## _wp10268

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---

### I. Framing and core mechanism
- Two major U.S. crises (Great Depression starting in 1929; Great Recession starting in 2007) were both preceded by:
  - sharp increases in income and wealth inequality;
  - sharp increases in debt-to-income ratios among lower- and middle-income households.
- Core endogenous crisis mechanism:
  - A shock raises relative bargaining power of investors (χ = 5% of population) and reduces that of workers (95% of population).
  - Investors use increased income to buy financial assets backed by loans to workers.
  - Workers borrow to limit consumption declines; workers’ debt-to-income ratios rise and financial fragility increases.
  - Increased saving at the top and increased borrowing at the bottom raises demand for financial intermediation; size of the financial sector (banks’ liabilities to GDP) increases.
  - Crisis outcome: large-scale household debt defaults and abrupt output contraction.
- Policy implication highlighted:
  - Redistribution policies that prevent excessive household indebtedness and reduce crisis-risk ex-ante can be preferable from a macro stabilization viewpoint to ex-post policies such as bailouts or debt restructurings.

### II. Stylized facts documented
- Income shares and household debt:
  - Top 5% share of total income (excluding capital gains):
    - rose from 24% in 1920 to 34% in 1928;
    - rose from 22% in 1983 to 34% in 2007.
  - Aggregate household debt-to-GNP/GDP:
    - almost doubled between 1920 and 1932;
    - almost doubled between 1983 and 2007, reaching higher levels in 2007 than in 1932.
- Wage and earnings changes, 1967–2005:
  - Real hourly wages cumulative changes:
    - top 10 percentile: +70%;
    - around the median: −5%;
    - bottom 10 percentile: around −25%.
  - Annual earnings exhibit even larger widening than hourly wages.
- Income vs. consumption inequality, 1980–2006:
  - Rise in income inequality has been much more pronounced than the rise in consumption inequality (ratio of disposable incomes and of non-durable consumption: top 10% vs. bottom 10%).
- Income mobility:
  - Short-term mobility (5-year horizon) and lifetime mobility stable or slightly worsening since the 1950s.
  - Surge in top earnings not due to increased mobility into the top 1%; probability of staying in top 1% after 1, 3 or 5 years shows no overall trend.
  - Variance increases in annual earnings largely reflect increases in permanent earnings variance.
- Wealth inequality and group debt-to-income ratios, 1983–2007:
  - 1983: top wealth group somewhat more indebted than bottom group (gap ~15 percentage points).
  - 2007: bottom group debt-to-income ratio ≈ 140% (about twice the top group), top group fluctuating around 70%.
  - Almost all aggregate increase in debt-to-income ratio comes from the bottom group.
- Size of the U.S. financial sector, 1980–2007:
  - Private credit by deposit banks and other financial institutions to GDP increased from 90% in 1981 to 210% in 2007.
  - Financial sector share of GDP almost doubled between 1981 and 2007 and reached 8% of U.S. GDP (Philippon (2010) measure).
- Mortgage debt and risk:
  - Majority of the bottom 95% debt increase came from mortgage debt.
  - Growth in share of subprime loans indicates increased riskiness.
  - Mortgage delinquency/default risk increased after 2007 to magnitudes unprecedented since the Great Depression.

### III. Model structure and equilibrium (key elements)
- Population groups:
  - Investors: share χ (calibrated at 5%), utility from consumption and wealth.
  - Workers: share 1−χ (calibrated at 95%), utility from consumption and inelastic labor supply.
- Investors’ utility and constraints (selected expressions preserved):
  - Lifetime utility:
    U_i^0 = E_0 ∑_{t=0}^∞ β_i^t [ (c_i_t − ̃c_i^min)^{(1−1/σ_i)}/(1−1/σ_i) + ξ_d log(d_t(1−(1−γ_ℓ)π_t)) + ξ_k log(k + k_t(1−(1−γ_k)π_t)) ].
  - Capital law of motion:
    k_t = (1−δ)∆^k_t k_{t−1} + I_t, where ∆^k_t = γ_k < 1 in a crisis, and 1 otherwise.
  - Budget constraint:
    d_t q_t = ∆^ℓ_t d_{t−1} + r^k_t ∆^k_t k_{t−1} − c_i_t − I_t.
- Workers’ utility and constraints:
  - Lifetime utility:
    U_w^0 = E_0 ∑_{t=0}^∞ β_w^t (c_w_t − ̃c_w^min)^{(1−1/σ_w)}/(1−1/σ_w).
  - Budget constraint:
    ℓ_t q_t = ∆^ℓ_t ℓ_{t−1} + c_w_t − w_t.
  - Default probability (logistic, increasing in leverage):
    π_t = exp(φ_0 + φ_1 (ℓ_t / (w_t − (1/q_{t−1}) ℓ_t))) / (1 + exp(φ_0 + φ_1 (ℓ_t / (w_t − (1/q_{t−1}) ℓ_t)))).
- Technology and bargaining:
  - Aggregate production:
    y_t = A (χ ∆^k_t k_{t−1})^α (1−χ)^{1−α}.
  - Wage via Nash bargaining:
    w_t = η_t f^h_t, where f^h_t = (1−α) y_t / (1−χ).
  - Bargaining power process:
    η_t = (1−ρ) η + ρη_{t−1} + e^η_t.
- Equilibrium conditions:
  - Goods market clearing:
    y_t = χ(c_i_t + I_t) + (1−χ) c_w_t.
  - Financial claims clearing:
    (1−χ) ℓ_t = χ d_t.
- Crisis events destroy portions of loan and capital stocks by factors (1−γ_ℓ) and (1−γ_k).

### IV. Calibration (annual) — baseline choices
- Preferences and steady-state targets:
  - σ_i = σ_w = 0.5.
  - Subsistence consumption level equals 50% of initial steady-state consumption.
  - Steady-state real interest rate ((1/q) − 1) fixed at 5% per annum.
  - β_w endogenized to meet the real interest rate; β_i = 0.9.
  - ξ_d chosen to impose initial steady-state loans-to-income ratio for workers of 64% (consistent with U.S. 1983).
  - ξ_k chosen to impose initial steady-state gross financial return to capital of 15% per annum.
  - Depreciation rate δ = 10% per annum.
  - Stone-Geary constant for physical capital: k = −3.
- Population shares:
  - χ = 5%; 1−χ = 95%.
- Technology and steady state:
  - Steady-state output normalized to one via parameter A.
  - Capital share α = 0.27, generating steady-state investment-to-GDP ratio of 18%.
  - Initial steady-state income share of investors equals 29.8%.

### V. Calibration of crisis probability and crisis sizes
- Logistic coefficients: φ_0 = −7.5 and φ_1 = 3.
- Baseline crisis probability: 0.38% at leverage of 64%; reaches almost 5% at leverage of 150%.
- Crisis sizes:
  - Capital destruction γ_k = 0.9 (capital destroyed equals 10% of pre-existing capital stock), generating an output collapse of around 2.7% given α = 0.27.
  - Alternative γ_k = 0.99 (capital destruction equals 1%).
  - Loan default parameter γ_ℓ = 0.9 (motivated by mortgage past due rates approaching 10% in 2007).

### VI. Solution methods and scenarios
- Global methods motivated by large discrete crisis events and high persistence of endogenous state variables.
- Monotone map method: functional iteration on discretized state space (three continuous state variables: capital, loans, bargaining power; one binary: crisis/no crisis).
- Perfect foresight method: implemented in TROLL using Newton-based stacking; shocks over first 10 years unanticipated.
- Standardized simulated impulse responses:
  - 50-year horizon; initial shock: decline in workers’ bargaining power from η = 1 over 10 years; crisis in year 30.
  - Comparison: perfect foresight = black solid line; monotone map = red dashed line in figures.

### VII. Baseline simulation outcomes (numerical)
- Shock: cumulative 7.5% decline in workers’ bargaining power over first 10 years (one half of one standard deviation per year); ρ = 0.96; crisis in year 30 with γ_ℓ = γ_k = 0.9.
- Key outcomes (monotone map and perfect foresight largely similar):
  - Real wage over initial decade collapses by close to 6%.
  - Return to capital increases by over 2 percentage points initially.
  - Workers’ consumption declines by around two thirds of the decline in wage income as workers borrow to smooth consumption.
  - Over 30 years prior to crisis:
    - Loans more than double; workers’ leverage from 64% to around 140%.
    - Crisis probability in year 30 exceeds 3%.
    - Loan interest rate up to 2 percentage points above initial value for much of initial period.
  - Investors’ outcomes:
    - Investors’ income share increases from <30% to over 35%.
    - Investors’ consumption eventually increases by over 20% prior to crisis.
    - Capital investment increases by over 15%; physical capital stock also increases.
    - Output increases by eventually close to 4%.
    - Loans increase by over 100%; investors’ consumption share increases by ~2 percentage points vs. 5 percentage points for income share.
  - Rates of return:
    - Initial gain in investors’ rate of return of >2 percentage points is later reduced by increased investment and gradual return of workers’ bargaining power.
    - By year 30 profitability declines below initial level; crisis temporarily raises return to capital via capital destruction.
  - Debt service and leverage at crisis:
    - Interest portion of debt service increases from ~3% to 6% of workers’ income at crisis.
    - Crisis reduces loans by 10% via default, but wage collapse and higher real interest rate on remaining debt raise debt servicing to 9% of income.
    - Net effect: leverage barely moves at crisis and can increase later; by year 50 leverage may be above pre-crisis level with very slow reduction thereafter.

### VIII. Uncertainty vs. perfect foresight — distinctions
- Under uncertainty (monotone map) relative to perfect foresight:
  - Investors briefly reduce consumption to boost capital investment and support faster capital stock growth.
  - Loans initially increase faster under uncertainty.
  - Long-run workers’ leverage under uncertainty ≈ 90% rather than 64%; about one third of pre-crisis leverage increase due to convergence to higher stochastic long-run value, two thirds due to realized shocks to η.
  - At 50-year horizon, capital and especially loans are lower under uncertainty due to convexity of crisis probability raising expected crisis probability at very high debt/capital levels.
  - Perfect foresight returns to leverage 64% in very long run; economy under uncertainty settles at leverage ≈ 90%.

### IX. Aggravating factors and sensitivity
- Baseline leverage and crisis probability:
  - Leverage ≈ 135% by crisis time (125% under uncertainty).
  - Crisis probability hovering around 3.5% for several decades (2% under uncertainty).
- Aggravating factor 1 — reduced productive investment by investors:
  - Change: set k̄ = −3^3 instead of k̄ = −30 (reduces capital accumulation by one third over first 30 years).
  - Consequences:
    - Output growth reduced accordingly.
    - Investors’ consumption share increases by 2 percentage points.
    - Leverage reaches ≈ 145% by crisis and keeps growing to 175% by year 50 under perfect foresight; stays near 135% under uncertainty.
    - Long-run crisis probabilities: almost 8% by year 50 under perfect foresight, 3% under uncertainty.
  - Interpretation: if additional investor income is consumed or invested in loans rather than productive capital ("financialization"), higher debt becomes less sustainable.
- Aggravating factor 2 — near-permanent decline in workers’ bargaining power:
  - Change: ρ = 0.99 (half-life = 80 years instead of 27 years).
  - Consequences:
    - Smaller increase in leverage and crisis probability by year 30, but leverage keeps increasing thereafter.
    - Crisis probability may not peak until 50 years after first crisis under uncertainty.
  - Interpretation: virtually no prospects of restoring workers’ earnings make high leverage and high crisis risk almost permanent.
- Aggravating factor 3 — higher subsistence consumption level:
  - Change: subsistence level ˜c_x_t = 0.8 * ˇc_x_t with moving average ˇc_x_t and ψ = 4.
  - Consequences (perfect foresight):
    - Households borrow more aggressively to avoid consumption drop.
    - Leverage reaches 155% at crisis and close to 170% around year 40.
    - Crisis probability peaks at 8%.
    - Over longer run, moving average adjusts and stabilizes leverage.
- Sensitivity to crisis probability calibration:
  - Even doubling perceived crisis probability around year 30 produces qualitatively identical results; quantitative changes are very small (a 2 percentage point increase in crisis probability at 10% default adds ~10–20 basis points to real interest rates).

### X. Deleveraging and policy-relevant scenarios
- Two forms of deleveraging in model:
  - Debt reduction (orderly debt reduction).
  - Increase in workers’ earnings (restoration of bargaining power) to allow gradual deleveraging.
- Orderly debt reduction scenario:
  - Change: γ_k = 0.99 (capital destruction = 1%) instead of 0.9; other baseline unchanged.
  - Consequences:
    - Debt reduction not accompanied by significant income reduction; real wage drops very little.
    - Leverage drops by 13.5 percentage points (vs. 3 percentage points in baseline).
  - Policy implication: minimizing spillovers from financial sector to real economy during debt restructuring is critical.
- Bailouts:
  - A bailout principally benefits creditors of financial institutions (investors in the model) by compensating them for loan losses.
  - Financing for bailouts comes from higher future general tax revenue used to service higher government debt, falling to a very significant extent on workers.
  - In an extended model with fiscal authorities, workers’ indebtedness should include present discounted value of future taxes; beneficial effects of debt default on leverage would then be mostly offset by negative effects of higher future taxes.
- Restoration of workers’ bargaining power (alternative):
  - Scenario: at period 30 workers’ bargaining power immediately and permanently returns to η = 1, sufficient to head off a crisis.
  - Immediate outcomes:
    - Real wage jumps to about 4% above its value in period 0.
    - Leverage drops by 8 percentage points on impact (both under perfect foresight and under uncertainty).
  - Medium/long run:
    - Under loan restructuring leverage and default probability resume upward trajectory over decades.
    - Under bargaining power restoration both leverage and default probability immediately go onto a declining path.
    - By year 50 leverage ≈ 20 percentage points lower under bargaining power solution than under loan restructuring.
  - Interpretation: permanent flow adjustment (higher worker incomes) is more successful for long-run sustainability than a stock adjustment (debt restructuring) unless the latter is extremely large.

### XI. Policy implications, challenges, and extensions
- Key policy implications:
  - Reducing income inequality could markedly reduce crisis likelihood.
  - Ex-ante redistribution or policies limiting excessive household indebtedness can lower crisis risk and may be preferable to ex-post bailouts or restructurings.
- Implementation challenges and tax policy considerations:
  - Downward pressure on wages driven by powerful international forces (e.g., competition from China).
  - Switching from labor to capital income taxes may drive investment abroad.
  - Taxes on economic rents (land, natural resources, financial sector rents) are less likely to drive investment relocation.
  - Strengthening workers’ bargaining power faces political and structural challenges to be weighed against risks of further financial and real crises.
- Suggested model extensions:
  - Add open-economy dimension with net foreign assets and foreign savings preferences.
  - Add contractionary technology and investment demand shocks to generate large persistent post-crisis investment declines.
  - Add shocks to workers’ labor supply to capture mechanisms like higher female labor force participation and longer hours.

### XII. Key conclusions
- Mechanism demonstrated: higher income advantage of high-income households → recycling of top income into loans to poorer households → temporary consumption maintenance, substantial loan growth, leverage growth, and elevated crisis probability.
- Debt restructuring limitations:
  - Unless loan defaults in a crisis are extremely large and accompanying real contraction very small, effect on leverage and on probability of further crisis is limited.
- Restoration of bargaining power:
  - Restoring poor and middle income households’ bargaining power can be very effective, producing sustained reduction in leverage and lower probability of further crisis.
- Open-economy implication:
  - Mechanism extends to foreign households, generating capital account surpluses in the borrowing country and current account deficits abroad, potentially explaining global current account imbalances triggered by increasing income inequality in surplus countries.

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp10268.pdf*

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

### _wp10268 - References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

### I. Introduction — framing and core mechanism
- Two major U.S. crises: the Great Depression starting in 1929 and the Great Recession starting in 2007 were both preceded by:
  - sharp increases in income and wealth inequality;
  - sharp increases in debt-to-income ratios among lower- and middle-income households.
- Core endogenous crisis mechanism (summarized):
  - A shock raises the relative bargaining power of investors (χ = 5% of population) and reduces that of workers (95% of population).
  - Investors use increased income to buy financial assets backed by loans to workers.
  - Workers borrow to limit consumption declines; workers’ debt-to-income ratios rise and financial fragility increases.
  - Increased saving at the top and increased borrowing at the bottom raises demand for financial intermediation; size of the financial sector (banks’ liabilities to GDP) increases.
  - Crisis outcome: large-scale household debt defaults and abrupt output contraction (as in 2007).
- Policy implication highlighted:
  - Redistribution policies that prevent excessive household indebtedness and reduce crisis-risk ex-ante can be preferable from a macro stabilization viewpoint to ex-post policies such as bailouts or debt restructurings.

### II. Stylized facts documented
- Income inequality and household debt, 1920s vs. 1980s–2007:
  - Top 5% share of total income (excluding capital gains):
    - rose from 24% in 1920 to 34% in 1928;
    - rose from 22% in 1983 to 34% in 2007.
  - Aggregate household debt-to-GNP/GDP:
    - almost doubled between 1920 and 1932;
    - almost doubled between 1983 and 2007, reaching higher levels in 2007 than in 1932.
- Wage and earnings changes, 1967–2005 (Heathcote, Perri and Violante (2010)):
  - Real hourly wages cumulative changes:
    - top 10 percentile: +70%;
    - around the median: −5%;
    - bottom 10 percentile: around −25%.
  - Annual earnings exhibit even larger widening than hourly wages.
- Income vs. consumption inequality, 1980–2006:
  - Rise in income inequality has been much more pronounced than the rise in consumption inequality (ratio of disposable incomes and of non-durable consumption: top 10% vs. bottom 10%).
- Income mobility:
  - Kopczuk, Saez and Song (2010): short-term mobility (5-year horizon) and lifetime mobility have been stable or slightly worsening since the 1950s.
  - Surge in top earnings is not due to increased mobility into the top 1% (probability of staying in top 1% after 1, 3 or 5 years shows no overall trend).
  - Variance increases in annual earnings largely reflect increases in permanent earnings variance, with modest increases in transitory variance (Figure 4).
  - These findings support modeling choice of two income groups with essentially fixed memberships.
- Wealth inequality and group debt-to-income ratios, 1983–2007 (ranked by wealth):
  - 1983: top wealth group somewhat more indebted than bottom group (gap ~15 percentage points).
  - 2007: bottom group debt-to-income ratio ≈ 140% (about twice the top group), top group fluctuating around 70%.
  - Almost all aggregate increase in debt-to-income ratio comes from the bottom group.
- Size of the U.S. financial sector, 1980–2007:
  - Private credit by deposit banks and other financial institutions to GDP:
    - increased from 90% in 1981 to 210% in 2007.
  - Philippon (2010) measure: financial sector share of GDP almost doubled between 1981 and 2007 and reached 8% of U.S. GDP.
- Mortgage debt and risk, pre- and post-2007:
  - Majority of the bottom 95% debt increase came from mortgage debt.
  - Growth in share of subprime loans indicates increased riskiness.
  - Mortgage delinquency/default risk increased after 2007 to magnitudes unprecedented since the Great Depression.
  - Default probabilities increasing with leverage and default rates at observed magnitudes are key model ingredients.

### III. The model — agents, technology, equilibrium
- Population groups:
  - Investors: share χ (calibrated at 5%), derive utility from consumption and wealth.
  - Workers: share 1−χ (calibrated at 95%), derive utility from consumption and supply inelastic labor.
- Preferences and utility specifications:
  - Investors:
    - Consumption utility: CRRA with intertemporal elasticity σ_i and subsistence level ̃c_i^min.
    - Wealth in utility captures “capitalist spirit”; wealth forms: physical capital k_t and financial deposits d_t.
    - Utility from deposits: log-form adjusted for expected losses from crisis probability π_t.
    - Utility from capital: Stone-Geary form with constant k.
    - Lifetime utility (equation (1)):
      U_i^0 = E_0 ∑_{t=0}^∞ β_i^t [ (c_i_t − ̃c_i^min)^{(1−1/σ_i)}/(1−1/σ_i) + ξ_d log(d_t(1−(1−γ_ℓ)π_t)) + ξ_k log(k + k_t(1−(1−γ_k)π_t)) ].
  - Investors’ capital law of motion (equation (2)):
    k_t = (1−δ)∆^k_t k_{t−1} + I_t,
    where ∆^k_t = γ_k < 1 in a crisis, and 1 otherwise.
  - Investors’ budget constraint (equation (3)):
    d_t q_t = ∆^ℓ_t d_{t−1} + r^k_t ∆^k_t k_{t−1} − c_i_t − I_t.
  - First-order conditions for investors (equations (4)–(6)) specify Euler conditions for consumption, deposits, and capital.
  - Workers:
    - Lifetime utility (equation (7)):
      U_w^0 = E_0 ∑_{t=0}^∞ β_w^t (c_w_t − ̃c_w^min)^{(1−1/σ_w)}/(1−1/σ_w).
    - Budget constraint (equation (8)):
      ℓ_t q_t = ∆^ℓ_t ℓ_{t−1} + c_w_t − w_t.
    - Workers default on loans with probability π_t that is increasing in leverage via a logistic function (equation (9)):
      π_t = exp(φ_0 + φ_1 (ℓ_t / (w_t − (1/q_{t−1}) ℓ_t))) / (1 + exp(φ_0 + φ_1 (ℓ_t / (w_t − (1/q_{t−1}) ℓ_t)))).
    - Optimality conditions for workers (equations (10)–(11)) give consumption and loan Euler conditions.
- Technology and bargaining:
  - Aggregate production (equation (12)):
    y_t = A (χ ∆^k_t k_{t−1})^α (1−χ)^{1−α}.
  - Wage determined by Nash bargaining with workers’ bargaining power η_t:
    - First-order condition (equation (15)): w_t = η_t f^h_t, where f^h_t = (1−α) y_t / (1−χ).
  - Workers’ bargaining power process (equation (16)):
    η_t = (1−ρ) η + ρη_{t−1} + e^η_t.
- Crisis probability and linkage to leverage:
  - Crisis probability π_t bounded in [0,1] via logistic function, convex in leverage.
  - Crisis events imply partial destruction of loan and capital stocks: factors (1−γ_ℓ) and (1−γ_k).

- Equilibrium conditions:
  - Goods market clearing (equation (18)):
    y_t = χ(c_i_t + I_t) + (1−χ) c_w_t.
  - Financial claims clearing (equation (19)):
    (1−χ) ℓ_t = χ d_t.

### IV. Calibration and parameter choices (annual frequency)
- Preferences and baseline steady-state targets:
  - Intertemporal elasticities: σ_i = σ_w = 0.5.
  - Subsistence consumption level: equals 50% of initial steady-state consumption.
  - Steady-state real interest rate ((1/q) − 1) fixed at 5% per annum.
  - Time preference:
    - β_w is endogenized to meet the real interest rate;
    - β_i = 0.9 (lower than β_w because of capitalist spirit terms).
  - Utility weight on financial wealth ξ_d: chosen to impose initial steady-state loans-to-income ratio for workers of 64% (consistent with U.S. 1983).
  - Utility weight on physical capital ξ_k: chosen to impose initial steady-state gross financial return to capital of 15% per annum (sum of real interest rate and depreciation).
  - Depreciation rate δ = 10% per annum.
  - Stone-Geary constant for physical capital: k = −3 (affects elasticity of capital response to bargaining power shocks).
- Population shares and calibration:
  - Investors share χ calibrated at 5%; workers share 1−χ = 95%.

### V. Model uses and implications
- The model is calibrated to broadly replicate documented stylized facts:
  - Rising top income shares, increasing debt-to-income ratios concentrated among the bottom group, expansion of the financial sector, rise in mortgage and subprime lending, and sharply higher default probabilities when leverage is high.
- Mechanisms highlighted:
  - Between-group inequality (top 5% vs. bottom 95%) can generate increased demand for financial intermediation as top wealth flows into assets backed by bottom borrowing.
  - Leverage-dependent crisis probability creates a nonlinear build-up of financial fragility that can culminate in a crisis with both financial and real losses.
- Policy relevance:
  - Ex-ante redistribution or policies limiting excessive household indebtedness can lower crisis risk and may be preferable to ex-post debt restructurings or bailouts in terms of macroeconomic stabilization.

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp10268.pdf*

### 0. We will experiment with alternative values for

### _wp10268 - 0. We will experiment with alternative values for

### Calibration and baseline parameter choices
- Technology and steady state:
  - Steady-state output normalized to one through parameter A.
  - Capital share parameter α =   0.27, generating a steady-state investment-to-GDP ratio of 18%.
  - Initial steady-state income share of investors equals 29.8%.
  - United States income share: 22% in the early 1980s and 34% in recent times (as mentioned in Section II).
- Bargaining power and shock processes:
  - Mean bargaining power η =  1 (replicates competitive outcome).
  - Standard deviation of bargaining power shocks σ_η = 0.015.
  - Also consider perfect foresight case σ_η = 0 to infer implications of intermediate σ_η.
- Crisis probability function (logistic):
  - Coefficients φ_0 = −7.5 and φ_1 = 3.
  - Baseline crisis probability: 0.38% at a leverage of 64%; reaches almost 5% at a leverage of 150%.
  - Range compared with literature: Barro (2006) 1%-2.5%; Rancière, Tornell and Westermann (2008) estimate 4% for 1980-2000.
- Crisis sizes (capital and loan collapses):
  - Capital destruction parameter γ_k = 0.9 (interpreted as capital destroyed equals 10% of pre-existing capital stock), generating an output collapse of around 2.7% given α =   0.27.
  - Alternative scenario: γ_k = 0.99 (capital destruction equals 1%).
  - Loan default parameter γ_ℓ = 0.9 (motivated by mortgage past due rates approaching 10% in the 2007 crisis).
- Other calibration notes:
  - The model cannot generate large output collapses via unemployment increases.
  - Parameter choices intended to be consistent with IMF (2009) average output reductions of 3.4% associated with major financial crises that coincided with real crises.

### Solution methods
- Motivation for global methods:
  - Presence of large, discrete crisis events producing jumps in state variables up to 10%.
  - High persistence of endogenous state variables (capital and loans) combined with large bargaining power shocks.
- Monotone map method (functional iteration on discretized state space):
  - Uses Coleman (1991) monotone map method.
  - Model state variables: three continuous (capital, loans, bargaining power) and one binary (crisis/no crisis).
  - Initial conjectures from DYNARE first-order approximation with small σ_η and narrow grid; then increase σ_η and grid width sequentially.
  - Numerical integration used to compute expectations.
  - Evidence of local uniqueness: perturb converged decision rules and check convergence back.
- Perfect foresight method:
  - Implemented in TROLL using a Newton-based stacking algorithm.
  - Bargaining power shocks over first 10 years are unanticipated; subsequent evolution expected with certainty.
  - Simulation approach: simulate infinite horizon for each year sequentially through year 10; no further shocks expected after year 10; final infinite horizon simulation at crisis time (period 30) taking given γ_k k_29 and γ_ℓ ℓ_29.
- Simulated impulse responses:
  - Present 50-year impulse responses for a standardized realization: initial decline in workers’ bargaining power from η =  1 over 10 years, gradual return to η =  1, crisis event in year 30.
  - Sensitivity analysis varies: size of decline in bargaining power over first 10 years, speed of reversal after year 10, size of crisis event, perceived probability of a crisis, elasticity of capital accumulation w.r.t. bargaining power shocks, and form (fixed or variable) of subsistence consumption.

### Simulated scenarios — presentation conventions
- Figures 10-15 show baseline and alternatives; perfect foresight = black solid line, monotone map = red dashed line.
- Horizontal axis: time with shock in year 1 and final period year 50.
- Simulations initiated at deterministic steady-state vector.
- Vertical axis conventions:
  - Percent deviations from initial deterministic steady state for real stock and flow variables.
  - Percentage point deviations for rates of return, leverage, crisis probability, interest expense to income ratio, income and consumption shares of investors.
  - Simple ratios for relative per capita income and consumption of investors and workers.

### Baseline scenario — numerical outcomes and dynamics
- Shock specification:
  - Cumulative 7.5% decline in workers’ bargaining power over first 10 years (corresponds to a shock of one half of one standard deviation in each year).
  - Autoregressive parameter ρ =  0.96 (very slow reversal).
  - Crisis in year 30 with γ_ℓ = γ_k = 0.9.
- Key simulated outcomes (monotone map and perfect foresight largely similar):
  - Real wage over initial decade collapses by close to 6%.
  - Return to capital increases by over 2 percentage points initially.
  - Workers’ consumption declines by around two thirds of the decline in wage income (workers borrow shortfall from investors).
  - Over 30 years prior to crisis:
    - Loans more than double; workers’ leverage from 64% to around 140%.
    - Crisis probability in year 30 exceeds 3%.
    - Loan interest rate up to 2 percentage points above initial value for much of initial period.
  - Investors’ income and allocation:
    - Investors’ share of the economy’s income increases from initially less than 30% to over 35%.
    - Investors’ consumption eventually increases by over 20% prior to crisis.
    - Capital investment increases by over 15%; physical capital stock also increases.
    - Output increases by eventually close to 4%.
    - Loans increase by over 100%; investors’ consumption share increases by only around 2 percentage points versus 5 percentage points for their income share.
  - Rates of return dynamics:
    - Initial gain in investors’ rate of return of more than 2 percentage points is later reduced by increased investment (lower marginal product of capital) and gradual return of workers’ bargaining power.
    - By year 30 profitability declines below initial level.
    - Crisis in year 30 temporarily raises return to capital via capital destruction, but respite is temporary given ongoing recovery in workers’ bargaining power.
  - Debt service and leverage at crisis:
    - Interest portion of debt service increases from initially around 3% to 6% of workers’ income at crisis.
    - Crisis reduces loans by 10% via default, but wage collapse and higher real interest rate on remaining debt raise debt servicing to 9% of income.
    - Net effect: leverage barely moves at crisis and can increase later; by year 50 leverage may be above pre-crisis level with very slow reduction thereafter.
  - Sensitivity remark: result depends critically on relative sizes of loan default versus collapse in real economy; if crisis mainly affects loans, workers receive more relief.

### Uncertainty versus perfect foresight — distinctions and effects
- Behavioral differences under uncertainty (monotone map) relative to perfect foresight:
  - Investors briefly but sharply reduce consumption initially to boost capital investment and support faster capital stock growth.
  - Loans initially increase faster under uncertainty.
- Mechanism and long-run differences:
  - Under uncertainty, investors prefer higher capital and loan stocks to reduce consumption risk from volatile bargaining power.
  - Long-run workers’ leverage under uncertainty is around 90% rather than 64%; about one third of pre-crisis leverage increase is due to convergence to higher stochastic long-run value, two thirds due to realized shocks to η.
  - At 50-year horizon, capital and especially loans are noticeably lower under uncertainty due to convexity of crisis probability function raising expected crisis probability at very high debt/capital levels; investors less willing to be exposed.
  - Very long run: perfect foresight economy returns to leverage of 64% while economy under uncertainty settles at leverage around 90%.

### High leverage — aggravating factors and sensitivity
- Baseline leverage and crisis probability:
  - Leverage around 135% by crisis time (125% under uncertainty).
  - Crisis probability hovering around 3.5% for several decades (2% under uncertainty).
- Aggravating factor 1 — reduced productive investment by investors:
  - Change: set k̄ = −3^3 instead of k̄ = −30 (reduces capital accumulation by one third over first 30 years).
  - Consequences:
    - Output growth reduced accordingly.
    - Investors’ consumption share increases by 2 percentage points.
    - Leverage reaches around 145% by crisis and keeps growing to 175% by year 50 under perfect foresight; stays near 135% under uncertainty.
    - Crisis characterized by small increase in leverage and crisis probability.
    - Long-run crisis probabilities: almost 8% by year 50 under perfect foresight, 3% under uncertainty.
  - Interpretation: if additional investor income is consumed or invested in loans rather than productive capital, higher debt becomes less sustainable and system more unstable ("financialization").
- Aggravating factor 2 — near-permanent decline in workers’ bargaining power:
  - Change: ρ =  0.99 (half-life of bargaining power = 80 years instead of 27 years).
  - Initial smaller drop in η to 0.95 by year 10 (versus 0.925 in baseline).
  - Consequences:
    - Smaller increase in leverage and crisis probability by year 30, but leverage keeps increasing thereafter (including under uncertainty).
    - Crisis probability does not peak until 50 years after first crisis under uncertainty, and another 30 years later under perfect foresight.
  - Interpretation: virtually no prospects of restoring workers’ earnings make high leverage and high crisis risk almost permanent.
- Aggravating factor 3 — higher subsistence consumption level:
  - Change: subsistence level ˜c_x_t = 0.8 * ˇc_x_t with moving average ˇc_x_t defined by (21) and ψ = 4 (moving average reflects more than 90% of any permanent changes within approximately 4 years).
  - Computational note: adds state variables (model has five continuous and one binary state variable), so only perfect foresight simulation reported.
  - Consequences (perfect foresight):
    - Households borrow more aggressively to avoid drop in consumption.
    - Leverage reaches 155% at crisis time and close to 170% around year 40.
    - Crisis probability peaks at 8%.
    - Over longer run, subsistence level comes down as moving average adjusts, stabilizing leverage and avoiding near explosive debt.
- Sensitivity to alternative calibrations of crisis probability function:
  - Even when perceived probability of crisis around year 30 and beyond is twice baseline, qualitative results remain identical and quantitative changes are very small.
  - Rationale: a 2 percentage point increase in crisis probability at 10% default rate adds at most around 10 to 20 basis points to real interest rates — small relative to overall changes in real interest rates in scenarios.

### High leverage — solutions and policy-relevant scenarios
- Two forms of deleveraging in the model:
  - Debt reduction (ideally "orderly" debt reduction).
  - Increase in workers’ earnings to allow gradual deleveraging.
- Orderly debt reduction scenario (Figure 14):
  - Change: capital destruction at crisis time equals 1% (γ_k = 0.99) instead of 10% (γ_k = 0.9); all other baseline calibration unchanged.
  - Consequences:
    - Debt reduction not accompanied by significant income reduction; real wage drops very little.
    - Leverage drops by 13.5 percentage points (compared to 3 percentage points in baseline).
  - Policy implication: minimizing spillovers from financial sector to real economy during large-scale debt restructuring is critical to success of restructuring.
- Note on bailouts:
  - A financial sector bailout (such as recently performed in the United States) does not represent a debt restructuring in the sense of the orderly debt-reduction scenario discussed above.

*Italic line: Source: _wp10268 - 0. We will experiment with alternative values for*

### 14. A bailout principally benefits the creditors of financial institutions, in other words the

### 14. A bailout principally benefits the creditors of financial institutions, in other words the

### Effects of a bailout and fiscal incidence
- A bailout principally benefits the creditors of financial institutions, in other words the investors of the model, by compensating them for loan losses.
- The financing for such a bailout comes from higher future general tax revenue that will be used to service higher government debt, and will therefore fall to a very significant extent on workers.
- In an extended model including the fiscal authorities, the proper definition of workers’ indebtedness would include the present discounted value of future taxes.
- In such a world the beneficial effects of debt default on leverage would then be mostly offset by the negative effects of higher future taxes.

### Restoration of workers’ bargaining power (alternative to debt restructuring) — summary of Figure 15 results
- Scenario: at period 30 a program is implemented whereby workers’ bargaining power immediately and permanently returns to η= 1. The assumption is that this is sufficient to head off a crisis event.
- Immediate and on-impact outcomes:
  - Real wage jumps to about 4% above its value in period 0, due to the now much higher capital stock.
  - Leverage drops by 8 percentage points on impact (both under perfect foresight and under uncertainty).
  - The leverage decline here is driven by a higher income level rather than a lower, restructured loan stock.
- Medium to long run outcomes:
  - Following period 30, under a loan restructuring leverage and default probability resume an upward trajectory for several additional decades.
  - Under the bargaining power solution both leverage and default probability immediately go onto a declining path.
  - By year 50 leverage is around 20 percentage points lower under the bargaining power solution than under the loan restructuring solution.
- Interpretation:
  - For long-run sustainability a permanent flow adjustment, giving workers the means to repay their obligations over time, is much more successful than a stock adjustment, unless the latter is extremely large.

### Policy implications and challenges
- Reducing income inequality could be very useful to reduce the likelihood of future crises.
- Policy implementation difficulties and trade-offs:
  - Downward pressure on wages is driven by powerful international forces such as competition from China.
  - A switch from labor to capital income taxes might drive investment to other jurisdictions.
  - A switch from labor income taxes to taxes on economic rents, including on land, natural resources and financial sector rents, is not subject to the same problem.
  - Strengthening workers’ bargaining power faces difficulties that must be weighed against the potentially disastrous consequences of further deep financial and real crises if current trends continue.

### Model design choices and possible extensions (Further Discussion)
- The model is deliberately simple to identify the key transmission channel: higher income inequality → higher leverage → higher probability of crises, and for computational tractability.
- Suggested model extensions to improve predictive detail:
  - Add an open economy dimension with net foreign assets as an additional state variable and foreign savings preferences as an additional shock to better replicate consumption booms facilitated by foreign savings.
  - Add contractionary technology and investment demand shocks to generate large and persistent post-crisis reductions in investment (as observed in the United States after 2007).
  - Add a shock to workers’ labor supply to account for coping mechanisms households employed in response to higher income inequality, notably higher female labor force participation and longer hours (as emphasized by Reich (2010)), which helped replace some lost income and limit additional borrowing.

### Key conclusions
- The paper presents stylized facts and a theoretical framework exploring the nexus between increases in the income advantage of high income households, higher debt leverage among poor and middle income households, and vulnerability to financial crises.
- Mechanism: increases in the bargaining power of high income households lead to recycling of part of their additional income back to the rest of the population via loans, enabling temporary consumption maintenance but fueling loan growth, leverage growth, and elevated crisis probability.
- Debt restructuring effects:
  - Unless loan defaults in a crisis are extremely large by historical standards, and unless the accompanying real contraction is very small, the effect on leverage and on the probability of a further crisis is quite limited.
- Restoration of bargaining power:
  - Restoration of poor and middle income households’ bargaining power can be very effective, potentially leading to a sustained reduction in leverage and a lower probability of a further crisis.
- Open-economy implication:
  - The same mechanism would extend to foreign households in an open economy, generating capital account surpluses in the borrowing country and current account deficits abroad, providing a potential mechanism to explain global current account imbalances triggered by increasing income inequality in surplus countries.

*Source: Excerpt from the provided IMF chapter/section content.*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp10268.pdf_