Final answer:
The profit-maximizing quantity (qm) for the monopolist is 160 units and the profit-maximizing price (pm) is $60 per unit, obtained by equating the marginal revenue with the marginal cost.
Step-by-step explanation:
The question asks us to determine the profit-maximizing price (pm) and quantity (qm) for a monopolist producing a new computer chip. To find these values, we need to equate the monopolist's marginal revenue (MR) to marginal cost (MC). The demand function provided is p = 100 - 0.25q. Since total revenue (TR) is price times quantity (p × q), we can determine the MR by differentiating TR with respect to q. To maximize profits, we set MR equal to the given MC of $20.
Marginal revenue is calculated by differentiating TR. If TR = p × q, substituting the demand equation yields TR = (100 - 0.25q)q = 100q - 0.25q2. Taking the derivative of TR with respect to q gives MR = 100 - 0.5q. Equating this to MC, we get 100 - 0.5q = 20, which simplifies to q = 160. Substituting this quantity back into the demand equation, we find the profit-maximizing price pm = 100 - 0.25(160) = $60.
Therefore, the profit-maximizing quantity qm is 160 units and the profit-maximizing price pm is $60 per unit. To graph the market for the computer chip, plot the demand curve, the MC line, and indicate the point where qm and pm are located. Lastly, calculate total revenue, total cost, and profit to analyze the monopolist's profitability.