Explain the rigidity of prices in an oligopoly in terms of the discontinuous MR curve.

An oligopoly is a market structure where a small number of firms dominate the market, with each firm having a significant share. These firms are interdependent, meaning that the actions of one firm directly affect the others, and vice versa. Oligopolies are characterized by barriers to entry, which make it difficult for new firms to enter the market and compete with existing firms.

In an oligopoly, the demand curve for a firm's product is typically kinked. The demand curve is more elastic (flatter) above the kink, where the firm can increase its sales by lowering its prices without affecting its rivals' prices. Below the kink, the demand curve is less elastic (steeper), as the rivals are expected to match any price reduction, so the firm cannot gain a significant increase in sales by lowering its prices. Now, let's construct the MR curve. The marginal revenue is the additional revenue a firm receives from selling one more unit of its product. It can be found by taking the derivative of the total revenue with respect to output (quantity sold). A key characteristic of a kinked demand curve is that it results in a discontinuous MR curve.

Due to the kinked demand curve in an oligopoly, we have two different segments – the upper segment, which is flatter and more elastic, and the lower segment, which is steeper and less elastic. For the upper segment, the MR curve is the tangent to the demand curve at the kink, but at a lower level, as the MR is always less than the corresponding price in the elastic segment. For the lower segment, the MR curve is below the demand curve and follows the same slope as the demand curve, since we assume that the rivals will match any price decrease. Now, there is a gap between the two segments of the MR curve resulting from the kink in the demand curve. This gap is known as the discontinuous MR curve.

Due to the discontinuous MR curve, any change in the firm's marginal costs (MC) that occurs within the gap does not affect the equilibrium price and output level. This is because when the MC curve intersects the MR curve within this gap, the MC curve could move up or down within the gap, and the intersection point would still be the same, implying the same quantity and price as before. Therefore, even if the cost conditions change and the MC curve shifts within the gap, the firms maintain their equilibrium price and output level, making prices rigid in an oligopoly market. This is consistent with the observation that prices in oligopolistic markets tend to be sticky and do not change frequently.

The rigidity of prices in an oligopoly can be explained in terms of the discontinuous MR curve. Due to the kinked demand curve, the MR curve is discontinuous. Hence, when a firm's marginal costs change, the firm can maintain the same equilibrium price and output level, as long as the change in costs remains within the discontinuous gap of the MR curve. This phenomenon leads to price rigidity in an oligopolistic market, where prices remain stable and do not fluctuate frequently.