Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that...

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Eco 529: Brunnermeier & Sannikov Macro, Money and Finance Lecture 06: Money versus Debt Markus Brunnermeier, Lars Hansen, Yuliy Sannikov Princeton, Chicago, NYU, UPenn, Northwestern, EPFL, Stanford, Chicago Fed Spring 2019

Transcript of Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that...

Page 1: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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Macro, Money and FinanceLecture 06: Money versus Debt

Markus Brunnermeier, Lars Hansen, Yuliy Sannikov

Princeton, Chicago, NYU, UPenn, Northwestern, EPFL, Stanford, Chicago Fed Spring 2019

Page 2: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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Towards the I Theory of Money One sector model with idio risk - β€œThe I Theory without I”

(steady state focus) Store of value Insurance role of money within sector

Money as bubble or not Fiscal Theory of the Price Level Medium of Exchange Role β‡’ SDF-Liquidity multiplier β‡’ Money bubble

2 sector/type model with money and idio risk Generic Solution procedure (compared to lecture 03) Real debt vs. Money Implicit insurance role of money across sectors

The curse of insurance Reduces insurance premia and net worth gains

I Theory with Intermediary sector Intermediaries as diversifiers

Welfare analysis Optimal Monetary Policy and Macroprudential Policy

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Page 3: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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Two Sector Model w/ Outside Equity & Money

Expert sector Household sector

Experts must hold fraction πœ’πœ’π‘‘π‘‘ β‰₯ π›Όπ›Όπœ“πœ“π‘‘π‘‘ (skin in the game constraint)

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A L

Capitalπœ“πœ“π‘‘π‘‘π‘žπ‘žπ‘‘π‘‘πΎπΎπ‘‘π‘‘

Outside equity𝑁𝑁𝑑𝑑

Real Debt

A L

Capital1 βˆ’ πœ“πœ“π‘‘π‘‘ π‘žπ‘žπ‘‘π‘‘πΎπΎπ‘‘π‘‘

Equity

Net worth𝑁𝑁𝑑𝑑

Real DebtClaims

β‰₯ 𝛼𝛼

Expanded on Handbook of Macroeconomics 2017, Chapter 18- Includes now money and idiosyncratic risk

Money Money/Nominal Debt

Households hold portfolio of experts’ outside equity and diversify idio risk away

Outside money

Diversification

Money/Nominal Debt

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Two Sector Model Setup

Expert sectorOutput: 𝑦𝑦𝑑𝑑 = π‘Žπ‘Žπ‘˜π‘˜π‘‘π‘‘ Consumption rate: 𝑐𝑐𝑑𝑑 Investment rate: πœ„πœ„π‘‘π‘‘π‘‘π‘‘π‘˜π‘˜π‘‘π‘‘

��𝚀

π‘˜π‘˜π‘‘π‘‘οΏ½οΏ½πš€

= Ξ¦ πœ„πœ„π‘‘π‘‘ βˆ’π›Ώπ›Ώ 𝑑𝑑𝑑𝑑+πœŽπœŽπ‘‘π‘‘π‘π‘π‘‘π‘‘+οΏ½πœŽπœŽπ‘‘π‘‘ οΏ½π‘π‘π‘‘π‘‘οΏ½οΏ½πš€+𝑑𝑑Δ𝑑𝑑

π‘˜π‘˜,��𝚀

𝐸𝐸0[∫0∞ π‘’π‘’βˆ’πœŒπœŒπ‘‘π‘‘π‘π‘π‘‘π‘‘

1βˆ’π›Ύπ›Ύ

1βˆ’π›Ύπ›Ύ 𝑑𝑑𝑑𝑑]

Friction: Can only issue Risk-free debt Equity, but most hold πœ’πœ’π‘‘π‘‘ β‰₯ 𝛼𝛼 4

Household sectorOutput: 𝑦𝑦𝑑𝑑 = π‘Žπ‘Žπ‘˜π‘˜π‘‘π‘‘ Consumption rate: 𝑐𝑐𝑑𝑑 Investment rate: πœ„πœ„π‘‘π‘‘π‘‘π‘‘π‘˜π‘˜π‘‘π‘‘

��𝚀

π‘˜π‘˜π‘‘π‘‘οΏ½οΏ½πš€

= Ξ¦ πœ„πœ„π‘‘π‘‘ βˆ’π›Ώπ›Ώ 𝑑𝑑𝑑𝑑+πœŽπœŽπ‘‘π‘‘π‘π‘π‘‘π‘‘+οΏ½πœŽπœŽπ‘‘π‘‘ οΏ½π‘π‘π‘‘π‘‘οΏ½οΏ½πš€+𝑑𝑑Δ𝑑𝑑

π‘˜π‘˜,��𝚀

𝐸𝐸0[∫0βˆžπ‘’π‘’βˆ’πœŒπœŒπ‘‘π‘‘π‘π‘π‘‘π‘‘

1βˆ’π›Ύπ›Ύ

1βˆ’π›Ύπ›Ύ 𝑑𝑑𝑑𝑑]

π‘Žπ‘Ž β‰₯ π‘Žπ‘Ž

𝜌𝜌 β‰₯ 𝜌𝜌

𝛿𝛿 ≀ π›Ώπ›ΏοΏ½πœŽπœŽ ≀ �𝜎𝜎

Page 5: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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Solving MacroModels Step-by-Step0. Postulate aggregates, price processes & obtain return processes1. For given SDF processes static

a. Real investment πœ„πœ„, (portfolio 𝜽𝜽, & consumption choice of each agent) Toolbox 1: Martingale Approach

b. Asset/Risk Allocation across types/sectors & asset market clearing Toolbox 2: β€œprice-taking social planner approach” – Fisher separation theorem

2. Value functions backward equationa. Value fcn. as fcn. of individual investment opportunities πœ”πœ”

Special casesb. De-scaled value fcn. as function of state variables πœ‚πœ‚

Digression: HJB-approach (instead of martingale approach & envelop condition)

c. Derive 𝜍𝜍 price of risk, 𝐢𝐢/𝑁𝑁-ratio from value fcn. envelop condition

3. Evolution of state variable πœ‚πœ‚ forward equation Toolbox 3: Change in numeraire to total wealth (including SDF) β€œMoney evaluation equation” πœ‡πœ‡πœ—πœ—

4. Value function iteration & goods market clearinga. PDE of de-scaled value fcn.b. Value function iteration by solving PDE 5

Page 6: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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0. Postulate Aggregates Individual capital evolution:

π‘‘π‘‘π‘˜π‘˜π‘‘π‘‘π‘–π‘–,��𝚀

π‘˜π‘˜π‘‘π‘‘π‘–π‘–,��𝚀 = Ξ¦ πœ„πœ„π‘–π‘–,��𝚀 βˆ’ 𝛿𝛿 𝑑𝑑𝑑𝑑 + πœŽπœŽπ‘‘π‘‘π‘π‘π‘‘π‘‘ + οΏ½πœŽπœŽπ‘–π‘–π‘‘π‘‘ �𝑍𝑍𝑑𝑑

𝑖𝑖,��𝚀 + π‘‘π‘‘Ξ”π‘‘π‘‘π‘˜π‘˜,𝑖𝑖,��𝚀

Where Ξ”π‘‘π‘‘π‘˜π‘˜,��𝚀,𝑖𝑖 is the individual cumulative capital purchase process

Capital aggregation: Within sector 𝑖𝑖: 𝐾𝐾𝑑𝑑𝑖𝑖 ≑ ∫ π‘˜π‘˜π‘‘π‘‘

𝑖𝑖,οΏ½οΏ½πš€π‘‘π‘‘ 𝚀𝚀 Across sectors: 𝐾𝐾𝑑𝑑 ≑ βˆ‘π‘–π‘– 𝐾𝐾𝑑𝑑𝑖𝑖 Capital share: πœ“πœ“π‘‘π‘‘π‘–π‘– ≑ 𝐾𝐾𝑑𝑑𝑖𝑖/𝐾𝐾𝑑𝑑

𝑑𝑑𝐾𝐾𝑑𝑑𝐾𝐾𝑑𝑑

= ∫ Ξ¦ πœ„πœ„π‘–π‘– βˆ’ 𝛿𝛿 𝑑𝑑𝑖𝑖 𝑑𝑑𝑑𝑑 + πœŽπœŽπ‘‘π‘‘π‘π‘π‘‘π‘‘ Networth aggregation: Within sector 𝑖𝑖: 𝑁𝑁𝑑𝑑𝑖𝑖 ≑ ∫ 𝑛𝑛𝑑𝑑

𝑖𝑖,οΏ½οΏ½πš€π‘‘π‘‘ 𝚀𝚀 Across sectors: 𝑁𝑁𝑑𝑑 ≑ βˆ‘π‘–π‘– 𝑁𝑁𝑑𝑑𝑖𝑖 Wealth share: πœ‚πœ‚π‘‘π‘‘π‘–π‘– ≑ 𝑁𝑁𝑑𝑑𝑖𝑖/𝑁𝑁𝑑𝑑

Value of capital: π‘žπ‘žπ‘‘π‘‘πΎπΎπ‘‘π‘‘ Value of money: 𝑝𝑝𝑑𝑑𝐾𝐾𝑑𝑑

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Page 7: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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0. Postulate Processes

Value of capital: π‘žπ‘žπ‘‘π‘‘πΎπΎπ‘‘π‘‘ Value of money: 𝑝𝑝𝑑𝑑𝐾𝐾𝑑𝑑 Postulate

π‘‘π‘‘π‘žπ‘žπ‘‘π‘‘/π‘žπ‘žπ‘‘π‘‘ = πœ‡πœ‡π‘‘π‘‘π‘žπ‘žπ‘‘π‘‘π‘‘π‘‘ + πœŽπœŽπ‘‘π‘‘

π‘žπ‘žπ‘‘π‘‘π‘π‘π‘‘π‘‘π‘‘π‘‘π‘π‘π‘‘π‘‘/𝑝𝑝𝑑𝑑 = πœ‡πœ‡π‘‘π‘‘

𝑝𝑝𝑑𝑑𝑑𝑑 + πœŽπœŽπ‘‘π‘‘π‘π‘π‘‘π‘‘π‘π‘π‘‘π‘‘

π‘‘π‘‘πœ‰πœ‰π‘‘π‘‘π‘–π‘–/πœ‰πœ‰π‘‘π‘‘π‘–π‘– = οΏ½πœ‡πœ‡π‘‘π‘‘πœ‰πœ‰ π‘‘π‘‘π‘‘π‘‘β‰‘βˆ’π‘Ÿπ‘Ÿπ‘‘π‘‘

+ οΏ½πœŽπœŽπ‘‘π‘‘πœ‰πœ‰π‘–π‘–

β‰‘βˆ’πœπœπ‘‘π‘‘π‘–π‘–

𝑑𝑑𝑍𝑍𝑑𝑑 + οΏ½οΏ½πœŽπœŽπœ‰πœ‰π‘–π‘–

β‰‘βˆ’οΏ½πœπœπ‘‘π‘‘π‘–π‘–π‘‘π‘‘ οΏ½π‘π‘π‘‘π‘‘οΏ½οΏ½πš€

Derive return processes

π‘‘π‘‘π‘Ÿπ‘Ÿπ‘‘π‘‘πΎπΎ,𝑖𝑖,��𝚀 =

π‘Žπ‘Žπ‘–π‘– βˆ’ πœ„πœ„π‘‘π‘‘οΏ½οΏ½πš€

π‘žπ‘žπ‘‘π‘‘+ Ξ¦ πœ„πœ„π‘‘π‘‘οΏ½οΏ½πš€ βˆ’ 𝛿𝛿 + πœ‡πœ‡π‘‘π‘‘

π‘žπ‘ž + πœŽπœŽπœŽπœŽπ‘‘π‘‘π‘žπ‘ž 𝑑𝑑𝑑𝑑 + 𝜎𝜎 + πœŽπœŽπ‘‘π‘‘

π‘žπ‘ž 𝑑𝑑𝑍𝑍𝑑𝑑 + οΏ½πœŽπœŽπ‘–π‘–π‘‘π‘‘ οΏ½π‘π‘π‘‘π‘‘οΏ½οΏ½πš€

π‘‘π‘‘π‘Ÿπ‘Ÿπ‘‘π‘‘π‘€π‘€ = Ξ¦ πœ„πœ„π‘‘π‘‘ βˆ’ 𝛿𝛿 + πœ‡πœ‡π‘‘π‘‘π‘π‘ + πœŽπœŽπœŽπœŽπ‘‘π‘‘

𝑝𝑝 βˆ’ πœ‡πœ‡π‘€π‘€ 𝑑𝑑𝑑𝑑 + 𝜎𝜎 + πœŽπœŽπ‘‘π‘‘π‘π‘ 𝑑𝑑𝑍𝑍𝑑𝑑

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Solving MacroModels Step-by-Step0. Postulate aggregates, price processes & obtain return processes1. For given SDF processes static

a. Real investment πœ„πœ„, (portfolio 𝜽𝜽, & consumption choice of each agent) Toolbox 1: Martingale Approach

b. Asset/Risk Allocation across types/sectors & asset market clearing Toolbox 2: β€œprice-taking social planner approach” – Fisher separation theorem

2. Value functions backward equationa. Value fcn. as fcn. of individual investment opportunities πœ”πœ”

Special casesb. De-scaled value fcn. as function of state variables πœ‚πœ‚

Digression: HJB-approach (instead of martingale approach & envelop condition)

c. Derive 𝜍𝜍 price of risk, 𝐢𝐢/𝑁𝑁-ratio from value fcn. envelop condition

3. Evolution of state variable πœ‚πœ‚ forward equation Toolbox 3: Change in numeraire to total wealth (including SDF) β€œMoney evaluation equation” πœ‡πœ‡πœ—πœ—

4. Value function iteration & goods market clearinga. PDE of de-scaled value fcn.b. Value function iteration by solving PDE 8

Page 9: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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1a. Agent Choice of πœ„πœ„, πœƒπœƒ, 𝑐𝑐 Portfolio Choice: Martingale Approach Let π‘₯π‘₯𝑑𝑑𝐴𝐴 be the value of a β€œself-financing trading strategy”

(reinvest dividends) Theorem: πœ‰πœ‰π‘‘π‘‘π‘₯π‘₯𝑑𝑑𝐴𝐴 follows a Martingale, i.e. drift = 0.

Let 𝑑𝑑𝑑𝑑𝑑𝑑𝐴𝐴

𝑑𝑑𝑑𝑑𝐴𝐴= πœ‡πœ‡π‘‘π‘‘π΄π΄π‘‘π‘‘π‘‘π‘‘ + πœŽπœŽπ‘‘π‘‘π΄π΄π‘‘π‘‘π‘π‘π‘‘π‘‘ + οΏ½πˆπˆπ‘‘π‘‘π΄π΄π‘‘π‘‘οΏ½π’π’π‘‘π‘‘,

Recall π‘‘π‘‘πœ‰πœ‰π‘‘π‘‘π‘–π‘–

πœ‰πœ‰π‘‘π‘‘π‘–π‘– = βˆ’π‘Ÿπ‘Ÿπ‘‘π‘‘π‘‘π‘‘π‘‘π‘‘ βˆ’ πœπœπ‘‘π‘‘π‘–π‘–π‘‘π‘‘π‘π‘π‘‘π‘‘ βˆ’ �𝝇𝝇𝑑𝑑𝑖𝑖𝑑𝑑�𝒁𝒁𝑑𝑑𝑖𝑖

By Ito product rule𝑑𝑑 πœ‰πœ‰π‘‘π‘‘

𝑖𝑖𝑑𝑑𝑑𝑑𝐴𝐴

πœ‰πœ‰π‘‘π‘‘π‘–π‘–π‘‘π‘‘π‘‘π‘‘π΄π΄

= βˆ’π‘Ÿπ‘Ÿπ‘‘π‘‘ + πœ‡πœ‡π‘‘π‘‘π΄π΄ βˆ’ πœπœπ‘‘π‘‘π‘–π‘–πœŽπœŽπ‘‘π‘‘π΄π΄ βˆ’ �𝝇𝝇𝑑𝑑𝑖𝑖 οΏ½πˆπˆπ‘‘π‘‘π΄π΄ 𝑑𝑑𝑑𝑑=0

+volatility terms

Expected return: πœ‡πœ‡π‘‘π‘‘π΄π΄ = π‘Ÿπ‘Ÿπ‘‘π‘‘ + πœπœπ‘‘π‘‘π‘–π‘–πœŽπœŽπ‘‘π‘‘π΄π΄ + �𝝇𝝇𝑑𝑑𝑖𝑖 οΏ½πˆπˆπ‘‘π‘‘π΄π΄

For risk-free asset, i.e. πœŽπœŽπ‘‘π‘‘π΄π΄ = οΏ½πˆπˆπ‘‘π‘‘π΄π΄ = 0:π‘Ÿπ‘Ÿπ‘‘π‘‘π‘“π‘“ = π‘Ÿπ‘Ÿπ‘‘π‘‘

Excess expected return to risky asset B: πœ‡πœ‡π‘‘π‘‘π΄π΄ βˆ’ πœ‡πœ‡π‘‘π‘‘π΅π΅ = πœπœπ‘‘π‘‘π‘–π‘–(πœŽπœŽπ‘‘π‘‘π΄π΄ βˆ’ πœŽπœŽπ‘‘π‘‘π΅π΅) + �𝝇𝝇𝑑𝑑𝑖𝑖 (οΏ½πˆπˆπ‘‘π‘‘π΄π΄ βˆ’ οΏ½πˆπˆπ‘‘π‘‘π΅π΅)

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Page 10: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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Solving MacroModels Step-by-Step0. Postulate aggregates, price processes & obtain return processes1. For given SDF processes static

a. Real investment πœ„πœ„, (portfolio 𝜽𝜽, & consumption choice of each agent) Toolbox 1: Martingale Approach

b. Asset/Risk Allocation across types/sectors & asset market clearing Toolbox 2: β€œprice-taking social planner approach” – Fisher separation theorem

2. Value functions backward equationa. Value fcn. as fcn. of individual investment opportunities πœ”πœ”

Special casesb. De-scaled value fcn. as function of state variables πœ‚πœ‚

Digression: HJB-approach (instead of martingale approach & envelop condition)

c. Derive 𝜍𝜍 price of risk, 𝐢𝐢/𝑁𝑁-ratio from value fcn. envelop condition

3. Evolution of state variable πœ‚πœ‚ forward equation Toolbox 3: Change in numeraire to total wealth (including SDF) β€œMoney evaluation equation” πœ‡πœ‡πœ—πœ—

4. Value function iteration & goods market clearinga. PDE of de-scaled value fcn.b. Value function iteration by solving PDE 10

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1b. Asset/Risk Allocation across Types Price-Taking Planner’s Theorem:

A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍𝑑𝑑, and risk allocation πŒπŒπ‘‘π‘‘,οΏ½πŒπŒπ‘‘π‘‘, that coincides with the choices implied by all individuals’ portfolio choices.

Planner’s problemmax

{𝝍𝝍𝑑𝑑,πŒπŒπ‘‘π‘‘,οΏ½πŒπŒπ‘‘π‘‘}𝐸𝐸𝑑𝑑[π‘‘π‘‘π‘Ÿπ‘Ÿπ‘‘π‘‘

�𝑁𝑁 πœ“πœ“π‘‘π‘‘ ] βˆ’ π‡π‡π‘‘π‘‘πœŽπœŽ 𝝍𝝍𝑑𝑑,πŒπŒπ‘‘π‘‘ βˆ’ �𝝇𝝇𝑑𝑑 �𝜎𝜎(𝝍𝝍𝒕𝒕, οΏ½πŒπŒπ‘‘π‘‘)

subject to friction: 𝐹𝐹 𝝍𝝍𝑑𝑑,πŒπŒπ‘‘π‘‘, οΏ½πŒπŒπ‘‘π‘‘ ≀ 0 Example:

1. πœ’πœ’π‘‘π‘‘ = πœ“πœ“π‘‘π‘‘ (if one holds capital, one has to hold risk)2. πœ’πœ’π‘‘π‘‘ β‰₯ π›Όπ›Όπœ“πœ“π‘‘π‘‘ (skin in the game constraint, outside equity up to a limit)

11

= π‘‘π‘‘π‘Ÿπ‘ŸπΉπΉ in equilibrium

𝝇𝝇𝑑𝑑 = πœπœπ‘‘π‘‘1, … , πœπœπ‘‘π‘‘πΌπΌπŒπŒπ’•π’• = πœ’πœ’π‘‘π‘‘1, … ,πœ’πœ’π‘‘π‘‘πΌπΌ

𝝈𝝈 𝝍𝝍𝑑𝑑,πŒπŒπ‘‘π‘‘ = πŒπŒπ‘‘π‘‘1πœŽπœŽοΏ½π‘π‘(πœ“πœ“π‘‘π‘‘), … ,πŒπŒπ‘‘π‘‘πΌπΌπœŽπœŽ

�𝑁𝑁(πœ“πœ“π‘‘π‘‘)�𝝈𝝈 𝝍𝝍𝑑𝑑,πŒπŒπ‘‘π‘‘ = οΏ½πœŽπœŽπ‘›π‘›1(𝝍𝝍𝑑𝑑, οΏ½πŒπŒπ‘‘π‘‘), … , οΏ½πœŽπœŽπ‘›π‘›πΌπΌ(𝝍𝝍𝒕𝒕, οΏ½πŒπŒπ‘‘π‘‘)

Note: By holding a portfolio of various experts’ outside equity HH can diversify idio risk away

Return on total wealth (including money)

Page 12: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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1b. Allocation of Capital, πœ“πœ“, and Risk, πœ’πœ’

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Cases πœ’πœ’π‘‘π‘‘ β‰₯ π›Όπ›Όπœ“πœ“π‘‘π‘‘ πœ“πœ“π‘‘π‘‘ ≀ 1 π‘Žπ‘Ž βˆ’ π‘Žπ‘Žπ‘žπ‘žπ‘‘π‘‘

β‰₯ 𝛼𝛼 πœπœπ‘‘π‘‘ βˆ’ πœπœπ‘‘π‘‘ 𝜎𝜎 + πœŽπœŽπ‘‘π‘‘π‘žπ‘ž

+ 𝛼𝛼 πœπœπ‘‘π‘‘ βˆ’ πœπœπ‘‘π‘‘ �𝜎𝜎

πœπœπ‘‘π‘‘ 𝜎𝜎 + πœŽπœŽπ‘‘π‘‘π‘žπ‘ž + πœπœπ‘‘π‘‘ �𝜎𝜎

> πœπœπ‘‘π‘‘ 𝜎𝜎 + πœŽπœŽπ‘‘π‘‘π‘žπ‘ž

1a = < = >

1b = = > >

2a > = > =

Impossible

Note that HH, which hold a portfolio of different experts’ outside equitycan diversify idiosyncratic risk away

If you shift one capital unit from HH to experts- Dividend yield rises by LHS, - Change the aggregate required

risk premium (alpha fraction dueto skin of the game constraint)

- HH reduce their risk by one unit, sell back 1 βˆ’ 𝛼𝛼 and diversified away

Page 13: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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1b. Allocation of Capital, πœ“πœ“, and Risk, πœ’πœ’

)

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Cases πœ’πœ’π‘‘π‘‘ β‰₯ π›Όπ›Όπœ“πœ“π‘‘π‘‘ πœ“πœ“π‘‘π‘‘ ≀ 1 π‘Žπ‘Ž βˆ’ π‘Žπ‘Žπ‘žπ‘žπ‘‘π‘‘

β‰₯ 𝛼𝛼 πœπœπ‘‘π‘‘ βˆ’ πœπœπ‘‘π‘‘ 𝜎𝜎 + πœŽπœŽπ‘‘π‘‘π‘žπ‘ž

+ 𝛼𝛼 πœπœπ‘‘π‘‘ βˆ’ πœπœπ‘‘π‘‘ �𝜎𝜎

πœπœπ‘‘π‘‘ 𝜎𝜎 + πœŽπœŽπ‘‘π‘‘π‘žπ‘ž + πœπœπ‘‘π‘‘ �𝜎𝜎

> πœπœπ‘‘π‘‘ 𝜎𝜎 + πœŽπœŽπ‘‘π‘‘π‘žπ‘ž

1a = < = >

1b = = > >

2a > = > =

Impossible

Case 1a Case 1b Case 2aπœ‚πœ‚

Experts’ skin in the game constraint binds, πœ’πœ’π‘‘π‘‘ = π›Όπ›Όπœ“πœ“π‘‘π‘‘

HHs’ short-sale constraint of capital binds, πœ“πœ“π‘‘π‘‘ = 1

Page 14: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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Solving MacroModels Step-by-Step0. Postulate aggregates, price processes & obtain return processes1. For given SDF processes static

a. Real investment πœ„πœ„, (portfolio 𝜽𝜽, & consumption choice of each agent) Toolbox 1: Martingale Approach

b. Asset/Risk Allocation across types/sectors & asset market clearing Toolbox 2: β€œprice-taking social planner approach” – Fisher separation theorem

2. Value functions backward equationa. Value fcn. as fcn. of individual investment opportunities πœ”πœ”

Special casesb. De-scaled value fcn. as function of state variables πœ‚πœ‚

Digression: HJB-approach (instead of martingale approach & envelop condition)

c. Derive 𝜍𝜍 price of risk, 𝐢𝐢/𝑁𝑁-ratio from value fcn. envelop condition

3. Evolution of state variable πœ‚πœ‚ forward equation Toolbox 3: Change in numeraire to total wealth (including SDF) β€œMoney evaluation equation” πœ‡πœ‡πœ—πœ—

4. Value function iteration & goods market clearinga. PDE of de-scaled value fcn.b. Value function iteration by solving PDE 14

Page 15: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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2b. CRRA Value Fcn: Isolating Idio. Risk

Rephrase the conjecture value function as

𝑉𝑉𝑑𝑑 =1πœŒπœŒπœ”πœ”π‘‘π‘‘π‘›π‘›π‘‘π‘‘ 1βˆ’π›Ύπ›Ύ

1 βˆ’ 𝛾𝛾=

1𝜌𝜌(1 βˆ’ 𝛾𝛾)

πœ”πœ”π‘‘π‘‘π‘π‘π‘‘π‘‘πΎπΎπ‘‘π‘‘

1βˆ’π›Ύπ›Ύ

=:𝑣𝑣𝑑𝑑

𝑛𝑛𝑑𝑑𝑁𝑁𝑑𝑑

1βˆ’π›Ύπ›Ύ

=: οΏ½πœ‚πœ‚π‘‘π‘‘οΏ½οΏ½πš€ 1βˆ’π›Ύπ›Ύ

𝐾𝐾𝑑𝑑1βˆ’π›Ύπ›Ύ

𝑣𝑣𝑑𝑑 depends only on aggregate state πœ‚πœ‚π‘‘π‘‘ Ito’s quotation rule𝑑𝑑 οΏ½πœ‚πœ‚π‘‘π‘‘οΏ½οΏ½πš€

οΏ½πœ‚πœ‚π‘‘π‘‘οΏ½οΏ½πš€=𝑑𝑑 𝑛𝑛𝑑𝑑/𝑁𝑁𝑑𝑑𝑛𝑛𝑑𝑑/𝑁𝑁𝑑𝑑

= πœ‡πœ‡π‘‘π‘‘π‘›π‘› βˆ’ πœ‡πœ‡π‘‘π‘‘π‘π‘ + πœŽπœŽπ‘‘π‘‘π‘π‘ 2 βˆ’ πœŽπœŽπ‘π‘πœŽπœŽπ‘›π‘› 𝑑𝑑𝑑𝑑 + πœŽπœŽπ‘‘π‘‘π‘›π‘› βˆ’ πœŽπœŽπ‘‘π‘‘π‘π‘ 𝑑𝑑𝑍𝑍𝑑𝑑 + οΏ½πœŽπœŽπ‘›π‘›π‘‘π‘‘ οΏ½π‘π‘π‘‘π‘‘οΏ½οΏ½πš€ = οΏ½πœŽπœŽπ‘›π‘›π‘‘π‘‘ οΏ½π‘π‘π‘‘π‘‘οΏ½οΏ½πš€

Ito’s Lemma𝑑𝑑 οΏ½πœ‚πœ‚π‘‘π‘‘οΏ½οΏ½πš€

1βˆ’π›Ύπ›Ύ

οΏ½πœ‚πœ‚π‘‘π‘‘οΏ½οΏ½πš€1βˆ’π›Ύπ›Ύ = βˆ’

12𝛾𝛾 1 βˆ’ 𝛾𝛾 οΏ½πœŽπœŽπ‘›π‘› 2𝑑𝑑𝑑𝑑 + 1 βˆ’ 𝛾𝛾 οΏ½πœŽπœŽπ‘›π‘›π‘‘π‘‘ οΏ½π‘π‘π‘‘π‘‘οΏ½οΏ½πš€

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Page 16: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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2b. CRRA Value Function

𝑑𝑑𝑉𝑉𝑑𝑑𝑉𝑉𝑑𝑑

=𝑑𝑑 𝑣𝑣𝑑𝑑 οΏ½πœ‚πœ‚π‘‘π‘‘οΏ½οΏ½πš€

1βˆ’π›Ύπ›ΎπΎπΎπ‘‘π‘‘1βˆ’π›Ύπ›Ύ

𝑣𝑣𝑑𝑑 οΏ½πœ‚πœ‚π‘‘π‘‘οΏ½οΏ½πš€1βˆ’π›Ύπ›ΎπΎπΎπ‘‘π‘‘

1βˆ’π›Ύπ›Ύ

By Ito’s product rule

= πœ‡πœ‡π‘‘π‘‘π‘£π‘£ + 1 βˆ’ 𝛾𝛾 Ξ¦ πœ„πœ„ βˆ’ 𝛿𝛿 βˆ’12𝛾𝛾 1 βˆ’ 𝛾𝛾 𝜎𝜎2 + οΏ½πœŽπœŽπ‘›π‘› 2 + 1 βˆ’ 𝛾𝛾 πœŽπœŽπœŽπœŽπ‘‘π‘‘π‘£π‘£ 𝑑𝑑𝑑𝑑

+ π‘£π‘£π‘£π‘£π‘£π‘£π‘Žπ‘Žπ‘‘π‘‘π‘–π‘–π‘£π‘£π‘–π‘–π‘‘π‘‘π‘¦π‘¦ π‘‘π‘‘π‘’π‘’π‘Ÿπ‘Ÿπ‘‘π‘‘π‘‘π‘‘

Recall by consumption optimality π‘‘π‘‘π‘‰π‘‰π‘‘π‘‘π‘‰π‘‰π‘‘π‘‘βˆ’ πœŒπœŒπ‘‘π‘‘π‘‘π‘‘ + 𝑐𝑐𝑑𝑑

𝑛𝑛𝑑𝑑𝑑𝑑𝑑𝑑 follows a martingale

Hence, drift above = 𝜌𝜌 βˆ’ 𝑐𝑐𝑑𝑑𝑛𝑛𝑑𝑑

16Still have to solve for πœ‡πœ‡π‘‘π‘‘π‘£π‘£, πœŽπœŽπ‘‘π‘‘π‘£π‘£

Poll 16: Why martingale?a) Because we can β€œprice”

networth with SDFb) because 𝜌𝜌 and 𝑐𝑐𝑑𝑑/𝑛𝑛𝑑𝑑

cancel out

Page 17: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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2b. CRRA Value Fcn BSDE Only conceptual interim solution

We will transform it into a PDE in Step 4 below From last slideπœ‡πœ‡π‘‘π‘‘π‘£π‘£ + 1 βˆ’ 𝛾𝛾 Ξ¦ πœ„πœ„ βˆ’ 𝛿𝛿 βˆ’

12𝛾𝛾 1 βˆ’ 𝛾𝛾 𝜎𝜎2 + οΏ½πœŽπœŽπ‘›π‘› 2 + 1 βˆ’ 𝛾𝛾 πœŽπœŽπœŽπœŽπ‘‘π‘‘π‘£π‘£

=:πœ‡πœ‡π‘‘π‘‘π‘‰π‘‰

= 𝜌𝜌 βˆ’π‘π‘π‘‘π‘‘π‘›π‘›π‘‘π‘‘

Can solve for πœ‡πœ‡π‘‘π‘‘π‘£π‘£, then 𝑣𝑣𝑑𝑑 must follow𝑑𝑑𝑣𝑣𝑑𝑑𝑣𝑣𝑑𝑑

= 𝑓𝑓 πœ‚πœ‚π‘‘π‘‘ , 𝑣𝑣𝑑𝑑,πœŽπœŽπ‘‘π‘‘π‘£π‘£ 𝑑𝑑𝑑𝑑 + πœŽπœŽπ‘‘π‘‘π‘£π‘£π‘‘π‘‘π‘π‘π‘‘π‘‘with

𝑓𝑓 πœ‚πœ‚π‘‘π‘‘ , 𝑣𝑣𝑑𝑑,πœŽπœŽπ‘‘π‘‘π‘£π‘£ = 𝜌𝜌 βˆ’π‘π‘π‘‘π‘‘π‘›π‘›π‘‘π‘‘βˆ’ 1 βˆ’ 𝛾𝛾 Ξ¦ πœ„πœ„ βˆ’ 𝛿𝛿 +

12𝛾𝛾 1 βˆ’ 𝛾𝛾 𝜎𝜎2 + οΏ½πœŽπœŽπ‘›π‘› 2 βˆ’ 1 βˆ’ 𝛾𝛾 πœŽπœŽπœŽπœŽπ‘‘π‘‘π‘£π‘£

Together with terminal condition 𝑣𝑣𝑇𝑇 (possibly a constant for 1000 periods ahead), this is a backward stochastic differential equation (BSDE)

A solution consists of processes 𝑣𝑣 and πœŽπœŽπ‘£π‘£

Can use numerical BSDE solution methods (as random objects, so only get simulated paths)

To solve this via a PDE we also need to get state evolution

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Page 18: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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Solving MacroModels Step-by-Step0. Postulate aggregates, price processes & obtain return processes1. For given SDF processes static

a. Real investment πœ„πœ„, (portfolio 𝜽𝜽, & consumption choice of each agent) Toolbox 1: Martingale Approach

b. Asset/Risk Allocation across types/sectors & asset market clearing Toolbox 2: β€œprice-taking social planner approach” – Fisher separation theorem

2. Value functions backward equationa. Value fcn. as fcn. of individual investment opportunities πœ”πœ”

Special casesb. De-scaled value fcn. as function of state variables πœ‚πœ‚

Digression: HJB-approach (instead of martingale approach & envelop condition)

c. Derive 𝜍𝜍, 𝜍𝜍 price of risk, 𝐢𝐢/𝑁𝑁-ratio from value fcn. envelop condition

3. Evolution of state variable πœ‚πœ‚ forward equation Toolbox 3: Change in numeraire to total wealth (including SDF) β€œMoney evaluation equation” πœ‡πœ‡πœ—πœ—

4. Value function iteration & goods market clearinga. PDE of de-scaled value fcn.b. Value function iteration by solving PDE 18

Page 19: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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2c. Get 𝜍𝜍s from Value Function Envelop Experts value function HH’s value function

𝑣𝑣𝑑𝑑𝐾𝐾𝑑𝑑1βˆ’π›Ύπ›Ύ

1βˆ’π›Ύπ›ΎοΏ½πœ‚πœ‚π‘‘π‘‘οΏ½οΏ½πš€

1βˆ’π›Ύπ›Ύ 𝑣𝑣𝑑𝑑𝐾𝐾𝑑𝑑1βˆ’π›Ύπ›Ύ

1βˆ’π›Ύπ›ΎοΏ½πœ‚πœ‚π‘‘π‘‘οΏ½οΏ½πš€

1βˆ’π›Ύπ›Ύ

To obtain πœ•πœ•π‘‰π‘‰π‘‘π‘‘ π‘›π‘›πœ•πœ•π‘›π‘›π‘‘π‘‘

use 𝐾𝐾𝑑𝑑 = π‘π‘π‘‘π‘‘πœ‚πœ‚π‘‘π‘‘ π‘žπ‘žπ‘‘π‘‘+𝑝𝑝𝑑𝑑

= 1οΏ½πœ‚πœ‚π‘‘π‘‘οΏ½οΏ½πš€

π‘›π‘›π‘‘π‘‘πœ‚πœ‚π‘‘π‘‘ π‘žπ‘žπ‘‘π‘‘+𝑝𝑝𝑑𝑑

𝑉𝑉𝑑𝑑 𝑛𝑛 = 𝑣𝑣𝑑𝑑𝑛𝑛𝑑𝑑1βˆ’π›Ύπ›Ύ/(πœ‚πœ‚π‘‘π‘‘ π‘žπ‘žπ‘‘π‘‘+𝑝𝑝𝑑𝑑 )1βˆ’π›Ύπ›Ύ

1βˆ’π›Ύπ›Ύβ€¦

Envelop condition πœ•πœ•π‘‰π‘‰π‘‘π‘‘ π‘›π‘›πœ•πœ•π‘›π‘›π‘‘π‘‘

= πœ•πœ•πœ•πœ• π‘π‘π‘‘π‘‘πœ•πœ•π‘π‘π‘‘π‘‘

π‘£π‘£π‘‘π‘‘π‘›π‘›π‘‘π‘‘βˆ’π›Ύπ›Ύ

(πœ‚πœ‚π‘‘π‘‘ π‘žπ‘žπ‘‘π‘‘+𝑝𝑝𝑑𝑑 )1βˆ’π›Ύπ›Ύ= 𝑐𝑐𝑑𝑑

βˆ’π›Ύπ›Ύ …

Using 𝐾𝐾𝑑𝑑 = 1οΏ½πœ‚πœ‚π‘‘π‘‘οΏ½οΏ½πš€

π‘›π‘›π‘‘π‘‘πœ‚πœ‚π‘‘π‘‘ π‘žπ‘žπ‘‘π‘‘+𝑝𝑝𝑑𝑑

, 𝐢𝐢𝑑𝑑 = 1οΏ½πœ‚πœ‚π‘‘π‘‘οΏ½οΏ½πš€ 𝑐𝑐𝑑𝑑

π‘£π‘£π‘‘π‘‘πœ‚πœ‚π‘‘π‘‘ π‘žπ‘žπ‘‘π‘‘+𝑝𝑝𝑑𝑑

πΎπΎπ‘‘π‘‘βˆ’π›Ύπ›Ύ = 𝐢𝐢𝑑𝑑

βˆ’π›Ύπ›Ύ …

πœŽπœŽπ‘‘π‘‘π‘£π‘£ βˆ’ πœŽπœŽπ‘‘π‘‘πœ‚πœ‚ βˆ’ πœŽπœŽπ‘‘π‘‘

π‘žπ‘ž+𝑝𝑝 βˆ’ π›Ύπ›ΎπœŽπœŽ = βˆ’π›Ύπ›ΎπœŽπœŽπ‘‘π‘‘π‘π‘ , πœŽπœŽπ‘‘π‘‘π‘£π‘£ βˆ’ πœŽπœŽπ‘‘π‘‘

πœ‚πœ‚βˆ’ πœŽπœŽπ‘‘π‘‘

π‘žπ‘ž+𝑝𝑝 βˆ’ π›Ύπ›ΎπœŽπœŽ = βˆ’π›Ύπ›ΎπœŽπœŽπ‘‘π‘‘π‘π‘

= βˆ’πœπœπ‘‘π‘‘ = βˆ’πœπœπ‘‘π‘‘ 19

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2c. Get 𝜍𝜍s from Value Function Envelop Experts value function HH’s value function

𝑣𝑣𝑑𝑑𝐾𝐾𝑑𝑑1βˆ’π›Ύπ›Ύ

1βˆ’π›Ύπ›ΎοΏ½πœ‚πœ‚π‘‘π‘‘οΏ½οΏ½πš€

1βˆ’π›Ύπ›Ύ 𝑣𝑣𝑑𝑑𝐾𝐾𝑑𝑑1βˆ’π›Ύπ›Ύ

1βˆ’π›Ύπ›ΎοΏ½πœ‚πœ‚π‘‘π‘‘οΏ½οΏ½πš€

1βˆ’π›Ύπ›Ύ

To obtain πœ•πœ•π‘‰π‘‰π‘‘π‘‘ π‘›π‘›πœ•πœ•π‘›π‘›π‘‘π‘‘

use 𝐾𝐾𝑑𝑑 = π‘π‘π‘‘π‘‘πœ‚πœ‚π‘‘π‘‘ π‘žπ‘žπ‘‘π‘‘+𝑝𝑝𝑑𝑑

= 1οΏ½πœ‚πœ‚π‘‘π‘‘οΏ½οΏ½πš€

π‘›π‘›π‘‘π‘‘πœ‚πœ‚π‘‘π‘‘ π‘žπ‘žπ‘‘π‘‘+𝑝𝑝𝑑𝑑

𝑉𝑉𝑑𝑑 𝑛𝑛 = 𝑣𝑣𝑑𝑑𝑛𝑛𝑑𝑑1βˆ’π›Ύπ›Ύ/(πœ‚πœ‚π‘‘π‘‘ π‘žπ‘žπ‘‘π‘‘+𝑝𝑝𝑑𝑑 )1βˆ’π›Ύπ›Ύ

1βˆ’π›Ύπ›Ύβ€¦

Envelop condition πœ•πœ•π‘‰π‘‰π‘‘π‘‘ π‘›π‘›πœ•πœ•π‘›π‘›π‘‘π‘‘

= πœ•πœ•πœ•πœ• π‘π‘π‘‘π‘‘πœ•πœ•π‘π‘π‘‘π‘‘

π‘£π‘£π‘‘π‘‘π‘›π‘›π‘‘π‘‘βˆ’π›Ύπ›Ύ

(πœ‚πœ‚π‘‘π‘‘ π‘žπ‘žπ‘‘π‘‘+𝑝𝑝𝑑𝑑 )1βˆ’π›Ύπ›Ύ= 𝑐𝑐𝑑𝑑

βˆ’π›Ύπ›Ύ …

Using 𝐾𝐾𝑑𝑑 = 1οΏ½πœ‚πœ‚π‘‘π‘‘οΏ½οΏ½πš€

π‘›π‘›π‘‘π‘‘πœ‚πœ‚π‘‘π‘‘ π‘žπ‘žπ‘‘π‘‘+𝑝𝑝𝑑𝑑

, 𝐢𝐢𝑑𝑑 = 1οΏ½πœ‚πœ‚π‘‘π‘‘οΏ½οΏ½πš€ 𝑐𝑐𝑑𝑑

π‘£π‘£π‘‘π‘‘πœ‚πœ‚π‘‘π‘‘ π‘žπ‘žπ‘‘π‘‘+𝑝𝑝𝑑𝑑

πΎπΎπ‘‘π‘‘βˆ’π›Ύπ›Ύ = 𝐢𝐢𝑑𝑑

βˆ’π›Ύπ›Ύ …

πœŽπœŽπ‘‘π‘‘π‘£π‘£ βˆ’ πœŽπœŽπ‘‘π‘‘πœ‚πœ‚ βˆ’ πœŽπœŽπ‘‘π‘‘

π‘žπ‘ž+𝑝𝑝 βˆ’ π›Ύπ›ΎπœŽπœŽ = βˆ’π›Ύπ›ΎπœŽπœŽπ‘‘π‘‘π‘π‘ , πœŽπœŽπ‘‘π‘‘π‘£π‘£ βˆ’ πœŽπœŽπ‘‘π‘‘

πœ‚πœ‚βˆ’ πœŽπœŽπ‘‘π‘‘

π‘žπ‘ž+𝑝𝑝 βˆ’ π›Ύπ›ΎπœŽπœŽ = βˆ’π›Ύπ›ΎπœŽπœŽπ‘‘π‘‘π‘π‘

= βˆ’πœπœπ‘‘π‘‘ = βˆ’πœπœπ‘‘π‘‘ 20

By Ito’s Lemmaπ‘žπ‘žπ‘‘π‘‘ + 𝑝𝑝𝑑𝑑 πœŽπœŽπ‘‘π‘‘

π‘žπ‘ž+𝑝𝑝 = π‘žπ‘žπ‘‘π‘‘πœŽπœŽπ‘‘π‘‘π‘žπ‘ž + π‘π‘π‘‘π‘‘πœŽπœŽπ‘‘π‘‘

𝑝𝑝

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2c. Get 𝜍𝜍s from Value Function Envelop Experts risk-premia HH’s risk premia

𝑣𝑣𝑑𝑑𝐾𝐾𝑑𝑑1βˆ’π›Ύπ›Ύ

1βˆ’π›Ύπ›ΎοΏ½πœ‚πœ‚π‘‘π‘‘οΏ½οΏ½πš€

1βˆ’π›Ύπ›Ύ 𝑣𝑣𝑑𝑑𝐾𝐾𝑑𝑑1βˆ’π›Ύπ›Ύ

1βˆ’π›Ύπ›ΎοΏ½πœ‚πœ‚π‘‘π‘‘οΏ½οΏ½πš€

1βˆ’π›Ύπ›Ύ

πœπœπ‘‘π‘‘ = π›Ύπ›ΎπœŽπœŽπ‘‘π‘‘π‘π‘ = πœπœπ‘‘π‘‘ = π›Ύπ›ΎπœŽπœŽπ‘‘π‘‘π‘π‘ =

βˆ’πœŽπœŽπ‘‘π‘‘π‘£π‘£ + πœŽπœŽπ‘‘π‘‘πœ‚πœ‚ + πœŽπœŽπ‘‘π‘‘

π‘žπ‘ž+𝑝𝑝 + π›Ύπ›ΎπœŽπœŽ βˆ’πœŽπœŽπ‘‘π‘‘π‘£π‘£ βˆ’ πœ‚πœ‚π‘‘π‘‘πœŽπœŽπ‘‘π‘‘

πœ‚πœ‚

1βˆ’πœ‚πœ‚π‘‘π‘‘+ πœŽπœŽπ‘‘π‘‘

π‘žπ‘ž+𝑝𝑝 + π›Ύπ›ΎπœŽπœŽ

For πœπœπ‘‘π‘‘note that from

π‘£π‘£π‘‘π‘‘π‘›π‘›π‘‘π‘‘βˆ’π›Ύπ›Ύ

(πœ‚πœ‚π‘‘π‘‘ π‘žπ‘žπ‘‘π‘‘+𝑝𝑝𝑑𝑑 )1βˆ’π›Ύπ›Ύ= 𝑐𝑐𝑑𝑑

βˆ’π›Ύπ›Ύ follows οΏ½πœŽπœŽπ‘‘π‘‘π‘›π‘› = οΏ½πœŽπœŽπ‘‘π‘‘π‘π‘

Hence, πœπœπ‘‘π‘‘ = 𝛾𝛾 οΏ½πœŽπœŽπ‘‘π‘‘π‘›π‘› πœπœπ‘‘π‘‘ = 𝛾𝛾 οΏ½πœŽπœŽπ‘‘π‘‘π‘›π‘›

Recall Martingale approach asset pricing21

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2c. Get 𝐢𝐢𝑑𝑑𝑁𝑁𝑑𝑑

, 𝐢𝐢𝑑𝑑𝑁𝑁𝑑𝑑

from Value Function Envelop

Experts Households

Recall π‘£π‘£π‘‘π‘‘π‘›π‘›π‘‘π‘‘βˆ’π›Ύπ›Ύ

(πœ‚πœ‚π‘‘π‘‘ π‘žπ‘žπ‘‘π‘‘+𝑝𝑝𝑑𝑑 )1βˆ’π›Ύπ›Ύ= 𝑐𝑐𝑑𝑑

βˆ’π›Ύπ›Ύ

𝑐𝑐𝑑𝑑𝑛𝑛𝑑𝑑

= (πœ‚πœ‚π‘‘π‘‘(π‘žπ‘žπ‘‘π‘‘+𝑝𝑝𝑑𝑑))1/π›Ύπ›Ύβˆ’1

𝑣𝑣𝑑𝑑1/𝛾𝛾

𝐢𝐢𝑑𝑑𝑁𝑁𝑑𝑑

= (πœ‚πœ‚π‘‘π‘‘ π‘žπ‘žπ‘‘π‘‘+𝑝𝑝𝑑𝑑 )1/π›Ύπ›Ύβˆ’1

𝑣𝑣𝑑𝑑1/𝛾𝛾

𝐢𝐢𝑑𝑑𝑁𝑁𝑑𝑑

= ((1βˆ’πœ‚πœ‚π‘‘π‘‘) π‘žπ‘žπ‘‘π‘‘+𝑝𝑝𝑑𝑑 )1/π›Ύπ›Ύβˆ’1

𝑣𝑣𝑑𝑑1/𝛾𝛾

𝐢𝐢𝑑𝑑 + 𝐢𝐢𝑑𝑑𝑁𝑁𝑑𝑑 + 𝑁𝑁𝑑𝑑

= πœ‚πœ‚π‘‘π‘‘πΆπΆπ‘‘π‘‘π‘π‘π‘‘π‘‘

+ (1 βˆ’ πœ‚πœ‚π‘‘π‘‘)𝐢𝐢𝑑𝑑𝑁𝑁𝑑𝑑

Plug in from above

22

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Solving MacroModels Step-by-Step0. Postulate aggregates, price processes & obtain return processes1. For given SDF processes static

a. Real investment πœ„πœ„, (portfolio 𝜽𝜽, & consumption choice of each agent) Toolbox 1: Martingale Approach

b. Asset/Risk Allocation across types/sectors & asset market clearing Toolbox 2: β€œprice-taking social planner approach” – Fisher separation theorem

2. Value functions backward equationa. Value fcn. as fcn. of individual investment opportunities πœ”πœ”

Special casesb. De-scaled value fcn. as function of state variables πœ‚πœ‚

Digression: HJB-approach (instead of martingale approach & envelop condition)

c. Derive 𝜍𝜍 price of risk, 𝐢𝐢/𝑁𝑁-ratio from value fcn. envelop condition

3. Evolution of state variable πœ‚πœ‚ forward equation Toolbox 3: Change in numeraire to total wealth (including SDF) β€œMoney evaluation equation” πœ‡πœ‡πœ—πœ—

4. Value function iteration & goods market clearinga. PDE of de-scaled value fcn.b. Value function iteration by solving PDE 23

Page 24: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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3. πœ‡πœ‡πœ‚πœ‚Drift of Wealth Share: Two Types Asset pricing formula (relative to benchmark asset)

πœ‡πœ‡π‘‘π‘‘πœ‚πœ‚ +

𝐢𝐢𝑑𝑑𝑁𝑁𝑑𝑑

βˆ’ πœ—πœ—π‘‘π‘‘πœ‡πœ‡π‘€π‘€ βˆ’ π‘Ÿπ‘Ÿπ‘‘π‘‘π‘€π‘€ = 𝜍𝜍 βˆ’ πœŽπœŽπ‘π‘ πœŽπœŽπœ‚πœ‚ βˆ’ πœŽπœŽπ‘€π‘€ + 𝜍𝜍 οΏ½πœŽπœŽπ‘‘π‘‘π‘›π‘›

Add up across types (weighted), (capital letters without superscripts are aggregates for total economy)

(πœ‚πœ‚π‘‘π‘‘πœ‡πœ‡π‘‘π‘‘πœ‚πœ‚ + 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœ‡πœ‡π‘‘π‘‘

πœ‚πœ‚)

=0

+πΆπΆπ‘‘π‘‘οΏ½π‘π‘π‘‘π‘‘βˆ’ πœ—πœ—π‘‘π‘‘πœ‡πœ‡π‘€π‘€ βˆ’ π‘Ÿπ‘Ÿπ‘‘π‘‘π‘€π‘€ =

πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ βˆ’ πœŽπœŽπ‘‘π‘‘οΏ½π‘π‘ πœŽπœŽπ‘‘π‘‘

πœ‚πœ‚ βˆ’ πœŽπœŽπ‘‘π‘‘π‘€π‘€ + 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ βˆ’ πœŽπœŽπ‘‘π‘‘οΏ½π‘π‘ πœŽπœŽπ‘‘π‘‘

πœ‚πœ‚βˆ’ πœŽπœŽπ‘‘π‘‘π‘€π‘€ + πœ‚πœ‚π‘‘π‘‘ 𝜍𝜍 οΏ½πœŽπœŽπ‘‘π‘‘π‘›π‘› + 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ οΏ½πœŽπœŽπ‘‘π‘‘

𝑛𝑛

Subtract from each other yields wealth share driftπœ‡πœ‡π‘‘π‘‘πœ‚πœ‚ = 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ βˆ’ πœŽπœŽπ‘‘π‘‘

�𝑁𝑁 πœŽπœŽπ‘‘π‘‘πœ‚πœ‚ βˆ’ πœŽπœŽπ‘‘π‘‘π‘€π‘€ βˆ’ (1 βˆ’ πœ‚πœ‚π‘‘π‘‘) πœπœπ‘‘π‘‘ βˆ’ πœŽπœŽπ‘‘π‘‘

�𝑁𝑁 πœŽπœŽπ‘‘π‘‘πœ‚πœ‚βˆ’ πœŽπœŽπ‘‘π‘‘π‘€π‘€

+ 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ οΏ½πœŽπœŽπ‘‘π‘‘π‘›π‘› βˆ’ 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ οΏ½πœŽπœŽπ‘‘π‘‘π‘›π‘› βˆ’ 𝐢𝐢𝑑𝑑

π‘π‘π‘‘π‘‘βˆ’ 𝐢𝐢𝑑𝑑+𝐢𝐢𝑑𝑑

π‘žπ‘žπ‘‘π‘‘+𝑝𝑝𝑑𝑑 𝐾𝐾𝑑𝑑 24

Seignorage due to money supply growth leads to transfers

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3. πœŽπœŽπœ‚πœ‚ Volatility of Wealth Share Since πœ‚πœ‚π‘‘π‘‘π‘–π‘– = 𝑁𝑁𝑑𝑑𝑖𝑖/�𝑁𝑁𝑑𝑑,

πœŽπœŽπ‘‘π‘‘πœ‚πœ‚π‘–π‘– = πœŽπœŽπ‘‘π‘‘π‘π‘

𝑖𝑖 βˆ’ πœŽπœŽπ‘‘π‘‘οΏ½π‘π‘ = πœŽπœŽπ‘‘π‘‘π‘π‘

𝑖𝑖 βˆ’οΏ½π‘–π‘–β€²πœ‚πœ‚π‘‘π‘‘π‘–π‘–

β€²πœŽπœŽπ‘‘π‘‘π‘π‘π‘–π‘–β€²

= 1 βˆ’ πœ‚πœ‚π‘‘π‘‘π‘–π‘– πœŽπœŽπ‘‘π‘‘π‘π‘π‘–π‘– βˆ’ οΏ½

π‘–π‘–βˆ’β‰ π‘–π‘–

πœ‚πœ‚π‘‘π‘‘π‘–π‘–βˆ’πœŽπœŽπ‘‘π‘‘π‘π‘

π‘–π‘–βˆ’

Note for 2 types exampleπœŽπœŽπ‘‘π‘‘πœ‚πœ‚ = 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœŽπœŽπ‘‘π‘‘π‘›π‘› βˆ’ πœŽπœŽπ‘‘π‘‘

𝑛𝑛

πœŽπœŽπ‘‘π‘‘π‘›π‘› = 𝜎𝜎 + πœŽπœŽπ‘‘π‘‘π‘π‘ + πœ’πœ’π‘‘π‘‘

πœ‚πœ‚π‘‘π‘‘(1 βˆ’ πœ—πœ—)

=πœƒπœƒπ‘˜π‘˜+πœƒπœƒπ‘œπ‘œπ‘œπ‘œ

(πœŽπœŽπ‘‘π‘‘π‘žπ‘ž βˆ’ πœŽπœŽπ‘‘π‘‘

𝑝𝑝), πœŽπœŽπ‘‘π‘‘π‘›π‘› = 𝜎𝜎 + πœŽπœŽπ‘‘π‘‘

𝑝𝑝 + 1βˆ’πœ’πœ’π‘‘π‘‘1βˆ’πœ‚πœ‚π‘‘π‘‘

(1 βˆ’ πœ—πœ—)(πœŽπœŽπ‘‘π‘‘π‘žπ‘ž βˆ’ πœŽπœŽπ‘‘π‘‘

𝑝𝑝)

Hence, πœŽπœŽπ‘‘π‘‘πœ‚πœ‚ = πœ’πœ’π‘‘π‘‘βˆ’πœ‚πœ‚π‘‘π‘‘

πœ‚πœ‚π‘‘π‘‘(1 βˆ’ πœ—πœ—)(πœŽπœŽπ‘‘π‘‘

π‘žπ‘ž βˆ’ πœŽπœŽπ‘‘π‘‘π‘π‘)

=βˆ’πœŽπœŽπ‘‘π‘‘πœ—πœ—

Note also, πœ‚πœ‚π‘‘π‘‘πœŽπœŽπ‘‘π‘‘πœ‚πœ‚ + 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœŽπœŽπ‘‘π‘‘

πœ‚πœ‚= 0 β‡’ πœŽπœŽπ‘‘π‘‘

πœ‚πœ‚= βˆ’ πœ‚πœ‚π‘‘π‘‘

1βˆ’πœ‚πœ‚π‘‘π‘‘πœŽπœŽπ‘‘π‘‘πœ‚πœ‚

25

Change in notation in 2 type settingType-networth is 𝑛𝑛 = 𝑁𝑁𝑖𝑖

Apply Ito’s Lemma on πœ—πœ—

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Solving MacroModels Step-by-Step0. Postulate aggregates, price processes & obtain return processes1. For given SDF processes static

a. Real investment πœ„πœ„, (portfolio 𝜽𝜽, & consumption choice of each agent) Toolbox 1: Martingale Approach

b. Asset/Risk Allocation across types/sectors & asset market clearing Toolbox 2: β€œprice-taking social planner approach” – Fisher separation theorem

2. Value functions backward equationa. Value fcn. as fcn. of individual investment opportunities πœ”πœ”

Special casesb. De-scaled value fcn. as function of state variables πœ‚πœ‚

Digression: HJB-approach (instead of martingale approach & envelop condition)

c. Derive 𝜍𝜍 price of risk, 𝐢𝐢/𝑁𝑁-ratio from value fcn. envelop condition

3. Evolution of state variable πœ‚πœ‚ forward equation Toolbox 3: Change in numeraire to total wealth (including SDF) β€œMoney evaluation equation” πœ‡πœ‡πœ—πœ—

4. Value function iteration & goods market clearinga. PDE of de-scaled value fcn.b. Value function iteration by solving PDE 26

Page 27: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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3. β€œMoney evaluation equation” πœ‡πœ‡πœ—πœ—

Recall πΆπΆπ‘‘π‘‘οΏ½π‘π‘π‘‘π‘‘βˆ’ π‘Ÿπ‘Ÿπ‘‘π‘‘π‘€π‘€ βˆ’ πœ—πœ—π‘‘π‘‘πœ‡πœ‡π‘€π‘€ =

πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ βˆ’ πœŽπœŽπ‘‘π‘‘οΏ½π‘π‘ πœŽπœŽπ‘‘π‘‘

πœ‚πœ‚ βˆ’ πœŽπœŽπ‘‘π‘‘π‘€π‘€+ 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ βˆ’ πœŽπœŽπ‘‘π‘‘

�𝑁𝑁 πœŽπœŽπ‘‘π‘‘πœ‚πœ‚βˆ’ πœŽπœŽπ‘‘π‘‘π‘€π‘€ + πœ‚πœ‚π‘‘π‘‘ 𝜍𝜍 οΏ½πœŽπœŽπ‘‘π‘‘π‘›π‘›

+ 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ οΏ½πœŽπœŽπ‘‘π‘‘π‘›π‘›

If benchmark asset is money Replace π‘Ÿπ‘Ÿπ‘‘π‘‘π‘€π‘€ = πœ‡πœ‡π‘‘π‘‘πœ—πœ— βˆ’ πœ‡πœ‡π‘€π‘€ and πœŽπœŽπ‘‘π‘‘π‘€π‘€ = πœŽπœŽπ‘‘π‘‘πœ—πœ— (in the total wealth numeraire)

βˆ’πœ‡πœ‡π‘‘π‘‘πœ—πœ—= βˆ’(1 βˆ’ πœ—πœ—)πœ‡πœ‡π‘€π‘€ βˆ’πΆπΆπ‘‘π‘‘οΏ½π‘π‘π‘‘π‘‘

+ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ βˆ’ πœŽπœŽπ‘‘π‘‘οΏ½π‘π‘ πœŽπœŽπ‘‘π‘‘

πœ‚πœ‚ βˆ’ πœŽπœŽπ‘‘π‘‘πœ—πœ—

+ 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ βˆ’ πœŽπœŽπ‘‘π‘‘οΏ½π‘π‘ πœŽπœŽπ‘‘π‘‘

πœ‚πœ‚βˆ’ πœŽπœŽπ‘‘π‘‘πœ—πœ— + πœ‚πœ‚π‘‘π‘‘ 𝜍𝜍 οΏ½πœŽπœŽπ‘‘π‘‘π‘›π‘› + 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ οΏ½πœŽπœŽπ‘‘π‘‘

𝑛𝑛

Why is return on money in new numeriare π‘‘π‘‘πœ—πœ—π‘‘π‘‘πœ—πœ—π‘‘π‘‘βˆ’ πœ‡πœ‡π‘€π‘€?

πœ—πœ—π‘‘π‘‘ = 𝑝𝑝𝑑𝑑𝐾𝐾𝑑𝑑/𝑁𝑁𝑑𝑑 is the value of money stock in total networth units27

Page 28: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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Solving MacroModels Step-by-Step0. Postulate aggregates, price processes & obtain return processes1. For given SDF processes static

a. Real investment πœ„πœ„, (portfolio 𝜽𝜽, & consumption choice of each agent) Toolbox 1: Martingale Approach

b. Asset/Risk Allocation across types/sectors & asset market clearing Toolbox 2: β€œprice-taking social planner approach” – Fisher separation theorem

2. Value functions backward equationa. Value fcn. as fcn. of individual investment opportunities πœ”πœ”

Special casesb. De-scaled value fcn. as function of state variables πœ‚πœ‚

Digression: HJB-approach (instead of martingale approach & envelop condition)

c. Derive 𝜍𝜍 price of risk, 𝐢𝐢/𝑁𝑁-ratio from value fcn. envelop condition

3. Evolution of state variable πœ‚πœ‚ forward equation Toolbox 3: Change in numeraire to total wealth (including SDF) β€œMoney evaluation equation” πœ‡πœ‡πœ—πœ—

4. Value function iteration & goods market clearinga. PDE of de-scaled value fcn.b. Value function iteration by solving PDE 28

Page 29: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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4. PDE Value Function Iteration

Postulate 𝑣𝑣𝑑𝑑 = 𝑣𝑣(πœ‚πœ‚π‘‘π‘‘, 𝑑𝑑) By Ito’s Lemma

𝑑𝑑𝑣𝑣𝑑𝑑𝑣𝑣𝑑𝑑

=πœ•πœ•π‘‘π‘‘π‘£π‘£π‘‘π‘‘+πœ•πœ•πœ‚πœ‚π‘£π‘£π‘‘π‘‘πœ‚πœ‚πœ‡πœ‡π‘‘π‘‘

πœ‚πœ‚+12πœ•πœ•πœ‚πœ‚πœ‚πœ‚π‘£π‘£π‘‘π‘‘ πœ‚πœ‚π‘‘π‘‘πœŽπœŽπ‘‘π‘‘πœ‚πœ‚ 2

𝑣𝑣𝑑𝑑𝑑𝑑𝑑𝑑 + πœ•πœ•πœ‚πœ‚π‘£π‘£π‘‘π‘‘πœ‚πœ‚πœŽπœŽπ‘‘π‘‘

πœ‚πœ‚

𝑣𝑣𝑑𝑑𝑑𝑑𝑍𝑍𝑑𝑑

That is,

πœ‡πœ‡π‘‘π‘‘π‘£π‘£π‘£π‘£π‘‘π‘‘ = πœ•πœ•π‘‘π‘‘π‘£π‘£π‘‘π‘‘ + πœ•πœ•πœ‚πœ‚π‘£π‘£π‘‘π‘‘πœ‚πœ‚πœ‡πœ‡π‘‘π‘‘πœ‚πœ‚ + 1

2πœ•πœ•πœ‚πœ‚πœ‚πœ‚π‘£π‘£π‘‘π‘‘ πœ‚πœ‚π‘‘π‘‘πœŽπœŽπ‘‘π‘‘

πœ‚πœ‚ 2

πœŽπœŽπ‘‘π‘‘π‘£π‘£π‘£π‘£π‘‘π‘‘ = πœ•πœ•πœ‚πœ‚π‘£π‘£π‘‘π‘‘πœ‚πœ‚πœŽπœŽπ‘‘π‘‘πœ‚πœ‚

Plugging in previous slides drift equation β‡’β€œgrowth equation”

πœ•πœ•π‘‘π‘‘π‘£π‘£π‘‘π‘‘ + πœ‚πœ‚πœ‡πœ‡π‘‘π‘‘πœ‚πœ‚ + 1 βˆ’ 𝛾𝛾 πœŽπœŽπœ‚πœ‚π‘‘π‘‘πœŽπœŽπ‘‘π‘‘

πœ‚πœ‚π‘£π‘£π‘‘π‘‘ πœ•πœ•πœ‚πœ‚π‘£π‘£π‘‘π‘‘ +12πœ•πœ•πœ‚πœ‚πœ‚πœ‚π‘£π‘£π‘‘π‘‘ πœ‚πœ‚π‘‘π‘‘πœŽπœŽπ‘‘π‘‘

πœ‚πœ‚ 2 =

= 𝜌𝜌 βˆ’ 1 βˆ’ 𝛾𝛾 Ξ¦ πœ„πœ„ βˆ’ 𝛿𝛿 +12𝛾𝛾(1 βˆ’ 𝛾𝛾)(𝜎𝜎2 + οΏ½πœŽπœŽπ‘›π‘› 2) 𝑣𝑣𝑑𝑑 βˆ’

𝑐𝑐𝑑𝑑𝑛𝑛𝑑𝑑𝑣𝑣𝑑𝑑

29

πœ‡πœ‡π‘‘π‘‘π‘£π‘£ πœŽπœŽπ‘‘π‘‘π‘£π‘£

Short-hand notation:πœ•πœ•π‘‘π‘‘π‘“π‘“ for πœ•πœ•π‘“π‘“/πœ•πœ•π‘₯π‘₯

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4a. Algorithm

Dynamic steps involves now iterating 𝑣𝑣(πœ‚πœ‚), 𝑣𝑣 πœ‚πœ‚ , and πœ—πœ—(πœ‚πœ‚)

Static step only involve planner’s conditions (which implicitly includes asset market clearing), Solve everything in terms of πœ—πœ— πœ‚πœ‚

π‘žπ‘ž πœ‚πœ‚ and 𝑝𝑝 πœ‚πœ‚ can be easily derived since we have it as a function of πœ—πœ— and πœ“πœ“ in closed form

π‘žπ‘žπ‘‘π‘‘ = 1 βˆ’ πœ—πœ—π‘‘π‘‘1+πœ…πœ…π΄π΄(πœ“πœ“π‘‘π‘‘)1βˆ’πœ—πœ—π‘‘π‘‘+πœ…πœ…πœπœπ‘‘π‘‘

, where πœπœπ‘‘π‘‘ ≔ 𝐢𝐢𝑑𝑑/𝑁𝑁𝑑𝑑

𝑝𝑝𝑑𝑑 = πœ—πœ—π‘‘π‘‘1+πœ…πœ…π΄π΄(πœ“πœ“π‘‘π‘‘)1βˆ’πœ—πœ—π‘‘π‘‘+πœ…πœ…πœπœπ‘‘π‘‘

Remark:One can obtain the moneyless equilibrium with πœ—πœ— πœ‚πœ‚ = 0by setting πœŽπœŽπ‘π‘ = βˆ’πœŽπœŽ (in models with real risk-free debt) Why? recall π‘‘π‘‘π‘Ÿπ‘Ÿπ‘€π‘€ = [ Ξ¦ πœ„πœ„ βˆ’ 𝛿𝛿 βˆ’ πœ‡πœ‡π‘€π‘€]𝑑𝑑𝑑𝑑 + 𝜎𝜎 + πœŽπœŽπ‘π‘ 𝑑𝑑𝑍𝑍

We never used the drift to solve the model. To make money, risk-free asset we have to set πœŽπœŽπ‘π‘ = βˆ’πœŽπœŽ 30

Page 31: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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Roadmap

Changes in solution procedure in a setting with idiosyncratic risk and money Compare to lecture 03 without idiosyncratic risk and money

Simple two sector model1. Real Debt

2. Money/Nominal Debt

31

Page 32: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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Two Sector Model Setup

Expert sectorOutput: 𝑦𝑦𝑑𝑑 = π‘Žπ‘Žπ‘˜π‘˜π‘‘π‘‘ Consumption rate: 𝑐𝑐𝑑𝑑 Investment rate: πœ„πœ„π‘‘π‘‘π‘‘π‘‘π‘˜π‘˜π‘‘π‘‘

��𝚀

π‘˜π‘˜π‘‘π‘‘οΏ½οΏ½πš€

= Ξ¦ πœ„πœ„π‘‘π‘‘ βˆ’π›Ώπ›Ώ 𝑑𝑑𝑑𝑑+πœŽπœŽπ‘‘π‘‘π‘π‘π‘‘π‘‘+οΏ½πœŽπœŽπ‘‘π‘‘ οΏ½π‘π‘π‘‘π‘‘οΏ½οΏ½πš€+𝑑𝑑Δ𝑑𝑑

π‘˜π‘˜,��𝚀

𝐸𝐸0[∫0∞ π‘’π‘’βˆ’πœŒπœŒπ‘‘π‘‘ log 𝑐𝑐𝑑𝑑 𝑑𝑑𝑑𝑑]

Friction: Can only issue Risk-free debt Equity, but most hold πœ’πœ’π‘‘π‘‘ β‰₯ 𝛼𝛼 32

Household sectorOutput: 𝑦𝑦𝑑𝑑 = π‘Žπ‘Žπ‘˜π‘˜π‘‘π‘‘ Consumption rate: 𝑐𝑐𝑑𝑑 Investment rate: πœ„πœ„π‘‘π‘‘π‘‘π‘‘π‘˜π‘˜π‘‘π‘‘

��𝚀

π‘˜π‘˜π‘‘π‘‘οΏ½οΏ½πš€

= Ξ¦ πœ„πœ„π‘‘π‘‘ βˆ’π›Ώπ›Ώ 𝑑𝑑𝑑𝑑+πœŽπœŽπ‘‘π‘‘π‘π‘π‘‘π‘‘+οΏ½πœŽπœŽπ‘‘π‘‘ οΏ½π‘π‘π‘‘π‘‘οΏ½οΏ½πš€+𝑑𝑑Δ𝑑𝑑

π‘˜π‘˜,��𝚀

𝐸𝐸0[∫0βˆžπ‘’π‘’βˆ’πœŒπœŒπ‘‘π‘‘ log 𝑐𝑐𝑑𝑑 𝑑𝑑𝑑𝑑]

π‘Žπ‘Ž = π‘Žπ‘Ž

𝜌𝜌 = 𝜌𝜌

𝛿𝛿 = π›Ώπ›ΏοΏ½πœŽπœŽ ≀ �𝜎𝜎

Page 33: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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Two Sector Model with & without Money

Idio risk without money and real debt

Experts’ idio risk �𝜎𝜎 can depress π‘Ÿπ‘Ÿπ‘‘π‘‘π‘“π‘“, which hurts households

33

A L

Capitalπœ“πœ“π‘‘π‘‘π‘žπ‘žπ‘‘π‘‘πΎπΎπ‘‘π‘‘

𝑁𝑁𝑑𝑑

A L

Capital1 βˆ’ πœ“πœ“π‘‘π‘‘ π‘žπ‘žπ‘‘π‘‘πΎπΎπ‘‘π‘‘

Net worth𝑁𝑁𝑑𝑑

Real DebtClaimsReal Debt

Poll 33: Increasing experts idiosyncratic risk �𝜎𝜎a) Lowers experts wealth share drift πœ‡πœ‡πœ‚πœ‚b) Increases experts wealth share drift πœ‡πœ‡πœ‚πœ‚, as

they earn some extra risk premiumc) Hurts the households, as it depresses π‘Ÿπ‘Ÿπ‘‘π‘‘

𝑓𝑓

Page 34: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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Two Sector Model with & without Money

Idio risk without money and real debt

Idio risk with money (and nominal short-term debt)

Value of money covaries with 𝐾𝐾-shocks β‡’ implicit insurance 34

A L

Capitalπœ“πœ“π‘‘π‘‘π‘žπ‘žπ‘‘π‘‘πΎπΎπ‘‘π‘‘

𝑁𝑁𝑑𝑑

A L

Capital1 βˆ’ πœ“πœ“π‘‘π‘‘ π‘žπ‘žπ‘‘π‘‘πΎπΎπ‘‘π‘‘

Net worth𝑁𝑁𝑑𝑑

Outside Money Outside Money

A L

Capitalπœ“πœ“π‘‘π‘‘π‘žπ‘žπ‘‘π‘‘πΎπΎπ‘‘π‘‘

𝑁𝑁𝑑𝑑

A L

Capital1 βˆ’ πœ“πœ“π‘‘π‘‘ π‘žπ‘žπ‘‘π‘‘πΎπΎπ‘‘π‘‘

Net worth𝑁𝑁𝑑𝑑

Real DebtClaimsReal Debt

Nominal DebtClaims

Inside Money/Nominal Debt

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Solution Procedure for Both Settings

Goods market clearing𝜌𝜌 𝑝𝑝𝑑𝑑 + π‘žπ‘žπ‘‘π‘‘ = π‘Žπ‘Ž βˆ’ πœ„πœ„π‘‘π‘‘ divide by π‘žπ‘ž and use π‘žπ‘ž = 1 + πœ…πœ…πœ„πœ„

𝜌𝜌1

1 βˆ’ πœ—πœ—π‘‘π‘‘=π‘Žπ‘Ž βˆ’ 𝑖𝑖1 + πœ…πœ…πœ„πœ„

πœ„πœ„π‘‘π‘‘ = 1βˆ’πœ—πœ—π‘‘π‘‘ π‘Žπ‘Žβˆ’πœŒπœŒ1βˆ’πœ—πœ—π‘‘π‘‘+πœ…πœ…πœŒπœŒ

π‘žπ‘žπ‘‘π‘‘ = 1 + πœ…πœ…πœ„πœ„π‘‘π‘‘ = 1 βˆ’ πœ—πœ—π‘‘π‘‘1+πœ…πœ…π‘Žπ‘Ž

1βˆ’πœ—πœ—π‘‘π‘‘+πœ…πœ…πœŒπœŒ

𝑝𝑝𝑑𝑑 = π‘žπ‘žπ‘‘π‘‘πœ—πœ—π‘‘π‘‘

1βˆ’πœ—πœ—π‘‘π‘‘= πœ—πœ—π‘‘π‘‘

1+πœ…πœ…π‘Žπ‘Ž1βˆ’πœ—πœ—π‘‘π‘‘+πœ…πœ…πœŒπœŒ

Capital market clearing (defines πœ“πœ“π‘‘π‘‘ for planner)1 βˆ’ πœƒπœƒπ‘‘π‘‘ = πœ“πœ“π‘‘π‘‘

πœ‚πœ‚π‘‘π‘‘(1 βˆ’ πœ—πœ—π‘‘π‘‘) 1 βˆ’ πœƒπœƒπ‘‘π‘‘ = 1βˆ’πœ“πœ“π‘‘π‘‘

1βˆ’πœ‚πœ‚π‘‘π‘‘(1 βˆ’ πœ—πœ—π‘‘π‘‘)

Money market clearing by Walras law35

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Solution Procedure for Both Settings

Goods market clearing𝜌𝜌 𝑝𝑝𝑑𝑑 + π‘žπ‘žπ‘‘π‘‘ = π‘Žπ‘Ž βˆ’ πœ„πœ„π‘‘π‘‘ divide by π‘žπ‘ž and use π‘žπ‘ž = 1 + πœ…πœ…πœ„πœ„

𝜌𝜌1

1 βˆ’ πœ—πœ—π‘‘π‘‘=π‘Žπ‘Ž βˆ’ 𝑖𝑖1 + πœ…πœ…πœ„πœ„

πœ„πœ„π‘‘π‘‘ = 1βˆ’πœ—πœ—π‘‘π‘‘ π‘Žπ‘Žβˆ’πœŒπœŒ1βˆ’πœ—πœ—π‘‘π‘‘+πœ…πœ…πœŒπœŒ

π‘žπ‘žπ‘‘π‘‘ = 1 + πœ…πœ…πœ„πœ„π‘‘π‘‘ = 1 βˆ’ πœ—πœ—π‘‘π‘‘1+πœ…πœ…π‘Žπ‘Ž

1βˆ’πœ—πœ—π‘‘π‘‘+πœ…πœ…πœŒπœŒ

𝑝𝑝𝑑𝑑 = π‘žπ‘žπ‘‘π‘‘πœ—πœ—π‘‘π‘‘

1βˆ’πœ—πœ—π‘‘π‘‘= πœ—πœ—π‘‘π‘‘

1+πœ…πœ…π‘Žπ‘Ž1βˆ’πœ—πœ—π‘‘π‘‘+πœ…πœ…πœŒπœŒ

Capital market clearing (defines πœ“πœ“π‘‘π‘‘ for planner)1 βˆ’ πœƒπœƒπ‘‘π‘‘ = πœ“πœ“π‘‘π‘‘

πœ‚πœ‚π‘‘π‘‘(1 βˆ’ πœ—πœ—π‘‘π‘‘) 1 βˆ’ πœƒπœƒπ‘‘π‘‘ = 1βˆ’πœ“πœ“π‘‘π‘‘

1βˆ’πœ‚πœ‚π‘‘π‘‘(1 βˆ’ πœ—πœ—π‘‘π‘‘)

Money market clearing by Walras law36

Poll 36: How would equations change if π‘Žπ‘Ž β‰  π‘Žπ‘Ža) Replace π‘Žπ‘Ž with 𝐴𝐴 πœ“πœ“π‘‘π‘‘b) Nothing c) Whole approach has to be different.

Page 37: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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Solution Procedure for Both Settings

Goods market clearing𝜌𝜌 𝑝𝑝𝑑𝑑 + π‘žπ‘žπ‘‘π‘‘ = π‘Žπ‘Ž βˆ’ πœ„πœ„π‘‘π‘‘ divide by π‘žπ‘ž and use π‘žπ‘ž = 1 + πœ…πœ…πœ„πœ„

𝜌𝜌1

1 βˆ’ πœ—πœ—π‘‘π‘‘=π‘Žπ‘Ž βˆ’ 𝑖𝑖1 + πœ…πœ…πœ„πœ„

πœ„πœ„π‘‘π‘‘ = 1βˆ’πœ—πœ—π‘‘π‘‘ π‘Žπ‘Žβˆ’πœŒπœŒ1βˆ’πœ—πœ—π‘‘π‘‘+πœ…πœ…πœŒπœŒ

π‘žπ‘žπ‘‘π‘‘ = 1 + πœ…πœ…πœ„πœ„π‘‘π‘‘ = 1 βˆ’ πœ—πœ—π‘‘π‘‘1+πœ…πœ…π‘Žπ‘Ž

1βˆ’πœ—πœ—π‘‘π‘‘+πœ…πœ…πœŒπœŒ

𝑝𝑝𝑑𝑑 = π‘žπ‘žπ‘‘π‘‘πœ—πœ—π‘‘π‘‘

1βˆ’πœ—πœ—π‘‘π‘‘= πœ—πœ—π‘‘π‘‘

1+πœ…πœ…π‘Žπ‘Ž1βˆ’πœ—πœ—π‘‘π‘‘+πœ…πœ…πœŒπœŒ

Capital market clearing (defines πœ“πœ“π‘‘π‘‘ for planner)1 βˆ’ πœƒπœƒπ‘‘π‘‘ = πœ“πœ“π‘‘π‘‘

πœ‚πœ‚π‘‘π‘‘(1 βˆ’ πœ—πœ—π‘‘π‘‘) 1 βˆ’ πœƒπœƒπ‘‘π‘‘ = 1βˆ’πœ“πœ“π‘‘π‘‘

1βˆ’πœ‚πœ‚π‘‘π‘‘(1 βˆ’ πœ—πœ—π‘‘π‘‘)

Money market clearing by Walras law37

Page 38: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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Solution Procedure for Both Settings

Price-taking Social Planner Problemmax

πœ“πœ“π‘‘π‘‘πœ—πœ—π‘‘π‘‘πΈπΈ π‘Ÿπ‘Ÿπ‘‘π‘‘π‘€π‘€ + 1 βˆ’ πœ—πœ—π‘‘π‘‘ 𝐸𝐸 π‘Ÿπ‘Ÿπ‘‘π‘‘πΎπΎ

βˆ’(πœπœπ‘‘π‘‘πœ“πœ“π‘‘π‘‘ + πœπœπ‘‘π‘‘(1 βˆ’ πœ“πœ“π‘‘π‘‘))(𝜎𝜎 + πœŽπœŽπ‘‘π‘‘π‘π‘+π‘žπ‘ž)

βˆ’( πœπœπ‘‘π‘‘πœ“πœ“π‘‘π‘‘ �𝜎𝜎 + 𝜍𝜍 1 βˆ’ πœ“πœ“π‘‘π‘‘ �𝜎𝜎)

FOC: πœπœπ‘‘π‘‘πœŽπœŽ + πœπœπ‘‘π‘‘ �𝜎𝜎 = πœπœπ‘‘π‘‘πœŽπœŽ + 𝜍𝜍 �𝜎𝜎 (prices of risks adjust for interior solution)

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Poll 38: Does 𝐸𝐸𝑑𝑑[π‘Ÿπ‘Ÿπ‘‘π‘‘πΎπΎ] depend on πœ“πœ“?a) Yesb) No

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Solution Procedure for Both Settings

Price-taking Social Planner Problemmax

πœ“πœ“π‘‘π‘‘πœ—πœ—π‘‘π‘‘πΈπΈ π‘Ÿπ‘Ÿπ‘‘π‘‘π‘€π‘€ + 1 βˆ’ πœ—πœ—π‘‘π‘‘ 𝐸𝐸 π‘Ÿπ‘Ÿπ‘‘π‘‘πΎπΎ

βˆ’(πœπœπ‘‘π‘‘πœ“πœ“π‘‘π‘‘ + πœπœπ‘‘π‘‘(1 βˆ’ πœ“πœ“π‘‘π‘‘))(𝜎𝜎 + πœŽπœŽπ‘‘π‘‘π‘π‘+π‘žπ‘ž)

βˆ’( πœπœπ‘‘π‘‘πœ“πœ“π‘‘π‘‘ �𝜎𝜎 + 𝜍𝜍 1 βˆ’ πœ“πœ“π‘‘π‘‘ �𝜎𝜎)

FOC: πœπœπ‘‘π‘‘πœŽπœŽ + πœπœπ‘‘π‘‘ �𝜎𝜎 = πœπœπ‘‘π‘‘πœŽπœŽ + 𝜍𝜍 �𝜎𝜎 (prices of risks adjust for interior solution)

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Page 40: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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1. Real Debt Setting: π‘žπ‘ž, 𝜍𝜍, Planner’s prob.

Set πœ—πœ—π‘‘π‘‘ = 0, β‡’ 𝑝𝑝 = 0

π‘žπ‘ž = 1+πœ…πœ…π‘Žπ‘Ž1+πœ…πœ…πœŒπœŒ

βˆ€π‘‘π‘‘ β‡’ πœŽπœŽπ‘žπ‘ž = πœŽπœŽπ‘π‘+π‘žπ‘ž = 0 (as in Basak Cuoco)

Prices of Risk

πœπœπ‘‘π‘‘ = πœŽπœŽπ‘‘π‘‘π‘›π‘› = 1 βˆ’ πœƒπœƒπ‘‘π‘‘ 𝜎𝜎 = πœ“πœ“π‘‘π‘‘πœ‚πœ‚π‘‘π‘‘πœŽπœŽ, πœπœπ‘‘π‘‘ = πœŽπœŽπ‘‘π‘‘

𝑛𝑛 = 1 βˆ’ πœƒπœƒπ‘‘π‘‘ 𝜎𝜎 = 1βˆ’πœ“πœ“π‘‘π‘‘1βˆ’πœ‚πœ‚π‘‘π‘‘

𝜎𝜎

πœπœπ‘‘π‘‘ = οΏ½πœŽπœŽπ‘‘π‘‘π‘›π‘› = 1 βˆ’ πœƒπœƒπ‘‘π‘‘ �𝜎𝜎 = πœ“πœ“π‘‘π‘‘πœ‚πœ‚π‘‘π‘‘οΏ½πœŽπœŽ, πœπœπ‘‘π‘‘ = οΏ½πœŽπœŽπ‘‘π‘‘

𝑛𝑛 = 1 βˆ’ πœƒπœƒπ‘‘π‘‘ �𝜎𝜎 = 1βˆ’πœ“πœ“π‘‘π‘‘1βˆ’πœ‚πœ‚π‘‘π‘‘

�𝜎𝜎

Plug in planners FOC: πœπœπ‘‘π‘‘πœŽπœŽ2 + πœπœπ‘‘π‘‘ �𝜎𝜎 = πœπœπ‘‘π‘‘πœŽπœŽ + 𝜍𝜍 �𝜎𝜎

πœ“πœ“π‘‘π‘‘ =πœ‚πœ‚π‘‘π‘‘(𝜎𝜎2 + �𝜎𝜎2)

𝜎𝜎2 + 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ �𝜎𝜎2 + πœ‚πœ‚π‘‘π‘‘ �𝜎𝜎2

πœ‡πœ‡π‘‘π‘‘πœ‚πœ‚ = β‹― ,πœŽπœŽπ‘‘π‘‘

πœ‚πœ‚ = β‹―40

Page 41: Macro, Money and Finance - PrincetonΒ Β· Price-Taking Planner’s Theorem: A social planner that takes prices as given chooses an physical asset and money allocation, 𝝍𝝍. 𝑑𝑑,

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1. Real Debt Setting: πœ‚πœ‚-Evolution

πœŽπœŽπ‘‘π‘‘πœ‚πœ‚ = 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœŽπœŽπ‘‘π‘‘π‘π‘ βˆ’ πœŽπœŽπ‘‘π‘‘π‘π‘ = 1 βˆ’ πœ‚πœ‚π‘‘π‘‘

πœ“πœ“π‘‘π‘‘πœ‚πœ‚π‘‘π‘‘βˆ’ 1βˆ’πœ“πœ“π‘‘π‘‘

1βˆ’πœ‚πœ‚π‘‘π‘‘πœŽπœŽ =

= πœ“πœ“π‘‘π‘‘βˆ’πœ‚πœ‚π‘‘π‘‘πœ‚πœ‚π‘‘π‘‘

𝜎𝜎

πœ‡πœ‡π‘‘π‘‘πœ‚πœ‚ = 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ βˆ’ πœŽπœŽπ‘‘π‘‘

�𝑁𝑁 πœŽπœŽπ‘‘π‘‘πœ‚πœ‚ βˆ’ πœŽπœŽπ‘‘π‘‘π‘€π‘€ βˆ’ (1 βˆ’ πœ‚πœ‚π‘‘π‘‘) πœπœπ‘‘π‘‘ βˆ’ πœŽπœŽπ‘‘π‘‘

�𝑁𝑁 πœŽπœŽπ‘‘π‘‘πœ‚πœ‚βˆ’ πœŽπœŽπ‘‘π‘‘π‘€π‘€

+ 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ οΏ½πœŽπœŽπ‘‘π‘‘π‘›π‘› βˆ’ 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ οΏ½πœŽπœŽπ‘‘π‘‘π‘›π‘› βˆ’ 𝐢𝐢𝑑𝑑

π‘π‘π‘‘π‘‘βˆ’ 𝐢𝐢𝑑𝑑+𝐢𝐢𝑑𝑑

π‘žπ‘žπ‘‘π‘‘+𝑝𝑝𝑑𝑑 𝐾𝐾𝑑𝑑

Benchmark asset is risk-free asset in 𝑁𝑁-numeraire πœŽπœŽπ‘‘π‘‘π‘€π‘€ = βˆ’πœŽπœŽ because πœŽπœŽπ‘‘π‘‘

�𝑁𝑁 = 𝜎𝜎 (since πœŽπœŽπ‘žπ‘ž = 0), 𝐢𝐢𝑁𝑁

= 𝜌𝜌

πœ‡πœ‡π‘‘π‘‘πœ‚πœ‚ = 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ βˆ’ 𝜎𝜎 πœŽπœŽπ‘‘π‘‘

πœ‚πœ‚ + 𝜎𝜎 βˆ’ (1 βˆ’ πœ‚πœ‚π‘‘π‘‘) πœπœπ‘‘π‘‘ βˆ’ 𝜎𝜎 πœŽπœŽπ‘‘π‘‘πœ‚πœ‚

+ 𝜎𝜎+ 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ οΏ½πœŽπœŽπ‘‘π‘‘π‘›π‘› βˆ’ 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ οΏ½πœŽπœŽπ‘‘π‘‘

𝑛𝑛

πœ‡πœ‡π‘‘π‘‘πœ‚πœ‚ = πœ“πœ“π‘‘π‘‘βˆ’πœ‚πœ‚π‘‘π‘‘

πœ‚πœ‚π‘‘π‘‘

πœ“πœ“π‘‘π‘‘βˆ’2πœ‚πœ‚π‘‘π‘‘πœ“πœ“π‘‘π‘‘+πœ‚πœ‚π‘‘π‘‘2

πœ‚πœ‚π‘‘π‘‘(1βˆ’πœ‚πœ‚π‘‘π‘‘)𝜎𝜎2 + 1 βˆ’ πœ‚πœ‚π‘‘π‘‘

πœ“πœ“π‘‘π‘‘πœ‚πœ‚π‘‘π‘‘

2�𝜎𝜎2 βˆ’ 1βˆ’πœ“πœ“π‘‘π‘‘

1βˆ’πœ‚πœ‚π‘‘π‘‘

2�𝜎𝜎2

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1. Real Debt Setting: risk free rate

π‘Žπ‘Žβˆ’πœ„πœ„π‘žπ‘ž

+ Ξ¦ πœ„πœ„ βˆ’ 𝛿𝛿 = π‘Ÿπ‘Ÿπ‘‘π‘‘π‘“π‘“ + πœπœπ‘‘π‘‘πœŽπœŽ + πœπœπ‘‘π‘‘ �𝜎𝜎

π‘Ÿπ‘Ÿπ‘‘π‘‘π‘“π‘“ = 𝜌𝜌 + Ξ¦ πœ„πœ„ βˆ’ 𝛿𝛿 βˆ’ πœ“πœ“π‘‘π‘‘

πœ‚πœ‚π‘‘π‘‘πœŽπœŽ2 + �𝜎𝜎2 , where πœ“πœ“π‘‘π‘‘

πœ‚πœ‚π‘‘π‘‘= (𝜎𝜎2+�𝜎𝜎2)

𝜎𝜎2+ 1βˆ’πœ‚πœ‚π‘‘π‘‘ �𝜎𝜎2+πœ‚πœ‚π‘‘π‘‘οΏ½πœŽπœŽ2

π‘Ÿπ‘Ÿπ‘‘π‘‘π‘“π‘“ = 𝜌𝜌 + Ξ¦ πœ„πœ„ βˆ’ 𝛿𝛿 βˆ’

𝜎𝜎2 + �𝜎𝜎2 𝜎𝜎2 + �𝜎𝜎2

𝜎𝜎2 + 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ �𝜎𝜎2 + πœ‚πœ‚π‘‘π‘‘ �𝜎𝜎2

Proposition: π‘Ÿπ‘Ÿπ‘‘π‘‘π‘“π‘“ is decreasing in �𝜎𝜎2

HH suffer from experts’ idiosyncratic risk exposure via a lower π‘Ÿπ‘Ÿπ‘‘π‘‘π‘“π‘“

Experts have more idio risk, but benefit from lower π‘Ÿπ‘Ÿπ‘‘π‘‘π‘“π‘“

(since they have to earn risk premium for idio risk)

Difference to Basak-Cuoco: limited participation πœ“πœ“ = 1,

HH fully at mercy of experts’ ability to hedge idio risk Here: HH participate in capital holding

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2. Money/Nominal Debt Setting: 𝜍𝜍s

Experts’ price of risk

πœπœπ‘‘π‘‘ = πœŽπœŽπ‘‘π‘‘π‘›π‘› = 𝜎𝜎 + πœŽπœŽπ‘‘π‘‘π‘π‘ + 1 βˆ’ πœƒπœƒπ‘‘π‘‘ πœŽπœŽπ‘‘π‘‘

π‘žπ‘ž βˆ’ πœŽπœŽπ‘‘π‘‘π‘π‘

= 𝜎𝜎 + πœŽπœŽπ‘‘π‘‘π‘π‘ +

πœ“πœ“π‘‘π‘‘πœ‚πœ‚π‘‘π‘‘

1 βˆ’ πœ—πœ—π‘‘π‘‘ πœŽπœŽπ‘‘π‘‘π‘žπ‘ž βˆ’ πœŽπœŽπ‘‘π‘‘

𝑝𝑝

πœπœπ‘‘π‘‘ = πœŽπœŽπ‘‘π‘‘οΏ½π‘›π‘› = 1 βˆ’ πœƒπœƒπ‘‘π‘‘ �𝜎𝜎 = πœ“πœ“π‘‘π‘‘πœ‚πœ‚π‘‘π‘‘

(1 βˆ’ πœ—πœ—π‘‘π‘‘) �𝜎𝜎

Households’ price of risk

πœπœπ‘‘π‘‘ = πœŽπœŽπ‘‘π‘‘π‘›π‘› = 𝜎𝜎 + πœŽπœŽπ‘π‘ + 1 βˆ’ πœƒπœƒπ‘‘π‘‘ πœŽπœŽπ‘‘π‘‘

π‘žπ‘ž βˆ’ πœŽπœŽπ‘π‘

= 𝜎𝜎 + πœŽπœŽπ‘π‘ +1 βˆ’ πœ“πœ“π‘‘π‘‘1 βˆ’ πœ‚πœ‚π‘‘π‘‘

1 βˆ’ πœ—πœ—π‘‘π‘‘ πœŽπœŽπ‘‘π‘‘π‘žπ‘ž βˆ’ πœŽπœŽπ‘π‘

πœπœπ‘‘π‘‘ = οΏ½πœŽπœŽπ‘‘π‘‘π‘›π‘› = 1 βˆ’ πœƒπœƒπ‘‘π‘‘ �𝜎𝜎 = 1βˆ’πœ“πœ“π‘‘π‘‘

1βˆ’πœ‚πœ‚π‘‘π‘‘(1 βˆ’ πœ—πœ—π‘‘π‘‘) �𝜎𝜎

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2. Money Setting: Planner’s Problem

Conjecture: πœŽπœŽπ‘‘π‘‘π‘žπ‘ž = πœŽπœŽπ‘‘π‘‘

𝑝𝑝 = 0 βˆ€π‘‘π‘‘β‡’ 𝜍𝜍 = 𝜎𝜎 = 𝜍𝜍 = 𝜎𝜎

Proposition: Aggregate risk is perfectly shared! Via inflation risk Stable inflation (targeting) would ruin risk-sharing

Example: Brexit uncertainty. Use inflation reaction to share risks within UK

Planner’s FOC: 𝜍𝜍 �𝜎𝜎 = 𝜍𝜍 οΏ½πœŽπœŽπœ“πœ“π‘‘π‘‘πœ‚πœ‚π‘‘π‘‘

1 βˆ’ πœ—πœ—π‘‘π‘‘ �𝜎𝜎2 = 1βˆ’πœ“πœ“π‘‘π‘‘1βˆ’πœ‚πœ‚π‘‘π‘‘

1 βˆ’ πœ—πœ—π‘‘π‘‘ �𝜎𝜎2

πœ“πœ“(πœ‚πœ‚,πœ—πœ—) does not depend on πœ—πœ—

πœ“πœ“ πœ‚πœ‚ =πœ‚πœ‚ �𝜎𝜎2

1 βˆ’ πœ‚πœ‚ �𝜎𝜎2 + πœ‚πœ‚ �𝜎𝜎244

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2. Money Setting: πœ‚πœ‚-Evolution

πœŽπœŽπ‘‘π‘‘πœ‚πœ‚ = 1 βˆ’ πœ‚πœ‚π‘‘π‘‘

πœ“πœ“π‘‘π‘‘πœ‚πœ‚π‘‘π‘‘βˆ’ 1βˆ’πœ“πœ“π‘‘π‘‘

1βˆ’πœ‚πœ‚π‘‘π‘‘πœ“πœ“π‘‘π‘‘βˆ’πœ‚πœ‚π‘‘π‘‘πœ‚πœ‚π‘‘π‘‘

1 βˆ’ πœ—πœ—π‘‘π‘‘ πœŽπœŽπ‘‘π‘‘π‘žπ‘ž βˆ’ πœŽπœŽπ‘π‘

If πœŽπœŽπ‘žπ‘ž = πœŽπœŽπ‘π‘ = 0, then πœŽπœŽπ‘‘π‘‘πœ‚πœ‚ = 0 βˆ€π‘‘π‘‘, if πœŽπœŽπ‘‘π‘‘

πœ‚πœ‚ = 0, then πœŽπœŽπ‘žπ‘ž = πœŽπœŽπ‘π‘ = 0By Ito’s lemma on π‘žπ‘ž(πœ‚πœ‚) and 𝑝𝑝(πœ‚πœ‚)

πœ‡πœ‡π‘‘π‘‘πœ‚πœ‚ = 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ βˆ’ πœŽπœŽπ‘‘π‘‘

�𝑁𝑁 πœŽπœŽπ‘‘π‘‘πœ‚πœ‚ βˆ’ πœŽπœŽπ‘‘π‘‘π‘€π‘€ βˆ’ (1 βˆ’ πœ‚πœ‚π‘‘π‘‘) πœπœπ‘‘π‘‘ βˆ’ πœŽπœŽπ‘‘π‘‘

�𝑁𝑁 πœŽπœŽπ‘‘π‘‘πœ‚πœ‚βˆ’ πœŽπœŽπ‘‘π‘‘π‘€π‘€

+ 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ οΏ½πœŽπœŽπ‘‘π‘‘π‘›π‘› βˆ’ 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ οΏ½πœŽπœŽπ‘‘π‘‘π‘›π‘› βˆ’ 𝐢𝐢𝑑𝑑

π‘π‘π‘‘π‘‘βˆ’ 𝐢𝐢𝑑𝑑+𝐢𝐢𝑑𝑑

π‘žπ‘žπ‘‘π‘‘+𝑝𝑝𝑑𝑑 𝐾𝐾𝑑𝑑

Benchmark asset is risk-free asset in 𝑁𝑁-numeraire πœŽπœŽπ‘‘π‘‘π‘€π‘€ = 0 and πœŽπœŽπ‘‘π‘‘

�𝑁𝑁 = 𝜎𝜎, 𝐢𝐢𝑁𝑁

= 𝜌𝜌

πœ‡πœ‡π‘‘π‘‘πœ‚πœ‚/ 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ = πœπœπ‘‘π‘‘ βˆ’ 𝜎𝜎 πœŽπœŽπ‘‘π‘‘

πœ‚πœ‚ βˆ’ πœπœπ‘‘π‘‘ βˆ’ 𝜎𝜎 πœŽπœŽπ‘‘π‘‘πœ‚πœ‚

+ πœπœπ‘‘π‘‘πœ“πœ“π‘‘π‘‘πœ‚πœ‚π‘‘π‘‘

(1 βˆ’ πœ—πœ—π‘‘π‘‘) �𝜎𝜎 βˆ’ πœπœπ‘‘π‘‘1βˆ’πœ“πœ“π‘‘π‘‘1βˆ’πœ‚πœ‚π‘‘π‘‘

(1 βˆ’ πœ—πœ—π‘‘π‘‘) �𝜎𝜎

πœ‡πœ‡π‘‘π‘‘πœ‚πœ‚ = 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ 1 βˆ’ πœ—πœ—π‘‘π‘‘ 2 πœ“πœ“π‘‘π‘‘

πœ‚πœ‚π‘‘π‘‘

2�𝜎𝜎2 βˆ’ 1βˆ’πœ“πœ“π‘‘π‘‘

1βˆ’πœ‚πœ‚π‘‘π‘‘

2�𝜎𝜎2

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2. Money Setting: Money Evaluation

Recall βˆ’πœ‡πœ‡π‘‘π‘‘πœ—πœ—= βˆ’(1 βˆ’ πœ—πœ—)πœ‡πœ‡π‘€π‘€ βˆ’πΆπΆπ‘‘π‘‘οΏ½π‘π‘π‘‘π‘‘

+ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ βˆ’ πœŽπœŽπ‘‘π‘‘οΏ½π‘π‘ πœŽπœŽπ‘‘π‘‘

πœ‚πœ‚ βˆ’ πœŽπœŽπ‘‘π‘‘πœ—πœ—

+ 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ βˆ’ πœŽπœŽπ‘‘π‘‘οΏ½π‘π‘ πœŽπœŽπ‘‘π‘‘

πœ‚πœ‚βˆ’ πœŽπœŽπ‘‘π‘‘πœ—πœ— + πœ‚πœ‚π‘‘π‘‘ 𝜍𝜍 οΏ½πœŽπœŽπ‘‘π‘‘π‘›π‘› + 1 βˆ’ πœ‚πœ‚π‘‘π‘‘ πœπœπ‘‘π‘‘ οΏ½πœŽπœŽπ‘‘π‘‘

𝑛𝑛

Plug in πœ‡πœ‡π‘€π‘€ = 0,𝐢𝐢𝑑𝑑�𝑁𝑁𝑑𝑑

= 𝜌𝜌, πœπœπ‘‘π‘‘ = πœπœπ‘‘π‘‘ = 𝜎𝜎, πœŽπœŽπ‘‘π‘‘οΏ½π‘π‘ = 𝜎𝜎

𝜍𝜍 οΏ½πœŽπœŽπ‘‘π‘‘π‘›π‘› = 1 βˆ’ πœ—πœ—π‘‘π‘‘ 2 πœ“πœ“π‘‘π‘‘πœ‚πœ‚π‘‘π‘‘

2

�𝜎𝜎2, πœπœπ‘‘π‘‘ = οΏ½πœŽπœŽπ‘‘π‘‘π‘›π‘› = 1 βˆ’ πœ—πœ—π‘‘π‘‘ 2 1 βˆ’ πœ“πœ“π‘‘π‘‘

1 βˆ’ πœ‚πœ‚π‘‘π‘‘

2

�𝜎𝜎2

βˆ’πœ‡πœ‡π‘‘π‘‘πœ—πœ—= βˆ’πœŒπœŒ + 1 βˆ’ πœ—πœ—π‘‘π‘‘ 2 πœ‚πœ‚π‘‘π‘‘πœ“πœ“π‘‘π‘‘πœ‚πœ‚π‘‘π‘‘

2�𝜎𝜎2 + 1 βˆ’ πœ‚πœ‚π‘‘π‘‘

1βˆ’πœ“πœ“π‘‘π‘‘1βˆ’πœ‚πœ‚π‘‘π‘‘

2�𝜎𝜎2

where πœ“πœ“π‘‘π‘‘ = πœ‚πœ‚π‘‘π‘‘οΏ½πœŽπœŽ2

1βˆ’πœ‚πœ‚π‘‘π‘‘ �𝜎𝜎2+πœ‚πœ‚π‘‘π‘‘οΏ½πœŽπœŽ2

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2. Money Setting: Adding Real Debt

Adding Real Debt does not alter the equilibrium, since Markets are complete w.r.t. to aggregate risk

(perfect aggregate risk sharing) Markets are incomplete w.r.t. to idiosyncratic risk only

Note: Result relies on absence of price stickiness

Both Settings: Real Debt and Money/Nominal Debt converge in the long-run to the β€œI Theory without I” steady state model of Lecture 05.

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Example: Real vs. Nominal Debt/Money

π‘Žπ‘Ž = .15,𝜌𝜌 = .03,𝜎𝜎 = .1, πœ…πœ… = 2, 𝛿𝛿 = .03, �𝜎𝜎 = .2, �𝜎𝜎 = .3

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Blue: real debt modelRed: nominal model

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Towards the I Theory of Money One sector model with idio risk - β€œThe I Theory without I”

(steady state focus) Store of value Insurance role of money within sector

Money as bubble or not Fiscal Theory of the Price Level Medium of Exchange Role β‡’ SDF-Liquidity multiplier β‡’ Money bubble

2 sector/type model with money and idio risk Generic Solution procedure (compared to lecture 03) Real debt vs. Money Implicit insurance role of money across sectors

The curse of insurance Reduces insurance premia and net worth gains

I Theory with Intermediary sector Intermediaries as diversifiers

Welfare analysis Optimal Monetary Policy and Macroprudential Policy

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