Price Equilibria in a Spatial Competition with Captive Buyers

TL;DR

This paper extends spatial price competition by combining a uniform distribution of informed consumers with linear transportation costs and captive buyers at the line endpoints, building on Varian and Nakagawa. It finds multiple pure equilibria in region-specific locations $(z_1,z_2)$ and three distinct mixed-strategy equilibria (M1–M3) characterized by price distributions with island supports and a common width parameter $w$ that governs undercutting. The analysis shows how captive buyers alter pricing incentives, often preserving profits from captive demand while moderating competition in the informed segment, and it compares equilibrium profits to the Osborne–Pitchik framework. A central region near the market center remains analytically unresolved and is explored numerically, with implications for price dispersion and competitive strategy in spatial markets with captive demand.

Abstract

This paper explores price competition with exogenous product differentiation in a spatial model similar to that of Nakagawa (2023). Nakagawa examines product differentiation within the framework of Varian (1980). Nakagawa integrates Varian's concept of uninformed consumers, who lack complete price information, into a spatial model based on Hotelling (1929). While Nakagawa placed informed consumers at the center of the Hotelling line and used quadratic transportation costs, our study employs a uniform distribution of informed consumers and linear transportation costs. This approach enables a more direct comparison with established spatial competition literature, particularly Osborne and Pitchik (1987). We classify equilibrium candidates and characterize the parameter regions corresponding to each equilibrium. There is no pure equilibrium in the region where we construct mixed strategy equilibria. Furthermore, we compare the expected profit in the equilibrium of our model with the findings of Osborne and Pitchik (1987). Finally, we discuss the impact of captive buyers on the nature of spatial competition.

Figures (13)

• Figure 1: Regions of different types of price equilibria in the $(z_1,z_2)$-plane.
• Figure 2: Equilibrium profits $\pi_1^\star$ and $\pi_2^\star$ are plotted as functions of $(z_1,z_2)$. The contours are spaced by 0.1. The thick contours are drawn at $\pi_j^\star = 0.5$ and $1$.
• Figure 3: Schematic for the pure strategy equilibrium P1, drawn for $(z_1,z_2)=(0.4, 0.7)$.
• Figure 4: Schematic for the pure strategy equilibrium P2, drawn for $(z_1,z_2)=(0.25, 0.85)$.
• Figure 5: Schematic for the pure strategy equilibrium P3, drawn for $(z_1,z_2)=(0.7, 0.8)$.

Theorems & Definitions (46)

• Lemma 1
• Lemma 2
• Proposition 1
• Proposition 2: Pure equilibrium $1$, P1
• Proposition 3: Pure equilibrium $2$, P2
• Proposition 4: Pure equilibrium $3$, P3
• Proposition 5: Pure equilibrium $4$, P4
• Definition 1: Exponential integral
• Lemma 3
• Definition 2