The Encyclopedia of Economic Principals

Chapter 81

Land, Housing, Transport, Congestion, and Urban Form

Access, distance and land quality set what a site is worth.

Four of the chapter's worked examples, made interactive: transit access and job offers, the bid-rent gradient, Ricardian rent as price moves, and scale economies on a large estate. Change one value at a time and watch the figure, the numbers and the hand calculation respond.

Every example here is a constructed teaching example: it uses the hypothetical numbers of the chapter's worked examples, plus a few values added for comparison and labelled as such in each panel. Nothing here measures a real market, firm or household.

Demonstration 1 of 4

Bus access, job offers and net daily pay

How much does a faster, cheaper trip raise both the chance of an offer and the value of the job?

Access works on two margins at once. Cheaper, shorter trips let more applications fit the same time and cash budget, which raises the chance of at least one offer. A cheaper daily commute raises the value of the job once it is offered.

Equation, written in LaTeX: 1-(1-0.20)^5=1-0.8^5=0.67232.

Equation, written in LaTeX: 2(\$6)+\$10=\$22,

Equation, written in LaTeX: 1-0.8^7=0.7902848.

Scroll sideways for the whole equation

p is the chance that one completed application yields an offer, with independent outcomes. The weekly search budget is 15 hours and $60. Commute cost values travel time at $6 per hour and adds the fare. The job pays $18 per hour for 8 hours.

Predict first. Is the bus's gain in offer chances larger at p = 0.10 than at the book's p = 0.20?

Your prediction

Choose an example

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Figure: Bus access, job offers and net daily pay. Left: chance of at least one offer for 1 to 8 applications at offer probability 0.20; the first 5 bars fit the budget and reach 0.67232. Right: the $144 daily wage split into $22 of commute cost and $122 of net pay.
Offer probability per application: 0.20, Transit access: Baseline
Constructed example: the chapter's hypothetical worker (wage $18, p = 0.20, budget 15 hours and $60, baseline and direct-bus trips); offer probabilities 0.10 and 0.30 are added for comparison.

Calculated values

Transit
Baseline
Applications that fit
5
P(at least one offer)
0.67232
Commute cost per day
$22
Net daily pay
$122

With baseline transit, each application takes 3 hours and $12, so 5 whole applications fit the budget. The chance of at least one offer is 1 - 0.80^5 = 0.67232. The commute costs $22 a day, leaving $122 of the $144 wage.

Worked steps

  1. Applications = min(15 / 3, 60 / 12) rounded down = 5
  2. P = 1 - (1 - 0.20)^5 = 1 - 0.32768 = 0.67232
  3. Daily pay = 18 x 8 = $144
  4. Commute cost = 2($6) + $10 = $22
  5. Net daily pay = $144 - $22 = $122

Use the idea

When judging a transit change, count suitable reachable vacancies and the applications a worker can actually complete, not raw job counts.

Where the conclusion applies

Independent applications with the same offer probability, a fixed weekly budget and a fixed value of time. If the reachable jobs need a credential the worker lacks, p is zero and access changes nothing.

Check your understanding: With the bus and offer probability 0.30, what is the chance of at least one offer?
7 applications fit; 1 - 0.7^7 = 1 - 0.0823543 = 0.9176457.

Chapter 81 source: section "Spatial-mismatch hypothesis".

Demonstration 2 of 4

Bid rent and the price of a commuting mile

How fast must a household's housing bid fall with distance to keep it as well off?

Each extra mile costs the household m dollars a year, so to stay as well off it can pay m less for housing. Spread over a fixed floor area, that gives a straight bid-rent line whose slope is the commuting cost per mile per square foot. Cheaper transport, such as rail at $1,000 per mile, flattens the line.

Equation, written in LaTeX: \frac{\$2{,}000}{1{,}000\text{ square feet}}=\$2

Equation, written in LaTeX: 4(\$2{,}000)=\$8{,}000.

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The household uses a fixed 1,000 square feet. Commuting costs m dollars per extra mile per year. At the 2-mile reference site the bid is $30 per square foot. The gradient is m divided by floor area.

Predict first. If commuting costs $3,000 per mile, is the 8-mile bid above or below half of the 2-mile bid?

Your prediction

Choose an example

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Figure: Bid rent and the price of a commuting mile. Bid-rent lines from 2 to 8 miles for commute costs of $1,000, $2,000 and $3,000 per mile, starting at $30 per square foot. The active line is $2,000 per mile; at 6 miles the bid is $22.
Commute cost per extra mile: $2,000, Site distance (miles): 6
Constructed example: the chapter's hypothetical household (1,000 square feet, $30 at 2 miles, $2,000 per mile and the $1,000 rail case); a $3,000 cost and the 4- and 8-mile sites are added.

Calculated values

Gradient ($ per sq ft per mile)
$2
Extra miles beyond 2
4
Extra commute burden
$8,000
Bid per sq ft
$22
Annual housing bid
$22,000

Bid per square foot = 30 - 4 x 2 = 22. Moving 4 miles beyond the reference site adds 4 x $2,000 = $8,000 of commuting a year. To keep other consumption unchanged the household's bid falls from $30,000 to $22,000, or $22 per square foot.

Worked steps

  1. Gradient = $2,000 / 1,000 sq ft = $2 per sq ft per mile
  2. Extra burden = 4($2,000) = $8,000
  3. Annual bid = $30,000 - $8,000 = $22,000
  4. Bid per sq ft = $22,000 / 1,000 = $22

Use the idea

To judge a site's housing bid, subtract the extra yearly commuting cost from the bid at a reference site and divide by the floor area.

Where the conclusion applies

Fixed floor area, a single central job, identical households and no amenity differences between sites. With variable floor space the gradient also reflects substitution toward larger homes.

Check your understanding: At $3,000 per mile, what is the bid per square foot at 8 miles?
Extra burden 6 x $3,000 = $18,000; $30,000 - $18,000 = $12,000, or $12 per square foot.

Chapter 81 source: section "Alonso-Muth-Mills bid-rent gradient".

Demonstration 3 of 4

Ricardian rent as price moves the margin

Which plots earn rent, and how does a higher crop price spread rent across them?

Competition bids each plot's surplus over the margin into rent. The worst plot worth farming sets the zero point. A higher price raises surplus on every plot in proportion to its yield, so the best land gains most, and a poorer plot can be brought into use.

Equation, written in LaTeX: S_A=10(30)-140=160,

Equation, written in LaTeX: S_B=10(22)-140=80,

Equation, written in LaTeX: S_C=10(14)-140=0.

Equation, written in LaTeX: S_D=12(12)-144=0

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S is pre-rent surplus per hectare: basket price times yield minus delivered cost. Plots A, B, C and D yield 30, 22, 14 and 12 baskets. Every plot needs the same nonland cost; D also pays 4 of transport.

Predict first. When price rises from 12 to 14, does rent on A rise by more or less than rent on C?

Your prediction

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Figure: Ricardian rent as price moves the margin. Bars of pre-rent surplus for plots A to D at price 10 and cost 140: A 160, B 80, C 0, D -24. Negative bars are grey and unused.
Basket price: 10, Nonland cost per hectare: 140
Constructed example: the chapter's hypothetical plots (yields 30, 22, 14 and 12, cost 140, D's transport 4, prices 10 and 12); price 14 and nonland costs 120 and 160 are added for comparison.

Calculated values

S_A
160
S_B
80
S_C
0
S_D
-24
Plots in use
A, B, C

At price 10 and nonland cost 140 per hectare (D adds 4 of transport), surplus is price times yield minus cost: A 10 x 30 - 140 = 160; B 10 x 22 - 140 = 80; C 10 x 14 - 140 = 0; D 10 x 12 - 144 = -24. Plot C earns exactly zero and is the no-rent margin, so rents on the better plots equal their surpluses. Plot D would lose money at this price and stays out of use.

Worked steps

  1. S_A = 10(30) - 140 = 160
  2. S_B = 10(22) - 140 = 80
  3. S_C = 10(14) - 140 = 0
  4. S_D = 10(12) - 144 = -24

Use the idea

To find land rent, compute each site's revenue minus delivered cost and measure it against the site that just breaks even.

Where the conclusion applies

Identical nonland cost per hectare, fixed yields, one crop price and competitive bidding for land. Delivered net value, not yield alone, ranks the plots.

Check your understanding: At price 14 and cost 140, what are the surpluses on A, B, C and D?
14(30) - 140 = 280; 14(22) - 140 = 168; 14(14) - 140 = 56; 14(12) - 144 = 24. A gains 60 over price 12, C gains 28.

Chapter 81 source: section "Ricardian Rent".

Demonstration 4 of 4

Scale, coordination loss and utilization

When does a large press give the estate lower cost per ton than a small producer?

A large fixed cost is cheap per ton only when spread over many tons. Coordination loss raises the cost of the last tons, and a poor harvest leaves the press underused, so the scale advantage depends on throughput and organization.

Equation, written in LaTeX: TC_S=300+30(20)=900,

Equation, written in LaTeX: TC_L=1{,}200+25(100)=3{,}700,

Equation, written in LaTeX: 1{,}200+25(80)+34(20)=3{,}880.

Scroll sideways for the whole equation

TC is total cost and AC = TC / q average cost per ton. The estate's press costs 1,200 a year and inputs cost 25 per ton up to 80 tons, then v per ton. The small producer's average cost is 45.

Predict first. At 50 tons, is the estate cheaper or dearer per ton than the 20-ton producer?

Your prediction

Choose an example

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Figure: Scale, coordination loss and utilization. Estate average cost from 30 to 120 tons with variable cost 25 beyond 80 tons, against the small producer's 45 and the price 42. At 100 tons average cost is 37.0.
Estate throughput (tons): 100, Variable cost beyond 80 tons: 25, Crop price: 42
Constructed example: the chapter's hypothetical estate (press 1,200, 25 per ton, 34 beyond 80 tons, price 42, 100 and 50 tons) and small producer; 80 tons and price 46 are added.

Calculated values

Total cost
3,700
Average cost
37.0
Revenue
4,200
Profit before land cost
500
Versus small producer AC 45
cheaper

TC = 1,200 + 25(80) + 25(20) = 3,700, so average cost is 37.0; 37.0 is below the small producer's 45, a saving of 8.0 per ton. Revenue 42 x 100 = 4,200 leaves 500 before land cost. The last 20 tons cost 25 each, the same as the first 80.

Worked steps

  1. TC = 1,200 + 25(80) + 25(20) = 3,700
  2. AC = 3,700 / 100 = 37.0
  3. Revenue = 42(100) = 4,200
  4. Profit = 4,200 - 3,700 = 500

Use the idea

Compare average cost at the throughput you expect, not at capacity, and price in the extra cost of monitoring large operations.

Where the conclusion applies

One crop and price, a fixed press cost and constant variable cost within each tier. Land cost is left out of profit.

Check your understanding: With coordination loss, 100 tons and price 46, what is profit before land cost?
TC = 1,200 + 25(80) + 34(20) = 3,880; revenue 46 x 100 = 4,600; profit 720.

Chapter 81 source: section "Economies of Scale and the Latifundia".