The total cost of production, $C(Q)$, can be calculated by adding the firm's total fixed costs, $C(0)$, to its total variable costs. Total variable costs are found by integrating the marginal cost function from zero up to the quantity $Q$. This integral represents the area under the marginal cost curve. The complete equation is expressed as: $C(Q) = C(0) + \int_{0}^{Q} C'(x)dx$.

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Given that marginal cost is the derivative of the total cost function, the total cost for producing a specific quantity, Q, can be calculated by integrating the marginal cost function from an output of zero up to the quantity Q.

Calculating Total Cost by Integrating Marginal Cost

To cite the ebook, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/.

To cite a unit, use The CORE Econ Team 2023 The Economy 2.0: Microeconomics Open access e-text https://core-econ.org/the-economy/ Unit [unit number] (url).

CORE Econ - The Economy 2.0: Microeconomics

Total Cost as the Sum of Fixed Costs and Total Variable Costs

The total consumer surplus can be precisely calculated by treating the quantity (Q) as a continuous variable, which allows for the use of integration. For any given price (P₀) and quantity (Q₀), whether at market equilibrium or not, the consumer surplus is found by integrating the individual surpluses—the difference between the inverse demand function, f(q), and the price, P₀—from zero to the total quantity sold, Q₀. The integral form is: $\int_{0}^{Q_0} (f(q) - P_0) \,dq$. This can also be expressed as $F(Q_0) - P_0Q_0$, where F(Q) represents the integral of the inverse demand function, f(q), from 0 to Q. This alternative method involves finding the total area under the demand curve up to the equilibrium quantity ($Q^*$) and then subtracting the total expenditure ($P^* \times Q^*$). For a more comprehensive understanding of integration, section 19.1 of 'Mathematics for Economists: An Introductory Textbook' by Pemberton and Rau is a recommended resource.

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