Indeterminate Output Allocations in Collusive Equilibria and Multi-plant Firms

If n colluding oligopolists all have the cost function C(q_i) = c * q_i, then it will not be possible to uniquely allocate among the firms the monopoly output that maximizes their joint profit. Similarly, if all plants of an n-plant firm have the cost function C(q_i) = c * q_i, then it will not be possible to uniquely allocate the firm's optimal output among the n plants. This paper identifies the necessary and sufficient conditions for such "allocative indeterminacies" to occur (i.e., the conditions to be avoided).