## 2.1 The Model

## Source details

**Canonical URL:** [2.1 The Model](https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021102-print-pdf.pdf)

## Other formats

- [Markdown version](/-/media/files/publications/wp/2021/english/wpiea2021102-print-pdf.pdf.md)
- [Structured JSON version](/-/media/files/publications/wp/2021/english/wpiea2021102-print-pdf.pdf.json)

---

### Environment and timing
- Two stages:
  - Stage 1: bankers and households make portfolio decisions. Fixed supply of real asset capital denoted ̄K, tradable at price q. Each unit of capital provides flow z of good. Banks borrow via deposits.
  - Stage 2: run game among households.
- Single source of uncertainty: endogenous risk of a systemic run. In equilibrium a systemic run occurs with probability λ (endogenously determined).
- Banks and households each hold capital (qk_b and qk_h), net worth (n_b and n_h) and deposits (δ).

### Households: portfolio choice, returns, and pricing
- Mass one of identical households. First-stage portfolio choice:
  - n_h = q k_h + ∑_b δ_b
- Returns:
  - Each unit of capital: flow income z and capital gains dq per unit of capital.
  - Each unit of deposit: interest income r_b.
  - If bank b is liquidated in a run, household recovers proportion ψ_b of deposits at bank b.
- Convex cost of managing capital:
  - f(k_h) = α/2 (k_h)^2
- End-of-second-period net worth:
  - No systemic run: c = ∑_b (1 + r_b) δ_b + (q_NR + z) k_h − f(k_h)
  - Systemic run: c_R = ∑_b δ_b ψ_b + (q_R + z) k_h − f(k_h)
- Objective (linear utility, discount rate ρ):
  - max_{ {δ_b}_{b=1}^B , k_h } − n_h + 1/(1 + ρ) [ (1 − λ) c + λ c_R ]
  - subject to n_h = ∑_b δ_b + q k_h
- Household F.O.C. / pricing equations (risk-neutral case presented later):
  - q_NR − q + z − f′(k_h) = − λ (q_R − q_NR) + ρ q  (equation (6))
  - ∀ b ∈ {1,...,B} : r_b = λ (1 + r_b − ψ_b) + ρ  (equation (7))

### Banks: objectives, constraints, and recovery
- Finite number B of banks. Banker utility:
  - u_b(d) = d^{1−γ} / (1 − γ) with γ ∈ [0,1)
- Bank choices: investment in real assets k_b and deposits δ_b (Cournot behavior; internalize impact on returns, prices, and λ). Enter with net worth n_b.
- Budget constraint: q k_b = n_b + δ_b
- Net worth at end if no run:
  - (q + z) k_b − (1 + r_b) δ_b
- Net worth at end if run:
  - n_b − k_b (q − q_R)
- Recovery rate of deposits at bank b when liquidated:
  - ψ_b = max( q_R k_b / δ_b , 1 ) + z k_b / δ_b
- Bank problem (v_R = 0 in static section):
  - max_{δ,k} (1 − λ) v(d_b) + λ v_R with d_b = (q′ + z) k_b − (1 + r_b) δ_b
  - constraint q k_b = n_b + δ_b
- Banks internalize impact of capital on q, r_b and λ.

### Run game (stage 2) and multiple equilibria
- Households decide whether to withdraw all deposits from all banks; symmetric equilibria: either no one runs or all run.
- If everyone runs, asset liquidation moves asset price; banks may or may not be able to honor deposits:
  - If banks cannot honor debt: banks liquidated; liquidation value shared only among households who ran. Non-runners lose all deposits.
- Aggregate definitions: Δ (sum of deposits), q_R, N_R, K_R for after-run; q, N, K before-run.
- Multiple equilibria exist when system’s aggregate equity if liquidated is negative:
  - N_R ≡ N + (q_R − q) K < 0
- Aggregate recovery Ψ:
  - Ψ = q_R K / (q K − N) = q_R K / Δ < 1
- Sunspot coordination: assume sunspot with probability ̄λ coordinates depositors on run equilibrium whenever multiple equilibria exist.
- Unconditional probability of a run:
  - λ = ̄λ 1_{Ψ < 1}
- From perspective of individual bank b when others symmetric:
  - λ(k, k_{−b}, n, n_{−b}) = ̄λ 1_{ ((B − 1) n_{−b} + n_b + (q_R − q) ((B − 1) k_{−b} + k_b )) < 0 }  (equation (5))

### Market clearing (first stage)
- Deposits market clearing: ∫ δ_{b,h} dh = δ_b for all b ∈ {1,...,B}
- Capital market clearing:
  - ∑_{b=1}^B k_b + ∫ k_h dh = ̄K

### 2.2 Properties of the Static Equilibrium

### Bank dividends and investment (no-run case)
- Dividends in no-run case (using household F.O.C. and recovery def.):
  - d = 1/(1 − λ(k,n)) [ (1 + ρ) n + f′( ̄K − ∑_{j=1}^B k_j ) k_b ]  (equation (8))
- Bank maximization (v_R = 0):
  - max_{k} (1 − λ(·)) v( 1/(1 − λ(·)) [ (1 + ρ) n + f′( ̄K − ∑_j k_j ) k_b ] )  (equation (9))
- Objective piecewise concave in k_b with downward step where λ jumps — two candidate optima: interior k_I and corner k_C.

### Interior solution k_I
- F.O.C. for interior solution:
  - f′( ̄K − ∑_{j=1}^B k_j ) − f′′( ̄K − ∑_{j=1}^B k_j ) k_b = 0  (equation (10))
- With f(k) = α/2 k^2 and symmetry:
  - k_I = ̄K / (B + 1)
  - Each bank holds fraction 1/(B + 1) of capital stock; remaining fraction held by households.

### Corner solution k_C (safe corner)
- Highest capital such that λ = 0:
  - k_C(n, n_{−b}, k_{−b}) = 1/(q − q_R) ( ∑_{j ≠ b} n_j + n ) − ∑_{j ≠ b} k_j = N/(q − q_R) − ∑_{j ≠ b} k_j
- In symmetric equilibrium:
  - k_C = n / (q − q_R)
- As q_R − q → 0 (recovery rate → 1), k_C → ∞; ratio q_R/q measures aggregate illiquidity.

### Tradeoff and selection between interior and corner
- Net return on capital:
  - R(k_b, k_{−b}) = f′( ̄K − ∑_{j ≠ b} k_j − k_b )  (equation (11))
- Global optimum compares:
  - max [ v( (1 + ρ) n + R(k_C(·), k_{−b}) k_C(·) ), (1 − λ(·))^γ v( (1 + ρ) n + R(k_I, k_{−b}) k_I ) ]  (equation (12))
- Tradeoff: interior yields higher current profits R(k_I,·) k_I > R(k_C,·) k_C but higher run risk (loss of franchise value).

### 2.3 Symmetric Cournot Equilibria

### Asset prices and price-drop channel
- Asset price when all banks play interior or corner (from equation (6)):
  - q_I = ( z + λ q_R + (1 − λ) q_NR − α/2 ( ̄K/(B + 1) )^2 ) / (1 + ρ)  (equation (13))
  - q_C(n) defined implicitly:
    - q_C(n) = ( z + q_NR − α/2 ( ̄K − B n / ( q_C(n) − q_R ) )^2 ) / (1 + ρ)  (equation (14))
- Lemma 1: q_I − q_R and q_C − q_R are increasing in number of banks B.
  - Interpretation: larger B raises normal-period asset prices; with q_R independent of competition in static model, the drop q − q_R is larger when B is larger.

### Equilibrium regimes as function of net worth n
- There exist unique thresholds { n_C1, n_I1 } and n_SS with n_C1 ≤ n_I1 < n_SS such that:
  - 0 ≤ n < n_C1: only symmetric Cournot equilibrium: all banks play interior k_I.
  - n_C1 ≤ n < n_I1: two symmetric equilibria: all play interior k_I or all play corner k_C(n).
  - n_I1 ≤ n ≤ n_SS: only symmetric Cournot equilibrium: all banks play corner k_C(n).
- Define a threshold n_1 ∈ [n_C1, n_I1] and k(n):
  - k(n) = { k_I if n < n_1 (Run zone); k_C(n) if n_1 ≤ n < n_SS (Safe zone); k_I if n_SS ≤ n (Steady-state zone) }
- Probability of systemic run λ(n):
  - λ(n) = { ̄λ if n < n_1 ; 0 if n ≥ n_1 }

### Concentration effects: franchise-value and price-drop channels
- Define price drop dq(B) = q_C(B) − q_R(B). Required aggregate net worth for safety:
  - N_C1(B, dq) = B n_C1(B, dq)
  - N_I1(B, dq) = B n_I1(B, dq)
- Proposition 2.2.1:
  1. Franchise value channel: holding dq = dq(B) fixed, aggregate thresholds increase with B. If B′ > B then:
     - N_C1(B′, dq(B)) > N_C1(B, dq(B)) and N_I1(B′, dq(B)) > N_I1(B, dq(B))
  2. Price-drop channel: allowing asset prices to change amplifies the effect. If B′ > B then:
     - N_C1(B′, dq(B′)) > N_C1(B′, dq(B)) and N_I1(B′, dq(B′)) > N_I1(B′, dq(B))
- Net result: as number of banks B increases, system requires more aggregate equity to avoid run risk.

### Output and efficiency at interior solution
- Total output per unit of time net of management cost in the run zone (interior Cournot equilibrium):
  - ̄K z − f( ̄K − B k_I ) = ̄K z − α 2 ( ̄K B + 1 ) 2.
- Total output is increasing in the number of banks B.
- Total output is decreasing in α (convexity of households’ cost f).

### Dynamic model overview

### Timing and shocks
- Time continuous with two stages within an instant.
- Two sources of uncertainty:
  - Productivity z can take two values, z < ̄z; transition rates ζ(̄ζ). ζ(z) = ζ if z = z and ζ(z) = ̄ζ if z = ̄z.
  - Endogenous risk of a run: systemic runs follow a Poisson process with arrival rate λ per unit of time, determined endogenously.
- Dynamic setting allows feedback of future run risk into current asset prices and endogenous accumulation of net worth.

### Households: law of motion and HJB (aggregate)
- Household net worth law (no shocks):
  - .n_h = ( ̇q + z ) k_h + ∑_b δ_b r_b − f(k_h) − c_h
- Jumps:
  - Productivity change Δz: Δn_h = Δq_z k_h with post-z-shock asset price q_z.
  - Systemic run: Δn_h = Δq_R k_h + ∑_b δ_{b,h} (ψ_b − 1)
- Household HJB (aggregate state variables n, z, B, taking q and λ* as given) presented in text.

### Banks: law of motion and HJB
- Bank instantaneous utility: HARA with parameters γ, θ:
  - u_b(d) = ξ ( d − γ θ )^{1−γ} / (1−γ) with d > 0, γ ∈ [0,1), ξ > 0, θ ≥ 0.
- Law of motion of bank equity:
  - n = q k − δ
  - dn = (dq + z) k − r δ − d
  - When neither run nor productivity shocks occur: ̇n = ( ̇q + z ) k − r δ − d
  - If a run occurs, Δn = k Δq_R.
- Bank HJB (state summarized by vectors n, z, B; taking other banks’ vectors as given) presented in text.

### Equilibrium definition and run-price consistency
- Recovery rate at bank b:
  - ψ_b = max( q_R k_b / δ_b , 1 )
- Markets clear: goods, deposits, capital.
- Entry after a run: B′ households drawn with intensity η to become new bankers; entrants endowed with equity n_E = 0. Special cases: η → +∞ (immediate entry); η = 0 (no entry).
- Run price consistency:
  - q_R(n,z,B) = q(0,z,0)
- Equilibrium: functions H, {δ_{b,h}}, c, k_h, v, k, δ, d, λ*, q satisfying household and bank optimality, market clearing, run price condition, and λ* consistent with run-game equilibrium.

### Symmetric equilibrium with systemic runs (dynamic)

### Pricing and bank law of motion (risk-neutral ζ(.) = 0)
- Asset pricing:
  - ̇q + z − f′(k_h) = −λ Δq_R + ρ q  (equation 17)
- Deposit Euler:
  - ∀ b ∈ {1,...,B} r_b = λ (1 − ψ_b) + ρ  (equation 18)
- Bank law of motion (no run):
  - ̇n = ( λ(k,n) + ρ ) n + f′( ̄K − ∑_{j=1}^B k_j ) k_b − d  (equation 19)

### Assumption 1 and existence of cutoffs
- Equilibrium paths satisfying:
  1. All banks have same net worth.
  2. F* (n) differentiable with (F*)′ > 0.
  3. Exists ε > 0 s.t. k_C* (n) differentiable with ∂k_C* / ∂n > ε.
  4. Banks’ capital investment strategies Markovian, symmetric and stationary.
- Under Assumption 1, thresholds n_C1, n_I1 exist with the same ordering as static case.

### Interior steady-state and dividends
- Interior steady-state (Lemma 2):
  - k_SS = ̄K / (B + 1)
  - q_SS = ( z − f′( ̄K − B k_SS ) ) / ρ
  - n_SS = k_SS ( q_SS − q_R )
  - d_SS = ρ n_SS + f′( ̄K − B k_SS ) k_SS
- Dividends and saving policy along symmetric equilibrium path (Lemma 3):
  - For n ∈ [0, max(n_{−1}, 0)]: ̇d = 0 and d(n) = (ρ + ̄λ) n + R(k_I) k_I
  - For n ∈ ( max(n_{−1}, 0), n_1 ): ̇d = 0 and d(n) = d(n_1)
  - For n ∈ [ n_1, n_SS ): d solves: γ ̇d / ( d − γ θ ) = k_C*1(n) α ( ̄K − (B + 1) k_C* (n) )
  - For n ≥ n_SS: d(n) = d_SS
  - For all n, saving path: ̇n = ( ρ + λ*(n) ) n + α ( ̄K − B k*(n) ) k*(n) − d

### Safe-zone Euler equation and interpretation
- In safe zone marginal value of net worth strictly positive: postponing dividends and building net worth increases future investment capacity and leads to safe interior investment.
- k_C^*(1)(n) (derivative of corner capital w.r.t. own net worth) is:
  - positive;
  - depends negatively on price drop q(n) − q_R and on q^*(1)(n).
- Transition path analyzed via a coupled differential system for (n,q,d) with boundary conditions; local linearization provides uniqueness and convergence results (Lemma 4.1).

### Full transition path characterization (Proposition 3.1)
- Four possible regimes for initial n_0 ≥ 0:
  1. Run zone with certain death: n_0 ∈ [0, max(n_{−1}, 0)] → k^*(n) = k_I, λ^*(n) = λ̄, n constant, eventual run.
  2. Run zone with escape: n_0 ∈ (max(n_{−1}, 0), n_1) → λ^*(n) = λ̄, n increases and reaches n_1 in finite time then enters safe zone.
  3. Safe zone: n_0 ∈ [n_1, n_SS) → variables increase and converge to steady-state; no run.
  4. Safe steady-state: n_0 ≥ n_SS → k = k_SS and n = n_0 forever.

### Financial accelerator and two-stage amplification
- Leverage rule in safe corner:
  - k_C^*(n) = n/(q − q_R) ⇔ q k_C^*(n)/n = 1/(1 − q_R/q)
- Unlike some literature, the leverage multiplier depends on endogenous q − q_R.
- Dynamics by shock size:
  - Small shocks: economy stays in safe zone; traditional financial accelerator operates.
  - Large shocks: economy enters run zone with high leverage and run risk; relationship between net worth and investment breaks down.
- Two-stage amplification over time: immediate run-zone amplification and delayed safe-zone accelerator.

### Recapitalization channel and convergence speed
- Banks in more concentrated systems earn higher profits and may recapitalize faster; effect depends on dividend policy and utility parameters.
- For η = 0, speed of convergence local expression given in text (exact expression preserved there); three channels through which B affects speed:
  1. More banks increases price-drop effect → slows convergence.
  2. More banks raise incentive to accumulate net worth (k_SS q_SS^1 term) → faster convergence.
  3. θ γ (B + 1) term: concentrated markets may converge faster if dividends not distributed too early.
- Role of γ and θ: γ = 0 (linear) implies strong recapitalization as profits saved; γ > 0 implies dividend smoothing and weaker recapitalization channel.

### Entry channel, contestability, and runs (Proposition 3.2)
- Increasing entry speed η of new banks raises run-price q_R and stabilizes system by shrinking run zone.
- Extreme case η → +∞ (immediate entry): runs do not happen in equilibrium and banks play interior solution without run risk:
  - As η → +∞: (n_1, n_SS) → (0, 0) and ∀ n ∈ [0, +∞):
    - λ^*(n) = 0
    - k^*(n) = k_SS
    - q^*(n) = q_SS
    - d(n) = ρ n + α (k_I)^2
- First-order approximation of q_R for finite η given in text: q_R = (η/(ρ + η)) q^*(0) + (z − α K̄)/(ρ + η); price-drop at n = 0 decreases with η.

### Numerical calibration results and comparative statics

### Calibration highlights
- Period: one year.
- Discount rate ρ = 5%.
- γ set to 0 in baseline.
- α calibrated to match a 2% interest margin.
- Benchmark number of banks B = 5 (HHI ≈ 2000).
- v_R = 0.
- λ̄ set to 3% to obtain cumulative probability of run at beginning of transition of 55%.

### Key numerical findings
- Increasing concentration (B = 1, 3, 5):
  - Threshold n_1 similar across concentration levels due to offsetting price-drop and franchise-value effects.
  - More concentrated systems reach safe zone and steady-state faster: going from B = 5 to 1 can reduce time spent in crisis by a factor 4 and cumulative probability of a run by a factor 3 in the example.
- Increasing entry speed η (benchmark η = 0; examples η = .05, η = .1):
  - Higher η raises q_R, reduces price-drop, shrinks run zone, speeds convergence, raises welfare, reduces instability.
- Dividends preference (γ effect):
  - Baseline γ = 0 (save profits) → strong recapitalization channel and faster escape.
  - γ = .1 (distribute dividends) → slower escape; competitive systems disproportionately affected (example: cumulative run probability for B = 5 increases from 50% to 80% while monopoly unchanged).
- Smooth λ(.) specification tested (quadratic form with λ̄ = .002) yields qualitatively similar dynamics: investment no longer features a downward jump at safe-zone entry, but patterns for dividends, prices, saving and leverage similar.
- Exogenous productivity uncertainty ("paradox of safety"):
  - Two-state calibration: z̄ = 5, z = 4.25 (GDP drop 15%), ζ̄ = .02, ζ = .08.
  - Stationary distribution of aggregate net worth shifts left for more concentrated systems in both states: concentrated systems accumulate less aggregate net worth relative to assets over long tranquil periods because price-drop is lower, making them more vulnerable to adverse exogenous shocks.

### Policy implications, trade-offs, and extensions

### Main policy implications
- Competition vs. stability:
  1. More competition (lower concentration) is not unambiguously welfare-improving in environments with run risk and freely traded assets — the price-drop channel can make lower concentration destabilizing.
  2. Fast entry or ready public recapitalization (raising q_R) stabilizes the system; contestability and public backstops can make pro-competitive policies welfare-improving.
  3. Shadow banks should face similar regulation and lender-of-last-resort access as banks to avoid shifting instability into unregulated sectors; in a dual system, increasing competition in the regulated sector or tougher capital requirements can increase run likelihood by altering the unregulated sector’s size and competition.

### Empirical predictions and research directions
- State-dependence of banks’ behavior: small shocks → safe strategy; large shocks → runs with delayed lending restriction.
- Two channels warrant empirical study:
  - Recapitalization channel: how quickly concentrated systems recapitalize after shocks.
  - Price-drop channel: concentration’s effect on asset-price drops during runs.

### D. A Dual System of Regulated and Shadow Banks

### Framework and assumptions
- System contains R regulated banks and S shadow banks with S + R = B. Both invest in same asset.
- Regulated banks face capital requirement:
  - k_r = κ n_r  (equation 25)
  - Leverage ratio binding: λ_r = 1  (equation 26)
  - Assumption: κ < 1 − ∆q_R so regulated banks remain solvent if a run occurs.
- Focused case: no entry after a run (η = 0), so after run S' = 0 and R' = R; run price:
  - q_R(n_s,n_r,S) = q(0,n_r + ∆q_R k(n_r),0)

### Run condition and sectoral independence
- Run on shadow banking sector occurs if:
  - N_S + ∆q_R K_S < 0  (equation 30)
- Condition independent of regulated banks’ net worth because regulated banks’ deposits insured and cannot transfer liquidity to shadow banks when runs occur.

### Counterfactuals and policy experiments
- Competition counterfactuals (examples):
  - Decrease regulated banks R from 3 to 2 (keep S = 2): slight stabilizing effect via franchise/recapitalization channel but price-drop effect counteracts.
  - Decrease shadow banks (R,S) = (3,1): stabilizes via franchise, recapitalization, and mitigated price-drop channels.
- Macroprudential counterfactuals on κ:
  - Comparing κ = 1 and κ = .25: a more tightly regulated sector can be more stable if regulated banks start better capitalized; however, imposing tighter κ starting from same steady-state can increase run probability due to larger price-drop and slower recapitalization dominating franchise effects.
- Net finding: interaction between regulated and shadow sectors yields non-trivial trade-offs; policy effects depend on steady-state positions and dynamic recapitalization.

*Source: wpiea2021102-print-pdf - 2.1 The Model and subsequent sections.*

### 2.1  The Model

### 2.1 The Model

### Environment and timing
- Two stages:
  - Stage 1: bankers and households make portfolio decisions. Fixed supply of real asset capital denoted ̄K, tradable at price q. Each unit of capital provides flow z of good. Banks borrow via deposits.
  - Stage 2: run game among households.
- Single source of uncertainty: endogenous risk of a systemic run. In equilibrium a systemic run occurs with probability λ (endogenously determined).
- Figure 1 (schema): banks and households each hold capital (qk_b and qk_h), net worth (n_b and n_h) and deposits (δ).

### Households
- Mass one of identical households. First-stage portfolio choice:
  - n_h = q k_h + ∑_b δ_b
- Returns:
  - Each unit of capital: flow income z and capital gains dq per unit of capital.
  - Each unit of deposit: interest income r_b.
  - If bank b is liquidated in a run, household recovers proportion ψ_b of deposits at bank b.
- Convex cost of managing capital (as in GK):
  - f(k_h) = α/2 (k_h)^2
  - Convex cost makes direct household capital management increasingly costly at the margin.
- End-of-second-period net worth:
  - If no systemic run:
    - c = ∑_b (1 + r_b) δ_b + (q_NR + z) k_h − f(k_h)
  - If systemic run:
    - c_R = ∑_b δ_b ψ_b + (q_R + z) k_h − f(k_h)
- Objective (linear utility, discount rate ρ):
  - max_{ {δ_b}_{b=1}^B , k_h } − n_h + 1/(1 + ρ) [ (1 − λ) c + λ c_R ]
  - subject to n_h = ∑_b δ_b + q k_h

### Run game (stage 2)
- Households decide whether to withdraw all deposits from all banks, taking others’ behavior as given.
- Symmetric equilibria focus: either no one runs or all run.
- If an atomistic individual withdraws while others do not, required sales by banks do not move asset price; cost is forgone interest.
- If everyone runs, asset liquidation moves asset price; banks may or may not be able to honor deposits:
  - If banks can honor despite price decline: not running saves interest.
  - If banks cannot honor debt: banks liquidated; liquidation value shared only among households who ran. Non-runners lose all deposits.

### Banks
- Finite number B of banks indexed b ∈ {1,...,B}. Bankers maximize end-of-period dividends with instantaneous utility u_b(d), assumed bounded below, increasing, concave. In this section:
  - u_b(d) = d^{1−γ} / (1 − γ) with γ ∈ [0,1)
- Concavity implies equity issuance is costly; net worth matters.
- Bank choices: investment in real assets k_b and deposits δ_b (Cournot behavior: internalize impact on returns, prices, and λ). No exogenous limit on deposit issuance. Enter with net worth n_b.
- Net worth at end if no run:
  - (q + z) k_b − (1 + r_b) δ_b
- Net worth at end if run:
  - n_b − k_b (q − q_R)
- After a successful run, banks are liquidated; banker exit value v_R set to 0 in this section.
- Bank problem:
  - max_{δ,k} (1 − λ) v(d_b) + λ v_R with d_b = (q′ + z) k_b − (1 + r_b) δ_b
  - constraint q k_b = n_b + δ_b
- Banks internalize impact of capital on q, r_b and λ.

### Recovery rate
- Recovery rate of deposits at bank b when liquidated:
  - ψ_b = max( q_R k_b / δ_b , 1 ) + z k_b / δ_b

### Market clearing (first stage deposits and capital)
- For all b ∈ {1,...,B}:
  - ∫ δ_{b,h} dh = δ_b
- Capital market:
  - ∑_{b=1}^B k_b + ∫ k_h dh = ̄K

### Asset prices at end of second stage
- Run-price q_R (when there is a run) equals value if households operate all capital stock forever:
  - q_R = (z − α/2 ̄K^2) / ρ
- Non-run price q_NR (when no run) equals value if financial system operates without run risk forever:
  - q_NR = (z − α/2 (̄K/(B+1))^2) / ρ
- These prices are treated as exogenous in this static section; later shown as equilibrium prices in fully dynamic model under stated conditions.

### 2.2 Properties of the Static Equilibrium

### Run game and probability of a run
- Aggregate definitions: Δ (sum of deposits), q_R, N_R, K_R for after-run; q, N, K before-run.
- Multiple equilibria exist when system’s aggregate equity if liquidated is negative:
  - N_R ≡ N + (q_R − q) K < 0
- Denoting aggregate recovery Ψ, second inequality becomes:
  - Ψ = q_R K / (q K − N) = q_R K / Δ < 1
- Lower equity-over-capital ratio (higher leverage) increases likelihood of multiple equilibria.
- Assume a sunspot with probability ̄λ coordinates depositors on run equilibrium whenever multiple equilibria exist.
- Unconditional probability of a run:
  - λ = ̄λ 1_{Ψ < 1}
- From perspective of individual bank b when others symmetric:
  - λ(k, k_{−b}, n, n_{−b}) = ̄λ 1_{ ((B − 1) n_{−b} + n_b + (q_R − q) ((B − 1) k_{−b} + k_b )) < 0 }  (equation (5))

### Households’ optimality and pricing equations (F.O.C.)
- Using households’ F.O.C., asset pricing and Euler condition:
  - q_NR − q + z − f′(k_h) = − λ (q_R − q_NR) + ρ q  (equation (6))
  - ∀ b ∈ {1,...,B} : r_b = λ (1 + r_b − ψ_b) + ρ  (equation (7))
- Interpretation:
  - (6) is the asset pricing equation including run-probability adjustment.
  - (7) is Euler for deposits adjusted for run probability and recovery.

### Bank’s capital investment and deposit decisions
- Rewriting bank dividends using household F.O.C. and recovery definition, dividends in no-run case:
  - d = 1/(1 − λ(k,n)) [ (1 + ρ) n + f′( ̄K − ∑_{j=1}^B k_j ) k_b ]  (equation (8))
  - Aggregate capital K = ∑_j k_j; λ depends on k_b discontinuously via equation (5).
- Bank maximization (v_R = 0 normalization):
  - max_{k} (1 − λ(·)) v( 1/(1 − λ(·)) [ (1 + ρ) n + f′( ̄K − ∑_j k_j ) k_b ] )  (equation (9))
- Objective is piecewise concave in k_b with downward step where λ jumps up — two candidate optima: interior k_I and corner k_C.

### Interior solution k_I
- F.O.C. for interior solution:
  - f′( ̄K − ∑_{j=1}^B k_j ) − f′′( ̄K − ∑_{j=1}^B k_j ) k_b = 0  (equation (10))
- In symmetric Cournot equilibrium, with f(k) = α/2 k^2:
  - k_I = ̄K / (B + 1)
  - Each bank holds fraction 1/(B + 1) of capital stock; remaining fraction held by households. k_I independent of banks’ net worth in this static setup.

### Corner solution k_C
- Corner solution is highest capital such that λ = 0 (safe corner):
  - k_C(n, n_{−b}, k_{−b}) = 1/(q − q_R) ( ∑_{j ≠ b} n_j + n ) − ∑_{j ≠ b} k_j = N/(q − q_R) − ∑_{j ≠ b} k_j
- In symmetric equilibrium (all banks same n and k):
  - k_C = n / (q − q_R)
- As q_R − q → 0 (recovery rate → 1), k_C → ∞; ratio q_R/q measures aggregate illiquidity.

### Choice between interior and corner (tradeoff)
- Denote net return on capital:
  - R(k_b, k_{−b}) = f′( ̄K − ∑_{j ≠ b} k_j − k_b )  (equation (11))
- Global optimum compares:
  - max [ v( (1 + ρ) n + R(k_C(·), k_{−b}) k_C(·) ), (1 − λ(·))^γ v( (1 + ρ) n + R(k_I, k_{−b}) k_I ) ]  (equation (12))
- Tradeoff: interior yields higher current profits R(k_I,·) k_I > R(k_C,·) k_C but higher run risk (loss of franchise value).

### 2.3 Symmetric Cournot Equilibria

### Asset prices and price-drop channel
- Asset price when all banks play interior or corner (from households’ F.O.C. equation (6)):
  - q_I = ( z + λ q_R + (1 − λ) q_NR − α/2 ( ̄K/(B + 1) )^2 ) / (1 + ρ)  (equation (13))
  - q_C(n) defined implicitly:
    - q_C(n) = ( z + q_NR − α/2 ( ̄K − B n / ( q_C(n) − q_R ) )^2 ) / (1 + ρ)  (equation (14))
- Lemma 1: q_I − q_R and q_C − q_R are increasing in number of banks B.
  - Interpretation: more competitive (larger B) systems intermediate more capital, raising normal-period asset prices; with q_R independent of competition in this static model, the drop q − q_R is larger when B is larger.

### Equilibria as a function of net worth n
- Existence of unique thresholds { n_C1, n_I1 } and n_SS with n_C1 ≤ n_I1 < n_SS such that:
  - If 0 ≤ n < n_C1: only symmetric Cournot equilibrium: all banks play interior k_I.
  - If n_C1 ≤ n < n_I1: two symmetric equilibria possible: all play interior k_I or all play corner k_C(n).
  - If n_I1 ≤ n ≤ n_SS: only symmetric Cournot equilibrium: all banks play corner k_C(n).
- Figure 3 (best-response stylized): for low n, risky interior preferred; for high n, safe corner preferred. For intermediate n both sustainable.
- When multiplicity, choose threshold n_1 ∈ [n_C1, n_I1] and define:
  - k(n) = { k_I if n < n_1 (Run zone); k_C(n) if n_1 ≤ n < n_SS (Safe zone); k_I if n_SS ≤ n (Steady-state zone) }
- Probability of systemic run λ(n):
  - λ(n) = { ̄λ if n < n_1 ; 0 if n ≥ n_1 }

### Concentration effects: franchise-value and price-drop channels
- Degree of concentration B reduces run risk via two channels: franchise value and price-drop.
- Define price drop dq(B) = q_C(B) − q_R(B). Required aggregate net worth for safety:
  - N_C1(B, dq) = B n_C1(B, dq)
  - N_I1(B, dq) = B n_I1(B, dq)
- Proposition 2.2.1:
  1. Franchise value channel: holding dq = dq(B) fixed, aggregate thresholds increase with B. If B′ > B then:
     - N_C1(B′, dq(B)) > N_C1(B, dq(B)) and N_I1(B′, dq(B)) > N_I1(B, dq(B))
  2. Price-drop channel: allowing asset prices to change amplifies the effect. If B′ > B then:
     - N_C1(B′, dq(B′)) > N_C1(B′, dq(B)) and N_I1(B′, dq(B′)) > N_I1(B′, dq(B))
- Intuitions:
  - Franchise-value channel: higher concentration → larger, more profitable banks → higher franchise value → banks willing to take less risk → less aggregate equity required for safety.
  - Price-drop channel: lower concentration → larger price drop upon run (because normal-period prices are higher due to more intermediation) → banks need more net worth to reach a given capital investment → less appealing to play interior; overall, higher B increases aggregate equity required for safety.
- Net result: as number of banks B increases, system requires more aggregate equity to avoid run risk.

*Source: wpiea2021102-print-pdf - 2.1 The Model (IMF Working Paper chapter).*

### section is stabilizing but it is also less efficient when it invests at the interior solution

### section is stabilizing but it is also less efficient when it invests at the interior solution

### Run-zone output and comparative statics
- Total output per unit of time net of management cost in the run zone (banks coordinate on the interior Cournot equilibrium) is:
  - ̄K z − f( ̄K − B kI ) = ̄K z − α 2 ( ̄K B + 1 ) 2.
- Total output is increasing in the number of banks: the more banks there are, the more competition and the less the distortion from market power.
- Total output is decreasing in α, the degree convexity of the households’ cost function: the higher α, the more costly it is at the margin for households to manage capital, the higher the output loss for a given amount of capital holding.

### Dynamic model overview
- Time is continuous with two stages within an instant as in the static setting.
- Two sources of uncertainty:
  - Productivity z can take two values, z < ̄z; when low (resp. high) it switches to low (resp. high) with arrival rate ζ( ̄ζ ). ζ is defined as ζ(z) = ζ if z = z and ζ(z) = ̄ζ if z = ̄z.
  - Endogenous risk of a run: systemic runs follow a Poisson process with arrival rate λ per unit of time, determined endogenously.
- Dynamic setting allows analysis of:
  - How future run risk feeds back into current asset prices and run risk.
  - How accumulation of net worth (dependent on endogenous dividends) shapes time spent in the run zone, which determines franchise value and asset prices.
- Focus on two additional channels: the recapitalization channel and the entry channel. Model allows for productivity shocks; analysis focuses on transition path following a productivity shock that depletes banks’ net worth.

### Households: law of motion and HJB
- Law of motion of household net worth when no shocks:
  - .n_h = ( ̇q + z ) k_h + ∑_b δ_b r_b − f(k_h) − c_h
- Discrete productivity change Δz triggers asset-price and net-worth jumps: Δn_h = Δq_z k_h with post-z-shock asset price q_z.
- When a systemic run occurs, households’ net worth change:
  - Δn_h = Δq_R k_h + ∑_b δ_{b,h} (ψ_b − 1) where Δq_R = q_R − q and ψ_b is recovery rate per unit of deposit at bank b.
- Household HJB (aggregate state variables: banks’ net worth vector n, productivity z, number of banks B) — taking asset price q and equilibrium run intensity λ* as given:
  - ρ H(n_h,n,z,B) = max_{c,{δ},k_h} [ h(c) + H_n ( ( ̇q+z ) k_h + ∑_b δ_b r_b − f(k_h) − c ) 
    + λ*(n,z,B) ( H(n_h + Δq_R k_h + ∑_b δ_{b,h} (ψ_b − 1), 0, z, 0) − H(n_h,n,z,B) )
    + ζ(z) ( H(n_h + Δq_z k_h, n + Δq_z k, z′, B) − H(n_h,n,z,B) )
    + ∑_b H_{n_b} . ̇n_b + η ( H(n_h, n_E, z, B′) − H(n_h, 0, z, 0) ) 1_{B=0} ]
- Notation: n_h = q k_h + ∑_b δ_{b,h}.

### Banks: preferences, law of motion, and HJB
- Instantaneous utility: HARA with parameters γ, θ:
  - u_b(d) = ξ ( d − γ θ )^{1−γ} / (1−γ) with d > 0, γ ∈ [0,1), ξ > 0, θ ≥ 0. (γ → +∞ gives CARA; θ = 0 gives CRRA; θ > 0, θ < +∞ gives decreasing relative risk aversion.)
- Bank chooses dividends d_b and real-asset investment k_b, issues deposits δ_b, behaves à la Cournot internalizing effects on returns, asset prices, and run probability.
- Law of motion of bank equity (qk financed by δ or n):
  - n = q k − δ
  - General law: dn = (dq + z) k − r δ − d
  - When neither run nor productivity shocks occur: ̇n = ( ̇q + z ) k − r δ − d
  - If a run occurs, Δn = k Δq_R.
- In equilibrium, after a successful run all banks are liquidated and each banker exits and gets v_R (calibration sets v_R = 0).
- Bank HJB (state summarized by vectors n, z, B; taking other banks’ vectors as given):
  - ρ v(n, n_{−b}, z, B) = max_{d,k} [ u(d) + v_n (( ̇q+z) k − r δ − d) + ∑_{j≠b} v_{n_{−j}} ̇n_{−j} + λ(k, k_{−b}, n, n_{−b}, z, B) ( v_R − v(n, n_{−b}, z, B) ) + ζ(z) ( v(n + Δq_z k, n_{−b} + Δq_z k_{−b}, z′, B) − v(n, n_{−b}, z, B) ) ]

### General equilibrium conditions
- Recovery rate at bank b:
  - ψ_b = max( q_R k_b / δ_b , 1 )
- Market clearing: goods, deposits, and capital markets must clear.
- Entry after a run: B′ households are drawn with intensity η to become new bankers; η and B′ are parameters. Entry occurs only after a run; in normal times B remains constant. Assumption B′ = B (number of new banks equals pre-run number). Entrants are endowed with equity n_E = 0.
  - Special cases: "immediate entry" η → +∞; "no entry" η = 0.
- Run price consistency:
  - q_R(n,z,B) = q(0,z,0)
- Equilibrium defined as collection of functions H, {δ_{b,h}}, c, k_h, v, k, δ, d, λ*, q such that i) households’ problems solved taking prices as given, ii) banks’ problems solved taking others as given, iii) markets clear, iv) run price condition (q_R) holds, v) λ*(n,z,B) consistent with run-game equilibrium.

### Symmetric equilibrium with systemic runs (section 3.2)
- Restrict to symmetric net-worth paths, abstract from productivity shocks in analytical derivations; productivity risk considered numerically.
- Run game identical to static framework; λ denotes run arrival rate.
- Households risk neutral and ζ(.) = 0: first-order conditions yield pricing equations:
  - Asset pricing: ̇q + z − f′(k_h) = −λ Δq_R + ρ q (equation 17).
  - Deposit Euler: ∀ b ∈ {1,...,B} r_b = λ (1 − ψ_b) + ρ (equation 18).
- Using households’ F.O.C., bank law of motion (no run) can be written:
  - ̇n = ( λ(k,n) + ρ ) n + f′( ̄K − ∑_{j=1}^B k_j ) k_b − d (equation 19).

### Equilibrium restrictions (Assumption 1)
- Consider equilibrium paths satisfying:
  1. All banks have the same amount of net worth.
  2. F* (n) is differentiable with (F*)′ > 0.
  3. There exists ε > 0 such that k_C* (n) is differentiable with ∂k_C* / ∂n > ε.
  4. Banks’ capital investment strategies are Markovian, symmetric and stationary.
- Under Assumption 1, there exist two thresholds n_C1, n_I1 such that:
  - If banks’ net worth < n_C1 only the interior equilibrium is sustainable.
  - If banks’ net worth > n_I1 only the corner equilibrium is sustainable.
  - Both equilibria are sustainable for intermediate net worth levels.

### Steady-state and dividend/accumulation policies
- Interior steady-state with positive net worth (Lemma 2):
  - k_SS = ̄K / (B + 1)
  - q_SS = ( z − f′( ̄K − B k_SS ) ) / ρ
  - n_SS = k_SS ( q_SS − q_R )
  - d_SS = ρ n_SS + f′( ̄K − B k_SS ) k_SS
  - Note: these steady-state expressions are functions of primitives only when q_R is exogenous, which is the case when there is no entry after a systemic run (η = 0), q_R = z − α 2 ̄K 2 / z. q_R is generally endogenous.
- Dividends and saving policy along symmetric equilibrium path (Lemma 3, with utility 15 and k*(n) of the form in corollary 1):
  - For all n ∈ [0, max(n_{−1}, 0)]:
    - ̇d = 0 and d(n) = (ρ + ̄λ) n + R(k_I) k_I
    - where n_{−1} = ( d(n_1) − R(k_I) k_I ) / ( ρ + ̄λ )
  - For all n ∈ ( max(n_{−1}, 0), n_1 ):
    - ̇d = 0 and d(n) = d(n_1)
  - For all n ∈ [ n_1, n_SS ):
    - d solves: γ ̇d / ( d − γ θ ) = k_C*1(n) α ( ̄K − (B + 1) k_C* (n) )
    - with k_C* (n) = n / ( q(n) − q_R ), k_C*1 (n) = (1 − k_C*(n) q*1(n)) / ( q(n) − q_R )
  - For all n ≥ n_SS:
    - d(n) = d_SS
  - For all n, saving path:
    - ̇n = ( ρ + λ*(n) ) n + α ( ̄K − B k*(n) ) k*(n) − d
- Interpretation:
  - Below n_1 (run zone), dividends are constant at d(n_1) across [ max(n_{−1},0), n_1 ), because current interior Cournot investment is disconnected from current net worth and banks have no local incentive to change dividends; concavity of utility pins continuity at n_1.
  - For n < n_{−1}, banks set dividends to keep net worth constant; this lower zone exists iff d(n_1) < ( ρ + ̄λ ) n + R(k_I) k_I.
  - Dividends are increasing in the safe zone because ( ̄K − (B + 1) k_C* (n) ) > 0.

*Italic: Source — wpiea2021102-print-pdf - section is stabilizing but it is also less efficient when it invests at the interior solution*

### 0. In

### 0. In

### Safe zone dynamics and dividends Euler equation
- In the safe zone the marginal value of net worth is strictly positive: postponing dividend payments and building net worth allow banks to increase their investment capacity in the future and eventually reach the level of net worth for which it is safe to invest at the interior Cournot capital investment. At this point dividends stop increasing.
- The Euler equation for dividends in the safe zone (equation 21) features the derivative of the corner capital investment with respect to a bank own net worth, k_C^*(1)(n). This derivative is:
  - positive;
  - depends negatively on the size of the price drop q(n) − q_R: the larger the price drop, the smaller the amount of additional capital a marginal increase in net worth permits and the slower the increase in dividends over time;
  - depends negatively on the derivative of asset prices with respect to a bank own net worth, q^*(1)(n), because accumulation of net worth raises asset prices in equilibrium and tempers capital investment increases.
- The derivative q^*(1)(n) must be consistent with the equilibrium asset price function; the paper focuses on that equilibrium next.

### Transition path — setup and safe zone system
- The transition path is analyzed in two steps: (i) dynamics in the safe zone n ∈ [n_1, n_SS] via a system of differential equations and local linearization; (ii) full transition path including the run zone, studied tractably under η → 0 (no entry after a run), with discussion of η > 0 in subsection 3.2.
- In the safe zone capital investment is k_C^*(n). The coupled differential system (system 22) for net worth, dividends and asset prices is:
  - ̇n = ρ n + α( K̄ − B k_C^*(n) ) k_C^*(n) − d
  - ̇q = ρ q − z + α( K̄ − B k_C^*(n) ) γ
  - ̇d/(d − γ θ) = k_C^*(1)(n) α( K̄ − (B + 1) k_C^*(n) )
  - with k_C^*(n) = n/(q − q_R), k_C^*(1)(n) = [1 − k_C^*(n) q^*(1)(n)]/(q − q_R), q^*(1)(n) = ̇q/(B ̇n)
- Initial condition n(t_0) = n_1 and terminal conditions:
  - k_SS = K̄/(B + 1)
  - q_SS = z − α( K̄ − B k_SS )/ρ
  - d_SS = ρ n_SS + α( K̄ − B k_SS ) k_SS

### Monotonicity, existence/uniqueness and convergence (Lemma 4.1)
- Lemma 4.1 establishes:
  1. Along any transition path solution to initial-terminal value problem 22, all variables (d(t), n(t), k_C(t), q(t)) are increasing.
  2. There exists a unique solution (d(t), n(t), k(t), q(t)) to the linear approximation of system 22 around (d_SS, n_SS, k_SS, q_SS).
- The (negative of the) rate of convergence of capital to its steady-state is given by:
  - `-λ = z − ρ q_R − sqrt((z − ρ q_R)^2 + 4 γ (1 − k_SS q_SS^1) (d_SS − θ γ) (B + 1) α^2 (q_SS − q_R))`
- Phase diagram explanation (Figure 4): in the (k, q̃) plane with q̃ = q − q_R, the ̇q = 0 locus determines whether asset prices increase or decrease; capital is always increasing in the safe zone. Numerical checks confirm existence and uniqueness for the non-linear system under the paper’s parametrization.

### Full transition path (Proposition 3.1)
- Any transition path starting at n_0 ≥ 0 and satisfying assumption 1 is characterized by:
  1. Run zone with certain death: if n_0 ∈ [0, max(n_{−1}, 0)], k^*(n) = k_I, the system operates in the run zone, λ^*(n) = λ̄, n is constant over time, the system is in steady-state, banks eventually experience a run.
  2. Run zone with escape: if n_0 ∈ (max(n_{−1}, 0), n_1), k^*(n) ≠ k_I, the system operates in the run zone, λ^*(n) = λ̄, n increases and reaches in finite time n_1, at which point the system enters the safe zone.
  3. Safe zone: if n_0 ∈ [n_1, n_SS), then all variables increase and converge to their steady-state value: k^*(n) = k_C^*(n) → k_SS and n → n_SS, q → q_SS and the system is not subject to run.
  4. Safe Steady-State: if n_0 ≥ n_SS, then k = k_SS and n = n_0 forever; it is a steady-state.
- On the equilibrium path, capital investment jumps at n_1 from k_SS to k_C^*(n_1) < k_SS.

### Financial accelerator and two-stage amplification
- The model identifies a fourth mechanism linking net worth and lending: banks optimally restrict leverage to avoid run risk, yielding the leverage rule:
  - k_C^*(n) = n/(q − q_R) ⇔ q k_C^*(n)/n = 1/(1 − q_R/q)
- Comparison to literature:
  - Unlike Gertler and Kiyotaki (2015) where the leverage multiplier is constant, here it depends on the endogenous difference q − q_R.
- Dynamics by shock size:
  - Small shocks: economy remains in safe zone and features a traditional financial accelerator due to voluntary restrictions on capital investment.
  - Large adverse shocks: economy enters run zone where relationship between net worth and investment breaks down as banks keep leverage high.
  - Amplification is twofold over time: run-zone amplification (possibility of runs lowering output) and delayed safe-zone accelerator effect (restriction of lending occurs later).
- Empirical tie: explains runs during 2007–2009 and lending restrictions in 2009–2010.

### Recapitalization channel and speed of convergence
- Banks in more concentrated systems earn higher profits and can potentially recapitalize more quickly; whether they do depends on dividend policy and the utility specification.
- For η = 0 (quasi-analytical case), denoting z̃ = z − ρ q_R, the expression for the speed of convergence is:
  - `λ = z̃ − sqrt(z̃^2 + 4 γ (1 − k_SS q_SS^1) (K̄^2 α − θ γ (B + 1)) α^2 α B K̄ ρ (B + 1))` (expression as given in text)
- Three channels through which number of banks affects speed of convergence:
  1. Increasing number of banks increases denominator → slows convergence (price-drop effect: larger drop in asset prices in more competitive systems).
  2. Increasing B increases speed via the k_SS q_SS^1 term in numerator: more competitive banks have more incentive to accumulate net worth because they move asset prices less.
  3. θ γ (B + 1) term: more concentrated markets make higher profits and, if dividends are not distributed "too early", converge more quickly.
- Role of utility parameters:
  - Linear utility (γ = 0) with non-negativity constraint on dividends ⇒ dividends zero until steady-state and speed of convergence varies one-for-one with profits and competition.
  - γ > 0 (dividend smoothing): strength/sign of recapitalization channel depends on θ (how intertemporal elasticity changes with dividends). If θ = 0 (CRRA), recapitalization channel has no effect on local speed of convergence.
- Empirical importance: how fast banks distribute dividends while recapitalizing crucially determines how quickly the system escapes the run zone and the associated output costs.

### Run price, entry channel and contestability (Proposition 3.2)
- Main take-away: increasing the speed of entry η of new banks raises the run-price q_R and stabilizes the system by shortening the run zone.
- Extreme case η → +∞ (immediate entry): runs don’t happen in equilibrium and banks play the interior solution without taking run risk. Proposition 3.2 states:
  - If η → +∞ then (n_1, n_SS) → (0, 0) and for all n ∈ [0, +∞):
    - λ^*(n) = 0
    - k^*(n) = k_SS
    - q^*(n) = q_SS
    - d(n) = ρ n + α (k_I)^2
- Intuition: expectations of very fast entry or ready public recapitalization raise q_R, eliminating price drops and runs (contestable markets effect). This is consistent with empirical findings that contestability increases stability.
- For finite η, first-order approximation of run price q_R = (η/(ρ + η)) q^*(0) + (z − α K̄)/(ρ + η); a first-order approximation of price-drop at n = 0 is given and shows price-drop decreases with η, increasing interior investment and reducing required net worth to reach safe strategy.

### Competition, entry, and trade-offs for stability vs. efficiency
- Two dimensions of competition: level of competition (concentration B) and speed of entry η.
  - Entry channel: faster entry after a run (or available public recapitalization) reduces price-drop, shrinks run zone, and stabilizes the system.
  - Recapitalization channel: more concentrated systems recapitalize faster due to higher profits and may escape the run zone faster, improving stability and efficiency once in safe zone — strength and sign depend on dividend policy.
- Welfare trade-off:
  - Market-power distortion: concentrated systems intermediate less capital (oligopolistic distortion), reducing output.
  - If entry is immediate, concentration has no effect on stability and a social planner should increase competition to reduce oligopolistic distortion.
  - If entry is slow, stability concerns make competition less appealing.

### Numerical examples — setup and key calibration choices
- Welfare measure: with linear household utility and zero weight on bankers, expected discounted lifetime income is natural; asset price equals expected discounted sum of future flow of output, hence asset price is sufficient statistic for welfare.
- Instability measure: cumulative probability of a run (probability that a run happens by the end of the transition).
- Calibration choices (benchmarks described in the text):
  - Period is a year.
  - Discount rate ρ = 5%.
  - γ set to 0 in baseline in absence of clear estimates.
  - α calibrated to match a 2% interest margin.
  - Benchmark number of banks B = 5 (HHI ≈ 2000).
  - v_R = 0.
  - λ̄ set to 3% to obtain a cumulative probability of run at the beginning of the transition path of 55%.

### Numerical findings: concentration, entry speed, dividends preference, and smooth λ
- Increasing concentration (comparative examples for B = 1, 3, 5):
  - Policy rules and transition path figures (Figure 5) show three zones: low-net-worth interior high investment with high/decreasing leverage; at n_1 capital investment drops to safe level; high-net-worth interior investment becomes safe.
  - Threshold n_1 similar across concentration levels due to (1) price-drop effect and (2) franchise value effect keeping net worth thresholds comparable.
  - More concentrated systems reach the safe zone and steady-state faster: going from B = 5 to 1 can reduce time spent in crisis by a factor 4 and cumulative probability of a run by a factor 3 in the example.
- Increasing speed of entry η (Figure 6; benchmark η = 0, middle gray η = .05, light gray η = .1):
  - As η increases, run price q_R increases, price-drop decreases, run zone shrinks, convergence to safe zone and steady-state is faster, welfare increases, instability decreases.
- Dividends policy (γ effect):
  - Baseline γ = 0 implies strong recapitalization channel because profits are saved.
  - With γ = .1, banks distribute dividends and save less; escaping the run zone and convergence to steady-state take longer. Competitive systems are disproportionately affected: at time 0, cumulative run probability for B = 5 increases from 50% to 80% while monopoly unchanged in the example.
- Smooth arrival rate λ(.) functional form tested:
  - λ = λ̄/2 · (max( Σ_b k_b − Σ_b n_b/(q − q_R), 0 ))^2 with λ̄ = .002 yields qualitatively similar results; investment no longer features a downward jump at safe-zone entry, but patterns for dividends, prices, saving and leverage similar. Instantaneous run probability decreases over time as net worth accumulates.
- Exogenous productivity uncertainty — paradox of safety:
  - Two-productivity-state calibration: z̄ = 5, z = 4.25 (GDP drop 15% in recession), transition rates ζ̄ = .02, ζ = .08 (average time low state 12 years, high state 50 years).
  - Stationary distribution of aggregate net worth (Figure 7) shifts left for more concentrated systems in both high and low states: concentrated systems can accumulate less aggregate net worth relative to assets over long tranquil periods because price-drop is lower, making them more vulnerable to adverse exogenous shocks — a "paradox of safety" akin to paradox of volatility in Brunnermeier and Sannikov (2014).

### Conclusion and policy implications
- Model summary:
  - Dynamic general equilibrium with finite number of large intermediaries subject to endogenous run risk and Cournot interaction.
  - Near steady-state, banks optimally restrict investment as function of net worth (accelerator effect). Large shocks put system in run zone with high leverage and risk of runs; escape time determines instability.
  - Competition affects instability through franchise value, price-drop, and recapitalization channels; contestability (speed of entry) provides the entry channel stabilizing effect.
- Policy implications:
  1. More competition (lower concentration) is not unambiguously welfare-improving in environments with run risk and freely traded assets — the price-drop channel can make lower concentration destabilizing.
  2. Fast entry or ready public recapitalization (raising q_R) stabilizes the system; contestability and public backstops can make pro-competitive policies welfare-improving.
  3. Shadow banks should face similar regulation and lender-of-last-resort access as banks to avoid shifting instability into unregulated sectors; in a dual system, increasing competition in the regulated sector or tougher capital requirements can increase run likelihood by altering the unregulated sector’s size and competition.
- Empirical predictions and research directions:
  - State-dependence of banks’ behavior matters: safe strategy for small shocks vs. runs after large shocks with delayed lending restriction.
  - Two overlooked channels warrant empirical study: recapitalization channel (how quickly concentrated systems recapitalize) and price-drop channel (concentration’s effect on asset-price drops during runs).

*Italicized: Source — wpiea2021102-print-pdf - 0. In*

### References

### wpiea2021102-print-pdf - References (appendix proofs excerpt)

### Main analytical results
- Existence and uniqueness of equilibrium thresholds:
  - There exists a unique pair (n_C1, n_I1) ∈ [0, n_SS]^2 such that NS(n_C1) = 0 and NR(n_I1) = 0.
- Monotonicity with respect to number of banks B:
  - q_NR is increasing in B.
  - q_R is independent of B.
  - q_I is increasing in B.
  - Both B n_C1 and B n_I1 increase with B (concentration raises risk via franchise-value and price-drop channels).
- Price-drop effect:
  - The partial effect of B on B n_k through the drop in asset prices dq amplifies the franchise-value effect: ∂B n_k/∂dq * ∂dq/∂B > 0 for k = 1, 2.
  - From Lemma 1, ∂dq/∂B > 0.

### First-order conditions and pricing relations
- Household (risk-neutral) no-arbitrage conditions:
  - ̇q + z − f′(k_h) = −λ ∆q_R − ζ(z) ∆q_z + ρ q
  - ∀b: r_b = −λ(ψ_b − 1) + ρ
- Pricing equation (from F.O.C. with Lagrange multiplier μ and substitution for c and c_R):
  - q_NR − q + z − f′(k_h) = −λ(q_R − q_NR) + ρ q
  - ∀b ∈ {1,...,B}: r_b = λ(1 + r_b − ψ_b) + ρ
- In the household F.O.C. together with market clearing (K = ∑_B j k_j), the net excess return:
  - R(K) = ̇q + z − r_b q = f′(β − ∑_j k_j) − λ_b ∆q_R^b + λ_b q(ψ_b − 1) − ζ(z) ∆q_z
- Law of motion for a bank’s net worth (general):
  - ̇n = r n + (̇q + z − r q) k − d
  - Rewriting using aggregate dependence:
    - ̇n = (λ_b + ρ − λ_b ψ_b) n + ( f′(β − ∑_j k_j) − λ_b ∆q_R^b + λ_b q(ψ_b − 1) − ζ(z) ∆q_z ) k − d
  - Using ψ_b = q_R^b k_b δ_b = q_R^b k_b / (q k_b − n_b):
    - ̇n = (λ_b + ρ) n + ( f′( ̄K − ∑_j k_j ) − ζ(z) ∆q_z ) k − d
  - Particular case without productivity shock:
    - ̇n = (λ_b + ρ) n + ( f′( ̄K − ∑_j k_j ) ) k − d

### Strategic payoff definitions and deviation payoffs (symmetric Cournot game)
- Definitions for payoffs when banks choose safe (C) or risky (I) strategies (functions of n and k_C):
  - SS(n,k_C) = u[ (1 + ρ) n + α * ( ̄K − B * k_C ) * k_C ]
  - RS(n,k_C) = (1 − λ)/γ * u[ (1 + ρ) n + α * ( ̄K − (B − 1) * k_C / 2 )^2 ]
  - SR(n,k_C) = u[ (1 + ρ) n + α * ( ̄K − B * k_C ) * ( B k_C − (B − 1) k_I ) ]
  - RR(n) = (1 − λ)/γ * u[ (1 + ρ) n + α * ( ̄K / (B + 1) )^2 ]
- Net payoffs of playing safe:
  - NS(n,k_C) = SS(n,k_C) − RS(n,k_C)
  - NR(n,k_C) = SR(n,k_C) − RR
- Auxiliary payoff components:
  - SSp(k_C) = α * ( ̄K − B * k_C ) * k_C
  - RSp(k_C) = α * ( ̄K − (B − 1) * k_C / 2 )^2
  - SRp(k_C) = α * ( ̄K − B * k_C ) * ( B k_C − (B − 1) k_I )
  - RRp = α * ( ̄K / (B + 1) )^2
- Monotonicity properties used in existence proofs:
  - SR(n,k_C) is increasing in n on [0,+∞) and in k_C on (0,k_I]; SR(n,k_C(n)) is increasing in n.
  - RR is independent of k_C and increasing in n.
  - NR(n) is increasing in n.
  - SS(n,k_C) is increasing in n on [0,+∞) and in k_C on (0,k_I]; SS(n,k_C(n)) is increasing in n.
  - RS is decreasing in k_C and increasing in n; NS is increasing in n and in k_C.
  - NS(0) < 0 and NR(0) < 0; NS(n_SS, k_I) > 0 and NR(n_SS, k_I) > 0 where k_C = k_I defines n = n_SS.

### Existence and uniqueness arguments (sketches and polynomial conditions)
- Existence and uniqueness of q_C(n):
  - LHS of equation (14) is continuous, strictly increasing and linear in q_C.
  - RHS of equation (14) is continuous and strictly decreasing in q_C for q_C ∈ (q_R + B n / ̄K, q_NR].
  - RHS(q_R + B n / ̄K) > z + q_NR / (1 + ρ) > q_NR > q_R + B n / ̄K ensures a unique solution q_C.
  - q_C(n) is continuous and strictly increasing in n and in B.
- Concentration and risk channels (holding dq = q_C − q_R constant):
  - Rewriting SS, RS, SR, RR as functions of X = B n and dq isolates franchise value channel.
  - Showing B α * ( ̄K − ((B − 1)/B) * X dq^−1 / 2 )^2 is increasing in B for X in [0, dq ̄K / (B + 1)] implies B n_C1 increases with B.
  - For NR, evaluating ∂NR/∂B and using concavity of u and bounds yields a sufficient condition for ∂NR/∂B < 0; thus X = B n_I1 must increase to maintain NR = 0.
- Polynomial condition used in sufficiency:
  - Study P(x) = x^2 − [3 − 2/B] x + 2; for B > 1 and x ∈ [0,1] one has P(x) > P(1) = 2/B > 0, which supports the sign conclusions used.

### Equilibrium selection and dynamic decision rule
- Banks choose between interior vs corner solution according to:
  - max [ λ(k,n) ( n − v(n,n_−b) − v_R / v_n(n,n_−b) ) + R(k_I, k_−b) k_I, R(k_C(n,k_−b,n_−b), k_−b) k_C(n,k_−b,n_−b) ]
- Payoff definitions in equilibrium and deviations (symmetric case, k_I = ̄K/(B+1), k_C* = n − ∆q_R, etc.):
  - P_C^C(k_C*) = α ( ̄K − B k_C* ) k_C*
  - P_I^C(k_C*) = α ( ̄K − (B − 1) k_C* / 2 )^2
  - P_I^I(k_C*) = α ( ̄K − (B − 1) k_I / 2 )^2
  - P_C^I(k_C*) = α ( ̄K − B k_C* ) ( B k_C* − (B − 1) k_I )
  - F_I(k_C*) = P_I^I(k_C*) − P_C^I(k_C*)
  - F_C(k_C*) = P_I^C(k_C*) − P_C^C(k_C*)
- Existence lemma (Lemma 5.1):
  - The four functions P_I^I, P_I^C, P_C^I, P_C^C of k_C are parabolas.
  - For all k_C \ { ̄K/(B+1) }: P_C^C(k_C) < P_I^C(k_C) and P_C^I(k_C) < P_I^I(k_C).
  - F_I and F_C are strictly decreasing for k_C* ∈ [0, k_I] and strictly increasing for k_C* > k_I. They are equal to 0 at k_C* = k_I.

*Source: Excerpt from appendix proofs and online publication material of "Competition vs. Stability: Oligopolistic Banking System with Run Risk" (wpiea2021102-print-pdf)*

### 4. And atk

### 4. And atk

### Key lemmas and comparative-statics (existence and uniqueness of cutoffs)
- Equilibrium identities at k = ̄K/(B+1): P_C_C(̄K_B+1) = P_I_C(̄K_B+1) and P_C_I(̄K_B+1) = P_I_I(̄K_B+1).
- Lemma (ordering of distribution functions): ∀k_C^∗ ∈ [0,k_I] F_I(k_C^∗) > F_C(k_C^∗) with equality for k_C^∗ = k_I.
- Proof sketch reduces F_C(x) − F_I(x) algebraically to
  - −(α/2)(B−1)(k_C^∗ − k_I)^2 (B+1 + (B−1)/2) < 0, establishing strict inequality for k_C^∗ ≠ k_I.
- Reparameterization: functions are redefined in terms of net worth n over [0,n_I] using bijection k_C(n) with k_C(0)=0, k_C(n_I)=k_I.
- Define ̃P_C^k(n) = P_C^k(n) + F^∗(n) for k = I,C.
- Lemma 6 (cutoff existence under v_R = 0): If F^∗ is differentiable with (F^∗)' > 0, there exists a unique {n_I1, n_C1} ∈ [0,n_I] such that
  - ̃P_C_C(n_C1) = P_I_C(n_C1)
  - ̃P_C_I(n_I1) = P_I_I(n_I1)
- Construction uses H_I(n) = F_I(n) − F^∗(n) and H_C(n) = F_C(n) − F^∗(n) with continuity and strict monotonicity on [0,n_I] to locate unique zeros; three cases for signs at n=0 determine corner/interior placements (conventions: if H_C(0) ≤ 0 set n_C1 = 0; if H_I(0) ≤ 0 set n_C1 = n_I1 = 0).

### Corollary (n1 > 0)
- To ensure n1 > 0, show contradiction if n1 = 0:
  - If n1 = 0, all banks play safe for n ≥ 0 and at n = 0 safe strategy yields zero profit ⇒ v^∗(0) = 0.
  - But v^∗(0) = 0 implies H_C(0) > 0 so deviation from safe strategy is optimal at n = 0, contradicting Cournot equilibrium at n = 0.
  - Hence n1 > 0.

### Dividend policy and steady-state (Lemmas 2 and 3)
- In absence of aggregate shock ζ(z) = 0 and interior dividends (CRRA utility), first-order condition and envelope theorem in the safe zone yield
  - u_d(d) = v_n(n,n_−b) ⇒ −γ ̇d/d = [v_nn(n,n_−b) ̇n + Σ_{j≠b} v_{n n_j}(n,n_−b) ̇n_j] / v_n(n,n_−b)
  - Using Cournot assumption and CRRA, derive
    - γ ̇d/d = k_C_n(n,n_−b) α(̄K − (B+1) k_C(n,n_−b))    (equation (24))
- When capital investment is interior (marginal effect of net worth on capital is null),
  - γ ̇d/d = k_C_n(n,n_−b) α(̄K − (B+1) k) = 0
- Continuity of dividend policy implies d(n) = d(n1) for n_−1 < n < n1 where
  - n_−1 = [ d(n1) − R(k_I) k_I ] / [ ρ + ̄λ ]
  - If n_−1 > 0, constant dividends chosen so net worth is constant: d(n) = (ρ + ̄λ) n + R_I k_I.
  - If dividends set higher, net worth decreases to 0 in finite time; if lower, net worth increases to n_−1 then requires discontinuous jump to d(n1), violating continuity and concavity.
- Steady-state conditions:
  - Set equation (24) to 0 ⇒ k_SS = ̄K/(B+1).
  - Use asset price equation with ̇q = 0 to solve for q_SS as function of k_SS.
  - Since steady-state capital investment is safe, n_SS = (q_SS − q_R) k_SS.
  - d_SS found by equating ̇n = 0.

### Equilibrium with immediate entry and η → +∞ (Proposition 3.2)
- Contradiction argument: suppose with immediate entry there is equilibrium of type in proposition 3.1 with 0 ≤ n1 < +∞ then ∀n ∈ [0,n1), λ(n) = ̄λ and k^∗(n) = k_I.
- As η → +∞, run price q_R = q(0). For n = 0, q(n) = q(0) ⇒ q(n) = q_R and n/(q(n) − q_R) → +∞ > k_I ⇒ corner investment n/(q(n) − q_R) > k_I at 0 and by continuity in neighborhood ⇒ interior investment k_I is safe ⇒ λ(n) = 0 in a neighborhood of 0, contradiction.
- Candidate equilibrium when η → +∞: ∀n ≥ 0
  - λ(n) = 0
  - k^∗(n) = k_I
  - d(n) = ρ n + α (k_I)^2
  - q(n) = (z − α k_I)/ρ = q_SS
- Verification:
  - q(n) − q_R = 0 ⇒ n/(q(n) − q_R) infinite ⇒ k_I always safe ⇒ k^∗(n) = k_I ⇒ λ(n) = 0.
  - From dividend optimality, ̇d = 0 ⇒ d(n) = ρ n + α (k_I)^2.
  - Asset price P.D.E. ̇q = α k_I − z + ρ q with final condition q(n) = q_SS has unique stable solution q(n) = (z − α k_I)/ρ = q_SS for all n ≥ 0.
  - Same proof applies for individual bank run case.

### Phase diagram, global dynamics, and saddle path (Lemma 4)
- Change variables with ̃q = q − q_R and k = n / ̃q. For η = 0:
  - ̇k = (1/̃q) (k(z − ρ q_R) − d) = (1/̃q) (k α ̄K − d)
  - ̇q = ρ ̃q + ρ q_R − z + α(̄K − B k)
  - γ ̇d/d = k (1/α)(̄K − (B+1) k)
- Phase diagram insights:
  - ̇k = 0 locus is strictly increasing because ̃q_R < z/ρ.
  - In (k,d)-plane, steady-state is intersection of ̇k = 0 and ̇d = 0 lines; if k1 > 0 saddle path must have d and k moving in same direction.
  - For initial net worth below steady-state, both k and d increase over time toward steady-state, implying k1 > 0.
  - 3D phase diagram: ̇q = 0 locus is a vertical plane, ̇d = 0 is orthogonal plane, ̇k = 0 is a ramp; steady-state is their intersection.
  - Saddle-path must lie left of ̇d = 0 plane and below ̇k = 0 plane, within zone where arrows point to steady-state.
  - Initial condition constraint: green line in (k, ̃q)-plane given by k(t0) ̃q(t0) = n1; along equilibrium path ̃q increases over time.

### Local determinacy and linearization (uniqueness of saddle path)
- Linearize system around (d_SS, n_SS, q_SS) with deviations denoted ˆ.
- Matrix form: ̇X = A X where X = (ˆn, ˆ̃q, ˆd)' and explicit A is given in the text (entries include ρ, α, ̃q_SS, n_SS, B, k^∗_1(n_SS), d_SS, θ, γ).
- Eigenvalue sign restrictions:
  - Tr A = 2ρ + α/̃q_SS (̄K − B k_SS) = l1 + l2 + l3 > 0.
  - det A = −ρ (d_SS − θ γ) k^∗_1(n_SS) (B+1) α / (γ ̃q_SS) = l1 × l2 × l3 < 0.
  - Hence one eigenvalue negative and two positive ⇒ system with two jump variables and one initial condition for n admits one and only one saddle-path solution.
- Speed of convergence around steady-state (k,d submatrix):
  - Linearized 2×2 system with A matrix entries:
    - Tr A = (z − ρ q_R)/̃q_SS = `1 + `2 > 0
    - det A = −(d_SS − θ γ) k_SS 1 α γ ̃q_SS (B+1) = `1 `2 < 0
  - There is one negative root `− governing convergence rate:
    - `− = (z − ρ q_R)/̃q_SS − sqrt((z − ρ q_R)^2 + 4 γ (1 − k_SS q_SS^1) (d_SS − θ γ) (B+1) α^2 (q_SS − q_R))   (exact expression as in text)

### Expressions for k^∗_1(n_SS) and numerical solution procedure
- k^∗_1(n) = [1 − k_C^∗(n) q^∗_1(n)/ (q(n) − q_R)].
- Need q^∗_1(n_SS): start from ̇q(n) = ρ q(n) − z + α (̄K − B k_C(n)) and differentiate w.r.t. a single bank’s net worth to obtain
  - q_1(n) = ̇q(n)/(B ̇n(n)).
- Use L’Hospital rule as n → n_SS to handle indeterminate form:
  - lim_{n→n_SS} q_1(n) = lim_{n→n_SS} ̇q(n) / (B ̇n(n))
  - This yields an expression involving q_1(n), q^∗_1(n), k_C^∗(n), q(n) − q_R, α, and derivatives; the limit depends on d_n(n_SS) = lim_{n→n_SS} ̇d(n)/̇n(n).
- Similarly obtain expression for d_n(n_SS) in terms of q_1(n_SS) and other steady-state objects.
- The two equations in unknowns {d_n(n_SS), q_1(n_SS)} can be solved numerically; generally no closed-form solution.
- Monopoly case B = 1 simplification:
  - q_1(n_SS) = (1/k_I) ρ/(ρ + d_n) d_n/(ρ − d_n) = −2 d_SS (1 − k_I q_1(n_SS)/(q(n_SS) − q_R))^2
  - Eliminating k_I q_1 yields polynomial of degree 3 in d_n with exactly one solution d_n > ρ.

### Existence, uniqueness challenges for nonlinear system and final proposition remarks
- Existence/uniqueness of global solution to nonlinear system (system 22) is difficult:
  - Local existence could follow from Cauchy-Peano and uniqueness from Picard-Lindelöf with restrictions, but problem has both an initial condition for n and terminal conditions for other variables complicating standard existence/uniqueness arguments.
- Proof sketch for Proposition 3.1 (middle zone n_−1 < n0 < n1):
  - At n_−1 < n0 < n1, optimal interior investment k = k_I (corollary 1).
  - Lemma 4 implies k, q, n, d grow in safe zone, so right-limit dividends at n1 are low enough that bank saves; jump in profits at n1 implies accumulation approaching n1 from left ⇒ local saving property holds.
  - If n_−1 ∈ (0,n1) and n ∈ [0,n_−1) then dividends constant to keep net worth constant (lemma 3) and bank reaches n1 in finite time since savings are strictly positive.
  - Third point in proof is an implication of lemma 4 (global dynamics and saddle-path structure).

*Italic: Source: wpiea2021102-print-pdf - 4. And atk*

### 4. Immediate

### 4. Immediate

### Verification of equilibrium assumptions and properties
- The function k_C(n) is positive, continuous, strictly increasing (lemma 4) with derivative with respect to n bounded away from 0.
- q − q_R is always well-defined for n > 0 and strictly positive:
  - q_R = η/(ρ+η) q(0) + (z−α ̄K)/(ρ+η).
  - In the safe zone, q − q_R > 0.
  - In the run zone:
    - If q is increasing, then
      - q(n) − q_R ≥ q(0) − q_R = ρ/(ρ+η) q(0) − (z−α ̄K)/(ρ+η) = ρ [ q(0) − (z−α ̄K)/ρ ]/(ρ+η) > 0,
      - where (z−α ̄K)/ρ is the smallest price possible (no banks operate).
    - If q is decreasing, then for n ∈ [0,n1], q(n) − q_R ≥ q(n1) − q_R > 0 by lemma 4 and continuity at n1.
  - Hence −∆q_R is well defined and strictly positive.
- The assumptions regarding F* and the value function in lemma 1 hold; the value is admitted twice differentiable.
  - (F*)' expressions (as in source) imply (F*)' > 0 because:
    - v*_n_b(n) > 0 for all b (another bank’s net worth increases the value).
    - v*_{1,n_b}(n) < 0 for all b (diminishing marginal returns).
- Conclusion: monotonicity and differentiability assumptions used to characterize the equilibrium are validated.

### Additional figures (descriptive summary)
- Figures illustrate policy rules, asset prices, transition dynamics, leverage, savings, dividends, cumulative probability of run, and λ across different calibrations.
- Key calibration labels and cases shown in figures:
  - Figure 10: Policy rules and asset price as a function of net worth in benchmark model for B = 1,3,5.
  - Figure 11: Transition path as a function of time for B = 1,3,5 (panels for All Bank Investment, Leverage, Saving, Asset Price, Dividends, CumProbRun).
  - Figure 12: Transition path as a function of time for B = 1,3,5 (panels include λ).
  - Figures 14–16: Transition paths for counterfactuals and regulatory experiments (described below).

*Source: IMF Working Paper content unit "wpiea2021102-print-pdf - 4. Immediate".*

---

### D. A Dual System of Regulated and Shadow Banks

### Motivation and overview
- Modern financial systems contain both regulated banks (benefit from deposit insurance / discount window access) and shadow banks (subject to systemic runs).
- Since the Great Recession: three of the big five investment banks disappeared (Bear Stearns acquired by JP Morgan Chase; Merrill Lynch by Bank of America; Lehman Brothers liquidated); many remaining investment banks converted to bank holding companies and are subject to stricter rules.
- The model generalizes Section 3 to include:
  - R regulated banks (protected, subject to capital and liquidity requirements).
  - S unregulated (shadow) banks (subject to systemic runs).
  - Total institutions: S + R = B.
  - Both types invest in the same asset.
- Figure 13 (schema) maps qk_s, n_s, δ_s for shadow banks and qk_r, n_r, δ_r for regulated banks; households and capital are included.

### Regulated banks: constraints and problem
- Regulated banks choose capital investment, dividends, net worth subject to:
  - Capital requirement constraint: k_r = κ n_r (equation 25).
  - Leverage ratio binding: λ_r = 1 (equation 26).
  - Assumption: κ < 1 − ∆q_R so that if a run occurs, net worth of regulated banks remains positive.
- Aggregate investment by regulated banks: K_R = κ N_R = κ R n_R.
- Regulated bank HJB (taking k_{−r}, k_s, ̇n_{−r}, ̇n_s as given):
  - ρ v(n,n_{−r},n_s,S) = max_{d,k} [ u(d) + v_n((̇q+z) k − r δ − d) + ∑_{j≠r} v_{n_j} ̇n_j + ∑_s v_{n_s} ̇n_s + λ( v(n+ ∆q_R k, n_{−r}+ ∆q_R k_{−r},0,0) − v(n,n_{−r},n_s,S) ) ]
  - Subject to k_r = κ n_r.

### Shadow banks: problem and interaction
- Shadow banks solve a similar HJB taking regulated banks’ investment as given:
  - ρ v(n,n_{−s},n_r,S) = max_{d,k} [ u(d) + v_n((̇q+z) k − r δ − d) + ∑_{j≠s} v_{n_j} ̇n_j + ∑_r v_{n_r} ̇n_r + λ( v_R − v(n,n_{−b},n_r,S) ) ].

### Market clearing and run price
- Capital market clearing:
  - ∑_{s=1}^S k_s + ∑_{r=1}^R k_r + ∫ k_h dh = ̄K (equation 27).
- Focused case: no entry (η = 0) so after a run on shadow banks the system remains populated only by regulated banks:
  - S' = 0 (equation 28).
  - R' = R (equation 29).
- Run price consistency:
  - q_R(n_s,n_r,S) = q(0,n_r + ∆q_R k(n_r),0), i.e., price when no shadow banks operate and all capital held by regulated banks and households consistent with S' = 0.

### Symmetric equilibrium, run game, and behavior
- Symmetry is assumed among regulated banks and among shadow banks, but asymmetry allowed between groups.
- Run condition on the shadow banking sector (second stage, depositors decide):
  - N_S + ∆q_R K_S < 0 (equation 30), where N_S = S n_s and K_S = S k_s.
  - Condition is independent of regulated banks’ net worth because regulated banks’ deposits are insured and they cannot transfer liquidity to shadow banks when runs occur (leverage constraint binding). No pooling of liquidity between sectors.
  - Note: a more general condition N_S + N_R + ∆q_R (K_S + K_R) < 0 would imply pooling of equity and acquisition of shadow banking system by regulated sector (extreme case).
- Shadow banks’ behavior parallels the unregulated equilibrium but with ̄K replaced by:
  - ̃K(n_R) = ̄K − R min( κ n_R, k_SS ).
- Regulated banks accumulate net worth until optimal Cournot interior:
  - k_SS = ̄K_{S+R+1} (as stated in source).

### Competition policy counterfactuals and findings
- Two counterfactuals to examine impact of competition:
  1. Decrease number of regulated banks from 3 to 2, keep shadow banks at 2: (R,S) = (2,2) with baseline (3,2) and alternative (3,1).
  2. Keep regulated at 3, decrease shadow banks to 1: (R,S) = (3,1).
- Transition path experiment: economy hit by a shock bringing net worth of shadow banks to 0. Transition paths plotted in Figure 14:
  - Original economy: black; first counterfactual: dark gray; second: light gray.
  - Panels (a),(b): solid line = shadow banks, dashed = regulated banks. Panel (c): solid = asset price, dashed = run price.
- Findings:
  - Decreasing competition in the regulated sector (fewer regulated banks) has a slight stabilizing effect via higher profits for shadow banks (franchise value channel and recapitalization channel), which yields less risk-taking and faster travel through crisis zone (observable in regulated net worth path).
  - However, price-drop effect counteracts: fewer regulated banks imply more severe price drop after a run on the entire shadow banking system (panel (c)), enlarging the crisis zone.
  - The larger drop results from market power (less banks → lower output and asset prices) and larger drop in regulated banks’ net worth due to relatively larger exposure.
  - Decreasing the number of shadow banks (R,S) = (3,1) stabilizes via franchise value channel, recapitalization channel, and mitigated price-drop channel.

### Macroprudential policies in the regulated sector and findings
- Counterfactuals on capital requirement κ:
  - Compare κ = 1 and κ = .25; both economies hit by shock bringing shadow banks’ net worth to 0. Black lines: benchmark κ = 1.
- Findings when regulated sector starts from different steady-states:
  - If regulated banks start better capitalized (κ = .25 lower κ meaning tighter capital? — preserved as presented in source), a more tightly regulated banking sector implies a more stable overall financial system in the example:
    - Regulated banks are better capitalized; after a run and disappearance of shadow banks, capital investment by intermediaries drops less.
    - This maintains a higher run-price when κ = .25 (Figure 15, panel (c)).
  - Alternative experiment: both economies begin at the same steady-state with κ = .25, and after the shock one economy tightens capital requirement to κ = 1 (counterfactual mirroring post-Great Recession tightening).
    - Surprisingly, the more tightly regulated system shows a higher probability of a run when starting from the same point after the shock.
    - Two offsetting mechanisms:
      1. Larger price-drop in the more tightly regulated system because regulated system takes much longer to recapitalize, depressing asset prices if they alone intermediate funds; this enlarges the run zone.
      2. Recapitalization/franchise-channel: stricter capital requirements for regulated banks reduce competition from regulated banks, increasing shadow banks’ profits, savings, and speed of recovery.
    - In the presented example the first effect (larger price-drop and slower recapitalization) dominates, producing higher run probability.
- Figure references for these experiments:
  - Figure 15: Transition path for κ = 1 and κ = .25 (panels: Ind Bank Investment, Saving, Asset Price, CumProbRun).
  - Figure 16: Transition path for κ = 1, .25 and common starting net worth (panels: Ind Bank Investment, Saving, Asset Price, CumProbRun).

*Source: IMF Working Paper content unit "wpiea2021102-print-pdf - 4. Immediate".*

---


_Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021102-print-pdf.pdf_