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Spin-off from: market equilibrium quantity $\ne$ firm profit maximising quantity?

Consider a perfectly competitive market with equilibrium price $P_{eq}$ and quantity $Q_{eq}$ and firm with profit maximising quantity $Q_f$ as illustrated below:


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Is the MC for each firm in the market the same?

Can we deduce such from any of the assumptions listed in the PC Wiki page?

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  • $\begingroup$ To clarify: Do you mean the $MC_i(q_i)$ functions or the $MC_i(q_i^*)$ equilibrium values? $\endgroup$ – Giskard Aug 17 '16 at 19:10
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    $\begingroup$ To expound on what @denesp has asked, I do think that this differentiation is pivotal to answering your question. If we are asking whether equilibrium values of marginal cost are the same, then clearly they are because marginal revenue is price, so all firms produce until their marginal cost is equal to the market price. However if we are asking whether the marginal cost function is the same for every firm, then the answer need not be yes. $\endgroup$ – DornerA Aug 17 '16 at 20:53
  • $\begingroup$ @denesp $MC_i(q_i)$. I'm asking about the curve and not the marginal cost of producing a certain $q_i$ i.e. $q_i*$ $\endgroup$ – BCLC Aug 18 '16 at 6:28
  • $\begingroup$ @DornerA ditto. $\endgroup$ – BCLC Aug 18 '16 at 6:28
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No, the marginal cost curves are not necessarily the same for each firm in the market. However the values of marginal costs are.

To disprove the general claim that "The marginal cost curve of each firm in a competitive market is the same" we simply need to find one counter-example, such as the one given below:

Suppose there are two firms in the market and the market inverse demand function is given by $P=7 - q^m = 7 - (q_1 + q_2)$

Here $P$ represents the price, $q^m$ the market supply and $q_1, q_2$ represent the respective supplies of the firms 1 and 2.

Suppose the marginal cost curves for each firm are given by:

$MC_1 = 2q_1$,

$MC_2 = 4q_2$

Note that these are not the same! Since we are looking for a competitive equilibrium, the firms behave competitively and hence we must assume that they do not realize they have market power. Hence they each set:

$P = MC_i, \; i\in\{1,2\}$

From this, solving for quantity we have the respective Supply functions of each firm:

$S_1 = q_1 = 0.5P$

$S_2 = q_2 = 0.25P$

Inserting these two supply functions into the inverse demand function we have:

$P=4$, $q_1=2$, $q_2=1$.

Is this an equilibrium with perfect competition? Well we found it by combining Supply and Demand assuming perfectly competitive firms, so it must be. To verify:

$Market \; Demand = 7 - p = 3$

$Market \; Suppy=q_1 + q_2 = 3$

So the market is indeed in equilibrium. Are both firms setting $P=MC$ as they should in perfect competition?

$MC_1 = 2*q_1 = 4 = P$

$MC_2 = 4*q_2 = 4 = P$

So indeed we have a market equilibrium with perfect competition and different marginal cost curves. QED

Note however that the values of the marginal cost curves at the equilibrium quantities are the same for each firm (and must be the same for each firm in equilibrium).

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The marginal cost is the same for any firm in a perfectly competitive market at equilibrium.

Now, let's prove it.

Suppose there is at least one firm (firm 1) that has a higher marginal cost, $MC_1$, than the remaining firms, at $MC_0$. The price, $P$, is the same for every firm because goods are homogenous. Rational consumers with perfect information would never buy the same good at a higher price. As firms maximize their profits and the technology does not have increasing returns to scale, their margional costs must be equal to the equilibrium price, $P=MC_0=MC_1$. This contradicts our initial assumption $MC_1>MC_0$. We have just shown our result.

An interesting case to get a complementary intuition is the Bertrand competition. Suppose the technology has constant returns to scale. In other words, producing a quantity $Q$ for firm 1 costs $MC_1 \times Q$, the marginal cost is constant. Imagine many firms able to produce the same homogenous good but with different marginal costs. As consumers would go for the cheapest price. Firms with the lowest marginal cost (the most efficient firms) can prevent the other firms for making a positive profit by setting a price low enough. The less efficient firms would leave the market and only the firms with the lowest marginal cost would exist at equilibrium. The equilibrium price would be equal to this lowest marginal cost. The result holds because of the same assumptions: the ability of consumers to choose the cheapest good, homogeneous goods, enough market competition.

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    $\begingroup$ This shows that the values of MC for each firm are the same. However the curves could be different and firms with different cost (and marginal cost) functions could produce different amounts in equilibrium and still be in equilibrium (have MC=p). In the comments OP clarified he is asking about the MC curve and not values of MC. $\endgroup$ – BB King Aug 19 '16 at 17:46
  • $\begingroup$ I just want to make clear that I am not the one who downvoted you. I think your answer is not bad, you just misunderstood OP's question. $\endgroup$ – BB King Aug 19 '16 at 19:37
  • $\begingroup$ I am the downvoter, my reasons are explained in BB King's comments. $\endgroup$ – Giskard Aug 22 '16 at 18:24

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