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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.

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    $\begingroup$ 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?...) $\endgroup$ – Bertrand Dec 11 '20 at 7:39
  • $\begingroup$ 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. $\endgroup$ – Apprentice Dec 11 '20 at 21:10
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    $\begingroup$ 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 $(K,L)$. $\endgroup$ – Bertrand Dec 12 '20 at 11:00
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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 ?

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  • $\begingroup$ 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. $\endgroup$ – Apprentice Dec 13 '20 at 20:32
  • $\begingroup$ 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]. $\endgroup$ – Jesper Hybel Jan 13 at 22:01
  • $\begingroup$ 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). $\endgroup$ – Jesper Hybel Jan 13 at 22:02
  • $\begingroup$ @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. $\endgroup$ – Alecos Papadopoulos Jan 14 at 0:29
  • $\begingroup$ @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. $\endgroup$ – Jesper Hybel Jan 14 at 8:29

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