An economics student is trying to algebraically prove that a competitive equilibrium maximizes total surplus. They define total surplus as N(Q) = F(Q) - C(Q), where F(Q) is the integral of the inverse demand function (total willingness to pay) and C(Q) is the total cost function. They correctly derive the first-order condition for maximization as F'(Q) = C'(Q). However, they get stuck interpreting this result. Their final, incorrect conclusion is: 'Surplus is maximized when the rate of change of total willingness to pay is equal to the marginal cost.' What is the fundamental flaw in the student's interpretation of the condition F'(Q) = C'(Q)?

Verifying Surplus Maximization in a Specific Market

A formal proof is used to show that the quantity produced in a competitive market equilibrium is the same quantity that maximizes the total gains from trade (total surplus). Arrange the key steps of this algebraic proof in the correct logical order.

Sufficiency of the First-Order Condition for Surplus Maximization

According to the formal algebraic proof, total surplus is maximized at the competitive equilibrium quantity because at this specific quantity, the total societal benefit from consumption, represented by the integral of the inverse demand function, is exactly equal to the total societal cost of production.

Critique of the Surplus Maximization Proof

A formal proof for why a competitive market maximizes total surplus relies on calculus. Match each mathematical expression from this proof with its correct economic interpretation. Assume F(Q) is the integral of the inverse demand function and C(Q) is the total cost function.

Analyzing Inefficiency Below Equilibrium

Analyzing Market Interventions with Calculus

In the algebraic proof that a competitive equilibrium maximizes total surplus, the first-order condition identifies a quantity where the slope of the total surplus function is zero. To confirm this quantity represents a maximum rather than a minimum, a second-order condition must be satisfied. Which statement correctly explains why this second-order condition (that the second derivative of the total surplus function is negative) holds in a typical competitive market?