Optimal time-consistent taxation with default Anastasios G. Karantounias Federal Reserve Bank of Atlanta

May 2018

Motivation

Motivation • Sovereign debt models (Arellano (2008), Aguiar and Gopinath (2006),

etc):

Motivation • Sovereign debt models (Arellano (2008), Aguiar and Gopinath (2006),

etc): • Government debt is held by foreigners.

Motivation • Sovereign debt models (Arellano (2008), Aguiar and Gopinath (2006),

etc): • Government debt is held by foreigners. • Ignore fiscal dimensions of the decision to default.

Motivation • Sovereign debt models (Arellano (2008), Aguiar and Gopinath (2006),

etc): • Government debt is held by foreigners. • Ignore fiscal dimensions of the decision to default.

However: • On average a large fraction of government debt is held domestically.

Motivation • Sovereign debt models (Arellano (2008), Aguiar and Gopinath (2006),

etc): • Government debt is held by foreigners. • Ignore fiscal dimensions of the decision to default.

However: • On average a large fraction of government debt is held domestically. • Domestic sovereign default is more frequent than we think: Reinhart

and Rogoff (2011).

Motivation • Sovereign debt models (Arellano (2008), Aguiar and Gopinath (2006),

etc): • Government debt is held by foreigners. • Ignore fiscal dimensions of the decision to default.

However: • On average a large fraction of government debt is held domestically. • Domestic sovereign default is more frequent than we think: Reinhart

and Rogoff (2011). • Governments decide jointly on default, distortionary taxes and new

debt issuance.

Motivation • Sovereign debt models (Arellano (2008), Aguiar and Gopinath (2006),

etc): • Government debt is held by foreigners. • Ignore fiscal dimensions of the decision to default.

However: • On average a large fraction of government debt is held domestically. • Domestic sovereign default is more frequent than we think: Reinhart

and Rogoff (2011). • Governments decide jointly on default, distortionary taxes and new

debt issuance. • Question: How should we design fiscal policy when there is a default

option?

What this paper does • Incorporate the option to default in a closed economy with incomplete

markets and distortionary taxes (domestic sovereign default).

What this paper does • Incorporate the option to default in a closed economy with incomplete

markets and distortionary taxes (domestic sovereign default). • Default is a tool for: 1 avoiding distortionary taxation. 2

increasing the state-contingency of debt.

What this paper does • Incorporate the option to default in a closed economy with incomplete

markets and distortionary taxes (domestic sovereign default). • Default is a tool for: 1 avoiding distortionary taxation. 2

increasing the state-contingency of debt.

• Analyze optimal time-consistent fiscal policy: Markov-perfect

equilibrium. • Cannot commit to tax policies designed in the past. • Cannot commit to repay debt.

What this paper does • Incorporate the option to default in a closed economy with incomplete

markets and distortionary taxes (domestic sovereign default). • Default is a tool for: 1 avoiding distortionary taxation. 2

increasing the state-contingency of debt.

• Analyze optimal time-consistent fiscal policy: Markov-perfect

equilibrium. • Cannot commit to tax policies designed in the past. • Cannot commit to repay debt.

• Basic Questions:

What this paper does • Incorporate the option to default in a closed economy with incomplete

markets and distortionary taxes (domestic sovereign default). • Default is a tool for: 1 avoiding distortionary taxation. 2

increasing the state-contingency of debt.

• Analyze optimal time-consistent fiscal policy: Markov-perfect

equilibrium. • Cannot commit to tax policies designed in the past. • Cannot commit to repay debt.

• Basic Questions: 1 What are the incentives to issue debt without commitment?

What this paper does • Incorporate the option to default in a closed economy with incomplete

markets and distortionary taxes (domestic sovereign default). • Default is a tool for: 1 avoiding distortionary taxation. 2

increasing the state-contingency of debt.

• Analyze optimal time-consistent fiscal policy: Markov-perfect

equilibrium. • Cannot commit to tax policies designed in the past. • Cannot commit to repay debt.

• Basic Questions: 1 What are the incentives to issue debt without commitment? 2

What is the behavior of optimal taxes when there is the option to default and lack of commitment? • Smoothing over states and dates? • Drifts? Accumulation or decumulation of debt in the long-run? Cycles? • Endogenous borrowing constraints due to the default option.



Related literature

Mechanisms in the model • How does lack of commitment and default risk affect optimal debt

issuance?

Mechanisms in the model • How does lack of commitment and default risk affect optimal debt

issuance? • Generalized Euler Equation (GEE): equates the MC of more debt

(more future taxes) to the MB (more revenue = less current taxes).

Mechanisms in the model • How does lack of commitment and default risk affect optimal debt

issuance? • Generalized Euler Equation (GEE): equates the MC of more debt

(more future taxes) to the MB (more revenue = less current taxes). • Two opposite mechanisms making debt cheaper/more expensive:

Mechanisms in the model • How does lack of commitment and default risk affect optimal debt

issuance? • Generalized Euler Equation (GEE): equates the MC of more debt

(more future taxes) to the MB (more revenue = less current taxes). • Two opposite mechanisms making debt cheaper/more expensive: • Markov-perfect: more debt issuance ⇒ consumption of future policy maker ↓ ⇒ price of debt sold ↑

Mechanisms in the model • How does lack of commitment and default risk affect optimal debt

issuance? • Generalized Euler Equation (GEE): equates the MC of more debt

(more future taxes) to the MB (more revenue = less current taxes). • Two opposite mechanisms making debt cheaper/more expensive: • Markov-perfect: more debt issuance ⇒ consumption of future policy maker ↓ ⇒ price of debt sold ↑ • ⇒ cheaper to issue new debt currently and postpone taxation.

Mechanisms in the model • How does lack of commitment and default risk affect optimal debt

issuance? • Generalized Euler Equation (GEE): equates the MC of more debt

(more future taxes) to the MB (more revenue = less current taxes). • Two opposite mechanisms making debt cheaper/more expensive: • Markov-perfect: more debt issuance ⇒ consumption of future policy maker ↓ ⇒ price of debt sold ↑ • ⇒ cheaper to issue new debt currently and postpone taxation. • Lack of commitment leads to an incentive for back-loading tax

distortions and debt.

Mechanisms in the model • How does lack of commitment and default risk affect optimal debt

issuance? • Generalized Euler Equation (GEE): equates the MC of more debt

(more future taxes) to the MB (more revenue = less current taxes). • Two opposite mechanisms making debt cheaper/more expensive: • Markov-perfect: more debt issuance ⇒ consumption of future policy maker ↓ ⇒ price of debt sold ↑ • ⇒ cheaper to issue new debt currently and postpone taxation. • Lack of commitment leads to an incentive for back-loading tax

distortions and debt. • Default risk: more debt issuance ⇒ increase the probability of default

⇒ price of debt sold ↓ ⇒ more expensive to issue debt

Mechanisms in the model • How does lack of commitment and default risk affect optimal debt

issuance? • Generalized Euler Equation (GEE): equates the MC of more debt

(more future taxes) to the MB (more revenue = less current taxes). • Two opposite mechanisms making debt cheaper/more expensive: • Markov-perfect: more debt issuance ⇒ consumption of future policy maker ↓ ⇒ price of debt sold ↑ • ⇒ cheaper to issue new debt currently and postpone taxation. • Lack of commitment leads to an incentive for back-loading tax

distortions and debt. • Default risk: more debt issuance ⇒ increase the probability of default

⇒ price of debt sold ↓ ⇒ more expensive to issue debt • ⇒ tax more today vs tomorrow (front-loading of tax distortions and

debt).

Mechanisms in the model • How does lack of commitment and default risk affect optimal debt

issuance? • Generalized Euler Equation (GEE): equates the MC of more debt

(more future taxes) to the MB (more revenue = less current taxes). • Two opposite mechanisms making debt cheaper/more expensive: • Markov-perfect: more debt issuance ⇒ consumption of future policy maker ↓ ⇒ price of debt sold ↑ • ⇒ cheaper to issue new debt currently and postpone taxation. • Lack of commitment leads to an incentive for back-loading tax

distortions and debt. • Default risk: more debt issuance ⇒ increase the probability of default

⇒ price of debt sold ↓ ⇒ more expensive to issue debt • ⇒ tax more today vs tomorrow (front-loading of tax distortions and

debt). • ⇒ the possibility of default mitigates the time-consistency problem.

Roadmap

• Economy.

Roadmap

• Economy. • Fiscal policy problem.

Roadmap

• Economy. • Fiscal policy problem. • A Generalized Euler Equation (GEE) with default: short-term debt.

Roadmap

• Economy. • Fiscal policy problem. • A Generalized Euler Equation (GEE) with default: short-term debt. • (A GEE with default: long-term debt).

Roadmap

• Economy. • Fiscal policy problem. • A Generalized Euler Equation (GEE) with default: short-term debt. • (A GEE with default: long-term debt). • Preliminary numerical results: • Markov-perfect taxation without default in a risk-averse economy. • Markov-perfect taxation with default in a risk-averse economy. • Default limits tax-smoothing: taxes become more volatile.

Economy • Closed economy without capital and exogenous stochastic gt

ct + gt = ht

Economy • Closed economy without capital and exogenous stochastic gt

ct + gt = ht • Government issues non-contingent debt to finance government

expenditures and taxes labor income: Bt = τt wt ht − gt + qt Bt+1

Economy • Closed economy without capital and exogenous stochastic gt

ct + gt = ht • Government issues non-contingent debt to finance government

expenditures and taxes labor income: Bt = τt wt ht − gt + qt Bt+1 • Government can default on debt and run a balanced budget,

τtd wtd hdt = gt

Economy • Closed economy without capital and exogenous stochastic gt

ct + gt = ht • Government issues non-contingent debt to finance government

expenditures and taxes labor income: Bt = τt wt ht − gt + qt Bt+1 • Government can default on debt and run a balanced budget,

τtd wtd hdt = gt • If default: • Direct default cost: ct + gt = zht ,

z<1

• Indirect default cost as in Arellano (2008): Collapse of debt market for

a stochastic number of periods.

Household

• Household works, consumes, pays taxes and trades with the

government: ct + qt bt+1 = (1 − τt )wt ht + bt

Household

• Household works, consumes, pays taxes and trades with the

government: ct + qt bt+1 = (1 − τt )wt ht + bt

• If the government defaults:

cdt = (1 − τtd )wtd hdt .

Household

• Household works, consumes, pays taxes and trades with the

government: ct + qt bt+1 = (1 − τt )wt ht + bt

• If the government defaults:

cdt = (1 − τtd )wtd hdt . P∞ • Household’s utility: E0 t=0 β t u(ct , 1 − ht )

Competitive equilibrium • Labor supply

ul = (1 − τ )w uc

Competitive equilibrium • Labor supply

ul = (1 − τ )w uc • Euler equation with default

qt = βEt

uc,t+1 (1 − dt+1 ) uct

• dt = 1: government defaults dt = 0: government repays.

Competitive equilibrium • Labor supply

ul = (1 − τ )w uc • Euler equation with default

qt = βEt

uc,t+1 (1 − dt+1 ) uct

• dt = 1: government defaults dt = 0: government repays. • Default premium: Probx (default) rt − rtF Et xt+1 dt+1 = = F Et xt+1 (1 − dt+1 ) Probx (repayment) 1 + rt • xt+1 ≡ uc,t+1 /Et uc,t+1 : risk-adjusted measure.

Time-consistent planner • CE with policy (τ, B, d): prices (q, w) and an allocation (c, h, b) such

that a) household maximizes utility b) firms maximize profits b) markets clear, bt = Bt .

Time-consistent planner • CE with policy (τ, B, d): prices (q, w) and an allocation (c, h, b) such

that a) household maximizes utility b) firms maximize profits b) markets clear, bt = Bt . • Optimal policy: choice of (τ, B, d). • Follow primal approach.

Time-consistent planner • CE with policy (τ, B, d): prices (q, w) and an allocation (c, h, b) such

that a) household maximizes utility b) firms maximize profits b) markets clear, bt = Bt . • Optimal policy: choice of (τ, B, d). • Follow primal approach. • Planner is a large player: takes into account how debt affects interest

rates.

Time-consistent planner • CE with policy (τ, B, d): prices (q, w) and an allocation (c, h, b) such

that a) household maximizes utility b) firms maximize profits b) markets clear, bt = Bt . • Optimal policy: choice of (τ, B, d). • Follow primal approach. • Planner is a large player: takes into account how debt affects interest

rates. • State variable of Markov-perfect equilibrium: (B, g).

Time-consistent planner • CE with policy (τ, B, d): prices (q, w) and an allocation (c, h, b) such

that a) household maximizes utility b) firms maximize profits b) markets clear, bt = Bt . • Optimal policy: choice of (τ, B, d). • Follow primal approach. • Planner is a large player: takes into account how debt affects interest

rates. • State variable of Markov-perfect equilibrium: (B, g).

• Timing 1 Repay debt or not? 2

Act as a Stackelberg leader within the period: tax and debt decisions.

Time-consistent planner • CE with policy (τ, B, d): prices (q, w) and an allocation (c, h, b) such

that a) household maximizes utility b) firms maximize profits b) markets clear, bt = Bt . • Optimal policy: choice of (τ, B, d). • Follow primal approach. • Planner is a large player: takes into account how debt affects interest

rates. • State variable of Markov-perfect equilibrium: (B, g).

• Timing 1 Repay debt or not? 2

Act as a Stackelberg leader within the period: tax and debt decisions.

• Time-consistency: 1 Take into account the optimal reaction of the future policy-maker ⇒ will affect the optimal choice of debt B 0 .

Time-consistent planner • CE with policy (τ, B, d): prices (q, w) and an allocation (c, h, b) such

that a) household maximizes utility b) firms maximize profits b) markets clear, bt = Bt . • Optimal policy: choice of (τ, B, d). • Follow primal approach. • Planner is a large player: takes into account how debt affects interest

rates. • State variable of Markov-perfect equilibrium: (B, g).

• Timing 1 Repay debt or not? 2

Act as a Stackelberg leader within the period: tax and debt decisions.

• Time-consistency: 1 Take into account the optimal reaction of the future policy-maker ⇒ will affect the optimal choice of debt B 0 . 2

Not bound by past promises ⇒ will be reflected in the current τ .

Value of default • Value function

V (B, g) = max{V d (g), V r (B, g)} • V d : value of default, V r : value of repayment.

Value of default • Value function

V (B, g) = max{V d (g), V r (B, g)} • V d : value of default, V r : value of repayment. • Default value function

  V d (g) = max u(c, 1 − h) + βEg0 |g αV (0, g 0 ) + (1 − α)V d (g 0 ) c,h

subject to uc c − ul h = c+g

=

0

(Balanced budget)

zh (default cost, z < 1)

Value of default • Value function

V (B, g) = max{V d (g), V r (B, g)} • V d : value of default, V r : value of repayment. • Default value function

  V d (g) = max u(c, 1 − h) + βEg0 |g αV (0, g 0 ) + (1 − α)V d (g 0 ) c,h

subject to uc c − ul h = c+g

=

0

(Balanced budget)

zh (default cost, z < 1)

• α controls the harshness of default.

Value of default • Value function

V (B, g) = max{V d (g), V r (B, g)} • V d : value of default, V r : value of repayment. • Default value function

  V d (g) = max u(c, 1 − h) + βEg0 |g αV (0, g 0 ) + (1 − α)V d (g 0 ) c,h

subject to uc c − ul h = c+g

=

0

(Balanced budget)

zh (default cost, z < 1)

• α controls the harshness of default. • α = 0: permanent “exclusion” from debt markets ⇒ autarky.

Default and repayment sets • Default/repayment sets: D(B) = {g ∈ G|V d (g) > V r (B, g)}

and

A(B) = {g ∈ G|V d (g) ≤ V r (B, g)}

Default and repayment sets • Default/repayment sets: D(B) = {g ∈ G|V d (g) > V r (B, g)}

and

A(B) = {g ∈ G|V d (g) ≤ V r (B, g)}

• Property I: B ≤ B 0 ⇒ D(B) ⊆ D(B 0 ): Default sets increase with debt.

Default and repayment sets • Default/repayment sets: D(B) = {g ∈ G|V d (g) > V r (B, g)}

and

A(B) = {g ∈ G|V d (g) ≤ V r (B, g)}

• Property I: B ≤ B 0 ⇒ D(B) ⊆ D(B 0 ): Default sets increase with debt. • Property II: If g 0 > g and g ∈ D(B) ⇒ g 0 ∈ D(B): Default incentives

increase with fiscal shocks. d

r

• Equivalent to | dV | < | ∂V |. dg ∂g • Depends on parameters of the model (asymmetric default costs etc).

Typically holds numerically.

Default and repayment sets • Default/repayment sets: D(B) = {g ∈ G|V d (g) > V r (B, g)}

and

A(B) = {g ∈ G|V d (g) ≤ V r (B, g)}

• Property I: B ≤ B 0 ⇒ D(B) ⊆ D(B 0 ): Default sets increase with debt. • Property II: If g 0 > g and g ∈ D(B) ⇒ g 0 ∈ D(B): Default incentives

increase with fiscal shocks. d

r

• Equivalent to | dV | < | ∂V |. dg ∂g • Depends on parameters of the model (asymmetric default costs etc).

Typically holds numerically. • Threshold value of debt such that planner is indifferent:

V d (g) = V r (ω(g), g) ⇒ default if B > ω(g), repay if B ≤ ω(g).

Default and repayment sets • Default/repayment sets: D(B) = {g ∈ G|V d (g) > V r (B, g)}

and

A(B) = {g ∈ G|V d (g) ≤ V r (B, g)}

• Property I: B ≤ B 0 ⇒ D(B) ⊆ D(B 0 ): Default sets increase with debt. • Property II: If g 0 > g and g ∈ D(B) ⇒ g 0 ∈ D(B): Default incentives

increase with fiscal shocks. d

r

• Equivalent to | dV | < | ∂V |. dg ∂g • Depends on parameters of the model (asymmetric default costs etc).

Typically holds numerically. • Threshold value of debt such that planner is indifferent:

V d (g) = V r (ω(g), g) ⇒ default if B > ω(g), repay if B ≤ ω(g). • If Property II is true: ω(g) monotonically decreasing.

Default and repayment sets • Default/repayment sets: D(B) = {g ∈ G|V d (g) > V r (B, g)}

and

A(B) = {g ∈ G|V d (g) ≤ V r (B, g)}

• Property I: B ≤ B 0 ⇒ D(B) ⊆ D(B 0 ): Default sets increase with debt. • Property II: If g 0 > g and g ∈ D(B) ⇒ g 0 ∈ D(B): Default incentives

increase with fiscal shocks. d

r

• Equivalent to | dV | < | ∂V |. dg ∂g • Depends on parameters of the model (asymmetric default costs etc).

Typically holds numerically. • Threshold value of debt such that planner is indifferent:

V d (g) = V r (ω(g), g) ⇒ default if B > ω(g), repay if B ≤ ω(g). • If Property II is true: ω(g) monotonically decreasing. • Then A(B) = [g, ω −1 (B)] and D(B) = (ω −1 (B), g¯]

Default/repayment sets

.

Value of repayment • Let C, H: policy functions for next period.

Value of repayment • Let C, H: policy functions for next period. • Value of repayment

V r (B, g) = max0 u(c, 1 − h) + βEg0 |g V (B 0 , g 0 ) c,h,B

subject to   uc B = uc c − ul h + βEg0 ∈A(B 0 )|g uc (C(B 0 , g 0 ), 1 − H(B 0 , g 0 )) · B 0 | {z } | {z } surplus

c+g =h c ≥ 0, h ∈ [0, 1]

price × new debt

Value of repayment • Let C, H: policy functions for next period. • Value of repayment

V r (B, g) = max0 u(c, 1 − h) + βEg0 |g V (B 0 , g 0 ) c,h,B

subject to   uc B = uc c − ul h + βEg0 ∈A(B 0 )|g uc (C(B 0 , g 0 ), 1 − H(B 0 , g 0 )) · B 0 | {z } | {z } surplus

price × new debt

c+g =h c ≥ 0, h ∈ [0, 1]

• MPE: value functions V d , V r and policy functions c, h, B 0 such that

c = C, h = H.

Value of repayment • Let C, H: policy functions for next period. • Value of repayment

V r (B, g) = max0 u(c, 1 − h) + βEg0 |g V (B 0 , g 0 ) c,h,B

subject to   uc B = uc c − ul h + βEg0 ∈A(B 0 )|g uc (C(B 0 , g 0 ), 1 − H(B 0 , g 0 )) · B 0 | {z } | {z } surplus

price × new debt

c+g =h c ≥ 0, h ∈ [0, 1]

• MPE: value functions V d , V r and policy functions c, h, B 0 such that

c = C, h = H. • Let Φ be the multiplier on IC: excess burden of taxation.

Optimal choice of debt: Generalized Euler Equation (GEE) I • Let Qt ≡ uct · qt denote the price of debt in current MU units

Q(B 0 , g) ≡ β

Z g

ω −1 (B 0 )

uc (C(B 0 , g 0 ), 1 − H(B 0 , g 0 ))dF (g 0 |g)

Optimal choice of debt: Generalized Euler Equation (GEE) I • Let Qt ≡ uct · qt denote the price of debt in current MU units

Q(B 0 , g) ≡ β

Z

ω −1 (B 0 )

uc (C(B 0 , g 0 ), 1 − H(B 0 , g 0 ))dF (g 0 |g)

g

• The GEE takes the form:

−β |

∂EV (B 0 , g 0 ) ∂B 0 {z }

MC: average taxes tomorrow

 ∂Q 0  = Φ · Q(B 0 , g) + B 0 {z ∂B } | MR from debt issuance

Optimal choice of debt: Generalized Euler Equation (GEE) I • Let Qt ≡ uct · qt denote the price of debt in current MU units

Q(B 0 , g) ≡ β

Z

ω −1 (B 0 )

uc (C(B 0 , g 0 ), 1 − H(B 0 , g 0 ))dF (g 0 |g)

g

• The GEE takes the form:

−β |

∂EV (B 0 , g 0 ) ∂B 0 {z }

MC: average taxes tomorrow

 ∂Q 0  = Φ · Q(B 0 , g) + B 0 {z ∂B } | MR from debt issuance

• LHS: MC of issuing more debt: costly due to more taxes tomorrow.

Optimal choice of debt: Generalized Euler Equation (GEE) I • Let Qt ≡ uct · qt denote the price of debt in current MU units

Q(B 0 , g) ≡ β

Z

ω −1 (B 0 )

uc (C(B 0 , g 0 ), 1 − H(B 0 , g 0 ))dF (g 0 |g)

g

• The GEE takes the form:

−β |

∂EV (B 0 , g 0 ) ∂B 0 {z }

MC: average taxes tomorrow

 ∂Q 0  = Φ · Q(B 0 , g) + B 0 {z ∂B } | MR from debt issuance

• LHS: MC of issuing more debt: costly due to more taxes tomorrow. • RHS: Marginal revenue of new debt issuance × social value of relaxing

the constraint.

Optimal choice of debt: Generalized Euler Equation (GEE) I • Let Qt ≡ uct · qt denote the price of debt in current MU units

Q(B 0 , g) ≡ β

Z

ω −1 (B 0 )

uc (C(B 0 , g 0 ), 1 − H(B 0 , g 0 ))dF (g 0 |g)

g

• The GEE takes the form:

−β |

∂EV (B 0 , g 0 ) ∂B 0 {z }

MC: average taxes tomorrow

 ∂Q 0  = Φ · Q(B 0 , g) + B 0 {z ∂B } | MR from debt issuance

• LHS: MC of issuing more debt: costly due to more taxes tomorrow. • RHS: Marginal revenue of new debt issuance × social value of relaxing

the constraint. • Time-consistency: planner affects interest rates through next period C.

This shows up in ∂Q/∂B 0 .

GEE II • The GEE becomes Z ω−1 (B 0 ) h  dω −1 ∂V r −β dF = Φ · Q + β uc (B 0 , ω −1 (B 0 ))f (ω −1 (B 0 )|g) 0 ∂B dB 0 } g | {z (-) default region ↑

ω −1 (B 0 )

Z +

(u0cc − u0cl )

g

|

{z

(+)due to MU ↑

∂C dF · B 0 ∂B 0 } 

i

GEE II • The GEE becomes Z ω−1 (B 0 ) h  dω −1 ∂V r −β dF = Φ · Q + β uc (B 0 , ω −1 (B 0 ))f (ω −1 (B 0 )|g) 0 ∂B dB 0 } g | {z (-) default region ↑

ω −1 (B 0 )

Z +

(u0cc − u0cl )

g

|

{z

(+)due to MU ↑

• Two price effects:

∂C dF · B 0 ∂B 0 } 

i

GEE II • The GEE becomes Z ω−1 (B 0 ) h  dω −1 ∂V r −β dF = Φ · Q + β uc (B 0 , ω −1 (B 0 ))f (ω −1 (B 0 )|g) 0 ∂B dB 0 } g | {z (-) default region ↑

ω −1 (B 0 )

Z +

(u0cc − u0cl )

g

|

{z

(+)due to MU ↑

∂C dF · B 0 ∂B 0 } 

i

• Two price effects: 1

Standard Eaton-Gersovitz-Arellano effect: Q ↓ due to an increase in the −1 default region ( dω < 0). dB 0

GEE II • The GEE becomes Z ω−1 (B 0 ) h  dω −1 ∂V r −β dF = Φ · Q + β uc (B 0 , ω −1 (B 0 ))f (ω −1 (B 0 )|g) 0 ∂B dB 0 } g | {z (-) default region ↑

ω −1 (B 0 )

Z +

(u0cc − u0cl )

g

|

{z

(+)due to MU ↑

∂C dF · B 0 ∂B 0 } 

i

• Two price effects: 1

Standard Eaton-Gersovitz-Arellano effect: Q ↓ due to an increase in the −1 default region ( dω < 0). dB 0

2

Markov-Perfect effect: increase in B 0 reduces consumption next period ⇒ increases Q.

GEE II • The GEE becomes Z ω−1 (B 0 ) h  dω −1 ∂V r −β dF = Φ · Q + β uc (B 0 , ω −1 (B 0 ))f (ω −1 (B 0 )|g) 0 ∂B dB 0 } g | {z (-) default region ↑

ω −1 (B 0 )

Z +

(u0cc − u0cl )

g

|

{z

(+)due to MU ↑

∂C dF · B 0 ∂B 0 } 

i

• Two price effects: 1

Standard Eaton-Gersovitz-Arellano effect: Q ↓ due to an increase in the −1 default region ( dω < 0). dB 0

2

Markov-Perfect effect: increase in B 0 reduces consumption next period ⇒ increases Q.

3

Opposite effects.

A primer on tax-smoothing: excess burden of taxation Φt • Commitment/complete markets/no default: Lucas and Stokey (1983):

Φt+1 = Φt

A primer on tax-smoothing: excess burden of taxation Φt • Commitment/complete markets/no default: Lucas and Stokey (1983):

Φt+1 = Φt • Commitment/incomplete markets/no default: Aiyagari et al. (2002):

Et xt+1 Φt+1 = Φt ,

xt+1 ≡

uc,t+1 Et uc,t+1

A primer on tax-smoothing: excess burden of taxation Φt • Commitment/complete markets/no default: Lucas and Stokey (1983):

Φt+1 = Φt • Commitment/incomplete markets/no default: Aiyagari et al. (2002):

Et xt+1 Φt+1 = Φt ,

xt+1 ≡

uc,t+1 Et uc,t+1

• MPE/ Complete markets/ no default: Krusell et al. (2004), Debortoli

and Nunes (2013): Φt+1 = Φt · [1 +

ucc,t+1 − ucl,t+1 ∂C · Bt+1 ]. uc,t+1 ∂Bt+1

A primer on tax-smoothing: excess burden of taxation Φt • Commitment/complete markets/no default: Lucas and Stokey (1983):

Φt+1 = Φt • Commitment/incomplete markets/no default: Aiyagari et al. (2002):

Et xt+1 Φt+1 = Φt ,

xt+1 ≡

uc,t+1 Et uc,t+1

• MPE/ Complete markets/ no default: Krusell et al. (2004), Debortoli

and Nunes (2013): Φt+1 = Φt · [1 +

ucc,t+1 − ucl,t+1 ∂C · Bt+1 ]. uc,t+1 ∂Bt+1

• MPE/ Incomplete markets/no default:   ucc,t+1 − ucl,t+1 ∂C Et xt+1 Φt+1 = Φt · 1 + Et xt+1 · Bt+1 uc,t+1 ∂Bt+1

GEE and tax-smoothing • Use the envelope condition ⇒ GEE in terms of the excess burden Φt : h Et mt+1 Φt+1 = Φt · 1 +  i (ucc,t+1 − ucl,t+1 ) ∂C  dω −1 ftm (ω −1 (Bt+1 )) + Et mt+1 · Bt+1 dBt+1 uc,t+1 ∂Bt+1

GEE and tax-smoothing • Use the envelope condition ⇒ GEE in terms of the excess burden Φt : h Et mt+1 Φt+1 = Φt · 1 +  i (ucc,t+1 − ucl,t+1 ) ∂C  dω −1 ftm (ω −1 (Bt+1 )) + Et mt+1 · Bt+1 dBt+1 uc,t+1 ∂Bt+1 • mt+1 ≡

(1−dt+1 )uc,t+1 Et (1−dt+1 )uc,t+1 :

default-and-risk-adjusted change of measure.

• ftm ≡ mt+1 · f (.|gt ): default-and-risk-adjusted conditional density.

GEE and tax-smoothing • Use the envelope condition ⇒ GEE in terms of the excess burden Φt : h Et mt+1 Φt+1 = Φt · 1 +  i (ucc,t+1 − ucl,t+1 ) ∂C  dω −1 ftm (ω −1 (Bt+1 )) + Et mt+1 · Bt+1 dBt+1 uc,t+1 ∂Bt+1 • mt+1 ≡

(1−dt+1 )uc,t+1 Et (1−dt+1 )uc,t+1 :

default-and-risk-adjusted change of measure.

• ftm ≡ mt+1 · f (.|gt ): default-and-risk-adjusted conditional density. • Drifts? Depend on size of negative (default) versus positive (MU) price

effect.

GEE and tax-smoothing • Use the envelope condition ⇒ GEE in terms of the excess burden Φt : h Et mt+1 Φt+1 = Φt · 1 +  i (ucc,t+1 − ucl,t+1 ) ∂C  dω −1 ftm (ω −1 (Bt+1 )) + Et mt+1 · Bt+1 dBt+1 uc,t+1 ∂Bt+1 • mt+1 ≡

(1−dt+1 )uc,t+1 Et (1−dt+1 )uc,t+1 :

default-and-risk-adjusted change of measure.

• ftm ≡ mt+1 · f (.|gt ): default-and-risk-adjusted conditional density. • Drifts? Depend on size of negative (default) versus positive (MU) price

effect. 1

No default or small default price effect ⇒ positive drift: postpone taxes for the future (back-loading), Et mt+1 Φt+1 ≥ Φt .

GEE and tax-smoothing • Use the envelope condition ⇒ GEE in terms of the excess burden Φt : h Et mt+1 Φt+1 = Φt · 1 +  i (ucc,t+1 − ucl,t+1 ) ∂C  dω −1 ftm (ω −1 (Bt+1 )) + Et mt+1 · Bt+1 dBt+1 uc,t+1 ∂Bt+1 • mt+1 ≡

(1−dt+1 )uc,t+1 Et (1−dt+1 )uc,t+1 :

default-and-risk-adjusted change of measure.

• ftm ≡ mt+1 · f (.|gt ): default-and-risk-adjusted conditional density. • Drifts? Depend on size of negative (default) versus positive (MU) price

effect. 1

No default or small default price effect ⇒ positive drift: postpone taxes for the future (back-loading), Et mt+1 Φt+1 ≥ Φt .

2

Strong default price effect ⇒ tax on average more today, (front-loading of taxes), Et mt+1 Φt+1 ≤ Φt

GEE and tax-smoothing • Use the envelope condition ⇒ GEE in terms of the excess burden Φt : h Et mt+1 Φt+1 = Φt · 1 +  i (ucc,t+1 − ucl,t+1 ) ∂C  dω −1 ftm (ω −1 (Bt+1 )) + Et mt+1 · Bt+1 dBt+1 uc,t+1 ∂Bt+1 • mt+1 ≡

(1−dt+1 )uc,t+1 Et (1−dt+1 )uc,t+1 :

default-and-risk-adjusted change of measure.

• ftm ≡ mt+1 · f (.|gt ): default-and-risk-adjusted conditional density. • Drifts? Depend on size of negative (default) versus positive (MU) price

effect. 1

No default or small default price effect ⇒ positive drift: postpone taxes for the future (back-loading), Et mt+1 Φt+1 ≥ Φt .

2

Strong default price effect ⇒ tax on average more today, (front-loading of taxes), Et mt+1 Φt+1 ≤ Φt

• Similar mechanism with long-term debt: PV of the MU and default

effect

GEE with long-term debt

.

Optimal tax • No commitment: Choice of current tax rate is not bound by past

promises.

Optimal tax • No commitment: Choice of current tax rate is not bound by past

promises. • Proposition: The optimal tax rate is

  Φ cc (1 − B/c) + ch + hh + hc (1 − B/c) τ (B, g) = 1 + Φ(1 + hh + hc (1 − B/c)) • i,j : elasticitities of MU of consumption and leisure.

Optimal tax • No commitment: Choice of current tax rate is not bound by past

promises. • Proposition: The optimal tax rate is

  Φ cc (1 − B/c) + ch + hh + hc (1 − B/c) τ (B, g) = 1 + Φ(1 + hh + hc (1 − B/c)) • i,j : elasticitities of MU of consumption and leisure. • Example: U =

c1−ρ 1−ρ

1+φh

− ah h1+φH τ=

Φ(ρ(1 − B/c) + φh ) 1 + Φ(1 + φh )

Optimal tax • No commitment: Choice of current tax rate is not bound by past

promises. • Proposition: The optimal tax rate is

  Φ cc (1 − B/c) + ch + hh + hc (1 − B/c) τ (B, g) = 1 + Φ(1 + hh + hc (1 − B/c)) • i,j : elasticitities of MU of consumption and leisure. • Example: U =

c1−ρ 1−ρ

1+φh

− ah h1+φH τ=

Φ(ρ(1 − B/c) + φh ) 1 + Φ(1 + φh )

• Lack of commitment⇒ incentive to devalue current debt: For a given Φ

a low τ increases consumption and reduces the value of debt in current MU units ⇒ decrease interest rates.

Numerical results • Utility function: constant Frisch

U=

c1−ρ − 1 h1+φh − ah 1−ρ 1 + φh

• (β, φh ) = (.9, 1). • g ∼ U [0, 0.2 × FB output] • Risk aversion: ρ = 2. 1

MPE without default (only MU effect on prices).

2

MPE with default. Both effects present.

• Limit of the finite horizon economy. • Numerical difficulties: MPE leads to non-convexities and jumps in

policy functions: Krusell et al. (2004) • Uncertainty smooths these jumps.

computational issues

.

MPE with no default, ρ = 2: Price schedule and Revenue

Price schedule in MU units, Q(b′)

8

Revenue from debt issuance in MU units

1.2

7.9

1

7.8 0.8 7.7 0.6 7.6 0.4 7.5

0.2

7.4

7.3

0 0

0.02

0.04

0.06

0.08

b′

0.1

0.12

0.14

0

0.02

0.04

0.06

0.08

b′

0.1

0.12

0.14

MPE with no default: Consumption/tax/debt policies

Consumption policy function

0.38

Tax policy function

30

low average high

0.37

Debt policy function

0.14

low average high

low average high

0.12

25 0.36

0.1 20 0.35

b′

%

0.08 0.34

15

0.06 0.33 10 0.04 0.32 5

0.02

0.31

0.3

0 0

0.05

0.1

b

0 0

0.05

0.1

b

0

0.05

0.1

b

MPE with default: price schedule and debt Laffer curve

Price schedule in MU units, Q(b′)

8

Revenue from debt issuance in MU units

0.25

7 0.2 6

5

0.15

4 0.1

3

2 0.05 1

0

0 0

0.02

0.04

0.06

0.08

b′

0.1

0.12

0.14

0

0.02

0.04

0.06

0.08

b′

0.1

0.12

0.14

MPE with default: Default and repayment sets

Default and repayment sets Repay Default

0.08 0.07 0.06

g

0.05 0.04 0.03 0.02 0.01 0 0

0.02

0.04

0.06

0.08

b

Return

0.1

0.12

0.14

MPE with default: Consumption/tax/debt policies

Consumption policy function

Tax policy function

low average high

0.37

Debt policy function

0.06

low average high

25

low average high

0.05 0.36

20 0.04

15

0.34

10

0.33

5

0.32

b′

%

0.35

0.02

0.01

0 0

0.05

0.1

b

0.03

0 0

0.05

0.1

b

0

0.05

0.1

b

MPE without and with default

Shock path

0.065

Tax rate path

14

Debt path

0.045

no default option default option

0.04

13 0.06

0.035 12

0.03

0.055 no default option default option

b

%

0.025 11

0.02 0.05 10

0.015 0.01

0.045 9

0.005 0.04

8 0

2

4

6

t

8

10

0 0

2

4

6

8

10

0

t

• With default the tax rate is used more to absorb the shock.

2

4

6

t

8

10

Debt issuance under default

Government expenditure shock

0.09

1

Debt issuance and default

0.025

0.9

0.08

0.8

0.07

0.02

0.7 0.06 0.6

0.015

g

0.5

Debt

0.05

0.04 0.4

0.01

0.03 0.3 0.02

0.2

0.005 issuance effective

0.01

0.1

0 0

5

10

15

t

20

25

0 30

0 0

5

10

15

t

20

25

30

Consumption and taxes

Consumption

0.38

Tax rate

1

20

0.9

18

1

0.9

0.8

16

0.8

0.7

14

0.7

0.6

12

0.6

10

0.5

0.4

8

0.4

0.3

6

0.3

0.2

4

0.2

0.1

2

0.1

0.37

0.5

0.35

%

c

0.36

0.34

0.33

0.32 0

5

10

15

t

20

25

0 30

0 0

5

10

15

t

20

25

0 30

Stationary moments Tax rate in %

MPE

MPE with default

Mean St. Dev Corr with g Autocorrelation Debt-output in % Mean St. Dev Autocorrelation

12.64 3.8 0.6033 0.54

10.44 4.82 0.9369 0.2316

21 11.4 0.85

1.62 2 0.2

• 5% probability of default. • E(g/y) = 10%, std(g/y) = 6%. • With default debt is not used anymore to absorb shocks ⇒ large volatility of tax rates.

Future steps

• Further quantitative analysis (persistent shocks, larger state space). • Explore front-loading versus back-loading of taxes. • Explore various fiscal rules that have been proposed to countries with

default risk and contrast with the optimal rules.

THANK YOU!

GEE with long-term debt • Long-term debt as in Hatchondo and Martinez (2009):

qt = βEt

  uc,t+1 (1 − dt+1 ) 1 + (1 − δ)qt+1 uct

GEE with long-term debt • Long-term debt as in Hatchondo and Martinez (2009):

qt = βEt

  uc,t+1 (1 − dt+1 ) 1 + (1 − δ)qt+1 uct

• The GEE becomes

where nt+1

  Et nt+1 Φt+1 = Φt · 1 + (Bt+1 − (1 − δ)Bt )ηt   (1−dt+1 ) uc,t+1 +(1−δ)Qt+1 ∂Qt 1   , ηt ≡ ∂B ≡ : price Q Et (1−dt+1 ) uc,t+1 +(1−δ)Qt+1

semi-elasticity.

t+1

t

GEE with long-term debt • Long-term debt as in Hatchondo and Martinez (2009):

qt = βEt

  uc,t+1 (1 − dt+1 ) 1 + (1 − δ)qt+1 uct

• The GEE becomes

where nt+1

  Et nt+1 Φt+1 = Φt · 1 + (Bt+1 − (1 − δ)Bt )ηt   (1−dt+1 ) uc,t+1 +(1−δ)Qt+1 ∂Qt 1   , ηt ≡ ∂B ≡ : price Q Et (1−dt+1 ) uc,t+1 +(1−δ)Qt+1

t+1

t

semi-elasticity. • ηt follows the recursion:

ηt

=

 n −1 dω −1 ucc,t+1 − ucl,t+1 ∂Ct+1  f (ω (Bt+1 )|gt ) t+1 + Et nt+1 dBt+1 uc,t+1 + (1 − δ)Qt+1 ∂Bt+1 Qt+1 ∂Kt+1 +(1 − δ)Et nt+1 ηt+1 uc,t+1 + (1 − δ)Qt+1 ∂Bt+1

• C, K: policy functions for next period. ηt : PV of two opposing effects. Return

Issues about computation of the model • Non-convexity leads to discontinuous policy functions for consumption

and debt. • Problem severe without uncertainty: Krusell et al. (2004).

KMRR

• Similar behavior as in the hyperbolic discounting literature. • It is an open issue if there exists 1 a continuous and differentiable MPE. 2

multiple discontinuous MPEs or none.

• Uncertainty smooths out discontinuities: Uniform shocks with large

support. • Use lotteries to convexify? (Luttmer-Mariotti 2003,

Chatterjee-Eyigungor 2014). Return

Krusell et. al 2005

Consumption

Policy function for debt next period

0.335

0.14

0.12 0.33 0.1

0.325 0.08

0.06 0.32

0.04 0.315 0.02

0.31 0

0.02

0.04

0.06

0.08 B

Return

0.1

0.12

0.14

0 0

0.02

0.04

0.06

0.08 B

0.1

0.12

0.14

Related literature

• Optimal taxation: Lucas and Stokey (1983), Aiyagari et al. (2002),

Krusell et al. (2004), Klein et al. (2008), Martin (2009), Debortoli and Nunes (2013). • Sovereign default: Eaton and Gersovitz (1981), Arellano (2008), Aguiar

and Gopinath (2006), Hatchondo and Martinez (2009), Chatterjee and Eyigungor (2012). • Optimal taxation and default: Cuadra et al. (2010), Pouzo and Presno

(2014), D’Erasmo and Mendoza (2016). Return

Aguiar, Mark and Gita Gopinath. 2006. Defaultable debt, interest rates and the current account. Journal of International Economics 69 (1):64–83. Aiyagari, S. Rao, Albert Marcet, Thomas J. Sargent, and Juha Seppala. 2002. Optimal Taxation without State-Contingent Debt. Journal of Political Economy 110 (6):1220–1254. Arellano, Cristina. 2008. Default Risk and Income Fluctuations in Emerging Economies. American Economic Review 98 (3):690–712. Chatterjee, Satyajit and Burcu Eyigungor. 2012. Maturity, Indebtedness, and Default Risk. American Economic Review 102 (6):26742699. Cuadra, Gabriel, Juan M. Sanchez, and Horacio Sapriza. 2010. Fiscal policy and default risk in emerging markets. Review of Economic Dynamics 13:452–469. Debortoli, Davide and Ricardo Nunes. 2013. Lack of commitment and the level of debt. Journal of the European Economic Association 11 (5):1053–1078. D’Erasmo, Pablo and Enrique G. Mendoza. 2016. Optimal Domestic Sovereign Default. Mimeo, University of Pennsylvania. Eaton, Jonathan and Mark Gersovitz. 1981. Debt with Potential Repudiation: Theoretical and Empirical Analysis. The Review of Economic Studies 48 (2):289–309. Hatchondo, Juan Carlos and Leonardo Martinez. 2009. Long-duration bonds and sovereign defaults. Journal of International Economics

Optimal time-consistent taxation with default

... distortionary taxation. 2 increasing the state-contingency of debt. ..... G|V d(g) ≤ V r(B, g)}. • Property I: B ≤ B ⇒ D(B) ⊆ D(B ): Default sets increase with debt.

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