<p>Profits of a price taking firm take the form
<span class="math-container">$$ p q - c,$$</span>
where <span class="math-container">$p$</span> is the price, <span class="math-container">$q$</span> is the output and <span class="math-container">$c$</span> is the cost. Of course there is a relation between the output <span class="math-container">$q$</span> and the cost <span class="math-container">$c$</span> in the sense that higher <span class="math-container">$q$</span> will correspond to higher <span class="math-container">$c$</span>. As such, these two quantities cannot be chosen independently.</p>

<p>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.</p>

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

<p>The various functions are related by the identity:
<span class="math-container">$$w x \equiv c(f(x)).$$</span></p>

CC BY-SA 4.0
2020-05-27T11:12:39.097Z
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 $wx$?
Yorgos
CC BY-SA 4.0
2020-05-27T18:14:45.467Z
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 $c = w(x) x$.
tdm
CC BY-SA 4.0
2020-05-27T06:24:01.693Z
2020-05-27T06:24:01.693Z
tdm
<p>Let's assume that there is a firm that produces a single good, <span class="math-container">$q=f(x)$</span>, where <span class="math-container">$x$</span> is a single input. The firm can sell it on the market at a  price <span class="math-container">$p$</span>. It's production cost is given by a cost function <span class="math-container">$c(x)$</span>.</p>

<p>In the most microeconomics textbooks the profit maximization problem is expressed as </p>

<p><span class="math-container">$maxΠ=max\{pq-c(x)\}$</span>, where <span class="math-container">$q$</span> is assumed to be linear function of <span class="math-container">$x$</span>. </p>

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

<p><span class="math-container">$Π=pf(x)-c(x)$</span>, where <span class="math-container">$f$</span> is concave in <span class="math-container">$x$</span> and <span class="math-container">$c$</span> is convex in <span class="math-container">$x$</span>.</p>

CC BY-SA 4.0
2020-05-26T11:33:37.570Z
This is almost always the case with $c(x)=w'x$
Bertrand
CC BY-SA 4.0
2020-05-26T19:08:29.500Z
what do you mean by $w'$?
Yorgos
CC BY-SA 4.0
2020-05-26T22:15:37.143Z
$w$ denotes the vector of input prices
Bertrand
CC BY-SA 4.0
2020-05-27T11:17:42.860Z
@Bertrand i know that it is almost the case, but i am wondering why... why you cannot have something like $c(x)$=wx^2$
Yorgos
<microeconomics><production-function><cost-functions>
Revenues and cost functions
Yorgos
# Revenues and cost functions

(Asked by: username_0 on 2020-05-26)

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 $x$.

**Tags:** microeconomics, production-function, cost-functions
> **Comments:**
> **username_1**: This is almost always the case with $c(x)=w'x$  
> **username_0**: what do you mean by $w'$?  
> **username_1**: $w$ denotes the vector of input prices  
> **username_0**: @Bertrand i know that it is almost the case, but i am wondering why... why you cannot have something like $c(x)$=wx^2$  

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## 1 answers

### Answer by username_2 on 2020-05-27. Score: 1

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:
$$w x \equiv c(f(x)).$$

> **Comments:**
> **username_0**: 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 $wx$?  
> **username_2**: 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 $c = w(x) x$.  

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