Neutral Property Taxation Under Uncertainty

Abstract

In a framework where no uncertainty arises, Arnott (J Publ Econ Theor 7:27–50, 2005) investigates a neutral property taxation policy that will not affect a landowner’s choices of capital intensity and timing of development. We investigate the same issue, but allow rents on structures to be stochastic over time. We assume that a regulator implements taxation on capital, vacant land, and post-development property so as to expropriate a certain ratio of pre-tax site value as well as to achieve neutrality. We find that the optimal taxation policy is to tax capital and subsidize properties before and after development. We also investigate how this optimal policy changes in response to changes in several exogenous forces related to demand and supply conditions of the real estate market.

Appendices

Appendix 1

Differentiating k* and x* in Eqs. 23 and 24, respectively, with respect to τ _k yields

$$\frac{{{\text{d}}k^* }}{{{\text{d}}\tau _k }} = \frac{{k^* }}{{\left( {c + \tau _k } \right)}} <0,$$

(34)

$$\frac{{dx^* }}{{d\tau _k }} = \frac{{\gamma x^* }}{{\left( {c + \tau _k } \right)}} >0.$$

(35)

Differentiating k* and x* with respect to τ _b yields

$$\frac{{{\text{d}}k^* }}{{{\text{d}}\tau _b }} = \frac{{ - k^* }}{{\left( {1 - \frac{1}{{\beta _{1t} }} - \gamma } \right)\beta _{1t} ^2 }}\frac{{\partial \beta _{1t} }}{{\partial \tau _b }} <0,$$

(36)

$$\frac{{{\text{d}}x^* }}{{{\text{d}}\tau _b }} = \frac{{ - \left( {1 - \gamma } \right)x^* }}{{\left( {1 - \frac{1}{{\beta _{1t} }} - \gamma } \right)\beta _{1t} ^2 }}\frac{{\partial \beta _{1t} }}{{\partial \tau _b }} <0.$$

(37)

Differentiating k* and x* with respect to τ _a yields

$$\frac{{{\text{d}}k^* }}{{{\text{d}}\tau _a }} = 0,$$

(38)

$$\frac{{dk^* }}{{d\tau _a }} = \frac{{x^* }}{{\tau _a }} >0.$$

(39)

QED.

Appendix 2

Define $g\left( Z \right) = E_t \int_t^{T^* } {Z\left( {x\left( s \right)} \right)e^{ - \rho \left( {s - t} \right)} ds}$. Substituting A ₁ in Eq. 19 and A ₂ in Eq. 20 into Eq. 14 yields

$$Z\left( {x\left( s \right)} \right) = \frac{{k^{*\gamma } x^* }}{{\left( {\rho + \tau _a - \alpha } \right)\beta _{1t} }}\left( {\frac{{x\left( s \right)}}{{x^* }}} \right)^{\beta _{1t} } .$$

(40)

Given that x(s) evolves according to the geometric Brownian motion given by Eq. 1, Z(s) evolves according to a geometric Brownian motion given by

$$\frac{{{\text{d}}Z\left( s \right)}}{{Z\left( s \right)}} = \left[ {\alpha \beta _{1t} + \frac{{\sigma ^2 }}{2}\beta _{1t} \left( {\beta _{1t} - 1} \right)} \right]{\text{d}}t + \sigma \beta _{1t} d\Omega \left( s \right).$$

(41)

Thus, g(Z) will satisfy the differential equation

$$\frac{1}{2}\sigma ^2 \beta _{1t} ^2 g\prime \left( Z \right) + \left[ {\alpha \beta _{1t} + \frac{{\sigma ^2 }}{2}\beta _{1t} \left( {\beta _{1t} - 1} \right)} \right]Zg\prime \left( Z \right) - \rho g\left( Z \right) + Z = 0.$$

(42)

This has the general solution

$$g\left( Z \right) = B_1 Z^{\theta _1 } + B_2 Z^{\theta _2 } + \frac{Z}{{\rho - \alpha \beta _{1t} - \frac{{\sigma ^2 }}{2}\beta _{1t} \left( {\beta _{1t} - 1} \right)}},$$

(43)

where θ ₁ and θ ₂ are, respectively, the larger and smaller roots of the quadratic equation

$$ - \frac{1}{2}\sigma ^2 \beta _{1t} ^2 \theta \left( {\theta - 1} \right) - \left[ {\alpha \beta _{1t} + \frac{{\sigma ^2 }}{2}\beta _{1t} \left( {\beta _{1t} - 1} \right)} \right]\theta + \rho = 0.$$

(44)

The boundary conditions are given by

$$g\left( 0 \right) = 0\;{\text{and}}\;g\left( {Z\left( {x^* } \right)} \right) = 0.$$

(45)

Therefore,

$$B_2 = 0\;{\text{and}}\;B_1 = \frac{{ - Z\left( {x^* } \right)^{1 - \theta _1 } }}{{\rho - \alpha \beta _{1t} - \frac{{\sigma ^2 }}{2}\beta _{1t} \left( {\beta _{1t} - 1} \right)}}.$$

(46)

After detailed calculations, this yields

$$g\left( {Z\left( x \right)} \right) = \frac{{f\left( {\frac{x}{{x^* }}} \right)^{\beta _{1t} } \left[ {1 - \left( {\frac{x}{{x^* }}} \right)^{\beta _{1t} \left( {\theta _1 - 1} \right)} } \right]}}{{ - \tau _b \left[ {\left( {1 - \gamma } \right)\beta _{1t} - 1} \right]}}.$$

(47)

Furthermore,

$$\begin{aligned} & \quad E_{t} {\int_{T^{*} }^\infty {{\left[ {W_{a} {\left( {x{\left( s \right)},k} \right)} - {\left( {c + \tau _{k} } \right)}k - f} \right]}e^{{ - \rho {\left( {s - t} \right)}}} } }{\text{d}}s \\ & = E_{t} e^{{ - \rho {\left( {T^{*} - t} \right)}}} {\int_{T^{*} }^\infty {{\left[ {\frac{{x{\left( s \right)}k^{\gamma } }}{{{\left( {\rho + \tau _{a} - \alpha } \right)}}} - {\left( {1 + \tau _{k} } \right)}ck - f} \right]}e^{{ - \rho {\left( {s - T^{*} } \right)}}} {\text{d}}s,} } \\ \end{aligned} $$

(48)

where

$$\begin{aligned} & \quad \quad {\int_{T^{*} }^\infty {{\left[ {\frac{{x{\left( s \right)}k^{\gamma } }}{{{\left( {\rho + \tau _{a} - \alpha } \right)}}} - {\left( {1 + \tau _{k} } \right)}ck - f} \right]}e^{{ - \rho {\left( {s - T^{*} } \right)}}} {\text{d}}s} } \\ & \quad \quad = \frac{{x^{*} k^{{*\gamma }} }}{{{\left( {\rho + \tau _{a} - \alpha } \right)}{\left( {\rho - \alpha } \right)}}} - \frac{{{\left( {1 + \tau _{k} } \right)}ck^{*} }}{\rho } - \frac{f}{\rho } \\ & \quad \quad = \frac{f}{{{\left( {1 - \frac{1}{{\beta _{{1t}} }} - \gamma } \right)}}}{\left( {\frac{1}{{\rho - \alpha }} - \frac{\gamma }{\rho }} \right)} - \frac{f}{\rho } \\ & {\text{and}}\;E_{t} e^{{ - \rho {\left( {T^{*} - t} \right)}}} = {\left( {\frac{x}{{x^{*} }}} \right)}^{{\beta _{{10}} }} . \\ \end{aligned} $$

(49)

Finally, the pre-tax site value at development, discounted back to the current time t is given by

$$\begin{aligned} V = E_t e^{ - \rho \left( {T^* - t} \right)} \left[ {\frac{{x^* k^{*\gamma } }}{{\left( {\rho - \alpha } \right)}} - \left( {ck^* + f} \right)} \right] \\ \quad = \left( {\frac{x}{{x^* }}} \right)^{\beta _{10} } {\kern 1pt} \frac{f}{{\beta _{10} \left( {1 - \gamma } \right) - 1}}. \\ \end{aligned} $$

(50)

QED.

Appendix 3

Differentiating k* and k ₀ with respect to σ yields

$$\frac{{\partial k^ * }}{{\partial \sigma }} = \frac{{2\left( {\beta _{1t} - 1} \right)k^ * }}{{\sigma \left[ {\left( {1 - \gamma } \right)\beta _{1t} - 1} \right]\left( {2\beta _{1t} - 1} \right)}} >0,$$

(51)

$$\frac{{\partial k_0 }}{{\partial \sigma }} = \frac{{2\left( {\beta _{10} - 1} \right)k_0 }}{{\sigma \left[ {\left( {1 - \gamma } \right)\beta _{10} - 1} \right]\left( {2\beta _{10} - 1} \right)}} >0.$$

(52)

Differentiating x* and x ₀ with respect to σ yields

$$\frac{{\partial x^ * }}{{\partial \sigma }} = \frac{{2\left( {1 - \gamma } \right)\left( {\beta _{1t} - 1} \right)k^ * }}{{\sigma \left[ {\left( {1 - \gamma } \right)\beta _{1t} - 1} \right]\left( {2\beta _{1t} - 1} \right)}} >0,$$

(53)

$$\frac{{\partial x_0 }}{{\partial \sigma }} = \frac{{2\left( {1 - \gamma } \right)\left( {\beta _{10} - 1} \right)k_0 }}{{\sigma \left[ {\left( {1 - \gamma } \right)\beta _{10} - 1} \right]\left( {2\beta _{10} - 1} \right)}} >0.$$

(54)

Differentiating k* and k ₀ with respect to α yields

$$\frac{{\partial k^ * }}{{\partial \alpha }} = \frac{{2k^ * }}{{\sigma \left[ {\left( {1 - \gamma } \right)\beta _{1t} - 1} \right]\left( {2\beta _{1t} - 1} \right)}} >0,$$

(55)

$$\frac{{\partial k_0 }}{{\partial \sigma }} = \frac{{2k_0 }}{{\sigma \left[ {\left( {1 - \gamma } \right)\beta _{10} - 1} \right]\left( {2\beta _{10} - 1} \right)}} >0.$$

(56)

Differentiating x* and x ₀ with respect to α yields

$$\frac{{\partial x^ * }}{{\partial \alpha }} = \frac{{ - x^ * }}{{\left( {\rho + \tau _a - \alpha } \right)}} + \frac{{2\left( {1 - \gamma } \right)x^ * }}{{\sigma \left[ {\left( {1 - \gamma } \right)\beta _{1t} - 1} \right]\left( {2\beta _{1t} - 1} \right)}}\frac{ >}{ <}0,$$

(57)

$$\frac{{\partial x_0 }}{{\partial \alpha }} = \frac{{ - x_0 }}{{\left( {\rho - \alpha } \right)}} + \frac{{2\left( {1 - \gamma } \right)x_0 }}{{\sigma \left[ {\left( {1 - \gamma } \right)\beta _{10} - 1} \right]\left( {2\beta _{10} - 1} \right)}}\frac{ >}{ <}0.$$

(58)

Differentiating k* and x ₀ with respect to γ yields

$$\frac{{\partial k^ * }}{{\partial \gamma }} = \frac{{k^ * }}{\gamma } + \frac{{k^ * }}{{1 - \frac{1}{{\beta _{1t} }} - \gamma }} >0,$$

(59)

$$\frac{{\partial k_0 }}{{\partial \gamma }} = \frac{{k_0 }}{\gamma } + \frac{{k_0 }}{{1 - \frac{1}{{\beta _{10} }} - \gamma }} >0.$$

(60)

Differentiating x* and x ₀ with respect to γ yields

$$\frac{{\partial x^ * }}{{\partial \gamma }} = \left[ {\frac{{ - \left( {c + \tau _k } \right)}}{f} + \frac{{\left( {\gamma - 1} \right)}}{{\left( {1 - \frac{1}{{\beta _{1t} }} - \gamma } \right)}} - \ln \left( {1 - \frac{1}{{\beta _{1t} }} - \gamma } \right) + \frac{f}{{\left( {c + \tau _k } \right)}}\ln \frac{{f\gamma }}{{\left( {c + \tau _k } \right)}}} \right]x^ * \frac{ >}{ <}0,$$

(61)

$$\frac{{\partial x_0 }}{{\partial \gamma }} = \left[ {\frac{{ - c}}{f} + \frac{{\left( {\gamma - 1} \right)}}{{\left( {1 - \frac{1}{{\beta _{10} }} - \gamma } \right)}} - \ln \left( {1 - \frac{1}{{\beta _{10} }} - \gamma } \right) + \frac{f}{c}\ln \frac{{f\gamma }}{c}} \right]x_0 \frac{ >}{ <}0.$$

(62)

Differentiating k* and k ₀ with respect to c yields

$$\frac{{\partial k^{ * } }}{{\partial c}} = \frac{{ - k^{ * } }}{c} < 0,$$

(63)

$$\frac{{\partial k_0 }}{{\partial c}} = \frac{{ - k_0 }}{c} <0.$$

(64)

Differentiating x* and x ₀ with respect to c yields

$$\frac{{\partial x^{ * } }}{{\partial c}} = \frac{{\gamma x^{ * } }}{c} > 0,$$

(65)

$$\frac{{\partial x_0 }}{{\partial c}} = \frac{{\gamma x_0 }}{c} >0.$$

(66)

Differentiating k* and k ₀ with respect to f yields

$$\frac{{\partial k^ * }}{{\partial f}} = \frac{{k^ * }}{f} >0,$$

(67)

$$\frac{{\partial k_0 }}{{\partial f}} = \frac{{k_0 }}{f} >0.$$

(68)

Differentiating x* and x ₀ with respect to f yields

$$\frac{{\partial x^ * }}{{\partial f}} = \frac{{\left( {1 - \gamma } \right)x^ * }}{f} >0,$$

(69)

$$\frac{{\partial x_0 }}{{\partial f}} = \frac{{\left( {1 - \gamma } \right)x_0 }}{f} >0.$$

(70)

Given that k*=k ₀, x*=x ₀, and β ₁₀ > β _1t , it then follows that ${{\partial k^* } \mathord{\left/ {\vphantom {{\partial k^* } {\partial \sigma >{{\partial k_0 } \mathord{\left/ {\vphantom {{\partial k_0 } {\partial \sigma }}} \right. \kern-\nulldelimiterspace} {\partial \sigma }}}}} \right. \kern-\nulldelimiterspace} {\partial \sigma >{{\partial k_0 } \mathord{\left/ {\vphantom {{\partial k_0 } {\partial \sigma }}} \right. \kern-\nulldelimiterspace} {\partial \sigma }}}}$, ${{\partial x^* } \mathord{\left/ {\vphantom {{\partial x^* } {\partial \sigma >{{\partial x_0 } \mathord{\left/ {\vphantom {{\partial x_0 } {\partial \sigma }}} \right. \kern-\nulldelimiterspace} {\partial \sigma }}}}} \right. \kern-\nulldelimiterspace} {\partial \sigma >{{\partial x_0 } \mathord{\left/ {\vphantom {{\partial x_0 } {\partial \sigma }}} \right. \kern-\nulldelimiterspace} {\partial \sigma }}}}$, ${{\partial k^* } \mathord{\left/ {\vphantom {{\partial k^* } {\partial \alpha >{{\partial k_0 } \mathord{\left/ {\vphantom {{\partial k_0 } {\partial \alpha }}} \right. \kern-\nulldelimiterspace} {\partial \alpha }}}}} \right. \kern-\nulldelimiterspace} {\partial \alpha >{{\partial k_0 } \mathord{\left/ {\vphantom {{\partial k_0 } {\partial \alpha }}} \right. \kern-\nulldelimiterspace} {\partial \alpha }}}}$, ${{\partial k^* } \mathord{\left/ {\vphantom {{\partial k^* } {\partial \gamma >{{\partial k_0 } \mathord{\left/ {\vphantom {{\partial k_0 } {\partial \gamma }}} \right. \kern-\nulldelimiterspace} {\partial \gamma }}}}} \right. \kern-\nulldelimiterspace} {\partial \gamma >{{\partial k_0 } \mathord{\left/ {\vphantom {{\partial k_0 } {\partial \gamma }}} \right. \kern-\nulldelimiterspace} {\partial \gamma }}}}$, ${{\partial x^* } \mathord{\left/ {\vphantom {{\partial x^* } {\partial \gamma >{{\partial x_0 } \mathord{\left/ {\vphantom {{\partial x_0 } {\partial \gamma }}} \right. \kern-\nulldelimiterspace} {\partial \gamma }}}}} \right. \kern-\nulldelimiterspace} {\partial \gamma >{{\partial x_0 } \mathord{\left/ {\vphantom {{\partial x_0 } {\partial \gamma }}} \right. \kern-\nulldelimiterspace} {\partial \gamma }}}}$, ${\partial k^{*} } \mathord{\left/ {\vphantom {{\partial k^{*} } {\partial c = {\partial k_{0} } \mathord{\left/ {\vphantom {{\partial k_{0} } {\partial c}}} \right. \kern-\nulldelimiterspace} {\partial c}}}} \right. \kern-\nulldelimiterspace} {\partial c = {\partial k_{0} } \mathord{\left/ {\vphantom {{\partial k_{0} } {\partial c}}} \right. \kern-\nulldelimiterspace} {\partial c}}$, ${\partial x^{*} } \mathord{\left/ {\vphantom {{\partial x^{*} } {\partial c = {\partial x_{0} } \mathord{\left/ {\vphantom {{\partial x_{0} } {\partial c}}} \right. \kern-\nulldelimiterspace} {\partial c}}}} \right. \kern-\nulldelimiterspace} {\partial c = {\partial x_{0} } \mathord{\left/ {\vphantom {{\partial x_{0} } {\partial c}}} \right. \kern-\nulldelimiterspace} {\partial c}}$, ${\partial k^{*} } \mathord{\left/ {\vphantom {{\partial k^{*} } {\partial f = {\partial k_{0} } \mathord{\left/ {\vphantom {{\partial k_{0} } {\partial f}}} \right. \kern-\nulldelimiterspace} {\partial f}}}} \right. \kern-\nulldelimiterspace} {\partial f = {\partial k_{0} } \mathord{\left/ {\vphantom {{\partial k_{0} } {\partial f}}} \right. \kern-\nulldelimiterspace} {\partial f}}$, and ${\partial x^{*} } \mathord{\left/ {\vphantom {{\partial x^{*} } {\partial f = {\partial x_{0} } \mathord{\left/ {\vphantom {{\partial x_{0} } {\partial f}}} \right. \kern-\nulldelimiterspace} {\partial f}}}} \right. \kern-\nulldelimiterspace} {\partial f = {\partial x_{0} } \mathord{\left/ {\vphantom {{\partial x_{0} } {\partial f}}} \right. \kern-\nulldelimiterspace} {\partial f}}$.

QED.

Appendix 4

Suppose that we explicitly write x* and k* as a function of τ _a , τ _b , τ _k , and σ, that is, as x*(τ _a , τ _b , τ _k , σ) and k*(τ _a , τ _b , τ _k , σ), respectively, and x ₀ and k ₀ as a function of σ, that is, as x ₀(σ) and k ₀(σ), respectively. Suppose that at $\sigma = \overline \sigma$, $k^* \left( {\tau _a ,\tau _b ,\tau _k ,\overline \sigma } \right) = k_0 \left( {\overline \sigma } \right)$ and $x^* \left( {\tau _a ,\tau _b ,\tau _k ,\overline \sigma } \right) = x_0 \left( {\overline \sigma } \right)$. Also suppose that $\overline \sigma$ is increased to $\sigma \prime$. Then Proposition 3(1) indicates that $k^* \left( {\tau _a ,\tau _b ,\tau _k ,\sigma \prime } \right) = k_0 \left( {\sigma \prime } \right)$ and $x^* \left( {\tau _a ,\tau _b ,\tau _k ,\sigma \prime } \right) = x_0 \left( {\sigma \prime } \right)$. Our problem then reduces to finding a combination of $\left( {\tau _a \prime ,\tau _b \prime ,\tau _k \prime } \right)$ such that $k^* \left( {\tau _a \prime ,\tau _b \prime ,\tau _k \prime ,\sigma \prime } \right) = k_0 \left( {\sigma \prime } \right)$ and $x^* \left( {\tau _a \prime ,\tau _b \prime ,\tau _k \prime ,\sigma \prime } \right) = x_0 \left( {\sigma \prime } \right)$. However, we are unable to ascertain the relationship between τ _a and $\tau _a \prime$, τ _b and $\tau _b \prime$, and τ _k and $\tau _k \prime$. We can only rule out several possibilities. For example, we can rule out the combination such that $\tau _a \prime >\tau _a$, $\tau _k \prime >\tau _k$, and $\tau _b <\tau _b \prime$ because $x^{*} {\left( {\tau _{a} \prime ,\tau _{b} \prime ,\tau _{k} \prime ,\sigma \prime } \right)} > x^{*} {\left( {\tau _{a} ,\tau _{b} ,\tau _{k} ,\sigma \prime } \right)} > x_{0} {\left( {\sigma \prime } \right)}$. We can also rule out the combination such that $\tau _b \prime <\tau _b$, $\tau _k \prime >\tau _k$ and either $\tau _a \prime >\tau _a$ or $\tau _a \prime <\tau _a$ because $k^{*} {\left( {\tau _{a} \prime ,\tau _{b} \prime ,\tau _{k} \prime ,\sigma \prime } \right)} > k^{*} {\left( {\tau _{a} ,\tau _{b} ,\tau _{k} ,\sigma \prime } \right)} > k_{0} {\left( {\sigma \prime } \right)}$.

QED.