## _wp09105

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### I. INTRODUCTION — purpose and framing
- Purpose: Evaluate general equilibrium implications of the “charter value hypothesis” (CVH) for competition, financial stability, and welfare in models with unobservable risky investment choices (moral hazard).
- Framing:
  - CVH intuition: under limited liability and unobservable risk choices, higher funding costs erode a borrower’s charter value, prompting riskier investments.
  - Paper maps bank market power rents to economy-wide investment and consumption allocations independent of a specific mechanism generating rents.
  - Two model versions: (1) banks as coalitions of entrepreneurs financed by depositors (banks and depositors only), (2) extended model including firms financed by banks funded with deposits (firms, banks, and depositors).
- Main high-level result:
  - Lower banks’ market power rents lead to lower economy-wide risk, lower capital ratios, more efficient production plans, and Pareto-ranked real allocations.
  - Perfect competition (absence of banks’ market power rents) supports a (second) best allocation and optimal levels of bank risk and capitalization.
  - General equilibrium forces can reverse partial-equilibrium CVH predictions because funding amounts are endogenous.

### Model setup: time, agents, endowments, preferences, technologies
- Dates: 0, 1 and 2.
- Agents: continuum indexed q ∈ [0,1]; type-q agent has endowment qW units of date 0 good and labor l.
- Aggregate date 0 goods: ∫_0^1 W q dq = 1/2 W, with 1W> (as stated in source).
- Preferences: risk neutral over date 2 consumption and disutility from work U(c) − V(l) with U(c) = c and V(l)=l^2/(2α).
- Entrepreneurial technology: invest all date 0 endowment to obtain kW units of date 1 goods with k ∈ (0, ). Project p ∈ [0,1] yields X per unit invested with probability p and 0 otherwise; labor required l = p; V(p) = p^2/(2α).
- Information regimes: moral hazard (p privately observed by banks) and no-moral hazard (p publicly observed).
- Debt contracts: pay R̂ per unit invested if success, 0 otherwise. R̂ treated as exogenous parameter indexing bank market power rents.

### Equilibrium with banks and depositors — moral hazard
- Equilibrium characterization (triplet (p*, I*, q̂*)):
  - First-order (interior) condition: X R I − p α + X k = 0 (presented in source form).
  - Indifference condition for marginal agent q̂: *() I p R q̂ W Π =  (equation (5) form in source).
  - Market-clearing deposit supply: I = ∫_{q̂}^1 q W dq = (1/2) (1 − q̂^2) W (integral (8) in source).
- Existence/uniqueness:
  - Proposition 1: In the moral hazard economy with banks and depositors an equilibrium with intermediation exists and it is unique for every ( , ] 2 X k R X W ∈.
- Comparative statics (Proposition 2 — signs preserved):
  - dI/dR > 0
  - d q̂ / dR < 0
  - d p* / dR > 0
  - d K* / dR < 0
  - d p* / d k > 0
- Interpretation (source text):
  - As R increases, bank profits decline ceteris paribus, inducing more agents to become depositors (q̂ falls), increasing total funds I. The increase in I can offset profit declines from higher R, leading banks to choose safer projects (p increases). Bank capital ratio K declines with R, but p goes up — higher aggregate capital ratios need not imply lower bank risk in general equilibrium.

### Equilibrium with banks and depositors — no moral hazard
- Characterization:
  - With observable p, debt contract pays R_p per unit invested; bank choice of p does not depend directly on R (no bank risk shifting).
  - Equilibrium conditions analogous to moral hazard case with R_p replacing R.
- Existence/uniqueness:
  - Proposition 3: In the no-moral hazard economy with banks and depositors an equilibrium with intermediation exists and it is unique for every 2222 (2) (,) 24 Xk X Wk R WW α α + ∈.
- Comparative statics (Proposition 4 — signs preserved):
  - dI/dR > 0
  - d q̂ / dR < 0
  - d p* / dR > 0
  - d K* / dR < 0
  - d p* / d k > 0

### Optimality, intermediary rents, and welfare implications
- Planner’s (first-best) problem:
  - Planner maximizes aggregate consumption net of entrepreneur disutility; objective (21) strictly concave in q and p.
  - First-best closed-form: p^O and q^O given in source form (e.g., "2(2) 5 O XkW p α + =  and  2 5 O kW q W + =").
- Moral hazard and Pareto support:
  - Proposition 5: In the moral hazard economy with banks and depositors, there is no equilibrium with intermediation that supports the Pareto optimal allocation.
- Second-best under moral hazard:
  - Define R* solving the constrained maximization (25) subject to (26).
  - Proposition 6: R* is an equilibrium with intermediation that supports the second best allocation (moral hazard).
- First-best under no-moral hazard:
  - Proposition 7: In the no-moral hazard economy with banks and depositors, there exists a unique 0 R such that the corresponding equilibrium with intermediation supports the (first best) Pareto optimal allocation.
- Definition of market power rents ρ:
  - Rents ρ are the difference between the perfectly competitive outcome (R* in moral hazard, 0 R in no-moral hazard) and an equilibrium value of R strictly lower than the corresponding perfectly competitive outcome.
  - Planning objective strictly concave in R; outcomes with 0 < ρ are Pareto-ranked (if ρ_2 > ρ_1 > 0 then allocations associated with ρ_2 Pareto-dominate those with ρ_1).
- Convergence as rents vanish (Proposition 8):
  - As market power rents vanish (ρ → 0):
    - * ˆ p p ↑ (equilibrium ˆ p converges to optimal p* from below)
    - * ˆ K K ↓ (equilibrium ˆ K converges to optimal K* from above)
    - Equilibrium converges to a best allocation.

### Extension: firms, intermediaries, depositors; correlated vs independent projects
- Extended model: agents choose to be firms or banks (intermediaries who monitor and intermediate funds). Banks raise funds from depositors at rate R^D and lend to firms at rate R^L.
- Perfectly correlated projects:
  - Equilibrium conditions reduce to prior intermediary-only model when R^D = R.
  - Lending rate R^L is increasing in D R, decreasing in I; displays partial-equilibrium CVH effect where risk declines with total loans.
- Independent projects:
  - Diversification benefits from intermediation; first-order conditions and profits give analogs to the no-moral hazard base model when D R = R.
- Source conclusion: earlier derivations and propositions carry over to the extended model in both perfectly correlated and independent project cases.

### Policy-relevant conclusions and empirical relevance
- Principal theoretical implication:
  - General equilibrium analysis can yield implications opposite to some partial-equilibrium constructs: in general equilibrium, greater bank competition (lower market power rents) lowers economy-wide risk, lowers bank capitalization, and improves production efficiency and welfare.
  - Policy inference: inefficiencies or instabilities attributed to "excessive" competition should be modeled in a general equilibrium framework; systemic and economy-wide funding endogeneity matters for optimal capital regulation design.
- Empirical findings cited (source):
  - Jayaratne and Strahan (1998): branch deregulation resulted in a sharp decrease in loan losses.
  - Restrictions on entry/activity negatively associated with some measures of bank stability: Barth, Caprio and Levine (2004); Beck (2006a and 2006b).
  - Positive linkages between bank competition and economic growth: Cetorelli and Gambera (2001); Cetorelli and Strahan (2006).
  - Positive association between financial stability and stronger bank competition: Boyd, De Nicoló and Jalal (2006, 2009); De Nicolò and Loukoianova (2007); Boyd, De Nicoló and Loukoianova (2009).
- Policy recommendation (source emphasis):
  - Develop general equilibrium setups where the nexus between bank capital and risk is defined at the system level; general equilibrium modeling of intermediation is essential for illuminating desirable levels of systemic risk and for designing optimal capital regulations.

### Technical propositions and key mathematical conditions (selected)
- Domains and equilibrium conditions preserved from source:
  - q ∈ (0,1)
  - Existence conditions: e.g., 2 Xk R W ≥ ; 22 2 Xk R W α ≥ ; 22 (2) 4 XW k R W α + ≤ (as presented in source).
  - Comparative static signs explicitly: dp/dR > 0, dI/dR > 0, dq/dR < 0, dK/dR < 0, dp/dk > 0.
- Selected proposition summaries (verbatim structure preserved):
  - Proposition 1: Existence and uniqueness in moral hazard economy for every (,]2XkRXW∈.
  - Proposition 2: Moral hazard comparative statics with signs *0 R I >, *ˆ 0 R q <, *0 R p >, *0 R K <, *0 k p >.
  - Proposition 3: Existence and uniqueness in no-moral hazard economy for every 2222 (2) (,)24 Xk X Wk R WW α α + ∈.
  - Proposition 4: No-moral hazard comparative statics identical in sign to moral hazard case.
  - Proposition 5: No equilibrium with intermediation supports Pareto optimum under moral hazard.
  - Proposition 6: R* supports second-best allocation under moral hazard.
  - Proposition 7: Unique *0 R supports first-best allocation under no-moral hazard.
  - Proposition 8: As 0 ρ →, * ˆ p p ↑ and * ˆ K K ↓; equilibrium converges to a best allocation.

*Source: _wp09105 - References; Section VI; Propositions and Appendix (all expressions and numeric conditions preserved as in source).*

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

### _wp09105 - References..............................................................................................................

### I. INTRODUCTION — purpose and framing
- Evaluates the general equilibrium implications of the “charter value hypothesis” (CVH) for competition, financial stability, and welfare in models where the choice of risky investment is unobservable by investors (moral hazard).
- Motivation: the CVH has strongly influenced financial policy and bank regulation (e.g., capital requirements).
- Approach: map bank market power rents to economy-wide investment and consumption allocations independent of a specific mechanism generating rents.

### The charter value hypothesis (CVH) — mechanism and standard partial-equilibrium implication
- Core intuition: under limited liability and unobservable risk choices, higher funding costs erode a borrower’s charter value (expected profits), prompting riskier investments because upside is retained while downside is limited.
- Implementation in literature: banks modeled as entities raising funds from insured depositors and choosing portfolio risk; increased competition raises deposit costs and thus incentives for banks to take more risk.
- Representative literature cited: Allen and Gale (2000 and 2004a), Keeley (1990), Matutes and Vives (1996), Hellmann, Murdock and Stiglitz (2000), Cordella and Levi-Yeyati (2002), Repullo (2004).
- Note: CVH is a basic tenet underlying many models rationalizing risk-taking under deposit insurance and underlies bank regulation design.1

### Interacting loan and deposit competition — alternative partial-equilibrium result
- Boyd and De Nicolò (2005): when banks compete in both loan and deposit markets, firms and banks jointly determine risk choices.
  - Increased bank competition reduces loan rates to firms (increasing firms’ charter value) but increases banks’ funding costs.
  - Firms’ higher charter value leads them to choose safer investments, producing a safer loan portfolio for banks.
  - Under standard assumptions, equilibrium features lower loan rates as competition increases; the naive CVH trade-off can vanish.
- Extensions aiming to delineate conditions for the CVH trade-off: Boyd, De Nicolò and Jalal (2006 and 2009); De Nicolò and Loukoianova (2007); Martinez-Miera and Repullo (2008).

### General equilibrium results contrasting with partial equilibrium
- Allen and Gale (2004b): in a Diamond and Dybvig (1983)-type general equilibrium economy, perfect competition among intermediaries is Pareto optimal under complete markets and constrained Pareto optimal under incomplete markets, with financial “instability” as a necessary condition of optimality.
- Boyd, De Nicolò and Smith (2004): analogous general equilibrium results under low inflation in a monetary economy with aggregate liquidity risk.
- Caveat: these general equilibrium models do not feature the specific moral hazard in investment choices central to partial-equilibrium CVH models.2

### This paper’s contributions and key results
- Model setup:
  - Agents specialize at start date into firm-entrepreneurs, bank-entrepreneurs, or depositors; later they make investment and financing decisions.
  - Two model versions:
    1. Banks as coalitions of entrepreneurs financed by depositors (banks and depositors only).
    2. Extended model including firms financed by banks funded with deposits (firms, banks, and depositors).
- Methodology:
  - Identify equilibriums parameterized by a given interest rate on debt contracts.
  - Characterize first best and constrained optimality.
  - Define a perfectly competitive equilibrium as the equilibrium debt contract interest rate that supports a core allocation.
  - Define bank market power rents as the deviation of an equilibrium debt contract interest rate from the debt contract interest rate prevailing at a core (perfectly competitive) allocation.
- Main findings (both model versions):
  - Lower banks’ market power rents lead to:
    - Lower economy-wide risk.
    - Lower capital ratios.
    - More efficient production plans.
    - Pareto-ranked real allocations.
  - Perfect competition (absence of banks’ market power rents) supports a (second) best allocation and optimal levels of bank risk and capitalization.
  - General equilibrium with moral hazard yields implications identical to Allen and Gale (2004b) and Boyd, De Nicolò and Smith (2004) despite including unobservable investment choices.
- Reconciliation with partial-equilibrium CVH:
  - In partial equilibrium, higher funding costs for a given funding amount lead borrowers to choose riskier investments (charter value effect).
  - In general equilibrium, specialization and funding decisions are endogenous: an increase in funding cost also increases the amount of funding available for investment, which can offset the charter-value erosion and lead borrowers to choose lower risk.
  - Conclusion: the CVH correctly describes incentives but may not predict equilibrium outcomes when general equilibrium forces operate.

### Paper organization
- Section II: basic version of model with banks and depositors only.
- Section III: existence of equilibriums under moral hazard and no-moral hazard for given interest rates on debt contracts; comparative statics.
- Section IV: characterization of optimality, definition of bank market power rents, and demonstration of an inverse relation between bank market power rents and optimality.
- Section V: extended model with firms, banks, and depositors; shows the robustness of prior results in the extended setting.

*Source: _wp09105 - References..............................................................................................................*

### Section VI concludes, pointing out fruitful research’s avenues and  the empirical relevance of

### _wp09105 - Section VI concludes, pointing out fruitful research’s avenues and  the empirical relevance of

### Model setup: time, agents, endowments, preferences, technologies
- Three dates: 0, 1 and 2.
- Continuum of agents indexed q ∈ [0,1]. A type-q agent has endowment qW units of date 0 good and labor l.
- Total date 0 goods: ∫_0^1 W q dq = 1/2 W, with 1W> (as stated).
- Types are private information and not credibly communicable to others at feasible resource cost.
- Preferences: agents are risk neutral over date 2 consumption and incur disutility from work. Utility represented by U(c) − V(l), where U(c) = c and V(l) = l^2/(2α) (written as V(l)=l^2/(2α) in the source form).
- Entrepreneurial technology: requires investing all date 0 endowment to obtain kW units of date 1 goods with k ∈ (0, ). Resource cost for agent q to become entrepreneur = qW.
- Projects indexed by p ∈ [0,1]; a p-project yields X per unit invested with probability p and 0 otherwise. Labor required: l = p. Disutility from choosing p: V(p) = p^2/(2α).
- Two information regimes analyzed: moral hazard (p privately observed by banks) and no-moral hazard (p publicly observed).
- Debt contracts between banks and depositors pay R̂ per unit invested if success, 0 otherwise. R̂ is treated as an exogenous parameter indexing bank market power rents.

### Equilibrium with banks and depositors — moral hazard (key equilibrium conditions and comparative statics)
- Given depositors’ supply I, a coalition of bank-entrepreneurs of positive measure maximizes:
  - (1) maximize over p: λ [ p X I + X k − (p^2)/(2α) ] (source presents analogous expression leading to first-order condition).
- Interior optimality condition (moral hazard):
  - (2) *p satisfies: X R I − p α + X k = 0  (presented in source as *()XRIXk p α −+ = )
  - Profits per bank-entrepreneur (Π*): Π* = (2) [ X R I + X k − (p^2)/(2α) ] (presented in source as equation (3) form).
- Agent q chooses to become bank-entrepreneur if:
  - (4) *p I p R q W Π ≥  (source expression: *()IpRqWΠ≥)
  - Indifferent agent q̂ solves: *() I p R q̂ W Π =  (equation (5) in source).
- Fractions: bank-entrepreneurs = q̂; depositors = 1 − q̂.
- Equilibrium with intermediation defined by triplet (p*, I*, q̂*) satisfying:
  - (6) first-order p condition (as above),
  - (7) indifference condition for q̂,
  - (8) market-clearing deposit supply: I = ∫_{q̂}^1 q W dq = (1/2) (1 − q̂^2) W (presented in source as integral (8)).
- Existence/uniqueness:
  - Proposition 1: In the moral hazard economy with banks and depositors an equilibrium with intermediation exists and it is unique for every ( , ] 2 X k R X W ∈. (Preserve source formatting: proposition statement as in source.)
- Comparative statics (Proposition 2, moral hazard):
  - dI/dR > 0  (written in source as *0 R I>,)
  - d q̂ / dR < 0  (source: *ˆ 0 R q<,)
  - d p* / dR > 0  (source: *0 R p>,)
  - d K* / dR < 0  (source: *0 R K<)
  - d p* / d k > 0  (source: *0 k p>.)
- Interpretation from source:
  - As R increases, bank profits decline ceteris paribus, inducing more agents to become depositors (q̂ falls), increasing total funds I to banks. The increase in I can offset profit declines from higher R, leading banks to choose safer projects (higher p). Bank capital ratio K declines with R, but p goes up — so higher aggregate capital ratios need not imply lower bank risk in general equilibrium.

### Equilibrium with banks and depositors — no moral hazard (key equilibrium conditions and comparative statics)
- With observable p, debt contract pays R_p per unit invested and bank choice of p follows:
  - (12) *p solves: X I + k / α + ... (source states: *()XI k p α + = — preserving structure).
  - In no-moral hazard, choice of p does not depend directly on R; no bank risk shifting.
- Profits per bank-entrepreneur in no-moral hazard: equation (13) in source.
- Equilibrium conditions (equations (15)–(17) in source) analogous to the moral hazard case but with observable p and R_p.
- Existence/uniqueness:
  - Proposition 3: In the no-moral hazard economy with banks and depositors an equilibrium with intermediation exists and it is unique for every 2222 (2) (,) 24 Xk X Wk R WW α α + ∈. (Preserve source format.)
- Comparative statics (Proposition 4, no moral hazard) — identical comparative statics signs to moral hazard:
  - dI/dR > 0, d q̂ / dR < 0, d p* / dR > 0, d K* / dR < 0, d p* / d k > 0 (source lists these as Proposition 4).

### Optimality, intermediary rents, and welfare implications
- Planner’s problem (first best): maximize aggregate consumption net of entrepreneur disutility subject to resource constraints; reduces to maximizing total output net of p-disutility:
  - Planning objective (21) in source: strictly concave in q and p.
  - First-order conditions yield closed-form first-best solutions:
    - (24) p^O = 2(2) / (5) * X k W / α + (as in source) and q^O = 2/5 * k W / (W + k) (presented in source form: "2(2) 5 O XkW p α + =  and  2 5 O kW q W + =").
- Moral hazard: no equilibrium with intermediation supports the Pareto optimal allocation.
  - Proposition 5: In the moral hazard economy with banks and depositors, there is no equilibrium with intermediation that supports the Pareto optimal allocation.
- Second-best: choose R to maximize equilibrium welfare V(p̂(R), q̂(R)) over admissible R:
  - Define R* solving problem (25) subject to (26). Proposition 6: R* is an equilibrium with intermediation that supports the second best allocation (moral hazard).
- No moral hazard: there exists a unique R^0 such that corresponding equilibrium supports the first best allocation.
  - Proposition 7: In the no-moral hazard economy with banks and depositors, there exist a unique 0 R such that the corresponding equilibrium with intermediation supports the (first best) Pareto optimal allocation.
- Definition of market power rents ρ:
  - Rents ρ are the difference between the perfectly competitive outcome (R* in moral hazard, 0 R in no-moral hazard) and an equilibrium value of R strictly lower than the corresponding perfectly competitive outcome. (Source defines ρ accordingly.)
  - The planning objective is strictly concave in R; all outcomes with 0 < ρ are Pareto-ranked: if ρ_2 > ρ_1 > 0 then allocations associated with ρ_2 Pareto-dominate those with ρ_1 (source text).
- Main comparative result (Proposition 8, source):
  - As market power rents vanish (ρ → 0):
    - Equilibrium p̂ converges to optimal p* from below: p̂ ↑ p* (source: *ˆ pp↑)
    - Equilibrium K̂ converges to optimal K* from above: K̂ ↓ K* (source: *ˆ KK↓)
    - Equilibrium converges to a best allocation.
  - Summary statement from source: lower market power rents result in lower economy-wide risk, lower bank capitalization, more efficient production plans and Pareto-ranked real allocations. Perfect competition (absence of rents) supports a best allocation and optimal levels of risk and bank capitalization in model variants.

### Extension: firms, intermediaries, depositors; correlated vs independent projects
- Extended model allows entrepreneurs to choose between being firms or banks (intermediaries who monitor and intermediate funds).
- Banks raise funds from depositors at rate R^D and lend to firms at rate R^L. Deposit-side bargaining sets R^D; R^L determined by entrepreneurs’ specialization decisions.
- Perfectly correlated projects case:
  - Firms’ first-order condition and expressions yield equilibrium conditions analogous to the basic intermediary-only model; when setting R^D = R (as in the base model), expressions reduce to prior equilibrium conditions (equations (32) and (33) in source).
  - Lending rate R^L is increasing in D R, decreasing in I; shows partial-equilibrium CVH effect where risk declines with total loans.
- Independent projects case:
  - Diversification benefits accrue via intermediaries; entrepreneurs’ first-order condition (35) and profit expressions (36) give equilibrium analogs to the no-moral hazard base model when D R = R.
- Source conclusion: all earlier derivations and propositions carry over to the extended model in both the perfectly correlated and independent project cases.

### Policy-relevant conclusions and empirical relevance (from Section VI)
- Principal theoretical implication:
  - General equilibrium analysis yields implications opposite to some partial-equilibrium constructs: in general equilibrium, greater bank competition (lower market power rents) lowers economy-wide risk, lowers bank capitalization, and improves production efficiency and welfare.
  - Therefore, any identified inefficiency or instability attributed to "excessive" competition should be modeled in a general equilibrium framework.
- Empirical relevance (selected findings cited in source):
  - Jayaratne and Strahan (1998): branch deregulation resulted in a sharp decrease in loan losses.
  - Restrictions on banks’ entry and activity found negatively associated with some measures of bank stability: Barth, Caprio and Levine (2004); Beck (2006a and 2006b).
  - Linkages between bank competition and economic growth supportive of positive role of bank competition: Cetorelli and Gambera (2001); Cetorelli and Strahan (2006).
  - Studies finding positive association between financial stability and stronger bank competition: Boyd, De Nicoló and Jalal (2006, 2009); De Nicolò and Loukoianova (2007); Boyd, De Nicoló and Loukoianova (2009).

*Source: _wp09105 - Section VI concludes, pointing out fruitful research’s avenues and  the empirical relevance of our results, and arguing for the importance of general equilibrium modeling for financial policy. Proofs of all propositions are in the Appendix.*

### Conclusions derived from partial equilibrium modeling that are not robust to general

### Conclusions derived from partial equilibrium modeling that are not robust to general equilibrium extensions are likely to result in unwarranted welfare implications

### General conclusion and policy relevance
- Conclusions from partial equilibrium modeling can yield unwarranted welfare implications when not robust to general equilibrium extensions.
- Developing general equilibrium set ups in which the nexus between bank capital and risk is defined not only relative to individual institutions, but at a system level, is highly relevant for understanding the implications of systemic risk for the design of optimal capital regulations.
- A general equilibrium perspective of bank regulation has been scarcely explored theoretically and has been absent in policy discourse.
- The current financial crisis is cited as a stark example of the dichotomy between partial and general equilibrium views and their real consequences, showing that serious difficulties have arisen in financial institutions adhering to minimum capital requirements designed—in a partial equilibrium perspective—to control individual banks’ risk taking incentives.
- General equilibrium modeling of intermediation appears an essential tool to illuminate the desirable level of systemic risk in the economy and how it could be attained.
- The authors aim to contribute to this research agenda in the future.

### Technical results and propositions (existence, uniqueness, comparative statics, optimality)
- Proposition 1
  - In the moral hazard economy with banks and depositors an equilibrium with intermediation exists and it is unique for every (,]2XkRXW∈.
  - Key algebraic relations: (A1) 1 2 () RWq Xk Iq I XR − ≡ = − ; (A2) 2 2 () (1) 2 W Iq Iq ≡ = − .
  - Existence requires 2 Xk R W ≥ so 1 I is non-negative for every value of (0,1)q∈.
  - A unique value * (0,1)q∈ exists such that *() 0Fq= with Fq defined in text; *I is determined using (A1) or (A2); *1*()pXRI Xkα − ⎡⎤ = − + ⎣⎦.

- Proposition 2 (moral hazard comparative statics)
  - In the moral hazard economy with banks and depositors:
    - *0 R I >,
    - *ˆ 0 R q <,
    - *0 R p >,
    - *0 R K < and
    - *0 k p >.
  - Total differentiations yield (A3) 22 XRX dIRWdqqWdRdk − − = − and (A4) 0 dIqWdq +=.
  - Determinant Δ = 0 2 XR qWRW − + >. Cramer’s rule gives (A5) 2 1 ()0 R dI IqW dR ==> Δ and (A6) 1 0 R dq qqW dR ≡ = − < Δ.
  - Relations: (A7) 2 RWq p α = ; (A8) expression for dp/dR showing dp/dR > 0; (A9) and (A11) give signs for dK/dR and dp/dk respectively.

- Proposition 3
  - In the no-moral hazard economy with banks and depositors an equilibrium with intermediation exists and it is unique for every 2222 (2) (,)24 Xk X Wk R WW αα + ∈.
  - Rearranged equations: (A9) 22 1 () () 2 XIkI qI q RWWα + ≡ = − ; (A10) 2 2 () 1 I qI q W ≡ = − .
  - Existence and uniqueness require inequalities (A11) and (A12) which imply 22 2 Xk R W α ≥ and 22 (2) 4 XW k R W α + ≤.

- Proposition 4 (no-moral hazard comparative statics)
  - In the no-moral hazard economy with banks and depositors:
    - *0 R I >,
    - *ˆ 0 R q <,
    - *0 R p >,
    - *0 R K < and
    - *0 k p >.
  - Total differentiations (A13)-(A14) and determinant Δ = 2 () (1)0 XIk qWRWq α + − > lead to Cramer results (A15) and (A16).
  - Differentiation of (15) gives (A18) 0 RR dpXI pI dR α ≡ => showing dp/dR > 0.
  - Additional relations (A20)-(A21) characterize responses of I and p to changes in k and R.

- Proposition 5 (no Pareto support under moral hazard)
  - In the moral hazard economy with banks and depositors, there is no equilibrium with intermediation that supports the Pareto optimal allocation.
  - Substituting (OO) pq into equilibrium conditions yields (A22) and (A23), and combining leads to a contradiction: 12 4 0 2 4220 5 XW RR kW Wk W W = ≠ = + − + . Hence no R yields an equilibrium with ( , ) ( , ) OO p q p q =.

- Proposition 6 (second-best under moral hazard)
  - In the moral hazard economy with banks and depositors, * R is an equilibrium with intermediation that supports the second best allocation.
  - Existence and uniqueness follow from maximization of continuous strictly concave function V(.) in (25) over compact set (26); p̂(R) and q̂(R) are uniquely determined by equilibrium conditions (8)-(10), implying a unique * R.

- Proposition 7 (first-best under no-moral hazard)
  - In the no-moral hazard economy with banks and depositors, there exists a unique *0 R such that the corresponding equilibrium with intermediation supports the (first best) Pareto optimal allocation.
  - Equations (A24)-(A26) determine pO and * R. Verification requires inequalities (A27)-(A28), which reduce to (A29)-(A30) and hold given assumptions 1k <, 1W > and 01 q ≤ ≤, so * R is an equilibrium supporting the Pareto optimum.

- Proposition 8 (convergence as market power rents vanish)
  - In both the moral-hazard and no-moral hazard economies with banks and depositors, the equilibrium level of risk ˆ p converges to the optimal level of risk * p from below, to the optimal level of bank capitalization * K from above, and to a best allocation as market power rents vanish, i.e.
    - * ˆ p p ↑, and
    - * ˆ K K ↓ as 0 ρ →.
  - Proof outline: by propositions 2 and 4, ˆ p increases monotonically in R. By propositions 6 and 7, there exists a unique optimal * R that supports * p. Therefore ** ˆ ˆ () ( ) pR pR p ↑ = and ** ˆ ˆ () ( ) KR KR K ↓ = as * 0 R R ρ = − →.

### Key mathematical and logical conditions referenced
- Equilibrium variables and domains explicitly preserved from source: (0,1)q∈ ; (0,]2 W I ∈ ; (,]2 Xk R X W ∈ ; 2222 (2) (,)24 Xk X Wk R WW αα + ∈.
- Necessary algebraic inequalities and equalities are preserved verbatim where presented (e.g., 2 Xk R W ≥ ; equations (A1)-(A11), (A13)-(A21), (A22)-(A30) as cited).
- Comparative static signs explicitly stated: dp/dR > 0, dI/dR > 0, dq/dR < 0, dK/dR < 0, dp/dk > 0 (as given across propositions).

*Source: _wp09105 - Conclusions derived from partial equilibrium modeling that are not robust to general*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2009/_wp09105.pdf_