Chapter One Land Water - Institut national de la recherche ... · Introduces the concept of 'opportunity cost' associated with these ... Land as a capital good Appendix : The concept

Natural resources economics M1

LandandWater

Chapter One

2

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

5

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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Actual land markets

High level of 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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Price dispersion around a city

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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 , ...}