Which of the following is a feature of Lösch’s economic landscape that is absent from Christaller’s scheme?

A separate hexagon net for each good, with city-rich and city-poor sectors and no rigid hierarchy

Hexagonal market areas based on the idea that consumers always buy at the nearest centre offering the good

A separate hexagon net for each good, with city-rich and city-poor sectors and no rigid hierarchy

An isotropic plain with population and purchasing power spread evenly in all directions

A fixed K value that stays the same at every level of the hierarchy of central places

Which of the following statements on regions are correct?

A river-basin area chosen for integrated development is a planning region

A semi-arid tract with uniform rainfall and soils is a formal (uniform) region

The commuting catchment of a metropolis is a functional (nodal) region

A river-basin area chosen for integrated development is a planning region

A region is functional when it is bounded by uniformity in a single trait such as soil

A semi-arid tract with uniform rainfall and soils is a formal (uniform) region

The commuting catchment of a metropolis is a functional (nodal) region

A country’s largest city has about five times the population of its second city, and dominates its politics, finance and culture, with few intermediate cities. This is best described as

a primate city with a high primacy index

a city that conforms closely to the rank-size rule

a primate city with a high primacy index

a Lösch metropolis with six city-rich sectors

a functional region delineated by commuter flows

Which of the following statements on Burgess’s concentric zone model are correct?

It assumes a single centre on a uniform plain with an even transport surface

Growth proceeds by invasion and succession, pushing each zone outward

The zone in transition lies just outside the central business district and has decaying housing and new immigrants

It assumes a single centre on a uniform plain with an even transport surface

Growth proceeds by invasion and succession, pushing each zone outward

High-rent housing extends in wedge-shaped sectors along rail lines

The zone in transition lies just outside the central business district and has decaying housing and new immigrants

A city’s high-rent residential area began on one side near a river road and has since extended outward in the same direction, while industry has followed the rail lines. Which model best describes this pattern?

Burgess’s concentric zone model

Harris and Ullman’s multiple nuclei model

Lowry’s activity allocation model

Under Reilly’s law of retail gravitation, the breaking point between a large city and a small city lies

nearer the smaller city, so the larger city’s trade area reaches further

exactly midway between the two cities, whatever their sizes

nearer the large city, since it has more shops to offer

nearer the smaller city, so the larger city’s trade area reaches further

at distances divided between the cities in direct proportion to their populations

Under Weber’s least-cost theory, which industry is most likely to locate close to its market rather than its raw-material source?

a bottling plant that mixes ubiquitous water with a little concentrate

a sugar mill crushing cane, which loses most of its weight in processing

a plant concentrating a low-grade ore, which discards most of the rock

a bottling plant that mixes ubiquitous water with a little concentrate

a cement works using heavy limestone that loses carbon dioxide on burning

Which of the following are assumptions of Weber’s least-cost location theory?

Localised raw materials occur at known points

There is a single market of fixed demand at a known point

Transport cost is proportional to weight times distance

The firm varies its price and market area to maximise profit

Localised raw materials occur at known points

There is a single market of fixed demand at a known point

Transport cost is proportional to weight times distance

Two sellers of the same product with fixed prices compete on a straight beach with customers spread evenly. According to Hotelling, they will locate

together at the centre, though the quarter points would give a shorter average walk

at the two ends of the beach, as far apart as the beach allows them to be

at the quarter points, which is both the equilibrium and the social optimum

wherever raw materials are cheapest, whatever the position of the rival seller

A large, fast-growing industry with strong forward and backward linkages that stimulates other firms to grow around it is, in Perroux’s terms,

a propulsive industry at the heart of a growth pole

a footloose industry with a material index of one

a marginal industry on the edge of the critical isodapane

a lower-order centre inside a K = 3 hexagon

Which of the following statements on the rank-size rule and primacy are correct?

A strongly primate pattern has the largest city well above the rank-size line

With q = 1, the second-largest city has half the population of the largest

With q = 1, the product of a city’s rank and its population is constant

A strongly primate pattern has the largest city well above the rank-size line

Under the rank-size rule the two-city primacy index equals 1

With q = 1, the second-largest city has half the population of the largest

With q = 1, the product of a city’s rank and its population is constant

Which pairing of a Christaller principle with its K value is correct?

administrative principle, K = 7

administrative principle, K = 3

administrative principle, K = 7

Urban Geography, Growth and Location: Central Places, Regions, Growth Models and Industrial Location

This chapter answers the middle of AR Part B2, Section B2.1: “Theories of urban geography; Regional delineation; Settlement hierarchy; Urban growth models; Industrial location theories”. It teaches Christaller’s central place theory (K = 3, 4, 7) and Lösch; the formal, functional and planning region; the settlement hierarchy with the rank-size rule and the primate city; the Burgess, Hoyt, Harris–Ullman and Lowry models; the gravity model and Reilly’s law; and the industrial location theories of Weber, Hotelling, Lösch, Isard and Perroux. The numericals are hexagon counts, rank-size populations, Reilly breaking points, Weber weight and isodapane arithmetic, and the Lowry multiplier.

1. Central place theory: Christaller and Lösch

Walter Christaller (1933, from a study of southern Germany) asked why towns differ in size and why they are spaced as they are. His answer: a central place exists to supply goods and services to the surrounding area. Higher-order goods (university, specialist hospital, jewellery) are needed rarely, so only a few large places offer them; lower-order goods (groceries, primary school) are needed daily and are offered everywhere. Two ideas control everything. The range of a good is the maximum distance people will travel for it; the threshold is the minimum market needed to keep the seller in business. A good is offered only where the population within its range exceeds its threshold.

Assumptions: an isotropic plain (uniform in every direction); evenly spread population and purchasing power; transport cost proportional to distance and equal in every direction; consumers rational, buying from the nearest centre that offers the good; sellers earn only normal profit, so no area is left unserved.
Why hexagons: circles around centres leave gaps or overlap; squares and triangles are less compact. The hexagon is the shape closest to a circle that tiles the plane with no gap and no overlap.

The K value is the number of next-lower-order market areas inside the market area of a higher-order centre, counting the one at its own site. Counting is the whole trick: a lower-order centre wholly inside the higher-order hexagon counts 1, one shared between two higher-order hexagons counts one half, and one shared between three counts one third.

The three Christaller principles
Principle	K	Where the six surrounding lower-order centres sit	Count of market areas	Market areas by order
Marketing	3	At the corners of the higher-order hexagon, each shared by three higher-order areas	1 + 6 × ⅓ = 3	1, 3, 9, 27
Transport	4	At the midpoints of its edges, each shared by two; main roads join higher places through them	1 + 6 × ½ = 4	1, 4, 16, 64
Administrative (separation)	7	Wholly inside it, none shared, so administrative boundaries are not split	1 + 6 × 1 = 7	1, 7, 49, 343

Counting centres. With one top-order city and n orders in all, the number of centres of the r-th order (r = 1 is the top) is (K − 1) × K^(r − 2) for r ≥ 2. So K = 3 gives 1, 2, 6, 18; K = 4 gives 1, 3, 12, 48; K = 7 gives 1, 6, 42, 294. Every centre also serves the lowest-order market, so the lowest-order market areas total K^(n − 1): 27 for K = 3 with four orders (1 + 2 + 6 + 18), 64 for K = 4, 49 for K = 7 with three orders. Sizes: a market area is K times the area one order below, and the spacing between centres of an order is √K times the spacing one order below, so 5 km between villages becomes 5√3 ≈ 8.7 km, 10 km or 5√7 ≈ 13.2 km between the towns for K = 3, 4, 7.

Lösch (1940) starts from demand rather than supply. Each good has its own hexagonal market area, so the hexagon nets of different goods can take any K of the form a² + ab + b²: 3, 4, 7, 9, 12, 13 and so on. The nets are rotated about one common metropolis, giving six city-rich and six city-poor sectors. Unlike Christaller, places of the same size can offer different sets of goods, and there is no rigid nested hierarchy. Both theories assume an even plain and ignore agglomeration, terrain and transport networks, and they fit rural, agricultural regions best.

⚠️ Which K means what

K = 3 (marketing) gives the most centres for an area, because each lower-order centre is shared among three higher places. K = 7 (administrative) gives the fewest, because lower centres are wholly enclosed. K = 4 (transport) is the one with centres on the main roads. Do not mix up the K value, which counts market areas, with the number of centres of a given order, which excludes the higher place itself.

2. Regional delineation and settlement hierarchy

Planning needs a bounded area with a shared problem or potential. A region is an area with some unity, and the basis of that unity gives three kinds. Boudeville’s three-fold scheme (homogeneous, polarised, planning space) is the same classification in other words.

Types of region
Type	Basis	Boundary found by	Example
Formal (uniform, homogeneous)	Sameness in one or more traits: rainfall, soil, crop, language, income	Map overlay, composite index, cluster or factor analysis; boundaries are zones of transition	A semi-arid tract with uniform rainfall and soils; a tribal belt
Functional (nodal)	Interaction and flows organised around a node	Commuting, trade, bus, telephone or media flows; boundary where the pull of another node takes over (breaking point, gravity model)	Commuter-shed of a metropolis; hinterland of a port
Planning (programming)	Defined for policy and administration; often blends the two types above	Administrative units (districts, states), because data and execution follow them	The National Capital Region; a metropolitan region; a river basin; a backward-area programme region

The NCR is an Indian example (the National Capital Region Planning Board Act of 1985), and the Metropolitan Planning Committee created under the 74th Constitutional Amendment is the legal mechanism for planning a metropolitan area as one unit. Regions also nest: macro (a group of states), meso (state or sub-state) and micro (district or block).

Settlement hierarchy ranks places by size and by the functions they perform: hamlet, village, market town, town, city, metropolis. In regional planning the hierarchy is read from the functions present, using scalogram analysis (settlements as rows, functions as columns, a mark where a function exists) and a weighted centrality score. The Census of India separates statutory towns (a municipality, corporation, cantonment board or notified town area committee) from census towns, which need a population of at least 5,000, at least 75 percent of the male main workers in non-agricultural work, and a density of at least 400 per sq km. Towns are classed by size, Class I being 100,000 and above, down to Class VI below 5,000; a city of one million or more is a million-plus city.

Rank-size rule (Auerbach; Zipf): the n-th largest city has a population P_n = P_1 / n^q, and with q = 1 the product of rank and size is constant. If the largest city has 12 million, the second has 6, the third 4, the fourth 3 and the fifth 2.4; the four largest together have 12 × (1 + ½ + ⅓ + ¼) = 25 million. On a log-log plot the cities fall on a straight line of slope −1. A primate city (Mark Jefferson, 1939) is far larger than the second and dominates national life: the two-city index P_1 / P_2 is about 2 under the rank-size rule, and values well above 2 signal primacy, while the four-city index P_1 / (P_2 + P_3 + P_4) is 12/13 ≈ 0.92 under the rule. Primacy is common in small, centralised or formerly colonial countries with a single port or capital, whose gateway city absorbed investment.

⚠️ Law or regularity

The rank-size rule is an empirical regularity, not a law of nature. A country can fit it well, which is taken as a sign of a mature, integrated urban system, or break it with a primate city or with several equal-sized cities. Check which exponent q the question gives before dividing: P_n = P_1 / n only when q = 1.

3. Urban growth models: Burgess, Hoyt, Harris–Ullman and Lowry

Three classic models describe how an American-style city arranges its land uses, each correcting the one before. All three are descriptions, built from North American cities of their time.

Burgess (1925), concentric zones, from Chicago. Growth proceeds outward from the centre by invasion and succession, and the city forms five rings: (1) the central business district; (2) the zone in transition, with factories, decaying housing and new immigrants, waiting to be absorbed by the business district; (3) the zone of workers’ homes; (4) better residential area; (5) the commuter zone. The pattern assumes one centre, flat uniform land and an even transport surface; rent falls with distance from the centre.

Hoyt (1939), sector model, from rent data on 142 American cities. Land uses spread in wedge-shaped sectors along radial transport routes: once a high-rent residential sector begins on one side of the city it extends outward in the same direction, low-rent housing lies in other sectors, and industry follows rail lines and rivers. The explanation is that high-income groups choose the best land and leave older housing to be filtered down to poorer households.

Harris and Ullman (1945), multiple nuclei. With the car and growth of the city, activities cluster round several nuclei, not one: the central business district, wholesale and light industry, heavy industry, residential districts of different classes, outlying business centres and suburbs. Four forces make the nuclei: activities need specialised facilities (a rail terminal for wholesaling); like activities gain by clustering (agglomeration); some uses repel each other (disassociation, as with heavy industry and high-class housing); and some cannot afford the highest rents at the centre.

The three classical models
Model	Shape	Driver	Main limitation
Burgess 1925	Rings round one centre	Invasion and succession; rent gradient	Ignores terrain, routes and several centres
Hoyt 1939	Wedges along routes	Radial transport; high-income choice and filtering	Still assumes one centre
Harris–Ullman 1945	Patchwork of nuclei	Specialised facilities, clustering, repulsion, rent	Descriptive: no fixed pattern to predict

The Lowry model (Ira Lowry, 1964, for the Pittsburgh region) is not a descriptive pattern but a working land use model. Its three sectors are basic employment (export industry, whose location is given), service employment (retail and local services) and households. Basic jobs generate households, placed round the jobs by a gravity-type rule; households generate demand, so service jobs are placed near households by another gravity rule; the service jobs bring more households, and the loop repeats until it converges. For regional totals the loop has a closed form. If each job supports α persons and each person supports β service jobs, total employment is E = B / (1 − αβ) and population P = αE. Take B = 10,000 basic jobs, α = 2.5 and β = 0.2: E = 10,000 / (1 − 0.5) = 20,000 jobs and P = 50,000 persons, of whom the service jobs are 0.2 × 50,000 = 10,000.

⚠️ Sector is not zone

The two names are easy to swap: Burgess gives zones (rings, one city centre), Hoyt gives sectors (wedges along transport routes), Harris and Ullman give nuclei (several centres). If a question says that the high-rent area has grown in one direction along a main road and railway, the model is Hoyt, whatever the rings suggest.

4. Spatial interaction: the gravity model and Reilly’s law

The gravity model borrows Newton: interaction between two places (trips, trade, migration, calls) rises with their sizes and falls with the distance between them, T_ij = k P_i P_j / d_ij^b. The exponent b measures the friction of distance; with b = 2, doubling the distance cuts the interaction to a quarter. The model is the basis of spatial interaction analysis, of the trip-distribution step in transport planning and of the Lowry model above.

Reilly’s law of retail gravitation (1931) applies the model to shopping. Two cities A and B attract trade from an intermediate town in direct proportion to their populations and in inverse proportion to the square of their distances from the town: B_A / B_B = (P_A / P_B) × (d_B / d_A)². The breaking point is where the two pulls are equal, so the ratio is 1: P_A / d_A² = P_B / d_B², hence d_B / d_A = √(P_B / P_A). With d_A + d_B = D, the distance between the cities, the breaking point lies at

Breaking-point formulas
Distance measured from	Formula	Reading
The larger city A	d_A = D / (1 + √(P_B / P_A))	Larger than D/2: A’s sphere extends further
The smaller city B	d_B = D / (1 + √(P_A / P_B))	Smaller than D/2: the point lies near B

Worked example 1. P_A = 900,000, P_B = 100,000, D = 120 km. √(P_B / P_A) = √(1/9) = 1/3, so d_A = 120 / (1 + 1/3) = 90 km and d_B = 30 km. Worked example 2. P_A = 250,000, P_B = 160,000, D = 81 km: √0.64 = 0.8, so d_A = 81 / 1.8 = 45 km and d_B = 36 km. Worked example 3 (trade share). A town lies 30 km from A (P_A = 400,000) and 20 km from B (P_B = 100,000). B_A / B_B = 4 × (20/30)² = 16/9, so A draws 16/25 = 64 percent of its retail trade and B 36 percent.

⚠️ The two common slips

First, the breaking point is not at a distance proportional to population: sharing 120 km in the ratio 9 : 1 would give 108 km, but the square root gives 90 km. Second, check which city the formula measures from. The point always lies nearer the smaller city, so a d_A smaller than D/2 for the larger city is a sign of an error.

Limits: straight-line distance replaces real road distance and time; shops differ in quality and price; and the exponent 2 is an assumption (Huff’s later model gives probabilities, not a sharp line). The result still serves for first-cut market areas, hospital and mall catchments and for drawing the boundary of a functional region.

5. Industrial location I: Weber’s least-cost theory

Alfred Weber (1909) asked where a single plant will locate to minimise cost. Three factors pull it: transport cost, which depends on weight and distance, labour cost and agglomeration. His assumptions: a uniform plain; a single market of fixed demand at a known point; raw materials either ubiquitous (found everywhere, such as water) or localised (found at known points); labour at fixed points with unlimited supply at a fixed wage; perfect competition; and transport cost proportional to weight × distance at a uniform rate.

The key tool is the material index, MI = weight of localised materials ÷ weight of the finished product.

Material index and the pull of the plant
MI	Process	Plant tends to locate	Typical case
Greater than 1	Weight-losing: much of the material is lost in processing	At the raw-material source	Ore concentration, sugar from cane
Equal to 1	Pure material: all its weight stays in the product	Anywhere on the line between source and market	Transport cost is the same everywhere on the line
Less than 1	Weight-gaining: ubiquitous materials such as water are added	At the market	Bottled drinks, bakery goods

The locational triangle. With a market M and two sources R1 and R2, give each corner a weight: the tonnes of that material needed per tonne of product at the sources, and 1 tonne of product at the market. The least-transport-cost point is where the three pulls balance (a mechanical frame with weights on strings, the Varignon frame, finds it). One shortcut decides many questions: if one corner’s weight is equal to or greater than the sum of the other two, the plant sits at that corner. Example: 3 t of material from R1 and 1 t from R2 make 1 t of product, so the corner weights are 3, 1 and 1. Since 3 is greater than 1 + 1 = 2, the plant goes to R1. With R1–R2 = 60 km, R1–M = 90 km and a rate of ₹2 per tonne-km, the cost per tonne of product is 2 × (1 × 60 + 1 × 90) = ₹300, because the 3 t from R1 do not travel at all.

Labour and isodapanes. An isotim joins points of equal transport cost to a point; an isodapane joins points of equal additional transport cost over the least-cost point. A cheap-labour town pulls the plant away only if its labour saving is greater than the extra transport it causes, that is, if the town lies inside the critical isodapane, the one whose extra cost equals the labour saving. Weber measured labour’s power by the labour coefficient, labour cost per tonne of product divided by the total weight moved: the higher it is, the more easily labour pulls. Example: the weight moved is 3 t per tonne of product at ₹2 per tonne-km, so each kilometre of deviation costs 3 × 2 = ₹6 per tonne of product; if the labour saving is ₹90, the critical isodapane lies at 90 / 6 = 15 km. Agglomeration works the same way: if the savings from clustering exceed the extra transport cost, firms cluster inside their overlapping critical isodapanes; congestion costs act as the reverse (deglomeration).

🎯 Why weight matters most

Weber put transport first because every tonne has to be carried: the cheapest place is the one that moves the least weight over the least distance. Weight-losing industries therefore stand at the source, which is why sugar mills sit among cane fields, and weight-gaining ones at the market. Labour and agglomeration are only deviations from that base.

Criticism. The theory is normative and supply-side: it ignores demand, market area and competition, assumes one plant and one product, fixed wages, uniform transport rates and perfect knowledge, and says nothing of governments or of how real managers decide. Hotelling, Lösch and the behavioural school fill these gaps.

6. Industrial location II: Hotelling, Lösch, behavioural views, Isard and growth poles

Hotelling (1929) adds competition. Customers are spread evenly along a linear beach (a street, a highway) and buy from the nearer of two sellers with fixed prices. Each seller gains by moving next to the rival to take more customers, so the sellers end up back to back at the centre: the principle of minimum differentiation. This is poor for customers: with both at the centre of a beach of length L the average walk is L/4, but with the sellers at the quarter points (L/4 and 3L/4) each serves one half and the average walk is L/8, half as much. The same logic explains why rival shops and parties’ programmes look alike.

Lösch (1940) locates for maximum profit, not minimum cost: the plant should stand where the market area it can capture, given a spatial demand cone (quantity bought falls as price plus transport rises), is large enough to earn the most profit. Weber’s question is the cost of producing and moving; Lösch’s is how much a place can sell, and competitive entry shrinks market areas into hexagons until excess profit vanishes.

The behavioural view (Pred, 1967; Herbert Simon’s idea of satisficing) says that real firms have incomplete information and limited ability to use it, so they choose a location that is good enough, not the optimum. David Smith’s spatial margins of profitability show the same thing: any site between the two margins where cost equals revenue is profitable, and the manager’s choice within that area is shaped by personal and other non-economic factors.

Isard (1956) generalised location theory by treating transport as one input that can be substituted for the others (more transport against less land, labour or materials), and, with colleagues, developed industrial complex analysis: a set of technically interdependent activities at one site, such as a refinery, petrochemical and fertiliser plants, whose joint location saves transport and storage. Growth poles come from François Perroux (1950s). A propulsive industry, large, fast-growing, innovative and strongly linked forward and backward, makes other firms grow around it and polarises growth. Boudeville gave the idea geographical meaning, and regional planners use it as the growth-centre strategy: invest in a few chosen centres and hope the benefits spread. Hirschman expected such trickle-down spread effects; Myrdal warned of backwash effects, in which the pole drains labour and capital from the periphery (cumulative causation).

Location theories at a glance
Theory	Orientation	Chooses the site that	Key variable
Weber 1909	Cost, supply side	minimises transport, labour and agglomeration cost	Weight × distance; material index
Hotelling 1929	Competition	captures the larger share of customers against a rival	Rival’s location; fixed prices
Lösch 1940	Demand, profit	maximises profit through its market area	Spatial demand cone
Behavioural (Pred, Smith)	Decision-maker	is good enough within the margins of profitability	Information and ability
Isard; Perroux	Linkage and growth	clusters interdependent activities; stimulates growth round a propulsive industry	Linkages; polarisation

Key takeaways

Central place theory: range and threshold, hexagonal market areas; K = 3 marketing, K = 4 transport, K = 7 administrative; centres of the r-th order = (K − 1) K^(r − 2); market area × K and spacing × √K per order. Lösch allows any K = a² + ab + b² and gives city-rich and city-poor sectors.
Regions are formal (uniform), functional (nodal, flows) or planning (policy and administrative units). A town is statutory by its local body and a census town by 5,000 persons, 75 percent male non-agricultural main workers and 400 per sq km density.
Rank-size: P_n = P_1 / n (for q = 1); the four largest cities with a 12 million top city total 25 million. A primate city has a two-city index well above 2.
Burgess = concentric zones, Hoyt = sectors along routes, Harris–Ullman = multiple nuclei. Lowry: basic jobs drive households and service jobs; E = B / (1 − αβ).
Reilly: B_A / B_B = (P_A / P_B)(d_B / d_A)²; breaking point d_A = D / (1 + √(P_B / P_A)), always nearer the smaller city. Example: 900,000 and 100,000 at 120 km gives 90 km and 30 km.
Weber: material index above 1 means near the source, below 1 near the market; a corner whose weight is at least the sum of the others takes the plant; critical isodapane = labour saving ÷ extra cost per km. Hotelling: minimum differentiation, optimum at the quarter points; Lösch: profit; Perroux: growth poles; Isard: industrial complexes.

Practice questions (20)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

In a Christaller landscape following the administrative principle (K = 7) there is one top-order city, and centres of the second, third and fourth orders below it. The number of centres of the third order is ____.
Numerical answer — type the value.
Show answer
Answer: 42
The centres of the r-th order number (K − 1) K^(r − 2): for r = 2 that is 6, and for r = 3 it is 6 × 7 = 42. Check: the whole four-order system has 1 + 6 + 42 + 294 = 343 = 7³ lowest-order market areas.
In Christaller’s central place theory, the minimum market (number of customers or purchasing power) needed to keep a seller of a good in business is called its
1. range
2. threshold
3. hinterland
4. market potential
Show answer
Answer: B — threshold
The threshold is the minimum demand that makes supply viable. The range is the maximum distance customers will travel for the good; a good appears in a place only if the population within its range exceeds its threshold. Range and threshold are the pair most often confused.
Which of the following is a feature of Lösch’s economic landscape that is absent from Christaller’s scheme?
1. Hexagonal market areas based on the idea that consumers always buy at the nearest centre offering the good
2. A separate hexagon net for each good, with city-rich and city-poor sectors and no rigid hierarchy
3. An isotropic plain with population and purchasing power spread evenly in all directions
4. A fixed K value that stays the same at every level of the hierarchy of central places
Show answer
Answer: B — A separate hexagon net for each good, with city-rich and city-poor sectors and no rigid hierarchy
Lösch lets every good have its own market-area net (K = 3, 4, 7, 9, 12 and so on) and rotates the nets about a metropolis, so places of the same size can offer different goods. Hexagonal areas and an even plain are shared with Christaller, and a single fixed K at every level is Christaller’s feature, not Lösch’s.
Which of the following statements on regions are correct?
1. A river-basin area chosen for integrated development is a planning region
2. A region is functional when it is bounded by uniformity in a single trait such as soil
3. A semi-arid tract with uniform rainfall and soils is a formal (uniform) region
4. The commuting catchment of a metropolis is a functional (nodal) region
Show answer
Answer: A — A river-basin area chosen for integrated development is a planning region; C — A semi-arid tract with uniform rainfall and soils is a formal (uniform) region; D — The commuting catchment of a metropolis is a functional (nodal) region
The river-basin, semi-arid-tract and commuting-catchment statements match the definitions: delineation for policy (planning), uniformity (formal) and flows round a node (functional). The statement about uniformity in a single trait such as soil describes a formal region, not a functional one.
In a country that follows the rank-size rule with exponent q = 1, the largest city has 12 million people. The combined population of its four largest cities, in millions, is ____.
Numerical answer — type the value.
Show answer
Answer: 25
P_n = 12 / n gives 12, 6, 4 and 3 million. Sum = 25 million. Check: 12 × (1 + 1/2 + 1/3 + 1/4) = 12 × 25/12 = 25.
A country’s largest city has about five times the population of its second city, and dominates its politics, finance and culture, with few intermediate cities. This is best described as
1. a city that conforms closely to the rank-size rule
2. a primate city with a high primacy index
3. a Lösch metropolis with six city-rich sectors
4. a functional region delineated by commuter flows
Show answer
Answer: B — a primate city with a high primacy index
The two-city index here is 5, far above the value of 2 expected under the rank-size rule, which is the signature of a primate city. A city that conformed to the rank-size rule would have about half the largest city’s population in the second city.
Which of the following statements on Burgess’s concentric zone model are correct?
1. It assumes a single centre on a uniform plain with an even transport surface
2. Growth proceeds by invasion and succession, pushing each zone outward
3. High-rent housing extends in wedge-shaped sectors along rail lines
4. The zone in transition lies just outside the central business district and has decaying housing and new immigrants
Show answer
Answer: A — It assumes a single centre on a uniform plain with an even transport surface; B — Growth proceeds by invasion and succession, pushing each zone outward; D — The zone in transition lies just outside the central business district and has decaying housing and new immigrants
The single centre on a uniform plain, invasion and succession, and the zone in transition are features of the Burgess model. Wedge-shaped sectors of high-rent housing along transport routes belong to Hoyt’s sector model.
A city’s high-rent residential area began on one side near a river road and has since extended outward in the same direction, while industry has followed the rail lines. Which model best describes this pattern?
1. Hoyt’s sector model
2. Burgess’s concentric zone model
3. Harris and Ullman’s multiple nuclei model
4. Lowry’s activity allocation model
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Answer: A — Hoyt’s sector model
Directional growth of high-rent housing along a radial route, with industry on rail lines, is Hoyt’s sector pattern. Burgess would show closed rings, Harris and Ullman several separate centres, and Lowry is a working model that allocates jobs and households, not a pattern of this type.
Two cities A and B are 120 km apart. A has 900,000 people and B has 100,000. Using Reilly’s law, the distance of the breaking point from the smaller city B, in km, is ____.
Numerical answer — type the value.
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Answer: 30
d_B = D / (1 + √(P_A / P_B)) = 120 / (1 + √9) = 120 / 4 = 30 km. Check from A: d_A = 120 / (1 + √(1/9)) = 120 / (4/3) = 90 km, and 90 + 30 = 120. Sharing the distance in proportion to population (12 km from B) is the common slip.
Under Reilly’s law of retail gravitation, the breaking point between a large city and a small city lies
1. exactly midway between the two cities, whatever their sizes
2. nearer the large city, since it has more shops to offer
3. nearer the smaller city, so the larger city’s trade area reaches further
4. at distances divided between the cities in direct proportion to their populations
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Answer: C — nearer the smaller city, so the larger city’s trade area reaches further
Setting P_A / d_A² = P_B / d_B² makes d_B / d_A = √(P_B / P_A), which is less than 1 when B is smaller, so the point is nearer B and A’s sphere extends further. The tempting reading, that the bigger city holds the point close to itself, reverses this.
A town is 30 km from city A (population 400,000) and 20 km from city B (population 100,000). Using Reilly’s law, the percentage of the town’s retail trade drawn by city A is ____.
Numerical answer — type the value.
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Answer: 64
B_A / B_B = (400,000 / 100,000) × (20/30)² = 4 × 4/9 = 16/9. A’s share = 16 / (16 + 9) = 16/25 = 0.64, which is 64 percent. B draws the other 36 percent.
Under Weber’s least-cost theory, which industry is most likely to locate close to its market rather than its raw-material source?
1. a sugar mill crushing cane, which loses most of its weight in processing
2. a plant concentrating a low-grade ore, which discards most of the rock
3. a bottling plant that mixes ubiquitous water with a little concentrate
4. a cement works using heavy limestone that loses carbon dioxide on burning
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Answer: C — a bottling plant that mixes ubiquitous water with a little concentrate
The bottler’s product is heavier than the localised input because water, found everywhere, is added, so the material index is below 1 and carrying the product is the costlier haul: the plant goes to the market. The other three are weight-losing, with an index above 1, and stand at the source.
A plant makes 1 tonne of product from 3 tonnes of material at source R1 and 1 tonne at source R2. The market M is 90 km from R1; R1 and R2 are 60 km apart; M and R2 are 120 km apart. Transport costs ₹2 per tonne-km. At the least-transport-cost location, the transport cost per tonne of product in rupees is ____.
Numerical answer — type the value.
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Answer: 300
The corner weights are 3 (R1), 1 (R2) and 1 (M). The weight at R1 is 3, which is not less than 1 + 1 = 2, so the plant sits at R1. Then the 3 t from R1 do not move; the 1 t from R2 travels 60 km and the 1 t of product travels 90 km: 2 × (60 + 90) = ₹300. At M the cost would be 2 × (3 × 90 + 1 × 120) = ₹780.
In a Weber problem the plant moves 3 tonnes of locational weight per tonne of product at ₹2 per tonne-km. Toward a cheap-labour town each kilometre away from the least-cost point adds one kilometre of haulage for every tonne moved. The labour saving at the town is ₹90 per tonne of product. The distance of the critical isodapane from the least-cost point in that direction, in km, is ____.
Numerical answer — type the value.
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Answer: 15
Extra transport cost per km = 3 t × 1 km × ₹2 = ₹6 per tonne of product. The critical isodapane is where this equals the labour saving: 6x = 90, so x = 15 km. A cheap-labour town inside 15 km pulls the plant; one beyond it does not.
Which of the following are assumptions of Weber’s least-cost location theory?
1. The firm varies its price and market area to maximise profit
2. Localised raw materials occur at known points
3. There is a single market of fixed demand at a known point
4. Transport cost is proportional to weight times distance
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Answer: B — Localised raw materials occur at known points; C — There is a single market of fixed demand at a known point; D — Transport cost is proportional to weight times distance
Weber fixes the market and demand and minimises cost, with weight × distance transport. Choosing price and market area to maximise profit is the Lösch approach, which is the demand-side correction of Weber.
Two sellers of the same product with fixed prices compete on a straight beach with customers spread evenly. According to Hotelling, they will locate
1. together at the centre, though the quarter points would give a shorter average walk
2. at the two ends of the beach, as far apart as the beach allows them to be
3. at the quarter points, which is both the equilibrium and the social optimum
4. wherever raw materials are cheapest, whatever the position of the rival seller
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Answer: A — together at the centre, though the quarter points would give a shorter average walk
Each seller can take customers from the other by moving toward the centre, so they meet back to back there: minimum differentiation. The quarter points are the social optimum (average walk L/8 against L/4), but they are not stable because either seller gains by edging inward. The quarter-point answer wrongly equates equilibrium and optimum.
In a Lowry-type economic base model for a region, basic employment is 10,000. Each job supports 2.5 persons in the population, and each person supports 0.2 service jobs. The total population of the region at equilibrium is ____.
Numerical answer — type the value.
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Answer: 50000
Total jobs E = B + service jobs = B + 0.2 P, and P = 2.5 E. So E = 10,000 + 0.2 × 2.5 E = 10,000 + 0.5 E, giving E = 20,000 and P = 2.5 × 20,000 = 50,000. Check: service jobs = 0.2 × 50,000 = 10,000, and 10,000 + 10,000 = 20,000.
A large, fast-growing industry with strong forward and backward linkages that stimulates other firms to grow around it is, in Perroux’s terms,
1. a propulsive industry at the heart of a growth pole
2. a footloose industry with a material index of one
3. a marginal industry on the edge of the critical isodapane
4. a lower-order centre inside a K = 3 hexagon
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Answer: A — a propulsive industry at the heart of a growth pole
Perroux’s growth pole is built round a propulsive industry whose linkages make other activities grow near it. Material index, isodapane and K hexagon belong to Weber and Christaller, not to growth-pole theory.
Which of the following statements on the rank-size rule and primacy are correct?
1. A strongly primate pattern has the largest city well above the rank-size line
2. Under the rank-size rule the two-city primacy index equals 1
3. With q = 1, the second-largest city has half the population of the largest
4. With q = 1, the product of a city’s rank and its population is constant
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Answer: A — A strongly primate pattern has the largest city well above the rank-size line; C — With q = 1, the second-largest city has half the population of the largest; D — With q = 1, the product of a city’s rank and its population is constant
With P_n = P_1 / n the second city is half the first, rank × size is the constant P_1, and a primate city lies far above the line. The two-city index P_1 / P_2 is 2 under the rule, not 1.
Which pairing of a Christaller principle with its K value is correct?
1. marketing principle, K = 4
2. transport principle, K = 7
3. administrative principle, K = 3
4. administrative principle, K = 7
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Answer: D — administrative principle, K = 7
The marketing principle gives K = 3, the transport principle K = 4 and the administrative (separation) principle K = 7, in which six lower-order centres lie wholly inside the higher-order hexagon (1 + 6 = 7). The other pairings each exchange two of these values.