The I Theory of Money - Woodrow Wilson School of Public...
Transcript of 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
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
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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
Friction OLG
deterministic endowment riskborrowing constraint
Only money Samuelson
With capital Diamond
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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
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
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
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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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
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”
Bru
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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
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
Each households can only operate one firm• Physical capital
• Output
Demand for money
𝑑𝑘𝑡𝑘𝑡= (Φ 𝜄𝑡 − 𝛿)𝑑𝑡 + 𝜎
𝑎𝑑𝑍𝑡𝑎 + 𝜎𝑑 𝑍𝑡
𝑎
sector idiosyncratic risk
A LA L
A LA L
𝐴1
Money
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Outside Money
𝑦𝑡 = 𝐴𝑘𝑡
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
Each households can only operate one firm• Physical capital
• Output
Demand for money
𝑑𝑘𝑡𝑘𝑡= (Φ 𝜄𝑡 − 𝛿)𝑑𝑡 + 𝜎
𝑎𝑑𝑍𝑡𝑎 + 𝜎𝑑 𝑍𝑡
𝑎
sector idiosyncratic risk
A LA L
A LA L
𝐴1
Money
Net
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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
𝑦𝑡 = 𝐴𝑘𝑡
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Demand with 𝐸 0∞𝑒−𝜌𝑡 log 𝑐𝑡 𝑑𝑡
Technologies 𝑎
A LA L
A LA L
Money
Net
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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
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
Net
wo
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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
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
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
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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Market clearing Technologies 𝑎
A LA L
A LA L
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
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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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
Moneyless equilibrium Money equilibrium
𝑝0 = 0 𝑝 = 𝜎− 𝜌𝜌𝑞
𝑞0 =𝜅𝐴+1𝜅𝜌+1
𝑞 = 𝜅𝐴+1𝜅 𝜌 𝜎+1
𝑔0 𝑔
welfare0 welfare<
>
>
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Welfare analysis
Moneyless equilibrium Money equilibrium
𝑝0 = 0 𝑝 = 𝜎− 𝜌𝜌𝑞
𝑞0 =𝜅𝐴+1𝜅𝜌+1
𝑞 = 𝜅𝐴+1𝜅 𝜌 𝜎+1
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
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
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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
𝑎 + 𝜎𝑑 𝑍𝑡𝑎
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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
Technologies 𝑎 Technologies 𝑏
SwitchSwitch technology
A LA L
A LA L
𝐴1
Money
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rth
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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
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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
ort
h
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
h
Technologies 𝑎
A LA L
A LA L
𝐴1
Money
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Technologies 𝑏
A L
Ris
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laim
A LR
isky
Cla
im
Add intermediaries
…
Net worth
A L
Ris
ky C
laim
Ris
ky C
laim
Ris
ky C
laim
Outside Money
A L
𝐵1
Money
Ris
ky C
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
Ris
ky C
laim
A LR
isky
Cla
im
Add intermediaries Technologies 𝑏
…
Net worth
Inside Money(deposits)
A L
Outside Money Pass through
Ris
ky C
laim
Ris
ky C
laim
Ris
ky C
laim
Outside Money
Intermediaries• Can hold outside equity
& diversify within sector 𝑏• Monitoring• Create inside money• Maturity/liquidity
transformation
A L
𝐵1
Money
Ris
ky C
laim
Insi
de
equ
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
Ris
ky C
laim
A L
Shock impairs assets: 1st of 4 steps Technologies 𝑎
…
Net worth
Inside Money(deposits)
A L
Outside Money Pass through
Ris
ky C
laim
Ris
ky C
laim
Ris
ky C
laim
Losses
A L
𝐴1
Money
Ris
ky C
laim
Insi
de
equ
ity
𝐵1
A LA L
A LA L
𝐴1
Money
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Technologies 𝑏
Shrink balance sheet: 2nd of 4 steps
A
Technologies 𝑎
Inside Money(deposits)
…
Net worth
Inside Money(deposits)
A L
Pass through
Losses
Deleveraging Deleveraging
…
Ris
ky C
laim
Ris
ky C
laim
Ris
ky C
laim
Outside Money
Switch
A L
Ris
ky C
laim
A LA L
𝐴1
Money
Ris
ky C
laim
Insi
de
equ
ity
𝐵1
A LA L
A LA L
𝐴1
Money
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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
Ris
ky C
laim
Insi
de
equ
ity
𝐵1
Inside Money(deposits)
Outside Money
…
Net worth
Inside Money(deposits)
A L
Pass through
Ris
ky C
laim
Ris
ky C
laim
Ris
ky C
laim
Losses
Deleveraging Deleveraging
A LA L
A LA L
𝐴1
Money
HH
Net
wo
rth
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Technologies 𝑎 Technologies 𝑏
Disinflationary spiral: 4th of 4 steps
A LA L
A LA L
𝐴1
Money
HH
Net
wo
rth
A L
Ris
ky C
laim
A LA L
𝐴1
Money
Ris
ky C
laim
Insi
de
equ
ity
𝐵1
Inside Money(deposits)
Outside Money
…
Net worth
Inside Money(deposits)
A L
Pass through
Ris
ky C
laim
Ris
ky C
laim
Ris
ky C
laim
Losses
Deleveraging Deleveraging
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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
Ris
ky C
laim
A LR
isky
Cla
im
Capital/risk shares Technologies 𝑏
…
Net worth
Inside Money(deposits)
A L
Outside Money Pass through
A L
𝜓𝑡𝑞𝑡𝐾𝑡
Money
Ris
ky C
laim
Insi
de
equ
ity
A LA L
A LA L
Money
HH
Net
wo
rth
χ𝑡 1 − χ𝑡
1 − χ𝑡 𝜓𝑡𝑞𝑡𝐾𝑡
𝑁𝑡(1 − 𝜓𝑡)𝑞𝑡𝐾𝑡
Fraction 𝛼𝑡 of HH
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Technologies 𝑎
A L
Ris
ky C
laim
A LR
isky
Cla
im
Capital/risk shares Technologies 𝑏
…
Net worth
Inside Money(deposits)
A L
Outside Money Pass through
A L
𝜓𝑡𝑞𝑡𝐾𝑡
Money
Ris
ky C
laim
Insi
de
equ
ity
A LA L
A LA L
Money
HH
Net
wo
rth
χ𝑡 1 − χ𝑡
1 − χ𝑡 𝜓𝑡𝑞𝑡𝐾𝑡
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
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
Inside Money(deposits)
A L
Outside Money Pass through
𝑁𝑡
Bonds 𝑏𝑡𝐾𝑡
1 − χ𝑡 𝜓𝑡𝑞𝑡𝐾𝑡
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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
Numerical example with monetary policy
Steady state
Less volatile
Recall𝑏𝑡𝑝𝑡
𝐵′ 𝜂
𝐵 𝜂= 𝛼𝑡𝜋′ 𝜂
𝜋(1 − 𝜋)
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Numerical example with monetary policy
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
ho
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