Biosight: Quantitative Methods for Policy Analysis- Introduction to GAMS, Linear Programming

Published on: **Mar 3, 2016**

Published in:
Education

- 1. Day 1: Introduction to GAMS, Linear Programming, and PMP Day 1 NotesHowitt and Msangi 1
- 2. Understand basic GAMS syntax Calibrate and run regional or farm models from minimal datasets Calculate regional water demands Calculate elasticity of water demand Estimate the value of rural water demand for water policy Day 1 NotesHowitt and Msangi 2
- 3. Linear Models Linear Programming: Primal Positive Mathematical Programming Day 1 NotesHowitt and Msangi 3
- 4. Day 1 NotesHowitt and Msangi 4
- 5. We can typically specify a model as a constrained or unconstrained maximization Consider the general production function ◦ The price of the output q is p per unit output, and the cost per unit x is w. Define profit Π . Consider the profit maximization problem Which we can write and solve as Day 1 NotesHowitt and Msangi 5 1 2( , )q f x x= 1 1 2 2Max pq x w x wΠ= − − 1 2subject to ( , )q f x x= ( )1 1 2 2 1 2( , )L pq x w x w q f x xλ= − − − −
- 6. Let’s assume linear production technology (Leontief) so we can write We can rewrite the linear model with one output as Or, in more compact matrix notation Day 1 NotesHowitt and Msangi 6 1 2 1 1 2 2( , )f x x a x a x= + 1 1 2 2Max pq x w x wΠ= − − 1 1 2 2subject to 0q a x a x− − = 1 2[ , , ]p w w= − −c' 1 2[1, , ]a a= − −a' 1 2' [ , , ]q x x=x Max 'c x subject to ' 0=a x⇒
- 7. We will modify this example to include multiple outputs and derive the LP problem Linear Programming ◦ Output levels and input availability should be specified as inequality constraints ◦ Given a set of m inequality constraints in n variables ( x ), we want to find the non-negative values of a vector x which satisfies the constraints and maximizes an objective function Define as the quantity available for each input (or “resource”) i Resources can be used in the production of multiple outputs (i), reflected in technical coefficients Day 1 NotesHowitt and Msangi 7 ib ija
- 8. Let’s define the matrix of technical coefficients and vector of available inputs And we can write the general LP as Note that we have 2 (constrained) inputs and 2 outputs in our example, but this notation generalizes to any number. Day 1 NotesHowitt and Msangi 8 11 12 21 22 a a a a = A1 2 b b = b Max 'c x subject to ≤Ax b
- 9. The Machakos example: Machakos_Primal_Day1.gms Leontief technology 5 Crops: Inter Cropped, Maize, Beans, Tomato, Grass 4 inputs (constrained): land, labor, chemicals, and seed We will formulate the model Day 1 NotesHowitt and Msangi 9 Max 'c x subject to ≤Ax b
- 10. Day 1 NotesHowitt and Msangi 10 [ ]1 2 3 4 5' [ ]x x x x x Inter Cropped Maize Beans Tomato Grass= = −x 1 2 3 4 Land (hectares) 2.78 Labor (person days) 250 Chemicals (kg) 6,000 Seed (kg) 6,000 b b b b =≡ = b 11 12 13 14 15 21 22 23 24 25 31 32 33 34 35 41 42 43 44 45 1 1 1 1 1 40.3 159 126.5 136 0 8.75 83.9 12.03 181.3 30 43 44.6 50.3 22 0 a a a a a a a a a a a a a a a a a a a a = A [ ]1 2 3 4 5' [ ] 13,563 8,350 31,125 37,704 24,980c c c c c=c
- 11. Let’s multiply out a constraint and interpret Constraint 3: Interpretation: total use of chemicals in the production of all crops must be less than or equal to the total available chemicals Numerically: We will formulate and solve the model during the afternoon session Day 1 NotesHowitt and Msangi 11 31 1 32 2 33 3 34 4 35 5 3a x a x a x a x a x b+ + + + ≤ 1 2 3 4 58.75 83.9 12.03 181.3 30 6,000x x x x x kg+ + + + ≤
- 12. Minimizing the cost of inputs subject to a minimum output level is equivalent to maximizing profit subject to production technology and the total input available For every Primal Problem there exists a Dual Problem which has the identical optimal solution. ◦ Primal question: what is the maximum value of firm's output? ◦ Dual question: what is the minimum acceptable price that I can pay for the firm's assets? The “dual” or “shadow” value has economic meaning: ◦ It is the marginal value (or marginal willingness to pay) of another unit of a given resource. Day 1 NotesHowitt and Msangi 12 ( )iλ
- 13. Dual objective function ◦ Equal to the sum of the imputed values of the total resource stock of the firm (amount of money that you would have to offer a firm owner for a buy-out). Dual Constraints ◦ Set of prices for the fixed resources (or assets) of the firm that would yield at least an equivalent return to the owner as producing a vector of products ( x ), which can be sold for prices ( c ), from these resources. Where do these values come from? Day 1 NotesHowitt and Msangi 13 Max 'c x subject to ( )≤Ax b λ
- 14. Day 1 NotesHowitt and Msangi 14
- 15. Linear Programming shortfalls ◦ Overspecialization ◦ Will not reproduce an observed allocation without restrictive constraints ◦ Tendency for “jumpy” response to policy Questions ◦ How do we calibrate to observed but limited data? ◦ How do we use these models for policy analysis? ◦ How do we introduce rich resource constraints? Perennial crops Climate change Technology Regulations Day 1 NotesHowitt and Msangi 15
- 16. Behavioral Calibration Theory ◦ We need our calibrated model to reproduce observed outcomes without imposing restrictive calibration constraints Nonlinear Calibration Proposition ◦ Objective function must be nonlinear in at least some of the activities Calibration Dimension Proposition ◦ Ability to calibrate the model with complete accuracy depends on the number of nonlinear terms that can be independently calibrated Day 1 NotesHowitt and Msangi 16
- 17. Let marginal revenue = KSh 500/hectare Average cost = KSh 300/hectare Observed acreage allocation = 50 hectares Introduce calibration constraint to estimate residual cost needed to calibrate crop acreage to 50 Day 1 NotesHowitt and Msangi 17 Max500 300x x− subject to 50x ≤ 2λ
- 18. We need to introduce a nonlinear term in the objective function to achieve calibration. Here we introduce a quadratic total cost function. This is a common approach in PMP. Under unconstrained optimization, MR=MC ◦ For this condition to hold at x*=50 it must be that is the difference at the constrained calibration value (MR-AC). ◦ We know that MR=MC ◦ Therefore , since we require MR=MC at x*=50 Day 1 NotesHowitt and Msangi 18 2 0.5TC x xα γ= + 2λ 2 MC - ACλ =
- 19. We can now calculate the slope and intercept of the nonlinear cost function which will allow us to calibrate the mode without constraints and , thus We can calculate the cost slope coefficient Given the slope, the intercept follows from the AC equation Verify that Day 1 NotesHowitt and Msangi 19 MC xα γ= + 0.5AC xα γ= + 2( 0.5 )MC AC x xα γ α γ λ− = + − + = 2 * 2 x λ γ = 0.5 *AC xα γ= + 8 and 100γ α= =
- 20. Combine this information and introduce the calibrated cost function into an unconstrained problem Verify that we get the observed allocation as the optimal solution through standard unconstrained maximization ◦ We see that x=50, which is our observed allocation and we have verified that the model calibrates Day 1 NotesHowitt and Msangi 20 2 500 0.5Max x x xα γΠ= − − 2 500 100 0.5(8)Max x x xΠ= − − 2 400 4Max x xΠ= −
- 21. Now the model can be used for policy simulations The unconstrained profit maximization problem reproduces the observed base year We can introduce changes and evaluate the response without restrictive calibration constraints The method extends to multiple crops Day 1 NotesHowitt and Msangi 21
- 22. The PMP method extends to multiple crops ◦ PMP example: Machakos_QuadraticPMP_Day1.gms There are three stages to PMP 1. Constrained LP model is used to derive the dual values for both resource and calibration constraints 2. The calibration constraint dual values are used to derive the calibration cost function parameters 3. The cost function parameters are used with the base year data to specify the PMP model Day 1 NotesHowitt and Msangi 22
- 23. 2 Crop example: wheat and oats Observed Data: 2 ha oats and 3 ha wheat (total farm size of 5 hectares) Day 1 NotesHowitt and Msangi 23
- 24. We maintain the assumption of Leontief production technology and assume that land (input i=1) is the binding calibrating constraint We can write the calibrated problem as PMP calibration proceeds in three stages Day 1 NotesHowitt and Msangi 24 ( ) 3 2 0.5i i i i i i i j ij i i j Max p y x x x w a xα γ = − + −∑ ∑ subject to and= ≥Ax b x 0
- 25. Stage 1 Formulate and solve the constrained LP and note the dual values () We introduce a perturbation term to decouple resource and calibration constraints Day 1 NotesHowitt and Msangi 25 2 1 1 max ( ) ( ) 5 ( ) 3 ( ) 2 ( ) w w w w o o o o w o w w o o y p w x y p w x subject to x x x x λ ε λ ε λ Π= − + − + ≤ ≤ + ≤ +
- 26. The optimal solution is when the wheat calibration constraint is binding at 3.01 (wheat is the most valuable crop), and the resource constraint ensures oats at 1.99 Store the dual values for use in stage 2 Day 1 NotesHowitt and Msangi 26 2 1 1 max ( ) ( ) 5 ( ) 3 ( ) 2 ( ) w w w w o o o o w o w w o o y p w x y p w x subject to x x x x λ ε λ ε λ Π= − + − + ≤ ≤ + ≤ +
- 27. Stage 2 Derive the parameters of the quadratic total cost function ◦ Use same logic as in the single crop example Notice two types of crops in the problem depending on which constraint is binding ◦ Calibrated crops ◦ Marginal crops Calculate the cost intercept and slope for the calibrated wheat crop Day 1 NotesHowitt and Msangi 27
- 28. Graphically Day 1 NotesHowitt and Msangi 28
- 29. Stage 3 No restrictive calibration constraints Calibration checks ◦ Hectare allocation (all input allocation) ◦ Input cost = Value Marginal Product Can use the model for policy simulation Day 1 NotesHowitt and Msangi 29 ( ) 3 2 0.5 ,i i i i i i i j ij i i j Max p y x x x w a x where i o wα γ = − + − =∑ ∑ 0 5wx x+ ≤
- 30. We have covered a range of topics ◦ Linear models ◦ Linear Programming Primal Dual ◦ Positive Mathematical Programming Single crop mathematical derivation Multiple crop generalization This afternoon we will revisit these topics in GAMS ◦ Intro.gms ◦ Machakos_Primal_Day1.gms ◦ Machakos_Dual_Day1.gms ◦ Machakos_QuadraticPMP_Day1.gms Day 1 NotesHowitt and Msangi 30