Consider a single-price monopoly that produces electrical cars. Suppose you are told that the monopoly has the following cost curves where TC is total cost measured in dollars, Q is the quantity of electrical cars, and P is the price per electrical car in dollars:Total Cost: 18+5Q+0.5Q2Market demand curve is also given by following equationMarket Demand: P=95-QIf the monopoly aims to maximize its profit, what price it will charge for each electrical car? (Round your answer to two decimal places.)
Question
Consider a single-price monopoly that produces electrical cars. Suppose you are told that the monopoly has the following cost curves where TC is total cost measured in dollars, Q is the quantity of electrical cars, and P is the price per electrical car in dollars:Total Cost: 18+5Q+0.5Q2Market demand curve is also given by following equationMarket Demand: P=95-QIf the monopoly aims to maximize its profit, what price it will charge for each electrical car? (Round your answer to two decimal places.)
Solution
To find the price that the monopoly will charge for each electrical car, we first need to find the quantity that maximizes the monopoly's profit.
Step 1: Find the Monopoly's Marginal Cost (MC) The marginal cost is the derivative of the total cost function with respect to quantity. The total cost function is given as TC = 18 + 5Q + 0.5Q^2. The derivative of this function is MC = 5 + Q.
Step 2: Find the Monopoly's Marginal Revenue (MR) The marginal revenue is the derivative of the revenue function with respect to quantity. The revenue function is PQ, and P is given as 95 - Q. So, the revenue function is R = Q(95 - Q) = 95Q - Q^2. The derivative of this function is MR = 95 - 2Q.
Step 3: Set MC = MR and Solve for Q To maximize profit, a firm will produce up to the point where marginal cost equals marginal revenue. So, we set MC = MR and solve for Q:
5 + Q = 95 - 2Q 3Q = 90 Q = 30
Step 4: Substitute Q into the Demand Function to Find P Finally, we substitute Q = 30 into the demand function P = 95 - Q to find the price that the monopoly will charge:
P = 95 - 30 = $65
So, the monopoly will charge $65 for each electrical car to maximize its profit.
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