Brunnermeier & Sannikov
The I Theory of Money
Markus K. Brunnermeier & Yuliy Sannikov
Princeton University
CSEF-IGIER Symposium
Capri, June 24th, 2015
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
Brunnermeier & Sannikov
 Endogenous money multiplier = f(capitalization of critical sector)
• Value of money goes up
• Fire-sales of assets
– Disinflation spiral a la Fisher (1933)
– Liquidity spiral
 Flight to safety
 Time-varying risk premium and
endogenous volatility dynamics
Some literature
 Macro-friction models without money
• Kiyotaki & Moore, BruSan2014, He & Krishnamurthy, DSS2015
 “Money models” without intermediaries
Brunnermeier & Sannikov
• 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
Some literature
 Macro-friction models without money
• Kiyotaki & Moore, BruSan2014, He & Krishnamurthy, DSS2015
 “Money models” without intermediaries
Brunnermeier & Sannikov
• 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
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
Friction
OLG
Brunnermeier & Sannikov
deterministic
Only money
Samuelson
With capital
Diamond
endowment risk
borrowing constraint
 With intermediaries/inside money
• “Money view” (Friedman & Schwartz) vs. “Credit view”(Tobin)
 New Keynesian Models: BGG, Christian et al., … money in utility function
Some literature
 Macro-friction models without money
• Kiyotaki & Moore, BruSan2014, He & Krishnamurthy, DSS2015
 “Money models” without intermediaries
Brunnermeier & Sannikov
• Store of value: Money pays no dividend and is a bubble
Friction
OLG
Incomplete Markets + idiosyncratic risk
Risk
deterministic
endowment risk
borrowing constraint
Only money
Samuelson
Bewley
With capital
Diamond
Ayagari, Krusell-Smith
 With intermediaries/inside money
• “Money view” (Friedman & Schwartz) vs. “Credit view”(Tobin)
 New Keynesian Models: BGG, Christian et al., … money in utility function
Some literature
 Macro-friction models without money
• Kiyotaki & Moore, BruSan2014, He & Krishnamurthy, DSS2015
 “Money models” without intermediaries
Brunnermeier & Sannikov
• Store of value: Money pays no dividend and is a bubble
Friction
OLG
Incomplete Markets + idiosyncratic risk
Risk
deterministic
endowment risk
borrowing constraint
Only money
Samuelson
Bewley
With capital
Diamond
Ayagari, Krusell-Smith
investment risk
Basic “I Theory”
 With intermediaries/inside money
• “Money view” (Friedman & Schwartz) vs. “Credit view”(Tobin)
 New Keynesian Models: BGG, Christian et al., … money in utility function
Some literature
 Macro-friction models without money
• Kiyotaki & Moore, BruSan2014, He & Krishnamurthy, DSS2015
 “Money models” without intermediaries
Brunnermeier & Sannikov
• Store of value: Money pays no dividend and is a bubble
Friction
OLG
Incomplete Markets + idiosyncratic risk
Risk
deterministic
endowment risk
borrowing constraint
Only money
Samuelson
Bewley
With capital
Diamond
Ayagari, Krusell-Smith
investment risk
Basic “I Theory”
 With intermediaries/inside money
• “Money view” (Friedman & Schwartz) vs. “Credit view”(Tobin)
 New Keynesian Models: BGG, Christian et al., … money in utility function
Some literature
 Macro-friction models without money
• Kiyotaki & Moore, BruSan2014, He & Krishnamurthy, DSS2015
 “Money models” without intermediaries
Brunnermeier & Sannikov
• Store of value: Money pays no dividend and is a bubble
Friction
OLG
Incomplete Markets + idiosyncratic risk
Risk
deterministic
endowment risk
borrowing constraint
Only money
Samuelson
Bewley
With capital
Diamond
Ayagari, Krusell-Smith
investment risk
Basic “I Theory”
 With intermediaries/inside money
• “Money view” (Friedman & Schwartz) vs. “Credit view”(Tobin)
 New Keynesian Models: BGG, Christian et al., … money in utility function
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
Brunnermeier & Sannikov
 Model with macro-prudential policy
One sector basic model
 Technologies 𝑎
Brunnermeier & Sannikov
𝐴1
 Each households can only
operate one firm
• Physical capital
𝑑𝑘𝑡
= (Φ 𝜄𝑡 − 𝛿)𝑑𝑡 + 𝜎 𝑎 𝑑𝑍𝑡𝑎 + 𝜎𝑑𝑍𝑡𝑎
𝑘𝑡
• Output
𝑦𝑡 = 𝐴𝑘𝑡
 Demand for money
sector idiosyncratic
risk
Adding outside money
Outside Money
 Technologies 𝑎
 𝑞𝑡 𝐾𝑡 value of physical capital
𝑏
𝑏
𝑏
• Postulate constant 𝑞𝑡 𝐴−𝜄
𝑑𝑡
+
Φ(𝜄
−
𝛿)
𝑑𝑡
+
𝜎
𝑑𝑍
+
𝜎𝑑
𝑍
𝑡
𝑡
𝑞
• Postulate value of money changes proportional to 𝐾𝑡
A
L
A
A
Money
Brunnermeier & Sannikov
𝐴1
L
L
L
Net worth
A
 𝑝𝑡 𝐾𝑡 value of outside money
 Each households can only
operate one firm
• Physical capital
𝑑𝑘𝑡
= (Φ 𝜄𝑡 − 𝛿)𝑑𝑡 + 𝜎 𝑎 𝑑𝑍𝑡𝑎 + 𝜎𝑑𝑍𝑡𝑎
𝑘𝑡
• Output
𝑦𝑡 = 𝐴𝑘𝑡
 Demand for money
sector idiosyncratic
risk
Adding outside money
Outside Money
 Technologies 𝑎
 𝑞𝐾𝑡 value of physical capital
𝑎
𝑎
𝑎
• 𝑑𝑟 𝑎 = 𝐴−𝜄
𝑑𝑡
+
Φ(𝜄
−
𝛿)
𝑑𝑡
+
𝜎
𝑑𝑍
+
𝜎𝑑
𝑍
𝑡
𝑡
𝑞
A
 𝑝𝐾𝑡 value of outside money
• 𝑑𝑟
𝑀
= Φ(𝜄 − 𝛿) 𝑑𝑡 + 𝜎
𝑎
𝑑𝑍𝑡𝑎
A
L
A
A
L
L
L
Money
Brunnermeier & Sannikov
𝐴1
Net worth
𝑔
 Each households can only
operate one firm
• Physical capital
𝑑𝑘𝑡
= (Φ 𝜄𝑡 − 𝛿)𝑑𝑡 + 𝜎 𝑎 𝑑𝑍𝑡𝑎 + 𝜎𝑑𝑍𝑡𝑎
𝑘𝑡
• Output
𝑦𝑡 = 𝐴𝑘𝑡
 Demand for money
sector idiosyncratic
risk
Demand with 𝐸
∞ −𝜌𝑡
𝑒
0
log 𝑐𝑡 𝑑𝑡
 𝑞𝐾𝑡 value of physical capital
Outside Money
 Technologies 𝑎
• 𝑑𝑟 𝑎 = 𝐴−𝜄
𝑑𝑡 + Φ(𝜄 − 𝛿) 𝑑𝑡 + 𝜎 𝑎 𝑑𝑍𝑡𝑎 + 𝜎𝑑 𝑍𝑡𝑎
𝑞
 𝑝𝐾𝑡 value of outside money
• 𝑑𝑟
𝑀
= Φ(𝜄 − 𝛿) 𝑑𝑡 + 𝜎
𝑎
A
𝑑𝑍𝑡𝑎
A
L
A
A
Money
 Consumption demand:
𝜌 𝑝 + 𝑞 𝐾𝑡 = 𝐴 − 𝜄 𝐾𝑡
 Asset (share) demand 𝑥 𝑎 :
𝑎
Brunnermeier & Sannikov
𝐸
𝑑𝑟 𝑎
−
𝑑𝑟 𝑀
/𝑑𝑡 =
𝐶𝑜𝑣[𝑑𝑟 𝑎
− 𝑑𝑟 𝑀 ,
𝑑𝑛𝑡
𝑛𝑡𝑎
𝑑𝑟 𝑀 +𝑥 𝑎 𝑑𝑟 𝑎 −𝑑𝑟 𝑀
𝑥𝑎
=
 Investment rate:
𝐸 𝑑𝑟 𝑎 −𝑑𝑟 𝑀 /𝑑𝑡
(𝐴−𝜄)/𝑞
𝑞
=
=
𝑞+𝑝
𝜎2
𝜎2
(Tobin’s q)
Φ′ 𝜄 = 1/𝑞
• For Φ 𝜄 = 𝜅1 log(𝜅𝜄 + 1) ⇒ 𝜄∗ = 𝑞−1
𝜅
𝐴1
] = 𝑥𝑎𝜎2
Net worth
𝑔
L
L
L
Demand with log-utility
Outside Money
 Technologies 𝑎
 𝑞𝐾𝑡 value of physical capital
• 𝑑𝑟 𝑎 = 𝐴−𝜄
𝑑𝑡 + Φ(𝜄 − 𝛿) 𝑑𝑡 + 𝜎 𝑎 𝑑𝑍𝑡𝑎 + 𝜎𝑑 𝑍𝑡𝑎
𝑞
 𝑝𝐾𝑡 value of outside money
• 𝑑𝑟
𝑀
= Φ(𝜄 − 𝛿) 𝑑𝑡 + 𝜎
𝑎
A
𝑑𝑍𝑡𝑎
A
L
A
A
Money
 Consumption demand:
𝜌 𝑝 + 𝑞 𝐾𝑡 = 𝐴 − 𝜄 𝐾𝑡
 Asset (share) demand 𝑥 𝑎 :
𝑎
Brunnermeier & Sannikov
𝐸
𝑑𝑟 𝑎
−
𝑑𝑟 𝑀
/𝑑𝑡 =
𝐶𝑜𝑣[𝑑𝑟 𝑎
− 𝑑𝑟 𝑀 ,
𝑑𝑛𝑡
𝑛𝑡𝑎
𝑑𝑟 𝑀 +𝑥 𝑎 𝑑𝑟 𝑎 −𝑑𝑟 𝑀
𝑥𝑎
=
 Investment rate:
𝐸 𝑑𝑟 𝑎 −𝑑𝑟 𝑀 /𝑑𝑡
(𝐴−𝜄)/𝑞
𝑞
=
=
𝑞+𝑝
𝜎2
𝜎2
(Tobin’s q)
Φ′ 𝜄 = 1/𝑞
• For Φ 𝜄 = 𝜅1 log(𝜅𝜄 + 1) ⇒ 𝜄∗ = 𝑞−1
𝜅
𝐴1
] = 𝑥𝑎𝜎2
Net worth
𝑔
L
L
L
Demand with log-utility
Outside Money
 Technologies 𝑎
 𝑞𝐾𝑡 value of physical capital
• 𝑑𝑟 𝑎 = 𝐴−𝜄
𝑑𝑡 + Φ(𝜄 − 𝛿) 𝑑𝑡 + 𝜎 𝑎 𝑑𝑍𝑡𝑎 + 𝜎𝑑 𝑍𝑡𝑎
𝑞
 𝑝𝐾𝑡 value of outside money
• 𝑑𝑟
𝑀
= Φ(𝜄 − 𝛿) 𝑑𝑡 + 𝜎
𝑎
A
𝑑𝑍𝑡𝑎
A
L
A
A
Money
 Consumption demand:
𝜌 𝑝 + 𝑞 𝐾𝑡 = 𝐴 − 𝜄 𝐾𝑡
 Asset (share) demand 𝑥 𝑎 :
𝑎
Brunnermeier & Sannikov
𝐸
𝑑𝑟 𝑎
−
𝑑𝑟 𝑀
/𝑑𝑡 =
𝐶𝑜𝑣[𝑑𝑟 𝑎
− 𝑑𝑟 𝑀 ,
𝑑𝑛𝑡
𝑛𝑡𝑎
𝑑𝑟 𝑀 +𝑥 𝑎 𝑑𝑟 𝑎 −𝑑𝑟 𝑀
𝑥𝑎
=
 Investment rate:
𝐸 𝑑𝑟 𝑎 −𝑑𝑟 𝑀 /𝑑𝑡
(𝐴−𝜄)/𝑞
𝑞
=
=
𝑞+𝑝
𝜎2
𝜎2
(Tobin’s q)
Φ′ 𝜄 = 1/𝑞
• For Φ 𝜄 = 𝜅1 log(𝜅𝜄 + 1) ⇒ 𝜄 = 𝑞−1
𝜅
𝐴1
] = 𝑥𝑎𝜎2
Net worth
𝑔
L
L
L
Market clearing
Outside Money
 Technologies 𝑎
 𝑞𝐾𝑡 value of physical capital
• 𝑑𝑟 𝑎 = 𝐴−𝜄
𝑑𝑡 + Φ(𝜄 − 𝛿) 𝑑𝑡 + 𝜎 𝑎 𝑑𝑍𝑡𝑎 + 𝜎𝑑 𝑍𝑡𝑎
𝑞
 𝑝𝐾𝑡 value of outside money
• 𝑑𝑟
𝑀
= Φ(𝜄 − 𝛿) 𝑑𝑡 + 𝜎
𝑎
A
𝑑𝑍𝑡𝑎
A
L
A
A
Money
 Consumption demand:
𝜌 𝑝 + 𝑞 𝐾𝑡 = 𝐴 − 𝜄 𝐾𝑡
 Asset (share) demand 𝑥 𝑎 :
𝑎
Brunnermeier & Sannikov
𝐸
𝑑𝑟 𝑎
−
𝑑𝑟 𝑀
/𝑑𝑡 =
𝐶𝑜𝑣[𝑑𝑟 𝑎
− 𝑑𝑟 𝑀 ,
𝑑𝑛𝑡
𝑛𝑡𝑎
𝑑𝑟 𝑀 +𝑥 𝑎 𝑑𝑟 𝑎 −𝑑𝑟 𝑀
𝑥𝑎
=
 Investment rate:
𝐸 𝑑𝑟 𝑎 −𝑑𝑟 𝑀 /𝑑𝑡
(𝐴−𝜄)/𝑞
𝑞
=
=
𝑞+𝑝
𝜎2
𝜎2
(Tobin’s q)
Φ′ 𝜄 = 1/𝑞
• For Φ 𝜄 = 𝜅1 log(𝜅𝜄 + 1) ⇒ 𝜄 = 𝑞−1
𝜅
𝐴1
] = 𝑥𝑎𝜎2
Net worth
𝑔
L
L
L
Equilibrium
Moneyless equilibrium
𝑝0 = 0
𝜅𝐴+1
𝑞0 = 𝜅𝜌+1
Money equilibrium
>
𝑞0
𝜎− 𝜌
𝑝= 𝜌 𝑞
𝑞 = 𝜅 𝜅𝐴+1
𝜌𝜎+1
𝑝
Brunnermeier & Sannikov
𝑞
0
𝜌
𝜎
Welfare analysis
Moneyless equilibrium
𝑝0 = 0
𝜅𝐴+1
𝑞0 = 𝜅𝜌+1
Brunnermeier & Sannikov
𝑔0
welfare0
Money equilibrium
>
>
<
𝜎− 𝜌
𝑝= 𝜌 𝑞
𝑞 = 𝜅 𝜅𝐴+1
𝜌𝜎+1
𝑔
welfare
Welfare analysis
Moneyless equilibrium
𝑝0 = 0
Money equilibrium
𝜎− 𝜌
𝑝= 𝜌 𝑞
𝑞 = 𝜅 𝜅𝐴+1
𝜌𝜎+1
𝜅𝐴+1
𝑞0 = 𝜅𝜌+1
Brunnermeier & Sannikov
welfare0
<
welfare
𝑝
 What ratio nominal to total wealth 𝑞+𝑝
maximizes welfare?
• Force agents to hold less 𝑘 & more money
𝑝
• Raise
if and only if 𝜎(1 − 𝜅𝜌) ≤ 2 𝜌
𝑞+𝑝
• Lowers 𝑞 ⇒ higher E[𝑑𝑟 𝑎 − 𝑑𝑟 𝑀 ] = 𝐴−𝜄
𝑑𝑡
𝑞
pecuniary externality
• Create 𝑞-risk to make precautionary money savings more attractive
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
Brunnermeier & Sannikov
 Model with macro-prudential policy
Outline of two sector model
 Technologies 𝑏
 Technologies 𝑎
𝐴
𝐴1 1
𝐵1
𝐴1
SwitchSwitch
technology
Brunnermeier & Sannikov
 Households have to
• Specialize in one subsector
sector specific + idiosyncratic risk
for one period
𝑑𝑘𝑡
= ⋯ 𝑑𝑡 + 𝜎 𝑏 𝑑𝑍𝑡𝑏 + 𝜎𝑑𝑍𝑡𝑏
𝑘𝑡
• Demand for money
𝑑𝑘𝑡
= ⋯ 𝑑𝑡 + 𝜎 𝑎 𝑑𝑍𝑡𝑎 + 𝜎𝑑𝑍𝑡𝑎
𝑘𝑡
Add outside money
 Technologies 𝑎
L
Money
𝐵1
L
L
A
Net worth
A
A
L
A
A
Money
𝐴1
SwitchSwitch
technology
 Households have to
Brunnermeier & Sannikov
A
• Specialize in one subsector
for one period
• Demand for money
Net worth
 Technologies 𝑏
A
Outside Money
L
L
L
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
Brunnermeier & Sannikov
 Model with macro-prudential policy
Add intermediaries
 Technologies 𝑏
Outside Money
 Technologies 𝑎
A
L
Net worth
L
Money
Brunnermeier & Sannikov
𝐵1
L
L
• Risk can be
partially sold off
to intermediaries
A
A
L
A
A
Money
𝐴1
L
L
Net worth
A
Net worth
A
A
• Risk is
not contractable
(Plagued with
moral hazard
problems)
L
Add intermediaries
 Technologies 𝑏
Outside Money
 Technologies 𝑎
A
L
Net worth
L
Money
𝐵1
L
L
A
L
A
A
Money
𝐴1
 Intermediaries
Brunnermeier & Sannikov
A
• Can hold outside equity
& diversify within sector 𝑏
• Monitoring
Net worth
A
Net worth
A
A
L
L
L
Add intermediaries
…
A
L
A
A
Money
Net worth
𝐴1
 Intermediaries
Brunnermeier & Sannikov
A
• Can hold outside equity
& diversify within sector 𝑏
• Monitoring
Net worth
𝐵1
L
Risky Claim
Money
Risky Claim
Risky Claim
Risky Claim
L
L
L
Risky Claim
A
Inside equity
A
A
 Technologies 𝑎
A
Risky Claim
 Technologies 𝑏
Outside Money
L
L
L
Add intermediaries
…
L
Pass through
Inside Money
(deposits)
A
L
A
A
Money
Net worth
𝐴1
 Intermediaries
Brunnermeier & Sannikov
A
• Can hold outside equity
& diversify within sector 𝑏
• Monitoring
• Create inside money
• Maturity/liquidity
transformation
HH Net worth
𝐵1
L
Risky Claim
Money
Risky Claim
Risky Claim
Risky Claim
L
L
Risky Claim
A
Inside equity
A
A
 Technologies 𝑎
A
Outside Money
Risky Claim
 Technologies 𝑏
Outside Money
L
L
L
Shock impairs assets: 1st of 4 steps
Brunnermeier & Sannikov
…
 Technologies 𝑎
L
Pass through
Inside Money
(deposits)
A
A
L
A
A
Money
Net worth
Losses
𝐴1
HH Net worth
𝐵
𝐴11
L
Risky Claim
Money
Risky Claim
Risky Claim
L
L
Risky Claim
A
Inside equity
A
A
A
Outside Money
Risky Claim
 Technologies 𝑏
L
L
L
Shrink balance sheet: 2nd of 4 steps
 Technologies 𝑎
A
L
Deleveraging
𝐵1
𝐴1
…
Risky Claim
Outside Money
Risky Claim
L
Risky Claim
Money
L
Risky Claim
Risky Claim
L
Inside equity
A
A
A
Deleveraging
Pass through
Inside Money
Inside Money
(deposits)
(deposits)
A
L
A
A
Money
Net worth
Losses
Switch
Brunnermeier & Sannikov
A
𝐴1
HH Net worth
 Technologies 𝑏
L
L
L
Liquidity spiral: asset price drop: 3rd of 4
 Technologies 𝑎
A
L
Deleveraging
…
Risky Claim
Outside Money
Risky Claim
𝐵1
𝐴1
L
Risky Claim
Money
L
Risky Claim
Risky Claim
L
Inside equity
A
A
A
Deleveraging
Pass through
Inside Money
Inside Money
(deposits)
(deposits)
A
L
A
A
Money
Net worth
Losses
Switch
Brunnermeier & Sannikov
A
𝐴1
HH Net worth
 Technologies 𝑏
L
L
L
Disinflationary spiral: 4th of 4 steps
 Technologies 𝑎
A
L
Deleveraging
Brunnermeier & Sannikov
…
Risky Claim
Outside Money
Risky Claim
𝐵1
𝐴1
L
Risky Claim
Money
L
Risky Claim
Risky Claim
L
Inside equity
A
A
A
Deleveraging
Pass through
Inside Money
Inside Money
(deposits)
(deposits)
A
A
L
A
A
Money
Net worth
Losses
𝐴1
HH Net worth
 Technologies 𝑏
L
L
L
Formal model: capital & output
Technologies
𝑏
𝑎
Physical capital 𝐾𝑡
- Capital share
𝜓𝑡
1 − 𝜓𝑡
Output goods
Aggregate good (CES)
-
Consumed or invested
numeraire
Brunnermeier & Sannikov
Price of goods
𝑌𝑡𝑏
𝑌𝑡 =
=
𝐴𝑘𝑡𝑏
Imperfect
substitutes
𝑌𝑡𝑎 = 𝐴𝑘𝑡𝑎
𝑠/(𝑠−1)
1 𝑏 (𝑠−1)/𝑠 1 𝑎 (𝑠−1)/𝑠
𝑌
+ 𝑌𝑡
2 𝑡
2
𝑃𝑡𝑏 = 12
𝑌𝑡
𝑌𝑡𝑏
1/𝑠
𝑃𝑡𝑎
 Model setup in paper is more general: 𝑌𝑡 = 𝐴 𝜓𝑡 𝐾𝑡
=
1 𝑌𝑡 1/𝑠
2 𝑌𝑡𝑎
Formal model: risk
 When 𝑘𝑡 is employed in sector 𝑎 by agent 𝑗
𝑑𝑘𝑡 = Φ 𝜄𝑡 − 𝛿 𝑘𝑡 𝑑𝑡 + 𝜎 𝑎 𝑘𝑡 𝑑𝑍𝑡𝑎 + 𝜎 𝑗 𝑘𝑡 𝑑𝑍𝑡𝑎
Investment rate (per unit of 𝑘𝑡 )
sectorial
idiosyncratic
independent Brownian motions
(fundamental cash flow risk)
e.g. log[ 𝜅𝜄𝑡 + 1 /𝜅]
Brunnermeier & Sannikov
• Φ 𝜄𝑡 is increasing and concave,
• All 𝑑𝑍 are independent of each other
 Intermediaries can diversify within sector 𝑏
• Face no idiosyncratic risk
 Households cannot become intermediaries or vice versa
Asset returns on money
 Money: fixed supply in baseline model, total value 𝑝𝑡 𝐾𝑡
• Return = capital gains (no dividend/interest in baseline model)
𝒑
𝑝
• If 𝑑𝑝𝑡 /𝑝𝑡 = 𝜇𝑡 𝑑𝑡 + 𝝈𝒕 𝑑𝒁𝒕 ,
• 𝑑𝐾𝑡 /𝐾𝑡 = Φ 𝜄𝑡 − 𝛿 𝑑𝑡 + 1 − 𝜓𝑡 𝜎 𝑎 𝑑𝑍𝑡𝑎 + 𝜓𝑡 𝜎 𝑏 𝑑𝑍𝑡𝑏
𝑇
(𝝈𝑲
𝑡 ) 𝑑𝒁𝑡
Brunnermeier & Sannikov
𝑑𝑟𝑡𝑀
 𝜋𝑡 =
= Φ 𝜄𝑡 − 𝛿
𝑝𝑡
𝑞𝑡 +𝑝𝑡
𝑝
+ 𝜇𝑡
+
𝒑 𝑇 𝑲
𝝈𝒕 𝝈𝒕
𝒑
𝑑𝑡 + 𝝈𝒕 + 𝝈𝑲
𝒕 𝑑𝒁𝒕
fraction of wealth in form of money
Capital/risk shares
A
Brunnermeier & Sannikov
𝜓𝑡 𝑞𝑡 𝐾𝑡
L
Risky Claim
Risky Claim
Risky Claim
Money
L
Inside equity
A
A
χ𝑡 1 − χ𝑡
A
Outside Money
L
…
1 − χ𝑡 𝜓𝑡 𝑞𝑡 𝐾𝑡
L Technologies 𝑎
Fraction 𝛼𝑡 of HH
Pass through
Inside Money
(deposits)
A
A
L
A
A
Money
Net worth 𝑁𝑡
(1 − 𝜓𝑡 )𝑞𝑡 𝐾𝑡
HH Net worth
 Technologies 𝑏
L
L
L
Capital/risk shares
A
Brunnermeier & Sannikov
𝜓𝑡 𝑞𝑡 𝐾𝑡
L
Risky Claim
Risky Claim
Risky Claim
Money
L
Inside equity
A
A
A
Outside Money
L
…
1 − χ𝑡 𝜓𝑡 𝑞𝑡 𝐾𝑡
L Technologies 𝑎
Fraction 𝛼𝑡 of HH
Pass through
Inside Money
(deposits)
A
A
L
A
A
Money
Net worth 𝑁𝑡
(1 − 𝜓𝑡 )𝑞𝑡 𝐾𝑡
χ𝑡 1 − χ𝑡
 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
HH Net worth
 Technologies 𝑏
L
L
L
Allocation
 Equilibrium is a map
Histories of shocks
prices 𝑞𝑡 , 𝑝𝑡 , 𝜆𝑡 , allocation
𝛼𝑡 , 𝜒𝑡 & portfolio weights (𝑥𝑡 , 𝑥𝑡𝑎 , 𝑥𝑡𝑏 )
𝒁𝜏 , 0 ≤ 𝜏 ≤ 𝑡
wealth distribution
𝜂𝑡 =
𝑁𝑡
(𝑝𝑡 +𝑞𝑡 )𝐾𝑡
∈ 0,1
Brunnermeier & Sannikov
intermediaries’ wealth share
• All agents maximize utility
 Choose: portfolio, consumption, technology
• All markets clear
 Consumption, capital, money, outside equity of 𝑏
Numerical example: capital shares
𝜌 = 5%, 𝐴 = .5, 𝜎 𝑎 = 𝜎 𝑏 = .4, 𝜎 𝑗 = .9, 𝜎 𝑎 = .6, 𝜎 𝑎 = 1.2, 𝑠 = .8,
log 𝜅𝜄 + 1
Φ 𝜄 =
, 𝜅 = 2, 𝜒 = .001
𝜅

intermediaries
Brunnermeier & Sannikov
1−
technology 𝑏 HH
𝜒𝜓
technology 𝑎 HH
Numerical example: prices
Disinflation
spiral
𝑝
Brunnermeier & Sannikov
𝑞
Liquidity
spiral
Numerical example: prices
Disinflation
spiral
𝑝
𝜋 = 𝑝+𝑞
𝑝
Brunnermeier & Sannikov
𝑞
Liquidity
spiral
Numerical example: dynamics of 𝜂
fundamental volatility
𝑥𝑡 (𝜎 𝑏 1𝑏 − 𝜎𝑡𝐾 )
𝜂
𝜎𝑡 =
1−
𝜓𝑡 (1−𝜒𝑡 )−𝜂 −𝜋′ 𝜂
𝜂
𝜋/𝜂
elasticity
leverage
Brunnermeier & Sannikov
amplification
Steady state
Overview
 No monetary economics
• Fixed outside money supply
• Amplification/endogenous risk through
 Liquidity spiral
 Disinflationary spiral
asset side of intermediaries’ balance sheet
liability side
 Monetary policy
• Aside: Money vs. Credit view (via helicopter drop)
• Wealth shifts by affecting relative price between
Brunnermeier & Sannikov
 Long-term bond
 Short-term money
• Risk transfers – reduce endogenous aggregate risk
 Macroprudential policy
A
Outside Money
L
Pass through
Bonds 𝑏𝑡 𝐾𝑡
Inside Money
(deposits)
…
1 − χ𝑡 𝜓𝑡 𝑞𝑡 𝐾𝑡
Brunnermeier & Sannikov
Net worth 𝑁𝑡
 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
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
Brunnermeier & Sannikov
 Redistribution of aggregate risk, mitigates risk that an
essential sector can become undercapitalized
 Affects earnings distribution, rents that different sectors
get in equilibrium
Monetary policy and endogenous risk
 Intermediaries’ risk (borrow to scale up)
fundamental risk
𝑥𝑡 (𝜎 𝑏 1𝑏 − 𝜎𝑡𝐾 )
𝜂
𝜎𝑡 =
1−
𝜓−𝜂 −𝜋′ 𝜂
𝜂
𝜋/𝜂
+
𝜓(1−𝜒)−𝜂
𝜂
amplification
Brunnermeier & Sannikov
 Example:
𝑏𝑡 𝐵 ′ 𝜂
𝑝𝑡 𝐵 𝜂
=
+
1−𝜓
mitigation
𝜋′ 𝜂
𝛼𝑡
𝜋(1−𝜋)
 Intuition:
with right monetary policy
bond price 𝐵(𝜂) rises as 𝜂 drops
“stealth recapitalization”
𝜋
𝜂
• Can reduce liquidity and disinflationary spiral
𝛼(𝜂)
𝑏𝑡 −𝐵 ′ 𝜂
𝑝𝑡 𝐵(𝜂)/𝜂
Numerical example with monetary policy
 Allocations
Prices
Higher intermediaries’
capital share (1 − 𝜒)𝜓
Brunnermeier & Sannikov
𝑝 less disinflation
𝑞 is more stable
𝜓 𝑎 Less production of
good 𝑎
Numerical example with monetary policy
Less volatile
Brunnermeier & Sannikov

Recall
𝑏𝑡 𝐵′ 𝜂
𝜋′ 𝜂
= 𝛼𝑡
𝑝𝑡 𝐵 𝜂
𝜋(1 − 𝜋)
Steady state
𝜂
𝜎𝑡
=
𝑥𝑡 (𝜎 𝑏 𝟏𝑏 − 𝜎𝑡𝐾 )
−𝜋′ 𝜂
1 − 𝜓−𝜂
𝜂
𝜋/𝜂
+
𝜓−𝜂
𝜂
+
𝜋
𝜂
1−𝜓
𝑏𝑡 −𝐵′ 𝜂
𝑝𝑡 𝐵 𝜂 /𝜂
Numerical example with monetary policy
Brunnermeier & Sannikov
 Welfare:
HH and Intermediaries
Sum
Monetary policy … in the limit
 full risk sharing of all aggregate risk
𝑥𝑡
𝜂
 𝜎𝑡 =
′
1−
𝜓−𝜂 −𝜋 𝜂
𝜋
𝜂
𝜂
+
𝜓 1−𝜒 −𝜂 𝜋
+𝜂
𝜂
1−𝜓
𝑏𝑡 −𝐵′ 𝜂
𝑝𝑡 𝐵(𝜂)/𝜂
−∞
(𝜎 𝑏 𝟏𝑏 − 𝜎𝑡𝐾 )
Brunnermeier & Sannikov
 𝜂 is deterministic and converges over time towards 0
Monetary policy … in the limit
 full risk sharing of all aggregate risk
 Aggregate risk sharing
makes 𝑞 determinisitic
 Like in benchmark toy
model
Brunnermeier & Sannikov
• Excessive 𝑘-investment
• Too high 𝑞
(pecuniary externality)
 Lower capital return
 Endogenous risk
corrects pecuniary
externality
Overview
 No monetary economics
• Fixed outside money supply
• Amplification/endogenous risk through
 Liquidity spiral
 Disinflationary spiral
asset side of intermediaries’ balance sheet
liability side
 Monetary policy
• Wealth shifts by affecting relative price between
 Long-term bond
 Short-term money
Brunnermeier & Sannikov
• Risk transfers – reduce endogenous aggregate risk
 Macroprudential policy
• Directly affect portfolio positions
MacroPru policy
 Regulator can control
• Portfolio choice 𝜓s, 𝑥s
cannot control
⋅ investment decision
⋅ consumption decision
Brunnermeier & Sannikov
of intermediaries and households
𝜄 𝑞
𝑐
MacroPru policy
 Regulator can control
• Portfolio choice 𝜓s, 𝑥s
cannot control
⋅ investment decision
⋅ consumption decision
𝜄 𝑞
𝑐
of intermediaries and households
• De-facto controls 𝑞 and 𝑝 within some range
• De-factor controls wealth share 𝜂
distorts
Brunnermeier & Sannikov
 Force agents to hold certain assets and generate capital gains
 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 1st best
30
optimal macroprudential
25
20 policy that removes endogenous risk
household welfare
Brunnermeier & Sannikov
15
no policy
10
5
0
-5
-10
-15
-20
-20
-15
-10
-5
0
5
intermediary welfare
10
15
20
25
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
Brunnermeier & Sannikov
 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