## 1. Modeling the spending process (and related results)

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---

### Overview and motivation
- The recession beginning in December 2007 prompted fiscal stimulus including the $787 billion American Recovery and Reinvestment Act (ARRA) of 2009.
- Nearly two thirds of the ARRA is government spending and transfers; infrastructure provisions include $44 billion for water quality, transportation, and housing, and $88 billion in federal spending on energy, innovative technology, and federal buildings (Congressional Budget Office (2009)).
- The ARRA was estimated to add about $720 billion stimulus between fiscal years 2009 and 2011, roughly 5 percent of GDP in 2009.
- The Economic Stimulus Act of 2008 and the Worker, Homeownership, and Business Assistance Act of 2009 were estimated to add about $190 billion stimulus between fiscal years 2008 and 2011.
- Key modeling objectives:
  - Incorporate implementation delays (time-to-build) and expected fiscal financing adjustments into a neoclassical growth model to analyze short- and long-run effects of government investment.
  - Highlight how implementation delays and fiscal adjustments can attenuate or reverse stimulative effects of government investment.

### Time-to-build formulation and spending implementation
- Distinction:
  - Authorized government investment (A_t): the “stock” or budget authority granted by legislation.
  - Implemented government investment (G^I_t): the “flow” of spending outlays that actually occur over time.
- Public capital law of motion:
  - K^G_{t-1} = (1−δ_G) K^G_{t-2} + A_{t−N}
  - N is the number of quarters between granting budget authority and completing a project.
  - Example: funding authorized at time t−12 for a highway that takes three years to build (N = 12) cannot be used in production until time t.
- Implemented outlays parameterization:
  - G^I_t = sum_{n=0}^{N−1} φ_n A_{t−n}, with constraint sum_{n=0}^{N−1} φ_n = 1.
  - The φ sequence {φ_0, φ_1, ..., φ_{N−1}} denotes spending rates from authorization (0) to period before completion (N−1).
- Authorized investment process:
  - ln A_t = ρ_A ln A_{t−1} + σ_A ε^A_t, ε^A_t ∼ N(0,1).
- Modeling choice:
  - The model distinguishes authorized stock from implemented flow, unlike models that assume immediate implementation and immediate productivity.

### Empirical motivation for φ structure and examples
- Authorized appropriations often translate into outlays over several years; contemporaneous authorized amounts can differ substantially from contemporaneous outlays.
- CBO examples:
  - For ARRA: $27.5 billion authorized for highway construction in 2009, estimated outlays $2.75 billion for fiscal year 2009; bulk of outlays occur over next six years, with nearly half after fiscal year 2011.
  - For the proposed National Highway Bridge Reconstruction and Inspection Act of 2008 (not enacted): first-year outlays estimated at 27 percent of total budget authority; cumulative outlays at end of second year about 67 percent.
- Historical note: by end of fiscal year 2009, ARRA outlays for infrastructure spending were less than 10 percent of the budget authority granted for infrastructure in that year (Congressional Budget Office (2010c)).

### Macroeconomic implications of implementation delays
- Short-run effects:
  - Implementation delays determine the rate of spending outlays and hence the speed of stimulus delivery; slower implementation can reduce or postpone short-run stimulative effects.
  - Compared to little-delay scenarios, delays can cause private investment to fall more while labor and output rise less or even decline slightly on impact.
  - Mechanism: anticipation of future productive public capital generates a positive wealth effect that discourages current work effort; slow outlays can let this positive wealth effect dominate negative wealth effects from higher government purchases, producing small or negative short-run changes in labor and output.
  - Private investment typically falls initially and may not rebound until public capital becomes productive and raises private input productivity.
- Longer-run effects and fiscal financing:
  - How deficit spending is ultimately financed matters for longer-run effects; expected fiscal adjustments to preserve sustainability can dampen growth effects of government investment.
  - The model allows transfers, government consumption, and income taxes to adjust with a two-year lag to rising government debt; adjustment process estimated from post-1960 data.
  - Empirical estimates indicate reliance on cutting government consumption and transfers and raising income taxes to stabilize debt in the post-1960 sample.
  - When public capital is weakly productive, government investment can be contractionary at longer horizons because distortionary fiscal adjustments deter work and private investment more than productivity gains stimulate them.
  - Speed of fiscal adjustment matters: rapid retirement of debt via distortionary instruments mitigates stimulative impacts of deficit-financed government investment.

### Relation to fiscal foresight literature
- Delays in public investment are analogous to phased-in tax cuts where expectations of future policy changes influence current labor and production decisions.
- The model incorporates fiscal foresight: debt-financed expansions trigger expected adjustments that influence current decisions.

---

### 2. Debt financing, fiscal rules, estimation, and calibration

### Fiscal rules specification
- Fiscal instruments (log-linearized) react to debt-to-output ratio with an eight-quarter lag:
  - ˆZ_t = −ψ_Z ˆY_t − γ_Z ˆsB_{t−8} + u_Z_t, ˆu_Z_t = ρ_Z ˆu_Z_{t−1} + σ_Z ε_Z_t
  - ˆτ_i_t = ψ_i ˆY_t + γ_i ˆsB_{t−8} + u_i_t, ˆu_i_t = ρ_i ˆu_i_{t−1} + σ_i ε_i_t, i ∈ {K, L}
  - ˆτ_C_t = u_C_t, ˆu_C_t = ρ_C ˆu_C_{t−1} + σ_C ε_C_t
  - ˆG^C_t = −γ_G ˆsB_{t−8} + u_G_t, ˆu_G_t = ρ_G ˆu_G_{t−1} + σ_G ε_G_t
  - where sB_{t−8} ≡ B_{t−8} / Y_{t−8} and ε_t’s ∼ N(0,1).
- A positive γ implies expected cuts to transfers or government consumption, or increases in income taxes to stabilize debt growth.
- Transfers and income tax rates also respond to within-period output fluctuations (automatic stabilizers).
- An eight-quarter implementation lag reflects empirical plausibility for federal government.

### Estimation approach and data
- U.S. quarterly data from 1960Q1 to 2008Q1.
- Ten observables: consumption, investment, hours worked, consumption tax revenue, capital tax revenue, labor tax revenue, government consumption, government investment, government transfers, and debt.
- Fiscal data consist of federal and state and local governments.
- Model log-linearized, solved by Sims’s (2001) method, estimated by Bayesian techniques as in An and Schorfheide (2007).
- Priors follow Leeper et al. (2010) choices.
- Posterior draws: one million draws; first 50,000 discarded; sample thinned by every 200 draws.
- Convergence diagnostics ensured convergence to a unique posterior.
- Except for γ_L, 90-percent posterior intervals for all parameters do not contain zero.
- Government investment authorization persistence: mean estimate ρ_A = 0.94.

### Calibration (selected values)
- Discount factor β = 0.99 (implying an annual real interest rate of 4 percent).
- Capital income share α = 0.36.
- Steady-state depreciation rate of private capital δ_0 = 0.025.
- Ratio of public to private capital K_G / K = 0.31 (historical average 1960–2007).
- Steady-state fiscal variables calibrated to sample means:
  - G_I / Y = 0.038
  - G_C / Y = 0.144
  - τ_K = 0.384
  - τ_L = 0.214
  - τ_C = 0.095
  - Ratio of government debt to annual output = 0.381
- Given G_I / Y and K_G / K, model implies δ_G = 0.02.

### Productivity of public capital (α_G)
- Literature estimates vary widely (examples cited):
  - Aschauer (1989) core infrastructure elasticity = 0.24.
  - Other studies find no effect or negative effects in some sectors.
- This analysis explores two values:
  - α_G = 0.05 (benchmark from Baxter and King (1993))
  - α_G = 0.1

### Implementation-delay scenarios and spending-rate assumptions
- Three delay scenarios for government investment:
  - N = 12 (three-year delay) for large infrastructure projects.
  - N = 4 (one-year delay) for maintenance or smaller projects.
  - N = 1 (one-quarter delay) as typically assumed in the literature.
- Spending-rate assumptions:
  - N = 12: φ_0 = 0, φ_1 = φ_2 = φ_3 = 0.25/3, φ_4 = ... = φ_11 = 0.75/8 (25 percent spent by end of first year; remaining budget spent equally over remaining eight quarters).
  - N = 4: φ_0 = 0 and φ_1 = φ_2 = φ_3 = 1/3.
  - N = 1: φ_0 = 1.
- Justification: CBO notes initial outlays usually total less than 25 percent for major federal infrastructure projects.

---

### 3. Impacts of government investment, implementation delays, and fiscal financing

### General mechanisms
- If government spending is unproductive (government consumption), dominant effect is negative wealth effect: raises labor and decreases consumption (neoclassical view).
- If government spending is productive (α_G > 0), additional effects:
  - Positive wealth effect from higher future public capital, dampening labor increase from negative wealth effect.
  - As public capital builds, it raises marginal product of private inputs, inducing more labor and private capital accumulation over time.

### Implementation delays — key findings
- Short-run dynamics:
  - N = 1 (one-quarter delay): short-run responses align with neoclassical view—consumption and investment fall but output and labor rise immediately.
  - N = 4 or N = 12: immediate output and labor responses are slightly negative on impact and muted in early periods.
  - Longer delays imply the government absorbs fewer goods each period, reducing competition for goods; consumption falls less and labor rises less on impact.
  - Positive wealth effect is identical across delay scenarios because total authorized investment is the same.
- Private investment:
  - Longer delays delay the rise in private investment (e.g., under N = 12 it takes two years longer).
  - Longer delays lead to more negative private investment responses in the short run because the marginal product of private capital rises more slowly.

### Fiscal adjustments — financing method and speed

A. Financing method — key findings
- Scenarios compared:
  - All instruments adjust per estimated fiscal parameters (mean estimates).
  - Only lump-sum transfers adjust (γ_Z = 0.155 and γ_GC = γ_K = γ_L = 0).
  - Only income taxes adjust (γ_K = 0.143, γ_L = 0.077, γ_GC = γ_Z = 0); note γ_K and γ_L scaled by 1.5 to ensure equilibrium when other instruments set to 0.
- Main results:
  - Choice of financing instrument substantially affects long-horizon outcomes regardless of α_G.
  - Government investment most expansionary when financed by cutting non-distorting lump-sum transfers.
  - Government investment can be least expansionary—or contractionary—when financed by raising income tax rates.
  - When α_G = 0.05 and income taxes alone adjust, consumption, investment, and output are persistently negative at long horizons.
  - When α_G = 0.1 and income taxes alone adjust, government investment can expand output over most of the horizon (except initial negative response due to implementation delays).
- Caveat:
  - Financing via reductions in transfers may have adverse distributional consequences since transfers often go to low-income households.

B. Financing speed — key findings
- Faster debt stabilization (larger γ coefficients and earlier onset of debt retirement) brings forward negative impacts of distorting financing.
- Example comparison:
  - Baseline: debt retirement begins two years after initial debt increase.
  - Alternative: responses to debt twice as large and debt retirement begins one year after increase.
  - Result: speeding up debt retirement damps short-run expansionary effects—labor and output rise less and private investment falls more; output can turn negative earlier and by larger magnitude.
- Policy implication:
  - Retiring debt too soon can defeat the short-run stimulative purpose of deficit-financed government investment; financing speed matters for effectiveness of countercyclical fiscal measures.

### Present-value multiplier definition and computation
- Present-value multiplier k quarters after an increase in government spending:
  - kPV multiplier = [∑_{i=0}^k (∏_{j=0}^i r_{t+j}^{−1}) 4Y_{t+i}] / [∑_{i=0}^k (∏_{j=0}^i r_{t+j}^{−1}) 4G^I_{t+i}]
  - 4Y_{t+i} and 4G^I_{t+i} are level changes relative to steady state; discount factors r’s are model-based.
  - k in (16) set to 1000 to capture full dynamics.
- Multipliers computed for output, consumption, and private investment following Mountford and Uhlig (2009).
- Multipliers computed for α_G = 0.05 and α_G = 0.1 under three implementation delays.
- Present-value multipliers and 90-percent posterior intervals reported in source tables; conditional on α_G and delay, multipliers are tightly estimated.
- Dominant determinant: productivity of public capital (α_G) — multipliers uniformly larger when α_G = 0.1 than when α_G = 0.05 for a given implementation delay.

---

### 4. Quantitative findings, multipliers, and robustness

### Present-value cumulative multipliers (selected values from Table 3)
- For α_G = 0.05:
  - 1Q delay: Y = 0.39 (0.01,0.65); C = −0.07 (−0.16,0.005); I = −0.35 (−0.59,−0.19)
  - 1Y delay: Y = 0.40 (0.09,0.63); C = −0.08 (−0.16,−0.02); I = −0.36 (−0.56,−0.21)
  - 3Y delay: Y = 0.31 (−0.03,0.57); C = −0.11 (−0.19,−0.05); I = −0.40 (−0.62,−0.24)
- For α_G = 0.1:
  - 1Q delay: Y = 1.14 (0.90,1.34); C = 0.43 (0.35,0.52); I = −0.17 (−0.32,−0.06)
  - 1Y delay: Y = 1.10 (0.92,1.30); C = 0.40 (0.33,0.49); I = −0.20 (−0.32,−0.10)
  - 3Y delay: Y = 0.90 (0.68,1.11); C = 0.32 (0.26,0.41); I = −0.31 (−0.45,−0.20)

### Present-value mean output multipliers at various horizons (selected from Table 4)
- α_G = 0.05
  - 1Q: 1Y after = 0.51 (0.47,0.57); 3Y after = 0.42 (0.37,0.49); cumulative = 0.39 (0.01,0.65)
  - 1Y: 1Y after = 0.43 (0.39,0.49); 3Y after = 0.37 (0.32,0.44); cumulative = 0.40 (0.09,0.63)
  - 3Y: 1Y after = 0.33 (0.22,0.49); 3Y after = 0.31 (0.24,0.41); cumulative = 0.31 (−0.03,0.57)
- α_G = 0.1
  - 1Q: 1Y after = 0.52 (0.47,0.57); 3Y after = 0.46 (0.40,0.53); cumulative = 1.14 (0.90,1.34)
  - 1Y: 1Y after = 0.38 (0.33,0.43); 3Y after = 0.35 (0.30,0.41); cumulative = 1.11 (0.92,1.30)
  - 3Y: 1Y after = 0.10 (0.03,0.21); 3Y after = 0.16 (0.11,0.23); cumulative = 0.90 (0.68,1.11)

### Lump-sum financing (no distortionary fiscal responses)
- With γ_GC = γ_K = γ_L = 0 and all other parameters at mean estimates:
  - For α_G = 0.05: present-value cumulative output multipliers are 0.93 (one-quarter delay), 0.78 (one-year delay), and 0.72 (three-year delay), compared to 0.39, 0.40, and 0.31 in Table 3.
  - For α_G = 0.1: multipliers are 1.39 (one-quarter delay), 1.30 (one-year delay), and 1.15 (three-year delay), compared to 1.14, 1.11, and 0.90 in Table 3.
- Implication: models assuming lump-sum financing could significantly over-estimate cumulative output multipliers.

### Additional quantitative insights and robustness
- Present-value consumption multipliers can be positive when positive wealth effects eventually dominate short-run negative consumption responses.
- Present-value investment multipliers remain negative in all examined cases; for a given implementation delay, larger α_G implies a less negative investment multiplier.
- Longer implementation delays lead to more negative investment multipliers because of larger short-run dips in investment.
- In a standard neoclassical growth model, cumulative output multiplier can be larger than 1 even under distortionary financing and modest public capital productivity.
- To generate positive consumption multipliers over horizons of 1 to 3 years, α_G must be between 40 percent and 110 percent higher than the maximum value used here (depending on whether implementation delays are 3 years or 1 year).
- Estimation uncertainty: 90-percent posterior intervals reflect parameter uncertainty conditional on the model; they do not account for model uncertainty.
- Conventional macro time series provide limited information on implementation delays and public capital productivity; further progress requires fresh data on implementation delays, productivity of public capital, and fiscal financing schemes.

### Data construction (brief)
- Primary data source: National Income and Product Accounts (NIPA) compiled by the Bureau of Economic Analysis.
- Key variable definitions:
  - Consumption Ct: personal consumption expenditure on nondurable goods and services.
  - Investment It: sum of personal consumption expenditure on durable goods and gross private domestic investment.
  - Hours worked Lt: Lt = Ht Nt / 100, where Ht is average weekly hours (seasonally adjusted, nonfarm business, all persons) and Nt is civilian employment 16+.
  - Government consumption GCt: government consumption expenditure and net purchases of non-produced assets, minus government consumption of fixed capital.
  - Government investment GIt: gross government investment.
  - Transfers Zt: net current transfers plus net capital transfers and subsidies, minus the tax residual.
  - Government debt Bt: sum of net borrowing at t and government debt at t−1 less seigniorage; net borrowing = GCt + GIt + Zt + interest payment − (TCt + TKt + TLt).
- Series scaled by population and transformed: X = ln(x / p) × 100, where p is civilian noninstitutional population (ages 16+, seasonally adjusted).

*Source: _wp10229 - IMF Working Paper content provided.*

### 1. Modeling the spending process ..........................................................................7

### 1. Modeling the spending process

### Overview
- Context and motivation:
  - The recession beginning in December 2007 prompted fiscal stimulus including the $787 billion American Recovery and Reinvestment Act (ARRA) of 2009.
  - Nearly two thirds of the ARRA is government spending and transfers; infrastructure provisions include $44 billion for water quality, transportation, and housing, and $88 billion in federal spending on energy, innovative technology, and federal buildings (Congressional Budget Office (2009)).
  - The ARRA was estimated to add about $720 billion stimulus between fiscal years 2009 and 2011, roughly 5 percent of GDP in 2009. In addition, the Economic Stimulus Act of 2008 and the Worker, Homeownership, and Business Assistance Act of 2009 were estimated to add about $190 billion stimulus between fiscal years 2008 and 2011.
- Key modeling objectives:
  - Incorporate implementation delays (time-to-build) and expected fiscal financing adjustments into a neoclassical growth model to analyze short- and long-run effects of government investment.
  - Highlight how implementation delays and fiscal adjustments can attenuate or reverse stimulative effects of government investment.

### Time-to-build formulation and key equations
- Distinction introduced:
  - Authorized government investment (A_t): the “stock” or budget authority granted by legislation.
  - Implemented government investment (G^I_t): the “flow” of spending outlays that actually occur over time.
- Time-to-build law of motion for public capital:
  - K^G_{t-1} = (1−δ_G) K^G_{t-2} + A_{t−N}
  - Interpretation: N is the number of quarters between granting budget authority and completing a project; projects authorized at time t−N contribute to the productive public capital stock at time t−1.
  - Example: If funding is authorized at time t−12 for a highway that takes three years to build (N = 12), the highway cannot be used in production until time t (i.e., K^G_{t−1} becomes productive at t).
- Implemented government investment (spending outlays) parameterization:
  - G^I_t = sum_{n=0}^{N−1} φ_n A_{t−n}
  - Constraint: sum_{n=0}^{N−1} φ_n = 1
  - The {φ_0, φ_1, φ_2, ..., φ_{N−1}} sequence denotes spending rates from the date funding is authorized (0) to the period before project completion (N−1).
  - Authorized investment process: ln A_t = ρ_A ln A_{t−1} + σ_A ε^A_t, ε^A_t ∼ N(0,1).
- Modeling choices:
  - The model distinguishes authorized stock from implemented flow, unlike models that assume immediate implementation and immediate productivity of authorized spending.

### Spending rates, empirical motivations, and examples
- Empirical observation motivating φ structure:
  - Authorized appropriations often translate into outlays over several years; contemporaneous authorized amounts can differ substantially from contemporaneous outlays.
- Congressional Budget Office examples and implications:
  - For the ARRA: Congress authorized $27.5 billion for highway construction in 2009, but estimated outlays were $2.75 billion for fiscal year 2009; the bulk of outlays occur over the next six years, with nearly half of estimated outlays occurring after fiscal year 2011.
  - For the proposed National Highway Bridge Reconstruction and Inspection Act of 2008 (not enacted): planned outlays would extend more than four years into the future; first-year outlays were estimated to account for only 27 percent of total budget authority, and cumulative outlays at the end of the second year were about 67 percent.

### Macroeconomic implications of implementation delays
- Short-run effects:
  - Implementation delays determine the rate of spending outlays and hence the speed at which stimulus is delivered; slower implementation can reduce or postpone short-run stimulative effects.
  - Compared to little-delay scenarios, implementation delays can cause private investment to fall more while labor and output rise less or even decline slightly in the short run.
  - Mechanism: anticipation of future productive public capital generates a positive wealth effect that discourages current work effort; if implemented spending is slow, this positive wealth effect may dominate the negative wealth effects from higher government purchases, producing small or negative short-run changes in labor and output.
  - Private investment typically falls initially and may not rebound until public capital becomes productive and raises private input productivity.
  - Historical note: by the end of fiscal year 2009, ARRA outlays for infrastructure spending were less than 10 percent of the budget authority granted for infrastructure in that year (Congressional Budget Office (2010c)), despite claims that many projects were “shovel ready.”
- Longer-run effects and fiscal financing:
  - How deficit spending is ultimately financed matters for longer-run effects; expected fiscal adjustments (to preserve sustainability) can dampen growth effects of government investment.
  - The model allows transfers, government consumption, and income taxes to adjust with a two-year lag to rising government debt; the adjustment process is estimated from post-1960 data.
  - Empirical estimates indicate systematic reliance on cutting government consumption and transfers and raising income taxes to stabilize debt in the post-1960 sample.
  - When public capital is weakly productive, government investment can be contractionary at longer horizons because distortionary fiscal adjustments deter work and private investment more than productivity gains stimulate them.
  - The speed of fiscal adjustment matters: rapid retirement of debt via distortionary instruments mitigates stimulative impacts of deficit-financed government investment.
- Relationship to fiscal foresight literature:
  - Delays in public investment are analogous to phased-in tax cuts where expectations of future policy changes can influence current labor and production decisions (e.g., House and Shapiro (2006)).
  - The model incorporates fiscal foresight: debt-financed expansions trigger expected adjustments that influence current decisions.

### Implications for multiplier estimates
- Differences from prior literature:
  - Existing multiplier estimates vary widely and typically focus on unproductive government consumption. This model emphasizes three aspects often overlooked:
    - Whether spending is productive (public capital productivity, α_G).
    - Implementation delays (time-to-build, φ sequence, N).
    - Longer-run impacts of fiscal financing (adjustment instruments and speed).
- Key qualitative results:
  - Present-value cumulative multipliers for output can exceed 1 in a standard neoclassical growth model with distorting financing if public capital is sufficiently productive.
  - Under implementation delays, the multiplier for government investment can be much smaller in the short run than in the long run, in contrast to the typical pattern of initially large multipliers that decline over time.

*Source: _wp10229 - 1. Modeling the spending process (excerpt).*

### 2.  Debtfi

### 2.  Debtfinancing

### Fiscal rules
- Fiscal instruments are specified in log-linearized form with an eight-quarter lag before reacting to changes in the debt-to-output ratio:
  - ˆZ_t = −ψ_Z ˆY_t − γ_Z ˆsB_{t−8} + u_Z_t, ˆu_Z_t = ρ_Z ˆu_Z_{t−1} + σ_Z ε_Z_t
  - ˆτ_i_t = ψ_i ˆY_t + γ_i ˆsB_{t−8} + u_i_t, ˆu_i_t = ρ_i ˆu_i_{t−1} + σ_i ε_i_t, i ∈ {K, L}
  - ˆτ_C_t = u_C_t, ˆu_C_t = ρ_C ˆu_C_{t−1} + σ_C ε_C_t
  - ˆG^C_t = −γ_G ˆsB_{t−8} + u_G_t, ˆu_G_t = ρ_G ˆu_G_{t−1} + σ_G ε_G_t
  - where sB_{t−8} ≡ B_{t−8} / Y_{t−8} and ε_t’s ∼ N(0,1).
- A positive γ implies that the government is expected to cut transfers or government consumption, or increase income taxes to stabilize debt growth.
- Transfers and income tax rates respond to within-period output fluctuations, capturing automatic stabilizers.
- An eight-quarter implementation lag reflects empirical plausibility given that the federal government is not subject to year-to-year balanced budget rules.

### Estimation
- Data and observables:
  - U.S. quarterly data from 1960Q1 to 2008Q1.
  - Ten observables: consumption, investment, hours worked, consumption tax revenue, capital tax revenue, labor tax revenue, government consumption, government investment, government transfers, and debt.
  - Fiscal data consist of federal and state and local governments.
- Model solution and estimation:
  - Model log-linearized, solved by Sims’s (2001) method, estimated by Bayesian techniques as in An and Schorfheide (2007).
  - Prior distributions follow Leeper et al. (2010) (LPT) choices.
  - One million draws from the posterior using a random walk Metropolis-Hastings algorithm; first 50,000 draws discarded; sample thinned by every 200 draws.
  - Convergence diagnostics and checks for multiple modes ensured convergence to a unique posterior.
  - Table 2 (in source) reports priors and posterior means, 5th and 95th percentiles, and standard deviations. Except for γ_L, the 90-percent posterior intervals for all parameters do not contain zero.
- Government investment authorization persistence:
  - Mean estimate for ρ_A = 0.94, indicating persistent government investment decisions.

### Calibration
- Calibrated parameter values:
  - Discount factor β = 0.99 (implying an annual real interest rate of 4 percent).
  - Capital income share α = 0.36.
  - Steady-state depreciation rate of private capital δ_0 = 0.025.
  - Ratio of public to private capital K_G / K = 0.31 (historical average 1960–2007).
  - Steady-state fiscal variables calibrated to sample means:
    - G_I / Y = 0.038
    - G_C / Y = 0.144
    - τ_K = 0.384
    - τ_L = 0.214
    - τ_C = 0.095
    - Ratio of government debt to annual output = 0.381
  - Given G_I / Y and K_G / K, the model implies δ_G = 0.02.

### Productivity of public capital
- Literature shows diverse estimates for public capital productivity α_G:
  - Example estimates: Aschauer (1989) core infrastructure elasticity = 0.24.
  - Contrasting results: Holtz-Eakin (1994) finds no effect; Evans and Karras (1994) find often negative productivity; Kamps (2004) finds no significant effects; Nadiri and Mamuneas (1994) find significant effects in manufacturing industries.
- This analysis explores two values: α_G = 0.05 (benchmark from Baxter and King (1993)) and α_G = 0.1.

### Spending rates and implementation delays
- Three implementation-delay scenarios for government investment:
  - N = 12 (three-year delay) for large infrastructure projects.
  - N = 4 (one-year delay) for maintenance or smaller new projects.
  - N = 1 (one-quarter delay) as typically assumed in the literature.
- Spending-rate assumptions:
  - For N = 12: φ_0 = 0, φ_1 = φ_2 = φ_3 = 0.25/3, and φ_4 = ... = φ_11 = 0.75/8 (25 percent spent by end of first year; remaining authorized budget spent equally over remaining eight quarters).
  - For N = 4: φ_0 = 0 and φ_1 = φ_2 = φ_3 = 1/3.
  - For N = 1: one-quarter delay, φ_0 = 1 (equivalent to φ_0 = 1, φ delay N = 1).
- Justification: Congressional Budget Office (2008, p. 19) notes initial outlays usually total less than 25 percent for major federal infrastructure projects.
- Implementation delays are critical for the short-run stimulative effects of government investment.

### Impacts of government investment

- General mechanisms:
  - If government spending is unproductive (like government consumption), the dominant effect is a negative wealth effect: raises labor and decreases consumption (neoclassical view).
  - If government spending is productive (α_G > 0), two additional effects:
    - Positive wealth effect from higher future public capital, dampening the labor increase from the negative wealth effect.
    - As public capital builds, it raises the marginal product of private inputs, inducing more labor and private capital accumulation over time.

A. Implementation delays — key findings
- Short-run dynamics:
  - Under one-quarter delay (N = 1), short-run responses align with the neoclassical view: consumption and investment fall but output and labor rise immediately.
  - With longer delays (N = 4 or N = 12), immediate output and labor responses are slightly negative on impact and muted in early periods.
  - Longer delays imply the government absorbs fewer goods each period, reducing competition for goods and therefore causing consumption to fall less and labor to rise less on impact.
  - The positive wealth effect is identical across delay scenarios because the total authorized investment is the same.
- Private investment:
  - Longer implementation delays delay the rise in private investment (e.g., under N = 12 it takes two years longer).
  - Longer delays lead to more negative private investment responses in the short run because the marginal product of private capital rises more slowly.

B. Fiscal adjustments — financing method and speed

1. Financing method — key findings
- Scenarios compared:
  - All instruments adjust per estimated fiscal parameters (mean estimates).
  - Only lump-sum transfers adjust (γ_Z = 0.155 and γ_GC = γ_K = γ_L = 0).
  - Only income taxes adjust (γ_K = 0.143, γ_L = 0.077, γ_GC = γ_Z = 0); note mean estimates of γ_K and γ_L were scaled by 1.5 to ensure an equilibrium exists when other instruments set to 0.
- Main results:
  - Choice of financing instrument substantially affects long-horizon outcomes regardless of α_G.
  - Government investment is most expansionary when financed by cutting non-distorting lump-sum transfers.
  - Government investment can be least expansionary—or contractionary—when financed by raising income tax rates.
  - When α_G = 0.05 and income taxes alone adjust, consumption, investment, and output are persistently negative at long horizons.
  - When α_G = 0.1 and income taxes alone adjust, government investment can expand output over most of the horizon (except initial negative response due to implementation delays).
- Caveat:
  - Financing via reductions in transfers may have adverse distributional consequences since transfers often go to low-income households.

2. Financing speed — key findings
- Faster debt stabilization (larger γ coefficients and earlier onset of debt retirement) brings forward the negative impact of distorting financing (higher tax rates or reduced government consumption).
- Example:
  - Comparing a baseline where debt retirement begins two years after the initial debt increase to a scenario where responses to debt are twice as large and debt retirement begins one year after the increase:
    - Speeding up debt retirement dampens short-run expansionary effects: labor and output rise less and private investment falls more.
    - Output can turn negative earlier and by a larger magnitude when debt is stabilized more rapidly.
- Policy implication:
  - Retiring debt too soon can defeat the short-run stimulative purpose of deficit-financed government investment; financing speed matters for the effectiveness of countercyclical fiscal measures.

### Present-value multipliers
- Definition:
  - Present-value multiplier k quarters after an increase in government spending:
    - kPV multiplier = [∑_{i=0}^k (∏_{j=0}^i r_{t+j}^{−1}) 4Y_{t+i}] / [∑_{i=0}^k (∏_{j=0}^i r_{t+j}^{−1}) 4G^I_{t+i}]
    - where 4Y_{t+i} and 4G^I_{t+i} are level changes relative to steady state; discount factors r’s are model-based.
  - k in (16) set to 1000 to capture full dynamics.
- Computation and precision:
  - Present-value multipliers computed for output, consumption, and private investment following Mountford and Uhlig (2009).
  - Multipliers computed for α_G = 0.05 and 0.1 under three implementation delays.
  - Table 3 (in source) reports cumulative present-value multipliers, their 90-percent posterior intervals computed from the posterior distribution; conditional on α_G and delay, multipliers are tightly estimated.
- Dominant determinant:
  - Productivity of public capital (α_G) is the dominant factor: multipliers are uniformly larger when α_G = 0.1 than when α_G = 0.05 for a given implementation delay.
  - If government spending transforms into productive public capital, cumulative output multipliers can be substantially larger (table details in source).

*Source: _wp10229 - 2.  Debtfi*

### 1.3 whenα

### _wp10229 - 1.3 whenα

### Key findings on government investment multipliers
- Present-value consumption multipliers can be positive when the positive wealth effect eventually dominates the short-run negative consumption response.
- Present-value investment multipliers remain negative for all cases examined; for a given length of implementation delay, a larger α_G implies a less negative investment multiplier.
- Longer implementation delays lead to more negative investment multipliers because there is a larger short-run dip in investment.
- In a standard neoclassical growth model, the cumulative multiplier for output can be larger than 1 even under distortionary financing and with a modest degree of productivity of public capital.
- Different government spending categories are likely to have very different multipliers, depending on the productivity of the spending.

### Effects of implementation delays and productivity (α_G)
- Typical pattern in existing literature: stimulative effect of government spending is highest on impact and declines gradually afterwards; however, with implementation delays and productive government spending, output multipliers can be relatively small on impact.
- When there is a one-quarter delay to build public capital and government investment is weakly productive (α_G = 0.05): the output multiplier declines over time, mainly due to subsequent expected fiscal adjustments.
- With longer delays, output multipliers change very little over time; initial reductions reflect implementation delays that produce slightly negative or muted responses in output.
- When government investment is more productive (α_G = 0.1):
  - Patterns are generally reversed: short-run output multipliers are much smaller than long-run multipliers.
  - Output multipliers after one year are smaller than those when α_G = 0.05.
  - More productive government investment generates stronger positive wealth effects, so labor rises less, investment falls more, and output rises less in the short run.
- To generate positive consumption multipliers over horizons of 1 to 3 years, α_G must be between 40 percent and 110 percent higher than the maximum value used here (depending on whether implementation delays are 3 years or 1 year).

### Quantitative results (selected table values and comparisons)
- Table 3: Present-value cumulative multipliers for an increase in government investment (mean and 90-percent intervals)
  - For α_G = 0.05:
    - 1Q delay: Y = 0.39 (0.01,0.65); C = −0.07 (−0.16,0.005); I = −0.35 (−0.59,−0.19)
    - 1Y delay: Y = 0.40 (0.09,0.63); C = −0.08 (−0.16,−0.02); I = −0.36 (−0.56,−0.21)
    - 3Y delay: Y = 0.31 (−0.03,0.57); C = −0.11 (−0.19,−0.05); I = −0.40 (−0.62,−0.24)
  - For α_G = 0.1:
    - 1Q delay: Y = 1.14 (0.90,1.34); C = 0.43 (0.35,0.52); I = −0.17 (−0.32,−0.06)
    - 1Y delay: Y = 1.10 (0.92,1.30); C = 0.40 (0.33,0.49); I = −0.20 (−0.32,−0.10)
    - 3Y delay: Y = 0.90 (0.68,1.11); C = 0.32 (0.26,0.41); I = −0.31 (−0.45,−0.20)
- Table 4: Present-value mean output multipliers at various horizons (mean and 90-percent intervals)
  - α_G = 0.05
    - 1Q: 1Y after = 0.51 (0.47,0.57); 3Y after = 0.42 (0.37,0.49); cumulative = 0.39 (0.01,0.65)
    - 1Y: 1Y after = 0.43 (0.39,0.49); 3Y after = 0.37 (0.32,0.44); cumulative = 0.40 (0.09,0.63)
    - 3Y: 1Y after = 0.33 (0.22,0.49); 3Y after = 0.31 (0.24,0.41); cumulative = 0.31 (−0.03,0.57)
  - α_G = 0.1
    - 1Q: 1Y after = 0.52 (0.47,0.57); 3Y after = 0.46 (0.40,0.53); cumulative = 1.14 (0.90,1.34)
    - 1Y: 1Y after = 0.38 (0.33,0.43); 3Y after = 0.35 (0.30,0.41); cumulative = 1.11 (0.92,1.30)
    - 3Y: 1Y after = 0.10 (0.03,0.21); 3Y after = 0.16 (0.11,0.23); cumulative = 0.90 (0.68,1.11)
- Lump-sum financing (no distortionary fiscal responses: γ_GC = γ_K = γ_L = 0) — all other parameters set to their mean estimates:
  - For α_G = 0.05: present-value cumulative multipliers for output are 0.93 (one-quarter delay), 0.78 (one-year delay), and 0.72 (three-year delay), compared to 0.39, 0.40, and 0.31 in Table 3.
  - For α_G = 0.1: multipliers are 1.39 (one-quarter delay), 1.30 (one-year delay), and 1.15 (three-year delay), compared to 1.14, 1.11, and 0.90 in Table 3.
  - These comparisons suggest models assuming lump-sum financing could significantly over-estimate cumulative output multipliers.

### Fiscal financing and adjustment dynamics
- Distorting fiscal adjustments matter: the choice of fiscal adjustment instruments is important for minimizing negative effects from stabilizing government debt.
- Faster fiscal adjustment speeds reduce negative effects; slower adjustment speeds magnify short-run contractionary responses (illustrated in impulse response figures).
- Specific financing scenarios shown in impulse responses:
  - Mean estimated debt financing parameters (as in Table 2).
  - Only transfers adjusting (γ_Z = 0.154, γ_GC = γ_K = γ_L = 0).
  - Only income taxes adjusting (γ_K = 0.142, γ_L = 0.077, γ_GC = γ_Z = 0).
- Multipliers and impulse responses vary substantially across these financing schemes.

### Estimation uncertainty and implications for policy
- The 90-percent posterior intervals reflect estimation uncertainty conditional on the model specification in Section II; they do not account for model uncertainty.
- Alternative specifications considered in related work (Leeper et al. (2009)) include:
  - Utility from government consumption.
  - Private capital subject to time-to-build.
  - A government production section employing workers and purchasing goods.
- General messages hold across calibrated alternatives: the more productive government investment is, the more favorable the growth effects and the more likely the cumulative consumption multiplier will be positive; the shorter the implementation delay, the less negative the cumulative investment multiplier.
- Conventional macroeconomic time series tell us very little about the two critical pieces of the puzzle: implementation delays and the productivity of public capital.
- Further progress requires bringing fresh data to bear on implementation delays, productivity of public capital, and fiscal financing schemes.

### Data construction (brief)
- Primary data source: National Income and Product Accounts (NIPA) compiled by the Bureau of Economic Analysis.
- Key variable definitions (selected):
  - Consumption Ct: personal consumption expenditure on nondurable goods and services.
  - Investment It: sum of personal consumption expenditure on durable goods and gross private domestic investment.
  - Hours worked Lt: Lt = Ht Nt / 100, where Ht is average weekly hours (seasonally adjusted, nonfarm business, all persons) and Nt is civilian employment 16+.
  - Government consumption GCt: government consumption expenditure and net purchases of non-produced assets, minus government consumption of fixed capital.
  - Government investment GIt: gross government investment.
  - Transfers Zt: net current transfers plus net capital transfers and subsidies, minus the tax residual.
  - Government debt Bt: sum of net borrowing at t and government debt at t−1 less seigniorage; net borrowing = GCt + GIt + Zt + interest payment − (TCt + TKt + TLt).
- All series are scaled by population and transformed: X = ln(x / p) × 100, where p is civilian noninstitutional population (ages 16+, seasonally adjusted).

*Source: _wp10229 - 1.3 whenα (IMF Working Paper content provided).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp10229.pdf_