📈 economics

From Horizontal to Vertical Differentiation: Global Markov Equilibria and Regime Transitions

This paper develops a dynamic model of quality competition to characterize global Markov-perfect equilibria, demonstrating how endogenous quality investments drive regime transitions between horizontal, vertical, and intermediate competition forms, with the intermediate regime potentially emerging as a stable long-run outcome under specific transportation cost conditions.

Original authors: Didier Laussel

Published 2026-09-18

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Didier Laussel

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of business, companies rarely compete on just one thing. They often try to stand out in two distinct ways. First, they might offer products that are simply different in style or location, catering to specific tastes; this is like choosing between a red car and a blue car, where neither is objectively better, just different. Second, they might compete on quality, where one product is undeniably superior to the other, like a luxury watch versus a basic timepiece. For decades, economists have studied these two types of competition separately, treating them as static snapshots. They have also developed a general rule of thumb suggesting that when firms compete, they will try to maximize their differences in style while minimizing their differences in quality, effectively avoiding a direct battle over who makes the "best" product.

However, the real world is rarely static. Companies constantly invest in improving their products, and those improvements change the nature of the competition over time. A firm might start by competing on location, but as it pours money into making its product significantly better, the competition shifts to a battle of quality. This dynamic evolution raises a difficult question: how do these shifts happen, and what does the long-term balance look like when firms are constantly adjusting their strategies? Understanding this requires looking beyond a single moment in time to see the entire journey of how competition evolves.

Didier Laussel, an economist at Aix-Marseille Université, tackled this problem by building a mathematical model that simulates how two competing firms invest in quality while simultaneously setting prices. He did not treat the different types of competition as separate, isolated scenarios. Instead, he created a single, continuous system where the rules of the game change automatically based on how far apart the firms are in terms of quality. In his model, the state of the market can be "horizontal," where differences in style dominate; "vertical," where differences in quality dominate; or "intermediate," where both factors matter equally. The core of his work is to trace the path a market takes as it moves between these states, rather than just calculating the outcome at a single point.

The study reveals that finding a stable long-term outcome is much more complex than simply checking if a firm is winning or losing at a specific moment. The researchers found that for a market to settle into a stable pattern, the transition between these different states must be perfectly smooth. If the path from a quality-focused market to a style-focused market has a sudden jump or a break, the entire scenario falls apart and cannot exist in reality. This means that the existence of a stable outcome depends on the ability to connect these different worlds seamlessly. The model shows that simply having a stable point in one area does not guarantee that the market will actually reach it; the path leading there must be valid and unbroken.

When the cost for consumers to travel between different product options is low, the market tends to settle into a state where quality differences are the main driver. In this scenario, one firm eventually pulls ahead, and the competition becomes purely about who has the better product. However, as the cost of travel increases, this straightforward path breaks down. The model shows that at a certain point, the smooth connection between the high-quality state and the low-quality state disappears. Even though a high-quality outcome is still theoretically possible, the market cannot reach it through a stable, continuous path. This creates a gap where no clear long-term equilibrium exists, suggesting that the market might be in a state of constant flux or instability.

Surprisingly, the study finds that the "middle ground" can become the permanent home for the market. In many traditional economic theories, the intermediate state—where firms compete on both style and quality—is seen only as a temporary phase, a stepping stone between two extremes. Laussel's simulations show that under specific conditions, this intermediate state can become the final destination. When transportation costs reach a certain level, the market settles into a stable pattern where both firms maintain a moderate quality difference and compete on both dimensions. This outcome is not a fleeting moment but a durable, long-run equilibrium where the firms coexist in a balanced, mixed form of competition.

The research identifies four critical thresholds for transportation costs that dictate these shifts. Below the first threshold, the market is dominated by quality competition. Between the first and second thresholds, the smooth path to a quality-dominated future vanishes, leaving the market without a clear global solution. Between the second and third thresholds, a new, stable intermediate state emerges, but it cannot yet connect smoothly to the other states. It is only when costs rise past the third threshold that this intermediate state becomes the organizing center of the entire market, creating a valid path that links the low-quality, intermediate, and high-quality worlds. Finally, if costs rise too high, even this intermediate path breaks down, and the market reverts to a local, unstable state.

These findings challenge the long-held belief that firms will always try to maximize their differences in style while minimizing differences in quality. The model demonstrates that when firms invest dynamically over time, they can end up in a stable state where they are neither maximally different nor minimally different. The intermediate state, often dismissed as a transient phase in static models, proves to be a robust and attractive long-term outcome in a dynamic world. The study suggests that the history of how firms invest and the specific costs consumers face determine whether the market ends up in a race for quality, a battle of styles, or a stable middle ground.

The robustness of these results was tested by changing the rates at which product quality wears out and how much firms value future profits. While the exact numbers for the critical thresholds shifted with these changes, the overall pattern remained the same. The sequence of transitions—from a quality-dominated world to a broken path, then to a stable intermediate world, and finally to a breakdown of that stability—held true across different scenarios. This suggests that the phenomenon is not a fluke of a specific set of numbers but a fundamental feature of how quality competition evolves. The study concludes that to truly understand market competition, economists must look at the entire journey of the market, ensuring that the path between different states is smooth and unbroken, rather than just analyzing the destination.

Technical Summary: From Horizontal to Vertical Differentiation: Global Markov Equilibria and Regime Transitions

Problem Statement
This paper investigates a dynamic model of quality competition between two firms located antipodally on a circle. The firms simultaneously compete in prices and continuously invest in product quality. The core problem addresses how the nature of competition evolves endogenously as the quality differential ( $\Delta$ ) between firms changes relative to transportation costs ( $\tau$ ). While static literature often treats horizontal (location-based) and vertical (quality-based) differentiation as distinct or alternative strategies, this model posits that the market can shift between three distinct regimes:

Horizontal Regime: Small quality differentials where competition is driven primarily by location.
Intermediate Regime: Moderate differentials where both dimensions interact.
Vertical Regime: Large differentials where competition is driven primarily by quality.

The central challenge is that equilibrium conditions are regime-dependent. The static profit functions, and consequently the dynamic equations governing investment, change across these regions. The paper seeks to characterize the global equilibrium dynamics generated by endogenous transitions across these regimes, rather than merely analyzing local steady states within isolated regimes.

Methodology
The study employs a differential game framework to model the continuous time evolution of the quality differential.

State and Control: The state variable is the quality differential $\Delta(t)$ . Firms choose quality adjustment efforts ( $i_A, i_B$ ) to maximize the discounted stream of net profits, subject to a depreciation rate $\mu$ .
Markov Perfect Equilibrium (MPE): The analysis focuses on Markov strategies where investment decisions depend only on the current state $\Delta$ . The solution is derived using the Hamilton-Jacobi-Bellman (HJB) equations.
Unified Dynamic System: A key methodological insight is that the system of coupled nonlinear differential equations governing the marginal values (costates $p$ and $q$ ) is identical across all three regimes. The regimes differ only in the specific functional forms of the profit derivatives ( $\Phi$ and $\Psi$ ).
Global Admissibility Conditions: To define a valid global equilibrium, the paper imposes two critical conditions at the regime boundaries:
1. Smooth Pasting: The costates (marginal values of quality) must be continuous across the boundaries separating the horizontal, intermediate, and vertical regimes. This ensures continuous investment policies and state dynamics.
2. No-Bubbles/Transversality: The solution must satisfy rational expectations, ensuring that the present value of marginal future profits converges appropriately (ruling out explosive trajectories or bubbles).

Key Contributions

Global Equilibrium Continuation: The paper argues that the relevant equilibrium object is not a collection of local MPEs but a "global equilibrium continuation" that connects the horizontal, intermediate, and vertical regions of the state space. The equilibrium selection problem is reframed as a problem of global continuation rather than local stability.
Regime-Dependent Dynamics: It demonstrates that the dynamic problem can be represented by a unified system of differential equations whose coefficients depend on the prevailing competition regime, allowing for the analysis of transitions between fundamentally different forms of competition.
Endogenous Transitions: The analysis reveals how endogenous quality investment can induce transitions between horizontal, intermediate, and vertical competition forms, challenging the static view that intermediate configurations are merely transient.

Results
The numerical analysis identifies four critical thresholds for the transportation cost parameter ( $\tau$ ) that divide the parameter space into five qualitatively distinct regions:

Low $\tau$ ( $\tau < \tau_1$ ): A unique globally admissible equilibrium continuation originates from the vertical steady state. This trajectory smoothly connects the vertical, intermediate, and horizontal regimes. The vertical steady state acts as the organizing center of the dynamics.
Intermediate $\tau$ ( $\tau_1 < \tau < \tau_2$ ): The global continuation generated by the vertical steady state disappears due to a singularity within the intermediate regime, even though the vertical steady state itself remains locally well-defined. No globally admissible continuation is identified in this range.
Emergence of Intermediate Steady State ( $\tau_2 < \tau < \tau_3$ ): An interior steady state emerges within the intermediate regime. While locally stable, its associated trajectory fails to connect smoothly to the horizontal regime, making it only locally attractive.
Global Attractiveness ( $\tau_3 < \tau < \tau_4$ ): The intermediate steady state becomes the organizing center of the dynamics. A globally admissible trajectory connects the horizontal, intermediate, and vertical regimes. In this range, the intermediate regime constitutes a genuine long-run equilibrium outcome.
High $\tau$ ( $\tau > \tau_4$ ): The intermediate steady state reverts to being a local attractor. Its continuation into the vertical regime violates the no-bubble condition (the trajectory moves away from the vertical steady state rather than converging to it), rendering it globally inadmissible.

The paper also notes that for low transportation costs, the results contradict the "max-min principle" of Neven and Thisse (1990) and Irmen and Thisse (1998), which suggests firms differentiate maximally horizontally and minimally vertically. Here, an intermediate steady state with neither maximal nor minimal vertical differentiation can be the long-run outcome.

Significance and Claims
The paper claims that the existence of a steady state is insufficient to characterize the long-run equilibrium of an industry; what matters is whether the local dynamics can be extended into a globally admissible continuation satisfying smooth-pasting and transversality conditions.

Methodological Significance: The analysis highlights that in dynamic games with regime-dependent dynamics, equilibrium selection is fundamentally a global problem. Local stability analysis alone cannot predict the emergence of intermediate regimes as long-run outcomes.
Robustness: The qualitative organization of the equilibrium dynamics (the sequence of transitions and the existence of globally admissible continuations) remains stable across variations in discount rates and depreciation rates, despite significant shifts in the numerical values of the critical thresholds.
Theoretical Implication: The study suggests that intermediate configurations, often viewed as transient in static models, may become the stable, long-run equilibrium in dynamic settings where quality investment is endogenous.

The paper concludes by noting that a complete analytical characterization of the critical thresholds remains an open direction for future research, as does extending the framework to more firms or multidimensional product characteristics.

Technical Summary: From Horizontal to Vertical Differentiation: Global Markov Equilibria and Regime Transitions

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