Kinked Demand Curve and Marginal Revenue

Introduction

In our previous lesson, we explored the conceptual underpinnings of the kinked demand curve model, establishing why oligopolistic firms might be hesitant to change prices. We saw how the assumption of an asymmetric response from rivals—matching price cuts but not price hikes—creates a "kink" in a firm's demand curve. The key consequence, as we saw graphically, is a vertical gap in the marginal revenue (MR) curve, which is the reason prices can remain stable even when costs change.

Building directly on that foundation, this lesson moves from the conceptual to the quantitative. Your goal is to calculate the segmented marginal revenue curve implied by a kinked demand curve. This is a crucial skill for your CB2 exam, as it allows you to precisely determine the range of marginal costs over which a firm will not change its price or output. Leveraging your background in statistics and quantitative methods, we will work through the algebra of deriving the MR function from a piecewise demand function.

From Demand to Marginal Revenue: A Mathematical Review

Before tackling the kinked curve, let's briefly review the mathematical relationship between demand and marginal revenue for a standard linear demand curve. As you know from your prior studies on monopoly, this is a core building block.

A linear inverse demand curve is typically expressed in the form:

P = a - b Q

where $P$ is price, $Q$ is quantity, $a$ is the price intercept, and $b$ is the absolute value of the slope.

Total Revenue (TR) is price multiplied by quantity:

T R = P \times Q = (a - b Q) \times Q = a Q - b Q^{2}

Marginal Revenue (MR) is the first derivative of Total Revenue with respect to quantity, representing the change in total revenue from selling one additional unit. Using calculus:

M R = \frac{d ( T R )}{d Q} = \frac{d}{d Q} (a Q - b Q^{2}) = a - 2 b Q

Notice a simple but powerful rule emerges: for a linear demand curve $P = a - b Q$ , the corresponding marginal revenue curve is $M R = a - 2 b Q$ . The MR curve has the same price intercept ( $a$ ) but twice the slope ( $- 2 b$ ). This rule is the primary tool we will use to solve today's problem.

Analyzing the Kinked Demand Curve Numerically

In the previous lesson, we saw that the kinked demand curve is actually a combination of two different demand curves: a more elastic one for price increases and a more inelastic one for price decreases. This means we are dealing with a piecewise function.

Let's use a concrete numerical example based on the following diagram, which illustrates a firm in an oligopoly facing a kink in its demand curve (D) at the prevailing price of $1 and quantity of 10,000 units.

This graph shows a kinked demand curve (D) at the point (Q=10,000, P=\$1). The corresponding marginal revenue curve (MR) has a vertical discontinuity at Q=10,000.

Our task is to find the mathematical equations that describe the MR curve, including the gap. To do this, we must first define the two segments of the demand curve.

1. Define the Piecewise Demand Function

Based on the graph, let's define the two linear functions that constitute the kinked demand curve.

Upper Segment (Elastic): This segment applies to quantities less than or equal to 10,000 (i.e., for any potential price increase from $1). From the graph, this line appears to pass through the points (Q=0, P=$1.50) and (Q=10,000, P=$1).
- The intercept is $a_{1} = 1.5$ .
- The slope is $b_{1} = \frac{Δ P}{Δ Q} = \frac{1 - 1.5}{10 , 000 - 0} = \frac{- 0.5}{10 , 000} = - 0.00005$ .
- So, the equation for this segment is: $P = 1.5 - 0.00005 Q$ .
Lower Segment (Inelastic): This segment applies to quantities greater than 10,000 (i.e., for any price decrease from $1). This line is steeper. Let's assume it passes through (Q=10,000, P=$1) and would hit the quantity axis at Q=20,000 (where P=0).
- The slope is $b_{2} = \frac{Δ P}{Δ Q} = \frac{0 - 1}{20 , 000 - 10 , 000} = \frac{- 1}{10 , 000} = - 0.0001$ .
- Using the point-slope form with (10,000, 1): $P - 1 = - 0.0001 (Q - 10, 000)$ .
- $P = - 0.0001 Q + 1 + 1$ , which simplifies to $P = 2 - 0.0001 Q$ .

Combining these, we get our piecewise demand function:

P = {1.5 - 0.00005 Q 2 - 0.0001 Q for 0 \leq Q \leq 10, 000 for Q > 10, 000

2. Derive the Segmented Marginal Revenue Curve

Now, we apply the "twice the slope" rule to each segment of the demand function separately.

Upper MR Segment:
- For the demand segment $P = 1.5 - 0.00005 Q$ , the corresponding marginal revenue is:
- $M R_{1} = 1.5 - 2 (0.00005) Q = 1.5 - 0.0001 Q$ .
- This equation is valid for the range $0 \leq Q \leq 10, 000$ .
Lower MR Segment:
- For the demand segment $P = 2 - 0.0001 Q$ , the corresponding marginal revenue is:
- $M R_{2} = 2 - 2 (0.0001) Q = 2 - 0.0002 Q$ .
- This equation is valid for the range $Q > 10, 000$ .

So, the full segmented marginal revenue curve is:

M R = {1.5 - 0.0001 Q 2 - 0.0002 Q for 0 \leq Q \leq 10, 000 for Q > 10, 000

3. Calculate the Marginal Revenue Gap

The discontinuity, or vertical gap, in the MR curve exists at the kink quantity, $Q = 10, 000$ . To find the size and location of this gap, we evaluate each MR function at this quantity.

Top of the Gap: We use the upper MR function, $M R_{1}$ , and find its value as $Q$ approaches 10,000 from below.
$M R_{1} (10, 000) = 1.5 - 0.0001 (10, 000) = 1.5 - 1 = $0.50$
Bottom of the Gap: We use the lower MR function, $M R_{2}$ , and find its value as $Q$ approaches 10,000 from above.
$M R_{2} (10, 000) = 2 - 0.0002 (10, 000) = 2 - 2 = $0$

At the quantity of 10,000 units, the marginal revenue curve drops vertically from $0.50 to $0. This vertical segment is the gap.

The profound implication is that if the firm's marginal cost (MC) is anywhere between $0 and $0.50, the profit-maximizing rule (MC=MR) is met at the quantity $Q = 10, 000$ . The firm will therefore continue to produce 10,000 units and charge the kink price of $1. This calculation gives a precise quantitative meaning to the price rigidity predicted by the model.

Conclusion

In this lesson, we have moved beyond the graphical intuition of the kinked demand curve model to perform the underlying calculations. By defining the piecewise demand function around the kink, you learned how to derive the corresponding segmented marginal revenue curve and precisely calculate the vertical gap between its segments.

Key Takeaways:

A kinked demand curve is a piecewise function composed of two linear demand segments with different slopes.
The marginal revenue curve is also piecewise, derived by applying the $M R = a - 2 b Q$ rule to each demand segment separately.
To find the MR gap, you must evaluate the upper MR function and the lower MR function at the kink quantity ( $Q^{*}$ ). The difference between these two values defines the vertical discontinuity.
This calculated gap represents the range within which the firm's marginal cost can fluctuate without causing a change in the profit-maximizing price or quantity.

Having analyzed a non-cooperative model of oligopoly that explains price stability, we will next turn our attention to cooperative models. In the following lesson, we will investigate cartels, where firms attempt to collude to act like a monopolist. We will use game theory and payoff matrices to analyze why such collusive agreements are inherently unstable.