AG RISK ANALYDEC MAK
AG RISK ANALYDEC MAK AEB 6182
Popular in Course
verified elite notetaker
Popular in Agricultural Economics And Business
This 5 page Class Notes was uploaded by Weldon Rau I on Friday September 18, 2015. The Class Notes belongs to AEB 6182 at University of Florida taught by Charles Moss in Fall. Since its upload, it has received 17 views. For similar materials see /class/206591/aeb-6182-university-of-florida in Agricultural Economics And Business at University of Florida.
Reviews for AG RISK ANALYDEC MAK
Report this Material
What is Karma?
Karma is the currency of StudySoup.
Date Created: 09/18/15
AEB 6182 Lecture XI Professor Charles Moss Value of Information Lecture XI 1 Decision Making and Bayesian Probabilities A Traditionally Bayesian analysis involves a procedure whereby new information is integrated into a prior distribution to generate an updated or posterior distribution B At times the concept of Bayesian probability theory is con rsed with the subjective probability theory where an individual has an intuition regarding the probability of an event instead of a frequency view of probability theory which strives for an objective version of probability C At the base of Bayesian inference is Bayes s eqaution Pa b P b where Palb is the probability of the event a occurring such that event a has already occurred Pab is the joint probability of both event a and event b occurring and Pb is the marginal probability of event b occurring or the probability of event b occurring such that event a has been integrated from the distribution D This basic concept of prior and posterior probabilities can be used to develop one manifestation of the value of information Speci cally imagine the state space Palbl 0 l1 P01017 o E O prod 5 POZIQFNII AEB 6182 Lecture XI Professor Charles Moss 1 This diagram depicts the potential outcomes of two random events each of which has two potential outcomes a The rst event yields outcomes 01 and 02 each of which occurs with probability 5 b The second event results in Om and 02H if 01 occurred in the rst event and Om and 0m given that 02 occurred in the rst event c Intuitively we can picture Om and Om as the same event with a di erent intervening event Similarly 02H and 0m are the same event d What does change with the intervening event is the relative probability that each outcome will occur i Given 01 the probability of outcome 1 in the second stage Om is 7 compared with a probability of outcome 2 in the second stage Om of 3 ii The di erence in the probability of the payolTs given the outcome in the rst stage gives rise to the value of information 2 Next we want to introduce two altematives A1 which pays 10 in state 1 and 0 in state 2 Similarly A2 pays 5 in state 1 and 4 in state 2 a To determine the expected value of the investment we must rst determine the probabilities that state 1 and state 2 will occur For states 1 and 2 the total probabilities are PS1 P01P0m P02P01 2 5753 5 PS2 P01POm P02 P02 2 5357 5 The expected value of each altemative is then EA1 PS1 VA1 51 PSZVA1lSZ 51050 500 EA2 PS1 VA2lSl PSZVA2SZ 5554 450 Thus in the absence of risk aversion the decision maker would choose A1 with an expected value of 500 The next question is What is the initial signal worth Starting from the last node and working backward assume that 01 has occurred what is the optimum decision EA1Ol POl VA1S1 P02 1VA1S271030 700 Fquot EA2 01 POmVA2S1 P02 1VA2S2 7534 470 Thus just like the scenario without the intervening event we choose A1 However the result is somewhat diiTerent given that 02 occurs Speci cally AEB 6182 Lecture XI Professor Charles Moss EA1102 POI ZVA1S1P02 2VA11S231070 300 EA2OZ POI ZVA2S1P02 2VA21S23574 430 Under this scenario we would choose A2 over A1 The decision rule is then to choose A1 if event 01 occurs and action A2 if event 02 occurs The expected value of this strategy is VStrategy P01EA101 P02 EA2OZ 57005430 565 The value of the information is then the di erence between the conditioned and unconditioned decision VInformation 565 500 65 II Chavas JeanPaul and Rulan Pope Information Its Measurement and Valuation American Journal ongricultural Economics 661984 705 10 A The objective of the paper is to discuss the measurement and economic valuation of information B Concepts of Information 1 One way to define information is to focus on the entropy of the signal Following Shannon and Weaver the entropy of a signal is defined as H Zp1np 11 Intuitively as H increases the value of the signal decreases Further H decreases as one event becomes increasing likely In the limit as the probability of one event approaches one the natural log of one goes to zero and the value of the entropy measure approaches zero a From the example in the previous lecture the unconditioned probability has equally likely events The value of the information in that distribution function is State Probabilitj pi lnpi S1 05 03466 82 05 03466 H 06931 Note that EXpH 2 Alternatively the value of either conditioned distribution function is State Probabilitj pi lnpi SI 03 03612 82 07 02497 H 06109 Note that EXpH 184202 AEB 6182 Lecture XI Professor Charles Moss N Thus we conclude that random events with less uniform distributions have a lower level of entropy and contain more information The next level of complexity is added by the concepts of prior and posterior probabilities Assume that we have two signals p and q each of which with two potential outcomes One question is what is the value of the information in signal q given the information already observed in signal p Theil gives the value of such as signal as Fquot 17 q Zip 1namp 11 qt Following the distributions for 01 in the preceding example Probability p Probability q pi lnpiqi 07 0 1682 03 02554 State S1 05 S2 05 I 00872 Ifthe intervening event would have been uninforrnative Pi 1nPi Ti 00000 00000 State S1 05 S2 05 Probability p Probability q 05 05 I 00000 So in this case we get the result that the value of I increases as the intervening event adds more information Unfortunately there is in general no relationship between the entropy measure for information and the value of information in a particular decisionmaking process Another statistical based definition of information comes from the estimation process via the likelihood function One estimation procedure involves choosing the value of parameters which maximizes the likelihood of a particular sample Under normality the natural log of the likelihood for a linear regression can be written as T iv a a1xt2 T L ln H 2 O const Finding the parameters oco and x1 which maximize the likelihood function then yields an estimate of these parameters This maximization is found by taking the derivative of the lnction with respect to each parameter and AEB 6182 Lecture XI Professor Charles Moss 4 setting it equal to zero The matrix of second moments is referred to as the information matrix This matrix is the basis for statistical inference and yields such things as the standard error of the parameter estimates Information can also be defined as a message which alters tastes or perceptions which are certain Finally information can be defined as a message which alters probabilistic perceptions of random even B A Model of Information 1 N E 4 Back to Equations First assume that economic agents possess a concave utility Jnction Uyx1x2e1e2 There goal is to choose the level of XI and xz which maximizes that level of utility max E1Uyx1 x e1ez 161162 where y is the initial level of income and el and e2 are random variables The notation E1 denotes expectations taken in the first period in time Making all decisions at time 1 corresponds to the open loop solution where no learning takes place Given that e1 can be observed before decisions on xz have to be made the second phase of decision process can be rewritten as Vx1yael HESXEZUCV x1 9 x2 991992 In other words this process represents the optimum selection of xz conditioned on the new information or e1 Given this formulation x1 can then be selected to maximize the expectation of this value function max E1Vx1 ye1 1C1 Taking that it can be shown that E1 maXVx1yaer Z maXElVx1 y er x1 61 we can define a maximum bid price B1 which equates the two E1 maXVx1 9y Br 9 81 maXE1Vx1 y 91 x1 I1 Taking a Taylor series expansion around B1 yields the value of the information as B 1 82V all1 82V 1 2 Belaxl 9x12 axlael 9V Varel