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?

• You did not define $p$, $w$ and "production plan", so this is quite difficult to answer. – Giskard Jan 23 '17 at 16:31
• @denesp better? – EconJohn Jan 23 '17 at 17:57

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.