Biosight: Quantitative Methods for Policy Analysis: CES Production Function and Exponential PMP Cost Function

1. Day 2: CES Production Function and
Exponential PMP Cost Function
Day 2 NotesHowitt and Msangi 1

2.  Understand the effect on policy models of a
quadratic PMP cost function.
 Understand this formulation has a production
function which enables us to measure
adjustments at the intensive margin.
 Run and interpret the PMP CES Machakos
model, and Calculate the elasticities of supply
and input demand.
Day 2 NotesHowitt and Msangi 2

3.  PMP Calibration
◦ Calibration Checks
◦ Livestock PMP Machakos Model
◦ CES Production Function
◦ Machakos CES PMP Model
◦ Calibrating Demands with Limited Data
◦ Endogenous Prices
Day 2 NotesHowitt and Msangi 3

4. Day 2 NotesHowitt and Msangi 4

5.  I. All positive net returns
 II. LP estimated acreage is close to observed base acreage
 III. Difference between marginal PM cost at base land allocation from
corresponding dual calibration constraint value
 IV. First-order conditions hold
 V. Verify calibrated non-linear model reproduces observed base
solution
Day 2 NotesHowitt and Msangi 5
0≥c
*
100 tolerance
 −
⋅ ≤ 
 
x XBASE
XBASE
( ) ( )2
2
100 tolerancei i i i i
i i
XBASE adj
adj
α γ λ
λ
+ − +
⋅ ≤
+
( ) ( )
( )
1 2
1 2
VMP
100 tolerance
ij ij j i i
ij j i i
w adj
w adj
λ λ
λ λ
− + + +
⋅ ≤
+ + +
*
100 tolerance
 −
⋅ ≤ 
 
xn XBASE
XBASE

6. 1. Base
Dataset
PMP Calibration Stages and Tests
2. Calibrated
Linear Program
3. CES
Analytical
Derivation
4. PMP Least
Squares
Solution
5. Demand
Calibration
6. CES & PMP
Endogenous
Price
Net
Returns
% Diff
from Base
VMP vs.
Opportunity
Cost
% Diff
PMP
Price
Check
% Diff
from Base
Policy runs
Tests
Stages
Day 2 NotesHowitt and Msangi 6

7. Day 2 NotesHowitt and Msangi 7
max i i i ij ij
i ij
v y x a csΠ= −∑ ∑
subject to
i ix XBASE iε≤ + ∀
i i
i i
x XBASE≤∑ ∑
CATT,HAYLINK GRASS,HAYLINK 0CATT GRASSa x a x+ =
0ix ≥
 Livestock PMP Machakos Example: Machakos_Cattle_PMP_Day2.gms
 Same Primal LP problem, except with calibration constraints:

8. Day 2 NotesHowitt and Msangi 8
subject to
( )0.5i i i i i i i
i
Max v y xn xn xnα γ− +∑
i i
i i
xn XBASE≤∑ ∑
CATT,HAYLINK GRASS,HAYLINK 0CATT GRASSa xn a xn+ =
 After introducing the “IL” set, allowing us to exclude
cattle, we can define the calibrated non-linear program:

9. Day 2 NotesHowitt and Msangi 9

10.  Assume Constant Returns to Scale
 Assume the Elasticity of Substitution is known
from previous studies or expert opinion.
◦ In the absence of either, we find that 0.17 is a
numerically stable estimate that allows for limited
substitution
 CES Production Function
Day 2 NotesHowitt and Msangi 10
/
1 1 2 2 ...
i
i i i
gi gi gi gi gi gi gij gijy x x x
υ ρρ ρ ρ
 = τ β +β + +β 

11.  Consider a single crop and region to illustrate
the sequential calibration procedure:
 Define:
 And we can define the corresponding farm profit
maximization program:
Day 2 NotesHowitt and Msangi 11
1σ −
ρ =
σ
/
max .
j
j j j j j
x
j j
v x x
υ ρ
ρ 
π= τ β − ω 
 
∑ ∑

12.  Constant Returns to Scale requires:
 Taking the ratio of any two first order conditions
for optimal input allocation, incorporating the CRS
restriction, and some algebra yields our solution
for any share parameter:
Day 2 NotesHowitt and Msangi 12
1.j
j
β =∑
( )
( )
1 1
1
1
1
1
1
1 l
l l
letting l all j
x
x
− σ
− σ
β= = ≠
 ω
+  
ω  
∑
( )
( )
1
1
11
11
1
1
1
.
1
l
l
ll
l l
x
xx
x
− σ
− σ− σ
− σ
ω
β =
ω ω
+  
ω  
∑

13.  As a final step we can calculate the scale parameter
using the observed input levels as:
Day 2 NotesHowitt and Msangi 13
/
( / )
.i
land land
j j
j
yld x x
x
υ ρ
ρ
⋅
τ =
 
β 
 
∑
 

14.  To avoid unbalance coefficients, we can scale
input costs into units of the same order of
magnitude for the program, and then de-scale
inputs back into standard units.
Day 2 NotesHowitt and Msangi 14
( )
( )
1
1 1
1
1
l
l
l
x
x
− σ
− σ
ω β
β =
ω
jβ

15.  Machakos CES PMP Example: Machakos_CES_Crops_PMP_Day2.gms
 Specify model with same data used for Primal LP and
Quadratic PMP with Leontief production technology.
subject to
 One important difference: Input constraints
Day 2 NotesHowitt and Msangi 15
( )0.5i i i i i i i
i
Max v y xn xn xnα γ− +∑
i i
i i
xn XBASE≤∑ ∑
/
1 1 2 2 ...
i
i i i
i i i i i i ij ijy x x x
υ ρρ ρ ρ
 = τ β +β + +β 

16.  When only equilibrium price and quantity in the model
base year, and an estimate of elasticity are available, we
can follow these steps to derive a demand function:
◦ Assume linear form specification:
◦ Recall demand elasticity:
◦ Rearrange the flexibility relationship:
◦ Derive the intercept:
Day 2 NotesHowitt and Msangi 16
inti i i iv yδ= +
i i
i
i i
y v
v y
η
∂
=
∂
i i
i
i
v
y
µ
δ =
inti i i iv yδ= −

17.  Redefine the non-linear profit maximization
program with endogenous prices and CES
production, except we include the demand function
as an additional constraint in the model:
Day 2 NotesHowitt and Msangi 17
( )0.5i i i i i i i
i
Max v y xn xn xnα γ− +∑
subject to
i i
i i
xn XBASE≤∑ ∑
/
1 1 2 2 ...
i
i i i
i i i i i i ij ijy x x x
υ ρρ ρ ρ
 = τ β +β + +β 
inti i i iv yδ= +