## _wp13247

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---

### Model setup and assumptions
- Financial sector is sole intermediary; actors termed “bankers.”
- Two financial imperfections:
  - “Skin in the game” commitment: bankers must hold sufficient capital to intermediate (minimum aggregate bank capital required to intermediate first-best credit).
  - Incomplete insurance markets: extreme assumption that holdings of bank equity are concentrated in bankers.
- Consequence: if aggregate bank capital falls below threshold, financial constraints bind and credit must be cut back.

### Mechanisms, equilibrium implications, and key quantitative relations
- Binding financial constraints produce pecuniary externalities that:
  - Reduce output and wages.
  - Increase lending spreads.
  - Benefit bankers via higher returns/spreads while harming workers (empirically illustrated by US data around 2008/09).
- Risk-taking incentives and distribution:
  - Bankers trade off higher expected returns from risk-taking against risk of binding constraints but do not internalize negative externalities.
  - Bankers always choose strictly positive risk-taking if E[Ã] > 1; workers prefer limiting risk-taking to stabilize credit supply.
  - Pareto-frontier: higher risk-taking increases bankers’ welfare and decreases workers’ welfare.
- First-best allocation (planner):
  - `x = 1` in period 0 since E[Ã] > 1.
  - `\ell* = 1`.
  - `k* = (αA)^{1/(1−α)}` solves F_k(k*,1) = 1.
  - `R* = 1`.
  - Social surplus S* = (1 − α) F(k*,1) + E[Ã].
- Financial constraint (commitment/shirking):
  - Depositors impose rd ≤ φ R k with φ ∈ [0,1]; equivalent shirking interpretation φ = 1 − B/Δ.
- Period 1 equilibrium as function of aggregate bank equity e:
  - Unconstrained if e ≥ e* = (1 − φ) k*; then r = R = 1 and wage w* = (1 − α) F(k*,1).
  - Constrained if e < e*: equilibrium capital k̂(e) solves k = e + φ k F_k(k,1); k(e) = min{ k̂(e), k* }.
  - R(e) = α F(k(e),1) / k(e); w(e) = (1 − α) F(k(e),1).
- Marginal effects when constraint binds (e < e*):
  - k′(e) = 1 / [1 − φ α F_k] > 1.
  - s′(e) = 1 + (F_k − 1) k′(e).
  - w′(e) = (1 − α) F_k k′(e).
  - π′(e) = 1 + (α F_k − 1) k′(e).
- Redistributive effect at margin around e* (Lemma 1):
  - Marginal tightening around e* has first-order redistributive effects but only second-order efficiency costs.
  - lim_{ε→0} [ π′(e* − ε) + 1 ] = (1 − α) k′(e*).
  - lim_{ε→0} w′(e* − ε) = − (1 − α) k′(e*).
  - First-order effect on total surplus s′ = π′ + w′ is zero.

### Determination of period-0 risk allocation (Section 3.3)
- Bankers choose x_i ∈ [0,1] to maximize expected profits subject to ei = (1 − xi) + ÃA xi.
- First-order condition at interior optimum: E[π1(ei,e) (ÃA − 1)] = 0, where π1(ei,e) = 1 + [R(e) − 1] k1 with k1 = 1/(1 − φR) when constrained.
- Definitions and regimes:
  - Well-capitalized: e* ≤ 1. First-best intermediation k* reached with x = 0; bankers choose xLF > 1 − e* so constraint binds in low states.
  - Under-capitalized: e* > 1. Economy constrained even if bankers invest all endowment safely; ̄A(x) strictly decreasing from ∞ to e* as x increases.
- Pareto frontier and comparative statics (Proposition 3):
  - Define xB = arg max Π(x), xW = max arg max W(x). Then xW < xB; over [xW, xB], W(x) strictly decreasing, Π(x) strictly increasing.
  - xLF < xB; if e* ≤ 1 then xW < xLF < xB.
  - Movement along frontier: increasing x benefits bankers and harms workers via higher returns and redistribution through binding constraints.
- Market incompleteness and conflict:
  - Removing period-1 constraint or completing risk markets eliminates distributive conflict (xW = xB = xFB = 1).
  - Special case (deposits d = 0, Cobb-Douglas): constant shares produce no distributive conflict despite pecuniary externalities.
- Regulatory instruments affecting x:
  1. Ceiling xi ≤ ̄x (capital adequacy analog).
  2. Tax τx on risk-taking: modifies FOC to E[π1·(ÃA − τx − 1)] = 0; tax revenue rebated lump-sum to bankers.
- Corollary 4:
  - Tightening regulation (lower ̄x or higher τx) increases worker welfare and reduces banker welfare for ̄x ∈ [xW, xLF].
  - Deregulation redistributes from workers to bankers.
- Scope for Pareto-improving deregulation:
  - Planner can provide lump-sum or state-contingent transfers; efficient compensation requires overcoming at least one market imperfection (mitigate pledgeable income constraint or risk-market incompleteness).
  - Without ability to improve these imperfections, Pareto-improving deregulation scope is limited.

### Agency problems: asymmetric compensation schemes
- Extension: owners hire managers whose incentive payments create payoff asymmetry (parameter δ > 1 amplifies managers’ upside).
- Manager payoff p(ei,e) = ε min{π(ei,e), π(e*,e*)} + δε max{0, ei − e*}.
- Marginal benefit p1(e,e) = ε π1(e,e) for e < e*, p1 = δε for e ≥ e*.
- Proposition 5:
  - Managers choose higher risk-taking than xLF if δ > 1.
  - Worker expected welfare declines in δ.
- Intuition: manager payoff convexity and upside skew increase risk-taking and negative externalities on workers.

### Financial institutions with market power (Section 5.2)
- Finite n identical bankers, each mass 1/n; banker internalizes effect of own x_i on aggregate e.
- Symmetric equilibrium capital:
  - k_{∗,n} = (1 − 1/n (1 − α))^{1/(1−α)} k_∗.
  - e_{∗,n} = (1 − φ 1 − 1/n (1 − α)) k_{∗,n}.
- Marginal valuation ordering under market power:
  - π′ < π_{i,n}^1 < π_i^1.
  - Optimality condition: Π_{i,n}^1 = Π^1(x) + 1/n (Π′ − Π^1) = 0.
- Proposition 6:
  - Optimal risk allocation x_n is declining in number n of banks.
  - x_1 = x_B ≥ x_∞ = x_{LF}, strict inequality except corner cases.
- Intuition: greater competition lowers bankers’ internalization of aggregate scarcity rents; market power reduces precautionary incentives and increases risk-taking, redistributing towards bankers.

### Financial innovation (Section 5.3)
- Innovation expands bankers’ investable risky assets (introduces stochastic return Ã).
- Example where e_∗ < 1 (safe return suffices pre-innovation):
  - Pre-innovation x = 0 maximizes worker welfare.
  - Post-innovation bankers choose x_{LF} > 1 − e_∗ and face risk of binding constraints in low states.
- Result:
  - Innovation increases banker welfare and reduces worker welfare.
  - Policy implication: restrictions on bank risk-taking (e.g., analogous to the Volcker rule) can serve as second-best devices to complete markets.

### Bailouts (Section 5.4): endogenous nature, optimal policy, and incentive effects
- Workers may optimally provide ex-post bailouts because bank capital scarcity creates severe real-economy losses.
- Lemma 7 — Optimal bailout policy:
  - If aggregate bank capital e in period 1 satisfies e < ˆe (< e_∗), workers optimally provide lump-sum transfer t = ˆe − e.
  - Threshold ˆe solves w′(ˆe) = 1 and equals ˆe = (1 − α) [1 − (1 − φ) α]^{α/(1−α)} e_∗.
- Assumption 1: parameters α, φ, A such that ˆe < 1 (no bailout needed if safe project invested).
- Period-1 equilibrium with bailouts (e < ˆe):
  - w_{BL}(e) = (1 − α) F(e + t(e), 1) − t(e).
  - π_{BL}(e) = α F(ˆe, 1).
  - Within bailout region 1 + t′(e) = 0 so w_{BL}′(e) = 1 and π_{BL}′(e) = 0.
  - Interpretation: marginal unit of bank equity reduces required bailout dollar-for-dollar (worker marginal benefit = 1); bankers’ marginal benefit = 0 within region but bankers obtain “bailout rents” at margin ˆe since π′(ˆe) = α/(1 − α) > 0.
- Period-0 risk-taking under bailouts:
  - Discretionary bailouts cap market interest R_{BL}(e) ≤ R(ˆe) and weaken precautionary incentives — increases bankers’ optimal risk-taking (wealth effect).
  - Bailouts conditional on individual bank capital amplify moral hazard (substitution effect).
- Bailout specification:
  - t(e_i, e; γ) = 0 if e ≥ ˆe; t = ˆe − (1 − γ) e − γ e_i if e < ˆe; γ ∈ [0,1].
  - x_{BL}(γ) denotes period-0 risk-taking under bailouts.
- Proposition 8:
  - (i) Introducing bailout transfers increases period-0 risk-taking x_{BL}(γ) > x_{LF} for any γ ≠ 0.
  - (ii) x_{BL}(γ) increases in γ.
- Welfare decomposition and Corollary 9:
  - Welfare changes split into market-completion effect (for given x, bailouts are Pareto-improving) and incentive effect (bailouts raise x and can harm workers).
  - (i) Bankers always benefit from introducing bailouts; Π_{BL}(γ) increases in γ.
  - (ii) Workers benefit from market completion but are hurt by incentive effects when e_∗ < 1; absolute magnitudes increase in γ.
- Graphical implication: bailouts shift Pareto frontier outward at left end but bias gains toward bankers (bailouts constitute banker-biased technological change).

### Extensions, equivalences, and policy tools (Technical Appendix highlights)
- Emergency lending and equity injections are isomorphic to lump-sum transfers in net present value if they relax bankers’ financial constraint; only transfers that relax the constraint in net present value expand intermediation (Lemma 10).
- Optimal instruments require government ability to provide unconditional/subsidized transfers or superior enforcement to extract repayments/dividends.
- Recapitalizations forced by regulator can be part of toolkit to maintain credit supply and protect real economy even if private bankers bear costs.
- Policy trade-off for regulator:
  - Prioritize real economy → tighter regulation, higher capital requirements on risky activities, limit payouts that endanger capitalization, separate risky activities, reduce incentives for risk-taking (limit market power, asymmetric pay schemes, bailout expectations), force recapitalizations when necessary.
  - Prioritize bankers’ welfare → rollback regulation, reduce capital requirements.

### Parameterization and data notes (figures and empirical motivation)
- Parameter values used in Figures 3 and 5:
  - α 1/3
  - A 10
  - φ 0.5
- Parameter values in Figures 4 and 6 (Ã lognormal truncated to [0,2], mean μ and variance σ2):
  - α 1/5
  - A 8.1335
  - φ 0.5
  - μ 1.04
  - σ2 0.5
- Data sources for empirical illustrations:
  - Federal Reserve Bank of St. Louis FRED database.
  - Bank equity: market value of equity from Federal Reserve Flow of Funds (series FL - 79 - 41900 and 41940), deflated by GDPDEF.
  - Spread on risky borrowing: BAA − DGS10.
  - Real wage bill: W209RC1 deflated by GDPDEF.

*Source: _wp13247 - 2.1 Model Setup; Sections 3.3, 5.2, 5.3, 5.4; Technical Appendix and parameterization.*

### 2.1    Model Setup    .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .9

### 2.1    Model Setup

### Model assumptions
- The financial sector is the only sector that can engage in financial intermediation and channel capital into productive investments; all actors in this broad financial sector are referred to as “bankers.”
- Two financial imperfections are introduced:
  - Bankers suffer from a commitment problem and need to have sufficient capital in order to engage in financial intermediation (“skin in the game” constraint).
  - Insurance markets between bankers and the rest of society are incomplete; the extreme assumption is made that holdings of bank equity are concentrated in the hands of bankers (a sufficient condition more generally is that holdings of bank equity are not proportionally distributed across the financial elite and the rest of society).
- Because of the “skin in the game” constraint, a minimum level of aggregate bank capital is required to intermediate the first-best level of credit and achieve optimal output; if aggregate bank capital falls below this threshold, financial constraints bind and bankers must cut back on credit.

### Mechanisms and equilibrium implications
- Binding financial constraints produce pecuniary externalities that harm the real economy but benefit bankers:
  - Credit crunches reduce output and wages, and increase lending spreads; these effects are illustrated by US data around the 2008/09 financial crisis (Figure 1: decline in bank equity; increase in spread on risky borrowing; decline in real wage bill; sample timestamps include 2006Q3, 2008Q4, 2011Q1, 2013Q2).
- Risk-taking incentives:
  - Bankers trade off higher expected returns from risk-taking against the risk of being financially constrained, but do not internalize negative externalities on the rest of the economy.
  - Bankers always choose a strictly positive level of risk-taking in the model to earn superior returns; workers are averse to fluctuations in bank capital and prefer limiting risk-taking to stabilize credit supply.
- Distributional outcome:
  - A Pareto-frontier is characterized along which higher risk-taking implies higher welfare for bankers and lower welfare for workers; financial regulation/deregulation moves the economy along this frontier.
- Counterfactuals where the distributive conflict disappears:
  - If bankers were not financially constrained, they could intermediate the optimal amount and their risk-taking would not affect the real economy.
  - If risk markets were complete, bankers and the rest of the economy would share both downside and upside of risk-taking.
- Analogy:
  - Financial deregulation is compared to relaxing safety rules on nuclear power plants: it reduces costs and increases industry profits in most states of nature but raises the risk of large negative externalities in bad states.

### Extensions analyzed
- Financial innovation:
  - Broadening bankers’ investment menu that allows more risk-taking always benefits bankers but can increase volatility and externalities on workers; examples show workers can be unambiguously worse off.
- Asymmetric compensation schemes:
  - Managerial pay asymmetries induce higher risk-taking and larger negative externalities on the real economy.
- Financial institutions with market power:
  - Market power reduces bankers’ precautionary incentives because bankers internalize that losses reduce aggregate bank capital and raise lending rates, which mitigates losses; this increases risk-taking and benefits bankers at workers’ expense, highlighting welfare losses from concentrated banking systems.
- Bailouts:
  - Ex-post, workers collectively find it optimal to provide bailouts when aggregate bank capital is sufficiently scarce because bailouts ease credit crunches and mitigate wage declines; this makes commitment not to bail out difficult.
  - Ex-ante, bailouts reduce bankers’ precautionary incentives and increase risk-taking even if provided lump-sum; bailouts conditional on individual institutions’ capital amplify distortions (traditional moral hazard).
  - A novel channel: bailouts increase expected bank profits via higher risk-taking and raise the incidence/severity of credit crunches that harm workers; in expectation, redistribution via higher risk-taking can exceed the direct transfers from bailouts.
- Recapitalizations:
  - For a regulator focused on the real economy, forcing recapitalizations when necessary (even imposing private costs on bankers) is part of policy options to maintain credit supply.

### Policy implications (regulator objectives and instruments)
- Regulators face a trade-off between:
  - Greater efficiency in the financial sector (aided by deregulation and risk-taking), and
  - Greater efficiency in the real economy (aided by tighter regulation and a stable credit supply).
- If regulators prioritize the real economy, desirable measures include:
  - (i) separating risky activities, such as proprietary trading, from traditional financial intermediation;
  - (ii) imposing higher capital requirements on risky activities, in particular on those that do not directly contribute to lending to the real economy;
  - (iii) limiting payouts if they endanger a sufficient level of capitalization in the financial sector;
  - (iv) using structural policies that reduce incentives for risk-taking, e.g., by limiting market power, asymmetric managerial incentive contracts, financial innovations that increase risk-taking, and bailout expectations;
  - (v) forcing recapitalizations when necessary, even if they impose private costs on bankers.
- If regulators place higher welfare weight on bankers, they will tend to relax these constraints (rollback regulation, reduce capital requirements).

### Related literature and empirical motivation
- The model connects to a literature on financial frictions that amplify shocks and cause macroeconomic fluctuations (examples cited in the source include Bernanke and Gertler, Kiyotaki and Moore, Gertler and Karadi).
- It relates to literature arguing regulation should internalize pecuniary externalities under incomplete markets (examples in the source include Lorenzoni, Jeanne and Korinek, Korinek, Gersbach and Rochet).
- The paper highlights a gap in the literature on redistribution between financial intermediaries and the rest of the economy during crises and contributes an endogenous rationale for why workers may choose bailouts ex-post.
- Empirical patterns motivating the setup:
  - Growth and deregulation of the financial sector preceding the 2008/09 crisis (references in the source include Abiad et al., Philippon and Reshef).
  - Declines in labor share and increases in financial-sector rents observed in recent decades (references include Karabarbounis and Neiman, Kaplan and Rauh, Philippon and Reshef).
  - Evidence that financial intermediaries’ capital positions have strong real effects (reference: Adrian et al., 2010).

*Source: _wp13247 - 2.1    Model Setup*

### Section 3 analyzes the determination of equilibrium and how changes in bank

### _wp13247 - Section 3 analyzes the determination of equilibrium and how changes in bank

### Model setup: agents, timing, technologies
- Time periods: t = 0, 1, 2.
- Agents: unit mass each of bankers and workers.
- Single good serves as consumption and capital.
- Bankers:
  - Born in period 0 with one unit of the consumption good.
  - Choose risky investment fraction x ∈ [0,1] in period 0.
  - Risky project delivers payoff à with distribution G(Ã) on [0,∞), density g(Ã), and expected value E[Ã] > 1.
  - Hold remainder (1 − x) in storage with gross return 1.
  - Bank equity after realization: e = xà + (1 − x). (Bank capital is used synonymously with bank equity e.)
  - In period 1 bankers raise deposits at gross deposit rate r and lend k ≤ d + e to firms at gross interest rate R.
  - Period 2 profits: π = Rk − rd (linear utility over profits).
- Workers:
  - Born in period 1 with large endowment m; lend deposits d to bankers at rate r and hold remainder in storage with gross return 1.
  - No-arbitrage implies r = 1.
  - In period 2 supply inelastic labor ` = 1 and earn wage w.
  - Worker utility normalized to u = w` (with m subtracted).
- Firms:
  - Competitive neoclassical firms produce in period 2; production F(k,`) = A k^α `^{1−α} with α ∈ (0,1).
  - Firms rent capital k from bankers at rate R and hire labor ` at wage w; zero-profit in equilibrium.
  - First-order conditions: R = F_k = αA k^{α−1} `^{1−α}; w = F_` = (1 − α)A k^α `^{−α}.
- Market incompleteness remark:
  - Workers are born in period 1 after realization of à and cannot insure against xà chosen in period 0; risk xà must be borne by bankers.

### First-best allocation
- Planner maximizes aggregate surplus subject to resource constraints:
  - Variables: x, e, k, `.
  - Constraints: e = xà + (1 − x); k ≤ e + m; x ∈ [0,1]; ` ∈ [0,1].
- Optimal period-2 choices:
  - `* = 1.
  - k* = (αA)^{1/(1−α)} solves F_k(k*,1) = 1.
  - R* = 1.
- Planner chooses x = 1 in period 0 (since E[Ã] > 1) to maximize E[e].
- First-best net social surplus:
  - S* = (1 − α) F(k*,1) + E[Ã].

### Financial constraint (commitment/shirking interpretation)
- Bankers can divert fraction (1 − φ) of gross revenue; φ ∈ [0,1] is enforceable fraction.
- Depositors impose constraint: rd ≤ φ R k.
- Alternative interpretation: shirking yields private benefit B per unit of period-2 revenue, failure probability Δ; constraint equivalent to φ = 1 − B/Δ to prevent shirking.
- Binding financial constraints reduce supply (or alternatively demand) and thus hurt the real economy.

### Laissez-faire equilibrium: definitions and solution approach
- Equilibrium: prices {r, R, w} and allocation {x, e, d, k} (all except x contingent on Ã) such that agents optimize and markets clear.
- Solve by backward induction:
  - First solve period 1 equilibrium for given aggregate bank equity e.
  - Then solve period 0 portfolio choice of bankers to determine e.

### Period 1 equilibrium as function of aggregate bank equity e
- Employment always ` = 1 (wages flexible).
- Unconstrained (“normal times”): e ≥ e* = (1 − φ) k*.
  - Deposit and lending rates r = R = 1.
  - Bankers earn zero returns on lending; wage w* = (1 − α) F(k*,1).
- Constrained (“credit crunch”): e < e*.
  - Depositors supply d = φ R k / r with r = 1.
  - Lending rate R = F_k(k,1).
  - Equilibrium capital k̂(e) solves implicitly:
    - k = e + φ k F_k(k,1)
    - Unique positive solution for any e ≥ 0.
  - Aggregate capital k(e) = min{ k̂(e), k* }.
  - k(e) is strictly positive, strictly increasing in e on e ∈ [0, e*) and constant at k* for e ≥ e*.
- Lending rate and wage as functions of e:
  - R(e) = α F(k(e),1) / k(e).
  - w(e) = (1 − α) F(k(e),1).
- Individual banker i with equity e_i:
  - k(e_i,e) = min{ k*, e_i / (1 − φ R(e)) }.
  - π(e_i,e) = e_i + [R(e) − 1] · k(e_i,e).
  - In symmetric equilibrium e_i = e, aggregate banking profits:
    - π(e) = e + α F(k(e),1) − k(e).
  - Worker utility: w(e) = (1 − α) F(k(e),1).
  - Total utilitarian surplus: s(e) = w(e) + π(e) = e + F(k(e),1) − k(e).

### Marginal value of bank capital and distributional effects
- In constrained region e < e*, a marginal increase in e raises capital intermediation k more than one-for-one:
  - k′(e) = 1 / [1 − φ α F_k] > 1 for e < e*.
- For e ≥ e*, k′(e) = 0.
- Marginal effect on total surplus:
  - s′(e) = 1 + (F_k − 1) k′(e).
    - First term: direct consumption value of extra unit of wealth to bankers.
    - Second term: additional real investment k′(e) earning marginal return (F_k − 1).
- Distribution between workers and bankers:
  - w′(e) = (1 − α) F_k k′(e).
  - π′(e) = 1 + (α F_k − 1) k′(e).
- Regime differences:
  - Unconstrained (e ≥ e*): k′(e) = 0 → w′(e) = 0 and π′(e) = 1. Additional bank equity benefits only bankers.
  - Constrained (e < e*): additional equity increases k and output; fraction (1 − α) of additional output accrues to workers via higher wages, fraction α (net of added capital) accrues to bankers. Binding constraint creates pecuniary externalities that bankers do not internalize.

### Equity shortages, redistribution, and efficiency
- Small shortages of bank equity have first-order redistributive effects but only second-order efficiency costs around e*.
- Consider marginal tightening removing infinitesimal ε of bank capital in period 1 and returning in period 2:
  - Payoffs change to π(e* − ε) + ε for bankers and w(e* − ε) for workers.
- Lemma 1 (Redistributive Effects of Equity Shortages):
  - A marginal tightening around e* has first-order redistributive effects but only second-order efficiency costs.
  - Left-sided derivatives at e* satisfy:
    - lim_{ε→0} [ π′(e* − ε) + 1 ] = (1 − α) k′(e*).
    - lim_{ε→0} w′(e* − ε) = − (1 − α) k′(e*).
  - The first-order effect on total surplus s′ = π′ + w′ is zero.
  - Intuition: marginal tightening reduces workers’ wages and increases bankers’ returns via higher lending spreads by equal amounts; redistribution is one-to-one from workers to bankers when the financial constraint binds.
- Additional remarks:
  - Decline in wages for e < e* arises because labor is complementary to capital.
  - Increase in lending rates for e < e* arises because constrained supply pushes R(e) = F_k(k(e),1) > 1; spread R(e) − 1 transfers scarcity rents to bankers and signals social value of carrying capital into scarce states.

*Source: _wp13247 - Section 3 analyzes the determination of equilibrium and how changes in bank*

### 3.3    Determination of Period 0 Risk Allocation

### 3.3    Determination of Period 0 Risk Allocation

### Determination of Period 0 risk allocation and banker optimality
- Bankers take the lending rate R as given and perceive the deposit constraint d ≤ φRk as a leverage limit.
- When constrained, a marginal increase in bank capitalei increases intermediation activity by k1(ei,e) = 1/(1−φR), implying an increase in bank profits by
  - π1(ei,e) = 1 + [R(e)−1]k1(ei,e) (equation (3)).
- In period 0, bankers choose xi ∈ [0,1] (fraction of endowment to the risky project) taking aggregate x and e as given to maximize
  - max xi∈[0,1],ei Πi(xi;x) = E[π(ei,e)] s.t. ei = (1−xi) + ÃA xi (equation (4)).
- First-order condition at an interior optimum:
  - E[π1(ei,e) (ÃA−1)] = 0 (equation (5)): the risk-adjusted return on the stochastic payoff ÃA equals the return of the safe storage technology.
- π1 is strictly declining in e as long as e < e* and constant at 1 otherwise.
- In symmetric equilibrium ei = e and xi = x; equilibrium x solves (5).
- If E[ÃA] > 1 then optimal allocation to the risky project satisfies x > 0; if expected return is sufficiently high equilibrium may be the corner x = 1; otherwise x uniquely pinned down by (5).
- Laissez-faire allocation: denote xLF; resulting welfare levels:
  - ΠLF = E[π(1−xLF + ÃA xLF)]
  - WLF = E[w(1−xLF + ÃA xLF)].
- Define ̄A(x) as the threshold of ÃA above which bank capitale is sufficiently high to support first-best production; can be expressed as ̄A(x) = 1 + (e* −1)/x.

### Well-capitalized versus under-capitalized banking systems
- Well-Capitalized (e* ≤ 1):
  - Safe return suffices to avoid financial constraint; first-best intermediation k* reached with x = 0.
  - Risky project ÃA interpreted as diversion from main intermediation.
  - Bankers choose xLF > 1−e*, i.e. take on sufficient risk so constraint binds at least for low realizations: ̄A(x) > 0.
  - For e* < 1, ̄A(x) strictly increasing from ̄A(1−e*) = 0 to ̄A(1) = e*.
- Under-Capitalized (e* > 1):
  - Economy would be constrained even if bankers invest all endowment in safe return.
  - ̄A(x) strictly decreasing from limx→0 ̄A(x) = ∞ to ̄A(1) = e*.

### Pareto frontier: mapping risk-taking to welfare
- Collective choices:
  - xB = arg max x∈[0,1] E[π(ÃA x + 1−x)] (bankers' collective preference).
  - xW = max { arg max x∈[0,1] E[w(ÃA x + 1−x)] } (workers' collective preference).
- In well-capitalized system (e* ≤ 1):
  - Workers prefer x ∈ [0,1−e*] to keep constraints loose; indifference implies the Pareto-frontier point is xW = 1−e*.
- In under-capitalized system (e* > 1):
  - Workers optimally choose positive risk-taking xW > 0 because risk increases expected availability of finance in period 1.
- Definition 2 (Pareto Frontier): set of pairs (Π(x), W(x)) for x ∈ [xW, xB].
- Assumption for non-degenerate frontier: xW < 1 and xLF < 1.
- Proposition 3 (Characterization of Pareto Frontier):
  - (i) xW < xB.
  - (ii) Over [xW, xB], W(x) is strictly decreasing in x; Π(x) is strictly increasing in x.
  - (iii) xLF < xB. If e* ≤ 1 then xW < xLF < xB.
- Movement along the Pareto frontier: increasing x moves upwards and left — bank profits rise (higher returns and redistribution from workers to bankers via binding constraints), worker welfare falls.

### Market incompleteness and distributive conflict
- Removing the financial constraint in period 1:
  - Bankers' profits/losses do not affect capital intermediated; workers indifferent about x; xW = xB = xFB = 1: distributive conflict disappears.
- Introducing complete insurance market in period 0 but keeping period-1 constraint:
  - Workers insure bankers; all agents invest first-best xW = xB = xFB = 1: distributive conflict disappears.
- If either the period-1 constraint or risk-market incompleteness is mitigated, workers can transmit risk preferences to bankers; risk-sharing removes distributive conflict.
- Special case: ‘Marxist’ two-sector framework (deposits d = 0 and no storage), Cobb-Douglas technology:
  - π(e) = α F(e,1), w(e) = (1−α) F(e,1); identical relative risk aversion implies arg maxx E[π(e)] = arg maxx E[F(e,1)] = arg maxx E[w(e)]: no distributive conflict despite pecuniary externalities because constant shares replicate perfect risk-sharing.
- In benchmark with both imperfections reintroduced:
  - Negative pecuniary externalities occur only on the downside; once bank capital exceeds threshold, it is irrelevant to workers. Distributive conflict generated by occasionally binding constraints plus lack of risk-sharing.

### Financial regulation: instruments and redistributive effects
- Regulatory instruments affecting x:
  1. Ceiling on individual bankers' risk-taking xi ≤ ̄x. Binding if ̄x < xLF. Corresponds to capital adequacy regulations.
  2. Tax τx on risk-taking xi modifying banker FOC to E[π1·(ÃA − τx − 1)] = 0. Tax revenue rebated lump-sum to bankers for simplicity. Can implement any x ∈ [0,1].
- Regulators can implement any ̄x ≤ xLF by ceiling ̄x or equivalent tax τx ≥ 0.
- Corollary 4 (Redistributive Effects of Financial Regulation):
  - Tightening regulation by lowering ̄x or raising τx increases worker welfare and reduces banker welfare for any ̄x ∈ [xW, xLF].
  - Conversely, deregulation (increasing ̄x) redistributes from workers to bankers.

### Scope for Pareto-improving deregulation
- Planner providing uncontingent lump-sum transfers from bankers to workers:
  - Marginal benefit to workers is 1 − E[w′(e)] if transfer in period 1 or 1 − φ E[w′(e)] if in period 2.
  - Transfers entail efficiency costs from tightening constraints on bankers (reducing bank capital or pledgeable income).
  - Efficient compensation without imposing tightening costs requires planner to extract payments in excess of financial constraint (superior enforcement).
- Planner providing state-contingent transfers to workers contingent on states where bankers are unconstrained (i.e., high profits from ÃA):
  - No efficiency costs but requires state-contingent transactions not available in private markets; corresponds to proportional/progressive profit taxation.
- Conclusion:
  - Efficient compensation mechanisms require getting around at least one market imperfection (mitigate constraint (1) or risk-market incompleteness).
  - If planner cannot improve these imperfections or transfers are distortionary, Pareto-improving deregulation scope limited; constrained Pareto frontier is enveloped by unconstrained frontier in Figure 4.

### Risk-taking and redistribution: overview
- Factors increasing risk-taking (market power, agency problems, safety nets, financial innovation) can be expected to redistribute welfare from workers to bankers by increasing bank capital volatility.
- The paper illustrates this for managerial agency problems, institutions with market power, financial innovation, and safety nets.

### Asymmetric compensation schemes (agency problems)
- Extend model: bank owners hire managers who choose unobservable xi in period 0. Owners observe ei in period 1 and can instruct allocation up to e* to intermediation; excess max{0, ei − e*} placed in storage.
- Managers can threaten to withdraw monitoring in period 1; if they withdraw, returns diminished by ε for intermediation and δε for storage, with δ > 1 (financial investments more sensitive to managerial effort).
- Managers have bargaining power; their negotiated incentive payment:
  - p(ei,e) = ε min{π(ei,e), π(e*,e*)} + δε max{0, ei − e*}.
- Marginal benefit of bank equity for manager:
  - p1(e,e) = ε π1(e,e) for e < e* and p1(e,e) = δε π1(e,e) = δε for e ≥ e*.
- Managers' payoff relatively more convex than bank payoff π(e,e); managers benefit disproportionately from high realizations.
- Π(x) remains joint surplus; Π(x) and W(x) unchanged but chosen x differs when managers act.
- Managers choose x to maximize E[p(ei,e)] with ei = ÃA xi + 1−xi; optimality: E[(ÃA − 1) p1(e,e)] = 0.
- Proposition 5 (Agency Problems and Risk-Taking):
  - (i) Managers' optimal risk-taking exceeds xLF in benchmark if payoff asymmetry δ > 1.
  - (ii) Expected welfare of workers is a declining function of δ.
- Proof sketch:
  - p(ei,e) = ε π(ei,e) + ε(δ−1)(ei − e) I_{ei ≥ e*}.
  - Preferred choice xA satisfies P1(x) = Ε[(ÃA − 1) p1(ei,e)] ≥ 0 and P1(x) = ε Π1(x) + ε(δ−1) Ε[(ÃA − 1) I_{ei ≥ e*}].
  - Second term strictly positive ⇒ P1(x) > Π1(x) so xA > xLF.
  - xA strictly increasing in δ because derivative of P1(x) wrt δ is ε Ε[(ÃA − 1) I_{ei ≥ e*}] > 0; since xW < xLF < xA, increasing δ lowers worker welfare.

*Source: _wp13247 - 3.3    Determination of Period 0 Risk Allocation*

### 5.2    Financial Institutions with Market Power

### 5.2    Financial Institutions with Market Power

### Strategic setting and banker problem
- Economy with a finite number n of identical bankers, each of mass 1/n.
- A banker i internalizes that his period 0 risk-taking decision x_i affects aggregate bank capital e = 1/n e_i + (n−1)/n e_−i, where e_−i captures capital of other bankers.
- For a given e, bankers charge the competitive market interest rate R(e) in period 1.
- Bankers who partially internalize their effect on interest rates and are not subject to a leverage constraint solve
  - max_{k_i} {(F_k(k,1)−1) k_i}
  - where k = 1/n k_i + (n−1)/n k_−i
- First-order condition (solution) satisfies
  - 1/n F_{kk} k_i + F_k = 1
- Assuming symmetry, the equilibrium k is
  - k_{∗,n} = (1 − 1/n (1−α))^{1/(1−α)} k_∗
- Corresponding equity level achieved:
  - e_{∗,n} = (1 − φ 1 − 1/n (1−α)) k_{∗,n}
- If e < e_{∗,n}, the deposit constraint binds and bankers receive the same profits as before.

### Marginal valuation of bank capital under market power
- Marginal valuation of bank capital for banker i when there are n banks:
  - π_{i,n}^1(e_i,e_−i) = { 1/n π′(e) + (n−1)/n π_i^1(e_i,e) for e < e_{∗,n}; 1 for e ≥ e_{∗,n} }
- Ordering: π′ < π_{i,n}^1 < π_i^1.
- Alternative representation:
  - π_{i,n}^1(e_i,e) = π_i^1 + 1/n (π′ − π_i^1)
- Optimality condition for one of n large firms:
  - Π_{i,n}^1 = Π^1(x) + 1/n (Π′ − Π^1) = 0
  - Special cases:
    - n = 1 reduces to Π′ = 0 with solution x_B.
    - n → ∞ reduces to Π^1 = 0 with solution x_{LF} < x_B.

### Comparative statics in n and Proposition
- For x_n ∈ (x_{LF}, x_B), differentiating the optimality condition w.r.t. n:
  - d/dn Π_{i,n}^1 = −1/n^2 (Π′ − Π^1)
- Since Π^1 and Π′ are strictly decreasing in x and zero at x_{LF} and x_B respectively, in (x_{LF}, x_B) we have Π^1 < Π′, hence d/dn Π_{i,n}^1 < 0, so x_n decreases in n.
- Proposition 6:
  - The optimal risk allocation x_n of bankers is a declining function of the number n of banks in the market.
  - In particular, x_1 = x_B ≥ x_∞ = x_{LF}, with strict inequality unless corner solutions.

### Intuition and implication
- Bankers with market power internalize that additional equity benefits the rest of the economy (relieving the credit crunch), reducing their scarcity rents and lowering incentives for precautionary behavior.
- Non-competitive behavior by banks can manifest as socially excessive risk-taking.

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### 5.3    Financial Innovation

### Modeling financial innovation
- Financial innovation expands the set of risky assets available to bankers in period 0 (e.g., adds risky project with stochastic return Ã).
- Example setup: pre-innovation bankers can only access safe projects; financial innovation introduces Ã.
- Assume e_∗ < 1 so safe return in period 0 generates sufficient period 1 equity for bankers to intermediate the first-best level of capital.
- Pre-innovation equilibrium corresponds to x = 0, which maximizes worker welfare in the benchmark.

### Example 1 — Distributive effects
- Expanding investable projects to include Ã:
  - Increases banker welfare but reduces worker welfare.
- After innovation bankers allocate strictly positive fraction x_{LF} > 1 − e_∗ to the risky project and face risk of being financially constrained in low states.
- Reasoning:
  - Expected return E[Ã] > 1 gives first-order benefit over safe return.
  - Bankers perceive cost of being marginally constrained as second-order because π_1(e_i,e) is continuous at e_∗.
  - Workers unambiguously lose: they gain nothing from bank capital exceeding e_∗ and suffer first-order losses if bank capital falls below e_∗ (reducing investment and wages).
- Policy implication mentioned in text:
  - Restrictions on bank risk-taking (e.g., Volcker rule) may benefit workers as a second-best device to complete markets in this example.

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### 5.4    Bailouts

### Overview and endogenous nature of bailouts
- Bailouts are explicit transfers and have redistributive effects that can be subtle: ex post they can generate Pareto improvements by mitigating credit crunches; ex ante bailout expectations increase risk-taking and redistribute surplus from workers to bankers.
- Model of endogenous bailouts:
  - Workers optimally coordinate to provide bailouts (emergency lending or equity injections) to mitigate severe capital shortages because bank capital is essential for the real economy.
  - What matters is total resources (subsidies) given to bankers to relax binding financial constraints; focus on direct transfers for simplicity.

### Lemma 7 — Optimal bailout policy
- If aggregate bank capital in period 1 is below a threshold 0 < ˆe < e_∗, workers find it collectively optimal to provide lump-sum transfer t = ˆe − e to bankers.
- Threshold ˆe determined by w′(ˆe) = 1 or
  - ˆe = (1 − α) [1 − (1 − φ) α]^{α/(1−α)} e_∗
  - (expression as given in the source: ˆe= (1−α) [1−(1−φ)α]^{α/(1−α)} e∗)
- Proof sketch in text:
  - Worker welfare when providing transfer t is w(e+t) − t; interior optimum satisfies w′(e+t) = 1. Define ˆe = e + t.
  - w′(e) declines from w′(0) = 1/φ > 1 to w′(e_∗) = (1−α)/(1−φα) < 1, so ˆe uniquely defined.
  - If e < ˆe workers provide the transfer; if e ≥ ˆe optimal t = 0.

### Assumption for analysis
- Assumption 1: Parameters α, φ and A are such that ˆe < 1.
  - Guarantees no bailout needed if safe project invested; bailouts not desirable in states where risky project yields higher returns than safe project.

### Period 1 equilibrium with bailouts
- For e ≥ ˆe, w(e) and π(e) unchanged.
- For e < ˆe, welfare expressions modified:
  - w_{BL}(e) = (1 − α) F(e + t(e), 1) − t(e)
  - π_{BL}(e) = α F(ˆe, 1)
- Qualitative features (Figure 5 described in text):
  - t(e) > 0 but decreases to zero on [0, ˆe]; within this interval transfers stabilize banker profits at π(ˆe).
  - Worker welfare increases at slope 1 in this region because each additional dollar of bank capital reduces bailout by one dollar.
  - Bailouts make payoff functions less concave and bankers' payoffs locally convex.

### Marginal value of bank capital in bailout region (e < ˆe)
- 1 + t′(e) = 0 in this region, so:
  - w_{BL}′(e) = (1 − α) F_k [1 + t′(e)] − t′(e) = 1
  - π_{BL}′(e) = α F_k [1 + t′(e)] = 0
- Interpretation:
  - Marginal benefit for workers w_{BL}′(e) = 1 within bailout region since each additional dollar of bank equity reduces required bailout by a dollar.
  - For the last marginal unit of bailout, workers' benefit w′(ˆe) − 1 = 0; marginal benefit to bankers π′(ˆe) = α/(1−α) > 0, implying “bailout rents” to bankers.

### Period 0 risk-taking incentives under bailouts
- Optimal discretionary bailouts impose a ceiling on market interest rate R_{BL}(e) ≤ R(ˆe) = 1/(1−α) since they ensure k ≥ ˆe at all times; this mitigates precautionary incentives and increases bankers' optimal risk-taking (a “wealth effect”).
- If bailouts are conditional on individual bank capital e_i, bankers get additional incentives to increase risk-taking to raise expected bailout rents (a “substitution effect”).

### Bailout allocation specification and risk-taking effects
- Specification of bailout to banker i:
  - t(e_i, e; γ) = { 0 if e ≥ ˆe; ˆe − (1 − γ) e − γ e_i if e < ˆe }
  - γ ∈ [0,1] captures extent to which bailout depends on individual bank equity (γ = 0: aggregate-only; γ = 1: individual-only).
- Define x_{BL}(γ) as amount allocated to risky project in period 0 under bailouts.
- Proposition 8:
  - (i) Introducing bailout transfers increases period 0 risk-taking x_{BL}(γ) > x_{LF} for any γ ≠ 0.
  - (ii) Risk-taking x_{BL}(γ) is an increasing function of γ.
- Intuition:
  - (i) Bailouts reduce constraint tightness and returns in low states, lowering precautionary incentives.
  - (ii) When γ > 0 bankers internalize that losses increase their bailout by γ dollars, raising moral hazard and risk-taking.

### Redistributive effects and decomposition
- Welfare change from introducing bailouts decomposed into market completion and incentive effects:
  - ∆Π = [Π_{BL}(x_{BL}) − Π(x_{BL})] + [Π(x_{BL}) − Π(x_{LF})]
  - ∆W = [W_{BL}(x_{BL}) − W(x_{BL})] (market completion) + [W(x_{BL}) − W(x_{LF})] (incentive effect)
- Corollary 9:
  - (i) Bankers always benefit from introducing bailouts; expected profits Π_{BL}(γ) increase in γ.
  - (ii) Workers benefit from market completion effect but are hurt by incentive effects of bailouts if e_∗ < 1. The absolute magnitude of both effects increases with γ.
- Explanation:
  - First term (market completion) is positive for both agents because for given x bailouts generate Pareto improvement.
  - Second term (incentive effect) is positive for bankers because Π′(x) > 0 and negative for workers when e_∗ < 1 because W′(x) < 0.
  - Higher γ increases risk-taking and thus increases bailout frequency/magnitude, amplifying all terms.
- Graphical implication (Figure 6 described in text):
  - Introducing bailouts (γ = 0 example) shifts Pareto frontier outward at its left end and biases the shift toward bankers; introduction constitutes banker-biased technological change.
  - Risk-taking increases significantly even for γ = 0; banker welfare rises by ∆Π while worker welfare falls by ∆W.

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*Source: 5.2 Financial Institutions with Market Power; sections 5.3 Financial Innovation; 5.4 Bailouts from provided IMF content unit.*

### References

### _wp13247 - References

### Technical Appendix A — Proofs and Main Analytical Results
- Marginal functions Π′(x), Π1(xi,x), and W′(x) are shown to be strictly decreasing in x: Π′′(x) < 0, d/dx Π1(xi,x) < 0, and W′′(x) = [(1−α)/(1−φ)α] Π′′(x) < 0.
- Ordering of equilibrium risk-taking thresholds:
  - If xW < xB, part (ii) follows immediately.
  - At an interior solution the proof shows xLF < xB.
  - If e∗ ≤ 1 then xW = 1−e∗ and xLF > 1−e∗, therefore xW < xLF.
  - For interior solutions the proof establishes xW < xB by showing Π′(xW) > 0 since W′(xW) = 0 and Π′(x)−[(1−φ)α/(1−α)]W′(x) = ∫̄A 0 (Ã−1) dG(Ã) > 0.
- Proposition 8:
  - (i) With bailouts (parameter γ > 0) individual bankers choose xBL > xLF; ΠBL1(xLF,xLF;γ) > Π1(xLF,xLF) = 0.
  - (ii) Increasing γ raises the incentive for bankers to take risk: dΠBL1/dγ = −π1(ê,ê) ∫êA 0 (Ã−1) dG(Ã) > 0, given êA < 1.
- Notation and key conditions used in proofs are preserved exactly (e.g., Π1, Π′, Π′′, ΠBL, xLF, xB, xW, xBL, γ, ê, ˆA).

### A.2 Period 0 Production Function — Extension and Implications
- Periods t = 1 and 2 production: [Ãt xt + 1−xt] F(kt, `t) with symmetric Cobb-Douglas across periods.
- Bankers choose fraction xt allocated to risky projects; firms choose capital kt before shock Ãt is realized.
- Period 0: bankers supply initial equity e0 so k0 = e0. In period 1 productivity shock Ã1 realized; firms hire ` = 1 and produce Ã1 F(e0,1).
- Period 1 factor shares:
  - e1 = α [Ã1 x1 + 1−x] F(e0,1)
  - w1 = (1−α) [Ã1 x1 + 1−x] F(e0,1)
  - Equation (7) is law-of-motion for bank equity from period 0 to 1.
- Aggregate welfare functions as functions of period 0 risk-taking x1:
  - Π(x1) = E{π(e1)}
  - W(x1) = E{w1 + w(e1)}
- Key implications:
  - Period 1 wages depend positively on x1 because wages are (1−α) of output and E[Ã1] > 1.
  - Bankers do not internalize externalities on period 1 and period 2 wages.
  - Under interior solution and assuming φ > α and ˆA < 1, workers prefer less risk-taking than bankers (W′(xB1) < 0), so distributive conflict persists when accounting for production in both periods.
- Intuition: asymmetry of binding financial constraints — workers are hurt by constraints but do not benefit from extra bank capital; bankers benefit from extra capital via dividends — implies workers prefer less risk than bankers.

### A.3 Variants of Bailouts — Emergency Lending and Equity Injections
- Emergency lending: a loan dBL at interest rate rBL (often subsidized rBL ≤ 1) constitutes a transfer of (rBL − 1) dBL in net present value terms. Such lending is subject to constraint rd + rBL dBL ≤ φRk (constraint (1’)).
- Equity injections: provide additional bank equity q in exchange for dividend distribution D; transfer of q − D from workers to bankers in net present value. Subject to rd + D ≤ φRk (constraint (1”)).
- Equivalence result (Lemma 10):
  - Both emergency loans and equity injections matter only to the extent they provide a subsidy (outright transfer in expected value) to constrained bankers that relaxes their financial constraint.
  - Workers and bankers are indifferent between bailouts via subsidized emergency loans such that (1−rBL) dBL = t or via subsidized equity injections such that q − D = t.
  - Emergency lending and/or equity injections that do not represent a transfer in net present value terms are ineffective in this model.
- Mechanics and isomorphism:
  - Emergency loan (rBL, dBL) is isomorphic to lump-sum transfer t = (1−rBL) dBL; bankers intermediate k = e + (1−rBL)dBL / (1−φR(k)) = k(e + (1−rBL)dBL).
  - An unsubsidized emergency loan (rBL = 1) reduces private deposits by ∆d = −dBL and does not affect real capital k via constraint (1’); similarly q = D for equity injections crowds out private deposits by ∆d = −D.
  - A transfer of one dollar allows bankers to raise an additional φR/(1−φR) dollars of deposits and expand intermediation by 1/(1−φR) dollars in total.
  - If government cannot relax the commitment problem, repayments on emergency loans or dividends reduce bankers’ share in the same way as repaying depositors; only transfers that relax the constraint in net present value terms expand intermediation.
  - If government has superior enforcement to extract repayments/dividends, instruments with such capabilities are preferred.

### Model Parameterization (Appendix B) — Exact Parameter Values Used in Figures
- Figures 3 and 5 (period 1 welfare and marginal values of bank equity):
  - α 1/3
  - A 10
  - φ 0.5
- Figures 4 and 6 (period 0 welfare and equilibrium), with à distributed lognormally truncated to [0,2], mean μ and variance σ2:
  - α 1/5
  - A 8.1335
  - φ 0.5
  - μ 1.04
  - σ2 0.5
- Notes on parameter choices:
  - For illustrative purposes α was decreased and A chosen so as to leave e∗ unchanged compared to the previous set of figures.
  - Decrease in α raises bailout threshold ê so effects of bailouts on period 0 risk-taking decisions are more pronounced.

### Data Sources (Appendix C) — Figure 1 Data and Series Definitions
- Data source: Federal Reserve Bank of St. Louis FRED database (Federal Reserve Economic Data), unless otherwise noted.
- Panel 1:
  - Bank equity = difference between series "Total Liabilities and Equity" and "Total Liabilities" in "Financial Business" category from Federal Reserve Flow of Funds data (series FL - 79 - 41900 and 41940).
  - Market value of equity is used. Series deflated by "Gross Domestic Product: Implicit Price Deflator" (FRED series GDPDEF).
- Panel 2:
  - The spread on risky borrowing in Panel 4 is the difference between "Moody’s Seasoned Baa Corporate Bond Yield" (FRED series BAA) and "10-Year Treasury Constant Maturity Rate" (FRED series DGS10).
- Panel 3:
  - The real wage bill is "Compensation of employees, received" (FRED series W209RC1) deflated by "Gross Domestic Product: Implicit Price Deflator" (FRED series GDPDEF).

*Source: _wp13247 - References*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp13247.pdf_