Optimal Time-Consistent Macroprudential Policy

Optimal Time-Consistent Macroprudential Policy Javier Bianchi Minneapolis Fed & NBER Enrique G. Mendoza Univ. of Pennsylvania, NBER & PIER

Why study macroprudential policy? MPP has gained relevance as a tool aimed at hampering credit booms that precede financial crises (booms occur with 2.8% prob., but a third of them end in a crisis (Mendoza & Terrones (2012)) Theoretical work highlights inefficiencies that justify ex-ante intervention, but is loosely connected with MPP practice and ignores commitment issues MPP analysis needs a quantitative framework capable of: (1) Matching crisis dynamics and capturing prudential mechanisms (2) Evaluating effectiveness (frequency & magnitude of crises) (3) Addressing inability to commit

What is in this paper? 1 A theoretical and quantitative analysis of optimal, time-consistent MPP in a Fisherian model of financial crises: Occasionally-binding collateral constraint causes crises Collateral valued at marked prices introduces pecuniary externality Forward-looking asset pricing makes MPP time-inconsistent under commitment, so we solve for optimal policy without commitment

Main Theoretical Findings 1 Optimal MPP under commitment is time-inconsistent: Via pricing kernel, future consumption affects current prices When constraint binds, reducing future consumption raises current asset prices & borrowing capacity If government re-optimizes ex post, it ignores the costs of current consumption over previous collateral values and borrowing capacity 2 Constrained-efficient allocations (with or without commitment) are implementable with a state-contingent debt tax

Main Quantitative Findings 1 Optimal, time-consistent MPP is very effective: Probability of crises falls from 4% to 0.02% Asset Prices fall 39 ppts less (44% v. 5%) Equity Premium decreases by a factor of 6 (from 5% to 0.8%) 2 Tax on debt averages 3.6%, with 0.7 corr. with leverage

Related Literature Pecuniary Externalities and MPP: Caballero-Krishnamurthy (2001), Lorenzoni (2008), Bianchi (2011), Bianchi-Mendoza(2010), Jeanne-Korinek (2011), Benigno et al. (2010), Stein (2012) Quantitative Macro-Finance Models: Financial Accelerator Models: Bernanke-Gertler-Gilchrist (1999), Kiyotaki-Moore (1997), Jermann-Quadrini (2012), Gertler-Kiyotaki (2010), Christiano, Motto and Rostagno (2013)... Non-Linear Systemic Risk Models: Mendoza (2010), Bianchi (2012), He-Krishnamurthy (2012), Brunnermeier-Sannikov (2011)... Markov Perfect Equilibrium: Klein-Krusell-Rios-Rull (2008), Judd (2004), Krusell-Smith (2003)

Outline 1 Analytics of Pecuniary Externality and Time Inconsistency (in a simplified model for presentation) Aggregate collateral, endowment economy 2 Model for Quantitative Analysis Individual collateral, production, working capital 3 Quantitative Findings

Decentralized Equilibrium without Policy Households solve: max {c t,k t+1,b t+1 } t 0 E t β t u(c t ) t=0 s.t. c t + q t k t+1 + b t+1 R b t+1 R = k t (q t + z t ) + b t (λ t ) κq t k t (µ t ) z t follows a Markov process Aggregate capital in unit fixed supply used as collateral One-period, non-state-contingent bonds, exog. interest rate R Recursive DE

Excess Returns and Asset Pricing Binding constraint increases excess returns E t [R k t+1] R = µ t Cov t (βu (c t+1 ), R k t+1 R) βe t u (c t+1 )

Excess Returns and Asset Pricing Binding constraint increases excess returns E t [R k t+1] R = µ t Cov t (βu (c t+1 ), R k t+1 R) βe t u (c t+1 ) causing asset prices to fall z t+j+1 q t = E t j j=0 E t+i Rt+1+i k i=0 tightening further the constraint and feeding back to asset prices

Normative Analysis Constrained-efficient regulator (planner) chooses debt and transfers borrowed resources facing the same credit constraint Households choose c t+1, k t+1 (asset market remains competitive) Asset Euler eq. becomes implementability constraint: q t u (c t ) = βe t u (c t+1 ) (z t+1 + q t+1 ) Without commitment, the regulator at date t takes into account how its decisions affect the regulator s plans at t+1, and thereby c t+1, q t+1 and thus q t

Time-Consistent Regulator s Problem Taking as given future regulator s C and Q, the planner solves: subject to [ ] V (b, z) = max u(c) + βe z c,b,q zv (b, z ) c + b R = b + z (λ) b R κq (µ ) q = βeu (C(b, z ))(Q(b, z ) + z ) u (c) (ξ)

Via q t (when µ t > 0): Pecuniary Externalities c t : λ t = u (c t ) κµ t q t u (c t ) u (c t ) }{{} Extra Benefits from c t

Via q t (when µ t > 0): Pecuniary Externalities c t : λ t = u (c t ) κµ t q t u (c t ) u (c t ) }{{} Extra Benefits from c t b t+1 : λ t = βre t λ t+1 + ( ξ t βe t u (c t+1 )C b (t + 1)(Q t+1 (t + 1)) + z t+1 ) + Q b (t + 1)u (c t+1 ) ) + µ t }{{} Effects of Future Policies on Current Asset Prices

Optimal Time-Consistent Debt Tax Proposition: The regulator s equilibrium can be decentralized with a state-contingent debt tax (i.e. bond prices become 1/[R(1 + τ t )]) with its revenue rebated as a lump-sum transfer and a tax rate such that: 1 + τ t = 1 E t u (t + 1) E [ t u (t + 1) ξ t+1 u ] (t + 1)Q t+1 + ξ t Ω t+1 1 [ + ξt βre t u u ] (t)q t (t + 1) MP debt tax: If µ t = 0 ande[µ t+1 ] > 0, the tax reduces to: τ MP t = E t κµ t+1 u (C(b t+1,z t+1 )) u (C(b t+1, z t+1 ))Q(b t+1, z t+1 ) E t u (C(b t+1, z t+1 ))

Equity Premia in the DE and SP Decentralized equilibrium R ep t = µ t u (t)e t m t+1 }{{} Liquidity E t (φ t+1 m t+1 ) E t m t+1 } {{ } Collateral m t+1 βu (c t+1) u (c t), φ t+1 κ µt+1 q t+1 u (c t) q t. cov t(m t+1, R q t+1 ) E t [m t+1 ] }{{} Risk Social planner R ep t = µ t + ξ t u (t)q t + βre t ξ t Ω t+1 u E ( ) t φ t+1 m t+1 (t)e t m t+1 E t m t+1 }{{}}{{} Liquidity Colllateral cov t(m t+1, R q t+1 ) E t m t+1 } {{ } Risk βr te t (ξ t+1 u (t + 1)Q t+1 ) } u (t)e t m t+1 {{ } Externality

Comparison with Commitment c t : λ t = u (c t ) ξ t q t u (c t ) + u (c t )ξ t 1 (q t + z t ) t > 0 q t :: ξ t = ξ t 1 + µ t κ u (c t ) t > 0 b t+1 :: λ t = βr t E t λ t+1 + µ t t 0 Higher current consumption still raises current asset prices

Comparison with Commitment c t : λ t = u (c t ) ξ t q t u (c t ) + u (c t )ξ t 1 (q t + z t ) t > 0 q t :: ξ t = ξ t 1 + µ t κ u (c t ) t > 0 b t+1 :: λ t = βr t E t λ t+1 + µ t t 0 But now lower current consumption raises previous asset prices

Optimal Macroprudential Debt Tax Without commitment (Markov stationary): τ M t = E t κµ t+1 u (c t+1 ) u (c t+1 )q t+1 E t u (c t+1 ) With commitment (Ramsey): τ R t = E t κµ t+1 u (c t+1 ) u (c t+1 )q t+1 + ξ t 1 (E t u (c t+1 )z t+1 z t u (c t )) E t u (c t+1 ) Taxes differ if a collateral constraint was binding in the past τt R τt M if output is high relative to the future Quantitatively, asset prices are higher (lower) with (without) commitment than without regulation

Decentralized Eq. v. MPP with Commitment -66 Value Function 1.5 Consumption -66.5 1-67 0.5-67.5-0.8-0.6-0.4-0.2 0 b Asset Prices 0.8 0.6 0.4 0.2 0-0.8-0.6-0.4-0.2 0 b 0-0.8-0.6-0.4-0.2 0 b Bond Policy Function 0-0.1-0.2-0.3 Ramsey DE -0.8-0.6-0.4-0.2 0 b

Quantitative Analysis Introduce firms, labor supply, intermediate goods Add working capital for purchases of intermediate goods Assume capital has individual value as collateral Introduce TFP, interest-rate and financial shocks

Representative Firm-Household Problem Maximize: subject to: [ ] E 0 β t u(c t G(n t )) t=0 q t k t+1 + c t + b t+1 R t = q t k t + b t + [z t F(k t, v t, n t ) p v v t ] b t+1 R + θp vv t κ t q t k t

Representative Firm-Household Problem Maximize: subject to: [ ] E 0 β t u(c t G(n t )) t=0 q t k t+1 + c t + b t+1 R t = q t k t + b t + [z t F(k t, v t, n t ) p v v t ] with functional forms: b t+1 R + θp vv t κ t q t k t u(c G(h)) = ) 1 σ (c χ h1+ω 1+ω 1 1 σ ω > 0, σ > 1 F(k, h, v) = e z k α k v αv h α h, α k, α v, α h 0 α k + α v + α v 1

Calibration to OECD & U.S. Data Parameters set independently Value Source/Target Risk aversion σ = 1. Standard value Share of inputs in gross output α v = 0.45 Cross country average OECD Share of labor in gross output α h = 0.352 OECD GDP Labor share = 0.64 Labor disutility coefficient χ = 0.352 Normalization (mean h = 1) Frisch elasticity 1/ω = 2 Keane and Rogerson (2012) Working capital coefficient θ = 0.16 U.S. WK/GDP ratio=0.133 Tight credit regime κ L = 0.75 U.S. post-crisis LTV ratios Normal credit regime κ H = 0.90 U.S. pre-crisis LTV ratios Interest rate R = 1.1%, ρr = 0.68 U.S. 90-day T-Bills σ R = 1.86% Parameters set by simulation Value Target TFP shock ρ z = 0.78, σ z = 0.01 GDP sd. & autoc. (OECD average) Share of assets in gross output α k = 0.008 Value of collateral matches total credit Discount factor β = 0.95 Private NFA = 25 percent Transition prob. κ H to κ L P H,L = 0.1 4 crises every 100 years (Appendix E2) Transition prob. κ L to κ L P L,L = 0. 1 year duration of crises (Appendix E2) More calibration

Comparing DE and SP Decision Rules Positive Crisis Probability Region Next-Period Bond Holdings (B ) -0.15-0.2-0.25 Constrained Credit Region B SP (B,s) B DE (B,s) B DE (B,s) B SP (B,s) Stable Credit Region -0.25-0.2-0.15-0.1 Current Bond Holdings (B)

Financial Amplification in DE -0.2 Next-Period Bond Holdings -0.25-0.3 A -0.3-0.25-0.2 Current Bond Holdings Stationary bond choice at t with good shock

Financial Amplification in DE Cont. -0.2 A Next-Period Bond Holdings -0.25-0.3 A -0.3-0.25-0.2 Current Bond Holdings Response to a bad shock at t + 1

Financial Amplification for the Planner -0.2 A Next-Period Bond Holdings -0.25-0.3 B A -0.3-0.25-0.2 Current Bond Holdings SP s bond choice at t for same initial condition

Financial Amplification for the Planner Cont. -0.2 A Next-Period Bond Holdings -0.25-0.3 B B A -0.3-0.25-0.2 Current Bond Holdings SP s response to SAME bad shock at t + 1

Effectiveness of MPP: Summary Statistics DE SP Crisis Statistics Probability of crisis 4.0 0.02 Asset Price Drop -43.7-5.4 Equity Premium 4.8 0.7 Mean tax and welfare gains Macroprudential Debt Tax 3.6 Welfare Gains 0.30

10 Financial Crises with and without Policy 0 40 35 (a) Credit 10 0 (b) Asset Price -10-10 30-20 25-30 -40-20 20 0 t-3 t t+3 (c) Output -50 5 t-2 t t+2 (d) Consumption -30-2 0-5 -4-10 -40-6 -15-20 -8-25 -50-10 t-2 t t+2-30 t-2 t t+2 Decentralized Equilibrium Social Planner

Optimal MP Taxes around Financial Crises (b) Dynamics around Crises Events 12 9 % 6 3 0 t-2 t t+2

Distributions of Asset Returns 1 0.8 Probability 0.6 0.4 Social Planner 0.2 Decentralized Equilibrium 0-45 -35-25 -15-5 5 15 Asset Returns (in percentage)

Asset Pricing in Good Times 10 assetp 0 25 20 (d) Volatility Asset Return 0.1 0.08 (e) Risk Premium 15 0.06-10 10 0.04 5 0.02-20 -30-40 -50 0-0.3-0.25-0.2 Current Bond Holdings 30 27 24 21 18 15 12 9 6 3 (a) Expected Return on Assets 0-0.3-0.25-0.2 Current Bond Holdings 0-0.3-0.25-0.2 Current Bond Holdings 0.48 0.46 0.44 0.42 0.4 0.38 0.36 0.34 Asset Price (b) 0.32-0.3-0.25-0.2 Current Bond Holdings

Simple Macroprudential Policy Rules 1 Fixed debt tax across time and states 2 Financial Taylor Rule: τ = max[0, τ 0 (b t+1 / b) η b 1] Both are set to maximize average welfare gain γ(b, s)dπ 0 (b, s), where π 0 (b, s) is DE s cum. ergodic distribution and γ(b, s) is the welfare gain at state (b, s) defined as: E 0 t=0 β t u(c DE t (1 + γ) G(h DE t )) = E 0 t=0 β t u(c SP t G(h SP t ))

Comparing Optimal TC-MPP with Simple Rules Decentralized Optimal Best Best Equilibrium Policy Taylor Fixed Welfare Gains (%) 0.30 0.09 0.03 Crisis Probability (%) 4.0 0.02 2.2 3.6 Drop in Asset Prices (%) 43.7 5.4 36.3 41.3 Equity Premium (%) 4.8 0.77 3.9 4.3 Tax Statistics Mean 3.6 1.0 0.6 Std relative to GDP 0.5 0.2 Correlation with Leverage 0.7 0.3

Fixed Taxes, Crisis Probability & Welfare 4 (a) Crisis Probability 3.5 Percentage 3 2.5 2 1.5 1 0 0.5 1 1.5 2 Tax (%)

Fixed Taxes, Crisis Probability & Welfare Percentage 4 3.5 3 2.5 2 1.5 (a) Crisis Probability Percentage Points 0.3 0.2 0.1 0-0.1-0.2 (b) Welfare Gains max average min 1 0 0.5 1 1.5 2 Tax (%) -0.3 0 0.5 1 1.5 2 Tax (%)

0-10 Effects of Simple Policies on Crises 40-20 (a) Credit/GDP (b) Asset Price 35-30 0-10 30-20 25-40 -30-40 20-50 t-3 t t+3-50 t-2 t t+2 Decentralized Equilibrium Optimal Tax Simple Rule Fixed Tax

Conclusions 1 Optimal MPP under commitment is not credible 2 Optimal, time-consistent MPP is very effective at reducing frequency & severity of crises, and increasing welfare 3 Simple rules reduce frequency of crises but are otherwise much less effective and can reduce welfare 4 MPP faces other serious hurdles: adapting to financial innovation and imperfect information (Bianchi, Boz & Mendoza, 2012), coordination with monetary policy, debtor heterogeneity, etc. 5 Ongoing agenda: MPP with heterogeneous agents and nominal rigidities, value of commitment in MPP

Commitment: Recursive Problem Time t > 0 problem V (b, J, z) = max b,j (z ),c u(c) + βev (b, J (z ), z ) b R + c = b + z b κq q = βe zj (z ) u (c) J = u (c)z + βe z J ( z )

Commitment: Recursive Problem Time t = 0 problem: V 0 (b, J, z) = max b,j (z ),c u(c) + βev (b, J (z ), z ) b R + c = b + z b κq q = βe zj (z ) u (c) J = u (c)z + βe z J ( z ) model

Recursive Competitive Equilibrium A RCE is defined by a pricing function q(b, z), a law of motion B, and policy functions with associated value function such that: { 1 V, ˆb, ˆk }, ĉ solve: V (b, k, B, z) = max b,k,c u(c) + βe z zv (b, k, B, z ) s.t. q(b, z)k + c + b = k (q(b, z) + z) + b R with B = B(B, z) b κq(b, z) R 2 Rational Expectations: B(B, z) = ˆb(B, 1, B, z). 3 Asset market clears ˆk (B, 1, B, z) = 1 model

Microfoundation for Collateral Constraint Households enter period with outstanding debt, repay and then issue new debt Opportunity to default on new issuances at the end of the period Upon default: HH loses (1 κ) of value of assets, but can immediately raise new debt HH makes take it or leave it offer to creditors accepted if b R κqk

Value of Default V d ( d, b, k, X) = max b,k,c u(c) + βe s sv (b, k, B, z ) s.t. q(b, z)k + c + b R = d + q(b, z)k(1 κ) + b + zk Household defaults if b R κqk model b κq(b, z)k R

Asset Pricing Statistics (1) (2) (3) (4) (5) (6) (7) (8) (9) Expected Risk-free Equity Liquidity Collateral Risk Price Return Plus Tax Premium Premium Effect Premium of Risk σ t (R q t+1 ) SR t Decentralized Equilibrium Unconditional 6.0 1.2 4.8 4.7 1.4 1.5 14.6 9.1 0.5 Constrained 85.6 1.2 84.4 84.1 0.0 0.2 4.1 6.2 13.7 Unconstrained 1.3 1.2 0.1 0.0 1.5 1.6 15.3 9.3 0.0 Social Planner Unconditional 4.1 3.3 0.8 1.8 1.2 0.2 5.2 3.8 0.2 Constrained 6.9-21.8 28.7 28.5 0.0 0.2 5.1 3.8 7.6 Unconstrained 3.9 5.0-1.2 0.0 1.3 0.2 5.2 3.8-0.3 assetpg

Calibration Strategy Industrialized economies for post-financial globalization period (1984:Q1 2010:Q2): Preferences and production parameters set independently to match standard targets TFP and interest rates estimated as a VAR(1) Financial shocks are assumed to be independent and follow a { } two-state Markov chain κ L, κ H with transition matrix P P calibrated to match frequency and duration of financial crises (crisis defined as a fall in credit of more than 2SD) back calibration