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
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:
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= +
