Optimal Fiscal Policy Lecture 1 a Winter School 2019 Delhi School of Economics

Optimal Fiscal Policy Lecture 1 a

Winter School 2019 Delhi School of Economics

V.V. Chari

University of Minnesota Federal Reserve Bank of Minneapolis

December 2019

Outline

Take government expenditures as given Take available tax system as given

Design taxes to maximize representative agent’s welfare Representative agent

Heterogeneous agents

Representative Agent

Ramsey Taxation

Use primal approach, develop recursive methods Findings

I No intertemporal distortions in steady state

I No intertemporal distortions with standard macro preferences

I Production efficiency is optimal

I Show multiple implementation

Ramsey Approach: Tax Instruments Given

We assume rich tax system: taxes commonly used worldwide; labor, capital, and dividend income, consumption

Private behavior modeled as competitive equilibrium Ramsey equilibrium is best competitive equilibrium Ramsey approach yields wedges

I Taxes not necessarily pinned down

I Multiple implementations

Need to take stand on initial policies or promises

I Assume value of wealth cannot be below benchmark level

Preferences and Technology

Preferences

U =

∞

X

t=0

β^tu(ct,nt)

Technology

c_t+g_t+x_t ≤F(k_t,n_t) (1) xt=kt+1−(1−δ)kt

ct: consumption of final good nt: labor input

gt: exogenous government consumption kt: capital

xt: investment

Benchmark Tax System

Consumption tax: τ_t^c Labor income tax: τ_tⁿ Tax on initial wealth: `₀ Tax on dividends: τ_t^d

Competitive Equilibrium: Households

max

∞

X

t=0

β^tu(c_t,n_t) subject to

∞

X

t=0

qt

(1 +τ_c^t)ct−(1−τ_tⁿ)wtnt

≤(1−`0)

"

b0+

∞

X

t=0

qt(1−τ_t^d)dt

#

qt: intertemporal price wt: wage rate

`0: initial wealth tax b0: initial government debt dt: dividends

Competitive Equilibrium: Firms

max

∞

X

t=0

qt(1−τ_t^d)dt

subject to

dt =F(kt,nt)−wtnt−τ_t^k[F(kt,nt)−wtnt−δkt]−[kt+1−(1−δ)kt] dt can be interpreted either as dividends or as payments on debt Miller-Modigliani theorem applies here

Competitive Equilibrium: Government

Budget constraint

∞

X

t=0

q_t

τ_t^cc_t+τ_tⁿw_tn_t+τ_t^dd_t−g_t +b₀

"

`₀+

∞

X

t=0

q_t(1−τ_t^d)d_t

#

= 0

Competitive and Ramsey Equilibrium

CE is a set of allocation{c_t,n_t,k_t+1,d_t}, prices {q_t,w_t}, and policies τ_t^c, τ_tⁿ, τ_t^d, τ_t^k, `0 such that:

I HH solves their problem

I Firms solve their problem

I Markets clear (Resource constraint holds)

Ramsey equilibrium is CE with highest utility for households

Characterization: Necessary Conditions

Lemma 1: Any CE must satisfy the implementability constraint

∞

X

t=0

β^t[c_tu_ct+n_tu_nt] =W0, (IC) where

W₀= (1−`0)uc0

1 +τ₀^c [(1−δ+F_k0)k₀+b₀] Proof:

I Substitute first order conditions of households into households’ budget constraint.

I Use first order conditions of firm to show that

∞

X

t=0

q_t(1−τ_t^d)d_t = (1−τ₀^d)

1 + (1−τ₀^k)(F_k0−δ)k₀

Left as exercise.

Wedges and Multiple Implementations

Key first order conditions

I Intratemporal wedge

−unt

u_ct = (1−τ_tⁿ)Fnt

(1 +τ_t^c) (2)

I Intertemporal wedge (1−τ_t^d) (1 +τ_t^c)

uct

βuc,t+1

= (1−τ_t+1^d ) 1 +τ_t+1^c

1 + (1−τ_t+1^k )(Fk,t+1−δ) (3) Two of these taxes are redundant

Will say there are no intertemporal distortions if 3 is undistorted

Characterization: Sufficient Conditions

Lemma 2: Given any allocations and date zero policies that satisfy implementability constraint and resource constraint, there exist prices and policies such that the collection is a competitive equilibrium.

Proof: Setq0= 1

qt=β^tuct

u_c0

(1 +τc0) (1 +τ_ct) w_t =F_nt

−unt

uct

= wt(1−τ_tⁿ) 1 +τ_t^c qt(1−τ_t^d) =qt+1(1−τ_t+1^d )

1 + (1−τ_t+1^k )(Fk,t+1−δ)

Characterization Theorem

Proposition 1: Any CE satisfies IC and RC. If an allocation and period 0 policies satisfy IC and RC it is implementable as a CE.

I Proof: Follows from Lemma 1 and 2.

Define a tax system as complete if it satisfies second part of proposition 1.

Incomplete Tax System and Restriction

Supposeτ_t^c =τ_t^d= 0 and 0≤τ_t^k ≤1.

Then sufficiency still holds.

Need to add restriction (from intertemporal condition) that u_ct

βuc,t+1

−1

F_k,t+1−δ ≥0 (4)

Necessity now says if allocation satisfies IC, RC and 4 it is implementable.

Multiple Implementations

Diamond-Mirrlees system: taxes all final consumption goods, all primary inputs, here labor, and fixed factors here initial wealth: τ_t,τ_tⁿ,`₀.

I Such a tax system is always complete.

Abel system: tax dividends and labor.

I Withτ₀^d≤1 system may be incomplete.

Chamley-Judd system: onlyτ_tⁿandτ_t^k.

I Ifτ_t^k ≤1, system may be incomplete.

Wealth Restriction

Will impose wealth restriction that W0≥W¯ That is

∞

X

t=0

β^t(ctuct+ntunt)≥W¯

Spirit is that the value of wealth in utility term cannot be reduced below ¯W.

Ramsey Problem

Using proposition 1 we have that Ramsey problem is to choose (c_t,n_t,k_t+1) and period zero policies to solve

max

∞

X

t=0

β^tu(ct,nt) subject to

∞

X

t=0

β^t(ctuct+ntunt) =W0, where

W₀= (1−`0)uc0

1 +τ₀^c

b₀+ (1−τ₀^d) 1 + (1−τ₀^k)(F_k0−δ) k₀

W0≥W¯

Characterization of Ramsey Solution

LetW =u(c,n) +λ[cuc+nun]

−W_nt Wct

=F_nt

Wct =βWc,t+1[1 + (Fk,t+1−δ)]

Hereλis the Lagrange multiplier on the resource constraint.

Proposition 2: If economy converges to a steady state, no intertemporal wedges.

Proposition 3: Ifu(c,n) = c^1−σ

1−σ−φn^ξ+1

ξ+ 1, thenWc ∝Uc,Wn∝Un, so never intertemporal distortion, constant intratemporal distortion.

Key Property of Ramsey Allocation

Resource constraint holds with equality. That is we have production efficiency.

Production efficiency is requirement that allocation are at the boundary of the production set.

Implies intermediate goods (like investment) production decisions are not distorted.

Recursive Formulation

Consider functional equation

V_t(k,W) = maxu(c,n) +βV_t+1(k⁰,W⁰) subject to

βW⁰+cuc+nun=W

c+gt+k⁰−(1−δ)k ≤F(k,n)

Ramsey problem is equivalent to functional equation ifβ^tV_t→0.

Extension to Uncertainty

Letst be random variable

Lets^t be the history of shocks with probabilityµ(s^t) Preferences are U=P∞

t=0

P

s^tβ^tu(c(s^t),n(s^t)) Budget constraint becomes

∞

X

t=0

X

s^t

q(s^t)

(1 +τ^c(s^t))c(s^t)−(1−τⁿ(s^t))n(s^t)

= (1−`₀)

"

b₀+

∞

X

t=0

X

s^t

q(s^t)(1−τ^d(s^t))d_t

#

Resource constraint becomes

Extensions

Implementability constraint becomes X

t

X

s^t

β^tµ(s^t)

c(s^t)uc(s^t) +n(s^t)un(s^t)

=(1−`0)uco

1 +τ₀^c

b0+ (1−τ₀^d) 1 + (1−τ₀^k)(Fk0−δ) k0

Ramsey problem is to maximize expected utility subject to IC and RC.

Can also set up problem with households making investment decisions rather than firms. Trivial to show equivalence. (See paper for details.)

Ramsey with Heterogeneous Agents

Heterogeneous Agents Model

Continuum of agents indexed byθdrawn from distribution G(θ) Allocations now indexed by type of agentθ

Preferences

u(θ) =

∞

X

t=0

β^tu(ct(θ),nt(θ))

Resource constraint Z

ct(θ)dG(θ) +gt+kt+1−(1−δ)kt ≤AtF

kt, Z

θnt(θ)dG(θ)

Planner’s preferences

Z

u(θ)dλ(θ)dG(θ)

Ramsey Problem

Assume initially taxes can depend onθ Implementability constraint

∞

X

t=0

β^t[ct(θ)uct(θ) +nt(θ)unt(θ)]

= (1−`0(θ)) [b0(θ) + (1 + (1−τko(θ))) (Fk0−δ)k0(θ)]

Ramsey problem

max Z

u(θ)dG(θ)dλ(θ) subject to IC and RC and LHS of IC ≥ W(θ)

Ramsey Solution

If the economy converges to a steady state, no intertemporal distortion in the steady state

If preferences are of standard macro form, no intertemporal distortion ever and constant intratemporal distortion (Same for all agents.)

That is, outcome can be implemented with constant tax rates on consumption that are the same for all agents

Restricted Tax System if Taxes Cannot Depend on Θ

Then CE must satisfy

−u_n(θ)

uc(θ) =θFn(θ)(1−τ) −u_n(ˆθ)

uc(ˆθ) = ˆθFn(ˆθ)(1−τ) Then must add restriction to characterization theorem

θuct(ct(θ),nt(θ)) u_nt(c_t(θ),n_t(θ)) =

θuˆ ct

ct(ˆθ),nt(ˆθ) unt

ct(ˆθ),nt(ˆθ) Similarly, from the intertemporal condition

uct(ct(θ),nt(θ)) βuc,t+1(ct+1(θ),nt+1(θ)) =

u_ct

c_t(ˆθ),n_t(ˆθ) βu_c,t+1

c_t+1(ˆθ),n_t+1(ˆθ)

Recursive Formulation without Restriction

State variables arek,H(W, θ): distribution over promised wealth, andθ.

s= (W, θ).

V(k,H) = max Z

u(c(s),n(s))dλ(θ)dH(s) +βV(k⁰,H⁰)

subject to

c+g+k⁰−(1−δ)k =AF

k, Z

n(s)dG(θ)

βW⁰(θ) +uc(s)c(s⁰) +n(s)un(s) =W(θ)

H⁰(W⁰, θ) = Z

B(W,θ)

dH(W, θ) where

B(W⁰, θ) ={W;h(W, θ)≤ W⁰}

Recursive Formulation with Restricted Tax System

Now need to add last period’s marginal utility as a state variable s= (W,U, θ)

H is distribution over s Add restrictions

θu_c(s)

un(s) =θuˆ _c(ˆs)

un(ˆs) ∀θ,s,θ,ˆ sˆ and

uc(s)

U =uc(ˆs)

Uˆ s,s,ˆ U,Uˆ

Recursive Formulation with Restricted Tax System

V(k,H) = max Z

u(c(s),n(s))dλ(θ)dH(s) +βV(k⁰,H⁰)

subject to resource constraint

βW⁰(θ) +uc(s)c(s⁰) +n(s)un(s) =W(θ)

H⁰(W⁰,U⁰, θ) = Z

B(W,U,θ)

dH(W, θ) where

B(W⁰,U⁰, θ) ={(W,U);h(W,U, θ)≤ W⁰}