Why is a perfectly competitive firm called a price taker and a monopolist a price maker?

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A perfectly competitive firm operates within a structure that is defined by five criteria: a) the sale of products that are identical to their competition b) the inability of firms to control market price c) the ownership of a relatively small market share by each firm d) complete transparency regarding products and prices and e) freedom of entry and exit. A perfectly competitive firm would be characterized as a "price taker" due to its inability to influence market price. In a perfectly competitive market, the price of the products are fixed since each firm is producing just enough to stay in business. Therefore, a perfectly competitive firm is essentially given a price at which to sell their products since they have no motivation to sell their products at prices that differ from the "perfect" market demand. 

In contrast, a monopoly firm has full control over the market in which it operates, and as such, this type of firm can manipulate market price purely by changing the price and quantity of their product. A monopoly firm is called a "price maker" because it determines market price and the rate of supply. Under these conditions, there must be roadblocks to entry and exit within the market, imperfect knowledge on behalf of the buyers, and only one producer within the market. Therefore, a monopoly firm's domination is dependent upon the elasticity of the market. The less elastic, the greater degree to which a firm can act as a "price maker" and manipulate market prices. 

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The above answer is mostly correct, but I would like to add a little to it to tell you WHY a firm in perfect competition can not influence the market --- WHY it must be a price taker.

The major reason for this is that the product it sells is identical to that of its competitors.  Therefore, it can not, for example, raise its price above that of its competitors and claim that its product is superior.  If it raises its prices above those of its competitors, it will simply be unable to sell any goods.

Take, for example, eggs which, in America at least, are graded by size and quality.  One Grade A egg is essentially identical to another.  Therefore, one egg supplier can not possibly claim its eggs are superior to those of other suppliers.

This is why a firm in perfect competition must be a price taker.

One small quibble with the above answer:

In perfect competition, firms may easily enter the market.  Because of this, firms in this market structure, by definition, make zero economic profit in the long term.  This is why they do not lower prices -- it's not because they have no incentive to do so -- it's because if they do they make negative profit.

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