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Is there a way to understand if its a monopoly by just looking the quantities (Q), marginal costs (MC), marginal revenue (MR), and average costs (AC) of a firm?

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What you can actually detect is market power, not monopoly per se.

Monopolies have market power, but there are other ways to get market power such as oligopoly and monopolistic competition. But these are also similar in many ways to monopoly, so in a broad sense you can detect "monopolistic situations" just by looking at marginal revenue and marginal cost.

Specifically, a firm with market power will set its marginal revenue equal to its marginal cost:

MR = MC

Compare this to a competitive firm, which would set its price equal to its marginal cost:

P = MC

The marginal revenue is typically smaller than the price, because increasing the quantity sold will also decrease the price as long as the demand curve is downward-sloping.

Specifically:

MR = d[PQ]/dQ = dP/dQ Q + P

If the demand curve is downward-sloping (as they usually are), dP/dQ is negative and thus MR < P.

Therefore we expect a monopoly (or more generally a firm with market power) to set a price higher than its marginal cost and sell a quantity smaller than the optimal quantity a competitive firm would sell.

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