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Answer :
Final answer:
The equation for this monopolist's marginal revenue is MR = 5P/4 - 25.
Explanation:
To find the equation for a monopolist's marginal revenue, we need to understand the relationship between price, quantity demanded, and marginal revenue. The demand curve facing a monopolist is given by the equation Qd = 100-4P, where Qd represents the quantity demanded and P represents the price of the product.
Now, to find the monopolist's marginal revenue, we need to differentiate the total revenue function with respect to quantity. Total revenue (TR) is given by the product of price (P) and quantity demanded (Qd), so TR = P * Qd.
To find the marginal revenue (MR), we differentiate the total revenue function with respect to quantity (Qd). In this case, the derivative of TR with respect to Qd is the marginal revenue (MR).
Let's differentiate TR = P * Qd with respect to Qd:
MR = d(TR)/d(Qd) = d(P * Qd)/d(Qd) = P + Qd * dP/d(Qd)
Since the demand curve is given by Qd = 100-4P, we can substitute this into the equation for MR:
MR = P + (100-4P) * dP/d(Qd)
Now, to simplify the equation, we need to find dP/d(Qd). Taking the derivative of the demand curve with respect to P:
dP/d(Qd) = -1/4
Substituting this back into the equation for MR:
MR = P + (100-4P) * (-1/4)
Simplifying further:
MR = P - (100-4P)/4
MR = P - 25 + P/4
Combining like terms:
MR = 5P/4 - 25
Therefore, the equation for this monopolist's marginal revenue is MR = 5P/4 - 25.
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