# Algebraic demand curve in a perfectly competitive market for a firm

I've had a few years of economics courses already but still there are some basic things in the field that I just can not wrap my head around... (let's say economics is not my favorite field of study)

The problem that I'm having is about an algebraic representation of a demand curve in a perfectly competitive market.

Say that there is a set of N suppliers (with |N| possible tending towards infinity, hence a perfectly competitive market; let's just assume free entrance to the market too), consider a supplier i. The market demand function in terms of the output of supplier i can now be written as:

Here what is in the curly brackets is just the total market output. Now here's my problem, if you take the derivative of this function wrt. q_i, then you find that this derivative equals beta. However, I've also heard that the derivative of market demand with respect to a firm's output, in a perfectly competitive environment, is 0; as in perfect competition, a firm cannot control the price through its output this effect is very very small because of the many other suppliers.

Where in my reasoning did I go wrong?

If we're considering the canonical case of one firm facing perfect competition, then we're faced with a demand curve (for each firm's goods) that's perfectly flat- regardless of how many units any given firm produces, consumers are willing to pay a certain market price $p$ and no more (all of this is just a restatement of what you've mentioned in the question, of course).
What we can do is then consider how to have a model that incorporates this definition of perfect competition with your given definition of the demand facing firm $i$. As you point out, the derivative of the demand as you've defined it is $\beta$. (Again, all a restatement of what you've pointed out).
The next question to ask, then, is "what is $\beta$?" Effectively, it's the slope of the demand curve each firm faces. If we assume perfect competition, then we assume a flat demand curve, which leads us to conclude that $\beta=0$. I'd suggest that your reasoning didn't go wrong anywhere, just that the two threads need to be directly connected!
• @WBM I think you're mixing up the demand faced by an individual firm $i$ vs the market demand overall. If you're differentiating your market demand function with respect to an individual firm's output, you're solving for the first order conditions for that individual firm. In perfect competition, the demand function faced by every individual firm is horizontal- no single firm can impact market price by changing their quantity. Differentiating with respect to q (no subscript) gives you the overall market equilibrium. But differentiating with respect to only one firm's quantity gives you (1/2) – AndrewC Jun 12 '18 at 17:46
• @WBM gives you the $\beta_i$ for each firm. – AndrewC Jun 12 '18 at 17:48