WAPM and CRS across all production plans In Hal Varian's Book "Microeconomic analysis" on page 35 he gives the following description of a profit maximising firm.


  ...If the firm is maximising profits, then the
  observed net output choice at price pt must have a level of profit at least as
  great as the profit at any other net output the firm could have chosen. We
  don't know all the other choices that are feasible in this situation, but we
  do know some of them-namely, the other choices $y^s$ for $s = 1,. . . , T$ that
  we have observed. Hence, a necessary condition for profit maximisation is
  that
  $$p^ty^t\ge p^ty^s$$
  for all $t$ and $s=1,...,T$ 
  
  We will refer to this condition as the Weak Axiom of Profit Maximisation
  (WAPM).


In the case of a firm which has a production function with one input and one output where $p=w$, such that $p$ is the price the output is sold and $w$ is the price of the input across all production possibilities sets. Can such a firm be called profit maximising at any level of output?
Sure it can, because the maximal profit is 0, and that level is attained at all output levels.

A similar problem is a profit maximizing firm with constant returns to scale. If such a firm has a maximal profit it can attain then that profit level is 0. If the maximal profit level was something else, it could be increased by doubling or halving production.