I'm working on a problem where $N$ agents want to trade $M$ goods. Each agent $j$ has an initial allocation of goods $\omega_j = (\omega_{j1},...,\omega_{jN})$, and---through trade---ends up with a final allocation $C_j = (C_{j1},...,C_{jN})$. Each agent has a strictly concave and differentiable utility function $U_j(C_{j1},...,C_{jN})$.
In a standard setting, the optimal allocation can be found by solving the Lagrange Optimization Problem
$$\max_{C} \sum_{j=1}^{M} U_j(C_{j1},...,C_{jN})$$ $$\text{Subject to: } \sum_{j=1}^{M} C_{ji} \leq \sum_{j=1}^{M} \omega_{ji} \text{ for all } i.$$
The solution yields an allocation $C$ satisfying the first order conditions $$\frac{\partial U_j}{\partial C_i} = \lambda_i$$ where we can interpret $\lambda_i$ as the market price of good $i$.
I have a twist of this problem where some fraction $\tau$ of the goods are lost when they are traded. That is, if agent $j$ starts with $\omega_{ji}$ units of good $i$ and ends up with $C_{ji}$ units of the good, then $\tau |\omega_{ji}-C_{ji}|$ units are lost in transit. Finding the social optimum is equivalent to solving the convex optimization problem
$$\max_{C} \sum_{j=1}^{M} U_j(C_{j1},...,C_{jN})$$ $$\text{Subject to: } \underbrace{\sum_{j=1}^{M} C_{ji}}_{\text{consumed}} + \underbrace{\tau \sum_{j=1}^{M} |\omega_{ji}-C_{ji}|}_{\text{lost in transit}} \leq \sum_{j=1}^{M} \omega_{ji} \text{ for all } i.$$
Because the constraint is no longer differentiable, I can no longer derive first order conditions for this problem. Is there a way I can write Lagrange Multipliers for this problem that I can interpret as prices?
