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So I hear this distinction between price-taking firms and price-making firms in regards to perfect competition. Price-taking firms are firms that accept the price determined by supply and demand. But by that definition, isn’t every firm a price-taker? Even if there were a monopoly, the firm couldn’t just set any price they wanted (assuming they want to sell all their output). The price would be determined by what people are willing to pay. So can any firm truly be a price-maker?

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No, not all firms are price takers. You seem to be confused about demand firm faces for its product and market demand. On a perfectly competitive market price will be determined by market demand and market supply but firm-specific demand is simply perfectly elastic (i.e. flat), regardless of downward sloping market demand, which is what makes firm price taker. However, in other market structures such as imperfect competition firm specific demand will be (typically) downward sloping and hence firm won't be price taker.

Empirically firm can be said to be price taker when:

$$\frac{P-MC}{P} =0$$

Where $P$ is price and $MC$ marginal costs. People would also accept values $\approx 0$ as evidence that firm is price taker, but for any values above taht we would say the firm has market power, where 1 would represent firm with highest possible market power that can set prices arbitrarily.

The above can be also rewritten in terms of elasticity of firm specific demand since:

$$\frac{P-MC}{P} =- \frac{1}{\epsilon_d}$$

For perfectly competitive firm $\epsilon_d = -\infty$.

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