# Returns to scale in perfect competition

In a perfectly competitive market, is it necessary for the production functions to have decreasing returns to scale? Can it have a constant or increasing returns to scale production function like $$q(l,k) = l^2k$$? Further, is homogeneity of the production function a necessity?

If the price of the output good is positive, there can be no profit-maximizing profit plan for the production function $$q(l,k) = l^2k$$, which means that this production function is not compatible with perfect competition.
However, it is in principle possible to have a production function with increasing returns to scale under perfect competition- provided producing positive amounts is never optimal. Here is an example: Let the output price be $$p=1$$, the input price be $$w=1$$ and the production function be given by $$q(l)=l-\log(1+l)$$. This production function is strictly increasing and strictly convex, which means it also has increasing returns to scale. But the slope is always smaller than $$1$$, so the unique optimal production plan is to produce zero output using zero input.