How Do the Location, Size and Budget of Open Space Conservation Affect Land Values?

Abstract

In this article we present a model to examine the optimal location, size, and budget of open space conservation and the resulting impact on land values and local fiscal conditions in an urban area. Results indicate that open space conservation can transform the defining features of an urban landscape. A well-designed open space conservation program can improve municipal services, increase total property values, and attract households to the city without substantially increasing tax burdens, while an improperly designed open space program can have the opposite effects. Results also reveal the key parameters that determine the optimal location and size of open space conservation and their fiscal and land value effects.

Appendix

Proof of Proposition 1

When households have homogeneous preferences, the maximization problem (6) can be simplified to

$$ \underset{\left(\tau, g\right)}{Max}\kern1em \frac{g^{\mu }}{c+\tau}\kern0.37em \mathrm{s}.\ \mathrm{t}.\kern0.5em T{C}_s+T{C}_o\le TR. $$

Substituting Eq. (1) into Eqs. (2)–(4), the total tax revenue and the total costs of municipal services and open space conservation can be written as:

$$ TR=\frac{\tau {g}^{\mu }}{\left(c+\tau \right)}{\displaystyle \underset{D-S}{\iint }A{\left[y\left(u,v\right)-tx\right]}^{\beta_i}a{\left(u,v;S\right)}^{\gamma_i} dudv}\equiv \frac{\tau {g}^{\mu }}{\left(c+\tau \right)}\tilde{R}, $$

(A1)

$$ T{C}_s=g{N}^{\lambda }={g}^{1+\lambda \mu \delta }{\left[{\displaystyle \underset{D-S}{\iint }{A}^{\delta }{\left[y\left(u,v\right)-tx\right]}^{\beta_i\delta }a{\left(u,v;S\right)}^{\gamma_i\delta } dudv}\right]}^{\lambda}\equiv {g}^{1+\lambda \mu \delta }{\tilde{N}}^{\lambda }, $$

(A2)

$$ T{C}_o=\frac{c{g}^{\mu }}{\left(c+\tau \right)}{\displaystyle \underset{S}{\iint }A{\left[y\left(u,v\right)-tx\right]}^{\beta_i}a{\left(u,v;S\right)}^{\gamma_i} dudv}\equiv \frac{c{g}^{\mu }}{\left(c+\tau \right)}\tilde{G}, $$

(A3)

where y(u, v) denotes the income of the household located at (u, v). Using (A1)–(A3), the city’s budget constraint can be written as:

$$ \frac{c{g}^{\mu }}{c+\tau}\le \frac{M(g)}{\tilde{R}+\tilde{G}}, $$

(A4)

where $M(g)\equiv {g}^{\mu}\tilde{R}-{g}^{1+\lambda \mu \delta }{\tilde{N}}^{\lambda }$. At the optimum, the constraint must hold with equality; otherwise, a smaller τexists that satisfies the budget constraint and improves the objective function. Substituting (A4) into the objective function, the maximization problem (5) can be transformed to

$$ \underset{\left(\tau, g\right)}{Max}\ \frac{M(g)}{\tilde{R}+\tilde{G}}, $$

(A5)

with τ being determined by the budget constraint that holds with equality. Note that M(g) is a concave function and reaches its maximum at

$$ {g}^{\ast }={\left[\frac{\mu \tilde{R}}{\left(1+\lambda \mu \delta \right){\tilde{N}}^{\lambda }}\right]}^{\frac{1}{1-\left(1-\lambda \delta \right)\mu }}, $$

(A6)

which gives Eq. (8).

Thus, if $\overline{G}<M\left({g}^{\ast}\right)$, τ ^* defined by the budget constraint at g* is positive, and (τ ^∗, g ^∗) is the optimal solution of (A1). Substituting g* into the budget constraint and solving for τ, we obtain:

$$ {\tau}^{\ast }=\frac{c\left(1+\lambda \mu \delta \right)}{1-\left(1-\lambda \delta \right)\mu}\left(1+\frac{\tilde{G}}{\tilde{R}}\right)-c. $$

(A7)

By definition,

$$ 1-\rho =\frac{T{C}_o}{TR}=\frac{c\tilde{G}}{\tau \tilde{R}}\Rightarrow \kern0.5em \frac{\tilde{G}}{\tilde{R}}=\left(1-\rho \right)\frac{\tau }{c} $$

(A8)

Substituting (A8) into (A7) and solving for τ, we obtain Eq. (6).

If $\overline{G}<M\left({g}^{\ast}\right),{\tau}^{\ast }$ defined by the budget constraint at g* is positive, and (τ ^∗, g ^∗) is the optimal solution of (A5). If $\overline{G}\ge M\left({g}^{\ast}\right)$, no (τ, g) combination would satisfy the budget constraint, and (A5) has no solution.

Proof of Corollary 1

Differentiating $\log \left(\tilde{G}/\tilde{R}\right)$ with respect to S (i.e., expanding the boundary S parallally to all directions by an infinitesimal amount) gives:

$$ \begin{array}{l}\frac{d}{dS} \log \left(\frac{\tilde{G}}{\tilde{R}}\right)=\frac{1}{\tilde{G}}\left[{\left.\frac{\partial \tilde{G}}{\partial S}\right|}_a+{\displaystyle \underset{S}{\iint }A{\left(y-tx\right)}^{\beta_i}\gamma {a}^{\gamma_i}\frac{\partial a}{\partial S} dudv}\right]\hfill \\ {}\kern5em -\frac{1}{\tilde{R}}\left[{\left.\frac{\partial \tilde{R}}{\partial S}\right|}_a+{\displaystyle \underset{D-S}{\iint }A{\left(y-tx\right)}^{\beta_i}\gamma {a}^{\gamma_i}\frac{\partial a}{\partial S} dudv}\right]\hfill \\ {}\kern5em =\left[{\varepsilon}_S^G+\gamma {\varepsilon}_S^a\right]-\left[-{\varepsilon}_S^R+\gamma {\varepsilon}_S^a\right]\hfill \\ {}\kern5em ={\varepsilon}_S^G+{\varepsilon}_S^R\ge 0\hfill \end{array}. $$

(A9)

Likewise, differentiating $\log \left(\tilde{R}/{\tilde{N}}^{\lambda}\right)$ with respect to S gives

$$ \begin{array}{l}\frac{d}{dS} \log \left(\frac{\tilde{R}}{{\tilde{N}}^{\lambda }}\right)=\left[-{\varepsilon}_S^R+\gamma {\varepsilon}_S^a\right]-\lambda \left[-{\varepsilon}_S^N+\gamma \delta {\varepsilon}_S^a\right]\hfill \\ {}\kern6em =\left(\lambda {\varepsilon}_S^N-{\varepsilon}_S^R\right)+\gamma {\varepsilon}_S^a\left(1-\lambda \delta \right)\hfill \end{array}, $$

(A10)

which is greater than or equal to zero if and only if (13) holds.

Proof of Corollary 2

Differentiating $\log \left({\tilde{R}}^{1+\lambda \delta \mu }/{\tilde{N}}^{\lambda \mu}\right)$ with respect to S, we obtain:

$$ \begin{array}{c}\hfill \frac{d}{dS} \log \left(\frac{{\tilde{R}}^{1+\lambda \delta \mu }}{{\tilde{N}}^{\lambda \mu }}\right)=\left(1+\lambda \delta \mu \right)\left[-{\varepsilon}_S^R+\gamma {\varepsilon}_S^a\right]-\lambda \mu \left[-{\varepsilon}_S^N+\gamma \delta {\varepsilon}_S^a\right]\hfill \\ {}\hfill =\left[\lambda \mu {\varepsilon}_S^N-\left(1+\lambda \delta \mu \right){\varepsilon}_S^R\right]+\gamma {\varepsilon}_S^a\hfill \end{array} $$

(A11)

which is non-negative if and only if (15) holds.

Proof of Proposition 2

Suppose the high-income households are the majority in the city. The maximization problem (6) can be written as

$$ \underset{\left(\tau, g\right)}{Max}\kern0.5em \frac{g^{\mu_h}}{\left(c+\tau \right)}\kern0.75em \mathrm{s}.\mathrm{t}.\kern0.5em \frac{M(g)-\overline{G}}{\left({\tilde{R}}_l{g}^{\mu_l}+{\tilde{R}}_h{g}^{\mu_h}\right)}\ge \frac{1}{c+\tau }, $$

(A12)

where $M(g)=\left({\tilde{R}}_l{g}^{\mu_l}+{\tilde{R}}_h{g}^{\mu_h}\right)-g{\left({\tilde{N}}_l{g}^{\mu_l}+{\tilde{N}}_h{g}^{\mu_h}\right)}^{\lambda }.$ At the optimum, the constraint must hold with equality. Substituting the budget constraint into the objective function, the maximization problem can be transformed to

$$ \underset{\left(\tau, g\right)}{Max}\ \frac{M(g)-\overline{G}}{\widehat{R}(g)}\kern0.75em \mathrm{s}.\mathrm{t}.\kern0.5em \frac{M(g)-\overline{G}}{g^{\mu_h}\widehat{R}(g)}=\frac{1}{c+\tau }, $$

(A13)

where $\tilde{R}(g)=\left({\tilde{R}}_l{g}^{\mu_l-{\mu}_h}+{\tilde{R}}_h\right)$. Note that the objective function is continuous and bounded in $\left[0,\overline{g}\right]$, where $\overline{g}$ is the largest g defined by $M\left(\overline{g}\right)=0$, beyond which M(g) < 0. Thus, the objective function has a globe maximum in $\left[0,\overline{g}\right]$. Denote the maximum point by g* and G ^∗≡M(g ^∗). If $\overline{G}<M\left({g}^{\ast}\right)$, τ ^∗ defined by the budget constraint at gi is positive, and (τ ^∗, g ^∗) is the optimal solution of (A13).

Denote the solution of (A13) by (τ ^∗ _i , g ^∗ _i ) when income group i is the majority. (τ ^∗ _h , g ^∗ _h ) is indeed the majority group if and only if

$$ {N}_h^{\ast}\left({\tau}_h^{\ast },{g}_h^{\ast}\right)\ge {N}_l^{\ast}\left({\tau}_h^{\ast },{g}_h^{\ast}\right). $$

(A14)

Likewise, (τ ^∗ _l , g ^∗ _l ), is indeed the majority group if and only if

$$ {N}_l^{\ast}\left({\tau}_l^{\ast },{g}_l^{\ast}\right)\ge {N}_h^{\ast}\left({\tau}_l^{\ast },{g}_l^{\ast}\right). $$

(A15)

We now prove that either (A14) or (A15) must hold. If (A14) does not hold, then N ^∗ _h (τ ^∗ _h , g ^∗ _h ) < N ^∗ _l (τ ^∗ _h , g ^∗ _h ). In this case, (A15) must hold because N ^∗ _h (τ ^∗ _l , g ^∗ _l ) ≤ N ^∗ _h (τ ^∗ _h , g ^∗ _h ) < N ^∗ _l (τ ^∗ _h , g ^∗ _h ) ≤ N ^∗ _l (τ ^∗ _l , g ^∗ _l ) . Similarly, we can prove that if (A15) does not hold, (A14) must hold. This proves that a majority equilibrium must exist in the city if $\overline{G}<M\left({g}^{\ast}\right)$.

To derive the equilibrium property tax rate, we use the transformation $g={\left[u\left(c+\tau \right)\right]}^{1/{\mu}_h}$ and τ = τ to transform the maximization problem (A13) into:

$$ \underset{\left(\tau, u\right)}{ \max}\kern1em u\kern0.5em \mathrm{s}.\mathrm{t}.\kern0.5em T{C}_s+\overline{G}=T{R}_h+T{R}_l. $$

(A16)

where

$$ T{C}_s=g{\left({N}_l+{N}_h\right)}^{\lambda }={\left(c+\tau \right)}^{\frac{1}{\mu_h}}{u}^{\frac{1}{\mu_h}}{\left({N}_h+{N}_l\right)}^{\lambda }, $$

(A17)

$$ {N}_h={\left[\left(c+\tau \right)u\right]}^{\delta }{\tilde{N}}_h,\kern0.5em {N}_l={\left(c+\tau \right)}^{\frac{\delta {\mu}_l}{\mu_h}}{u}^{\frac{\delta {\mu}_l}{\mu_h}}{\tilde{N}}_l, $$

(A18)

$$ T{R}_h=\tau u{\tilde{R}}_h,\kern1em T{R}_l={\left(c+\tau \right)}^{\frac{\mu_l-{\mu}_h}{\mu_h}}{u}^{\frac{\mu_l}{\mu_h}}{\tilde{R}}_l, $$

(A19)

The Lagrangian function for the maximization problem is

$$ L\left(\tau, u\right)=u+\xi \left[T{R}_h+T{R}_l-T{C}_s-T{C}_o\right]. $$

(A20)

Where ξ is the Lagarangian multiplier. Differentiating (A20) with respect to τ and setting it equal zero, we obtain the following first-order condition:

$$ \frac{\partial L}{\partial \tau }=0\Rightarrow \frac{\partial T{C}_s}{\partial \tau }=\frac{\partial T{R}_h}{\partial \tau }+\frac{\partial T{R}_l}{\partial \tau }. $$

(A21)

Assume household Differentiating (A17) with respect to τ and using (A18) gives

$$ \begin{array}{l}\frac{\partial T{C}_s}{\partial \tau }=\frac{1}{\mu_h}{\left(c+\tau \right)}^{\frac{1}{\mu_h}-1}{u}^{\frac{1}{\mu_h}}{\left({N}_h+{N}_l\right)}^{\lambda }+{\left(c+\tau \right)}^{\frac{1}{\mu_h}}{u}^{\frac{1}{\mu_h}}\lambda {\left({N}_h+{N}_l\right)}^{\lambda -1}\left(\frac{\partial {N}_h}{d\tau }+\frac{\partial {N}_l}{d\tau}\right)\hfill \\ {}\kern2.5em =\frac{T{C}_s}{\left(c+\tau \right)}\frac{1}{\mu_h}+\frac{\lambda T{C}_s}{N}\left(\frac{\partial {N}_h}{\partial \tau }+\frac{\partial {N}_l}{\partial \tau}\right)\hfill \end{array}. $$

(A22)

Differentiating (A10) with respect to τ and substituting the results into (A22), we obtain

$$ \frac{\partial T{C}_s}{\partial \tau }=\frac{T{C}_s}{\left(c+\tau \right)}\left[\frac{1}{\mu_h}+\frac{\lambda \delta }{N}\left({N}_h+\frac{\mu_{{}_l}}{\mu_h}{N}_l\right)\right]. $$

(A23)

Differentiating (A19) with respect to τ gives

$$ \frac{\partial T{R}_h}{\partial \tau }=\frac{T{R}_h}{\tau } $$

(A24)

$$ \frac{\partial T{R}_l}{\partial \tau }=\frac{T{R}_l}{\tau }+\frac{\mu_{{}_l}-{\mu}_h}{\mu_h}\frac{T{R}_l}{\left(c+\tau \right)} $$

(A25)

Substituting (A23)–(A25) into (A21) and noting TR = TR _h  + TR _l and TC _s  + TC _o  = TR gives

$$ \frac{T{C}_s}{\left(c+\tau \right)}\left[\frac{1}{\mu_h}+\frac{\lambda \delta }{N}\left(N-{N}_l+\frac{\mu_{{}_l}}{\mu_h}{N}_l\right)\right]=\frac{TR}{\tau }+\frac{\mu_{{}_l}-{\mu}_h}{\mu_h}\frac{T{R}_l}{\left(1+\tau \right)} $$

(A26)

$$ \rho \left[\frac{1}{\mu_h}+\lambda \delta \left(1+\frac{\mu_{{}_l}-{\mu}_h}{\mu_h}{r}_l\right)\right]=\frac{\left(1+\tau \right)}{\tau }+\frac{\mu_{{}_l}-{\mu}_h}{\mu_h}{m}_l, $$

(A27)

where ρ = TC _s /TR, r _l  = N _l /N, m _l  = TR _s /TR. Solving (A27) for τ gives (18). The result for the case where low-income households are the majority can be similarly derived.