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xp90

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Which of the following is the correct representation of a profit-maximising monopoly earning positive economic profits?

1. P = MR = MC, and P > AVC
2. ATC = MR, and P > AVC
3. MC = MR, and P > ATC
4. P = MC, and P > ATC

Économiste Monétaire

Answer is 3. MC = MR, and P > ATC.

Économiste Monétaire

I should probably expand on that, otherwise you won't see why it's the right answer. Denote the cost function by C(Q), the demand function by P(Q), and total revenue is P(Q) x Q, so the profit maximisation problem is

[latex] \displaystyle \max_{Q} \Pi = P(Q) \times Q - C(Q) [/latex]

FOC:

[latex] \displaystyle P(Q) + (P'(Q) \times Q) - C'(Q) = 0 [/latex]

or

[latex] \displaystyle P(Q) + (P'(Q) \times Q) = C'(Q) [/latex]

Note that [latex] \displaystyle MC = C'(Q)[/latex] and [latex] \displaystyle MR = \frac{d(P(Q) \times Q)}{dQ} = P(Q) + (P'(Q) \times Q) [/latex], so MC = MR.

Profit can be expressed as

[latex] \displaystyle \Pi = Q \times (P - ATC) [/latex]

so for the firm to have positive profits it must be true that P > ATC (and Q > 0).

A numerical example would look like this: Cost function given by,

[latex] \displaystyle C(Q) = 40 + Q^{2} [/latex]

Demand function:

[latex] \displaystyle P(Q) = 30 - Q [/latex]

Profit maximisation problem:

[latex] \displaystyle \max_{Q} \Pi = P(Q) \times Q - C(Q) [/latex]

[latex] \displaystyle P(Q) + (P'(Q) \times Q) - C'(Q) = 0 [/latex]

[latex] \displaystyle Q = \frac{C'(Q) - P(Q)}{P'(Q)} [/latex]

[latex] \displaystyle Q = \frac{2Q - (30 - Q)}{-1} [/latex]

[latex] \displaystyle Q = 30 - 3Q \Rightarrow Q^{*} = 7.5 [/latex]

[latex] \displaystyle \Rightarrow P = 30 - 7.5 = 22.5 [/latex]

[latex] \displaystyle ATC = \frac{40 + Q^{2}}{Q} = \frac{40}{Q} + Q [/latex]

When Q = 7.5

[latex] \displaystyle ATC = \frac{40}{7.5} + 7.5 = 12.833 < P[/latex]

Profit is: [latex] \displaystyle \Pi = 22.5 \times 7.5 - (40 + 7.5^{2} = 168.75 - 96.25 = 72.5 [/latex]

So, MC = MR and P > ATC.