International Monetary Theory: Mundell Fleming Redux...v International Monetary Theory: Mundell Fleming Redux by Markus K. Brunnermeier and Yuliy Sannikov Princeton and Stanford University

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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

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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