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Difficulty: Very hardOligopoly: Characteristics, Types, and Price Interdependence

In a non-collusive oligopolistic market, a leading firm faces a kinked demand curve with two distinct price-demand relationships: for price increases above the current equilibrium, the demand curve is P1=280−2QP_1 = 280 - 2Q; for price cuts below the current equilibrium, the demand curve is P2=400−5QP_2 = 400 - 5Q, where PP is price in Naira (₦) and QQ is output in units. What is the value of the vertical discontinuity (gap) in the firm's marginal revenue curve at the kink equilibrium quantity?

Answer: 120 Naira

Answer

The vertical discontinuity (gap) in the firm's marginal revenue curve at the kink equilibrium quantity is 120 Naira.
At the kink quantity Q=40Q = 40, the marginal revenue curve experiences a vertical jump (discontinuity) because the slope of the demand curve changes abruptly from −2-2 (for price increases) to −5-5 (for price cuts). The upper segment marginal revenue at Q=40Q = 40 is MR1=280−4(40)=120MR_1 = 280 - 4(40) = 120, while the lower segment marginal revenue at Q=40Q = 40 is MR2=400−10(40)=0MR_2 = 400 - 10(40) = 0. Subtracting the lower value from the upper value yields a vertical gap of 120120 Naira.

Step-by-Step Solution

1
Equate the two demand equations to find the kink equilibrium quantity (Q0Q_0).
280−2Q=400−5Q  ⟹  3Q=120  ⟹  Q0=40 units280 - 2Q = 400 - 5Q \implies 3Q = 120 \implies Q_0 = 40\text{ units}.
The kink occurs where the upper elastic demand segment intersects the lower inelastic demand segment.
2
Derive the marginal revenue function for the segment above the kink (MR1MR_1) and evaluate it at Q0=40Q_0 = 40.
TR1=P1⋅Q=280Q−2Q2  ⟹  MR1=dTR1dQ=280−4QTR_1 = P_1 \cdot Q = 280Q - 2Q^2 \implies MR_1 = \frac{dTR_1}{dQ} = 280 - 4Q. At Q=40Q = 40, MR1=280−4(40)=120 NairaMR_1 = 280 - 4(40) = 120\text{ Naira}.
Determines the upper bound of the marginal revenue gap at the kink output.
3
Derive the marginal revenue function for the segment below the kink (MR2MR_2) and evaluate it at Q0=40Q_0 = 40.
TR2=P2⋅Q=400Q−5Q2  ⟹  MR2=dTR2dQ=400−10QTR_2 = P_2 \cdot Q = 400Q - 5Q^2 \implies MR_2 = \frac{dTR_2}{dQ} = 400 - 10Q. At Q=40Q = 40, MR2=400−10(40)=0 NairaMR_2 = 400 - 10(40) = 0\text{ Naira}.
Determines the lower bound of the marginal revenue gap at the kink output.
4
Compute the vertical discontinuity (gap) in the marginal revenue curve.
\text{Gap} = MR_1 - MR_2 = 120 - 0 = 120\text{ Naira}.
The gap represents the range within which marginal cost can fluctuate without altering the firm's optimal price or output level under price rigidity.

Key Concept

Kinked Demand Curve and Marginal Revenue Discontinuity in Oligopoly
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Topics

Oligopoly: Characteristics, Types, and Price Interdependence

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