Final answer:
A monopoly with an inverse demand of P(Q)=1-Q and a marginal cost of zero will have an equilibrium quantity of 0.5 units, a price of 0.5, and profits of 0.25 when using the profit-maximizing condition that marginal revenue equals marginal cost.
Step-by-step explanation:
When considering a monopoly with an inverse demand function P(Q) = 1 - Q and a marginal cost (MC) of zero, we use the process described to find the monopoly's equilibrium quantity, price, and profit.
In Step 1, the monopolist selects the quantity where marginal revenue (MR) equals MC, which is zero in this case. As MR is the derivative of total revenue and given the linear demand curve, MR will also be linear with the same intercept but twice the slope: MR = 1 - 2Q. Setting MR equal to MC, we get: 1 - 2Q = 0, hence Q₁ = 0.5 (the profit-maximizing level of output).
In Step 2, the monopoly decides the price to charge for output level Q₁. By substituting Q₁ = 0.5 into the demand function, we find the price P(Q) = 1 - 0.5 = 0.5 (the profit-maximizing price, P₁). In Step 3, we calculate profit by subtracting total cost from total revenue. Since MC is zero, total cost is zero, and total revenue will simply be Q₁ multiplied by P₁, which equals 0.5 * 0.5 = 0.25. Therefore, the monopoly's profit is 0.25.