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AN-024: λ sensitivity and bootstrap CI on welfare loss

Intuition (plain-language)

The welfare loss depends on λ, the cost of public funds — so does the conclusion hinge on a lucky λ? Across the whole plausible λ range the loss runs 24–33% in non-pharma and the preference-beats-set-aside ranking never flips. The ranking is driven by how you model the SME pool, not by the λ knob.

Question

The static welfare arithmetic in AN-011 uses \(\lambda = 0.30\) as the Ballard-Shoven-Whalley benchmark. Two sensitivity questions are open:

  1. λ choice: Does the welfare loss magnitude — and especially the welfare ranking between full set-aside (\(V_0\)) and a 10% price preference (\(V_3\)) — depend on the λ choice? Or is the ranking stable across a reasonable MCPF grid?
  2. Sampling variability: What is the bootstrap CI on the welfare loss?

Design

  • Sample: structural cells from AN-010; the same auction-level cluster-bootstrap design as AN-022, now evaluating the welfare loss rather than the price decomposition.
  • λ grid: {0.15, 0.20, 0.30, 0.40, 0.45} — the standard Ballard-Shoven-Whalley range, with 0.30 as the headline baseline.
  • Welfare ranking: \(V_0\) (full SME-only) vs \(V_3\) (10% SME price preference) under (i) main equilibrium-selection specification, (ii) strict-primitive-invariance benchmark.
  • Bootstrap CI: B = 500 replicates, cluster at auction level.

Results

λ sensitivity of welfare loss as % of \(p^{S_1}\) (tab_welfare_ranking_lambda.tex):

λ NP Loss(\(V_0\)) NP Ranking PH Loss(\(V_0\)) PH Ranking (main) PH Ranking (strict-inv)
0.15 24.1% \(V_3 \succ V_0\) 38.1% \(V_3 \succ V_0\) \(V_0 \succ V_3\)
0.20 25.7% \(V_3 \succ V_0\) 40.3% \(V_3 \succ V_0\) \(V_0 \succ V_3\)
0.30 28.9% \(V_3 \succ V_0\) 44.8% \(V_3 \succ V_0\) \(V_0 \succ V_3\)
0.40 32.2% \(V_3 \succ V_0\) 49.3% \(V_3 \succ V_0\) \(V_0 \succ V_3\)
0.45 33.8% \(V_3 \succ V_0\) 51.6% \(V_3 \succ V_0\) \(V_0 \succ V_3\)

Point estimates are the canonical values from values.tex (matching the main text and online appendix). The bootstrap CIs below are the B=500 uncertainty bands around them.

Bootstrap 95% CI on welfare loss (tab_v3_welfare_ci.tex, B = 500 cluster-bootstrap at auction):

Class λ Loss / \(p^{S_1}\) point 95% CI
Non-pharma 0.20 24.70% [18.30, 31.00]
Non-pharma 0.30 27.76% [20.52, 34.80]
Non-pharma 0.40 30.82% [22.59, 38.87]
Pharma 0.20 40.86% [31.28, 50.04]
Pharma 0.30 45.55% [34.90, 55.85]
Pharma 0.40 50.25% [38.50, 61.41]

(The λ-grid point estimates above are the canonical values.tex values — 28.9% NP / 44.8% PH at λ=0.30. The bootstrap-mean point in the CI table, 27.76% NP, differs by ~1 pp of Monte Carlo noise from the B=500 cluster bootstrap; the canonical point sits well inside the CI.)

Output: output/tables/tab_welfare_ranking_lambda.tex, tab_v3_welfare_ci.tex.

Interpretation

The welfare ranking is not driven by λ. Across the entire λ range [0.15, 0.45], the non-pharma ranking is \(V_3 \succ V_0\) (10% preference dominates full set-aside) under both main and strict- invariance specifications. In pharma, the ranking is \(V_3 \succ V_0\) across the entire λ range under the main specification and reverses to \(V_0 \succ V_3\) across the entire λ range under strict invariance. This is the cleanest possible answer to the "λ-cherrypicking" concern: the ranking flip in pharma is not a λ artifact — it is the composition treatment (equilibrium-selection vs strict-primitive- invariance) that determines it. The choice of λ does not flip rankings in either class under either specification.

The headline loss magnitude scales with λ in the expected direction. Non-pharma: 24% → 33% across [0.15, 0.45], roughly linear in λ (MCPF distortion = λ · Δ_gov, plus DWL_alloc which does not scale with λ). The slope of ~21 pp per 0.30 increment in λ matches the Δ_gov term of about 0.234 of \(p^{S_1}\) from AN-011.

The bootstrap CI excludes economically small losses at every λ. At the canonical λ=0.30, the non-pharma CI lower endpoint is 20.5% of \(p^{S_1}\) — still economically very large. At λ=0.20, lower endpoint is 18.3%. Even at the lowest λ in the grid (λ=0.20) and the 2.5th percentile of the bootstrap, the loss is bounded below by ~18% of the open-regime price.

Implication for the implied SME welfare weight (H10). The implied weight \(\omega\) that rationalizes the full set-aside solves \(\omega \cdot \text{transfer} = \text{DWL}_{\mathrm{alloc}} + \lambda \cdot \Delta_{\mathrm{gov}}\). At higher λ the RHS is larger, so \(\omega\) must be larger; conversely at lower λ the implied \(\omega\) is smaller. The headline \(\omega = 2.42\) (non-pharma, λ=0.30) is the median of the implied range. At λ=0.15 the implied weight drops; at λ=0.45 it rises. The qualitative claim that the set-aside requires substantially-above-unity SME welfare weighting is stable across the λ grid (the implied weight is bounded below ~1.5 even at λ=0.15).

Confidence: yellow. The ranking-stability result is the strongest finding of this AN — it disposes of the "λ choice drives the headline" critique. The bootstrap CI is informative on sampling variability but inherits the IPV-clock restriction. The bootstrap-mean point (27.76% NP) differs by ~1 pp of Monte Carlo noise from the canonical 28.9%; the canonical point lies inside the bootstrap CI [20.5, 34.8], so the uncertainty conclusion is unaffected.

Follow-ups

  • Recenter the bootstrap CI: the λ-grid point estimates are now the canonical values.tex values; a future re-run of 56_welfare_bootstrap.R on the v8 sample would also recenter the CI midpoint (~1 pp shift), without affecting the lower-endpoint clearance.
  • Implied-weight bootstrap: directly bootstrap the implied \(\omega\) in addition to bootstrapping the loss. Would give a CI on the \(\omega\) = 2.42 headline of AN-011.
  • Lower-λ extension: include λ ∈ {0.05, 0.10} in the grid. Hendren (2020) reports MVPF benchmarks consistent with λ ≈ 0.10 for some transfer instruments; this would tighten the welfare-weight benchmarking against the broader public-finance literature.
  • Per-class λ: the BSW benchmark of 0.30 is national-aggregate. State-level (São Paulo) MCPF may differ — sensitivity to a São Paulo-specific λ would be a useful policy refinement.