The I Theory of Money & On the Optimal Inflation Rate

The I Theory of Money & On the Optimal Inflation Rate Markus Brunnermeier & Yuliy Sannikov Princeton Initiative 2017 Princeton, Sept. 8 th, 2017

Money and Banking (in macro-finance) Money Banking store of value/safe asset diversifier holds risky assets, issues inside money mplification/endogenous risk dynamics Value of capital declines due to fire-sales Flight to safety Value of money rises iquidity spiral Disinflation spiral a la Fisher Demand for money rises less idiosyncratic risk is diversified Supply for inside money declines less creation by intermediaries Endogenous money multiplier = f(capitalization of critical sector) Paradox of Thrift (in risk terms) Monetary Policy (redistributive)

Money and Banking (in macro-finance) Money Banking store of value/safe asset diversifier holds risky assets, issues inside money mplification/endogenous risk dynamics Value of capital declines due to fire-sales Flight to safety Value of money rises iquidity spiral Disinflation spiral a la Fisher Demand for money rises less idiosyncratic risk is diversified Supply for inside money declines less creation by intermediaries Endogenous money multiplier = f(capitalization of critical sector) Paradox of Prudence Paradox of Thrift (in risk terms) Monetary Policy (redistributive)

Some literature Roles of money Unit of account Medium of exchange (Clower, agos & Wright) Store of value (Samuelson, Bewley, iyagari, Scheinkman & Weiss, Kiyotaki & Moore) Models without inside money imply inflation in downturns ess money needed to perform fewer transactions Money view (Friedman & Schwartz) Downturns Bank liabilities decrease Credit view Downturns equity capital bank cuts assets/credit BGG, Kiyotaki & Moore, He & Krishnamurthy, BruSan2014, Drechsler, Jeanne & Korinek, Savov & Schnabl Financial Stability Diamond & Rajan 2010, Curdia & Woodford 2010, Stein 2012

Monetary Policy Transmission Channel Consumption Boost approach to Bottleneck approach (New) Keynesian Demand Management Stimulate aggregate consumption Substitution effect I Theory of Money Risk (premium) management lleviate balance sheet constraints Income/wealth effect Woodford Tobin (1982) BruSan Price stickiness Perfect capital markets Representative gent Cut i Reduces r due to price stickiness Consumption c rises Both Heterogeneous gents Financial Frictions Incomplete markets - -

Monetary Policy Transmission Channel Consumption Boost approach to Bottleneck approach (New) Keynesian Demand Management Stimulate aggregate consumption Substitution effect I Theory of Money Risk (premium) management lleviate balance sheet constraints Income/wealth effect Woodford Tobin (1982) BruSan Price stickiness Perfect capital markets Representative gent Cut i Reduces r due to price stickiness Consumption c rises Both Cut i Changes bond prices Redistributes from low MPC to high MPC consumers Heterogeneous gents Financial Frictions Incomplete markets - -

Monetary Policy Transmission Channel Consumption Boost approach to Bottleneck approach (New) Keynesian Demand Management I Theory of Money Risk (premium) management Stimulate aggregate consumption Substitution effect lleviate balance sheet constraints Income/wealth effect Woodford Tobin (1982) BruSan Price stickiness Perfect capital markets Both Financial Frictions Incomplete markets Representative gent Heterogeneous gents Cut i Reduces r due to price stickiness Consumption c rises Cut i Changes bond prices Redistributes from low MPC to high MPC consumers Cut i Changes asset prices Ex-post: Redistributes to balance sheet impaired sector QE - -

iterature Without intermediaries: Money as store of value = bubble \Friction OG Incomplete Markets + idiosyncratic risk Risk deterministic endowment risk borrowing constraint investment risk Only money Samuelson Bewley With capital Diamond iyagari ngeletos Inefficiency r < r, K > K Risk tied up with individual capital

iterature Without intermediaries: Money as store of value = bubble \Friction OG Incomplete Markets + idiosyncratic risk Risk deterministic endowment risk borrowing constraint investment risk Only money Samuelson Bewley With capital Diamond iyagari f k = r, Dynamic inefficiency r < r, K > K (money) bubbles if r < g bel et al. vs. Geerolf Inefficiency r < r, K > K Basic I Theory cash flow shock Pecuniary externality Inefficiency r > r, K < K r m = g

Roadmap Model without intermediaries Fixed (outside) money supply Optimal money growth rate On the optimal inflation rate (inflation target) Model with intermediaries Fixed outside money supply Monetary Policy Macro-prudential policy Intermediaries with market power The Reversal Interest Rate: The Effective ower Bound

Model without intermediaries Technologies a 1 Each household can only operate one firm Physical capital dk t = (Φ ι k t δ)dt + σ a dz a a t + σd Z t t Output y t = k t Demand for money sector idiosyncratic risk

Net worth dding outside money q t K t value of physical capital Postulate constant q t ι q dt + Φ(ι δ) dt + σb dz t b + σd Z t b p t K t value of outside money Postulate value of money changes proportional to K t Outside Money Technologies a Money 1 Each household can only operate one firm Physical capital dk t = (Φ ι k t δ)dt + σ a dz a a t + σd Z t t Output y t = k t Demand for money sector idiosyncratic risk

Solving 1. Postulate Price processes dp t /p t = μ t p dt + σ p dz t, dq t /q t = Portfolio processes dx t a /x t a 2. Derive return processes dr Ka = Φ ι δ dt + ι dt + σd Z q t dr M = dt (μ M μ Mi )dt money supply growth rate that is NOT distributed via interest payment Set μ Mi = 0 3. Optimality conditions & Market clearing conditions 4. Solve undetermined coefficients (μ x s t, σ x (s t )) Solving ODE with boundary conditions Solve for constants p, q

Solving 1. Postulate Price processes dp t /p t = μ t p dt + σ p dz t, dq t /q t = Portfolio processes dx t a /x t a 2. Derive return processes dr Ka = Φ ι δ dt + ι q dt + σd Z t dr M = Φ ι δ dt (μ M μ Mi )dt money supply growth rate that is NOT distributed via interest payment Set μ Mi = 0 3. Optimality conditions & Market clearing conditions 4. Solve undetermined coefficients (μ x s t, σ x (s t )) Solving ODE with boundary conditions Solve for constants p, q

Optimality (=) for E 0 e ρt log c t dt ) Investment rate, ι Tobin s q: Φ ι = 1 q For Φ ι = 1 log(κι + 1) κι = q 1 κ (static problem) Portfolio choice, x a E dr Ka dr M /dt = Cov[dr Ka dr M, x a = E drka dr M /dt (σ) 2 dn t ] = x ด a ( n σ)2 t dr M +x a dr K dr M = ( ι)/q+μm (σ) 2 = q q+p Dividend yield on capital must be ρ Consumption, c t Demand ρn t = ρ q + p K t = a ι K t Supply q = q q + p =x a (a ι)/ρ

Optimality (=) & market clearing (=) Investment rate, ι Tobin s q: Φ ι = 1 q For Φ ι = 1 log(κι + 1) κι = q 1 κ (static problem) Portfolio choice, x a E dr Ka dr M /dt = Cov[dr Ka dr M, x a = E drka dr M /dt (σ) 2 dn t ] = x ด a ( n σ)2 t dr M +x a dr K dr M = ( ι)/q+μm (σ) 2 = q q+p Dividend yield on capital must be ρ Consumption, c t Demand ρn t = ρ q + p K t = a ι K t Supply q = q q + p =x a (a ι)/ρ

Optimality (=) & market clearing (=) Investment rate, ι Tobin s q: Φ ι = 1 q For Φ ι = 1 log(κι + 1) κι = q 1 κ (static problem) Portfolio choice, x a E dr Ka dr M /dt = Cov[dr Ka dr M, x a = E drka dr M /dt (σ) 2 Dividend yield on capital must be ρ Consumption, c t Output market clearing Demand ρn t = ρ q + p K t = ι K t dn t ] = x ด a ( n σ)2 t dr M +x a dr K dr M = ( ι)/q+μm (σ) 2 = q q+p Capital market clearing Supply q = q q + p =x a ( ι)/ρ

Equilibrium Moneyless equilibrium p 0 = 0 Money equilibrium p = σ ρ ρ q q 0 = κ+1 κρ+1 > q = κ+1 κ ρσ+1 q 0 p q 0 ρ σ

Welfare analysis Moneyless equilibrium p 0 = 0 Money equilibrium p = σ ρ ρ q q 0 = κ+1 κρ+1 > q = κ+1 g 0 welfare 0 > < κ ρσ+1 g welfare

Equilibrium Collecting the three equations: Equilibrium solved in terms of μ x k μ (monotone transformation)

Welfare Plug in FOC in value function Plug in equilibrium ll households start symmetrically Expected Utility of an individual household

Optimal inflation rate Money growth μ affects (steady state) inflation in two ways π = μ M (Φ ι μ M δ) Proposition: If σ ρ > 2(κ+1) 1+2κρ, welfare maximizing money growth rate μ > 0. Market outcome is not even constrained Pareto efficient Economic growth rate, g > r m, is also higher Growth maximizing μ g μ M, s.t. p g = 0, Tobin (1965) g increasing in μƹ Corollary: No super-neutrality of money Nominal money growth rate affects real economy No price/wage rigidity, no monopolistic competition

Optimal inflation rate: Emerging markets Proposition: (comparative static) μ M does not depend on depreciation rate δ, but inflation does μ M is strictly increasing in idiosyncratic risk σ Emerging markets should have higher inflation target

Main results HH portfolio choice Physical capital: w/ idiosyncratic risk + dividend Money: w/o idiosyncratic risk + no dividend (bubble) Tilted inefficiently towards money Money supply growth inflation tax on money lowers real interest rate tilts portfolio choice boosts physical investment higher economic growth raises real interest rate (partially undoes inflation tax) Pecuniary externality: individual households do not take this GE effect into account. Planner who can print money and distribute seignorage can improve growth + Pareto welfare. Derive optimal money growth rate/inflation rate

Outline of two sector model Technologies b Technologies a 1 B 1 1 1 Households have to Switch Switch technology Specialize in one subsector Demand for money sector specific + idiosyncratic risk for one period dk t = dt + σ b dz b b dk t k t + σd Z t = dt + σ a dz a a t + σd Z t t k t

Net worth Net worth dd outside money Technologies b Outside Money Technologies a Money Money B 1 1 Switch Switch technology Households have to Specialize in one subsector for one period Demand for money

Net worth Net worth dd intermediaries Technologies b Outside Money Technologies a Net worth Money Money B 1 1 Risk can be partially sold off to intermediaries Risk is not contractable (Plagued with moral hazard problems)

Net worth Net worth dd intermediaries Technologies b Outside Money Technologies a Net worth Money Money B 1 1 Intermediaries Can hold outside equity & diversify within sector b Monitoring

Inside equity Risky Claim Risky Claim Risky Claim Net worth Risky Claim Risky Claim Risky Claim dd intermediaries Technologies b Outside Money Technologies a Money Money B 1 Net worth 1 Intermediaries Can hold outside equity & diversify within sector b Monitoring

Inside equity Risky Claim Risky Claim Risky Claim HH Net worth Risky Claim Risky Claim Risky Claim dd intermediaries Technologies b Outside Money Pass through Outside Money Technologies a Money Inside Money (deposits) B 1 Net worth Money 1 Intermediaries Can hold outside equity & diversify within sector b Monitoring Create inside money Maturity/liquidity transformation

Inside equity Risky Claim Risky Claim HH Net worth Risky Claim Risky Claim Risky Claim Shock impairs assets: 1 st of 4 steps Technologies b Outside Money Pass through Technologies a Money Inside Money (deposits) B 1 Net worth osses Money 1

Inside equity Risky Claim Risky Claim HH Net worth Risky Claim Risky Claim Risky Claim Shrink balance sheet: 2 nd of 4 steps Technologies b Money Deleveraging Deleveraging Outside Money Pass through Inside Money Inside Money (deposits) (deposits) Technologies a B 1 1 Net worth osses Money 1 Switch Paradox of Prudence

Inside equity Risky Claim Risky Claim HH Net worth Risky Claim Risky Claim Risky Claim iquidity spiral: asset price drop: 3 rd of 4 Technologies b Money Deleveraging Outside Money Deleveraging Pass through Inside Money Inside Money (deposits) (deposits) Technologies a B 1 1 Net worth osses Money 1 Switch

Inside equity Risky Claim Risky Claim HH Net worth Risky Claim Risky Claim Risky Claim Disinflationary spiral: 4 th of 4 steps Technologies b Money Deleveraging Deleveraging Outside Money Pass through Inside Money Inside Money (deposits) (deposits) Technologies a B 1 1 Net worth osses Money 1

after an adverse shock Intermediaries are hit and shrink their balance sheets inducing sset side liquidity spiral financial stability iability side disinflation spiral price stability Response of intermediaries to adverse shock leads to endogenous risk mplification Persistence Other sectors can also be undercapitalized Japan 1980: US 2000s: corporate sector household sector

Formal model: capital & output Technologies b a Physical capital K t - Capital share ψ t 1 ψ t Output goods Y b b t = k t Y a a substitutes t = k t ggregate good (CES) - Consumed or invested - numeraire Price of goods Y t = P t b = 1 2 1 2 Y t b (s 1)/s + 1 s/(s 1) 2 Y a (s 1)/s t Y t Y t b 1/s Imperfect P t a = 1 2 Y t Y t a 1/s Model setup in paper is more general: Y t = ψ t K t

Formal model: risk When k t is employed in sector a by agent j dk t = Φ ι t δ k t dt + σ a k t dz t a + σ j k t d Z t a Investment rate (per unit of k t ) sectorial idiosyncratic independent Brownian motions (fundamental cash flow risk) Φ ι t is increasing and concave, e.g. log[ κι t + 1 /κ] ll dz are independent of each other Intermediaries can diversify within sector b Face no idiosyncratic risk Households cannot become intermediaries or vice versa

Financing constraints Technologies b a Equity issuance - Special case Inside equity χ t χ χ = 0% (no inside equity) Inside equity only Households risk dz b & d Z b dz a & d Z a sector & idiosyncratic sector & idiosyncratic Intermediaries risk dz b can diversify idiosyncratic risk

Inside equity Risky Claim Risky Claim Risky Claim HH Net worth Capital/risk shares Technologies b Outside Money Pass through Technologies a Fraction α t of HH Money Inside Money (deposits) χ t ψ t q t K t Money Net worth N t ψ t q t K t (1 ψ t )q t K t 1 χ t χ t

Formal model: preferences ll agents have logarithmic utility with discount rate ρ E න e ρt log c t dt 0 Implies Consumption = ρ * net worth Equilibrium Sharpe ratio Covariance with net worth

Solution steps 1. Postulate endogenous processes dq t /q t = μ t q dt + σ t q,a dz t a + σ t q,b dz t b Returns from holding capital dp t /p t = μ t p dt + σ t p,a dz t a + σ t p,b dz t b 2. Equilibrium conditions gents optimization Internal investment (new capital formation) Optimal portfolio choice Sharpe ratio Cov. with net worth Optimal consumption ρ networth Market clearing conditions 3. aw of motion of state variable wealth (share) distribution η t 4. Express in ODEs of state variable 61

sset returns on technology b Physical capital: (in technology b) also earns dividend yield Vector dz t a, dz t b If dq t /q t = μ q q t dt + (σ t ) T dz t, dk t /k t = Φ ι t δ dt + σ b dz b t + σ j b,j dz t dr b t = P t b ι t q q dt+ Φ ι q t δ+μ t t +(σt ) T σ i 1 b q dt+ σ t +σ a 1 b T dz t +σ j b,j dz t dr a t = (analogous) Return on outside equity held by intermediaries dr t I = dr t b λ t dt risk premium Return on inside equity (fraction χ t ) held by a-hh dr t χ = drt b + 1 χ t χ t λ t dt

sset returns on technology b Physical capital: (in technology b) also earns dividend yield If dq t /q t = μ q q t dt + (σ t ) T dz t, dk t /k t = Φ ι t δ dt + σ b dz b t + σ j b,j dz t dr b t = P t b ι t q q dt+ Φ ι q t δ+μ t t +(σt ) T σ i 1 b q dt+ σ t +σ a 1 b T dz t +σ j b,j dz t dr a t = (analogous) Dividend yield Expected capital gains Return on outside equity held by intermediaries dr t I = dr t b λ t dt risk premium Return on inside equity (fraction χ t ) held by a-hh dr t χ = drt b + 1 χ t χ t λ t dt

sset returns on technology b Physical capital: (in technology b) also earns dividend yield If dq t /q t = μ q q t dt + (σ t ) T dz t, dk t /k t = Φ ι t δ dt + σ b dz b t + σ j b,j dz t dr b t = P t b ι t q q dt+ Φ ι q t δ+μ t t +(σt ) T σ1 b q dt+ σ t +σ1 b T dz t +σ j b,j dz t dr a t = (analogous) χ t dr t χ + 1 χt dr t I = dr t b Return on outside equity held by intermediaries dr t I = dr t b λ t dt risk premium Return on inside equity (fraction χ t ) held by b-hh dr t χ = drt b + 1 χ t χ t λ t dt

sset returns on money Money: fixed supply in baseline model, total value p t K t Return = capital gains (no dividend/interest in baseline model) If dp t /p t = μ p p t dt + σ t dzt, dk t /K t = Φ ι t δ dt + 1 ψ t σ a dz a t + ψ t σ b b dz t (σ t K ) T dz t dr t M = Φ ι t δ + μ t p + σ t p T σt K dt + σ t p + σt K dz t θ t = p t q t +p t fraction of wealth in form of money

llocation Equilibrium is a map Histories of shocks prices q t, p t, λ t, allocation Z τ, 0 τ t α t, χ t & portfolio weights (x t, x t a, x t b ) wealth distribution η t = N t (p t +q t )K t 0,1 intermediaries wealth share ll agents maximize utility Choose: portfolio, consumption, technology ll markets clear Consumption, capital, money, outside equity of b

Numerical example: prices 3 2.5 2 p, q 1.5 iquidity spiral q 1 0.5 Disinflation spiral p 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Numerical example: prices 3 2.5 2 θ = p p+q p, q 1.5 iquidity spiral q 1 0.5 Disinflation spiral p 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Numerical example: dynamics of η fundamental volatility η x t (σ b 1 b σ K t ) σ t = 1 x t 1 1 θ t leverage θ η t θ/η t elasticity amplification Steady state

Numerical example: dynamics of η Steady state

Welfare analysis Challenge: Heterogeneous agents with idiosyncratic risks Inefficiencies in Production Investment Risk sharing -20-25 log( (1 - ))/ + U H ( ) households -30-35 log( )/ + U I ( ) -40 household welfare in autarky -45-50 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Roadmap Model without intermediaries Fixed (outside) money supply Optimal money growth rate On the optimal inflation rate (inflation target) Model with intermediaries Fixed outside money supply - mplification/endogenous risk iquidity spiral Disinflationary spiral asset side of intermediaries balance sheet liability side Monetary Policy Macro-prudential policy Intermediaries with market power The Reversal Interest Rate: The Effective ower Bound

Monetary Policy: Ex-post perspective Money view Friedman-Schwartz Restore money supply Replace missing inside money with outside money im: Reduce deflationary spiral but banks extent less credit & diversify less idiosyncratic risk away as households have to hold more idiosyncratic risk, money demand rises Undershoots inflation target Credit view Tobin Restore credit im: Switch off deflationary spiral & liquidity spiral I Theory: Stealth recapitalization of impaired sector Interest policy and OMO affect asset prices 76

Policy Monetary Policy Introduce long-term bond Central bank s actions change money supply/transfer risk Interest rate cuts in downturns raise the value of long-term bonds Change relative price between long-term bond and short-term money Risk transfer (ex-post redistribution) Macro-prudential policy 1. everage upper bounds 2. ffect agents portfolio choice directly

Introducing ong-term Gov. Bond Introduce long-term (perpetual) bond No default held by intermediaries in equilibrium Value b t K t Perpetual bonds: pay in money (at unit rate) endogenous price B t (in money) Money Capital Value p t K t Value q t K t Value of long-term bond is endogenous db t /B t = μ t B dt + σ t B dz t

Redistributive MoPo: Ex-post perspective Outside Money Pass through Bonds b t K t χ t ψ t q t K t Inside Money (deposits) Net worth N t dverse shock value of risky claims drops Monetary policy Interest rate cut long-term bond price sset purchase asset price stealth recapitalization - redistributive risk premia iquidity & Deflationary Spirals are mitigated

Monetary policy and endogenous risk Intermediaries risk (borrow to scale up) η x t (1 b σ b σ K t ) σ t = 1 + χ t ψ t η 1 η x η t t + θ t t η t amplification θ η t θ/η t fundamental risk b t p t mitigation B η t B(η t )/η t MoPo works through B η t B(η t )/η t with right monetary policy bond price B(η) rises as η drops stealth recapitalization Switch off liquidity and disinflationary spiral Example: Remove amplification s.t. σ t η = xt (1 b σ b σ t K )

Numerical example with monetary policy Prices 2 1.8 1.6 1.4 1.2 q, with policy q, without policy q is more stable p, q 1 0.8 0.6 p, without policy 0.4 0.2 p, with policy p less disinflation 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Numerical example with monetary policy Drift and volatility of η 0.08 0.06 total volatility without policy drift and volatility of 0.04 0.02 0-0.02 total = fundamental volatility with policy drift without policy drift with policy -0.04-0.06 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Observations s interest rate are cut in downturns, bonds held by intermediaries appreciate, this protects intermediaries against shocks increases the supply of asset that can be used as storage (weakens disinflation) Ex-post stabilization iquidity spiral Disinflationary spiral Ex-ante Higher leverage (shift in steady state)

Monetary policy in the limit full risk sharing of all aggregate risk σ t η = x t (1 b σ b σ t K ) 1 χψ η η θ η θ/η ψχ η + 1 θ η +θ1 η η bt B η ptb(η)/η η is deterministic and converges over time towards 0

Monetary policy: 3 versions No MoPo No mplification ggregate risk sharing

Monetary Policy Transmission Channel Consumption Boost approach to Bottleneck approach (New) Keynesian Demand Management Stimulate aggregate consumption Substitution effect I Theory of Money Risk (premium) management lleviate balance sheet constraints Income/wealth effect Woodford Tobin (1982) BruSan Price stickiness Perfect capital markets Representative gent Cut i Reduces r due to price stickiness Consumption c rises Both Cut i Changes bond prices Redistributes from low MPC to high MPC consumers Heterogeneous gents Financial Frictions Incomplete markets - -

Overview No monetary economics Fixed outside money supply mplification/endogenous risk through iquidity spiral Disinflationary spiral asset side of intermediaries balance sheet liability side Monetary policy Wealth shifts by affecting relative price between ong-term bond Short-term money Risk transfers reduce endogenous aggregate risk Macroprudential policy Directly affect portfolio positions

MacroPru MacroPru complements MoPo Not substitutes Good MacroPru enables more aggressive MoPo More redistribution ex-post More risk-transfers/insurance ex-ante ower q reduces cost to repurchase capital after shock owers importance of idiosyncratic shocks

MacroPru policy Regulator can control cannot control Portfolio choice ψs, xs investment decision ι q consumption decision c of intermediaries and households

MacroPru policy Regulator can control cannot control Portfolio choice ψs, xs investment decision ι q consumption decision c of intermediaries and households De-facto controls q and p within some range De-factor controls wealth share η Force agents to hold certain assets and generate capital gains distorts In sum, regulator maximizes sum of agents value function Choosing ψ b, q, η

MacroPru policy: Welfare frontier Stabilize intermediaries net worth and earnings Control the value of money to allow HH insure idiosyncratic risk (investment distortions still exists, otherwise can get 1 st best

Conclusion Unified macro Money and Banking model to analyze Financial stability Monetary stability - iquidity spiral - Fisher disinflation spiral Exogenous risk & Sector specific idiosyncratic Endogenous risk Time varying risk premia flight to safety Capitalization of intermediaries is key state variable Monetary policy rule Risk transfer to undercapitalized critical sectors Income/wealth effects are crucial instead of substitution effect Reduces endogenous risk better aggregate risk sharing Self-defeating in equilibrium excessive idiosyncratic risk taking Macro-prudential policies MacroPru + MoPo to achieve superior welfare optimum

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