Final answer:
To determine the short-run supply function, we consider the cost function and market structure. In a perfectly competitive market, the supply function is where marginal cost equals market price. In a monopoly or monopolistically competitive market, it is where marginal revenue equals marginal cost.
Step-by-step explanation:
The student has asked how to determine the short-run supply function for a firm with the cost function C(qi) = 100 + 50qi - 4qi2 + qi3 under three different market structures: a perfectly competitive market, a monopoly, and a monopolistically competitive business. The short-run supply function differs in each case based on how the firm perceives its demand curve and sets prices.
In a perfectly competitive market, the firm considers the demand curve to be flat, implying that the firm can sell its product at one market price, regardless of the quantity. The firm will select the quantity where marginal cost (MC) equals price (P). To find MC, we differentiate the cost function with respect to qi, set MC equal to P, and solve for qi to define the short-run supply function.
For a monopoly, the firm's demand curve is the market demand curve, which is typically downward sloping. The monopolist aims to maximize profit by producing at a level where marginal revenue (MR) equals MC. The short-run supply function in this case is determined by equating MR and MC and solving for the quantity where this occurs.
Monopolistic competition behaves similarly to a monopoly in the short-run, as each firm faces a downward-sloping demand curve and must consider MR in setting its output level, but the presence of close substitutes tempers the firm's pricing power. The firm will also equate MR to MC to find the profit-maximizing quantity.
In the case of market regulation, a point may be chosen where MC crosses the demand curve, benefitting both consumers and the broader social interest by setting price equal to MC.