Rural land development under hyperbolic discounting: a real option approach

Abstract

This article presents a simple model of land development under uncertainty and hyperbolic discounting. Land kept in rural use pays an uncertain rent, while net returns from land development are known and constant. The landowner is viewed here as a sequence of selves with time-inconsistent preferences. We solve the underlying timing game under both naïve and sophisticated beliefs about the landowner’s time-inconsistency and show that (i) land development is accelerated due to his present-bias and (ii) a higher acceleration is associated with sophistication.

Appendix

1.1 Exponential discounting

As standard, to guarantee optimality,¹⁵ the solution of the differential Eq. (3.1) must meet the following value-matching and smooth-pasting conditions:

$$\begin{aligned} V(R^{*})=P, \end{aligned}$$

(A.1.1)

$$\begin{aligned} V^{\prime }(R^{*})=0. \end{aligned}$$

(A.1.2)

A candidate solution for Eq. (3.1) takes the form¹⁶:

$$\begin{aligned} V(R)=kR^{\gamma }+R/(\rho -\alpha ), \end{aligned}$$

(A.1.3)

where k is a constant to be determined and $\gamma$ is the negative root of the characteristic equation $\Lambda (\gamma )=(1/2)\sigma ^{2}\gamma (\gamma -1)+\alpha \gamma -\rho =0$.

Substituting Eq. (A.1.3) into Eq. (A.1.1) and Eq. (A.1.2) yields:

$$\begin{aligned} \left\{ \begin{array} [c]{c} kR^{*\gamma }+R^{*}/(\rho -\alpha )=P,\\ k\gamma R^{*\gamma -1}+1/(\rho -\alpha )=0. \end{array} \right. \end{aligned}$$

Solving this system gives k and $R^{*}$. Plugging k into Eq. (A.1.3) yields Eq. (5).

1.2 Hyperbolic discounting

Equations (6.1) and (9.1) are technically similar. Proceed by solving first the underlying common problem and then characterising the solution according to the assumed beliefs concerning future selves’ time preferences.

The equation to be solved is:

$$\begin{aligned}&(1/2)\sigma ^{2}R^{2}V^{q\,\prime \prime }\left( R\right) +\alpha RV^{q\;\prime }\left( R\right) -(\rho +\lambda )V^{q}(R)\nonumber \\&\qquad =-\{R[1+\lambda \beta /(\rho -\alpha )]+\lambda \beta \{[P-\overline{R}^{q} /(\rho -\alpha )](R/\overline{R}^{q})^{\gamma }\} \end{aligned}$$

(A.2.1)

where

Suppose that the particular solution for Eq. (A.2.1) takes the form $V^{q}(R)=c_{1}R^{\gamma }+c_{2}R.$ Substituting this candidate form and its first two derivatives, $\partial V^{q}(R)/\partial R=c_{1}\gamma R^{\gamma -1}+c_{2}$ and $\partial ^{2}V^{q}(R)/\partial R^{2}=c_{1}\gamma (\gamma -1)R^{\gamma -2}$ into Eq. (A.2.1) yields:

$$\begin{aligned}&\{[(1/2)\sigma ^{2}\gamma \left( \gamma -1\right) +\alpha \gamma -\rho ]-\lambda \}c_{1}R^{\gamma }+\left[ \alpha -\left( \rho +\lambda \right) \right] c_{2}R\nonumber \\&\quad =-\{R[1+\lambda \beta /(\rho -\alpha )]+\lambda \beta \{[P-\overline{R}^{q} /(\rho -\alpha )](R/\overline{R}^{q})^{\gamma }\} \end{aligned}$$

(A.2.2)

The coefficients $c_{1}$ and $c_{2}$ can be determined by solving the following two equations:

$$\begin{aligned} \{[(1/2)\sigma ^{2}\gamma \left( \gamma -1\right) +\alpha \gamma -\rho ]-\lambda \}c_{1}&=-\lambda \beta [P-\overline{R}^{q}/(\rho -\alpha )](1/\overline{R}^{q})^{\gamma }\\ \left[ \alpha -\left( \rho +\lambda \right) \right] c_{2}&=-[1+\lambda \beta /(\rho -\alpha )] \end{aligned}$$

Solving both equations yields:

$$\begin{aligned} c_{1}&=\beta [P-\overline{R}^{q}/(\rho -\alpha )](1/\overline{R} ^{q})^{\gamma }\\ c_{2}&=[1+\lambda \beta /(\rho -\alpha )]/\left( \rho +\lambda -\alpha \right) =\eta /(\rho -\alpha ) \end{aligned}$$

where $\eta =(\rho +\lambda \beta -\alpha )/(\rho +\lambda -\alpha )\le 1.$

The general solution then takes the form¹⁷:

$$\begin{aligned} V^{q}(R;\overline{R}^{q})=kR^{\theta }+\beta [P-\overline{R}^{q} /(\rho -\alpha )](R/\overline{R}^{q})^{\gamma }+\eta [R/(\rho -\alpha )] \end{aligned}$$

(A.2.3)

where k is a constant to be determined and $\theta$ is the negative root of the characteristic equation $\Omega (\theta )=(1/2)\sigma ^{2}\theta (\theta -1)+\alpha \theta -(\rho +\lambda )=0$.

At the critical threshold value, $g(\overline{R}^{q})$, optimality requires that the following value-matching and smooth-pasting conditions hold:

$$\begin{aligned}&V^{q}(R^{q};\overline{R}^{q})=P, \end{aligned}$$

(A.2.4)

$$\begin{aligned}&\left. \partial V^{q}(R;\overline{R}^{q})/\partial R\right| _{R=g(\overline{R}^{q})}=0 \end{aligned}$$

(A.2.5)

Solving the system [A.2.4–A.2.5] yields:

$$\begin{aligned} k=\{P-\beta [P-\overline{R}^{q}/(\rho -\alpha )](g(\overline{R} ^{q})/\overline{R}^{q})^{\gamma }+\eta [g(\overline{R}^{q})/(\rho -\alpha )]\}(1/g(\overline{R}^{q}))^{\theta } \end{aligned}$$

(A.2.6)

and

$$\begin{aligned}&\beta [(\theta -\gamma )/(\theta -1)][P-\overline{R}^{q}/(\rho -\alpha )](g(\overline{R}^{q})/\overline{R}^{q})^{\gamma }\nonumber \\&\quad +\eta [g(\overline{R}^{q})/(\rho -\alpha )]-[\theta /(\theta -1)]P=0 \end{aligned}$$

(A.2.7)

1.2.1 Naïve beliefs

Substituting $q=n$, $g(\overline{R}^{n})=R^{n}$ and $\overline{R}^{n}=R^{*}$ into Eqs. (A.2.6) and (A.2.7) yields:

$$\begin{aligned} k=\{P-\beta [P-R^{*}/(\rho -\alpha )](R^{n}/R^{*})^{\gamma } -\eta [R^{n}/(\rho -\alpha )]\}(1/R^{n})^{\theta } \end{aligned}$$

and

$$\begin{aligned}&\beta [(\theta -\gamma )/(\theta -1)][P-R^{*}/(\rho -\alpha )](R^{n} /R^{*})^{\gamma }\\&\quad +\eta [R^{n}/(\rho -\alpha )]-[\theta /(\theta -1)]P=0 \end{aligned}$$

Plugging k into Eq. (A.2.3) gives Eq. (8).

1.2.2 Sophisticated beliefs

Substituting $q=s$ and $\overline{R}^{n}=\widetilde{R}$ into Eqs. (A.2.6) and (A.2.7) yields:

$$\begin{aligned} k=\{P-\beta [P-\widetilde{R}/(\rho -\alpha )](g(\widetilde{R} )/\widetilde{R})^{\gamma }-\eta [g(\widetilde{R})/(\rho -\alpha )]\}(1/g(\widetilde{R}))^{\theta }, \end{aligned}$$

and

$$\begin{aligned}&\beta [(\theta -\gamma )/(\theta -1)][P-\widetilde{R}/(\rho -\alpha )](g(\widetilde{R})/\widetilde{R})^{\gamma }\\&\quad +\eta [g(\widetilde{R} )/(\rho -\alpha )]-[\theta /(\theta -1)]P=0. \end{aligned}$$

Then, imposing the stationarity condition $g(\widetilde{R})=\widetilde{R}=R^{s}$ yields:

$$\begin{aligned} k=\{P-\beta [P-R^{s}/(\rho -\alpha )]-\eta [R^{s}/(\rho -\alpha )]\}(1/R^{s})^{\theta }, \end{aligned}$$

and

$$\begin{aligned} R^{s}=\frac{\theta +\beta (\gamma -\theta )}{\eta (\theta -1)+\beta (\gamma -\theta )}P(\rho -\alpha ). \end{aligned}$$

Plugging k into Eq. (A.2.3) gives Eq. (11).

1.3 Timing thresholds: properties

Define the function $Z(x)=\beta [(\theta -\gamma )/(\theta -1)][P-R^{*}/(\rho -\alpha )](x/R^{*})^{\gamma }+\eta [x/(\rho -\alpha )]$ $-[\theta /(\theta -1)]P$. Note that Z(x) is convex in x,$\ Z(R^{n})=0$ and $Z(R^{*})<0$. Optimality requires that $Z^{\text { }\prime }(R^{n})>0$. Hence, it follows that $R^{*}<R^{n}$. Then, as $\theta >\gamma$, it can easily be shown that $[\gamma /(\gamma -1)]<[\theta +\beta (\gamma -\theta )]/[\eta (\theta -1)+\beta (\gamma -\theta )]<[\theta /(\theta -1)]$. This in turn implies that $R^{*}<R^{s}<R^{**}.$

Finally, define the function $Z(x;\widetilde{x})=\beta [(\theta -\gamma )/(\theta -1)][P-\widetilde{x}/(\rho -\alpha )](x/\widetilde{x})^{\gamma }+\eta [x/(\rho -\alpha )]$ $-[\theta /(\theta -1)]P$. Denote by $g(\widetilde{x})\$the solution of the equation $Z(g(\widetilde{x});\widetilde{x})=0.$ Totally differentiating with respect to $\widetilde{x}$ yields:

$$\begin{aligned}&\beta [(\theta -\gamma )/(\theta -1)]\{-1/(\rho -\alpha )\\&\quad +[P-\widetilde{x}/(\rho -\alpha )][(1/g(\widetilde{x}))\partial g(\widetilde{x})/\partial \widetilde{x}-(\gamma /\widetilde{x})]\}(g(\widetilde{x})/\widetilde{x})^{\gamma }\\&\quad +\eta (\partial g(\widetilde{x})/\partial \widetilde{x})/(\rho -\alpha )=0 \end{aligned}$$

Rearranging gives:

$$\begin{aligned}&\partial g(\widetilde{x})/\partial \widetilde{x}\\&\quad =\frac{\beta [(\theta -\gamma )/(\theta -1)](g(\widetilde{x})/\widetilde{x})^{\gamma } \widetilde{x}(\gamma -1)}{\beta [(\theta -\gamma )/(\theta -1)][P(\rho -\alpha )-\widetilde{x}](g(\widetilde{x})/\widetilde{x})^{\gamma -1} +\eta \widetilde{x}}\left( \frac{R^{*}}{\widetilde{x}}-1\right)<0\text {, for }R^{*}<\widetilde{x} \end{aligned}$$

Note that:

$$\begin{aligned} Z(R^{n};R^{*})=Z(R^{s};R^{s})=0 \end{aligned}$$

Hence, as $R^{*}<R^{s}$, it follows that $R^{n}<R^{s}$.