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Introduction and overview
The fundamental distinction between resourcesRenewable resources
Provide an infinite duration flow of services'correctly' managed (in a sustainable way)Examples : land, water, wind, solar energy, forests, crops and cattle, biological resourcesAvailable resource flow submited to some 'limits':
Physical limitsTechnical limitsCultural and religious constraintsPolitical and institutional limits (transboundary sharing of rivers)
The fundamental distinction between resources
Non renewable (or exhaustible) resourcesProvide only a finite duration flow of servicesExamples : coal, oil, iron, copper, and other mineral resourcesCombine limits upon the flow of services and upon the stock of services (of limited size)
Two important remarksA distinction economically based (concept of 'service')A common characteristic: the existence of 'limits'
Makes room for 'scarcity' considerationsIntroduces the concept of 'opportunity cost' associated with these 'limits'Equivalence between 'opportunity cost' and 'resource rent'
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Overview of Chapter One
Principles of Land EconomicsEconomics of land use : the concept of 'rent'
A starter : a Leontiev exampleLand rent when inputs are subsituableLand of differing qualities : the 'Ricardian rent'Von Thunen like land rentLand as a capital goodAppendix : The concept of present value
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Overview of Chapter One
Water economics Water problems in the world todayWater as a resourceWater quality
A 'damage function' approachAn 'environmental benefits' approach
Water scarcity rentsSharing a riverAppendix : water issues in France
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Land economics
An elementary modelAn agricultural good produced from land and labour
Limited supply of land Land and labour of homogeneous qualityPrice of the consumption good : pWage rate : wThe landlord demand for labour and supply of consumption good 'small' with respect to the market (exogeneous prices)
QF A , LA
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A Leontiev example
The technology is of the Leontiev class:
Efficient use of inputs requires:
The landlord profit as a function of land use:
A necessary condition for economic activity:Under this condition all the available land is put in use.
F A , L=minaL L ,aA A
aL L=a AA
A= paA A−waLa AA=a A p−
wa L
A
pw /aL
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Land rent in the Leontiev case
Increase of profit with land availability increase ('marginal' rent):
Operating cost:
The total land rent is identical to the total profit.
=aA p−w /aL
C=aAa Lw A
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Land rent in the Leontiev case
Fig1 : Land rent in the Leontiev example
Slope : w aA/ aLa A p
w aA/ aL
A
Rent
Cost
Unit cost function C/A
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Land rent when inputs are substituable
No subsituability in the Leontiev caseWith substituability, the land rent becomes a function of the land plot size
Questions:How the landlord can determine the profit maximizing level of workers ?What would be the level of the land rent ?
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Land rent as a difference between average and marginal productivity
Average productivity (production per capita)Marginal productivity in value :
AP L= pF A , L/ L
MP L= p ∂F A , L∂ L
Fig 2 : Land rent as a difference between average and marginal productivity
APMP
L
wRent
Cost
Ouput in value pQ
Labour
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The profit maximizing landlord
Gamma appears as the marginal rent of landIt is also the 'marginal opportunity cost' of the land constraintIt is also the marginal willingness to pay (WTP) of the landlord for
one extra unit of land (the land 'value')
Max= pF A , L−wL s: t AAThe Lagrangian of this problem is:
L= p F A , L−wLA−AAnd the first order conditions give:
p∂ F A , L/∂ L=wp∂ F A , L/∂ A=
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From marginal rent to total rent=A≡ pF A , L A−w LA
The marginal profit in terms of land is given by :dA/d A= p∂ F A , L/∂ A p∂ F A , L/∂ L−w dL/dA
But the necessary conditions implies that p∂F A , L/∂ L=w , hence:
dA/d A= p∂ F A , L/∂ A=≡AIntegrating over [0, A] , we get :
=A=∫0
AdA/dAdA=∫0
AAd A≡A
NB : A consequence of the envelope theorem which states that: d V /d =∂V /∂ ,
where V stands for the value function of the optimization program
(the profit in our case), is some parameter (here A ).
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Marginal land rent as a function of land size
Totally differentiating :pF AA , L=⇒ p F AAA , LdAF ALA , LdL /dAdA=d
Differentiating the optimality condition with respect to labour:pF LA , L=w⇒ p F LA A , LdAF L LA , LdL /dAdA=0
gives:dLA/dA=−F AL A , L/F L LA , L .
Plugging into the above:d Ad A
= p F A A−F ALF ALF L L
= pF L L
F AAF L L−F AL2
Joint strict concavity assumption :F L L0 F A A0 F L L F AA−F AL
2 0Implies that:
d AdA
0
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Conclusion
For a land of homogeneous quality:● The total land rent is equal to the profit of the landlord, that is the difference betweensales and the labour cost.● When the labour and land inputs are substitutable, the land rent depends upon thesize of the land plot.● The marginal land rent is the extra profit (or marginal profit) over an extra land unitabove the landlord property. ● The marginal land rent is equal to the marginal opportunity cost of land, that is theLagrange multiplier associated to the land constraint.● The total rent is equal to the total opportunity cost, that is the integral of themarginal opportunity costs over the land size range.● If the production technology exhibits marginal decreasing returns on inputs, that is ifthe production function is jointly concave in the land and labour inputs, then themarginal rent is a decreasing function of the size of the land plot.
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Lands of differing 'qualities'
The concept of Ricardian RentA family of land plots of differing 'qualities''Quality' is here identified to the productivity of labour
The technolgy is of the linear class:
Ranking by labour productivity index : Total profit on plot Ai:
Unit rent on plot Ai:In the linear case : unit rent (or average rent) = marginal rent
F A={A1 , A2, ... , Ai , ... , AI }
ai labour units 1 output unit on plot Ai
i= pQi−wLi= p−waiQip−wai
a1a2...ai...a I
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Ricardian rents in the linear case
Fig 4 : The ranking of the Ricardian rent
p-w a1
p-w a2
p-w aM
p-w aM+1
0
Rent
A1 A2 AM AM+1
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Ricardian rents
The plot AM is called the marginal landThe marginal land is the least productive land plot earning a positive profitIn the linear case, the marginal land rent (or here the unit rent) does not depend upon the land sizeThe marginal land rents are ranked in order of marginal productivities of landThe total land rents may be ranked in totally different order (depending of the land sizes)The meaningful economic concept is those of marginal land rent
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Ricardian rent (general case)
A family of technological constraints
A corresponding family of marginal land rents:
The technology is assumed concave in (A,L)The marginal land rents are hence decreasing functions of AThe marginal land rents are ranked by strictly decreasing order:
{F i L , Ai i∈{1, ... , I }}
{i A , AAi , i∈{1, ... , I }}
i Aii10 , i∈{1, ... , I }
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The ranking of ricardian rents
Fig 5 : An example of ranking of the marginal rents
A
A1
A2
A3
Marginal rent
Land A1 A2 A3 AM
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Conclusion
For lands of differing qualities:● Lands with different labour productivity exhibit the so-called 'differential'rents of 'Ricardian rents' property. That is land with higher labour productivityhave higher marginal land rents.● With linear technologies, the ranking of the marginal and unit rents areidentical and determined by the ranking of the labour productivity coefficients.● Under more general technological assumptions, the marginal and total rentswill depend upon the respective sizes of the land plots. The ranking of marginalrents and total rents may be completely different. ● The concept of marginal Ricardian rent provides the sound economic basisto the determination of the economic value of land.● The marginal land rent is the WTP of the land owner to get on extra unit ofland of the same 'quality'.
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Von Thunen like land rents
Introduce the idea of distance to some 'centre'First developed by Von Thunen to explain land prices differentialsThe very basic model of housing price determinationThe spatial land model:
Fig 6 : a simple spatial land model
A0x
A
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Spatial land rents
The model (continued)x(A) : distance form the centre of the point AT(Q,x(A)) = c x(A) Q : transportation cost of output from AThe technology is of the Leontiev class
Under efficiency :
The profit at the distance x from the centre:
Q=mina L L ,aA
Q=aA , L=a A/aL
x = pQ−wL−T Q , x = pa A−wa A/aL−cxa A=a A p−w /a L−cx
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Differential rents in the spatial model
Fig 7 : Differential land rent in a spatial model
a A p−w /a l
a A p−w/a l−aA c x
x ADistance x
Rent
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Land specialization
Two activities with different marginal spatial rents:
Fig 8 : specialization of land use and land pricing
X
Rent
Q1
Q2
X Distance x
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Specialization of activities
In the Leontiev case, the unit rents are equal to the marginal rents (as a function of the distance)The highest unit rents are the most profitable activitiesThe curves Q1 and Q2 also stand for the spot demand functionsThe marginal rents are the spot land prices (hiring prices)The land spot price as a distance function is the upper contour of the marginal land rents.On actual land markets, the price depends upon transaction costs
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Conclusion
Lands plots may be differing by their access costs (or distance costs) from somelocation in space (the 'centre' ):
● The land rent in this case is a decreasing function of the distance betweenthe land plot and the 'centre'● The distance cost introduces the same differential element between rentsas observed in ranked land quality models (the so-called Ricardian rentmodels)● When multiple activities may be undertaken on the land, the activities shouldbe ranked by decreasing order with the distance from the centre. That ishighest rent activities should occur besides the centre and less profitableactivities should occur farther in distance from the centre.● The equilibrium spot land price upon the land market should be equal to themarginal land rent and decrease with the distance from the centre.
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Land as a capital
Land 'rent' is usually identified to a flow of wealth from land propertyPricing land as an asset : the concept of present value
A rent flow from a land plot over an infinite duration:
The present value of the rent flow:
With i the interest rate. The present value gives the land price as an asset in a perfect land market equilibrium.
{r0, r1, ... , r t , ....}
V=∑t=0
∞
[ 11i t
] rt
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Conclusion
The land price of the land, as an asset (capital good), should be equal in a perfect equilibrium without transaction costs to the present value of the perpetual flow of rents obtained from an efficient exploitation of the land plot
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Appendix : The concept of present value
A property right over a money amount R to be received in 10 yearsWhat is the smallest amount of money v you would accept now if you decide to sell this property right ?Assume you invest this money amount on the capital market at an interest rate i:
After one year get : v1 = v + iv = (1+i)vAfter two years get: After 10 years get: The money amount v must make you indifferent between getting R in 10 years or receiving the capitalized value of v in 10 years, that is:
v2=v1i v1=1iv1=1i 2 vv10=1i 10 v
R=v10 ⇒ v= R1i10
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Appendix : the concept of present value
Now, let us consider an asset producing an infinte stream of money amounts :To each money amount, compute the present money equivalent:
The money amount V making someone indifferent between receiving the flow of money amounts and selling the asset today is the sum of these present money equivalents.
V is the present value of the infinite stream of money
{R0,R1, ... , Rt , ...}
vt=Rt
1i t
V=∑t=0
∞ Rt1it
{R0,R1, ... , Rt , ...}