A monopolist faces the same demand curve above and has a total cost function 𝑪(𝒒) = 𝟔𝒒. Compare profits, consumer surplus, and deadweight loss under two scenarios: a- The seller charges a single price to all customers; and b- The seller can perfectly price discriminate.

In the given scenario, a monopolist faces the same demand curve and has a total cost function of C(q) = 6q. Under scenario a, the monopolist charges a single price to all customers. To find the profit, we need to determine the monopolist's revenue and cost functions. Since the monopolist faces the same demand curve, the revenue function is given by: R(q) = p(q) * q = (30 - q) * q = 30q - q^2 The monopolist's profit function is given by: π(q) = R(q) - C(q) = 30q - q^2 - 6q = 24q - q^2 To find the monopolist's profit-maximizing quantity, we take the derivative of the profit function with respect to q and set it equal to zero: dπ(q)/dq = 24 - 2q = 0 Solving for q, we get: q = 12 Therefore, the monopolist's profit-maximizing quantity is 12 units. The monopolist's profit is: π(12) = 24(12) - 12^2 = 192 To find the consumer surplus, we need to determine the area under the demand curve and above the price. Since the monopolist charges a single price to all customers, the price is equal to the monopolist's marginal revenue, which is given by: MR(q) = 30 - 2q Setting MR(q) = 0, we get: **q = 15** Therefore, the monopolist's profit-maximizing quantity is 15 units, and the price is: p = MR(15) = 0 The consumer surplus is given by the area of the triangle formed by the demand curve, the price, and the quantity: CS = (1/2) * (30 - 0) * (15 - 0) = 225 To find the deadweight loss, we need to determine the area between the demand curve and the marginal cost curve. Since the total cost function is given by C(q) = 6q, the marginal cost function is: MC(q) = dC(q)/dq = 6 Therefore, the deadweight loss is given by the area of the triangle formed by the marginal cost curve, the price, and the quantity: DWL = (1/2) * (6 - 0) * (15 - 12) = 9 Under scenario b, the monopolist can perfectly price discriminate. In this case, the monopolist charges each customer the maximum price they are willing to pay. Since the monopolist faces the same demand curve, the price charged to each customer is equal to their marginal willingness to pay, which is given by: MWTP(q) = 30 - q The monopolist's profit is equal to the sum of the profits earned from each customer. To find the monopolist's profit-maximizing quantity for each customer, we take the derivative of the marginal willingness to pay with respect to q and set it equal to the monopolist's marginal cost: dMWTP(q)/dq = -1 Setting -1 = MC(q) = 6, we get: q = 6 Therefore, the monopolist's profit-maximizing quantity for each customer is 6 units. The monopolist's profit is: π = MWTP(6) + MWTP(5) + MWTP(4) + MWTP(3) + MWTP(2) = 30 + 25 + 20 + 15 + 10 = 100 Since the monopolist can perfectly price discriminate, there is no consumer surplus or deadweight loss. Therefore, under scenario a, the monopolist's profit is 192, the consumer surplus is 225, and the deadweight loss is 9. Under scenario b, the monopolist's profit is 100, and there is no consumer surplus or deadweight loss.