Land Use in a Long, Narrow City: The Economics of Urban Stretch

Picture a city shaped like a ribbon—long and narrow, stretching out like a path through the wilderness. There’s a center, sure, but not much width to spread. In such cities, like coastal strips, riverbanks, or transport corridors, land use isn’t random—it’s a dance of distance, economics, and optimization.

The Basic Setup

Let’s define the spatial layout: the city is a line segment of length L, and each point x ∈ [0, L] represents a location along this line. People and businesses make decisions based on:

• Distance from city center (x = 0)
• Land rent at each location
• Commuting costs—they increase with distance
• Utility from location

What emerges is a spatial equilibrium. Each resident or firm chooses where to locate based on trade-offs. The further you go, the cheaper the land—but the pricier the commute.

Mathematical Formulation

Suppose we want to optimize total utility of land use along the city:

Maximize: ∫₀ᴸ [ U(x) - C(x) ] ρ(x) dx  
Subject to: ∫₀ᴸ ρ(x) dx = P  (fixed population)  
           ρ(x) ≥ 0  (non-negative density)
  

Where:

• U(x) is the utility gained at point x (depends on land, proximity, etc.)
• C(x) is the commuting or congestion cost at x
• ρ(x) is the population density at x
• P is total population

Why Density Isn’t Uniform

In a long, narrow city, high densities tend to cluster around economic hubs (usually the midpoint or boundary, depending on the setup). Farther out, land gets cheaper—but fewer people want the inconvenience. This leads to falling density profiles—sharper in cities with higher transport costs.

Euler Equation in Land Use Optimization

This is a problem for the calculus of variations. Define the Lagrangian:

L = [ U(x) - C(x) ] ρ(x) + λ (P - ∫ρ(x) dx)
  

Take the functional derivative. Solve the Euler–Lagrange condition. The result tells us the shape of ρ(x). Sensitivity analysis—by shifting parameters like commuting cost—reveals how the optimal population shifts.

Real World: From Hong Kong to Valparaíso

Urban corridors like Hong Kong’s north shore or Chile’s Valparaíso fit this model. Land is constrained. Choices are compressed into a one-dimensional layout. Public transit, rent pricing, and zoning decisions all hinge on how population density distributes optimally.

Conclusion: Cities on a Wire

Land use in narrow cities isn’t an accident. It’s an optimization problem—woven with economics, constrained by space, and powered by calculus. The math beneath the urban form reveals a hidden order.

Want to plan a better city? Start by solving for equilibrium. The edge isn’t just physical—it’s mathematical.