Land Rent. Economics 312 Martin Farnham

Transcription

1 Land Rent Economics 312 Martin Farnham

2 Valuing Land Like any other asset, value of land should reflect present discounted value of stream of future earnings (from rents, value added in production, etc.) PV = n R = R + (1 + i) t t =0 R (1 + i) + R (1 + i) R 2 (1+ i) n i denotes the real interest rate, t denotes year, and n denotes the asset s productive life in years Money tomorrow worth less than today (because money today can put in bank today to earn interest; same amount of money tomorrow can t earn us interest today) 2

3 Value of future income (1+i) is amount that 1 dollar invested today will yield after 1 year in bank. ex: At 5% interest rate (i=0.05), $1 invested today will yield $1.05 in a year; $x invested today will yield $x(1.05) in a year. Suppose I want to buy $1000 in a year. I can find out how much a bank will charge me by solving x(1.05)= $1000. x=$ So my max WTP (and everyone else s) for $1000 in a year should be no more than $ And if I offer less, someone else will outbid me. 3

4 Valuing Land Note that the expression 2 slides back assumes the asset has finite life. Land arguably has an infinite stream of earnings attached to it (especially residential land) It s actually easier to calculate the value of an infinitely-lived asset PV = t =0 R (1+ i) t = R i The proof for this involves a simple rearrangement of an infinite series 4

5 Rent and Value R is annual rental payment for land Should reflect highest private use value of land If R is less than private use value, someone would offer the owner a higher rent If R is greater than private use value, people would refuse to rent it. Land rent can be thought of as price of land over a fixed period of time; just like wage is price of labour for an hour or week, and rental cost of capital is price of capital for a month or year. Dividing annual rent (assuming it will remain fixed) by interest rate gives market value of piece of land 5

6 History of Land Valuation David Ricardo (Mr. Comparative Advantage) argued price of land was function of fertility Rent increasing in agricultural yield Consider a simple model Output and input prices fixed Free entry into farming 3 types of land by degree of fertility Price of land allowed to vary No transport costs 6

7 Land Rent and Fertility P Corn Market Farm Production S High Profit Medium Profit MC APC APC MC APC MC Negative Profit D Q High Fertility Medium Fertility Low Fertility 7

8 Land Rent and Fertility Each farmer is willing to pay as much as annual profits in annual rent Recall economic profits are normal profits Opportunity cost of the farmer s time are taken into account in MC/APC curves Zero profits imply the farmer does as well as she could at best alternative vocation Competition drives land rent up to exactly farmers willingness to pay. 8

9 Land Rent and Fertility High fertility (low cost) land comes with highest willingness to pay==>highest rent Medium fertility (medium cost) land comes with lower WTP==>lower rent Low fertility land is not worth farming (WTP is negative). Won t be put to agricultural use, unless the price of corn rises sufficiently 9

10 Land Rent and Fertility Note that we ve assumed corn prices are fixed. Price of corn will also affect rent, by affecting farmer profits, and thus willingness to pay for land Corn Laws in England (Ricardo talked lots about these) charged tariffs on European grain which kept price of corn high. Kept rents high on British land. Landowners loved Corn Laws. Tenant farmers weren t necessarily crazy about them (on the one hand, the Corn Laws kept grain prices high, and on the other it drove up equilibrium rents). Arguably, tenant farmers should have been indifferent to the Corn Laws. Urban dwellers were also hurt by corn laws (kept price of food high, which kept real wage low) 10

11 Land Rent and Fertility Leftover principle: Competition for land means landowner gets leftovers (any prerent profits of land user) Any excess of TR over non-land TC goes to landlord This assumes farmers all face same costs Otherwise lowest cost farmer could hang onto some profits (amount of difference between her profits and next-best farmer s profits) Suggests that incidence of fertility-improving policy falls on landowner Why? Because land is in fixed supply 11

12 Policy and Land Rent Policy that improves fertility or raises price of agricultural output (locally) will benefit landowners, not farmer-renters Why? Because by lowering cost of farming, improvements raise WTP for land Higher WTP translates directly into higher rent Note that if policy raises output on a large enough scale (so the supply curve for corn shifts), price of output will fall, at least partly reversing gains for landlord. 12

13 Policy and Land Value If policy increases rents on land, this automatically increases land value, by our formula, PV=R/i. We call this process capitalization Value of land will change to reflect changes in expected future rents Can apply same principle to residential property e.g. Arguably increased property taxes are capitalized into home values This is why people become so touchy about city politics and policy when they become homeowners Wasteful municipal government should be capitalized into home prices (causing them to fall) 13

14 Residential Housing Market Similar ideas apply Rent reflects consumption value of an apartment Renter willingness to pay should be fully captured by landlords Changes in local amenities, taxation should be capitalized into house value e.g. improvements in local schools increase renter WTP==>increase rents==>increase market value of apartments Landlords win/lose depending on the policy change. By this theory, renters don t benefit from improvements in local services (might explain why renters tend to vote less in municipal elections) 14

15 Manufacturers Bid-Rent Function Bid-rent (WTP) function for manufacturers Assume fixed factors of production, fixed price, fixed output, competitive markets, freight cost WTP for land will be function of distance from highway; transport costs of firm are lower, the closer it is to highway. Profit function for firm: π(x) = P Q C t Q x R(x) Where x is distance from highway, C is cost of nonland inputs, P is price, Q is units of output, t is shipping rate (per mile or km), and R is rent. 15

16 Manufacturer s Bid-Rent (WTP) Function If there is free entry, manufacturers will compete for land, drive up rents until they make zero profits in the LR. Recalling leftover principle, π(x) = P Q C t Q x R(x) = 0, implies R(x) = P Q C t Q x This says that manufacturers pay whatever is leftover of (TR-nonland costs) in rent 16

17 Numerical Example Suppose firm produces 50 tons of output using 1 acre and $1000 of nonland inputs. P=$60/ton, t=$4/ton, R(x) = P Q C t Q x Bid-rents (WTP) decline with distance R(x=0)=$3000-($1000)=$2000 R(x=1)=$3000-($1000+$200)=$1800 R(x=2)=$3000-($1000+$400)=$

18 Manufacturer s Bid-Rent Thus far, we ve considered a firm that uses 1 acre of land in production More generally WTP per acre is R(x) = P Q C t Q x LotSize Bid-rent declines with distance from highway; reflects increasing freight costs Slope is change in WTP per unit increase in distance from highway, ΔR / Δx 18

19 Manufacturer s Bid-Rent Bid-rent declines with distance from highway; reflects increasing freight costs With fixed proportions, bid-rent function is linear in distance, with slope reflecting freight costs With factor substitution (non-fixed proportions) bid-rent function is convex Flexible firms (those that can produce same output with less land) outbid inflexible firms 19

20 Bid-Rent Function for Info Sector Office firms differ from manufacturers in a couple of ways Information based Require lots of face-to-face contact Consider a financial services firm: Based in office Travel to city center to serve clients Price of client services is fixed Competitive market (zero profits) 20

21 Info-Sector Bid-Rent Office firms will locate near clients to minimize travel cost You have to pay workers wage, while they go to meet clients Want to minimize these travel expenditures Note travel time tends to increase more rapidly as distance increases; this will affect shape of bidrent Worker Travel Time as function of office distance from clients Travel Time 0 Location: Distance from Clients 21

22 Median Location Locating in the middle (spatially) of your clients will minimize transport costs This is an important principle of firm location (and another reason firms cluster!) This means office firms will tend to most highly value land at the center of the Central Business District (CBD); this is where most of their clients locate. Office rents should decline as we move out of the CBD. 22

23 Annual profit given by Info-Sector Bid-Rent π = P Q cap cost other prod cost t (x) M w R(x) T T is the amount of land used; tbar is the average travel time (in hours) per meeting (which is an increasing function of the average distance travelled per meeting, xbar), w is the hourly wage, M is the number of meetings per year Applying the leftover principle (π=0) we get the following WTP for rent (bid-rent): R(x) = P Q (capital + prodcosts) travel cost T This is just like for mfg. firm, except transport costs are a function of the worker s wage (to reflect opportunity cost of time spent getting to meetings). 23

24 Info-Sector Bid-Rent Just as with the manufacturing bid-rent we expect this function to be downward sloping (i.e. decreasing in x). Office firms will bid lower rents for land that is further away from their clients. This tends to lead them to locate near the city center (or near parts of the city with lots of clients) Note that proximity to the highway isn t as important for office firms as for manufacturers. Means we ll expect to see mfg firms near highways and office firms near city center. 24

25 Info-Sector Bid-Rent without Factor Assume that each office firm needs a fixed amount of land Bid-rent will slope down; decrease at increasing rate This is because travel time increases at increasing rate in distance Shape of bid-rent will be different, if office firms can respond to higher rents by substituting away from land (i.e. if land input is not fixed) Substitution Office Bid-Rent Without Factor Substitution $ Average distance from client 25

26 Info-Sector Bid-Rent with Factor Substitution Seems unrealistic to suppose that an office firm would always use the same amount of land, regardless of rents Might expect firm to locate in 1-floor building in Lethbridge; in highrise if located in downtown Vancouver. As rents rise, expect firm to make do with less land. If firm uses less of an input as price rises, we say it practices factor substitution. 26

27 Recall Isoquants Isoquants show the different combinations of inputs (say land and non-land inputs) that yield the same level of overall output, q. Isoquants reflect substitutability of inputs 27

28 Degrees of Substitutability Nonland Inputs Fixed proportions (no substitutability) Isoquant Perfect Substitutes (other extreme) Nonland Inputs Isoquant Land Land 28

29 Degrees of Substitutability With fixed proportions, no flexibility With perfect substitutes, can produce with zero land Technology with some, but not perfect substitutability, more realistic Nonland Inputs Something in between Isoquant Land 29

30 Bid-Rent and Factor Substitution Factor substitution allows firms to lower production costs, by switching to cheaper inputs This raises profitability per hectare and hence willingness to pay for land (on a per hectare basis) The more useful (profitable) small bits of land are, the more firms are willing to pay per hectare Ironically, this means ability to substitute away from land causes rents to rise Bid-rent per hectare ($) Bid-Rent Function with and w/o factor substitution w/factor substitution w/fixed inputs Avg. Distance from client 30

31 Bid-Rent and Factor Substitution Recall the leftover principle. Pre-rent profits accrue to the landlord, due to competitive bidding for land. Substituting to capital raises a firms nonland costs and lowers its total pre-rent profits. But a firm will only substitute away from land if it s profitable to do so (taking rent into account), so this substitution to capital raises pre-rent profits per acre Hence WTP per acre rises 31

32 Office Firm Bid-Rent Office firms can easily substitute from land to capital They do this by locating in skyscrapers The closer they get to the CBD (where most of their clients are), the higher the building they use Skyscrapers are expensive, but land costs justify building costs Bid-rent per hectare ($) Office firm bid-rent w/factor substitution Avg. Distance from Client 32

33 Office Firm Bid-Rent If most office firm clients are located in central business district (CBD), then office firms bid-rents will be highest in CBD and declining as distance from CBD increases This implies skyscrapers in CBD Lower height buildings as distance from CBD increases This pattern is typically observed in cities 33

34 Housing Prices Thus far we ve thought about WTP for land by manufacturers and firms in the info-sector (office firms). Manufacturers want to locate near highways Office firms want to locate near clients Another major user of land in cities is the residential user (households) 34

35 Residential Land Use Basic model assumptions Cost of commuting is $t per km One commuter in each HH==>CBD or manufacturing district Non-commuting travel is costless Local fiscal characteristics (taxes/services) and environment identical at all locations 35

36 Residential Housing Price Function (linear case) If we assume households do not adjust their housing size according to the rent, house price is linear in distance to employment district Example: HH has fixed budget of $1000 for housing and commuting Commuting costs $25/km/month WTP for rent: ($1000-$25*km) per month km: km from work 36

37 Residential Housing Price Function (linear case) Spz all houses are 1000 sq feet WTP near employment districts (office and mfg) is $1000 total or $1 per sq foot, because commuting costs are zero 10 km out, WTP would be $1000-$25 (10km)=$750, or $0.75 per sq foot Rents will be higher near employment districts, and will decline as HHs move out 37

38 Linear Residential Housing Price Households willing to pay more to live near jobs Function must slope down to induce people to use housing stock outside employment district Higher commuting costs must be rewarded with lower rents Slope will be determined by commuting costs (higher costs==>steeper curve) Function Monthly house rent per square ft. ($) Residential Housing Price Curve slope = ΔP Δx Distance of house from employment 38 district

39 Housing Price Gradient Housing rents fall as distance from employment districts increases Suppose a renter considers two identical houses; One is Δx kms further away from employment district. The additional commuting cost will be Δx*t, (where t is per km monthly commuting cost) For the renter to be indifferent between the nearer house and farther house, the decline in rent, ΔP*h (where h is # square feet and P is monthly rent per square foot) must exactly offset the increase in commuting cost associated with picking the farther house. 39

40 Housing Price Gradient For potential renter to be indifferent, need ΔP h + Δx t = 0 or ΔP h = Δx t We can rearrange this to get an expression for the slope of the houseprice function ΔP Δx = t h which shows us that the slope of the housing price function is a function of commuting costs. 40

41 Housing Price Gradient Note that if this indifference condition doesn t hold, potential renters will all want to live in same place For instance, if the drop in rent from location A to location B doesn t compensate for increased commuting cost, then everyone will want to locate in location A. If it overcompensates, everyone will want to live in B. Excess demand or supply will force rents to adjust accordingly until indifference condition holds. 41

42 Consumer Substitution and Housing Price Curve Thus far we have assumed demand for housing is perfectly inelastic i.e. that people all demand identical house (size-wise) regardless of rent per sq ft. This is clearly unrealistic Most people respond to higher rents by renting smaller houses. i.e. they substitute away from housing in response to higher prices. More flexible consumers will outbid less flexible for small housing near jobs 42

43 Consumer Substitution and Housing Spz that the amount of house consumed is not a constant, h, but is increasing in distance (due to falling rents per sq ft as distance increases), Housing consumption is h(x) Indiff condition is now ΔP Δx = t h(x) Price Curve Monthly house rent per square ft. ($) Residential Housing Price Curve w/ consumer substitution w/o consumer substitution slope = ΔP Δx Distance of house from employment 43 district

44 Residential Bid-Rent Note, we have not been talking about consumers WTP for land, instead we ve been talking about WTP for square footage of housing It s an inconvenient fact that houses must sit on land So we can translate the housing-price function into a bid-rent for land. 44

45 Residential Bid-Rent Producers of housing will have different WTP for land in different locations Producers will pay land rent equal to (TRnonland costs) TR is decreasing in distance from employment districts, because the housing price function is decreasing in distance from employment districts 45

46 Residential Bid Rent R(x) = [P(x) Q(x) C(x)] / T (x) At some point, x*, TR<Nonland Costs Bid-rent becomes negative at that point Expect no housing built beyond that distance from the employment district. Bid-rent function convex due to convexity of housing price function (shows up in TR) Even more convex if housing producers substitute capital for land: C=C(x), T=T(x) Build taller apartment buildings as land gets more expensive 46

47 Housing Price and Bid-Rent Functions $ TR=P(x) Q(x) C=Nonland Costs R(x)=TR(x)-C x* Distance from jobs 47

48 Residential Density Population density increases as move towards employment districts Consumers substitute toward smaller units (consumer substitution) Producers substitute toward less landintensive dwellings (factor substitution) Build up Build on smaller plots Note new condo projects in downtown Victoria, Vancouver 48

49 Other Determinants of Rent We ve assumed only commuting costs matter; unrealistic People pick location based on proximity to waterfront, good schools, good restaurants If property taxes vary across communities, people may try to avoid high taxes, ceteris paribus Houses near freeways, railroad tracks, airports, generally less desirable 49

50 Bid-rents and Land Use Patterns In a competitive land market, land use will be determined by the high-bid Whoever has the highest bid-rent for a particular location Near highway, expect this to be manufacturers Near center city, expect this to be office firms Elsewhere, expect this to be residential users What if highway runs through center of city? Who wins bidding war, office firms or manufacturers? Transporting people more expensive than transporting goods, so office firms will likely win. 50

51 Land Use Patterns Figure 6-9 in text gives a nice illustration of bid-rent surfaces (3 dimensional view) Can stick to a two-dimensional representation like we ve seen before, but combine bid-rents for different users 51

52 Land Use Patterns Highway Location Bid Rent ($) Office firm bid-rent Office worker bid-rent Mfg. worker bid-rent Mfg. firm bid-rent Mfg. worker bid-rent Agricultural bid-rent City Centre Office District White-collar Blue-collar neighborhood Mfg. neighborhood District Blue-collar neighborhood Farm land Distance 52

53 Land Use Patterns Some things to notice Position of highway is critical for location of mfg. sector and blue-collar workers Office firms/workers tend to locate near middle Equilibrium rents aren t monotonically declining in distance; due to location of highway, they fall as we move from city centre, then rise, then fall again As distance from the city centre gets very large, winner bidders for land are farmers. There are no suburbs! Or at least, the suburbs are occupied by manufacturing workers. This seems unrealistic (more on this later). 53