Learning about Fiscal Policy and the Effects of Policy Uncertainty

Learning about Fiscal Policy and the Effects of Policy Uncertainty Josef Hollmayr and Christian Matthes Deutsche Bundesbank and Richmond Fed

What is this paper about? What are the effects of subjective uncertainty about fiscal policy on economic outcomes? To have something to say about this in a general equilibrium setting, we have to take a stand on how to model uncertainty

Approach we set up a simple real business cycle model with a rich fiscal sector (similar to Leeper et al.(2))

Approach we set up a simple real business cycle model with a rich fiscal sector (similar to Leeper et al.(2)) Agents learn about fiscal policy rule coefficients

Approach we set up a simple real business cycle model with a rich fiscal sector (similar to Leeper et al.(2)) Agents learn about fiscal policy rule coefficients - This governs the evolution of policy uncertainty in our model

Approach we set up a simple real business cycle model with a rich fiscal sector (similar to Leeper et al.(2)) Agents learn about fiscal policy rule coefficients - This governs the evolution of policy uncertainty in our model Simulate a crisis and how fiscal policy makers may react to it Does learning matter for outcomes?

Approach we set up a simple real business cycle model with a rich fiscal sector (similar to Leeper et al.(2)) Agents learn about fiscal policy rule coefficients - This governs the evolution of policy uncertainty in our model Simulate a crisis and how fiscal policy makers may react to it Does learning matter for outcomes? In other words: are previous results in the literature robust to the specification of expectation formation?

What to take away Learning matters!

What to take away Learning matters! - Outcomes of key macro variables (GDP, consumption and capital etc.) can be several percentage points lower for substantial periods of under learning Very robust trade-off between either lower outcomes or higher business cycle volatility

Related Papers Strand of learning Literature Giannitsarou (26) Eusepi and Preston - multiple papers 2 Uncertainty in the economy Bloom et al. (22) Born and Pfeifer (2) Fernandez-Villaverde, Guerron, Kuester and Rubio-Ramirez (22)

Model Learning Algorithm Households Each period, households maximize the infinite sum of discounted utility with the period utility function: subject to the budget constraint: U t = C t σ σ L+φ t + φ () C t( + τt C ) + B t + I t = W tl t( τt L ) + ( τt K )Rt K K t + R t B t + Z t (2) and the law of motion for capital: K t = ( δ)k t + I t (3) and subject to their beliefs about fiscal policy (their perceived policy rule)

Model Learning Algorithm Firms The production function follows the standard Cobb-Douglas type: Y t = A t K α t L α t (4) with the exogenous process for technology given by: A t = ρ a A t + ɛ A t (5)

Model Learning Algorithm Fiscal Sector The government is allowed to run deficits over. The corresponding budget constraint is then given by: B t = B t R t R K t K tτ K t W tl tτ L t C tτ C t + G t + Z t (6)

Model Learning Algorithm Fiscal Sector The government is allowed to run deficits over. The corresponding budget constraint is then given by: B t = B t R t R K t K tτ K t W tl tτ L t C tτ C t + G t + Z t (6) The fiscal rules follow Leeper et al. (2) - except for timing: Government Expenditure: Transfers: log(g t) = G c ρ g,y log(y t ) ρ g,b log(b t ) + ɛ G t (7) log(z t) = Z c ρ z,y log(y t ) ρ z,b log(b t ) + ɛ Z t, (8)

Model Learning Algorithm Fiscal Sector The government is allowed to run deficits over. The corresponding budget constraint is then given by: B t = B t R t R K t K tτ K t W tl tτ L t C tτ C t + G t + Z t (6) The fiscal rules follow Leeper et al. (2) - except for timing: Government Expenditure: Transfers: log(g t) = G c ρ g,y log(y t ) ρ g,b log(b t ) + ɛ G t (7) log(z t) = Z c ρ z,y log(y t ) ρ z,b log(b t ) + ɛ Z t, (8) Consumption Tax Rate Rule: Other Tax Rate Rules: log(τ C t ) = τ c c + ɛ C t (9) log(τ i t ) = τ i c + ρ i,y log(y t ) + ρ i,b log(b t ) + ɛ i t () with i=l,k

Model Learning Algorithm Calibration of Parameters Description Parameter Value impatience β.99 Capital share α.33 Depreciation rate δ.25 CES utility Consumption σ 2 CES utility labor φ 2 coeff. on Y in gov. exp. rule ρg,y.34 coeff. on B in gov. exp. rule ρ g,b.23 coeff. on Y in transfer rule ρz,y.3 coeff. on B in transfer rule ρ z,b.5 coeff. on Y labor tax rule ρ L,y.36 coeff. on B labor tax rule ρ L,b.49 coeff. on Y capital tax rule ρ K,y.7 coeff. on B capital tax rule ρ K,b.39 AR parameter technology ρa.9 Std.deviation technology σa.62 Std.deviation gov. spending σg.3 Std.deviation transfers σz.34 Std.deviation cons.tax σc.4 Std.deviation labor tax σ l.3 Std.deviation capital tax σ k.44 Table : Calibrated Parameters of the model Calibrated parameters of the real economy in line with the literature on US economy. The fiscal parameters are taken from the estimation of Leeper et al. (2).

Model Learning Algorithm What do agents know and what do they learn about? Agents know the structure of the economy - this includes the variables entering the fiscal policy rules

Model Learning Algorithm What do agents know and what do they learn about? Agents know the structure of the economy - this includes the variables entering the fiscal policy rules The only thing private agents do not know is the vector of coefficients in the fiscal policy rules

Model Learning Algorithm Estimation some notation: Ω t represents the vector of coefficients of all fiscal policy rules and τ t the vector of fiscal policy instrument at t firms and households learn via the Kalman filter the state space system is given by: τ t = X t Ω t + η t () Ω t = Ω t + t ν t (2) We assume agents know the volatility of the innovations in the policy rules

Model Learning Algorithm What happens under learning? - An illustration Let s focus on one policy rule for simplicity: log(g t ) = G c ρ g,y log(y t ) ρ g,b log(b t ) + ɛ G t

Model Learning Algorithm What happens under learning? - An illustration Let s focus on one policy rule for simplicity: log(g t ) = G c ρ g,y log(y t ) ρ g,b log(b t ) + ɛ G t This is the true policy rule

Model Learning Algorithm What happens under learning? - An illustration Let s focus on one policy rule for simplicity: log(g t ) = G c ρ g,y log(y t ) ρ g,b log(b t ) + ɛ G t This is the true policy rule agents are going to estimate the coefficients. their perceived policy rule is given by: log(g t ) = G c,t ρ g,y,t log(y t ) ρ g,b,t log(b t ) + ɛ G t

Model Learning Algorithm What happens under learning? - An illustration Let s focus on one policy rule for simplicity: log(g t ) = G c ρ g,y log(y t ) ρ g,b log(b t ) + ɛ G t This is the true policy rule agents are going to estimate the coefficients. their perceived policy rule is given by: log(g t ) = G c,t ρ g,y,t log(y t ) ρ g,b,t log(b t ) + ɛ G t note the form of the perceived policy shock ɛ G t : ɛ G t = ɛ G t +G c G ct (ρ g,y ρ g,y,t ) log(y t ) (ρ g,b ρ g,b,t ) log(b t )

Model Learning Algorithm Within-period timing Private agents enter period t with beliefs (or point estimate) Ω t, which is the posterior mean coming out of the Kalman filter. They treat estimated parameters as if they were known with certainty and formulate plans accordingly (Kreps 998 calls this anticipated utility ). The fiscal authority sets the systematic part of policy at the beginning of the period. Then shocks are realized, and agents observe the technology shock and the perceived fiscal policy rule shocks. Outcomes are determined in accordance with beginning-of-period plans. After observing outcomes, private agents update estimates and carry them forward to t +.

Model Learning Algorithm Perceived Equilibrium Dynamics - How the economy evolves according to private agents A(Ω t )Y t = B(Ω t )E t Y t+ + C(Ω t )Y t + Dε t log-linearized around perceived steady state Conditional on beliefs, this equation can be solved using standard algorithms (e.g. Gensys) Y t = S(Ω t )Y t + G(Ω t )ε t

Model Learning Algorithm Solving for actual equilibrium dynamics Some more notation: we modify C(Ω t ) to now include the true policy coefficients. We call this matrix C true (Ω t ) A(Ω t )Y t = B(Ω t )E t Y t+ + C true (Ω t )Y t + Dε t Y t = H(Ω t )Y t + G(Ω t )ε t H(Ω t ) = S(Ω t ) + (A(Ω t ) B(Ω t )S(Ω t )) (C true (Ω t ) C(Ω t ))

Simulations Outline of Setup of our experiments Effects of Learning/Uncertainty vs. RE benchmark What would be the effects of different fiscal measures? Do results hold under different specifications?

Simulations Setup of the Computational Experiment We start off our simulations at the original steady state In period 9, there is a negative technology shock that puts technology 5% below steady state The following period the government permanently increases government spending by % of initial steady state GDP (unanticipated) The government does so by changing the intercept in the policy rule for government spending policy change represents a 2 standard deviation shock according to agents views

Simulations Rational Expectations - Outcomes.75.7.65.35.4.45.5.55 2.7 2.65 2.6.5.6.7 GDP 2 4 6 8 Cap. Tax.42.4.38.24.26 2 4 6 8 Capital.4.5.6 2 4 6 8 Gov. Spending.25.3.35.4.45 2 4 6 8 Consumption 3.52 3.54 3.56 3.58 2 4 6 8 Labor 2 4 6 8 Debt 2 4 6 8 Transfers 2 4 6 8.2.65 Cons. Tax 2 4 6 8 Investment 2 4 6 8 Labor Tax.7 2 4 6 8 x 3 Interest Rate 9 2 4 6 8

Simulations Learning - Outcomes: Benchmark 5 2 5 Cons. GDP 5 Hours Investment Captax Debt 5 Capital Gov.Spending 2 LE std / RE std, log C.5 LE std / RE std, log GDP LE std / RE std, log debt.2.5 LE std / RE std, log capital..5.98.95.99 G c ρ gy ρ gb.6.36.24.7 Median.34 Upper.235.23 Lower.8.32.225 5 Actual

Simulations A Partial Explanation.255.25.245.24.235.23 perceived SS median learning original SS new SS.225 2 3 4 5 6 7 8 9

Simulations Learning - Outcomes: Standard Dev. 5 2 2 Cons. GDP Hours Investment Captax 2 Debt Capital Gov.Spending 2 LE std / RE std, log C.5 5 LE std / RE std, log GDP LE std / RE std, log debt.5.5 4 LE std / RE std, log capital.2.5.95.95.98 G c ρ gy ρ gb.6.38.24.7.36 Median Upper.34.23 Lower.8.32.22 5 Actual

Simulations Learning - Outcomes: 3 Standard Dev. 2 5 2 5 2 Cons. GDP 5 Hours Investment Captax Debt 5 Capital Gov.Spending 4 LE std / RE std, log C.2 LE std / RE std, log GDP LE std / RE std, log debt LE std / RE std, log capital.2.5.5.8.98.95.995 G c ρ gy ρ gb.6.35.235.7 Median.34.23 Upper Lower.8 5 Actual.33.225

Simulations Including Irrelevant Regressors 2 5 2 5 2 Cons. GDP 5 Hours Investment Captax Debt 5 Capital Gov.Spending 4 LE std / RE std, log C.5 LE std / RE std, log GDP LE std / RE std, log debt.2.2 LE std / RE std, log capital.2.5.98.98.98 G c ρ gy ρ gb.6.38.25.7.36 Median Upper.34.24.23 Lower.8.32.22 5 Actual

Simulations The Case Of No Policy Changes.5 2.5.5 Cons. GDP Hours Investment Captax Debt Capital Gov.Spending.5.5 LE std / RE std, log C.5 LE std / RE std, log GDP LE std / RE std, log debt.2..5 LE std / RE std, log capital.2.5.98.99.98 G c ρ gy ρ gb.65.38.26.7.75.36 Median Upper.34 Lower.8.32 5 Actual.24.22.2

Simulations Concluding Thoughts Changes in fiscal policy are not as effective under learning as under rational expectations

Simulations Concluding Thoughts Changes in fiscal policy are not as effective under learning as under rational expectations 2 Differences can be substantial: either large average differences or increased volatility

Simulations Concluding Thoughts Changes in fiscal policy are not as effective under learning as under rational expectations 2 Differences can be substantial: either large average differences or increased volatility 3 robust across a large variety of simulation exercises

Different Utility Functions 5 3 5 Cons. GDP Hours Investment 2 Captax 5 Debt Capital Gov.Spending 2 LE std / RE std, log C 2 5 LE std / RE std, log GDP LE std / RE std, log debt LE std / RE std, log capital..5.4.5.2.9.95 G c ρ gy ρ gb.4.6.4.6.5 Median Upper.4.2 Lower.8.3 5 Actual

Different Utility Function II 2 2 2 2 2 Cons. GDP 2 Hours Investment Captax 2 Debt Capital Gov.Spending 4 4 5 5 LE std / RE std, log C. 5 4 5 LE std / RE std, log GDP LE std / RE std, log debt LE std / RE std, log capital.5.5.5.9.95 5 5.95.995 5 5 G c ρ gy ρ gb.6.38.24.7.36 Median Upper.34.23 Lower.8.32.22 5 Actual 5 5

Learning only about G 2 5 2 5 2 Cons. GDP 5 Hours Investment Captax Debt 5 Capital Gov.Spending 4 LE std / RE std, log C LE std / RE std, log GDP LE std / RE std, log debt LE std / RE std, log capital.5.4.5.2.2.998.95.998.95.996 G c ρ gy ρ gb.6.38.25.7.36 Median Upper.34.24.23 Lower.8.32.22 5 Actual

Capital Tax Decrease Suppose now that capital tax respond by a % (of GDP) decrease 5 2 Cons. GDP Hours Investment Captax Debt Capital Gov.Spending 5 2 2 LE std / RE std, log C LE std / RE std, log GDP LE std / RE std, log debt LE std / RE std, log capital..5.5.5.99.995.995.995 Tax kc ρ ky ρ kb 2.38.72.4 2.4 2.42 Median Upper.7.395.39 Lower 2.44.68.385 5 Actual