When a Small Capital Change Becomes a Crisis
The Federal Reserve model's next puzzle is not why capital matters, but why its effects are discontinuous. This chapter examines how a convex bank payoff turns a gradual change in returns, volatility, or capital requirements into a switch between safe and risky lending. That distinction lets the reader separate a threshold-driven crisis from an ordinary credit fluctuation and understand what the paper's no-failure requirement is actually trying to hold constant.
The bank has already been given a reason to value risky exposure: limited liability protects equity holders from the worst losses, while deposit insurance keeps their funding relatively cheap. The missing piece is the shape of the choice. If a bank could smoothly adjust from safe loans to risky loans, a regulator might expect a small capital change to produce a small change in risk. In the model, the payoff is curved in a way that makes the extremes more attractive than the middle.
From smooth adjustment to a discrete regime
Let the bank choose the share of its loan portfolio exposed to the risky firm. A higher share lowers the expected return on the portfolio because the risky firm's additional shock has a negative mean. At the same time, it increases exposure to the volatility that creates favorable upside states. Limited liability means that sufficiently bad outcomes are cut off from the equity holder's payoff: once the bank is insolvent, the equity claim is worth zero rather than a further negative amount.
Those two forces create the bank's trade-off. More risky exposure makes the portfolio worse on average, but it also makes the limited-liability option more valuable. The bank participates in good realizations and is protected in sufficiently bad ones. In the paper's derivation, expected bank dividends are convex in the portfolio's risk share. Convexity matters because the value of an intermediate choice lies below the payoff implied by the two extremes.
The consequence is stark. Within the model's permitted range, banks choose either the minimum or the maximum risky share rather than an interior mix. The minimum-risk choice is called a safe bank; the maximum-risk choice is called a risky bank. The numerical implementation keeps those limits close to, but not exactly at, zero and one. This small technical allowance lets the model track both safe and risky loans, but it does not change the economic point: banks do not gradually diversify their way through the incentive boundary.
A mixed regime is possible only at a knife-edge condition. If the expected returns on safe and risky bank equity are exactly equal, households can hold equity in both types of banks, and the economy can contain a mixture. The authors allow for this case in the numerical solution, although their simulations do not find a situation in which safe and risky banks coexist in equilibrium. It is therefore useful to distinguish a possible equality point from the typical simulated outcomes on either side of it.
The three labels describe choices, not moral categories. A safe bank is one whose portfolio stays at the minimum permitted risky exposure; it does not fail in the model. A risky bank takes the maximum permitted exposure and can fail when the idiosyncratic shock to risky returns is bad enough to make net worth negative. The distinction also does not mean that every loan made by a risky bank is individually doomed. It means that the bank has chosen the portfolio with the greatest sensitivity to the protected upside of the risky project.
The switch is governed by expected dividends, not by a single observable statistic. In the safe state, the bank compares the return from safe lending with the off-equilibrium payoff it could obtain by becoming risky. In the risky state, the comparison runs in the other direction. Since a bank cannot short one type of loan to finance the other in the model, the relevant boundary appears through the constraints preventing negative loans. When risky lending would be more profitable, the safe-loan constraint binds; when safe lending is more profitable, the risky-loan constraint binds.
Several forces can move the comparison toward risky lending.
A lower expected return on safe assets. If an aggregate shock makes safe capital less productive, the expected return on a safe loan falls. The risky project does not need to become better in social terms. It only needs to look less unattractive relative to a safe project whose return has weakened. The bank's limited-liability bonus then becomes more important in its private calculation.
Higher risky-return volatility. An increase in the standard deviation of the risky firm's idiosyncratic return expands the favorable tail. The downside is still truncated by limited liability, while the upside becomes more valuable to the bank's residual claim. In the model, this can trigger risk-taking even if the broader business cycle barely moves.
A smaller penalty for risky lending. The risky firm's average idiosyncratic penalty is represented by a parameter whose lower value makes risky projects more appealing. This is a change in the project's expected return, not a change in the insurance contract, but it pushes the same bank comparison toward the risky regime.
A looser capital requirement. Lower capital permits more deposit funding and leaves less equity at risk. It reduces the bank's funding cost and weakens the amount of “skin in the game” that makes the equity holders internalize losses. The safe and risky expected-dividend comparison can therefore cross even without an unusually high credit-to-GDP ratio.
These triggers can occur together or offset one another. That is why the paper avoids treating a business-cycle label as a sufficient diagnosis. A recession may lower safe returns, but an expansion can also increase the capital stock enough to lower the marginal return on safe lending. A financial volatility shock can alter the bank's payoff without producing a large movement in GDP. The common object is the relative attractiveness of safe and risky lending.
The state diagram is not a claim that the economy literally pauses in a separate “boundary” regime. It marks the threshold at which the two portfolio choices exchange their advantage. The mixed state is the equality case; the usual movement is from safe to risky or back again. A capital rule that stays just on the safe side has a very different task from one that tries to fine-tune the bank's risk share inside the risky state, because the model offers no stable interior share for that rule to target.
The safe-side threshold and its asymmetric costs
Once a regime boundary exists, the policy question can be stated precisely: how much capital is enough to keep the bank from crossing it? The paper calls the answer the no-failure capital requirement. At each date, it sets capital at the lowest level that, given the realized shocks, prevents the economy from entering the risk-taking regime. “No-failure” describes the model's outcome, not a promise that regulators can observe the exact threshold in real time.
Why not set capital higher? Above the boundary, additional equity does not create a new safety benefit in this model if the bank is already choosing safe lending. It does, however, replace deposits. Because households value deposits for their liquidity services, that replacement lowers welfare. The marginal cost of excess capital is continuous: a little more equity means a little less deposit liquidity and a higher funding cost.
Why not set capital lower? Below the boundary, the immediate gain is also easy to identify. Banks can use more cheap deposits, and households can hold more of the liquid asset. But if the lower requirement makes risky lending the bank's preferred choice, the economy does not merely move a little farther along a smooth path. Risky firms produce less on average, some risky banks fail, equity is destroyed, and the government must cover insured deposits through taxes. Output and income fall, and the loss of productive capacity can be much larger than the incremental deposit benefit.
This difference explains the asymmetry between tightening and loosening. In a safe equilibrium, a tighter requirement mainly changes the funding mix. The paper's calibration finds that a one-percentage-point tightening has tiny effects on consumption, investment, and output, even though it produces sizable financial-sector effects and lowers welfare through reduced deposits. A symmetric one-percentage-point loosening can consume a modest static buffer and cross into the risky regime. Once it does, the output and welfare losses are disproportionate to the size of the policy change.
The asymmetry is endogenous, not an artifact of assigning different shocks to the two experiments. The capital shocks are equal in size and opposite in sign. What differs is the state of the bank's payoff. A tightening moves farther into the safe region, where the main cost is lost liquidity. A loosening can move through the boundary, activate limited liability's protected downside, and create a crisis. Linear models often produce impulse responses that look like mirror images because they approximate behavior around one regime. This model is designed to show what that approximation leaves out.
The effect also propagates beyond banks. When risky lending becomes prevalent, the risky firms' lower expected productivity reduces output and household income. Failed banks destroy equity. Deposit insurance prevents depositors from taking the direct loss, but the government must raise the taxes that fund the payout. Households therefore face the same event through several channels: less production, lower income, lower bank equity, and a fiscal cost. Consumption can be smoother than output because households adjust saving, but smoothing does not erase the loss; it changes how the loss is distributed over time.
Deposits can move sharply because they are both a funding source and a household asset. During a regime shift, households may adjust deposits in anticipation of future income losses and insurance payments. The paper notes that discontinuities are especially visible in deposits and total output, while consumption is smoother. That pattern is economically intelligible. Consumption-smoothing motives spread the effect of a discrete shock, but the underlying deposit services and productive capacity have still changed.
The threshold is calibrated rather than universal. In the paper's steady state, a capital requirement of 7.2 percent is just sufficient to remain in the safe regime. The baseline calibrated requirement is 7.8 percent, leaving a 0.6-percentage-point static buffer above that threshold. A grid search over steady-state requirements finds welfare peaking at the 7.2 percent safe-side boundary. At higher requirements, deposit balances and consumption decline monotonically, so welfare falls as the liquidity cost accumulates.
The risky side of the grid is more complicated. Within the risky regime, the interaction among output, investment, capital, consumption, and deposit liquidity is not monotonic. The paper identifies a local welfare peak around a capital requirement of roughly 5.2 percent before consumption's response changes direction. This does not overturn the safe-side result. It shows instead that once the economy has crossed into risky lending, raising capital can have several competing effects and the welfare landscape may contain local features. The paper's static conclusion is that the overall peak occurs at the safe boundary, not that every point in the risky region behaves monotonically.
That qualification helps define what the no-failure policy does and does not establish. It identifies a compelling local rule: do not pay for more equity than is needed to prevent the discrete crisis, but do not step below the boundary for a proportional increase in deposit liquidity. The steady-state grid provides evidence for global optimality within the static version of the model. The dynamic version is harder. The threshold moves with aggregate and firm-specific shocks, and determining it requires knowledge of the shocks and the structure that translates them into returns.
The authors test the dynamic benchmark against flexible linear rules later in the paper. Those tests make the no-failure rule look optimal within a specified class, but they do not make it operationally observable. A regulator who knew every relevant shock could adjust capital just enough to keep the safe choice attractive. A regulator who sees only delayed, noisy aggregates may have to choose between a static cushion and a rule whose apparent precision is not backed by reliable identification.
The central diagnosis is therefore complete. A small capital change becomes a crisis when it crosses an incentive boundary created by convex bank payoffs, not because the number itself is large. The safe threshold protects against the bank's ability to privatize upside and externalize downside. But the threshold is a moving object: a productivity shock, an investment boom, and a volatility increase can all move it differently. The next test is whether the right capital response can nevertheless be inferred from the direction of the economic cycle.