## _wp1032

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---

### 1.1 Related Literature

- Context and motivation
  - Monetary policy in many countries is conducted by Monetary Policy Committees (MPCs) composed of a small number of appointed members (examples noted: the Federal Reserve, the European Central Bank, the Bank of England, the Bank of Japan, and the Bank of Sweden).
  - Much of the canonical monetary policy literature assumes a single, infinitely-lived policymaker; this paper departs from that assumption by modeling MPC members with finite, overlapping tenures.
  - Overlapping tenures imply that at any time the committee contains both “old” and “young” members; each member’s loss function is defined over their finite tenure and penalizes deviations of inflation and output gap from targets.

- Model setup and mechanism (brief)
  - Framework: New Keynesian model with standard NK Phillips Curve and IS equation; nominal rigidities can be derived from Calvo (1983), Taylor (1980), or Rotemberg (1982) formulations.
  - Phillips curve specification used:
    - π_t = β E_t π_{t+1} + k x_t + v_t
    - v_t = ρ v_{t-1} + ε_t
    - x_t = E_t x_{t+1} − σ (i_t − E_t π_{t+1} − r^n_t)
  - Each MPC member can commit to a path of future state-contingent policies for their own tenure but cannot commit the actions of successors; incoming members choose policy sequentially in their first period under rational expectations.
  - Disagreement between old and young arises because a young member optimally wants to commit to strong future inflation responses while an old member has already made state-contingent plans in their first-period tenure.

- Decision rule within the committee
  - Differences between old and young members are resolved via a utilitarian bargaining mechanism (maximizing the sum of members’ objective functions), which here coincides with averaging the desired inflation rates proposed by each MPC member.
  - Extension to n-member committee:
    - A fixed proportion (1−β) of members retire each period (the churning rate is 1−β); proportion β are old and (1−β) are young in each period.
    - If no members ever retire (β = 1), the committee is equivalent to a single infinitely-lived policymaker with commitment.
    - If churning is complete (entire committee replaced each period), the committee is equivalent to a single policymaker acting under discretion.
    - Intermediate churning rates yield monetary outcomes between discretion and full commitment (links to concepts: quasi commitment, loose commitment, imperfect credibility).

- Key findings and sensitivity
  - Slower churning rates (larger share of old members) increase social welfare; the closer the committee composition is to a majority of old members, the closer outcomes are to the commitment optimum.
  - Welfare gains from commitment are sensitive to calibration:
    - Under the benchmark calibration (Woodford 1999), gains from commitment are close to linear in the churning rate.
    - Results reported in Schaumburg and Tambalotti (2007)—that small departures from discretion bridge most of the welfare gap—do not hold generally in this model; calibration matters.
  - When churning is high, utilitarian bargaining is preferred to voting; when the majority of members are old, voting replicates commitment outcomes, whereas if young members are in majority, voting replicates discretion.

- Relation to existing literature
  - Builds on and contrasts with literature on MPC heterogeneity:
    - Prior papers assume heterogeneity in preferences (Aksoy, De Grauwe, and Dewachter (2002); Hefeker (2003); Sibert (2003)), skill differences (Hahn and Gersbach (2001)), or information asymmetries (Gerlach-Kristen (2006)).
    - This paper is, to the author’s knowledge, the first to analyze MPCs with overlapping finite tenures.
  - Connects to literature on imperfect credibility / quasi commitment:
    - Provides an institutional origin for imperfect commitment (churning of MPC membership) rather than imposing an exogenous probability of reneging (as in Schaumburg and Tambalotti (2007)).
    - Related concepts: imperfect commitment (Kara (2003)), quasi commitment (Schaumburg and Tambalotti (2007)), loose commitment (Debortoli and Nunes (2007)).
  - Links to fiscal-policy literature on commitment (e.g., Judd (1985), Chari and Kehoe (1990), Persson, Persson, and Svensson (2006), Lucas and Stokey (1983)) where commitment affects optimal long-run choices (e.g., zero optimal long-run capital tax with commitment).

- Policy implications and recommendations
  - Institutional design: Keeping the MPC churning rate low (i.e., ensuring a majority of old members at any point in time) improves welfare whether decisions are made by voting or utilitarian bargaining.
  - Decision procedure: If churning is high, utilitarian bargaining yields higher welfare than simple majority voting.
  - Caution: Magnitude of welfare gains from reduced churning depends critically on model calibration; empirical or country-specific calibration is important for evaluating institutional reform.

- Contribution and organization
  - Contributions:
    - Introduces overlapping finite-tenure MPC members into a New Keynesian framework, producing an endogenous form of imperfect commitment.
    - Provides welfare comparisons of institutional structures (churning rate, decision rule).
  - Paper structure:
    - Section 2: model and benchmarks (commitment vs. discretion).
    - Section 3: overlapping generations of MPC members and utilitarian bargaining; extension to n-member committees.
    - Section 4: comparison of utilitarian bargaining vs. voting.
    - Section 5: calibration and impulse-response analysis.
    - Section 6: conclusions.

### 2.2 Optimal Response under Discretion

- Discretionary outcome (optimal response under discretion)
  - In the absence of a commitment technology, the monetary authority takes agents’ expectations as given, producing the discretionary outcome and the stabilization bias.
  - The period-by-period optimization problem reduces to minimizing L_t = (π_t^2 + λ x_t^2) subject to the NKPC (2.1) and (2.3), with constraint (2.3) ignored because it does not enter the objective.
  - First-order condition:
    - π_t + λ k x_t = 0 (2.5)
  - Combining (2.5) with the NKPC (2.1) yields:
    - π_t = λ / (2 + λ(1−βψ)) v_t (2.6)
    - x_t = −k / (2 + λ(1−βψ)) v_t (2.7)
  - Key implications:
    - Both inflation and output gap are functions of the current period cost-push shock v_t.
    - The monetary authority brings inflation back to its zero target immediately under discretion.
    - Impulse responses of inflation and output gap to a one standard deviation cost-push shock are illustrated in Figure (9.1).

- Optimal policy under commitment (summary)
  - With commitment technology the monetary authority influences expectations by committing to future policy rules; the problem becomes dynamic and the Lagrangian is:
    - L_c = E_0 Σ_{t=0}^∞ [π_t^2 + λ x_t^2 + 2 μ_t(π_t − β E_t π_{t+1} − k x_t − v_t)] (2.8)
  - First-order conditions and initial condition μ_{−1} = 0 yield:
    - π_t − μ_t + μ_{t−1} = 0 and λ x_t + k μ_t = 0 for t ≥ 0
  - Eliminating μ_t gives an inflation rule implementing optimal policy:
    - π_0 = −λ k x_0 (2.9)
    - π_t + λ k [x_t − x_{t−1}] = 0 for t > 0 (2.10)
  - Timeless-perspective variant (Woodford (1999)):
    - π_t + λ k [x_t − x_{t−1}] = 0 for t ≥ 0 (2.11)
  - Combining (2.11) with the NKPC (2.1) yields:
    - x_{t+1} − [(β + 1 + k^2/λ)/β] x_t + (1/β) x_{t−1} = (k/β λ) v_t (2.12)
  - Stationary solution for x_t and π_t:
    - x_t = c1 x_{t−1} − (k/λ β)[c2 − ψ] v_t (2.13), where c1 < 1 and c2 > 1 are roots of the characteristic equation
    - π_t = (λ k (1 − c1)) x_{t−1} + (1/β)[c2 − ψ] v_t (2.14)

- Comparative implications (discretion vs. commitment)
  - Discretion: policy reoptimizes one-shot each period, reacting only to current shocks and expectations taken as given; immediate return of inflation to target.
  - Commitment: ability to influence expectations leads to an intertemporal rule (2.11) and more persistent, forward-looking dynamics (second-order difference equations (2.12)).
  - Under commitment the output-gap response to shocks is more persistent (roots c1, c2), whereas under discretion the response is contemporaneous as in (2.6)–(2.7).

### 5.1 Impulse Responses to Independent Shocks

- Setup and normalization
  - Impulses are normalized to produce an annualized one percentage point increase in inflation on impact for given expectations.
  - The economy starts in the steady state with zero inflation and no output gap.
  - The actual increase in inflation is a function of the forecasted response of the equilibrium policy to the shock because the model is forward-looking.
  - No shocks to the natural interest rate are assumed.

- Impulse responses: commitment versus discretion (i.i.d. shock)
  - Under discretion:
    - The monetary authority moves its instrument with the shock and returns the economy to the steady state as soon as the effects of the shock have faded.
    - With an i.i.d. impulse, the economy is driven into a sharp recession, accompanied by high inflation, but only for one period.
    - Interest rates: raised heavily in the period of the shock and brought back to steady state immediately after.
  - Under commitment:
    - The monetary authority exploits the possibility of influencing inflation expectations by promising a protracted mild recession accompanied by deflation in periods following the shock.
    - This is achieved with a relatively limited movement in the interest rate compared to the discretion case.
    - Interest rates: initial rise is much lower but the return to steady state is very persistent.
  - Steady state values of endogenous variables under discretion (in absence of new shocks and inflation bias) coincide with those under optimal policy with commitment.

- Monetary Policy Committee (MPC) outcomes for independent shocks
  - Voting interpretation:
    - If the level of inflation is decided by simple majority vote, paths of inflation, output gap and interest rates equal those under commitment when a majority of MPC members are old; equal those under discretion when the majority are young.
  - Committee as averaging of preferred inflation levels:
    - Two-member committee (presented as dashed-plus line in figures): paths of inflation, output gap, and interest rates are the same as for any committee with β = 1=2.
    - The inflation response for a two-member committee is roughly halfway between commitment and discretion.
    - A committee with three-quarters of members old (β = 3=4) produces responses much closer to commitment: very roughly, responses are three-quarters of the way between discretionary and commitment responses.
  - Interest rate response under committees:
    - For β = 1=2 the initial hike in interest rates is roughly halfway between discretionary and commitment responses.
    - For β = 3=4 the response is much closer to that under commitment.

- Figures referenced (i.i.d. shock)
  - Figure (9.1): Dynamic responses to a one standard deviation, uncorrelated cost-push shock under commitment (dashed line) and discretion (dotted line) using benchmark calibration.
  - Figure (9.2): Dynamic responses to the same shock in an MPC with various majorities. Dotted-plus line corresponds to β = 0:5; dotted-cross line corresponds to β = 0:75. Commitment and discretion responses are dashed and dotted lines respectively. The two-person committee (β = 0:5) lies roughly halfway between commitment and discretion; β = 0:75 is much closer to commitment.

- Implications emphasized
  - Without commitment technology, the protracted-interest-rate path is time inconsistent because the monetary authority would want to return to zero inflation and output gap once the shock disappears.
  - A committee’s composition (proportion of old vs. young members) systematically interpolates outcomes between the discretionary and commitment extremes; the degree of interpolation is tied to the share of old members.

*Source: _wp1032 (IMF working paper excerpts).*

### 1.1   Related Literature

### 1.1   Related Literature

### Context and motivation
- Monetary policy in many countries is conducted by Monetary Policy Committees (MPCs) composed of a small number of appointed members (examples noted: the Federal Reserve, the European Central Bank, the Bank of England, the Bank of Japan, and the Bank of Sweden).
- Much of the canonical monetary policy literature assumes a single, infinitely-lived policymaker; this paper departs from that assumption by modeling MPC members with finite, overlapping tenures.
- Overlapping tenures imply that at any time the committee contains both “old” and “young” members; each member’s loss function is defined over their finite tenure and penalizes deviations of inflation and output gap from targets.

### Model setup and mechanism (brief)
- Framework: New Keynesian model with standard NK Phillips Curve and IS equation; nominal rigidities can be derived from Calvo (1983), Taylor (1980), or Rotemberg (1982) formulations.
- Phillips curve specification used (as in the source):
  - π_t = β E_t π_{t+1} + k x_t + v_t
  - v_t = ρ v_{t-1} + ε_t
  - x_t = E_t x_{t+1} − σ (i_t − E_t π_{t+1} − r^n_t)
- Each MPC member can commit to a path of future state-contingent policies for their own tenure but cannot commit the actions of successors; incoming members choose policy sequentially in their first period under rational expectations.
- Disagreement between old and young arises because a young member optimally wants to commit to strong future inflation responses while an old member has already made state-contingent plans in their first-period tenure.

### Decision rule within the committee
- Differences between old and young members are resolved via a utilitarian bargaining mechanism (maximizing the sum of members’ objective functions), which here coincides with averaging the desired inflation rates proposed by each MPC member.
- Extension to n-member committee:
  - A fixed proportion (1−β) of members retire each period (the churning rate is 1−β); proportion β are old and (1−β) are young in each period.
  - If no members ever retire (β = 1), the committee is equivalent to a single infinitely-lived policymaker with commitment.
  - If churning is complete (entire committee replaced each period), the committee is equivalent to a single policymaker acting under discretion.
  - Intermediate churning rates yield monetary outcomes between discretion and full commitment (links to concepts: quasi commitment, loose commitment, imperfect credibility).

### Key findings and sensitivity
- Slower churning rates (larger share of old members) increase social welfare; the closer the committee composition is to a majority of old members, the closer outcomes are to the commitment optimum.
- Welfare gains from commitment are sensitive to calibration:
  - Under the benchmark calibration (Woodford 1999), gains from commitment are close to linear in the churning rate.
  - Results reported in Schaumburg and Tambalotti (2007)—that small departures from discretion bridge most of the welfare gap—do not hold generally in this model; calibration matters.
- When churning is high, utilitarian bargaining is preferred to voting; when the majority of members are old, voting replicates commitment outcomes, whereas if young members are in majority, voting replicates discretion.

### Relation to existing literature
- Builds on and contrasts with literature on MPC heterogeneity:
  - Prior papers assume heterogeneity in preferences (Aksoy, De Grauwe, and Dewachter (2002); Hefeker (2003); Sibert (2003)), skill differences (Hahn and Gersbach (2001)), or information asymmetries (Gerlach-Kristen (2006)).
  - This paper is, to the author’s knowledge, the first to analyze MPCs with overlapping finite tenures.
- Connects to literature on imperfect credibility / quasi commitment:
  - Provides an institutional origin for imperfect commitment (churning of MPC membership) rather than imposing an exogenous probability of reneging (as in Schaumburg and Tambalotti (2007)).
  - Related concepts: imperfect commitment (Kara (2003)), quasi commitment (Schaumburg and Tambalotti (2007)), loose commitment (Debortoli and Nunes (2007)).
- Links to fiscal-policy literature on commitment (e.g., Judd (1985), Chari and Kehoe (1990), Persson, Persson, and Svensson (2006), Lucas and Stokey (1983)) where commitment affects optimal long-run choices (e.g., zero optimal long-run capital tax with commitment).

### Policy implications and recommendations
- Institutional design: Keeping the MPC churning rate low (i.e., ensuring a majority of old members at any point in time) improves welfare whether decisions are made by voting or utilitarian bargaining.
- Decision procedure: If churning is high, utilitarian bargaining yields higher welfare than simple majority voting.
- Caution: Magnitude of welfare gains from reduced churning depends critically on model calibration; empirical or country-specific calibration is important for evaluating institutional reform.

### Contribution and organization
- Contributions:
  - Introduces overlapping finite-tenure MPC members into a New Keynesian framework, producing an endogenous form of imperfect commitment.
  - Provides welfare comparisons of institutional structures (churning rate, decision rule).
- Paper structure (as organized in the source):
  - Section 2: model and benchmarks (commitment vs. discretion).
  - Section 3: overlapping generations of MPC members and utilitarian bargaining; extension to n-member committees.
  - Section 4: comparison of utilitarian bargaining vs. voting.
  - Section 5: calibration and impulse-response analysis.
  - Section 6: conclusions.

*Source: _wp1032 - 1.1   Related Literature (IMF working paper chapter).*

### 2.2  Optimal Response under Discretion

### 2.2  Optimal Response under Discretion

### Discretionary outcome (optimal response under discretion)
- In the absence of a commitment technology, the monetary authority takes agents’ expectations as given, producing the discretionary outcome and the stabilization bias.
- The period-by-period optimization problem reduces to minimizing Lt = (πt^2 + λ xt^2) subject to the NKPC (2.1) and (2.3), with constraint (2.3) ignored because it does not enter the objective.
- First-order condition:
  - πt + λ k xt = 0 (2.5)
- Combining (2.5) with the NKPC (2.1) yields:
  - πt = λ / (2 + λ(1−βψ)) vt (2.6)
  - xt = −k / (2 + λ(1−βψ)) vt (2.7)
- Key implications:
  - Both inflation and output gap are functions of the current period cost-push shock vt.
  - The monetary authority brings inflation back to its zero target immediately under discretion.
  - Impulse responses of inflation and output gap to a one standard deviation cost-push shock are illustrated in Figure (9.1) (figure referenced in source).

### Optimal policy under commitment (summary from adjacent section 2.3)
- With commitment technology the monetary authority influences expectations by committing to future policy rules; the problem becomes dynamic and the Lagrangian is:
  - Lc = E0 Σt=0^∞ [πt^2 + λ xt^2 + 2μt(πt − β Et πt+1 − k xt − vt)] (2.8)
- First-order conditions (differentiating (2.8) wrt πt and xt) and initial condition μ−1 = 0 yield:
  - πt − μt + μt−1 = 0 and λ xt + k μt = 0 for t ≥ 0
- Eliminating μt gives an inflation rule implementing optimal policy:
  - π0 = −λ k x0 (2.9)
  - πt + λ k [xt − xt−1] = 0 for t > 0 (2.10)
- Timeless-perspective variant (Woodford (1999)) imposes optimality in all periods, including t = 0:
  - πt + λ k [xt − xt−1] = 0 for t ≥ 0 (2.11)
- Combining (2.11) with the NKPC (2.1) yields a second-order difference equation for xt:
  - xt+1 − [(β + 1 + k^2/λ)/β] xt + (1/β) xt−1 = (k/β λ) vt (2.12)
- Stationary solution for xt and πt:
  - xt = c1 xt−1 − (k/λ β)[c2 − ψ] vt (2.13), where c1 < 1 and c2 > 1 are roots of the characteristic equation
  - πt = (λ k (1 − c1)) xt−1 + (1/β)[c2 − ψ] vt (2.14)

### Comparative implications (discretion vs. commitment)
- Discretion: policy reoptimizes one-shot each period, reacting only to current shocks and expectations taken as given; immediate return of inflation to target.
- Commitment: ability to influence expectations leads to an intertemporal rule (2.11) and more persistent, forward-looking dynamics (second-order difference equations (2.12)).
- Under commitment the output-gap response to shocks is more persistent (roots c1, c2), whereas under discretion the response is contemporaneous as in (2.6)–(2.7).

*Italic: Source — _wp1032 - 2.2  Optimal Response under Discretion (excerpt) from the provided IMF PDF content.*

### 5.1  Impulse Responses to Independent Shocks

### 5.1  Impulse Responses to Independent Shocks

### Setup and normalization
- Impulses are normalized to produce an annualized one percentage point increase in ináation on impact for given expectations.
- The economy starts in the steady state with zero ináation and no output gap.
- The actual increase in ináation is a function of the forecasted response of the equilibrium policy to the shock because the model is forward-looking.
- No shocks to the natural interest rate are assumed.

### Impulse responses: commitment versus discretion (i.i.d. shock)
- Under discretion:
  - The monetary authority moves its instrument with the shock and returns the economy to the steady state as soon as the effects of the shock have faded.
  - With an i.i.d. impulse, the economy is driven into a sharp recession, accompanied by high ináation, but only for one period.
  - Interest rates: raised heavily in the period of the shock and brought back to steady state immediately after.
- Under commitment:
  - The monetary authority exploits the possibility of influencing ináation expectations by promising a protracted mild recession accompanied by deáation in periods following the shock.
  - This is achieved with a relatively limited movement in the interest rate compared to the discretion case.
  - Interest rates: initial rise is much lower but the return to steady state is very persistent.
- Steady state values of endogenous variables under discretion (in absence of new shocks and ináation bias) coincide with those under optimal policy with commitment.

### Monetary Policy Committee (MPC) outcomes for independent shocks
- Voting interpretation:
  - If the level of ináation is decided by simple majority vote, paths of ináation, output gap and interest rates equal those under commitment when a majority of MPC members are old; equal those under discretion when the majority are young.
- Committee as averaging of preferred ináation levels:
  - Two-member committee (presented as dashed-plus line in figures): paths of ináation, output gap, and interest rates are the same as for any committee with = 1=2.
  - The ináation response for a two-member committee is roughly halfway between commitment and discretion.
  - A committee with three-quarters of members old (= 3=4) produces responses much closer to commitment: very roughly, responses are three-quarters of the way between discretionary and commitment responses.
- Interest rate response under committees:
  - For = 1=2 the initial hike in interest rates is roughly halfway between discretionary and commitment responses.
  - For = 3=4 the response is much closer to that under commitment.

### Figures referenced (i.i.d. shock)
- Figure (9.1): Dynamic responses to a one standard deviation, uncorrelated cost-push shock under commitment (dashed line) and discretion (dotted line) using benchmark calibration.
- Figure (9.2): Dynamic responses to the same shock in an MPC with various majorities. Dotted-plus line corresponds to = 0:5; dotted-cross line corresponds to = 0:75. Commitment and discretion responses are dashed and dotted lines respectively. The two-person committee (= 0:5) lies roughly halfway between commitment and discretion; = 0:75 is much closer to commitment.

### Implications emphasized
- Without commitment technology, the protracted-interest-rate path is time inconsistent because the monetary authority would want to return to zero ináation and output gap once the shock disappears.
- A committee’s composition (proportion of old vs. young members) systematically interpolates outcomes between the discretionary and commitment extremes; the degree of interpolation is tied to the share of old members.

*Source: _wp1032 - 5.1  Impulse Responses to Independent Shocks*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp1032.pdf_