-State B: government debt value reported as "153.7% of GDP".
- Dynamics: initial price shocks and persistence
  - Experiment: initial value of b_1 perturbed by 3% and real initial wealth of non-generic generations perturbed by 3% (a "3% shock to the initial price level").
    - R_2 restricted to equal its steady state value; all other elements of Z_0 allowed to respond to keep interest rates and debt on a convergent path.
    - Choice of 3% is arbitrary; other admissible equilibrium paths exist.
  - Findings:
    - Return to steady state from an arbitrary initial condition is extremely slow.
    - The model can endogenously generate prolonged periods of negative real interest rates.
    - Small deviations of initial conditions can have long-lasting effects; real interest rate may be negative for periods well in excess of ten years.
- Empirical relevance:
  - Long-run low-frequency real rate cycles generated by the model are similar to documented long-run real interest rate behavior in G7 countries (Figure 4 reproduced from Yi and Zhang (2017)).
  - Long-run real interest rates in Yi and Zhang (2017) are computed as 11-year centered moving averages of annual real interest rates.

### Robustness to alternative calibrations (Section 7)
- Table 2 explores 40 parameterizations: risk aversion parameters (RA) ranging from 5 to 9 and annual discount factors ranging from 0.91 to 0.98, holding the calibrated income profile fixed.
- For each calibration, Table 2 reports:
  - number of steady-state equilibria in the interval R̄ ∈ [0.5,1.5],
  - number of degrees of indeterminacy at the golden-rule steady state,
  - value of debt at the monetary steady state.
- Aggregate summary from Table 2:
  - 15 parameterizations in which the golden-rule steady state displays one degree of indeterminacy.
  - 12 parameterizations in which the golden-rule steady state displays two degrees of indeterminacy; in all 12 of these parameterizations debt has positive value in the steady state.
- Example row excerpts (exact entries preserved):
  - RA = 5 row entries for Annual Discount Factor 0.91..0.98:
    - # Steady States: 4 4 4 4 4 2 2 2
    - Degree of Indeterminacy: 1 1 1 1 1 2 0 0
    - Value of Debt: -2.8 -2.2 -1.5 -0.8 -0.2 0.5 1.1 1.8
  - RA = 6 row entries for Annual Discount Factor 0.91..0.98:
    - # Steady States: 4 4 4 4 4 4 2 2
    - Degree of Indeterminacy: 1 1 1 1 2 2 0 0
    - Value of Debt: -1.9 -1.3 -0.7 -0.2 0.4 0.9 1.5 2.0
  - RA = 9 row entries for Annual Discount Factor 0.91..0.98:
    - # Steady States: 3 3 3 3 3 3 3 3
    - Degree of Indeterminacy: 1 2 2 2 2 0 0 0
    - Value of Debt: -0.2 0.1 0.5 0.9 1.3 1.6 2.0 2.4
- Interpretation:
  - Two degrees of indeterminacy implies both the price level and the real interest rate can be indeterminate.
  - Only one degree of indeterminacy is required for violations of FTPL; this occurs more frequently than second-degree indeterminacy in the parameter grid.

### Fiscal and monetary policy implications (Section 8)
- Fisher parity identity and bounded inflation under peg:
  - lim_{T→∞} Π_T = lim_{T→∞} (1 + ī) / R̄ = 1 + ī / R̄, implying bounded inflation under interest rate peg if real rates are bounded.
- Passive fiscal policy (raising taxes in proportion to real debt):
  - modeled as d_t = −δ b_t with δ ≥ 0, leading to b_{t+1} = [R_{t+1} − δ] b_t.
  - For values of [R̄ − δ] < 1, passivity introduces an additional stability mechanism and increases the degree of indeterminacy at each steady state when δ is large enough.
  - Conclusion: Passive fiscal policy makes indeterminacy more likely.
- Taylor rule for monetary policy: 1 + i_t = (R̄ / Π̄^η) Π_t^{1+η}, t = 1,...,∞.
  - Taylor rule passive if −1 ≤ η ≤ 0; active if η > 0.
- Passive Taylor rule (η in passive range):
  - Linearized inflation law: ̃Π_{t+1} = (1 + η) ̃Π_t − κ ̃R_{t+1}, with κ ≡ Π̄ / R̄.
  - Iteration shows inflation bounded whenever R_t is bounded (Equation (11)/(C3)).
  - No additional equilibrium restrictions beyond those needed for real variables.
- Active Taylor rule (η > 0):
  - Forward-looking restriction on initial price level: p_1 = p_0 ( Π̄ + κ Σ_{s=1}^∞ (1/(1 + η))^s (R_{1+s} − R̄) ).
  - This restriction is additional to p_1 = B_0 / b_1 used to generate equilibrium sequences.
  - Active monetary policy eliminates one degree of indeterminacy (removes nominal indeterminacy), but does not eliminate real indeterminacy.
  - If model has one degree of indeterminacy under passive-active policy, active-active can yield a unique solution; if two-degree indeterminacy exists, active policy removes nominal freedom but leaves many possible initial real interest rates and associated equilibrium paths.

### Analytical appendix highlights
- Appendix A: analytic solutions for excess demand and consumption with CES preferences.
  - Generic optimization Problem 1 solution (consumption allocation formula): ˆc_{t}^{t−1+k} = a_ηk (Σ_{i=1}^T (Q_{t−1+i}^t ̃w_i)) (Q_{t−1+k}^t)^η / Σ_{i=1}^T (Q_{t−1+i}^t)^{1−η} a_ηi, k = 1,...,T.
  - Non-generic optimization Problem 2 solution for truncated lifetimes provided (Equation (A6)).
- Appendix B: conditions for local uniqueness of equilibrium via Jordan decomposition J = Q Λ Q^{−1} and restrictions from unstable eigenvalues.
  - K = number of eigenvalues of J outside the unit circle imposes K linear restrictions Q_u^{−1} X_{T−2} = 0.
  - Sufficient condition for unique equilibrium: matrix M = [G_X  G_Y; Q_u^{−1}  0] is square and nonsingular.
  - If K > T − 1, generally no solution; if K = T − 1 and M full rank, unique solution; if K ≤ T − 1, can append additional linear restrictions (e.g., b_1 = b̄_1) to pin down equilibrium.
- Appendix C: derivations for inflation dynamics under passive and active Taylor rules; key equations:
  - Passive Taylor rule linearized (C2): ̃Π_{t+1} = (1 + η) ̃Π_t − κ ̃R_{t+1}.
  - Active Taylor rule initial price-level restriction (C6): p_1 = p_0 ( Π̄ + κ Σ_{s=1}^∞ (1/(1 + η))^s (R_{1+s} − R̄) ).

### Conclusions and policy messages
- OLG models differ fundamentally from RA models: government debt need not net out within a lifetime, so changes in real interest rates are redistributive across generations and can induce self-stabilizing fluctuations in bond demand.
- Golden-rule steady state in calibrated sixty-two generation model can be dynamically efficient and second-degree indeterminate; price level and real interest rate may not be uniquely pinned down by fundamentals even with rational agents and rational expectations.
- Policy implications:
  - Monetary policy that pegs interest rates cannot necessarily rely on fiscal policy to uniquely anchor the price level in an OLG setting.
  - Active fiscal policy can be pursued without necessarily causing explosive debt dynamics even when monetary policy is active; fiscal policy can be stabilizing in OLG frameworks.
  - Active monetary policy removes nominal indeterminacy but may leave real indeterminacy; additional restrictions (or policies) would be required to select among multiple real equilibria.

*Source: IMF Working Paper wpiea2019219-print-pdf (content as provided).*

### 25445. Farmer reserves a special thanks for Costas Azariadis, who suggested that he work on this problem more

### wpiea2019219-print-pdf - 25445. Farmer reserves a special thanks for Costas Azariadis, who suggested that he work on this problem more

### Introduction
- New Keynesian classification:
  - Monetary policy active if the central bank raises the interest rate more than one-for-one in response to inflation; passive if less than one-for-one.
  - Fiscal policy active if the fiscal authority borrows to finance an arbitrary path of expenditure and taxes; passive if it adjusts expenditures and the tax rate to ensure fiscal solvency for all possible paths of the real interest rate.
- NK model uniqueness requirement:
  - Either monetary policy active and fiscal policy passive, or monetary policy passive and fiscal policy active.
  - Passive monetary policy with active fiscal policy → fiscal theory of the price level (FTPL).
- Post-2008 observation:
  - Central banks maintained passive interest rate policies with a constant nominal interest rate peg for more than a decade; standard theory predicts indeterminate price level under a peg.
  - FTPL advocates claim price level is nevertheless uniquely determined when interest rate is pegged via reinterpretation of government debt accumulation as a debt valuation equation.
- Paper’s contribution:
  - Studies FTPL implications in Overlapping Generations (OLG) model rather than representative agent (RA) model.
  - Provides an algorithm to construct steady-state equilibria for a class of T-period OLG models and studies local properties of dynamic equilibria around steady states.
  - Shows OLG models can possess indeterminate steady-state equilibria where money has value; provides a calibrated sixty-two generation example matching U.S. endowment profile.
- Key calibrated example results (sixty-two generation):
  - When monetary policy is passive and fiscal policy is active, the steady state displays two degrees of indeterminacy: both initial price level and real interest rate are not uniquely determined.
  - If monetary and fiscal policy are both active, degree of indeterminacy reduces from two to one: a linear combination of the real interest rate and the price level is determinate, but price level remains indeterminate.
  - Fiscal policy that does not respond to endogenous fluctuations in debt can, within parameter limits, be pursued without leading to exploding debt.

### Relationship to Previous Literature
- Core mechanism:
  - Local dynamics of prices near any steady-state in OLG models may be indeterminate of arbitrary degree (Kehoe and Levine, 1985).
- Differences from prior work:
  - Focus on a single good per period and a single type of person living T-periods; government issues interest-bearing nominal liability rather than fixed stock of money.
  - First long-lived example calibrated to U.S. micro data showing indeterminate steady-state when both monetary and fiscal policy are active.
  - Results not driven by global or non-linear dynamics; analysis restricted to linearized system around monetary steady state (contrast with Benhabib et al. (2001)).
- Relation to other strands:
  - Political-economy OLG literature often ignores indeterminacy or rules it out by calibration; Aiyagari (1985) result about convergence to RA model considered limiting curiosity here.
  - Distinct from literature on wealth distribution in stochastic OLG models and on bubbly/non-fundamental equilibria with credit constraints; here the model is complete-markets OLG without credit constraints and contains ‘bubbly equilibria’.
  - Contrasts with Bassetto and Cui (2018) and others: indeterminacy here does not rely on dynamic inefficiency, risk premia or liquidity effects; exists in dynamically efficient equilibria with no frictions other than lifecycle births/deaths.
  - Poses challenge to methods assuming unique transition paths (e.g., Eggertsson et al. (2019) study of fifty-six generation model with sticky prices).

### The Fiscal Theory of the Price Level (FTPL) — Mechanism and OLG Failure
- Government debt accumulation (nominal bonds, lump-sum taxes, consumption good price definitions):
  - Nominal discount bond price Q_t, gross nominal interest rate i_t ≡ 1/Q_t − 1, gross inflation Π_{t+1} ≡ p_{t+1}/p_t.
  - Real debt b_t ≡ B_{t−1}/p_t; real primary deficit a^d_t ≡ g_t − τ_t.
  - Gross real return R_{t+1} ≡ (1 + i_t)/Π_{t+1} (Fisher parity).
- Real-form government budget equation:
  - b_{t+1} = R_{t+1} (b_t + d_t).  (Equation (2))
- Forward iteration (debt valuation):
  - B_0 / p_1 = − Σ_{t=1}^∞ Q^1_t d_t + lim_{T→∞} Q^1_T b_T.  (Equation (3))
- FTPL claim:
  - Treat Equation (3) as debt valuation equation that determines initial price level p_1 as function of fiscal policy path {−d_t}.
  - Under passive monetary policy, FTPL implies unique p_1; other initial price levels infeasible because they lead to unbounded debt paths.
- OLG counterpoint in this paper:
  - In calibrated example, equilibrium real interest rates are not uniquely pinned down; Equation (3) may hold for more than one p_1.
  - Concludes FTPL logic does not extend to OLG model generally; indeterminacy can be more prevalent.

### Equilibria in the T-Generation OLG Model — Structure and Characterization
- Model features:
  - Government issues nominal liability B_t to finance real primary deficit d_t and pay interest at constant rate ī (bar i).
  - Inflation recovered via Fisher parity: p_{t+1}/p_t = (1 + ī)/R_{t+1}, t = 1,...,∞.
  - Fiscal policy not responsive to outstanding debt → active fiscal policy; constant interest rate → passive monetary policy.
- Two sets of equilibrium equations:
  - Generic equations: characterize market clearing in period T−1 and later (apply when all agents are generic).
  - Non-generic equations: characterize market clearing in first T−2 periods due to presence of T−1 non-generic generations born before period 1 with varying horizons and initial dollar financial assets/liabilities.
- Generic consumer problem:
  - Generation t maximizes lifetime utility U_t(c^t_t,...,c^t_{t+T−1}) subject to Σ_{k=t}^{t+T−1} Q^k_t (c^t_k − ẇ_{k−t+1}) ≤ 0.
  - Defines T excess demand functions ˜x^t_k(R_{t+1},...,R_{t+T−1}) ≡ ĉ^t_k − ẇ_{k−t+1}, and T−1 savings functions s^t_k computed recursively:
    - s^t_t = −˜x^t_t;
    - s^t_{k+1} = R_{k+1} s^t_k − ˜x^t_{k+1}, k = t,...,t+T−3.
  - Dependence notation s^t_k(R_{t+1},...,R_{t+T−1}).
- Non-generic consumers:
  - Indexed by birth date 1−j, j = 1,...,T−1; initial real wealth λ_{1−j} B_0 / p_1 ≡ ν_{1−j}.
  - Non-generic generations have excess-demand functions depending on initial wealth distribution L ≡ {λ_0, λ_{−1},..., λ_{2−T}} (shares sum to 1 given B_0 ≠ 0).
- Generic equilibrium equations:
  - Asset market clearing represented by f(R_{t−T+3},...,R_{t+T−1}) = R_{t+1}^{−1} b_{t+1}.  (Equation (6))
  - Combined with government accumulation b_{t+1} = R_{t+1} (b_t + d_t).  (Equation (2))
- Non-generic equilibrium equations:
  - For T=4 illustration, G_4(Z^4_0) = 0 produces three restrictions on six elements of Z^4_0 = [R_5,R_4,R_3,R_2,b_2,b_1]'.
  - General T case: unknown vector Z^T_0 contains 3T−6 elements; function G_T(Z^T_0) has 2T−5 rows and imposes 2T−5 restrictions on Z^T_0, leaving T−1 non-predetermined elements of X_{T−2}.

### Determinacy Properties of Equilibria — Local Dynamics and Blanchard-Kahn Conditions
- Linearization around steady state { R̄, b̄ }:
  - Define deviations ˜R_t ≡ R_t − R̄ and ˜b_t ≡ b_t − b̄.
  - State vector X_t ≡ [R_{t+T−1}, R_{t+T−2},...,R_{t−T+4}, b_t]'.
  - Function F(X_t, X_{t−1}) with partial derivatives J_1 and J_2; linear difference equation:
    - J_1 ˜X_t = J_2 ˜X_{t−1}, t = T−1,...  (Equation (8))
  - Matrix J ≡ J_1^{−1} J_2; local stability depends on eigenvalues of J.
- Proposition 1 (Blanchard-Kahn specialized):
  - Let K denote number of eigenvalues of J with modulus greater than 1.
  - If K > T−1: no bounded sequences satisfy equilibrium near X̄ → equilibrium does not exist.
  - If K = T−1: unique bounded sequence exists and converges to steady state (R̄, b̄) → steady state determinate.
  - If K ∈ {0,...,T−2}: there is a T−1−K dimensional subspace of initial conditions satisfying equilibrium equations; all sequences converge asymptotically to steady state → steady state indeterminate with degree T−1−K.
- Implication:
  - In the model with active fiscal and passive monetary policy (fiat money), FTPL predicts unique initial price level; however, the calibrated example shows money can have value while FTPL fails — both initial price level and initial real interest rate can be indeterminate.

### A Sixty-Two Generation Example — Calibration and Key Features
- Construction:
  - Sixty-two generation model where each generation begins economic life at age 18 and a period corresponds to one year.
  - Preferences: people maximize u(c^t_t,...,c^t_{t+61}) = Σ_{i=1}^{62} β^{i−1} ([c^t_{t+i−1}]^α − 1)/α (utility form stated in text).
  - Model inspired by Kehoe and Levine (1983) three-generation example with hump-shaped income profile and high risk aversion.
- Motivation and findings:
  - Calibrated income profile matches U.S. micro data; preference weights adjusted for constant discount rate.
  - Key features that generate indeterminacy: hump-shaped income profile and coefficient of relative risk aversion of 6 (noted as within bounds in macro-finance literature).
  - In the sixty-two generation calibration:
    - Existence of a steady-state equilibrium where money has positive value.
    - Under passive monetary policy and active fiscal policy, steady state displays two degrees of indeterminacy (both initial price level and initial real interest rate indeterminate).
    - There exist indeterminate equilibria that are dynamically efficient (interest rate ≥ growth rate).
- Robustness and additional analysis (as outlined for subsequent sections):
  - Section 7 checks robustness to alternative calibrations for the discount rate and coefficient of risk aversion.
  - Section 8 discusses changes when monetary and fiscal policy are both passive or both active.
  - Section 9 concludes.

*Source: IMF Working Paper wpiea2019219-print-pdf (content as provided).*

### Appendix A.

### Appendix A.

### Calibrated income profile and baseline parameters
- Representative generation: enters labour force at age 18, retires at age 66, lives to age 79.
- Working-age income profile: Guvenen et al. (2015) data used for ages 25 to 60; ages 18–24 and 61–66 extrapolated by log-linear interpolation.
- Retirement income: U.S. Social Security Administration data used for retirement portion.
- Supplemental Security Income (SSI) in the U.S. estimated at 0.137% of GDP.
- Baseline model parameters:
  - budget deficit d_t = 0
  - annual discount rate β = 0.953
  - elasticity of substitution α = 0.17 (corresponding to α = −5 and Arrow-Pratt risk aversion ρ = 6)
- For this calibrated income profile and parameter choice, the model exhibits four steady-state equilibria.

### Steady states and their properties
- Four steady-state equilibria found (labelled A, B, C, D).
- Nature of equilibria:
  - Steady-States A, C, D: three equilibria in which steady-state government debt equals zero (autarkic steady-state equilibria).
  - Steady-State B: the golden-rule steady-state equilibrium; government debt positive at the golden-rule steady state.
- Dynamic efficiency:
  - Steady-States B, C, D: associated with a non-negative interest rate and are dynamically efficient.
  - Steady-State A: associated with a negative interest rate of −47.5% and is dynamically inefficient.
- Table 1 reported equilibrium statistics (values and stability properties). Selected reported numeric entries:
  - Steady-State A: real interest factor value 0.525060611.
  - Steady-State C: real interest factor value 1.022060611.
  - Steady-State D: real interest factor value 1.13061610.
  - Steady-State B: associated government debt value reported as "153.7% of GDP" in the Table 1 layout.

### Dynamics: initial price shocks and persistence
- Experiment: initial value of b_1 perturbed by 3% and real initial wealth of non-generic generations perturbed by 3% (a "3% shock to the initial price level").
  - R_2 restricted to equal its steady state value; all other elements of Z_0 allowed to respond to keep interest rates and debt on a convergent path.
  - Note: choice of 3% is arbitrary; other admissible equilibrium paths exist.
- Findings from Figure 3 and discussion:
  - Return to steady state from an arbitrary initial condition is extremely slow.
  - The model can endogenously generate prolonged periods of negative real interest rates.
  - Small deviations of initial conditions can have long-lasting effects; real interest rate may be negative for periods well in excess of ten years.

### Empirical relevance: comparison to long-run real rates
- The long-run low-frequency real rate cycles generated by the model are similar to documented long-run real interest rate behavior in G7 countries (Figure 4 reproduced from Yi and Zhang (2017)).
- Long-run real interest rates in Yi and Zhang (2017) are computed as 11-year centered moving averages of annual real interest rates.

### Robustness to alternative calibrations (Section 7)
- Table 2 explores 40 parameterizations: risk aversion parameters (RA) ranging from 5 to 9 and annual discount factors ranging from 0.91 to 0.98, holding the calibrated income profile fixed.
- For each calibration, Table 2 reports:
  - number of steady-state equilibria in the interval R̄ ∈ [0.5,1.5],
  - number of degrees of indeterminacy at the golden-rule steady state,
  - value of debt at the monetary steady state.
- Aggregate summary from Table 2:
  - 15 parameterizations in which the golden-rule steady state displays one degree of indeterminacy.
  - 12 parameterizations in which the golden-rule steady state displays two degrees of indeterminacy; in all 12 of these parameterizations debt has positive value in the steady state.
- Example row excerpts from Table 2 (preserving exact entries):
  - RA = 5 row entries for Annual Discount Factor 0.91..0.98:
    - # Steady States: 4 4 4 4 4 2 2 2
    - Degree of Indeterminacy: 1 1 1 1 1 2 0 0
    - Value of Debt: -2.8 -2.2 -1.5 -0.8 -0.2 0.5 1.1 1.8
  - RA = 6 row entries for Annual Discount Factor 0.91..0.98:
    - # Steady States: 4 4 4 4 4 4 2 2
    - Degree of Indeterminacy: 1 1 1 1 2 2 0 0
    - Value of Debt: -1.9 -1.3 -0.7 -0.2 0.4 0.9 1.5 2.0
  - RA = 9 row entries for Annual Discount Factor 0.91..0.98:
    - # Steady States: 3 3 3 3 3 3 3 3
    - Degree of Indeterminacy: 1 2 2 2 2 0 0 0
    - Value of Debt: -0.2 0.1 0.5 0.9 1.3 1.6 2.0 2.4
- Interpretation:
  - Two degrees of indeterminacy implies both the price level and the real interest rate can be indeterminate.
  - Only one degree of indeterminacy is required for violations of the Fiscal Theory of the Price Level (FTPL); this occurs more frequently than second-degree indeterminacy in the parameter grid.

### Fiscal and monetary policy implications (Section 8)
- Fisher parity identity used: lim_{T→∞} Π_T = lim_{T→∞} (1 + ī) / R̄ = 1 + ī / R̄, implying bounded inflation under interest rate peg if real rates are bounded.
- Passive fiscal policy (raising taxes in proportion to real debt):
  - modeled as d_t = −δ b_t with δ ≥ 0, leading to b_{t+1} = [R_{t+1} − δ] b_t.
  - For values of [R̄ − δ] < 1, passivity introduces an additional stability mechanism and increases the degree of indeterminacy at each steady state when δ is large enough.
  - Conclusion: Passive fiscal policy makes indeterminacy more likely.
- Taylor rule for monetary policy: 1 + i_t = (R̄ / Π̄^η) Π_t^{1+η}, t = 1,...,∞.
  - Taylor rule passive if −1 ≤ η ≤ 0; active if η > 0.
- Passive Taylor rule (η in passive range):
  - Linearized inflation law: ̃Π_{t+1} = (1 + η) ̃Π_t − κ ̃R_{t+1}, with κ ≡ Π̄ / R̄.
  - Iteration shows inflation bounded whenever R_t is bounded (Equation (11)/(C3)).
  - No additional equilibrium restrictions beyond those needed for real variables.
- Active Taylor rule (η > 0):
  - Forward-looking restriction on initial price level: p_1 = p_0 ( Π̄ + κ Σ_{s=1}^∞ (1/(1 + η))^s (R_{1+s} − R̄) ).
  - This restriction is additional to p_1 = B_0 / b_1 used to generate equilibrium sequences.
  - Active monetary policy eliminates one degree of indeterminacy (removes nominal indeterminacy), but does not eliminate real indeterminacy.
  - If model has one degree of indeterminacy under passive-active policy, active-active can yield a unique solution; if two-degree indeterminacy exists, active policy removes nominal freedom but leaves many possible initial real interest rates and associated equilibrium paths.

### Analytical appendix highlights
- Appendix A: analytic solutions for excess demand and consumption with CES preferences.
  - Generic optimization Problem 1 solution (consumption allocation formula): ˆc_{t}^{t−1+k} = a_ηk (Σ_{i=1}^T (Q_{t−1+i}^t ̃w_i)) (Q_{t−1+k}^t)^η / Σ_{i=1}^T (Q_{t−1+i}^t)^{1−η} a_ηi, k = 1,...,T.
  - Non-generic optimization Problem 2 solution for truncated lifetimes provided (Equation (A6)).
- Appendix B: conditions for local uniqueness of equilibrium via Jordan decomposition J = Q Λ Q^{−1} and restrictions from unstable eigenvalues.
  - K = number of eigenvalues of J outside the unit circle imposes K linear restrictions Q_u^{−1} X_{T−2} = 0.
  - Sufficient condition for unique equilibrium: matrix M = [G_X  G_Y; Q_u^{−1}  0] is square and nonsingular.
  - If K > T − 1, generally no solution; if K = T − 1 and M full rank, unique solution; if K ≤ T − 1, can append additional linear restrictions (e.g., b_1 = b̄_1) to pin down equilibrium.
- Appendix C: derivations for inflation dynamics under passive and active Taylor rules; key equations:
  - Passive Taylor rule linearized (C2): ̃Π_{t+1} = (1 + η) ̃Π_t − κ ̃R_{t+1}.
  - Active Taylor rule initial price-level restriction (C6): p_1 = p_0 ( Π̄ + κ Σ_{s=1}^∞ (1/(1 + η))^s (R_{1+s} − R̄) ).

### Conclusions and policy messages
- Overlapping generations (OLG) models differ fundamentally from representative agent (RA) models: government debt need not net out within a lifetime, so changes in real interest rates are redistributive across generations and can induce self-stabilizing fluctuations in bond demand.
- The golden-rule steady state in the calibrated sixty-two generation model can be dynamically efficient and second-degree indeterminate; price level and real interest rate may not be uniquely pinned down by fundamentals even with rational agents and rational expectations.
- Implications for policy design:
  - Monetary policy that pegs interest rates cannot necessarily rely on fiscal policy to uniquely anchor the price level in an OLG setting.
  - Active fiscal policy can be pursued without necessarily causing explosive debt dynamics even when monetary policy is active; fiscal policy can be stabilizing in OLG frameworks.
  - Active monetary policy removes nominal indeterminacy but may leave real indeterminacy; additional restrictions (or policies) would be required to select among multiple real equilibria.

*Source: Appendix A of wpiea2019219-print-pdf.*

### References

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### Bubbles, asset prices, and macroeconomic fluctuations
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### Computational methods and mathematical tools
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### Secular stagnation, long-run interest rates, and macro evidence
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*Source: References section of wpiea2019219-print-pdf*

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