Equivalence of Producer Surplus Formulas
A firm's total producer surplus from selling a quantity Q₀ at a market price P₀ can be calculated using two different expressions:
Expression A: The integral of the difference between the market price and the marginal cost (C'(q)) from zero to Q₀, represented as ∫[0 to Q₀] (P₀ - C'(q)) dq.
Expression B: The firm's total revenue (P₀ * Q₀) minus its total variable costs.
Explain mathematically why these two expressions are equivalent. Your explanation must clarify the relationship between a firm's total cost function (C(q)), its marginal cost, and its variable costs.
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