# Essay on Calculus of Profit Maximization

Any profit-maximizing firm chooses inputs and outputs to maximize economic profits. By definition, maximization of economic profits entails maximization of the difference between the firm's total revenue and its total cost.

• A firm's total revenue is defined as the quantity, Q, sold at a price, P(q): TR(q) = P(q) ∙ Q

• A firm's total costs are defined as the quantity of capital, K, used multiplied by the price of capital, v, plus the quantity of labor, L, used multiplied by the price of labor (wage rate), w. TC = vK + wL

• Therefore, economic profits (π), are defined as the difference between total revenue and total

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furthermore

4. P = MC

Equations 3 and 4 are the new profit maximization equations for a competitive firm.

Equation 4 also serves as the supply curve for the competitive firm, where the firm will supply only those points along the marginal cost curve where (1) the marginal cost curve is rising, and (2) the point above where the marginal cost curve intersects the average variable cost curve. A point on the marginal cost curve below the average variable cost curve implies that the firm's profits cannot cover its AVC, and thus it is better off producing nothing and losing only its fixed costs instead of losing both; this point is the firm's shutdown price. (for a geometric interpretation of this, refer to section 8.2)

Practice Problems: section 8.c.

1.Q) A competitive firm faces a price of $6 and a total cost function of TC = 10 +2q – 0.2q2 + 0.01q3. a) Find MC, AVC, ATC, FC b) Find the optimal quantity produced and π* c) Find the shutdown price

1.A) a) MC = (dTC)/(dq) MC = 2 – 0.4q + 0.03q2

AVC = (VC)/(q) VC = 2q – 0.2q2 + 0.01q3 AVC = (2q/q) – (0.2q2 / q) + (0.01q3 / q) AVC = 2 – 0.2q + 0.01q2

ATC = (TC)/(q) ATC = (10/q) +(2q/q) – (0.2q2 /q) +