to that change in quantity sold. This can be formulated as: This can also be represented as a derivative when the change in quantity sold becomes arbitrarily small. Define the revenue function to be where Q is output and P(Q) is the inverse demand function of customers. By the product rule, marginal revenue is then given by where the prime sign indicates a derivative. For a firm facing perfect competition, price does not change with quantity sold so marginal revenue is equal to price. For a monopoly, the price decreases with quantity sold so marginal revenue is less than price for positive (see Example 1). Example 1: If a firm sells 20 units of books (quantity) for $50 each (price), this earns total revenue: P*Q = $50*20 = $1000 Then if the firm increases quantity sold to 21 units of books at $49 each, this earns total revenue: P*Q = $49*21 = $1029 Therefore, using the marginal revenue formula (MR) = Example 2: If a firm's total revenue function is written as Then, by first order derivation, marginal revenue would be expressed as Therefore, if Q = 40, MR = 200 − 2(40) = $120 Marginal revenue curve The marginal revenue curve is affected by the same factors as the demand curve – changes in income, changes in the prices of complements and substitutes, changes in populations, etc. These factors can cause the MR curve to shift and rotate. Marginal revenue curve differs under perfect competition and imperfect competition (monopoly). Under perfect competition, there are multiple firms present in the market. Changes in the supply level of a single firm does not have an impact on the total price in the market. Firms follow the price determined by market equilibrium of supply and demand and are price takers. The marginal revenue curve is a horizontal line at the market price, implying perfectly elastic demand and is equal to the demand curve. Under monopoly, one firm is a sole seller in the market with a differentiated product. The supply level (output) and price is determined by the monopolist in order to maximise profits, making a monopolist a price maker. The marginal revenue for a monopolist is the private gain of selling an additional unit of output. The marginal revenue curve is downward sloping and below the demand curve and the additional gain from increasing the quantity sold is lower than the