## _wp1214

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---

### I. INTRODUCTION — context and definitions
- Central banks implemented unconventional operations by accumulating risky assets to mitigate the financial turmoil that began in 2008 and the Euro zone fiscal crises.
- These operations altered central banks’ balance sheets in both size and substance; the magnitude was significant.
- Definition used: a quasi-fiscal policy is any policy action that affects the central banks’ balance sheets, with the exception of the traditional monetary policy used to stabilize inflation by controlling the policy interest rate.
- Example: credit easing operations alter the composition of central banks’ asset accounts and therefore are considered quasi-fiscal policies.
- If losses are incurred on central bank assets and fiscal authorities decide not to compensate these losses, that fiscal decision is also a quasi-fiscal policy because it decreases the central bank’s capital account.
- Model purpose and key questions:
  - (i) Do quasi-fiscal policies and the central banks’ balance sheets affect inflation?
  - (ii) Do the central banks’ balance sheets have implications for policy interactions between the central banks and fiscal authorities?
- Empirical and institutional motivation:
  - Concerns that quasi-fiscal policies may undermine central bank independence and ability to stabilize inflation (Sims, 2003; Goodfriend, 2011).
  - Goodfriend (2011) proposed accord principles between central banks and fiscal authorities to insulate central banks from quasi-fiscal shocks.
  - Klüh and Stella (2008) found financially weak central banks were ineffective in stabilizing inflation.
  - Stella and Lönnberg (2008) surveyed 135 central banks and found laws did not always guarantee fiscal authorities’ responsibility for central bank liabilities and fiscal authorities are not always prompt in recapitalizing central banks.
- Model innovation:
  - Separates the central bank’s flow budget constraint from the fiscal authority’s flow budget constraint.
  - Models the fiscal authority’s transfer rule for recapitalizing the central bank as the key quasi-fiscal policy in the benchmark model.

### Key mechanism: balance sheet shocks, portfolio adjustment, and inflation
- Mechanism summary:
  - Central bank balance sheet shocks affect inflation through private agents’ portfolio adjustment when fiscal authorities do not financially support central banks.
  - Price level is determined by a ratio between the nominal and real value of the central bank’s net liability in that policy regime.
- Illustrative scenario:
  - A central bank with negative capital suffers additional losses from long-term bond holdings while attempting to increase short-term interest rates to counter inflationary pressure.
  - As the real value of the central bank’s net liability falls below its equilibrium with the negative return shock, private agents decrease holdings of the central bank’s liability (nominal money balance) and increase consumption.
  - Result: the general price level increases and the central bank passively increases nominal money supply to satisfy real money balance demand.
  - Paradox: a hike in the policy rate (deflationary monetary policy) induces inflation in this case.
- Implication: the central bank cannot play an active role in stabilizing inflation when fiscal support is absent; fiscal backing is a pre-condition for effective monetary policy when the central bank performs additional roles (e.g., financial stability).

### Policy interactions and comparison with conventional FTPL models
- Conventional FTPL literature consolidates central bank and fiscal authority budget constraints, effectively assuming fiscal authorities acknowledge central bank assets/liabilities and automatically compensate central bank losses; under that approach the central bank’s budget constraint does not impose additional restrictions on equilibrium.
- This paper departs by explicitly modeling separated flow budget constraints and the fiscal transfer rule for recapitalization; these generate additional intertemporal equilibrium conditions.
- Assumption: the real values of central bank and fiscal authority liabilities have finite upper bounds (expected present values of their future earnings), excluding Ponzi-type equilibria.
- Count of equilibrium conditions and policy instruments:
  - This paper’s public sector model: two intertemporal equilibrium conditions (from separated flow budget constraints) and three policy instruments (monetary, fiscal, quasi-fiscal).
  - Conventional FTPL: one intertemporal equilibrium condition (from consolidated flow budget constraint) and two policy instruments (monetary and fiscal).

### Policy regime taxonomy and definitions
- Active quasi-fiscal policy regime (AQFP) defined:
  - Quasi-fiscal policy is “active” while monetary and fiscal policies are “passive.”
  - “Active quasi-fiscal policy” means fiscal authorities do not stabilize the central bank’s real capital and do not increase fund transfers to the central bank when losses are incurred.
  - In this regime, monetary and fiscal policies are passively adjusted and equilibrium inflation is uniquely determined by the central bank’s net liability valuation formula.
- Relationship to FTPL:
  - The model extends FTPL by incorporating an additional intertemporal condition and a quasi-fiscal instrument.
  - Distinctive result: a new type of equilibrium can exist even if the policy interest rate does not depend on the central bank’s balance sheet status (central bank follows a simple Taylor rule and will not generate seigniorage in response to balance-sheet concerns).

### Numerical illustration (Federal Reserve System asset-account change)
- Asset account was $894 billion at the end of 2007.
- Asset account increased to $2,266 billion at the end of 2008.

### Model structure and core equilibrium conditions (Sections II–III; summary)
- Closed-economy DSGE with money in the utility function; output given by exogenous endowment process.
- Representative agent yields:
  - Money demand equation (Euler): Equation (2).
  - Fisher equation: Equation (3).
  - Transversality / No-Ponzi condition: Equations (4) and (5).
- Market clearing:
  - Goods: Equation (6).
  - Nominal money: Equation (7).
  - Government bonds (including central bank demand): Equation (8).
- After market clearing, equilibrium conditions under standard preferences:
  - Real money market: Equation (9).
  - Fisher relation in equilibrium: Equation (10).
- Public sector:
  - Separated central bank and fiscal authority flow budget constraints: Equations (12) and (13).
  - Separated intertemporal equilibrium conditions (IECs):
    - Central bank net liability valuation formula: Equation (19).
    - Fiscal authority liability valuation formula: Equation (20).
  - Core separation enabled by boundedness conditions on real liabilities: Conditions (15) and (16) (equivalently (17) and (18)).

### Policy rules and quasi-fiscal instrument (Section III)
- Monetary policy: simple Taylor-type rule for gross nominal interest rate: Equation (23); central bank does not target its capital via interest-rate response.
- Quasi-fiscal policy: transfer from fiscal authority to central bank rule: Equation (24).
  - Key parameter Qg: responsiveness of transfer to deviation of central bank real capital from target cap.
  - When Qg = 0, transfer is not responsive to central bank capital (quasi-fiscal policy can be active).
- Fiscal surplus rule: Equation (25) with parameter Fg and reaction to real debt Atb.
- Clarification: quasi-fiscal policy defined as any non-monetary action affecting central bank balance sheet; in the paper the transfer tk is the representative quasi-fiscal instrument.
- Active vs passive designation follows Leeper (1991) logic—activeness defined by whether instrument responds to the relevant state variable.

### Linearized system and regimes of policy interaction (Section III)
- Log utility and constant output simplify to closed-form linearized equilibrium: Equations (26)–(30).
- Linearized dynamic system in form (31); endogenous vector includes deviations of central-bank and fiscal debt and price (ˆ(, , )CA ttt xbbp´).
- Proposition 2: three regions of policy parameter space deliver unique stationary solutions; only one policy should be active for a unique solution:
  - AMP regime (active monetary policy): satisfies the inequalities stated in the proposition.
  - AFP regime (active fiscal policy): satisfies the inequalities stated in the proposition.
  - AQFP regime (active quasi-fiscal policy): satisfies the inequalities stated in the proposition.
- Intuition for regimes:
  - AMP: monetary policy pins down inflation; transfer tk must support central bank liability via IEC (19); fiscal policy passive.
  - AQFP: transfer tk is exogenous in IEC (19) and pins down inflation; monetary and fiscal policies passive.
  - AFP: fiscal policy pins down inflation via unified IEC (14); monetary and quasi-fiscal policies passive.
- Passive monetary policy adjusts nominal money balances to satisfy money market clearing given price level determined elsewhere; it does not need to raise seigniorage.

### Active Quasi-Fiscal Policy (AQFP) equilibrium — mechanism and quantitative implications
- In AQFP, quasi-fiscal shocks (including central bank earning shocks) affect surprise inflation through private-agent portfolio adjustments.
- Special illustrative parameterization (Equation (35)): 1, , , , 0, 1. QFqxy ag r r rb g - = - <.
- Under this specification surprise inflation equals (Equation (36)):
  - 2 , tt C Rbm x p h e = - .
- Full inflation decomposition in AQFP (Equation (37)):
  - 2 1 ˆ . ttt C Rbm qx p p b e e - = + - (where 1tq be - is expected inflation).
- Sign of quasi-fiscal shock effect depends on steady-state central bank capital:
  - If central bank steady-state capital is negative (net liability), a negative transfer shock can induce inflation (private agent consumes more, holds less money).
  - If central bank steady-state capital is positive, the same negative transfer shock can induce deflation (private agent consumes less, holds more money).
- Observational equivalence: AQFP outcomes (inflation, nominal money balance, fiscal surplus adjustments) can be observationally identical to AMP or AFP regimes despite different causal directions.

### Application to exit strategy (Section IV)
- Exit strategy: plan to reverse unconventional quasi-fiscal operations by reducing central bank balance sheet (selling acquired assets) and “reduce excess reserve balances” (Bernanke, 2009).
- Extended model includes short- and long-term nominal bonds (1,tB and 2,tB), with gross nominal rates 1,tR and 2,tR and market pricing of premature long-term bonds.
- Agent’s optimization revised: Equation (40); additional Euler condition enforces expectations theory of term structure: Equation (41).
- Market clearing for short- and long-term bonds: Equations (42) and (43).
- Central bank and fiscal flow budget constraints with long-term bonds: Equations (44) and (45).
- Revised quasi-fiscal and fiscal rules with bond composition: Equations (46) and (47); maturity structure specified by (48) and (49) with policy parameter d and maturity shock tm.
- Key analytical result for AQFP special case (as in (35)) yields surprise inflation (Equation (50)):
  - 2 1 11 , (1) C ttt C b Rbm x q d p h e e p d æö ÷ ç ÷ ç = - ÷ ç ÷ ç ÷ + - èø .
- Central insight for exit:
  - Increasing short-term policy rate during exit can devalue central bank long-term bond holdings (valuation loss).
  - In AQFP, when central bank steady-state capital is negative (11 (1)0 C Rbmd + - <), a contractionary monetary policy shock (positive tq e) can induce surprise inflation rather than reduce inflation.
  - Mechanism: valuation loss reduces central bank net worth; private agent shifts portfolio toward consumption and away from nominal money, inducing inflation; central bank then passively increases nominal money balance to clear money market.
  - Paradox: intended tightening to reduce inflation may instead increase inflation unless fiscal authority recapitalizes the central bank.
- Policy implication for exit strategy:
  - Systematic fiscal recapitalization of central bank is the straightforward remedy to avoid paradoxical inflationary effects during exit.
  - Coordination between fiscal authority and central bank is essential to isolate perverse effects of quasi-fiscal shocks on inflation.

### Conclusion and policy recommendations (Section V)
- Main finding: quasi-fiscal policies that alter central bank balance sheets can determine inflation in the AQFP regime where quasi-fiscal policy is active and monetary and fiscal policies are passive.
- The central bank’s net liability valuation formula (an asset-pricing relation) can bind the equilibrium price level when the real value of central bank net liabilities is bounded by present value of seigniorage and transfers.
- Economic mechanism: private-agent portfolio adjustments in response to quasi-fiscal shocks or central bank asset-return shocks transmit to inflation.
- Practical recommendations:
  - Fiscal–monetary coordination is indispensable: fiscal authority should stabilize/target central bank capital to prevent quasi-fiscal shocks from affecting inflation.
  - During exit strategies, absent fiscal support, tightening monetary policy can have unintended inflationary consequences if the central bank has negative capital; therefore systematic recapitalization plans should accompany asset unwinding.
- Institutional note: assumption that central bank net liability has finite upper bound (present value of seigniorage and transfers) is argued to be reasonable historically and under modern central bank independence.

### Appendices: technical proofs and solution method (APPENDIX I and II)
- APPENDIX I:
  - Shows equivalence between Conditions (15)/(16) and Conditions (A10)/(A11).
  - Summing flow budget constraints (12) and (13) and taking discounted expected values yields (A1) and (A2).
  - From Conditions (15) and (16) derive (A3)-(A6) and near-bound conditions (A7)-(A9), concluding (A10) and (A11).
  - Converse argument: (A10) and (A11) imply (15) and (16).
  - References used: Equations (5), (12), (13), (A1)-(A11).
- APPENDIX II:
  - Linearization of Fisher equation (28) and budget constraints (12) and (13) forms dynamic system (A12).
  - Substitutions: money market equilibrium (27) and policy rules (23), (24), (25).
  - Stacked form transforms (A12) into (A13); coefficient matrices denoted 010 ,,GGF and 1 F.
  - Eigen-decomposition: A = P L P-1; P-1 given in (A14) with explicit block entries.
  - Solution method: eliminate explosive eigenvalues by enforcing 0 j t PY = γ for explosive eigenvalues; linear mappings from exogenous shocks to endogenous forecast errors given by 0 j t PC z = γ.
  - References: Equations (23), (24), (25), (27), (28), (A12)-(A14).

*Source: _wp1214 - References; Sections I–V; APPENDIX I; APPENDIX II (as provided in the content unit).*

### References .............................................................................................................

### _wp1214 - References .............................................................................................................

### I. INTRODUCTION — context and definitions
- Central banks implemented unconventional operations by accumulating risky assets to mitigate the financial turmoil that began in 2008 and the Euro zone fiscal crises.
- These operations altered central banks’ balance sheets in both size and substance; the magnitude was significant.2
- Definition used in the paper: a quasi-fiscal policy is any policy action that affects the central banks’ balance sheets, with the exception of the traditional monetary policy used to stabilize inflation by controlling the policy interest rate.3
- Example: credit easing operations alter the composition of central banks’ asset accounts and therefore are considered quasi-fiscal policies.
- If losses are incurred on central bank assets and fiscal authorities decide not to compensate these losses, that fiscal decision is also a quasi-fiscal policy because it decreases the central bank’s capital account.

### Model purpose and key questions
- The paper proposes a simple dynamic stochastic general equilibrium (DSGE) model to address:
  - (i) Do quasi-fiscal policies and the central banks’ balance sheets affect inflation?
  - (ii) Do the central banks’ balance sheets have implications for policy interactions between the central banks and fiscal authorities?

### Empirical motivation and institutional detail
- Concerns exist that quasi-fiscal policies may undermine central bank independence and ability to stabilize inflation (Sims, 2003; Goodfriend, 2011).
- Goodfriend (2011) proposed accord principles between central banks and fiscal authorities to insulate central banks from quasi-fiscal shocks.
- Klüh and Stella (2008) found financially weak central banks were ineffective in stabilizing inflation.
- Stella and Lönnberg (2008) surveyed 135 central banks and found laws did not always guarantee fiscal authorities’ responsibility for central bank liabilities, and fiscal authorities are not always prompt in recapitalizing central banks.
- The paper relaxes conventional assumptions by:
  - Separating the central bank’s flow budget constraint from the fiscal authority’s flow budget constraint.
  - Modeling the fiscal authority’s transfer rule for recapitalizing the central bank as the key quasi-fiscal policy in the benchmark model.

### Key mechanism: balance sheet shocks, portfolio adjustment, and inflation
- The model predicts central bank balance sheet shocks affect inflation through private agents’ (households’) portfolio adjustment when fiscal authorities do not financially support central banks.
- Price level determination: in that policy regime, the price level is determined by a ratio between the nominal and real value of the central bank’s net liability.
- Illustrative scenario from the text:
  - A central bank with negative capital suffers additional losses from long-term bond holdings while attempting to increase short-term interest rates to counter inflationary pressure.
  - As the real value of the central bank’s net liability falls below its equilibrium with the negative return shock, private agents attempt to decrease holdings of the central bank’s liability (nominal money balance) and increase consumption.
  - Result: the general price level increases and the central bank passively increases nominal money supply to satisfy real money balance demand. Paradox: a hike in the policy rate (deflationary monetary policy) induces inflation in this case.
- Implication: the central bank is confined and cannot play an active role in stabilizing inflation when fiscal support is absent.
- Fiscal backing is a pre-condition for effective monetary policy when the central bank is engaged in other policy roles such as maintaining financial stability.

### Policy interactions and comparison with conventional FTPL models
- Conventional literature (Sargent and Wallace (1981), Leeper (1991), Sims (1994)) consolidates central bank and fiscal authority budget constraints, effectively assuming fiscal authorities acknowledge central bank assets/liabilities and automatically compensate central bank losses.
- Under that conventional approach, the central bank’s budget constraint does not impose additional restrictions on equilibrium.
- This paper departs by explicitly modeling separated flow budget constraints and the fiscal transfer rule for recapitalization; these details generate additional intertemporal equilibrium conditions.
- The paper also assumes the real values of central bank and fiscal authority liabilities have finite upper bounds (the expected present values of their future earnings), which excludes Ponzi-type equilibria where real value of central bank capital grows or shrinks infinitely.
- Number of equilibrium conditions and policy instruments:
  - This paper’s public sector model: two intertemporal equilibrium conditions (from separated flow budget constraints) and three policy instruments (monetary, fiscal, quasi-fiscal).
  - Conventional FTPL: one intertemporal equilibrium condition (from consolidated flow budget constraint) and two policy instruments (monetary and fiscal).

### Policy regime taxonomy and results
- Definition: In the relevant regime, quasi-fiscal policy is “active,” while monetary and fiscal policies are “passive.”4
  - “Active quasi-fiscal policy” means fiscal authorities do not stabilize the central bank’s real capital and do not increase fund transfers to the central bank when losses are incurred.
  - Monetary and fiscal policies are passively adjusted to satisfy other equilibrium conditions, so equilibrium inflation is uniquely determined by the central bank’s net liability valuation formula in this regime.
- The model is an extension of the Fiscal Theory of Price Level (FTPL) literature (e.g., Leeper (1991)) by incorporating an additional intertemporal condition and a quasi-fiscal instrument.

### Relation to prior theoretical results
- Jeanne and Svensson (2007): if central bank losses push capital below a fixed level, central bank commitment to escape a liquidity trap is more credible.
- Sims (2003): central bank balance sheet concerns might undermine the ability to prevent inflation.
- Berriel and Bhattarai (2009): optimal monetary policy differs when central bank and fiscal authority budget constraints are separate; if central banks place higher effective weight on inflation in the loss function, inflation variation decreases.
- Distinction of this paper: a new type of equilibrium can exist even if the policy interest rate does not depend on the central bank’s balance sheet status. The central bank here follows a simple Taylor rule and will not generate seigniorage in response to its balance sheet concerns.

### Numerical illustration from the source
- Example asset-account change for the Federal Reserve System cited in the text:
  - asset account was $894 billion at the end of 2007
  - increased to $2,266 billion at the end of 2008

### Paper organization (as stated in the provided content)
- Section II: builds the model for rational-expectations general equilibrium in exact nonlinear forms.
- Section III: linearizes equilibrium conditions around the deterministic steady state to derive analytic solutions and explains the equilibrium in the active quasi-fiscal policy regime.
- The benchmark model is extended to include the exit strategy (continuation beyond provided excerpt).

*Source: _wp1214 - References .............................................................................................................*

### Section IV, and Section V concludes the paper.

### _wp1214 - Section IV, and Section V concludes the paper.

### Model structure and key equilibrium conditions
- Closed-economy DSGE with money in the utility function; output given by exogenous endowment process.
- Representative agent optimization yields:
  - Money demand equation (Euler): Equation (2).
  - Fisher equation: Equation (3).
  - Transversality / No-Ponzi condition: Equations (4) and (5).
- Market clearing:
  - Goods: Equation (6).
  - Nominal money: Equation (7).
  - Government bonds (including central bank demand): Equation (8).
- After market clearing, equilibrium conditions under standard preferences:
  - Real money market: Equation (9).
  - Fisher relation in equilibrium: Equation (10).
- Public sector modeled with separated central bank and fiscal authority budget constraints (Equations (12) and (13)), leading under additional expectation assumptions to separated intertemporal equilibrium conditions (IECs) for:
  - Central bank net liability valuation formula: Equation (19).
  - Fiscal authority liability valuation formula: Equation (20).
- Core assumption enabling separation: the real values of central bank and fiscal authority liabilities are bounded by expected present value of future earnings (Conditions (15) and (16); equivalently (17) and (18)). The paper argues these are empirically and institutionally plausible.

### Policy rules and quasi-fiscal policy
- Monetary policy: simple Taylor-type rule for gross nominal interest rate (Equation (23)); central bank does not target its capital via interest-rate response.
- Quasi-fiscal policy (transfer from fiscal authority to central bank) rule: Equation (24). Key parameters:
  - Qg: responsiveness of transfer to deviation of central bank real capital from target cap.
  - When Qg = 0, transfer is not responsive to central bank capital (quasi-fiscal policy can be active).
- Fiscal surplus rule: Equation (25) with parameter Fg and fiscal reaction to real debt Atb.
- Clarification: quasi-fiscal policy defined as any non-monetary action affecting central bank balance sheet; in the paper the transfer tk is modeled as representative quasi-fiscal instrument. Active vs passive policy concept follows Leeper (1991) logic—activeness defined by whether instrument responds to the relevant state variable.

### Linearized system and regimes of policy interaction
- Log utility, constant output simplification yields closed-form linearized equilibrium (Equations (26), (27), (28), (29), (30)).
- Linearized dynamic system put in form (31); endogenous vector includes deviations of central-bank and fiscal debt and price (ˆ(, , )CA ttt xbbp´).
- Proposition 2: three regions of policy parameter space deliver unique stationary solutions. Only one policy should be active for a unique solution:
  - AMP regime (active monetary policy): condition (inequalities stated in proposition).
  - AFP regime (active fiscal policy): condition (inequalities stated in proposition).
  - AQFP regime (active quasi-fiscal policy): condition (inequalities stated in proposition).
- Intuition:
  - If monetary policy active (1ab>), monetary policy pins down inflation; transfer tk must support central bank liability via IEC (19); fiscal policy passive.
  - If quasi-fiscal policy active (1
Q
bg
-
->), the transfer tk is exogenous in IEC (19) and thus pins down inflation; monetary and fiscal policies passive.
  - If fiscal policy active, it pins down inflation via unified IEC (14); monetary and quasi-fiscal policies passive.
- Key point: passive monetary policy does not need to raise seigniorage to balance intertemporal budgets; instead it adjusts nominal money balances to satisfy money market clearing given the price level determined elsewhere.

### Active Quasi-Fiscal Policy (AQFP) equilibrium — mechanism and implications
- In AQFP, quasi-fiscal shocks (including central bank earning shocks) affect surprise inflation through private-agent portfolio adjustments.
- Special illustrative parameterization (Equation (35)): 1, , , , 0, 1. QFqxy ag r r rb g - = - <. Under this specification surprise inflation equals (Equation (36)):
  - 2 , tt C Rbm x p h e = - .
- Full inflation decomposition in AQFP (Equation (37)):
  - 2 1 ˆ . ttt C Rbm qx p p b e e - = + - (where 1tq be - is expected inflation).
- Sign of effect of a quasi-fiscal shock depends on steady-state central bank capital:
  - If central bank steady-state capital is negative (net liability), a negative transfer shock can induce inflation (private agent consumes more, holds less money).
  - If central bank steady-state capital is positive, the same negative transfer shock can induce deflation (private agent consumes less, holds more money).
- Observational equivalence: AQFP equilibrium outcomes (inflation, nominal money balance, fiscal surplus adjustments) can be observationally identical to AMP or AFP regimes, though causal directions differ.

### Application to exit strategy (Section IV)
- Exit strategy defined as plan to reverse unconventional quasi-fiscal operations by reducing central bank balance sheet (selling acquired assets) and “reduce excess reserve balances” (Bernanke, 2009).
- Extended model includes short- and long-term nominal bonds (1,tB and 2,tB), with gross nominal rates 1,tR and 2,tR and market pricing of premature long-term bonds.
- Agent’s optimization revised (Equation (40)); additional Euler condition enforces expectations theory of term structure (Equation (41)).
- Market clearing for short- and long-term bonds: Equations (42) and (43).
- Central bank and fiscal flow budget constraints with long-term bonds: Equations (44) and (45).
- Revised quasi-fiscal and fiscal rules with bond composition: Equations (46) and (47); maturity structure specified by (48) and (49) with policy parameter d and maturity shock tm.
- Key analytical result (Equation (50)) for AQFP special case (as in (35)) yields surprise inflation:
  - 2 1 11 , (1) C ttt C b Rbm x q d p h e e p d æö ÷ ç ÷ ç = - ÷ ç ÷ ç ÷ + - èø .
- Central insight for exit strategy:
  - Increasing short-term policy rate during exit can devalue central bank long-term bond holdings (valuation loss).
  - In AQFP, and when central bank steady-state capital is negative (11 (1)0 C Rbmd + - <), a contractionary monetary policy shock (positive tq e) can induce surprise inflation rather than reduce inflation.
  - Mechanism: valuation loss reduces central bank net worth; private agent adjusts portfolio toward consumption and away from nominal money, inducing inflation. Central bank then must passively increase nominal money balance to clear money market.
  - Paradox: intended tightening to reduce inflation may instead increase inflation unless fiscal authority recapitalizes the central bank.
- Policy implication from exit-strategy application:
  - Systematic fiscal recapitalization of central bank is the straightforward remedy to avoid paradoxical inflationary effects during exit.
  - Coordination between fiscal authority and central bank is essential to isolate perverse effects of quasi-fiscal shocks on inflation.

### Conclusion and policy recommendations (Section V)
- Main finding: quasi-fiscal policies that alter central bank balance sheets can determine inflation in a specific policy regime (AQFP) where quasi-fiscal policy is active and monetary and fiscal policies are passive.
- The central bank’s net liability valuation formula (an asset-pricing relation) can bind the equilibrium price level when the real value of central bank net liabilities is bounded by present value of seigniorage and transfers.
- Economic mechanism: private-agent portfolio adjustments in response to quasi-fiscal shocks or central bank asset-return shocks transmit to inflation.
- Practical recommendations:
  - Fiscal–monetary coordination is indispensable: fiscal authority should stabilize/target central bank capital to prevent quasi-fiscal shocks from affecting inflation.
  - During exit strategies, absent fiscal support, tightening monetary policy can have unintended inflationary consequences if the central bank has negative capital; therefore systematic recapitalization plans should accompany asset unwinding.
- Institutional note: assumption that central bank net liability has finite upper bound (present value of seigniorage and transfers) is argued to be reasonable historically and under modern central bank independence.

*Source: _wp1214 - Section IV, and Section V concludes the paper.*

### APPENDIX I. PART OF THE PROOF FOR PROPOSITION 1

### APPENDIX I. PART OF THE PROOF FOR PROPOSITION 1

### Equivalence between Conditions (15)/(16) and Conditions (A10)/(A11)
- Conditions (15) and (16) imply Conditions (17) and (18).
- Summing the flow budget constraints (12) and (13) from t to 1T- and taking the discounted expected value at period t yields Equations (A1) and (A2) for all 1Tt³+.
  - Equation (A1): (displayed as in source) (A1)
  - Equation (A2): (displayed as in source) (A2)
- From Conditions (15) and (16), equalities (A3) and (A4) hold:
  - (A3) (as in source)
  - (A4) (as in source)
- Taking T¥ in (A1) and (A2) gives limits (A5) and (A6):
  - (A5) (as in source)
  - (A6) (as in source)
- Equations (A3)-(A6) imply the near-bound conditions:
  - (A7): ,lim    []    0, C ttTT T EX d ¥ -¥ <£
  - (A8): ,lim    []    0. F ttTT T EX d ¥ -¥ <£
- Using the agent’s transversality condition (5) and the identity ACF TTT wdd=+:
  - (A9): ,lim    []    lim    []    0, CF t   tTTt   tTT TT EX dEX d ¥¥ +=
- Therefore limits equal zero:
  - (A10): ,lim    []    0, C ttTT T EX d ¥ =
  - (A11): ,lim    []    0. F ttTT T EX d ¥ =
- Conversely, Conditions (A10) and (A11) imply assumptions (15) and (16):
  - Starting from budget constraints (A1) and (A2) and taking T¥ yields (A3) and (A4) by equality.
  - Since Ct d and Ft d are finite, Conditions (15) and (16) necessarily hold.
- Conclusion: Conditions (15) and (16) are equivalent to Conditions (A10) and (A11).

*Proof references: Equations (5), (12), (13), (A1)-(A11) in the source text.*

---

### APPENDIX II. SOLUTION OF THE DYNAMIC SYSTEM IN SECTION III

### Linearization and system construction
- The model linearizes:
  - the Fisher equation (28),
  - the budget constraints ((12) and (13)).
- These three linearized equations form the dynamic system displayed as matrix equation (A12).
  - First row of (A12): from the Fisher equation.
  - Second row of (A12): from the central bank budget constraint.
  - Third row of (A12): from the fiscal authority budget constraint.
- Substitutions into (A12):
  - Money market equilibrium condition (27) and policy rules ((23), (24) and (25)) are substituted.
- Coefficient matrices in (A12) are denoted 010 ,,GGF and 1 F, where variables without time index refer to steady state values.
- Full system as presented: (A12) (see source for the full matrix expression and coefficient definitions).

### Stacked form and matrix A
- Stacking vectors t x and t z transforms (A12) into Equation (A13).
  - (A13) given in the source (full stacked matrix representation).
- Numerical/structural definitions appearing in the source:
  - Scalar/vector definitions and combinations noted: 1 62173 12 1 81425938 ,,,()    ,.
  - Full coefficient expressions shown in the source.

### Eigen-decomposition and solution method
- The eigenvalues of transition matrix A can be read off from A’s diagonal elements.
- Eigenvector decomposition: A = P L P-1 where:
  - L is diagonal with eigenvalues.
  - Columns of P are right eigenvectors of A.
- Inverse P-1 is expressed as matrix (A14) in the source:
  - (A14) shown explicitly (1000 0 / 100 / 010 / ... structure as in source).
- Definitions for P-1 block entries are given in the source (QQQ, FFF, ppp, etc.) with exact algebraic combinations preserved there.
- Solution approach:
  - Eliminate explosive eigenvalues.
  - If the j’th eigenvalue is explosive, enforce 0 j t PY = γ where j P γ refers to the j’th row of P-1.
  - The linear mappings from the exogenous shock to the endogenous forecast error are given by 0 j t PC z = γ.
- The source provides the explicit matrices and mapping relationships used to derive the non-explosive solution (see (A12)–(A14) for detailed expressions).

*Proof and solution references: Equations (23), (24), (25), (27), (28), (A12), (A13), (A14) in the source text.*

*Source: IMF staff paper appendix text (APPENDIX I and APPENDIX II) as provided in the content unit.*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp1214.pdf_