<p>The &quot;hint&quot; is wrong and misleading.</p>
<p>It is wrong, because first-order conditions are sufficient for a maximum when the objective function is <em>strictly</em> concave, or at least strictly-quasi-concave, under some conditions. For example, a linear function is also concave (as well as convex), and in such a case, the first-order conditions are not sufficient for a maximum.</p>
<p>It is misleading, because concavity of a multivariate function depends on the signs of its Hessian matrix of <em>second-order</em> derivatives.  So the OP should compute the second derivatives of the profit function, with respect to capital and labor, since I guess the presumption here is that we maximize with respect to input quantities only, treating prices as exogenous constants.</p>
<p>Both of the above are standard material in many microeconomics books or &quot;math for economics&quot; books, so the OP should look them up there.</p>
<p>The proper &quot;hint&quot; would be that costs are linear in capital and labor...</p>
<p>...so the profit function has a non-linear part that relates to production function, and a linear part that relates to costs. And what happens to the second derivative when linearity is present ?</p>

CC BY-SA 4.0
2020-12-13T20:32:07.317Z
Thank you for pointing out the problem with the hint. Just to clarify, we have not worked on Hessian part in the class. We are expected to prove concavity only using first order conditions(that is the reason for the hint in the first place). And also you have mentioned in your answer that this thing can be found in any book. Can you please recommend at least one where the concavity of profit function leads to the concavity of production function.
Apprentice
CC BY-SA 4.0
2021-01-13T22:01:47.043Z
Linear function $ax+b$ differentiated wrt. x gives a, hence FOC equal to 0 iff. a=0, seems to be pretty much sufficiant for a max. however this is not unique. See THM. 2.5 of Nocedal and Wright: When f is convex, any local minimizer x is a global minimizer of f . If in addition f is differentiable, then any stationary point x is a global minimizer of f. [page 17].
Jesper Hybel
CC BY-SA 4.0
2021-01-13T22:02:10.290Z
Also calculating second order derivative seems overkill if profit is $\pi = pf(x) - w'x$ which is clearly concave if $f(x)$ is concave since $-w'x$ is linear and concave and $p>0$ so profit is linear combination of concave function by positive coefficients (p,1).
Jesper Hybel
CC BY-SA 4.0
2021-01-14T00:29:36.470Z
@JesperHybel The "hint is misleading", because, say in a classic $K,L$ production function, when the profit function is just concave, then the first order conditions do not give "the vector that maximizes profits", they give an infinite number of vectors that maximize  profits, since they only give us the optimal ratio of $K/L$ given prices. And uniqueness of solution is of central concern and importance in economics. If this student carries on thinking "ok with a convex profit function I can determine the optimal vector", they will be in for unpleasant surprises down the road.
Alecos Papadopoulos
CC BY-SA 4.0
2021-01-14T08:29:51.533Z
@AlecosPapadopoulos FOC of concave function is sufficient for a maximum not for the maximum. You can find many texts in economics that says that FOC are sufficient when the objective function is concave. I was merely suggesting that a formulation such as "concavity is not sufficient for a unique maximum" instead of simply speaking of sufficiency for "a maximum" would make it easier to understand what you are trying to say.
Jesper Hybel
CC BY-SA 4.0
2020-12-13T18:57:33.023Z
2020-12-13T18:57:33.023Z
Alecos Papadopoulos
<p>Show that concavity of firm's production function implies concavity of its profit function.
(Hint: For a concave function, first order conditions gives the vector that maximizes the function)</p>
<p>Confusing point: How do I relate the concavity of production function to the profit function. And where do I begin this proof mathematically.</p>

CC BY-SA 4.0
2020-12-11T07:39:53.607Z
It would be useful to give the arguments wrt which concavity has to be satisfied, and a definition of the profit function is also required for anwering your question (is it a short-run profit function? is there market power?...)
Bertrand
CC BY-SA 4.0
2020-12-11T21:10:35.263Z
By arguments I suppose you mean Labor and Capital. As for definition of Profit function it is that: pf(K,L) - wL - rk. Also it is not specified whether the Profit function is short run or long run. I have written the whole question as it is. The profit function that I gave is based on class notes.
Apprentice
CC BY-SA 4.0
2020-12-12T11:00:38.513Z
In this case you should begin your proof with $\pi(K,L;p,w,r) = pf(K,L) - wL - rk$ and show that it is concave in $(K,L)$.
Bertrand
<microeconomics><mathematical-economics><production-function><producer-theory><theory-of-the-firm>
If production function is concave, then demonstrate that profit function will also be concave
Apprentice
# If production function is concave, then demonstrate that profit function will also be concave

(Asked by: username_0 on 2020-12-11)

Show that concavity of firm's production function implies concavity of its profit function.
(Hint: For a concave function, first order conditions gives the vector that maximizes the function)

Confusing point: How do I relate the concavity of production function to the profit function. And where do I begin this proof mathematically.

**Tags:** microeconomics, mathematical-economics, production-function, producer-theory, theory-of-the-firm

> **Comments:**
> **username_1**: It would be useful to give the arguments wrt which concavity has to be satisfied, and a definition of the profit function is also required for anwering your question (is it a short-run profit function? is there market power?...)  
> **username_0**: By arguments I suppose you mean Labor and Capital. As for definition of Profit function it is that: pf(K,L) - wL - rk. Also it is not specified whether the Profit function is short run or long run. I have written the whole question as it is. The profit function that I gave is based on class notes.  
> **username_1**: In this case you should begin your proof with $\pi(K,L;p,w,r) = pf(K,L) - wL - rk$ and show that it is concave in $(K,L)$.  

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## 1 answers

### Answer by username_2 on 2020-12-13. Score: -1

The "hint" is wrong and misleading.

It is wrong, because first-order conditions are sufficient for a maximum when the objective function is *strictly* concave, or at least strictly-quasi-concave, under some conditions. For example, a linear function is also concave (as well as convex), and in such a case, the first-order conditions are not sufficient for a maximum.

It is misleading, because concavity of a multivariate function depends on the signs of its Hessian matrix of *second-order* derivatives. So the OP should compute the second derivatives of the profit function, with respect to capital and labor, since I guess the presumption here is that we maximize with respect to input quantities only, treating prices as exogenous constants.

Both of the above are standard material in many microeconomics books or "math for economics" books, so the OP should look them up there.

The proper "hint" would be that costs are linear in capital and labor...

...so the profit function has a non-linear part that relates to production function, and a linear part that relates to costs. And what happens to the second derivative when linearity is present ?

> **Comments:**
> **username_0**: Thank you for pointing out the problem with the hint. Just to clarify, we have not worked on Hessian part in the class. We are expected to prove concavity only using first order conditions(that is the reason for the hint in the first place). And also you have mentioned in your answer that this thing can be found in any book. Can you please recommend at least one where the concavity of profit function leads to the concavity of production function.  
> **username_3**: Linear function $ax+b$ differentiated wrt. x gives a, hence FOC equal to 0 iff. a=0, seems to be pretty much sufficiant for a max. however this is not unique. See THM. 2.5 of Nocedal and Wright: When f is convex, any local minimizer x is a global minimizer of f . If in addition f is differentiable, then any stationary point x is a global minimizer of f. [page 17].  
> **username_3**: Also calculating second order derivative seems overkill if profit is $\pi = pf(x) - w'x$ which is clearly concave if $f(x)$ is concave since $-w'x$ is linear and concave and $p>0$ so profit is linear combination of concave function by positive coefficients (p,1).  
> **username_2**: @JesperHybel The "hint is misleading", because, say in a classic $K,L$ production function, when the profit function is just concave, then the first order conditions do not give "the vector that maximizes profits", they give an infinite number of vectors that maximize  profits, since they only give us the optimal ratio of $K/L$ given prices. And uniqueness of solution is of central concern and importance in economics. If this student carries on thinking "ok with a convex profit function I can determine the optimal vector", they will be in for unpleasant surprises down the road.  
> **username_3**: @AlecosPapadopoulos FOC of concave function is sufficient for a maximum not for the maximum. You can find many texts in economics that says that FOC are sufficient when the objective function is concave. I was merely suggesting that a formulation such as "concavity is not sufficient for a unique maximum" instead of simply speaking of sufficiency for "a maximum" would make it easier to understand what you are trying to say.  

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