# Revenues and cost functions

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 $$x$$.

• This is almost always the case with $c(x)=w'x$ May 26 '20 at 11:33
• what do you mean by $w'$? May 26 '20 at 19:08
• $w$ denotes the vector of input prices May 26 '20 at 22:15
• @Bertrand i know that it is almost the case, but i am wondering why... why you cannot have something like $c(x)$=wx^2$May 27 '20 at 11:17 ## 1 Answer 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: $$w x \equiv c(f(x)).$$ • 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$wx$? May 27 '20 at 11:12 • 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
May 27 '20 at 18:14