The right hand side of (3) is the marginal cost per unit remaining land, which is the sum
of cost spent on purchasing an extra unit of open space and the value lost that would have
been gained otherwise from this extra unit of land, divided by the amount of the
remaining land. The left hand side of (3) represents the marginal benefit per unit
remaining land at the optimal preservation. From the perspective of the capitalization of
open space amenity in property value at market equilibrium, this marginal land price at
the optimal preservation represents residents’ willingness to pay for per unit preserved
open space. Because residents’ willingness to pay may be dependent on the price level
considered, we divide both sides of (3) by the post-preservation equilibrium land price:
P'(a* + a0) = 1 (1 + P(a 0) )
(4)
P(a* + a0) (L - a0 - a Y P (a* + a0)
We use this equation to identify the condition for preserving more open space to be
socially efficient, namely the condition under which a* > 0. The left hand side of
equation (4) is the marginal change rate in land price with respect to the amount of open
space, and which can be regarded as the standardized marginal benefit per unit land of
open space preservation. Let g(a) = P’(a + a0)/P(a + a0), which describes how local
residents’ standardized willingness to pay (WTP) changes with preserved open space.
Differentiate g(a) with respect to a,
P"(a+a0) P'(a+a0)
g(a)=7(a+√0)- PO+aa0ÿ
(5)
Because P’(a + a0) > 0 and P”(a + a0) < 0, g’(a) < 0, which means residents’ standardized
WTP is decreasing with preserved open space.