Day 2: CES Production Function and
Exponential PMP Cost Function
Day 2 NotesHowitt and Msangi 1
 Understand the effect on policy models of a
quadratic PMP cost function.
 Understand this formulation has a production
...
 PMP Calibration
◦ Calibration Checks
◦ Livestock PMP Machakos Model
◦ CES Production Function
◦ Machakos CES PMP Model
◦...
Day 2 NotesHowitt and Msangi 4
 I. All positive net returns
 II. LP estimated acreage is close to observed base acreage
 III. Difference between margi...
1. Base
Dataset
PMP Calibration Stages and Tests
2. Calibrated
Linear Program
3. CES
Analytical
Derivation
4. PMP Least
Sq...
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≤∑ ∑
CA...
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 ...
Day 2 NotesHowitt and Msangi 9
 Assume Constant Returns to Scale
 Assume the Elasticity of Substitution is known
from previous studies or expert opinio...
 Consider a single crop and region to illustrate
the sequential calibration procedure:
 Define:
 And we can define the ...
 Constant Returns to Scale requires:
 Taking the ratio of any two first order conditions
for optimal input allocation, i...
 As a final step we can calculate the scale parameter
using the observed input levels as:
Day 2 NotesHowitt and Msangi 13...
 To avoid unbalance coefficients, we can scale
input costs into units of the same order of
magnitude for the program, and...
 Machakos CES PMP Example: Machakos_CES_Crops_PMP_Day2.gms
 Specify model with same data used for Primal LP and
Quadrati...
 When only equilibrium price and quantity in the model
base year, and an estimate of elasticity are available, we
can fol...
 Redefine the non-linear profit maximization
program with endogenous prices and CES
production, except we include the dem...
of 17

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

Biosight: Quantitative Methods for Policy Analysis: CES Production Function and Exponential PMP Cost Function
Published on: Mar 3, 2016
Published in: Education      
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Transcripts - 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= +