I have the following maximization formula of an adapted Monti-Klein model of Banking constructed by Borio et al. (2015):
\begin{equation} \max_{\{L_{j},D_{j}\}}\pi_{j} = [l(L_{j}+L_{-j})-\tau]L_{j}-[d(D_{j}+D_{-j})-\omega]D_{j}\tag{4} \end{equation}
\begin{equation} where~L_{j} = \sum_{h=1, h\neq j}^{N}L_{h}, \quad D_{j} = \sum_{h=1, h\neq j}^{N}D_{h}, \quad \tau=(1-\rho)r + \mu(r,\theta) + \psi(\theta) + \gamma_{L}>0\quad and \quad \omega = (1-\alpha)r - \gamma_{D}. \end{equation}
The variables are defined as followed:
l is the loan rate, d is the deposit rate, L_(j) is the loan volume of bank j, D_(j) is the deposit volume of bank j. The variables roh and omega include restrictions like minimum reserves, loan loss provisions and costs of loan and deposit creation in addition to the market rate r set by the central bank.
My question is what is the meaning of the negative subscript in the maximization equation. To my understanding it indicates the loan volume of all other banks. But if this would be the case, the two equations of the sum are missing a negative subscript since it is the total sum of of the loan volume of all banks excluding the one of bank j, therefore it would have to be equal L_(-j) and D_(-j). Is there any known definition in economics or mathematics in general that defines the meaning of a negative subscript? Since this is from an official economics paper I kinda doubt that there would be such a mistake in the notation.
Thanks in advance.
The 'minus index' is notation shorthand used commonly to refer to 'not that index'. For example, in your case $L_j$ is the loan of firm $j$, whereas $L_{-j}$ is the loan of the other firm. Or, sometimes, all the other firms that are not indexed by $j$.