1. 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
2. Introduction Data Model Identification Estimation Results
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)
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
3. Introduction Data Model Identification Estimation Results
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
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
4. Introduction Data Model Identification Estimation Results
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
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
5. Introduction Data Model Identification Estimation Results
Why go through a structural exercise?
Apart from the pure value of estimating a less restricted
production function...
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
6. Introduction Data Model Identification Estimation Results
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
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
7. Introduction Data Model Identification Estimation Results
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
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)
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
8. Introduction Data Model Identification Estimation Results
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
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
9. Introduction Data Model Identification Estimation Results
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
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
10. Introduction Data Model Identification Estimation Results
Plan of the talk
• Data set
• Model basics
• Identification of shock densities
• Estimation
• Results
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
11. Introduction Data Model Identification 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
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
12. Introduction Data Model Identification Estimation Results
Measuring subjective distributions
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Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
13. Introduction Data Model Identification Estimation Results
Evolution of subjective price distributions
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Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
14. Introduction Data Model Identification Estimation Results
Evolution of subjective yield distributions
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Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
15. Introduction Data Model Identification Estimation Results
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
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
16. Introduction Data Model Identification Estimation Results
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.
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
17. Introduction Data Model Identification Estimation Results
Model assumptions
Important:
1. Farmers are dynamically consistent (will relax, if we have time)
2. Independence of shocks across time
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
18. Introduction Data Model Identification Estimation Results
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
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
19. Introduction Data Model Identification Estimation Results
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
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
20. Introduction Data Model Identification Estimation Results
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)
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
21. Introduction Data Model Identification Estimation Results
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
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
22. Introduction Data Model Identification Estimation Results
Identification of gt (θt )
We need measures of gt (θt ) in order to proceed
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
23. Introduction Data Model Identification Estimation Results
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
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
24. Introduction Data Model Identification Estimation Results
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 )
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
25. Introduction Data Model Identification Estimation Results
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 )
We can use those to directly estimate gt (θt ), within the context of
the model
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
26. Introduction Data Model Identification Estimation Results
Timeline of decisions, realizations, and data collection
y reported
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
27. Introduction Data Model Identification Estimation Results
Timeline of decisions, realizations, and data collection
!3 realized
y reported
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
28. Introduction Data Model Identification Estimation Results
Timeline of decisions, realizations, and data collection
(l3 , p3) chosen
!3 realized
y reported
"3(y) reported
incl:
g3(!3)
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
29. Introduction Data Model Identification Estimation Results
Timeline of decisions, realizations, and data collection
(l3 , p3) chosen
!2 realized !3 realized
y reported
"3(y) reported
incl:
g3(!3)
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
30. Introduction Data Model Identification Estimation Results
Timeline of decisions, realizations, and data collection
(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)
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
31. Introduction Data Model Identification Estimation Results
Timeline of decisions, realizations, and data collection
(l2 , p2) chosen (l3 , p3) chosen
!1 realized !2 realized !3 realized
y reported
"2(y) reported
incl:
g2(!2) "3(y) reported
g3(!3) incl:
(l3* , p3*) g3(!3)
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
32. Introduction Data Model Identification Estimation Results
Timeline of decisions, realizations, and data collection
(l1 , p1) chosen (l2 , p2) chosen (l3 , p3) chosen
!1 realized !2 realized !3 realized
y reported
"1(y) reported
incl:
g1(!1) "2(y) reported
g2(!2) incl:
g3(!3) g2(!2) "3(y) reported
(l2* , p2*) g3(!3) incl:
(l3* , p3*) (l3* , p3*) g3(!3)
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
33. Introduction Data Model Identification Estimation Results
Timeline of decisions, realizations, and data collection
(l1 , p1) chosen (l2 , p2) chosen (l3 , p3) chosen
!0 realized !1 realized !2 realized !3 realized
y reported
"1(y) reported
incl:
g1(!1) "2(y) reported
g2(!2) incl:
g3(!3) g2(!2) "3(y) reported
(l2* , p2*) g3(!3) incl:
(l3* , p3*) (l3* , p3*) g3(!3)
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
34. Introduction Data Model Identification Estimation Results
Timeline of decisions, realizations, and data collection
(l1 , p1) chosen (l2 , p2) chosen (l3 , p3) chosen
!0 realized !1 realized !2 realized !3 realized
"0(y) reported
incl: y reported
g0(!0) "1(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)
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
35. Introduction Data Model Identification Estimation Results
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 )
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
36. Introduction Data Model Identification Estimation Results
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
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
37. Introduction Data Model Identification Estimation Results
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 (·)
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
38. Introduction Data Model Identification Estimation Results
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
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
39. Introduction Data Model Identification Estimation Results
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
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
40. Introduction Data Model Identification Estimation Results
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
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
41. Introduction Data Model Identification Estimation Results
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
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
42. Introduction Data Model Identification Estimation Results
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
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
43. Introduction Data Model Identification Estimation Results
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
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
44. Introduction Data Model Identification Estimation Results
Thanks!
Brian Dillon Estimation of a Dynamic Agricultural Production Model with O