The Encyclopedia of Economic Principals

Chapter 70

Fiscal Federalism, Local Public Finance, and Tiebout Sorting

Match services to places, follow the movers, and price the package into houses.

Four of the chapter's worked examples, made interactive: tailored versus uniform service, Tiebout sorting between two towns, tax capitalization into house prices, and Wagner's law. 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

Tailored versus uniform service

How much does letting districts choose their own service level gain, and when does a spillover undo it?

Decentralization gains when preferences differ across places and benefits stay inside each jurisdiction. The gain grows with the square of the distance between peaks. A spillover makes the local choice too small, because the district ignores its neighbour's benefit.

Equation, written in LaTeX: W^D=49+49+49=147.

Equation, written in LaTeX: W^U=40+49+40=129,

Equation, written in LaTeX: 147-129=18.

Scroll sideways for the whole equation

Each district's net surplus is S_i(q) = 49 - (q - peak)^2. W^D is total surplus when each chooses its peak, W^U when one uniform q applies to all. A spillover adds s benefit units to District 2 per unit District 1 supplies.

Predict first. If the three districts had identical peaks, what would the tailoring gain be?

Your prediction

Choose an example

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Figure: Tailored versus uniform service. Left: three district surplus parabolas peaking at 1, 4, 7, with the uniform quantity 4 marked. Right: District 1's social contribution with spillover 2, 51 at its local choice and 52 at q = 2.
Distance of outer peaks from middle: 3, Spillover to District 2 per unit of District 1 service: 2
Constructed example: the chapter's hypothetical districts (peaks 1, 4, 7 and a spillover of 2); outer distances 0 and 1 and spillovers 0 and 4 are added for comparison.

Calculated values

Tailored welfare W^D
147
Uniform welfare W^U
129
Tailoring gain
18
District 1 contribution at local choice
51
Best q for District 1
2
Contribution at best q
52

Tailored choices give each district 49, so W^D = 49 + 49 + 49 = 147. A uniform rule at q = 4 leaves the outer districts 3 units from their peaks, so W^U = 40 + 49 + 40 = 129 and tailoring gains 18. With a spillover of 2 per unit, District 1's social contribution peaks at q = 2 (52) rather than its own choice q = 1 (51), so it underprovides by 1 unit.

Worked steps

  1. W^D = 49 + 49 + 49 = 147
  2. Uniform q^U = 4: outer districts get 49 - 3^2 = 40
  3. W^U = 40 + 49 + 40 = 129; gain = 147 - 129 = 18
  4. Spillover: -2(q - 1) + 2 = 0 gives q = 2
  5. At q = 1: 49 + 2 x 1 = 51; at q = 2: 49 - 1^2 + 2 x 2 = 52

Use the idea

Before assigning a service to local government, ask how much preferences differ across districts and how much of the benefit crosses district lines.

Where the conclusion applies

Quadratic surplus, identical costs already netted out, no economies of scale and a linear spillover only from District 1.

Check your understanding: With peaks 1, 4, 7 and spillover 4 per unit, what quantity maximizes District 1's social contribution, and how much is lost at q = 1?
-2(q - 1) + 4 = 0 gives q = 3; at 3 the contribution is 49 - 4 + 12 = 57, at 1 it is 49 + 4 = 53, so 4 is lost.

Chapter 70 source: section "Oates decentralization theorem".

Demonstration 2 of 4

Voting with your feet

Which households sort into which town, and what blocks the move?

Households compare the net value of each town's tax and service package. With free mobility they sort by taste, so each town serves people who want its package. Moving costs and rent premiums can block the move or capture the package's value.

Equation, written in LaTeX: 12-5=7

Equation, written in LaTeX: 3-2=1

Equation, written in LaTeX: 4-5=-1

Equation, written in LaTeX: (12-5)-(3-2)-m=6-m.

Scroll sideways for the whole equation

Harbor Town supplies a package worth 12 to high-value households and 4 to low-value ones for a tax of 5. Plain Town supplies a service worth 3 for a tax of 2. m is a one-time moving cost and the rent premium is an extra price of living in Harbor.

Predict first. Will a high-value household in Plain move if moving costs 7?

Your prediction

Choose an example

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Figure: Voting with your feet. Left: net payoffs, high type 7 in Harbor and 1 in Plain, low type -1 and 1. Right: the move decision, fiscal gap 6 less rent 0 and moving cost 0 gives 6.
Moving cost: 0, Harbor rent premium: 0
Constructed example: the chapter's hypothetical towns (values 12 and 4, taxes 5 and 2, service 3, moving costs 0 and 7, rent premium 4); a moving cost of 4 and a premium of 8 are added.

Calculated values

High type in Harbor
7
High type in Plain
1
Low type in Harbor
-1
Low type in Plain
1
Gain from moving
6
Move happens
Yes

A high-value household nets 12 - 5 = 7 in Harbor against 3 - 2 = 1 in Plain, so it prefers Harbor. A low-value household nets -1 in Harbor and 1 in Plain, so it prefers Plain. For a high-value household now in Plain the gain from moving is 7 - 1 - 0 = 6, so it moves to Harbor.

Worked steps

  1. High in Harbor: 12 - 5 = 7; in Plain: 3 - 2 = 1
  2. Low in Harbor: 4 - 5 = -1; in Plain: 1
  3. Gain from moving: (12 - 5) - (3 - 2) - 0 = 6
  4. The move happens only if the gain is positive: 6 means the household moves to Harbor

Use the idea

When comparing places, add up service value minus taxes, then subtract housing premiums and the cost of moving before calling one place better.

Where the conclusion applies

Six households, three homes per town, fixed packages, and housing consumption otherwise identical across towns.

Check your understanding: With rent premium 8 and zero moving cost, where does a high-value household prefer to live?
Harbor nets 12 - 5 - 8 = -1, below Plain's 1, so it prefers Plain; the premium takes more than the package's fiscal advantage of 6.

Chapter 70 source: section "Tiebout sorting".

Demonstration 3 of 4

Taxes and services priced into houses

How much does a new local tax change house prices once the services it buys are counted?

Buyers pay for a house the present value of what living there brings. A tax above the value of its services lowers the price by the discounted net cost; the shorter the expected duration, the smaller the capitalized change.

Equation, written in LaTeX: \Delta f=1{,}500-2{,}400=-900.

Equation, written in LaTeX: \Delta P=\frac{-900}{0.045}=-20{,}000.

Equation, written in LaTeX: -900\frac{1-(1.045)^{-10}}{0.045}\approx -7{,}121.

Equation, written in LaTeX: \Delta P=\frac{400}{0.045}\approx 8{,}889.

Scroll sideways for the whole equation

A recurring property tax of $2,400 a year funds services worth the stated amount to the marginal buyer. Delta f is the annual net fiscal change, discounted at r = 0.045 into the price change Delta P, for a perpetual change or a fixed number of years.

Predict first. If the tax is expected to last only 10 years, is the price drop closer to $7,000 or $20,000?

Your prediction

Choose an example

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Figure: Taxes and services priced into houses. Left: the capitalized price change against how long the -$900 annual change lasts, approaching the perpetual value. Right: comparison house $360,000 and affected house $340,000.
Expected service value ($/yr): $1,500, Expected duration: Perpetual
Constructed example: the chapter's hypothetical houses ($360,000, tax $2,400, service $1,500 or $2,800, perpetual or 10 years); a service value of $2,400 and a 20-year horizon are added.

Calculated values

Annual net fiscal change
-$900
Capitalized price change
-$20,000
Affected house price
$340,000

Net fiscal value changes by 1,500 - 2,400 = -900 a year. Discounted at 4.5 percent over a perpetual change, Delta P = -900 / 0.045 = -$20,000, so the affected house sells for $340,000. The tax outweighs the service, so the owner at the time of the surprise absorbs the loss.

Worked steps

  1. Delta f = 1,500 - 2,400 = -900
  2. Delta P = -900 / 0.045 = -20,000
  3. Price = 360,000 + (-20,000) = 340,000

Use the idea

Before blaming or crediting a local tax for a price move, net out the expected value of the services and ask how long buyers expect the package to last.

Where the conclusion applies

No change in housing supply, risk or other amenities, a 4.5 percent discount rate and a marginal buyer whose valuation sets the price.

Check your understanding: With service value $1,500 and a 20-year horizon, what is the capitalized change?
-900 x (1 - 1.045^-20) / 0.045 = -900 x 13.0079 = -$11,707.

Chapter 70 source: section "Oates capitalization effect".

Demonstration 4 of 4

Spending share as income grows

When does public spending's share of income rise as the economy grows?

With a constant elasticity, spending grows by the income ratio raised to eps. An elasticity above 1 makes spending grow faster than income, so its share rises; below 1 the level still rises but the share falls.

Equation, written in LaTeX: \frac{96}{800}=0.12.

Equation, written in LaTeX: G_1=96(\frac{1{,}200}{800})^{1.25}=96(1.5)^{1.25}\approx159.36.

Equation, written in LaTeX: \frac{159.36}{1{,}200}\approx0.1328,

Scroll sideways for the whole equation

Y is real income and G real public spending, starting at Y_0 = 800 and G_0 = 96. The income elasticity eps gives G_1 = 96 (Y_1/800)^eps. The strong form of Wagner's law needs eps above 1, so the share G/Y rises.

Predict first. At elasticity exactly 1, what happens to the share?

Your prediction

Choose an example

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Figure: Spending share as income grows. Log-log plot of spending against income with elasticity 1.25 and the constant-share line. At income 1,200 spending is 159.36, a share of 13.28%.
Income elasticity of spending: 1.25, Later income: 1,200
Constructed example: the chapter's hypothetical economy (Y 800 to 1,200, G_0 = 96, elasticities 1.25 and 0.75); elasticities 1.0 and 1.5 and later incomes 1,000 and 1,600 are added.

Calculated values

Later spending G_1
159.36
Share G_1/Y_1
13.28%
Spending growth
66.0%
Income growth
50.0%
Share versus 12%
Rises

G_1 = 96 x 1.5000^1.25 = 96 x 1.660023 = 159.36, a share of 13.28% against 12% at the start: the share rises, matching the strong form. Spending grows 66.0% while income grows 50.0%.

Worked steps

  1. Y_1 / Y_0 = 1,200 / 800 = 1.5000
  2. G_1 = 96 x 1.5000^1.25 = 96 x 1.660023 = 159.36
  3. Share = 159.3622 / 1,200 = 0.1328
  4. Spending growth 66.0% against income growth 50.0%

Use the idea

To test Wagner's law in data, check whether the spending share rises with income, not merely whether spending rises.

Where the conclusion applies

A constant long-run elasticity and an unchanged institutional factor; real values throughout.

Check your understanding: With elasticity 1.0 and Y_1 = 1,200, what are G_1 and the share?
G_1 = 96 x 1.5 = 144; share = 144 / 1,200 = 0.12, unchanged, so the level rises but the strong form fails.

Chapter 70 source: section "Wagner's law".