Final answer:
The calculation for Craig's Red Sea Restaurant's maximum monopoly profit involves determining the profit-maximizing quantity where MR equals MC and then calculating profit by subtracting total cost from total revenue. The correct answer none of the options.
Step-by-step explanation:
The question involves calculating the maximum monopoly profit for Craig's Red Sea Restaurant, which specializes in Ethiopian food in Columbia, South Carolina. To find this, we first need to determine the profit-maximizing quantity (Q) where marginal revenue (MR) equals marginal cost (MC). The demand function given is Q = 25 - P, where P is the price, and the total cost (TC) is given by TC = 25 + Q + 5Q².
Since we're dealing with a monopoly, MR is not equal to the price. To find MR, we need to derive it from the demand equation which is MR = d(TR)/dQ = d(P*Q)/dQ. For the linear demand curve, this simplifies to MR = P - (1/slope of demand curve)*Q. The slope of the demand curve is -1, hence MR = P - Q. Since P = 25 - Q from the demand function, this simplifies to MR = 25 - 2Q.
To find the MC, we take the first derivative of TC with respect to Q, which is MC = d(TC)/dQ = 1 + 10Q. Setting MR = MC to find the profit-maximizing quantity, we have 25 - 2Q = 1 + 10Q which simplifies to Q = 2.4. We substitute Q into the demand function to find P, which turns out to be P = 25 - 2.4 = $22.60.
The profit (π) is calculated as π = TR - TC. So, TR at the profit-maximizing quantity is TR = P*Q = 22.60*2.4 = $54.24. Next, we calculate the TC at Q = 2.4; TC = 25 + 2.4 + 5*(2.4)² = 25 + 2.4 + 5*5.76 = 25 + 2.4 + 28.8 = $56.20. Finally, the maximum monopoly profit is π = TR - TC = $54.24 - $56.20 = -$1.96, which is approximately -$2 when rounded to the nearest dollar. Thus, none of the options provided is correct as the profit is a loss of approximately $2.