the i theory of money - woodrow wilson school of public...

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The I Theory of Money

Markus K. Brunnermeier & Yuliy Sannikov

Princeton University

CSEF-IGIER Symposium Capri, June 24th, 2015

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Motivation

Framework to study monetary and financial stability

Interaction between monetary and macroprudential policy

Connect theory of value and theory of money

Intermediation (credit)• “Excessive” leverage and liquidity mismatch

Inside money – as store of value• Demand for money rises with endogenous volatility• In downturns, intermediaries create less inside money

Endogenous money multiplier = f(capitalization of critical sector)

• Value of money goes up – Disinflation spiral a la Fisher (1933)• Fire-sales of assets – Liquidity spiral

Flight to safety

Time-varying risk premium and endogenous volatility dynamics

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Some literature

Macro-friction models without money• Kiyotaki & Moore, BruSan2014, He & Krishnamurthy, DSS2015

“Money models” without intermediaries• Money pays no dividend and is a bubble – store of value

With intermediaries/inside money• “Money view” (Friedman & Schwartz) vs. “Credit view”(Tobin)

New Keynesian Models: BGG, Christian et al., … money in utility function

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Some literature


“Money models” without intermediaries• Store of value: Money pays no dividend and is a bubble



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Some literature





Friction OLG

deterministic endowment riskborrowing constraint

Only money Samuelson

With capital Diamond

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Some literature





Friction OLG Incomplete Markets + idiosyncratic risk

Risk deterministic endowment riskborrowing constraint

Only money Samuelson Bewley

With capital Diamond Ayagari, Krusell-Smith

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Some literature





Friction OLG Incomplete Markets + idiosyncratic risk

Risk deterministic endowment riskborrowing constraint

investment risk

Only money Samuelson Bewley

With capital Diamond Ayagari, Krusell-Smith Basic “I Theory”

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Roadmap

Model absent monetary policy• Toy model: one sector with outside money

• Two sector model

• Adding intermediary sector and inside money

Model with monetary policy

Model with macro-prudential policy

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One sector basic model

𝐴1

Technologies 𝑎

Each households can only operate one firm• Physical capital

• Output

Demand for money

sector idiosyncratic risk

𝑦𝑡 = 𝐴𝑘𝑡

𝑑𝑘𝑡𝑘𝑡= (Φ 𝜄𝑡 − 𝛿)𝑑𝑡 + 𝜎

𝑎𝑑𝑍𝑡𝑎 + 𝜎𝑑 𝑍𝑡

𝑎

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Adding outside money Technologies 𝑎


• Output

Demand for money



𝑎


A LA L

A LA L

𝐴1

Money

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Outside Money


𝑞𝑡𝐾𝑡 value of physical capital• Postulate constant 𝑞𝑡

𝐴−𝜄

𝑞𝑑𝑡 + Φ(𝜄 − 𝛿) 𝑑𝑡 + 𝜎𝑏𝑑𝑍𝑡

𝑏 + 𝜎𝑑 𝑍𝑡𝑏

𝑝𝑡𝐾𝑡 value of outside money• Postulate value of money changes proportional to 𝐾𝑡

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Adding outside money Technologies 𝑎


• Output

Demand for money



𝑎


A LA L

A LA L

𝐴1

Money

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Outside Money

𝑞𝐾𝑡 value of physical capital• 𝑑𝑟𝑎 = 𝐴−𝜄

𝑞𝑑𝑡 + Φ(𝜄 − 𝛿) 𝑑𝑡 + 𝜎𝑎𝑑𝑍𝑡

𝑎 + 𝜎𝑑 𝑍𝑡𝑎

𝑝𝐾𝑡 value of outside money• 𝑑𝑟𝑀 = Φ(𝜄 − 𝛿)

𝑔

𝑑𝑡 + 𝜎𝑎𝑑𝑍𝑡𝑎


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Demand with 𝐸 0∞𝑒−𝜌𝑡 log 𝑐𝑡 𝑑𝑡

Technologies 𝑎

A LA L

A LA L

Money

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Outside Money





𝑔


Consumption demand:𝜌 𝑝 + 𝑞 𝐾𝑡 = 𝐴 − 𝜄 𝐾𝑡

Asset (share) demand 𝑥𝑎:

𝐸 𝑑𝑟𝑎 − 𝑑𝑟𝑀 /𝑑𝑡 = 𝐶𝑜𝑣[𝑑𝑟𝑎 − 𝑑𝑟𝑀,

𝑑𝑛𝑡𝑎

𝑛𝑡𝑎

𝑑𝑟𝑀+𝑥𝑎 𝑑𝑟𝑎−𝑑𝑟𝑀

] = 𝑥𝑎 𝜎2

𝑥𝑎 =𝐸 𝑑𝑟𝑎−𝑑𝑟𝑀 /𝑑𝑡

𝜎2= (𝐴−𝜄)/𝑞 𝜎2= 𝑞𝑞+𝑝

Investment rate: (Tobin’s q) Φ′ 𝜄 = 1/𝑞• For Φ 𝜄 = 1

𝜅log(𝜅𝜄 + 1) ⇒ 𝜄∗ = 𝑞−1

𝜅

𝐴1

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Demand with log-utility Technologies 𝑎

A LA L

A LA L

Money

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Outside Money





𝑔





𝑑𝑛𝑡𝑎

𝑛𝑡𝑎


] = 𝑥𝑎 𝜎2




𝜅log(𝜅𝜄 + 1) ⇒ 𝜄∗ = 𝑞−1

𝜅

𝐴1

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Demand with log-utility Technologies 𝑎

A LA L

A LA L

Money

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Outside Money





𝑔





𝑑𝑛𝑡𝑎

𝑛𝑡𝑎


] = 𝑥𝑎 𝜎2




𝜅log(𝜅𝜄 + 1) ⇒ 𝜄 = 𝑞−1

𝜅

𝐴1

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Market clearing Technologies 𝑎

A LA L

A LA L

Money

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Outside Money





𝑔





𝑑𝑛𝑡𝑎

𝑛𝑡𝑎


] = 𝑥𝑎 𝜎2




𝜅log(𝜅𝜄 + 1) ⇒ 𝜄 = 𝑞−1

𝜅

𝐴1

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Equilibrium

Moneyless equilibrium Money equilibrium

𝑝0 = 0 𝑝 = 𝜎− 𝜌𝜌𝑞

𝑞0 =𝜅𝐴+1𝜅𝜌+1

𝑞 = 𝜅𝐴+1𝜅 𝜌 𝜎+1

𝜎𝜌

𝑝

𝑞

0

𝑞0

>

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Welfare analysis


𝑝0 = 0 𝑝 = 𝜎− 𝜌𝜌𝑞



𝑔0 𝑔

welfare0 welfare<

>

>

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Welfare analysis


𝑝0 = 0 𝑝 = 𝜎− 𝜌𝜌𝑞



welfare0 welfare

What ratio nominal to total wealth 𝑝𝑞+𝑝

maximizes welfare?

• Force agents to hold less 𝑘 & more money

• Raise 𝑝

𝑞+𝑝if and only if 𝜎(1 − 𝜅𝜌) ≤ 2 𝜌

• Lowers 𝑞 ⇒ higher E[𝑑𝑟𝑎 − 𝑑𝑟𝑀] = 𝐴−𝜄𝑞𝑑𝑡

• Create 𝑞-risk to make precautionary money savings more attractive

<

pecuniary externality

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Roadmap


• Two sector model with outside money




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𝐴1𝐴1

Outline of two sector model

𝐴1

𝐵1

SwitchSwitch technology

Technologies 𝑎 Technologies 𝑏

Households have to • Specialize in one subsector sector specific + idiosyncratic risk

for one period

• Demand for money

𝑑𝑘𝑡𝑘𝑡= ⋯𝑑𝑡 + 𝜎𝑏𝑑𝑍𝑡

𝑏 + 𝜎𝑑 𝑍𝑡𝑏 𝑑𝑘𝑡

𝑘𝑡= ⋯𝑑𝑡 + 𝜎𝑎𝑑𝑍𝑡


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Add outside money

Households have to • Specialize in one subsector

for one period

• Demand for money

Outside Money

A LA L

A

Money

𝐵1

LN

et w

ort

h


SwitchSwitch technology

A LA L

A LA L

𝐴1

Money

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Roadmap


• Two sector model with outside money




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Technologies 𝑏

• Risk can bepartially sold off to intermediaries

Add intermediaries

Net worth

A L

Outside Money

A LA L

A

Money

𝐵1

LN

et w

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Technologies 𝑎

• Risk is not contractable(Plagued with moral hazard problems)

A LA L

A LA L

𝐴1

Money

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Technologies 𝑏

Add intermediaries

Net worth

A L

Outside Money

Intermediaries • Can hold outside equity

& diversify within sector 𝑏• Monitoring

A LA L

A

Money

𝐵1

LN

et w

ort

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Technologies 𝑎

A LA L

A LA L

𝐴1

Money

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Technologies 𝑏

A L

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Add intermediaries

…

Net worth

A L

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Outside Money

A L

𝐵1

Money

Ris

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laim

Insi

de

equ

ity

Intermediaries • Can hold outside equity

& diversify within sector 𝑏• Monitoring

Technologies 𝑎

A LA L

A LA L

𝐴1

Money

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A L

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Add intermediaries Technologies 𝑏

…

Net worth

Inside Money(deposits)

A L

Outside Money Pass through

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ky C

laim

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Outside Money

Intermediaries• Can hold outside equity

& diversify within sector 𝑏• Monitoring• Create inside money• Maturity/liquidity

transformation

A L

𝐵1

Money

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laim

Insi

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ity

A LA L

A LA L

𝐴1

Money

HH

Net

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Technologies 𝑎

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Technologies 𝑏

A L

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A L

Shock impairs assets: 1st of 4 steps Technologies 𝑎

…

Net worth


A L


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Losses

A L

𝐴1

Money

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Insi

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ity

𝐵1

A LA L

A LA L

𝐴1

Money

HH

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Technologies 𝑏

Shrink balance sheet: 2nd of 4 steps

A

Technologies 𝑎


…

Net worth


A L

Pass through

Losses

Deleveraging Deleveraging

…

Ris

ky C

laim

Ris

ky C

laim

Ris

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Outside Money

Switch

A L

Ris

ky C

laim

A LA L

𝐴1

Money

Ris

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Insi

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ity

𝐵1

A LA L

A LA L

𝐴1

Money

HH

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Technologies 𝑏

Liquidity spiral: asset price drop: 3rd of 4 Technologies 𝑎

Switch

A L

Ris

ky C

laim

A LA L

𝐴1

Money

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ky C

laim

Insi

de

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ity

𝐵1


Outside Money

…

Net worth


A L

Pass through

Ris

ky C

laim

Ris

ky C

laim

Ris

ky C

laim

Losses


A LA L

A LA L

𝐴1

Money

HH

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Disinflationary spiral: 4th of 4 steps

A LA L

A LA L

𝐴1

Money

HH

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A L

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A LA L

𝐴1

Money

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Insi

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ity

𝐵1


Outside Money

…

Net worth


A L

Pass through

Ris

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laim

Ris

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laim

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Losses


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Formal model: capital & output

Model setup in paper is more general: 𝑌𝑡 = 𝐴 𝜓𝑡 𝐾𝑡

Technologies 𝑏 𝑎

Physical capital 𝐾𝑡- Capital share 𝜓𝑡 1 − 𝜓𝑡

Output goods 𝑌𝑡𝑏 = 𝐴𝑘𝑡

𝑏 𝑌𝑡𝑎 = 𝐴𝑘𝑡

𝑎

Aggregate good (CES)- Consumed or invested- numeraire

𝑌𝑡 =12 𝑌𝑡𝑏(𝑠−1)/𝑠

+12 𝑌𝑡𝑎 (𝑠−1)/𝑠

𝑠/(𝑠−1)

Price of goods 𝑃𝑡𝑏 = 12

𝑌𝑡𝑌𝑡𝑏

1/𝑠

𝑃𝑡𝑎 = 12

𝑌𝑡𝑌𝑡𝑎

1/𝑠

Imperfectsubstitutes

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Formal model: risk

When 𝑘𝑡 is employed in sector 𝑎 by agent 𝑗

𝑑𝑘𝑡 = Φ 𝜄𝑡 − 𝛿 𝑘𝑡𝑑𝑡 + 𝜎𝑎𝑘𝑡𝑑𝑍𝑡

𝑎 + 𝜎𝑗𝑘𝑡𝑑 𝑍𝑡𝑎

• Φ 𝜄𝑡 is increasing and concave, e.g. log[ 𝜅𝜄𝑡 + 1 /𝜅]

• All 𝑑𝑍 are independent of each other

Intermediaries can diversify within sector 𝑏• Face no idiosyncratic risk

Households cannot become intermediaries or vice versa

Investment rate (per unit of 𝑘𝑡)sectorial idiosyncratic independent Brownian motions(fundamental cash flow risk)

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Asset returns on money

Money: fixed supply in baseline model, total value 𝑝𝑡𝐾𝑡• Return = capital gains (no dividend/interest in baseline model)

• If 𝑑𝑝𝑡/𝑝𝑡 = 𝜇𝑡𝑝𝑑𝑡 + 𝝈𝒕

𝒑𝑑𝒁𝒕,

• 𝑑𝐾𝑡/𝐾𝑡= Φ 𝜄𝑡 − 𝛿 𝑑𝑡 + 1 − 𝜓𝑡 𝜎𝑎𝑑𝑍𝑡𝑎 + 𝜓𝑡𝜎

𝑏𝑑𝑍𝑡𝑏

(𝝈𝑡𝑲)𝑇𝑑𝒁𝑡

𝑑𝑟𝑡𝑀 = Φ 𝜄𝑡 − 𝛿 + 𝜇𝑡

𝑝+ 𝝈𝒕𝒑 𝑇𝝈𝒕𝑲 𝑑𝑡 + 𝝈𝒕

𝒑+ 𝝈𝒕𝑲 𝑑𝒁𝒕

𝜋𝑡 =𝑝𝑡

𝑞𝑡+𝑝𝑡fraction of wealth in form of money

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Technologies 𝑎

A L

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Capital/risk shares Technologies 𝑏

…

Net worth


A L


A L

𝜓𝑡𝑞𝑡𝐾𝑡

Money

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Insi

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A LA L

A LA L

Money

HH

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χ𝑡 1 − χ𝑡

1 − χ𝑡 𝜓𝑡𝑞𝑡𝐾𝑡

𝑁𝑡(1 − 𝜓𝑡)𝑞𝑡𝐾𝑡

Fraction 𝛼𝑡 of HH

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Technologies 𝑎

A L

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laim

A LR

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Capital/risk shares Technologies 𝑏

…

Net worth


A L


A L

𝜓𝑡𝑞𝑡𝐾𝑡

Money

Ris

ky C

laim

Insi

de

equ

ity

A LA L

A LA L

Money

HH

Net

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χ𝑡 1 − χ𝑡


𝑁𝑡(1 − 𝜓𝑡)𝑞𝑡𝐾𝑡

If 𝜒𝑡 > 𝜒, inside and outside equity

earn same returns (as portfolio of b-technology and money).

If the equity constraint 𝜒𝑡 = 𝜒 binds,

inside equity earns a premium λ

Fraction 𝛼𝑡 of HH

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Allocation

Equilibrium is a map

Histories of shocks prices 𝑞𝑡 , 𝑝𝑡, 𝜆𝑡, allocation𝒁𝜏, 0 ≤ 𝜏 ≤ 𝑡 𝛼𝑡, 𝜒𝑡 & portfolio weights (𝑥𝑡, 𝑥𝑡

𝑎 , 𝑥𝑡𝑏)

wealth distribution

𝜂𝑡 =𝑁𝑡

(𝑝𝑡+𝑞𝑡)𝐾𝑡∈ 0,1

intermediaries’ wealth share

• All agents maximize utility Choose: portfolio, consumption, technology

• All markets clear Consumption, capital, money, outside equity of 𝑏

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Numerical example: capital shares

𝜌 = 5%,𝐴 = .5, 𝜎𝑎 = 𝜎𝑏 = .4, 𝜎𝑗 = .9, 𝜎𝑎 = .6, 𝜎𝑎 = 1.2, 𝑠 = .8,

Φ 𝜄 =log 𝜅𝜄 + 1

𝜅, 𝜅 = 2, 𝜒 = .001

technology 𝑎 HH

technology 𝑏 HH

1 −

intermediaries

𝜒𝜓

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Numerical example: prices

𝑝

𝑞

Disinflationspiral

Liquidityspiral

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Numerical example: prices

𝑝

𝑞

𝜋 = 𝑝𝑝+𝑞Disinflation

spiral

Liquidityspiral

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Numerical example: dynamics of 𝜂

amplification

leverage elasticity

Steady state

𝜎𝑡𝜂=𝑥𝑡(𝜎𝑏1𝑏 − 𝜎𝑡

𝐾)

1 − 𝜓𝑡(1−𝜒𝑡)−𝜂𝜂−𝜋′ 𝜂𝜋/𝜂

fundamental volatility

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Overview

No monetary economics• Fixed outside money supply• Amplification/endogenous risk through

Liquidity spiral asset side of intermediaries’ balance sheet Disinflationary spiral liability side

Monetary policy• Aside: Money vs. Credit view (via helicopter drop)• Wealth shifts by affecting relative price between

Long-term bond Short-term money

• Risk transfers – reduce endogenous aggregate risk

Macroprudential policy

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Adverse shock value of risky claims drops

Monetary policy response: cut short-term interest rate• Value of long-term bonds rises - “stealth recapitalization”

Liquidity & Deflationary Spirals are mitigated

…

Net worth


A L


𝑁𝑡

Bonds 𝑏𝑡𝐾𝑡


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Effects of policy

Effect on the value of money (liquid assets) – helps agents hedge idiosyncratic risks, but distorts investment • We saw this in the toy model with one sector

Redistribution of aggregate risk, mitigates risk that an essential sector can become undercapitalized

Affects earnings distribution, rents that different sectors get in equilibrium

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Monetary policy and endogenous risk

Intermediaries’ risk (borrow to scale up)

𝜎𝑡𝜂=

𝑥𝑡(𝜎𝑏1𝑏 − 𝜎𝑡

𝐾)

1 − 𝜓−𝜂𝜂−𝜋′ 𝜂𝜋/𝜂+ 𝜓(1−𝜒)−𝜂

𝜂+ 𝜋𝜂1 − 𝜓

𝑏𝑡𝑝𝑡

−𝐵′ 𝜂

𝐵(𝜂)/𝜂

Example: 𝑏𝑡

𝑝𝑡

𝐵′ 𝜂

𝐵 𝜂= 𝛼𝑡

𝜋′ 𝜂

𝜋(1−𝜋)

Intuition: with right monetary policy bond price 𝐵(𝜂) rises as 𝜂 drops “stealth recapitalization”• Can reduce liquidity and disinflationary spiral

fundamental risk

amplification mitigation

𝛼(𝜂)

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Numerical example with monetary policy

Allocations Prices

Higher intermediaries’capital share (1 − 𝜒)𝜓

𝜓𝑎 Less production of good 𝑎

𝑞 is more stable

𝑝 less disinflation

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𝜎𝑡𝜂=

𝑥𝑡(𝜎𝑏𝟏𝑏 − 𝜎𝑡

𝐾)

1 − 𝜓−𝜂𝜂−𝜋′ 𝜂𝜋/𝜂+ 𝜓−𝜂𝜂 +

𝜋𝜂 1 − 𝜓

𝑏𝑡𝑝𝑡

−𝐵′ 𝜂𝐵 𝜂 /𝜂


Steady state

Less volatile

Recall𝑏𝑡𝑝𝑡

𝐵′ 𝜂

𝐵 𝜂= 𝛼𝑡𝜋′ 𝜂

𝜋(1 − 𝜋)

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Welfare: HH and Intermediaries Sum

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Monetary policy … in the limit

full risk sharing of all aggregate risk

𝜎𝑡𝜂=

𝑥𝑡

1−𝜓−𝜂

𝜂

−𝜋′ 𝜂𝜋𝜂

+𝜓 1−𝜒 −𝜂

𝜂+𝜋

𝜂1−𝜓

𝑏𝑡

𝑝𝑡

−𝐵′ 𝜂

𝐵(𝜂)/𝜂

(𝜎𝑏𝟏𝑏 − 𝜎𝑡𝐾)

𝜂 is deterministic and converges over time towards 0−∞

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Monetary policy … in the limit

full risk sharing of all aggregate risk

Aggregate risk sharing makes 𝑞 determinisitic

Like in benchmark toy model• Excessive 𝑘-investment

• Too high 𝑞(pecuniary externality) Lower capital return

Endogenous risk corrects pecuniary externality

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Overview

No monetary economics• Fixed outside money supply• Amplification/endogenous risk through

Liquidity spiral asset side of intermediaries’ balance sheet Disinflationary spiral liability side

Monetary policy• Wealth shifts by affecting relative price between

Long-term bond Short-term money

• Risk transfers – reduce endogenous aggregate risk

Macroprudential policy• Directly affect portfolio positions

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MacroPru policy

Regulator can control cannot control

• Portfolio choice 𝜓s, 𝑥s ⋅ investment decision 𝜄 𝑞

⋅ consumption decision 𝑐

of intermediaries and households

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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

In sum, regulator maximizes sum of agents value function• Choosing 𝜓𝑏, 𝑞, 𝜂

distorts

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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 1st best

intermediary welfare-20 -15 -10 -5 0 5 10 15 20 25

ho

use

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ld w

elfare

-20

-15

-10

-5

0

5

10

15

20

25

30

no policy

policy that removes endogenous risk

optimal macroprudential

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Conclusion

Unified macro model to analyze• Financial stability - Liquidity spiral• Monetary stability - 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

the i theory of money - woodrow wilson school of public...

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