preserved open space without referring the relative location r, can be regarded as a
special case of the spatial model where A = A(a) = a.
Consider a circular residential community with a radius r0, which, as will be seen,
is not essential to the model. Assume the community has already preserved some land of
a0 units at the community center as a circular central community park, and is considering
to expand the range of the park outward further, with its radius changing from ra0 to ra1.
If the area of the planned open space increment is a, the total value of the remaining land
after preservation is∫r0 P(A(a0 + a, r))2πrdr , where ra1 is the radius of the post-
ra1
a0 + a
preservation area of open space, with ra1 = J------. The expected cost for preserving a
∏ π
units of incremental open space would be ∫ra1P(A(a0,r))2πrdr , which is approximately
ra0
equal to P(A(a0, ra0))a when the involved variation in the radius of preserved open space
is limited. Consequently, the land manager’s model of using property tax increment to
finance socially efficient open space preservation becomes
Max π = ∫r0 P(A(a0 + a, r))2πrdr - P(A(a0,r0))a (16)
a ra 1
Tr r
s.t. ∫ (τ∫ P(A(a0 + a, r))2πrdr -τ∫ P(A(a0, r))2πrdr)we-δt dt ≥ P(A(a0,r0))a (17)
0 ra1 ra 0
Compared to the non-spatial model, the model accounting for the spatial pattern of
equilibrium land price is complicated by the integral of land value over the remaining
land. This complicating, however, can be simplified using the average value theorem.
Specifically, the total value of the remaining land after preservation
∫r0 P(A(a0 + a, r))2πrdr = π∫r0 P(A(a0 + a,r))dr2 = πP~(a0 + a)(r02 - ra2) (18)
ra 1 ra 1
20