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Optimal Fiscal and MonetaryPolicy

Outline

(1) Background: Phelps-Friedman Debate

(2) Some Ideas from Public Finance -Ramsey Theory

– Policy– Private Sector Equilibrium– Private Sector Allocation Rule– Ramsey Problem– Ramsey Equilibrium– Implementability Constraint– Ramsey Allocation Problem– Ramsey Allocations

(3) Simple One-Period Example

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(4) Evaluating Phelps-Friedman DebateUsing Lucas-Stokey Cash-Credit GoodModel

(a) General Remarks(b) Model(c) Ramsey Problem, Ramsey Allocation

Problem(d) Surprising Result:

Friedman is “Right” for Lots of Pa-rameterizations (Used Homotheticityand Separability).

(5) Interpretation of Result(a) Uniform Taxation Result in Pub-

lic Finance for Non-MonetaryEconomies

(b) Homotheticity and SeparabilityCorresponds to Unit ConsumptionElasticity of Money Demand

(c) What Happens When You Don’tHave Unit Elasticity?

(d) Who Is Right, Friedman or Phelps?

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(6) What Happens When g, z Are Random?(Answer: Make P Random)

(7) Financing a War: Barro versus Ramsey.

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Friedman-Phelps Debate

• Money Demand:

M

P= exp[−αR]

• Friedman:(a) Efforts to Economize Cash Balances

when R High is Socially Wasteful(b) Set R as Low As Possible: R = 1.(c) Since R = 1 + r + π, Friedman

Recommends π = −r.(i) r ∼ exogenous (net) real interest

rate rate(ii) π ∼ inflation rate, π = (P −

P−1)/P−1

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• Phelps:(a) Inflation Acts Like a Tax on Cash

Balances -

Seignorage =Mt −Mt−1

Pt=Mt

Pt− Pt−1Pt

Mt−1Pt−1

≈ M

P

π

1 + π

(b) Use of Inflation Tax Permits ReducingSome Other Tax Rate

(c) Extra Distortion in Economizing CashBalances Compensated by ReducedDistortion Elsewhere.

(d) With Distortions a Convex Functionof Tax Rates, Would Always Want toTax All Goods (Including Money) AtLeast A Little.

(e) Inflation Tax Particularly Attractive ifInterest Elasticity of Money DemandLow.

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Question: Who is Right,Friedman or Phelps?

• Answer: Friedman Right SurprisinglyOften

• Depends on Income Elasticity of Demandfor Money

• Will Address the Issue From a StraightPublic Finance Perspective, In the Spiritof Phelps.

• Easy to Develop an Answer, Exploiting aBasic Insight From Public Finance.

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Some Basic Ideas fromRamsey Theory

• Policy, π, Belonging to the Set of ‘BudgetFeasible’ Policies, A.

• Private Sector Equilibrium Allocations,Equilibrium Allocations, x, Associatedwith a Given π; x ∈ B.

• Private Sector Allocation Rule, mappingfrom π to x (i.e., π : A→ B).

• Ramsey Problem: Maximize, w.r.t. π,U(x(π)).

• Ramsey Equilibrium: π∗ ∈ A and x∗,such that π∗ solves Ramsey Problemand x∗ = x(π∗). ‘Best Private SectorEquilibrium’.

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• Ramsey Allocation Problem: Solve,x = argmax U(x) for x ∈ B

• Alternative Strategy for Solving theRamsey Problem:

(a) Solve Ramsey Allocation Problem, toFind x.

(b) Execute the Inverse Mapping, π =x−1(x).

(c) π and x Represent a Ramsey Equilib-rium.

• Implementability Constraint: Equationsthat Summarize Restrictions on Achiev-able Allocations, B, Due to DistortionaryTax System.

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Policy, π

Set, A, of Budget-Feasible Policies

Private sector Allocation Rule, x(π)

Set, B, of Private Sector Allocations Achievable by Some Budget-Feasible Policy

Private Sector Equilibrium

Allocations, x

Utility

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Example

• Households:

maxc,lu(c, l)

c ≤ z(1− τ )l,

z ∼ wage rateτ ∼ labor tax rate

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• Household Problem Implies Private SectorAllocation Rules, l(τ ), c(τ ), defined by:

ucz(1− l) + ul = 0, c = (1− τ )zl

l(τ) l

u[z(1-τ)l,l] ucz(1-τ)+ul=0

Private Sector Allocation Rules: l(τ), c(τ) = z(1-τ)l

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• Ramsey Problem:

maxτu(c(τ ), l(τ ))

subject to g ≤ zl(τ )τ

• Ramsey Equilibrium: τ ∗, c∗, l∗ such that(a) c∗ = c(τ ∗), l∗ = l(τ ∗)

‘Private Sector Allocations are aPrivate Sector Equilibrium’

(b) τ ∗ Solves Ramsey Problem‘Best Private Sector Equilibrium’

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Analysis of RamseyEquilibrium

• Simple Utility Specification:

u(c, l) = c− 12l2

• Two Ways to Compute the RamseyEquilibrium

(a) Direct Way: Solve Ramsey Problem(In Practice, Hard)

(b) Indirect Way: Solve Ramsey Alloca-tion Problem (Can Be Easy)

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Direct Approach

• Private Sector Allocation Rules:

ucz(1− τ ) + ul = 0, c = (1− τ )zl

=⇒ z(1− τ ) = l(τ )

=⇒ c(τ ) = z(1− τ )l(τ ) = z2(1− τ )2

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• Ramsey Problem:

maxτ

1

2z2(1− τ )2

subject to : g ≤ τzl(τ ) = τz2(1− τ ).

¼z2

g

τz2(1-τ) ‘Laffer Curve’

0 1 τ

½z2(1-τ)2

τ1 τ2

τ* = τ1 =½ -½[ 1 – 4 g/z2 ]½ τ2 =½+½[ 1 – 4 g/z2 ]½

l(τ*) =½{ z+[ z2 – 4 g ]½ }

A

Government Preferences

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Indirect Approach

• Approach: Solve Ramsey AllocationProblem, Then ‘Inverse Map’ Back intoPolicies

• Problem: Would Like a Characterizationof B that Only Has (c, l), Not the Policies

B = {c, l : ∃τ , with ucz(1− τ ) + ul = 0,

c = (1− τ )zl, g ≤ τzl}

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• Solution: Rearrange Equations in B, SoThat Only (c, l) Appears

(a) Multiply Household Budget Equationby uc :

ucc− uc(1− τ )zl = 0

(b) Substitute out for uc(1 − τ )z FromLabor First Order Condition:

(∗) ucc + ull = 0.

(c) Combine Household Budget EquationWith Government Budget Constraint:

(∗∗) c + g ≤ zl.

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• Previous Manipulations Suggest Candi-date Alternative Representation of B :

D =

(c, l) : c + g ≤ zl| {z }resource constraint

, ucc + ull = 0| {z }implementability constraint

• Want,D = B,

(a) Have Shown (c, l) ∈ B → (c, l) ∈ D(b) Still Need to Verify (c, l) ∈ D →

(c, l) ∈ B

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Proof that (c, l) ∈ D⇒ (c, l) ∈ B

• Suppose (c, l) ∈ D, i.e., ucc + ull = 0,c + g ≤ zl

• Need to show:

∃τ s.t. (i) uc(1−τ )z+ul = 0, (ii) c = (1−τ )zl, (iii) g ≤ τzl

• Set τ so that

1− τ =−ulucz

, so (i) holds.

• Multiply Both Sides by lz and rewrite:

(1− τ ) lz =−ulluc

= c, so (ii) holds.

• (iii) follows (ii) and c + g ≤ zl.

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• Conclude:

B = D

• Express Ramsey Allocation Problem:

maxc,lu(c, l)

s.t. ucc + ull = 0, c + g ≤ zl

or

maxll2

s.t. l2 + g ≤ zl

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Ramsey Allocation Problem:

Max ½l2 Subject to l2 + g ≤ zl Solution: l2 = ½{ z + [ z2 - 4g ]½ } Same Result as Before!

l2 - zl +g = 0

½z l1 l2

g - ¼z2

½l2

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Lucas-Stokey Cash-CreditGood Model

• Households

• Firms

• Government

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Households

• Household Preferences:

∞Xt=0

βtu(c1t, c2t, lt),

c1t ˜ cash goods, c2t ˜ credit goods, lt ˜ labor

• Distinction Between Cash and CreditGoods:

– All Goods Paid With Cash At the SameTime, After Goods Market, in AssetMarket

– Cash Good: Must Carry Cash In PocketBefore Consuming It

Mt ≥ Ptc1t

– Credit Good: No Need to Carry CashBefore Purchase.

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Household Participation inAsset and Good Markets

• Asset Market: First Half of Period,When Household Settles Financial ClaimsArising Activities in Previous AssetMarket and in Previous Goods Market.

• Goods Market: Second Half of Period,Goods are Consumed, Labor Effort isApplied, Production Occurs.

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t t+1

Asset Market Goods Market

Sources of Cash for Household: • Md

t-1 - Pt-1c1,t-1 - Pt-1c2,t-1 • Rt-1Bd

t-1 • (1-τt-1)zlt-1

Uses of Cash • Bonds, Bd

t • Cash, Md

t

• c1,t , c2,t Purchased • lt Supplied • Production Occurs • Md

t Not Less Than Ptc1,t

• Constraint On Households in AssetMarket (Budget Constraint)

Mdt +B

dt

≤ Mdt−1 − Pt−1c1t−1 − Pt−1c2t−1

+Rt−1Bdt−1 + (1− τ t−1)zlt−1

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Household First OrderConditions

• Cash versus Credit Goods:

u1tu2t= Rt

• Cash Goods Today versus Cash GoodsTomorrow:

u1t = βu1t+1RtPtPt+1

• Credit Goods versus Leisure:

u3t + (1− τ t)zu2t = 0.

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Firms

• Technology: y = zl

• Competition Guarantees Real Wage = z.

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Government

• Inflows and Outflows in Asset Market(Budget Constaint):

Mst −Ms

t−1 +Bst| {z }

Sources of Funds

≥Rt−1Bst−1 + Pt−1gt−1 − Pt−1τ t−1zlt−1| {z }Uses of Funds

• Policy:

π = (Ms0 ,M

s1 , ..., B

s0, B

s1, ..., τ 0, τ 1, ...)

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Ramsey Equilibrium

• Private Sector Allocation Rule:

For each policy, π ∈ A, there is a Private Sector Equilibrium:x = ({c1t} , {c2t} , {lt} , {Mt} , {Bt})

p = ({Pt} , {Rt})Mt =M

st =M

dt

Bt = Bst = B

dt

Rt ≥ 1 (i.e., u1t/u2t ≥ 1)

• Ramsey Problem:

maxπ∈A

U(x(π))

• Ramsey Equilibrium:

π∗, x(π∗), p(π∗),

Such that π∗ Solves Ramsey Problem.

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Finding The RamseyEquilibrium By Solving theRamsey Allocation Problem

max{c1t,c2t,lt}∈D

∞Xt=0

βtu(c1t, c2t, lt),

where D is the set of allocations, c1t, c2t, lt,t = 0, 1, 2, ..., such that

∞Xt=0

βt[u1tc1t + u2tc2t + u3tlt] = u2,0a0,

c1t + c2t + g ≤ zlt,u1tu2t≥ 1,

a0 =R−1B−1P0

∼ real value of initial government debt

Assumption: B−1 = 0.

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Lagrangian Representation ofRamsey Allocation Problem:

• There is a λ ≥ 0, Such that the Solutionto the RA Problem and the FollowingProblem Coincide:

max{c1t,c2t,lt}

∞Xt=0

βtW (c1t, c2t, lt;λ)

subject to : c1t + c2t + g ≤ zlt, u1tu2t≥ 1,

W (c1t, c2t, lt;λ) = u(c1t, c2t, lt) + λ[u1tc1t + u2tc2t + u3tlt].

• Note that If You Could Ignore u1tu2t≥ 1,

Optimization Implies

W1t

W2t= 1

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Restricting the UtilityFunction

• Utility Function:

u(c1, c2, l) = h(c1, c2)v(l),

h ∼ homogeneous of degree kv ∼ strictly decreasing.

• Then, u1c1 + u2c2 + u3l = h [kv + v0], so

W (c1, c2, l;λ) = hv + λh [kv + v0]= h(c1, c2)Q(l,λ).

• Conclude - Homogeneity and SeparabilityImply:

W1(c1, c2, l;λ)

W2(c1, c2, l;λ)=u1(c1, c2, l)

u2(c1, c2, l).

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Surprising Result: Friedmanis Right More Often Than

You Might Expect

• Suppose You Can Ignore u1t/u2t ≥ 1Constraint. Then, Necessary Condition ofSolution to Ramsey Allocation Problem:

W1(c1, c2, l;λ)

W2(c1, c2, l;λ)= 1.

• This, In Conjunction with Homogeneityand Separability, Implies:

u1(c1, c2, l)

u2(c1, c2, l)= 1.

• Note: u1t/u2t ≥ 1 is Satisfied, So Restric-tion is Redundant Under Homogeneityand Separability.

• Conclude: R = 1, So Friedman Right!

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Generality of the Result

• Result is True for the Following MoreGeneral Class of Utility Functions:

u(c1, c2, l) = V (h(c1, c2), l),

where h is homothetic.

• Analogous Result Holds in ‘Money inUtility Function’ Models and ‘Transac-tions Cost’ Models (Chari-Christiano-Kehoe, Journal of Monetary Economics,1996.)

• Actually, strict homotheticity and separa-bility are not necessary.

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Interpretation of the Result

• ‘Looking Beyond the Monetary Veil’ -The Connection Between The R = 1Result and the Uniform Taxation Resultfor Non-Monetary Economies

• The Importance of Homotheticity

• The Link Between Homotheticity andSeparability, and The ConsumptionElasticity of Money Demand.

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Uniform Taxation Resultfrom Public Finance For

Non-Monetary Economies

• Households:

maxc1,c2,l

u(c1, c2, l)

s.t. zl ≥ c1(1 + τ 1) + c2(1 + τ 2)

⇒ c1 = c1(τ 1, τ 2), c2 = c2(τ 1, τ 2), l = l(τ 1, τ 2).

• Ramsey Problem:

maxτ 1,τ 2

u(c1(τ 1, τ 2), c2(τ 1, τ 2), l(τ 1, τ 2))

s.t. g ≥ c1(τ 1, τ 2)τ 1 + c2(τ 1, τ 2)τ 2

• Uniform Taxation Result:

if u = V (h(c1, c2), l), h ∼ homotheticthen τ 1 = τ 2.

Proof : trivial! (just study Ramsey Allocation Problem)

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Similarities to MonetaryEconomy

• Rewrite Budget Constraint:

zl

1 + τ 2≥ c11 + τ 1

1 + τ 2+ c2.

• Similarities:

1

1 + τ 2∼ 1− τ ,

1 + τ 11 + τ 2

∼ R.

• Positive Interest Rate ‘Looks’ Like aDifferential Tax Rate on Cash and CreditGoods.

• Have the Same Ramsey AllocationProblem, Except Monetary EconomyAlso Has:

u1u2≥ 1.

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What Happens if You Don’tHave Homotheticity?

• Utility Function:

u(c1, c2, l) =c1−σ1

1− σ+c1−δ2

1− δ+ v(l)

• ‘Utility Function’ in Ramsey AllocationProblem:

W (c1, c2, l) = [1 + (1− σ)λ]c1−σ1

1− σ

+ [1 + (1− δ)λ]c1−δ2

1− δ+ v(l) + λv0(l)l

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• Marginal Rate of Substitution in RamseyAllocation Problem That Ignores u1/u2 ≥1 Condition:

1 =W1(c1, c2, l;λ)

W2(c1, c2, l;λ)=1 + (1− σ)λ

1 + (1− δ)λ× u1u2,

or, since u1/u2 = R :

R =1 + (1− δ)λ

1 + (1− σ)λ

• Finding:

δ = σ ⇒ R = 1 (homotheticity case)δ > σ ⇒ R ≥ 1 Binds, so R = 1δ < σ ⇒ R > 1.

Note: Friedman Right More OftenThan Uniform Taxation Result, Becauseu1/u2 ≥ 1 is a Restriction on theMonetary Economy, Not the BarterEconomy.

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Consumption Elasticity ofDemand

• Homotheticity and Separability Corre-spond to Unit Consumption Elasticity ofMoney Demand.

• Money Demand:

R =u1u2=h1h2= f

µc2c1

¶= f

Ãc− M

PMP

!= f

µc

M/P

¶.

• Note: Holding R Fixed, Doubling cImplies DoublingM/P

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Elasticity of Money Demandand Failure of Homotheticity

• Money Demand:

R =u1u2=c−σ1c−δ2

=

¡MP

¢−σ¡c− M

P

¢−δ• Taylor Series Approximation About

Steady State (m ≡M/P in steady state) :

m =1

mc +

σδ

¡1− m

c

¢| {z }×cConsumption Money Demand Elasticity, εM

− 1

δ mc−m + σ| {z }×RInterest Elasticity

• Can Verify:Utility Function Non-Monetary MonetaryParameters εM Economy Economyδ > σ εM > 1 τ 2 ≥ τ 1 R = 1δ < σ εM < 1 τ 2 < τ 1 R > 1δ = σ εM = 1 τ 1 = τ 2 R = 1

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Bottom Line:

• Friedman is Right (R = 1) When Con-sumption Elasticity of Money Demand isUnity or Greater

• Implicitly, High Interest Rates TaxSome Goods More Heavily that Others.Under Homotheticity and SeparabilityConditions, Want to Tax Goods at SameRate.

• What is Consumption Elasticity in theData?

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What To Do, When g, z AreRandom?

• Ramsey Principle: Minimize Tax Distor-tions

• If There is A Low Elasticity Item, Tax It

• If a Bad Shock Hits: Tax Capital(i.e., hit things that reflect past decisionslike physical capital)

• Important ..... If a Good Shock Hits:Subsidize Capital(that minimizes ex ante distortions tocapital accumulation)

• Movements in P May Be Best Thing (seeSimulations)This Conclusion Will Be Dependent onDegree of Price Stickiness

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Financing War: Barro versusRamsey

When War (or Other Large Financing Need)Suddenly Strikes:

• Barro:– Raise Labor and Other Tax Rates a

Small Amount So That When HeldConstant at That Level, Expected Valueof War is Financed

– This Minimizes Intertemporal Substi-tution Distortions

– Involves a Big Increase in Debt inShort Run

– Prediction for Labor Tax Rate: RandomWalk.

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• Ramsey:– Tax Existing Capital Assets (Human,

Physical, etc) For Full Amount ofExpected Value of War. Do This at theFirst Sign of War.

– This Minimizes Intertemporal andIntratemporal Distortions (Don’tChange Tax Rates on Income at all).

– Example:∗ Suppose War is Expected to Last

Two Periods, Cost: $1 Per Period∗ Suppose Gross Rate of Interest is

1.05 (i.e., 5%)∗ Tax Capital 1 + 1/1.05 = 1.95 Right

Away.∗ Debt Falls $0.95 in Period When

War Strikes.– Involves a Reduction of Outstanding

Debt in Short Run.– Prediction for Labor Tax Rate: Roughly

Constant.