These derivations are not part of the official forthcoming version of Vasilaky and Leonard

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1 Target Input Model with Learning, Derivations Kathryn N Vasilaky These derivations are not part of the official forthcoming version of Vasilaky and Leonard 06 in Economic Development and Cultural Change. Rather, they are supplementary derivations, particularly for those who are unfamiliar with the target input model, and who have not derived the intermediary steps shown here. The syntax and lettering does not exactly correspond to what is used in the published version. We changed some of the notation in the final version for ease of reading. Target Input Model There are two main sources for the target input model. Udry and Bardhan s Development Microeconomics Handbook, as well as Foster and Rosenzweig 995. They use different notations. Farmer i chooses an input level or, time to apply inputs: θ it, at time t, to maximize profits. The ideal input is θ it, and farmer s profits are larger the closer is θ it to θ it. Profits q for farmer i in time t: q it θ it θ it Choose: θ it ; Target: θ it θ it is determined by θ it θ + µ it µ it N 0, ϑ µ Note: Udry/Bardhan Book Version κ it κ + µ it θ N θ t, ϑ θit Maximization of expected profit implies that: θ it θ it 0

2 θ it E t θit θ t E t q it ϑ θit ϑ µ 3 Expected profits increase as ϑ θit decreases or as the individual learns about true θ.

3 a Derivation of E t q it ϑ θit ϑ µ E t q it E t θ it + E t θ it θit E t θ it Now θ it E t θit θ t from Maxmization E t θit E t θ t And E t θ it θit E t θt θ it E t θ t θ + µ it Plugging in θ t E t θ + θ t E µ it θt + 0 θt And E t θ it E t θ + µ it by E t θ + E t θ µ it + E t µ it E t θ + E t µ it E t θ + ϑ µ 3 Putting 3 together: E t q it E t θ t + θt E t θ + ϑ µ θt + θt E t θ ϑ µ [ E t θ E θ E θ + E θ ] ϑ µ ϑ θit ϑ µ which is Equation 3, pg 55 in Development Microeconomics By Pranab Bardhan, Christopher Udry. 3

4 Farmer i s variance of her beliefs about θ in period t- is ϑ θi,t, after observing θ i,t and applying Baye s rule for a Normal distribution with Normal prior. b For observation model θ it θ N θ, ϑµ and priors on θit N θ i,t, ϑ θi,t The posterior variance of θ, ϑ θit after one update is: b ϑ θit ϑ θi,t ϑ µ 4 ϑ θi,t + ϑ µ if we plug in for t- ϑ θi,t + ϑ µ + ϑ µ ϑ θi,t + ϑ µ ϑ θi,t + ϑ µ We can see that if we continue to iterate until until period 0: + N t ϑ ϑ θio µ where N t represents the number of iterations from period t- to period 0, Define precision as P io and P µ, ϑ θio ϑ µ ϑ θit lim N ϑ θi,t 0 lim N E q it ϑ µ P io + N t ϑ µ 4 See Derivation on next page 4. 4

5 Here we do a side derivation for what the posterior distribution with normally distributed data with normal priors, with a change of notation. Observation model y µ Nµ, ϑ, Normal prior N m, S We consider the a normal prior of mean m and variance s. fµ πs e s µm The posterior is proportional to the prior times the likelihood. Baye s rule f µ y f y µ f µ. e y µ ϑ e µ m S e e y µ e ϑ y m S { µ µy + y + m mµ + µ } ϑ S S µ µys + y S + ϑ m mµϑ + µ ϑ ϑ S ϑ S Drop { } y S + ϑ m since constants known ϑ S So f µ y e e e S µ µys mµϑ + µ ϑ ϑ S + ϑ S S + ϑ µ µ ys + mϑ S ϑ µ µ ys + mϑ S + ϑ S ϑ S + ϑ 5

6 µ ys + mϑ e S + ϑ S ϑ S + ϑ The nd term is a constant when you square this out, and can be dropped. So f µ y is normal with moments: mean: ys + mϑ S + ϑ sd: S ϑ S + ϑ We used the above posterior sd in equation 4. Now back to the target input model. 6

7 From Pg 3, if N θ t, 0. lim N ϑ θit 0, then in the limit i s beliefs about θ are distributed as Learning From Others Now suppose person i can observe person j s input choice, she observes person j s choice θ jt + ε jt, or θ + µ jt + ε jt. For the moment, we assume that information flow, and the errors µ it and µ jt, are independent Covµ it, µ jt 0. Now farmer i has an additional update regarding θ it using her neighbors plot. A second update takes the following form. Let ϑ v ϑ θi,t ϑ µ ϑ θi,t + ϑ µ st update on the priors. Let ϑ z ϑ µ + ϑ ɛ, which is our additional data coming from an update from peers. where θ jt N θ, ϑ z Following our derivation on Pg 4, the nd update takes the following form: ϑ θi,t ϑ µ ϑ zϑ θi,t ϑ µ ϑ zϑ v ϑ z + ϑ v ϑ z ϑ θi,t + ϑ µ ϑ θi,t ϑ ϑ µ z + ϑ θi,t + ϑ µ ϑ θi,t + ϑ µ ϑ zϑ θi,t + ϑ zϑ µ + ϑ θi,t ϑ µ ϑ θi,t + ϑ µ ϑ zϑ θi,t ϑ µ ϑ zϑ θi,t + ϑ zϑ µ + ϑ θi,t ϑ µ ϑ µ + + ϑ θi,t ϑ z ϑ µ + + ϑ θi,t ϑ µ + ϑ ɛ Let P v. ϑ µ + ϑ ɛ Now if we substitute in for ϑ θi,t, the farmer will update her priors using her own observations of θ it and her, S, neighbors θ jt, and the variance of her beliefs about θ, between time t and time 0 are: 7

8 ϑ θi,t P io + N t P µ + S t P v For the remainder of the model found in Vasilaky and Leonard 06, see the Appendix in the paper. The pre-print version can be found on Columbia University s Academic Commons. 8