## Appendix B
Then the inequality expressed by Equation (13) holds, i.e., case (i) occurs, and since according to Equation (9) \(2\delta-b_{fr}k_{fb}=0\), we immediately conclude that Equation (7) is satisfied only for volume \(V_{b0}=0\). Thus, if for \(\varepsilon_{0}\) Equation (A10) holds, the optimal bale dimensions cannot be near-equal for any \(\varepsilon<\varepsilon_{0}\). Assume now, that for \(\varepsilon_{0}\) the inequality expressed by Equation (13) is satisfied. Let us consider bale volume \(V_{b0}(\varepsilon_{0})\) uniquely given by Equation (7). Differentiating Equation (7) on both sides with respect to \(\varepsilon_{0}\) results in  
\[\frac{2p_{u}(1+\varepsilon_{0})}{\Omega\Big{(}k_{f}\Big{)}}\sqrt[3]{\frac{4V _{00}}{\pi}}+\frac{p_{b}}{1-k_{fb}}\Big{(}2\delta-b_{fr}k_{fb}\Big{)}\left( \sqrt[3]{1+\varepsilon_{0}}\right)^{2}-\sqrt[3]{\frac{4V_{00}}{\pi}}\Bigg{[} \frac{p_{b}}{1-k_{fb}}+\frac{p_{u}}{\Omega\Big{(}k_{f}\Big{)}}\Bigg{]}=0,\]1  
whence, after algebraic manipulations, we obtain  
\[\frac{2p_{u}(1+\varepsilon_{0})}{\Omega\Big{(}k_{f}\Big{)}}\sqrt[3]{\frac{4V _{00}}{\pi}}+\frac{p_{b}}{1-k_{fb}}\Big{(}2\delta-b_{fr}k_{fb}\Big{)}\left( \sqrt[3]{1+\varepsilon_{0}}\right)^{2}-\sqrt[3]{\frac{4V_{00}}{\pi}}\Bigg{[} \frac{p_{b}}{1-k_{fb}}+\frac{p_{u}}{\Omega\Big{(}k_{f}\Big{)}}\Bigg{]}=0,\]2 (A11)  
Thus, monotonicity of the function \(V_{b0}(\varepsilon_{0})\) depends on the sign of the expression \(\Big{(}2\delta-b_{fr}k_{fb}\Big{)}\). First, consider case (a) of the proposition assuming that inequality from Equation (12) holds. Then the denominator of the left hand side of Equation (7) is also positive, and if inequality given by Equation (13) holds for \(\varepsilon_{0}\), then it holds also for any \(0\leq\varepsilon\leq\varepsilon_{0}\). Function \(V_{b0}(\varepsilon)\) is monotonically increasing for \(0\leq\varepsilon\leq\varepsilon_{0}\). For \(\varepsilon_{0}=\varepsilon\), by Equations (5), (6) and (A8), the optimal bale dimensions \(D_{b}^{*}(\varepsilon)\), \(H_{b}^{*}(\varepsilon)\) can be equivalently expressed by Equations (16) and (17). Thus, \(D_{b}^{*}(\varepsilon)\), \(H_{b}^{*}(\varepsilon)\) are also monotonically increasing for \(0\leq\varepsilon\leq\varepsilon_{0}\). In case (b), when inequality expressed by Equation (14) is satisfied, function \(V_{b0}(\varepsilon_{0})\) decreases with \(\varepsilon_{0}\) and, by Equation (7), inequality from Equation (13) takes the form  
\[\frac{2p_{u}(1+\varepsilon_{0})}{\Omega\Big{(}k_{f}\Big{)}}\sqrt[3]{\frac{4V _{00}}{\pi}}+\frac{p_{b}}{1-k_{fb}}\Big{(}2\delta-b_{fr}k_{fb}\Big{)}\left( \sqrt[3]{1+\varepsilon_{0}}\right)^{2}-\sqrt[3]{\frac{4V_{00}}{\pi}}\Bigg{[} \frac{p_{b}}{1-k_{fb}}+\frac{p_{u}}{\Omega\Big{(}k_{f}\Big{)}}\Bigg{]}=0,\]3  
If, simultaneously, inequality expressed by Equation (15) holds, then for any \(0\leq\varepsilon\leq\varepsilon_{0}\) there exists \(V_{b0}(\varepsilon)\) fulfilling Equation (7). Otherwise, there exists \(0<\overline{\varepsilon}<\varepsilon_{0}\) such that Equation (18) holds and only for \(\overline{\varepsilon}<\varepsilon\leq\varepsilon_{0}\) Equation (7) is satisfied for some \(V_{b0}(\varepsilon)\). In both cases, for \(0\leq\varepsilon\leq\varepsilon_{0}\) or \(\overline{\varepsilon}<\varepsilon\leq\varepsilon_{0}\), by Equation (16) we have  
\[\frac{2p_{u}(1+\varepsilon_{0})}{\Omega\Big{(}k_{f}\Big{)}}\sqrt[3]{\frac{4V _{00}}{\pi}}+\frac{p_{b}}{1-k_{fb}}\Big{(}2\delta-b_{fr}k_{fb}\Big{)}\left( \sqrt[3]{1+\varepsilon_{0}}\right)^{2}-\sqrt[3]{\frac{4V_{00}}{\pi}}\Bigg{[} \frac{p_{b}}{1-k_{fb}}+\frac{p_{u}}{\Omega\Big{(}k_{f}\Big{)}}\Bigg{]}=0,\]4  
Thus, in view of the inequality from Equation (14), optimal bale diameter decreases with increasing \(\varepsilon\). The analysis of Equation (17) yields analogous property of \(H_{b}^{*}(\varepsilon)\). Proposition 2 is proved.