The Encyclopedia of Economic Principals

Chapter 66

Tax Incidence, Elasticity, Salience, and Revenue

Who pays a tax, how much it raises, and who should collect it.

Four of the chapter's worked examples, made interactive: a tax buyers only partly notice, the split of a tax between buyers and sellers in the short and long run, a revenue curve with and without a peak, and a tax farm against a salaried collector. Change one value at a time and watch the figure, the numbers and the hand calculation respond.

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Attention changes the tax that buyers react to

If buyers notice only part of a tax at the shelf, how many units do they buy and what does the tax raise?

The legal tax and the cash price stay the same; only the price buyers react to changes. A less visible tax keeps quantity, and so the tax base, higher. Making the tax visible moves the decision price up the demand line toward the cash price.

Equation, written in LaTeX: q=90-2\widetilde p.

Equation, written in LaTeX: R_0=6(27)=162.

Scroll sideways for the whole equation

The pretax price is 30 currency units and t is the per-unit tax. theta is the share of the tax buyers notice when deciding, so the decision price is p-tilde = 30 + theta t. Demand q depends on p-tilde, but each unit is paid at the cash price 30 + t. Revenue R = t q.

Predict first. If an inclusive shelf label moves attention from 0.25 to 0.75 with a tax of 6, by how much does revenue fall?

Your prediction

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Figure: Attention changes the tax that buyers react to. Demand line q = 90 - 2p with the pretax price 30, the decision price 31.5 and the cash price 36 drawn as horizontal lines. Buyers choose 27 units; the shaded revenue rectangle is 6 high and 27 wide, 162.
Share of tax noticed (theta): 0.25, Per-unit tax: 6
Constructed example: the chapter's hypothetical shop (pretax price 30, tax 6, demand 90 - 2p, attention 0.25, 0.75 and 1); attention 0.5 and taxes of 4 and 8 are added for comparison.

Calculated values

Decision price
31.5
Quantity
27
Cash price paid
36
Tax revenue
162
Revenue at full attention
108

The decision price is 30 + 0.25 x 6 = 31.5, so q = 90 - 2 x 31.5 = 27. Every unit still costs 36 in cash and raises 6 of tax, so revenue is 162. Buyers notice only 25 percent of the tax, so they buy 9 more units than at full attention (18) and revenue is 162 instead of 108.

Worked steps

  1. Decision price = 30 + 0.25(6) = 31.5
  2. q = 90 - 2(31.5) = 27
  3. Cash price = 30 + 6 = 36
  4. Revenue = 6(27) = 162
  5. Full attention: q = 90 - 2(36) = 18, revenue 6(18) = 108

Use the idea

When a tax is shown separately at checkout, estimate demand with the price buyers attend to, not the statutory price, before predicting quantity or revenue.

Where the conclusion applies

The pretax price is held fixed, as in the chapter's teaching comparison; with responsive supply it would move too. Demand is linear and the attention weight is taken as given.

Check your understanding: With a tax of 6 and full attention (theta = 1), what quantity and revenue result?
p-tilde = 30 + 6 = 36, q = 90 - 72 = 18 and R = 6(18) = 108, against 162 when only a quarter of the tax is noticed.

Chapter 66 source: section "Tax salience".

Demonstration 2 of 4

Who pays depends on how fast supply can respond

Sellers remit the tax, but how much of it do buyers end up paying?

The tax drives a wedge between the two prices. The side that can adjust quantity more easily moves its price less, so the less elastic side bears more of the tax, whoever writes the check.

Equation, written in LaTeX: Q_D=180-3p_b, Q_S=30+p_s.

Equation, written in LaTeX: 180-3(p_s+8)=30+p_s.

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p_b is the price buyers pay and p_s the price sellers keep, so p_b = p_s + t. Demand is Q_D = 180 - 3p_b. Short-run supply is Q_S = 30 + p_s; long-run supply, after entry, is Q_S = -45 + 3p_s, which passes through the same pretax point (67.5, 37.5).

Predict first. When supply becomes as price-responsive as demand, what share of the 8 tax do buyers bear?

Your prediction

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Figure: Who pays depends on how fast supply can respond. Left: demand and short-run supply crossing at (67.5, 37.5), with the 8 tax wedge at quantity 61.5 between seller price 31.5 and buyer price 39.5. Right: buyers bear 2.0 and sellers 6.0 per unit.
Supply horizon: Short run, 30 + p, Tax per unit: 8
Constructed example: the chapter's hypothetical market (demand 180 - 3p, supply 30 + p in the short run and -45 + 3p in the long run, tax 8); taxes of 4 and 12 are added for comparison.

Calculated values

Seller price p_s
31.5
Buyer price p_b
39.5
Quantity
61.5
Buyer share
1/4 (25%)
Tax revenue
492.0

With supply 30 + p_s, the buyer price rises from 37.5 to 39.5 and the seller price falls to 31.5: buyers bear 2.0 and sellers 6.0 of the 8 tax. Supply elasticity at the pretax point is 5/9 against demand's 5/3, so sellers, the less responsive side, bear three quarters. Revenue is 8 x 61.5 = 492.0.

Worked steps

  1. 180 - 3(p_s + 8) = 30 + p_s
  2. 4p_s = 156 - (30) = 126, so p_s = 31.5
  3. p_b = 31.5 + 8 = 39.5; Q = 61.5
  4. Buyer share = (39.5 - 37.5) / 8 = 2.0 / 8 = 1/4
  5. Revenue = 8(61.5) = 492.0

Use the idea

Before saying who pays a tax, ask how quickly each side can change quantity over the horizon you care about; the answer can differ between the first year and later years.

Where the conclusion applies

Linear schedules, a competitive market and a per-unit tax. The shares from elasticities are local; with linear schedules they match the exact price changes here.

Check your understanding: In the long run with a tax of 8, what are p_b, p_s and revenue?
180 - 3(p_s + 8) = -45 + 3p_s gives 6p_s = 201, p_s = 33.5, p_b = 41.5, Q = 55.5 and revenue 8(55.5) = 444; each side bears 4 of the 8.

Chapter 66 source: section "Tax Incidence & Elasticity".

Demonstration 3 of 4

A Laffer peak appears only when the base is sensitive

When does cutting a tax rate raise revenue?

A higher rate takes more from each unit of base but shrinks the base. Revenue peaks where the base elasticity reaches 1. With a weakly responsive base the peak lies above any feasible rate; with a strongly responsive base it moves inside.

Equation, written in LaTeX: B_L(t)=1200(1-0.4t).

Equation, written in LaTeX: B_H(t)=1200(1-0.8t).

Equation, written in LaTeX: R_H(0.625)=0.625(600)=375.

Equation, written in LaTeX: \varepsilon_B(0.80)=\frac{0.80(960)}{432}=\frac{16}{9}>1.

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t is the tax rate between 0 and 1. The untaxed base is 1,200 and k measures how strongly the base shrinks as the rate rises, B(t) = 1200(1 - kt). Revenue is R = tB. The base elasticity is t(1200k)/B; revenue falls with the rate when it exceeds 1.

Predict first. With base sensitivity 0.8, does cutting the rate from 0.80 to 0.625 raise or lower revenue?

Your prediction

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Figure: A Laffer peak appears only when the base is sensitive. Revenue curve R(t) = 1200 t (1 - 0.8 t) for rates 0 to 1, with the chosen rate 0.8 marked at revenue 345.6 and the peak at 0.625.
Base sensitivity k: 0.8, Tax rate t: 0.80
Constructed example: the chapter's hypothetical bases B_L and B_H (sensitivity 0.4 and 0.8, rates 0.625, 0.80 and 1); sensitivity 0.6 and the rate 0.5 are added for comparison.

Calculated values

Base B(t)
432.0
Revenue R(t)
345.6
Base elasticity
1.778
Revenue-maximizing rate
0.625
Reading
lower rate raises revenue

At t = 0.8 the base is 432.0 and revenue 345.6. The base elasticity 1.778 is above 1, so this rate is on the inverse side: a lower rate raises revenue. The revenue peak is at t = 1/(2 x 0.8) = 0.625, where revenue is 375.0.

Worked steps

  1. B = 1200(1 - 0.8 x 0.8) = 1200(0.36) = 432.0
  2. R = 0.8 x 432.0 = 345.6
  3. Elasticity = 0.8(960) / 432.0 = 768.0 / 432.0 = 1.778
  4. Peak rate = 1 / (2 x 0.8) = 0.625

Use the idea

Before claiming a rate cut pays for itself, estimate how much the base responds at the current rate; the claim needs an elasticity above 1 there.

Where the conclusion applies

A linear base response chosen for exposition. The low and high sensitivities are two different curves, not points on one observed curve.

Check your understanding: With base sensitivity 0.4, where is the revenue peak and what is revenue at t = 1?
The peak would be at t = 1/0.8 = 1.25, outside 0 to 1, and R(1) = 1200(0.6) = 720.

Chapter 66 source: section "The Laffer Curve & the Tax Base".

Demonstration 4 of 4

Tax farm or salaried collector

Should the state sell the right to collect for a fixed fee or pay a salaried collector it monitors?

The fixed fee makes the farmer the residual claimant, so the farmer works hard and bears the risk. A salary removes the risk but also the reason to work hard, unless the state pays to monitor. Cheaper monitoring tips the choice toward direct administration.

Equation, written in LaTeX: \mathbb{E}[Y_H]=0.5(130)+0.5(170)=150.

Equation, written in LaTeX: 150-110-12=28.

Equation, written in LaTeX: 150-15-30=105.

Equation, written in LaTeX: 150-15-12=123.

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Low effort collects 120 at no cost; high effort costs the collector 12 and collects 130 or 170 with equal probability, 150 on average. A salaried collector earns 15. A tax farmer pays the fixed fee F and keeps the rest. Monitoring that enforces high effort costs the state m.

Predict first. At what monitoring cost does direct administration tie the 110 farm?

Your prediction

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Figure: Tax farm or salaried collector. Bars of expected net receipts: unmonitored salary 105, tax farm 110, monitored salary 105 at monitoring cost 30. The tax farm arrangement nets the state the most, 110.
Monitoring cost: 30, Farm fee F: 110
Constructed example: the chapter's hypothetical collection (yields 120, 130 and 170, effort cost 12, salary 15, fee 110, monitoring 30 and 12); a monitoring cost of 20 and fees of 100 and 120 are added for comparison.

Calculated values

Unmonitored salary
105
Tax farm
110
Monitored salary
105
Farmer expected profit
28
Farmer outcomes
8 or 48
Best for the state
Tax farm

A salaried collector without monitoring shirks and nets 105. The farmer paying 110 chooses high effort, expecting 150 - 110 - 12 = 28 with outcomes 8 or 48. Monitoring at cost 30 nets 150 - 15 - 30 = 105. The tax farm arrangement nets the state the most, 110. Monitoring would tie the farm at a cost of 25.

Worked steps

  1. E[Y_H] = 0.5(130) + 0.5(170) = 150
  2. Unmonitored salary: low effort, 120 - 15 = 105
  3. Farm: high effort 28 = 150 - 110 - 12 beats low effort 120 - 110 = 10, so the state gets 110
  4. Monitored salary: 150 - 15 - 30 = 105
  5. Monitoring cost that ties the farm: 150 - 15 - 110 = 25

Use the idea

Compare a fixed-fee contract with monitored direct provision by their expected net receipts, and find the monitoring cost at which they tie.

Where the conclusion applies

Risk-neutral collectors, an honored farm contract and receipts as the only criterion. The chapter notes that taxpayer harm, renegotiation and the value of public records can reverse the ranking.

Check your understanding: With monitoring cost 12 and farm fee 120, which arrangement nets the state more?
Monitored administration nets 150 - 15 - 12 = 123, above 120. The farmer would still choose high effort, since 150 - 120 - 12 = 18 beats 120 - 120 = 0.

Chapter 66 source: section "Tax Farming & the Principal-Agent Problem".