Natural resources economics M1 Chapter One Land and Water
Introduction and overview The fundamental distinction between resources Renewable 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 resources Available resource flow submited to some 'limits': Physical limits Technical limits Cultural and religious constraints Political and institutional limits (transboundary sharing of rivers) 2
The fundamental distinction between resources Non renewable (or exhaustible) resources Provide only a finite duration flow of services Examples : coal, oil, iron, copper, and other mineral resources Combine limits upon the flow of services and upon the stock of services (of limited size) Two important remarks A distinction economically based (concept of 'service') A common characteristic: the existence of 'limits' Makes room for 'scarcity' considerations Introduces the concept of 'opportunity cost' associated with these 'limits' Equivalence between 'opportunity cost' and 'resource rent'
Overview of Chapter One Principles of Land Economics Economics of land use : the concept of 'rent' A starter : a Leontiev example Land rent when inputs are subsituable Land of differing qualities : the 'Ricardian rent' Von Thunen like land rent Land as a capital good Appendix : The concept of present value 4
Overview of Chapter One Water economics Water problems in the world today Water as a resource Water quality A 'damage function' approach An 'environmental benefits' approach Water scarcity rents Sharing a river Appendix : water issues in France 5
Land economics An elementary model An agricultural good produced from land and labour Q F A , L A Limited supply of land Land and labour of homogeneous quality Price of the consumption good : p Wage rate : w The landlord demand for labour and supply of consumption good 'small' with respect to the market (exogeneous prices) 6
A Leontiev example The technology is of the Leontiev class: F A, L = min a L L ,a A A Efficient use of inputs requires: a L L = a A A The landlord profit as a function of land use: A = pa A A − w a L A = a A p − w A a A a L A necessary condition for economic activity: p w / a L Under this condition all the available land is put in use. 7
Land rent in the Leontiev case Increase of profit with land availability increase ('marginal' rent): = a A p − w / a L Operating cost: C = a A w A a L The total land rent is identical to the total profit. 8
Land rent in the Leontiev case Fig1 : Land rent in the Leontiev example Slope : w a A / a L a A p Rent Unit cost w a A / a L function C/A Cost A 9
Land rent when inputs are substituable No subsituability in the Leontiev case With 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 ? 10
Land rent as a difference between average and marginal productivity Average productivity (production per capita) AP L = pF A , L / L MP L = p ∂ F A , L Marginal productivity in value : ∂ L Fig 2 : Land rent as a difference between average and marginal productivity Ouput in value pQ Rent w MP AP Cost Labour L 11
The profit maximizing landlord Max = pF A, L − wL s:t A A The Lagrangian of this problem is: L = p F A , L − wL A − A And the first order conditions give: p ∂ F A, L /∂ L = w p ∂ F A, L /∂ A = Gamma appears as the marginal rent of land It is also the 'marginal opportunity cost' of the land constraint It is also the marginal willingness to pay (WTP) of the landlord for one extra unit of land (the land 'value') 12
From marginal rent to total rent = A ≡ pF A ,L A − w L A The marginal profit in terms of land is given by : d A / 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: d A / d A = p ∂ F A ,L /∂ A =≡ A Integrating over [ 0, A ] , we get : A A A = ∫ 0 d A / dA dA = ∫ 0 = A d 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 ). 13
Marginal land rent as a function of land size Totally differentiating : pF A A , L =⇒ p F AA A , L dA F AL A, L dL / dA dA = d Differentiating the optimality condition with respect to labour: pF L A , L = w ⇒ p F LA A, L dA F L L A , L dL / dA dA = 0 gives: dL A / dA =− F AL A, L / F L L A, L . Plugging into the above: d A F AL = p 2 = p F A A − F AL F AA F L L − F AL d A F L L F L L Joint strict concavity assumption : 2 0 F L L 0 F A A 0 F L L F AA − F AL Implies that: d A 0 dA 14
Conclusion For a land of homogeneous quality: ● The total land rent is equal to the profit of the landlord, that is the difference between sales and the labour cost. ● When the labour and land inputs are substitutable , the land rent depends upon the size of the land plot. ● The marginal land rent is the extra profit (or marginal profit) over an extra land unit above the landlord property. ● The marginal land rent is equal to the marginal opportunity cost of land, that is the Lagrange multiplier associated to the land constraint. ● The total rent is equal to the total opportunity cost , that is the integral of the marginal opportunity costs over the land size range. ● If the production technology exhibits marginal decreasing returns on inputs, that is if the production function is jointly concave in the land and labour inputs, then the marginal rent is a decreasing function of the size of the land plot. 15
Lands of differing 'qualities' The concept of Ricardian Rent A family of land plots of differing 'qualities' 'Quality' is here identified to the productivity of labour F A ={ A 1 , A 2, ... , A i , ... , A I } The technolgy is of the linear class: a i labour units 1 output unit on plot A i a 1 a 2 ... a i ... a I Ranking by labour productivity index : Total profit on plot Ai: i = pQ i − wL i = p − w a i Q i p − w a i Unit rent on plot Ai: In the linear case : unit rent (or average rent) = marginal rent 16
Ricardian rents in the linear case Fig 4 : The ranking of the Ricardian rent Rent p-w a1 p-w a2 p-w aM 0 A1 A2 AM AM+1 p-w aM+1 17
Ricardian rents The plot AM is called the marginal land The marginal land is the least productive land plot earning a positive profit In the linear case, the marginal land rent (or here the unit rent) does not depend upon the land size The marginal land rents are ranked in order of marginal productivities of land The 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 18
Ricardian rent (general case) A family of technological constraints { F i L , A i i ∈{ 1,... , I }} A corresponding family of marginal land rents: { i A , A A i , i ∈{ 1,... , I }} The technology is assumed concave in (A,L) The marginal land rents are hence decreasing functions of A The marginal land rents are ranked by strictly decreasing order: i A i i 1 0 , i ∈{ 1,... , I } 19
The ranking of ricardian rents Fig 5 : An example of ranking of the marginal rents Marginal rent A A 1 A 2 A 3 A1 A2 A3 AM Land 20
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 productivity have higher marginal land rents. ● With linear technologies, the ranking of the marginal and unit rents are identical and determined by the ranking of the labour productivity coefficients. ● Under more general technological assumptions, the marginal and total rents will depend upon the respective sizes of the land plots. The ranking of marginal rents and total rents may be completely different. ● The concept of marginal Ricardian rent provides the sound economic basis to the determination of the economic value of land. ● The marginal land rent is the WTP of the land owner to get on extra unit of land of the same 'quality'. 21
Von Thunen like land rents Introduce the idea of distance to some 'centre' First developed by Von Thunen to explain land prices differentials The very basic model of housing price determination The spatial land model: Fig 6 : a simple spatial land model A A 0 x 22
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