## wp18194 ## Source details **Canonical URL:** [wp18194](https://www.imf.org/-/media/files/publications/wp/2018/wp18194.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2018/wp18194.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2018/wp18194.pdf.json) --- ### 2.1 Setup — Technology, Preferences, and Securities - Technology and markets - Production: y_t = n_t; w_t = 1. Housing supply fixed and normalized to 1; e_t denotes housing price. - Household preferences - Lifetime utility: E_0 ∑_{t=0}^∞ β^t [ ∫_0^1 v_it log(c_it) di + ̄η log(h_t) − 1/(1+ν) n_t^{1+ν} ]. - Taste shifter v_it ≥ 1 drawn i.i.d. Pareto: Pr(v_it ≤ v) = 1 − v^{−α}. Parameter constraint: α > 1. - Long-term perpetuity security - Coupon payments decay geometrically at rate γ. Seller issues one unit at price q_t repaying 1 in t+1, γ in t+2, γ^2 in t+3, ... - Coupon payments satisfy b_{t+1} = ∑_{i=0}^∞ γ^i l_{t−i} = l_t + γ b_t. - q_t a_t = household total asset holdings; q_t b_t = outstanding debt. - One-period realized return: R_{t+1} = (1 + γ q_{t+1}) / q_t. - Liquidity, budget and borrowing constraints - Liquidity constraint: c_it ≤ x_t. - Flow budget: x_t + e_t (h_{t+1} − h_t) = w_t n_t + q_t l_t − b_t + (1 + γ q_t) a_t. - Borrowing constraint: q_t l_t ≤ m_t e_t h_{t+1}; log m_t = (1−ρ_m) log ̄m + ρ_m log m_{t−1} + σ_m ε_{m,t}. - Credit limit restricts new loans only; maturity γ affects equilibrium through q_t b_{t+1} ≤ γ q_t γ b_t + m_t e_t h_{t+1}. - Savings and timing - Savings: q_t a_{t+1} = x_t − ∫_0^1 c_it di. - Period t timing: start with a_t, h_t, b_t; m_t realized; choose n_t, h_{t+1}, b_{t+1}, x_t; v_it realized; consumption purchases; end with pooling unspent funds to buy a_{t+1}. - First-order conditions and multipliers - Multipliers: μ_t (flow budget), ξ_t(v) (liquidity constraint), λ_t (borrowing constraint). - Transfers x_t: μ_t = β E_t [ μ_{t+1} R_{t+1} ] + ∫_0^1 ξ_t(v) dF(v). - Debt choice b_{t+1}: μ_t = β E_t [ μ_{t+1} R_{t+1} ] + λ_t − β γ E_t [ λ_{t+1} q_{t+1} / q_t ]. - Housing h_{t+1}: e_t μ_t − β E_t [ μ_{t+1} e_{t+1} ] = β E_t [ ̄η / h_{t+1} ] + λ_t m_t e_t. - Individual consumption (log utility): c_t(v) = min[ v / (β E_t μ_{t+1} R_{t+1}), x_t ]. - Equilibrium conditions and aggregates - Asset market clearing: a_{t+1} = b_{t+1}. - Labor supply: n_t^{ν} = μ_t w_t. - Define minimum-member consumption c_t = 1 / (β E_t μ_{t+1} R_{t+1}). - Relation (transfers vs spread): (1/(α−1)) (x_t / c_t)^{−α} = (β E_t [ μ_{t+1} / μ_t R_{t+1} ])^{−1} − 1 ≈ ρ_t − r_t, where ρ_t = − log β E_t ( μ_{t+1} / μ_t ), r_t = log E_t (R_{t+1}). - Total consumption: c_t / c_t = α/(α−1) [ 1 − (1/α) (x_t / c_t)^{1−α} ]. - Savings-to-consumption ratio: q_t a_{t+1} / c_t = [ α/(α−1) [(α−1) ∆_t]^{1/α} − ∆_t ]^{−1} −1, with ∆_t = (β E_t [ μ_{t+1} / μ_t R_{t+1} ])^{−1} − 1. - Unbounded taste shocks imply positive expected multiplier on liquidity constraint and λ_t > 0; uncertainty pushes interest rate below rate of time preference. - Steady state (constant m_t = ̄m) - Steady debt: q b = 1/(1−γ) ̄m e h. - Reduced-form Euler for housing: e h = (̄η / μ) [ 1 / ( (1 − ̄m/(1−βγ)) ρ + (̄m/(1−βγ)) r ) ]. - Intuition: house value discounted by weighted average of ρ and r; weight depends on ̄m. If ρ < r, increase in ̄m raises house prices. - Impulse responses to a credit shock - One-time negative ε_{m,t} leads to gradual household debt reduction; equilibrium interest rate falls after credit tightening, more so with high idiosyncratic uncertainty. - Employment barely falls on impact despite magnified consumption-leisure distortions. - Aggregate Euler: (1 + ∆_t) β E_t [ μ_{t+1} / μ_t R_{t+1} ] = 1. - Debt/income ratio expression: q_t b_{t+1} / y_t = [ α/(α−1) [(α−1) ∆_t]^{1/α} − ∆_t ]^{−1} −1. - Decline in debt limits raises ∆_t (average multiplier), especially when idiosyncratic volatility α^{−1} is high, acting like an endogenous natural rate shock. ### 3.2 Monetary Policy — Aggregation, Policy Rule, and ZLB - Aggregates and notation - Aggregate output: y_t = ∫_0^1 p_t(s) y_t(s) / p_t ds; p_t = ∫_0^1 p_t(s) ds. - Inflation: π_t = p_t / p_{t−1}. - Nominal rate: 1 + i_t = E_t R_{t+1}. - First-order approximation of pricing: log(π_t / ̄π) = ̄β E_t log(π_{t+1} / ̄π) + (1−λ_p)(1−λ_p ̄β) / λ_p (log(w_t) − log(z_t)) + θ_t. - Monetary policy rule - Taylor rule (ZLB not binding): 1 + i_t = (1 + i_{t−1})^{α_r} [ (1 + ̄ı) π_t^{α_π} (y_t / ̄y)^{α_y} exp(ε_t^r) ]^{1−α_r} (y_t / y_{t−1})^{α_x}. - ε_t^r monetary policy shock; α_r persistence; α_π, α_y, α_x responses to inflation, output gap, output growth. - When ZLB binds: i_t = 0. - The Fed can manipulate expectations of the path of interest rates (forward guidance-type effects); survey data used to discipline expected ZLB duration (2009–2015). - Monetary union and clearing - Islands are measure zero; policy does not react to island-specific disturbances. - Closed monetary union: ∫_0^1 a_{t+1}(s) ds = ∫_0^1 b_{t+1}(s) ds. - Calibration and steady-state numeric targets (preserved exactly) - Steady-state inflation ̄π = 2% per year. - Period = one quarter. - Frisch elasticity = 1/2, so ν = 2. - Debt-duration parameter γ = 0.985 (Macaulay duration ≈ 13 years). - ω = 0.7; σ = 0.5; κ = 4. - ψ chosen to ensure a wage markup of 5%. - Steady-state real interest rate targeted at 2% per year via ̄β. - ̄η_h chosen so housing/(annual) income = 2.5. - Steady-state LTV such that aggregate debt to housing = 0.29 → aggregate debt to (annual) income = 2.5×0.29 = 0.725. - Policy implications - ZLB presence crucial: example where a large gradual credit tightening causes ZLB to bind for five years and employment falls by 6.5 percent on impact; absent ZLB, Fed could set negative nominal rates and employment would fall very little. - Price rigidities alone insufficient for state-level credit shocks to affect aggregate employment—monetary policy constraint (ZLB) matters. - Fed’s ability to affect expectations about zero-rate duration materially affects macro outcomes. - Expectations and identification - Survey-based sequences of expected ZLB durations used as inputs (Blue Chip 2009–2010; NY Fed 2011–2015). - Island-level responses conditional on aggregate states are time-invariant; simplifies likelihood construction. ### 4.3 Parameter Estimates — Modal/Posteriors and Variance Decompositions - Pareto tail parameter α - Modal estimate α = 3.68, implying a 0.3% spread between subjective discount rate and interest rate at the annual frequency. - Posterior 10th and 90th percentiles: 3.2 and 4.1. - Identification driven by comovement of state-level household debt, consumption, and employment. - Nominal rigidities - Modal λw = 0.85; modal λp = 0.97. High λp required to match inflation stability around the Great Recession. - Simulation evidence (Table 3) - Data correlations: (∆ emplt, ∆debtt income t) = 0.17; (∆ const, ∆debtt income t) = 0.23. - Baseline model (α = 3.68) reproduces correlations: 0.19 and 0.36. - α = 2 → correlations 0.58 and 0.70; α = 10 → correlations 0.00 and 0.06. - Variance decompositions (Table 4) - Panel A (state-level, 1999–2015) - Employment: Credit/LTV 10.3%; Housing 0.2%; Productivity 8.8%; Leisure 13.5%; Discount 47.2%. - Consumption: Credit/LTV 18.0%; Housing 0.3%; Productivity 0.1%; Leisure 3.2%; Discount 78.5%. - Panel B (aggregate-level, 1984–2008) - Employment: Credit/LTV 0.2%; Housing 0.0%; Productivity 32.3%; Leisure 0.3%; Discount 24.4%; Fed Funds 32.5%; Markup 10.3%. - Consumption: Credit/LTV 0.2%; Housing 0.0%; Productivity 0.9%; Leisure 0.5%; Discount 35.7%; Fed Funds 47.6%; Markup 15.1%. - Interpretation - Estimated α implies limited aggregate amplification of credit shocks: credit shocks explain much cross-sectional variation but little aggregate movement without other shocks. - High λp reconciles stable aggregate inflation with regional volatility; state-only estimation implies lower λp. ### Robustness Exercises and Key Quantitative Conclusions - Robustness experiments (selected) - Lower idiosyncratic uncertainty (α from 3.68 to 5) - Credit shocks produce almost no relative movements in employment across states and virtually no aggregate employment drop even at ZLB. - Lower mortgage duration (γ from 0.985 to 0.965; implied duration ≈ 6 years) - Re-estimated α = 2.97; credit shocks still explain regional employment variation similar to baseline; slightly larger aggregate employment drop during recovery. - Lower elasticity of substitution between labor varieties (ψ from 21 to 5) - α falls to 3.2; model assigns greater role to credit shocks in state-level employment volatility and in delaying aggregate employment recovery, but credit shocks alone still generate a small initial aggregate employment drop. - Construction sector added - χ = 0.37, δ_h = 0.012 to match 4.9% construction employment share. - Credit shocks account for about 40% of state-level employment variation, 20–35% of consumption variation, and 70–80% of construction employment variation; aggregate implications unchanged. - Alternative Taylor rule estimates - Estimates similar to Justiniano and Primiceri (2008); model implications largely unchanged. - Main quantitative conclusions (Section 7) - Credit shocks account for about 40% of the differential rise and fall in state-level employment and consumption during the boom and bust. - Credit shocks alone generate a modest 0.8% drop in aggregate employment from 2008 to 2010 (about one-tenth of the observed 7% drop), despite the ZLB. - Gradual decline in household credit caused a gradual decline in the natural rate of interest; persistence of deleveraging increases importance of credit shocks over time. - By 2015, credit shocks account for about one half of the employment gap. - Tightening in household-level credit limits helps account for the slow employment recovery after the Great Recession. - Scope and limitations - Focus on household leverage mechanism; abstracts from other forces (constraints on financial intermediaries, demographic trends) which were captured by reduced-form shocks; interactions with these factors left for future research. *Source: wp18194 (IMF Working Paper extract)* ### 2.1 Setup ### 2.1 Setup ### Technology and preferences - Production: - Competitive firms produce output y_t with labor n_t subject to y_t = n_t. (equation (1)) - Competition pins down the real wage so w_t = 1. - Supply of housing is fixed and normalized to 1; e_t denotes the price of housing. Consumption good is the numeraire. - Household preferences (representative household): - Lifetime utility: E_0 ∑_{t=0}^∞ β^t [ ∫_0^1 v_it log(c_it) di + ̄η log(h_t) − 1/(1+ν) n_t^{1+ν} ]. (equation (2)) - v_it ≥ 1 is an i.i.d. taste shifter drawn from a Pareto distribution: Pr(v_it ≤ v) = F(v) = 1 − v^{−α}. (equation (3)) - Parameter constraints: α > 1 (lower α implies more uncertainty). ### Securities - Long-term perpetuity security: - Coupon payments decay geometrically at rate γ. - Seller issues one unit at price q_t in period t and repays 1 in t+1, γ in t+2, γ^2 in t+3, and so on. - Household holdings recorded by coupon payments b_t to be paid in period t. If l_t is amount sold in period t, then coupon payments satisfy b_{t+1} = ∑_{i=0}^∞ γ^i l_{t−i} = l_t + γ b_t. (equation (4)) - a_t denotes coupon payments household is entitled to receive in period t; b_t denotes amount it must repay. - q_t a_t = household total asset holdings (savings); q_t b_t = outstanding debt. - Long-term security one-period realized return: R_{t+1} = (1 + γ q_{t+1}) / q_t. (equation (10)) ### Liquidity, budget and borrowing constraints - Liquidity constraint (individual consumption limited by transfers): c_it ≤ x_t. (equation (5)) - x_t: funds each individual member has available for consumption when entering goods market. - Flow budget constraint: - x_t + e_t (h_{t+1} − h_t) = w_t n_t + q_t l_t − b_t + (1 + γ q_t) a_t. (equation (6)) - Interpretation: uses = transfers x_t and housing expenditure; resources = labor income w_t n_t, proceeds from issuing l_t at price q_t net of coupon payments b_t, and market value of a_t securities where each unit pays coupon 1 and can be sold at price γ q_t. - Borrowing constraint (limits new loan issuance): - q_t l_t ≤ m_t e_t h_{t+1}. (equation (7)) - m_t follows AR(1) and is the only source of aggregate uncertainty in the simplified model: - log m_t = (1−ρ_m) log ̄m + ρ_m log m_{t−1} + σ_m ε_{m,t}. (equation (8)) - Credit limit restricts issuance of new loans but does not force prepayment of old debt; maturity γ affects equilibrium through its impact on borrower’s overall credit limit q_t b_{t+1} ≤ γ q_t γ b_t + m_t e_t h_{t+1}. - Comparison to cash-in-advance models: - Household can access date-t labor income w_t n_t immediately. - Agents can save in interest-bearing assets at end of shopping period. - Distortion arises from household’s inability to borrow to smooth marginal utility across members. ### Savings and timing - Household savings (unspent funds of shoppers): - q_t a_{t+1} = x_t − ∫_0^1 c_it di. (equation (9)) - Timing summary (period t): - Begin with a_t, h_t, b_t. m_t realized (credit limit). - Household chooses n_t, h_{t+1}, b_{t+1}, x_t. - Individual v_it realized; members purchase c_it subject to liquidity constraint. - End of period: unspent funds pooled to purchase a_{t+1} units of security. ### Decision rules (first-order conditions and multipliers) - Multipliers: - μ_t: shadow value of wealth (multiplier on flow budget constraint (6)). - ξ_t(v): multiplier on individual liquidity constraint (5) for realization v. - λ_t: multiplier on borrowing constraint (7). - Choice of transfers x_t (chosen prior to realization of v; x_t not measurable with respect to v): - μ_t = β E_t [ μ_{t+1} R_{t+1} ] + ∫_0^1 ξ_t(v) dF(v). (equation (11)) - Interpretation: μ_t = discounted expected value of unspent funds used to buy long-term assets plus expected liquidity service (expected multiplier on liquidity constraint). - Choice of debt b_{t+1}: - μ_t = β E_t [ μ_{t+1} R_{t+1} ] + λ_t − β γ E_t [ λ_{t+1} q_{t+1} / q_t ]. (equation (12)) - Borrowing benefits (μ_t) versus cost (β E_t μ_{t+1} R_{t+1}); borrowing tightens current constraint λ_t but relaxes next period’s constraint because credit limit applies to new debt. - Choice of housing h_{t+1}: - e_t μ_t − β E_t [ μ_{t+1} e_{t+1} ] = β E_t [ ̄η / h_{t+1} ] + λ_t m_t e_t. (equation (13)) - LHS: user cost (purchase price minus discounted next-period selling price); RHS: marginal utility of housing services plus collateral value of housing λ_t m_t e_t. - Individual member consumption with logarithmic preferences: - c_t(v) = min[ v / (β E_t μ_{t+1} R_{t+1}), x_t ]. (equation (14)) ### Equilibrium conditions - Asset market clearing: a_{t+1} = b_{t+1}. (equation (15)) - Labor supply condition: n_t^{ν} = μ_t w_t. (equation (16)) - Firm optimization implies w_t = 1 and housing stock fixed. ### Model mechanics and aggregates - Define minimum-member consumption: - c_t = 1 / (β E_t μ_{t+1} R_{t+1}). (equation (17)) (consumption of member with v = 1) - Relation linking transfers and spread between discount rate and interest rate: - (1/(α−1)) (x_t / c_t)^{−α} = (β E_t [ μ_{t+1} / μ_t R_{t+1} ])^{−1} − 1 ≈ ρ_t − r_t. (equation (18)) - ρ_t = − log β E_t ( μ_{t+1} / μ_t ) is household subjective discount rate. - r_t = log E_t (R_{t+1}) is the interest rate. - Interpretation: RHS is spread between discount rate and interest rate; LHS proportional to fraction of constrained members with v > x_t / c_t. Larger spread → transfers fall relative to consumption → more constrained members. - Total household consumption: - c_t / c_t = α/(α−1) [ 1 − (1/α) (x_t / c_t)^{1−α} ]. (equation (19)) - Lower spread between discount and interest rates → smaller fraction constrained → larger mean/min consumption ratio. - Savings-to-consumption ratio: - q_t a_{t+1} / c_t = [ α/(α−1) [(α−1) ∆_t]^{1/α} − ∆_t ]^{−1} −1, where ∆_t = (β E_t [ μ_{t+1} / μ_t R_{t+1} ])^{−1} − 1. (equations (20), (21)) - Ratio increases as ∆_t decreases and is steeper for higher ∆_t. - Borrowing constraint multipliers: - Because taste shocks are unbounded, expected multiplier on liquidity constraint is positive, implying λ_t > 0. Uncertainty pushes interest rate below rate of time preference generating desire to borrow. ### Steady state equilibrium interest rate (constant m_t = ̄m) - Debt in steady state proportional to house value: - q b = 1/(1−γ) ̄m e h. - Euler equation for housing in steady state (reduced form): - e h = (̄η / μ) [ 1 / ( (1 − ̄m/(1−βγ)) ρ + (̄m/(1−βγ)) r ) ]. (equation (22)) - Interpretation: house value = ̄η / μ discounted by weighted average of rate of time preference ρ and interest rate r, weight depends on loan-to-value ̄m. If ρ < r, increase in ̄m reduces effective discount rate and raises house prices. - Graphical intuition (Figure 3 described): - Intersection of upward-sloping savings curve and downward-sloping debt curve determines equilibrium interest rate. - Tightening debt limit reduces demand for debt → reduces interest rate; effect larger when idiosyncratic uncertainty is high. - With low idiosyncratic uncertainty, agents save less and equilibrium interest rate is higher; asset supply curve flatter so decline in debt limit produces smaller reduction in interest rate. ### Impulse response to a credit shock - One-time negative shock to credit ε_{m,t}: - Leads to gradual reduction in household debt due to long-term securities and credit limit applying to new loans only. - Equilibrium interest rate falls after a credit tightening, more so when idiosyncratic uncertainty is high. - Employment barely falls despite magnified consumption-leisure distortions (similar to cash-in-advance models). - Aggregate Euler equation representation: - (1 + ∆_t) β E_t [ μ_{t+1} / μ_t R_{t+1} ] = 1. (equation (23)) - Using asset market clearing, q_t b_{t+1} / y_t = debt / income = [ α/(α−1) [(α−1) ∆_t]^{1/α} − ∆_t ]^{−1} −1. - Decline in debt limits raises average multiplier on liquidity constraints ∆_t, more so when idiosyncratic volatility (α^{−1}) is high, acting like a natural rate shock. Natural rate is endogenous and responds to credit limits. *Source: wp18194 - 2.1 Setup (IMF working paper extract)* ### 3.2 Monetary Policy ### 3.2 Monetary Policy ### Aggregate definitions and notation - Total real output: Lety t = ∫_0^1 p_t(s) y_t(s) / p_t ds, where p_t = ∫_0^1 p_t(s) ds is the aggregate price index. - Inflation: π_t = p_t / p_{t−1}. - Expected nominal return on the long-term security (nominal interest rate): 1 + i_t = E_t R_{t+1} (Equation (41)). - Aggregation of individual pricing implies, up to a first-order approximation: - log(π_t / ̄π) = ̄β E_t log(π_{t+1} / ̄π) + (1−λ_p)(1−λ_p ̄β) / λ_p (log(w_t) − log(z_t)) + θ_t, - where θ_t is an AR(1) disturbance to firms’ desired markups, - ̄β is the steady state discount factor, - ̄π is the steady-state level of inflation. ### Monetary policy rule and the zero lower bound (ZLB) - Taylor-rule specification (when ZLB does not bind): - 1 + i_t = (1 + i_{t−1})^{α_r} [ (1 + ̄ı) π_t^{α_π} (y_t / ̄y)^{α_y} exp(ε_t^r) ]^{1−α_r} (y_t / y_{t−1})^{α_x}, - ε_t^r is a monetary policy shock. - α_r determines persistence of the nominal rate. - α_π, α_y, and α_x determine responses to inflation, output gap, and output growth respectively. - ̄ı is set to ensure a steady state inflation level of ̄π. - When the ZLB binds: - i_t = 0. - The interest rate can be at zero either because aggregate shocks cause the ZLB to bind or because the Fed commits to keeping i_t at 0 for a longer period than implied by the constraint. - The Fed is assumed able to manipulate expectations of the path of interest rates (as in Eggertsson and Woodford (2003) and Werning (2015)). - Survey data from the New York Federal Reserve is used to discipline the expected duration of the zero interest rate regime during the 2009 to 2015 period. ### Monetary union and market clearing - Individual islands are measure zero; monetary policy does not react to island-specific disturbances. - Closed monetary union constraint (aggregate savings equal aggregate debt): - ∫_0^1 a_{t+1}(s) ds = ∫_0^1 b_{t+1}(s) ds (Equation (42)). ### Calibration and steady-state numeric targets (preserved exactly) - Steady-state inflation ̄π is set equal to 2% per year. - The period is one quarter. - Frisch elasticity of labor supply assumed 1/2, so ν = 2. - Debt-duration parameter γ = 0.985, implying the Macaulay duration of debt in the model equals that of mortgage debt in the data, approximately 13 years. - Weight on non-traded goods in consumption basket ω = 0.7. - Elasticity of substitution between traded and non-traded goods σ = 0.5. - Elasticity of substitution between varieties of traded goods κ = 4. - ψ chosen to ensure a wage markup of 5%. - Steady-state real interest rate targeted at 2% per year (via choice of ̄β). - Steady state weight of housing in preferences ̄η_h chosen so that aggregate housing to (annual) income ratio = 2.5 (computed from the 2001 SCF). - Steady state LTV ratio chosen so aggregate debt to housing ratio = 0.29 (computed from the SCF). - Implication: aggregate debt to (annual) income ratio = 2.5×0.29 = 0.725. ### Key implications for policy and modeling - Monetary policy responses and the presence of the ZLB crucially affect how credit shocks translate into real activity: - Example impulse response: a large gradual credit tightening causes the ZLB to bind for five years, with employment falling by 6.5 percent on impact. Absent the ZLB, the Fed would reduce the nominal rate below zero and employment would fall very little. - Price rigidities alone are not sufficient for state-level credit shocks to affect aggregate employment — monetary policy must be constrained (e.g., by the ZLB) for large aggregate effects. - The Fed’s ability to affect expectations about the duration of zero rates matters for macro outcomes. ### Role of expectations and identification - The model uses survey-based sequences of expected ZLB durations (Blue Chip Financial Forecasts 2009–2010; New York Fed Survey of Primary Dealers 2011–2015) as inputs for estimation. - Individual islands take aggregate prices, including the interest rate, as given; island-level responses to idiosyncratic shocks are time-invariant conditional on aggregate states (used to simplify likelihood construction). *Source: wp18194 - 3.2 Monetary Policy* ### 4.3 Parameter Estimates ### 4.3 Parameter Estimates ### Modal and posterior estimates of structural parameters - The modal estimate of the Pareto tail parameter α is equal to 3.68, implying a 0.3% spread between the subjective discount rate and interest rate at the annual frequency. - The posterior distribution for α is relatively tight around its mode, with a 10th and 90th percentile of 3.2 and 4.1, respectively. - Identification of α largely comes from the comovement of state-level household debt, consumption and employment, which are highly correlated in the data. - If α is too high, fluctuations in credit have no consequences for real variables because an island’s asset holdings adjust one-for-one with credit, leaving net asset positions unchanged; the model then cannot replicate the observed correlation between credit and real variables and the likelihood is low. - If α is too low, the model implies too strong a relationship between household debt and consumption and employment, which also reduces the value of the likelihood. - Simulation evidence (Table 3): - Data correlations (changes in employment and household debt; changes in consumption and household debt): 0.17 and 0.23, respectively. - Baseline model reproduces these correlations reasonably well. - Imposing α = 2 increases these correlations to 0.58 and 0.7, respectively. - Simulating with α = 10 reduces these correlations to zero. - Nominal rigidities: - Modal estimate of wage stickiness λw is 0.85. - Modal estimate of price stickiness λp is 0.97. - High price stickiness is required to account for the stability of inflation around the Great Recession. - Estimating the model using state-level data alone implies a much lower degree of price stickiness (consistent with Beraja, Hurst and Ospina (2015)); implications of that alternative parameterization are explored in Robustness. ### Persistence and volatility of shocks; variance decompositions (state and aggregate) - Posterior estimates of persistence and volatility of state and aggregate shocks, and theoretical forecast error variance decompositions, are reported in the Appendix (used to interpret shock estimates). - Filtered shocks are used to calculate contributions of each shock to the variance of observed consumption and employment. - Panel A (state-level variance decomposition, deviations from aggregate counterparts): - State-level credit shocks account for about 10% of the variation of relative employment. - State-level credit shocks account for about 18% of the variation in relative household spending. - State-level productivity shocks account for about 29% of the variation in relative employment and a negligible fraction of consumption volatility. - Shocks to the disutility from work account for about 14% of the volatility of employment and 3% of the volatility of consumption. - Shocks to individual states’ discount rates account for about 50% of the volatility of relative employment and 80% of the volatility of relative consumption. - Shocks to the preference for housing have a negligible impact on state-level real variables. - Panel B (aggregate variance decomposition, pre-ZLB period 1984 to 2008): - Household credit shocks account for only 0.2% of the volatility of aggregate consumption and employment. - Productivity shocks account for about 30% of employment volatility and 1 percent of consumption volatility. - Discount rate, monetary policy and markup shocks each account for a substantial fraction of movements in consumption and employment. - Shocks to the disutility from work or preference for housing have little effect on aggregate consumption and employment. - Note: non-linearities make decomposition during ZLB years less straightforward to interpret. ### Interpretation and role of parameters in matching moments - The estimated α implies limited macroeconomic amplification of credit shocks in the aggregate because: - Household debt evolves gradually in the data; the estimated α implies a relatively low elasticity of the natural rate to changes in debt. - Thus, while credit shocks explain a large fraction of cross-sectional (state-level) variation, they explain little of aggregate movements in the natural rate and aggregate employment absent other shocks. - Sticky prices (high λp) are necessary to reconcile stable aggregate inflation with regional volatility; regional-only estimation implies lower nominal stickiness, suggesting prices respond more to large shocks than to small shocks (consistent with menu cost and rational inattention models). _italic: Source: wp18194 - 4.3 Parameter Estimates (PDF chapter)_ ### 0.72 during the recession), the slope coefficient is much smaller (0.26 during the boom and ### wp18194 - 0.72 during the recession), the slope coefficient is much smaller (0.26 during the boom and ### 6.2 Lower Degree of Idiosyncratic Uncertainty - Changed α from 3.68 (baseline) to 5 and re-estimated all other parameters. - Findings: - Credit shocks produce almost no relative movements in employment across states (Panel B of Figure 14). - Absent large idiosyncratic uncertainty, agents adjust the asset side of their balance sheet with little consequence for consumption and employment. - Credit shocks generate virtually no aggregate employment drop even at the ZLB (Panel B of Figure 15). - Interpretation: - Results indicate that the role of credit shocks at state-level and aggregate levels derives from the data richness used to estimate the model, not from modeling artefacts. ### 6.3 Lower Duration of Mortgage Contracts - Changed γ from 0.985 (baseline) to 0.965, implying mortgage duration of 6 years versus 13 years in baseline, and re-estimated the model. - Parameter estimates: - Degree of price and wage rigidity similar to baseline. - Estimate of α becomes 2.97 vs. 3.68 in baseline. - Findings: - Credit shocks explain almost as much of the variation in employment across regions as in the baseline (Panel C of Figure 14). - Volatility and persistence of shocks, especially credit shocks, adjust mechanically when γ is reduced to match household credit dynamics across regions. - Credit shocks account for a slightly larger drop in aggregate employment during the recovery (Panel C of Figure 15). ### 6.4 Lower Elasticity of Substitution Between Labor Varieties - Set ψ from 21 (baseline) to 5 and re-estimated all other parameters. - Parameter responses: - Estimation favors a greater degree of nominal wage and price stickiness to offset removal of the real rigidity. - Modal estimate of α falls to 3.2 from baseline 3.7. - Findings: - Model assigns a greater role to credit shocks in explaining state-level employment variation (Panel D of Figure 14). - Credit shocks play a larger role in delaying the aggregate employment recovery (Panel D of Figure 15). - Nonetheless, credit shocks alone still have a small role in generating the initial aggregate employment drop at recession onset. ### 6.5 Construction Sector - Introduced construction sector; added construction employment as observable. - Housing stock law: h_{t+1}(s) = (1−δ_h) h_t(s) + y^H_t(s). - Production for new housing: y^H_t(s) = z^H_t(s) (n^H_t(s))^χ. - Firm profit specification includes adjustment cost term w_t(s) ξ/2 (n^H_t(s)− ̄n^H)^2. - Calibration and estimation: - Set χ = 0.37 and δ_h = 0.012 to match 4.9% share of construction employment in total employment. - Estimated z^H_t(s) process and other parameters using original variables plus construction employment data. - Structural parameter estimates very similar to baseline (Panel E of Table 5). - Findings (Figure 16 and Panels): - Credit shocks alone account for about 40% of state-level employment variation. - Credit shocks account for about 20-35% of consumption variation. - Credit shocks account for about 70-80% of variation in construction employment. - Construction employment is much more volatile than non-construction employment but small in share, so aggregate implications are unchanged. - Model’s implications for aggregate employment dynamics remain unchanged (Panel E of Figure 15). ### 6.6 Alternative Estimates of the Taylor Rule - Baseline used Taylor rule parameters from Justiniano and Primiceri (2008) (pre-Great Recession). - Authors also estimated policy parameters using a longer sample inclusive of 2009 to 2015. - Findings: - Reported estimates in the Appendix are similar to Justiniano and Primiceri (2008). - Model implications are largely unchanged under the alternative Taylor rule estimates. ### 7 Conclusions - Model introduces household credit limits and precautionary savings driven by idiosyncratic uncertainty to study aggregate implications of regional comovement between household debt and employment/consumption. - Estimation approach: - Bayesian likelihood methods using state-level and aggregate data. - Novel methodology exploiting model structure to identify parameters from relative variation of state-level variables and U.S. aggregates. - Allows efficient likelihood computation with many regional variables and non-linear ZLB dynamics. - Key quantitative conclusions: - Credit shocks account for about 40% of the differential rise and fall in state-level employment and consumption during the boom and bust. - Credit shocks alone generate a modest 0.8% drop in aggregate employment from 2008 to 2010, about one-tenth of the observed 7% drop, despite the ZLB constraint. - The gradual decline of household credit in the data caused a gradual decline in the natural rate of interest. - Persistence of household deleveraging increases the importance of credit shocks over time: credit shocks account for about one half of the employment gap in 2015. - Tightening in household-level credit limits helps account for the slow recovery of employment after the Great Recession. - Scope and limitations: - Analysis focuses on the ‘household leverage’ mechanism and abstracts from other forces (constraints on financial intermediaries, demographic trends), which were captured in reduced-form shocks. - 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αy 0.05 Taylor coefficient output Justiniano and Primiceri (2008) - αx 0.21 Taylor coefficient output growth Justiniano and Primiceri (2008) - Parameters chosen to match steady-state target - −ln(β) 2.31% Annual discount rate 2% real rate - η 0.077 Weight on housing Housing-to-income ratio of 2.5 - ̄m 0.0044 Credit limit Debt-to-housing ratio of 0.29 - Table 2: Estimated Structural Parameters - Columns: Prior, Posterior - Parameter — Dist — Median — 10% — 90% — Mode — Median — 10% — 90% - αN 2.6 1.5 3.8 3.68 3.60 3.18 4.11 - λp B 0.5 0.2 0.8 0.97 0.96 0.94 0.98 - λw B 0.5 0.2 0.8 0.85 0.85 0.82 0.88 - Table 3: Moments from State-Level Simulations - Moment — Data — α= 3.68 — α= 2 — α= 10 - Correlation (∆ emplt, ∆debtt income t) 0.17 0.19 0.58 0.00 - Correlation (∆ const, ∆debtt income t) 0.23 0.36 0.70 0.06 - Table 4: Variance of Consumption and Employment Due to Each Filtered Shock, % - Variable — Shock — LTV — Housing — Productivity — Leisure — Discount — Fed Funds — Markup - A. State-level, 1999–2015 - Employment 10.3 0.2 8.8 13.5 47.2 −− - Consumption 18.0 0.3 0.1 3.2 78.5 −− - B. Aggregate-level, 1984–2008 - Employment 0.2 0.0 32.3 0.3 24.4 32.5 10.3 - Consumption 0.2 0.0 0.9 0.5 35.7 47.6 15.1 - Table 5: Estimated Structural Parameters: Robustness - A. Regional Data; B. α= 5; C. γ= 0.965; D. ψ= 5; E. Construction Model — parameter modes and 10%/90% intervals as listed - Example entries: - A. Regional Data — α Mode 3.60 10% 3.35 90% 4.27; λp 0.93 0.89 0.96; λw 0.58 0.46 0.66 - B. α= 5 — λp Mode 0.97 10% 0.94 90% 0.98; λw 0.86 0.83 0.89 - C. γ= 0.965 — α Mode 2.97 10% 2.69 90% 3.68; λp 0.96 0.94 0.98; λw 0.86 0.82 0.88 - D. ψ= 5 — α Mode 3.15 10% 2.95 90% 3.88; λp 0.97 0.94 0.98; λw 0.93 0.91 0.94 - E. Construction Model — α Mode 3.60 10% 2.98 90% 3.80; λp 0.97 0.94 0.98; λw 0.86 0.82 0.88 - Figures (captions as given) - Figure 1: State Debt, Employment, and Spending - Figure 2: Timing of the Model - Figure 3: Equilibrium Real Interest Rate in Steady State - Figure 4: Impulse Response to a Credit Tightening. Simple Model - Figure 5: Response to State-Specific Credit Shock. Modal Estimates - Figure 6: Response to State-Specific Credit Shock. Low Idiosyncratic Uncertainty (α= 5) - Figure 7: Response to State-Specific Credit Shock. Flexible Prices and Wages - Figure 8: Aggregate Impulse Response to Credit Shock - Figure 9: Dynamics of Sectoral Employment During the Recession - Figure 10: Dynamics of the Mortgage Rate - Figure 11: Effect of Credit Shocks on State Employment and Consumption - Figure 12: Effect of Credit Shocks Only on Aggregate Variables - Figure 13: Effect of Credit Shocks on Aggregate Employment - Figure 14: Effect of Credit Shocks on State Employment: Robustness - Figure 15: Effect of Credit Shocks on Aggregate Employment: Robustness - Figure 16: Effect of Credit Shocks on State Employment: Model with Construction *Content excerpted from "References" page of wp18194 (PDF).* --- _Source: https://www.imf.org/-/media/files/publications/wp/2018/wp18194.pdf_