## 1. Sequence of Events in the Basic Model

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---

### I. Introduction — purpose and setup
- Objective: study the relationship between bank competition, financial stability, and welfare in general equilibrium banking models with moral hazard where choice of “systematic” risk by banks or firms is unobservable.
- Agents: risk-neutral, specialize at date 0 as entrepreneurs, bankers, or depositors; make financing and investment decisions at later dates.
- Welfare criterion: total surplus, defined as total output net of effort costs.
- Two model versions:
  - Basic: the bank is a coalition of entrepreneurs financed by depositors.
  - Extended: the firm is a coalition of entrepreneurs financed by the bank, which is a coalition of bankers financed by depositors.
- Intermediation technology specifications:
  - Constant returns to scale (CR): effort cost of screening/monitoring proportional to investment size.
  - Increasing returns to scale (IR): effort cost independent of investment size.

### III. Key mechanisms and comparative-static results (basic model)
- Basic setup:
  - Bank chooses (systematic) risk P, unobservable to outsiders.
  - Competition in the deposit market indexed by depositors’ opportunity cost max[0,]ρρ∈ (denoted maxρ for “perfect” deposit market competition).
- Results under (CR):
  - As competition in the deposit market increases:
    - bank risk increases,
    - bank capital declines,
    - welfare is maximized for some intermediate degree of competition.
  - Perfect deposit market competition is sub-optimal (excessive risk-taking and sub-optimally low capitalization).
  - Allocating large shares of surplus (rents) to banks is not optimal (leads to sub-optimally low risk-taking and excessive capitalization).
- Results under (IR):
  - As competition increases:
    - bank risk declines,
    - capitalization increases,
    - perfect deposit market competition is optimal,
    - the lowest feasible level of bank risk is best.
  - Mechanism: increased competition raises funding costs (ceteris paribus) which tends to increase bank risk, but higher supply of funds reduces intermediation technology costs (due to increasing returns), offsetting the funding-cost effect and inducing lower bank risk.
- Implication: a positive relationship between bank competition and financial stability can arise without modeling loan market competition.

### III. The basic model — timing, agents, technologies, and contracts
- Timing: three dates, 0, 1 and 2.
- Population: continuum of agents on [0,1] indexed by q∈[0,1].
- Agent choices at date 0:
  - Become investor: exert effort at date 0 to obtain qW units of intermediate good at date 1; this can be reinvested at date 1 or lent to entrepreneurs.
  - Become entrepreneur: forgo qW (the opportunity cost of becoming an entrepreneur).
- Cutoff rule: there exists *q such that agents with q<*q choose to become entrepreneurs and those with q>*q choose investors.
- The bank:
  - Formed as a coalition of entrepreneurs; collects funds from investors (depositors) and distributes profits equally to members.
  - Can operate a risky project indexed by P∈[0,1]; one unit investment yields XW> with probability P, and 0 otherwise.
  - Interprets ability to choose P as intermediation technology (screening/monitoring).
- Intermediation technology cost functions (as specified):
  - CR: 2()( ) CPPZ α = 
  - IR: 2()( ) CPP α = 
- Bank capital technology:
  - By choice of k≥0 the bank generates total capital kZ at effort cost 2 2 kZ β .
  - Bank capital ratio: 1 (1)Kk k − = + .
- Contracts, information, and competition:
  - Depositors provide simple debt contracts paying fixed amount R per unit invested if outcome positive, 0 otherwise.
  - Moral hazard: bank choices of P and k not observable by depositors; depositors anticipate bank’s optimal choices when accepting terms.
  - Degree of deposit market competition indexed by depositors’ opportunity cost max[0,]ρρ∈; higher ρ forces bank to raise deposit remuneration.
  - Term maxρ denotes maximal (perfect) deposit market competition.

### Table: Sequence of Events in the Basic Model (chronology)
- t=0:
  - Agents choose to become entrepreneurs or investors.
  - Entrepreneurs form a coalition called the bank.
  - Variable: *q: measure (fraction) of entrepreneurs.
- t=1:
  - Debt contract terms between the bank and depositors are determined.
  - Depositors deliver funds to the bank.
  - The bank chooses the riskiness of projects and capital.
  - Variables: R, Z, P, k.
- t=2:
  - Project’s output is realized and agents’ consumption follows.

### IV. Equilibrium (definition sketch)
- Given max[0,]ρρ∈, an equilibrium is a level of bank risk *(0,1]P∈, a capitalization rate *k, a deposit rate *R, total investment *Z, and a cut-off level *q such that (model conditions and optimality at each stage hold).

### Date 1 — profit maximization and constraints
- Profit maximization (as presented):
  - 2
    (,, , ) [()]()
    2
    B
    Pk RZP X   R  Xkk  Z  CP
    
            (1)
- Constraint on R*:
  - *
    PR
            (2)
- Condition for Z:
  - *
    1
    **2
    (1)
    2
    q
    W
    ZW   qdqq
    
    
      (3)

### Date 0 — model setup, surplus, and basic equilibrium characterization
- Bank chooses P, k to maximize expected profits (Equation (1)) as of the end of date 1. Given these choices, R is determined by depositors’ participation constraint (Equation (2)). Equation (3) is the deposit market equilibrium.
- Welfare metric (Surplus): total expected output net of effort costs. Definition (as preserved):
  - *2 ****()(1)( ) k Y P X k Z C P β ρ  = + − −  (Equation (5))
- Characterization of basic equilibrium (Proposition 1):
  - Under (CR):
    - P* = * * 2 () min{,1} XR P X β α β − = −
    - k* = * * 2 () min{,1} X R X k X β α β β − = −
    - R* = 2 * 2 4()()(1 1) 2 XX R X ρ α β ρ β − = − − − if P* < 1; R* = ρ if P* = 1.
  - Under (IR):
    - P* = * * 2 () min{,1} XRZ P XZ β α β − = −
    - k* = * * 2 () min{,1} XRZ X k XZ β α β β − = −
    - R* = 2 * 2 4( / ) (, )(1 1) 2 X Z X RZ X ρ α β ρ β − = − − − if P* < 1; R* = ρ if P* = 1.
  - Assumptions: 2 X W α β > and 2 01 X W X W β α β < < − ensure P* ∈ (0,1) for all ρ ∈ max[0, ρ].

### Comparative statics in the basic model (Proposition 3) and intuition
- Under (CR):
  - ∂( )/∂ρ for {Π**, Pk} : **0;0 Pk ρ ρ < < ; implying bank risk increases (P* declines) and capitalization declines as deposit market competition increases.
  - q* decreases with ρ (∂q*/∂ρ < 0).
  - Z* increases with ρ (∂Z*/∂ρ > 0).
- Under (IR):
  - **0;0 Pk ρ ρ > > ; bank risk declines (P* increases) and capitalization increases as deposit market competition increases.
  - q* decreases with ρ (∂q*/∂ρ < 0).
  - Z* increases with ρ (∂Z*/∂ρ > 0).
- Mechanism summary:
  - Under (CR) higher deposit rate R* (increasing in ρ up to R_max = 2/X) induces banks to increase risk and reduce capital.
  - Under (IR) increasing Z* offsets higher cost of funds, so bank reduces risk (P* up) and raises capital (k* up).

### Welfare properties (Proposition 4)
- Under (CR): there exists ρ̂ ∈ (0, ρ_max) such that Y(ρ̂) ≥ Y(ρ) for all ρ ∈ max[0, ρ]. Optimal competition is imperfect; the optimal bank risk is intermediate.
- Under (IR): Y(ρ) is strictly increasing in ρ for ρ ∈ max(0, ρ]; perfect competition (ρ = ρ_max) is welfare-optimal and supports the lowest feasible bank risk.
- Analytical expressions (preserved forms):
  - Derivative under (CR) (Equation (6)): ** ** *2*2**2* 11 [()()] 2 RR Y RZPXXP ZPXXP ρ ρ α β α β β β  = − − + − − — first term positive, second negative; trade-off depends on Z*.
  - Under (IR) (Equation (7)): *2 *** 2 ()() 2 P Y Z P X ZX ρ α β β = + − — both terms positive, so Y(ρ) increasing.

### Extended model — motivation, setup, and main comparative-statics insights
- Extended model features:
  - Competition in both lending and deposit markets.
  - Bank risk jointly determined by borrowers and banks.
  - Intermediation technology entails set-up costs.
  - Competition indexed by depositors’ opportunity cost (as in basic model) and the firm’s opportunity cost to be financed by the bank.
- Main comparative-statics insights (summary):
  - If intermediation technology is relatively inefficient (high monitoring/screening costs, low set-up costs): an optimal level of competition lower than perfect competition exists, corresponding to an “intermediate” optimal level of bank risk.
  - If intermediation technology is relatively efficient (low monitoring/screening costs, large set-up costs): perfect competition is optimal and the optimal level of bank risk is the lowest attainable.
  - These results are independent of whether intermediation technology exhibits CR or IR in screening/monitoring effort.
- Policy relevance: what matters for financial stability is not rents per se, but their sources and how banks exploit them; implications for supporting bank profits or “charter value” are nuanced.

### Extended model — timing, technologies, and equilibrium characterization (high-level)
- Two agent sets: F (firms) and B (banks/bankers); investors can choose to be entrepreneurs or financiers.
- Firms choose P_F ∈ [0,1], firm capital k_F, with effort cost labelled (M): 2 (F) FFFF CPPZ α .
- Bank incurs fixed set-up cost B ρ to set up intermediation technology and chooses B P, with effort cost functions (CR) and (IR) preserved in source.
- Market-clearing, participation constraints, and first-order conditions (preserved forms):
  - Firm profit (Equation (8)): Π_F ≡ [P (X − R_L + X k) − β/2 k^2] Z − α/2 P^2 Z
  - Bank profit and constraints (Equations (9)–(13)) preserved as in source.
- Key Propositions (preserved statements):
  - Proposition 5: closed-form expressions for P_F*, k_F*, k_B*, R_L, R_D with conditions involving 2 2 4() X X β ρ α β ≡ −.
  - Proposition 6: comparative-statics regimes depending on relative loan-market competition threshold 2 2 4() FD X X β ρ ρ α β − < − (low) versus > − (high); implications for ∂P_F/∂ρ_F, ∂K_F/∂ρ_F, ∂K_B/∂ρ_F.
  - Existence and forms for B P solutions under (CR) and (IR) (Equations (22) and (23), preserved).

### Aggregate equilibrium, welfare, and policy implications (summary)
- Aggregate equilibrium four-tuple (Z_B*, Z_F*, q_F*, q_B*) satisfies F B B Z k Z = + and closed-form expressions differ under (CR) vs (IR) as in Proposition 7.
- Policy and efficiency findings (preserved qualitative conclusions):
  - Relatively efficient intermediation technology: as competition increases, bank risk declines under both (CR) and (IR); perfect competition is optimal; increases in competition raise supply of funds B Z* and F Z*; firm capital increases and bank capital declines when competition increases from non-low levels.
  - Relatively inefficient intermediation technology: imperfect (intermediate) competition is welfare-optimal; maximum welfare attained at intermediate ρ and intermediate bank risk; results hold under both (CR) and (IR).
- Normative implications:
  - Under (CR): increased deposit-market competition tends to raise bank risk and lower capital; an imperfect level of deposit-market competition (ρ̂ ∈ (0, ρ_max)) is optimal.
  - Under (IR): increased competition reduces bank risk and raises capital; perfect deposit-market competition is optimal.
  - In the extended model, relative efficiency of intermediation technology determines whether perfect competition is welfare-optimal.
- Policy takeaway:
  - Supporting bank profitability (charter values) via rents that are independent of bank actions to improve efficiency can be unwarranted. Competitive pressures can incentivize banks to adopt more efficient intermediation technologies; rents should be earned through investment in efficiency rather than freely granted market power.

*Source: _wp11295 - 1. Sequence of Events in the Basic Model (source PDF).*

### 1. Sequence of Events in the Basic Model ...............................................................................

### 1. Sequence of Events in the Basic Model

### I. Introduction — purpose and setup
- Objective: study the relationship between bank competition, financial stability, and welfare in general equilibrium banking models with moral hazard where choice of “systematic” risk by banks or firms is unobservable.
- Agents: risk-neutral, specialize at date 0 as entrepreneurs, bankers, or depositors; make financing and investment decisions at later dates.
- Welfare criterion: total surplus, defined as total output net of effort costs.
- Two model versions:
  - Basic: the bank is a coalition of entrepreneurs financed by depositors.
  - Extended: the firm is a coalition of entrepreneurs financed by the bank, which is a coalition of bankers financed by depositors.
- Two intermediation technology specifications:
  - Constant returns to scale (CR): effort cost of screening/monitoring proportional to investment size.
  - Increasing returns to scale (IR): effort cost independent of investment size.

### III. Key mechanisms and comparative-static results (basic model)
- In the basic model:
  - Bank chooses (systematic) risk P, unobservable to outsiders.
  - Competition in the deposit market is indexed by depositors’ opportunity cost max[0,]ρρ∈ (denoted maxρ for “perfect” deposit market competition).
- Results under constant returns to scale (CR):
  - As competition in the deposit market increases:
    - bank risk increases,
    - bank capital declines,
    - welfare is maximized for some intermediate degree of competition.
  - Perfect deposit market competition is sub-optimal (excessive risk-taking and sub-optimally low capitalization).
  - Allocating large shares of surplus (rents) to banks is not optimal (leads to sub-optimally low risk-taking and excessive capitalization).
- Results under increasing returns to scale (IR):
  - As competition increases:
    - bank risk declines,
    - capitalization increases,
    - perfect deposit market competition is optimal,
    - the lowest feasible level of bank risk is best.
  - Mechanism: increased competition raises funding costs (ceteris paribus) which tends to increase bank risk, but higher supply of funds reduces intermediation technology costs (due to increasing returns), offsetting the funding-cost effect and inducing lower bank risk.
- Implication: a positive relationship between bank competition and financial stability can arise without modeling loan market competition.

### IV. Motivation for the extended model (overview)
- Extended model features:
  - Competition in both lending and deposit markets.
  - Bank risk jointly determined by borrowers and banks.
  - Intermediation technology entails set-up costs.
  - Competition indexed by depositors’ opportunity cost (as in basic model) and the firm’s opportunity cost to be financed by the bank.
- Main comparative-statics insights (summary):
  - If intermediation technology is relatively inefficient (high monitoring/screening costs, low set-up costs): an optimal level of competition lower than perfect competition exists, corresponding to an “intermediate” optimal level of bank risk.
  - If intermediation technology is relatively efficient (low monitoring/screening costs, large set-up costs): perfect competition is optimal and the optimal level of bank risk is the lowest attainable.
  - These results are independent of whether intermediation technology exhibits CR or IR in screening/monitoring effort.
- Policy relevance: what matters for financial stability is not rents per se, but their sources and how banks exploit them; implications for supporting bank profits or “charter value” are nuanced.

### III. The basic model — timing, agents, technologies, and contracts
- Timing: three dates, 0, 1 and 2.
- Population: continuum of agents on [0,1] indexed by q∈[0,1].
- Agent choices at date 0:
  - Become investor: exert effort at date 0 to obtain qW units of intermediate good at date 1; this can be reinvested at date 1 or lent to entrepreneurs.
  - Become entrepreneur: forgo qW (the opportunity cost of becoming an entrepreneur).
- Cutoff rule: there exists *q such that agents with q<*q choose to become entrepreneurs and those with q>*q choose investors.
- The bank:
  - Formed as a coalition of entrepreneurs; collects funds from investors (depositors) and distributes profits equally to members.
  - Can operate a risky project indexed by P∈[0,1]; one unit investment yields XW> with probability P, and 0 otherwise.
  - Interprets ability to choose P as intermediation technology (screening/monitoring).
- Intermediation technology cost functions (as specified):
  - CR: 2()( ) CPPZ α =  (i.e., C(P,Z)=α P^2 Z^2 ? — preserve text as given)
  - IR: 2()( ) CPP α =  (i.e., C(P)=α P^2 ? — preserve text as given)
- Bank capital technology:
  - By choice of k≥0 the bank generates total capital kZ at effort cost 2 2 kZ β .
  - Bank capital ratio: 1 (1)Kk k − = + . (preserve expression as presented)
  - Capital is endogenous and depends on specialization choices, funds received, and bank risk choice.
- Contracts, information, and competition:
  - Depositors provide simple debt contracts paying fixed amount R per unit invested if outcome positive, 0 otherwise.
  - Moral hazard: bank choices of P and k not observable by depositors; depositors anticipate bank’s optimal choices when accepting terms.
  - Degree of deposit market competition indexed by depositors’ opportunity cost max[0,]ρρ∈; higher ρ forces bank to raise deposit remuneration.
  - Term maxρ denotes maximal (perfect) deposit market competition.

### Table: Sequence of Events in the Basic Model (chronology)
- t=0:
  - Agents choose to become entrepreneurs or investors.
  - Entrepreneurs form a coalition called the bank.
  - Variable: *q: measure (fraction) of entrepreneurs.
- t=1:
  - Debt contract terms between the bank and depositors are determined.
  - Depositors deliver funds to the bank.
  - The bank chooses the riskiness of projects and capital.
  - Variables: R, Z, P, k.
- t=2:
  - Project’s output is realized and agents’ consumption follows.

### IV. Equilibrium (definition sketch)
- Given max[0,]ρρ∈, an equilibrium is a level of bank risk *(0,1]P∈, a capitalization rate *k, a deposit rate *R, total investment *Z, and a cut-off level *q such that (model conditions and optimality at each stage hold). (Full formal definition and conditions follow in the paper.)

*Source: _wp11295 - 1. Sequence of Events in the Basic Model (source PDF).*

### 1. Date 1

### 1. Date 1

### Profit maximization problem
- Given Z and q, P*, R*, k* maximize the profits of the bank:
  - ΠB( , , , ) ≡ 2β [P k R Z P X() 2 B  -?]  (equation as presented)
  - Exact expression from source:
    - 2
      (,, , ) [()]()
      2
      B
      Pk RZP X   R  Xkk  Z  CP
      
              (1)

### Constraint on R*
- Subject to:
  - *
    PR
            (2)

### Condition for Z
- *
  Z satisfies:
  - *
    1
    **2
    (1)
    2
    q
    W
    ZW   qdqq
    
    
      (3)

*Source: _wp11295 - 1. Date 1 — _wp11295 - 1. Date 1 (PDF)*

### 2. Date 0

### _wp11295 - 2. Date 0

### Model setup and equilibrium definitions
- Bank chooses P, k to maximize expected profits (Equation (1)) as of the end of date 1. Given these choices, R is determined by depositors’ participation constraint (Equation (2)). Equation (3) is the deposit market equilibrium: bank demand for funds equals total funds supplied by depositors. Equation (4) determines agent specialization: q* is the indifferent agent between entrepreneur and depositor; fraction entrepreneurs = q*, fraction depositors = 1 - q*.
- Welfare metric (Surplus): total expected output net of effort costs. Total output in successful state: (1)XkZ + ** (notation preserved). Bank’s effort in project risk choice: C P(*) . Total cost of capital: *2 2 k β.
- Definition 2 (Surplus): given ρ ∈ max[0, ρ], expected total output net of effort costs:
  - *2 ****()(1)( ) k Y P X k Z C P β ρ  = + − − (Equation (5)) (notation and ordering preserved as in source).

### Characterization of basic equilibrium (Proposition 1)
- Under (CR):
  - P* = * * 2 () min{,1} XR P X β α β − = −  (preserved form)
  - k* = * * 2 () min{,1} X R X k X β α β β − = −
  - R* = 2 * 2 4()()(1 1) 2 XX R X ρ α β ρ β − = − − − if P* < 1; R* = ρ if P* = 1.
- Under (IR):
  - P* = * * 2 () min{,1} XRZ P XZ β α β − = −  (preserved form)
  - k* = * * 2 () min{,1} XRZ X k XZ β α β β − = −
  - R* = 2 * 2 4( / ) (, )(1 1) 2 X Z X RZ X ρ α β ρ β − = − − − if P* < 1; R* = ρ if P* = 1.
- Assumptions illustrated: 2 X W α β > and 2 01 X W X W β α β < < − ensure P* ∈ (0,1) for all ρ ∈ max[0, ρ].

### Comparative statics in the basic model (Proposition 3) and intuition
- Under (CR):
  - ∂( )/∂ρ for {Π**, Pk} : **0;0 Pk ρ ρ < < ; implying bank risk increases (P* declines) and capitalization declines as deposit market competition increases.
  - q* decreases with ρ (∂q*/∂ρ < 0).
  - Z* increases with ρ (∂Z*/∂ρ > 0).
- Under (IR):
  - **0;0 Pk ρ ρ > > ; bank risk declines (P* increases) and capitalization increases as deposit market competition increases.
  - q* decreases with ρ (∂q*/∂ρ < 0).
  - Z* increases with ρ (∂Z*/∂ρ > 0).
- Mechanism: under (CR) higher deposit rate R* (increasing in ρ up to R_max = 2/X) induces banks to increase risk and reduce capital; under (IR) increasing Z* offsets higher cost of funds, so bank reduces risk (P* up) and raises capital (k* up).

### Welfare properties (Proposition 4)
- Under (CR): there exists ρ̂ ∈ (0, ρ_max) such that Y(ρ̂) ≥ Y(ρ) for all ρ ∈ max[0, ρ]. Optimal competition is imperfect; the optimal bank risk is intermediate (between highest and lowest feasible).
- Under (IR): Y(ρ) is strictly increasing in ρ for ρ ∈ max(0, ρ]; perfect competition (ρ = ρ_max) is welfare-optimal and supports the lowest feasible bank risk.
- Analytical expressions:
  - Derivative under (CR) (Equation (6)): ** ** *2*2**2* 11 [()()] 2 RR Y RZPXXP ZPXXP ρ ρ α β α β β β  = − − + − − — first term positive, second negative; trade-off depends on Z*.
  - Under (IR) (Equation (7)): *2 *** 2 ()() 2 P Y Z P X ZX ρ α β β = + − — both terms positive, so Y(ρ) increasing.

### Extended model: firms and banks, sequence of events
- Two sets of continuum agents on [0,1]: set F (choose investor or entrepreneur) and set B (choose investor or banker). Investors use effort at date 0 to obtain q W units of intermediate good at date 1; autarkic technology returns ρ_D at date 2.
- Firms (successful entrepreneurs) choose project risk F P ∈ [0,1] and firm capital k_F; firm effort cost: C_F (P_F) = 2 (F) FFFF CPPZ α (constant returns in effort cost per external funding F Z) (labelled (M) in source).
- Firm also chooses capital technology cost: 2 2 FF k Z β .
- Firm alternative: risk-free technology returning ρ_F > ρ_D per unit borrowed directly from investors; competition in loan market indexed by ρ_F.
- Bank chooses B P at date 0, incurs fixed set-up cost B ρ to set up intermediation technology. Bank effort cost functions:
  - (CR): 2 () 2 BBBB CPP Z δ ρ = −
  - (IR): 2 () 2 BBB CPP δ ρ = −
- Bank capital technology: by choice k_B ≥ 0, bank generates B B k Z at cost 2 2 B B k Z β .
- Bank risk BF P P R = determined jointly: bank’s B P and firm’s F P (probabilities independent).
- Contracts: depositors finance banks; banks finance firms with simple debt. Moral hazard: choices (P,k) not observed by outsiders; outsiders anticipate optimal choices.

### Extended-model equilibrium definition (Definition 3) and first-order conditions
- Date 1 firm maximizes (Equation (8)):
  - Π_F ≡ [P (X − R_L + X k) − β/2 k^2] Z − α/2 P^2 Z  (preserved form)
- Date 1 bank maximizes (Equation (9)):
  - Π_B ≡ [P R_L − R_D + R_R (?) − β/2 k^2] Z − ??? (notation preserved as in source) subject to:
  - Firm participation constraint: Π_F ≥ ρ_F − PR (?) (Equation (11)) and depositors’ constraint (Equation (10)): P R_D ≥ ρ_D.
- Date 0 bank chooses B P to maximize (Equation (12)) subject to bank participation (Equation (13)).
- Market-clearing / specialization conditions:
  - Deposit market: 22 (1)(1) B FB WW Z q q = − + − − (Equation (14) as preserved).
  - Loan market: * (1) F B B Z k Z = + (Equation (15)).
  - Specialization via equations (16) and (17): BF P R q W q Π = (preserved).

### Characterization in extended model (Proposition 5 and 6)
- Proposition 5 (definitions preserved):
  - Define 2 2 4() X X β ρ α β ≡ −.
  - (a) Firm choices:
    - P_F* = β (X − R_L) / (α β − X^2) (preserved)
    - k_F* = X P_F* / β
    - k_B* = *2 () L L X R R k X α β − = − (preserved)
  - (b) Equilibrium loan rate and deposit rate:
    - R_L = 2 / X
    - R_D = 2 2 D X R X α β ρ β − = − if FD ρ ρ ρ − ≤  (preserved condition)
    - alternative expressions when FD ρ ρ ρ − > . (Preserved forms as in source.)
- Proposition 6 (comparative statics regimes):
  - When relative loan-market competition is “low” (2 2 4() FD X X β ρ ρ α β − < −), F P, F K, B K are constant and independent of (ρ_F, ρ_D).
  - When relative loan-market competition is “high” (2 2 4() FD X X β ρ ρ α β − > −), an increase in loan-market competition (ρ_F up) given deposit competition leads to:
    - ∂P_F/∂ρ_F > 0; ∂K_F/∂ρ_F > 0; ∂K_B/∂ρ_F < 0.
  - An increase in deposit-market competition given loan-market competition has opposite effects: firms increase risk and reduce capital; banks increase capital.
- Existence and forms for B P solutions:
  - Under (CR): P_B = 1 / (min{,1} B B P δ π −) (Equation (22) preserved).
  - Under (IR): P_B = 1 / (min{,1} B B B P Z δ π −) (Equation (23) preserved).

### Aggregate equilibrium and comparative results (Proposition 7, welfare implications)
- Equilibrium four-tuple (Z_B*, Z_F*, q_F*, q_B*) satisfies (1) F B B Z k Z = + and explicit closed-form expressions depending on (CR) vs (IR) (preserved formulas in Proposition 7).
- Policy and efficiency findings:
  - When intermediation technology is relatively efficient (example parameter configuration denoted (δ, ρ_B) with δ low and ρ_B low): as competition increases, bank risk declines under both (CR) and (IR); perfect competition is optimal; increases in competition raise supply of funds B Z* and F Z*; firm capital increases and bank capital declines when competition increases from non-low levels, but bank risk need not increase.
  - When intermediation technology is relatively inefficient: imperfect (intermediate) competition is welfare-optimal; maximum welfare attained at intermediate ρ and intermediate bank risk; results hold under both (CR) and (IR).
- Figures (described): Figure 1 (Relatively Efficient Intermediation Technology) and Figure 2 (Relatively Inefficient Intermediation Technology) illustrate R(ρ), Y(ρ), Z(B) and Z(F), q(B) and q(F) behavior under CR and IR with parameter sets specified in figures (parameters listed exactly in source figures).
- Summary of main normative implications:
  - Under (CR) of intermediation technology: increased deposit-market competition tends to raise bank risk and lower capital; an imperfect level of deposit-market competition (ρ̂ ∈ (0, ρ_max)) is optimal.
  - Under (IR): increased competition reduces bank risk and raises capital; perfect deposit-market competition is optimal.
  - In the extended model, the efficiency of intermediation technology (relative efficiency) determines whether perfect competition is welfare-optimal: efficient technology → perfect competition optimal; inefficient technology → intermediate competition optimal.
- Policy takeaway:
  - Supporting bank profitability (charter values) via rents that are independent of bank actions to improve efficiency can be unwarranted. Competitive pressures can incentivize banks to adopt more efficient intermediation technologies; rents should be earned through investment in efficiency rather than freely granted market power.

*Source: _wp11295 - 2. Date 0 (excerpt from provided IMF PDF content)*

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### _wp11295 - REFERENCES

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*Source: _wp11295 - REFERENCES*

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