Revenues and cost functions

Question

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 $x$.

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