## _wp14129 — 1.1 Related literature; Sections 4.1–7 (excerpts) ## Source details **Canonical URL:** [_wp14129 — 1.1 Related literature; Sections 4.1–7 (excerpts)](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2014/_wp14129.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2014/_wp14129.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2014/_wp14129.pdf.json) --- ### Leverage, liquidity trap, and mechanism - Leverage in the household sector rose before 2008 and deleveraging after 2008 is linked to job losses (Mian and SuÖ (2012)). - Representative-agent liquidity-trap models emphasize a lower bound on the nominal rate because hoarding cash is an alternative to holding bonds; sticky inflation expectations can prevent the real rate from declining and create a demand-driven recession. - Empirical illustration: short-term nominal and real interest rates in the US appear constrained since December 2008 (Figure coverage: Q3 1981–Q4 2013). - Key model feature: - Two household types h ∈ {b, l} with measure 1/2. - Borrowers have weakly lower discount factor: β_b ≤ β_l. - Debt notation: d^h_t; borrowing constraint for t ≥ 1: d^h_{t+1} ≤ ε with ε > 0; unconstrained choice at date 0. - Lower bound on real rate: r_{t+1} ≥ r̲_{t+1}; baseline normalizes r̲_{t+1} = 0 for t ≥ 1. - Anticipated deleveraging at date 1 can trigger a liquidity-trap-induced, demand-driven recession if outstanding debt exceeds a threshold: - Threshold condition (eq. (6) / text): d_1 > d̄_1 = ε + c̃^l_1 − ē. - When d_1 > d̄_1: r_2 = 0, e_1 = c̃^l_1 + ε − d_1 < ē (eqs. (7)–(8)). ### Aggregate demand externality vs. pecuniary externalities - Two regimes depending on aggregate debt D1: - If D1 ∈ [ε; d̄_1): deleveraging reduces the real interest rate and creates pecuniary redistribution externalities (negative for lenders, positive for borrowers); with complete markets these net out and equilibrium is constrained efficient (Lemma 1, part (i)). - If D1 > d̄_1: liquidity trap binds (r_2 = 0), pecuniary channels are muted, net income e_1 declines with leverage (@e1/@D1 < 0) and aggregate demand externalities hurt all agents (Lemma 1, part (ii)). - Mechanism: borrowers have higher marginal propensity to consume (MPC) out of liquid wealth than lenders; deleveraging transfers liquid wealth from borrowers to lenders, reducing aggregate demand when the real rate cannot fall. ### Main normative result — restraining leverage ex-ante (Proposition 2) - Private decision-making can yield excessive leverage ex-ante when aggregate demand externalities are possible. - Constrained-planner characterization: - Constrained efficiency requires e0 = ē and either: - (i) D1 < d̄_1 and Euler equation (9) holds; or - (ii) D1 = d̄_1 and the distorted Euler inequality (eq. (13)) holds: - Φ_l [u0 − c^l_1] / [u0 − c^l_0] ≥ Φ_b [u0 − c^b_1] / [u0 − c^b_0]. - Implementation: a uniform debt limit d^h_1 ≤ d̄_1 combined with an appropriate ex-ante transfer T0 can implement the constrained-efficient allocation. - Interpretation: lowering date-0 debt when the economy risks a liquidity trap yields first-order welfare benefits via demand externalities; the planner internalizes the externality via a debt limit and transfers. ### Underinsurance and mandatory insurance (uncertainty, Proposition 4) - Two-state setup from date 1: s ∈ {H, L} with debt limits ε_{t+1;L} = ε and ε_{t+1;H} = 1. - Date-0 trading of Arrow securities; full-insurance condition across states (eq. (18)): - q_{1;H}/q_{1;L} = π^l_H u0(c^l_{1;H}) / π^l_L u0(c^l_{1;L}) = π^b_H u0(c^b_{1;H}) / π^b_L u0(c^b_{1;L}). - Proposition 4 (Underinsurance): - A constrained-efficient allocation ((c^h_0,n^h_0)_h; (D_{1;s})_s) with D_{1;L} ≥ ε satisfies e0 = ē and efficiency in substitution to state H, and either: - (i) D_{1;L} < d̄_1 and full-insurance eq. (18) holds; or - (ii) D_{1;L} = d̄_1 and the distorted insurance inequality holds (eq. (20)): - π^l_H u0(c^l_{1;H}) / π^l_L u0(c^l_{1;L}) ≥ π^b_H u0(c^b_{1;H}) / π^b_L u0(c^b_{1;L}). - Implementation: can be achieved by mandatory insurance requirement d^h_{1;L} ≤ D_{1;L} plus an ex-ante transfer T0. - Interpretation: borrowers buy too little insurance (underinsurance) because they fail to internalize the positive aggregate demand externalities of insurance; mandatory insurance is a justified policy instrument. ### MPC heterogeneity and magnitude of inefficiency (Section 4.3) - Extension: two borrower types differing in MPC at date 1; fraction β ∈ [0,1] are b_high with MPC = 1, remaining b_low have MPC = 1 − Φ (with Φ = β? in text). - Log-utility calibration: - MPC_l1 = MPC_b_low1 = 1 − Φ (eq. (14)). - MPC_b1 = β + (1 − β)(1 − Φ) (eq. (15)). - Comparative static (eq. (16)): - ∂e1/∂d1 = − [ MPC_b1 − MPC_l1 ] / [ 2 − MPC_b1 − MPC_l1 ]. - Implication: the decline in net income from higher leverage is increasing in MPC differences between borrowers and lenders; larger MPC gaps imply larger aggregate demand externalities. - Empirical note: Baker (2013) suggests a one standard deviation increase in household debt-to-asset ratio raises MPC by about 20% (~7 percentage points from baseline 37 percentage points), which can guide calibration for policy design. ### Fire-sale externalities and endogenous debt limits (appendix extensions) - Endogenizing debt limits via collateralized assets introduces fire-sale externalities: - Higher leverage lowers asset prices during deleveraging, reducing borrowers’ debt capacity and increasing distress. - Fire-sale externalities operate in the same direction as aggregate demand externalities and exacerbate recessions via feedback between output and asset prices. - Extensions show these channels amplify the desirability of ex-ante macroprudential constraints. ### Preventive monetary policies — changing inflation target and contractionary policy (Section 6) - Raising the inflation target: - Taylor-rule microfoundation: positive inflation target ϑ > 0 lowers the bound on the real rate: r_{t+1} = − ϑ/(1 + ϑ) for t ≥ 1 (text). - Consequence: greater leverage is required to trigger a demand-driven recession; raising the inflation target can reduce incidence of liquidity traps but must be weighed against costs of higher steady-state inflation. - Contractionary monetary policy (Section 6.2, Proposition 5): - Policymaker sets r1 > natural rate at date 0, causing e0 < ē. - Under conditions (u''(x) = −u(x) weakly decreasing; d0 sufficiently large; liquidity trap at date 1), Proposition 5 establishes: - e0′(r1) < 0 and d1′(r1) > 0. - Interpretation: raising r1 reduces current net income and can perversely increase equilibrium leverage d1, worsening subsequent deleveraging and recession. - Log-utility illustrative closed forms (eq. (22)) show partial-equilibrium substitution effects can be dominated by general-equilibrium income and wealth-transfer effects. - Policy message: interest-rate tightening is an inferior instrument for addressing excessive household leverage compared with macroprudential limits or insurance mandates. ### Separable preferences and robustness (A.5 and related) - Extension with separable preferences u(c) − v(n) (Appendix A.5) yields qualitatively similar results: - There exists a threshold d̄_1 such that r2 ≤ 0 only if d1 ≥ d̄_1. - For D1 > d̄_1 aggregate employment and output decline with leverage, generating aggregate demand externalities (eq. A.18). - Planner condition (A.19) yields an analogue of the distorted Euler inequality; constrained-efficient allocations can be implemented with debt limits plus transfers. - Difference from GHH baseline: with separable preferences planner may mitigate but not fully avoid the recession (D1 could be > d̄_1 in constrained-efficient allocations) because aggregate demand externalities depend on the labor wedge size. ### Proof sketches and implementation remarks (appendix summaries) - Proposition 1: thresholds on impatience and initial debt yield deleveraging-induced recession at date 1. - Lemma 1: signs of ∂V_h/∂D1 differ by regime (pecuniary vs. aggregate-demand). - Proposition 2 and 4: constrained-efficient allocations characterized and implementable by uniform debt limits and ex-ante transfers (or mandatory insurance for state L). - Proposition 3: under uncertainty, borrower impatience, indebtedness, or optimism can each trigger recession in state L. - Proposition 5: comparative statics showing contractionary policy can increase leverage use monotonicity and concavity arguments (Lemma 2). *Italic: Source — _wp14129 (excerpts from the provided PDF content).* ### 1.1 Related literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 ### 1.1 Related literature ### Leverage, liquidity trap, and the recent US recession - Leverage in the household sector rose dramatically before 2008 and was followed by deleveraging after 2008; county-level evidence links household deleveraging to much of the job losses between 2007 and 2009 (Mian and SuÖ (2012)). - Representative-agent liquidity-trap models (e.g., Hall (2011), Eggertsson and Krugman (2012), Guerrieri and Lorenzoni (2012)) emphasize that the nominal interest rate cannot fall below zero because hoarding cash is an alternative to holding bonds, producing a liquidity trap. - When inflation expectations are sticky, the lower bound on the nominal rate can prevent the real interest rate from declining, creating a demand-driven recession. - Empirical illustration: short-term nominal and real interest rates in the US appear constrained since December 2008; Figure 2 covers the period between the third quarter of 1981 and the fourth quarter of 2013. ### Scope for ex-ante macroprudential policy in debt markets - The paper analyzes ex-ante macroprudential policies in debt markets (e.g., debt limits, insurance requirements) as a response to episodes where tightening borrowing constraints leads to deleveraging and may trigger a liquidity trap. - Key modeling feature: a class of agents ("borrowers") endogenously accumulate leverage despite anticipating future tightening of borrowing constraints; if borrowers are sufficiently motivated (e.g., due to impatience), the economy can enter a liquidity trap and an anticipated demand-driven recession. ### Main normative result: restraining leverage ex-ante - Individual optimization leads borrowers to undertake excessive leverage from a social perspective in the run-up to a liquidity trap. - A simple macroprudential policy that restricts leverage (with appropriate ex-ante transfers) can make all agents better off, provided the liquidity trap cannot be fully alleviated by ex-post policies. - Mechanism: an aggregate demand externality operates when output is influenced by aggregate demand; deleveraging transfers liquid wealth from borrowers to lenders, but borrowers have a higher marginal propensity to consume (MPC) out of liquid wealth than lenders, so ex-ante leverage increases the depth of the recession and reduces aggregate output. - The strength of the inefficiency (and the size of the required intervention) depends on MPC differences between borrowers and lenders. ### Contrast with pecuniary externalities - When borrowers’ leveraging motives are weak and the real interest rate during deleveraging remains positive, the economy avoids a liquidity trap and ex-ante leverage generates pecuniary externalities by lowering the ex-post real interest rate. - In a complete-markets setting absent a liquidity trap, pecuniary externalities net out and the equilibrium is constrained efficient. - In a liquidity trap, pecuniary externalities are muted because the real interest rate is fixed at its lower bound; aggregate demand externalities dominate and hurt all agents by lowering incomes, opening the door for inefficiencies. ### Underinsurance and insurance mandates - Deleveraging episodes are highly uncertain ex-ante; borrowers are under-insured against deleveraging risk. - A mandatory insurance requirement (with ex-ante transfers) could make all households better off by transferring liquid wealth to borrowers during deleveraging, raising aggregate demand because borrowers have higher MPC. - Policy implication: supports instruments such as indexing mortgage liabilities to house prices (as proposed by Shiller and Weiss, 1999). ### Preventive monetary policies: mixed effects - Raising the interest rate in the run-up to leverage accumulation can perversely increase leverage in the model. - Higher interest rates reduce borrowers’ incentives to borrow in partial-equilibrium reasoning but also create a temporary recession, increase borrowers’ incentives to borrow to smooth consumption, and transfer wealth from borrowers to lenders — general equilibrium effects that can dominate. - These forces may help explain the continued increase in household leverage when the US Fed raised interest rates starting in June 2004 (Figures 1 and 2). - Even when raising the interest rate reduces leverage in some model versions, interest-rate policies are inferior to macroprudential policies for addressing excessive leverage because interest-rate policies affect all agents’ intertemporal incentives equally and create an unnecessary recession. - Raising the inflation target is a preventive monetary policy supported by the model because it would reduce the incidence of liquidity traps. ### Fire-sale externalities and endogenous debt limits - Endogenizing debt limits via collateralized financial assets introduces fire-sale externalities: - Higher leverage lowers asset prices during deleveraging, reducing borrowers’ debt capacity and increasing distress. - Fire-sale externalities operate in the same direction as aggregate demand externalities and exacerbate recessions. - Lower aggregate output further depresses asset prices, creating feedback that makes deleveraging episodes with asset fire-sales particularly severe. ### Paper structure and appendices (overview of content) - Section 2: environment and model setup. - Section 3: characterization of equilibrium with an anticipated demand-driven recession. - Section 4: aggregate demand externalities, contrast with pecuniary externalities, main result on excessive leverage, relation to MPC differences. - Section 5: uncertainty and underinsurance; second main result on underinsurance. - Section 6: role of preventive monetary policies. - Section 7: extension with endogenous debt limits and fire-sale externalities. - Section 8: conclusion. - Appendix: omitted proofs, microfoundations of the lower bound on the real interest rate, extensions (heterogeneous borrowers, fire-sales, separable preferences). *Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2014/_wp14129.pdf* ### 1.1 Related literature ### _wp14129 - 1.1 Related literature ### Relation to prior literature - Connects to long literature on the zero lower bound and liquidity traps: Hicks (1937); Krugman (1998); Eggertsson and Woodford (2003, 2004). - Adds to recent literature on optimal fiscal and monetary policy responses to liquidity traps (e.g., Eggertsson, 2011; Christiano et al., 2011; Werning, 2012; Correia et al., 2013) by focusing on debt market policies, mainly from an ex-ante perspective. - Closely related to Eggertsson and Krugman (2012) and Guerrieri and Lorenzoni (2012): models deleveraging between impatient borrowers and patient lenders and implications for liquidity traps. - Difference: introduces an ex-ante stage and investigates macroprudential policies and novel debt-market policy actions that differ from traditional policy responses to liquidity traps. - Methodological difference: uses a simple equilibrium concept with rationing in the goods market (rather than New-Keynesian framework), enabling sharp analytical characterization of inefficiencies in debt markets. ### Aggregate demand externality and related mechanisms - Builds on literature identifying aggregate demand externalities in firm price-setting contexts: Mankiw (1985), Akerlof and Yellen (1985), Blanchard and Kiyotaki (1987). - In those models, output deviates from first-best due to monopoly distortions and complementarities in firms’ demand. - In this paper, output deviates from first-best due to a liquidity trap, and aggregate-demand effects arise from agents’ debt choices (differences in marginal propensities to consume). - Related recent work analyzing aggregate demand externalities in similar contexts: Farhi and Werning (2012ab, 2013); Schmitt-Grohe and Uribe (2012abc). - Schmitt-Grohe and Uribe: negative aggregate-demand externalities from actions that increase wages during good times, leading to greater unemployment during bad times. - Farhi and Werning: output responds to aggregate demand because of sticky prices in currency unions; emphasize inefficiencies in cross-country insurance. - This paper: closed-economy focus on household leverage and externalities created by deleveraging in a liquidity trap. - Connects to literature on pecuniary externalities and excessive borrowing/risk-taking: Caballero and Krishnamurthy (2003), Lorenzoni (2008), Bianchi and Mendoza (2010), Jeanne and Korinek (2010ab), Korinek (2011). - Pecuniary externalities: agents do not internalize the impact of individual decisions on asset prices; planner can improve welfare by moving asset prices to relax financial constraints. - Distinction: the aggregate demand externality in this paper operates when a price—the real interest rate—is fixed (not when prices are volatile). Further differences are discussed in Section 4; interaction with fire-sale externalities illustrated in Section 7. ### Key model ingredients and environment - Time: infinite discrete time t ∈ {0,1,2,...}. - Agents: two types of households, borrowers and lenders, denoted h ∈ {b, l}, with equal measure normalized to 1/2. - Discount factors: borrowers have a weakly lower discount factor than lenders, β_b ≤ β_l. - Debt/asset notation: - d^h_t denotes outstanding debt (assets if negative) of household h at date t. - Initial holdings d^h_0 are given. - Households choose d^h_{t+1} each date facing one-period real interest rate r_{t+1}. - Borrowing constraint: - From date 1 onwards, households face d^h_{t+1} ≤ ε for each t ≥ 1, with ε > 0 an exogenous debt limit (as in Aiyagari (1994), Eggertsson and Krugman (2012)). Households can choose d^h_1 at date 0 without constraint. - The constraint captures a financial shock in reduced form (e.g., drop in loan-to-value ratios or collateral values) and generates leveraging at date 0 followed by deleveraging at date 1. - Baseline assumes the future financial shock is perfectly anticipated (no uncertainty); Section 5 introduces uncertainty. - Lower bound on the real interest rate: - r_{t+1} ≥ r̲_{t+1} for each t. - For baseline analysis, the lower bound is normalized to zero (r̲_{t+1} = 0); Section 6 analyzes effects of changing the lower bound. - In practice, the bound emerges from the zero lower bound on nominal rates combined with stickiness of inflation expectations (via the Fisher equation 1 + r_{t+1} = (1 + i_{t+1}) E_t[P_t/P_{t+1}]). - A Taylor-rule microfoundation with a constant inflation target leads to r_{t+1} equal to the negative of the inflation target; alternative microfoundations include sticky nominal prices/wages (New-Keynesian) or bounded rationality in inflation expectations (Malmendier and Nagel (2013)). The paper takes the lower bound as exogenous. ### Household preferences, market structure, and equilibrium notion - Preferences: state utility over consumption ~c^h_t and labor n^h_t takes form u(~c^h_t - v(n^h_t)); net consumption c_t = ~c^h_t - v(n^h_t). - Household problem (equation (2)): - max_{c^h_t, d^h_{t+1}, n^h_t} Σ_{t=0}^∞ -β_h^t u(c^h_t) - subject to c^h_t = e^h_t - d^h_t + d^h_{t+1}/(1 + r_{t+1}) for all t, - e^h_t = w_t n^h_t + Π_t - v(n^h_t), and d^h_{t+1} ≤ ε for each t ≥ 1. - Supply side: - Linear technology: one unit of labor → one unit of consumption. - Efficient net income ē = max_n n - v(n) (constant efficient output level due to GHH preferences). - Goods sector optimization with rationing (equation (3)): - Π_t = max_{n_t} n_t - w_t n_t subject to: - 0 ≤ n_t, if r_{t+1} > r̲_{t+1}; - 0 ≤ n_t ≤ (ĉ^b_t + ĉ^l_t)/2, if r_{t+1} = r̲_{t+1}. - When r_{t+1} is at the lower bound, supply cannot exceed aggregate demand (ĉ^b_t + ĉ^l_t)/2; available demand allocated symmetrically across firms; households hold equal ownership so Π_t = n_t - w_t n_t. - Equilibrium definition (Definition 1): path of allocations {c^h_t, d^h_{t+1}, n^h_t, e^h_t}_t and real prices/profits {w_t, r_{t+1}, Π_t}_t such that households solve (2), final good sector solves (3), and markets clear. - Equilibrium notion comparable to rationing equilibria in Clower (1965), Barro and Grossman (1971), Malinvaud (1977), Benassy (1986), and produces a similar rationing outcome as New-Keynesian models but abstracts from inflation dynamics. ### Anticipated demand-driven recession — decentralized equilibrium characterization - Simplifying notation and baseline normalizations: - Equilibrium labor supply identical across types: n^h_t = (v')^{-1}(w_t), common net income e_t = n_t - v(n_t). - Market clearing for debt: d^l_t = -d^b_t; denote borrower debt d^b_t = d_t, lender debt d^l_t = -d_t. - Normalize lower bound to zero for t ≥ 1: r̲_{t+1} = 0. - Set r_1 = -1 in the initial period to exclude a liquidity trap at date 0. - Preference conditions: u(·) and v(·) strictly increasing, u(·) strictly concave, v(·) strictly convex, with lim_{c→0} u'(c) = 1; v'(0) = 0; lim_{n→∞} v'(n) = 1. Also assume u'(2ē)/u'(ē + ε/(1 - β_l)) < β_l to allow the real-rate constraint to bind. - Steady state for t ≥ 2 when borrowers’ constraint binds (d_{t+1} = ε for t ≥ 1) and lenders’ constraints do not bind: - Real interest rate determined by lenders’ discount factor: r_{t+1} = 1/β_l - 1 > 0. - Wages given by w_t = v'(n_t) from problem (3). - Consumption for t ≥ 2 (equation (4)): - c^b_t = ē - ε/(1 - β_l) - c^l_t = ē + ε/(1 - β_l) - Deleveraging at t = 1: - Borrower consumption: c^b_1 = e_1 - [d_1 - ε]/(1 + r_2). - Lender consumption increases to absorb slack: c^l_1 = e_1 + [d_1 - ε]/(1 + r_2). - Lenders’ Euler equation: u'(c^l_1) β_l u'(c^l_2) = 1 + r_2, with c^l_2 given by steady-state in (4). - The lower bound on the real rate imposes an upper bound on lenders’ consumption in equilibrium, c̃^l_1, given by solution to (equation (5)): - u'(c̃^l_1) = β_l u'(ē + ε/(1 - β_l)). - Equilibrium outcome at t = 1 depends on comparison of: - Left term: d_1 - ε (amount of deleveraging forced on borrowers when borrowing limit falls to ε). - Right term: c̃^l_1 - ē (maximum demand unconstrained agents can absorb when r is at lower bound). - Two cases: - If d_1 - ε ≤ c̃^l_1 - ē, then r_2 ≥ 0 and e_1 = ē. Deleveraging effects on aggregate demand are offset by a reduction in the real interest rate; aggregate supply remains at its efficient level ē. (Range d_1 ≤ d_1 in Figure 3.) - If d_1 - ε > c̃^l_1 - ē, equivalently when outstanding debt exceeds threshold (equation (6)): - d_1 > d̄_1 = ε + c̃^l_1 - ē, then real rate constraint binds, r_2 = 0, and the interest rate cannot fall sufficiently to induce lenders to consume the efficient level. Households’ net consumption: - c^b_1 = e_1 - d_1 + ε - c^l_1 = c̃^l_1 - Firms’ demand for labor determined by aggregate demand n_1 = (ĉ^b_1 + ĉ^l_1)/2, so net income e_1 = n_1 - v(n_1) determined by aggregate demand. - Rearranged equilibrium net income (equation (7)): - e_1 = c̃^l_1 + ε - d_1 < ē (equation (8)). - Interpretation: demand shortage and rationing in goods market lower wages and employment → demand-driven recession. Greater outstanding debt leads to a deeper recession. ### Keynesian cross and multiplier in the model - Equation (7) illustrates a Keynesian cross: - An increase in borrowers’ liquid wealth by one unit increases aggregate demand by 1/2 unit (borrowers’ population share is 1/2 and their marginal propensity to consume out of liquid wealth is 1). - Feedback: the increase in aggregate demand raises net income which further increases borrowers’ liquid wealth, creating a multiplier sequence (1/2 + 1/4 + 1/8 + ...). - The feature borrowers’ MPC = 1 is used for sharpness of inefficiency results but is not necessary for the mechanism. ### Implications emphasized in the section - Deleveraging induced by shocks to borrowing constraints can transmit into aggregate demand shortfalls when the real interest rate is bounded below. - There exists a critical debt threshold d̄_1 = ε + c̃^l_1 - ē such that outstanding debt above this threshold triggers a liquidity-trap-induced recession at date 1. - The model highlights inefficiencies in ex-ante debt choices and motivates investigation of macroprudential debt-market policies as distinct from conventional monetary/fiscal responses to liquidity traps. *Source: _wp14129 - 1.1 Related literature* ### Section 4.3 shows that net income is declining in outstanding debt, ### _wp14129 - Section 4.3 shows that net income is declining in outstanding debt, ### Date 0 allocations and equilibrium characterization - Conjectured equilibrium: net income is at its efficient level, e0 = e. - Households are unconstrained at date0, so the Euler equations for both types hold: - 1 / (1 + r1) = [ l (u0 - cl1) / (u0 - cl0)] = [ b (u0 - cb1) / (u0 - cb0)]. (Eq. (9)) - The equilibrium debt level, d1, and the interest rate, r1, are determined by these Euler conditions. - There exist conditions under which households choose debt d1 > d1̄ that trigger a recession at date1: - Proposition 1: There is a deleveraging-induced recession at date1 if the borrower is sufficiently impatient or sufficiently indebted at date0. - For any debt level d0 there is a threshold impatience  b(d0) such that the economy experiences a recession at date1 if b <  b(d0). - For any impatience b there is a threshold debt level d0( b) such that the economy experiences a recession at date1 if d0 > d0( b). - Under these conditions the appendix establishes the economy experiences a demand driven recession and liquidity trap at date1. ### Aggregate demand externalities (Section 4.1) - Planner setup: - Constrained planner at date0 can affect individual debt d1 but cannot interfere for dates ≥1. - Focus on constrained efficient allocations with d1 ≥ , so conditional on d1 the date1+ behavior is as analyzed previously. - Definitions: - Vh(d1; D1) denotes utility of a household of type h conditional on entering date1 with individual debt d1 and aggregate debt D1. - Aggregate debt D1 affects household utility because it determines the interest rate and net income at date1. - Continuation utilities from date2 onward do not depend on d1 or D1 [cf. Eq. (4)]. - Private versus social marginal values: - Private marginal value of debt for an individual household: @Vh/@d1 = u0 - ch1. - Social marginal value: @Vh/@d1 + @Vh/@D1. Externalities are captured by @Vh/@D1. - Lemma 1: - (i) If D1 ∈ [; d1), then - @Vh/@D1 = ( - u0 - ch1 ) < 0, if h = l - @Vh/@D1 = (  u0 - ch1 ) > 0, if h = b - where  ∈ (0;1). - (ii) If D1 > d1, then - @Vh/@D1 = (@e1/@D1) u0 - ch1 = - u0 - ch1 < 0, for each h ∈ {b; l}. (Eq. (11)) - Interpretation: - For D1 low enough that output is not demand-influenced (e1(D1) = e), higher aggregate debt induces greater deleveraging at date1 which reduces the interest rate; this redistribution from lenders to borrowers creates pecuniary externalities (positive for borrowers, negative for lenders). With complete markets between date0 and date1, these net out ex-ante, so date0 equilibrium is constrained efficient in this region. - For D1 large enough (D1 > d1) the economy is in a liquidity trap: - Interest rate is fixed at r2(D1) = 0, so pecuniary externalities do not apply. - Net income is decreasing in leverage: @e1/@D1 < 0, via reduction in aggregate demand (see Figure 3 in source). - An increase in aggregate leverage reduces welfare of all agents — this is the aggregate demand externality. - Aggregate demand externalities hurt all agents because they operate through lowering incomes; unlike pecuniary externalities, they can lead to constrained inefficiencies. ### Excessive leverage and constrained inefficiency (Section 4.2) - Debt writedowns (ex-post) example: - If lenders forgive some borrower debt reducing leverage from d1 to threshold d1̄ (given by Eq. (6)), the recession is avoided and net income rises to its efficient level e. - Borrowers' net consumption and welfare increase. - Lenders' net consumption remains the same at the upper bound, cl1. - Direct effect on lenders' welfare is negative: - @Vl/@d1 = - u0 - cl1 < 0. - Indirect effect through aggregate demand externalities is positive: @Vl/@D1 = u0 - cl1 > 0. - These externalities can be sufficiently strong to fully counter the direct effect, leading to ex-post Pareto improvement. - Debt-writedowns are always associated with aggregate demand externalities, but they are not always sufficiently strong to yield Pareto improvement. - Ex-post writedowns face practical implementation challenges (legal restrictions, moral hazard, financial health of intermediaries), so the paper focuses on ex-ante prevention. - Ex-ante prevention via debt limits and transfers: - Suppose date0 leverage choices are subject to an additional constraint dh1 ≤ D1 (endogenous debt limit). - Allow a date0 wealth transfer T0 from lenders to borrowers so outstanding debt becomes d0 - T0. - Planner problem for constrained efficiency: - Planner chooses date0 allocations (ch0; nh0)h and debt level D1, leaving allocations from date1 onward to the market. - Allocation ((ch0; nh0)h; D1) is constrained efficient if it solves: - max_{(ch0; nh0)h; D1} Σh αh [ - u(ch0) + h Vh(D1; D1) ] subject to Σh ch0 = Σh nh0 - v(nh0). (Eq. (12)) - αh ≠ 0 captures relative welfare weights assigned to type h agents. - Main result — Proposition 2 (Excessive Leverage): - An allocation ((ch0; nh0)h; D1), with D1 ≥ , is constrained efficient iff: - Output at date0 is efficient: e0 = e; and - Consumption and debt allocations satisfy one of: - (i) D1 < d1 and the Euler equation (9) holds. - (ii) D1 = d1 and the inequality holds: - l [u0 - cl1] / [u0 - cl0] ≥ b [u0 - cb1] / [u0 - cb0]. (Eq. (13)) - Moreover, every constrained efficient allocation of this type can be implemented as a competitive equilibrium with debt limit dh1 ≤ d1 for each h, combined with an appropriate ex-ante transfer, T0. - Interpretation of Proposition 2: - Part (i) verifies that when D1 < d1 (pecuniary-externality region) equilibrium allocations are constrained efficient: pecuniary externalities alone do not generate inefficiencies. - Part (ii) is the main result: when D1 ≥ d1 and aggregate demand externalities are active, constrained efficient allocations are characterized by D1 = d1 and the “distorted Euler inequality” (Eq. (13)). - At the efficient allocation borrowers would like to borrow more (increasing date0 consumption and reducing date1 consumption) but are prevented by the planner. - These allocations can be implemented by a simple debt limit applied to all agents plus an appropriate ex-ante transfer. - Therefore the competitive equilibrium with d1 > d1 satisfying the Euler equation (9) is constrained inefficient. - Intuition: lowering debt when the economy is in a liquidity trap yields first-order welfare benefits via positive aggregate demand externalities; distorting date0 consumption away from private Euler conditions imposes only locally second-order losses. A debt limit internalizes the externality and leads to an ex-ante Pareto improvement. In the baseline setting the planner avoids the recession fully (it is never optimal to choose D1 > d1). - Generality and extensions: - The ex-ante inefficiency result applies quite generally, except the part that the recession is fully avoided (in general the planner mitigates but may not completely avoid the recession). - Appendix A.5 establishes an analogous result for separable preferences u(c) - v(n). - Section 4.3 (referenced) provides a different generalization useful to gauge the magnitude of inefficiency (details not included in the supplied excerpt). *Italic: Source — _wp14129 - Section 4.3 shows that net income is declining in outstanding debt,* ### 4.3 MPC di§erences and the magnitude of the ine¢ ciency ### 4.3 MPC di§erences and the magnitude of the ine¢ ciency ### Model extension: heterogeneous borrower MPCs - Introduces two groups of borrowers identical except for their MPCs at date 1. - A fraction 2[0;1] of borrowers (type b_high) have high MPC at date 1; the remaining fraction (type b_low) have lower MPC. - Type b_high borrowers face the exogenous debt limit, . Type b_low borrowers are unconstrained at all dates. - Borrowers as a group have average MPC lower than 1 and depending on parameter . ### Log-utility calibration and marginal propensities - With u(c) = log c and type b_low borrowers having the same discount factor as lenders starting date 1 ( _b_low = _l): - MPC_l1 = MPC_b_low1 = 1 . (Equation (14)) - MPC_b_high1 = 1 (since constrained borrowers consume all additional income). - Borrowers' group MPC: - MPC_b1 = + (1 ) (1 ). (Equation (15)) - Parameter calibrates MPC differences between borrowers and lenders. ### General equilibrium sensitivity to leverage and MPC differences - Maintain assumptions: borrowers do not know their types at date 0 and cannot trade type-contingent assets, so each borrower enters date 1 with the same outstanding debt d1. - There exists a threshold debt level d1 such that a liquidity trap occurs only if d1 > d1. - Comparative static (derivative result): - @e1/@d1 = 2 = MPC_b1 MPC_l1 2 MPC_b1 + MPC_l1  . (Equation (16)) - Key implications: - An increase in outstanding leverage at date 1 leads to a deeper recession. - The strength of the effect depends on MPC differences between borrowers and lenders: greater transfer of wealth from borrowers to lenders affects aggregate demand more when MPC differences are larger. ### Planner's constrained optimality condition with heterogeneous MPCs - Planner condition (analogue of Eq. (13)): - _l u0( c_l1 ) (1MPC_l1) u0( c_l0 ) = _b E0[ u0( c_b1 ) ] (1MPC_b1) u0( c_b0 ) for each D1 > d1. (Equation (17)) - Expectation E0[·] is taken over borrowers' types at date 1. - Interpretation: - Planner weighs type-h agents' consumption at date 1 by factor 1/(1MPC_h1). - Because borrowers have higher MPC, planner distorts Euler equations towards providing more consumption to borrowers at date 1. - The optimal intervention (wedge between borrowers' and lenders' Euler equations) depends on MPC differences. ### Empirical relevance - Empirical literature indicates borrowers' MPC was significantly greater than lenders' MPC in recent deleveraging: - Baker (2013): one standard deviation increase in household debt-to-asset ratio raises MPC by about 20% (about 7 percentage points from a baseline of 37 percentage points — sample median debt to asset ratio ≈ 0.4). - implication: empirical MPC differences can guide optimal macroprudential policy design. --- ### 5 Uncertainty and underinsurance ### Setup with state-dependent debt limits - Single borrower type; economy in one of two states s ∈ {H, L} from date 1 onwards. - State L: debt limit _{t+1;L} =  for each t ≥ 1 (deleveraging state). - State H: unconstrained, _{t+1;H} = 1 for each t ≥ 1. - Let ^h_s denote type-h households' belief for state s. Assume ^h_L > 0 for all h (deleverage anticipated). - From date 1 both types have same discount factor _b = _l = ; at date 0 borrowers may be more impatient: _b0 ≤ _l0. - Borrowers are (weakly) more optimistic than lenders: ^b_H ≥ ^l_H. ### Arrow securities and equilibrium across states - Households trade one-period ahead Arrow securities at date 0. q1;s denotes price of Arrow paying 1 unit in state s at date 1. - Real interest rate at date 0 satisfies 1 + r1 = 1/Σ_s q1;s. - Equilibrium in state L at date 1: interest rate is zero and a demand-driven recession occurs if outstanding debt large enough, d1;L > d1. - Equilibrium in state H jumps to steady state with 1 + r_{t+1} = 1/ > 0 and c_{h t;H} = e^{  (1 ) d_{h1;H} } for t ≥ 1. ### Full-insurance condition across states at date 0 - Date-0 allocations satisfy Euler equation and full-insurance equation: - q1;H / q1;L = ^l_H u0( c_l1;H ) / ^l_L u0( c_l1;L ) = ^b_H u0( c_b1;H ) / ^b_L u0( c_b1;L ). (Equation (18)) ### Conditions for a deleveraging-induced recession (Proposition 3) - There is a deleveraging-induced recession in state L of date 1 if the borrower is either: - (i) sufficiently impatient; or - (ii) sufficiently indebted; or - (iii) sufficiently optimistic at date 0. - Formally, for any two of the parameters ( _b0; d0; ^b_L) one can determine a threshold for the third such that d1;L > d1 if the threshold is crossed: - e.g., _b0 <  _b0(d0; ^b_L) or d0 > d0( _b0; ^b_L) or ^b_L < ^b_L( _b0; d0). - Intuition: - Impatience and high initial indebtedness induce leverage including in state L, even if it triggers recession. - Optimism (assigning low probability to state L relative to lenders) increases debt allocated to state L and can exacerbate outcome. ### Constrained efficiency and underinsurance (Proposition 4) - Planner chooses date-0 allocations and outstanding leverage at date 1, leaving remaining allocations to markets. Constrained planning problem written as: - max_{(c^h_0,n^h_0)_h,(D_{1;s})_s} Σ_h [ u( c^h_0 ) + ^h_0 Σ_s V^h_s( D_{1;s} ; D_{1;s} ) ] subject to Σ_h c^h_0 = Σ_h n^h_0 v( n^h_0 ). (Equation (19)) - Proposition 4 (Underinsurance): An allocation ((c^h_0,n^h_0)_h ; (D_{1;s})_s), with D_{1;L} ≥ , is constrained efficient iff: - Output at date 0 is efficient: e0 = e^{  }. - Households' substitution between date 0 and state H is efficient: - _l0 ^l_H u0( c_l1;H ) / u0( c_l0 ) = _b0 ^b_H u0( c_b1;H ) / u0( c_b0 ). - Remaining consumption and leverage allocations satisfy one of: - (i) D_{1;L} <  d1 and the full insurance equation (18) holds. - (ii) D_{1;L} =  d1 and the distorted insurance inequality holds: - ^l_H u0( c_l1;H ) / ^l_L u0( c_l1;L ) ≥ ^b_H u0( c_b1;H ) / ^b_L u0( c_b1;L ). (Equation (20)) - Implementation: - Every constrained efficient allocation of this type can be implemented as a competitive equilibrium with a mandatory insurance requirement d^h_{1;L} ≤ D_{1;L} for each h, combined with an appropriate ex-ante transfer T0. ### Interpretation: underinsurance and mandatory insurance - Main result with uncertainty: constrained efficient allocations display a distorted insurance inequality (underinsurance) and can be implemented via an endogenous limit on agents' outstanding debt in state L — interpreted as a mandatory insurance requirement restricting losses in the deleveraging state. - Competitive equilibrium with d1;L > d1 is constrained inefficient and Pareto improvable with such a mandatory insurance requirement. - Mechanism: - Borrowers not only take excessive leverage but also buy too little insurance against severe deleveraging — they do not internalize positive aggregate demand externalities from insurance purchases. - Policy implication: - Rationale for indexing mortgage liabilities to housing prices (home equity insurance) and for making such insurance mandatory for severe, economy-wide downturns in house prices. ### Role of optimism and complementarity with externalities - Homeowners' lack of interest in home equity insurance can be driven by borrower optimism. - Optimism generates a first source of underinsurance (privately optimal); it also increases leverage and the likelihood of aggregate demand externalities, which produce a second source of underinsurance that is socially inefficient. - Conclusion: optimism and aggregate demand externalities are complementary sources of underinsurance. ### Generalization to incomplete markets - Result generalizes when financial markets are incomplete and only noncontingent debt is available (imposing d1 = d1;L = d1;H). - Planner setting D1 must consider aggregate demand externalities in state L and pecuniary externalities in state H because marginal utilities across states are not equated. - With two continuation states, aggregate demand externalities are sufficiently powerful that the equilibrium always features too much leverage. --- ### 6 Preventive monetary policies ### Changing the inflation target - A Taylor rule with a higher inflation target lowers the bound on the real rate (Equation (1) bound). - With positive inflation target  > 0, the bound is given by r_{t+1} = /(1+) for each t ≥ 1. - Consequence: greater leverage is necessary to plunge the economy into a demand-driven recession — consistent with Blanchard, Dell'Ariccia and Mauro (BDM, 2010). - Additional insight: raising the inflation target might improve social welfare because aggregate demand externalities emerge only when the real rate is constrained; welfare benefits must be weighed against costs of higher steady-state inflation. --- *Source: _wp14129 - 4.3 MPC di§erences and the magnitude of the ine¢ ciency* ### 6.2 Contractionary monetary policy ### 6.2 Contractionary monetary policy ### Setup and key equations - Consider baseline setting with a single type of borrower and no uncertainty; conditions in Proposition 1 imply a liquidity trap at date 1. - Contractionary monetary policy: policymaker sets interest rate at date 0 to a level r1 higher than the “natural” interest rate, so equilibrium interest rate is r1 = r1 and agents’ net income falls to e0 < ē. - Agents’ Euler equations under policy-induced recession (equation (21)): - 1/(1 + r1) = Φl u0(e1 + (d1 − ε)) / u0[e0 + d0 − d1/(1 + r1)] = Φb u0(e1 − (d1 − ε)) / u0[e0 − (d0 − d1/(1 + r1))], - where e1 = c̄l1 − (d1 − ε) < ē as in (8). - These two equations determine e0(r1) and d1(r1) as functions of the policy rate r1. ### Proposition 5 and comparative statics - Proposition 5 (Contractionary Monetary Policy) — under: - u00(x)=u(x) weakly decreasing in x, - d0 sufficiently large so that d0 − d1(r1) /(1 + r1) > 0, - and equilibrium with a liquidity trap at date 1 and constrained interest rate r1 at date 0, - then e0′(r1) < 0 and d1′(r1) > 0. - Interpretation of result: - Increasing r1 decreases current net income e0. - Increasing r1 increases outstanding debt level d1. - Thus, raising the interest rate in the run-up to a deleveraging episode creates a recession and, under natural assumptions, increases equilibrium leverage, worsening the subsequent recession at date 1. ### Intuition and log-utility example - With u(c) = log c and ε = 0, closed-form conditional debt choices (equation (22)): - db1 = 1/(1 + Φb) [ − e1 − Φb (1 + r1)(e0 − d0) ] - dl1 = 1/(1 + Φl) [ − e1 − Φl (1 + r1)(e0 + d0) ] - Holding e0 and e1 constant, higher r1 reduces both db1 and dl1 (substitution effect: borrowers borrow less; lenders save more) creating excess demand in asset market (db1 + dl1 falls below 0) or a goods market shortage. - Equilibration requires output fall and net income e0 decline; as e0 falls, both db1 and dl1 increase (borrowers borrow more, lenders save less) to smooth consumption. - If d0 = 0, the reduction in e0 would exactly offset initial effects and equilibrium debt d1 would remain unchanged (with log utility). If d0 sufficiently large, higher r1 creates an additional wealth transfer from borrowers to lenders, increasing borrowers’ debt db1 and raising equilibrium debt level d1 = db1. - More generally, two general equilibrium effects dominate the partial-equilibrium substitution effect and can lead to higher leverage when interest rates rise. ### Policy implications and normative comparisons - Conventional wisdom that raising the interest rate decreases leverage can fail due to two general equilibrium effects: - Higher r1 creates a temporary recession reducing borrowers’ current income and inducing greater debt. - Higher r1 transfers wealth from borrowers to lenders, further increasing borrowers’ debt. - Variants where raising r1 decreases d1 are possible (e.g., borrowers’ intertemporal substitution more elastic than lenders’ as in Curdia and Woodford (2009)), but even then interest rate policy is not the optimal instrument to address excessive leverage. - Constrained efficient allocations (Proposition 2) do not feature a recession at date 0 and satisfy the distorted Euler inequality (13); raising r1 creates an inefficient recession and continues to satisfy Euler equations (21). - Interest rate policy targets a single wedge (intertemporal substitution) and thus “the wrong wedge” for addressing asymmetric incentives of borrowers and lenders; macroprudential policies (debt limits, insurance requirements) optimally internalize aggregate demand externalities created by leverage and are superior instruments to reduce excessive leverage. - Caveat: contractionary monetary policy could be desirable for other reasons outside this model (e.g., mitigate inefficient investment booms, fire-sale externalities, or discourage “search for yield”); the main point is that contractionary monetary policy is not the ideal instrument to reduce household leverage and may unintentionally raise leverage. *Source: _wp14129 - 6.2 Contractionary monetary policy* ### References ### _wp14129 - References ### A.1 Microfounding the lower bound on the real interest rate - Setup: cashless limit economy (Woodford (2003)). Denote P_t nominal price, i_{t+1} nominal interest. - Assumption (A1): There is a zero lower bound on the nominal interest rate: - i_{t+1} ≥ 0 for each t ≥ 0. (A.1) - Assumption (A2): Nominal interest rate set by a Taylor rule adjusted for the zero lower bound: - log (1 + i_{t+1}) = max { 0; log (1 + r^n_{t+1}) + (log (P_t / P_{t-1}) - log (1 + ϑ)) }, (A.2) - where 1 + r^n_{t+1} = min { 2f b; l g u_0 - c_h_t; u_0 - c_h_{t+1} } and > 1. (text preservation) - Implication: The Taylor rule implies P_{t+1} / P_t = 1 + ϑ for each t ≥ 1. (A.3) - Combining (A.1), (A.3) and the Fisher equation 1 + r_{t+1} = (1 + i_{t+1}) E_t[P_t / P_{t+1}] leads to the bound: - r_{t+1} ≥ r_{t+1} = - ϑ / (1 + ϑ) for each t ≥ 1. (text preserved) - Remarks: - Baseline normalizes ϑ = 0; Section 6 discusses ϑ ≥ 0. - With ϑ = 0 the Taylor rule is ex-post efficient and yields the lower bound absent commitment; commitment could circumvent the bound (Krugman (1998) literature). - Argument sketch for (A.3): For t ≥ 2 real interest rate constant r_{t+1} = 1/ϕ_l - 1 > 0; deviations P_t / P_{t-1} ≠ 1 + ϑ yield contradictions given Taylor coefficient > 1 (detailed iterative contradiction argument). ### A.2 Omitted proofs for the baseline model (summary of main logical steps and conclusions) - Proof of Proposition 1 (existence of recession at date 1 under conditions): - Construct interest rate ˜r_1(d_0) solving 1/(1+˜r_1) = ϕ_l (u_0 - ē + ˜d_1 - ε) / (u_0 - ē + d_0 - ˜d_1) = (1 + ˜r_1)^{-1}. - Show ˜r_1(d_0) decreasing in d_0; d_0 - ˜d_1 = (1 + ˜r_1(d_0)) increasing in d_0. - Condition (A.4) on marginal rates of substitution yields d_1 > ˜d_1 and hence a recession at date 1 when borrowers are sufficiently impatient or indebted at date 0. - Proof of Lemma 1 (signs of derivatives and subgradients): - Case d_1 > ˜d_1: de_1 / dd_1 = -1 (from Eq. (8)), and @V_h/@D_1 = - (u_0 - u_h^1) < 0 (from Eq. (10)). - Case d_1 < ˜d_1: differentiating lenders' Euler eq. (9) gives dr_2 / dd_1 < 0; consequences for dc_b1 / dd_1 and dc_l1 / dd_1 derived; establishes ϑ^2(0;1). - Proof of Proposition 2 (characterization of constrained-efficient allocations and subgradients): - Define r_sub V_h(d_1;D_1) as set of subgradients. - If D_1 ≠ d_1, V_h differentiable in second argument and subgradient unique (per Lemma 1). - If D_1 = d_1, kink yields subgradient interval (A.5) with explicit left/right derivative bounds preserved. - First-order conditions (A.6) use subgradients ε_h = r_sub V_h(D_1;D_1). - Cases: - D_1 < d_1: condition(A.6) reduces to Euler eq. (9). - D_1 > d_1: no constrained-efficient allocation (A.6) violated. - D_1 = d_1: allocations characterized by distorted Euler inequality (13) and consumption allocations (A.7). - Implementation: such allocations can be implemented with endogenous debt limit d^h_1 ≤ d_1 and transfer T_0 (equivalently altering initial debt ˜d_0 = d_0 - T_0). - Proof of Proposition 3 (existence of equilibrium with d_{1;L} ≥ d_1 and recession in state L): - Optimality conditions summarized in (A.8) — four equations in four unknowns (d_{1;H}, d_{1;L}, q_{1;L}, q_{1;H}). - For conditions (i) or (ii) (ψ^b_L = ψ^l_L), thresholds on parameters (ϕ_b, d_0) guarantee d_{1;L} ≥ d_1. - Under condition (iii), show d_{1;L} decreasing in ψ^b_L and lim_{ψ^b_L→0} d_{1;L} > d_1; therefore threshold function ˜ψ^b_L(ϕ_b0,d_0) exists such that d_{1;L} > d_1 when ψ^b_L < ˜ψ^b_L. - Proof of Proposition 4 (planner problem with state-dependent decisions): - First-order conditions for problem (19) give (A.9) with subgradients ε_l and ε_b. - Cases: - D_{1;L} < ˜d_1: condition reduces to (18). - D_{1;L} > ˜d_1: never optimal. - D_{1;L} = ˜d_1: insurance inequality (20) characterizes solution; implementable with mandatory insurance requirement d^h_{1;L} ≤ ˜d_1 plus transfer T_0. - Lemma 2 (monotonicity properties for strictly increasing, strictly concave u(·) with -u''(x)/u'(x) weakly decreasing): - For x,y ∈ R_+: - d/dx [ u'(x+y) / u'(x-y) ] ≥ 0 - d/dy [ u'(x+y) / u'(x-y) ] < 0 - Proof uses ratio derivatives and concavity/monotonicity of -u''/u'. - Proof of Proposition 5 (comparative statics on solutions of Eqs. (21)): - Define d_1(r_1), e_0(r_1) and y(r_1) = (d_0 - d_1(r_1)) / (1 + r_1) with y(r_1) > 0 by assumption. - Show e_0'(r_1) < 0 by contradiction using lenders' Euler (A.10) and borrowers' Euler. - Show d_1'(r_1) > 0 by contradiction and application of Lemma 2 to the ratio of marginal utilities; completes proof. ### A.3 Extension with heterogeneous borrowers (two borrower types b_high and b_low) - Date-1 equilibrium when all borrowers choose same d_1 and there is threshold ˜d_1 (liquidity-trap threshold). - If d_1 ≥ ˜d_1: - Type b_high forced into deleveraging: d_2 = ε and - c_{b_high,1} = e_1 - d_1 + ε - c_{b_high,2} = ē - ε(1 - ϕ) - Type b_low solves Euler equation: - u'(c_{b_low,1}) = ϕ u'(c_{b_low,2}). (A.11) - with c_{b_low,1} = e_1 - d_1 + d^{b_low}_2, c_{b_low,2} = ē - d^{b_low}_2 (1 - ϕ). - Aggregate borrowers' date-2 debt: - d_2 = β ε + (1 - β) d^{b_low}_2. (A.12) - Lenders' Euler equation: - u'(c_{l,1}) = ϕ u'(c_{l,2}). (A.13) - with c_{l,1} = e_1 + d_1 - d_2, c_{l,2} = ē + d_2 (1 - ϕ). - Equilibrium triple (e_1; d^{b_low}_2; d_2) solves (A.11), (A.12), (A.13). - Closed-form solution with log utility: - e_1 = ē / ϕ - β / (2 - 2β) [ d_1 - ε ] / ϕ. (A.14) (text preserved) - d_{low,2} = ϕ [ d_1 + β/(2 - 2β) ( d_1 - ε ) / ϕ ] - d_2 = β / (2 - β) ε + (1 - β) / 2 = β / 2 ϕ d_1 (text preserved as in source) - Liquidity trap condition: there is a trap with e_1 < ē and r_2 = 0 as long as: - d_1 > d_1 = ē [ (1/ϕ - 1) ]^{2 - β} β / (2 - β) + ε / ϕ . (A.15) (text preserved) - Date-0 equilibrium: conjecture e_0 = ē; both households' Euler equations hold with expectation over borrower types: - 1/(1 + r_1) = ϕ_l u'(c_{l,1}) / u'(c_{l,0}) = ϕ_b E_0[u'(c_{b,1})] / u'(c_{b,0}). - Existence of solution (r_1, d_1) follows; deleveraging-induced recession at date 1 (d_1 > ˜d_1) occurs when borrowers are sufficiently impatient or indebted at date 0. - Efficiency and externalities: - Define V_h(d_1; D_1) expected utility before type realization. - Aggregate demand externalities for D_1 > ˜d_1: - ∂V_l/∂D_1 = ∂e_1/∂D_1 u'(c_{l,1}) = - β / (2 - 2β) u'(c_{l,1}). (A.16) - ∂V_b/∂D_1 = ∂e_1/∂D_1 E_0[u'(c_{b,1})] = - β / (2 - 2β) E_0[u'(c_{b,1})]. - Planner first-order condition for D_1 > ˜d_1: - ϕ_l u'(c_{l,1}) / u'(c_{l,0}) = ϕ_b E_0[u'(c_{b,1})] / (1 - β) u'(c_{b,0}) - Plugging β yields Eq.(17) and an analogue of Proposition 2 (constrained-efficient insurance rule). - Implementation: analogous mandatory insurance / transfer instruments can implement planner allocation. ### A.4 Extension with fire sales - Section heading appears but full text of this extension is not included in the supplied content. *Italic: Source: _wp14129 - References (appendix and omitted proofs, supplied content).* ### 7. To characterize the conditionp ### 7. To characterize the conditionp ### Uniqueness condition for the solution to equation (24) - Expression for the derivative @MRS/@p1 (as given): - (1 )u00c b2 (1)u0c b1 2 u0c b2 (1)u00c b1 u0c b2 (1)u0c b1 + u0c b2  - Under approximation 1, all but the last term disappear from the numerator. - Approximation used: u0c b1  u0c b2  (holds exactly in the neighborhood where the constraint becomes binding). - Simplified condition: - @MRS/@p1 (1)u00c b1 u0c b1  = (1) c b1 < 1 p2 - equivalently: (1) <  c b1 p2 - where  is the intertemporal elasticity of substitution. - Interpretation: - The solution to equation (24) is unique and well-defined if the leverage parameter is sufficiently small compared to the consumption/asset price ratio. - If violated, an infinitesimal increase in date1 consumption would cause a discrete upward jump in asset price, relaxing the constraint by more than necessary and violating the assumption of a binding borrowing constraint. - This type of condition is common in models of financial amplification to guarantee uniqueness (see Lorenzoni, 2008; Jeanne and Korinek, 2010b). ### A.5 Extension with separable preferences (Online appendix) — setup - Preferences: households have separable preferences u(c)v(n), where u() and v() satisfy standard assumptions. - Ownership assumption: type h households hold all shares of firms in which they work (none of the shares of other firms) so their equilibrium income is always n h1. (Assumption made for simplicity.) - Date t ≥ 2: consumer debt constant at maximum permissible level d t =  and borrowers pay lenders interest 1 1 1+r t+1   = 1 l   at every date. - Households' labor supply conditions: - u0  n l +  1 l    = v0  n l  - u0  n b +  1 l    = v0  n b  - with n l < n b (since c l > c b). ### Equilibrium characterization at date 1 and liquidity trap threshold - There exists a threshold, d1, such that r20 only if d1  d1. - To characterize threshold: - Lenders' Euler at zero interest: v0n l1  = l v0n l1  which pins down n l1. - Intratemporal optimality: u0n l1 + d1  = l v0n l1  then pins down d1. - Equilibrium when d1 < d1 (standard; r2 > 0): - Variables r2 > 0; n l1; n b1 satisfy: - u0  n b1d1 +  1 + r2  = v0  n b1  - u0  n l1 + d1 1 + r2  = v0  n l1  = l (1 + r2) v0  n l1  - Comparative statics: - @r2/@d1 < 0 - @n l1/@d1 2(1;0) - @n b1/@d1 2(0;1) - An increase in debt leads to a reduction in the interest rate, a decrease in lenders' labor supply and an increase in borrowers' labor supply (wealth effect). Agents' labor supply responses are less than one-for-one. - Equilibrium when d1 > d1 (liquidity trap): - r2 = 0. - Lenders' Euler: u0  n l1 + d1  = l u0  n l +  1 l   . - Implies: n l1(d1) = n l1 + d1d1 ; analogue of Eq.(8). - Lenders' employment and output decrease one-to-one with leverage: - @n l1/@d1 = 1. - Equilibrium wage: - w1(d1) = v0n l1  u0n l1 + d1  = v0n l1  l v0(n l) = v0n l1  v0n l1  < 1. - Labor wedge strictly positive: - 1 = 1w1(d1) > 0 (d1 > d1 implies demand driven recession). - Elasticity of wage with respect to leverage: - @w1/@d1 w1 = d1 = 1  l d1 n l1 - where  l = v00(n l1) n l1 v0(n l1) is the Frish elasticity of lenders' labor supply. - Wage response influenced by size of debt relative to output and the Frish elasticity. - Borrowers' intratemporal condition: - v0n b1  = w1(d1) u0n b1(d1) . - Implicit differentiation yields: - @n b1/@d1 v00n b1 u00c b1  c b1 w1 u0c b1  = 1  l c b1 n l1 + 1  b , - where  b = 1 = u00(c b1) c b1 u0(c b1) is defined as lenders' intertemporal elasticity of consumption. - This implies @n b1(d1)/@d1 < 1. - Aggregate employment response: - @n l1/@d1 + @n b1/@d1 < 1 (Equation A.17). - Total employment decreases in leverage regardless of parameters. - Date 0 equilibrium characterized by Euler equations (9). Under conditions similar to Proposition 1, equilibrium features d1 > d1 and an anticipated recession. ### Efficiency properties and planner's problem with separable preferences - Define V h(d1; D1) as utility of household type h conditional on entering date1 with individual debt d1 and aggregate debt D1. - Example (borrowers): - V b(d1; D1) = u  n1(D1)d1 +  1 + r2(D1)  v(n1(D1)) + 1 X t=2  b  t u  c b t  - For D1 > d1, first-order condition with respect to D1: - @V h/@D1 = u0  c h1  @n h1/@D1 1 for D1 > d1 (Equation A.18). - This characterizes aggregate demand externalities; strength depends on employment response to leverage. - Planner's ex-ante constrained planning problem: W h(D1) = V h(d1; D1). - First-order conditions imply Eq.(A:6) also holds. - Using (A:18), planner's optimality condition: - l u0c l1   1 + @n l1/@D1 1  u0c l0  = b u0c b1   1 @n b1/@D1 1  u0c b0  for D1 > d1 (Equation A.19). - Recall from (A:17) aggregate employment always declines with leverage. - Combining with (A:19) shows the distorted Euler inequality (13) also holds in this case. ### Proposition 6 (Excessive Leverage with Separable Preferences) — characterization of constrained efficient allocations - Statement: - An allocation c h0 ; n h0  h ; D1  , with D1  , is constrained efficient iff labor supply and output at date 0 are efficient, i.e., u0c h0  = v0n h0  for each h; and the consumption and debt allocations satisfy one of: - (i) D1 <  d1 and Euler equation (9) holds. - (ii) D1   d1, and the planner's optimality condition in (A:19), and thus also the distorted Euler inequality in (13), holds. Moreover, every constrained efficient allocation of this type can be implemented as a competitive equilibrium with a debt limit, d h1  D1 for each h, combined with an appropriate ex-ante transfer, T0. - Interpretation and contrast with main text: - This result is the analogue of Proposition 2 for separable preferences. - Difference: debt levels strictly greater than  d1 can correspond to constrained efficient allocations if they satisfy planner's optimality condition (A:19). - In these cases the planner mitigates but does not completely alleviate the recession because aggregate demand externalities with separable preferences depend on size of labor wedge, 1 [cf. Eq.(A:18)]. - At D1 =  d1, labor wedge is small, 1 = 0, so planner might allow a mild recession to improve ex-ante risk sharing. - In contrast, with GHH preferences in main text, aggregate demand externalities are large regardless of labor wedge [cf. Eq.(11)], inducing planner to fully avoid the recession. *Source: _wp14129 - 7. To characterize the conditionp* --- _Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2014/_wp14129.pdf_