Mathematical programming and exercises

1. Torbjörn Jansson* CAPRICommon Agricultural Policy Regional Impact Mathematical programming and exercises CAPRI Training Session in Warzaw June 26-30, 2006 *Corresponding author +49-228-732323www.agp.uni-bonn.de Department for Economic and Agricultural Policy Bonn University Nussallee 21 53115 Bonn, Germany

2. Session outline: • A linear programming model • A quadratic programming model • Experiments with a linear and a quadratic model (exercise)

3. An aggregate LP model Margins m(yield*price-variable cost) Endogenousvariables,here activity levels Objective value ObjectiveFunction Constraints Shadow pricesof constraints I/O coefficients Constraint vector

4. Theory: Linear programming Lagrange function First order conditions (Kuhn - Tucker) Revenue Exhaustion (margin = opportunity costs) Constrains must hold Programming model

5. Three activities {1,2,3} and two resources {l,c} Kuhn-Tucker conditions: At most two of the inequalities on the left can be satisfied with equality (if matrix A has full rank) At most two activities can be non-zero At least one activity will have too small a margin m to pay for the fix resources at least as good as the other activities. Reaction to changed margins (I)

6. Numerical example DATA Activities: CERE, SUGB, POTA Resouces: Land, Capital Margins: CERE 575 SUGB 1000 POTA 500 Resource use matrix A = CERE SUGB POTA Land 1 1 1 Capital 100 300 280 Resource constraints B: Land = 10, Capital = 2540 FOC CERE: 575 - l - c100 ≤ 0  xc ≥ 0 SUGB: 1000 - l - c300 ≤ 0  xs ≥ 0 POTA: 500 - l - c280 ≤ 0  xp ≥ 0 (solve with algorithm…) l = 362.5 c = 2.125 CERE: 0 ≤ 0, xc = 2.3 SUGB: 0 ≤ 0, xs = 7.7 POTA: -457.5≤ 0, xp = 0.0

7. At m0POTA only SUGB and CERE take place Raise margin of the “zero activity”(POTA) and observe behaviour At mPOTA < m’ POTA only SUGB and CERE take place. At mPOTA> m’ POTA only activities POTA and CERE take place, a.s.o. Dual values change DEMO: Tuesday\LPQP.gms Reaction to changed margins (II) x SUGB POTA CERE m0POTA m’POTA mPOTA Land Capital

8. Conclusions LP • If there are k constraints, at most k activities will non-zero in the optimal solution • A linear model responds discontinuously (semicontinuously) to changes • Generally, it is not possible to set up the model to exactly reproduce observed activity levels

9. Theory: Quadratic programming I Programming model Lagrange function

10. Theory: Quadratic programming II Revenue Exhaustion (margin = opportunity costs) Constrains must hold Kuhn - Tucker conditions

11. How determine PMP-terms? • Howitt 1995  works, but wrong dual values, no information on price effects • Heckelei 2003  Estimate first order conditions. Difficult. • In CAPRI: Use exogenous supply elasticities.

12. A calibration method for a QP using exogenous own price elasticities If only non-zero activities are considered Solving for x yields Assumption 1: jk = 0 for j  k Assumption 2:  is constant and known  /m= 0 (with mj=pj-cj) and

13. Calibrate to own price elasticities of unity Raise price of output of POTA and observe behaviour DEMO: Tuesday\LPQP.gms Reaction to changes (III) x SUGB POTA CERE mPOTA

14. Exercises • Use tuesday\LPQP.gms • Task 1: Type the Kuhn-Tucker conditions of the NLP-model and solve them.Hints:- assume that all activities are non-zero,- define an equation z = 1 and solve system by max. z. • Task 2: Plot the relationship between exogenous exasticities and point elasticities of model.Hint: Use the existing loop and parameters to calibrate the QP to different own price elasticities, simulate a 1% margin-increase, compute the point elasticity and plot the results.