Algebraic Proof that Competitive Equilibrium Maximizes Total Surplus

Verifying a Surplus Maximum

An economist has determined the quantity, Q*, that satisfies the first-order condition for maximizing total surplus (i.e., where the first derivative of the total surplus function with respect to quantity is zero). To confirm that Q* truly represents a maximum rather than a minimum, what additional condition must be met, and why is it typically satisfied in this economic context?

Calculating the Surplus-Maximizing Quantity

The Rationale Behind the Second-Order Condition for Surplus Maximization

If an economist determines that at a specific quantity, Q', the first derivative of the total surplus function with respect to quantity is equal to zero, it is guaranteed that Q' is the quantity that maximizes total surplus.

Match each mathematical expression from the calculus-based method of finding the surplus-maximizing quantity with its correct economic interpretation or condition.

A microeconomist wants to use calculus to find and verify the exact quantity of a good that maximizes total surplus. Arrange the following steps into the correct logical sequence they should follow.

Analyzing Second-Order Conditions for Surplus Maximization

Consider a market where, at the current production level of 500 units, the value to the consumer of the 500th unit is $40, and the cost to the producer of making that 500th unit is $30. Assuming the standard conditions required for a unique surplus-maximizing quantity are met, what does this situation imply about the total surplus?

An economist is analyzing a market and identifies a quantity, Q', where the value of the last unit to the consumer is exactly equal to the cost of producing it. However, a further analysis reveals that for quantities immediately greater than Q', the cost of producing an additional unit is less than the value consumers place on it. Based on this information, what can be concluded about the total surplus at quantity Q'?