U PDATED ESTIMATE OF P In Case 2, HH updates estimate of P each - - PowerPoint PPT Presentation

THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

Alexander W. Richter

Auburn University

Nathaniel A. Throckmorton

College of William & Mary

MOTIVATION

• Agreement on benefits of central bank communication
• No consensus about conduct of fiscal policy
• Recently adopted fiscal rules:
• EU Stability and Growth Pact sets debt target equal to 60%
• Sweden 2010 Budget Act sets lending target of 1% of GDP
• NZ Fiscal Responsibility Act requires “prudent” debt level
• Canada committed to debt-to-GDP ratio of 25% by 2021
• 1985 U.S. Gramm-Rudman-Hollings Balanced Budget Act

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

U.S. BUDGET PROPOSALS

2013 2015 2017 2019 2021 55 60 65 70 75 80 85 90 95 100 U.S. Debt−to−GDP (%)

CBO Alternative CBO Pres Budget Fiscal Commission House Budget CBO Baseline

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

POLARIZATION OF THE U.S. CONGRESS

1953 1963 1973 1983 1993 2003 2013 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 Polarization Index

House Senate

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

MAIN RESULTS

• 1. An unknown debt target amplifies the effects of tax shocks.
• 2. Stark changes in fiscal policy lead to welfare losses.
• 3. The Bush tax cut debate may have slowed the recovery.

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

RBC MODEL

Household chooses {cj, nj, ij, bj}∞

j=t to maximize

Eℓ

t ∞

• j=t

βj−t

• log cj − χ n1+η

j

1 + η

• subject to

ct + it + bt = (1 − τt)(wtnt + rk

t kt−1) + rt−1bt−1 + ¯

z kt = it + (1 − δ)kt−1 P .C. firm produces yt = ¯ akα

t−1n1−α t

, and chooses {kt−1, nt} to maximize yt − wtnt − rk

t kt−1.

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

FISCAL POLICY

• Government budget constraint,

bt + τt(wtnt + rk

t kt−1) = rt−1bt−1 + ¯

g + ¯ z.

• State-dependent income tax rate policy,

τt = ¯ τ(st) + γ(bt−1/yt−1 − by(st)) + εt, where s is an m-state hidden Markov chain with transition matrix P, and ε ∼ N (0, σ2

ε).

• Signal extraction problem,

xt ≡ τt − γbt−1/yt−1 = ¯ τ(st) − γby(st) + εt, which has a mixed PDF of m normal distributions.

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

SOURCES OF LIMITED INFORMATION

• 1. Time-varying mean, not standard deviation
• 2. Unknown debt target state
• Bayesian updates conditional probabilities
• Expectations formation is rational/Bayesian
• Rational learning is embedded in optimization problem
• 3. Unknown transition matrix
• Bayesian updates transition matrix
• Expectations formation is adaptive
• Household must reoptimize given estimate

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

INFORMATION SETS

Full Information Limited Information Case 0 Case 1 Case 2 Current Debt Target State Known Unknown Unknown Debt Target Transition Matrix Known Known Unknown

E

• f(vt+1, zℓ

t+1, vt, zℓ t)|Ωℓ t

• = 0

vt ≡ (ct, nt, kt, it, bt) zℓ

t ≡

• (kt−1, rt−1bt−1, τt, st),

for ℓ = 0, (kt−1, rt−1bt−1, τt, qt−1), for ℓ ∈ {1, 2}, Ω0

t ≡ {M, Θ, z0 t, P}

Ω1

t ≡ {M, Θ, z1 t, P}

Ω2

t ≡ {M, Θ, z2 t, ˆ

Pt, xt} Θ ≡ (β, η, χ, δ, ¯ a, α, γ, {¯ τ(i)}m

i=1, {by(i)}m i=1, σ2 ε)

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

EXPECTATIONS FORMATION

E

• f(vt+1, zℓ

t+1, vt, zℓ t)|Ωℓ t

• =

m

j=1 pij

+∞

−∞ f(vt+1, z0 t+1, vt, z0 t)φ(εt+1)dεt+1

for ℓ = 0 m

i=1 qt(i) m j=1 pℓ ij

+∞

−∞ f(vt+1, zℓ t+1, vt, zℓ t)φ(εt+1)dεt+1

for ℓ ∈ {1, 2}

• For ℓ = 0, st = i is known.
• For ℓ ∈ {1, 2}, qt(i) ≡ Pr[st = i|xt].
• For ℓ ∈ {0, 1}, pij ∈ P is known.
• For ℓ = 2, pij ∈ ˆ

Pt are estimates.

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

RELATED LITERATURE

• 1. Recurring regime change: Aizenman and Marion (1993);

Bizer and Judd (1989); Dotsey (1990)

• 2. Current regime unobserved:
• Monetary: Andolfatto and Gomme (2003); Leeper and Zha

(2003); Schorfheide (2005)

• Fiscal: Davig (2004)
• 3. Other policy uncertainty: Davig and Leeper (2011); Davig

et al. (2010, 2011); Richter (2012); Davig and Forester (2014); Bi et al. (2013)

• 4. Learning papers:
• Adaptive: Kreps (1998); Cogley and Sargent (2008)
• Bayesian: Schorfheide (2005); Bianchi and Melosi (2012)
• 5. Stochastic Volatility: Bloom (2009); Bloom et al. (2012)

SV in Fiscal Policy: Fern´ andez-Villaverde et al. (2013); Born and Pfeifer (2014)

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

CALIBRATION AND SOLUTION

Low Debt Target by(1) 0.60 Mid Debt Target by(2) 0.75 High Debt Target by(3) 0.90 Fiscal Policy Rule Coefficient γ 0.30 Fiscal Noise Standard Deviation σε Estimated Prior Transition Matrix ¯ P Estimated

Debt targets are far apart so we use global nonlinear solution:

• Evenly spaced discretization
• Fixed-point policy function iteration
• Linear interpolation
• Gauss-Hermite integration
• 3-state Markov chain

Discretization RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

γ CALIBRATION

2013 2015 2017 2019 2021 60 65 70 75 80 Debt−to−GDP (%)

γ = 0.30 γ = 0.50 House Budget

(A) Path to long-run debt target

2013 2015 2017 2019 2021 19 20 21 22 23 24 25 26 27 28 Tax Rate (%)

γ = 0.30 γ = 0.50

(B) Short-run income tax adjustment

American Taxpayer Relief Act: Top marginal tax rate increased 4.6 pp, payroll tax increased 2 pp

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

DATA AND TAX RULE FIT

1961 1971 1981 1991 2001 2011 0.26 0.28 0.3 0.32 0.34 Tax Rate 1961 1971 1981 1991 2001 2011 0.3 0.4 0.5 0.6 0.7 Debt−to−GDP ratio 1961 1971 1981 1991 2001 2011 0.1 0.15 0.2 0.25 Observations/Intercepts 1961 1971 1981 1991 2001 2011 −0.02 0.02 0.04 Discretionary Tax Shock

Observations Intercepts RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

ESTIMATION RESULTS

• Estimating with Gibbs sampler gives ˆ

σε = 0.013.

• The sampled average transition matrix and 68% credible

interval are

P16 =   0.78 0.11 0.05 0.07 0.81 0.05 0.07 0.12 0.66   ¯ P =   0.81 0.12 0.07 0.08 0.84 0.08 0.10 0.18 0.72   P84 =   0.83 0.15 0.08 0.10 0.87 0.11 0.12 0.24 0.79  

CBO Projections RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

SIMULATION PROCEDURE

• 1. Fiscal authority chooses st and εt to set τt given bt−1/yt−1
• 2. HH observes xt = τt − γbt−1/yt−1 and in
• Case 1 updates qt−1 given xt with Bayes’ rule

more

• Case 2 also updates ˆ

P given xt with Gibbs sampler

more

• 3. In case 2, HH updates policy functions given ˆ

P

• 4. HH makes decisions conditional on information set, which

updates bt−1/yt−1

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

SIMULATION PATHS

50 100 −0.1 0.1 Output (%) 50 100 −0.4 −0.2 0.2 0.4 Consumption (%) 50 100 −0.4 −0.2 0.2 Capital (%) 50 100 −0.1 0.1 Labor Hours (%) Case 1 Case 2

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

EFFECTS OF UNKNOWN STATE

20 40 60 80 100 1 1.5 2 2.5 3 Average Debt Target Inference versus Truth Output (Case 1) 20 40 60 80 100 −0.1 0.1 Discretionary Tax Shocks 20 40 60 80 100 −0.04 −0.02 0.02

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

DIFFERENCES IN OUTPUT

Output (Case 1) 20 40 60 80 100 −0.1 0.1 Output (Case 2) 20 40 60 80 100 −0.1

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

MACROECONOMIC UNCERTAINTY

• yt represents output in the model
• The expected value of the forecast error is given by

Et[FEℓ

y,t+1] = Et[yt+1 − Etyt+1|Ωℓ t]

• The expected volatility of the forecast error is

σℓ

y,t ≡

• Et[(FEℓ

y,t+1 − Et[FEℓ y,t+1])2|Ωℓ t]

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

EXPECTED VOLATILITY OF OUTPUT

Output 20 40 60 80 100 −0.1 0.1 SD of Output FE (σℓ

y − σ0 y)

20 40 60 80 100 0.1 0.2 0.3 Case 1 Case 2

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

FULL INFORMATION IRFS

0 1 2 3 4 5 6 7 8 9 10 1 2 3 Debt Target State 0 1 2 3 4 5 6 7 8 9 10 −2 −1 Tax Rate (% Point) 0 1 2 3 4 5 6 7 8 9 10 1 2 Debt/Output (% Change) 0 1 2 3 4 5 6 7 8 9 10 −0.2 0.2 0.4 0.6 0.8 Output (% Change) 0 1 2 3 4 5 6 7 8 9 10 0.2 0.4 Consumption (% Change) 0 1 2 3 4 5 6 7 8 9 10 −0.15 −0.1 −0.05 Real Interest Rate (% Point) 0 1 2 3 4 5 6 7 8 9 10 −0.5 0.5 1 Labor Hours (% Change) 0 1 2 3 4 5 6 7 8 9 10 −0.2 0.2 Capital (% Change) 0 1 2 3 4 5 6 7 8 9 10 −2 2 Investment (% Change)

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

UNKNOWN STATE IRFS

0 1 2 3 4 5 6 7 8 9 10 1.8 2 2.2

• Avg. Debt Target Inference

0 1 2 3 4 5 6 7 8 9 10 −0.02 0.02 Tax Rate (% Point) 0 1 2 3 4 5 6 7 8 9 10 −0.1 0.1 Debt/Output (% Change) 0 1 2 3 4 5 6 7 8 9 10 −0.02 0.02 Output (% Change) 0 1 2 3 4 5 6 7 8 9 10 −0.1 0.1 Consumption (% Change) 0 1 2 3 4 5 6 7 8 9 10 −0.1 0.1 Real Interest Rate (% Point) 0 1 2 3 4 5 6 7 8 9 10 −0.04 −0.02 0.02 0.04 Labor Hours (% Change) 0 1 2 3 4 5 6 7 8 9 10 −0.05 0.05 Capital (% Change) 0 1 2 3 4 5 6 7 8 9 10 −0.5 0.5 Investment (% Change)

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

UPDATED ESTIMATE OF P

• In Case 2, HH updates estimate of P each period
• In period 1, their estimate is updated from

P = ˆ P0 =   0.90 0.05 0.05 0.05 0.90 0.05 0.05 0.05 0.90   to ˆ P1 =   0.8947 0.0506 0.0547 0.0492 0.8970 0.0538 0.0454 0.0459 0.9087   .

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

CASE 2 IRFS

0 1 2 3 4 5 6 7 8 9 10 1.8 2 2.2

• Avg. Debt Target Inference

0 1 2 3 4 5 6 7 8 9 10 −0.001 0.001 Tax Rate (% Point) 0 1 2 3 4 5 6 7 8 9 10 −0.005 0.005 Debt/Output (% Change) 0 1 2 3 4 5 6 7 8 9 10 −0.005 0.005 Output (% Change) 0 1 2 3 4 5 6 7 8 9 10 −0.02 0.02 Consumption (% Change) 0 1 2 3 4 5 6 7 8 9 10 −0.005 0.005 Real Interest Rate (% Point) 0 1 2 3 4 5 6 7 8 9 10 −0.005 0.005 Labor Hours (% Change) 0 1 2 3 4 5 6 7 8 9 10 −0.01 0.01 Capital (% Change) 0 1 2 3 4 5 6 7 8 9 10 −0.1 0.1 Investment (% Change)

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

WELFARE CALCULATION

• Treat limited info. cases as alternative to full info.
• Solve for λℓ that satisfies

Eℓ

tW(ct(z1 t−1), nt(z1 t−1)) =

              

m

• i=1

qt(i)E0

t W((1 − λ1)ct(z1 t−1|st = i), nt(z1 t−1|st = i))

Case 1

m

• i=1

qt(i)

m

• j=1

ˆ pijE0

t W((1 − λ2)ct(z2 t−1|st, st+1), nt(z2 t−1|st, st+1))

Case 2

• λℓ > 0 (λℓ < 0) represents a welfare loss (gain) in case ℓ

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

CASE 1 WELFARE DISTRIBUTION

(16-50-84 BANDS)

10 20 30 40 50 −0.1 −0.05 0.05 0.1 0.15

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

WELFARE GAINS AND LOSSES

τt = ¯ τ(st) + γ(bt−1/yt−1 − by(st)) + εt,

L

Belief Current State Gain Loss

M H L

Gain Loss

M H

Current State Belief Scenario 1 Scenario 2

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

CASE 2 WELFARE DISTRIBUTION

(16-50-84 BANDS)

10 20 30 40 50 −1 1 2 3

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

TAX CUT DEBATE

• Assumption: People expected Bush tax cuts to sunset

consistent with the goal of deficit reduction

• Reality: Tax cuts were largely extended (projected to add

$360B to annual deficit)

• Suppose true debt target had always been high, despite

Congress’ rally against debt

• Hypothesis: People’s expectations were misaligned with

the actual higher long-run debt target, which led to lower investment, output, and welfare loss

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

DEBT TARGET IS HIDDEN

10 20 30 40 50 −0.4 −0.2 Output (%) 10 20 30 40 50 −1.5 −1 −0.5 Capital (%) Unknown

(A) Contractionary paths

10 20 30 40 50 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

(B) Welfare costs

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

DEBT TARGET IS REVEALED

10 20 30 40 50 −0.4 −0.2 Output (%) 10 20 30 40 50 −1.5 −1 −0.5 Capital (%) Unknown Known

(A) Contractionary paths

10 20 30 40 50 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16

(B) Welfare costs

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

CONCLUSION

• 1. An unknown debt target amplifies the effect of tax shocks

through changes in expected tax rates

• 2. Unknown debt target leads to welfare losses on average
• 3. The Bush tax cut debate may have led to welfare losses

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

CBO BASELINE PROJECTIONS

1982 1987 1992 1997 2002 2007 2012 10 20 30 40 50 60 70 80 11 09 07 05 03 01 99 97 95 93 91 89 87 85 83 U.S. Debt−to−GDP (%)

RICHTER AND THROCKMORTON: THE CONSEQUENCES OF AN UNKNOWN DEBT TARGET

DISTRIBUTION OF DIFFERENCES

(25-50-75 QUANTILES) 1 2 3 4 5 6 7 8 9 10 −60 −50 −40 −30 −20 −10 10 20 Projection Year Difference from Actual Debt−to−GDP (%)

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DISCRETIZATION METHOD

3-STATE MARKOV CHAIN

• Define a projection g : R3 → R2,

g(qt) ≡ (qt − o)B = ξt, where o is the origin and

i qt(i) = 1.

• Apply the Gram-Schmidt process to obtain

b1 = ˜ b1 = [0, 1, −1], b2 = ˜ b2−projb1

• ˜

b2

• = [1, −1/2, −1/2],

so that B ≡ [bT

1 /||b1||, bT 2 /||b2||] is an orthonormal basis.

• The mapping becomes

ξt(1) = qt(2)(b21 − b11) + qt(3)(b31 − b11) ξt(2) = qt(2)(b22 − b12) + qt(3)(b32 − b12). where bij ∈ B.

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HAMILTON FILTER

• 1. Calculate the joint probability of (st = i, st−1 = j),

Pr[st = i, st−1 = j|xt−1] = Pr[st = i, st−1 = j] Pr[st−1 = j|xt−1].

• 2. Calculate the joint conditional density-distribution,

f(xt, st = i, st−1 = j|xt−1) = f(xt|st = i, st−1 = j, xt−1) Pr[st = i, st−1 = j|xt−1].

• 3. Calculate the likelihood of xt conditional on its history,

f(xt|xt−1) =

m

• i=1

m

• j=1

f(xt, st = i, st−1 = j|xt−1).

• 4. Calculate the joint probabilities of (st = j, st−1 = i) conditional on xt,

Pr[st = i, st−1 = j|xt] = f(xt, st = i, st−1 = j|xt−1) f(xt|xt−1) .

• 5. Calculate the output by summing the joint probabilities over the realizations st−1,

Pr[st = i|xt] =

m

• j=1

Pr[st = i, st−1 = j|xt].

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IMPORTANCE SAMPLER

• Posterior density is product of two independent Dirichlet

distributions: f(P|sT) ∝ 3

• j=1

Πj(P)1j 3

• i=1

3

• j=1

p

aij+mo

ij−1

ij

• where π is the stationary distribution of P and a are the

initial shaping parameters.

• Sample L draws, θℓ

ij, from Dirichlet distribution, then weight

them with wℓ ≡ 3

j=1 Πj(P ℓ t )1j

• ˆ

pij result from weighting procedure ˆ pij = L

ℓ=1 wℓθℓ ij

L

ℓ=1 wℓ

.

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GIBBS SAMPLER

• 1. Initialize sT = {s1, . . . , sT} by sampling from the prior, P.
• 2. For t ∈ {1, . . . , T} and j ∈ {1, 2, 3}, sample st
• If t = 1, then f(s1|xT , s−1) ∝ Πj(P)pjkf(x1|s1), where

s2 = k.

• If 1 < t < T, then f(st|xT , s−t) ∝ pijpjkf(xt|st), where

st−1 = i and st+1 = k.

• If t = T, then f(sT|xT , s−T ) ∝ Πj(P)pijf(xT |sT ), where

sT−1 = i.

Πj(P) is the jth element of the stationary distribution of P, f(xt|st) = exp {−ε2

t/(2σ2)} /

√ 2πσ2, where εt = xt − (¯ τ(st) − γby(st))) is the discretionary i.i.d. tax shock.

• 3. Use the importance sampler to draw P given sT .
• 4. Repeat steps 2 and 3 N times.

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