How Disagreement About Social Costs Leads to Inefficient Energy-Productivity Investment

Abstract

Public energy-productivity investment influences the amount of future energy consumption. If a present government expects its successor to value the social costs of fuel usage differently, this adds a strategic component to its investment considerations. We analyze this governmental time-inconsistency situation as a sequential game. In particular, we show how the expectation of a more conservative party taking over makes a “green” government choose an investment level that is inefficient, in that neither of the parties would prefer it to the investment level of a permanent green government. Under some circumstances, the opposition would even prefer the government definitely to stay in power: The gain from avoiding a strategic investment then outweighs the loss of not being able to regulate energy consumption. We also analyze the welfare gains from binding agreements.

Appendix

1.1 Second-Order Condition for Optimal Investment in the Reelection Case

To ensure that we have a maximum at $\tau _x^*, F_x^*$, it is necessary that the following condition holds at this point:

$$\begin{aligned} \frac{\partial ^2 W^x(F_x(\tau ), \tau ) }{ \partial \tau ^2 }&= - \frac{\partial ^2 T}{\partial \tau ^2} + \frac{\partial ^2 B}{\partial E^2} \cdot \left( \frac{\partial E_{x}}{\partial \tau } \right) ^2 - \kappa _x \cdot \frac{\partial ^2 Z}{\partial F^2} \cdot \left( \frac{\partial F_{x}}{\partial \tau } \right) ^2 \nonumber \\&\quad + \frac{\partial B}{\partial E} \cdot \frac{\partial ^2 E_{x}}{\partial \tau ^2} - \kappa _x \cdot \frac{\partial Z}{\partial F} \cdot \frac{\partial ^2 F_{x}}{\partial \tau ^2} < 0 . \end{aligned}$$

(6.1)

Substituting $\partial ^2 E_x / \partial \tau ^2 = 2 \cdot \partial F_x / \partial \tau + \tau \cdot \partial ^2 F_x / \partial \tau ^2$ and the optimal fuel use condition (3.6) yields:

$$\begin{aligned} \frac{\partial ^2 W^x(F_x(\tau ), \tau )}{\partial \tau ^2}&= - \frac{\partial ^2 T}{\partial \tau ^2} + \frac{\partial ^2 B}{\partial E^2} \cdot \left( \frac{\partial E_{x}}{\partial \tau } \right) ^2 \nonumber \\&\quad - \kappa _x \cdot \frac{\partial ^2 Z}{\partial F^2} \cdot \left( \frac{\partial F_{x}}{\partial \tau } \right) ^2 + 2 \cdot \frac{\partial B}{\partial E} \cdot \frac{\partial F_{x}}{\partial \tau } . \end{aligned}$$

(6.2)

In an energy-saving situation, $\partial F_x/\partial \tau$ is negative, so that all summands are negative; then the second-order condition is always fulfilled. Thus, we have to determine under which circumstances the second-order condition is fulfilled in a backfire situation. Factoring out $\partial B / \partial E$ and $F_x/\tau$ and substituting the first-order conditions (3.6) (for $\lambda = x$) and (4.3), we obtain:

$$\begin{aligned} \frac{\partial ^2 W^x(F_x^*, \tau _x^*)}{\partial \tau ^2}&= \frac{F_x^*}{\tau _{x}^{*}} \cdot \frac{\partial B}{\partial E} \nonumber \\&\quad \cdot \Biggl [ - \theta - \beta \cdot \left( \frac{\partial E_{x}}{\partial \tau } \cdot \frac{1}{F_{x}^{*}} \right) ^2 - \varphi \cdot \left( \frac{\partial F_{x}/F_{x}^{*}}{\partial \tau /\tau _{x}^{*}} \right) ^2 + 2 \cdot \frac{\partial F_{x} / F_{x}^{*}}{\partial \tau / \tau _{x}^{*}} \Biggr ] . \end{aligned}$$

Noting that $(\partial E_{x}/\partial \tau ) \cdot 1/F_{x} = 1 + \eta _{x}$, we can substitute the elasticities and simplify:

$$\begin{aligned} \frac{\partial ^2 W^x(F_{x}^{*}, \tau _{x}^{*})}{\partial \tau ^2}&= \frac{F_x^*}{\tau _{x}^{*}} \cdot \frac{\partial B}{\partial E} \cdot \left( 1 - \beta \right) \cdot \left( {\eta _{x}} - \frac{1}{\omega } \right) . \end{aligned}$$

(6.3)

Thus, in a backfire situation (where $1-\beta > 0$), the second-order condition is fulfilled if $\eta _{x} < 1/\omega$, which means that an $F, \tau$ combination can be a utility maximum if for these values the $F_x(\tau )$ function is less steep than the inverse of the $\tau ^*(F)$ function. In an energy-saving case (where $1 - \beta < 0$), $\eta _{x} > 1/\omega$ is implied, so that the $F_x(\tau )$ function is steeper than the $\tau ^*(F)$ function. Substituting $\eta _x$ and $\omega$ in (6.3), we see that the general requirement for the inequality to be fulfilled is

$$\begin{aligned} \frac{ 1 - 2 \cdot \beta - \theta \cdot \varphi - \theta \cdot \beta - \varphi \cdot \beta }{\beta + \varphi }&< 0 \end{aligned}$$

at the optimum. After further rearrangements, we can state this condition as

$$\begin{aligned} 1 - \beta < \frac{(1 + \varphi ) \cdot (1 + \theta )}{ (1 + \varphi ) + (1 + \theta ) }. \end{aligned}$$

(6.4)

In (3.10), we assumed $\varphi \ge 1, \theta \ge 1$; as $\beta > 0$, this is sufficient for (6.4) to be fulfilled and then, any extremum of the utility function is a maximum.

1.2 The Impact of Fuel Usage Cost Valuation on Investment in the Reelection Case

We differentiate (4.2) with respect to $\kappa _x$ and $\tau$. This yields:

$$\begin{aligned} 0&= \left( \frac{\partial B}{\partial E} \cdot \tau - \kappa _x \cdot \frac{\partial Z}{\partial F} \right) \cdot d\left( \frac{\partial F_{x}}{\partial \tau } \right) \nonumber \\&\quad + \Biggl \{ \frac{\partial ^2 B}{\partial E^2} \cdot \left[ {F_x}^2 + 2 \cdot E_x \cdot \frac{\partial F_{x}}{\partial \tau } + \tau ^2 \cdot \left( \frac{\partial F_x}{ \partial \tau } \right) ^2 \right] \nonumber \\&\quad - \kappa _x \cdot \frac{\partial ^2 Z}{\partial F^2} \cdot \left( \frac{\partial F_{x}}{\partial \tau } \right) ^2 + 2 \cdot \frac{\partial B}{\partial E} \cdot \frac{\partial F_x}{ \partial \tau } - \frac{\partial ^2 T}{\partial \tau ^2} \Biggr \} \cdot d \tau \nonumber \\&\quad + \Biggl [ \left( \frac{\partial B}{\partial E} + \frac{\partial ^2 B}{\partial E^2} \cdot E_x \right) \cdot \frac{\partial F_x}{ \partial \kappa _x } - \frac{\partial Z}{\partial F} \cdot \frac{\partial F_{x}}{\partial \tau } \nonumber \\&\quad + \left( \frac{\partial ^2 B}{\partial E^2} \cdot \tau ^2 - \kappa _x \cdot \frac{\partial ^2 Z}{\partial F^2} \right) \cdot \frac{\partial F_{x}}{\partial \tau } \cdot \frac{\partial F_x}{ \partial \kappa _x } \Biggr ] \cdot d \kappa _x . \end{aligned}$$

(6.5)

The first line of the right-hand side equals zero, because of the period-2 government’s optimality condition. The square-bracketed term in the second line can be replaced as

$$\begin{aligned} {F_x}^2 + 2 \cdot E_x \cdot \frac{\partial F_x}{\partial \tau } + \tau ^2 \cdot \left( \frac{\partial F_{x}}{\partial \tau } \right) ^2&= \left( \frac{\partial E_{x}}{\partial \tau } \right) ^2 . \end{aligned}$$

(6.6)

Doing so and rearranging yields:

$$\begin{aligned} 0&= \Biggl [ - \frac{\partial ^2 T}{\partial \tau ^2} + \frac{\partial ^2 B}{\partial E^2} \cdot \left( \frac{\partial E_x}{ \partial \tau } \right) ^2 - \kappa _x \cdot \frac{\partial ^2 Z}{\partial F^2} \cdot \left( \frac{\partial F_{x}}{\partial \tau } \right) ^2 + 2 \cdot \frac{\partial B}{\partial E} \cdot \frac{\partial F_x}{ \partial \tau } \Biggr ] \cdot \frac{ d \tau }{ d \kappa _x } \nonumber \\&\quad + \Biggl [ \frac{\partial B}{\partial E} \cdot \left( 1 - \beta \right) \cdot \frac{\partial F_x}{ \partial \kappa _x } - \frac{\partial Z}{\partial F} \cdot \frac{\partial F_{x}}{\partial \tau } \nonumber \\&\quad + \left( \frac{\partial ^2 B}{\partial E^2} \cdot \tau ^2 - \kappa _x \cdot \frac{\partial ^2 Z}{\partial F^2} \right) \cdot \frac{\partial F_x}{ \partial \kappa _x } \cdot \frac{\partial F_{x}}{\partial \tau } \Biggr ] . \end{aligned}$$

(6.7)

The square-bracketed term in the first line is equivalent to $\partial ^2 W^x/\partial \tau ^2$, as given in (6.2). We can also substitute the elasticities, factor out $\partial B/\partial E$ and $F_x/\kappa _x$, substitute the optimality condition (3.6) and rearrange to obtain:

$$\begin{aligned} \frac{ \partial \tau _x^{*} }{ \partial \kappa _x }&= \Biggl [ \frac{\partial ^2 W^x(\kappa _x, \tau )}{\tau ^2} \Biggr ]^{-1} \cdot \frac{F_x^{*}}{\kappa _{x}} \cdot \frac{\partial B}{\partial E} \cdot \eta _{x} . \end{aligned}$$

(6.8)

Now substitute $\partial ^2 W^x/\partial \tau ^2$ from (6.3):

$$\begin{aligned} \frac{ \partial \tau _x^{*} }{ \partial \kappa _x }&= \Biggl [ \frac{F_{x}^*}{\tau _{x}^*} \cdot \frac{\partial B}{\partial E} \cdot \left( 1 - \beta \right) \cdot \left( {\eta _{x}} - \frac{1}{\omega } \right) \Biggr ]^{-1} \cdot \frac{F_{x}^*}{\kappa _{x}} \cdot \frac{\partial B}{\partial E} \cdot \eta _{x}. \end{aligned}$$

(6.9)

Simplifying finally yields:

$$\begin{aligned} \psi _{x} \equiv \frac{ \partial \tau _x^{*}/\tau _x^{*} }{ \partial \kappa _x/\kappa _x }&= - \xi _{x} \cdot \Biggl ( {\eta _{x}} - \frac{1}{\omega } \Biggr )^{-1}. \end{aligned}$$

1.3 Second-Order Condition for Optimal Investment in the Voting-Out Case

To ensure a Party-$x$ utility maximum for $\tau _x^\#, F_y^\#$, it is necessary that the following condition holds at this point:

$$\begin{aligned} \frac{\partial ^2 W^x(F_{y}(\tau ), \tau )}{\partial \tau ^2}&= - \frac{\partial ^2 T}{\partial \tau ^2} + \frac{\partial ^2 B}{\partial E^2} \cdot \left( \frac{\partial E_{y}}{\partial \tau } \right) ^2 - \kappa _x \cdot \frac{\partial ^2 Z}{\partial F^2} \cdot \left( \frac{\partial F_{y}}{\partial \tau } \right) ^2 \nonumber \\&\quad + \frac{\partial B}{\partial E} \cdot \frac{\partial ^2 E_{y}}{\partial \tau ^2} - \kappa _x \cdot \frac{\partial Z}{\partial F} \cdot \frac{\partial ^2 F_{y}}{\partial \tau ^2} <0 . \end{aligned}$$

Substituting $\partial ^2 E_{y} / \partial \tau ^2 = 2 \cdot \partial F_{y} / \partial \tau + \tau \cdot \partial ^2 F_{y} / \partial \tau ^2$ yields:

$$\begin{aligned} \frac{\partial ^2 W^x(F_{y}(\tau ), \tau )}{\partial \tau ^2}&= - \frac{\partial ^2 T}{\partial \tau ^2} + \frac{\partial ^2 B}{\partial E^2} \cdot \left( \frac{\partial E_{y}}{\partial \tau } \right) ^2 - \kappa _{x} \cdot \frac{\partial ^2 Z}{\partial F^2} \cdot \left( \frac{\partial F_{y}}{\partial \tau } \right) ^2 \nonumber \\&\quad + 2 \cdot \frac{\partial B}{\partial E} \cdot \frac{\partial F_{y}}{\partial \tau } \nonumber \\&\quad + \left( \frac{\partial B}{\partial E} \cdot \tau - \kappa _x \cdot \frac{\partial Z}{\partial F} \right) \cdot \frac{\partial ^2 F_{y}}{\partial \tau ^2} . \end{aligned}$$

(6.10)

Substituting the first-order conditions (3.6) (for $\lambda = y$) and (4.9), and factoring out $\partial B/\partial E$ and $F_y/\tau$, we obtain:

$$\begin{aligned} \frac{\partial ^2 W^x(F_{y}(\tau _{x}^{\#}), \tau _{x}^{\#})}{\partial \tau ^2}&= \frac{F_{y}^{\#} }{ \tau _{x}^{\#} } \cdot \frac{\partial B}{\partial E} \cdot \Biggl \{ - \theta \cdot \left[ 1 + \left( 1 - \frac{\kappa _x}{\kappa _y} \right) \cdot \eta _y \right] \nonumber \\&\quad - \beta \cdot \left( \frac{\partial E_{y}}{\partial \tau } \cdot \frac{ 1 }{F_{y}^{\#}} \right) ^2 - \frac{\kappa _{x}}{\kappa _{y}} \cdot \varphi \cdot \left( \frac{\partial F_{y}/F_{y}^{\#}}{\partial \tau /\tau _{x}^{\#}} \right) ^2 \nonumber \\&\quad + 2 \cdot \frac{\partial F_{y}/F_{y}^{\#}}{\partial \tau /\tau _{x}^{\#}} + \tau _{x}^{\#} \cdot \left( 1 - \frac{ \kappa _x }{\kappa _y} \right) \cdot \frac{\partial ^2 F_{y}}{\partial \tau ^2} \cdot \frac{ \tau _{x}^{\#} }{F_{y}} \Biggr \}. \end{aligned}$$

(6.11)

For the second derivative of the period-2 government’s fuel consumption choice with respect to energy productivity, rearranging and differentiating (3.9) yields:

$$\begin{aligned} \frac{\partial ^2 F_{y}}{\partial \tau ^2}&= \frac{F_{y}}{\tau } \cdot \left[ \frac{\partial \eta _{y}}{\partial \tau } + \frac{1}{\tau } \cdot \eta _{y} \cdot \left( \eta _{y} - 1 \right) \right] . \end{aligned}$$

(6.12)

Substituting (6.12) and $(\partial E_{y}/\partial \tau ) \cdot 1/F_{y} = 1 + \eta _{y}$ into (6.11) and simplifying yields:

$$\begin{aligned} \frac{\partial ^2 W^x(F_{y}(\tau _{x}^{\#}), \tau _{x}^{\#})}{\partial \tau ^2}&= \frac{F_{y}^{\#}}{ \tau _{x}^{\#} } \cdot \frac{\partial B}{\partial E} \cdot \left( 1 - \beta \right) \cdot \Biggl \{ \eta _{y} - \frac{1}{\omega } \nonumber \\&\quad + \frac{1}{\beta + \varphi } \cdot \left( 1 - \frac{ \kappa _x }{\kappa _y} \right) \cdot \left[ \eta _{y} \cdot \left( 1 + \varphi \right) - 1 - \theta \right] \nonumber \\&\quad + \frac{1}{\beta + \varphi } \cdot \left( 1 - \frac{ \kappa _x }{\kappa _y} \right) \cdot \left( \frac{\partial \eta _{y}/\eta _{y}}{\partial \tau /\tau _{x}^{\#}} \right) \Biggr \} . \end{aligned}$$

With some rearrangements, we obtain:

$$\begin{aligned} \frac{\partial ^2 W^x(F_{y}(\tau _{x}^{\#}), \tau _{x}^{\#})}{\partial \tau ^2}&= \frac{F_{y}^{\#}}{ \tau _{x}^{\#} } \cdot \frac{\partial B}{\partial E} \cdot \left( 1 - \beta \right) \cdot \Biggl \{ \left( \eta _{y} - \frac{1}{\omega } \right) \cdot \left[ 1 + \eta _{y} \cdot \left( 1 - \frac{ \kappa _x }{\kappa _y} \right) \right] \nonumber \\&\quad + \frac{1}{\beta + \varphi } \cdot \left( 1 - \frac{ \kappa _x }{\kappa _y} \right) \cdot \left( \frac{\partial \eta _{y}/\eta _{y}}{\partial \tau /\tau _{x}^{\#}} \right) \Biggr \}. \end{aligned}$$

(6.13)

We assume that the elasticity of $\eta _y$ with respect to $\tau$ is negligible.¹⁷ Then, determining under which conditions this second derivative is smaller than zero is equivalent to finding out whether the following inequality holds:

$$\begin{aligned} \left( 1 - \beta \right) \cdot \left( \eta _{y} - \frac{1}{\omega } \right) \cdot \left[ 1 + \eta _{y} \cdot \left( 1 - \frac{ \kappa _x }{\kappa _y} \right) \right]&< 0 . \end{aligned}$$

(6.14)

From Sect. 6.1, we know that the product of the first two multiplicands is negative, so we now have to analyze whether the third multiplicand is positive. Note that $1 - \kappa _x/\kappa _y$ is bounded between $-\infty$ (for $\kappa _x \rightarrow \infty$ and $\kappa _y \rightarrow 0$) and 1 (for $\kappa _x \rightarrow 0$ and $\kappa _y \rightarrow \infty$) and that $1 > \eta _{y} > 0$ in the backfire case and $0 > \eta _{y}> -1$ in the energy-saving case, given our assumptions in Sect. 3.4.

First consider the energy-saving case in which $\eta _y < 0$. Then for $\kappa _x > \kappa _y$, no term in the square brackets is negative. For $\kappa _x < \kappa _y$, the square-bracketed term must be larger than $1 + (-1) \cdot 1 = 0$, so the energy-saving case is never a problem.

Now consider the backfire case with $\eta _{y} > 0$. For $\kappa _x < \kappa _y,\,1 - \kappa _x/\kappa _y$ is positive, so the whole term is again positive. However, for $\kappa _x > \kappa _y$, the square-bracketed term can take negative values. The condition for this not to happen is:

$$\begin{aligned} \frac{\kappa _y}{\kappa _x} > \frac{1 - \beta }{1 + \varphi }. \end{aligned}$$

To sum up, the second-order condition is always fulfilled with energy-saving benefit functions. For a backfire case, the second-order condition is fulfilled if the investing government is more conservative than its successor $(\kappa _x < \kappa _y)$, or if the future government will not raise fuel consumption too much in optimum.

1.4 The Impact of the Period-2 Government’s Fuel-Usage Cost Valuation on Investment

To analyze how a change in the period-2 government’s valuation parameter $\kappa _{y}$ changes the Party-$x$ government’s investment, we proceed along the lines of Sect. 6.2, but now it is not the period-1 government’s own cost valuation parameter that varies, but its successor’s. First, we differentiate the strategic first-order condition (4.8) with respect to $\tau$ and $\kappa _{y}$. Rearranging yields:

$$\begin{aligned} 0&= \Biggl [ - \frac{\partial ^2 T}{\partial \tau ^2} + \frac{\partial ^2 B}{\partial E^2} \cdot \left( \frac{\partial E_{y}}{\partial \tau } \right) ^2 - \kappa _{x} \cdot \frac{\partial ^2 Z}{\partial F^2} \cdot \left( \frac{\partial F_{y}}{\partial \tau } \right) ^2 \nonumber \\&\quad + 2 \cdot \frac{\partial B}{\partial E} \cdot \frac{\partial F_{y}}{\partial \tau } + \left( \frac{\partial B}{\partial E} \cdot \tau - \kappa _x \cdot \frac{\partial Z}{\partial F} \right) \cdot \frac{\partial ^2 F_{y}}{\partial \tau ^2} \Biggr ] \cdot \frac{ d \tau }{ d \kappa _y} \nonumber \\&\quad + \Biggl [ \frac{\partial B}{\partial E} \cdot \left( 1 - \beta \right) \cdot \frac{\partial F_{y}}{ \partial \kappa _{y} } + \left( \frac{\partial ^2 B}{\partial E^2} \cdot \tau ^2 - \kappa _x \cdot \frac{\partial ^2 Z}{\partial F^2} \right) \cdot \frac{\partial F_{y}}{\partial \tau } \cdot \frac{\partial F_{y}}{ \partial \kappa _{y} } \nonumber \\&\quad + \left( \frac{\partial B}{\partial E} \cdot \tau - \kappa _x \cdot \frac{\partial Z}{\partial F} \right) \cdot \frac{\partial ^2 F_{y}}{\partial \tau \partial \kappa _{y}} \Biggr ] . \end{aligned}$$

(6.15)

The square-bracketed term in the first two lines is equivalent to $\partial ^2 W^x(F_{y}(\tau ), \tau )/\partial \tau ^2$, as given in (6.10). From the definitions of $\xi _{\lambda }$ and $\eta _{\lambda }$ in (3.8) and (3.9), we can derive the following equivalence for $\partial ^2 F_{y}/\partial \tau \partial \kappa _y$:

$$\begin{aligned} \frac{\partial ^2 F_{y}}{\partial \tau \partial \kappa _{y} }&= \eta _{\lambda } \cdot \frac{F_{y}}{\tau \cdot \kappa _{y} } \cdot \left( \frac{ \partial \eta _{y}/\eta _{y} }{\partial \kappa _{y}/\kappa _{y}} + \xi _{y} \right) . \end{aligned}$$

(6.16)

Substituting both findings and rearranging yields:

$$\begin{aligned} \frac{ d \tau }{d \kappa _{y}}&= - \Biggl [ \frac{\partial ^2 W^x(F_{y}(\tau ), \tau )}{\partial \tau ^2} \Biggr ]^{-1} \nonumber \\&\quad \cdot \Biggl [ \frac{\partial B}{\partial E} \cdot \left( 1 - \beta \right) \cdot \frac{\partial F_{y}}{\partial \kappa _{y}} + \left( \frac{\partial ^2 B}{\partial E^2} \cdot \tau ^2 - \kappa _x \cdot \frac{\partial ^2 Z}{\partial F^2} \right) \cdot \frac{\partial F_{y} }{\partial \tau } \cdot \frac{\partial F_{y}}{\partial \kappa _{y}} \nonumber \\&\quad + \left( \frac{\partial B}{\partial E} \cdot \tau - \kappa _x \cdot \frac{\partial Z}{\partial F} \right) \cdot \eta _{\lambda } \cdot \frac{F_{y}}{\tau \cdot \kappa _{y} } \cdot \left( \frac{ \partial \eta _{y}/\eta _{y} }{\partial \kappa _{y}/\kappa _{y}} + \xi _{y} \right) \Biggr ]. \end{aligned}$$

(6.17)

We factor out $\partial B(E_{y})/\partial E$ and $F_{y}/\kappa _{y}$ and substitute the period-2 government’s optimality condition, (3.6) for $\lambda = y$:

$$\begin{aligned} \frac{ \partial \tau _x^{\#}}{\partial \kappa _{y}}&= - \Biggl [ \frac{\partial ^2 W^x(F_{y}(\tau ), \tau )}{\partial \tau ^2} \Biggr ]^{-1} \cdot \Biggl [ \varphi \cdot \xi _{y} + \left( \frac{ \partial \eta _{y}/\eta _{y} }{\partial \kappa _{y}/\kappa _{y}} + \xi _{y} \right) \Biggr ] \nonumber \\&\quad \cdot \frac{F_{y}}{\kappa _{y}} \cdot \frac{\partial B}{\partial E} \cdot \eta _{y} \cdot \left( 1 - \frac{\kappa _{x}}{\kappa _{y}} \right) . \end{aligned}$$

Now substitute $\partial ^2 W^x(F_{y}(\tau ), \tau )/\partial \tau ^2$ from (6.13), and assume that the elasticity of $\eta _y$ with respect to $\kappa _y$ (and, as we assumed before, with respect to $\tau$—see footnote 17) is neglectable:

$$\begin{aligned} \frac{ \partial \tau _x^{\#}}{\partial \kappa _{y}}&= - \Biggl \{ \frac{F_{y}}{ \tau _{x}^{\#} } \cdot \frac{\partial B}{\partial E} \cdot \left( 1 - \beta \right) \cdot \left( \eta _{y} - \frac{1}{\omega } \right) \cdot \left[ 1 + \eta _{y} \cdot \left( 1 - \frac{ \kappa _x }{\kappa _y} \right) \right] \Biggr \}^{-1} \nonumber \\&\quad \cdot \left[ \varphi \cdot \xi _{y} + \xi _{y} \right] \cdot \frac{\partial B}{\partial E} \cdot \frac{F_{y}}{\kappa _{y}} \cdot \eta _{y} \cdot \left( 1 - \frac{\kappa _{x}}{\kappa _{y}} \right) . \end{aligned}$$

(6.18)

Simplifying yields:

$$\begin{aligned} \mu \equiv \frac{ \partial \tau _x^{\#} / \tau _x^{\#}}{\partial \kappa _{y} / \kappa _{y} }&= \left( \xi _{y} \right) ^2 \cdot \left( \eta _{y} - \frac{1}{\omega } \right) ^{-1} \cdot \left( 1 + \varphi \right) \nonumber \\&\quad \cdot \left[ 1 + \eta _{y} \cdot \left( 1 - \frac{ \kappa _x }{\kappa _y} \right) \right] ^{-1} \cdot \left( 1 - \frac{\kappa _{x}}{\kappa _{y}} \right) . \end{aligned}$$