Information Exchange and Transnational Environmental Problems

Abstract

This paper analyzes information exchange in a model of transnational pollution control in which countries use private information in independently determining their domestic environmental policies. We show that countries may not always have an incentive to exchange their private information. However, for a sufficiently high degree of predictability of domestic environmental policy processes, the expected welfare from sharing information is greater than the expected welfare from keeping it private. The minimum degree of policy predictability for which information sharing occurs increases with the level of environmental risk. Intuitively, information exchange can help mitigate the perception of global uncertainty (both political and scientific) that surrounds transnational environmental problems and potentially improve welfare if policymaking processes are sufficiently aligned with evidence-based approaches (predictable).

Appendices

Appendix A: Expression of Probabilities in Eqs. (17) and (18)

Using the Bayes’ rule, The conditional probability of ${\upbeta }_{L}$ given the private signal $s_{L}^{1}$ is computed as:

$$\begin{aligned} P({\upbeta }_{L}|s_{L}^{1})= & {} \frac{P({\upbeta }_{L},s_{L}^{1})}{P({\upbeta }_{L})},\nonumber \\= & {} \frac{P(s_{L}^{1}|{\upbeta }_{L})P({\upbeta }_{L})}{P(s_{L}^{1}|{\upbeta }_{L})P ({\upbeta }_{L})+P(s_{L}^{1}|{\upbeta }_{H})P({\upbeta }_{H})}, \nonumber \\= & {} \frac{P(s_{L}^{1}|{\upbeta }_{L})P({\upbeta }_{L})}{P(s_{L}^{1}|{\upbeta }_{L})P ({\upbeta }_{L})+\left[ 1-P(s_{H}^{1}|{\upbeta }_{H})\right] P({\upbeta }_{H})}, \nonumber \\= & {} \frac{\frac{1}{2}\upsigma }{\frac{1}{2}\upsigma +\frac{1}{2}(1-\upsigma )},\nonumber \\= & {} \frac{\frac{1}{2}\upsigma }{\frac{1}{2}},\nonumber \\= & {} \upsigma . \end{aligned}$$

(35)

The conditional probability of ${\upbeta }_{H}$ given the private signal $s_{L}^{1}$ is computed by using the complement rule for conditional probabilities:

$$\begin{aligned} P({\upbeta }_{H}|s_{L}^{1})=1-P({\upbeta }_{L}|s_{L}^{1})=1-\upsigma . \end{aligned}$$

(36)

Using the multiplication rule (or chain rule) for conditional probabilities, the joint probability $P({\upbeta }_{L},s_{L}^{2}|s_{L}^{1})$ is computed as follows:

$$\begin{aligned} P({\upbeta }_{L},s_{L}^{2}|s_{L}^{1})= & {} \frac{P({\upbeta }_{L},s_{L}^{2},s_{L}^{1})}{P(s_{L}^{1})}\end{aligned}$$

(37)

$$\begin{aligned}= & {} \frac{P(s_{L}^{1},s_{L}^{2},{\upbeta }_{L})}{P(s_{L}^{1})},\nonumber \\= & {} \frac{P(s_{L}^{1},s_{L}^{2}|{\upbeta }_{L})P({\upbeta }_{L})}{P(s_{L}^{1})}\nonumber \\= & {} \frac{P(s_{L}^{1})|{\upbeta }_{L})P({\upbeta }_{L})P(s_{L}^{2}|{\upbeta }_{L})}{P(s_{L}^{1})}\nonumber \\&\qquad (since\,s_{L}^{1}\,and\,s_{L}^{2}\,are\,conditionally\,independent\,given\, {\upbeta }_{L}),\nonumber \\= & {} \frac{P(s_{L}^{1},{\upbeta }_{L})P(s_{L}^{2}|{\upbeta }_{L})}{P(s_{L}^{1})},\nonumber \\= & {} \frac{P({\upbeta }_{L}|s_{L}^{1})P(s_{L}^{1})P(s_{L}^{2}|{\upbeta }_{L})}{P(s_{L}^{1})},\nonumber \\= & {} P({\upbeta }_{L}|s_{L}^{1})P(s_{L}^{2}|{\upbeta }_{L}),\nonumber \\= & {} \upsigma \upsigma ,\nonumber \\= & {} \upsigma ^{2}. \end{aligned}$$

(38)

The computation of the following conditional probabilities follows the same reasoning.

$$\begin{aligned} \left\{ \begin{array}{ll} P({\upbeta }_{L},s_{H}^{2}|s_{L}^{L})=\upsigma (1-\upsigma ),\\ P({\upbeta }_{H},s_{L}^{2}|s_{L}^{1})=(1-\upsigma )^{2},\\ P({\upbeta }_{H},s_{H}^{2}|s_{L}^{1})=\upsigma (1-\upsigma ),\\ P({\upbeta }_{L},s_{L}^{2}|s_{H}^{1})=(1-\upsigma )\upsigma ,\\ P({\upbeta }_{L},s_{H}^{2}|s_{H}^{1})=(1-\upsigma )^{2},\\ P({\upbeta }_{H},s_{L}^{2}|s_{H}^{1})=\upsigma (1-\upsigma ),\\ P({\upbeta }_{H},s_{H}^{2}|s_{H}^{1})=\upsigma ^{2}.\\ \end{array} \right. \end{aligned}$$

(39)

Note also that $P({\upbeta }_{L}|s_{H})=1-\upsigma$ and $P({\upbeta }_{H}|s_{H})=\upsigma$

Appendix B: Expression of Probabilities in the First Order Eq. (29)

Using the Bayes’ rule, the conditional probability $P({\upbeta }_{L}|s_{L},s_{L})$ is computed as

$$\begin{aligned} P({\upbeta }_{L}|s_{L},s_{L})= & {} \frac{ P(s_{L},s_{L}|{\upbeta }_{L})P({\upbeta }_{L})}{ P(s_{L},s_{L})}, \end{aligned}$$

(40)

$$\begin{aligned}= & {} \frac{ P(s_{L}|{\upbeta }_{L})P(s_{L}|{\upbeta }_{L})P ({\upbeta }_{L})}{ P(s_{L},s_{L}|{\upbeta }_{L})P({\upbeta }_{L})+P(s_{L},s_{L}| {\upbeta }_{H})P({\upbeta }_{H})} , \end{aligned}$$

(41)

$$\begin{aligned}= & {} \frac{ P(s_{L}|{\upbeta }_{L})^{2}P({\upbeta }_{L})}{ P(s_{L}|{\upbeta }_{L})^{2} P({\upbeta }_{L})+P(s_{L}|{\upbeta }_{H})^{2}P({\upbeta }_{H})}, \end{aligned}$$

(42)

$$\begin{aligned}= & {} \frac{\frac{1}{2}\upsigma ^{2}}{\frac{1}{2}\upsigma ^{2}+\frac{1}{2}(1-\upsigma )^{2}}, \end{aligned}$$

(43)

$$\begin{aligned}= & {} \frac{\upsigma ^{2}}{\upsigma ^{2}+(1-\upsigma )^{2}} , \end{aligned}$$

(44)

A similar reasoning is used to compute the following probabilities:

$$\begin{aligned} \left\{ \begin{array}{ll} P({\upbeta }_{L}|s_{L},s_{H})=\frac{1}{2},\\ P({\upbeta }_{L}|s_{H},s_{L})=\frac{1}{2},\\ P({\upbeta }_{L}|s_{H},s_{H})=\frac{(1-\upsigma )^{2}}{\upsigma ^{2}+(1-\upsigma )^{2}},\\ P({\upbeta }_{H}|s_{L},s_{H})=\frac{1}{2},\\ P({\upbeta }_{H}|s_{L},s_{L})=\frac{(1-\upsigma )^{2}}{\upsigma ^{2}+(1-\upsigma )^{2}},\\ P({\upbeta }_{H}|s_{H},s_{L})=\frac{1}{2},\\ P({\upbeta }_{H}|s_{H},s_{H})=\frac{\upsigma ^{2}}{\upsigma ^{2}+(1-\upsigma )^{2}}.\\ \end{array} \right. \end{aligned}$$

(45)

Appendix C: Expression of Probabilities in the Expected Welfare Eqs. (23) and (32)

$$\begin{aligned} P({\upbeta }_{L},s_{L},s_{L})=P({\upbeta }_{L},s_{L}|s_{L})P(s_{L})=\frac{1}{2}\upsigma ^{2}. \end{aligned}$$

(46)

Following the same reasoning, the following probabilities are computed:

$$\begin{aligned} \left\{ \begin{array}{ll} P({\upbeta }_{L},s_{H},s_{L})=\frac{1}{2}\upsigma (1-\upsigma ),\\ P({\upbeta }_{H},s_{L},s_{L})=\frac{1}{2}(1-\upsigma )^{2},\\ P({\upbeta }_{H},s_{H},s_{L})=\frac{1}{2}\upsigma (1-\upsigma ),\\ P({\upbeta }_{L},s_{L},s_{H})=\frac{1}{2}(1-\upsigma )\upsigma ,\\ P({\upbeta }_{L},s_{H},s_{H})=\frac{1}{2}(1-\upsigma )^{2},\\ P({\upbeta }_{H},s_{L},s_{H})=\frac{1}{2}\upsigma (1-\upsigma ),\\ P({\upbeta }_{H},s_{H},s_{H})=\frac{1}{2}\upsigma ^{2}. \end{array} \right. \end{aligned}$$

(47)

Appendix D: The Value of Information Sharing

The value of information sharing is given by:

$$\begin{aligned}&E\left[ \widetilde{V}_{sharing}(e_{ij},\widetilde{\upbeta })\right] -E\left[ \widetilde{V}_{no\,sharing}(e_{i})\right] \nonumber \\&\quad =\frac{\upsigma ^{2}}{2}\left[ e_{LL}+e_{HH}-\frac{(1+\upalpha )^2}{2} ({\upbeta }_{L} e_{LL}^{2}+{\upbeta }_{H} e_{HH}^2)\right] \nonumber \\&\qquad + \frac{(1-\upsigma )^{2}}{2}\left[ e_{LL}+e_{HH}-\frac{(1+\upalpha )^2}{2} ({\upbeta }_{L} e_{HH}^{2}+{\upbeta }_{H} e_{LL}^{2})\right] \nonumber \\&\qquad + \upsigma (1-\upsigma )\left[ 2e_{LH}-\frac{(1+\upalpha )^2}{2}({\upbeta }_{L} +{\upbeta }_{H}) e_{LH}^{2})\right] -\log \left( \frac{1}{\upsigma }\right) \nonumber \\&\qquad - \frac{(1-\upalpha ^2)}{4}\left[ \upsigma ({\upbeta }_{L}e^2_{L}+{\upbeta }_{H}e^2_{H}) + (1-\upsigma )( {\upbeta }_{L}e^2_{H} +{\upbeta }_{H}e^2_{L}) \right] -\log \left( \frac{1}{\upsigma }\right) . \end{aligned}$$

(48)

where $e_{LL},\,e_{LH}$ and $e_{HH}$ are given by Eq. (30) while $e_{L}$ and $e_{H}$ are given by Eqs. (20)–(21).

Appendix E: Graph of the Value of Information Sharing, with $\upalpha =1$

As shown in Fig. 3 above, the value of information sharing can take either a positive or a negative sign depending on the values taken by the level of riskiness of the environmental damage and the degree of predictability of the environmental policy. This suggests that sharing information or keeping it private are plausible outcomes that may occur in analyzing information exchange in transnational environmental problems.

Fig. 3

The value of information sharing drawn on the $\left( {\upbeta }_{H}/ \upbeta _{H} \,,\,\upsigma \right)$ plane, with $\upalpha =1$ and ${\upbeta }_{L}=0.3$