# Perfect competition allocations

Suppose the market demand is $$P(Q) = \alpha - \beta(Q)$$ where $$Q = \sum q_1$$. Variable $$q_i$$ denotes the output of the $$i$$th firm and $$Q$$ is the total output. The marginal cost for each firm is $$c$$.

What's the quantity that each firm will set for themselves in a perfectly competitive market?

I reckon the (market) price will be $$c$$ which means the overall market size will be given by $$Q = \frac{\alpha - c}{\beta}$$. What next?

• "What next?", indeed - a great question! Show us your attempt (all the way to the answer) for problems that appear to be homework. Oct 19, 2022 at 21:16
• @RegressForward The first bit cracked me up but no, I didn't mean it that way. It isn't a homework; I just wanted to see if the firms produce all of $Q = (\alpha-c) / \beta$ regardless of their cost functions and productions. The current answer assumes the same cost function and nothing about the production function, so it back when I posted it, it didn't clear my doubt. Now I have learnt that the production function and cost functions mean the same (since we can derive one from the other) and hence, identical cost functions will mean identical production functions. Oct 20, 2022 at 7:41
• @RegressForward As for attaining the full quantity $Q = (\alpha - c) / \beta$, the firms will do again do so by definition of equilibrium. This the current answerer mentions as a comment if you see. That said, I agree that I should have been more clear with what I wanted as an answer. Oct 20, 2022 at 7:43

It looks like your question is implying all the firms are identical, so they have the same production function.

Then each firm must provide the same amount, because they are the same internally (in their production function) and face the same price in perfect competition (so face the same external environment too).

Then we can denote each $$q_i$$ simply as $$q$$ because they’re all the same.

Suppose you have N firms.

Then $$Q = \Sigma^N q = q + q + q$$ (N times) $$= N*q$$.

Use that to solve for $$q$$.

• How do you know that the firms have the same production function? I know the marginal costs are equal, but that doesn't mean the production functions will be the same I suppose. Let me ask something: Without further information about the production capabilities, can we say that $Q = \frac{\alpha - c}{\beta}$ will be reached? Can the total supply go above that? Or below that? Oct 8, 2022 at 16:09
• Q must be reached in equilibrium, because in equilibrium the market clears and demand = supply. While I don’t know for sure that all the firms are identical, that is usually assumed in these types of questions at this level, especially if the marginal costs are equal. Oct 8, 2022 at 17:27
• Alright, one final question: Since the profits will be zero, how do you know that each will produce $Q/n$ units of output? For them, producing any unit will be the same (since profit is always $0$), so what guarantees that each will produce the exact same amount? Oct 8, 2022 at 18:31