## wpiea2024003-print-pdf - 1.5 percent of GDP.

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---

### Key calibration parameters and baseline quantitative assumptions
- Capital-flow elasticity parameter μ: either 0.10 or 0.60.
  - When μ = 0.1: an increase in U.S. foreign debt from 40 to 50 percent of initial GDP raises the world market interest rate from 6 to 6.15 percent.
  - When μ = 0.6: the same debt increase raises the rate to 6.97 percent.
- BQ: ratio of net foreign debt to GDP in the U.S.
- Depreciation rate of public capital δg = 0.04.
- Public infrastructure investment ξg = 0.04 (4 percent of GDP) assumed needed to offset depreciation (reference: U.S. actual 2.4 percent of GDP; Europe average 5 percent of GDP).
- Rates of return on public capital Rg considered:
  - Low/conservative: 10 percent.
  - Normal/high: 15 percent.
- Public investment in higher education ξs = 1.3 percent of GDP.
- Depreciation rate for education capital δs = 0.03.
- Return on education Rs = 7 percent.
- Input-output coefficient φ backed out from Rs, δs, ξs, the skill premium, and θs at the initial steady state.
- Note on tax revenue context: Tax Policy Center estimates revenue loss from the corporate tax cut at approximately 1.1 percent of GDP ($200 billion a year). Other business tax cuts push the figure close to 1.5 percent of GDP.

### Remarks on model structure and important modeling choices
- Inelastic supply of skilled labor:
  - Skilled-labor supply treated as exogenous; inclusion of endogenous skilled-labor supply would not alter qualitative results unless supply response to skill premium is unusually large.
- Infrastructure specification:
  - Infrastructure modeled as factor-neutral (Hicks-neutral) in a Cobb-Douglas–style specification.
  - Acknowledged limitation: some infrastructure likely non–factor-neutral (e.g., power grid complements capital; rural broadband complements skilled labor and ICT capital), requiring more complex production functions.

### Calibration and estimation of elasticities of substitution (σi)
- Preferred three-level CES specification deemed most empirically plausible.
- Key calibration anchors and ranges:
  - σ1 (elasticity between composite inputs H(K,S) and V(K,L)):
    - Runs include σ1 = 0.5 - 1 and also solutions for σ1 = 1.
  - σ3 (elasticity between traditional capital and skilled labor in composite H):
    - Runs assume either σ3 = σ1 or σ3 = 0.5σ1.
  - σ2 (elasticity between “robots”/ICT capital and labor):
    - Runs let σ2 vary between 1.5 and 5.
    - Authors’ econometric estimations to 2020 point to σ2 between 2.2 and 2.5; baseline point estimate reported below is 2.54.
- Implications for σsl (elasticity between low- and high-skill labor):
  - For σ1 = σ3 = 1, σsl slightly larger than unity.
  - Authors’ estimates put σsl around 0.4 for the U.S. in preferred runs.

### Empirical estimates — baseline nested CES (U.S. data, 1967–2020)
- Main point estimates (baseline production function):
  - σ2 (EOS between ICT capital/“robots” Z and low-skill/routine labor L) = 2.54.
  - σ3 (EOS between non-ICT capital K and non-routine/high-skill labor S) = 0.32.
  - EOS between ICT/“robot” capital (Z) and non-routine/high-skill labor (S) — Hicks-EOS = 0.52.
  - EOS between non-ICT capital (K) and routine/low-skill labor (L) = 0.39.
  - σ1 (EOS between H(S,K) and V(L,Z)) = 0.36.
- Robustness:
  - σ2 remains above 2 across alternative specifications.
  - Baseline nested CES performs better empirically than two plausible alternatives (Alternative A and Alternative B).

### Stylized facts on capital and labor (empirical context)
- ICT capital:
  - Includes Communications, Software, PCs, Terminals, Semiconductors, and Storage devices.
  - Stock of ICT capital increased exponentially since the 2000s; ICT capital price declined and fell below non-ICT capital price.
  - Depreciation rate for ICT capital has increased since the early 1980s.
- Labor categories and trends:
  - Employment in non-routine (“Managerial, Professional, and Technical”) has increased over time, especially in last decade.
  - Routine labor relatively stable; small increase in “Operatives/Laborers”.
  - Real wages increased for both non-routine and routine groups; non-routine wage increases drove skill-premium dynamics.

### Policy experiments — setup
- Policies analyzed: corporate tax cuts (CTC), increased government spending on infrastructure (II), increased investment in education (IE); all financed by reductions in transfers to the skilled worker/capitalist.
- Analysis uses analytical comparative statics and numerical transition dynamics solved with Dynare 4.5.7.
- Key steady-state interest-rate relation: r = ρ 1−x + δ and ˆr = n dx 1−x, where n ≡ ρ ρ+δ(1−x) < 1 and ρ = (1−β)/β.

### A. Corporate Tax Cuts (CTC) — analytical and numerical findings
- Analytical comparative-statics highlights:
  - In “robot” economy with σ2 >> σ1 and σ1 > σ3, ˆZ > ˆK: robot investment responds more elastically than traditional investment.
  - Capital deepening via CTC can reduce the real wage paid to low-skill labor and worsen wage inequality.
  - Weighted average wage ˆω follows ˆω = − (θk + θz)/(θs + θl) * (n dx/(1−x)).
  - Likelihood that CTC will boost growth without exacerbating inequality described as "slim to none."
- Numerical/transition results (selected quantitative outcomes):
  - Cobb-Douglas case: GDP and both wages increase 4 percent.
  - Adding “robot” capital:
    - Runs with σ1 = σ3 = 1 and σ2 = 3−5 raise GDP by another 1−1.7 percentage points relative to Cobb-Douglas.
    - As σ2 rises to 1.5 and above, gains accrue disproportionately to capitalists and high-skill workers; total labor share θ declines, driven by decrease in low-skill income share θL.
  - Transition example (“robot” economy σ2 = 3; σ1 = σ3 = 0.5):
    - Rise in interest rate limited to seven basis points.
    - GDP increase = 1.1 percent at t = 10; 1.75 percent at t = 20.
    - Real wages: low-skill = 0.1 percent at t = 10 and 0.2 percent at t = 20; high-skill = 2.1 percent at t = 10 and 3.5 percent at t = 20.
    - Result: wage gap widens over time; CTC increases wage inequality.

### B. Investment in Infrastructure (II) — setup, analytical thresholds, and numerical results
- Fiscal equivalence: increase in Ig financed by an offsetting cut in transfers at t = 1; ˆIg = − (θk + θz)/ξg (n dx |t=1).
- Long-run analytical results (selected equations preserved in model notation):
  - Wages and quantities respond positively to infrastructure: ˆω = 1/(θs + θl) * η ˆG > 0; ˆwl, ˆws, ˆK, ˆZ, ˆQ expressions given in equations (48)–(50).
  - η ˆG = − (θk + θz) (Rg + δg)/δg (n dx) and ˆG = ˆIg in long run.
- Comparative thresholds (selected numerical guidance):
  - ˆω|II > ˆω|CTC iff Rg > R⋄g = (x/(1−x)) δg.
    - Example: small positive return ≈ 3 percent suffices when x = 0.40 and δg = 0.04.
  - Given parameter ranges, thresholds for ˆwl and ˆws comparisons produce modest Rg values; maximum threshold for R#g is 5 percent when δg = 0.04 and xold = 0.40.
  - (ˆws − ˆwl)|II > (ˆws − ˆwl)|CTC iff Rg > R* g; example numbers give R* g = 12.7 percent for a particular calibration.
- Comparison with CTC on private investment and output:
  - ˆQ|II > ˆQ|CTC iff Rg > R& g; for calibration in Table 1 and δg = 0.04, maximum R& g is 4 percent, minimum negative.
  - Interpretation: II often increases private investment and GDP more than CTC because government invests saved dollars and empirical estimates often find Rg > pre-tax private return.
- Numerical experiments (fiscal-equivalent II = 1.5 percent of initial GDP):
  - Table 5: Rg = 10 percent; Table 6: Rg = 15 percent.
  - II dominates CTC in both Cobb-Douglas and “robot” economies: II increases GDP, wages, and capital stocks more, especially for higher Rg.
  - As σ2 increases, CTC amplifies inequality more than II; II yields much higher low-skill wage and output growth than CTC across σ2 values examined.
- Transition dynamics and sensitivity:
  - With II, large capital inflows (new capital inflows reach 11 percent of GDP) help long-run positive effects dominate consumption-smoothing motive; interest rate rise limited to twenty basis points.
  - Across steady states, private capital stock increase under II equals 75 percent of the increase induced by CTC.
  - At 20-year horizon, gains in GDP and NI are 57 percent and 46 percent higher than on the path for the CTC, respectively.
  - When μ = 0.60, K and Z do not increase until year ten and another eight years elapse before w_s rises above the CTC path.

### Investment in Education (IE) — analytical and numerical results
- Long-run analytical results (preserved inequalities and thresholds):
  - ˆw_l > 0 and ˆw_s < 0 for skill-rebalancing IE with ψ = w_l/w_s < 1; weighted-average wage change cancels: ˆω = θ_s/(θ_s+θ_l) (1−ψ) ˆS.
  - Conditions for IE to outperform CTC expressed as thresholds on R_s (e.g., R_s > R^⋄_s = (x/(1−x)) δ_s) and other inequalities; numerical guidance indicates many conditions satisfied by R_s > 2 percent.
  - Range of maximum threshold values: R^*_s = 3.2-9.5 percent and R&_s = 0-1.8 percent for calibration values.
- Role of σ2:
  - Higher σ2 makes it more likely that IE increases low-skill wages and “robot” capital relative to CTC and reduces the skill premium.
  - Some thresholds (R^⋄_s and R^*_s) do not depend on σ2; conditions for Z and K comparisons become less stringent as σ2 rises.
- Numerical long-run findings (IE increase = 1.5 percent of initial GDP):
  - Huge increases in low-skill wage w_l, GDP, and Z observed.
  - Supply of low-skill labor decreases 25 percent in the long run; supplies of high-skill labor and traditional capital increase 25 percent and 7 - 21 percent, respectively.
  - Low-skill real wage rises 24 - 47 percent to eliminate large ex-ante excess demand.
  - “Robot” capital increases by 69 - 196 percent when σ2 increases from 3 to 5.
  - IE often delivers gains in GDP 3-8 percentage points larger than II (with Rg = 15 percent). Average increase in GDP across IE runs is 14.5 percent vs. 9.3 percent for II (excluding NA cases).
- Transition dynamics example (σ2 = 3, σ1 = σ3 = 0.5, R_s = 0.07):
  - Long-run advantage of IE in traditional capital accumulation emerges only after several decades, but output is higher from the beginning and increases as “robot” and human capital grow.
- Labor-share effects:
  - IE increases the unskilled labor share in the Cobb-Douglas case; as σ2 increases above 1, total and unskilled labor shares fall, but IE still increases the unskilled labor share by at least four percentage points (from 20 percent to 24-24.4 percent) in discussed calibrations.

### Welfare comparisons and policy rankings
- Social welfare function (baseline, no distributional weights):
  - SW = Σ_{t=0}^∞ β^t_sp [ c_t^{1−1/τ} / (1−1/τ) ].
- Benchmark welfare findings (σ2 = 3):
  - II always dominates CTC across calibrations (II reduces underinvestment more effectively; crowding-in of private capital is 75 percent as large as with CTC).
  - IE’s ranking depends on σ1, σ3, and β_sp:
    - IE scores best when σ1 and σ3 are small and β_sp is large.
    - For σ1 = σ3 = 1, IE’s welfare gain smaller than II’s; IE overtakes CTC only at β_sp = 0.97.
    - For σ1 = σ3 = 0.5, IE dominates CTC everywhere except at β_sp = β and beats II once β_sp > 0.957.
- Robot-economy sensitivity:
  - Welfare effects of II, CTC, and especially IE rise with σ2.
  - IE becomes preferred to CTC with high enough σ2 in most calibrations and can outperform II for σ2 > 1.5 with β_sp = 0.97 in some runs.
- Distributional-weighted welfare (proxy metric):
  - Extended SW: SW = Σ_{t=0}^∞ β^t_sp [(c_t + ζ w_t L_o)^{1−1/τ} / (1−1/τ)], ζ > 0.
  - With small positive ζ (ζ = 0.25 and 0.50) in benchmark σ2 = 3 calibration:
    - Welfare rankings change dramatically in favor of IE.
    - IE strongly dominates CTC and beats II in three of four runs with ζ > 0.

### Key policy-relevant conclusions (numeric ranges and comparative statements preserved)
- Presence of substitutable “robot” capital (higher σ2) materially changes policy effectiveness:
  - CTC can raise long-run GDP growth but may raise low-skill wages very little or cause them to fall when σ2 is high.
    - Long-run low-skill real wage ranges reported: under CTC −1.2 to 3.4 percent.
  - II yields larger wage gains across the skill spectrum than CTC; II long-run gains reported in aggregate ranges for outcomes are 4.1 to 8.5 percent for II (contextual range in source headings).
  - IE yields the largest increases in low-skill wages, GDP, and capital stocks in the “robot” economy; long-run low-skill real wage gains range from 25 to 47 percent under IE.
- General equilibrium effects can reverse partial-equilibrium rankings:
  - IE’s lower direct return (7 percent) can nevertheless produce larger aggregate capital accumulation and welfare gains than policies with higher direct returns when substitution patterns and complementarities amplify its effects.
- Robustness and caveats:
  - Results are robust to many caveats but model omits relevant factors; efficacy with which IE produces labor complementary to robots merits closer examination.
  - Even with much lower returns to IE (30 to 40 percent as high as returns on II and private capital), IE can remain appealing if policymakers care about helping the poor.

*Source: wpiea2024003-print-pdf - 1.5 percent of GDP.*

### 1.5 percent of GDP.

### wpiea2024003-print-pdf - 1.5 percent of GDP.

### Key calibration parameters and baseline quantitative assumptions
- Capital-flow elasticity parameter μ: either 0.10 or 0.60.
  - When μ = 0.1: an increase in U.S. foreign debt from 40 to 50 percent of initial GDP raises the world market interest rate from 6 to 6.15 percent.
  - When μ = 0.6: the same debt increase raises the rate to 6.97 percent.
- BQ: ratio of net foreign debt to GDP in the U.S.
- Depreciation rate of public capital δg = 0.04.
- Public infrastructure investment ξg = 0.04 (4 percent of GDP) assumed needed to offset depreciation (reference: U.S. actual 2.4 percent of GDP; Europe average 5 percent of GDP).
- Rates of return on public capital Rg considered:
  - Low/conservative: 10 percent (equal to pre-tax return on private capital).
  - Normal/high: 15 percent.
- Public investment in higher education ξs = 1.3 percent of GDP (U.S. value in 2017).
- Depreciation rate for education capital δs = 3 percent (δs = 0.03).
- Return on education Rs = 7 percent.
- The input-output coefficient φ is backed out from Rs, δs, ξs, the skill premium, and θs at the initial steady state.
- Note on tax revenue context: The Tax Policy Center estimates the revenue loss from the corporate tax cut at approximately 1.1 percent of GDP ($200 billion a year). Other business tax cuts push the figure close to 1.5 percent of GDP.

### Remarks on model structure and important modeling choices
- Inelastic supply of skilled labor:
  - Supply of skilled labor treated as exogenous, justified by Autor (2014) and Murphy and Topel (2016) showing little change in college-educated share despite large increases in the skill premium since 1980.
  - Including endogenous skilled-labor supply would not alter qualitative results; quantitative effects matter only if supply response to the skill premium is unusually large.
- Infrastructure specification:
  - Infrastructure is modeled as factor-neutral (Hicks-neutral technological progress) affecting a Cobb-Douglas–style specification in private inputs.
  - Acknowledged limitation: some infrastructure likely non–factor-neutral (e.g., power grid complements capital; rural broadband complements skilled labor and ICT capital), which would require a more complex production function with Genters as an additional argument in composite inputs H(K,S) and V(L,Z).

### Calibration and estimation of elasticities of substitution (σi)
- Overview conclusions from calibration and estimation:
  1. Preferred three-level CES specification is the most empirically plausible.
  2. Best estimates of elasticities support the model assumptions laid out.
  3. The elasticity of substitution between “robot” capital and labor (σ2) is above 2 and higher than other elasticities in both baseline and alternative specifications.
- Specific parameter calibration choices and empirical anchors:
  - σ1 (elasticity between composite inputs H(K,S) and V(K,L)):
    - Macroeconomic estimates typically close to unity.
    - Newer microeconomic estimates place σ1 between 0.4 and 0.6.
    - Authors’ recent macro estimates ≈ 0.35 (see Table 3).
    - Runs include σ1 = 0.5 - 1 and also solutions for σ1 = 1.
  - σ3 (elasticity between traditional capital and skilled labor in composite H):
    - Literature suggests σ3 is 20 - 60 percent smaller than σ1.
    - In estimation, σ3 is very close to σ1; runs assume either σ3 = σ1 or σ3 = 0.5σ1.
  - σ2 (elasticity between “robots”/ICT capital and labor):
    - No direct econometric estimates in literature; technology experts indicate substitution between robots/AI and human labor is much easier than between most primary inputs.
    - Eden and Gaggl (2018) calibration implies σ2 = 2.13 for 1950–2013 with the authors’ nesting.
    - Authors’ econometric estimations to 2020 point to σ2 between 2.2 and 2.5 and increasing since the 1990s (see Table 3).
    - Acemoglu and Restrepo (2019b) evidence (one robot directly eliminates 10.6 jobs, conditional on several parameters) suggests σ2 might be quite large and rising.
    - Runs let σ2 vary between 1.5 and 5 (reflecting expectation that automation capital will become more substitutable with labor).
- Implications for elasticity of substitution between low- and high-skill labor σsl:
  - For σ1 = σ3 = 1, associated σsl is slightly larger than unity.
  - In other preferred runs, σsl is below unity; authors’ estimates put σsl around 0.4 for the U.S. (see estimation discussion).
  - Nested-CES relation used: σsl = (a+b+c)(a/σ3 + b/σ2 + c/σ1)−1, with definitions of a, b, c given in text; for σ2 = 1.5 - 5 and σ1 = σ3 = 1, σsl = 1.008 − 1.020.
- Sensitivity analysis and comparison to other calibrations:
  - Authors test sensitivity using Krusell and others (2000b) calibration: σ1 = 1.69 and σ3 = 0.67, yielding σsl ≈ 1.40 (σsl ranges 1.35 - 1.37 for σ2 = 1.5 - 5).
  - Results using Krusell et al.’s values lie between the authors’ runs for σ1 = 1 and σ3 = 0.5, 1.
  - Concern with Krusell et al.’s calibration: a high EOS between low- and high-skill labor implies an equally high EOS between traditional capital and low-skill labor, contradicting many empirical estimates.
  - Authors solved alternative 3-tiered CES Q = F{K, H[S, J(L,Z)]} with σ1 ≤ 1 and σ4 (EOS between skilled labor S and composite J) = 1.5; analytical and numerical results for that specification are available in a longer version (Berg and others (2023)). For σ2 > σ4, quantitative results are slightly weaker.

### Empirical and literature context for elasticities
- Empirical evidence and meta-analyses:
  - Cantore, Ferroni, and Leon-Ledesma (2017) note a literature “consensus estimate” around 1.5 for EOS between workers.
  - Havranek and others (2020): estimates of σsl vary widely; 29.6 percent of estimates lie in the (0,1) interval and 34.7 percent in the (1,2) interval; after correcting for biases, mean σsl for the U.S. lies in range 0.6 - 0.9.
  - Authors favor calibrations implying lower σsl (e.g., ~0.4) and test sensitivity to higher σsl.
- Rationale for CES framework vs. task-based framework:
  - Task-based frameworks analyze micro mechanisms of robot–labor substitution and task reallocation; for the macro issues addressed, authors are confident CES framework gives similar implications, with σ2 capturing ease of task substitution.

*Source: wpiea2024003-print-pdf - 1.5 percent of GDP.*

### 0.53 to 0.65, whileσ

### wpiea2024003-print-pdf - 0.53 to 0.65, whileσ

### B. Estimation — overview
- Calibrated elasticities of substitution (“σi” for i=1, 2, 3) are checked against econometric estimates using U.S. data for 1967-2020.
- Key empirical finding: σ2 (EOS between L and Z) is greater than 2 and features a positive trend over recent decades, indicating increasing substitutability between low-skill labor and “robots” (ICT capital).
- Baseline nested CES production function (equations (22)–(25)) performs better empirically than two plausible alternatives (Alternative A and Alternative B).

### Stylized facts on capital and labor
- ICT capital:
  - Includes Communications, Software, PCs, Terminals, Semiconductors, and Storage devices.
  - The stock of ICT capital has increased exponentially since the 2000s.
  - ICT capital price has declined, falling below that of non-ICT capital.
  - The depreciation rate for ICT capital has increased from the early 1980s; the increase after the 2000s could reflect smartphone technologies, software, cloud computing, and faster internet connections.
- Non-ICT capital:
  - Price remained constant from the 1980s until the 2000s, and since then experienced a moderate increase.
- Labor categories:
  - Non-routine captures problem-solving, creativity, complex decision-making; follows aggregation from Eden and Gaggl (2018) based on Acemoglu and Autor (2011).
  - Employment in non-routine (“Managerial, Professional, and Technical”) has increased over time, especially in the last decade; “Services” and other groups plateaued since mid-2010s.
  - Routine labor has been relatively stable; small increase in “Operatives/Laborers”.
  - Real wages increased for both non-routine and routine groups, with non-routine wage increases driven by “Managerial, Professional, and Technical” and “Services”.

### Empirical approach and estimates of elasticities of substitution
- Baseline nested CES production function:
  - Top-tier: F[H(S,K), V(L,Z)] with F(•) a CES (equations (22)–(25)).
  - Nested aggregators: V(•) = [κ1 Lε2 + κ2 Zε2]1/ε2 and H(•) = [γ1 Sε3 + γ2 Kε3]1/ε3.
  - Parameters relate as: ι1=a1/σ1, ι2=(1−a)1/σ1, ε1=(σ1−1)/σ1, κ1=e1/σ2, κ2=(1−e)1/σ2, ε2=(σ2−1)/σ2, γ1=g1/σ3, γ2=(1−g)1/σ3, ε3=(σ3−1)/σ3.
- Estimation strategy:
  - Use first-order conditions to form three log-linear equations (26)–(28) relating income shares and normalized input ratios.
  - Estimate equations (26) and (27) by OLS to obtain ε2, ε3 and intercepts; construct V and H and estimate (28) to obtain ε1.
  - Convert εi to σi via EOS(L,Z)=σ2=1/(1−ε2), EOS(S,K)=σ3=1/(1−ε3), EOS(H,V)=σ1=1/(1−ε1).
  - Note: some series failed cointegration tests; results treated as stylized facts/parameters’ estimates.

### Key empirical estimates (baseline)
- Main point estimates (baseline production function):
  - σ2 (EOS between ICT capital/“robots” Z and low-skill/routine labor L) = 2.54.
    - Interpretation: high substitution between Z and L; σ2 has increased over time (see Figure 3).
  - σ3 (EOS between non-ICT capital K and non-routine/high-skill labor S) = 0.32 (smallest in baseline).
  - EOS between ICT/“robot” capital (Z) and non-routine/high-skill labor (S) — Hicks-EOS = 0.52 (low substitutability).
  - EOS between non-ICT capital (K) and routine/low-skill labor (L) = 0.39 (not a close substitute).
  - σ1 (EOS between H(S,K) and V(L,Z)) = 0.36.
- Hicks-EOS:
  - Calculated based on Sato (1967) harmonic-mean formula; lies between relevant elasticity bounds.

### Alternative production functions and robustness
- Alternative A: Eden and Gaggl (2018) specification F{K, G[L, W(S,Z)]} (outer nest Cobb-Douglas; equate Ks and Ke with K and Z).
  - Imposes elasticity between L and S similar to that between L and Z — potentially counterfactual.
  - Empirical drawback: EOS between K and G(•) around -0.13 (negative), leaving some Hicks-EOS undefined.
- Alternative B: F{K, J[S, V(L,Z)]}.
  - Also produces small/negative EOS between K and composite nests in some estimates.
- Across specifications:
  - Estimate for σ2 remains above 2 across specifications.
  - σ3 is smaller in the baseline (0.32) compared to alternative B.
  - σ(Z,S) is much lower under the baseline than under the two alternatives.
- Conclusion: baseline nested CES is the most plausible specification; estimates provide discipline for calibration ranges used in policy experiments.

### V. Policy experiments — setup
- Policies analyzed: corporate tax cuts (CTC), increased government spending on infrastructure and education (II), all financed by reductions in transfers to the skilled worker/capitalist.
- Analysis uses mix of analytical (small differential changes) and numerical (global nonlinear saddle path solved with Dynare 4.5.7) results.
- Key model steady-state relation for interest rate across steady states:
  - r = ρ 1−x + δ
  - ˆr = n dx 1−x, where n ≡ ρ ρ+δ(1−x) < 1 and ρ = (1−β)/β.

### A. Corporate Tax Cuts (CTC) — analytical results
- Log-differential expressions (small dx<0, transfers adjust):
  - ˆwl, ˆws, ˆK, ˆZ, ˆQ given in equations (35)–(39).
  - Composite parameters:
    - m ≡ q + p θl θs
    - p ≡ σ3 χk + σ1 χs
    - q ≡ σ2 αz + σ1 αl
    - Cost shares: χk = θk/(θk+θs), χs = θs/(θk+θs), αl = θl/(θl+θz), αz = θz/(θl+θz)
- Comparative statics key messages:
  - When σi = σ ∀i (non-nested CES), ˆwl = ˆws = − (θk + θz)/(θs + θl) * (n dx/(1−x)) > 0 and ˆK = ˆZ = −σ/(θs + θl) * (n dx/(1−x)) > 0.
  - In “robot” economy with σ2 >> σ1 and σ1 > σ3:
    - ˆZ > ˆK (robot investment response is more elastic than traditional investment).
    - Increases in Z tend to raise demand for skilled labor and can reduce demand for low-skill labor.
- Conditions:
  - ˆws > 0 iff σ1(θk + χk θl) − σ3 χk θl + αz[σ2(θk + θl + θz) − σ1 θk] > 0 (equation (41)).
  - ˆwl > 0 iff σ3 χk 1−θl/θs − αz(σ2 − σ1)(θk + θs) > 0 (equation (42)).
  - With θk = 0.36 and θs = 0.40, demand for low-skill labor decreases if σ2 > σ1 θk + θs ≈ 1.3 σ1.
  - Empirical context: σ1 estimates cluster between 0.4 and 1 (authors’ near lower bound); σ2 estimates north of two.
- Implications:
  - Even if “robot” capital accounts for at most 11 percent of aggregate capital stock, σ2 > 2 and ˆZ >> ˆK imply outsized effects on labor markets.
  - Capital deepening via CTC can reduce real wage paid to low-skill labor and worsen wage inequality.
  - Weighted average wage ω rises as in non-nested CES: ˆω = − (θk + θz)/(θs + θl) * (n dx/(1−x)) (equation (43)).
  - ˆws > ˆwl iff (σ2 − σ1) αz > (σ3 − σ1) χk (equation (44)).
  - Income share for low-skill labor ˆwl − ˆQ declines when (σ2 − σ1) αz θs + p θz (σ2 − 1) + σ3 θk (q − 1) > 0 (equation (45)).
  - Likelihood that CTC will boost growth without exacerbating inequality is described as "slim to none."

### A. Corporate Tax Cuts (CTC) — numerical / transition results
- Solution method: global nonlinear saddle path; Dynare 4.5.7; saddle-point stable system with two state variables Ka,t−1 and Bt−1 and two jump variables Ct and Ia,t.
- Table 4 findings (qualitative summary):
  - Cobb-Douglas case: GDP and both wages increase 4 percent.
  - Adding “robot” capital:
    - Growth impact of CTC increases in σ2: runs with σ1 = σ3 = 1 and σ2 = 3−5 raise GDP by another 1−1.7 percentage points.
    - As σ2 rises to 1.5 and above, gains accrue disproportionately to capitalists and high-skill workers; total labor share (θ) declines, driven by decrease in low-skill income share (θL).
- Transition example (Figure 4; “robot” economy with σ2 = 3 and σ1 = σ3 = 0.5):
  - Rise in interest rate limited to seven basis points (optimistic bias).
  - Slow adjustment: GDP increase = 1.1 percent at t = 10; 1.75 percent at t = 20.
  - Real wages:
    - Low-skill: 0.1 percent at t = 10; 0.2 percent at t = 20.
    - High-skill: 2.1 percent at t = 10; 3.5 percent at t = 20.
  - Result: wage gap widens over time; CTC increases wage inequality.

### B. Investment in Infrastructure (II) — setup and fiscal equivalence
- Fiscal equivalence imposed for apple-to-apple comparison: cut in transfers that would offset corporate tax revenue loss at t = 1 now finances increase in Ig,t.
  - dIg = dT |t=1 = −(r − δ) Ka dx |t=1.
  - Rewritten: ˆIg = − (θk + θz)/ξg (n dx |t=1), with θk + θz = r Ka/Q and ξg ≡ Ig/Q (equation (46)).
- Note: fiscal equivalence at t = 1 does not hold subsequently due to endogenous tax base evolution.

### B. II — analytical results (long run)
- Wages and quantities respond positively:
  - ˆω = 1/(θs + θl) * η ˆG > 0 (equation (47)).
  - ˆwl = p/(θs m) * η ˆG > 0; ˆws = q/(θs m) * η ˆG > 0 (equation (48)).
  - ˆK = σ3 q/(θs m) * η ˆG > 0; ˆZ = σ2 p/(θs m) * η ˆG > 0 (equation (49)).
  - ˆQ = [θk σ3 q + θz σ2 p]/(θs m) + 1 times η ˆG > 0 (equation (50)).
- Relation to return on infrastructure:
  - ∂Q/∂G = Rg + δg = η Q/G = η δg ξg.
  - η ˆG = − (θk + θz) (Rg + δg)/δg (n dx) (equation (51)); ˆG = ˆIg long run.
- Comparative thresholds (equations (52)–(55)):
  - ˆω|II > ˆω|CTC iff Rg > R⋄g = (x/(1−x)) δg.
    - Example: small positive return ≈ 3 percent suffices when x = 0.40 and δg = 0.04.
  - ˆwl|II > ˆwl|CTC iff Rg > R+g = [1−θs ((σ2 − σ1) αz − (σ3 − σ1) χk) / (p x (θk + θz))] (x/(1−x)) δg (equation (53)).
    - Under assumed parameter ranges, R⋄g condition often suffices for II to increase low-skill wage more than CTC.
  - ˆws|II > ˆws|CTC iff Rg > R#g = [1 + θl/(x (θk + θz)) (1 − p/q)] (x/(1−x)) δg (equation (54)).
    - For δg = 0.04, initial tax rate xold = 0.40, maximum threshold value for R#g is 5 percent given provided ranges.
  - (ˆws − ˆwl)|II > (ˆws − ˆwl)|CTC iff Rg > R* g = [1/((1−x)(θk + θz)) − 1] δg (equation (55)).
    - With θk + θz = 0.40, x = 0.040, δg = 0.04, R* g = 12.7 percent.
- Key analytical takeaways:
  - Labor across the skill spectrum benefits more from II than from CTC for modest Rg (e.g., ≈3 percent).
  - Wage inequality is still likely to worsen under II, especially for larger σ2; whether II raises inequality more than CTC depends on Rg.
  - For high rates of return, II can increase wage inequality more than CTC; threshold R* g = 12.7 percent is high but within some literature estimates for returns on core capital.

### B. II — comparison with CTC on private investment and growth
- Comparison condition for private investment and output (equation (56)):
  - ˆQ|II > ˆQ|CTC iff Rg > R& g = [σ3 θk q + σ2 θz p]/(u(1−x)(θk + θz) − 1) δg, where u ≡ q(σ3 θk + θs) + p(σ2 θz + θl).
  - For believable parameters, threshold R& g is relatively small; for calibration in Table 1 and δg = 0.04, maximum R& g is 4 percent, minimum negative.
- Interpretation:
  - II often increases private investment and GDP more than CTC because government invests saved dollars (rather than partial consumption by capital owners) and empirical estimates often find Rg > pre-tax private return.
  - Therefore, the presumption that CTCs are more effective at promoting private investment is weak.

### B. II — numerical and transition results
- Numerical experiments (Tables 5 and 6):
  - Fiscal-equivalent increase in II equals 1.5 percent of initial GDP.
  - Table 5: infrastructure return Rg = 10 percent (same as pre-tax return on private capital).
  - Table 6: Rg = 15 percent (low end of returns for core capital in Bom and Ligthart (2014)).
- Numerical findings:
  - In both Cobb-Douglas and “robot” economies, II dominates CTC: II increases GDP, wages, and capital stocks more, especially for higher Rg.
  - “Robot” capital amplifies inequality-inducing effects of CTC relative to II (Table 7):
    - As σ2 increases, high-skill wages increasingly outpace low-skill wages and output outpaces total wages for both CTC and II.
    - Differences in low-skill wage and output growth between II and CTC are substantial in magnitude—II yields much higher low-skill wage and output growth than CTC for all σ2 values examined.
  - Example note interrupted in text: "For example, when σ2 = 5, low-skill wages grow by only" — sentence is truncated in provided content.

*Source: wpiea2024003-print-pdf*

### 0.7 percentage points with CTC and 7.0 percentage points with II (andR

### wpiea2024003-print-pdf - 0.7 percentage points with CTC and 7.0 percentage points with II (andR

### II versus CTC: transition dynamics and comparative magnitudes
- Impulse-response comparison strongly favors Investment in Infrastructure (II) over the Corporate Tax Cut (CTC), but displays large quantitative and qualitative surprises in the paths for K and Z.
- CTC stimulates private investment immediately; II exerts conflicting short/medium-run effects:
  - Infrastructure increases future income of capitalists and high-skill labor, inducing consumption-smoothing that can temporarily reduce private investment.
  - Infrastructure also raises productivity of capital and reduces adjustment costs, encouraging immediate investment.
- Large capital inflows (new capital inflows reach 11 percent of GDP) help the long-run positive effects of II dominate the consumption-smoothing motive, preventing the interest rate from rising more than twenty basis points.
- With II:
  - Across steady states, the increase in the private capital stock (K+Z) equals 75 percent of the increase induced by the CTC.
  - The slow pace of private capital accumulation under II slows growth of GDP and national income (NI).
  - At the 20-year horizon, gains in GDP and NI are only 57 percent and 46 percent higher than on the path for the CTC, respectively.
  - For the high-skill labor wage, it takes twenty-seven years for the gap with the CTC path to become positive.
- Sensitivity to elasticity of interest-rate response to capital flows (μ):
  - When μ = 0.60, K and Z do not increase until year ten and another eight years elapse before w_s rises above the path associated with the CTC.

### Investment in Education (IE): long-run analytical results
- Steady-state differential effects (notation preserved from source):
  - ˆw_l = θ_l+θ_s ψ θ_l m  ˆS 
 > 0 and ˆw_s = − θ_l+θ_s ψ θ_s m  ˆS 
 < 0, where ψ = w_l / w_s < 1 and m ≡ q + p θ_l/θ_s > 0.
  - Weighted-average wage change cancels: ˆω = θ_s/(θ_s+θ_l) (1−ψ) ˆS.
- Capital and output steady-state responses:
  - ˆK, ˆZ, and ˆQ given by equations (58)-(60) (preserved in text).
- Comparative conditions for IE to outperform CTC (preserved inequalities and threshold returns):
  - ˆω|IE > ˆω|CTC iff R_s > R^⋄_s = ((x/(1−x)) δ_s).
  - ˆw_l|IE > ˆw_l|CTC if (i) (σ_2−σ_1) α_z > (σ_3−σ_1) χ_k and (ii) R_s > R^+_s = [θ_l p (1−ψ) / ((1−x)(θ_l+θ_s ψ)−1)] δ_s.
  - ˆK_IE > ˆK_CTC if (i) σ_1 ≥ σ_3 and (ii) R_s > R^*_s = [σ_3 (1−x)(θ_k+θ_z)−1] δ_s.
  - ˆZ_IE > ˆZ_CTC if (i) σ_2 > σ_1 [1 + (θ_l+θ_z)/θ_s (p/σ_1)] and (ii) R_s > R′_s = [p(1−ψ)/((1−x)(θ_k+θ_z)−1)] δ_s.
  - ˆQ_IE > ˆQ_CTC iff R_s > R&_s = [(θ_k σ_3 q + θ_z σ_2 p)(1−ψ)/(1−x)(q−ψp)(θ_k+θ_z)−1] δ_s.
- Threshold numerical guidance (for calibration values in Table 1):
  - Conditions in (61) and (62) are satisfied by R_s > 2 percent.
  - Range of maximum threshold values: R^*_s = 3.2-9.5 percent and R&_s = 0-1.8 percent.
- Role of substitutable “robot” capital (σ_2):
  - Higher σ_2 makes it more likely that IE increases low-skill wages and “robot” capital relative to CTC.
  - Higher σ_2 reduces the skill premium (w_s falls more as σ_2 rises).
  - R^⋄_s and R^*_s do not depend on σ_2; conditions (i) in (62) and (64) become less stringent as σ_2 rises.
  - A sufficient condition for ∂R^⋄_s/∂σ_2 < 0 is given by equation (66); for the parameter values in Table 1 this reduces to σ_3 > 0.49 σ_1.

### Investment in Education (IE): long-run and transition numerical results
- Policy experiment: increase in IE equal to 1.5 percent of initial GDP (fiscally-equivalent to the CTC transfer change).
- Key long-run numerical findings (Table 8 referenced):
  - Huge increases in low-skill wage w_l, GDP, and Z are observed.
  - Supply of low-skill labor decreases 25 percent in the long run.
  - Supplies of high-skill labor and traditional capital increase 25 percent and 7 - 21 percent, respectively.
  - Low-skill labor wage rises 24 - 47 percent to eliminate large ex-ante excess demand.
  - “Robot” capital increases by 69 - 196 percent when σ_2 increases from 3 to 5.
  - IE often delivers gains in GDP 3-8 percentage points larger than II (which pays a return of 15 percent). Across runs (excluding NA cases), average increase in GDP in Table 8 is 14.5 percent vs. 9.3 percent in Table 6.
  - Despite IE’s assumed return of 7 percent (pre-tax), the total investment impact is large enough to outperform II in many parameterizations.
- Interaction mechanisms:
  - IE reduces the supply of unskilled labor, raising the productivity of “robots” and incentivizing large investment in “robot” capital.
  - Larger “robot” capital sustains investment in traditional capital by mitigating decreases in low-skill labor services.
- Transition dynamics example:
  - For σ_2 = 3, σ_1 = σ_3 = 0.5, and R_s = 0.07, the long-run advantage of IE in traditional capital accumulation does not emerge for several decades, but output is higher from the beginning and increasingly so due to faster “robot” and human capital growth.
- Labor-share effects:
  - IE increases the share going to unskilled labor in the non-robot economy (Cobb-Douglas).
  - As σ_2 increases above 1, the total and unskilled labor shares fall, but IE still increases the unskilled labor share by at least four percentage points (from 20 percent to 24-24.4 percent) for the discussed calibrations.

### Social welfare comparisons and policy rankings
- Baseline social welfare function (no distributional weights):
  - SW = Σ_{t=0}^∞ β^t_sp [ c_t^{1−1/τ} / (1−1/τ) ], where β_sp is the social discount factor and c ≡ C + w_t L_t.
- Benchmark case (σ_2 = 3):
  - Runs for σ_1 = σ_3 = 1 and σ_1 = σ_3 = 0.5; social discount factor β_sp varied from the private discount factor 0.943 up to 0.97 - 0.99.
  - CTC reduces revenue by one percent of GDP at t = 0.
  - Robust result: II always dominates CTC across calibrations (II reduces underinvestment more effectively; crowding-in of private capital is 75 percent as large as with CTC).
  - IE’s ranking depends on elasticities and β_sp:
    - IE scores best when σ_1 and σ_3 are small and β_sp is large.
    - For σ_1 = σ_3 = 1, IE’s welfare gain is smaller than II’s and does not overtake the CTC until β_sp = 0.97.
    - For σ_1 = σ_3 = 0.5, IE dominates CTC everywhere except at β_sp = β (where it ties) and beats II once β_sp > 0.957.
- Robot-economy sensitivity (varying σ_2 up to 5):
  - Welfare effects of II, CTC, and especially IE rise with σ_2.
  - IE’s effect is amplified because it relieves skilled-labor scarcity and increases incentives for “robot” capital investment.
  - In particular, except when σ_1 = σ_3 = 1 and β_sp = 0.943, IE becomes preferred to CTC with high enough σ_2.
  - IE outperforms II in the CES economy with σ_2 > 1.5 and β_sp = 0.97.
- Incorporating distributional concerns (proxy metric):
  - Extended social welfare: SW = Σ_{t=0}^∞ β^t_sp [(c_t + ζ w_t L_o)^{1−1/τ} / (1−1/τ)], ζ > 0, where w_t L_o is real income of low-wage workers and 1+ζ is the marginal rate of substitution between consumption of the poor and the non-poor.
  - Acknowledged limitations: metric ignores income distribution within the saving class and undercounts consumption gains from IE for workers who move from low-skill to high-skill status.
  - With small positive ζ (ζ = 0.25 and 0.50) in the benchmark σ_2 = 3 calibration:
    - Welfare rankings change dramatically in favor of IE.
    - IE strongly dominates CTC and beats II in three of four runs with ζ > 0.
    - Ranking ambiguous only for σ_1 = σ_3 = 1 and ζ = 0.25, where IE is a close second to II before pulling ahead at β_sp = 0.97.

### Key policy-relevant conclusions
- The presence of substitutable “robot” capital (higher σ_2) materially changes policy effectiveness:
  - CTC can raise long-run GDP growth but may raise low-skill wages very little or cause them to fall when σ_2 is high.
  - II yields larger wage gains across the skill spectrum than CTC, and low-skill wage gains remain large as σ_2 increases from 1.5 to 5.
  - IE yields the largest increases in low-skill wages, GDP, and capital stocks in the “robot” economy; long-run low-skill real wage gains range from 25 to 47 percent under IE versus -1.2 to 3.4 percent for the CTC (long-run ranges preserved from source).
- General equilibrium effects can reverse partial-equilibrium rankings: IE’s lower direct return (7 percent) can nevertheless produce larger aggregate capital accumulation and welfare gains than policies with higher direct returns when substitution patterns and complementarities amplify its effects.
- Caveat on IE calibration: the parameter φ translating education spending into skilled labor was calibrated with historical estimates of the return to education; sustaining historical targeting/returns as technology evolves may be challenging.

*Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024003-print-pdf.pdf*

### 4.1 to 8.5 percent for II.

### wpiea2024003-print-pdf - 4.1 to 8.5 percent for II.

### Welfare rankings across policies
- IE (investment in education that produces labor complementary to “robots”) is near-dominant despite a noted negative bias towards IE in the welfare measurement.
- For plausible calibrations, II (infrastructure investment) dominates the CTC (child tax credit).
- IE tends to produce the highest welfare gains, especially when:
  - the elasticity of substitution between traditional capital (non-robot) and labor is low;
  - there are explicit distributional objectives;
  - the discount factor is high.
- Absent explicit distributional objectives:
  - IE benefits strongly from highly substitutable “robot” capital;
  - CTC and IT (income taxation?) benefit weakly from highly substitutable “robot” capital.
- II delivers larger welfare gains than IE under traditional production functions; CTC also tends to do so in that case.
- Once “robot” capital becomes highly substitutable with unskilled labor, IE overtakes both II and CTC in the welfare ranking.

### Role of elasticities and empirical inference
- The welfare rankings depend critically on elasticities of substitution, especially σ2.
- Introducing “robot” capital as a distinct factor (measured by ICT capital) implies that many empirical estimates in the literature are not directly comparable to the model’s parameters.
- By extracting data on capital and labor stocks, wages, and rates of return corresponding to the model’s production function, the authors infer implied elasticities for their specification and for two alternative CES nestings.
- Conclusion: the preferred specification is the most empirically plausible; for this baseline and reasonable alternative nestings, σ2 is above than 2, lending broad support to the paper’s key assumption.

### Robustness, caveats, and research agenda
- Main results are likely robust to wide-ranging caveats, but the simple model omits many relevant factors.
- Key assumptions—e.g., the efficacy with which additional IE produces labor complementary to “robots”—merit closer examination.
- New technology-related skepticism about the trickle-down effects of CTC, and the more positive effects of II and IE, are driven by simple underlying forces modeled here.
- General lessons:
  - Richer analysis of policy payoffs should consider implications of increasing automation.
  - Specific results may depend on exact modeling of technological change (initial indications that Large Language Models, such as ChatGPT, may look somewhat different).
  - Underlying results remain: general equilibrium effects are first-order; traditional production functions give the wrong answer; partial equilibrium rates of return may give the wrong welfare rankings.
- Additional note: Results with much lower returns to IE are available on request; surprisingly, the case for IE remains strong even if its direct return is only 30 to 40 percent as high as the direct returns on II and private capital, provided policymakers care a little bit about helping the poor.

### Appendix — derivation of long-run comparisons between CTC and IE
- Goal: re-express solutions in (57)–(60) in terms of fiscally-equivalent increases in IE and compare with CTC solutions in (35)–(39), focusing on dependence on the rate of return to investment in education R_s.
- Fiscally-equivalent increase in I_s:
  - ˆI_s = − (θ_k + θ_z) ξ_s (ndx) , analogous to (46) and given in (69), where ξ_s = I_s / Q.
- Across steady states:
  - ˆS = φ_Su S ˆS_u = φ_Su S ˆI_s, and since ˆS_u = ˆI_s, combining with (69) yields
  - ˆS = − [φ_Su S] (θ_k + θ_z) ξ_s (ndx) , equation (70).
- The return to investment in education R_s depends primarily on φ and the skill premium 1/ψ:
  - Using the marginal product of education capital,
    - ∂Q/∂S_u = R_s + δ_s = (w_s − w_l) dS/dS_u = (1−ψ) θ_s φ Q/S,
  - hence
    - R_s + δ_s = (1−ψ) θ_s δ_s ξ_s [φ S_u / S] , equation (71).
- Using (71) and (70) produces:
  - ˆS = − [ (θ_k + θ_z) / ((1−ψ) θ_s) ] (R_s + δ_s)/δ_s (ndx) , equation (72).
- Substituting ˆS into (57) and (59)–(60) and comparing to CTC solutions yields the conditions in (61)–(65).

*Source: wpiea2024003-print-pdf - 4.1 to 8.5 percent for II.*

### REFERENCES

### REFERENCES (wpiea2024003-print-pdf)

### Calibration and key model parameters
- Base case calibration (Table 1):
  - β 0.94
  - δ 0.06
  - τ 0.5
  - Ω 2 q-elasticity of investment
  - θ_k 0.36 (Capital’s cost share evaluated at the initial steady state)
  - θ_z 0.04 (Robots’s cost share evaluated at the initial steady state)
  - θ_s 0.4 (High-skill labor’s cost share evaluated at the initial steady state)
  - θ_l 0.2 (Low-skill labor’s cost share evaluated at the initial steady state)
  - w_l / w_s 0.5 (Inverse of the skill premium)
  - x_old 0.4 (Initial corporate profits tax rate)
  - x_new 0.36 (After-cut corporate profits tax rate)
  - μ 0.1, 0.6 (Elasticity to capital flows)
  - B/Q 0.4 (Net debt-to-GDP ratio)
  - σ1 0.5, 1 (Elasticity of substitution between the composite inputs H(•) and V(•))
  - σ2 1.5, 3, 5 (Elasticity of substitution between low-skill labor and robots)
  - σ3 0.25, 0.5, 1 (Elasticity of substitution between high-skill labor and traditional capital)
  - δ_g 0.04 (Depreciation rate of infrastructure)
  - ξ_g 0.04 (Ratio of infrastructure investment to GDP)
  - R_g 0.10, 0.15 (Return on infrastructure investment net of depreciation)
  - η 0.10, 0.15 (Elasticity of output with respect to infrastructure)
  - δ_s 0.03 (Depreciation rate of education capital)
  - ξ_s 0.013 (Ratio of investment in education to GDP)
  - R_s 0.07 (Return on investment in education net of depreciation)
  - φ 0.5 (Input-output coefficient that links education capital to the supply of skilled labor)
- Note: values for η and φ are derived from other parameter assignments (Table notes).

### Estimated coefficients and elasticities of substitution (EOS)
- Key estimated parameters (our data through 2020; Table 18 / Table 13):
  - ε2 0.462 [0.449, 0.475]
  - γ1 0.365 [0.337, 0.395]
  - ε3 0.609 (or 0.610 in some tables) [0.596, 0.623]
  - κ1 0.626 [0.603, 0.649]
  - ι1 0.979 (Table 13 alternative entries; confidence bands reported in tables)
- Estimated EOS and Hicks-EOS (selected; Table 3, Table 14, Table 15, Table 18):
  - EOS(S,Z) 1.859 [1.814, 1.904]
  - EOS(L,Z) 2.542 [2.457, 2.631]
  - EOS [H(S,K), V(Z,L)] 0.356 [0.347, 0.375]
  - Hicks-EOS (with Cobb-Douglas top tier) reports variations; e.g., EOS(S,Z) 0.93 (Table 15) under the imposed Cobb-Douglas top tier assumption.
- Comparison to Eden and Gaggl (data to 2013) (Table 17 / discussion):
  - Eden and Gaggl ε2 approx 0.275 [0.299, 0.275 reported ranges]
  - Eden and Gaggl ε3 approx 1.071 (their estimates differ from extended-sample estimates)

### Long-run impacts of alternative fiscal policies (overview, tables 4–12)
- Common experiment: corporate profits tax reduced from 27 percent to 20 percent (Tables 4 and 9; figures reported in percent).
  - Tables present long-run percentage changes in w_l, w_s, ω (average wage), GDP, K, Z, θ (labor share), θ_L (low-skilled share) under multiple production-function parameterizations (σ1, σ2, σ3).
  - Results depend strongly on the elasticities of substitution (σ1, σ2, σ3); tables provide matrices of numeric outcomes for many parameter combinations.
- Infrastructure Investment (II) scenarios:
  - Runs with R_g = 0.10 (Table 5 / Table 10) and R_g = 0.15 (Table 6 / Table 11). δ_g = 0.04 and initial I_g/Q = 0.04 in all runs; I_g increases by 1.5% of initial GDP in reported numerical experiments (notes).
  - Tables report long-run percent impacts on w_l, w_s, ω, GDP, K, Z, θ, θ_L across σi specifications.
- Investment in Education (IE) scenarios:
  - Runs with R_s = 0.07 and δ_s = 0.03, initial I_s/Q = 0.013; increases in I_s are fiscally-equivalent to transfers as in corporate tax cut experiments (Tables 8 and 12).
  - Tables report long-run percent impacts (including cases where w_s < w_l at the new steady state, marked NA).
- Comparative policy tables:
  - Table 7: Long-run impact of CTC vs II vs IE for different σ2 values (σ1 = σ3 = 1). Figures reported in percent for each policy and parameter combination.
  - Notes emphasize that the increases in I_g and I_s, in percent of initial GDP, are chosen to yield the same fiscally-equivalent change in transfers as the corporate tax cut example.
- Numerical notes and conventions:
  - Figures presented in percent.
  - w_l and w_s denote wages of unskilled and skilled labor; ω denotes the average wage.
  - θ and θ_L denote the labor share and the low-skilled labor share, respectively.

### Analytical conditions and comparative-statics (Online Appendix I / Appendix A)
- Corporate tax cut (CTC) vs Infrastructure Investment (II) comparative inequalities:
  - Wage inequality worsens under CTC when σ2 is sufficiently large relative to σ4. Analytical condition: w_l_hat < 0 iff σ2 > σ4 [1 + (θ_k + θ_z)/θ_s * α_z] (equation (82) in Appendix A).
  - For calibration θ_k + θ_z = .40, θ_s = .20, α_z = .167 the condition requires σ2 > 6.99 σ4 (Appendix A, footnote).
- Thresholds for II to outperform CTC on various outcomes (Appendix A):
  - II increases GDP and wages more than CTC provided return on infrastructure is not unusually low.
  - For parameters σ1 = .5, σ2 = 3, σ4 = 1.5 and the paper calibration, the threshold values of R_g that satisfy comparisons are .02, .016, and .036 for particular inequalities (Appendix A, text).
- Signs and monotonicities:
  - K, Z, and Q are increasing in σ2 under the analytical solutions presented (Appendix A).
  - II increases low-skill wage, high-skill wage, GDP and private capital components (equations (87)–(92)).

### Estimation data and series (Online Appendix II / Appendix C)
- Capital data:
  - ICT and non-ICT asset series from BEA detailed fixed asset accounts; ICT asset codes include EP, EN, RD2, RD4; categories include Communications, Software, PCs, Terminals, Semiconductors, Storage devices (Appendix C.1).
- Labor and earnings data:
  - CPS IPUMS-CPS ASEC ("March") supplement annual frequency (1967–2021) used for occupation and earnings series (Flood and others (2021) referenced).
  - Routine vs non-routine occupational aggregates follow Acemoglu and Autor (2011) and Eden and Gaggl (2018) mappings.
- Estimation approach:
  - OLS regressions on fitted log-linear trends for relative income-share series; results extended to 2020 (N = 54) unless otherwise noted (Appendix C.5).
  - Robustness: standard errors and bootstrap alternatives discussed; results reported with 95% confidence intervals where applicable.

### Selected numerical examples and highlights
- Corporate profits tax experiment:
  - Tax rate reduction: 27 percent to 20 percent (Tables 4 and 9).
  - Numerical solutions reported across alternative CES nests and σ parameterizations; tables report many percent changes for wages, GDP, capital stocks, and shares.
- Infrastructure return thresholds for II > CTC (Appendix A numeric example):
  - For σ1 = .5, σ2 = 3, σ4 = 1.5 and calibration in the paper, threshold R_g values cited: .02, .016, .036 (for specific comparative inequalities).
- EOS evolution:
  - Time-series evolution of σ2 shown for 1993–2020 sub-samples (Figure 3); estimated EOS(L,Z) = 2.542 [2.457, 2.631] reported in tables.

*Searching for Wage Growth: Policy Responses to the “New Machine Age” — Working Paper No. WP/2024/003 (References, Tables, and Appendices as contained in wpiea2024003-print-pdf)*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024003-print-pdf.pdf_