## 2.1 Decentralized Competitive Equilibrium ## Source details **Canonical URL:** [2.1 Decentralized Competitive Equilibrium](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12181.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12181.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12181.pdf.json) --- ### Model setup and key mechanisms - Framework: - Dynamic stochastic general equilibrium with a collateral constraint: − b_{t+1} / R_t ≤ κ_t q_t k_{t+1} (Equation (3)). - Agents maximize E_s0[∑_{t=0}^∞ β^t c_{t}^{1−σ}/(1−σ)] (Equation (1)), choose c_t, k_{t+1}, b_{t+1}; k_t = 1 in equilibrium. - Bonds price = 1/R. Land price q_t, fixed unit supply; land enters production ε_t Y(k_t). - Collateral and informational structure: - κ_t follows two-regime Markov process with values κ_h and κ_l; true transition probabilities F^a_{hh}, F^a_{ll} (with F^a_{hl}=1−F^a_{hh}, F^a_{lh}=1−F^a_{ll}). - Agents do not know F^a and learn via Bayesian updating (beta-binomial counters n_{ij}^t). Posterior means: - E_t[F^s_{hh}] = n_{hh}^t / (n_{hh}^t + n_{hl}^t) - E_t[F^s_{ll}] = n_{ll}^t / (n_{ll}^t + n_{lh}^t) (Equation (6)). - Financial innovation: - Modeled as introduction of the κ process with initial counters n_{ij}^0 close to zero; learning can produce large early swings in E_t[F^s_{hh}] and E_t[F^s_{ll}]. ### Learning, amplification, and asset pricing channels - Belief-driven amplification: - Optimistic beliefs → over-borrowing → higher q_t → relaxed collateral constraint → further borrowing. - Pessimistic beliefs → under-borrowing → lower q_t → tighter collateral constraint → amplified downturn (Fisherian deflation mechanism). - Land premium and constraint interaction: - Define R^q_{t+1} ≡ (ε_{t+1} Y_k(t+1) + q_{t+1}) / q_t. - E_s_t[ R^q_{t+1} − R ] = (1 − κ_t) μ_t − cov^s_t(β u′(c_{t+1}), R^q_{t+1}) / E_s_t[ β u′(c_{t+1}) ] (Equation (7)). - When collateral constraint binds (μ_t > 0) land premia and price sensitivity to debt grow; optimistic beliefs that understate switching to κ_l lower expected land premia and raise q_t relative to full information. ### Equilibrium definitions and solution method - Decentralized competitive equilibria: - DEL (with learning): allocations and prices using agents’ evolving beliefs E_t[F^s]. - DEF (full information): allocations and prices using true transition F^a. - Solution approach: - Two-stage recursive Anticipated Utility (AU) method: - Stage 1: generate posterior means via Equation (6). - Stage 2: solve date-t AUOP using date-t posterior means as the conjectured transition matrix (Equations (9)-(12)). - Recursive AU competitive equilibrium chains date-t AUOP solutions across dates: b_{t+1} = b′_t(b_t, ε_t, κ_t), etc. ### Pecuniary externality and macro-prudential rationale - Pecuniary externality: - Individual borrowers ignore how current borrowing affects future q (∂q/∂b′) and hence future collateral values; becomes especially steep when collateral constraint binds and fire-sale feedbacks operate. - Macro-prudential policy: - Pigouvian-style taxes on debt τ_b,t and on land dividends τ_l,t can decentralize planner allocations (Equations (20)-(21)). - Taxes computed state-contingently to reproduce planner allocations; when μ_t > 0 multiple tax representations exist and the implementation sets τ_b,t = 0 when constraint binds for simplicity. ### Planner information scenarios and policy design implications - Two planners: - SP1 (uninformed planner): shares private agents’ priors, prices collateral using q^DEL_t(b, ε, κ). - SP2 (fully informed planner): knows F^a and prices collateral using q^DEF(b, ε, κ). - Comparative insights: - SP1 underestimates persistent low-borrowing risk and hence accumulates larger debt than SP2; SP1 may assign near-zero probability to κ_h → κ_l during extreme optimism. - SP2 implements precautionary and portfolio-choice components: lowers borrowing at given prices and addresses mispricing via knowledge of true F^a. - Effectiveness of macro-prudential policy depends on planner’s information set, borrowing-constraint tightness, and pace of optimism buildup. ### Quantitative baseline experiment and key statistics - Experiment setup: - Learning period t = 1, ..., 48; ε = 1 held at mean; initial condition b0 = −0.345 (net credit market assets-GDP ratio of U.S. households in 1996Q4). - Simulations chain date-t AUOP decision rules for DEL, SP1, SP2. - Belief dynamics and calibration facts: - During optimistic phase E_t[F^s_hh] rises from 0.980 to 0.999 from t = 1 to t = 40. - At date 41 (first κ_l realization): E_41[F^s_hh] falls to 0.975; E_41[F^s_ll] rises sharply from 0.5 to 0.98. - True symmetric process: F^a_hh = F^a_ll = 0.95. - Mean durations implied: - E_1[F^s_hh] = 0.98 → 50 quarters. - E_40[F^s_hh] = 0.999 → 1,000 quarters. - True mean duration F^a_hh = 0.95 → 28 periods. - Aggregate outcomes: - DEL produces a large sustained increase in debt (decline in bonds) for first 40 periods; sharp correction at date 41. - DEL’s debt increase accounts for about 2/3rds of the observed rise in net credit liabilities of U.S. households in calibration. - DEL’s surge in risky asset price accounts for roughly 44 percent of the observed rise in U.S. residential land prices. - SP1 chooses only slightly smaller debt than DEL; bond holdings and asset prices nearly identical to DEL starting t = 7 in baseline. - SP2 chooses much smaller debt than SP1 and DEL during optimistic phase; on average μ_t = 0 for t = 1,...,40 under SP2 (avoids binding constraint). ### Crisis dynamics and cross-equilibrium comparisons - Crisis at t = 41: - SP2 (lowest debt) experiences smallest debt correction; DEL and SP1 (higher debt) experience much larger corrections. - SP1’s correction at date 41 more than twice as large as SP2’s. - Price declines: SP2 smallest, SP1 and DEL larger; differences shrink as beliefs converge. - Externality behavior: - Externality term E_t[κ_{t+1} μ_t(t+1) ∂q^i_t(t+1)/∂b′] weakens over optimistic phase for SP1 because perceived switching probability to κ_l falls near zero despite steeper pricing functions at higher debt. ### Macro-prudential tax magnitudes in baseline (reported paths) - SP1 tax policy: - Debt tax τ_b ≈ 2-3 percent in first seven periods after innovation. - Dividend subsidy τ_l up to 3 percent initially; later τ_b drops to zero and τ_l rises to about 2 percent. - For SP1 information and interaction tax components are zero; externality term accounts for full SP1 debt taxes, rising up to ≈ 3 percent before vanishing after period 7. - SP2 tax policy: - Debt tax τ_b increases gradually from ≈ 4 percent to 8 percent during optimistic phase, then cuts to zero as crisis erupts. - Dividend subsidy τ_l increased early and kept at ≈ 5 percent until crisis, then falls to almost zero. - Decomposition: SP2’s debt taxes driven mainly by information (≈ 3.3 percent) and interaction (≈ 4.2 percent) components; externality component relatively small and slightly positive throughout the optimistic phase. ### Welfare results (exact reported values) - Welfare comparisons (Table 2 values, percentages; preserved verbatim) using true probabilities (average (b^DEL_t, κ_t, ε_t), t = 1, t = 40): - (1) SP2 versus DEF: 0.052 0.05 - (2) SP2 versus DEL: 0.37 7.4 - (3) SP1 versus DEL: 0.17 0.03 - Welfare comparisons using subjective beliefs: - (4) SP1 versus DEL: 0.025 0.0 - (5) DEL versus SP2: -0.39 -2.7 - Interpretation: - Largest gains when comparing SP2 v. DEL using true probabilities: average gain ≈ 0.37 percent at t = 1 and 7.4 percent at peak optimism t = 40. - Welfare losses attributable to pecuniary externality alone are modest (e.g., SP2 v. DEF: 0.052 percent). - Using subjective-belief evaluation reduces measured DEL losses versus SP2 (e.g., DEL v. SP2 = -0.39 percent at t = 1 and -2.7 percent at t = 40). ### Sensitivity analysis: role of priors and robustness - Experiments modify initial counters n_{ij}^0 to generate: - Gradual Optimism: n_hh0 = 7.6, n_hl0 = 0.4, n_ll0 = 0.38, n_lh0 = 0.02 (date-0 posterior means equal true F^a). - Under Gradual Optimism: - DEL debt exceeds SP1 by about 4 percentage points of GDP in the run-up to the crisis. - During the crash asset prices are 16 percent higher for SP1 due to lower leverage at crisis. - Externality component of taxes rises sharply for both planners; SP2’s information and interaction components decline relative to baseline. - Heterogeneous priors: - SP3: planner’s counters → ∞ so planner’s beliefs converge to true F^a while pricing uses q^DEL_t. - SP3 debt taxes increase gradually from about 4 percent to close to 9 percent in optimistic phase, then drop to zero at crisis. - SP4: planner priors p_hh0 = p_ll0 = 0.2 (E_p0[F_hh]=E_p0[F_ll]=0.5). - SP4 perceives more risk, borrows less than DEL and SP1; collateral constraint not binding for SP4 in optimistic phase. - Interaction component of debt tax is largest for SP4 almost throughout experiment. - Cross-experiment synthesis: - Macro-prudential policy effectiveness depends critically on initial priors and planner information. - Gradual optimism or better-informed planners raise the externality and interaction terms, increasing optimal debt taxes and improving crisis outcomes. ### Planner tax decomposition (exact structure preserved) - Debt tax decomposition (equation (22)) into three terms: - Information term: [ Ei t[u′(t+1)] / Es t[u′(t+1)] − 1 ]. - Interaction term: [ Ei t[ κ_{t+1} μ_{t}(t+1) ∂q^i_t(t+1)/∂b′ ] − Es t[ κ_{t+1} μ_{t}(t+1) ∂q^i_t(t+1)/∂b′ ] ] / Es t[u′(t+1)]. - Externality term: [ Es t[ κ_{t+1} μ_{t}(t+1) ∂q^i_t(t+1)/∂b′ ] ] / Es t[u′(t+1)]. - Interpretation: - Information term captures belief differences between planner and private agents; zero when beliefs identical (e.g., baseline SP1). - Interaction term captures how planner versus private-agent beliefs alter valuation of the externality. - Externality term is the value of the pecuniary externality under private-agent beliefs. *Source: Content unit _wp12181 — sections 2.1, 2.5, baseline quantitative experiment, 3.4 Sensitivity Analysis, and related passages from the provided IMF PDF content unit.* ### 2.1 Decentralized Competitive Equilibrium ..........................................................................8 ### 2.1 Decentralized Competitive Equilibrium ### Model setup and key mechanisms - Dynamic stochastic general equilibrium model with a collateral constraint that limits borrowing to a fraction of the market value of collateral. - Financial innovation is modeled as the introduction of a new financial regime that changes agents’ ability to “collateralize,” and therefore loan-to-value ratios. - Agents do not have full information about the stochastic process driving credit conditions; they learn (Bayesian learning) about the transition probabilities of a Markov-switching process only as they observe realizations of high- and low-borrowing-ability regimes. - In the long run, absent further innovation, agents learn the true transition probabilities and form rational expectations; in the short run beliefs can display waves of optimism and pessimism depending on initial priors and observed market conditions. - The interaction between learning-driven beliefs and the collateral constraint generates a financial amplification feedback mechanism: - Optimistic beliefs → over-borrowing → higher risky asset prices → relaxed collateral constraint → further borrowing. - Pessimistic beliefs → under-borrowing → lower asset prices → tighter collateral constraint → amplified downturn. ### Learning, financial innovation, and equilibrium dynamics - The arrival of financial innovation can trigger an “optimistic phase”: a few observations of enhanced borrowing ability lead agents to understate asset risk, bid up risky assets, and amplify credit booms. - Conversely, observing a low-borrowing regime can trigger a “pessimistic phase,” leading agents to overstate tail risk, reduce debt, and amplify crashes through the collateral channel. - These belief-driven swings in debt and prices make decentralized dynamics materially different from full-information rational expectations outcomes. ### Pecuniary externality and macro-prudential rationale - The collateral constraint introduces a pecuniary externality: individual borrowers do not internalize the effect of their borrowing on asset prices, especially future prices in states of distress where the collateral feedback triggers crashes. - Because of this externality, decentralized equilibrium features more debt and financial crises that are more severe and frequent than would be socially desirable. - Macro-prudential policy (Pigouvian-style taxes on debt and dividends in the paper’s quantitative analysis) can be used to induce agents to internalize the externality and curb credit growth in good times. ### Planner information scenarios and implications for policy design - Two planner information regimes analyzed: - Uninformed planner: must learn about the true transition probabilities and faces feasible credit positions supported by the collateral values of the competitive equilibrium with learning. - Informed planner: knows the true transition probabilities and faces feasible credit positions consistent with collateral pricing in the full-information rational expectations competitive equilibrium. - Comparative insights: - The uninformed planner underestimates the probability of persistent low-borrowing regimes and thus has weaker incentives to accumulate precautionary savings; this leads the uninformed planner to allow larger debt positions than the informed planner. - The informed planner implements an optimal macro-prudential policy with a precautionary component that lowers borrowing at given asset prices and a portfolio-choice component that addresses mispricing effects on collateral values. - Even the uninformed planner has an incentive to use macro-prudential policy to tackle the pecuniary externality. - Effectiveness of macro-prudential policy depends on: - The planner’s information set (informed vs. uninformed). - The tightness of the borrowing constraint. - The pace at which optimism builds during early stages of financial innovation. - In the baseline calibration described in the paper: - Rapidly building optimism makes the borrowing constraint bind early for the uninformed planner, reducing the effectiveness of macro-prudential policy (i.e., debt positions and asset prices differ little between the DEL and the uninformed planner in that calibration). - When optimism builds more gradually, macro-prudential policy can be effective even for an uninformed planner with the same information set. ### Quantitative and welfare findings (high-level) - The combined effect of the pecuniary externality and agents’ optimistic subjective beliefs produces sizable welfare losses. - Maximum reported welfare loss: up to 7 percent in terms of a compensating variation in permanent consumption that equalizes the welfare of the informed planner with that of the decentralized equilibrium with learning (DEL). - Welfare losses attributable to the pecuniary externality alone are relatively small and decline at the peak of optimism. - The interaction of collateral constraints and learning produces larger amplification of credit booms and crashes than a comparable full-information environment, strengthening the case for macro-prudential intervention in many calibrations. ### Relation to existing literature (as framed in the source) - Builds on and extends Boz and Mendoza (2010) by incorporating social planning problems under alternative information sets and collateral pricing functions. - Complements Bianchi and Mendoza (2010) by introducing informational frictions (learning about transition probabilities) into the analysis of macro-prudential policy and the pecuniary externality. - Connects to the wider literature on financial amplification and crises (Fisher (1933), Minsky (1992), Bernanke, Gertler, and Gilchrist (1999), Kiyotaki and Moore (1997)) while departing from their full-information, rational expectations assumptions. - Contrasts with Gennaioli, Shleifer, and Vishny (2010) and Stein (2011) by focusing on model uncertainty about regime transition probabilities and Pigouvian tax instruments to address the pecuniary externality. *Source: 2.1 Decentralized Competitive Equilibrium, from the chapter “A Fisherian Model of Financial Innovation” in the provided IMF content unit.* ### 2.1 Decentralized Competitive Equilibrium ### 2.1 Decentralized Competitive Equilibrium ### Model setup and assets - Continuum of identical agents maximizing constant-relative-risk-aversion utility: - Preferences: E_s0[∑_{t=0}^∞ β^t c_{t}^{1−σ}/(1−σ)] (Equation (1)). - Agents choose consumption c_t, holdings of a risky asset (land) k_{t+1}, and a one-period discount bond b_{t+1}. - Land: - Traded in a competitive market, price q_t, fixed unit supply, identical agents imply market-clearing with identical holdings. - Land enters production Y(k_t) with productivity shock ε_t (finite-state, stationary Markov process, agents perfectly informed about ε). - Bonds: - Exogenous price equal to 1/R, where R is an exogenous gross real interest rate. - Interpretation: small open economy (b = net foreign assets, R = world interest rate) or partial equilibrium subset of borrowers in a closed economy (b = borrowers’ net credit market assets, R = economy’s risk free real interest rate). - Creditors supply funds at real rate R subject to collateral constraint (creditor behavior not modeled from first principles). ### Collateral constraint and stochastic collateral coefficient - Collateral constraint (agents’ debt limited to fraction κ_t of market value of individual land holdings): - − b_{t+1} / R_t ≤ κ_t q_t k_{t+1} (Equation (3)). - κ_t is stochastic and follows a Markov regime-switching process (two regimes κ_h and κ_l). - Information is imperfect about the true transition probability matrix of κ; agents learn about it by observing realizations of κ over time. - Constraint interpretation examples: limited enforcement of credit contracts, margin calls, loan-to-value limits, value-at-risk collateralization, mark-to-market capital requirements. ### Budget constraint and first-order conditions - Budget constraint: - q_t k_{t+1} + c_t + b_{t+1}/R_t = q_t k_t + b_t + ε_t Y(k_t) (Equation (2)). - Collateral multiplier μ_t is Lagrange multiplier of (3). - First-order conditions: - u′(t) = βR E_s_t[ u′(t+1) ] + μ_t (Equation (4)). - q_t ( u′(t) − μ_t κ_t ) = β E_s_t[ u′(t+1) ( ε_{t+1} Y_k(k_{t+1}) + q_{t+1} ) ] (Equation (5)). ### Equilibrium definitions - Decentralized competitive equilibrium with learning (DEL): - Sequence of allocations [c_t, k_{t+1}, b_{t+1}]_{t=0}^∞ and prices [q_t]_{t=0}^∞ that satisfy the above conditions using agents’ beliefs about κ evolution, collateral constraint (3), and market-clearing for goods and assets: - c_t + b_{t+1}/R_t = b_t + ε_t Y(k_t) - k_t = 1 - Decentralized competitive equilibrium with full information (DEF): - Defined identically but expectations use the true transition distribution of κ. ### Learning environment (Bayesian learning specification) - κ follows a two-point regime-switching Markov process with κ_h and κ_l. - True continuation transition probabilities: F^a_{hh} and F^a_{ll}; switching probabilities F^a_{hl} = 1 − F^a_{hh}, F^a_{lh} = 1 − F^a_{ll}. - Learning concerns beliefs about F^s_{hh} and F^s_{ll}; agents combine initial priors with observed κ realizations. - Bayesian learning model: beta-binomial probability model with exogenous initial priors determined by counters n_{ij}^0 (assumed independent). - Counters track transitions: [n_{hh}^t, n_{hl}^t, n_{ll}^t, n_{lh}^t]_{t=0}^T; update rule: n_{ij}^{t+1} = n_{ij}^t + 1 if κ_{t+1}=κ_j and κ_t=κ_i, else n_{ij}^{t+1} = n_{ij}^t. - Posterior means (key result used for solution method): - E_t[F^s_{hh}] = n_{hh}^t / (n_{hh}^t + n_{hl}^t) - E_t[F^s_{ll}] = n_{ll}^t / (n_{ll}^t + n_{lh}^t) (Equation (6)). - Implication: posterior means change only when the same regime is observed at date t; beliefs about persistence and mean durations update only upon observing κ_h or κ_l. ### Learning, debt, and price dynamics after financial innovation - Financial innovation experiment: introduction of a brand new environment switching between κ_h and κ_l, approximated by n_{ij}^0 close to zero. - First realizations of κ_h generate substantial optimism (sharp increase in E_t[F^s_{hh}] relative to F^a_{hh]); subsequent κ_h realizations have smaller incremental optimism. - First realizations of κ_l generate a pessimistic phase (E_t[F^s_{ll}] > F^a_{ll}). - Example numeric illustration: if n_{ij}^0 = 0.1 and observing five quarters of κ_h, E_t[F^s_{hh}] rises from 0.5 at t=0 to 0.98 at t=5 while E_t[F^s_{ll}] remains 0.5. - Land premium (expected excess return one period ahead): - Define R^q_{t+1} ≡ (ε_{t+1} Y_k(t+1) + q_{t+1}) / q_t. - E_s_t[ R^q_{t+1} − R ] = (1 − κ_t) μ_t − cov^s_t(β u′(c_{t+1}), R^q_{t+1}) / E_s_t[ β u′(c_{t+1}) ] (Equation (7)). - Mechanisms when collateral constraint binds: - Land premium rises due to: increased excess return from shadow value of collateral (limited to fraction (1−κ_t) μ_t), lower covariance between marginal utility and land returns, and increased expected marginal utility of future consumption (constraint hampers consumption smoothing). - Interaction of optimistic beliefs and collateral constraint: - In initial optimistic phase with κ_h current and μ_t > 0, optimistic beliefs (E_t[F^s_{hh}] > F^a_{hh}) lower perceived probability of switching to κ_l (which has higher land returns), reducing expected land premium E_s_t[R^q_{t+1}] and thus increasing land price q_t relative to full information. - Higher q_t raises collateral values, enabling more borrowing; as collateral values rise μ_t may fall, further lowering land premia and amplifying price increases — amplification is nonlinear and stops if constraint becomes nonbinding. - Reversal on observation of κ_l (Fisherian deflation mechanism): - First κ_l observation raises E_t[F^s_{ll}] above F^a_{ll}, increasing expected land premia and lowering asset prices relative to full information. - Falling asset prices (if κ_l is current) tighten the collateral constraint (μ_t rises), triggering fire sales, higher premia, and further price declines — feedback amplifies downturn. ### Recursive Anticipated Utility (AU) competitive equilibrium solution method - Two-stage solution approach: - Stage 1: Use Bayesian learning to generate sequence of posterior means via Equation (6). - Stage 2: Adopt Kreps’s Anticipated Utility approach to solve a chain of conditional AU optimization problems (AUOP) using date-t posterior means as the transition matrix beliefs for κ. - Date-t AUOP: - Agents observe κ_t, form posterior means E_t[F^s_{hh}] and E_t[F^s_{ll}], construct date-t beliefs about transition matrix E^s_t[κ′|κ] ≡ [[E_t[F^s_{hh}], 1−E_t[F^s_{hh}]]; [1−E_t[F^s_{ll}], E_t[F^s_{ll}]]]. - Solve for policy functions b′_t(b, ε, κ), c_t(b, ε, κ), μ_t(b, ε, κ) and pricing function q_t(b, ε, κ) that satisfy recursive equilibrium conditions: - u′( c_t(b, ε, κ) ) = βR [ ∑_{ε′∈E} ∑_{κ′∈{κ_h,κ_l}} E^s_t[κ′|κ] π(ε′|ε) u′( c_t(b′, ε′, κ′) ) ] + μ_t(b, ε, κ) (Equation (9)). - q_t(b, ε, κ) [ u′( c_t(b, ε, κ) ) − μ_t(b, ε, κ) κ ] = β [ ∑_{ε′∈E} ∑_{κ′∈{κ_h,κ_l}} E^s_t[κ′|κ] π(ε′|ε) u′( c_t(b′, ε′, κ′) ) [ ε′ Y(1) + q_t(b′, ε′, κ′) ] ] (Equation (10)). - c_t(b, ε, κ) + b′_t(b, ε, κ) / R = ε Y(1) + b (Equation (11)). - b′_t(b, ε, κ) / R ≥ − κ q_t(b, ε, κ) 1 (Equation (12)). - Time subscripts indicate date of beliefs used to form expectations; the AUOP solves for full set of optimal plans over (b, ε, κ) conditional on date-t beliefs (plans conjectured for the infinite future under those beliefs), but only the date-t plans are enacted — beliefs evolve and AUOP solutions change over time. - Recursive AU competitive equilibrium definition: - Given a T-period history κ_T = (κ_T, κ_{T−1}, ..., κ_1), a recursive AU competitive equilibrium is a sequence of decision rules [b′_t(b, ε, κ), c_t(b, ε, κ), μ_t(b, ε, κ)]_{t=1}^T and pricing functions [q_t(b, ε, κ)]_{t=1}^T such that: - (a) decision rules and pricing function for date t solve date-t AUOP conditional on E^s_t[κ′|κ]; - (b) E^s_t[κ′|κ] is the conjectured transition probability matrix produced by date-t posterior density of F^s determined by Bayesian passive learning (Equation (6)). - Actual equilibrium dynamics are obtained by chaining date-t AUOP solutions across dates, e.g., b_2 = b′_1(b, ε, κ), b_3 = b′_2(b, ε, κ), ..., b_{T+1} = b′_T(b, ε, κ). *Source: _wp12181 - 2.1 Decentralized Competitive Equilibrium* ### 2.5 Conditionally Efficient Planners’ Problems ### 2.5 Conditionally Efficient Planners’ Problems ### Definition of the planners and key distinction from the DEL - Two versions of an optimal policy problem: benevolent social planners who maximize agents’ utility subject to the resource constraint and the collateral constraint, but who internalize effects of borrowing on collateral asset prices. - Planners face the same borrowing ability at every state as agents in a competitive equilibrium; hence planners implement the same pricing function for valuation of collateral as in the decentralized equilibrium (DEL). They can, however, choose debt to alter future land values. - This formulation corresponds to the concept of conditional or financial efficiency (Kehoe and Levine(1993); Lustig(2000)) and follows Bianchi and Mendoza(2010). - Advantages of the conditional-efficiency formulation: - Time-consistent optimization problem. - Simpler characterization and decentralization via Pigouvian taxes on debt and dividends. - Even with constrained efficiency, correcting fire-sale externality can sharply reduce probability and severity of financial crises. ### Two planners: uninformed (SP1) and fully informed (SP2) - SP1 (uninformed planner): - Observes the same history κT and learns from date-0 priors (pij0) as private agents. - Prices collateral using DEL’s collateral pricing functions qDELt(b, ε, κ), ensuring same feasible credit positions as private agents in the DEL. - Baseline scenario sets pij0 = nij0 so SP1 and private agents have identical beliefs at all times. - SP2 (fully informed planner): - Knows Fa hh and Fa ll and prices collateral using the time-invariant pricing function of the DEF qDEF(b, ε, κ). - Conditional efficiency implies SP2 implements the same feasible credit positions as private agents in the DEF. ### Planners’ intertemporal optimization (as stated) - Objective for i = SP1, SP2: - E i 0 [ ∑∞ t=0 βt c1−σ t /(1−σ) ] (equation (13) as in source) - Constraints: - c t + b t+1 / R t = b t + ε t Y(1) (equation (14)) - − b t+1 / R t ≤ κ t q i t (b, ε, κ) (equation (15)) - qSP1 t = qDEL t and qSP2 t = qDEF - Recursive first-order conditions (Euler equation for bonds and budget/feasibility): - u′(c t(b, ε, κ)) − μ t(b, ε, κ) = βR [ ∑ε′∈E ∑κ′∈{κh,κl} Ei t[κ′|κ] π(ε′|ε) [ u′(c t(b′, ε′, κ′)) + κ′ μ t(b′, ε′, κ′) ∂q i t(b′, ε′, κ′)/∂b′ ] ] (equation (16) preserved) - c t(b, ε, κ) + b′ t(b, ε, κ)/R = ε Y(1) + b (equation (17)) - b′ t(b, ε, κ)/R ≥ − κ q i t(b, ε, κ) 1 (equation (18)) - Expectations: - For SP1: ESP1 t[κ′|κ] ≡ [ E t[Fg hh] 1−E t[Fg hh] ; 1−E t[Fg ll] E t[Fg ll] ] (as in source) - For SP2: Ei t[κ′|κ] ≡ [ Fa hh 1−Fa hh ; 1−Fa ll Fa ll ] for i = SP2 ### Equilibrium definitions (recursive) - SP1 Equilibrium: - Given DEL time-varying asset pricing functions [qDEL t(b, ε, κ)]T t=1, a recursive AU equilibrium for SP1 is a sequence of decision rules [b′ t(b, ε, κ), c t(b, ε, κ), μ t(b, ε, κ)]T t=1 such that: - (a) decision rules for date t solve SP1’s date-t AUOP conditional on Eg t[κ′|κ]; - (b) elements of Eg t[κ′|κ] are posterior means from the date-t posterior densities of Fg hh and Fg ll determined by Bayesian learning. - SP2 Equilibrium: - Given DEF time-invariant asset pricing function qDEF(b, ε, κ), a recursive AU equilibrium for SP2 is given by time-invariant decision rules [b′(b, ε, κ), c(b, ε, κ), μ(b, ε, κ)] that solve SP2’s date-t AUOP conditional on Ea[κ′|κ] for all t. ### Interpretation of the pecuniary externality - Key difference between planners’ FOCs and DEL agents’ FOCs is the pecuniary externality in the right-hand side of the planner Euler equation for bonds (eq. (16)): - Planners internalize how debt choice today b′ alters next-period price q i t via ∂q i t(b′, ε′, κ′)/∂b′ when the collateral constraint is expected to bind (μ t(b′, ε′, κ′) > 0). - The derivative represents response of land price tomorrow to changes in debt today; can be very steep when the collateral constraint binds because of Fisherian deflation. - Behavioral differences across planner and DEL: - SP1 shares private agents’ optimism if pij0 = nij0, faces optimistic collateral prices via qDEL, and therefore may assign very low probability to crash transitions κh → κl; this can reduce macro-prudential policy effectiveness when optimism builds quickly and collateral constraint binds tightly. - SP2, knowing true Fa hh and Fa ll, assigns higher probability to κh → κl transitions than DEL, prompting stronger precautionary savings and lower borrowing; SP2 generally acquires less debt and experiences lower land price booms than SP1 and DEL. ### Decentralization via Pigouvian taxes - Implementation device: Pigouvian taxes on debt (τ i b,t) and land dividends (τ i l,t) can fully implement planner allocations for i = SP1, SP2. - Private agents’ budget constraint with taxes (as in source; equation (19)): - q t k t+1 + c t + b t+1 / R t (1 + τ i b,t) = q t k t + b t + ε t Y(k t)(1 − τ i l,t) + T i t. - Ti t are lump-sum transfers rebating tax revenue (or lump-sum tax if negative rates). - Competitive equilibrium Euler equations with macro-prudential policy (equations (20) and (21) preserved): - u′(t) = βR(1 + τ i b,t) E s t[ u′(t+1) ] + μ t (equation (20)) - q t (u′(t) − μ t κ) = βE s t[ u′(t+1) ( ε t+1 Yk(k t+1)(1 − τ i l,t) + q t+1 ) ] (equation (21)) - Computation: - State-contingent, time-varying tax schedules τ i b,t(b, ε, κ) and τ i l,t(b, ε, κ) are computed by replacing each planner’s allocations into (20)-(21) and solving for tax rates so DEL with macro-prudential policy supports planner allocations and corresponding asset pricing functions. - Debt tax needed to replicate planner debt choices; dividends tax needed to support planner pricing functions. - Non-uniqueness when collateral constraint binds: - When μ t > 0, multiple tax schedule representations can implement planner allocations because b t+1 is determined by collateral constraint rather than bond Euler equation. - For simplicity, the chosen representation sets τ i b,t = 0 when the collateral constraint binds; then μ t = u′(t) − βR Es t[u′(t+1)] and τ i l,t follows from (21). ### Decomposition of the debt tax - Debt tax expression (equation (22) preserved) decomposed into three interpretable terms: - τ i b,t = - [ Ei t[u′(t+1)] / Es t[u′(t+1)] − 1 ] (information term) - + [ Ei t[ κ t+1 μ t(t+1) ∂q i t(t+1)/∂b′ ] − Es t[ κ t+1 μ t(t+1) ∂q i t(t+1)/∂b′ ] ] / Es t[u′(t+1)] (interaction term) - + [ Es t[ κ t+1 μ t(t+1) ∂q i t(t+1)/∂b′ ] ] / Es t[u′(t+1)] (externality term) - Interpretation: - Information term: deviation in one-period-ahead expected marginal utilities of planner and private agents due to differing information sets; vanishes when beliefs identical (e.g., baseline SP1), nonzero for SP2. - Interaction term: difference in expected value of the externality when evaluated with planner’s beliefs versus private agents’ beliefs; zero when either information sets are the same or DEL far from binding region. - Externality term: value of the externality evaluated using private agents’ beliefs. ### Role in quantitative analysis - The quantitative analysis (Section 3) uses the above planner formulations to: - Compare DEL with SP1 and SP2. - Quantify macro-prudential tax schedules that decentralize planner allocations and decompose taxes into the three components above. *Source: IMF Working Paper excerpt — "2.5 Conditionally Efficient Planners’ Problems" (as provided).* ### 47.6 basis points and the model matches it withn ### _wp12181 - 47.6 basis points and the model matches it withn ### Baseline Results: dynamics, beliefs, and asset/debt responses - Experiment setup: - Learning period t = 1, ..., 48; TFP held at mean (ε = 1); initial condition b0 = −0.345 corresponding to the net credit market assets-GDP ratio of U.S. households observed in 1996Q4. - Simulations chain date-t AUOP decision rules for each equilibrium (DEL, SP1, SP2). - Belief evolution (Panel (d) description): - E_t[F^s_hh] rises from 0.980 to 0.999 from t = 1 to t = 40 during the optimistic phase. - At date 41, when κ switches to κ_l for first time: - E_41[F^s_hh] falls to 0.975. - E_41[F^s_ll] rises sharply from 0.5 to 0.98. - True regime-switching transition probabilities: F^a_hh = F^a_ll = 0.95 (symmetric κ process). - Degree of optimism/pessimism measured by excess of E_t[F^s_hh] over F^a_hh (and F^a_ll over E_t[F^s_ll]). - Implications of small belief changes: - Expected mean duration of κ_h: - With E_1[F^s_hh] = 0.98 → 50 quarters. - With E_40[F^s_hh] = 0.999 → 1,000 quarters. - True mean duration implied by F^a_hh = 0.95 → 28 periods. - Coefficient of variation of κ based on date-40 beliefs ≈ 1/4 of that based on date-1 beliefs. - During optimistic phase, E_t[F^s_ll] = 1/2 implies projected mean duration for κ_l of 2 periods, while true mean duration is 28 periods. - Debt and land-price dynamics (Panel (a) and (b)): - DEL: large sustained increase in debt (decline in bonds) for first 40 periods; sharp correction at date 41. - This increase in debt accounts for about 2/3rds of the observed rise in net credit liabilities of U.S. households. - DEL: surge in risky asset price roughly equal to 44 percent the observed rise in U.S. residential land prices. - Comparison of equilibria: - SP1 chooses only slightly smaller debt (higher bonds) than DEL; bond holdings and asset prices nearly identical to DEL starting t = 7 in baseline. - SP2 chooses much smaller debt levels than SP1 and DEL during optimistic phase; on average avoids hitting borrowing constraint during t = 1,...,40 and thus μ_t = 0 for t = 1,...,40. - Mechanisms behind SP1 ≈ DEL in baseline: - Rapid surge in optimism makes collateral constraint bind tightly; households’ willingness to borrow implies high shadow value of relaxing constraint. - SP1 values current consumption highly and thus borrows up to the limit despite positive externality; externality is not strong enough to offset borrowing incentive. - Externality dynamics (Panel (e) and Figure2 description): - Externality term: E_t[κ_{t+1} μ_t(t+1) ∂q^i_t(t+1)/∂b′] for i = SP1, SP2. - At t = 40, q^DEL_{40}(b,1,κ_h) is relatively flat ⇒ ∂q^DEL_{40}(·,κ_h)/∂b′ small; q^DEL_{40}(·,κ_l) very steep but carries negligible weight because SP1 perceives switching probability as near zero. - Externality weakens over optimistic phase as perceived probability of κ_h → κ_l falls (E_t[F^s_hl] close to zero), despite steeper pricing function at higher debt. - SP2 versus SP1 and DEL: - SP2 uses true transition probabilities and perceives higher risk: stronger precautionary savings, chooses much lower debt, supports lower land prices. - SP2 prices lower than SP1 because SP1 faces DEF pricing function q_DEF(b′,1,κ_h) that is affected by optimistic beliefs; under full information the variability of κ dominates and land prices fall slightly for SP2. - Crisis episode (first κ_l realization at t = 41; Figure4 event window ±7 quarters): - SP2 (lowest debt) experiences smallest debt correction at t = 41; correction mainly reflects exogenous tightening due to realized lower κ and cannot be avoided even with full information. - DEL and SP1 (higher debt) experience much larger corrections; SP1’s correction at date 41 is more than twice as large as SP2’s. - Price declines rank: SP2 smallest, SP1 and DEL larger; differences across equilibria shrink as beliefs converge toward rational expectations. - Macro-prudential taxes as DEL implementations of planners (Figure5 and Figure6): - SP1 tax policy: - Debt tax τ_b ≈ 2-3 percent in first seven periods after innovation. - Dividend subsidy τ_l up to 3 percent initially; later τ_b drops to zero and τ_l rises to about 2 percent. - Information and interaction components of debt tax are zero for SP1; externality term accounts for full amount of SP1’s debt taxes, rising up to ≈ 3 percent before vanishing after period 7. - SP2 tax policy: - Debt tax τ_b increases gradually from ≈ 4 percent to 8 percent during optimistic phase, then cuts to zero as crisis erupts. - Dividend subsidy τ_l increased early and kept at ≈ 5 percent until crisis, then falls to almost zero. - Decomposition: SP2’s debt taxes driven mainly by large information and interaction components (stabilize at about 3.3 and 4.2 percent respectively); externality component relatively small and slightly positive throughout optimistic phase. - Comparison note: magnitude of externality tax component comparable to debt taxes estimated by Bianchi and Mendoza (2010) in a related model. ### Welfare Analysis and quantitative gains/losses - Welfare evaluation method: - Perceived lifetime welfare V_t(b, ε, κ) computed from date-t AU solution: V_t = u(c_t(b, ε, κ)) + β E_i[V(b′_t(b, ε, κ), ε′, κ′)]_{i=s,a}, where expectations computed using either true transition probabilities or subjective beliefs. - Convert V_t into constant consumption level ĉ solving V_t = Σ_{t=0}^∞ β^t ĉ^{1−σ}/(1−σ). - Report average welfare effects using perceived ergodic distribution of (b, ε, κ) at each date in the DEL. - Welfare gains/losses (Table 2 reported values, in percentage; preserved verbatim): - True probabilities (average (b^DEL_t, κ_t, ε_t), t = 1, t = 40): - (1) SP2 versus DEF: 0.052 0.05 - (2) SP2 versus DEL: 0.37 7.4 - (3) SP1 versus DEL: 0.17 0.03 - Subjective beliefs: - (4) SP1 versus DEL: 0.025 0.0 - (5) DEL versus SP2: -0.39 -2.7 - (Also reported variants fixing ε_t = E[ε] = 1 and κ_t = κ_h; results similar.) - Main welfare conclusions: - Largest gains when comparing SP2 v. DEL using true transition probabilities: - Average welfare gain ≈ 0.37 percent at t = 1. - Welfare gain = 7.4 percent at peak of optimism (t = 40). - Interpretation: SP2 internalizes pecuniary externality and corrects informational friction; large gains largely due to removing informational friction and associated financial amplification mechanism. - Using subjective beliefs in welfare computation (DEL assessed under its biased beliefs) yields: - DEL losses versus SP2 of about -0.39 percent at t = 1 and -2.7 percent at t = 40. - Absolute value of the t = 1 loss ≈ gain of SP2 over DEL under true probabilities; at t = 40 the subjective-belief loss is about 1/3 the gain under true probabilities. - Isolating benefits of internalizing pecuniary externality alone: - SP2 v. DEF using true probabilities and SP1 v. DEL using subjective beliefs yield modest welfare gains up to 0.052 percent. *Source: content unit _wp12181 - 47.6 basis points and the model matches it withn (IMF PDF chapter/section).* ### 3.4 Sensitivity Analysis ### 3.4 Sensitivity Analysis ### Purpose and setup - Studies how the parameterization of the initial priors affects baseline results. - Two sets of sensitivity experiments altering the initial priors: - Induce a gradual buildup of optimism (labeled "Gradual Optimism"). - Introduce heterogeneous priors between private agents and the social planner. - In all experiments, the social planner’s optimization problem is analogous to that solved by SP1 in problem (13). - Table 3 (as reported in source) summarizes the values of the initial counters that characterize the initial priors for the experiments. Reported entries (DEL & SP1 baseline and Gradual Optimism) include: - Baseline: n_hh0 = 0.02, n_hl0 = 0.02, n_ll0 = 0.02, n_lh0 = 0.02 - Gradual Optimism: n_hh0 = 7.6, n_hl0 = 0.4, n_ll0 = 0.38, n_lh0 = 0.02 - SP2 & SP3 → ∞ (indicated in table) - SP4: n_hh0 = 0.2, n_hl0 = 0.2, n_ll0 = 0.2, n_lh0 = 0.2 ### (a) Gradual Optimism — experiment design - Initial priors are the same for government and private sector but constructed so date-0 posterior means equal the true transition probabilities. - Initial counters are asymmetric across the four transitions: - n_hh0 = 7.6 (higher than baseline) - n_hl0 = 0.4 (set so that E0[F^s_hh] = F^a_hh = 0.95) - n_lh0 = 0.02 (kept as in baseline) - n_ll0 = 0.38 (set so that E0[F^s_ll] = F^a_ll = 0.95) - Learning behavior: - Learning starts from E0[F^s_hh] = 0.95 and rises gradually towards 1. - In baseline, learning starts at E0[F^s_hh] = 0.5 and jumps to 0.98 with the first κ_h observation. - Under gradual optimism, reach 0.98 after 12 observations of κ_h. - Private agents and SP1 still do not know true transition probabilities; beliefs can shift away from true values as realizations of κ arrive, converging in the long run. ### (a) Gradual Optimism — key quantitative outcomes and mechanisms - Debt and prices: - In the run-up to the crisis, debt levels in DEL reach about 4 percentage points of GDP more than in SP1. - During the crash, asset prices are 16 percent higher for SP1 due to lower leverage at the time of the crisis. - Mechanism: - Under baseline, rapid surge in optimism plus households’ impatience leads households and SP1 to borrow up to the limit and attain a high shadow value from relaxing the collateral constraint. - Under gradual optimism the collateral constraint is more likely to remain slack or marginally binding during the optimistic phase; SP1 accumulates less debt and macro-prudential policy is more effective even if planner and private agents face the same learning problem. - DEL and SP1 prices remain very similar because the DEL pricing function in the κ_h regime is relatively flat. - The externality is larger because the pricing function is steep in the κ_l regime and gradual optimism raises SP1’s assigned probability of switching to κ_l; SP1 levies larger debt taxes in this scenario than in the baseline; SP2 charges slightly lower debt taxes. - Debt tax components (qualitative): - Gradual optimism reduces the information component for SP2 (recall it is always zero for SP1). - The interaction term is smaller than baseline for SP2. - The externality component of the taxes rises sharply for both planners under gradual optimism, relative to baseline. ### (b) Heterogeneous Priors between Government and Private Agents — experiment design - Private agents’ priors kept as in baseline DEL. - Two planner variants: - SP3: planner’s initial counters go to infinity under conditions p_hl0 / p_hh0 = F^a_hl / F^a_hh and p_lh0 / p_ll0 = F^a_lh / F^a_ll, effectively making the planner’s beliefs converged to the true transition probabilities. - SP4: planner has initial priors p_hh0 = p_ll0 = 0.2 so that E_p0[F_hh] = E_p0[F_ll] = 0.5 (symmetric priors). - Both SP3 and SP4 still value collateral using the DEL land pricing functions q_DEL_t(b,1,κ), which are influenced by private agents’ beliefs. ### (b) Heterogeneous Priors — SP3 outcomes and mechanisms - Debt and prices: - SP3 chooses lower debt levels than SP1 and DEL during the optimistic phase. - SP3 still allows larger debt positions than SP2 because SP3 cannot correct agents’ mispricing of collateral under DEL beliefs. - Land prices of SP1 and SP3 are similar because both use q_DEL_t(b,1,κ) and ∂q_DEL_t(b′,1,κ_h)/∂b′ is small for t = 1, ..., 40. - Taxes and components: - SP3 actively uses macro-prudential taxes: debt taxes increase gradually from about 4 percent to close to 9 percent in the optimistic phase, then drop to zero as the financial crisis erupts and the collateral constraint binds. - Dividends tax policies: SP3 and SP2 qualitatively similar (subsidies rising during optimistic phase); quantitatively SP2 uses smaller subsidies because SP2 aims to support the DEF asset pricing functions, which are uniformly lower than DEL pricing functions supported by SP3. - Dynamics of tax components: - Externality component remains small for SP3. - Information and interaction components are large and rise gradually during the optimistic phase. - SP3 displays a lower information component than SP2, and a higher interaction component. - The higher interaction term for SP3 is attributed to higher expected externality terms under DEL pricing functions in κ_l relative to DEF. ### (b) Heterogeneous Priors — SP4 outcomes and mechanisms - Priors and perceived risk: - SP4 has higher initial counters for persistence of each regime than DEL or SP1; for example, at date t = 1 after first κ_h realization SP4 expects mean duration of κ_h to be about 6 quarters while private agents and SP1 expect mean duration of 50 quarters under baseline calibration n_hh0 = 0.0205. - SP4 perceives more riskiness and optimism builds more gradually for SP4 than for DEL and SP1. - Debt, constraint, and prices: - SP4 chooses lower debt positions than DEL and lower than SP1. - Under these forces, the collateral constraint is not binding for SP4 during the entire optimistic phase; SP4 hits the borrowing limit only when the economy switches to the κ_l state (shadow price almost always zero except period 41 and last few periods). - Externality term is uniformly higher for SP4 than SP1 because SP4’s beliefs are uniformly less optimistic and assign higher weight to κ_l where ∂q_DEL_t(b′,1,κ_l)/∂b′ is large. - Price dynamics of SP4 are very similar to DEL; lower debt choices do not translate into large price differences given flatness of pricing function; SP4 prices are slightly above those of DEL since lower debt positions are associated with higher land prices. - Taxes and components: - SP4 levies debt taxes higher than baseline and higher than in gradual optimism scenario. - Interaction component of the debt tax is the largest almost throughout the experiment for SP4 (as was the case for SP2 and SP3 in baseline). - The interaction of financial and information frictions remains key to macro-prudential policy design under heterogeneous planner beliefs. ### Cross-experiment synthesis — robustness and mechanisms - Baseline vs alternatives: - Baseline: SP1’s macro-prudential policy makes little difference during optimistic phase because rapid surge in optimism and impatience lead both households and SP1 to borrow up to limits; DEL and SP1 have similar debt and land price dynamics. - Gradual optimism and heterogeneous-prior scenarios: SP1 (or a planner with different initial priors) can choose materially different (lower) debt paths than DEL and improve outcomes at crisis, making macro-prudential policy more effective even when the planner faces similar information as private agents. - Common mechanisms highlighted: - Flatness of DEL pricing function in κ_h regime dampens price differences across planners even when debt choices differ. - Steepness of pricing function in κ_l regime amplifies the externality and raises the value of debt taxes when planners assign higher probability to κ_l. - Interaction term (between pecuniary externality and information frictions) is quantitatively important for the design and magnitude of macro-prudential taxes under multiple scenarios. - Quantitative markers preserved exactly where reported: - Debt gap in run-up: about 4 percentage points of GDP (DEL vs SP1 under gradual optimism). - Asset price gap during crash: 16 percent higher for SP1 (gradual optimism). - SP3 debt tax path: increases from about 4 percent to close to 9 percent in optimistic phase, then drops to zero at crisis. ### Policy implications from sensitivity analysis - Effectiveness of macro-prudential policy depends critically on initial priors and information sets: - If regulators (planners) operate with the same incomplete information as private agents, macro-prudential policies can be limited or negligible. - If regulators can acquire better information (e.g., from historical episodes), macro-prudential policy has stronger potential to contain boom-bust cycles. - The interaction between perception of risk (learning/priors) and the pecuniary externality introduced by collateral constraints is central to amplification of credit booms and crisis severity. - The information and interaction components of debt taxes can be large and sensitive to priors; externality component can rise sharply when beliefs assign more weight to the risky κ_l regime. _Italic: Source — Content unit: _wp12181 - 3.4 Sensitivity Analysis (extracted from provided PDF content)._ ### 3. Land prices satisfyq(B, ε) =E ### 3. Land prices satisfyq(B, ε) =E ### Equilibrium condition for land prices - Land price condition (as presented): - q(B, ε) = E_{ε′|ε} { β u′(ˆc(Γ(B,ε), ̄K, Γ(B,ε), ε′)) [ ε′ F_k( ̄K, ε′) + q(Γ_t(B,ε), ε′) ] u′(ˆc(B, ̄K, B, ε)) − κ_max [ 0, u′(ˆc(B, ̄K, B, ε)) − β R E_{ε′|ε} u′(ˆc(Γ(B,ε), ̄K, Γ(B,ε), ε′)) ] } - The expression preserves the exact structure and notation as in the source (expectations E_{ε′|ε}, consumption marginal utilities u′(·), policy function Γ(·), production derivative F_k(·), adjustment κ_max, and discount/return parameters β and R). ### Goods and asset markets clearing conditions - Goods and asset market clearing conditions (as presented): - ˆb′(B, ̄K, B, ε) R + c(B, ̄K, B, ε) = ε f( ̄K) + B_t - ˆk(B, ̄K, B, ε) = ̄K ### Figures — dynamics, scenarios, and tax decompositions (figure notes and labels preserved) - Figure 1: Dynamics in the Baseline Calibration - Panels: (a) Bonds; (b) Land Price; (c) Shadow Price; (d) Beliefs; (e) Externality - Notes clarify series labels: DEL, SP1, SP2 and denote what each series represents: - DEL: Imperfect information decentralized equilibrium - SP1: Social planner with imperfect information implementing the set of feasible credit positions of DEL - SP2: Social planner with full information implementing the set of feasible credit positions of DEF - Figure 2 and Figure 3: Period 40 and Period 41 Bond Holdings and Asset Prices - Panels: (a) Bond Holdings: b′(b,1,κ_h); (b) Asset Prices: q(b,1,κ_h); (c) Bond Holdings: b′(b,1,κ_l); (d) Asset Prices: q(b,1,κ_l) - Notes reiterate SP1 and SP2 definitions and indicate DEL and DEF series in asset price panels - Figure 4: Crisis Episode - Plots time series dynamics in periods 41±7 for: (a) Bonds; (b) Land Price; (c) Shadow Price; (d) Consumption - Series labels: DEL, SP1, SP2 - Figure 5: Taxes on Debt and Land Dividends - Panels: (a) Taxes on Debt; (b) Taxes on Dividends - Notes: "This figure plots the taxes on debt and on land dividends that support the corresponding planners allocations as competitive equilibrium." Series: SP1, SP2 - Figure 6: Decomposition of Taxes on Debt - Panels: Information; Interaction; Externality - Notes: Decomposition definitions: - "Information" arises due to the differences in the expectation of one period ahead consumption between private agents and the social planner - "Externality" captures the pecuniary externality - "Interaction" is due to the differences in the expectation of the one period ahead externality between private agents and the social planner - Series: SP1, SP2 - Figure 7: Priors - Plots E[f_hh] and E[f_ll] under scenarios labeled: DEL & SP1 Baseline; SP1 Gradual Learning; SP3; SP4 - Notes: Definitions for DEL, SP1, Gradual Learning, SP3, SP4 are preserved exactly - Figures 8–11: Gradual Optimism calibration results - Figure 8 panels: (a) Bonds; (b) Land Price; (c) Shadow Price; (d) Beliefs; (e) Externality - Figure 9 panels: Period 40 Bond Holdings and Prices under Gradual Optimism (same panel labels as Figures 2–3) - Figure 10 panels: Taxes on Debt and Land Dividends: Gradual Optimism (panels (a) and (b)) - Figure 11 panels: Decomposition of Taxes on Debt: Gradual Optimism (Information; Interaction; Externality) - Notes preserve DEL, SP1, SP2 labels and the interpretation of decomposed tax components - Figures 12–13: Asymmetric Priors calibration and taxes on debt - Figure 12 panels: (a) Bonds; (b) Land Price; (c) Shadow Price; (d) Agents' Beliefs; (e) Externality; (f) Planners' Beliefs - Series: DEL, SP3, SP4 (with SP3 and SP4 definitions preserved) - Figure 13 panels: Taxes on Debt for SP3 and SP4 decomposed into Total, Information, Interaction, Externality - Notes explicitly restate the decomposition interpretation and planner scenario definitions ### References cited in the content unit (titles and authors preserved as listed) - Aiyagari, R., and M. 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