Mathematical Modeling in Planning

Economic Base Model: Multipliers and Dependency Ratio Concepts

The Economic Base Model is a foundational tool in regional economic analysis. It distinguishes between two types of industries:

• Basic (Export) Industries: Those that generate income by exporting goods or services outside the region.

• Non-Basic (Local Serving) Industries: Those that serve the local population, recycling income within the region.

Multipliers:

• The multiplier effect quantifies how an initial injection of spending in basic industries circulates through the local economy. Mathematically, if the multiplier is k, then an initial basic export income of I results in a total income of k × I in the region.

• The multiplier is often derived from the ratio of total employment or output to basic employment or output, reflecting indirect and induced economic activities.

Dependency Ratio:

• This ratio represents the proportion of non-basic activities to basic activities. A higher dependency ratio suggests that the local economy is more reliant on local consumption, whereas a lower ratio indicates a strong export base.

• Formally, if NB represents non-basic output and B represents basic output, the dependency ratio might be defined as NB/B. This ratio informs planners about the region’s vulnerability and growth potential.

Economic Base Model: Location Quotient (LQ) Technique

The Location Quotient (LQ) is a relative measure used to identify the degree of specialization in a particular industry within a region compared to a larger reference area (often the national economy).

Mathematical Formulation:

• The LQ is calculated as:

• An LQ greater than 1 indicates that the region is specialized in that industry (often considered a basic industry), while an LQ less than 1 implies that the industry is less prominent locally than nationally.

Applications in Planning:

• Planners use LQ to identify key economic drivers and to develop targeted strategies for economic development, diversification, or enhancement of competitive strengths.

3. Economic Base Model: Minimum Requirements Method

Principles of the Method:

The Minimum Requirements Method is used to estimate the local or non-export demand component of an industry. It involves determining the minimum level of production necessary to satisfy local consumption needs.

Procedure:

• First, determine the “minimum requirements” by identifying the lowest level of output that supports local consumption.

• The residual, above this minimum, is considered as export or basic output.

• In practical terms, if total employment or output is known, the model subtracts the estimated non-basic (minimum) portion to derive the basic component.

Implications for Regional Growth:

• This method helps to clarify the extent to which additional growth in an industry would benefit the broader regional economy versus merely satisfying existing local demand.

Fixed Share/Variable Share Technique

This technique divides total economic activity into two parts:

• Fixed Share: The portion of output or employment that remains constant regardless of economic expansion. It reflects the baseline, local-serving activities.

• Variable Share: The part that can change in response to external demand (exports), representing the growth potential of the basic sector.

Application:

• By estimating these shares, planners can model how shifts in external economic conditions (like increased demand for exports) will affect overall regional economic performance.

• For example, if a region’s economic structure is such that 70% is fixed and 30% is variable, an increase in external demand will primarily affect the variable portion, amplifying the multiplier effect.

Introduction to Gravity Models

Foundational Concepts:

Gravity models in planning are analogous to Newton’s law of gravity in physics. They predict the interaction between two entities (cities, regions, or zones) based on their “mass” (e.g., population, employment) and the distance between them.

Mathematical Expression:

A basic gravity model can be expressed as:

where:

• Tij = interaction (e.g., trips, trade flows) between locations i and j,

• Pi and Pj = measures of “mass” (such as population or economic activity) at i and j,

• Dij​ = distance between i and j,

• k = constant,

• α, β, γ = parameters calibrated based on empirical data.

Uses in Planning:

• These models are applied to predict travel patterns, analyze retail catchment areas, or model trade flows between regions.

Single-Constraint Gravity Model for Retail Trade Location

Specific Focus:

This model variant is tailored for analyzing retail trade locations. It applies the gravity model under the constraint that either the supply (retail capacity) or the demand (consumer population) is fixed.

Mechanism:

• The model estimates the potential customer base for a given retail outlet by weighing the attractiveness (e.g., store size, variety) against the friction of distance.

• Typically, the constraint might be on the total available consumer population, ensuring that the sum of estimated customer flows across all outlets equals the known market size.

Application Example:

• Retail chains use this model to determine optimal store locations by identifying areas with high accessibility and sufficient customer density, while accounting for competitive overlaps.

Hansen Model

Developed by Walter Hansen, this model is a seminal approach in spatial interaction modeling. It focuses on the accessibility of opportunities (jobs, services) based on travel impedance.

Key Formulation:

• The Hansen accessibility index is typically formulated as:

where:

• Ai = accessibility of location i,

• Oj = opportunities at location j (e.g., employment),

• cij = cost or travel time between i and j,

• f(cij) = an impedance function that typically increases with cij​.

Implications for Site Selection:

• This model helps identify locations that maximize access to opportunities, crucial for retail, housing, or public facility placement.

Lowry-Garin Model

The Lowry-Garin model is an advanced urban land use and transportation model that integrates economic, demographic, and spatial dynamics.

Core Elements:

• It simulates the distribution of households and employment across an urban region, considering the trade-offs between accessibility, housing costs, and job opportunities.

• The model typically uses iterative algorithms to balance land use and transportation demand, capturing the feedback loops between them.

Relevance in Planning:

• By predicting how changes in transportation infrastructure or zoning regulations affect urban form, the Lowry-Garin model supports comprehensive urban planning and policy evaluation.

General Transportation Models

Fundamental Structure:

General transportation models are comprehensive frameworks used to simulate travel behavior and network performance. They typically consist of four interrelated components:

• Trip Generation: Estimating the number of trips originating and ending in different zones.

• Trip Distribution: Allocating these trips between origins and destinations (often using gravity models).

• Mode Choice: Determining the mode of travel (e.g., car, public transit) based on factors like cost, time, and convenience.

• Route Assignment: Assigning trips to specific paths through the transportation network, often using shortest path or equilibrium models.

Applications:

• These models guide infrastructure investments, inform transit planning, and help assess the impact of policy changes on congestion and environmental outcomes.

Dual-Constraint Gravity Trip Distribution Model

Enhanced Gravity Modeling:

The dual-constraint (or doubly constrained) gravity model refines the basic gravity model by ensuring that both the total number of trips produced by each origin and the total trips attracted by each destination are satisfied.

Mathematical Formulation:

• The model adjusts the basic gravity formula with balancing factors:

where:

• Tij = trips from origin i to destination j,

• Ai​ and Bj are balancing factors ensuring the row (production) and column (attraction) totals match observed values,

• Pi​ and Pj represent the trip generation potentials,

• f(cij) is the impedance function as before.

Practical Use:

• In urban transportation planning, this model is used to generate more accurate trip distribution matrices, which in turn inform traffic forecasting, transit service design, and road network improvements.

Concluding Synthesis

Mathematical models in planning provide a structured, quantitative approach to decision-making and policy evaluation. The Economic Base Model and its associated techniques (multipliers, dependency ratios, location quotient, minimum requirements, and fixed/variable share methods) help planners understand and forecast regional economic dynamics. Meanwhile, gravity models—including specialized versions like the single-constraint model for retail, the Hansen model, and the dual-constraint trip distribution model—are essential tools for analyzing spatial interactions and travel behavior. Models like the Lowry-Garin and general transportation frameworks integrate these insights into comprehensive planning tools that support sustainable urban development.

Collectively, these models enable planners to simulate complex systems, evaluate trade-offs, and design interventions that optimize resource allocation, accessibility, and economic growth, ensuring that urban and regional development is both efficient and responsive to emerging challenges.