Salop Circular City Model

The Salop circular city model is a canonical model of spatial competition, horizontal product differentiation, and monopolistic competition in business and economics theory, introduced by American economist Steven C. Salop in his 1979 Bell Journal of Economics paper "Monopolistic Competition with Outside Goods". Whereas Hotelling's linear city model (1929) places duopolists along a bounded line segment whose endpoints create asymmetric backyard hinterlands, Salop wrapped the characteristic or geographical space into a closed circle of unit circumference containing an outside competitive good. This circular geometry eliminates boundary asymmetries, enables a clean symmetric analysis of an arbitrary number n of differentiated firms competing with their immediate neighbors, and demonstrates the Excess Entry Theorem: under free entry with fixed setup costs, private profit maximization induces twice the socially cost-minimizing number of firms to enter the industry because entrants capture market share via business-stealing externalities without internalizing the fixed-cost duplication imposed on society.

Circular Product Space and Consumer Preferences

In the Salop model, the product space (representing physical geography, store locations, departure schedules, or a continuous spectrum of product attributes) is a circle of total circumference C = 1. A continuum of consumers of total mass L = 1 is distributed uniformly around the perimeter at unit density. Each consumer is indexed by an ideal location x ∈ [0, 1) along the circumference and demands either one unit of the differentiated good from one of the n active firms or one unit of a homogeneous outside good produced competitively.

Purchasing one unit of the differentiated good from Firm i, located at coordinate x_i on the circle and charging mill price p_i, yields net indirect utility to a consumer located at x:

where v > 0 denotes the consumer's gross intrinsic valuation (reservation price) for their ideal variety, |x − x_i| is the shortest arc distance around the circle between consumer address x and firm location x_i, and t > 0 is the unit transportation cost or characteristic mismatch parameter measuring the intensity of horizontal product differentiation. If the consumer abstains from purchasing the differentiated product and consumes only the outside good, they obtain a baseline reservation surplus s₀ ≥ 0. Defining net reservation valuation as v̂ = v − s₀, a consumer at x is willing to buy from Firm i only if v̂ − p_i − t · |x − x_i| ≥ 0.

Eliminating Boundary Effects of Hotelling's Line

In Hotelling's linear city model on the interval [0, 1], firms located near the endpoints 0 and 1 enjoy captive exterior hinterlands on one side while competing only on their interior flank, making multi-firm (n ≥ 3) location equilibria analytically asymmetric. By identifying the endpoints 0 and 1 so that the commodity space forms a compact manifold without boundary, Salop ensured that every point on the circle is geometrically identical.

Following the standard two-stage entry-then-price specification, suppose n identical firms enter the market and are spaced equidistantly around the unit circle at arc intervals of 1/n (so Firm i is located at x_i = i/n for i = 0, 1, ..., n − 1). Under quadratic transportation costs t · (x − x_i)², Economides (1989) confirmed that equidistant spacing 1/n emerges endogenously as a subgame-perfect location equilibrium because the strategic incentive to soften price competition pushes each firm midway between its left and right neighbors. Each Firm i faces identical production technology with a fixed entry cost F > 0 and constant marginal production cost c ≥ 0, yielding total cost TC(q_i) = F + c · q_i.

Local Monopoly, Kinked Super-Elastic, and Competitive Regions

A foundational contribution of Salop (1979) is the characterization of a firm's piecewise-linear demand curve across three distinct price regimes when all rival firms charge a common symmetric price p and Firm i varies its own price p_i:

1. Local Monopoly Region (High Prices, Uncovered Market): When p_i and p are sufficiently high relative to net valuation v̂ that Firm i's catchment area does not touch the catchment areas of its two neighbors at x_i ± 1/n, the marginal consumer at distance d_m on either side of Firm i is indifferent between buying from Firm i and consuming the outside good:

Summing both sides (for d_m < 1/(2n)) gives local monopoly demand q_i^m(p_i) = 2d_m = 2(v̂ − p_i) / t, with slope ∂q_i^m / ∂p_i = −2/t.

1. Competitive Region (Low Prices, Fully Covered Market): When v̂ is large relative to prices so that every consumer on the circle buys from one of the differentiated firms, Firm i competes directly at its left and right boundaries against its two immediate neighbors at arc distance 1/n. A consumer at distance d_c ∈ (0, 1/n) between Firm i and its neighbor (charging p) is indifferent when delivered prices are equal:

Summing the clockwise and counterclockwise segments of length d_c gives Firm i's competitive demand function:

Here the demand slope ∂q_i^c / ∂p_i = −1/t is half as steep in quantity as in the local monopoly region (−2/t), because each unit reduction in p_i must overcome the rival's diminishing distance advantage t(1/n − d) in addition to Firm i's own rising transport cost t · d.

1. Kinked Super-Elastic Corner Region: At the boundary where the local monopoly catchment radius d_m = (v̂ − p_i)/t meets the neighbor's monopoly boundary 1/n − (v̂ − p)/t, the market transitions from uncovered to covered. Because the demand slope changes discontinuously from −2/t (for price increases) to −1/t (for price cuts), the inverse demand curve exhibits a concave kink at q_i = 1/n, generating a vertical discontinuity in marginal revenue where the symmetric equilibrium price p* = v̂ − t/(2n) remains rigid across an intermediate range of marginal cost c.

Symmetric Bertrand–Nash Price Equilibrium

In the competitive regime (where v̂ > c + 3t/(2n), ensuring full market coverage and an interior equilibrium), Firm i chooses p_i to maximize operating profit less fixed setup cost F, taking rival price p and firm count n as given:

Taking the first-order condition with respect to p_i yields:

Prices are strategic complements (∂p_i/∂p = 1/2 > 0). Imposing symmetry (p_i = p = p*) yields the unique symmetric short-run Bertrand–Nash equilibrium price, output, and net profit:

Higher unit transport or mismatch cost t insulates each store from its neighbors and raises the equilibrium markup p* − c = t/n, whereas an increase in the number of firms n shrinks the inter-firm arc distance 1/n, intensifying localized price competition and driving the markup t/n toward zero as n → ∞.

Free-Entry Monopolistic Competition Equilibrium

In the long run, firms enter or exit the circular city until net economic profit π*(n) reaches zero. Setting π*(n^e) = t / (n^e)² − F = 0 yields the free-entry equilibrium number of firms n^e and long-run equilibrium price p^e:

When total population size is L > 0 (rather than normalized to 1), per-firm profit is π*(n) = L · t / n² − F, so n^e(L) = √(L · t / F) and p^e(L) = c + √(t · F / L). Larger markets (L ↑) support more product varieties (n^e ↑) at lower markups (p^e ↓), a pro-competitive scale effect central to the Krugman New Trade Model.

Social Planner's Cost-Minimizing Variety Choice

To evaluate the Pareto efficiency and social welfare properties of the free-entry equilibrium n^e = √(t/F), consider a social planner who chooses the number of equidistantly spaced firms n to maximize aggregate social surplus in the covered market. Because each consumer demands one unit inelastically and total variable production cost c · 1 = c is independent of n, maximizing social surplus is equivalent to minimizing the sum of aggregate fixed setup costs and aggregate consumer transportation (mismatch) costs:

Differentiating TC^soc(n) with respect to n balances the marginal social cost of adding one more firm (F) against the marginal social benefit of reduced consumer travel distance (t / (4n²)):

Business-Stealing vs. Variety Externalities

Comparing the decentralized free-entry equilibrium n^e = √(t/F) against the socially optimal firm count n^s = (1/2)√(t/F) establishes the Excess Entry Theorem (also formalized in Cournot oligopoly by Mankiw and Whinston, 1986):

Under free entry in the Salop circular city, twice the socially optimal number of firms enter the market. Total social cost under free entry is TC^soc(n^e) = (5/4)√(t · F), exceeding the planner's minimum cost TC^soc(n^s) = √(t · F) by 25%.

In contrast to the Dixit–Stiglitz model (1977), where every firm competes symmetrically with all other varieties, two opposing entry externalities interact on the Salop circle:

1. The Consumer-Surplus (Variety) Externality: A new firm reduces travel distance for nearby consumers, a benefit the entrant cannot capture under uniform mill pricing.
2. The Business-Stealing Externality: In a covered market, 100% of an entrant's sales q = 1/n are diverted from adjacent incumbents, depriving incumbents of the price-cost margin (p* − c) = t/n. Because the private gross profit of entry (t/n²) is four times the marginal social transport saving (t/(4n²)), business stealing dominates and drives excessive entry.

Quadratic Transport Costs and Spatial Price Discrimination

When linear transport cost t · |x − x_i| is replaced by quadratic transport cost t · (x − x_i)² around the circle, the symmetric equilibrium price is p* = c + t/n², per-firm profit is π* = t/n³ − F, and the free-entry firm count is n^e = (t/F)^1/3. Because aggregate social transport cost under quadratic disutility is t / (12*n*²), the planner's optimum is n^s = (t / (6F))^1/3, so excess entry persists with n^e / n^s = 6^1/3 ≈ 1.817. Under spatial price discrimination (Thisse and Vives, 1988), firms observe consumer locations x and quote delivered prices p(x) equal to the second-lowest full marginal cost c + t · d₂(x), lowering average consumer prices relative to uniform mill pricing.

Retail Zoning, Banking, and Labor Market Monopsony

The Salop circular city model serves as a workhorse analytical framework across several applied subfields:

• Banking and Financial Intermediation: Chiappori, Perez-Castrillo, and Verdier (1995) placed banks around a Salop circle to analyze how deposit rate regulation interacts with branch network proliferation and loan pricing.
• Monopsonistic Labor Markets: Bhaskar, Manning, and To (2002) placed workers with heterogeneous commuting or job-attribute preferences around the circle and employers at equidistant nodes, explaining why employers face upward-sloping labor supply curves and enjoy wage-setting monopsony power (w* = MRP_L − t/n).

Why did Salop use a circle instead of Hotelling's straight line?

On a bounded straight line [0, 1], firms near the two endpoints possess a captive outer segment where they face no rival store, making any market with n ≥ 3 firms asymmetric because interior firms are sandwiched between two rivals while peripheral firms have a protected flank. By bending the line into a closed circle of circumference 1, Salop removed endpoints altogether so that all n equidistantly spaced firms face identical competitive environments with two neighbors at distance 1/n.

Why does free entry lead to too many firms in the Salop model?

In a covered market where total industry output is fixed at 1, a newly entering firm does not expand total industry consumption; instead, it diverts customers away from adjacent incumbent firms (business stealing). The entrant decides to enter if its private revenue margin t/n² covers its fixed setup cost F, ignoring the profit loss inflicted on rival stores. From society's perspective, the only benefit of an additional firm is the reduction in consumer travel distance (t/(4n²)), which is one-quarter of the entrant's private gross profit (t/n²), resulting in n^e = 2n^s.

How does the Salop model differ from the Dixit–Stiglitz model of monopolistic competition?

In the Dixit–Stiglitz model (1977), a representative consumer has constant-elasticity-of-substitution (CES) preferences over all n varieties simultaneously, competition is non-localized, and the equilibrium markup over marginal cost depends on the constant elasticity of substitution σ regardless of n. In the Salop circular city model, each consumer buys a single ideal variety, competition is localized between immediate spatial neighbors, and the equilibrium markup p* − c = t/n declines pro-competitively as n grows.

1. Salop, S. C. (1979). "Monopolistic Competition with Outside Goods." The Bell Journal of Economics, 10(1), 141–156. https://doi.org/10.2307/3003323
2. Mankiw, N. G., & Whinston, M. D. (1986). "Free Entry and Social Inefficiency." The RAND Journal of Economics, 17(1), 48–58. https://doi.org/10.2307/2555627
3. Economides, N. (1989). "Symmetric Equilibrium Existence and Optimality in Differentiated Product Markets." Journal of Economic Theory, 47(1), 178–194. https://doi.org/10.1016/0022-0531(89)90108-790108-7)
4. Thisse, J.-F., & Vives, X. (1988). "On the Strategic Choice of Spatial Price Policy." American Economic Review, 78(1), 122–137. https://www.jstor.org/stable/1814702
5. Bhaskar, V., Manning, A., & To, T. (2002). "Oligopsony and Monopsonistic Competition in Labor Markets." Journal of Economic Perspectives, 16(2), 155–174. https://doi.org/10.1257/0895330027300