08.30.2012 - brian dillon

Introduction Data Model Identification Estimation Results

Estimation of a Dynamic Agricultural ProductionModel with Observed, Subjective Distributions

Brian Dillon

Cornell Universityand Harvard Kennedy School

August 30, 2012

Brian Dillon Estimation of a Dynamic Agricultural Production Model with Observed, Subjective Distributions


Motivation: crop production

To grow crops, farmers solve a dynamic resource allocation problem

The problem is not unlike many other dynamic choice problems:portfolio management, inventory management, human capitalinvestment

The solution to this problem can involve delay of some choices,distribution of activities across time, and updating of expectationsas new information arrives

Between-farmer variation in expectations clearly matters (Gine,Townsend, Vickery 2008)



What if we measure expectations?

Early literature in agricultural economics (Bessler and Moore 1979;Eales 1990)

Manski (2004) makes the case for measuring expectations

Nyarko and Schotter (2002) show that there is a big differencebetween observed and estimated expectations

Delavande et al (2010) review the recent development literaturethat uses subjective probabilities



What we get from measuring expectations

Two contributions to the estimation of dynamic choice models:

1. Allow us to relax the rational expectations assumptions thatare standard for these models (Wolpin 1987; Rust 1987, 1997;Fafchamps 1993)

2. There is a lot of information in a subjective distribution overan endogenous outcome



Why go through a structural exercise?

Apart from the pure value of estimating a less restrictedproduction function...

Production elasticities tell us something about resilience of theproduction process to shocks

What we know about shocks already largely deals with

• Consumption/asset smoothing (Townsend 1994, Morduch1995, Hoddinott 2006, Barrett and Carter 2006, Jacoby andSkoufias 1998, Fafchamps et al 1998)

• Human capital (Hoddinott and Kinsey 2001, Aguilar andVicarelli 2012)

• Two papers look at how farmers move labor across time,within a season: Fafchamps (1993) and Kochar (1999)



Why go through a structural exercise?

And we can also simulate important, relevant policies:

1. Insurance

2. Forward contracting

3. Improvements in information delivery

4. Changes in input supply



Contributions of this paper

We use a sequence of observed inputs, price expectations, andyield expectations to estimate an agricultural production function

Methodological contributions:

1. Develop a general method for estimating dynamic choicemodels with observed subjective distributions

2. Show how counterfactual choice data (“How much pesticidedid you want to apply last week?”) can be used in estimation

Substantive contributions:

1. Recover estimates of all elasticities of substitution betweeninputs (within and across periods)

2. Simulate the impact of insurance, forward contracting, andinformation provision policies



Plan of the talk

• Data set

• Model basics

• Identification of shock densities

• Estimation

• Results



Data set

• 195 cotton farmers in 15 villages in NW Tanzania

• Face-to-face agriculture and LSMS surveys conducted insummer 2009 and summer 2010

• From September 2009 - June 2010: investment, time use,shocks, agricultural input and output, and other datagathered every 3 weeks

• High frequency interviews also gathered subjective probabilitydistributions over end-of-season prices and yields, andqualitative distributions over pest pressure and rainfall atvarious points throughout the year



Measuring subjective distributions

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Evolution of subjective price distributions

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Bin numberSurvey period

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Evolution of subjective yield distributions!

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Smoothing of distributions

Let

• xi be a response vector

• d ∈ RN+1 be the interval boundaries

• z be the random variable in question

• k be the number of counters

We fit a four parameter beta CDF, Gi (z | a, b, ρ, κ), by solving:

(ai , bi , ρi , κi ) = arg infa,b,ρ,κ

N∑j=1

(∑jm=1 xj

k− G (dj+1 | a, b, ρ, κ)

)2



Sample summary statistics

Mean sd Min MaxHousehold size (people) 8.33 3.90 2 23Dependency ratio* 1.31 0.85 0 5.5Head age 46.85 14.69 20 100Head is male (%) 85.0 - - -Years of education (HH head) 4.19 3.46 0 11Radios 0.83 0.71 0 4Bicycles 1.19 1.00 0 10Dairy cattle 1.33 2.84 0 20Non-dairy cattle 3.87 7.89 0 60Goats 5.27 8.05 0 50Sheep 1.67 3.74 0 30Total acres 9.67 11.03 1 82Number of plots 2.71 1.17 1 7Number of crops grown 3.45 1.26 1 8Labor expenditure (TSH) 78,248 139,485 0 1,020,000Fertilizer expenditure (TSH) 21,149 81,359 0 715,000Animal labor expenditure (TSH) 33,497 92,724 0 750,000Transport expenditure (TSH) 10,333 20,049 0 144,000Other cultivation expenditure (TSH) 6,929 15,817 0 100,000Total cultivation expenditure (TSH) 150,156 254,863 0 1,514,700Notes: author's calculation from survey data; cultivation data refers to 2008-2009 cultivation of all crops; 1 USD ! 1 400 TSH; *Dependency ratio is number of persons aged < 15 or aged > 65 dividedcrops; 1 USD !"1,400 TSH; *Dependency ratio is number of persons aged < 15 or aged > 65 divided by number aged between 15 and 65.



Model assumptions

Important:

1. Farmers are dynamically consistent (will relax, if we have time)

2. Independence of shocks across time

Less fundamental:

1. Separable household model

2. Risk-neutral maximization of expected plot-level profits

3. All forms of labor are interchangeable

4. No binding credit constraints

5. Functional form choices



Crop evolution

Expanding on Fafchamps (1993), crops grow according to:

yi0 = φiAieθi0

yi1 = h1(yi0, li1, pi1)eθi1

yi2 = h2(yi1, li2, pi2)eθi2

yi = h3(yi2, li3, pi3)eθi3

where θit ∼ git(θit) for t = 0, . . . , 3

Ai is acreage

φi is a plot-specific yield shifter

li and pi are labor and pesticides



Crop evolution cont.

We use nested CES functions:

h1(y0, l1, p1 |α1, α2, γ) = [α1yγ0 + α2l

γ1 + (1− α1 − α2)pγ1 ]

1γ

h2(y1, l2, p2 |β1, β2, δ) =[β1y

δ1 + β2l

δ2 + (1− β1 − β2)pδ2

] 1δ

h3(y2, l3, p3 |κ1, κ2, ω) = [κ1yω2 + κ2l

ω3 + (1− κ1 − κ2)pω3 ]

1ω

Which gives us 9 production parameters to estimate:

• Share parameters (α1, α2, β1, β2, κ1, κ2)

• Transformed elasticity parameters (γ, δ, ω)



Farmer’s objective function

From the viewpoint of the first period:

maxli1,pi1

E [qc ]Eθi1θi2θi3

[(κ1

β1

[α1y

γi0+α2l

γi1+(1−α1−α2)pγi1

] δγ eδθi1

+ β2l∗δi2 + (1− β1 − β2)p∗δi2

ωδ

eωθi2 + κ2l∗ωi3

+ (1− κ1 − κ2)p∗ωi3

) 1ω

eθi3]−

3∑t=1

(ql lit + qppit)



Identification of gt(θt)

We need measures of gt(θt) in order to proceed

Nested fixed point method (Rust 1987): iterate between guesses ofproduction parameters and gt parameters until convergence

But we only observe subjective output distributionsΨ0(y), . . . ,Ψ3(y)

We can use those to directly estimate gt(θt), within the context ofthe model



Timeline of decisions, realizations, and data collection

y reported




y reported

!3 realized




y reported

!3 realized

"3(y) reported

(l3 , p3) chosen

incl:g3(!3)




y reported

!3 realized

"3(y) reported

(l3 , p3) chosen

!2 realized

incl:g3(!3)




y reported

!3 realized

"3(y) reported

(l3 , p3) chosen

!2 realized

"2(y) reported

(l2 , p2) chosen

incl:g3(!3)

incl:g2(!2) g3(!3) (l3* , p3*)




y reported

!3 realized

"3(y) reported

(l3 , p3) chosen

!2 realized

"2(y) reported

(l2 , p2) chosen

!1 realized

incl:g3(!3)

incl:g2(!2) g3(!3) (l3* , p3*)




y reported

!3 realized

"3(y) reported

(l3 , p3) chosen

!2 realized

"2(y) reported

(l2 , p2) chosen

!1 realized

"1(y) reported

(l1 , p1) chosen

incl:g3(!3)

incl:g2(!2) g3(!3) (l3* , p3*)

incl:g1(!1)g2(!2) g3(!3) (l2* , p2*)(l3* , p3*)




y reported

!3 realized

"3(y) reported

(l3 , p3) chosen

!2 realized

"2(y) reported

(l2 , p2) chosen

!1 realized

"1(y) reported

(l1 , p1) chosen

!0 realized

incl:g3(!3)

incl:g2(!2) g3(!3) (l3* , p3*)

incl:g1(!1)g2(!2) g3(!3) (l2* , p2*)(l3* , p3*)




y reported

!3 realized

"3(y) reported

(l3 , p3) chosen

!2 realized

"2(y) reported

(l2 , p2) chosen

!1 realized

"1(y) reported

(l1 , p1) chosen

!0 realized

"0(y) reported

incl:g3(!3)

incl:g2(!2) g3(!3) (l3* , p3*)

incl:g1(!1)g2(!2) g3(!3) (l2* , p2*)(l3* , p3*)

incl:g0(!0)g1(!1)g2(!2) g3(!3)(l1* , p1*) (l2* , p2*)(l3* , p3*)



Identification of g3(θ3)

Taking the normalization E [eθt ] = 1 for all t:

Pr[y < Y ] = Pr

[E [y |Ω3]eθ3 < Y

]= Pr

[θ3 ≤ ln

( Y

E [y |Ω3]

)]where Ω3 is the period 3 information set

⇒ g3(θ3) is constructed by transforming ψ3(y)



Key proposition (summarized)

Proposition

If h = H(θ1, θ2) is a function of two random variables, and

1. We know densities fh(h) and fθ2(θ2)

2. H is monotonic in θ1

then we can consistently estimate fθ1(θ1) by taking repeated drawsfrom fh and fθ2



Identification of g2(θ2)

Plugging (l∗3 , p∗3) into the definition of output allows us to write

output from the period 2 perspective as:

y = H2

(φ, α1, α2, β1, β2, κ1, κ2, γ, δ, ω;

A, l1, p1, l2, p2; ql , qp,E [qc ]; θ0, θ1)eθ2eθ3

And E [y |Ω2] =∫∞−∞ yψ2(y)dy = H2(·)



Identification of g2(θ2) cont.

This gives a method for numerically estimating g1(θ1) usingrepeated draws from ψ1(y) and g2(θ2)

Pr[θ2 < Θ2] =1

M

M∑m=1

I[

ln

(ym

E [y |Ω2]

)− θ3m ≤ Θ1

]



Estimation of θt and φ

Given any guess of the parameters, we find the realized values ofthe shocks:

• θ0, θ1, θ2 come from FOC of the farmer’s decision problem

• θ3 comes from realized output y and ψ3(θ3)

Lastly

φ =

∫∞−∞ yψ0(y)dy

A



Likelihood function

Then the joint likelihood for the observed inputs, output anddistributions is:

L(α1, α2, β1, β2, κ1, κ2, γ, δ, ω |

φ,A, l , p, y , ql , qp,E [qc ], θ0, θ1)

=P∏

i=1

gi0(θi0)gi1(θi

1)gi2(θi2)gi3(θi

3)

We maximize the log likelihood over the 9 production parametersand: α1, α2, β1, β2, κ1, κ2, γ, δ, ω



Results: shock densities

Summary statistics for gt(θt)

Variable Mean s.d.!0 lower bound -2.95 1.86!0 upper bound 2.43 1.70E[!0] -0.14 0.61!1 lower bound -2.49 1.84!1 upper bound 2.01 1.48E[!01] -0.01 0.58!2 lower bound -4.19 1.4807*!2 upper bound 3.19 1.4807*E[!2] -2.35 1.17N 212 212*SD of !2 upper and lower bounds is constant by construction, because both reflect variation in acreage



Conclusion

Separation of output equation into its dynamic and stochasticcomponents is not a necessary condition for this to work

But monotonicity in θt is necessary

Observation of shock densities reduces number of parameters to beestimated

But it also increases the pressure on the functional form, becausethe error variance does not adjust to increase the contribution ofvery low probability parameter contributions to the likelihood



Where things stand...

Ongoing work on this paper involves:

1. Embedding the farmer’s problem in a utility framework

2. Comparing results with those from the nested fixed pointmethod

3. Interpretation and simulations

4. Relaxing the dynamic consistency assumption?→ could use data on counterfactual, optimal pesticideapplication



Thanks!


08.30.2012 - brian dillon

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