<p>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.</p>

CC BY-SA 4.0
2020-02-27T11:15:59.673Z
2020-02-27T11:15:59.673Z
Matin Mahirli
<p>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:</p>

<ol>
<li>The profit function is the result of a profit maximization problem. Not the objective function in the maximization problem. A profit function <span class="math-container">$\pi^*(p, w, r)$</span> identifies maximum profit given the price levels (p, w, r).</li>
<li>In the profit maximization problem, the objective function <span class="math-container">$\pi = pf(k, l) - wl - kr$</span> is concave in <span class="math-container">$k$</span> and <span class="math-container">$l$</span>, the choice variables of the maximization problem.</li>
</ol>

CC BY-SA 4.0
2019-10-25T16:21:43.103Z
May I ask what is the convex sets you are referring to in the first sentence? Is it the production set?
Aqqqq
CC BY-SA 4.0
2019-10-26T21:25:52.547Z
@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.
Paul
CC BY-SA 4.0
2016-11-02T03:59:07.320Z
2019-10-26T21:20:22.173Z
2019-10-26T21:20:22.173Z
Paul
<p>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. </p>

<microeconomics>
convexity of the profit function for profit maximization
MHall
# convexity of the profit function for profit maximization

(Asked by: username_0 on 2016-11-01)

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.

**Tags:** microeconomics

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## 2 answers

### Answer by username_2 on 2016-11-02. Score: 4 | Accepted answer

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:

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.

> **Comments:**
> **username_3**: May I ask what is the convex sets you are referring to in the first sentence? Is it the production set?  
> **username_2**: @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.  

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### Answer by username_1 on 2020-02-27. Score: -2

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.


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