The Encyclopedia of Economic Principals

Chapter 67

Optimal Commodity, Income, and Capital Taxation

Where taxes do least harm: across goods, across people and across time.

Four of the chapter's worked examples, made interactive: how a tax on interest compounds over a saving horizon, the inverse-elasticity rule for commodity taxes, the incentive limit on redistribution between two workers, and the Ramsey social discount rate. 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

A capital tax compounds with the saving horizon

How much future consumption does a steady tax on interest take, and how does that depend on the horizon?

A tax on interest lowers the growth rate every year, so the taxed and untaxed paths diverge geometrically. The relative price of consumption far in the future rises with the horizon, which is the force behind the zero long-run capital tax result.

Equation, written in LaTeX: 1{,}500(1.045)^{35}\approx7{,}001.02.

Equation, written in LaTeX: 0.045(1-0.25)=0.03375,

Equation, written in LaTeX: 1{,}500(1.03375)^{35}\approx4{,}793.30.

Scroll sideways for the whole equation

A household saves 1,500 at a pre-tax return of 4.5 percent a year. A tax takes the share tau of interest each year, so the net return is 0.045(1 - tau). The shortfall is the percentage by which the taxed value falls below the untaxed value.

Predict first. With the 25 percent tax, is the 35-year shortfall about 3.5 times the 10-year shortfall, more, or less?

Your prediction

Choose an example

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Figure: A capital tax compounds with the saving horizon. Growth of 1,500 saved over 35 years at 4.5 percent and at the net return 3.375 percent. At year 35 the values are 7,001.02 and 4,793.30, a shortfall of 31.53 percent.
Tax on interest: 25%, Years saved: 35
Constructed example: the chapter's hypothetical saver (1,500 at 4.5 percent, 25 percent tax, 35 and 10 years); a 40 percent tax and a 20-year horizon are added for comparison.

Calculated values

Net return
3.375%
No-tax value
7,001.02
Taxed value
4,793.30
Ratio taxed / no tax
0.6847
Shortfall
31.53%

The net return is 0.045 x (1 - 0.25) = 0.03375. After 35 years 1,500 grows to 7,001.02 without the tax and 4,793.30 with it, a ratio of 0.6847. Each year the taxed path falls only 1.08% further behind, but the loss compounds: after 35 years the taxed saving is worth 31.53% less.

Worked steps

  1. Net return = 0.045(1 - 0.25) = 0.03375
  2. No tax: 1,500(1.045)^35 = 7,001.02
  3. Taxed: 1,500(1.03375)^35 = 4,793.30
  4. Ratio = 4,793.30 / 7,001.02 = 0.6847
  5. Shortfall = 1 - 0.6847 = 31.53%

Use the idea

To compare a capital-income tax with a tax that does not compound, compute the shortfall at the saving horizons that matter, not the annual rate alone.

Where the conclusion applies

A constant pre-tax return and tax rate, interest taxed every year and no change in the amount saved. The chapter notes the result says nothing about whether the replacement tax is costless.

Check your understanding: At a 10-year horizon with the 25 percent tax, what is the shortfall?
1,500(1.045)^10 = 2,329.45 and 1,500(1.03375)^10 = 2,090.48, so 1 - 2,090.48/2,329.45 = 10.26%.

Chapter 67 source: section "Chamley-Judd zero long-run capital-tax result".

Demonstration 2 of 4

Ramsey rule, tax the inelastic good more

If every taxed good must shrink by the same share, which good carries the higher rate?

A small tax cuts compensated demand by about the elasticity times the rate. Equal proportional contractions therefore need rates inversely proportional to the elasticities: the less elastic good gets the higher rate.

Equation, written in LaTeX: \frac{\Delta q_i^c}{q_i}\approx\varepsilon_i^c\tau_i,

Equation, written in LaTeX: \tau_b=\frac{0.012}{0.40}=0.03

Equation, written in LaTeX: \tau_d=\frac{0.012}{1.20}=0.01

Equation, written in LaTeX: R=0.60(988)+0.40(494)=790.4.

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epsilon is the compensated price elasticity of each service (basic -0.40, discretionary as set). tau is the proportional tax rate. Each rate is chosen so compensated demand falls by the same target share. Prices are 20 and 40 and quantities 1,000 and 500 before tax.

Predict first. If the discretionary elasticity doubles in size from 0.8 to 1.6, what happens to the basic rate relative to the discretionary rate?

Your prediction

Choose an example

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Figure: Ramsey rule, tax the inelastic good more. Left: tax rates of 3 percent on the basic service and 1 percent on the discretionary one. Right: revenue 592.80 and 197.60, total 790.40.
Discretionary elasticity: -1.2, Target contraction: 1.2%
Constructed example: the chapter's hypothetical services (elasticities -0.40 and -1.20, contraction 0.012, prices 20 and 40, quantities 1,000 and 500); elasticities -0.8 and -1.6 and contractions 0.006 and 0.024 are added for comparison.

Calculated values

Basic rate tau_b
0.03
Discretionary rate tau_d
0.01
Rate ratio
3
Quantities after tax
988 and 494
Unit taxes
0.60 and 0.40
Revenue
790.40

To cut each compensated demand by 1.2%, the basic service needs 0.03 and the discretionary service 0.01. The basic rate is 3 times the discretionary rate because its demand is 3 times less elastic. Revenue is 592.80 + 197.60 = 790.40.

Worked steps

  1. tau_b = 0.012 / 0.40 = 0.03
  2. tau_d = 0.012 / 1.20 = 0.01
  3. x_b' = 1,000(1 - 0.012) = 988; x_d' = 500(1 - 0.012) = 494
  4. Unit taxes: 0.03(20) = 0.60; 0.01(40) = 0.40
  5. R = 0.60(988) + 0.40(494) = 592.80 + 197.60 = 790.40

Use the idea

When goods are unrelated and distribution is set aside, use the inverse-elasticity rule as the efficiency benchmark, then add distributional weights explicitly.

Where the conclusion applies

Small taxes, independent demands, full pass-through and no distributional weights. The chapter stresses that the example does not recommend taxing necessities.

Check your understanding: With a target contraction of 0.024 and the book's elasticities, what are the two rates?
tau_b = 0.024/0.40 = 0.06 and tau_d = 0.024/1.20 = 0.02; the ratio stays 3. Revenue is 1.20(976) + 0.80(488) = 1,171.20 + 390.40 = 1,561.60.

Chapter 67 source: section "Ramsey inverse-elasticity rule".

Demonstration 3 of 4

How much can be redistributed before the able worker mimics

How large a transfer from the high earner to the low earner survives the high earner's option to pretend to be low?

The government sees earnings, not ability. Every unit transferred lowers the high type's honest utility and raises the payoff from choosing the low bundle, so the gap closes twice as fast as the transfer grows. The cap is where the two lines cross.

Equation, written in LaTeX: 78{,}000-x\ge44{,}000+x,

Equation, written in LaTeX: x\le17{,}000.

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The low type earns 48,000 at effort cost 18,000. The high type earns 96,000 at the same effort cost, or can earn 48,000 at a mimicking cost (4,000 in the book). x is the balanced transfer, taxed from the high earner and paid to the low earner. Utility is consumption minus effort cost.

Predict first. If mimicking became costless, would the largest feasible transfer rise or fall?

Your prediction

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Figure: How much can be redistributed before the able worker mimics. The high type's honest utility falls and mimicking utility rises with the transfer; they cross at 17,000. At the chosen transfer 18,000 honest utility is 60,000 and mimicking utility 62,000.
High type's cost of mimicking: 4,000, Transfer to low earner: 18,000
Constructed example: the chapter's hypothetical workers (earnings 48,000 and 96,000, effort cost 18,000, mimicking cost 4,000, transfers 18,000 and 17,000); mimicking costs 0 and 8,000 and a transfer of 15,000 are added for comparison.

Calculated values

High type honest
60,000
High type mimicking
62,000
Incentive constraint
violated
Largest feasible transfer
17,000
Consumption, low and high
66,000 and 78,000

With a transfer of 18,000, the high type's honest utility is 78,000 - 18,000 = 60,000 and mimicking gives 48,000 + 18,000 - 4,000 = 62,000. Mimicking pays 2,000 more, so the high type takes the low bundle and the plan fails. With a mimicking cost of 4,000 the largest feasible transfer is 17,000.

Worked steps

  1. Honest: 96,000 - 18,000 - 18,000 = 60,000
  2. Mimic: 48,000 + 18,000 - 4,000 = 62,000
  3. Cap: 78,000 - x = 44,000 + x gives x = 34,000 / 2 = 17,000
  4. Consumption: 48,000 + 18,000 = 66,000 and 96,000 - 18,000 = 78,000, total 144,000

Use the idea

Before proposing a transfer schedule, check that each type still prefers its own bundle; the cheapest way to imitate the lower earner sets how much can be redistributed.

Where the conclusion applies

Two types, observable earnings, quasi-linear utility and a balanced budget. The chapter omits participation, wage effects, administration and avoidance.

Check your understanding: At x = 17,000 with the book's mimicking cost, what are low and high consumption, and do totals balance?
Low consumption is 48,000 + 17,000 = 65,000 and high 96,000 - 17,000 = 79,000, summing to the 144,000 earned. The high type gets 61,000 honest and 61,000 mimicking, so the constraint just binds.

Chapter 67 source: section "Mirrlees optimal-income-tax tradeoff".

Demonstration 4 of 4

The social discount rate rides on growth

Does a project paying 120 million in 35 years justify 45 million today?

When future generations are expected to be richer, an extra unit of their consumption is worth less, so future benefits are discounted more. Lower expected growth lowers the rate and raises the present value of distant benefits.

Equation, written in LaTeX: r=0.8\%+1.4(1.8\%)=3.32\%.

Equation, written in LaTeX: PV=\frac{120}{(1.0332)^{35}}\approx38.26

Scroll sideways for the whole equation

The Ramsey rule sets r = rho + eta g: rho = 0.8 percent is pure time preference, eta the elasticity of marginal utility and g expected growth of consumption per person. PV is the present value of the 120 million benefit; the cost today is 45 million.

Predict first. If expected growth falls from 1.8 percent to 0.4 percent, does the project pass?

Your prediction

Choose an example

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Figure: The social discount rate rides on growth. Present value of 120 million received in 35 years against the discount rate, with the 45 million cost line. At r = 3.32 percent the present value is 38.26 million, so the project fails.
Expected consumption growth: 1.8%, Marginal-utility elasticity eta: 1.4
Constructed example: the chapter's hypothetical project (cost 45, benefit 120 in 35 years, rho 0.8 percent, eta 1.4, growth 1.8, 0.4 and -1 percent); eta 1.0 and 2.0 are added for comparison, and present values at -1 percent growth are computed, not printed in the chapter.

Calculated values

Discount rate r
3.32%
Present value (million)
38.26
Cost (million)
45
Project
fails

The Ramsey rate is 0.8% + 1.4 x 1.8% = 3.32%. Discounted over 35 years, the 120 million benefit is worth 38.26 million today, below the 45 million cost, so the project fails. Any rate below 2.84% would pass it.

Worked steps

  1. r = 0.8% + 1.4(1.8%) = 3.32%
  2. PV = 120 / (1.0332)^35 = 38.26
  3. 38.26 < 45, so the project fails

Use the idea

Report how a long-lived project's verdict depends on expected growth and on eta, rather than using one discount rate as if it were a fact.

Where the conclusion applies

Constant rates, certainty and benefits measured in consumption equivalents. The chapter calls these teaching assumptions, not forecasts.

Check your understanding: With growth of -1 percent and eta 1.4, what are r and the present value?
r = 0.8% + 1.4(-1%) = -0.6%, so PV = 120/(0.994)^35 = 148.14 million and the project passes. The chapter prints the rate; the present value is computed here.

Chapter 67 source: section "Ramsey social-discounting rule".