that $$\sum_{i = 2}^{N} {\widetilde}\tau^{k}_{i}= \frac{\sum_{i=2}^N \sum_{p=0}^{N-i}({\varepsilon}+1)^p}{\sum_{p=0}^{N-1}({\varepsilon}+1)^p}{\bar{\tau}}= \frac{(({\varepsilon}+1)^{N} - 1) - N{\varepsilon}}{{\varepsilon}(({\varepsilon}+1)^{N} - 1)}{\bar{\tau}}.$$ Then, the first element of the vector inside the norm in the expression of $V(\tau)$ is given as $$\begin{aligned}
&({\widetilde}\tau^{k}_{1} - \tau_{1}) - \frac{1}{N}\left(\frac{(({\varepsilon}+1)^{N} - 1 )- N{\varepsilon}}{{\varepsilon}(({\varepsilon}+1)^{N} - 1)}{\bar{\tau}}- \sum_{i=2}^{N} \tau_{i}\right) 
\\ & \qquad\qquad\qquad\qquad= -\frac{(({\varepsilon}+1)^{N} - 1) - N{\varepsilon}}{{\varepsilon}N(({\varepsilon}+1)^{N} - 1)}{\bar{\tau}}+ \frac{1}{N}\sum_{i=2}^{N} \tau_{i},\end{aligned}$$ while the elements with $m \in \{2,3,\ldots,N\}$ are given by

After the jump at $\tau$, since $G(\tau)$ is single valued, $V(G(\tau))$ is given by $$|G(\tau)|_{{\widetilde}\ell_{k'}} = \left|({\widetilde}\tau^{k'} - G(\tau)) - \frac{1}{N}(({\widetilde}\tau^{k'} - G(\tau))^{\top}{\mathbf 1}){\mathbf 1}\right|.$$ Note that $({\widetilde}\tau^{k'} - G(\tau))^{\top}{\mathbf 1}= \sum_{i = 1}^{N} {\widetilde}\tau^{k'}_{i} - \sum_{i = 1}^{N} g_{i}(\tau)$ reduces to $\sum_{i = 1}^{N} {\widetilde}\tau^{k'}_{i} - \sum_{i = 2}^{N} (1+ {\varepsilon})\tau_i$, since $g_{1}(\tau) = 0$ and $g_{i}(\tau) = (1+{\varepsilon})\tau_{i}$ for $i > 1$. Using [Lemmas \[lem:consum1\] and \[lem:consum2\]]{}, it follows that $$\begin{aligned}
\sum_{i = 1}^{N} {\widetilde}\tau^{k'}_{i} &= \frac{\sum_{i=1}^N \sum_{p=0}^{N-i}({\varepsilon}+1)^p}{\sum_{p=0}^{N-1}({\varepsilon}+1)^p}{\bar{\tau}}\\ &= \frac{({\varepsilon}+1)(({\varepsilon}+1)^{N} - 1) - N{\varepsilon}}{{\varepsilon}(({\varepsilon}+1)^{N} - 1)}{\bar{\tau}}\end{aligned}$$ which leads to $$({\widetilde}\tau^{k'} - G(\tau))^{\top}{\mathbf 1}= \frac{({\varepsilon}+1)(({\varepsilon}+1)^{N} - 1) - N{\varepsilon}}{{\varepsilon}(({\varepsilon}+1)^{N} - 1)}{\bar{\tau}}- \sum_{i = 2}^{N} (1+{\varepsilon})\tau_{i}.$$ The first element inside the norm in $V(G(\tau))$ is given by $$\begin{aligned}
&({\widetilde}\tau_{1}^{k'} - g_1(\tau)) - \frac{1}{N}\left(\frac{({\varepsilon}+1)(({\varepsilon}+1)^{N} - 1) - N{\varepsilon}}{{\varepsilon}(({\varepsilon}+1)^{N} - 1)}{\bar{\tau}}\right. \\ & \qquad 