Introduction to Land Use & Urban Economic Modeling

CIVE 990T

Introduction

Transportation & Land Use

Urban Structure

• Bourne (1982) defines urban structure as combination of:
  • Urban form - spatial configuration of fixed elements such as buildings, transportation network, & other infrastructure
  • Urban interactions - flows of goods, people, information, & money
  • Organizing principles - relationshihps between urban form & interactions such as travel cost, social status, & segregation

Spatial Firm Competition

Firm Competition

• Price

• Service

• Quality

• Location

Von Thünen’s Isolated State

Central Place Theory (CPT)

• Attributed to Christaller (1966) & Lösch (1954)
• Elements:
  • Single homogeneous good, \(g\), is sold
  • Good sold at fixed unit price, \(p\)
  • Travel possible in any direction at uniform cost, \(r\), $/mile
  • Total cost to buyer is good price + transport cost \[ C = p + rd\]
  • Stores free to locate anywhere on flat, featureless plane
  • Uniform population density over plane
  • Free market entry/exit
  • Constant production functions (costs) for all firms

CPT Demand Function

Market Area

Given uniform populationd density, \(\rho\), and travel in all directions at constant unit cost, one can integrate distance-based demand function to compute total sales at a location.

\[S = \int_{\theta=0}^{2\pi} \int_{x=0}^{D}f(d) \rho dx d\theta\]

For a store to remain in business: \(S > S_{min} =\) sales threshold for good \(g\)

Multiple Firms & Market Equilibrium

• Given free market entry, firms will continue to enter market as long as \(S > S_{min}\)
• CPT assumes firms locate to maximize profit
• Firms pack together into hexagonal lattice, each generating sales \(S_{min}\)
• Market equilibrium distribution - also minimizes average travel distance for consumers

Market Equilibrium Lattice Structure

Spatial Residential Competition

• Monocentric city model
• Elements:
  • r = commuting distance & residential location on a line or radius
  • z = amount of composite consumption good
  • T(r) = cost of commuting from r to 0 (and back)
  • R(r) = land rental rate at location r
  • s = amount of rented space
  • Y = income
• Fundamental household problem \[max_{z,s}U(z,s)\text{ such that } z+R(r)s = Y-T(r))\]
• All households have same utility at equilibrium (or would move)

Bid-Rent Approach

• Rather than solve system, use equal utility assumption to derive bid-rent function \(\Psi(r,u)\) - maximum willingness-to-pay by household for unit of space at location \(r\) given utility level \(u\)
• Invert budget constraint \[\Psi(r,u)=max_s \frac{Y-T(r)-z}{s(r,u)} \text{ such that U(z,s) = u}\]
• Equivalently \[\Psi(r,u)=max_s \frac{Y-T(r)-z(s(r,u),u)}{s(r,u)}\]

Bid-Rent Approach

• By envelope theorem \[\frac{\partial\Psi(r,u)}{\partial r}=\frac{T'(r)}{s(r,u)}\]
• Using implicit function theorem on \(U(s,z)=u\) \[\frac{\partial z(s,u)}{\partial s}=\frac{-U_s}{U_z}=-\Psi(r,u)\]
• Stochastic bid rent, assuming Type I errors gives \[Pr(h|r,u)=\frac{exp(\Psi_h(r,u))}{\sum_h^H exp(\Psi_h(r,u))}\]

Operational Land Use Models

Classic Models

• Lowry model (1964)
• DRAM/EMPAL (1967)
• TRANUS & MEPLAN (1970s)

Modern Models

• IRPUD (Germany)
• RELU-TRANS (USA)
• MUSSA (South America)
• UrbanSim (USA)
• PECAS (Canada)
• SILO (Germany)
• ILUTE (Canada)