A monopoly faces a demand curve given by P = 100 - 0.5Q, where P is the price and Q is the quantity. Calculate the total revenue and marginal revenue when the quantity sold is 40 units.
Question
A monopoly faces a demand curve given by P = 100 - 0.5Q, where P is the price and Q is the quantity. Calculate the total revenue and marginal revenue when the quantity sold is 40 units.
Solution
To calculate the total revenue, we need to multiply the price (P) by the quantity (Q). In this case, the price is given by P = 100 - 0.5Q and the quantity sold is 40 units.
Substituting Q = 40 into the demand curve equation, we get P = 100 - 0.5(40) = 100 - 20 = 80.
Therefore, the price when 40 units are sold is 80.
To calculate the total revenue, we multiply the price (80) by the quantity (40):
Total Revenue = Price * Quantity = 80 * 40 = 3200.
So, the total revenue when 40 units are sold is 3200.
To calculate the marginal revenue, we need to find the change in total revenue when the quantity sold increases by one unit.
Let's calculate the total revenue when the quantity sold is 41 units. Substituting Q = 41 into the demand curve equation, we get P = 100 - 0.5(41) = 100 - 20.5 = 79.5.
Therefore, the price when 41 units are sold is 79.5.
To calculate the marginal revenue, we subtract the total revenue when 40 units are sold (3200) from the total revenue when 41 units are sold.
Marginal Revenue = Total Revenue (Q = 41) - Total Revenue (Q = 40) = (Price (Q = 41) * Quantity (Q = 41)) - (Price (Q = 40) * Quantity (Q = 40)) = (79.5 * 41) - (80 * 40) = 3259.5 - 3200 = 59.5.
So, the marginal revenue when the quantity sold is 40 units is 59.5.
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