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the profit function is convex in prices and wages (output and input prices respectively). How does this interact with profit maximization since convexity implies tangents always lie below the curve I would have thought convexity would be necessary for minimization rather than maximization.

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It is true that we are usually interested in minimizing convex functions over convex sets. But I think you have two confusions:

  1. The profit function is the result of a profit maximization problem. Not the objective function in the maximization problem. A profit function $\pi^*(p, w, r)$ identifies maximum profit given the price levels (p, w, r).
  2. In the profit maximization problem, the objective function $\pi = pf(k, l) - wl - kr$ is concave in $k$ and $l$, the choice variables of the maximization problem.
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