Taking One for the Team: Is Collective Action More Responsive to Ecological Change?

Abstract

Debate persists around the timing of risk reduction strategies for large-scale ecological change for three reasons: risks are difficult to predict, they involve irreversibilities, and they impact multiple jurisdictions. The combination of these three factors creates a general class of filterable spatial-dynamic externalities (FSDE) in which one person’s risk reduction investments reduce or filter undesirable events experienced by others and underestimates the option value placed on being able to respond to new information about the consequences of ecological change. By focusing on the optimal intervention decision, we illustrate how and when the opposing forces created by an FSDE will lead to a divergence in private and collective risk reduction strategies. We use bioinvasions as our motivating example. The bioinvasion first hits one jurisdiction, and that jurisdiction’s risk reduction investment reduces the risk faced by all other jurisdiction. We find that efforts to internalize the full benefits of risk reduction investments may have unintended consequences on the responsiveness of environmental policy. There is a social welfare gain from asking the first jurisdiction to delay a risk reduction investment to internalize the option values of all at-risk jurisdictions.

Appendix

1.1 Continuous Mitigation Investment with Immediate Adoption

Assume agent 1 invests in mitigation immediately but may choose the magnitude of the investment $r^{p}$. The optimal magnitude of investment by agent 1 is:

$$\begin{aligned} \mathop {\min }\limits _{r^{p}} \left\{ {W^{I}\left( {D_1^0 ,r^{p}} \right) } \right\} =\frac{D_1^0 }{\left( {\rho -\alpha ^{p}} \right) }-\frac{\alpha ^{p}{\bar{D}} _1 }{\rho \left( {\rho -\alpha ^{p}} \right) {\bar{D}} _1^{\varepsilon ^{p}}}D_1^{0{^{\varepsilon ^{p}}}}+C_1 \end{aligned}$$

(14)

The first-order condition for this problem implicitly defines the private optimum mitigation investment $r^{{p}^{{\prime }}}\left( {D_1^0 } \right)$:

$$\begin{aligned} \frac{\theta D_1^0 }{\left( {\rho -\alpha ^{{p}^{\prime }}} \right) ^{2}}-\frac{\theta {\bar{D}} _1 }{\left( {\rho -\alpha ^{{p}^{\prime }}} \right) ^{2}}\left( {\frac{D_1^0 }{{\bar{D}} _1 }} \right) ^{\varepsilon ^{{p}^{\prime }}}\left[ {1-\left( {\ln \frac{D_1^0 }{{\bar{D}} _1 }} \right) {\Omega }} \right] =\frac{2C_1^{\prime }}{{r^{0}-r^{{p}^{\prime }}}} \end{aligned}$$

(15)

where ${\Omega }=\frac{\alpha ^{{p}^{\prime }}\left( {\rho -\alpha ^{{p}^{\prime }}} \right) }{\sigma ^{2}\rho }\left[ 1-\frac{\frac{\alpha ^{{p}^{\prime }}}{\sigma ^{2}}-\frac{1}{2}}{\sqrt{\left( {\frac{\alpha ^{{p}^{\prime }}}{\sigma ^{2}}}-\frac{1}{2} \right) ^{2}+\frac{2\rho }{\sigma ^{2}}}}\right] >0$ and $C_1^{\prime } =c{\bar{A}} _1 \left( {r^{0}-r^{{p}^{\prime }}} \right) ^{2}$ . The left (right) hand side is the marginal cost (benefit) of choosing less mitigation or higher $r^{p}$. The size of agent 1’s spatial jurisdiction influences the amount of mitigation through the second term on the left-hand side of (15).

The optimal amount of mitigation in jurisdiction 1 by the collective is:

$$\begin{aligned} \mathop {\min }\limits _{r^{p}} \left\{ {W^{I}\left( {D_1^0 ,r^{p}} \right) } \right\}= & {} \frac{D_1^0 }{\left( {\rho -\alpha ^{p}} \right) }-\frac{\alpha ^{p}{\bar{D}} _1 }{\rho \left( {\rho -\alpha ^{p}} \right) {\bar{D}} _1 ^{\varepsilon ^{p}}}D_1^{0 ^{\varepsilon ^{p}}}\nonumber \\&+\, C_1 +\mathop \sum \limits _{j=2}^N W_j e^{-\rho \left( {\frac{1}{\alpha ^{p}-\frac{1}{2}\sigma ^{2}}\ln \left( {\frac{{\bar{D}} _1 }{D_1^0 }} \right) +\mathop \sum \nolimits _{n=2}^{j-1} {\bar{t}} _n } \right) } \end{aligned}$$

(16)

The first-order condition implicitly defines the social optimum amount of mitigation $r^{{p}^{*}}\left( {D_1^0 } \right)$:

$$\begin{aligned}&\frac{\theta D_1^0 }{\left( {\rho -\alpha ^{{p}^{*}}} \right) ^{2}}-\frac{\theta {\bar{D}} _1 }{\left( {\rho -\alpha ^{{p}^{*}}} \right) ^{2}}\left( {\frac{D_1^0 }{{\bar{D}} _1 }} \right) ^{\varepsilon ^{{p}^{*}}}\left[ {1-\left( {\ln \frac{D_1^0 }{{\bar{D}} _1 }} \right) {\Omega }} \right] \nonumber \\&\quad +\frac{\mathop \sum \nolimits _{j=2}^N W_j e^{-\rho \left( {\mathop \sum \nolimits _{n=2}^{j-1} {\bar{t}} _n } \right) }\rho \theta }{\left( {\alpha ^{{p}^{*}}-\frac{1}{2}\sigma ^{2}} \right) ^{2}}\ln \left( {\frac{{\bar{D}} _1 }{D_1^0 }} \right) e^{-\rho \left( {\frac{1}{\alpha ^{{p}^{*}}-\frac{1}{2}\sigma ^{2}}\ln \left( {\frac{{\bar{D}} _1 }{D_1^0 }} \right) } \right) }=\frac{2C_1^{*} }{r^{0}-r^{{p}^{*}}} \end{aligned}$$

(17)

Once again the left (right) hand side is the marginal cost (benefit) of less mitigation or higher $r^{p}$. Compared to (15), internalizing the FSDE leads to an additional cost of less mitigation. This is arises because less mitigation allows the invasion to spread faster after the mitigation investment decreasing the time until agent 2’s damage is incurred.

It is straightforward to show $r^{{p}^{\prime }}>r^{{p}^{*}}$. Equation (15) can be rewritten as:

$$\begin{aligned} \theta D_1^0 =\frac{2C_1^{\prime } \left( {\rho -\alpha ^{{p}^{\prime }}} \right) ^{2}}{r^{0}-r^{{p}^{\prime }}}+\theta {\bar{D}} _1 \left( {\frac{D_1^0 }{{\bar{D}} _1 }} \right) ^{\varepsilon ^{{p}^{\prime }}}\left[ {1-\left( {\ln \frac{D_1^0 }{{\bar{D}} _1 }} \right) \Omega '} \right] \end{aligned}$$

(18)

Equation (17) can be rewritten as:

$$\begin{aligned} \theta D_1^0= & {} \frac{2C_1^{*} \left( {\rho -\alpha ^{{p}^{*}}} \right) ^{2}}{r^{0}-r^{{p}^{*}}}+\theta {\bar{D}} _1 \left( {\frac{D_1^0 }{{\bar{D}} _1 }} \right) ^{\varepsilon ^{{p}^{*}}}\left[ {1-\left( {\ln \frac{D_1^0 }{{\bar{D}} _1 }} \right) {\Omega }^{{*}}} \right] \nonumber \\&-\frac{\left( {\rho -\alpha ^{{p}^{*}}} \right) ^{2}\mathop \sum \nolimits _{j=2}^N W_j e^{-\rho \left( {\mathop \sum \nolimits _{n=2}^{j-1} {\bar{t}} _n } \right) }\rho \theta }{\left( {\alpha ^{{p}^{*}}-\frac{1}{2}\sigma ^{2}} \right) ^{2}}\ln \left( {\frac{{\bar{D}} _1 }{D_1^0 }} \right) e^{-\rho \left( {\frac{1}{\alpha ^{{p}^{*}}-\frac{1}{2}\sigma ^{2}}\ln \left( {\frac{{\bar{D}} _1 }{D_1^0 }} \right) } \right) }\qquad \end{aligned}$$

(19)

Setting these equal to one another

$$\begin{aligned}&\frac{2C_1^{\prime } \left( {\rho -\alpha ^{{p}^{\prime }}} \right) ^{2}}{r^{0}-r^{{p}^{\prime }}}+\theta {\bar{D}} _1 \left( {\frac{D_1^0 }{{\bar{D}} _1 }} \right) ^{\varepsilon ^{{p}^{\prime }}}\left[ {1-\left( {\ln \frac{D_1^0 }{{\bar{D}} _1 }} \right) {{\Omega }'}} \right] \nonumber \\&\qquad -\frac{2C_1^{*}(\rho -\alpha ^{p^{*}})^{2}}{r^{0}-r^{p^{*}}}-\theta {\bar{D}}_{1}\left( \frac{{D}_1^{0}}{{\bar{D}}_1}\right) ^{\varepsilon ^{p^{*}}}\left[ 1-\left( \ln \frac{D_{1}^{0}}{{\bar{D}}_{1}}\right) \Omega ^{*}\right] \nonumber \\&\quad =-\frac{\left( {\rho -\alpha ^{{p}^{*}}} \right) ^{2}\mathop \sum \nolimits _{j=2}^N W_j e^{-\rho \left( {\mathop \sum \nolimits _{n=2}^{j-1} {\bar{t}} _n } \right) }\rho \theta }{\left( {\alpha ^{{p}^{*}}-\frac{1}{2}\sigma ^{2}} \right) ^{2}}\ln \left( {\frac{{\bar{D}} _1 }{D_1^0 }} \right) e^{-\rho \left( {\frac{1}{\alpha ^{{p}^{*}}-\frac{1}{2}\sigma ^{2}}\ln \left( {\frac{{\bar{D}} _1 }{D_1^0 }} \right) } \right) }\qquad \end{aligned}$$

(20)

Since the right-hand side of (20) is negative we know:

$$\begin{aligned}&\frac{2C_1^{*} \left( {\rho -\alpha ^{{p}^{*}}} \right) ^{2}}{r^{0}-r^{{p}^{*}}}+\theta {\bar{D}} \left( {\frac{D_1^0 }{{\bar{D}} _1 }} \right) ^{\varepsilon ^{{p}^{*}}}\left[ {1-\left( {\ln \frac{D_1^0 }{{\bar{D}} _1 }} \right) {\Omega }^{{*}}} \right] \nonumber \\&\quad >\frac{2C_1^{\prime } \left( {\rho -\alpha ^{{p}^{\prime }}} \right) ^{2}}{r^{0}-r^{{p}^{\prime }}}+\theta {\bar{D}} _1 \left( {\frac{D_1^0 }{{\bar{D}} _1 }} \right) ^{\varepsilon ^{{p}^{\prime }}}\left[ {1-\left( {\ln \frac{D_1^0 }{{\bar{D}} _1 }} \right) {{\Omega }^{'}}} \right] \end{aligned}$$

(21)

which implies $r^{{p}^{\prime }}>r^{{p}^{*}}$ since the left- and right-hand sides of the above expression are functionally equivalent. The FSDE causes agent 1 to engage in less mitigation. This is intuitive and confirms previous results (Shogren and Crocker 1991).

1.1.1 Derivation of Collective and Individual Timing Threshold Conditions

The value matching condition is

$$\begin{aligned}&\quad \eta D_1^{{*} ^{\varepsilon ^{0}}}+\frac{D_1^{*} }{\left( {\rho -\alpha ^{0}} \right) }-Z_1^0 D_1^{{*} ^{\varepsilon ^{0}}}+\mathop \sum \limits _{j=2}^N W_j e^{-\rho \left( {\frac{1}{\alpha ^{0}-\frac{1}{2}\sigma ^{2}}\ln \left( {\frac{{\bar{D}} _1 }{D_1^{*} }} \right) +\mathop \sum \nolimits _{n=2}^{j-1} {\bar{t}} _n } \right) }\nonumber \\&\quad =C_1 +\frac{D_1^{*} }{\left( {\rho -\alpha ^{p}} \right) }-Z_1^p D_1^{{*} ^{\varepsilon ^{p}}}+\mathop \sum \limits _{j=2}^N W_j e^{-\rho \left( {\frac{1}{\alpha ^{p}-\frac{1}{2}\sigma ^{2}}\ln \left( {\frac{{\bar{D}} _1 }{D_1^{*} }} \right) +\mathop \sum \nolimits _{n=2}^{j-1} {\bar{t}} _n } \right) } \end{aligned}$$

(22)

The smooth pasting condition is

$$\begin{aligned}&\varepsilon ^{0}\eta D_1^{{*} ^{\varepsilon ^{0}-1}}+\frac{1}{\left( {\rho -\alpha ^{0}} \right) }-\varepsilon ^{0}Z_1^0 D_1^{{*} ^{\varepsilon ^{0}-1}}+\frac{\rho \mathop \sum \nolimits _{j=2}^N W_j e^{-\rho \left( {\mathop \sum \nolimits _{n=2}^{j-1} {\bar{t}} _n } \right) }}{D_1^{*} }\frac{e^{-\rho \left( {\frac{1}{\alpha ^{0}-\frac{1}{2}\sigma ^{2}}\ln \left( {\frac{{\bar{D}} _1 }{D_1^{*} }} \right) } \right) }}{\alpha ^{0}-\frac{1}{2}\sigma ^{2}}\nonumber \\&\quad =\frac{1}{\left( {\rho -\alpha ^{p}} \right) }-\varepsilon ^{p}Z_1^p D_1^{{*} ^{\varepsilon ^{p}-1}}+\frac{\rho \mathop \sum \nolimits _{j=2}^N W_j e^{-\rho \left( {\mathop \sum \nolimits _{n=2}^{j-1} {\bar{t}} _n } \right) }}{D_1^{*} }\frac{e^{-\rho \left( {\frac{1}{\alpha ^{p}-\frac{1}{2}\sigma ^{2}}\ln \left( {\frac{{\bar{D}} _1 }{D_1^{*} }} \right) } \right) }}{\alpha ^{p}-\frac{1}{2}\sigma ^{2}} \end{aligned}$$

(23)

Combining (22) and (23) we find

$$\begin{aligned}&\frac{\varepsilon ^{0}}{D_1^{*} }\left[ {{\begin{array}{l} {\mathop \sum \limits _{j=2}^N W_j e^{-\rho \left( {\mathop \sum \nolimits _{n=2}^{j-1} {\bar{t}} _n } \right) }\left[ {e^{-\rho \left( {\frac{1}{\alpha ^{p}-\frac{1}{2}\sigma ^{2}}\ln \left( {\frac{{\bar{D}} _1 }{D_1^{*} }} \right) } \right) }-e^{-\rho \left( {\frac{1}{\alpha ^{0}-\frac{1}{2}\sigma ^{2}}\ln \left( {\frac{{\bar{D}} _1 }{D_1^{*} }} \right) } \right) }} \right] } \\ {+C_1 +\frac{D_1^{*} }{\left( {\rho -\alpha ^{p}} \right) }-Z_1^p D_1^{{*} ^{\varepsilon ^{p}}}+Z_1^0 D_1^{{*} ^{\varepsilon ^{0}}}-\frac{D_1^{*} }{\left( {\rho -\alpha ^{0}} \right) }} \\ \end{array} }} \right] \nonumber \\&\quad +\frac{1}{\left( {\rho -\alpha ^{0}} \right) }-\varepsilon ^{0}Z_1^0 D_1^{{*} ^{\varepsilon ^{0}-1}}+\frac{\rho \mathop \sum \nolimits _{j=2}^N W_j e^{-\rho \left( {\mathop \sum \nolimits _{n=2}^{j-1} {\bar{t}} _n } \right) }}{D_1^{*} }\frac{e^{-\rho \left( {\frac{1}{\alpha ^{0}-\frac{1}{2}\sigma ^{2}}\ln \left( {\frac{{\bar{D}} _1 }{D_1^{*} }} \right) } \right) }}{\alpha ^{0}-\frac{1}{2}\sigma ^{2}}\nonumber \\&\quad =\frac{1}{\left( {\rho -\alpha ^{p}} \right) }-\varepsilon ^{p}Z_1^p D_1^{{*} ^{\varepsilon ^{p}-1}}+\frac{\rho \mathop \sum \nolimits _{j=2}^N W_j e^{-\rho \left( {\mathop \sum \nolimits _{n=2}^{j-1} {\bar{t}} _n } \right) }}{D_1^{*} }\frac{e^{-\rho \left( {\frac{1}{\alpha ^{p}-\frac{1}{2}\sigma ^{2}}\ln \left( {\frac{{\bar{D}} _1 }{D_1^{*} }} \right) } \right) }}{\alpha ^{p}-\frac{1}{2}\sigma ^{2}}\qquad \quad \end{aligned}$$

(24)

which can be rewritten as

$$\begin{aligned} 1=\frac{D_1^{*} \frac{\alpha ^{0}-\alpha ^{p}}{\left( {\rho -\alpha ^{0}} \right) \left( {\rho -\alpha ^{p}} \right) }-\frac{\left( {\varepsilon ^{p}-\varepsilon ^{0}} \right) }{\left( {\varepsilon ^{0}-1} \right) }Z_1^p D_1^{{*} ^{\varepsilon ^{p}}}+\frac{\varepsilon ^{0}}{\left( {\varepsilon ^{0}-1} \right) }X-\frac{D_1^{*} }{\left( {\varepsilon ^{0}-1} \right) }{\Delta }}{\frac{\varepsilon ^{0}}{\left( {\varepsilon ^{0}-1} \right) }C_1 } \end{aligned}$$

(25)

The solution to (25) must also ensure the option value is negative:

$$\begin{aligned} \eta =-\frac{C_1 }{\left( {\varepsilon ^{0}-1} \right) D_1^{{*} ^{\varepsilon ^{0}}}}-\frac{\left( {\varepsilon ^{p}-1} \right) }{\left( {\varepsilon ^{0}-1} \right) }Z_1^p D_1^{{*} ^{\varepsilon ^{p}}-\varepsilon ^{0}}+\frac{\hbox {X}-D_1^{*} {\Delta }}{\left( {\varepsilon ^{0}-1} \right) D_1^{{*} ^{\varepsilon ^{0}}}}+Z_1^0 <0 \end{aligned}$$

The critical investment threshold for individual timing is found using a similar procedure and must also ensure the option value is negative

$$\begin{aligned} \eta =-\frac{C_1 }{\left( {\varepsilon ^{0}-1} \right) D_1^{{\prime } ^{\varepsilon ^{0}}}}-\frac{\left( {\varepsilon ^{p}-1} \right) }{\left( {\varepsilon ^{0}-1} \right) }Z_1^p D_1^{{\prime } ^{\varepsilon ^{p}-\varepsilon ^{0}}}+Z_1^0 <0 \end{aligned}$$

1.1.2 Sensitivity Analysis

Our model shows how an FSDE can delay or prematurely initiate investments in damage mitigation. This section investigates the sensitivity of this result to the parameter values in the numerical example. Due to the complexity of the model, an analytical comparative analysis is not possible. Instead we report how the change in expected time of mitigation investment in jurisdiction 1 due to the FSDE is impacted by varying each parameter value (Table 1).

Table 1 Sensitivity of model results to a change in parameter values

1.1.3 Derivation of Compensated Individual Timing Threshold Condition and Payment

The payment from agent 2 changes agent 1’s value matching condition to

$$\begin{aligned}&\eta D_1^{{\prime } ^{\varepsilon ^{0}}}+\frac{D_1^{\prime }}{\left( {\rho -\alpha ^{0}} \right) }-Z_1^0 D_1^{{\prime } ^{\varepsilon ^{0}}}\nonumber \\&\quad =C_1 +\frac{D_1^{\prime }}{\left( {\rho -\alpha ^{p}} \right) }-Z_1^p D_1^{{\prime } ^{\varepsilon ^{p}}}-\delta \left[ \frac{1}{\alpha ^{0}-\frac{1}{2}\sigma ^{2}}\ln \left( \frac{D_1^{\prime }}{D^{0}} \right) +\frac{1}{\alpha ^{p}-\frac{1}{2}\sigma ^{2}}\ln \left( \frac{{\bar{D}} _1 }{D_1^{\prime }} \right) -\tau \right] \nonumber \\ \end{aligned}$$

(26)

and the smooth pasting condition becomes

$$\begin{aligned}&\varepsilon ^{0}\eta D_1^{{\prime } ^{\varepsilon ^{0}-1}}+\frac{1}{\left( {\rho -\alpha ^{0}} \right) }-\varepsilon ^{0}Z_1^0 D_1^{{\prime } ^{\varepsilon ^{0}-1}}\,\nonumber \\&\quad =\frac{1}{\left( {\rho -\alpha ^{p}} \right) }-\varepsilon ^{p}Z_1^p D_1^{{\prime } ^{\varepsilon ^{p}-1}}+\frac{\delta }{D_1^{\prime }}\left[ {\frac{\alpha ^{0}-\alpha ^{p}}{\left( {\alpha ^{0}-\frac{1}{2}\sigma ^{2}} \right) \left( {\alpha ^{p}-\frac{1}{2}\sigma ^{2}} \right) }} \right] \end{aligned}$$

(27)

Combining (26) and (27) we find

$$\begin{aligned}&\frac{\varepsilon ^{0}}{\left( {\varepsilon ^{0}-1} \right) }C_1 \,=D_1^{\prime } \frac{\alpha ^{0}-\alpha ^{p}}{\left( {\rho -\alpha ^{0}} \right) \left( {\rho -\alpha ^{p}} \right) }-\frac{\left( {\varepsilon ^{p}-\varepsilon ^{0}} \right) }{\left( {\varepsilon ^{0}-1} \right) }Z_1^p D_1^{{\prime } ^{\varepsilon ^{p}}}\nonumber \\&\quad +\frac{\delta }{\left( {\varepsilon ^{0}-1} \right) }\left[ \frac{\varepsilon ^{0}}{\alpha ^{0}-\frac{1}{2}\sigma ^{2}}\ln \left( \frac{D_1^{\prime }}{D^{0}} \right) +\!\frac{\varepsilon ^{0}}{\alpha ^{p}-\frac{1}{2}\sigma ^{2}}\ln \left( {\frac{{\bar{D}} _1 }{D_1^{\prime }}} \right) -\!\varepsilon ^{0}\tau +\frac{\alpha ^{0}-\alpha ^{p}}{\left( {\alpha ^{0}-\frac{1}{2}\sigma ^{2}} \right) {\left( {\alpha ^{p}-\frac{1}{2}\sigma ^{2}} \right) }}\right] \nonumber \\ \end{aligned}$$

(28)

The compensated timing threshold becomes

$$\begin{aligned} \frac{D_1^{\prime } \frac{\alpha ^{0}-\alpha ^{p}}{\left( {\rho -\alpha ^{0}} \right) \left( {\rho -\alpha ^{p}} \right) }-\frac{\left( {\varepsilon ^{p}-\varepsilon ^{0}} \right) }{\left( {\varepsilon ^{0}-1} \right) }Z_1^p D_1^{{\prime } ^{\varepsilon ^{p}}}+{\Lambda }}{\frac{\varepsilon ^{0}}{\left( {\varepsilon ^{0}-1} \right) }C_1}=1 \end{aligned}$$

(29)

where ${\Lambda }=\frac{\delta }{(\varepsilon ^{0}-1)} \left[ \frac{\varepsilon ^{0}}{\alpha ^{0} -\frac{1}{2}\sigma ^{2}}\ln \left( \frac{D_1^{\prime }}{D^{0}}\right) +\frac{\varepsilon ^{0}}{\alpha ^{p}-\frac{1}{2}\sigma ^{2}} \ln \left( \frac{{\bar{D}}_1}{D_1^{\prime }}\right) -\varepsilon ^{0}\tau +\frac{\alpha ^{0}-\alpha ^{p}}{\left( {\alpha ^{0}-\frac{1}{2}\sigma ^{2}}\right) \left( \alpha ^{p}-\frac{1}{2}\sigma ^{2}\right) }\right]$.

If $D_1^{\prime } >D_1^*$, agent 1’s payoff from an immediate investment in mitigation under this payment scheme is

$$\begin{aligned} W_1^I =C_1 +\frac{D_1^{\prime }}{\left( \rho -\alpha ^{p}\right) } -Z_1^{p} D_1^{{\prime }^{\varepsilon ^{p}}}-\delta \left[ \frac{1}{\alpha ^{0}-\frac{1}{2}\sigma ^{2}}\ln \left( \frac{D_1^{\prime }}{D^{0}}\right) +\frac{1}{\alpha ^{p}-\frac{1}{2}\sigma ^{2}}\ln \left( \frac{{\bar{D}}_{1}}{D_1^{\prime }}\right) -\tau \right] \nonumber \\ \end{aligned}$$

(30)

Comparing (8) and (29), $\delta$ is found by setting ${\Lambda }$ equal to $\frac{\varepsilon ^{0}X-D_1^{*} {\Delta }}{\left( {\varepsilon ^{0}-1} \right) }$ and solving.