Optimal statute of limitations under land development timing decisions

Baird and Jackson (1984) argued that, because land is permanent and easy to identify, it is thus easy for a regulator to set rules governing transfer based on a recording system. Also, because chains of title are long, those who acquire titles can trace their titles back very far. Moreover, the cost of maintaining a title system is small, compared to the relative value of the property involved. A filing system must have some organizing principle because potential purchasers need to know which of many files in many jurisdictions are relevant. Because one can require filings to be made in the jurisdiction in which the land is located, it is quite easy for purchasers to determine which file to check. The title system can also reduce the uncertainty surrounding a transfer of real property and can largely eliminate the problem of thieves in the chain of title.

This theory contrasts with that of Cribbet (1975), who argued that regulators employ statues of limitations to increase land transactions and land development.

While the Stackelberg game consists of two stages, it differs from the two-stage game of Hotelling’s model (van Leeuwen and Lijesen 2016), in which firms compete over locations in the first stage and then over prices in the second stage.

Our hierarchy game model abstracts from the conflict of interest between a current landowner and previous landowners. However, such a conflict is incorporated by a hazard function which captures the phenomenon that the current landowner faces the risk of losing title of land during the period of the statutory limitations. Implicitly, we assume that the regulator will impose the same length of statutory limitations when the title of land is transferred from the current landowner to the previous owner. See Eq. (4) in Sect. 2 for details. Future study may investigate the polar case in which a current landowner and a previous landowner compete in a game in which the length of statutory limitations for the former differs from that of the latter.

McFarlane (1999) argued that investment in land development will be fully irreversible if demolition costs are extremely high. Similarly, Riddiough (1997) suggested that irreversibility is a reasonable assumption for real estate, in which the physical asset is long lived, and the costs of switching to alternative uses are quite high. Turnbull (2005) argued that the irreversibility assumption might not be realistic, but it provides an analytically tractable solution.

If both \( y(t) \) and \( u(t) \) follow joint arithmetic Brownian motion, then \( x(t) \) follows arithmetic Brownian motion. We can derive \( x(t) \) in Eq. (2) by assuming that both \( y(t) \) and \( u(t) \) follow joint arithmetic Brownian motion. Suppose that \( y(t) = \alpha_{1} {\text{d}}t + \,\sigma_{1} {\text{d}}Z_{1} (t) \) and \( u(t)\, = \alpha_{2} {\text{d}}t\, + \sigma_{2} {\text{d}}Z_{2} (t)\,, \) where both \( {\text{d}}Z_{1} (t) \) and \( dZ_{2} (t) \) are increases in a standard Wiener process, and \( E\left( {{\text{d}}Z_{1} (t)\,{\text{d}}Z_{2} (t)} \right)\, = r_{12} {\text{d}}t, \) where \( - 1\, \le \,r_{12} \le 1. \) By Ito’s Lemma, it follows that \( \alpha = \alpha_{1} - \alpha_{2} \) and \( \sigma \, = ( {\sigma_{1}^{2} - 2r_{12} \sigma_{1} \sigma_{2} + \sigma_{2}^{2} } )^{{{1 \mathord{/ {\vphantom {1 2}} \kern-0pt} 2}}} . \) Thus, the stochastic evolution of land rents shown in Eq. (2) is due to the stochastic evolution of both income, \( y(t), \) and the exogenous utility, \( u(t)\).

Our specification is the same as that assumed by Capozza and Helsley (1990), Capozza and Li (1994), and Capozza and Sick (1994). The arithmetic Brownian motion of urban rents is supported by the empirical data, but it suffers from the shortcoming that urban rents can be negative. See a thorough discussion in Capozza and Li (1994). While there are some merits by assuming that urban rents following the geometric Brownian motion (Bar-Ilan and Strange 1996), which, however, derives a trivial result in which a landowner will develop his or her land either immediately or never.

The risks associated with wrongly dispossessed owners can reduce the risks of being adversely dispossessed by future claims, which can be captured by assuming that \( \lambda_{2} \) is an increasing function of \( \lambda_{0} \) and/or \( \lambda_{1} \), indicating that a lower probability that the current owner still retains the land by \( t > 0 \) will reduce the monitoring cost.

Baker et al. (2001) consider both non-developable land and developable land. They assume that the return to developed land is increasing over time. By contrast, we consider only developable land whose return follows the arithmetic Brownian motion.

On the right-hand side of Eq. (15), the derivative of the first two terms with respect to \( \bar{t} \) is negative, while the sign of the derivative of the last term with respect to \( \bar{t} \) is indefinite. Consequently, the sign of \( {{\partial^{2} U(\bar{t})} \mathord{/ {\vphantom {{\partial^{2} U(\bar{t})} {\partial \bar{t}^{2} }}} \kern-0pt} {\partial \bar{t}^{2} }} \) is indefinite. On the right-hand side of Eq. (16), the signs of the derivative of the three terms with respect to \( \bar{t} \) are all indefinite. Consequently, the sign of \( {{\partial^{2} F(x)} \mathord{/ {\vphantom {{\partial^{2} F(x)} {\partial \bar{t}^{2} }}} \kern-0pt} {\partial \bar{t}^{2} }} \) is indefinite for \( T < \bar{t}. \) If \( T \ge \bar{t}, \) differentiating \( {{\partial F(x)} \mathord{\left/ {\vphantom {{\partial F(x)} {\partial \bar{t}}}} \right. \kern-0pt} {\partial \bar{t}}} \) with respect to \( \bar{t} \) yields a positive sign, i.e., \( {{F^{2} (x)} \mathord{/ {\vphantom {{F^{2} (x)} {\partial \bar{t}^{2} }}} \kern-0pt} {\partial \bar{t}^{2} }} = - (\lambda_{0} + \lambda_{1} ) \cdot {{\partial F(x)} \mathord{/ {\vphantom {{\partial F(x)} {\partial \bar{t}}}} \kern-0pt} {\partial \bar{t}}} > 0. \) Pooling all of the above information yields an ambiguous sign for \( {{\partial^{2} (U(\bar{t}) + F(x))} \mathord{/ {\vphantom {{\partial^{2} (U(\bar{t}) + F(x))} {\partial \bar{t}^{2} }}} \kern-0pt} {\partial \bar{t}^{2} }} \). However, we must impose it as a negative sign to ensure \( \bar{t}^{*} \) as an interior solution.

At a type A site, the demanded density increases over time, while in both type B and type C sites, the demanded density decreases over time.

This result is in agreement with that of Miceli et al. (2003), indicating that the optimal statute of limitations generally varies across locations within an urban area.

Note that Baker et al. (2001) assumed that a landowner’s decision to develop his or her land is exogenously given when investigating the issue of the optimal statute of limitations. In contrast, similar to our article, Miceli et al. (2003) considered a landowner’s decision to develop his or her land when investigating the issue of the optimal statute of limitations.

That is, an increase in \( s \) in the monitor cost function \( sm(\bar{t}), \) where \( m(\bar{t}) = a_{0} - \lambda_{2} \bar{t} - \delta \).

According to Goetzmann and Ibbotson (1990), during the period from 1969 to 1989, the annual standard deviation for REITs on commercial property was equal to 15.4%, which is within our choice region.