Final answer:
The rate at which the profit-maximizing quantity of output changes with respect to the price in a competitive market is found using implicit differentiation on the profit-maximization condition P = C'(Q*). This requires understanding of calculus applied to economics, where firms aim to produce where price equals marginal cost.
Step-by-step explanation:
Your question revolves around the profit-maximizing decision of a firm in perfectly competitive markets and involves applying calculus to economic analysis. Given that a profit-maximizing firm produces in a way that the price per unit (P) equals marginal cost (C'(Q)), we want to find the rate at which the firm should adjust its output (Q*) in response to changes in the market price (P).
We start with the condition for profit maximization: P = C'(Q*). To find the expression for dQ*/dP, which is the rate at which the profit-maximizing quantity of output Q* changes with respect to the price P, you would use implicit differentiation on this condition with respect to P. Since we assume that C'(Q) and C''(Q) are both greater than zero, implying increasing marginal costs, you would differentiate both sides of the equation with respect to P and rearrange terms to isolate dQ*/dP.
In terms of economic implications, if the market price faced by the firm is higher than its average cost of production at the quantity produced, then the firm will make a profit. If the market price is lower, the firm will make a loss. However, the decision to continue producing depends on whether the market price covers the variable costs, not just the average total costs. For perfectly competitive firms, producing where marginal revenue (MR), which is the price, is equal to marginal cost (MC) is essential for profit maximization.