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

The right hand side of equation (4) is the standardized marginal cost per unit

1             P(a⁰)

remaining land. Let f (a) =------0----(1 +---—), which describes how the

(L - a⁰ - a)    P(a + a⁰)

standardized marginal cost per unit remaining land would change with preserved open

space. Take the first derivative of f(a) with respect to a

f '(a) =-----------¹-------— [ P(a + a ⁰)2 + P(a⁰) P(a + a⁰) - P(a⁰) P (a + a ⁰)(L - a⁰ - a)] (6)

(       (L - a⁰ - a)²P(a + a⁰)²

How f(a) changes with respect to a depends on the sign of the nominator of f’(a). Some
algebraic manipulations can show that

if P(a_) ≤ _2  , _f,(a) > 0 for 0 < a < l — a0;

P(a⁰)   L -a⁰

if ʃ*⁽a ) > —, f′(a) < 0 for 0 < a < ~ ; f′(a) > 0 for ~ < a < L - a⁰.

P(a⁰)   L -a⁰

where a^~is defined by

P'(~ + a⁰) =     1     ₍ P(~ + a⁰) ₊₁)                    ₍₇₎

P(~ + a⁰) L - a⁰ - α   P(a⁰)

These mathematical properties show that the standardized marginal cost per unit
remaining land, f(a), may decrease for up to a fixed amount of open space, increases
when a is large enough, and eventually goes to infinity as a is approaching the total
amount of the remaining land L - a⁰.

Equation (4) requires at the optimal level a^* of increment of open space, the
standardized marginal benefit equals the standardized marginal cost per unit remaining
land, which means the curve of residents’ standardized WTP g(a) crosses the curve of the
standardized marginal cost per unit remaining land f(a) at a = a^* (See figure 1). Since
resident’s standardized WTP g(a) monotonically decreases with open space, and the

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