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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2 Answers
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It is true that we are usually interested in minimizing convex functions or maximizing concave functions, typically over convex sets. But I think you have two confusions:
- 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).
- 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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$\begingroup$ May I ask what is the convex sets you are referring to in the first sentence? Is it the production set? $\endgroup$– AqqqqOct 25, 2019 at 16:21
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$\begingroup$ @Aqqqq, "convex sets" in my answer refers to the choice set of a maximization or minimization problem. In the profit maximization problem, the choice set is labor >=0, and capital >=0, which is the first quadrant of a 2-D plane, a convex set. $\endgroup$– PaulOct 26, 2019 at 21:25
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Profit functions are convex in output price that is : π (tp + (1- t)p′) ≤ tπ(p) + (1- t)π(p′) Intuitively this implies that if price of output increases by one unite the profit will rise by exactly or more than one unite.
