Motivation: Agriculture is confronted with the serious problem of being identified

1. On the Economic Modelling of Field Sizes, Landscape Patterns, and Nature Elements for Landscape ManagementIf we want it!E.-A. NuppenauJustus Liebig University Giessen, Germany

2. Motivation: • Agriculture is confronted with the serious problem of • being identified • as less multi-functional, • being a savage species that endangers bio-diversity, and • being an evil that tries to reshape cultural landscapes • purely according to the modern agronomic rationale of • mass production. • Note in this context, that farmers are not voluntarily doing • that, rather they feel under great pressure from international • commodity markets which forces them to minimise costs. • The society seems to put pressure on agriculture to enter into • the business of the provision of nature services.

3. Idea: • The paper examines a new way how to model the • provision of a landscape that shall be • - characterised by diversity, that • contains nature elements, and • where the intensity of farming is controlled. • as much as this is related to the provision of bio-diversity. • The task is to show how questions like • where to pay, • for what to pay, and • how participation is guaranteed • can be simultaneously answered. • The approach addresses heterogeneity in landscapes and • shows how to use non-linear programming.

4. Objectives: • 1. The approach shall enable governments to specify goal • functions for landscape management which • include ecologically retrievable criteria for payments, • help to determine payments, and • fit into farmers’ concerns to capture economies of • scale for maintaining competitiveness. • 2. We outline why this is a problem. • 3. We show how a geometrical presentation of land use helps us to specify interfaces between economic and ecological units. • 4. We elaborate on a nature production function as linked to landscape elements for which farmers can be paid. • 5. We demonstrate how a principal agent specification of objective functions for farmers can be established.

5. Concept Fig. 1a: Traditional Land use structure Fig. 1b: Modern Land use structure

6. Fig. 2: Stylized Structure of landscape a: stylized for mathematical presentation b: calculation of area

7. Modelling Concept: • An important thing is to structure planning and payment • instruments that create landscapes as spatially based on • “suitable” field sizes, • field and crop diversity, necessary • agronomic differentiation of behaviour on fields, etc. • -This basic concept enables us to visualize planning options. • The purpose is • - to have a tool that can predict changes in farm and • landscape composition along current trends of fully • using economies of scale and • - to have a tool that enables landscape planners to • determine compensation payments for ecological • services.

8. Farmers’ Objective Function where: pc = price q = quantity produced l = land size of a field r = input prices z = size of a farm w = wage i = intensity x = exogenous factors

9. Farmers’ income and labour constraint where additionally: mi = income aspirations wio= off-farm wage, exogenous wii = returns to labour on the farm, endogenous

10. Land: Technically we can use a Taylor expansion of second order around a reference point. li,j = ai bj = ai,0 bj,0 + ai,0 bi + ai,0 bj =A/nB/m+A/n(bj -B/m) + B/m(ai -A/n) We obtain a function that is linear in grid length for new steps. li,j = o + 1 (bj - bj,0) + 2 (ai - ai,0)

11. Ecological Objectives: • Db = s´ ln (s) • where: s = species vector • As habitats “h” relates to a probability matrix of a likely appearance of species “s”, habitats translate in species • s =  h • Then habitats may decompose into fields “b”. • Approximating, we represent species by • the areas set aside, i.e. buffers trips, “c” • A new landscape structure, i.e. deviations from economically optimal field sizes, “b” and • An indicator of yields in farming that reflects the intensity, “y” i.e. use of inputs, all as function: • s = 1[a0c +ua0c0] +2/A/B[a0(b-b0) +(a-a0)b0] + 3[1´a0[1´b] • +1´a[1´b0]]+ 4(y-y0) • where: u = additional labour vector for controlling vegetation on buffer strips

12. Payments: • Our concurrent problem is to specify payments. Payments require three aspects: • the payment criteria, • the determination of the size of the contribution per criteria, and • the volume of an over­all payment. • Our equation (1) gives a set of criteria that are derived from previous modelling, i.e. • 1. change in field size, • 2. size of buffer strips • 3. labour, and • 4. yields. • gi,k=g0,i,k+ g1,i,kaj,i (bj,i,k-bj,0,i,k)+ g2,i,ka,i,cj,i+ g2,i,k (yj,i,k-yj,0,i,k) + • g3,i,kuj,i,k + g4,i,kbj,0,i,kai (10)

13. Principal Agent Approach: Agent: Farmers in the Landscape maximise income form payments For reasons to keep the inter­change active and practically, a vector presentation fulfils the argument to capture structural and numerical components. With the basic arguments of vectors presenting field “b” and farm size “a” components, buffer strips “c”, yields (i.e. intensity) “y”, labours (u), we receive for the farmers in the landscape P(p,g,a,b,i,b,u,i)= py + ge- ´y - 0.5[y´+e] [y-e]+ z´[y-e] + ´ [[y+e] +g´I e + Zx] Reaction function: This would give approximately: e = [+]-1 [[0 I + I]g + y* +  z + Zx] where y* is a function of p: y*=-1[p++z´]+… determining structures, off-set by payments. Principal: Government maximises diversity by given farmers’ reaction “e” and assigned budget “g”: E = [1y + 2 e] ´ [I+ [1y + 2 e]] -´[gg - ge]

14. Summary • It was the objective of this paper to show how payments • for environmental services can be integrated in landscape • planning in order to facilitate a cost-effective provision • of biodiversity. • A goal function for a government has been derived that • captures economic aspects of land user concerns on the • one side and ecological aspects of public concerns for bio- • diversity conservation on the other side. • A necessary condition was that farmers are compensated • for interventions in their design of landscapes, as been • characterised by ecologically preferred field sizes, buffer • strips, nature elements and intensity of farming: