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Let's assume that there is a firm that produces a single good, $q=f(x)$, where $x$ is a single input. The firm can sell it on the market at a price $p$. It's production cost is given by a cost function $c(x)$.

In the most microeconomics textbooks the profit maximization problem is expressed as

$maxΠ=max\{pq-c(x)\}$, where $q$ is assumed to be linear function of $x$.

I am wondering if there is a textbook (or a resource in general) in which profit function is expressed as

$Π=pf(x)-c(x)$, where $f$ is concave in $x$ and $c$ is convex in $x$.

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    $\begingroup$ This is almost always the case with $c(x)=w'x$ $\endgroup$ – Bertrand May 26 '20 at 11:33
  • $\begingroup$ what do you mean by $w'$? $\endgroup$ – Yorgos May 26 '20 at 19:08
  • $\begingroup$ $w$ denotes the vector of input prices $\endgroup$ – Bertrand May 26 '20 at 22:15
  • $\begingroup$ @Bertrand i know that it is almost the case, but i am wondering why... why you cannot have something like $c(x)$=wx^2$ $\endgroup$ – Yorgos May 27 '20 at 11:17
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Profits of a price taking firm take the form $$ p q - c,$$ where $p$ is the price, $q$ is the output and $c$ is the cost. Of course there is a relation between the output $q$ and the cost $c$ in the sense that higher $q$ will correspond to higher $c$. As such, these two quantities cannot be chosen independently.

There are two possibilities to make this relation explicit: (i) you express everything in terms of output or (ii) you express everything in terms of inputs.

  1. In case (i), one usually defines a cost function $c = c(q)$ which gives the cost necessary to produce an amount $q$. In this case, the profit maximization problem takes the form: $$\max_q p q - c(q).$$
  2. In case (ii) everything is expressed in terms of inputs. If we call $x$ the amount of inputs then we can write $q = f(x)$ as the total output that can be produced using inputs $x$ and $c = w x$ as the total cost of using an amount $x$ of inputs. Here, $w$ is the unit cost per input (e.g. the wage rate if $x$ is labour). In this case the profit maximization problem is given by: $$\max_x p f(x) - w x.$$

The various functions are related by the identity: $$w x \equiv c(f(x)).$$

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  • $\begingroup$ my question, was why in the 2 case you mention, the cost is linear in $x$. is there any reason why you cannot have $wx^2$ instead of $wx$? $\endgroup$ – Yorgos May 27 '20 at 11:12
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    $\begingroup$ If you buy $x$ units of inputs and the cost per unit is $w$ then the total costs are $wx$. This is always a linear function. I guess that in some cases you could have costs that are non-linear in inputs, but this would assume that the firm, for example, has market power. If it is the main purchaser of input $x$, it might be the case that the input price $w$ is increasing in the amount of inputs demanded, so you would get something like $c = w(x) x$. $\endgroup$ – tdm May 27 '20 at 18:14

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