### Setup and Assumptions

- A single homogeneous good G, price denoted by $$p$$
- Aggregate demand: $$Q = Q(p)$$. Unless otherwise stated this is common knowledge of all firms, and is continuously differentiable and invertible to give a well-defined function p(Q).
- Firms $$F_{i}, i = 1, ... N$$ (allow $$N = \infty$$). May also have $$i \in [0,1]$$. Firms produce outputs $$q_{i}$$. Let $$Q = \sum q_{i}$$. We will often abuse notation by using $$q$$ for the vector of outputs $$(q_{1},...,q_{i},..., q_{N})$$.
- Firms have known cost functions: $$C_{i}(q)$$. Marginal cost is $$C'(q) = c(q)$$. A simple case is that of constant returns to scale (CRS): $$C(q) = cq$$ where c is unit cost.