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

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 function which enables us to measure adjustments at the intensive margin.

Run and interpret the PMP CES Machakosmodel, and Calculate the elasticities of supply and input demand.


PMP Calibration◦ Calibration Checks◦ Livestock PMP Machakos Model◦ CES Production Function◦ Machakos CES PMP Model◦ Calibrating Demands with Limited Data◦ Endogenous Prices



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


0≥c

*

100 tolerance −⋅ ≤

x XBASEXBASE

( ) ( )2

2

100 tolerancei i i i i

i i

XBASE adjadj

α γ λλ

+ − +⋅ ≤

+

( ) ( )( )

1 2

1 2

VMP100 toleranceij ij j i i

ij j i i

w adj

w adj

λ λ

λ λ

− + + +⋅ ≤

+ + +

*

100 tolerance −⋅ ≤

xn XBASEXBASE

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



max i i i ij iji ij

v y x a csΠ = −∑ ∑subject to

i ix XBASE iε≤ + ∀

i ii 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:


subject to

( )0.5i i i i i i ii

Max v y xn xn xnα γ− +∑

i ii 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:


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


/

1 1 2 2 ... ii i igi gi gi gi gi gi gij gijy x x x

υ ρρ ρ ρ = τ β +β + +β

Consider a single crop and region to illustrate the sequential calibration procedure:

Define:

And we can define the corresponding farm profit maximization program:


1σ−ρ =

σ

/

max .j

j j j j jx j jv x x

υ ρ

ρ π = τ β − ω

∑ ∑

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:


1.jjβ =∑

( )

( )

1 11

11

1 11 l

l l

letting l all jx

x

− σ

− σ

β = = ≠ ω

+ ω

∑

( )

( )

11

1111

11

1 .1

ll

ll

l l

xxx

x

− σ

− σ− σ

− σ

ωβ =

ω ω+

ω ∑

As a final step we can calculate the scale parameter using the observed input levels as:


/

( / ) .i

land land

j jj

yld x x

xυ ρ

ρ

⋅τ =

β

∑

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.


( )

( )

11 1

11

ll

l

xx

− σ

− σ

ω ββ =

ω

jβ

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



Max v y xn xn xnα γ− +∑

i ii i

xn XBASE≤∑ ∑/

1 1 2 2 ... ii i ii i i i i i ij ijy x x x


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:


inti i i iv yδ= +

i ii

i i

y vv y

η ∂=

∂

i ii

i

vyµδ =

int i i i iv yδ= −

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:



Max v y xn xn xnα γ− +∑subject to

i ii i

xn XBASE≤∑ ∑/

1 1 2 2 ... ii i ii i i i i i ij ijy x x x


inti i i iv yδ= +

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