Give a numerical example to show that a monopolist's marginal revenue can be upward-sloping over part of its range. Hint: The price on the demand curve is the producer's average revenue

Certainly! Let's consider a monopolist who has a demand curve represented by the equation Q = 100 - P, where Q is the quantity and P is the price.

To find the monopolist's average revenue (AR), we divide the total revenue (TR) by the quantity (Q). Total revenue is calculated by multiplying the price (P) by the quantity (Q), so TR = P * Q.

Now, the average revenue formula becomes AR = TR / Q, which further simplifies to AR = (P * Q) / Q = P.

Therefore, the monopolist's average revenue (AR) is simply equal to the price (P) on the demand curve.

To analyze marginal revenue (MR), we calculate the change in total revenue resulting from a one-unit increase in quantity (Q). In mathematical terms, MR = ΔTR / ΔQ.

To establish the relationship between marginal revenue and price, we differentiate the total revenue equation with respect to quantity (TR = P * Q) to obtain MR = dTR / dQ.

In our example, differentiating TR = P * Q gives us MR = (dP * Q + P * dQ) / dQ. Simplifying this gives MR = dP + P * (dQ / dQ) = dP + P.

Since the slope of the demand curve (P) is negative, the marginal revenue equation (MR = dP + P) implies that marginal revenue would be positive if the rate of change in price (dP) is greater than the absolute value of the price (P).

For instance, suppose the monopolist starts at a quantity of 40 and a price of $60. If, due to some change in market conditions such as increased demand, the monopolist increases the quantity to 41 units and the price to $59, the marginal revenue would be positive.

In this case, MR = (59 - 60) + 60 = -1 + 60 = 59.

Hence, we have an example where the monopolist's marginal revenue is upward-sloping over part of its range.