Tax Increment Financing for Optimal Open Space Preservation: an Economic Inquiry

ratio under the constraint of tax increment financing, given policy parameters and the
amount of open space to be preserved. We can derive the following properties for these
two functions:

(i)    Ψ(0) = Φ(0) = 1

P'(a+a⁰)                  P"(a + a⁰)

(ⁱⁱ)    ^ψ'(a) =  P(_a0)  > ^0, and^ψ"(^a) =   P(_a0)   < ⁰

lim ₀ φ'(a) = +∞ , and lim Ψ'(a) =
a →L -a⁰                        a→L-a⁰

The above properties suggest that both curves Ψ(a) and φ(a) start from the same point (0,
1), and increase with the amount of preserved open space a (see figure 2). However, the
bid ratio Ψ(a) increases with the amount of preserved open space at a decreasing rate,
while the critical value φ(a) increases at an increasing rate which goes to infinity when a
is approaching the total available land L - a⁰. Therefore, if the marginal bid ratio Ψ'(a) is
larger than the marginal critical value Φ'(a) at a = 0, their loci will cross each other for
some amount a^t of open space, where φ(a^t) = Ψ(a^t), because the marginal critical value
Φ'(a) goes to infinity when a is getting close to L - a⁰. Before a is reaching a^t, 0 < a < a^t,
the bid ratio Ψ(a) is greater than the critical value Φ(a), which implies property tax
increment is sufficient to cover the expenditure in open space, and vice versa. In this
case, a^t represents the maximum amount of open space that can be self-financed without
imposing a new tax or increasing the current property tax rate.

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