08.30.2012 - Brian Dillon

Introduction Data Model Identiﬁcation Estimation Results

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

Brian Dillon

Cornell University
and Harvard Kennedy School

August 30, 2012

Brian Dillon Estimation of a Dynamic Agricultural Production Model with O


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 capital
investment

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

Between-farmer variation in expectations clearly matters (Gin´,
e
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 diﬀerence
between observed and estimated expectations

Delavande et al (2010) review the recent development literature
that 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 that
are standard for these models (Wolpin 1987; Rust 1987, 1997;
Fafchamps 1993)
2. There is a lot of information in a subjective distribution over
an endogenous outcome



Why go through a structural exercise?
Apart from the pure value of estimating a less restricted
production function...



Production elasticities tell us something about resilience of the
production process to shocks



Production elasticities tell us something about resilience of the
production process to shocks
What we know about shocks already largely deals with
• Consumption/asset smoothing (Townsend 1994, Morduch
1995, Hoddinott 2006, Barrett and Carter 2006, Jacoby and
Skouﬁas 1998, Fafchamps et al 1998)
• Human capital (Hoddinott and Kinsey 2001, Aguilar and
Vicarelli 2012)
• Two papers look at how farmers move labor across time,
within a season: Fafchamps (1993) and Kochar (1999)




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, and
yield expectations to estimate an agricultural production function
Methodological contributions:
1. Develop a general method for estimating dynamic choice
models with observed subjective distributions
2. Show how counterfactual choice data (“How much pesticide
did you want to apply last week?”) can be used in estimation

Substantive contributions:
1. Recover estimates of all elasticities of substitution between
inputs (within and across periods)
2. Simulate the impact of insurance, forward contracting, and
information 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 in
summer 2009 and summer 2010
• From September 2009 - June 2010: investment, time use,
shocks, agricultural input and output, and other data
gathered every 3 weeks
• High frequency interviews also gathered subjective probability
distributions over end-of-season prices and yields, and
qualitative distributions over pest pressure and rainfall at
various points throughout the year



Measuring subjective distributions
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Evolution of subjective price distributions

5

4

3
Mean number
of stones 2

1

0

6
5
4 7
6
3 5
Survey period 4
2 3
1 2 Bin number
1



Evolution of subjective yield distributions
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G



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:

N j 2
m=1 xj
(ai , bi , ρi , κi ) = arg inf − G (dj+1 | a, b, ρ, κ)
a,b,ρ,κ k
j=1



Sample summary statistics
Mean sd Min Max
Household size (people) 8.33 3.90 2 23
Dependency ratio* 1.31 0.85 0 5.5
Head age 46.85 14.69 20 100
Head is male (%) 85.0 - - -
Years of education (HH head) 4.19 3.46 0 11
Radios 0.83 0.71 0 4
Bicycles 1.19 1.00 0 10
Dairy cattle 1.33 2.84 0 20
Non-dairy cattle 3.87 7.89 0 60
Goats 5.27 8.05 0 50
Sheep 1.67 3.74 0 30
Total acres 9.67 11.03 1 82
Number of plots 2.71 1.17 1 7
Number of crops grown 3.45 1.26 1 8
Labor expenditure (TSH) 78,248 139,485 0 1,020,000
Fertilizer expenditure (TSH) 21,149 81,359 0 715,000
Animal labor expenditure (TSH) 33,497 92,724 0 750,000
Transport expenditure (TSH) 10,333 20,049 0 144,000
Other cultivation expenditure (TSH) 6,929 15,817 0 100,000
Total cultivation expenditure (TSH) 150,156 254,863 0 1,514,700
Notes: 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 divided
!"1,400
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



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 = φi Ai e θi0
yi1 = h1 (yi0 , li1 , pi1 )e θi1
yi2 = h2 (yi1 , li2 , pi2 )e θi2
yi = h3 (yi2 , li3 , pi3 )e θi3

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:
1
γ γ γ
h1 (y0 , l1 , p1 | α1 , α2 , γ) = [α1 y0 + α2 l1 + (1 − α1 − α2 )p1 ] γ
1
δ δ δ δ
h2 (y1 , l2 , p2 | β1 , β2 , δ) = β1 y1 + β2 l2 + (1 − β1 − β2 )p2
1
ω ω ω
h3 (y2 , l3 , p3 | κ1 , κ2 , ω) = [κ1 y2 + κ2 l3 + (1 − κ1 − κ2 )p3 ] ω

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:

δ
γ γ γ
max E [qc ]Eθi1 θi2 θi3 κ1 β1 α1 yi0 +α2 li1 +(1−α1 −α2 )pi1 γ
e δθi1
li1 ,pi1
ω
δ
∗δ ∗δ ∗ω
+ β2 li2 + (1 − β1 − β2 )pi2 e ωθi2 + κ2 li3
1 3
ω
∗ω
+ (1 − κ1 − κ2 )pi3 e θi3 − (ql lit + qp pit )
t=1



Identiﬁcation of gt (θt )

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





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






But we only observe subjective output distributions
Ψ0 (y ), . . . , Ψ3 (y )






But we only observe subjective output distributions
Ψ0 (y ), . . . , Ψ3 (y )

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



Timeline of decisions, realizations, and data collection

y reported




!3 realized

y reported



(l3 , p3) chosen

!3 realized

y reported

"3(y) reported
incl:
g3(!3)



(l3 , p3) chosen

!2 realized !3 realized

y reported

"3(y) reported
incl:
g3(!3)



(l2 , p2) chosen (l3 , p3) chosen

!2 realized !3 realized

y reported

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



(l2 , p2) chosen (l3 , p3) chosen

!1 realized !2 realized !3 realized

y reported

"2(y) reported
incl:
g3(!3) incl:
(l3* , p3*) g3(!3)



(l1 , p1) chosen (l2 , p2) chosen (l3 , p3) chosen

!1 realized !2 realized !3 realized

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




!0 realized !1 realized !2 realized !3 realized

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




!0 realized !1 realized !2 realized !3 realized

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



Identiﬁcation of g3 (θ3 )

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

Pr[y < Y ] = Pr E [y |Ω3 ]e θ3 < Y

Y
= Pr θ3 ≤ ln
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 draws
from fh and fθ2



Identiﬁcation of g2 (θ2 )

∗ ∗
Plugging (l3 , p3 ) 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 e θ2 e θ3

∞
And E [y |Ω2 ] = −∞ y ψ2 (y )dy = H2 (·)



Identiﬁcation of g2 (θ2 ) cont.

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

M
1 ym
Pr[θ2 < Θ2 ] = I ln − θ3m ≤ Θ1
M E [y |Ω2 ]
m=1



Estimation of θt and φ

Given any guess of the parameters, we ﬁnd the realized values of
the shocks:
• θ0 , θ1 , θ2 come from FOC of the farmer’s decision problem
• θ3 comes from realized output y and ψ3 (θ3 )

Lastly
∞
ˆ −∞ y ψ0 (y )dy
φ=
A



Likelihood function

Then the joint likelihood for the observed inputs, output and
distributions is:

L(α1 , α2 , β1 , β2 , κ1 , κ2 , γ, δ, ω |
P
i i i i
φ, A, l, p, y , ql , qp , E [qc ], θ0 , θ1 = gi0 (θ0 )gi1 (θ1 )gi2 (θ2 )gi3 (θ3 )
i=1

We maximize the log likelihood over the 9 production parameters
and: 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 1.70
E[!0] -0.14 0.61
!1 lower bound -2.49 1.84
!1 upper bound 2.01 1.48
E[!01] -0.01 0.58
!2 lower bound -4.19 1.4807*
!2 upper bound 3.19 1.4807*
E[!2] -2.35 1.17
N 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 stochastic
components is not a necessary condition for this to work

But monotonicity in θt is necessary

Observation of shock densities reduces number of parameters to be
estimated

But it also increases the pressure on the functional form, because
the error variance does not adjust to increase the contribution of
very 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 point
method
3. Interpretation and simulations
4. Relaxing the dynamic consistency assumption?
→ could use data on counterfactual, optimal pesticide
application



Thanks!


08.30.2012 - Brian Dillon

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