1. Based on Urban Economics by Arthur O’Sullivan Notes by Austin Troy
Land Use Economics Lecture Notes: Land Rents, Accessibility and Urban Form
Based on Urban Economics by Arthur O’Sullivan
Notes by Austin Troy
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

2. Land Rent vs. Market Value
Market value: the present value of the stream of rental income generated by land
Rental Income: the amount the landowner charges to use land; equal to income from land minus costs
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

3. What is Present Value?
It is the maximum amount an investor would be willing to pay for something, given that the investor could safely make i percent returns on an alternative investment (for instance, a savings account, or T-bills).
It equals, the stream of income, discounted over time
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

4. How is PV discounted?
PV takes into account the fact that a dollar earned 5 years from now if worth less to us now than a dollar earned today
This is because income put off until later has opportunity cost associated with it.
A dollar invested in five years is worth less than a dollar invested today
PV takes into account lost opportunity from that alternative investment
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

5. How is PV calculated? For $20 yearly stream for 5 years at 10%
For a constant stream of income into infinity, rule simplifies to PV= R/i = $20/.1= $200
Non-constant income example:
PV= $20 + $24/1.1 + $29/ $34/1.33…etc.
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

6. Market value of land
Equals PV of annual maximum rental payments that the landowner can charge
For market value to equal PV: given yearly income R and alternative ROR of i , investor is indifferent between buying the land and investing that money elsewhere
From here out we talk of land rent in place of price, and assume users of land pay rent
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

7. Land Rent and Productivity
Value of land, and hence land rent derives from productivity
Earliest model of productivity comes from Ricardo (1821) who looked at land fertility
Assumptions: fixed inputs/output prices (price takers), zero profit, 3 levels of fertility, land to highest bidder, location (transpo costs) can be ignored, owners are not farmers
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

8. Ricardo model
On fertile land, a farmer can produce same amount of corn with fewer inputs
The price of this type of land is bid up
All profit accrues to the landowner in the form of rents
Payment to farmer is considered a cost
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

9. Ricardo model $ “A” land “B” land “C” landMC
ATC
ATC
$10
Price determined exogenously by supply and demand in market $
Profit=rent>>to landowner
Profit=rent>>to landowner $8 $4
ATC MC MC
Q=amt of corn
220
160
“A” land
“B” land
“C” land
“A” land has lowest production costs= highest rents
“C” land’s rent is 0 because costs are greater than revenue
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

10. Ricardo Model
Competition among farmers for good land bids up rents on that land until economic profits* =0 for farmer. All profits on land go to owner.
Economic profits: greater than “normal” profits required to pay for time of those doing the work
Rent for A land= TR-TC= $2200-$880=$1320
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

11. Leftover principle
In equilibrium, Rent= profits, or revenue over total nonland costs
Rent eats up whatever is “left over” because competition for land bids away any excess
That is, competition among farmers for land bids away excess profits until they are zero and landowner gets all surplus value
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

12. Exceptions to leftover principle
If there is restrictions on entry or on competition
E.g. if farmer (non-owners) owns patent to farming techniques that reduce costs, landlord cannot charge additional rents reflecting those additional profits because noone else would be willing to pay such high rents
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

13. Who benefits from improvement?
Example: irrigation project
If price of corn is fixed (exogenous) the landlord benefits because competition among farmers for land will bid away profit
Winner: land owner; loser: farmer
However, if the project affects the price of corn (price is endogenous), consumers gain with lower prices, while farmer pre rent profits are reduced, lowering land rents
Winner: consumer; loser; land owner
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

14. Scale of improvement
Who benefits is determined by scale of improvement
Smaller the area, the more the benefit goes to landowner; larger the area, more goes to consumers because of price endogeneity
Benefits from any improvement are capitalized into the value of land; a positive capitalization increases rents, which increases market value
Negative factors can be capitalized too
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

15. Accessibility
Now replace fertility of land with location as the prime determinant of land value--Von Thunen model (1826)
No longer assume that transportation is costless
This model explains why more “central” locations command higher rents and have higher market values than fringe areas
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

16. The Carrot Farmer
Assume: land is equally fertile, profits are zero, there is one central market, p is fixed and farmers use fixed factor production
Cost is now fn of distance
Transport Cost= cost/ton/mile*dist*Q
Profit= P*Q-PC-TC-Rent = 0
Rent= P*Q-PC-TC
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

17. Carrot Farmer’s bid rent function
Total revenue per acre (P*Q; Q/acre does not vary)
$300
$250
Total Cost
$190
Bid rent/acre
$50
Land rents
$110
Close Distance to market Far
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

18. Carrot farmer’s decision
Now, market-proximate land replaces fertile land as the most valuable type
However, competition for close land bids away surplus profit so, assuming farmers are identical, they are indifferent among all locations, as long as total revenue exceeds total cost
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

19. The farmer and factor substitution
What if farmers can be different? Then the bid-rent function becomes convex.
Under linear function, fixed amount of land and non-land inputs, no matter where
Under convex function, farmers engage in factor substitution: they increase non-land inputs (equipment, labor, technology) as land gets more central and expensive
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

20. Bid Rent fn for both farmers
Bid rent for flexible farmer
Rent/ acre
Bid rent for fixed-factor farmer
Distance to market U*
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

21. Bid rent of flexible farmer
Flexible farmer will outbid the inflexible farmer in all locations but u
That is, land will be used more intensively and, hence, more efficiently at central locations, and non-land inputs will be fewer far away
With inflexible farmers, land is used more inefficiently
Rents will still equal profits of highest bidder
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

22. Two competing land uses
Different land uses (e.g. SPAM factory and grain farm) may have different bid rent functions. The shapes of those functions will determine who will locate where
Steepness of fn determined per unit transport costs relative to per unit price
As usual, land goes to highest bidder
Market allocates land efficiently to usage with the most to gain from being close to the market
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

23. Determinants of bid rent slope
Per acre transportation costs. The more weight you produce/acre, the more transport will cost per acre cultivated. E.g. potatoes vs. cotton
Unit transport costs. The more a given unit weight costs to ship, the higher the transport costs. E.g. eggs vs. turnips
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

24. Bid Rent fn for both farmers
Heavy good
U’= where heavy use transitions to light use
Rent/ acre
Light good U’
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

25. Multiple land uses in the CBD
Let’s assume a traditional 19th Century city:
Central railroad freight terminal
Central market
Workers travel to center via streetcar
Goods go from factory to railroad via horse cart
Also assume fixed factor production
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

26. Bid Rent of Firms in the CBD
Profit fn looks same as in chapter 7
= PQ-NC-TC(d)-R(d)
Profit= price*quantity – nonland costs- transport costs (function of distance) – rents (function of distance);
TC(d)= cost/ton/mile* distance*quantity
Then R(d)= PQ-NC-TC(d)
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

27. Freight Costs and Rents
Freight costs decrease with proximity to city center
Through leftover principle, rents increase as transport costs decrease
Hence, there will be a downward sloping bid rent function; it will be linear for fixed factor producers and convex for flexible producers
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

28. Flexible versus fixed producers
Fixed: R(d)= P*Q-NC-TC(d)
Flexible: = P*Q-NC-TC(d)-R(d)*L(d),
Where L(d) is amount of land used at distance d; this results in rent function:
R(d)= (P*Q-NC-TC(d))/L(d)
Flexible farmer substitutes nonland for land input: spends more on equipment and labor as land gets more expensive
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

29. Flexible versus fixed producers
Flexible produce = factor substitution = lower costs* = higher profits
By leftover principle, higher profits= higher bid rents
Close to city center, land costs are lower; at periphery, freight costs are lower
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

30. Monocentric city firms’ bid rent function
flexible producer
The flexible firm outbids the fixed factor firm everywhere but point u’’. At u’, the fixed factor producer uses too much land
Bid Rent B
Fixed-factor producer A u’
u’’
Distance from export hub
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

31. Nonland versus land inputs for the Flexible producer
Non-land inputs B
flexible producer A
Land Amount
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

32. Most central firm type: offices
Office firms: require 1) lots of meetings and face to face contact, 2)ability to gather, process and distribute information quickly and 3)access to services, like printing, lawyers, designers, accountants, etc.
This type of firm will have a steep bid rent function because the travel cost of individuals is very high; travel cost is high because their pay rate is high, since it is generally skilled work
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

33. Land use in CBD
All firms are attracted to center, but only some will be willing to bid enough
Office firms have steepest bid rent fn, and will occupy the most central land
Market allocation is efficient, because the office industry has the most to gain from being in the center; manufacturing could gain too, but not as much, so it’s willing to locate a little further out.
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

34. Office vs. manufacturing
Suppose office firm 1 block from center and manufacturer 5 blocks. If they swapped locations, this would dramatically increase the office firm’s travel cost
Office firm TC= 3min/block*$4/min*200 meetings/month= $2400 per block/month
So the swap increase TC for office by $9,600/mo, but only saves the manufacturer $800/month in transportation costs (50 tons * $4/ton/block= $200/block).
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

35. Multiple land use rent gradient
Office Bid rent
Manufacturing
zone
Residential
zone
Bid Rent
Manuf. Bid rent
Residential bid rent
Office zone U’
U’’
Distance to center
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

36. Locational choices Residential Zone Manufacturing Zone Office Zone U’
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

37. Locational choices
So workers live on the periphery because they are cheap to transport (i.e. commuting costs are low) relative to cost of moving freight (for manufacturers). For offices, same problem, because of high price of moving executives around for meetings. If office is in suburbs, executive is constantly going to CBD
Residential Zone
Manufacturing Zone
Office Zone U’
U’’
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

38. Who occupies what?
Activities are arranged according to transport costs; those with the highest costs occupy the most central land
Activity with the highest transport cost will have the highest bid rent curve
All firms have tug-of-war between locating centrally to keep transport cheap, and locating in the suburbs to keep workers’ commute cost lower (and hence pay lower wages)
CBD wins because cost of freight hauling greater than cost of moving workers
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

39. Housing price function
In the monocentric model, residents will be attracted towards the center but be outbid by offices and manufacturing.
Assume no factor substitution, identical 1000 sq ft houses, fixed budget of $300/mo for housing+commuting, commute cost (CC)= $20/mi.
WTP for housing = $300- CC.
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

40. Housing price fuction
$.3
D for housing near center pushes up price until rent= budget-CC.
P housing/ sq ft
Residents now indifferent among all locations in city
$.18
$.06
6 mi
12 mi
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

41. Housing price function with consumer substitution
With consumer substitution, residents consume less land as price of land goes up; instead consume more local amenities
Hence, more central homes are smaller
Flexible residents will outbid fixed factor residents everywhere but tangency point
Change in P due to distance= , or negative of (tranpo cost/mi divided by amount of housing consumed)
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

42. Residential housing price function with consumer substitution
$.3
Price w consumer sub
P housing/ sq ft
$.18
$.06
6 mi
12 mi
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

43. Residential bid rent function and factor subsitution
Residents’ housing price fn with consumer substitution drive producers’ factor substitution
Factor Substitution: Housing producers substitute capital for land as move closer to center; higher density allows residential housing firms to pay the higher cost of land in more central locations. Different from consumer substitution
Producers’ bid rent functions are convex because of consumer substitution in the housing price function
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

44. Density increases towards center because:
Consumer substitution in housing pr fn.
P(housing) goes down away from center, so households consume more housing (i.e. larger dwellings) towards periphery
Factor substitution in producer bid fn
Price of land goes down as move away because of housing price function; housing production firms respond by using more land per unit of housing (less density) towards periphery; towards center they respond with greater density
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics

45. Changes to Residential Model
All the following changes in assumptions make cost of commuting greater, and housing price fn/residential bid rent fn steeper:
No time cost to commuting (only monetary cost)>> time cost to commuting (opportunity cost)
Amenities (shopping, recreation) etc. are evenly distributed across city>> Amenities concentrated in the CBD
All households one earner>>two earners commuting to center
Lecture by Austin Troy University of Vermont, based on Arthur O’Sullivan, Urban Economics