Question

How can a monopolist identify the profit-maximizing level of output if it knows its marginal revenue and marginal costs?

Short answer

A monopolist can identify the profit-maximizing level of output by setting its marginal revenue (MR) equal to its marginal cost (MC) and solving for the output level: \[ MR(q) = MC(q) \] To confirm that this output level maximizes profit, check that the second derivative of the profit function with respect to quantity (q) is negative, or analyze the region around the identified output level to ensure that increasing or decreasing output would result in a decrease in profit.

Step-by-step solution

Understanding Marginal Revenue and Marginal Cost

Marginal revenue (MR) refers to the additional revenue a monopolist earns by producing and selling one extra unit of output. Marginal cost (MC) refers to the additional cost incurred in producing that extra unit of output. In perfect competition, firms set their output levels where Price (P) equals Marginal Cost (MC) to maximize profit, but for a monopolist, the profit maximization occurs where Marginal Revenue (MR) equals Marginal Cost (MC).

Setting up the profit-maximization condition

For a monopolist to maximize its profit, the additional revenue earned from selling one more unit should be equal to the additional cost incurred in producing that unit. Mathematically, this condition can be represented as: \[ MR = MC \] This equation shows that the profit-maximizing level of output for a monopolist occurs where the marginal revenue equals the marginal cost.

Deriving the profit-maximizing level of output

Since the monopolist knows its marginal revenue (MR) and marginal costs (MC), it can determine the profit-maximizing level of output by finding the output level (q) where the MR and MC curves intersect or when their derivative are equal. 1. First, set the MR function equal to the MC function: \[ MR(q) = MC(q) \] 2. If available, take the derivative of both the MR and MC functions with respect to quantity (q), set them equal, and then solve for the profit-maximizing output level q*: \[ \frac{dMR(q)}{dq} = \frac{dMC(q)}{dq} \] 3. If necessary, use numerical or graphical methods to solve for the profit-maximizing output level (q*).

Confirming the profit-maximizing level of output

Once you have found the profit-maximizing output level (q*), you should verify that this level of output does indeed maximize profit. To do this, check that: 1. The second derivative of the profit function with respect to quantity (q) is negative, which confirms that the profit function is concave and has a maximum point at q*: \[ \frac{d^2\pi(q)}{dq^2} < 0 \] 2. Alternatively, analyze the region around q* to confirm that increasing or decreasing the output level from q* would result in a decrease in profit. This can be done by comparing the MR and MC at output levels slightly below and above q*. In conclusion, a monopolist can identify the profit-maximizing level of output by setting its marginal revenue equal to its marginal cost and solving for the output level. Confirm that this output level maximizes profit by analyzing the properties of the profit function or the behavior of MR and MC around the identified output level.