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+0.5Q2 Market demand curve is also given by following equation Market Demand: P=95-Q If the monopoly aims to maximize its profit, what price it will charge for each electrical car?
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+0.5Q2
Market demand curve is also given by following equation
Market Demand: P=95-Q
If the monopoly aims to maximize its profit, what price it will charge for each electrical car?
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.
The monopoly's profit is given by its total revenue minus its total cost. The total revenue (TR) is the price (P) times the quantity (Q), and the total cost (TC) is given by the equation 18 + 5Q + 0.5Q^2.
To maximize profit, we need to find the quantity where marginal revenue (MR) equals marginal cost (MC).
First, we find the marginal revenue. The demand curve is P = 95 - Q, so the total revenue is TR = P*Q = (95 - Q)*Q = 95Q - Q^2. The marginal revenue is the derivative of the total revenue with respect to quantity, which is MR = 95 - 2Q.
Next, we find the marginal cost. The total cost is TC = 18 + 5Q + 0.5Q^2, so the marginal cost is the derivative of the total cost with respect to quantity, which is MC = 5 + Q.
Setting MR = MC gives us 95 - 2Q = 5 + Q. Solving for Q gives us Q = 30.
Finally, we substitute Q = 30 into the demand equation to find the price: P = 95 - Q = 95 - 30 = 65.
So, the monopoly will charge $65 for each electrical car to maximize its profit.
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