The I Theory of Money Markus K. Brunnermeier & Yuliy Sannikov - - PowerPoint PPT Presentation

Brunnermeier & Sannikov

The I Theory of Money

Markus K. Brunnermeier & Yuliy Sannikov

Princeton University

CSEF-IGIER Symposium Capri, June 24th, 2015

Brunnermeier & Sannikov

Motivation

• Framework to study monetary and fin

financial stability

• Interaction between monetary and macroprudential policy
• Connect th

theory of f value and th theory of f 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

Brunnermeier & Sannikov

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

Brunnermeier & Sannikov

Some literature

• Macro-friction models without money
• Kiyotaki & Moore, BruSan2014, He & Krishnamurthy, DSS2015
• “Money models” without intermediaries
• Store of value: Money pays no dividend and is a bubble
• With intermediaries/inside money
• “Money view” (Friedman & Schwartz) vs. “Credit view”(Tobin)
• New Keynesian Models: BGG, Christian et al., … money in utility function

Brunnermeier & Sannikov

Some literature

• Macro-friction models without money
• Kiyotaki & Moore, BruSan2014, He & Krishnamurthy, DSS2015
• “Money models” without intermediaries
• Store of value: Money pays no dividend and is a bubble
• With intermediaries/inside money
• “Money view” (Friedman & Schwartz) vs. “Credit view”(Tobin)
• New Keynesian Models: BGG, Christian et al., … money in utility function

Fric Frictio ion OL OLG deterministic endowment risk borrowing constraint Only money Samuelson With capital Diamond

Brunnermeier & Sannikov

Some literature

• Macro-friction models without money
• Kiyotaki & Moore, BruSan2014, He & Krishnamurthy, DSS2015
• “Money models” without intermediaries
• Store of value: Money pays no dividend and is a bubble
• With intermediaries/inside money
• “Money view” (Friedman & Schwartz) vs. “Credit view”(Tobin)
• New Keynesian Models: BGG, Christian et al., … money in utility function

Fric Frictio ion OL OLG Inc Incomple lete Mark arkets + + idiosyncratic ic ri risk sk Risk deterministic endowment risk borrowing constraint Only money Samuelson Bewley With capital Diamond Ayagari, Krusell-Smith

Brunnermeier & Sannikov

Some literature

• Macro-friction models without money
• Kiyotaki & Moore, BruSan2014, He & Krishnamurthy, DSS2015
• “Money models” without intermediaries
• Store of value: Money pays no dividend and is a bubble
• With intermediaries/inside money
• “Money view” (Friedman & Schwartz) vs. “Credit view”(Tobin)
• New Keynesian Models: BGG, Christian et al., … money in utility function

Fric Frictio ion OL OLG Inc Incomple lete Mark arkets + + idiosyncratic ic ri risk sk Risk deterministic endowment risk borrowing constraint investment risk Only money Samuelson Bewley With capital Diamond Ayagari, Krusell-Smith Basic “I Theory”

Brunnermeier & Sannikov

Some literature

• Macro-friction models without money
• Kiyotaki & Moore, BruSan2014, He & Krishnamurthy, DSS2015
• “Money models” without intermediaries
• Store of value: Money pays no dividend and is a bubble
• With intermediaries/inside money
• “Money view” (Friedman & Schwartz) vs. “Credit view”(Tobin)
• New Keynesian Models: BGG, Christian et al., … money in utility function

Fric Frictio ion OL OLG Inc Incomple lete Mark arkets + + idiosyncratic ic ri risk sk Risk deterministic endowment risk borrowing constraint investment risk Only money Samuelson Bewley With capital Diamond Ayagari, Krusell-Smith Basic “I Theory”

Brunnermeier & Sannikov

Some literature

• Macro-friction models without money
• Kiyotaki & Moore, BruSan2014, He & Krishnamurthy, DSS2015
• “Money models” without intermediaries
• Store of value: Money pays no dividend and is a bubble
• With intermediaries/inside money
• “Money view” (Friedman & Schwartz) vs. “Credit view”(Tobin)
• New Keynesian Models: BGG, Christian et al., … money in utility function

Fric Frictio ion OL OLG Inc Incomple lete Mark arkets + + idiosyncratic ic ri risk sk Risk deterministic endowment risk borrowing constraint investment risk Only money Samuelson Bewley With capital Diamond Ayagari, Krusell-Smith Basic “I Theory”

Brunnermeier & Sannikov

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

Brunnermeier & Sannikov

One sector basic model

𝐵1

• Technologies 𝑏
• Each households can only
• perate one firm
• Physical capital
• Output
• Demand for money

sector idiosyncratic risk

𝑧𝑢 = 𝐵𝑙𝑢 𝑒𝑙𝑢 𝑙𝑢 = (Φ 𝜅𝑢 − 𝜀)𝑒𝑢 + 𝜏𝑏𝑒𝑎𝑢

𝑏 +

𝜏𝑒 𝑎𝑢

𝑏

Brunnermeier & Sannikov

Adding outside money

• Technologies 𝑏
• Each households can only
• perate one firm
• Physical capital
• Output
• Demand for money

𝑒𝑙𝑢 𝑙𝑢 = (Φ 𝜅𝑢 − 𝜀)𝑒𝑢 + 𝜏𝑏𝑒𝑎𝑢

𝑏 +

𝜏𝑒 𝑎𝑢

𝑏

sector idiosyncratic risk

A L A L A L A L

𝐵1 Money

Net worth

Outside Money 𝑧𝑢 = 𝐵𝑙𝑢

• 𝑟𝑢𝐿𝑢 value of physical capital
• Postulate constant 𝑟𝑢

𝐵−𝜅 𝑟 𝑒𝑢 + Φ(𝜅 − 𝜀) 𝑒𝑢 + 𝜏𝑐𝑒𝑎𝑢

𝑐 +

𝜏𝑒 𝑎𝑢

𝑐

• 𝑞𝑢𝐿𝑢 value of outside money
• Postulate value of money changes proportional to 𝐿𝑢

Brunnermeier & Sannikov

Adding outside money

• Technologies 𝑏
• Each households can only
• perate one firm
• Physical capital
• Output
• Demand for money

𝑒𝑙𝑢 𝑙𝑢 = (Φ 𝜅𝑢 − 𝜀)𝑒𝑢 + 𝜏𝑏𝑒𝑎𝑢

𝑏 +

𝜏𝑒 𝑎𝑢

𝑏

sector idiosyncratic risk

A L A L A L A L

𝐵1 Money

Net worth

Outside Money

• 𝑟𝐿𝑢 value of physical capital
• 𝑒𝑠𝑏 = 𝐵−𝜅

𝑟 𝑒𝑢 + Φ(𝜅 − 𝜀) 𝑒𝑢 + 𝜏𝑏𝑒𝑎𝑢

𝑏 +

𝜏𝑒 𝑎𝑢

𝑏

• 𝑞𝐿𝑢 value of outside money
• 𝑒𝑠𝑁 = Φ(𝜅 − 𝜀)

𝑕

𝑒𝑢 + 𝜏𝑏𝑒𝑎𝑢

𝑏

𝑧𝑢 = 𝐵𝑙𝑢

Brunnermeier & Sannikov

Demand with 𝐹

∞ 𝑓−𝜍𝑢 log 𝑑𝑢 𝑒𝑢

• Technologies 𝑏

A L A L A L A L

Money

Net worth

Outside Money

• 𝑟𝐿𝑢 value of physical capital
• 𝑒𝑠𝑏 = 𝐵−𝜅

𝑟 𝑒𝑢 + Φ(𝜅 − 𝜀) 𝑒𝑢 + 𝜏𝑏𝑒𝑎𝑢

𝑏 +

𝜏𝑒 𝑎𝑢

𝑏

• 𝑞𝐿𝑢 value of outside money
• 𝑒𝑠𝑁 = Φ(𝜅 − 𝜀)

𝑕

𝑒𝑢 + 𝜏𝑏𝑒𝑎𝑢

𝑏

• Consumption demand:

𝜍 𝑞 + 𝑟 𝐿𝑢 = 𝐵 − 𝜅 𝐿𝑢

• Asset (share) demand 𝑦𝑏:

𝐹 𝑒𝑠𝑏 − 𝑒𝑠𝑁 /𝑒𝑢 = 𝐷𝑝𝑤[𝑒𝑠𝑏 − 𝑒𝑠𝑁, 𝑒𝑜𝑢

𝑏

𝑜𝑢

𝑏 𝑒𝑠𝑁+𝑦𝑏 𝑒𝑠𝑏−𝑒𝑠𝑁

] = 𝑦𝑏 𝜏2

𝑦𝑏 = 𝐹 𝑒𝑠𝑏−𝑒𝑠𝑁 /𝑒𝑢

𝜏2

= (𝐵−𝜅)/𝑟

𝜏2

=

𝑟 𝑟+𝑞

• Investment rate:

(Tobin’s q)

Φ′ 𝜅 = 1/𝑟

• For Φ 𝜅 = 1

𝜆 log(𝜆𝜅 + 1) ⇒ 𝜅∗ = 𝑟−1 𝜆

𝐵1

Brunnermeier & Sannikov

Demand with log-utility

• Technologies 𝑏

A L A L A L A L

Money

Net worth

Outside Money

• 𝑟𝐿𝑢 value of physical capital
• 𝑒𝑠𝑏 = 𝐵−𝜅

𝑟 𝑒𝑢 + Φ(𝜅 − 𝜀) 𝑒𝑢 + 𝜏𝑏𝑒𝑎𝑢

𝑏 +

𝜏𝑒 𝑎𝑢

𝑏

• 𝑞𝐿𝑢 value of outside money
• 𝑒𝑠𝑁 = Φ(𝜅 − 𝜀)

𝑕

𝑒𝑢 + 𝜏𝑏𝑒𝑎𝑢

𝑏

• Consumption demand:

𝜍 𝑞 + 𝑟 𝐿𝑢 = 𝐵 − 𝜅 𝐿𝑢

• Asset (share) demand 𝑦𝑏:

𝐹 𝑒𝑠𝑏 − 𝑒𝑠𝑁 /𝑒𝑢 = 𝐷𝑝𝑤[𝑒𝑠𝑏 − 𝑒𝑠𝑁, 𝑒𝑜𝑢

𝑏

𝑜𝑢

𝑏 𝑒𝑠𝑁+𝑦𝑏 𝑒𝑠𝑏−𝑒𝑠𝑁

] = 𝑦𝑏 𝜏2

𝑦𝑏 = 𝐹 𝑒𝑠𝑏−𝑒𝑠𝑁 /𝑒𝑢

𝜏2

= (𝐵−𝜅)/𝑟

𝜏2

=

𝑟 𝑟+𝑞

• Investment rate:

(Tobin’s q)

Φ′ 𝜅 = 1/𝑟

• For Φ 𝜅 = 1

𝜆 log(𝜆𝜅 + 1) ⇒ 𝜅∗ = 𝑟−1 𝜆

𝐵1

Brunnermeier & Sannikov

Demand with log-utility

• Technologies 𝑏

A L A L A L A L

Money

Net worth

Outside Money

• 𝑟𝐿𝑢 value of physical capital
• 𝑒𝑠𝑏 = 𝐵−𝜅

𝑟 𝑒𝑢 + Φ(𝜅 − 𝜀) 𝑒𝑢 + 𝜏𝑏𝑒𝑎𝑢

𝑏 +

𝜏𝑒 𝑎𝑢

𝑏

• 𝑞𝐿𝑢 value of outside money
• 𝑒𝑠𝑁 = Φ(𝜅 − 𝜀)

𝑕

𝑒𝑢 + 𝜏𝑏𝑒𝑎𝑢

𝑏

• Consumption demand:

𝜍 𝑞 + 𝑟 𝐿𝑢 = 𝐵 − 𝜅 𝐿𝑢

• Asset (share) demand 𝑦𝑏:

𝐹 𝑒𝑠𝑏 − 𝑒𝑠𝑁 /𝑒𝑢 = 𝐷𝑝𝑤[𝑒𝑠𝑏 − 𝑒𝑠𝑁, 𝑒𝑜𝑢

𝑏

𝑜𝑢

𝑏 𝑒𝑠𝑁+𝑦𝑏 𝑒𝑠𝑏−𝑒𝑠𝑁

] = 𝑦𝑏 𝜏2

𝑦𝑏 = 𝐹 𝑒𝑠𝑏−𝑒𝑠𝑁 /𝑒𝑢

𝜏2

= (𝐵−𝜅)/𝑟

𝜏2

=

𝑟 𝑟+𝑞

• Investment rate:

(Tobin’s q)

Φ′ 𝜅 = 1/𝑟

• For Φ 𝜅 = 1

𝜆 log(𝜆𝜅 + 1) ⇒ 𝜅 = 𝑟−1 𝜆

𝐵1

Brunnermeier & Sannikov

Market clearing

• Technologies 𝑏

A L A L A L A L

Money

Net worth

Outside Money

• 𝑟𝐿𝑢 value of physical capital
• 𝑒𝑠𝑏 = 𝐵−𝜅

𝑟 𝑒𝑢 + Φ(𝜅 − 𝜀) 𝑒𝑢 + 𝜏𝑏𝑒𝑎𝑢

𝑏 +

𝜏𝑒 𝑎𝑢

𝑏

• 𝑞𝐿𝑢 value of outside money
• 𝑒𝑠𝑁 = Φ(𝜅 − 𝜀)

𝑕

𝑒𝑢 + 𝜏𝑏𝑒𝑎𝑢

𝑏

• Consumption demand:

𝜍 𝑞 + 𝑟 𝐿𝑢 = 𝐵 − 𝜅 𝐿𝑢

• Asset (share) demand 𝑦𝑏:

𝐹 𝑒𝑠𝑏 − 𝑒𝑠𝑁 /𝑒𝑢 = 𝐷𝑝𝑤[𝑒𝑠𝑏 − 𝑒𝑠𝑁, 𝑒𝑜𝑢

𝑏

𝑜𝑢

𝑏 𝑒𝑠𝑁+𝑦𝑏 𝑒𝑠𝑏−𝑒𝑠𝑁

] = 𝑦𝑏 𝜏2

𝑦𝑏 = 𝐹 𝑒𝑠𝑏−𝑒𝑠𝑁 /𝑒𝑢

𝜏2

= (𝐵−𝜅)/𝑟

𝜏2

=

𝑟 𝑟+𝑞

• Investment rate:

(Tobin’s q)

Φ′ 𝜅 = 1/𝑟

• For Φ 𝜅 = 1

𝜆 log(𝜆𝜅 + 1) ⇒ 𝜅 = 𝑟−1 𝜆

𝐵1

Brunnermeier & Sannikov

Equilibrium

Moneyless equil ilib ibriu ium Money equil ilib ibriu ium 𝑞0 = 0 𝑞 =

𝜏− 𝜍 𝜍 𝑟

𝑟0 = 𝜆𝐵+1

𝜆𝜍+1

𝑟 =

𝜆𝐵+1 𝜆 𝜍 𝜏+1 𝜏 𝜍

𝑞 𝑟

𝑟0

>

Brunnermeier & Sannikov

Welfare analysis

Moneyless equil ilib ibriu ium Money equil ilib ibriu ium 𝑞0 = 0 𝑞 =

𝜏− 𝜍 𝜍 𝑟

𝑟0 = 𝜆𝐵+1

𝜆𝜍+1

𝑟 =

𝜆𝐵+1 𝜆 𝜍 𝜏+1

𝑕0 𝑕 welfare0 welfare < > >

Brunnermeier & Sannikov

Welfare analysis

Moneyless equil ilib ibriu ium Money equil ilib ibriu ium 𝑞0 = 0 𝑞 =

𝜏− 𝜍 𝜍 𝑟

𝑟0 = 𝜆𝐵+1

𝜆𝜍+1

𝑟 =

𝜆𝐵+1 𝜆 𝜍 𝜏+1

welfare0 welfare

• What ratio nominal to total wealth 𝑞

𝑟+𝑞 maximizes welfare?

• Force agents to hold less 𝑙 & more money
• Raise 𝑞

𝑟+𝑞 if and only if

𝜏(1 − 𝜆𝜍) ≤ 2 𝜍

• Lowers 𝑟 ⇒ higher E[𝑒𝑠𝑏 − 𝑒𝑠𝑁] = 𝐵−𝜅

𝑟 𝑒𝑢

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

<

pecuniary externality

Brunnermeier & Sannikov

Roadmap

• Model absent monetary policy
• Toy model: one sector with outside money
• Two sector model with outside money
• Adding intermediary sector and inside money
• Model with monetary policy
• Model with macro-prudential policy

Brunnermeier & Sannikov

𝐵1 𝐵1

Outline of two sector model

𝐵1 𝐶1 Switch Switch technology

• Technologies 𝑏
• Technologies 𝑐
• Households have to
• Specialize in one subsector

sector specific + idiosyncratic risk

for one period

• Demand for money

𝑒𝑙𝑢 𝑙𝑢 = ⋯ 𝑒𝑢 + 𝜏𝑐𝑒𝑎𝑢

𝑐 +

𝜏𝑒 𝑎𝑢

𝑐

𝑒𝑙𝑢 𝑙𝑢 = ⋯ 𝑒𝑢 + 𝜏𝑏𝑒𝑎𝑢

𝑏 +

𝜏𝑒 𝑎𝑢

𝑏

Brunnermeier & Sannikov

Add outside money

• Households have to
• Specialize in one subsector

for one period

• Demand for money

Outside Money

A L A L A

Money 𝐶1

L

Net worth

• Technologies 𝑏
• Technologies 𝑐

Switch Switch technology

A L A L A L A L

𝐵1 Money

Net worth

Brunnermeier & Sannikov

Roadmap

• Model absent monetary policy
• Toy model: one sector with outside money
• Two sector model with outside money
• Adding intermediary sector and inside money
• Model with monetary policy
• Model with macro-prudential policy

Brunnermeier & Sannikov

• Technologies 𝑐
• Risk can be

partially sold off to intermediaries

Add intermediaries

Net worth A L Outside Money

A L A L A

Money 𝐶1

L

Net worth

• Technologies 𝑏
• Risk is

not contractable (Plagued with moral hazard problems)

A L A L A L A L

𝐵1 Money

Net worth

Brunnermeier & Sannikov

• Technologies 𝑐

Add intermediaries

Net worth A L Outside Money

• Intermediaries
• Can hold outside equity

& diversify within sector 𝑐

• Monitoring

A L A L A

Money 𝐶1

L

Net worth

• Technologies 𝑏

A L A L A L A L

𝐵1 Money

Net worth

Brunnermeier & Sannikov

• Technologies 𝑐

A L

Risky Claim

A L

Risky Claim

Add intermediaries

…

Net worth A L

Risky Claim Risky Claim Risky Claim

Outside Money

A L

𝐶1 Money

Risky Claim Inside equity

• Intermediaries
• Can hold outside equity

& diversify within sector 𝑐

• Monitoring
• Technologies 𝑏

A L A L A L A L

𝐵1 Money

Net worth

Brunnermeier & Sannikov

A L

Risky Claim

A L

Risky Claim

Add intermediaries

• Technologies 𝑐

…

Net worth Inside Money (deposits) A L Outside Money

Pass through

Risky Claim Risky Claim Risky Claim

Outside Money

• Intermediaries
• Can hold outside equity

& diversify within sector 𝑐

• Monitoring
• Create inside money
• Maturity/liquidity

transformation

A L

𝐶1 Money

Risky Claim Inside equity

A L A L A L A L

𝐵1 Money

HH Net worth

• Technologies 𝑏

Brunnermeier & Sannikov

• Technologies 𝑐

A L

Risky Claim

A L

Shock impairs assets: 1st of 4 steps

• Technologies 𝑏

…

Net worth Inside Money (deposits) A L Outside Money

Pass through

Risky Claim Risky Claim Risky Claim

Losses

A L

𝐵1 Money

Risky Claim Inside equity

𝐶1

A L A L A L A L

𝐵1 Money

HH Net worth

Brunnermeier & Sannikov

• Technologies 𝑐

Shrink balance sheet: 2nd of 4 steps

A

• Technologies 𝑏

Inside Money (deposits)

…

Net worth Inside Money (deposits) A L

Pass through

Losses

Deleveraging Deleveraging

… Risky Claim Risky Claim Risky Claim

Outside Money Switch

A L

Risky Claim

A L A L

𝐵1 Money

Risky Claim Inside equity

𝐶1

A L A L A L A L

𝐵1 Money

HH Net worth

Brunnermeier & Sannikov

• Technologies 𝑐

Liquidity spiral: asset price drop: 3rd of 4

• Technologies 𝑏

Switch

A L

Risky Claim

A L A L

𝐵1 Money

Risky Claim Inside equity

𝐶1 Inside Money (deposits) Outside Money

…

Net worth Inside Money (deposits) A L

Pass through

Risky Claim Risky Claim Risky Claim

Losses

Deleveraging Deleveraging A L A L A L A L

𝐵1 Money

HH Net worth

Brunnermeier & Sannikov

• Technologies 𝑏
• Technologies 𝑐

Disinflationary spiral: 4th of 4 steps

A L A L A L A L

𝐵1 Money

HH Net worth

A L

Risky Claim

A L A L

𝐵1 Money

Risky Claim Inside equity

𝐶1 Inside Money (deposits) Outside Money

…

Net worth Inside Money (deposits) A L

Pass through

Risky Claim Risky Claim Risky Claim

Losses

Deleveraging Deleveraging

Brunnermeier & Sannikov

Formal model: capital & output

• Model setup in paper is more general: 𝑍

𝑢 = 𝐵 𝜔𝑢 𝐿𝑢

Technologies 𝑐 𝑏 Physical capital 𝐿𝑢

• Capital share

𝜔𝑢 1 − 𝜔𝑢 Output goods 𝑍

𝑢 𝑐 = 𝐵𝑙𝑢 𝑐

𝑍

𝑢 𝑏 = 𝐵𝑙𝑢 𝑏

Aggregate good (CES)

• Consumed or invested
• numeraire

𝑍

𝑢 = 1 2 𝑍

𝑢 𝑐 (𝑡−1)/𝑡

+1 2 𝑍

𝑢 𝑏 (𝑡−1)/𝑡

𝑡/(𝑡−1)

Price of goods 𝑄𝑢

𝑐 = 1 2 𝑍

𝑢

𝑍

𝑢 𝑐 1/𝑡

𝑄𝑢

𝑏 = 1 2 𝑍

𝑢

𝑍

𝑢 𝑏 1/𝑡

Imperfect substitutes

Brunnermeier & Sannikov

Formal model: risk

• When 𝑙𝑢 is employed in sector 𝑏 by agent 𝑘

𝑒𝑙𝑢 = Φ 𝜅𝑢 − 𝜀 𝑙𝑢𝑒𝑢 + 𝜏𝑏𝑙𝑢𝑒𝑎𝑢

𝑏 + 𝜏𝑘𝑙𝑢𝑒

𝑎𝑢

𝑏

• Φ 𝜅𝑢 is increasing and concave, e.g. log[ 𝜆𝜅𝑢 + 1 /𝜆]
• All 𝑒𝑎 are independent of each other
• Intermediaries can diversify within sector 𝑐
• Face no idiosyncratic risk
• Households cannot become intermediaries or vice versa

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

Brunnermeier & Sannikov

Asset returns on money

• Money: fixed supply in baseline model, total value 𝑞𝑢𝐿𝑢
• Return = capital gains (no dividend/interest in baseline model)
• If 𝑒𝑞𝑢/𝑞𝑢 = 𝜈𝑢

𝑞𝑒𝑢 + 𝝉𝒖 𝒒𝑒𝒂𝒖,

• 𝑒𝐿𝑢/𝐿𝑢= Φ 𝜅𝑢 − 𝜀 𝑒𝑢 + 1 − 𝜔𝑢 𝜏𝑏𝑒𝑎𝑢

𝑏 + 𝜔𝑢𝜏𝑐𝑒𝑎𝑢 𝑐 (𝝉𝑢

𝑳)𝑈𝑒𝒂𝑢

𝑒𝑠

𝑢 𝑁 = Φ 𝜅𝑢 − 𝜀 + 𝜈𝑢 𝑞 + 𝝉𝒖 𝒒 𝑈𝝉𝒖 𝑳 𝑒𝑢 + 𝝉𝒖 𝒒 + 𝝉𝒖 𝑳 𝑒𝒂𝒖

• 𝜌𝑢 =

𝑞𝑢 𝑟𝑢+𝑞𝑢 fraction of wealth in form of money

Brunnermeier & Sannikov

• Technologies 𝑏

A L

Risky Claim

A L

Risky Claim

Capital/risk shares

• Technologies 𝑐

…

Net worth Inside Money (deposits) A L Outside Money

Pass through

A L

𝜔𝑢𝑟𝑢𝐿𝑢 Money

Risky Claim Inside equity

A L A L A L A L

Money

HH Net worth χ𝑢 1 − χ𝑢 1 − χ𝑢 𝜔𝑢𝑟𝑢𝐿𝑢

𝑂𝑢 (1 − 𝜔𝑢)𝑟𝑢𝐿𝑢

Fraction 𝛽𝑢 of HH

Brunnermeier & Sannikov

• Technologies 𝑏

A L

Risky Claim

A L

Risky Claim

Capital/risk shares

• Technologies 𝑐

…

Net worth Inside Money (deposits) A L Outside Money

Pass through

A L

𝜔𝑢𝑟𝑢𝐿𝑢 Money

Risky Claim Inside equity

A L A L A L A L

Money

HH Net worth χ𝑢 1 − χ𝑢 1 − χ𝑢 𝜔𝑢𝑟𝑢𝐿𝑢

𝑂𝑢 (1 − 𝜔𝑢)𝑟𝑢𝐿𝑢

• If 𝜓𝑢 > 𝜓, inside and outside equity

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

• If the equity constraint 𝜓𝑢 = 𝜓 binds,

inside equity earns a premium λ Fraction 𝛽𝑢 of HH

Brunnermeier & Sannikov

Allocation

• Equilibrium is a map

Histories of shocks prices 𝑟𝑢, 𝑞𝑢, 𝜇𝑢, allocation

𝒂𝜐, 0 ≤ 𝜐 ≤ 𝑢

𝛽𝑢, 𝜓𝑢 & portfolio weights (𝑦𝑢, 𝑦𝑢

𝑏, 𝑦𝑢 𝑐)

wealth distribution 𝜃𝑢 =

𝑂𝑢 (𝑞𝑢+𝑟𝑢)𝐿𝑢 ∈ 0,1

intermediaries’ wealth share

• All agents maximize utility
• Choose: portfolio, consumption, technology
• All markets clear
• Consumption, capital, money, outside equity of 𝑐

Brunnermeier & Sannikov

Numerical example: capital shares

𝜍 = 5%, 𝐵 = .5, 𝜏𝑏 = 𝜏𝑐 = .4, 𝜏𝑘 = .9, 𝜏𝑏 = .6, 𝜏𝑏 = 1.2, 𝑡 = .8, Φ 𝜅 = log 𝜆𝜅 + 1 𝜆 , 𝜆 = 2, 𝜓 = .001

• technology 𝑏 HH

technology 𝑐 HH

1 −

intermediaries

𝜓𝜔

Brunnermeier & Sannikov

Numerical example: prices

𝑞 𝑟 Disinflation spiral Liquidity spiral

Brunnermeier & Sannikov

Numerical example: prices

𝑞 𝑟 𝜌 =

𝑞 𝑞+𝑟

Disinflation spiral Liquidity spiral

Brunnermeier & Sannikov

Numerical example: dynamics of 𝜃

amplification

leverage elasticity Steady state

𝜏𝑢

𝜃 =

𝑦𝑢(𝜏𝑐1𝑐 − 𝜏𝑢

𝐿)

1 −

𝜔𝑢(1−𝜓𝑢)−𝜃 𝜃 −𝜌′ 𝜃 𝜌/𝜃

fundamental volatility

Brunnermeier & Sannikov

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

Brunnermeier & Sannikov

• 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 Inside Money (deposits) A L Outside Money

Pass through

𝑂𝑢 Bonds 𝑐𝑢𝐿𝑢

1 − χ𝑢 𝜔𝑢𝑟𝑢𝐿𝑢

Brunnermeier & Sannikov

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

Brunnermeier & Sannikov

Monetary policy and endogenous risk

• Intermediaries’ risk (borrow to scale up)

𝜏𝑢

𝜃 =

𝑦𝑢(𝜏𝑐1𝑐 − 𝜏𝑢

𝐿)

1 −

𝜔−𝜃 𝜃 −𝜌′ 𝜃 𝜌/𝜃 + 𝜔(1−𝜓)−𝜃 𝜃

+ 𝜌

𝜃 1 − 𝜔

𝑐𝑢 𝑞𝑢 −𝐶′ 𝜃 𝐶(𝜃)/𝜃

• Example:

𝑐𝑢 𝑞𝑢 𝐶′ 𝜃 𝐶 𝜃 = 𝛽𝑢 𝜌′ 𝜃 𝜌(1−𝜌)

• Intuition:

with right monetary policy bond price 𝐶(𝜃) rises as 𝜃 drops “stealth recapitalization”

• Can reduce liquidity and disinflationary spiral

fundamental risk amplification mitigation

𝛽(𝜃)

Brunnermeier & Sannikov

Numerical example with monetary policy

• Allocations

Prices

Higher intermediaries’ capital share (1 − 𝜓)𝜔

𝜔𝑏 Less production of

good 𝑏

𝑟 is more stable 𝑞 less disinflation

Brunnermeier & Sannikov

• 𝜏𝑢

𝜃 =

𝑦𝑢(𝜏𝑐𝟐𝑐 − 𝜏𝑢

𝐿)

1 −

𝜔−𝜃 𝜃 −𝜌′ 𝜃 𝜌/𝜃 + 𝜔−𝜃 𝜃 + 𝜌 𝜃 1 − 𝜔

𝑐𝑢 𝑞𝑢 −𝐶′ 𝜃 𝐶 𝜃 /𝜃

Numerical example with monetary policy

Steady state Less volatile

Recall 𝑐𝑢 𝑞𝑢 𝐶′ 𝜃 𝐶 𝜃 = 𝛽𝑢 𝜌′ 𝜃 𝜌(1 − 𝜌)

Brunnermeier & Sannikov

Numerical example with monetary policy

• Welfare:

HH and Intermediaries Sum

Brunnermeier & Sannikov

Monetary policy … in the limit

• full risk sharing of all aggregate risk
• 𝜏𝑢

𝜃 = 𝑦𝑢 1− 𝜔−𝜃

𝜃 −𝜌′ 𝜃 𝜌 𝜃

+ 𝜔 1−𝜓 −𝜃

𝜃

+𝜌

𝜃 1−𝜔

𝑐𝑢 𝑞𝑢 −𝐶′ 𝜃 𝐶(𝜃)/𝜃

(𝜏𝑐𝟐𝑐 − 𝜏𝑢

𝐿)

• 𝜃 is deterministic and converges over time towards 0

−∞

Brunnermeier & Sannikov

Monetary policy … in the limit

• full risk sharing of all aggregate risk
• Aggregate risk sharing

makes 𝑟 determinisitic

• Like in benchmark toy

model

• Excessive 𝑙-investment
• Too high 𝑟

(pecuniary externality)

• Lower capital return
• Endogenous risk

corrects pecuniary externality

Brunnermeier & Sannikov

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

Brunnermeier & Sannikov

MacroPru policy

• Regulator can control

cannot control

• Portfolio choice 𝜔s, 𝑦s

⋅ investment decision

𝜅 𝑟

⋅ consumption decision 𝑑

• f intermediaries and households

Brunnermeier & Sannikov

MacroPru policy

• Regulator can control

cannot control

• Portfolio choice 𝜔s, 𝑦s

⋅ investment decision

𝜅 𝑟

⋅ consumption decision 𝑑

• f intermediaries and households
• De-facto controls 𝑟 and 𝑞 within some range
• De-factor controls wealth share 𝜃
• Force agents to hold certain assets and generate capital gains
• In sum,

regulator maximizes sum of agents value function

• Choosing 𝜔𝑐, 𝑟, 𝜃

distorts

Brunnermeier & Sannikov

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,

• therwise can get 1st best

intermediary welfare

• 20
• 15
• 10
• 5

5 10 15 20 25

household welfare

• 20
• 15
• 10
• 5

5 10 15 20 25 30 no policy policy that removes endogenous risk

• ptimal macroprudential

Brunnermeier & Sannikov

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