Attitude Towards Risk and Production Decision: an Empirical Analysis on French Private Forest Owners

Abstract

This paper deals with the forest owner’s attitude towards risk and the harvesting decision in several ways. First, we propose to characterize and quantify the forest owner’s attitude towards risk. Second, we analyze the determinants of the forest owner’s risk attitude. Finally, we determine the impact of the forest owner’s risk attitude on the harvesting decision. The French forest owner’s risk attitude is tackled by implementing a questionnaire, including a context-free measure borrowed from experimental economics. The determinants of the forest owner’s risk attitude and harvesting decision are estimated through a recursive bivariate ordered probit model. We show that French forest owners are characterized by a relative risk aversion coefficient close to 1 with a DARA assumption. In addition, we find that the forest owner’s risk aversion is influenced positively and significantly by the level of risk exposure, the geographical location of the forest and the fact to be a forester, and negatively by the income. Finally, we obtain that the forest owner’s risk aversion has a positive and significant impact on the harvesting decision.

Appendix: Maximum likelihood estimation

Following Sajaia [50], we show that the probability that y _1i = j and y _2i = 1 or 0:

$$\begin{array}{@{}rcl@{}} \begin{array}{ll} Pr(y_{1i}\,=\,j, y_{2i}\,=\,0) &\,=\, Pr(y_{1i}^{*} \!<\!c_{j}, y_{2i}^{*} \!<\!0) \!\\&-\! Pr(y_{1i}^{*} \!<\!c_{j-1}, y_{2i}^{*} \!<\!0) \end{array} \end{array} $$

and

$$ \begin{array}{lll} Pr(y_{1i}\,=\,j, y_{2i}\,=\,1) &=& Pr(y_{1i}^{*} <c_{j}) - Pr(y_{1i}^{*} <c_{j-1}) \\&&- Pr(y_{1i}^{*} <c_{j}, y_{2i}^{*} < 0)\\ & &+ Pr(y_{1i}^{*} <c_{j-1}, y_{2i}^{*} < 0) \end{array} $$

The system of Eq. 1 can be estimated by the maximum likelihood method. Indeed, we assume that (𝜖 _1i ,𝜖 _2i ) ∼ N(0, Ω) with ${\Omega }= \left (\begin {array}{cc} 1 & \rho \\ \rho & 1 \end {array} \right )$; thus, we get:

$$\begin{array}{@{}rcl@{}} \begin{array}{lll} Pr(y_{1i}=j, y_{2i}=0) &=& Pr(\epsilon_{1i} <c_{j} - X_{1i}^{\prime} \beta_{1}, \gamma \epsilon_{1i} + \epsilon_{2i} \\&&\qquad\quad- \gamma X_{1i}^{\prime} \beta_{1} - X_{2i}^{\prime} \beta_{2} )\\ & & Pr(\epsilon_{1i} \!\!<c_{j-1} \,-\, X_{1i}^{\prime} \beta_{1}, \gamma \epsilon_{1i} \,+\, \epsilon_{2i} \\&&\qquad\quad- \gamma X_{1i}^{\prime} \beta_{1} - X_{2i}^{\prime} \beta_{2} ) \end{array} \end{array} $$

Given that $\left (\begin {array}{cc} \!1\! & \!0\! \\ \!\gamma \! & \!1\! \end {array} \right ) \left (\begin {array}{c} \!\epsilon _{1i\!} \\ \!\epsilon _{2i}\! \end {array} \right )\!\!\! \sim \!\! N\! \left (0, \left [ \begin {array}{cc} 1 & \gamma \,+\, \rho \\ \gamma \,+\, \rho & \gamma ^{2} \,+\, 2 \gamma \rho \,+\, 1 \end {array} \right ] \right )$ we have:

$$ \begin{array}{lll} Pr(y_{1i}\,=\,j, y_{2i}\,=\,0) &\,=\,& {\Phi}_{2}(c_{j} \,-\, X_{1i}^{\prime} \beta_{1}, (\!- \gamma X_{1i}^{\prime} \beta_{1} \,-\, X_{2i}^{\prime} \beta_{2})\zeta, \tilde{\rho} )\\ & & \!\!\!- {\Phi}_{2}(c_{j-1} \,-\, X_{1i}^{\prime} \beta_{1}, (\!- \gamma X_{1i}^{\prime} \beta_{1} \,-\, X_{2i}^{\prime} \beta_{2})\zeta, \tilde{\rho}) \end{array} $$

Similarly, we obtain:

$$ \begin{array}{lll} Pr(y_{1i}\,=\,j, y_{2i}\,=\,1) &\,=\,& {\Phi}(c_{j} \,-\, X_{1i}^{\prime} \beta_{1}) \,-\, {\Phi}(c_{j-1} \,-\, X_{1i}^{\prime} \beta_{1}) \\&&\!\!\!- {\Phi}_{2}(c_{j} \,-\, X_{1i}^{\prime} \beta_{1}, (- \gamma X_{1i}^{\prime} \beta_{1} \,-\, X_{2i}^{\prime} \beta_{2})\zeta, \tilde{\rho} )\\ & &\! \!\!+ {\Phi}_{2}(c_{j-1} \,-\, X_{1i}^{\prime} \beta_{1}, (- \gamma X_{1i}^{\prime} \beta_{1} \,-\, X_{2i}^{\prime} \beta_{2})\zeta, \tilde{\rho}) \end{array} $$

with $\tilde {\rho } = \gamma + \rho$, ζ = (γ ² + 2γρ + 1)^−1/2 and Φ and Φ₂ the univariate and bivariate standard cumulative distribution functions, respectively. If j = 1, then the probabilities above shrink to:

$$ \begin{array}{lll} Pr(y_{1i}\,=\,j, y_{2i}\,=\,0) &\,=\,& {\Phi}_{2}(c_{j} \,-\, X_{1i}^{\prime} \beta_{1}, (- \gamma X_{1i}^{\prime} \beta_{1} \,-\, X_{2i}^{\prime} \beta_{2})\zeta, \tilde{\rho} ) \\ Pr(y_{1i}\,=\,j, y_{2i}\,=\,1) &\,=\,& {\Phi}(c_{j} \,-\, X_{1i}^{\prime} \beta_{1}) \,-\, {\Phi}_{2}(c_{j} \,-\, X_{1i}^{\prime} \beta_{1}, (- \gamma X_{1i}^{\prime} \beta_{1} - X_{2i}^{\prime} \beta_{2})\zeta, \tilde{\rho} ) \end{array} $$

If j = J, then the probabilities above shrink to:

$$ \begin{array}{lll} Pr(y_{1i}=J, y_{2i}=0) &=& {\Phi}((- \gamma X_{1i}^{\prime} \beta_{1} - X_{2i}^{\prime} \beta_{2})\zeta) \\&&- {\Phi}_{2}(c_{j-1} - X_{1i}^{\prime} \beta_{1}, (- \gamma X_{1i}^{\prime} \beta_{1} - X_{2i}^{\prime} \beta_{2})\zeta, \tilde{\rho} ) \\ Pr(y_{1i}=J, y_{2i}=1) &=& 1- {\Phi}(c_{j-1} - X_{1i}^{\prime} \beta_{1}) \\&&- {\Phi}(- \gamma X_{1i}^{\prime} \beta_{1} - X_{2i}^{\prime} \beta_{2})\zeta) +\\ & & {\Phi}_{2}(c_{j-1} - X_{1i}^{\prime} \beta_{1}, (- \gamma X_{1i}^{\prime} \beta_{1} - X_{2i}^{\prime} \beta_{2})\zeta, \tilde{\rho} ) \end{array} $$

If we assume that the observations are independent, the log-likelihood function can be written as follows:

$$ \ln\ L= \sum\limits_{i=1}^{N} \sum\limits_{k=1}^{K} \sum\limits_{j=1}^{J} I (y_{1i}=j, y_{2i}=k)\ \ln Pr(y_{1i}=j, y_{2i}=k) $$

The maximum weighted likelihood estimator can be written as:

$$ \ln\ L=\sum\limits_{i=1}^{N} \sum\limits_{k=1}^{K}\sum\limits_{j=1}^{J} w_{i} \ I (y_{1i}=j, y_{2i}=k)\ \ln Pr(y_{1i}=j, y_{2i}=k) $$

where w is the weighting vector.