## _wp06256 — 1. Distribution of Agents in the Economy and Some Potential Coalition Structures

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---

### Overview and research question
- Two broad forms of monetary-policy delegation:
  - goal independence (GICB): central bank chooses targets and instruments.
  - instrument independence: central bank controls instruments to meet government-decided target.
- Core puzzle: why do some countries adopt goal independence while others adopt instrument independence?
- Core hypothesis: delegation to a GICB can be motivated by coalition-formation incentives. Delegation removes an intracoalition source of costly political conflict (monetary policy), reducing coalition formation costs when factions are of comparable size.
- Empirical implication: GICBs are less likely where preferences over two policy dimensions are highly correlated.
- Endogeneity implication: endogenous selection of goal independence biases OLS estimates of CBI effects on inflation; modeling selection yields a strong negative effect of CBI on inflation.

### Model setup (demographics, preferences, political-economy game)
- Population and preferences:
  - Large, even, finite set of agents N with dichotomous preferences over two policy dimensions {0,1} × {0,1}.
  - Each agent derives the same utility V_j, j ∈ {0,1} from implementation of each preferred policy; dimensions ordered so V_1 ≥ V_0.
  - Factional composition: proportions of agents with each ideal policy distributed so that a proportion 1/2 prefers each outcome on each dimension; a proportion 1/4 (and other fractions) appear in the text for specific faction sizes.
  - Correlation parameter σ: when σ = 1/2 there is no correlation; σ = 1 denotes full correlation.
- Representation and game stages:
  - Four faction-players represent the four preference-pairs, weighted by faction size.
  - Three-stage political-economy game:
    1. Coalitional subgame: players join a coalition.
    2. Coalition platform stage: each coalition chooses a policy platform; intracoalition disagreement resolved via a rent-seeking subgame.
    3. Electoral subgame: largest coalition has opportunity to set policy; winning probabilities increase in coalition size and decrease in sizes of others.

### Rent-seeking subgame (intracoalition lobbying)
- For any disputed policy dimension, two factions within the coalition (S for small, L for large) bid to control policy.
- Benefit of preferred policy on one dimension: V.
- Each faction J ∈ {S, L} makes a dissipated bid x_J; cost shared equally among members: per-member bid x_J / n_J.
- Probability coalition in question controls policy: ϕ ∈ [0,1].
- Expected payoffs for typical members (as in source):
  - E[V_iL] = V - x_L / n_L
  - E[V_iS] = V - x_S / n_S
- Mixed-strategy equilibrium bids (Hillman (1989) notation preserved):
  - Larger-faction expected per-member payoff:
    - E[V_iL] = ϕ ( n_L - n_S ) / n_L V
  - Smaller-faction expected per-member payoff:
    - E[V_iS] = 0
  - Bidding cdfs:
    - F_L(x) = x / (ϕ V n_S); x ∈ [0, ϕ V n_S]
    - F_S(x) = n_L - n_S / n_L + x / (ϕ V n_L); x ∈ [0, ϕ V n_S]
  - Associated pdfs:
    - f_L = 1 / (ϕ V n_S); x ∈ [0, ϕ V n_S]
    - f_S = 1 / (ϕ V n_L); x ∈ [0, ϕ V n_S]
- Probability larger faction’s policy is chosen (exact expression preserved):
  - Pr[x_S ≤ x_L] ≡ ϵ = ϕ V n_S ∫_{x_L=0}^{ϕ V n_S} ( n_L - n_S ) / n_L + x_L / (ϕ V n_L) × 1 / (ϕ V n_S) dx_L
  - Simplified expression given: ϵ = 1 - n_S / (2 n_L) = 1/2 (as presented)
- Rent-seeking insights:
  - Smaller faction dissipates all its rents; larger faction retains some gain from control.
  - Coalition membership incurs a dissipated bidding cost of n_S / (2 n_L) ϕ V = (1 - ϵ) ϕ V per agent.
  - Cost of disagreement falls as ϵ → 1; incentive to delegate is strongest when factions are similar-sized (high expected contest costs).

### Coalitional structure and strategic consequences
- Coalitions form in equilibrium based on unanimity along one policy dimension and disagreement on the other. Factions that disagree on both dimensions have no incentive to form a coalition.
- For any faction with ideal point (θ_i, φ_i), coalition choices limited to:
  - with (θ_i, 1 − φ_i) (a “φ-coalition”),
  - with (1 − θ_i, φ_i) (a “θ-coalition”),
  - or remain alone (a “0-coalition”).
- Electoral contest rule assumptions:
  - a θ- or φ-coalition with size 1/2 wins against a 0-coalition with maximum size σ / 2 < 1/2 with probability ϕ = 1.
  - contest between two equal-sized coalitions: each wins with probability 1/2.
  - in a four-way contest the larger two factions each win with probability 1/2 and smaller coalitions have zero probability of winning.

### Main theoretical predictions and propositions
- Incentive to delegate (adopt goal independence) is highest when coalitions are composed of roughly equal-sized factions (high intracoalition heterogeneity).
- Comparative statics via σ:
  - As σ increases (preferences more correlated across dimensions), coalition preferences become more homogeneous, one faction dominates, and cost of political campaigning falls; GICBs are less likely where σ is higher.
- Selection and estimation implication:
  - Endogenous choice of goal independence biases OLS estimates of CBI effects on inflation toward zero; explicitly modeling selection yields a strong negative effect of CBI on inflation (empirical results summarized later).

### Connections to existing literature
- Builds on Rogoff (1985) and Walsh (1995) distinction between goal and instrument independence.
- Incorporates political-science perspectives on heterogeneous groups and costly disputes (Bernhard (1998); Bernhard and Leblang (2002)).
- Uses rent-seeking framework (Hillman (1989)) to model coalition-formation costs.
- Complements time-inconsistency explanations (Keefer and Stasavage (2003), Moser (1999), Eggertsson and Le Borgne (2003)) by offering coalition-formation rationale.

---

### Appendix I — Coalition-formation subgame (equilibrium features)
- Sequential game structure and stationary perfect equilibrium per Bloch (1996) are used.
- Only the coalition structure with two opposing coalitions formed along the most contentious policy dimension is consistent with stationary perfect equilibrium.
- Equilibrium features:
  - Agents inside each coalition agree unanimously on the most contentious policy dimension.
  - Disagreement on the other dimension resolved via lobbying.
  - Each coalition wins the last stage and sets policy with probability 1/2.
  - Expected utility expressions (as given in source):
    - U_L = U_S = 1/2 [ V_θ + 3−1/2? V_φ ] (exact expression preserved as in source)
- Interpretation of policy dimensions:
  - θ, φ interpreted as monetary policy ∈ {0,1} and a second policy dimension ∈ {0,1}.
  - θ-coalition economy: θ = φ (second dimension most important).
  - φ-coalition economy: θ = φ̄ (monetary policy most important).
- Payoffs for coalitions:
  - θ-coalition: U = 1/2 [ V_θ + 3−1/2 V_φ ]  (equation (10) in source)
  - φ-coalition: U = 1/2 [ V_φ + 3−1/2 V_θ ]  (equation (11) in source)

### Predictions — Delegation of policy to a GICB
- Delegation stage: Agents may delegate monetary policy to a GICB prior to coalition formation; all agents have veto power; political control is fall-back.
- GICB uncertainty modeled:
  - Policy set at φ = 1 with probability p and φ = 0 with probability 1 − p, where p ∼ continuous distribution on [0,1] with cdf F(p), common knowledge.
- Consequences:
  - Delegation removes monetary policy dimension; coalitions form along remaining dimension.
  - If delegation occurs, coalitions have weight 1/2 and each has probability 1/2 of controlling the remaining dimension.
- Central proposition (Proposition 1):
  - As σ increases, the probability of policy delegation to a GICB falls. Probability of observing goal independence is negatively related to correlation between preferences over φ and θ.
- Key expressions preserved (θ-coalition economy):
  - U_P = 1/2 [ V_θ + 3−1/2 V_φ ]  (12)
  - U_C = 1/2 V_θ + b_{p,i} V_φ  (13), where b_{p,i} ∈ {1 − p, p} depends on agent i's preferred φ_i ∈ {0,1}.
  - Condition U_C ≥ U_P ⇔ b_{p,i} ≥ 3/4 − 1/4 σ  (14)
  - CBI chosen iff p ∈ [ (3/4 − 1/4 σ), (1/4 + 1/4 σ) ]  (15)
  - Probability P = F(p) − F(p̄)  (16)
  - ∂P/∂σ = −1/4 σ^2 − [ f(p) + f(p̄) ] < 0  (17)
- Analogous expressions for φ-coalition economy (preserved):
  - U_P = 1/2 [ V_φ + 3−1/2 V_θ ]  (18)
  - U_C = 1/2 V_θ + b_{p,i} V_φ  (19)
  - Condition U_C ≥ U_P ⇔ b_{p,i} ≥ 1/2 [ 1 − 1−σ / 2σ V_θ/V_φ ]  (20)
  - CBI chosen iff p in analogous interval depending on σ and V_θ/V_φ (21)
  - P = F(p) − F(p̄)  (22)
  - ∂P/∂σ = −1/4 σ^2 − [ f(p) + f(p̄) ] V_θ/V_φ < 0  (23)
- Nontechnical restatement: GICB more likely when σ is low (preferences over monetary policy and second dimension weakly correlated).

### III.B Inflation and endogenous central bank independence
- Rogoff (1985) benchmark: central bankers are dominated by conservative (inflation-averse) types; distribution of p places higher weight on low values so E[p] < 1/2.
- Mapping to inflation outcomes:
  - φ ∈ {0,1} → inflation rates π̄ ∈ {π_0, π_1} with π_1 > π_0.
  - Expected inflation ignoring endogeneity:
    - π̄_POL = π_0 + 1/2 (π_1 − π_0) > π̄_CBI = π_0 + E[p] (π_1 − π_0)  (24)
- Endogenous selection complication:
  - Under CBI, expected inflation depends on conditional average p given p in the endogenous selection range (expression for π̄_CBI provided in equation (25)).
  - No automatic link between unconditional E[p] and conditional E[p | p in critical range]; π̄_CBI can be > π̄_POL even if central bankers are conservative on average.
- Intuition: if central bankers are too likely inflation-averse, delegation may be blocked by inflation-tolerant segments, yielding higher observed inflation under CBI in cross-section.
- Econometric implication:
  - Endogeneity of delegation biases OLS estimates of CBI effect on inflation toward zero.
  - With p ∈ {0,1/2} and derivations preserved: true effect Δ = −(π_1 − π_0) [ 1/2 − b_p ]; OLS Δ_OLS = −(π_1 − π_0) [ 1/2 − b_p^1 ]; since b_p^1 > b_p, measured effect biased toward zero.
  - Empirical approach: control for endogenous selection via treatment-effects (TE) model; TE estimate of coefficient on CBI is negative and statistically significant in the paper.

### IV. Empirical tests — Data, measurement, and main findings
- Data sources and sample:
  - Central bank independence: Fry and others (2000) dataset; goal independence rating = 100% → goal independent; < 100% → not goal independent.
  - Inflation: IMF International Financial Statistics.
  - Real GDP (PPP): Penn World Tables v. 6.1 (Heston, Summers and Aten, 2002).
  - Political system: Polity IV database (Marshall and Jaggers, 2001).
  - Inflationary preferences: World Values Survey second and third waves (1990 and 1995–1998).
  - Data available for a total of 36 countries (joint availability of institutional and preferences data).
- Measurement of monetary-policy preference indices (π): four indices b'_i^j, j ∈ {1,2,3,4} preserved as defined in source (including b'_i3 = S_gi − S_πi / 2 ∈ {−1, −0.5, 0, 0.5, 1} (equation (27))).
- Proxies for second preference dimension θ: income percentile and social class.
- Eight measures of absolute correlation σ (four π measures × two θ proxies) used as regressors in probit regressions for goal independence, with country average autocracy score a as single control.

Empirical findings (preserving reported coefficients and statistics)
- Table 2 (Probit regressions, goal independence):
  - Panel A (income proxy b_θ1): all four π proxies statistically significant (three at 5% level, one at 10%), negative coefficient on σ. Point estimates (standard errors):
    - −11.5 (5.71)
    - −14.1 (6.49)
    - −10.7 (4.92)
    - −7.63 (4.16)
    - Control a coefficients: −2.44 (1.18), −3.79 (1.42), −3.04 (1.65), −2.77 (1.46) (as reported).
  - Panel B (social class proxy b_θ2): all four π proxies statistically significant (two at 1% level, one at 5% level, one at 10%), negative sign (coefficients reported similarly in source).
- Table 3 (Predictive ability):
  - Model predicts GICB = 0 well: success ≈ 85–90%.
  - Ability to predict GICB = 1 weaker: ≈ 30–50%.
  - Predictive power higher for advanced economies than for emerging market economies (panels and exact rates in source).
- Table 4 (Inflation performance and CBI):
  - OLS regressions for 78 countries:
    - GICB coefficient: 1.02 (3.03) in full sample (not significant).
    - Advanced sample: GICB = −1.32 (0.436) (significant).
    - EMEs: 0.979 (4.17).
    - gdp−10^−3 coefficient: −0.551 (0.161) (significant).
  - Treatment-effects (TE, MLE) models on restricted sample (35 countries with σ data):
    - TE (MLE) estimates show large negative and statistically significant coefficients on GICB:
      - TE (MLE) restricted: −19.23 (5.15) and −17.4 (4.76) in two specifications reported.
    - Hazard equation for GICB includes proxies for σ and control a; example coefficient reported: σ coefficient −7.76 (2.40).
    - Covariance between error terms (ρ) positive, consistent with OLS bias toward zero.
  - Conclusion: controlling for endogenous selection via TE model, CBI reduces inflation (TE estimates negative and significant). Observed negative correlation between independence and inflation arises mainly in advanced countries; endogenous selection explains weak aggregate OLS effects.

### V. Case studies (illustrative evidence)
- Germany: Bundesbank independence helped coalition formation between Social Democrats and Free Democrats in 1970s; attempts to gain authority over Bundesbank in 1950s and 1990s.
- Netherlands: independent central bank and stable ER may have facilitated Labour (PvdA)–Liberal (VVD) coalition in 1990s.
- United Kingdom: Bank of England dependence reflected homogeneous monetary preferences across major parties; autonomy increased only as disputes intensified (1980s/1990s).
- South Africa: SARB subordinate under pre-1990 apartheid regime; granted goal independence in 1994 and 1996 amid broad, potentially fractious coalition (ANC-led government).
- United States, nineteenth century: regional splits and shifting coalition structures influenced willingness to delegate monetary authority; episodes described (1832 veto, Independent Treasury 1846, 1892 episode, 1913 Fed creation) illustrate model implications.

### VI. Conclusions, implications, and limitations
- Main argument: policy delegation (goal independence) can reduce coalition-formation costs by reducing the dimensionality of political conflict; delegation more likely when correlation σ between preferences across dimensions is low.
- Empirical support:
  - World Values Survey and Fry et al. (2000) institutional data support Proposition 1.
  - Treatment-effects empirical approach finds that granting central banks independence reduces inflation once endogenous selection is accounted for.
- Broader implications:
  - Endogenous selection helps explain why cross-country OLS estimates of CBI on inflation are weak or mixed.
  - Model generalizable to other delegation contexts (e.g., trade policy, fiscal responsibilities).
- Limitations and further work:
  - Model relies on restrictive assumptions for tractability (dichotomous policies, two policy dimensions).
  - Further research should test predictions in other settings and generalize to less restrictive environments.

*Source: _wp06256 — 1. Distribution of Agents in the Economy and Some Potential Coalition Structures; Appendix I. Equilibrium Coalition Structure (content preserved from supplied PDF).*

### 1.       Distribution of Agents in the Economy and Some Potential Coalition Structures ..........8

### 1.       Distribution of Agents in the Economy and Some Potential Coalition Structures

### Overview and research question
- Politicians delegate monetary policy to independent central banks; delegation takes two broad forms:
  - goal independence (GICB): principal delegates full policymaking powers including target choice (example discussed: U.S. Federal Reserve).
  - instrument independence: central bank acts autonomously over instruments to pursue a government-decided target (example discussed: Bank of England post-1997).
- Central puzzle: why do some countries adopt goal independence while others adopt instrument independence?
- Core hypothesis: delegation to a GICB can be motivated by coalition-formation incentives. Delegation removes an intracoalition source of costly political conflict (monetary policy), reducing coalition formation costs when factions are of comparable size.
- Empirical implication: GICBs are less likely where preferences over two policy dimensions are highly correlated, because correlated preferences produce dominant factions and lower intracoalition lobbying costs.
- Endogeneity implication: endogenous selection of goal independence biases OLS estimates of CBI effects on inflation; when selection is modeled explicitly, a strong negative effect of CBI on inflation emerges.

### Model setup (demographics, preferences, political economy game)
- Population: large, even, finite set of agents N with dichotomous preferences over two policy dimensions {0,1} × {0,1}.
- Utility: each agent derives the same utility Vj, j ∈ {0,1} from implementation of each preferred policy; dimensions ordered so V1 ≥ V0.
- Factional composition parameter:
  - Proportion of agents with ideal policy f0,0g is /2 (followed by notation: 1/4; 1/2 appears in text).
  - Preferences over each policy dimension are distributed evenly so that a proportion 1/2 prefers each outcome on each dimension.
  - Correlation parameter : when  = 1/2 there is no correlation;  = 1 denotes full correlation.
- Factions (agents with same preference pair) are represented as four players in the coalition game, weighted by faction size.
- Three-stage political economy game:
  1. Coalitional subgame: players join a coalition.
  2. Coalition platform stage: each coalition chooses a policy platform; intracoalition disagreement over a policy dimension is resolved via a rent-seeking subgame.
  3. Electoral subgame: largest coalition has opportunity to set policy; winning probabilities are increasing in coalition size and decreasing in the sizes of others.

### Rent-seeking subgame (intracoalition lobbying)
- For any disputed policy dimension, two factions within the coalition (denoted S for small, L for large) bid to control policy.
- Benefit of having preferred policy implemented on a single dimension: V.
- Each faction J ∈ {S, L} makes a dissipated bid x_J; cost shared equally among members: per-member bid x_J / n_J.
- Probability coalition in question controls policy:  ∈ [0,1].
- Expected payoffs for typical members (equations exactly as in source):
  - E[V_iL] =  - S - x_L = V - x_L / n_L  (presented in source as equation (1) form; see exact model relations below)
  - E[V_iS] =  - L - x_S = V - x_S / n_S  (presented in source as equation (2) form)
- Mixed-strategy equilibrium bids (cdf notation _J(x) from Hillman (1989)):
  - Larger-faction expected per-member payoff (equation (3)):
    - E[V_iL] = V - V n_S / n_L =  ( n_L - n_S ) / n_L V
  - Smaller-faction expected per-member payoff (equation (4)):
    - E[V_iS] = V - V n_S / n_S = 0
  - Bidding cdfs (equations preserved):
    - _L(x)  = x / (V n_S); x ∈ [0, V n_S]
    - _S(x)  = n_L - n_S / n_L + x / (V n_L); x ∈ [0, V n_S]
  - Associated pdfs (equations preserved):
    - L = 1 / (V n_S); x ∈ [0, V n_S]
    - S = 1 / (V n_L); x ∈ [0, V n_S]
- Probability that the larger faction’s policy is chosen (), preserving the analytic expression (equation (9)):
  - Pr[x_S ≤ x_L] ≡  = V n_S ∫_{x_L=0}^{V n_S} ( n_L - n_S ) / n_L + x_L / (V n_L) × 1 / (V n_S) dx_L
  - Simplified expression given in source:  = 1 - n_S / (2 n_L) = 1/2 (as presented)
- Key rent-seeking insights:
  - Smaller faction dissipates all its rents; larger faction retains some gain from control.
  - Coalition membership incurs a dissipated bidding cost of n_S / (2 n_L) V = (1 - ) V per agent.
  - Cost of disagreement falls as outcome becomes more certain (i.e., as  → 1), driving the incentive to delegate when factions are similar-sized (high expected contest costs).

### Coalitional structure and strategic consequences
- Coalitions form in equilibrium based on unanimity along one policy dimension and disagreement on the other. Given the rent-seeking payoffs, factions that disagree on both dimensions have no incentive to form a coalition.
- For any faction with ideal point (θ_i, φ_i), coalition choices are limited to:
  - coalition with (θ_i, 1 - φ_i) (a “φ-coalition”),
  - coalition with (1 - θ_i, φ_i) (a “θ-coalition”),
  - or remain alone (a “0-coalition”).
- Electoral contest rule assumptions (as stated):
  - a θ- or φ-coalition with size 1/2 wins against a 0-coalition with maximum size  / 2 < 1/2 with probability  = 1.
  - contest between two equal-sized coalitions: each wins with probability 1/2.
  - in a four-way contest the larger two factions each win with probability 1/2 and smaller coalitions have zero probability of winning.

### Main theoretical predictions and propositions
- Incentive to delegate (adopt goal independence) is highest when coalitions are composed of roughly equal-sized factions (high intracoalition heterogeneity). Reason:
  - When factions are equal-sized, per-member lobbying costs are similar and contests are competitive, raising dissipative campaign costs and making removal of monetary policy from the political bargaining table desirable.
  - When one faction dominates (preferences across dimensions highly correlated), the dominant faction will likely win intracoalition disputes without costly lobbying; delegation incentives are minimal.
- Comparative-statics via correlation parameter :
  - As  increases (preferences more correlated across the two dimensions), coalition preferences become more homogeneous, one faction dominates, and cost of political campaigning falls. Therefore, GICBs are less likely where  is higher.
- Selection and estimation implication:
  - If goal independence is chosen endogenously, OLS estimates of CBI effects on inflation are biased upwards (towards zero if the true causal effect is negative) because goal independence will be chosen when the central banker is likely to be neither too “hard” nor too “soft” on inflation; conservative central bankers are on average more common, so delegation tends to rule out extreme “inflation nutters,” biasing naive estimates.
  - Modeling endogenous selection explicitly yields a strong negative effect of CBI on inflation (empirical results summarized in later sections of the paper; not reproduced here).

### Connections to existing literature (roles and mechanisms)
- Builds on macroeconomic literature distinguishing goal independence vs. instrument independence (Rogoff (1985); Walsh (1995)); clarifies how institutional forms shape political incentives.
- Incorporates political science perspectives emphasizing heterogeneous groups and costly political disputes (Bernhard (1998); Bernhard and Leblang (2002)); provides formal game-theoretic foundation for delegation-as-coalition-management arguments.
- Uses a rent-seeking framework (Hillman (1989)) to model coalition-formation costs and capture welfare dissipation from intracoalition contesting; this introduces a cost that makes Pareto-improving delegation possible.
- Related to models stressing time-inconsistency motives for delegation but offers coalition formation as an alternative rationale (complements Keefer and Stasavage (2003), Moser (1999), Eggertsson and Le Borgne (2003)).

*Source: _wp06256 - 1.       Distribution of Agents in the Economy and Some Potential Coalition Structures ..........8*

### Appendix I gives the solution to the coalition-formation subgame using the sequential

### _wp06256 - Appendix I gives the solution to the coalition-formation subgame using the sequential

### Appendix I — Coalition-formation subgame (equilibrium features)
- Uses sequential game structure and stationary perfect equilibrium developed by Bloch (1996).
- Only the coalition structure (two coalitions formed along the policy dimension) is consistent with stationary perfect equilibrium.
- Key equilibrium features:
  - Agents inside each coalition agree unanimously on the most contentious policy dimension.
  - Disagreement on the other policy dimension is overcome through a process of lobbying.
  - Each coalition wins the last stage and sets policy with probability 1/2.
  - Expected utility for the larger and smaller factions in each coalition (superscripts L, S) is:
    - U_L = U_S = 1/2 [ V_θ + 3−1/2? V_φ ] (exact expression as given in source)

- Interpretation of policy dimensions:
  - θ, φ interpreted as monetary policy ∈ {0,1} and a second policy dimension ∈ {0,1} (the latter: “all other policies” or an indicator of agent “type”).
  - Define a θ-coalition economy where θ = φ (second dimension most important) and a φ-coalition economy where θ = φ̄ (monetary policy most important).
- Payoffs for coalitions (both factions same expected payoff):
  - θ-coalition: U = 1/2 [ V_θ + 3−1/2 V_φ ]  (equation (10) in source)
  - φ-coalition: U = 1/2 [ V_φ + 3−1/2 V_θ ]  (equation (11) in source)

### III. Predictions — Delegation of policy to a GICB
- Delegation stage: Agents may delegate monetary policy to a GICB prior to coalition formation; all agents have veto power; political control is fall-back.
- Modeling GICB uncertainty:
  - Policy set at φ = 1 with probability p and φ = 0 with probability 1−p, where p ∼ continuous distribution on [0,1] with cdf F(p), common knowledge.
- Consequences of delegation:
  - Delegation removes the monetary policy dimension; coalitions form along the remaining policy dimension.
  - If delegation occurs, coalitions have weight 1/2 and each has probability 1/2 of controlling policy on the remaining dimension.
  - Delegation can change equilibrium coalition structure if, absent delegation, coalitions would form along the monetary dimension.
- Central proposition (Proposition 1):
  - As σ (correlation between preferences over φ and θ) increases, the probability of policy delegation to a GICB falls.
  - Implies probability of observing goal independence is negatively related to the correlation between preferences over φ and θ.
- Proof highlights and expressions:
  - For a θ-coalition economy:
    - U_P = 1/2 [ V_θ + 3−1/2 V_φ ]  (12)
    - U_C = 1/2 V_θ + b_p_i V_φ  (13), where b_p_i ∈ {1−p, p} depends on agent i's preferred φ_i ∈ {0,1}.
    - Condition U_C ≥ U_P ⇔ b_p_i ≥ 3/4 − 1/4 σ  (14)
    - Range of p satisfying CBI depends on σ; when σ = 1/2 range p ∈ [1/4, 3/4]; as σ increases range narrows; at σ = 1 only p = 1/2 allows CBI.
    - CBI chosen iff p ∈ [ (3/4 − 1/4 σ), (1/4 + 1/4 σ) ]  (15)
    - Probability P = F(p) − F(p̄) (16)
    - ∂P/∂σ = −1/4 σ^2 − [ f(p) + f(p̄) ] < 0 (17) (expression as given)
  - For a φ-coalition economy:
    - U_P = 1/2 [ V_φ + 3−1/2 V_θ ]  (18)
    - U_C = 1/2 V_θ + b_p_i V_φ  (19)
    - Condition U_C ≥ U_P ⇔ b_p_i ≥ 1/2 [ 1 − 1−σ / 2σ V_θ/V_φ ]  (20)
    - CBI chosen iff p in analogous interval depending on σ and V_θ/V_φ (21)
    - Probability expression and derivative:
      - P = F(p) − F(p̄) (22)
      - ∂P/∂σ = −1/4 σ^2 − [ f(p) + f(p̄) ] V_θ/V_φ < 0 (23)

- Nontechnical restatement:
  - Goal independence (GICB) more likely when correlation between preferences over monetary policy and the second policy dimension is low (σ low), i.e., when coalition formation is harder.
  - Empirical tests (in paper) find support for this prediction.

### III.B Inflation and endogenous central bank independence
- Standard Rogoff (1985) argument: central bankers are dominated by conservative (inflation-averse) types; implies distribution of p places higher weight on low values so E[p] < 1/2.
- Policy implications ignoring endogeneity:
  - Map dichotomous inflation policy φ ∈ {0,1} into inflation rates π̄ ∈ {π_0, π_1} with π_1 > π_0.
  - Expected inflation:
    - π̄_POL = π_0 + 1/2 (π_1 − π_0) > π̄_CBI = π_0 + E[p] (π_1 − π_0)  (24) (ignoring endogeneity)
- Endogenous selection complication:
  - Under CBI, expected inflation depends on conditional average p given p is within the endogenous selection range determined by σ.
  - Expression for π̄_CBI under endogenous selection provided (equation (25)) — depends on integrals over p and distribution of σ.
  - There is no automatic link between unconditional E[p] and conditional E[p | p in critical range]; counterfactuals possible with π̄_CBI > π̄_POL even if central bankers are conservative on average.
- Intuition:
  - If central bankers are too likely to be inflation-averse, delegation may be blocked by inflation-tolerant sections of society, yielding higher observed inflation under CBI in cross-section.
- Econometric implications:
  - Endogeneity of delegation (decision related to p) biases OLS estimates of CBI effect on inflation toward zero.
  - Assuming central bankers are low-inflation types with probability at least one-half (p ∈ {0,1/2}), the bias direction is shown:
    - True effect (holding p constant): Δ = −(π_1 − π_0) [ 1/2 − b_p ]
    - OLS estimate ignoring selection: Δ_OLS = −(π_1 − π_0) [ 1/2 − b_p^1 ]
    - Since b_p^1 > b_p, measured effect biased toward zero (equation (26) discussion).
  - Empirical approach: control for endogenous selection via treatment effects (TE) model; TE estimate of coefficient on CBI is negative and statistically significant in the paper.

### IV. Empirical tests — Data and measurement
- Data sources:
  - Central bank independence: Fry and others (2000) dataset; differentiates goal and instrument independence; goal independence rating = 100% → goal independent; < 100% → not goal independent.
  - Inflation: IMF International Financial Statistics.
  - Real GDP (PPP): Penn World Tables v. 6.1 (Heston, Summers and Aten, 2002).
  - Political system: Polity IV database (Marshall and Jaggers, 2001).
  - Inflationary preferences: World Values Survey second and third waves (1990 and 1995–1998).
- Sample size:
  - Data available for a total of 36 countries (joint availability of institutional and preferences data).
- Measurement of monetary policy preference indices (π):
  - Four alternative indices b'_i^j, j ∈ {1,2,3,4}:
    - b'_i1: binary, 0 if “fighting rising prices” is top priority; 1 otherwise.
    - b'_i2: binary, 0 if “fighting rising prices” is first or second choice; 1 otherwise.
    - b'_i3: five-point trade-off index combining two questions: S_gi − S_πi / 2 ∈ {−1, −0.5, 0, 0.5, 1}  (equation (27)).
    - b'_i4: revealed preference binary score (1 if high growth ranked above low inflation, 0 if low inflation ranked above high growth, : if no transitive ordering).
- Proxies for second preference dimension θ:
  - Two proxies:
    - Income percentile (estimated percentile of income distribution).
    - Social class (self-assigned).
- Eight different measures of absolute correlation σ (four π measures × two θ proxies) used as regressors in probit regressions for goal independence, with country average autocracy score a as single control.

### IV.D Central bank independence and preference dispersal — empirical findings
- Table 1: σ measures and GICB status for 36 countries (panel A: income-percentile proxy; panel B: social-class proxy). GICB coded 0 or 1. (Full table values in source.)
- Table 2 (Probit regressions, goal independence):
  - Panel A (income proxy b_θ1): all four π proxies statistically significant (three at 5% level, one at 10%), negative coefficient on σ (point estimates: −11.5 (5.71), −14.1 (6.49), −10.7 (4.92), −7.63 (4.16); control a coefficients: −2.44 (1.18), −3.79 (1.42), −3.04 (1.65), −2.77 (1.46); constants reported).
  - Panel B (social class proxy b_θ2): all four π proxies statistically significant (two at 1% level, one at 5% level, one at 10%), negative sign (coefficients reported similarly).
  - Robust standard errors in parentheses; significance denoted by symbols in source.
- Table 3 (Predictive ability):
  - Model predicts GICB = 0 well (success ~85–90%); ability to predict GICB = 1 weaker (30–50%).
  - Predictive power higher for advanced economies (Panels C and D) than for emerging market economies (Panels E and F).
- Table 4 (Inflation performance and CBI):
  - OLS regressions for 78 countries:
    - GICB coefficient: 1.02 (3.03) in full sample (not significant); in advanced sample GICB = −1.32 (0.436) (significant); EMEs: 0.979 (4.17).
    - gdp−10^−3 coefficient: −0.551 (0.161) (significant).
  - Treatment-effects (TE, MLE) models on restricted sample (35 countries with σ data):
    - TE (MLE) estimates show large negative and statistically significant coefficients on GICB:
      - TE (MLE) restricted: −19.23 (5.15) and −17.4 (4.76) in two specifications reported.
    - Hazard equation for GICB includes proxies for σ and control a; coefficients reported (e.g., σ coefficient −7.76 (2.40) etc.).
    - Covariance between error terms (ρ) positive, consistent with OLS bias toward zero.
  - Conclusion: when accounting for endogenous selection via TE model, CBI reduces inflation (TE estimates negative and significant).
  - Observed cross-sectional pattern: negative correlation between independence and inflation for advanced countries but not for other countries or full sample — consistent with literature (Eijffinger and De Haan, 1996) but endogenous selection resolves apparent weak aggregate effect.

### V. Case studies (supporting anecdotal evidence)
- Cross-country evidence:
  - Germany: Bundesbank independence helped coalition formation between Social Democrats and Free Democrats in 1970s; Christian Democratic attempts to gain authority over Bundesbank in 1950s and 1990s.
  - Netherlands: strong monetary commitments (independent central bank, stable ER) may have facilitated Labour (PvdA)–Liberal (VVD) coalition in 1990s by removing monetary policy as source of conflict.
  - United Kingdom: Bank of England dependence reflected homogeneous monetary preferences across Conservative and Labour parties; delegation/increased autonomy only emerged as monetary disputes intensified (1980s/1990s).
  - South Africa: SARB subordinate under pre-1990 apartheid regime (homogeneous elite); granted goal independence in 1994 and 1996 constitutions amid broad and potentially fractious coalition (ANC-led government) — consistent with model that delegation responds to coalition-building needs.
- United States, nineteenth century:
  - Regional split on monetary policy (Northeast vs South/West) corresponded to debtor/creditor interests and produced variation in willingness to delegate:
    - Jackson’s 1832 veto of BUS renewal reflected a unified Southern/Western coalition; delegation not preferred when coalition unified.
    - Mid-century: diverse coalitions led to attempts to delegate monetary responsibility (Independent Treasury 1846).
    - Return to gold and silver debates: political actors often kept monetary issues sectional rather than party issues until late 19th century; 1892 episode showed internal splits despite diverse presidential coalition.
    - Post-1896 regional polarization accompanied greater monetary activism and ultimately the 1913 Federal Reserve creation; governance retained political control consistent with relatively homogeneous governing coalition.
  - Figure 2 (source) plots regional specificity of winner’s support (measure described in source) across elections 1828–1912; four historical episodes illustrate model implications.

### VI. Conclusions and implications
- Main argument:
  - Policy delegation can reduce coalition-formation costs by reducing dimensionality of political conflict.
  - Delegation (goal independence) more likely when correlation of agents' preferences across policy dimensions is lower (σ low).
- Empirical support:
  - World Values Survey and Fry et al. (2000) institutional data support Proposition 1.
  - Treatment-effects empirical approach finds evidence that granting central banks independence reduces inflation once endogenous selection is accounted for.
- Broader implications:
  - Endogenous selection can explain why cross-country OLS estimates of CBI on inflation are weak or mixed.
  - Model is generalizable to other delegation contexts (e.g., trade policy in customs unions, fiscal responsibilities across government tiers).
- Limitations and further work:
  - Model relies on restrictive assumptions for tractability (dichotomous policies, two policy dimensions).
  - Further research should test predictions in other policy settings and generalize to less restrictive environments.

*Source: _wp06256 — Appendix I and related sections as provided in the supplied PDF content.*

### Appendix I.Equilibrium Coalition Structure

### Appendix I.Equilibrium Coalition Structure

### Game setup and definitions
- The sequential coalition formation game of Bloch (1996) is adopted, with four players (a representative member of each of the four factions).
- The game is defined over a valuation v that maps the set of coalition structures into (four-element) payoff vectors and a rule of order  that defines the order in which the four players move.
- Structure of play:
  - The first player (according to ) proposes the formulation of a coalition to which he belongs.
  - Every other prospective member of this coalition responds in the order defined by . If any prospective member rejects, he makes a counter-offer (to which the others respond in order). If all accept, the coalition forms and its members exit the game. The game continues among remaining players until all have exited.
- The valuation is determined by the intra-coalition bidding process discussed in the text; payoffs are endogenous for the game as a whole but exogenous to the coalition-formation subgame.

### Payoff definitions and normalization
- Payoff entries (only one symmetric set shown) and notation as given in the source:
  - Own Coalition:   0
  - Facing : n.a.AB J
  - Coalition 0C J
  - Where:
    - A 1 2 (V+ ) ;B S ;B L (1)
    - A  1 2 (V+ 1) ;B S V;B L (1)V
    - C S V;C L V+ (21)
    - C S 1;C L 1 + (21)V
    - A0  1 2 (V+ 1)
- Payoffs have been normalized: V  V >1;V  1.
- The variable  gives the probability of the disputed policy matching the preferred policy of the larger faction within the victorious two-faction coalition.
- From equation (9), the value of  is given exactly as printed:
  - = 1 n S 2n L = 31 2 2  1 2 ;1 

### Equilibrium concept
- Equilibrium notion adopted: Bloch’s stationary perfect equilibrium — a subgame perfect equilibrium that is also stationary.
- Stationarity: only payoff-relevant variables (any existing coalition offer, the players who have exited the game, and the coalition structure (partition) of this set of exited players) affect strategies.

### Main result (Proposition A.1)
- The game has only one coalition structure consistent with stationary perfect equilibrium:
  -  ;  (two opposing coalitions form along the most contentious policy dimension ).

### Proof sketch and key logical steps
- Proof by backward induction; derive equilibrium strategies for subgames with two players having exited and for subgames with no coalitions formed.
- Cases when two players have exited (two remain): possible partitions of exited players:
  - [0;0]; [];   
- Optimal response to an offered membership in a  coalition is to accept in the two-feasible-subgame cases because:
  1. (if the coalition    has formed and exited): A > B J
  2. (if the coalitions [0;0] have formed and exited): C J > A0
- Therefore, if a  coalition has formed, then  ;  is the only stationary perfect equilibrium coalition structure.
- When two players have formed [] or [0;0] and exited, acceptance of a  offer depends on parameter inequalities:
  - If coalition [] has formed and exited:
    - 1 V >  (player is ‘S’);
    - 1 V > 23 (player is ‘L’).
  - If coalitions [0;0] have formed and exited:
    - 1 V > 34 (player is ‘L’).
  - A small faction never accepts if coalitions [0;0] have formed and exited and the player is ‘S’.
- The coalition  cannot form when the coalitions [0;0] have formed and exited because the optimality conditions cannot hold simultaneously for both faction sizes fS; Lg.
- The condition 1 V >  implies 1 V > 23 for the valid parameter ranges 1 V 2[0;1], 2  1 2 ;1 .
  - Hence, when two players have exited in the [] subgame, making the offer  (which will be accepted) is optimal if 1 V > . Otherwise, or if [0;0] have formed, the optimal next action is to withdraw (0 coalition).
- Subgame where a single player has formed [0] and exited:
  - A player offered    accepts because payoff C J in the resulting coalition structure is higher than payoffs in any other structure; making the offer is optimal for the same reason.
  - When a single player has exited, the only feasible coalition structure is  0;0; .
  - But the exiting player in that case obtains payoff B J, which is his lowest possible payoff; hence withdrawing cannot be optimal when no coalitions have formed.
- Therefore the first coalition to exit must be a  coalition; initial moves in equilibrium are making the offer    or [].
- Responses to offers when no coalitions have formed:
  - If a player receives an offer of    and accepts, both    coalitions form and all players receive payoff A.
  - If a player receives an offer of [] and accepts, the other [] forms if 1 V > ; in this case all players receive A. If 1 V  then the resulting structure is [0;0; ] and the accepting player gets C J.
- If the player rejects an offer, expected payoff depends on probabilities p_i assigned to four possible first-coalition scenarios (i=1..4) with sum 1:
  - i=1: [] forms, player is member
  - i=2:    forms, player is member
  - i=3: [] forms, player is not a member
  - i=4:    forms, player is not a member
- Payoff from not agreeing given equilibrium-consistent expectations:
  1. If 1 V > : (p1+p3)A + (p2+p4)A2  A;A 
  2. If 1 V  : p1 C J + p3 B J + (p2+p4)A
- From the first case (1 V > ), since A  (p1+p3)A + (p2+p4)A  A, the player always accepts an offer of    and never accepts an offer of []; thus  ;  is the unique stationary perfect equilibrium coalition structure when 1 V > .
- In the second case (1 V  ), for any small faction:
  - C S = 1 B S = V  1 V V= 1 A= 1 2 (V+)  1 2  V+ 1 V  >1 for V >1
  - A small faction will never accept an offer of [].
  - A small faction can propose [] only if he believes it will be rejected; rational beliefs imply large faction always rejects [] proposals. Hence [] can never be offered and accepted.
  - Therefore, for 1 V  , the only coalition structure consistent with stationary perfect equilibrium is also  ; .
- Conclusion: In all parameter regions considered, the only stationary perfect equilibrium coalition structure is  ; .

*Source: Appendix I.Equilibrium Coalition Structure, _wp06256 - Appendix I.Equilibrium Coalition Structure*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2006/_wp06256.pdf_