Final answer:
To determine the profit-maximizing price for the monopoly software application in Europe, the firm will set output where marginal revenue equals marginal cost. By differentiating the total revenue function with respect to price and setting it equal to the marginal cost of 20, we compute the profit-maximizing price to be 50.
Step-by-step explanation:
The student is asking about a firm with a monopoly on a software application in Europe and how to determine the profit-maximizing price given a demand schedule where the demand curve is Q1 = 120 - P, and the marginal cost (MC) is constant at 20.
To maximize profits, the firm would need to set output where marginal revenue (MR) equals marginal cost (MR = MC). Algebraically, this involves finding the derivative of the total revenue (TR), which is P times Q (in this case TR = P * (120 - P)), setting its derivative equal to MC, and solving for P to find the profit-maximizing price. The firm would follow three steps: first, identify the output level where MR equals MC, then determine the highest price they can charge for this quantity based on the demand curve, and finally calculate profits by subtracting total costs from total revenue.
Applying calculus: MR is the derivative of TR, so MR = d(TR)/dQ = d(P*(120-P))/dQ. To simplify, we take the inverse function of Q (P = 120 - Q) and differentiate TR with respect to P. Therefore, MR = 120 - 2P. Equating MR to MC gives us 120 - 2P = 20, which we can solve for P to find the profit-maximizing price P1.
Let's solve:
120 - 2P = 20
100 = 2P
P = 50
The monopolist would thus choose a price P1 of 50 to maximize profits.