Public Finance, 2nd Edition

and

N  MRS

X; Y

h

 NC

NX

 NC

NN

dN

dX









supply

(30:16)

Equation (30.15) is one of the two conditions for a welfare optimum. It

remains to show that Eqs. (30.15) and (30.16) together imply the second

pareto-optimal condition, the standard public goods decision rule, (Eq. 30.9).

To see that they do imply Eq. (30.9), totally diVerentiate Eq. (30.15) to

obtain:

C

X

dX  C

N

dN  NC

NX

dX  NC

NN

dN  C

N

dN (30:17)

Rearranging terms:

C

X

 NC

NX

dX  NC

NN

dN (30:18)

C

X

 NC

NX



NC

NN



dN

dX









supply

(30:19)

Substituting Eq. (30.19) into (30.16) and simplifying yields:

N  MRS

X; Y

h

 C

X

(30:20)

as required. Tiebout's conjecture holds true in McGuire's Xuid, frontier

model of Wscal federalism.

FIXED COMMUNITIES AND HOUSING SITES: ADDING THE

HOUSING MARKET

McGuire's frontier model can reasonably ignore the housing market on the

grounds that supply of land is perfectly elastic in frontier regions where

jurisdictions are forming, breaking apart, and reforming, with each town

replicating all others in the Wnal equilibrium. The housing market cannot be

ignored in models with a Wxed number of jurisdictions, however, because then

property values are necessarily tied to the provision of public services. Sup-

pose some town oVers a particularly attractive public services±tax mix. The

demand for that town's services±tax mix might well cause property values

there to rise as people try to move in. A Wnal equilibrium cannot be achieved

until the relative attractiveness of the town's services±tax mix is fully capital-

ized into the value of the town's property.

The Pauly Model of the Housing Market

In general, the housing market has an important impact on both the nature of

the equilibrium and whether an equilibrium even exists. A model developed

866 FIXED COMMUNITIES AND HOUSING SITES: ADDING THE HOUSING MARKET

by Mark Pauly is instructive for exploring the various possibilities when the

jurisdictions are Wxed.

7

It lies at the opposite end of the spectrum from

McGuire's frontier model in assuming a Wxed number of communities

along with a Wxed number of housing sites.

Let:

X  a composite commodity whose price equals 1.

G  a bundle of public services, exclusive of taxes.

g the unit price of (tax for) a public service, assumed constant across all

jurisdictions.

R  the rental value of a standardized vector of property and housing

services.

Y  lump-sum consumer income, assumed Wxed for each individual.

Consumer utility is deWned over X and G. Thus consumers solve the

following ``as if '' maximization problemÐthat is, as if they could choose the

value of G:

max

(X

h

; G

h

)

U

h

X

h

,G

h



s:t: Y

h

 X

h

 gG

h

 Rh 1, ...,H

Assume initially that R is equal across all jurisdictions. The as-if maxim-

ization determines each person's most preferred amount of G which they will

try to match as closely as possible with the set of Gs oVered in the given

communities.

Suppose the maximization generates a G

h

*. Person h, and all other

consumers identical to h in terms of preferences and income, will want to

form a jurisdiction providing exacting exactly G

h

* of public services, replicat-

ing if necessary to avoid any increases in R. If they could do so, preferences

for public service bundles would be met exactly by homogeneous subgroup-

ings of the population, and there would be no capitalization. This is the

situation envisioned by the McGuire frontier model. Furthermore, g is

essentially a head tax so that the subgroups would generate a pareto-optimal

equilibrium.

Suppose, however, that there are a Wxed number of localities, `  1, ...,L,

with the following characteristics:

1. Each locality oVers a particular level of public services represented by

the vector

~

G  (G

1

, ...,G

`

, ...,G

L

) in ascending order of G

`

;

2. Each town has a Wxed number of properties, represented by the vector

~

H  (H

1

, ...,H

`

, ...,H

L

), such that

P

L

`1

H

`

equals the entire

population of individuals seeking a location; and

7

M. Pauly, ``A Model of Local Government Expenditure and Tax Capitalization,'' Journal

of Public Economics, October 1976.

30. OPTIMAL FEDERALISM: THE SORTING OF PEOPLE WITHIN THE FISCAL HIERARCHY 867