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Bru
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International Monetary Theory:Mundell Fleming Redux
by
Markus K. Brunnermeier and Yuliy SannikovPrinceton and Stanford University
Princeton Initiative Princeton, Sept. 9th, 2017
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Motivation
Global currency spillovers• “Flight to safety” - Dollar appreciation when risk rises
• Local currency: good store of value/hedge for idiosyncratic risk
• Global currency: good hedge for international competitiveness risk
When to peg to world currency? When to dollarize?
MoPo space: “Nuanced Mundell-Fleming Trilemma”• Local and global money have different risk profile (imperfect substitutes)
⇒ increases MoPo space
• Too high inflation: local citizens substitute local currency for global currency⇒ limits MoPo space
Reserve currency management – Irrelevance theorem
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Modelling Framework
Closed economy Open Economy
Static Hick’s IS-LM Mundell-Fleming
DynamicImpulse response
New Keynesian Obstfeld-Rogoff
Risk & Dynamicfinancial frictions
Samuelson BewleyDiamond Aiyagari X
I Theory of Money
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Modelling Framework
Closed economy Open Economy
Static Hick’s IS-LM Mundell-Fleming
DynamicImpulse response
New Keynesian Obstfeld-Rogoff
Risk & Dynamicfinancial frictions
Samuelson BewleyDiamond Aiyagari X
I Theory of Money
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Modelling Framework
\Friction OLG Incomplete Markets + idiosyncratic risk
Risk deterministic endowment riskborrowing constraint
investment risk
Only money Samuelson Bewley
Basic “I Theory”
With capital Diamond Aiyagari
𝑓′ 𝑘∗ = 𝑟∗, Dynamic inefficiency𝑟 < 𝑟∗, 𝐾 > 𝐾∗
Inefficiency𝑟 < 𝑟∗, 𝐾 > 𝐾∗
Pecuniary externalityInefficiency𝑟 > 𝑟∗, 𝐾 < 𝐾∗
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Frictions
Incomplete markets• Within country
only w.r.t. idiosyncratic risk 𝑑 ෨𝑍𝑡𝑖
(other risks can be shared within national economy)
• Across countries Only global money can be traded
Money is a bubble Like in Samuelson, Bewley
Price are fully flexible
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International setting
Small Economy
• Local currency Store of value Hedge against idiosyn-
cratic risk
• Consumption basket Non-tradable local good 𝑎𝐾𝑡 tradable good 1 𝑏1,𝑡𝐾𝑡 tradable good 2
Large Economy*
• Global currency* $ … Hedge for SOE’s citizens against
“international competitive risk”
• Consumption basket* Non-tradable good* 𝑎∗𝐾𝑡
∗
tradable global good 1 𝑏1,𝑡∗𝐾𝑡
∗
tradable global good 2 𝑏2,𝑡∗𝐾𝑡
∗
Exchangerate
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Large Economy*
• Global currency* $ … Hedge for SOE’s citizens against
“international competitive risk”
• Consumption basket* Non-tradable good* 𝑎∗𝐾𝑡
∗
tradable global good 1 𝑏1,𝑡∗𝐾𝑡
∗
tradable global good 2 𝑏2,𝑡∗𝐾𝑡
∗
International setting
Small Economy
• Local currency Store of value Hedge against idiosyn-
cratic risk
• Consumption basket Non-tradable local good 𝑎𝐾𝑡 tradable good 1 𝑏1,𝑡𝐾𝑡 tradable good 2 𝑏2,𝑡𝐾𝑡 𝑏2,𝑡
𝑏1,𝑡<𝑏2,𝑡∗
𝑏1,𝑡∗
Exchangerate
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Intuition
Purchase good 2 in exchange of good 1 (depends on ToT)
Hold global money as Net Foreign Asset Position
Value of money – money is safe asset• Local money is store of value with nice hedge against
idiosyncratic risk
• Global money ($) hedges better “export risk” (competitiveness = ToT + productivity)
• 2 money can coexist (even though both are “bubbles”) Different return-risk profile
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Overview
Large country• Portfolio choice between
Physical capital 𝑘𝑡∗
US Dollar, $ – “bubble in positive net supply”
• No state variable: due to scale invariance
Small country• Portfolio choice between
Physical capital 𝑘𝑡 Peso “hedge against idiosyncratic shocks”
US Dollar, $ “hedge against ToT + export productivity shocks”
• State variable 𝜈𝑡:Accumulation dynamics of foreign asset position (in $)
𝜇𝑡𝑥 = 𝜇𝑥 𝜈𝑡 , 𝜎𝑡
𝑥 = 𝜎𝑥(𝜈𝑡)
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Large country
𝐸 න0
∞
𝑒−𝜌𝑡 log 𝑐𝑡∗ 𝑑𝑡
Consumption Cobb-Douglas preferences(over 1 non-tradable one 2 tradable goods)
𝑐0,𝑡∗ (1−𝛼)
(𝑐1,𝑡∗ )𝛽(𝑐2,𝑡
∗ )(1−𝛽)𝛼
Investment rate 𝜄∗ in terms of non-tradable local good
Evolution of physical capital stock𝑑𝑘𝑡
∗
𝑘𝑡∗ = Φ 𝜄∗ −𝛿 𝑑𝑡
Output shocks per unit of capital 𝑎∗, 𝑏1,𝑡∗ , or 𝑏2,𝑡
∗
• Determines relative prices• … has to be indifferent Ito-processes
Idiosyncratic real cash flow shocks 𝜎𝑘𝑡∗𝑑 ෨𝑍𝑡
𝑖∗
Net worth dynamics: 𝑑𝑛𝑡
∗
𝑛𝑡∗ = 𝜃𝑡
∗𝑟𝑀∗𝑑𝑡 + (1 − 𝜃𝑡)𝑟𝐾∗𝑑𝑡 −
𝑐𝑡∗
𝑛𝑡∗ 𝑑𝑡 + 𝜎
𝑘𝑡∗
𝑛𝑡∗ 𝑑 ෨𝑍𝑡
𝑖∗ +𝜏𝑡∗
𝑛𝑡∗ 𝑑𝑡
Value of output of all goods produced 𝑎∗𝐾𝑡∗ N
um
erai
re is
no
n-
trad
able
loca
l go
od
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Non-tradeable to consumption basket
Tradeable, non-tradeable good 1 and 2• 𝑎∗ units of non-tradeable good buy
𝑏1,𝑡∗ of tradeable good, or
𝑎∗ 1−𝛼 𝑏1,𝑡∗ 𝛼𝛽
𝑏2,𝑡∗ 𝛼(1−𝛽)
units of the “aggregate good” (consumption basket).
• Hence, production of consumption basket is
𝑎∗ − 𝜄𝑡∗ 𝑏𝑡
∗ 𝛼𝐾𝑡∗, with 𝑏𝑡
∗ =𝑏1,𝑡∗ 𝛽
𝑏2,𝑡∗ 1−𝛽
𝑎∗
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Return on global money ($)
In terms of non-tradable local good (as numeraire)(which is used for investment rate 𝜄𝑡
∗)
𝑟∗𝑑𝑡 = Φ 𝜄∗ − 𝛿 𝑑𝑡 is risk free
Change of numeraireIn terms of tradable basket (change of numeraire)
𝑟𝑡𝐺 ≡ 𝑟∗𝑑𝑡 + 𝑑𝑏𝑡
∗
𝑏𝑡∗
• Where price of non-tradable good in terms of tradable basket
𝑏𝑡∗ =
𝑏1,𝑡∗ 𝛽
𝑏2,𝑡∗ 1−𝛽
𝑎∗
• Special case: 𝑏1,𝑡∗ 𝛽
𝑏2,𝑡∗ 1−𝛽
and hence 𝑏𝑡∗ is constant
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Solving
1. Postulate• Price processes 𝑑𝑝𝑡
∗/𝑝𝑡∗ = 𝜇𝑡
𝑝∗𝑑𝑡 + 𝜎𝑝∗𝑑𝑍𝑡
∗, 𝑑𝑞𝑡∗/𝑞𝑡
∗= ⋯• Portfolio processes 𝑑𝜃𝑡
∗/𝜃𝑡∗
2. Derive return processes• 𝑑𝑟𝐾∗ = Φ 𝜄∗ − 𝛿 𝑑𝑡 + 𝑎∗−𝜄∗
𝑞𝑑𝑡 + 𝜎
𝑞𝑑 ෨𝑍𝑡
• 𝑑𝑟𝑀∗ = Φ 𝜄∗ − 𝛿 𝑑𝑡 − (𝜇𝑀∗−𝜇𝑀𝑖∗)𝑑𝑡
3. Optimality conditions & Market clearing conditions
4. Solve “undetermined coefficients” (𝜇𝑥 𝑠𝑡 , 𝜎𝑥(𝑠𝑡))
• Solving ODE with boundary conditions• For large country: simply solve for constants
money supply growth rate that is NOT distributed via interest payment
Set 𝜇𝑀𝑖∗ = 0
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Solving
1. Postulate• Price processes 𝑝𝑡
∗, 𝑞𝑡∗
• Portfolio processes 𝜃𝑡∗
2. Derive return processes• 𝑑𝑟𝐾∗ = Φ 𝜄∗ − 𝛿 𝑑𝑡 + 𝑎∗−𝜄∗
𝑞𝑑𝑡 + 𝜎
𝑞𝑑 ෨𝑍𝑡
• 𝑑𝑟𝑀∗ = Φ 𝜄∗ − 𝛿 𝑑𝑡 − (𝜇𝑀∗−𝜇𝑀𝑖∗)𝑑𝑡
3. Optimality conditions & Market clearing conditions
4. Solve “undetermined coefficients” (𝜇𝑥 𝑠𝑡 , 𝜎𝑥(𝑠𝑡))
• Solving ODE with boundary conditions• For large country: simply solve for constants
Simply constants for large country
money supply growth rate that is NOT distributed via interest payment
Set 𝜇𝑀𝑖∗ = 0
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Optimality (=) & market clearing (=)
Investment rate, 𝜄∗
• Tobin’s q: Φ′ 𝜄∗ =1
𝑞∗(static problem)
For Φ 𝜄∗ = 1
𝜅log(𝜅𝜄∗ + 1) ⇒ 𝜅𝜄∗ = 𝑞∗ − 1
Portfolio choice, 𝜃∗
• 𝐸 𝑑𝑟𝐾∗ − 𝑑𝑟𝑀∗ /𝑑𝑡 = 𝐶𝑜𝑣[𝑑𝑟𝐾∗ − 𝑑𝑟𝑀∗,ด𝑑𝑛𝑡
∗
𝑛𝑡∗
𝑑𝑟𝑀∗+(1−𝜃∗) 𝑑𝑟𝐾∗−𝑑𝑟𝑀∗
] = (1 − 𝜃∗)( 𝜎∗/𝑞)2
1 − 𝜃∗ =𝐸 𝑑𝑟𝐾∗−𝑑𝑟𝑀∗ /𝑑𝑡
(𝜎∗/𝑞∗)2= (𝑎∗−𝜄∗)/𝑞∗+𝜇𝑀∗
(𝜎∗/𝑞∗)2= 𝑞∗
𝑞∗+𝑝∗
• Dividend yield on capital must be 𝜌
Consumption• Demand 𝜌𝑁𝑡
∗ = 𝜌 𝑞∗ + 𝑝∗ 𝐾𝑡∗ = 𝑎∗ − 𝜄∗ 𝐾𝑡
∗ Supply
𝑞∗ =𝑞∗
𝑞∗ + 𝑝∗
=1−𝜃∗
(𝑎∗ − 𝜄∗)/𝜌
Output market clearing
Capital market clearing
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Equilibrium
where Ƹ𝜇𝑀∗ = 1 − 𝜃∗
𝑝𝑜𝑟𝑡𝑓𝑜𝑙𝑖𝑜 𝑠ℎ𝑎𝑟𝑒
𝜇𝑀∗ (monotone transformation)
Numeraire is local good
Moneyless equilibrium Money equilibrium
𝑝0∗ = 0 𝑝∗ = 𝜎∗(1+𝜅𝜌)
𝜌+ෝ𝜇𝑀∗−(1+𝜅𝑎∗)
𝑞0∗ = 𝜎∗
𝜌+ෝ𝜇𝑀∗𝑞∗ = 1+𝜅𝑎∗ −
𝜅𝜌𝜎∗
𝜌+ෝ𝜇𝑀∗>
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Optimal Monetary Policy
Money growth 𝜇 affects inflation in two ways
𝜋 = 𝜇𝑀∗ − (Φ 𝜄∗ 𝜇𝑀∗ − 𝜇𝑀,𝑖∗ − 𝛿)
𝑔
MoPo can correct pecuniary externality • Citizens take real interest rate as given
when choosing their portfolio between money & physical capital• Money exists for 𝜎∗ > 𝜌• Money growth > 0 is optimal for 𝜎∗ > 2 𝜌 (for 𝜅 = 0)
• MoPo improves insurance provided by “safe asset” Constrained optimal! Incentive compatible
Money is neither neutral nor super-neutral (no price stickiness)
boosts growth like in Tobin (1965)
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Overview
Large country• Portfolio choice between
Physical capital 𝑘𝑡∗
US Dollar, $ – “bubble in positive net supply”
• No state variable: due to scale invariance
Small country• Portfolio choice between
Physical capital 𝑘𝑡 Peso “hedge against idiosyncratic shocks”
US Dollar, $ “hedge against ToT + export productivity shocks”
• State variable 𝜈𝑡:Accumulation dynamics of foreign asset position (in $)
𝜇𝑡𝑥 = 𝜇𝑥 𝜈𝑡 , 𝜎𝑡
𝑥 = 𝜎𝑥(𝜈𝑡)
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Small country
Small country cannot produce tradable good 2
tradable basket can be traded for global good 1 at rate
𝑏𝑡 ≔ ด𝑏1,𝑡Productivity
𝑏2,𝑡∗ /𝑏1,𝑡
∗ 1−𝛽
Terms of Trade
=𝑏1,𝑡𝑏1,𝑡∗ ( 𝑏1,𝑡
∗ 𝛽𝑏2,𝑡∗ 1−𝛽
)
• Short-cut thinking:one unit of capital produces 𝑏𝑡 units of tradable basket(while actually it produces only good 1 at rate 𝑏1 and trades some of them for tradable good 2)
𝑑𝑏𝑡
𝑏𝑡= 𝜇𝑏𝑑𝑡 + 𝜎𝑏𝑑𝑍𝑡
since all 𝑏1,𝑡, 𝑏2,𝑡∗ , 𝑏1,𝑡
∗ are (correlated) geometric Brownian.
Return on global money can be written as
𝑑𝑟𝑡𝐺 = 𝜇𝐺𝑑𝑡 + 𝜎𝐺𝑑𝑍𝑡 + 𝜎𝐺,∗𝑑𝑍𝑡
∗
call prefer
Ito product rule: 𝑑 𝑋𝑡𝑌𝑡 = 𝑑𝑋𝑡𝑌𝑡 + 𝑋𝑡𝑑𝑌𝑡 + 𝜎𝑋𝜎𝑌𝑑𝑡
Part of 𝑏𝑡∗ which is orthogonal to 𝑏𝑡
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Small country
Same preferences:
• 𝐸 0∞𝑒−𝜌𝑡 log 𝑐𝑡 𝑑𝑡
• 𝑐0,𝑡(1−𝛼)
𝑐1,𝑡𝛽𝑐2,𝑡1−𝛽 𝛼
ො𝛼𝑡𝐾𝑡 devoted to produce tradable good 1• Can be traded for tradable basket since
small county can’t produce tradable good 2 itself
𝜉𝑡𝑏𝑡𝐾𝑡 consumption of tradable goods basket
( ො𝛼𝑡−𝜉𝑡)𝑏𝑡𝐾𝑡 trade-imbalance (net export)
𝐺𝑡 > 0 Net foreign asset position (only global money)
(in tradable goods basket)
𝑑𝐺𝑡𝐺𝑡
= 𝑑𝑟𝑡𝐺 + (ෝ𝛼𝑡−𝜉𝑡)𝑏𝑡𝐾𝑡
𝐺𝑡𝑑𝑡
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State variable
Equilibrium is a map
Histories of shocks prices 𝑞𝑡 , 𝑝𝑡, allocation𝑍𝜏, 𝑍𝜏
∗, 0 ≤ 𝜏 ≤ 𝑡 ො𝛼𝑡, 𝜄𝑡, 𝜉𝑡 & portfolio (1 − 𝜃𝑡 − 𝜁𝑡, 𝜃𝑡, 𝜁𝑡)
net foreign asset position to tradable production potential
𝜈𝑡 =𝐺𝑡
𝑏𝑡𝐾𝑡
Evolution
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Portfolio choice & Asset pricing
Portfolio share (processes)
• Local money𝑑𝜃𝑡𝜃𝑡
=𝜇𝑡𝜃𝑑𝑡+𝜎𝑡
𝜃𝑑𝑍𝑡+𝜎𝑡𝜃,∗𝑑𝑍𝑡
∗
• Global money𝑑𝜁𝑡
𝜁𝑡= 𝜇𝑡
𝜁𝑑𝑡 + 𝜎𝑡
𝜁𝑑𝑍𝑡 + 𝜎𝑡
𝜁,∗𝑑𝑍𝑡
∗
Returns expressed with country networth, 𝑁𝑡, as numeraire• Return on individual networth 𝑑𝑟𝑡
𝑛 = 𝜌𝑑𝑡 + (1 − 𝜃𝑡 − 𝜁𝑡)𝜎𝑛
𝜎(𝑞𝑡)
• Return on local money 𝑑𝑟𝑡𝑀𝐿 =
𝑑𝜃𝑡
𝜃𝑡
• Return on global money ($) 𝑑𝑟𝑡𝑀𝐺 =
ෝ𝛼𝑡−𝜉𝑡
𝜈𝑡𝑑𝑡 +
𝑑𝜁𝑡
𝜁𝑡
Asset pricing equation (due to log utility)
𝐸 𝑑𝑟𝑡𝑛 − 𝑑𝑟𝑡
𝑀𝐺 = 𝐶𝑜𝑣[𝑑𝑟𝑡𝑛 − 𝑑𝑟𝑡
𝑀𝐺 , 𝑑𝑟𝑡𝑛] ⇒ 𝜌 −
ෝ𝛼𝑡−𝜉𝑡
𝜈𝑡− 𝜇𝑡
𝜁= 𝜎𝑛 2
𝐸 𝑑𝑟𝑡𝑛 − 𝑑𝑟𝑡
𝑀𝐿 = 𝐶𝑜𝑣[𝑑𝑟𝑡𝑛 − 𝑑𝑟𝑡
𝑀𝐿 , 𝑑𝑟𝑡𝑛] ⇒ 𝜌 − 𝜇𝑡
𝜃 = 𝜎𝑛 2
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Portfolio choice & Asset pricing
Portfolio share (processes)
• Local money𝑑𝜃𝑡𝜃𝑡
=𝜇𝑡𝜃𝑑𝑡+𝜎𝑡
𝜃𝑑𝑍𝑡+𝜎𝑡𝜃,∗𝑑𝑍𝑡
∗
• Global money𝑑𝜁𝑡
𝜁𝑡= 𝜇𝑡
𝜁𝑑𝑡 + 𝜎𝑡
𝜁𝑑𝑍𝑡 + 𝜎𝑡
𝜁,∗𝑑𝑍𝑡
∗
Returns expressed with country net worth 𝑁𝑡 as numeraire• Return on individual net worth 𝑑𝑟𝑡
𝑛 = 𝜌𝑑𝑡 + (1 − 𝜃𝑡 − 𝜁𝑡)𝜎𝑛
𝜎(𝑞𝑡)
• Return on local money 𝑑𝑟𝑡𝑀𝐿 =
𝑑𝜃𝑡
𝜃𝑡
• Return on global money ($) 𝑑𝑟𝑡𝑀𝐺 =
𝜉𝑡−ෝ𝛼𝑡
𝜈𝑡𝑑𝑡 +
𝑑𝜁𝑡
𝜁𝑡
Asset pricing equation (due to log utility)
𝐸 𝑑𝑟𝑡𝑛 − 𝑑𝑟𝑡
𝑀𝐺 = 𝐶𝑜𝑣[𝑑𝑟𝑡𝑛 − 𝑑𝑟𝑡
𝑀𝐺 , 𝑑𝑟𝑡𝑛] ⇒ 𝜌 −
ෝ𝛼𝑡−𝜉𝑡
𝜈𝑡− 𝜇𝑡
𝜁= 𝜎𝑛 2
𝐸 𝑑𝑟𝑡𝑛 − 𝑑𝑟𝑡
𝑀𝐿 = 𝐶𝑜𝑣[𝑑𝑟𝑡𝑛 − 𝑑𝑟𝑡
𝑀𝐿 , 𝑑𝑟𝑡𝑛] ⇒ 𝜌 − 𝜇𝑡
𝜃 = 𝜎𝑛 2
Money worth 𝜃 net worths
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Portfolio choice & Asset pricing
Portfolio share (processes)• Local money
𝑑𝜃𝑡𝜃𝑡
=𝜇𝑡𝜃𝑑𝑡+𝜎𝑡
𝜃𝑑𝑍𝑡+𝜎𝑡𝜃,∗𝑑𝑍𝑡
∗
• Global money𝑑𝜁𝑡
𝜁𝑡= 𝜇𝑡
𝜁𝑑𝑡 + 𝜎𝑡
𝜁𝑑𝑍𝑡 + 𝜎𝑡
𝜁,∗𝑑𝑍𝑡
∗
Returns expressed with country net worth 𝑁𝑡 as numeraire• Return on individual
net worth 𝑑𝑟𝑡𝑛 = 𝜌𝑑𝑡 + 1 − 𝜃𝑡 − 𝜁𝑡
𝜎𝑛
𝜎 𝑞𝑡 𝑑 ෨𝑍𝑡
• Return on local money 𝑑𝑟𝑡𝑀𝐿 =
𝑑𝜃𝑡
𝜃𝑡
• Return on global money ($) 𝑑𝑟𝑡𝑀𝐺 =
𝜉𝑡−ෝ𝛼𝑡
𝜈𝑡𝑑𝑡 +
𝑑𝜁𝑡
𝜁𝑡
Asset pricing equation (due to log utility)
𝐸 𝑑𝑟𝑡𝑛 − 𝑑𝑟𝑡
𝑀𝐿 = 𝐶𝑜𝑣[𝑑𝑟𝑡𝑛 − 𝑑𝑟𝑡
𝑀𝐿 , 𝑑𝑟𝑡𝑛] ⇒ 𝜌 − 𝜇𝑡
𝜃 = 𝜎𝑛 2
𝐸 𝑑𝑟𝑡𝑛 − 𝑑𝑟𝑡
𝑀𝐺 = 𝐶𝑜𝑣[𝑑𝑟𝑡𝑛 − 𝑑𝑟𝑡
𝑀𝐺 , 𝑑𝑟𝑡𝑛] ⇒ 𝜌 −
𝜉𝑡−ෝ𝛼𝑡
𝜈𝑡− 𝜇𝑡
𝜁= 𝜎𝑛 2
E[net worth – money return]
Price of risk × risk
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Consumption & Investment
Consumption Demand 𝜌 ณ𝐺𝑡𝜁𝑡
wealthin trad.basket
= 𝜉𝑡𝑏𝑡𝐾𝑡𝛼
Supply
• Cobb-Douglas ⇒constant consumption expenditure shares
𝜉𝑡𝑏𝑡𝐾𝑡𝛼
𝑃𝑡𝑔= [ 1−ෝ𝛼𝑡 𝑎−𝜄𝑡]
1−𝛼𝑃𝑡𝑙
Production allocation
Investment rate 𝜄𝑡• Depends on 𝑞𝑡
Output of non-tradableConsumption of tradables
Non-tradable
Trade-able
(incl. net exports)
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(Co-)Existence of Money
Proposition 1:• If 𝜎2 > 𝜌 and M ≤ Φ 𝜄𝜈=0 , then
local money has value and 𝜈 = 0 (no NFAP) is absorbing state
• Otherwise, if 𝜎2 − 𝜌 +M−Φ 𝜄𝜈=0 > 0, then global money has value for citizens in small country(and local money may or may not have value)
𝜎2𝜌
M −Φ 𝜄 =
𝜇𝐺 − 𝜇𝑏 + 𝜎𝑏 − 𝜎𝐺 𝜎𝑏
−Φ 𝜄 + 𝛿attractiveness of globalmoney
ONLYLOCAL MONEYin the long run
GLOBAL MONEY+ possiblyLOCAL MONEY
attractiveness of local money
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Numerical Example
𝜌 = 5%, 𝜎 = .3, 𝛼 = .2, 𝜇𝑏 = 1%, 𝜎𝑏 = .15, 𝜇𝐺 = 2.2%, 𝑎 = .13,𝛿 = 2%, 𝜎𝐺 = 𝜎𝐺,∗ = 0, 𝜅 = 2 ⇒ 𝑀 = .0545, 𝜎𝜈 = .15
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Exchange rate dynamics - UIP
𝑖𝑡∗ − 𝑖𝑡 = 𝐸𝑡 Δℰ + 𝜓𝑡
𝜓𝑡 = ถ−𝜎𝜃
>0
(𝜎𝜁 − 𝜎𝜃)>0
(risk premium in terms of Peso)
For 𝑖𝑡 = 𝑖𝑡∗ (= 0)
• foreign currency is expected to appreciate relative to local currency (whenever it is held in positive quantity). Local currency is a hedge, it appreciates relative to net worth when 𝜈 drops. Global currency is risky, so to be held in positive amount it must earn a risk
premium
UIP violation, 𝜓𝑡, depends whether money is “printed”• to pay interest 𝜇𝑀𝑖
No real changes (portfolio choice is not affected) Higher inflation 𝜋 = 𝜇𝑀𝑖 − (𝜙 𝜄 − 𝛿), 𝐸𝑡 Δℰ = 𝜇𝑀𝑖 (dollar appreciates)
• to generate seignorage (redistributed ∝ wealth share) (𝜇𝑀−𝜇𝑀𝑖) Affects portfolio choice, 𝑞, investment rate 𝜄, growth rate
risk premium 𝜓𝑡
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Flight to safety (into dollar)
Unanticipated increase in 𝜎𝑏
• E.g. ToT becomes more volatile
Portfolio share held in dollars increases Dollar valuation is higher
(increase in volatility of 𝜈)
Transition• Start with current (dollar holding) 𝐺
• Recalculate new state variable 𝜈𝑡• Our full dynamics also includes transition dynamics
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Spillover from lower 𝜇𝐺
• Higher money supply growth 𝜇𝑀∗ in large country• Lower growth Φ 𝜄∗ − 𝛿 in large country• Loss of competitive edge in global tradable basket
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Higher Peso inflation 𝜋𝑡
seniorage 𝜇𝑀 − 𝜇𝑀𝑖 is distributed ∝ capital holding
Store of value is less attractive • Pricing equation now
𝜌 + 𝜋𝑡 − 𝜇𝜃 = 𝜎𝑁 2
higher investment 𝜄𝑡 ⇒ boosts growth, but higher idio risk
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Mundell-Fleming Trilemma
Trilemma: Can only pick a 2 desiderata out of 3 –1 side
“Dilemma”: Pick only 1
Autonomous Monetary Policy
Fixed ex-change rate
Free Capital Flow
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Mundell-Fleming Trilemma
Trilemma: Can only pick a 2 desiderata out of 3 –1 side
“Dilemma”: Pick only 1
Autonomous Monetary Policy
Fixed ex-change rate
Free Capital Flow
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Floating Exchange Rate
With floating exchange rate & open capital account Still range of Monetary policy, since local and global money are imperfect substitutes
• Inflation boosts growth, but only possible up to a limit ത𝜋(𝜇𝐺). Beyond ത𝜋(𝜇𝐺) monetary policy has little bite Global money becomes too attractive
• Range is higher with higher inflation in large country (global money) Large country’s MoPo determines
range for small country
Policy range is largerif local money is backed by taxes(ത𝜋 depends on distribution of seignorage)
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Closed capital account
Range of Monetary policy is much larger,up ധ𝜋 = 𝜎2 − 𝜌 ≈ 4%(physical capital is risky store of value)
Total money holding is larger with closed capital account• Global money would be a better
hedge for “export risk”
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Fixed exchange rate regimes & no MoPo
Dollarization = (fully backed) Currency Board Xx
Exchange rate peg• Requires strong fiscal backing
(since no backing through holding of global reserves) After a string of adverse shocks, government must tax and use to
proceeds to remove some of the local currency in circulation
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Foreign Currency Reserves
Irrelevance Theorem:• If central bank holds global money (reserves)
• Citizens in small country will hold accordingly less
Remark:• If central banks holds more $-reserves than citizens would like to
hold, then agents borrow foreign currency from abroad.
• If local money is worthless (without foreign reserves), then the value of local money with reserves only derives from the latter (currency board)
• With fiscal backing of the local money, complicates analysis
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Optimal Monetary Policy
LARGE COUNTRYMoPo can correct pecuniary externality • Citizens take real interest rate as given
when choosing their portfolio between money & physical capital• Money exists for 𝜎∗ > 𝜌• Inflation is optimal for 𝜎∗ > 2 𝜌
• MoPo improves insurance provided by “safe asset” Constrained optimal! Incentive compatible
SMALL COUNTRY• Additional savings decision due open capital account
Generally, optimal monetary policy depends on control social planner has
(For 𝜅 = 0, no adjustment costs)
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Conclusion
Endogenous value of money (safe asset) in 2 countries• Local currency: better hedge for idiosyncratic risk (non-tradable consumption)
• Global currency: hedge against ToT + export productivity shocks
Spillover effects from US monetary policy
Flight to safety
When to peg? When to dollarize?
“Nuanced Mundell-Fleming Trilemma”• Local and global money have different risk profile (imperfect substitutes)
⇒ increases MoPo space • Too high Peso inflation:
local citizens substitute local currency for global currency⇒ limits MoPo space
Central Bank’s foreign reserves holding: Irrelevance Result
Optimal Monetary Policy • Idiosyncratic risk – correct pecuniary externality (real interest rate)• International savings