The Verification Venue · pointed at a choice regulators keep settling by taste
The Dial That Picks Your Instrument
Fix the price or fix the quantity? When the cost of abatement is uncertain, most answers are tribal. Under linear margins the answer is a theorem: a tax beats a cap exactly when the marginal benefit curve is flatter than the marginal cost curve. This page derives the sign of the welfare gap live, and hands you the slider that flips it.
Three sliders are the whole argument. The gold line is the expected welfare gap between the optimal tax and the optimal cap, plotted as the benefit slope sweeps across the cost slope. Drag the benefit slope past the cost slope and watch the verdict flip sign. Nothing here is asserted: every number is recomputed in front of you from the constants named in the check panel.
Recommended instrument
TAX
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Expected welfare gap, tax minus cap (welfare units)
+0.0000
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Drag it past the cost slope and the verdict flips to CAP. The gold line crosses zero exactly at b1 = c1.
A steeper cost curve widens the region where the tax wins, because quantity errors become more expensive to tolerate.
At σ = 0 the instruments tie exactly. The gap grows with the square of σ, so uncertainty is what makes the choice matter at all.
The quantity doing the work is the expected welfare gap. Write marginal benefit as b0 − b1·x and marginal cost as c0 + c1·x + u, where u is a cost shock with mean zero and variance σ². The optimal cap and the optimal tax both get average abatement right; they differ only in how the shock travels. Solving each instrument's problem and integrating the resulting quadratic loss against the shock distribution closes into one expression, with the intercepts cancelled:
A tax lets abatement move with the shock. That flexibility is cheap when the benefit curve is flat, because the quantity errors it admits cost little, and expensive when the benefit curve is steep. A cap freezes abatement and is the mirror image. The gap measures which mistake is cheaper.
Drag the benefit slope to 0 and the gap is positive at every σ: a flat benefit curve makes quantity errors nearly free, so the tax wins. Drag it above the cost slope and the gap goes negative: a steep benefit curve makes quantity errors costly, so the cap wins. At equality the instruments tie exactly, whatever the uncertainty. That crossing is the entire result. Everything else on this page is bookkeeping around it.
Why the answer differs by pollutant: two damage structures, one cost side
The sophisticated dismissal is that real benefit curves are not straight lines. Fair, and beside the point: the theorem says what follows from a slope, not that any real curve has one. The two structures below are computed from typed constants on the same cost side your sliders set. Each enters the computation only through its marginal-benefit slope, which is all the theorem consumes. One has a flat benefit curve, one a steep one, and the verdict inverts.
Stylised structures, not measurements of any real pollutant. The page claims only the conditional: if a benefit curve has this slope, the verdict follows. ↓
Accumulated-stock structure
TAX
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Hard-threshold structure
CAP
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This is the shape of the standard argument for prices under accumulated-stock pollutants and for quantities under threshold pollutants. It is not a preference. The damage structure sets the slope, and the slope picks the instrument. Whether any real pollutant's benefit curve is actually flat is an argued empirical question this page does not settle and does not pretend to; what it settles, exactly, is what follows either way.
The check · every number recomputed in front of you
Each row compares two routes to the same number. The closed-form column is algebra: the expression above, evaluated at that row's settings. The Simpson column is brute force: the quadratic loss integrated against a Gaussian shock by composite Simpson quadrature with 4000 intervals over ±8σ, the Gaussian density applied to the integrand and normalized, no algebra used. A row passes only if the two agree within 1e-6 relative, and the verdict cell is computed from that comparison, not asserted.
| case | b1 | c1 | σ | closed form | Simpson | |diff| | verdict |
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Your current setting, recomputed live:
What is exact: the closed form holds for any mean-zero shock with variance σ², Gaussian or not; the companion verifier confirms this against a two-point shock distribution summed by hand, with no quadrature. The sign of the gap equals the sign of c1 − b1 at every point of a swept grid, and that sign assertion is the check that matters: a gap that agreed in magnitude but flipped sign anywhere would look green and be worthless.
Run it yourself: node research/the-dial-that-picks-your-instrument/verify-the-dial-that-picks-your-instrument.mjs (add --mutate to watch it detect a corrupted engine).
What's idealised here, and what's exactly true
Exactly true. The closed forms are exact solutions of the stated linear-quadratic model, not simulations. The gap σ²(c₁ − b₁)/(2c₁²) holds for any shock distribution with mean zero and variance σ²; only the first two moments enter. The intercepts cancel exactly, which the verifier checks numerically by moving the headroom m and confirming the gap does not move.
Idealised. Linear marginal benefit and marginal cost. The shock enters the cost intercept additively and the benefit side is known with certainty. Abatement is unbounded below zero: a firm hit by a bad shock can "negative-abate", which real firms cannot, so with σ large relative to the headroom the model flatters both instruments. The regulator is risk-neutral and maximises expected welfare. There are no compliance, enforcement, market-power, or irreversibility frictions.
Representative, not universal. The two damage structures are typed illustrations built from the constants ρ = 0.03, δ = 0.005, ds = 1.0, dt = 2.5, and they enter the computation only through their marginal-benefit slopes, which is all the theorem consumes: exactly zero for the stock structure, steep and straight for the threshold one, whose curve reaches zero marginal benefit at x = m/dt. They are not measurements. Magnitudes of the welfare gap are normalization-dependent across the literature; the sign condition is the invariant. Genuinely nonlinear margins break the exact expression while preserving the local slope logic. No jurisdiction, statute, or policy is named anywhere on this page, because the mechanism, not any law, is the subject.