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

characterized by varying amounts of open space (public goods) a, which mimics the
prototype of a series of local towns depicted in Tiebout’s theory (1956) on local
expenditures. Following the traditional monocentric urban model, each household
chooses residential location (community) represented by (x, a), house size q in units of
land area in the selected residential community, and a numeraire good z to maximize their
utility U = U(z, q, a). Each household is subject to a budget constraint z + Rq + tx = y,
where R denotes land rent, t denotes transportation cost per unit distance, and y is
household income.

For given land rent R and transportation cost t, the utility-maximizing choice of
house size q and numeraire good z can be represented as q* = q(y, t, x, R, a) and z* = y -
tx - Rq(y, t, x, R, a), respectively. Substitute the optimal consumption bundle (z*, q*)
into the utility function, U = U[y-Rq(y, t, x, R, a)-tx, q(y, t, x, R, a), a]. For an open city
model, household utility U at equilibrium is exogenously determined when migration is
costless, which is equal to the maximum utility attainable elsewhere in the economy.
Denote the exogenous utility level by V, which is expressed as

V = U(y-Rq(y, t, x, R, a)-tx, q(y, t, x, R, a), a)                  (1)

For given income level, transportation cost, and residential community, land rent
R has to change such that U(y, t, x, a, R) = V. Solving equation (1) for R, we can derive
the equilibrium land rent R = R(y, t, x, V, a), which represents the bid rent of each
household for per unit land in community (x, a) at market equilibrium. We suppress all
arguments but open space area a, and express the equilibrium land rent R as a function of
preserved open space, R = R(a). Assume the utility function U(∙) is concave, and it can
be shown that

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