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

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- 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 +,-./0!1,!2,+3,! ! 4+567879:;<:=>5?! #! $! %! &! "! !! *! )! (! ! !! ! Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
- 13. Introduction Data Model Identiﬁcation Estimation Results Evolution of subjective price distributions 5 4 3 Mean number of stones 2 1 0 6 5 4 7 6 3 5 Survey period 4 2 3 1 2 Bin number 1 Brian Dillon Estimation of a Dynamic Agricultural Production Model with O
- 14. Introduction Data Model Identiﬁcation Estimation Results Evolution of subjective yield distributions ?34+)(*+),-.$ ?34+)(*+),-.% ! "!# "$ "$# "% "%# "% "$# "$ "!# ! !# ! # $! !# ! # $! &()*+),-./0,1.*2+3*4.5)46 &()*+),-./0,1.*2+3*4.5)46 70-4.8.9*-2:-)7,(;.<*-6=)6+:.8.!"#!$> 70-4.8.9*-2:-)7,(;.<*-6=)6+:.8.!"#@#$ ?34+)(*+),-.A E*0(F+ "> "D "A "> "% "% "$ ! ! !$! !# ! # $! !# ! # $! &()*+),-./0,1.*2+3*4.5)46 &()*+),-./0,1.*2+3*4.5)46 70-4.8.9*-2:-)7,(;.<*-6=)6+:.8.!"AB#C 70-4.8.9*-2:-)7,(;.<*-6=)6+:.8.!"%A## G 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: authors 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

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