optimal fiscal and monetary policy outlinelchrist/course/... · 2003. 4. 10. · find x.˜ (b)...
TRANSCRIPT
1
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
2
(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?
3
(6) What Happens When g, z Are Random?(Answer: Make P Random)
(7) Financing a War: Barro versus Ramsey.
4
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
5
• 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.
6
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.
7
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’.
8
• 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.
9
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
10
Example
• Households:
maxc,lu(c, l)
c ≤ z(1− τ )l,
z ∼ wage rateτ ∼ labor tax rate
11
• 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
12
• 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’
13
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)
14
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
15
• 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
16
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}
17
• 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.
18
• 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
19
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.
20
• 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
21
0
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
22
Lucas-Stokey Cash-CreditGood Model
• Households
• Firms
• Government
23
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.
24
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.
25
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
26
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.
27
Firms
• Technology: y = zl
• Competition Guarantees Real Wage = z.
28
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, ...)
29
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.
30
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.
32
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
33
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).
34
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!
35
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.
36
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.
37
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)
39
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.
41
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
42
• 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.
43
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
44
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
45
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?
46
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
47
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.
48
• 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.