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 6% Sweden 21 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 221 1985 U.S. Gramm-Rudman-Hollings Balanced Budget Act
U.S. BUDGET PROPOSALS U.S. Debt to GDP (%) 1 95 9 85 8 75 7 65 6 CBO Alternative CBO Pres Budget Fiscal Commission House Budget CBO Baseline 55 213 215 217 219 221
POLARIZATION OF THE U.S. CONGRESS Polarization Index 1.1 1.9.8.7.6.5.4 House Senate.3 1953 1963 1973 1983 1993 23 213
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.
RBC MODEL Household chooses {c j,n j,i j,b j } j=t to maximize subject to E l t { } β j t logc j χ n1+η j 1+η j=t 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 P.C. firm produces y t = ākt 1n α t 1 α, and chooses {k t 1,n t } to maximize y t w t n t rt kk t 1.
FISCAL POLICY Government budget constraint, b t +τ t (w t n t +r k tk t 1 ) = r t 1 b t 1 +ḡ + z. State-dependent income tax rate policy, τ t = τ(s t )+γ(b t 1 /y t 1 by(s t ))+ε t, where s is an m-state hidden Markov chain with transition matrix P, and ε N(,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.
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
INFORMATION SETS Full Information Limited Information Case Case 1 Case 2 Current Debt Target State Known Unknown Unknown Debt Target Transition Matrix Known Known Unknown E [ f(v t+1,z l t+1,v t,z l t ) Ωl t] = v t (c t,n t,k t,i t,b t ) { z l t (k t 1,r t 1 b t 1,τ t,s t ), for l =, (k t 1,r t 1 b t 1,τ t,q t 1 ), for l {1,2}, Ω t {M,Θ,z t,p} Ω 1 t {M,Θ,z1 t,p} Ω2 t {M,Θ,z2 t, ˆP t,x t } Θ (β,η,χ,δ,ā,α,γ,{ τ(i)} m i=1,{by(i)}m i=1,σ2 ε )
EXPECTATIONS FORMATION E [ f(v t+1,z l t+1,v t,z l t ) Ωl t] = { m j=1 p + ij f(v t+1,z t+1,v t,z t )φ(ε t+1)dε t+1 forl = m i=1 q t(i) m + j=1 pl ij f(v t+1,z l t+1,v t,z l t )φ(ε t+1)dε t+1 forl {1,2} For l =, s t = i is known. For l {1,2}, q t (i) Pr[s t = i x t ]. For l {,1}, p ij P is known. For l = 2, p ij ˆP t are estimates.
RELATED LITERATURE 1. Recurring regime change: Aizenman and Marion (1993); Bizer and Judd (1989); Dotsey (199) 2. Current regime unobserved: Monetary: Andolfatto and Gomme (23); Leeper and Zha (23); Schorfheide (25) Fiscal: Davig (24) 3. Other policy uncertainty: Davig and Leeper (211); Davig et al. (21, 211); Richter (212); Davig and Forester (214); Bi et al. (213) 4. Learning papers: Adaptive: Kreps (1998); Cogley and Sargent (28) Bayesian: Schorfheide (25); Bianchi and Melosi (212) 5. Stochastic Volatility: Bloom (29); Bloom et al. (212) SV in Fiscal Policy: Fernández-Villaverde et al. (213); Born and Pfeifer (214)
CALIBRATION AND SOLUTION Low Debt Target by(1).6 Mid Debt Target by(2).75 High Debt Target by(3).9 Fiscal Policy Rule Coefficient γ.3 Fiscal Noise Standard Deviation σ ε 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
γ CALIBRATION Debt to GDP (%) 8 75 7 65 γ =.3 γ =.5 House Budget 6 213 215 217 219 221 (A) Path to long-run debt target Tax Rate (%) 28 27 26 25 24 23 22 21 2 γ =.3 γ =.5 19 213 215 217 219 221 (B) Short-run income tax adjustment American Taxpayer Relief Act: Top marginal tax rate increased 4.6 pp, payroll tax increased 2 pp
DATA AND TAX RULE FIT.34 Tax Rate.25 Observations/Intercepts.32.3.28.26 1961 1971 1981 1991 21 211.2.15 Observations Intercepts.1 1961 1971 1981 1991 21 211.7 Debt to GDP ratio.4 Discretionary Tax Shock.6.5.4.2.3.2 1961 1971 1981 1991 21 211 1961 1971 1981 1991 21 211
ESTIMATION RESULTS Estimating with Gibbs sampler gives ˆσ ε =.13. The sampled average transition matrix and 68% credible interval are.78.11.5 P 16 =.7.81.5.7.12.66.81.12.7 P =.8.84.8.1.18.72.83.15.8 P 84 =.1.87.11.12.24.79 CBO Projections
SIMULATION PROCEDURE 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 updatesq t 1 given x t with Bayes rule more Case 2 also updates ˆP given x t with Gibbs sampler 3. In case 2, HH updates policy functions given ˆP 4. HH makes decisions conditional on information set, which updates b t 1 /y t 1 more
SIMULATION PATHS Case 1 Case 2 Output (%).1.1 5 1 Capital (%).2.2.4.4.2.2 Consumption (%).4 5 1.1.1 Labor Hours (%) 5 1 5 1
2.5 3 EFFECTS OF UNKNOWN STATE Average Debt Target Inference versus Truth 1.5 2 1 2 4 6 8 1 Output (Case 1).1.1 2 4 6 8 1 Discretionary Tax Shocks.2.2.4 2 4 6 8 1
DIFFERENCES IN OUTPUT Output (Case 1).1.1 2 4 6 8 1 Output (Case 2).1 2 4 6 8 1
MACROECONOMIC UNCERTAINTY y t represents output in the model The expected value of the forecast error is given by E t [FEy,t+1 l ] = E t[y t+1 E t y t+1 Ω l t ] The expected volatility of the forecast error is σy,t E l t [(FEy,t+1 l E t[fey,t+1 l ])2 Ω l t]
EXPECTED VOLATILITY OF OUTPUT.1 Case 1 Case 2 Output.1 2 4 6 8 1 SD of Output FE (σy l σ y ).3.2.1 2 4 6 8 1
FULL INFORMATION IRFS 3 2 Debt Target State 1 1 2 3 4 5 6 7 8 9 1 Output (% Change).8.6.4.2.2 1 2 3 4 5 6 7 8 9 1 Labor Hours (% Change) 1 2 Tax Rate (% Point) 1 2 3 4 5 6 7 8 9 1 Consumption (% Change).4.2 1 2 3 4 5 6 7 8 9 1 Capital (% Change) 2 1 Debt/Output (% Change) 1 2 3 4 5 6 7 8 9 1 Real Interest Rate (% Point).5.1.15 1 2 3 4 5 6 7 8 9 1 Investment (% Change) 1.5.5 1 2 3 4 5 6 7 8 9 1.2.2 1 2 3 4 5 6 7 8 9 1 2 2 1 2 3 4 5 6 7 8 9 1
UNKNOWN STATE IRFS 2.2 2 Avg. Debt Target Inference 1.8 1 2 3 4 5 6 7 8 9 1 Output (% Change).2.2 1 2 3 4 5 6 7 8 9 1 Labor Hours (% Change).4.2.2.4 1 2 3 4 5 6 7 8 9 1.2.2 Tax Rate (% Point) 1 2 3 4 5 6 7 8 9 1 Consumption (% Change).1.1 1 2 3 4 5 6 7 8 9 1 Capital (% Change).5.5 1 2 3 4 5 6 7 8 9 1.1.1.1.1 Debt/Output (% Change) 1 2 3 4 5 6 7 8 9 1 Real Interest Rate (% Point) 1 2 3 4 5 6 7 8 9 1 Investment (% Change).5.5 1 2 3 4 5 6 7 8 9 1
UPDATED ESTIMATE OF P In Case 2, HH updates estimate of P each period In period 1, their estimate is updated from P = ˆP.9.5.5 =.5.9.5.5.5.9 to.8947.56.547 ˆP 1 =.492.897.538..454.459.987
CASE 2 IRFS 2.2 2 Avg. Debt Target Inference 1.8 1 2 3 4 5 6 7 8 9 1 Output (% Change).5.5 1 2 3 4 5 6 7 8 9 1 Labor Hours (% Change).5.5 1 2 3 4 5 6 7 8 9 1.1.1 Tax Rate (% Point).2 1 2 3 4 5 6 7 8 9 1 Capital (% Change).5.5.5 Debt/Output (% Change) 1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 9 1 Consumption (% Change) Real Interest Rate (% Point).2.5.1.1 1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 9 1 Investment (% Change).1.1 1 2 3 4 5 6 7 8 9 1
WELFARE CALCULATION Treat limited info. cases as alternative to full info. Solve for λ l that satisfies E l t W(c t(z 1 t 1 ),n t(z 1 t 1 )) = m q t (i)e t W((1 λ1 )c t (z 1 t 1 s t = i),n t (z 1 t 1 s t = i)) Case 1 i=1 m m q t (i) ˆp ij E t W((1 λ2 )c t (z 2 t 1 s t,s t+1 ),n t (z 2 t 1 s t,s t+1 )) Case 2 i=1 j=1 λ l > (λ l < ) represents a welfare loss (gain) in case l
CASE 1 WELFARE DISTRIBUTION (16-5-84 BANDS).15.1.5.5.1 1 2 3 4 5
WELFARE GAINS AND LOSSES τ t = τ(s t )+γ(b t 1 /y t 1 by(s t ))+ε t, Scenario 1 L Gain M Loss H Scenario 2 L Loss M Gain Belief Current State H Current State Belief
CASE 2 WELFARE DISTRIBUTION (16-5-84 BANDS) 3 2 1 1 1 2 3 4 5
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 $36B 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
DEBT TARGET IS HIDDEN.2.4.5 1 Unknown Output (%) 1 2 3 4 5 Capital (%) 1.5 1 2 3 4 5 (A) Contractionary paths.16.14.12.1.8.6.4.2 1 2 3 4 5 (B) Welfare costs
DEBT TARGET IS REVEALED.2.4.5 1 Unknown Output (%) 1 2 3 4 5 Capital (%) Known 1.5 1 2 3 4 5 (A) Contractionary paths.16.14.12.1.8.6.4.2 1 2 3 4 5 (B) Welfare costs
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
CBO BASELINE PROJECTIONS 8 7 93 11 U.S. Debt to GDP (%) 6 5 4 3 2 83 85 87 89 1 99 1 1982 1987 1992 1997 22 27 212 91 95 97 9 5 7 3
DISTRIBUTION OF DIFFERENCES (25-5-75 QUANTILES) Difference from Actual Debt to GDP (%) 2 1 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 1 Projection Year back
DISCRETIZATION METHOD 3-STATE MARKOV CHAIN Define a projection g : R 3 R 2, g(q t ) (q t o)b = ξ t, where o is the origin and i q t(i) = 1. Apply the Gram-Schmidt process to obtain b 1 = b 1 = [,1, 1], b 2 = b 2 proj b1 ( b2 ) = [1, 1/2, 1/2], so that B [b T 1 / b 1,b T 2 / b 2 ] is an orthonormal basis. The mapping becomes back where b ij B. ξ t (1) = q t (2)(b 21 b 11 )+q t (3)(b 31 b 11 ) ξ t (2) = q t (2)(b 22 b 12 )+q t (3)(b 32 b 12 ).
HAMILTON FILTER 1. Calculate the joint probability of (s t = i,s t 1 = j), Pr[s t = i,s t 1 = j x t 1 ] = Pr[s t = i,s t 1 = j]pr[s t 1 = j x t 1 ]. 2. Calculate the joint conditional density-distribution, f(x t,s t = i,s t 1 = j x t 1 ) = f(x t s t = i,s t 1 = j,x t 1 )Pr[s t = i,s t 1 = j x t 1 ]. 3. Calculate the likelihood of x t conditional on its history, f(x t x t 1 ) = m m f(x t,s t = i,s t 1 = j x t 1 ). i=1 j=1 4. Calculate the joint probabilities of (s t = j,s t 1 = i) conditional on x t, Pr[s t = i,s t 1 = j x t ] = f(xt,st = i,s t 1 = j x t 1 ) f(x t x t 1. ) 5. Calculate the output by summing the joint probabilities over the realizations s t 1, back Pr[s t = i x t ] = m Pr[s t = i,s t 1 = j x t ]. j=1
IMPORTANCE SAMPLER Posterior density is product of two independent Dirichlet distributions: ( 3 )( 3 ) 3 f(p s T ) Π j (P) 1 j j=1 i=1 j=1 p a ij+m o ij 1 ij where π is the stationary distribution of P and a are the initial shaping parameters. Sample L draws, θij, l from Dirichlet distribution, then weight them with w l 3 j=1 Π j(pt l)1 j ˆp ij result from weighting procedure ˆp ij = L l=1 w lθij l L l=1 w. l back
GIBBS SAMPLER 1. Initialize s T = {s 1,...,s T } by sampling from the prior, P. 2. For t {1,...,T} and j {1,2,3}, sample s t If t = 1, then f(s 1 x T,s 1 ) Π j (P)p jk f(x 1 s 1 ), where s 2 = k. If 1 < t < T, then f(s t x T,s t ) p ij p jk f(x t s t ), where s t 1 = i and s t+1 = k. If t = T, then f(s T x T,s T ) Π j (P)p ij f(x T s T ), where s T 1 = i. Π j (P) is the jth element of the stationary distribution of P, f(x t s t ) = exp{ ε 2 t /(2σ2 )}/ 2πσ 2, where ε t = x t ( τ(s t ) γby(s t ))) is the discretionary i.i.d. tax shock. 3. Use the importance sampler to draw P given s T. 4. Repeat steps 2 and 3 N times. back