Final answer:
A profit-maximizing monopolist in this case will choose to produce a quantity of 20 and set the price at $60. The deadweight loss is the difference between the quantity and price chosen by the monopolist (20 and 60) and the quantity and price of the perfectly competitive market (80 and 30).
Step-by-step explanation:
To maximize profits, a profit-maximizing monopolist will choose the quantity where marginal revenue equals marginal cost (MR = MC). In this case, the monopolist faces a demand function of q = 50 – p/2 and a constant marginal cost of 20. By substituting the demand function into the marginal revenue equation, we can find that MR = 50 - 3p/2. Setting MR equal to MC, we have 50 - 3p/2 = 20. Solving for p, we find p = 60.
Substituting the value of p back into the demand function, we can find the quantity chosen by the monopolist. q = 50 - 60/2 = 50 - 30 = 20.
Therefore, the monopolist will choose to produce a quantity of 20 and set the price at $60.
The deadweight loss can be calculated by comparing the perfectly competitive price and quantity with those of the monopolist. In a perfectly competitive market, the quantity would be where marginal cost equals demand (MC = D) and the price would be at the level where quantity is demanded (P = D). In this case, the quantity would be 80 and the price would be 30. The deadweight loss is the difference between the quantity and price chosen by the monopolist (20 and 60) and the quantity and price of the perfectly competitive market (80 and 30).