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

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