## wpiea2019184-print-pdf ## Source details **Canonical URL:** [wpiea2019184-print-pdf](https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019184-print-pdf.pdf) ## Other formats - [Markdown version](/-/media/files/publications/wp/2019/wpiea2019184-print-pdf.pdf.md) - [Structured JSON version](/-/media/files/publications/wp/2019/wpiea2019184-print-pdf.pdf.json) --- ### Paper structure and focus - Sections overview: - Section 3 presents the baseline model. - Section 4 analyzes optimal policy. - Section 5 shows numerical results. - Section 6 concludes. - Main focus: design of optimal macroprudential policies in the presence of asset price bubbles and occasionally binding borrowing constraints, with emphasis on ex-ante macroprudential regulation that decentralizes the social planner’s allocations. ### Literature and contribution - Two main strands connected: - Rational asset price bubbles literature. - Optimal macroprudential policy literature. - Frictions motivating bubbles: financial frictions, informational frictions, agency problems. - Closest related model: Miao-Wang (stock-price bubble in production economy with borrowing restricted by firm value); this paper augments by allowing occasionally binding borrowing constraints. - Distinct contributions: - Full normative (Ramsey) analysis with time-consistent planner (no commitment). - Models full non-linear dynamics of stochastic bubbles under occasionally binding borrowing constraints. - Policy focus on financial regulation (tax on borrowing) rather than monetary policy. - Key conceptual insight: bubbles both relax borrowing constraints (potentially beneficial) and create pecuniary externalities (potentially harmful); optimal policy depends non-monotonically on existing debt levels. ### Baseline model environment and key modeling choices - Small open economy with fixed capital supply K_t = 1 and assumption βR < 1. - Agents: - Households: consume, supply labor, own firms, trade firm shares frictionlessly. - Firms: operate production combining capital, labor, intermediate goods; borrow internationally; do not trade shares of other firms. - Borrowing requires advance financing of fraction θ ≤ 1 of p_v v_t. - Firm market value includes a bubbly component b_t not tied to fundamentals, with collateral B_t = β E_t[ (U_{c,t+1} / U_{c,t}) b_{t+1} ]. - Borrowing constraint (limited commitment / pledgeability): - L_{t+1}/R + θ p_v v_t ≤ m_t E_t[ (β U_{c,t+1}/U_{c,t}) V_{t+1}(k_t,0,b_{t+1}) ] with m_t < 1. - Conjectured firm value: - V(k_t, L_t, b_t) = a_t k_t + s_t L_t + b_t with s_t = −1 and a_t, b_t defined by first-order conditions and (time-varying) equilibrium objects. ### Competitive equilibrium — firm optimality and bubble conditions (Proposition 1 & 2) - Representative firm optimality conditions (selected): - (i) 1 = β E_t[ (U_{c,t+1} / U_{c,t}) R ] + μ_t - (ii) q_t = β E_t[ (U_{c,t+1} / U_{c,t}) ( z_{t+1} F_{k,t+1} + q_{t+1} (1 + m_{t+1} μ_{t+1}) ) ] - (iv) p_v (1 + θ μ_t) = z_t F_{v,t} - (v) b_t = (1 + m_t μ_t) β E_t[ (U_{c,t+1} / U_{c,t}) b_{t+1} ] - (vi) b_t = (1 + m_t μ_t) B_t - (vii) Complementary slackness: μ_t [ m_t (q_t k_t + B_t) − L_{t+1}/R − θ p_v v_t ] = 0 - Bubble existence and transversality (Proposition 2): - Deterministic steady state cannot sustain b > 0; focus on stochastic bubbles that burst with probability 1 − π. - If min(m_t) ≤ m (threshold defined in appendix), the borrowing constraint binds infinitely often: lim_{t→∞} sup μ_t > 0, allowing support of a rational stochastic bubble. - Equilibrium selection: bubble law of motion b^*_{t+1} solves fixed-point condition derived from the bubble accumulation equation and agents’ expectations. ### Planner’s environment and instruments (time-consistent Ramsey planner) - Planner’s sole instrument: state-contingent tax on borrowing τ_t with tax revenues rebated lump-sum T_t = τ_{t-1} L_t. - Decentralized resource constraint with tax: - c_t + L_t(1 + τ_{t-1}) + p_v v_t ≤ z_t F(1,l_t,v_t) + L_{t+1} R + T_t - Planner solves time-consistent (no commitment) maximization of E_t Σ_{s=0}^{∞} β^s U(c_{t+s} − G(l_{t+s})) subject to resource constraint, borrowing constraint expressed using B_t = b_t/(1 + m_t μ(l_t,v_t)), forward-looking q condition, and labor FOC. - Planner internalizes how current choices affect future bubble growth b^*_{t+1} via the bubble accumulation condition and thereby internalizes three externalities absent in private decisions. ### Decomposition of planner’s consumption FOC — externalities (equation (34)) - λ^p_t = U_{c,t} + Fundamental-price externality (F_t q_t) + Bubble-accumulation externality (V_t βE_t (...) db^*_{t+1}/dc_t) + Bubble-pledgeability externality (W_t dB_t/dl_t) - Interpretations: - Fundamental-price externality F_t q_t > 0: standard pecuniary externality from higher current consumption raising q_t and relaxing future borrowing constraints. - Bubble-pledgeability externality W_t dB_t/dl_t > 0: production choices affect collateral B_t via μ(l_t,v_t); this amplifies the fundamental-price externality. - Bubble-accumulation externality V_t·…: captures how current choices influence future bubble growth and thus future valuations; sign ambiguous. ### Planner’s Euler for borrowing (equation (36)) and policy trade-offs - λ^p_t = βR E_t( λ^p_{t+1} + m_t μ^p_t U_{c,t} (dH_{q,t+1}/dL_{t+1}) ) + μ^p_t - Two key policy-relevant cases: - Case 1 — constraint not binding at t (may bind at t+1): externalities absent at t; planner internalizes future reductions in borrowing capacity; bubble-accumulation externality’s effect on marginal cost of borrowing ambiguous. - Case 2 — constraint binds at t (and may at t+1): all externalities present, planner trades off current valuation gains vs. future valuation losses; net effect ambiguous and depends on which forces dominate. ### Optimal borrowing tax in presence of bubble (Section 4.2) - General optimal tax expression combines planner and decentralized Euler equations (equation (37)); macroprudential component isolated under μ_t = μ^P_t = 0: - τ^{mp}_t = E_t F_{t+1} q_{t+1} E_t U_{c,t+1} + E_t W_{t+1} dB_{t+1}/dl_{t+1} E_t U_{c,t+1} + βE_t V_{t+1} dH_{q,t+2}/db^*_{t+2} db^*_{t+2}/dc_{t+1} E_t U_{c,t+1} (equation (38)) - Macroprudential tax components: - First: addresses fundamental-price externality — tends to call for positive tax (lean against imbalances). - Second: addresses bubble-pledgeability externality — amplifies first component. - Third: addresses bubble-accumulation externality — ambiguous sign. - Comparison with bubbleless BM economy yields three margins determining policy differences: - Current extensive margin: bubble can relax current constraint relative to bubbleless economy, tending to lower tax. - Future extensive margin: bubble can lower probability of future binding (lower tax) or induce more debt accumulation (higher tax) — ambiguous net effect. - Intensive margin: planner affects bubble size and future bubble-pledgeability/accumulation externalities — generally increases tax when these externalities are severe. ### Calibration highlights (preserved parameter values) - Global interest rate: R = 1.01. - Bubble calibration: - Average duration of an equity bubble = 2 years ⇒ survival probability π = 0.5. - Average deviation of equity prices from long-term trend in first year of bubble ≈ 9.2 percent. - Benchmark state: equity bubble = 9.2 percent of the average fundamental price (average fundamental price from ergodic distribution in bubbleless economy). - Preferences and technology forms specified; TFP AR(1) with ρ_z = 0.78 and σ_ϵ = 0.01. - Credit regime: m_t ∈ {m_l, m_h} with m_l = 0.75 (tight), m_h = 0.90 (normal); transition probabilities P_{h,l} = 0.1, P_{l,l} = 0. - Selected calibrated parameter values (as in Table 1): - σ = 1 - α_v = 0.45 - α_l = 0.352 - α_k = 0.008 - χ = 0.352 - Fischer elasticity 1/ω = 2 - θ = 0.16 - R = 1.01 - ρ_z = 0.78 - σ_ϵ = 0.01 - β = 0.95 - π = 0.5 ### Numerical solution method and grids - Global, non-linear solution algorithm with explicit equilibrium-selection routine for non-stationary stochastic bubble. - Discretization / grids: - 150 grid points for L_t over range [0,0.11]. - 60 total states: 3 productivity states × 2 credit states × 10 bubble states (one bubbleless state and 9 bubble sizes up to over 20 percent of fundamental price). - CE computed by iterating first-order conditions; SP via value function iteration; equilibrium selection uses three-step procedure to compute b^*_{t+1} consistent with expectations. ### Numerical results — decision rules, optimal tax and margins (Section 5.2) - SP vs CE decision rules: - SP chooses new borrowing L_{t+1} always lower than CE (internalizes pecuniary externalities). - Borrowing constraint binds for lower L_t in SP than in CE. - Effect of bubble on collateral and borrowing: - Bubble raises collateral (q_t + B_t) for both SP and CE, relaxing borrowing constraint. - Outstanding debt L_t can increase by about 40 percent more versus bubbleless case before constraint binds at t. - Optimal tax behavior as function of debt L_t (for z_t = z_l, m_t = m_h, bubble = 9.2%): - For very low L_t: optimal tax = 0, irrespective of bubble presence. - For low-to-medium L_t: optimal tax is lower when bubble present (future extensive margin negative dominates intensive margin). - Around L_t ≈ 0.052: borrowing constraint binds at t in bubbleless economy; in bubbly case optimal tax increases until L_t ≈ 0.07. - Maximum tax with bubble: about 8.8 percent; maximum tax without bubble: about 3 percent. - Margin contributions to tax differential (bubbly − bubbleless): - Current extensive margin: positive. - Intensive margin: positive and dominant at higher L_t. - Future extensive margin: negative but not always large enough to overturn others. - Optimal tax as a function of bubble size (for given L_t levels): - Tax normalized so no-bubble tax = 100 for each L_t. - For lower L_t (e.g., L_t = 0.035): tax decreases with bubble size. - For higher L_t (e.g., L_t = 0.055): tax increases with bubble size. - Effect of bubble survival probability π: - Benchmark π = 0.5; compared π = 0.4 and π = 0.6. - Optimal macroprudential tax is higher when probability of bursting is higher (i.e., smaller π): lower π increases future likelihood of binding and intensifies intensive margin; therefore higher tax today for riskier bubbles. ### Crisis simulations and policy effects - Simulation design: - CE simulated for 11 periods per path; 100,000 simulated 11-period paths. - Initial bubble ≈ 4% deviation relative to fundamental price of capital; π = 0.5; absorbing no-bubble state after bursting. - Events compared: (i) no bubble during binding episode; (ii) bubble persists during binding; (iii) bubble pops as constraint starts to bind. - Role of initial debt L_1: - Low L_1 = 0.045: bubble has positive effect on consumption and collateral when constraint binds. - High L_1 = 0.095: bubble amplifies negative effects; consumption and borrowing fall more when bubble persists; worst when bubble pops and constraint binds simultaneously. - SP outcomes: - Setting initial L_1 = long-run average L̄ = 0.07 (ergodic mean from bubbleless CE). - SP prevents excessive borrowing ex-ante and mitigates deleveraging effects on consumption, output, and collateral during binding episodes. ### Summary statistics and welfare (Table 2 and welfare results) - Competitive Economy: Event statistics (100,000 simulations of 11-period paths, initial debt set at 0.07) - CC binds and there is no bubble - Probability 7.4% - Collateral value drop −28.0% - CC binds and bubble persists - Probability 0.4% - Collateral value drop −22.9% - CC binds and bubble pops - Probability 0.7% - Collateral value drop −34.9% - Social Planner: Welfare and tax statistics - Minimum tax on debt 0.0% - Average tax on debt 1.7% - Maximum tax on debt 5.3% - Welfare gains 0.4% - Net welfare effect of the bubble (re-run in bubbleless economy with same exogenous paths and initial L_1 = 0.07): −0.1 percent (bubble’s negative effects outweigh relaxation benefits in this experiment). ### Simple rules evaluation and quantitative findings - Simple rules considered (motivation: practical implementability and sharing key features of SP): 1. Time-invariant tax ̄τ equal to average planner tax = 1.7 percent. 2. Debt-deviation rule: τ_t = (1+τ_0)[(L_t/ ̄L)^{η1}]^{−1}, with η1 ∈ {0,0.5,1}. 3. Debt-and-valuation rule: τ_t = (1+τ_0){(L_t/ ̄L)^{η1} · [(q_t+B_t)/ ̄q]^{η2}}^{−1}, with η2 ∈ {0,0.5,1}. - Welfare (compensating consumption variation) results — Table 3 values: - η1 = 0.0: 0.09% (η2=0.0); 0.19% (η2=0.5); 0.22% (η2=1.0) - η1 = 0.5: 0.13% (η2=0.0); 0.22% (η2=0.5); 0.24% (η2=1.0) - η1 = 1.0: 0.14% (η2=0.0); 0.18% (η2=0.5); 0.23% (η2=1.0) - Key takeaways: - Small invariant tax (1.7 percent) increases welfare modestly. - Including credit imbalances (η1 > 0) or valuation imbalances (η2 > 0) improves welfare over invariant tax. - Rules combining both credit and valuation imbalances deliver larger welfare gains and approach, but do not fully reach, optimal policy gain of 0.4 percent. ### Main policy implications and conclusions - Asset price overvaluations matter for optimal macroprudential regulation but do not automatically mandate leaning against bubbles. - Optimal borrowing tax response depends on outstanding debt L_t: - Low/moderate debt: bubble persistence reduces probability of tight future conditions and the optimal tax may be lower (accommodative policy can be optimal). - High debt: asset overvaluation amplifies future deleveraging and pecuniary externalities; optimal macroprudential tax is (much) higher in bubbly economy. - Bubbles introduce additional externalities (bubble-pledgeability and bubble-accumulation) that can necessitate more aggressive intervention when debt is high or bubble is risky (low π). - Practical implication: regulators should condition macroprudential tightening on both credit imbalances and bubbly valuations (simple rules incorporating both perform substantially better than invariant taxes). ### Appendix — technical mechanisms, equilibrium selection, and numerical implementation (selected highlights) - Bubble accumulation iteration and transversality (A.20–A.21); existence condition for bubbly equilibrium when borrowing constraint binds infinitely often; threshold m defined in (A.25). - Three-step equilibrium selection procedure (A.5): 1. Generate conceivable future bubble values ̃b_{t+1} for all future state combinations. 2. Solve fixed-point ̃b(·, b^*_{t+1}, ·) = b^*_{t+1} to obtain b^*_{t+1}. 3. Replace b_{t+1} with b^*_{t+1} in expectations and policy functions. - Planner’s FOCs and algebraic expressions for shadow values, multipliers, and externality terms (A.28–A.46); expressions for F_t, W_t, V_t (A.41–A.43) and decomposition λ^p_t = U_{c,t} + F_t q_t + W_t dB_t/dl_t + V_t ... (A.40). - Numerical algorithm details (A.7): - CE: grid of 150 L states, 6 exogenous ω states, 10 bubble states; iterative procedure to compute policy functions and expectation operators H_U, H_q; convergence tolerance ε = 10^{−4}. - SP: value function iteration with nested inner loop; candidate grid for ˜L up to 5000 values when borrowing constraint does not bind; same convergence tolerance ε = 10^{−4}. *Source: wpiea2019184-print-pdf — IMF Working Paper (selected sections 3–5 and Appendix A.4–A.7 as provided).* ### Section 3 presents the baseline model. We analyze optimal policy in section 4. We show the ### wpiea2019184-print-pdf - Section 3 presents the baseline model. We analyze optimal policy in section 4. We show the ### Paper structure and focus - Sections overview: - Section 3 presents the baseline model. - Section 4 analyzes optimal policy. - Section 5 shows numerical results. - Section 6 concludes. - Main focus: design of optimal macroprudential policies in the presence of asset price bubbles and occasionally binding borrowing constraints. ### Literature (major themes and connections) - Two main strands: - Literature on rational asset price bubbles. - Literature on optimal macroprudential policy. - Gap addressed: rigorous normative analysis linking asset price bubbles and optimal macroprudential policy under occasionally binding borrowing constraints. - Key distinctions across bubble literature: - Early literature (Samuelson, 1958; Tirole, 1985): bubbles arise due to dynamic inefficiency. - Abel, Mankiw, Summers and Zeckhauser (1989): argue real economies are dynamically efficient.8 - More recent work shifts focus to other frictions that support bubbles in equilibrium. - Frictions motivating bubbles: - Financial frictions: intrinsically worthless assets or bubbly components relax borrowing constraints (Kocherlakota, 1992; Santos and Woodford, 1997; Kocherlakota, 2009; Farhi and Tirole, 2012b; Martin and Ventura, 2012, 2016; Hirano and Yanagawa, 2017; Miao and Wang, 2018). - Informational frictions and agency problems (Allen and Gorton, 1993; Allen and Gale, 2000; Allen, Barlevy and Gale, 2018). - Closest related model: - Miao-Wang: stock-price bubble in a production economy with infinitely-lived agents; borrowing restricted by firm value; bubble provides liquidity premium and relaxes borrowing constraints.9 - This paper augments Miao-Wang by allowing occasionally binding borrowing constraints. - Additional related findings from the literature: - Hirano and Yanagawa (2017): effects of bubble bursts depend on pledgeability; bubbles can increase welfare via relaxing financial frictions. - Chauvin, Laibson and Mollerstrom (2011): bubbles without financial frictions are welfare-reducing due to magnified consumption cycles. - Martin and Ventura (2016): propose taxing/subsidizing borrowing to replicate an “optimal” bubble that maximizes output and consumption. - Aoki and Nikolov (2015), Ikeda and Phan (2016), Bengui and Phan (2018): bubbles affect risk-taking and financial stability. - Miao, Wang and Zhou (2015): loan-to-value limits and property transaction taxes reduce benefits of holding bubbly assets. - Miao and Wang (2015): increasing bank capital requirements can mitigate adverse consequences of bubbly bank valuations. - Distinct contributions of this paper relative to others: - Focus on ex-ante macroprudential regulation when constraints bind only occasionally (as opposed to ex-post interventions when constraints always bind). - Policy analysis conducted via a fully-fledged Ramsey problem rather than comparative statics. - Real economy modeled with a focus on financial regulation (rather than monetary policy) to influence bubble evolution. - Broader macroprudential literature context: - Macroprudential intervention motivated by financial frictions generating pecuniary externalities (Bianchi-Mendoza; Stein, 2012; Bianchi, 2011; Jeanne and Korinek, 2010; Davila and Korinek, 2018) or aggregate demand externalities (Eggertsson and Krugman, 2012; Korinek and Simsek, 2016; Farhi and Werning, 2016). - Bianchi-Mendoza: studies optimal macroprudential policy with commitment in a small open economy with occasionally binding borrowing constraints and pecuniary externalities, but without considering elevated asset prices. - Biswas, Hanson and Phan (2020): find policy should lean against bubbles because post-burst activity dips below pre-bubble trend; contrast with this paper’s finding that “leaning against bubbles” is optimal only when debt levels are high—otherwise policy should be accommodative. - Methodological note: - This paper models full non-linear dynamics of stochastic bubbles under occasionally binding borrowing constraints, enabling richer policy prescriptions than frameworks centered on “stationary” stochastic bubbles.7 ### Model economy (baseline) - Environment: - Small open economy. - Rational bubble on a productive asset. - Occasionally binding borrowing constraint. - Modeling choices: - Framework similar to Bianchi-Mendoza but augmented with a rational asset price bubble modeled as in Miao-Wang. - Competitive economy (CE) equilibrium allocations are outlined first. - Section 4 analyzes time-consistent optimal policy and derives an optimal macroprudential tax on borrowing that decentralizes the social planner’s (SP) allocations. ### Key conceptual implications and policy orientation - Bubbles can relax financial constraints and thus can be beneficial for borrowing capacity and consumption smoothing. - Bubbles also introduce inefficiencies through pecuniary externalities that can warrant macroprudential intervention. - Optimal policy analysis: - Focus is on ex-ante macroprudential regulation (e.g., taxes on borrowing) designed to replicate SP allocations under stochastic bubbles and occasionally binding constraints. - Policy prescriptions depend on debt levels: leaning against bubbles is optimal when debt is high; accommodative policy is optimal when debt is low. *Source: wpiea2019184-print-pdf* ### 3.1 Competitive economy with an asset price bubble ### 3.1 Competitive economy with an asset price bubble ### Model setup and agents - Economy populated by a continuum of mass one of two types of infinitely-lived representative agents: households and firms. - Households: consume, provide labor, own firms, and frictionlessly trade firm shares in the stock market. - Firms: operate a production technology combining capital, labor and intermediate goods; purchase and sell capital; borrow internationally in the inter-temporal debt market; do not own or trade shares of other firms. - Fixed supply of capital: K_t = 1. - Small open economy assumption: βR < 1. - Borrowing finance of intermediate goods requires advance financing of a fraction θ ≤ 1 of p_v v_t. ### Households — preferences and first-order conditions - Representative household maximizes expected utility over infinite periods: - Utility defined over the composite commodity c_t − G(l_t), with G(0) = 0; G is convex, strictly increasing and continuously differentiable. - Budget constraint links consumption, equity trades, and wage income. - Key first-order conditions: - Marginal utility: λ_t = U_{c,t}. - Labor supply: w_t = G_{l,t}. - Equity pricing (Euler): V_{j,t} = β E_t[ (U_{c,t+1} / U_{c,t}) V_{j,t+1} ] + D_{j,t}. - Standard transversality condition: lim_{T→∞} β^T (U_{c,T}/U_{c,t}) V_{j,T} = 0, using η^j_T = 1 for all T and j. ### Firms — value function, decisions, and constraints - Firm market value includes a bubbly component: V_t(k_t, L_t, b_t) where b_t is a bubble component not tied to fundamentals. - Firm decisions: k_{t+1}, L_{t+1}, v_t, l^d_t to maximize market value using household stochastic discount factor β U_{c,t+1}/U_{c,t}. - Dividends: - D_t = y_t − p_v v_t − w_t l^d_t + L_{t+1}/R − L_t + q_t k_t − q_t k_{t+1} - Output: y_t = z_t F(k_t, l^d_t, v_t) with Cobb-Douglas F(·), p_v exogenous. - Borrowing constraint (limited commitment / pledgeability): - L_{t+1}/R + θ p_v v_t ≤ m_t E_t[ (β U_{c,t+1}/U_{c,t}) V_{t+1}(k_t, 0, b_{t+1}) ] - Parameters/conditions: m_t < 1; creditors can seize the firm with probability m_t if diversion detected; restructured firm value enters right-hand side. - Collateral includes going-concern value that incorporates discounted future bubbles as resale value. ### Asset price bubble — form and dynamics - Conjectured firm value function via undetermined coefficients: - V(k_t, L_t, b_t) = a_t k_t + s_t L_t + b_t - Derived coefficients: - a_t = [ z_t F_{k,t} + q_t (1 + m_t μ_t) ] - s_t = −1 - b_t = (1 + m_t μ_t) β E_t[ (U_{c,t+1} / U_{c,t}) b_{t+1} ] - Interpretation of components in V_t: - Fundamental cashflow from production: z_t F_{k,t} k_t. - Capital value: q_t k_t. - Shadow value of relaxing borrowing constraint: m_t μ_t q_t k_t. - Outstanding debt reduces value by L_t. - Bubble increases firm value by b_t. - Collateral value of bubble: - B_t = β E_t[ (U_{c,t+1} / U_{c,t}) b_{t+1} ] - Bubble pledgeability condition linking stock market bubble and collateral: b_t = (1 + m_t μ_t) B_t. - Implication: when borrowing constraint binds (μ_t > 0), collateral value B_t is less than market bubble b_t. ### Competitive equilibrium — firm optimality conditions (Proposition 1) - Representative firm choices satisfy (i)-(vii): - (i) Euler equation for borrowing L_{t+1}: - 1 = β E_t[ (U_{c,t+1} / U_{c,t}) R ] + μ_t - (ii) Euler equation for capital k_{t+1}: - q_t = β E_t[ (U_{c,t+1} / U_{c,t}) ( z_{t+1} F_{k,t+1} + q_{t+1} (1 + m_{t+1} μ_{t+1}) ) ] - (iii) Labor optimality condition: - w_t = z_t F_{l,t} - (iv) Intermediate good optimality condition: - p_v (1 + θ μ_t) = z_t F_{v,t} - (v) Bubble accumulation condition: - b_t = (1 + m_t μ_t) β E_t[ (U_{c,t+1} / U_{c,t}) b_{t+1} ] - (vi) Bubble pledgeability condition: - b_t = (1 + m_t μ_t) B_t - (vii) Complementary slackness: - μ_t [ m_t (q_t k_t + B_t) − L_{t+1}/R − θ p_v v_t ] = 0 - Key observations: - Binding borrowing constraint distorts inter- and intra-temporal margins: - Intermediate good choice embeds collateral financing cost θ μ_t p_v. - Euler equations include extra benefit terms from relaxing borrowing constraint (m_{t+1} μ_{t+1} q_{t+1}). - Both fundamental capital price and bubble depend on endogenous choices, creating pecuniary externalities that are central to policy analysis. ### Nature and support of the bubble - Deterministic steady state cannot sustain a rational stock market bubble: - If b_t = b_{t+1} = b (and B_t = B), q_t = q_{t+1} = q, and U_{c,t+1} = U_{c,t} = U_c, then conditions (19), (22), (23) cannot hold for b, B > 0 and finite; only bubbleless steady state b = B = 0 exists. - In deterministic steady state, μ = 1 − β R > 0 (constraint binds). - Focus on stochastic bubbles that can burst with positive probability: - Initial bubble b_0 > 0; persistence probability π each period (burst with probability 1 − π). - Bubble growth must satisfy equation (13) (the b_t accumulation condition) to be rationally priced. - Authors restrict attention to beliefs yielding: - b_{t+1} = b_t U_{c,t} / [ β (1 + m_t μ_t) π U_{c,t+1} ] - Equilibrium selection replaces b_{t+1} with b^*_{t+1} solving: - b^*_{t+1} = b_t U_{c,t} / [ β (1 + m_t μ_t) π U^*_{c,t+1} ] - where U^*_{c,t+1} = U'( x(L_{t+1}, z_{t+1}, m_{t+1}, b^*_{t+1}) ). - This yields a law of motion b^*(L_{t+1}, z_{t+1}, m_{t+1}, L_t, z_t, m_t). ### Bubble existence conditions (Proposition 2) and transversality - Two requirements for a non-stationary stochastic rational bubble: - (i) Bubble pricing condition (equation (22)) must hold for every t. - (ii) Transversality condition (6) must be satisfied. - Because borrowing constraint binds only occasionally (μ_t > 0 sometimes and μ_t = 0 other times), transversality does not trivially rule out the bubble. - Proposition 2: - If min(m_t) ≤ m, then the borrowing constraint (15) binds infinitely often: lim_{t→∞} sup μ_t > 0, and a rational bubble can be supported in equilibrium. - Intuition: when pledgeable portion of firm value is low enough, borrowing constraint is more restrictive than the natural debt limit, and the constraint binds infinitely often, preserving bubble usefulness for mitigating financial frictions. ### Competitive equilibrium definition and solution approach - Given initial L_0 and b_0, probability π, and exogenous processes { z_t, m_t }_{t=1}^∞, a competitive equilibrium is a sequence of allocations { c_t, l_t, l^d_t, v_t, y_t }_{t=0}^∞, asset profile { k_{t+1}, L_{t+1} }_{t=0}^∞, bubble processes { b_t }_{t=1}^∞ and { B_t }_{t=0}^∞, and price system { q_t, w_t, p_v, R }_{t=0}^∞ such that: - (1) Households’ and firms’ optimality conditions (3)-(5) and (18)-(23) hold given prices and bubble processes. - (2) Markets clear: l^d_t = l_t, k_t = K_t = 1, η^j_t = 1 ∀ j,t. - (3) Resource constraint holds: c_t + L_t = L_{t+1}/R + z_t F(1, l_t, v_t) − p_v v_t. - Equilibrium characterization: - Initial bubble b_0 is a sunspot realization; B_0 is endogenous with B_0 ≤ b_0 depending on whether borrowing constraint binds. - Bubble evolves endogenously and affects all choices; different initial sunspot realizations b'_0 and b''_0 with same L_0 produce different equilibrium paths. - To compute equilibrium, stationary policy functions x_t = x(L_t, z_t, m_t, b_t) for x_t ∈ { c_t, v_t, l_t, L_{t+1}, μ_t, w_t, q_t, B_t } are needed together with a law of motion for the endogenous bubble state. - The equilibrium selection enforces consistency of expectations via fixed-point condition (25) for b^*_{t+1}. *Source: IMF Working Paper — section 3.1 "Competitive economy with an asset price bubble" (wpiea2019184-print-pdf).* ### Appendix presents the detailed steps of the equilibrium selection procedure. Thus, we replace ### wpiea2019184-print-pdf - Appendix presents the detailed steps of the equilibrium selection procedure. Thus, we replace ### Recursive representation of competitive equilibrium - State at t: (L_t,m_t,z_t,b_t). State at t+1: (L_{t+1},m_{t+1},z_{t+1},b^*_{t+1}). - Competitive equilibrium characterized by: - Policy functions x(L_t,m_t,z_t,b_t) for period-t endogenous variables x_t ∈ {c_t,v_t,l_t,L_{t+1},μ_t,w_t,q_t,B_t}. - Policy functions H_u and H_q for forward-looking terms in (A.26) and (A.27). - Values b^*_{t+1} that the bubble b_t grows into each future state, satisfying market clearing, the resource constraint, and optimality conditions (3)-(5), (20), (21), (23), (25), (26), and (27). - Equilibrium conditions in recursive form (using bubble-accumulation condition (25)): - U_{c,t}(1−μ_t) = βR E_t H_u(L_{t+1},z_{t+1},m_{t+1},b^*_{t+1},L_t,z_t,m_t,b_t) (26) - q_t U_{c,t} = βE_t H_q(L_{t+1},z_{t+1},m_{t+1},b^*_{t+1},L_t,z_t,m_t,b_t) (27) - Bubble accumulation condition (25) implicitly defines law of motion b^*(L_{t+1},z_{t+1},m_{t+1},L_t,z_t,m_t). ### Planner’s environment and instruments - Planner chooses state-contingent tax on borrowing τ_t as sole policy instrument; tax revenues rebated lump-sum with T_t = τ_{t-1} L_t (footnote 19). - Decentralized resource constraint with tax: - c_t + L_t(1 + τ_{t-1}) + p_v v_t ≤ z_t F(1,l_t,v_t) + L_{t+1} R + T_t (28) - With T_t = τ_{t-1} L_t. - Euler condition from atomistic agents with borrowing tax (using recursive representation (26)): - U_{c,t}(1−μ_t) = βR(1 + τ_t) E_t H_u(L_{t+1},z_{t+1},m_{t+1},b^*_{t+1},L_t,z_t,m_t,b_t) (29) - This condition need not be imposed as constraint in planner’s problem since planner can implement allocations via τ_t. ### Time-consistent planner’s problem (Ramsey, no commitment) - Planner cannot commit to future policies (footnote 20); solve for optimal time-consistent macroprudential policy. - Planner’s maximization at state (L_t,z_t,m_t,b_t): - max_{c_t,q_t,L_{t+1},v_t,l_t} E_t Σ_{t=0}^{∞} β^t U(c_t − G(l_t)) - s.t. resource constraint (30), borrowing constraint (31), forward-looking q condition (32), labor first-order condition (33): - c_t + L_t + p_v v_t ≤ z_t F(1,l_t,v_t) + L_{t+1} R (λ^p_t) (30) - L_{t+1} R + θ p_v v_t ≤ m_t ( q_t + b_t/(1 + m_t μ(l_t,v_t)) ) (μ^p_t) (31) - q_t U_{c,t} = βE_t H_q(L_{t+1},z_{t+1},m_{t+1},b^*_{t+1}(c_t,μ(l_t,v_t)),L_t,z_t,m_t,b_t) (ξ_t) (32) - z_t F_{l,t} = G_{l,t} (ζ_t) (33) - Complementary slackness: [L_{t+1}/R + θ p_v v_t − m_t(q_t + b_t/(1 + m_t μ(l_t,v_t)))] μ^p_t = 0. - Bubble pledgeability: B_t = b_t/(1 + m_t μ(l_t,v_t)) substituted into (31); b_t is state variable, B_t endogenous. - Planner internalizes dependence of b^*_{t+1} on current choices: b^*_{t+1} = b^*_{t+1}(c_t,μ(l_t,v_t)) via (25) substituted in (32). - In bubbleless economy (b_t = 0 and b^*_{t+1} = 0), (31) and (32) coincide with BM economy constraints; (33) redundant absent bubble (footnote 22). ### Planner’s optimality conditions — decomposition of consumption FOC - Consumption first-order condition expressed as (34): - λ^p_t = U_{c,t} ︸︷︷︸ Competitive equilibrium + Fundamental-price externality︷︸︸︷ F_t q_t + Bubble-accumulation externality︷︸︸︷ V_t βE_t (dH_{q,t+1}/db^*_{t+1}) (db^*_{t+1}/dc_t) + Bubble-pledgeability externality︷︸︸︷ W_t (dB_t/dl_t) - F_t, W_t, and V_t defined in (A.41), (A.42), and (A.43) in the Appendix. - Interpretation of terms: - Fundamental-price externality (F_t q_t): positive; planner internalizes that higher c_t reduces marginal utility and raises q_t via (32), relaxing borrowing constraint — the standard BM externality. - Bubble-accumulation externality (V_t βE_t ...): captures how c_t affects bubble growth db^*_{t+1}/dc_t via (25); b^*_{t+1} affects q_t through H_q in (32); private agents ignore this link. - Bubble-pledgeability externality (W_t dB_t/dl_t): captures how production choices affect collateral value B_t via μ(l_t,v_t) and thereby the borrowing constraint; W_t > 0 as shown in Appendix. - Bubble-pledgeability amplification (35): W_t dB_t/dl_t = F_t q_t · W'_t · dB_t/dl_t where W'_t < 0 (A.44) and dB_t/dl_t < 0, so bubble-pledgeability scales up the fundamental-price externality. ### Euler condition with respect to borrowing in planner’s problem - Planner’s Euler for borrowing (36): - λ^p_t = βR E_t( λ^p_{t+1} + m_t μ^p_t U_{c,t} (dH_{q,t+1}/dL_{t+1}) ) + μ^p_t - dH_{q,t+1}/dL_{t+1} appears because L_{t+1} is a state at t+1 and affects q_t via expectations in (32). - Planner’s condition incorporates the three externalities; comparison to private agents’ L_{t+1} condition depends on whether borrowing constraint binds. ### Policy-relevant scenarios and implications for marginal cost of borrowing - Case 1 — Borrowing constraint does not bind at t, may bind at t+1: - μ^p_t = μ_t = F_t = W_t = V_t = 0. Then (36) reduces to U_{c,t} = βR E_t λ^p_{t+1}. - Planner’s marginal cost of borrowing differs from private agents by βR E_t(λ^p_{t+1} − U_{c,t+1}), with λ^p_{t+1} from (34). - Planner internalizes that higher debt at t reduces future borrowing capacity and consumption when constraint binds at t+1, which tightens q_{t+1} and B_{t+1} via bubble-pledgeability and reduces future bubble growth via bubble-accumulation externality. - The bubble-accumulation externality’s sign on planner’s marginal cost of borrowing is ambiguous. - Case 2 — Borrowing constraint binds at t and may bind at t+1: - All three externalities present in both periods, affecting marginal cost of borrowing in opposing directions. - Planner considers how L_{t+1} affects q_t through E_t H_{q,t+1}; faces trade-off: increasing current valuations q_t and B_t vs. potentially decreasing future q_{t+1} and B_{t+1}. - Net effect on marginal cost of borrowing relative to competitive economy depends on which force dominates. *Source: wpiea2019184-print-pdf - Appendix presents the detailed steps of the equilibrium selection procedure. Thus, we replace (IMF Working Paper content).* ### 4.2 Optimal tax rate in the presence of a bubble ### 4.2 Optimal tax rate in the presence of a bubble ### Derivation of the optimal borrowing tax - The optimal tax on borrowing is obtained by combining the planner's Euler equation (36) with the agents' Euler equation incorporating the tax on borrowing (29), yielding expression (37): τ_t = − λ_p_t − U_{c,t} βRE_t U_{c,t+1} + μ_p_t − U_{c,t} μ(l_t,v_t) βRE_t U_{c,t+1} + m_t μ_p_t U_{c,t} E_t dH_{q,t+1}/dL_{t+1} E_t U_{c,t+1}︸︷︷︸ Targeting current valuation externalities + E_t(λ_p_{t+1} − U_{c,t+1}) E_t U_{c,t+1}︸︷︷︸ Targeting future valuation externalities. (37) - To isolate the macroprudential component, the authors consider μ_t = μ^P_t = 0 (but E_t μ^p_{t+1} > 0). Under this focus, the macroprudential tax τ^{mp}_t is given by (38): τ^{mp}_t = E_t F_{t+1} q_{t+1} E_t U_{c,t+1} + E_t W_{t+1} dB_{t+1}/dl_{t+1} E_t U_{c,t+1} + βE_t V_{t+1} dH_{q,t+2}/db^*_ {t+2} db^*_{t+2}/dc_{t+1} E_t U_{c,t+1}, (38) where F_{t+1}, W_{t+1}, V_{t+1}, and db^*_{t+2}/dc_{t+1} are given by (A.41), (A.42), (A.43), and (A.33) applied to period t+1, respectively. ### Macroprudential interpretation and components - The macroprudential tax τ^{mp}_t comprises three components: - First component (via q_{t+1}): addresses the fundamental-price externality and calls for a positive tax to lean against imbalances. - Second component (via B_{t+1}): addresses the bubble-pledgeability externality; this amplifies the fundamental-price externality and requires a more aggressive macroprudential response. - Third component (via b^*_{t+2}): addresses the bubble-accumulation externality; the direction of its effect on the macroprudential tax is ambiguous. ### Comparison with the bubbleless economy - The optimal tax in the bubbleless BM economy is denoted ̃τ_t and (after setting b_t = b_{t+1} = 0 and μ^P_t = λ^P_t μ(l_t,v_t)) is given by (39): ̃τ_t = − ̃F_t ̃q_t (1 + μ(̃l_t, ̃v_t)) βRE_t ̃U_{c,t+1} − m_t (̃U_{c,t} + ̃F_t ̃q_t) μ(̃l_t, ̃v_t) ̃U_{c,t} E_t d ̃H_{q,t+1}/d ̃L_{t+1} E_t ̃U_{c,t+1}︸︷︷︸ Tackling externalities at t + E_t ̃F_{t+1} ̃q_{t+1} E_t ̃U_{c,t+1}︸︷︷︸ Tackling externalities at t+1. (39) - The difference between the macroprudential tax in the bubbly economy and the total tax in the bubbleless economy is: τ^{mp}_t − ̃τ_t = ̃F_t ̃q_t (1 + μ(̃l_t, ̃v_t)) βRE_t ̃U_{c,t+1} + m_t (̃U_{c,t} + ̃F_t ̃q_t) μ(̃l_t, ̃v_t) ̃U_{c,t} E_t d ̃H_{q,t+1}/d ̃L_{t+1} E_t ̃U_{c,t+1} + E_t F_{t+1} q_{t+1} E_t U_{c,t+1} − E_t ̃F_{t+1} ̃q_{t+1} E_t ̃U_{c,t+1}︸︷︷︸ Future extensive margin + E_t W_{t+1} dB_{t+1}/dl_{t+1} E_t U_{c,t+1} + βE_t V_{t+1} dH_{q,t+2}/db^*_{t+2} db^*_{t+2}/dc_{t+1} E_t U_{c,t+1}︸︷︷︸ Intensive margin. (40) ### Margins of optimal policy (interpretation) - Current extensive margin: - The bubble at t can relax borrowing constraints that would bind in the bubbleless economy, removing pecuniary externalities at t and prompting a reduction in the borrowing tax needed to support the current price ̃q_t. - Future extensive margin: - The bubble can make the borrowing constraint slack in some future states (if the bubble persists with probability π), lowering the conditional probability of constraint binding and calling for a lower tax. - Conversely, the bubble may lead to greater debt accumulation, exacerbating pecuniary externalities when the bubble bursts; this operates through changes in capital, consumption, and other equilibrium values. - Intensive margin: - Captures how current choices affect bubble growth (bubble is a state variable). It tackles bubble-pledgeability and bubble-accumulation externalities when the constraint binds at t+1, but only when the bubble does not burst at t+1. - The planner cannot affect the exogenous persistence probability of the bubble but can affect bubble size to maintain value when borrowing constraints start binding in the future. - The intensive margin is distinct from adverse effects of a bursting bubble (which are captured in the future extensive margin). - Overall sign of the combined effect of the two extensive margins and the intensive margin is ambiguous; the paper proceeds to a calibrated numerical solution to determine which forces prevail. ### Non-equivalence with a bubbleless economy with higher fundamentals - The macroprudential tax in a bubbly economy is different from that in a bubbleless economy with simply higher fundamental prices q'_ {t+1} = q_{t+1} + B_{t+1}; even if q'_ {t+1} = q_{t+1} + B_{t+1}, debt levels, consumption, and other endogenous variables differ. - Specifically: F_{t+1} q_{t+1} + W_{t+1} dB_{t+1}/dl_{t+1} = F_{t+1} (q_{t+1} + B_{t+1} · Y_{t+1}) 6= F_{t+1} (q_{t+1} + B_{t+1}) = F_{t+1} q'_{t+1}, because Y_{t+1} = q_{t+1} W'_{t+1}/B_{t+1} dB_{t+1}/dl_{t+1} 6= 1. ### Calibration highlights (setup for quantitative analysis) - Global interest rate: - R = 1.01 (kept fixed at its long-term average). - Bubble calibration (based on Jordà, Schularick and Taylor (2015), post-World War II equity bubbles): - Average duration of an equity bubble = 2 years ⇒ survival probability π = 0.5. - Average deviation of equity prices from long-term trend in first year of bubble ≈ 9.2 percent. - Results presented focus on the state where the equity bubble = 9.2 percent of the average fundamental price (average fundamental price computed from ergodic distribution in the bubbleless economy). - Preferences and technology: - U = (c − χ l^{1+ω}/(1+ω))^{1−σ} − 1)/(1−σ), ω, χ > 0, σ > 1. - F = e^{z} k^{α_k} v^{α_v} l^{α_l}, α_k, α_v, α_l > 0, α_k + α_v + α_l ≤ 1. - TFP process: - z_t = ̄z + ρ_z z_{t−1} + ϵ_t, ϵ_t ∼ N(0, σ_ϵ). - Productivity discretized via Tauchen’s method with three realizations z_l, ̄z, z_h. - Credit regime: - m_t follows two-state regime switching Markov process {m_l, m_h} (m_l tight, m_h normal), independent of z process. - Selected calibrated parameter values (Table 1): - Risk aversion σ = 1 - Share of intermediate good in output α_v = 0.45 - Share of labor in output α_l = 0.352 - Share of assets in output α_k = 0.008 - Labor disutility coefficient χ = 0.352 - Fischer elasticity 1/ω = 2 - Working capital coefficient θ = 0.16 - Tight credit regime m_l = 0.75 - Normal credit regime m_h = 0.90 - Global interest rate R = 1.01 - TFP process ρ_z = 0.78 - σ_ϵ = 0.01 - Discount factor β = 0.95 - Transition probability, m_h to m_l P_{h,l} = 0.1 - Transition probability, m_l to m_l P_{l,l} = 0 - Bubble bursting probability π = 0.5 *Source: “4.2 Optimal tax rate in the presence of a bubble”, wpiea2019184-print-pdf* ### 5.2 Numerical results ### 5.2 Numerical results ### Numerical solution method - The model is solved using a global, non-linear solution algorithm. - The CE solution is obtained by iterating over the first-order conditions; the SP problem solution is obtained by applying a value function iteration algorithm. - The global solution algorithm is augmented to incorporate a non-stationary stochastic bubble (see Appendix A.7 for numerical details in the source). - In both CE and SP algorithms: - grid of 150 points for L_t over a range [0,0.11]; - 60 states (3 productivity states, 2 credit condition states, and 10 bubble states: one bubbleless state and 9 states with different bubble size of up to over 20 percent of the fundamental price). ### Policy functions and optimal macroprudential tax - Figures (described in text) plot new borrowing L_{t+1} and total collateral value (q_t + B_t) as functions of outstanding debt L_t for low productivity (z_t = z_l) and favorable financial conditions (m_t = m_h). - CE (red) vs SP (blue) decision rules: - SP chooses new borrowing L_{t+1} that is always lower than in the CE, independent of the bubble’s presence, because it internalizes pecuniary externalities and mitigates their negative impact on consumption, asset prices and welfare. - The borrowing constraint starts binding in the SP equilibrium for lower levels of debt than in the CE. - Effect of a bubble: - With a bubble, collateral value is higher than in the no-bubble case for both SP and CE, relaxing the borrowing constraint and allowing higher outstanding debt. - The borrowing constraint starts binding at higher L_t in both CE and SP when a bubble is present. - Outstanding debt L_t can increase by about 40 percent more compared to the bubbleless case before the borrowing constraint starts binding at t. - Trade-off highlighted: reducing new borrowing alleviates pressures on future valuations but intensifies pressures on current-period valuations; the bubble amplifies this trade-off. - Optimal tax on borrowing (Figure 3 description): - Left panel: optimal tax at t in absence and presence of a bubble equal to 9.2% of the average fundamental price in the bubbleless economy, shown as a function of L_t for z_t low and m_t high. - Right panel: contribution of current and future extensive and intensive margins to the difference between the macroprudential tax in the bubbly economy and the borrowing tax in the bubbleless economy (given by equation (40)), reported for the region where the tax in presence of a bubble takes the macroprudential interpretation (i.e., borrowing constraint does not bind at t in the bubbly economy). - Key observations on the optimal tax: - For very low levels of debt the tax on borrowing is zero, independently of bubble presence, because the borrowing constraint does not bind at t or t+1. - For low to medium levels of outstanding debt, the optimal tax is lower when the bubble is present: - In this region the borrowing constraint is slack at t for both economies. - The future extensive margin (negative) dominates the intensive margin (positive), pushing the tax down in the bubbly economy. The bubble’s benefit of reducing probability of future binding constraints outweighs bubble-pledgeability and bubble-accumulation externalities. - Around L_t ≈ 0.052 the borrowing constraint becomes binding at t in the absence of the bubble: - SP reduces the tax in the bubbleless economy to weigh the benefit of relaxing the constraint today against the cost of it binding in the future; the tax can become slightly negative in the calibration. - In the presence of the bubble the optimal tax continues to increase until L_t ≈ 0.07. - Maximum tax with bubble: about 8.8 percent; maximum tax without bubble: about 3 percent. - Margin contributions to the tax differential (bubbly minus bubbleless): - Current extensive margin: positive contribution. - Intensive margin: positive contribution as future bubble-pledgeability and bubble-accumulation externalities become more severe for higher L_t. - Future extensive margin: negative contribution (due mainly to higher future fundamental-price externalities in the bubbleless economy for higher L_t), but not strong enough to dominate current extensive + intensive margins. - Conclusion: asset overvaluations amplify credit imbalances when debt is high, calling for a (much) higher tax in the bubbly economy. - Net extensive vs intensive margins (Figure 4 description): - Left chart: sum of current and future extensive margins as a function of L_t for z_t low and m_t high; region reported where tax in presence of a bubble is macroprudential; vertical dashed line indicates L_t where constraint starts binding in bubbleless economy. - Right chart: intensive margin for same region. - Findings: - Net extensive margin remains negative but starts to increase at the point where the constraint binds in the bubbleless economy. - Tax differential between bubbly and bubbleless economies is largely driven by the intensive margin beyond the indebtedness level where the constraint binds in the bubbleless economy. - For higher indebtedness the net extensive margin eventually becomes positive, further contributing to the higher macroprudential tax with the bubble. - Optimal tax as function of bubble size (Figure 5 description): - Normalized optimal tax at t plotted against bubble size for three levels of L_t: L_t = 0.035 (dashed), L_t = 0.052 (dotted), L_t = 0.055 (solid), with z_t low and m_t high. - Tax is normalized so the no-bubble tax for each L_t equals 100. - Findings: - For the same exogenous states z_t and m_t and endogenous state L_t, the borrowing tax differs across bubble states b_t in all three cases. - Macroprudential regulation should consider fundamentals z_t and m_t, level of debt L_t, and bubbly valuations. - Whether bubble raises or lowers the tax depends on L_t: - For lower L_t positive aspects of bubble dominate and tax decreases with bubble size. - For higher L_t negative effects dominate and tax increases with bubble size. - Overall implication: - Asset price overvaluations matter for optimal regulation but may not by themselves justify tightening macroprudential policy. - For low/moderate credit levels bubble persistence reduces likelihood of tight future conditions and future deflation costs are low, so it is optimal not to lean against the bubble at low/moderate debt. - Once debt increases sufficiently, macroprudential policy should be stronger. ### Optimal tax and probability of bubble bursting - Investigated effect of bubble survival probability π on optimal policy. - Left panel Figure 6: macroprudential tax for π = 0.4, π = 0.5, and π = 0.6 as function of L_t when b_t = 9.2% of fundamental price (benchmark). - Key results: - Optimal macroprudential tax is higher when probability of bubble bursting is higher (i.e., when π is smaller) for all L_t. - Reasons: - Lower π increases likelihood that borrowing constraint will bind in future states, pushing future extensive margin up (i.e., makes its contribution less negative). - Intensive margin also increases: for lower π a bigger future bubble is required to support same current bubble valuation, leading to more severe future bubble externalities when constraint binds, prompting higher tax today. - Conclusion: optimal macroprudential policy is more aggressive for “riskier” stochastic bubbles (as measured by probability of bursting). ### Crisis scenario simulations - Simulation setup: - Competitive economy simulated for 11 periods; initial period dropped when computing average responses to reduce dependence on initial conditions. - Initial period: bubble with around 4% deviation relative to fundamental price of capital; bubble continuation probability π = 0.5. - 11-period path simulated 100,000 times. - Bubble does not re-emerge after bursting (absorbing no-bubble state); hence limited-period simulations used. - Events of interest: (i) no bubble before, during or after borrowing constraint binds; (ii) bubble persists throughout the period when borrowing constraint binds; (iii) bubble pops when borrowing constraint starts to bind. - Figure 7: average responses (deviations from averages across all simulations) of consumption c_t, new borrowing L_{t+1}, and collateral value q_t + B_t in CE for these three events. - Role of initial L_1: - Left panels: starting L_1 = 0.045 (low). - Right panels: starting L_1 = 0.095 (high). - Findings from simulations: - With low starting debt, the bubble has a positive effect on consumption and collateral values when the borrowing constraint binds. - With high starting debt, the bubble amplifies negative effects of a binding constraint: consumption and borrowing fall more when bubble persists compared to the no-bubble case. - Worst outcomes occur when the bubble pops and borrowing constraint binds simultaneously: biggest declines in variables. - Intuition: prior existence of bubble allows firms to build more debt, making effect of a binding constraint more severe. ### Welfare comparisons and Social Planner outcomes - To mitigate dependence on initial conditions for welfare analysis: - Initial outstanding debt L_1 set equal to long-term average L̄ = 0.07, obtained from ergodic distribution of debt in bubbleless economy (ergodic distribution from a simulation of the bubbleless competitive economy for 100,000 periods). - Figure 8: average responses of output y_t, new borrowing L_{t+1}, consumption c_t, and capital price q_t in CE (red solid) and SP (blue dashed) across states where borrowing constraint binds and bubble persists. - Findings: - Social planner prevents excessive borrowing ex-ante and considerably mitigates impact of binding constraint on debt deleveraging, consumption, output, and collateral values. - Welfare computation: - Welfare gain in planner’s economy measured as average compensating consumption variation equalizing expected utility between CE and SP. - Limitations: non-recurrent character of bubble and small number of simulated periods. - Average tax and welfare results summarized below. ### Summary statistics (Table 2) - Competitive Economy: Event Statistics - CC binds and there is no bubble - Probability 7.4% - Collateral value drop -28.0% - CC binds and bubble persists - Probability 0.4% - Collateral value drop -22.9% - CC binds and bubble pops - Probability 0.7% - Collateral value drop -34.9% - Social Planner: Welfare Effects - Minimum tax on debt 0.0% - Average tax on debt 1.7% - Maximum tax on debt 5.3% - Welfare gains 0.4% - Notes on Table 2: - Table reports summary statistics of 100,000 simulations of 11-period paths when initial period level of debt is set at 0.07. - CC stands for borrowing constraint. *Source: wpiea2019184-print-pdf - 5.2 Numerical results* ### 0.4 percent. While this number might seem small, it is important to note that the welfare effects ### wpiea2019184-print-pdf - 0.4 percent. While this number might seem small, it is important to note that the welfare effects ### Welfare effects of the bubble and optimal policy - The welfare gain from the optimal policy in the planner’s simulated economy is reported as 0.4 percent. - Re-running simulations in a bubbleless economy (for the same realized paths of exogenous variables and an initial debt level of 0.07) yields a net welfare effect of the bubble equal to −0.1 percent. - Interpretation: the negative effects from the bubble outweigh the positive effect from relaxing borrowing constraints in this experiment. - Optimal policy is implemented by a state-contingent tax (planner’s tax), which raises concerns about practical implementation. ### Simple rules considered and their motivation - Rationale: simple rules may perform well if they share characteristics of the optimal policy, such as leaning against credit imbalances and asset overvaluations. - Three types of simple rules examined: 1. Time-invariant tax ̄τ equal to the average state-contingent tax in the planner’s simulated economy (equal to 1.7 percent as reported in Table 2). 2. Debt-deviation rule: τt = (1+τ0)[(Lt/ ̄L)η1]−1, where ̄L is the average of the ergodic distribution of debt in the bubbleless economy. 3. Debt-and-valuation rule: τt = (1+τ0){(Lt/ ̄L)η1 · [(qt+Bt)/ ̄q]η2}−1, where ̄q is the ergodic average of valuations in the bubbleless economy. - Policy design parameters considered: η1 ∈ {0,0.5,1} and η2 ∈ {0,0.5,1}. ### Quantitative findings for simple rules (welfare gains) - Three key takeaways: - A small invariant tax (η1 = η2 = 0) can increase welfare in the simulated economy. - Incorporating credit imbalances (η1 > 0, η2 = 0) or valuation imbalances (η1 = 0, η2 > 0) improves upon the invariant tax. - Incorporating both credit and valuation imbalances (η1, η2 > 0) further increases welfare gains and brings them closer to, but still below, the optimal policy gain of 0.4 percent. - Table 3: Simple rules’ welfare gains (compensating consumption variations) - η2 = 0.0, 0.5, 1.0 across columns; η1 = 0.0, 0.5, 1.0 down rows. - Values: - η1 = 0.0: 0.09% (η2=0.0); 0.19% (η2=0.5); 0.22% (η2=1.0) - η1 = 0.5: 0.13% (η2=0.0); 0.22% (η2=0.5); 0.24% (η2=1.0) - η1 = 1.0: 0.14% (η2=0.0); 0.18% (η2=0.5); 0.23% (η2=1.0) ### Key conclusions and policy implications - Presence of a bubble generates additional externalities and, ceteris paribus, requires more aggressive macroprudential intervention to avoid bubble deflation in bad times (when the bubble relaxes borrowing constraints most). - The bubble also alters equilibrium allocations; by keeping collateral constraints slack in some shock realizations, it can lead to a macroprudential tax on borrowing that is lower relative to the bubbleless case. - The optimal policy response depends non-monotonically on the outstanding level of debt: - When credit imbalances are moderate: the optimal tax is lower in the presence of the bubble (bubble relaxes borrowing constraint and externalities are not severe). - When credit imbalances are high: the optimal tax is much higher in the presence of the bubble (to counter amplifying effects on future deleveraging from elevated credit imbalances and asset overvaluations). - Implication for regulators: asset overvaluations should not always be an argument to lean more aggressively against the wind; if existing credit imbalances are not high, regulators may not need to worry as much about price overvaluations. ### Appendix — mechanism and technical derivations (selected) - Derivation of borrowing constraint: - Debt renegotiation setup: after last period’s debt Lt is repaid but before production and investment, firm may renegotiate outstanding debt L_{t+1}/R + θ p_v v_t+. - Lenders’ outside option equals mt E_t[β U_{c,t+1}/U_{c,t} V_{t+1}(k_t,0,b_{t+1})]. - Incentive compatibility yields borrowing constraint reported as equation (9): - L_{t+1}/R + θ p_v v_t ≤ m_t E_t[β U_{c,t+1}/U_{c,t} V_{t+1}(k_t,0,b_{t+1})] (equation (A.3)). - Alternative assumption (selling restructured firm at t+1) would lead to L_{t+1}/R + θ p_v v_t ≤ m_t E_t[V_{t+1}(k_t,0,b_{t+1})/R], but authors choose the within-period sale to match the bubbleless BM economy and simplify solution. - Conjectured firm value function and verification: - Conjecture: V_t(k_t,L_t,b_t) = a_t k_t + s_t L_t + b_t, with time-varying coefficients a_t, s_t, b_t. - First-order conditions yield: - q_t = β E_t[U_{c,t+1}/U_{c,t} a_{t+1}] (equation (A.7)) - 1/R = μ_t R − β E_t[U_{c,t+1}/U_{c,t} s_{t+1}] (equation (A.8)) - w_t = z_t F_{l,t} (equation (A.9)) - p_v = z_t F_{v,t} / (1 + θ μ_t) (equation (A.10)) - Value function coefficients found: - a_t = z_t F_{k,t} + q_t (1 + m_t μ_t) (equation (A.13)) - s_t = −1 (equation (A.14)) - b_t = (1 + m_t μ_t) β E_t[U_{c,t+1}/U_{c,t} b_{t+1}] (equation (A.15)) - Resulting value function: V_t(k_t,L_t,b_t) = (z_t F_{k,t} + q_t (1 + m_t μ_t)) k_t − L_t + b_t (equation (A.16)) - Corresponding credit constraint: L_{t+1}/R + θ p_v v_t ≤ m_t (q_t k_t + B_t) (equation (A.17)) - Proof sketch for Proposition 1: - Using a_{t+1} and s_{t+1} expressions yields: - U_{c,t} q_t = β E_t[U_{c,t+1} (z_{t+1} F_{k,t+1} + q_{t+1} (1 + m_{t+1} μ_{t+1}))] (equation (A.18)) - μ_t = 1 − β R E_t[U_{c,t+1}/U_{c,t}] (equation (A.19)) - Together with other conditions, these yield the first-order conditions (18)-(22) in the main text. *Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019184-print-pdf.pdf* ### 1. To derive (23), we combine (A.15) with the definition ofB ### wpiea2019184-print-pdf - 1. To derive (23), we combine (A.15) with the definition ofB ### A.4 Proof of Proposition 2 — existence and exclusion of rational bubbles - Iteration of the bubble accumulation equation (13) yields: - b_t = (1 + m_t μ_t) β E_t[U_{c,t+1}/U_{c,t} b_{t+1}] = lim_{T→∞} E_t[β^T U_{c,t+T}/U_{c,t} b_{t+T} Π_{i=0}^{T−1} (1 + m_{t+i} μ_{t+i})]. (A.20) - Transversality condition: - lim_{T→∞} E_t[β^T U_{c,t+T}/U_{c,t} b_{t+T}] = 0. (A.21) - Two exclusion cases (no bubble can be supported): - If μ_{t′} = 0 for all t′ > t, then b_t = 0 via (A.20) and (A.21). - If there exists T′ > t such that μ_{t′} = 0 for t′ ≥ T′ (but possibly positive for t′ ∈ (t, T′)), then b_t = 0 because b_{t+T′} = 0 by (A.21). - If the borrowing constraint binds infinitely often (lim_{t′→∞} sup μ_{t′} > 0), then Π_{i=0}^{T−1} (1 + m_{t′+i} μ_{t′+i}) > 1 for all t′ and (A.21) cannot exclude rational bubbles. - Natural debt limit (Aiyagari): - ̄L ≡ R [zF_k]_{min} R^{−1}. (A.22) - For any arbitrarily tighter borrowing limit ̃L ≡ ̄L − δ (small δ), the ̃L/R limit will bind in some states as t → ∞ by an Aiyagari-type argument: for debt ≤ ̃L dividends D_t and consumption c_t are strictly positive so U_{c,t} finite; assuming the debt limit never binds leads to a contradiction because βR < 1. - Upper bounds on collateralizable values: - q_{max} = [zF_k]_{max} R^{−1}. (A.23) - Define B_{max} as the maximum bubble collateral value (finite and non-zero by super-martingale arguments). - Maximum borrowing under the borrowing constraint (loose v_t → 0): - ˆL_R ≡ m_t (q_{max} + B_{max}). (A.24) - Existence condition for a bubbly equilibrium: - ˆL ≤ ̃L is satisfied if: - m_t ≤ m ≡ ([zF_k]_{min} − δ(R − 1)) / ([zF_k]_{max} + (R − 1) B_{max}). (A.25) - Given B_{max} finite, m ∈ (0,1). Thus there is a threshold m for the collateralizable portion of firm value such that an equilibrium with a rational bubble exists. - Complementary argument: - If B_{max} were infinite (no maximum), the borrowing constraint would not bind and b_t = 0 by (A.21). But Aiyagari (1994) implies the borrowing constraint would bind infinitely often in the bubbleless economy; introducing a small bubble (smaller than B_{max}) leads to an equilibrium with μ_{t′} > 0 infinitely often — a contradiction unless B_{max} finite. Hence the borrowing constraint binds infinitely often and a stochastic bubble can be supported. ### A.5 Steps for equilibrium selection (three-step procedure) - Step 1 — generate conceivable future bubble values: - For current states (L_t, z_t, m_t, b_t) compute all conceivable ̃b_{t+1} = b_t U_{c,t} / (β π (1 + m_t μ_t) U_{c,t+1}) conditional on no bursting, across all future state combinations (L_{t+1}, z_{t+1}, m_{t+1}, b_{t+1}). - Denote ̃b_{t+1} ≡ ̃b(L_{t+1}, z_{t+1}, m_{t+1}, b_{t+1}, L_t, z_t, m_t, b_t). - Step 2 — fixed point (consistency of expectations): - Solve ̃b(L_{t+1}, z_{t+1}, m_{t+1}, b^*_{t+1}, L_t, z_t, m_t, b_t) = b^*_{t+1} for b^*_{t+1} ≡ b^*(L_{t+1}, z_{t+1}, m_{t+1}, L_t, z_t, m_t, b_t). - Existence argument: consider range [0, b_{max} + ε]; ̃b is continuous, > 0 as b_{t+1} → 0, and < b_{t+1} as b_{t+1} → b_{max} + ε. Hence a solution b^*_{t+1} ∈ (0, b_{max} + ε) exists; uniqueness holds if ̃b is not S-shaped (assumed in quantitative analysis). - If bubble bursts at t+1, set b^*_{t+1} = 0. - Step 3 — collapse expectations into policy functions: - Replace b_{t+1} with b^*_{t+1} in period-t+1 policy functions so expectation terms in (18) and (19) become: - E_t(U_{c,t+1}) ≡ E_t H_U(L_{t+1}, z_{t+1}, m_{t+1}, b^*_{t+1}, L_t, z_t, m_t, b_t). (A.26) - E_t{U_{c,t+1}[z_{t+1} F_{k,t+1} + q_{t+1} (1 + m_{t+1} μ_{t+1})]} ≡ E_t H_q(L_{t+1}, z_{t+1}, m_{t+1}, b^*_{t+1}, L_t, z_t, m_t, b_t). (A.27) ### A.6 Planner’s optimality conditions — first-order conditions and multipliers - Planner FOCs (indexed as in source): - wrt c_t: λ^p_t = U_{c,t} − ξ_t q_t U_{cc,t} + ξ_t β E_t dH_{q,t+1}/db^*_{t+1} · db^*_{t+1}/dc_t. (A.28) - wrt q_t: ξ_t U_{c,t} = m_t μ^p_t. (A.29) - wrt L_{t+1}: λ^p_t = βR E_t(λ^p_{t+1} + ξ_t dH_{q,t+1}/dL_{t+1}) + μ^p_t. (A.30) - wrt v_t and l_t: see (A.31) and (A.32) for full expressions involving λ^p_t, ξ_t, ζ_t, μ^p_t and derivatives dH_{q,t+1}/d(·). - Differentials of the bubble accumulation condition (25): - db^*_{t+1}/dc_t = b^*_{t+1}/(1 + b^*_{t+1}) · (U^*_{cc,t+1}/U^*_{c,t+1}) · dc^*_{t+1}/db^*_{t+1} · U_{cc,t}/U_{c,t}. (A.33) - db^*_{t+1}/dv_t and db^*_{t+1}/dl_t given in (A.34)-(A.35). - Internal consistency yields: - μ^p_t = λ^p_t μ(l_t, v_t) + ζ_t Z_t. (A.36) - ζ_t = − μ^p_t z_t F_{ll,t} − G_{ll,t} z_t F_{vl,t} / (θ p_v m_t (1 + m_t μ(l_t, v_t))) · (m_t U_{c,t} X_t + b_t/(1 + m_t μ(l_t, v_t))). (A.37) - Definitions: - Z_t = (z_t F_{ll,t} − G_{ll,t}) / θ p_v · (F_{vv,t} F_{vl,t} − z_t F_{vl,t}). (A.38) - X_t = β E_t dH_{q,t+1}/db^*_{t+1} · b^*_{t+1}/(1 + b^*_{t+1}) · U^*_{cc,t+1}/U^*_{c,t+1} · dc^*_{t+1}/db^*_{t+1}. (A.39) - Shadow value λ^p_t expression (equation (34) in main text): - λ^p_t = U_{c,t} + F_t q_t + W_t dB_t/dl_t + V_t β E_t dH_{q,t+1}/db^*_{t+1} · db^*_{t+1}/dc_t. (A.40) - With: - F_t = − m_t μ(l_t, v_t) U_{cc,t}/(1 + m_t μ(l_t, v_t)) · (U_{cc,t}/U_{c,t}) q_t. (A.41) - W_t = − m_t μ^p_t Z_t q_t/(z_t F_{ll,t} − G_{ll,t}) · U_{cc,t}/U_{c,t} · 1/(1 + m_t μ(l_t, v_t)) · (U_{cc,t}/U_{c,t}) q_t. (A.42) - V_t = (m_t/(1 + m_t μ(l_t, v_t)) z_t F_{vl,t}/(θ p_v) m_t q_t Z_t z_t F_{ll,t} − G_{ll,t} + 1) m_t μ^p_t U_{c,t} · 1/(1 + m_t μ(l_t, v_t)) · (U_{cc,t}/U_{c,t}) q_t. (A.43) - W_t can be written as W_t = F_t q_t W'_t with W'_t = μ^p_t μ(l_t, v_t) U_{c,t} Z_t/(z_t F_{ll,t} − G_{ll,t}) < 0. (A.44) - Signs and interpretation: - F_t > 0 because U_{cc,t} < 0 and necessary positivity of λ^p_t. - Z_t > 0 under Cobb-Douglas with parameters a_v and a_l and 1 − a_v − a_l > 0, hence W_t < 0 and W_t dB_t/dl_t > 0 because dB_t/dl_t < 0. - V_t sign ambiguous; third and fourth terms in (A.40) associated with the bubble vanish if b_t = 0. Externality terms vanish if the collateral constraint does not bind (μ_t = μ^p_t = W_t = V_t = 0). - System for λ^p_t and μ^p_t: - λ^p_t = U_{c,t} − m_t μ^p_t U_{c,t} q_t U_{cc,t} + m_t μ^p_t U_{c,t} U_{cc,t}/U_{c,t} X_t. (A.45) - μ^p_t = λ^p_t μ(l_t, v_t) / (1 + (m_t/(1 + m_t μ(l_t, v_t))) z_t F_{vl,t}/(θ p_v) (m_t X_t U_{c,t} + B_t) Z_t/(z_t F_{ll,t} − G_{ll,t})). (A.46) ### A.7 Numerical algorithm — computational implementation - Competitive equilibrium (CE) solution method: - Discretization: - 150 states for debt L. - 6 exogenous states ω (3 for productivity z, 2 for pledgeable fraction m). - 10 states for bubble b. - Solve for policy functions {˜L(L, ω, b), c(L, ω, b), q(L, ω, b), l(L, ω, b), v(L, ω, b), μ(L, ω, b), B(L, ω, b), H_U(L, ω, b), H_q(L, ω, b)} satisfying equations (A.47)–(A.55). - Core equilibrium constraints (selected): - Budget: c(L, ω, b) + L + p_v v(L, ω, b) = z F(1, v(L, ω, b), l(L, ω, b)) + ˜L(L, ω, b). (A.47) - Borrowing constraint: ˜L/R + θ p_v v ≤ m (q + B). (A.48) - Bubble relation: b = (1 + m μ(L, ω, b)) B(L, ω, b). (A.49) - Euler and q equations (A.50)–(A.51), first-order conditions (A.52)–(A.55). - Iteration steps: 1. Conjecture current policy functions. 2. Compute ̄b(L′, ω′, b′, L, ω, b) for all future combinations. 3. For each (L′, ω′, L, ω, b) set b^*(·) = argmin_{b′} |̄b(·) − b′|. 4. Generate H_U and H_q using b^*. 5. Solve current policy functions assuming either the borrowing constraint binds (check μ > 0 via (A.50)) or does not bind (set μ = 0). 6. Update conjectures and iterate until sup_{L,ω} ||x^i − x^{i−1}|| < ε with ε = 10^{−4}. - Social planner (SP) solution method: - Value function form: - V(L, ω, b) = max_{˜L, c, l, v, q, B, μ} (U(c − G(l)) + β E_{ω′|ω}[H_V(L′, ω′, b^*, L, ω, b)]) subject to (A.47)–(A.49) and (A.51)–(A.55). (A.56) - H_V(L′, ω′, b^*, L, ω, b) = V(L′, ω′, b^*). (A.57) - Nested algorithm: 1. Outer loop: take current policies V and policy functions (or CE solution for initialization); compute ̄b and b^*; form H_U, H_q, H_V. 2. Inner loop: for each (L, ω, b) solve maximization over ˜L (grid of 5000 candidate ˜L values when borrowing constraint does not bind) or impose borrowing constraint equality when it binds; solve for v, l via (A.52)-(A.53) and compute consumption from (A.47). 3. Update policies and iterate until sup_{L,ω} ||x^i − x^{i−1}|| < ε with ε = 10^{−4}. *Italicized source attribution: wpiea2019184-print-pdf - section A.4–A.7 (as provided).* --- _Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019184-print-pdf.pdf_