T HE C ONSEQUENCES OF AN U NKNOWN D EBT T ARGET Alexander W. Richter Auburn University Nathaniel A. Throckmorton College of William & Mary
M OTIVATION • 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 R ICHTER AND T HROCKMORTON : T HE C ONSEQUENCES OF AN U NKNOWN D EBT T ARGET
U.S. B UDGET P ROPOSALS 100 CBO Alternative 95 CBO Pres Budget Fiscal Commission U.S. Debt−to−GDP (%) 90 House Budget CBO Baseline 85 80 75 70 65 60 55 2013 2015 2017 2019 2021 R ICHTER AND T HROCKMORTON : T HE C ONSEQUENCES OF AN U NKNOWN D EBT T ARGET
P OLARIZATION OF THE U.S. C ONGRESS 1.1 House 1 Senate 0.9 Polarization Index 0.8 0.7 0.6 0.5 0.4 0.3 1953 1963 1973 1983 1993 2003 2013 R ICHTER AND T HROCKMORTON : T HE C ONSEQUENCES OF AN U NKNOWN D EBT T ARGET
M AIN R ESULTS 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. R ICHTER AND T HROCKMORTON : T HE C ONSEQUENCES OF AN U NKNOWN D EBT T ARGET
RBC M ODEL Household chooses { c j , n j , i j , b j } ∞ j = t to maximize ∞ � log c j − χ n 1+ η � � j E ℓ β j − t t 1 + η j = t subject to c t + i t + b t = (1 − τ t )( w t n t + r k t k t − 1 ) + r t − 1 b t − 1 + ¯ z k t = i t + (1 − δ ) k t − 1 t − 1 n 1 − α ak α P .C. firm produces y t = ¯ , and chooses { k t − 1 , n t } to t maximize y t − w t n t − r k t k t − 1 . R ICHTER AND T HROCKMORTON : T HE C ONSEQUENCES OF AN U NKNOWN D EBT T ARGET
F ISCAL P OLICY • Government budget constraint, b t + τ t ( w t n t + r k t k t − 1 ) = r t − 1 b t − 1 + ¯ g + ¯ z. • State-dependent income tax rate policy, τ ( s t ) + γ ( b t − 1 /y t − 1 − by ( s t )) + ε t , τ t = ¯ where s is an m -state hidden Markov chain with transition matrix P , and ε ∼ N (0 , σ 2 ε ) . • Signal extraction problem, x t ≡ τ t − γb t − 1 /y t − 1 τ ( s t ) − γby ( s t ) + ε t , = ¯ which has a mixed PDF of m normal distributions. R ICHTER AND T HROCKMORTON : T HE C ONSEQUENCES OF AN U NKNOWN D EBT T ARGET
S OURCES 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 R ICHTER AND T HROCKMORTON : T HE C ONSEQUENCES OF AN U NKNOWN D EBT T ARGET
I NFORMATION S ETS Full Information Limited Information Case 0 Case 1 Case 2 Current Debt Target State Known Unknown Unknown Debt Target Transition Matrix Known Known Unknown � f ( v t +1 , z ℓ t +1 , v t , z ℓ t ) | Ω ℓ � E = 0 t v t ≡ ( c t , n t , k t , i t , b t ) � ( k t − 1 , r t − 1 b t − 1 , τ t , s t ) , for ℓ = 0 , z ℓ t ≡ ( k t − 1 , r t − 1 b t − 1 , τ t , q t − 1 ) , for ℓ ∈ { 1 , 2 } , Ω 0 t ≡ { M, Θ , z 0 t , P } t , ˆ Ω 1 t ≡ { M, Θ , z 1 Ω 2 t ≡ { M, Θ , z 2 P t , x t } t , P } τ ( i ) } m i =1 , { by ( i ) } m i =1 , σ 2 Θ ≡ ( β, η, χ, δ, ¯ a, α, γ, { ¯ ε ) R ICHTER AND T HROCKMORTON : T HE C ONSEQUENCES OF AN U NKNOWN D EBT T ARGET
E XPECTATIONS F ORMATION � f ( v t +1 , z ℓ t +1 , v t , z ℓ t ) | Ω ℓ � E = t � + ∞ �� m −∞ f ( v t +1 , z 0 t +1 , v t , z 0 j =1 p ij t ) φ ( ε t +1 ) d ε t +1 for ℓ = 0 � + ∞ � m i =1 q t ( i ) � m j =1 p ℓ −∞ f ( v t +1 , z ℓ t +1 , v t , z ℓ t ) φ ( ε t +1 ) d ε t +1 for ℓ ∈ { 1 , 2 } ij • For ℓ = 0 , s t = i is known. • For ℓ ∈ { 1 , 2 } , q t ( i ) ≡ Pr[ s t = i | x t ] . • For ℓ ∈ { 0 , 1 } , p ij ∈ P is known. • For ℓ = 2 , p ij ∈ ˆ P t are estimates. R ICHTER AND T HROCKMORTON : T HE C ONSEQUENCES OF AN U NKNOWN D EBT T ARGET
R ELATED L ITERATURE 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) R ICHTER AND T HROCKMORTON : T HE C ONSEQUENCES OF AN U NKNOWN D EBT T ARGET
C ALIBRATION AND S OLUTION 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 0 . 30 γ Fiscal Noise Standard Deviation Estimated σ ε ¯ 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 R ICHTER AND T HROCKMORTON : T HE C ONSEQUENCES OF AN U NKNOWN D EBT T ARGET
γ C ALIBRATION 28 γ = 0 . 30 γ = 0 . 30 80 γ = 0 . 50 27 γ = 0 . 50 House Budget 26 Debt−to−GDP (%) 75 25 Tax Rate (%) 24 70 23 22 65 21 20 60 19 2013 2015 2017 2019 2021 2013 2015 2017 2019 2021 ( A ) Path to long-run debt target ( B ) Short-run income tax adjustment American Taxpayer Relief Act: Top marginal tax rate increased 4.6 pp, payroll tax increased 2 pp R ICHTER AND T HROCKMORTON : T HE C ONSEQUENCES OF AN U NKNOWN D EBT T ARGET
D ATA AND T AX R ULE F IT Tax Rate Observations/Intercepts 0.34 0.25 0.32 0.2 0.3 0.15 0.28 Observations Intercepts 0.26 0.1 1961 1971 1981 1991 2001 2011 1961 1971 1981 1991 2001 2011 Debt−to−GDP ratio Discretionary Tax Shock 0.7 0.04 0.6 0.02 0.5 0 0.4 0.3 −0.02 1961 1971 1981 1991 2001 2011 1961 1971 1981 1991 2001 2011 R ICHTER AND T HROCKMORTON : T HE C ONSEQUENCES OF AN U NKNOWN D EBT T ARGET
E STIMATION R ESULTS • Estimating with Gibbs sampler gives ˆ σ ε = 0 . 013 . • The sampled average transition matrix and 68% credible interval are 0 . 78 0 . 11 0 . 05 P 16 = 0 . 07 0 . 81 0 . 05 0 . 07 0 . 12 0 . 66 0 . 81 0 . 12 0 . 07 ¯ P = 0 . 08 0 . 84 0 . 08 0 . 10 0 . 18 0 . 72 0 . 83 0 . 15 0 . 08 P 84 = 0 . 10 0 . 87 0 . 11 0 . 12 0 . 24 0 . 79 CBO Projections R ICHTER AND T HROCKMORTON : T HE C ONSEQUENCES OF AN U NKNOWN D EBT T ARGET
S IMULATION P ROCEDURE 1. Fiscal authority chooses s t and ε t to set τ t given b t − 1 /y t − 1 2. HH observes x t = τ t − γb t − 1 /y t − 1 and in • Case 1 updates q t − 1 given x t with Bayes’ rule more P given x t with Gibbs sampler • Case 2 also updates ˆ more 3. In case 2, HH updates policy functions given ˆ P 4. HH makes decisions conditional on information set, which updates b t − 1 /y t − 1 R ICHTER AND T HROCKMORTON : T HE C ONSEQUENCES OF AN U NKNOWN D EBT T ARGET
S IMULATION P ATHS Case 1 Case 2 Output (%) Consumption (%) 0.4 0.1 0.2 0 0 −0.1 −0.2 −0.4 0 50 100 0 50 100 Capital (%) Labor Hours (%) 0.2 0.1 0 0 −0.2 −0.1 −0.4 0 50 100 0 50 100 R ICHTER AND T HROCKMORTON : T HE C ONSEQUENCES OF AN U NKNOWN D EBT T ARGET
E FFECTS OF U NKNOWN S TATE Average Debt Target Inference versus Truth 3 2.5 2 1.5 1 0 20 40 60 80 100 Output (Case 1) 0.1 0 −0.1 0 20 40 60 80 100 Discretionary Tax Shocks 0.02 0 −0.02 −0.04 0 20 40 60 80 100 R ICHTER AND T HROCKMORTON : T HE C ONSEQUENCES OF AN U NKNOWN D EBT T ARGET
D IFFERENCES IN O UTPUT Output (Case 1) 0.1 0 −0.1 0 20 40 60 80 100 Output (Case 2) 0 −0.1 0 20 40 60 80 100 R ICHTER AND T HROCKMORTON : T HE C ONSEQUENCES OF AN U NKNOWN D EBT T ARGET
M ACROECONOMIC U NCERTAINTY • y t represents output in the model • The expected value of the forecast error is given by E t [ FE ℓ y,t +1 ] = E t [ y t +1 − E t y t +1 | Ω ℓ t ] • The expected volatility of the forecast error is � σ ℓ y,t ≡ E t [( FE ℓ y,t +1 − E t [ FE ℓ y,t +1 ]) 2 | Ω ℓ t ] R ICHTER AND T HROCKMORTON : T HE C ONSEQUENCES OF AN U NKNOWN D EBT T ARGET
E XPECTED V OLATILITY OF O UTPUT Case 1 Case 2 Output 0.1 0 −0.1 0 20 40 60 80 100 SD of Output FE ( σ ℓ y − σ 0 y ) 0.3 0.2 0.1 0 0 20 40 60 80 100 R ICHTER AND T HROCKMORTON : T HE C ONSEQUENCES OF AN U NKNOWN D EBT T ARGET
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