Partial Interests in Recreational Property

Abstract

Changing demographics, growing real incomes, and friendly tax laws underlie the continuing growth in demand for recreational real estate in the US. The market for recreational property has undergone a major transformation over the past decades, with the refinement and deepening of markets for partial property ownership vehicles. This paper represents the first to analyze the factors underlying the demand for partial ownership interests. It develops a theory of partial ownership demand that focuses on the roles of familiarity and location-specific human capital in mediating the consumption uncertainty associated with particular recreation locations. Using private data from a survey of partial ownership participants, the empirical analysis yields results consistent with the theory: factors associated with greater site-specific recreation price, like distance between the primary residence and the recreation site and frequency of visits per week, reduce the share of ownership demanded, while factors associated with lower consumption risk tend to increase the share of ownership demanded.

Appendix

Consumption Risk Comparative Static Properties

PROPOSITION: Greater consumption risk decreases the consumer’s maximum expected utility.

Proof. The consumer’s problem under consumption risk is

$$ {\mathop {\max }\limits_R }\,E{\left[ {U{\left( {\omega _{{\text{o}}} + \omega - qR,\theta R} \right)}} \right]} $$

Denote the consumption risk distribution parameter for mean preserving spread of the distribution as σ. The solution to the consumer’s problem is $R{\left( {q,\omega _{{\text{o}}} + \omega ,\sigma } \right)}$ implicitly defined by Eq. 5, restated below for convenience as

$$ E{\left[ {\theta U_{R} } \right]} - qE{\left[ {U_{X} } \right]} = 0 $$

(A.1)

Substituting $R{\left( {q,\omega _{{\text{o}}} + \omega ;\sigma } \right)}$ into the expected utility function yields the indirect expected utility function which depicts the consumer’s maximum expected utility as a function of the exogenous parameters in the model:

$$ V{\left( {q,\omega _{{\text{o}}} + \omega ;\sigma } \right)} = E{\left[ {U{\left( {\omega _{{\text{o}}} + \omega - qR{\left( {q,\omega _{{\text{o}}} + \omega ;\sigma } \right)},\theta R{\left( {q,\omega _{{\text{o}}} + \omega ;\sigma } \right)}} \right)}} \right]} $$

Differentiating with respect to σ (where dσ > 0 indicates an increase in mean preserving spread)

$$ \frac{{{\text{d}}V}} {{{\text{d}}\sigma }} = E{\left[ {{\left( {\theta - 1} \right)}RU_{R} } \right]} = {\text{COV}}{\left[ {RU_{R} ,\theta - 1} \right]} < 0 $$

(A.2)

where the covariance sign follows from $U_{{RR}} < 0$ under risk aversion and the properties of similar/dissimilar orderings.

PROPOSITION: Greater consumption risk decreases the consumer’s demand for recreation under risk non-inferiority.

Proof. Differentiate Eq. A.1 to find the effect of an increase in mean preserving spread as

$$ \frac{{{\text{d}}R * }} {{{\text{d}}\sigma }} = R\frac{{E{\left[ {{\left( {\theta - 1} \right)}U_{R} } \right]}}} {D} - R\frac{{E{\left[ {{\left( {\theta U_{{RR}} - qU_{{XR}} } \right)}{\left( {\theta - 1} \right)}} \right]}}} {D} $$

(A.3)

using $\raise0.7ex\hbox{${d^{2} E{\left[ U \right]}}$} \!\mathord{\left/ {\vphantom {{d^{2} E{\left[ U \right]}} {dR^{2} = D < 0}}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{${dR^{2} = D < 0}$}$ from the SOC for a maximum. The series of papers Dardanoni (1988), Davis (1989), and Turnbull (1995, 1998) develop the general principles underlying the following method of decomposing risk effects for this type of model. The first term on the right hand side of Eq. A.3 is unambiguously negative by Eq. A.2. This term can be identified as the effect of an increase in the marginal riskiness of recreation consumption while holding the level of risk unchanged. The second term on the right hand side of Eq. A.3 can be interpreted as an additive risk effect, the consumer’s response to an increase in the level of uncertainty at each level of recreation consumption while holding the marginal effect of consumption on risk constant. Unlike the first term, this second term is ambiguous a priori, taking the sign of the numerator. Planned recreation demand exhibits risk normality (risk inferiority) when the demand decreases (increases) with greater additive risk. Risk normality is the uncertainty analogue to income normality in standard demand theory (Turnbull, 1995).

To better understand the additive risk effect on recreation demand, note that the numerator of the additive risk effect is

$$ E{\left[ {{\left( {\theta U_{{RR}} - qU_{{XR}} } \right)}{\left( {\theta - 1} \right)}} \right]} = COV{\left[ {{\left( {\theta U_{{RR}} - qU_{{XR}} } \right)},{\left( {\theta - 1} \right)}} \right]} $$

The covariance term takes the sign of

$$ \frac{{\partial {\left( {\theta U_{{RR}} - qU_{{XR}} } \right)}}} {{\partial \theta }} = U_{{RR}} + R{\left[ {\theta U_{{RRR}} - qU_{{XRR}} } \right]} $$

(A.4)

This expression is negative (positive) when recreation demand is risk normal (inferior). To further interpret what this requires, use the index of relative risk aversion for the multiple argument expected utility function,

$$ \rho = - \frac{{RU_{{RR}} }} {{U_{R} }} $$

Differentiating with respect to recreation consumption yields

$$ \frac{{{\text{d}}\rho }} {{{\text{d}}R}} = {\left( { - \frac{1} {{U^{2}_{R} }}} \right)}{\left[ {U_{R} {\left( {U_{{RR}} + R{\left[ {\theta U_{{RRR}} - qU_{{XRR}} } \right]}} \right)} + U_{{RR}} {\left( {qU_{{XR}} - \theta U_{{RR}} } \right)}} \right]} $$

(A.5)

Note that $qU_{{XR}} - \theta U_{{RR}} > 0$ under the assumption that both goods are normal goods. Thus, risk non-inferiority or a non-positive Eq. A.4 implies that Eq. A.5 is strictly positive. This is the usual characterization of increasing relative risk aversion (IRRA) in the two argument expected utility function.

Recreation Activity Skill as Location-Specific Human Capital

This section analyzes the recreation activity skill effects within the flexible location version of the model. The underlying notion is that some activities are location specific, and therefore individuals who have acquired skills in certain activities will also tend to prefer sites with those activities ex post. Keeping this notion in mind, we focus on how flexible the consumer’s tastes for the different sites is—i.e., the mean-preserving-spread of the ${\text{MRS}}_{{1,2}} = \theta$ term. Let $E{\left( \theta \right)} = 1$ as before. If the variance in this MRS between recreation locations, ${\text{VAR}}{\left[ \theta \right]}$, is low then the consumer is pretty sure that he or she is going to want to recreate at site one than at site two. Assuming that our skill-specific recreation (e.g., golfing, skiing) is site-specific (i.e., no snow skiing at the beach and no water-skiing in the mountains), then a lower ${\text{VAR}}{\left[ \theta \right]}$ reflects more stable preference for snow skiing relative to other types of recreation. The skill analysis on recreation demand now follows the uncertainty analysis outlined earlier for the other model: use the parameter ó to indicate an increase in mean-preserving-spread. Implicitly differentiating the optimality condition yields

$$ \frac{{\partial R}} {{\partial \sigma }} = \frac{{{\int\limits_{q/{\left( {q + f} \right)}}^{\theta _{u} } {{\left( {\theta - 1} \right)}{\left[ {qU_{{XR}} {\left( {.,\theta R} \right)} - \theta U_{{RR}} {\left( {.,\theta R} \right)}} \right]}{\text{d}}F{\left( \theta \right)}} }}} {D} - \frac{{{\int\limits_{q/{\left( {q + f} \right)}}^{\theta _{u} } {{\left( {\theta - 1} \right)}U_{R} {\left( {.,\theta R} \right)}{\text{d}}F{\left( \theta \right)}} }}} {D} $$

(A.6)

The first term reflects how the consumer responds to greater recreation consumption risk per se; this is the additive risk effect identified in the previous section.

As above, the second term in Eq. A.6 captures how the consumer responds to an increase in the marginal risk increase from additional recreation, holding total riskiness constant. The integral

$$ {\int\limits_{\theta _{I} }^{\theta _{u} } {{\left( {\theta - 1} \right)}U_{R} {\left( {.,\theta R} \right)}{\text{d}}f{\left( \theta \right)}} } $$

takes the sign of U _RR <0. The second term in the above comparative static result therefore unambiguously negative in this model when the cost of exercising flexibility is sufficiently high, f >> 0. When this second term in Eq. A.6 capturing the marginal risk effect is negative, risk non-inferiority is sufficient for the consumer to reduce partial interest demand in response to greater variation in recreation location preference, or

$$ \frac{{\partial R}} {{\partial \sigma }} < 0 $$

This result also holds for partial interests that have modest flexibility, i.e., f is large but not large enough to completely foreclose the alternative site as a viable option in all realized states.