[speaker001:] they're sensitive aren't they [LAUGHTER]
[speaker002:] Right before we er [clears throat] kick off the lecture... erm... we'll have a lecture on Wednesday right hopefully we'll finish the course then on Wednesday, we're running over a touch behind
[speaker001:] Will that be the last one?
[speaker002:] Yes, that will be the last one. We may finish today but I don't think so... erm Now I was talking to your Doctor who was wondering whether... we might want to go out for a few beers at some point going home for Christmas or possibly after the exams
[speaker001:] Both
[speaker002:] [LAUGHTER] so are you all going home fairly sharpish?
[speaker001:] Oh yeah tomorrow night
[speaker002:] Right... in that case its probably a better idea if we go out after the exams, if you want, I mean, don't you don't have to Yeah [LAUGHTER] right, well in that case erm... we'll put a notice up or something... erm... around the exam period and let you know when we we will go out and we'll try sneak a few beers in after your exams but before the start of the second semester... so erm We'll have to try figure out a time when you're all available. Right Okay, so on Friday we were looking at... erm a model of agricultural supply and response that incorporated... erm, a notion that farmers take some time... er to react to changes in er in prices... erm due to psychological... acquiring fixed factors... and so on and so forth. [clears throat]... Although hypothesis as a useful... er sort of description... of how farmers might erm... adjust... their supply... it's a very simple one... alright. If you recall, we said that the desired... level of output S R T simply... function of last year's prices... alright or more im,... or alternatively... right, you make... erm the planting decisions on the basis... right... of current prices... right, so when that supply comes onto the market in say a years time it's now a crop... right. That supply will be determined by, not current market prices but prices of a year ago when you made the planting decisions... right. Now if you do... recall your first year notes, this is the relationship underlines a cobweb model, right that prices were based on erm... or supply decisions were based on prices at planting... right and we showed you that cobweb model erm... farmers make systematic errors... right cos they never appreciate the cycle there is a cycle to prices, right so they're making systematic errors... right cos prices are high this year... as a result erm erm of plant a lot so that when the supply comes on the market next year right, prices are very low and you would thought farmers would er, would learn... but prices fluctuated... right. So we may want to introduce a more sophisticated... mechanism... for determining what the desired level of supply is... and what we'll do... is that we'll say this is actually a special case of a much more general, er more s sophisticated. Right what we're going to do is to introduce the idea of expectations... right, cos what this equation's really saying... right... is that farmers expect... price prevailing in T minus one... alright... to prevails in S T... right so that's what we're, right, in... their ex their expectation of prices in T, right... are based on er current prices in sort of T minus one... okay and that, that is what we would call erm... expectations... alright. But what we're going to do, that's gonna be a special case in a more general formulation that we're going to look at now... which is the adapted expectation hypothesis.... Right this... basically... it's too simplistic, right we know that farmers are more intelligent than the cobweb model suggests that they are, let's try erm... increase the sophistication of their, their supply response. But we'll do that using adaptive expectations... okay [clears throat]... right. All the adaptive expectations model says... is that farmers supply, right, S T... is some function... of the expected... price... right, erm. When the supply comes onto the market, right so... this thing here... is the expectation of price in P T, when the supply comes onto the market... right and that expectation is formed at T minus one, right so at the beginning of the year... farmers make some... forecast or some expectation of prices when the crop will come onto the market in T, right and then they form that expectation or that prediction, right, at the beginning of the period, or at the end of the last one, T minus one.... [clears throat]... right.... Let's call that equation one seven... er... in actual fact what we'll do is that we'll put a random... random error term on the end as well just to say that this is, rule is not perfect and there will be some fluctuations around it.... Okay [clears throat]... Now expect,... farmer's expectations of prices are assumed to be revised, right, each year, farmers revise their own expectations of future prices so that makes some sense. You wouldn't expect them to... forecast the same price all the years, right.... [clears throat] Now expectations model implies that the revision in expectations, right, year on year... alright, is proportional... right, to the error last year, right, the difference between actual and forecasted last year... right, if they were way out if their forecast was way out... for last year, right, they'll revise their expectations for, for the next year... okay. So [clears throat] symbolically... the expectation at T minus one... prices at T minus the expectation of T minus two... in prices at T minus one... i.e.... revision in expectations or how our expectations change from one year to a next... is dependent upon... gamma... into the actual price, [clears throat] T minus one minus farmer's expectations of it... okay.... So the revision of expectations year on year, alright is some function, alright, proportionate to [cough]... the er, the difference between their forecast for prices of P T minus one... [cough] to track it from what actually occurred... okay. [clears throat]... Now we can just rearrange that... expression to get the expectation... of prices for... the period T, right all we're doing is that we're adding... P T minus two, P T minus one to both sides, alright and we'll get gamma into... P T minus one... erm, plus... open brackets mi minus gamma... into expectation at T minus two... of prices at T minus one... okay. So we're just rearranging... the first expression, okay, and getting this one... right so now we've got er a decision rule to obtain a, farmers... will form their expectation of prices say next year, in T... right, on the basis of last year's price, last year's actual price... and last year's, the expectation of last year's actual price... okay. Right, [sniff]... so as you'll probably agree alright, gamma... scribble here... right, plays an important role in this model and gamma is the co-efficient of expectations, the expectations co-efficient.... Now, [clears throat]... when gamma equals zero then what does this model imply unde under different conditions?, alright. If gamma equals zero... right... what happens here? The expectation for next year's exactly the same as what we expected last year, alright... just turned, propped out... okay. Essentially there are no revisions... no revisions in expectations... if if gamma equals zero... right, this right hand side top here completely disappears and if you wanted t just leave the error turning then then any changes in expectations there's no systematic behaviour. But like I, I omitted... that just for simplicity, right. So when gamma equals zero there's no revision to expectation.... When gamma equals one... what happens now? When gamma equals one... this turn here drops out... alright, so... expected price for next year... exactly the same... as the current price, T minus one... alright. Right.... Scribble that... and poverty is er... that case is called naive expectations... alright. Where gamma equals one we have naive expectations... [clears throat]... so, d'you think that what that implies in the top equation, right the revision to expectations and gamma equals one... is the whole of the difference between actual predicted last year... okay. So... here you have no revision... under naive expectations we have four... right [clears throat]... Now the the closer that our expectations come to gamma, lies to one... the more weight... farmers attach to last years price... relative to prices in T minus two, T minus three, T minus four... [clears throat] right. When gamma equals one... all the weight is given to last year's price... all the weight is given to last year's price... okay... right, so past history, all past history is ignored... apart from the price at T minus one... right and the closer that gamma is our expectations co-efficient, closer gamma is to one, the more weight be attached to last year's price, relative to the previous history of prices... okay [clears throat]... right.... If we play about with this expectations... [clears throat] model... right this is the erm, this describes the expectation of prices in T... it can also use the same as a decision rule... alright to find out what prices would expectation of prices were, right in in P T minus one, okay T minus one, right here.... What we now want in T minus one was, cos that was the actual price... what we don't know... what was the expectation of prices in T minus one. Well we can use the same decision rule and just change the subscripts... by one, so you've got gamma into P T minus two... right, plus one minus gamma... the expectation of T minus three, prices... T minus two... okay, and we could substitute... right, the right hand side... here, right, into the expec, into that expectation up there.... We then have... to er, determine the expected value of T minus three and the price of T minus two and we then use the same rule substitute it in there, right, we could repeat, repeatedly substitute into the expected value, right in that manner... like we did in, in the er, [clears throat] the symmetry between adaptive expectations and... partial adjustment type offices.... But we just repeat, repeatedly substitute into this unobservable term...... [cough] right. I mean we come out with... erm... an equation that look like... so expected price and T is a function right... it's the sum of gamma into one minus gamma into J... into P T minus one minus J... and J goes to infinity. But we just repeatedly substitute... back... er for that unobservable expectation... right. So again we find out that erm, the expected price in T minus one, sorry in T, is a function of all previous prices... all previous prices, okay and it won't surprise you to find out that the weight giving to... to treating the historical prices... right becomes geometrically in exactly the same way... as the adapted, as the partial adjustment er co-efficient... or the partial adjustment model implied... alright. So the way it's given erm... t to previous prices decline geometrically... right. [clears throat]... Trouble is you've got a bit of a problem at the moment... right, if we want to... find out what the expected price in T is, right, this derivation tells us... that it's a weighted sum of all previous prices... right. We won't be able to estimate that, that model... right, simply because we don't have all the previous prices... alright. We cannot use this... expression up here either, because we've got... an unobservable variable in there, we've got the expectation of which we don't know... alright, we don't know what that is, what number that is... through time, so [clears throat] the analysis could reach an impasse here, right, this,... nice little theory,... nice hypothesis about how people form expectations but... it it's not tractable, it's intractable, right we can't use it.... However... we can apply something that will trick... known as the quoick transformation... right, that will enable us... right, to convert... what is an infinite stream... of previous prices... into something that we can estimate, right, just has a very few number of parameters... right. Now [clears throat]... right, those parameters that we... that we will learn will get this model in the end which contains only observable variables, only those, so it will contain... prices, actual prices and actual supply... alright. There's no ex... expectations in there, no unobservables. Now, what I was gonna do now was go through... this quoick transformation right, you're not gonna be asked t to produce it in an exam or anything like that, but the reason why I'm doing it now is that you will need it for your er, your Q M exam... not er your exam, your project... right, cos there is a similar applicat, it's quite a commonly used tool... you'll find... er, where wherever we have a, erm, an expression with an infinite number of erm possibilities or an in infinite number of arguments.... So you can go, go to sleep if you want for the next minute to erm... but because you will need to know this quoick transformation for your Q M project, you do want, may just want to note it down. [clears throat] So this is a digression... just gonna whizz through it very quickly, we don't really need to er,... [clears throat] spend much time on it. Let's think of the general model... right. Y T... alright, is a of constant alpha plus beta... into erm X T... plus lander X T Y is one... that's lander squared, X T minus two... plus lander cubed, X T minus three and so on and so on into infinity. Alright, so let's call that equation... er A... okay. [cough] Right, similarly at time T minus one we have... Y, Y T minus one equals alpha plus beta... X T minus one... plus lander plus lander X T minus two, plus lander... squared... X T minus three plus X T minus four... infinity again... we call that B and that's just the same equation but we just... erm shifted the subscripts and right, this explains why in T, this ex explains why in T minus one... right, all the subscripts have just been... just been changed.... [clears throat]... Right, well if we multiply both sides, we multiply... both sides of B... right, by this co-efficient lander... alright, we get... right, lander by T minus one, lander... erm [clears throat]... right then what we do is that we get beta into lander X T minus one plus... lander squared, X T minus two... lander cubed, X T minus three,... and lander to four... X T minus four and so on and so s so on until infinity... right. [cough] No notice, right... with these... are exactly the same as those now... alright [clears throat] so what we're going to do... is we're going to subtract... right, if we subtract from a, subtract b from a... all those terms that run off into infinity, we're gonna drop out, okay.... [clears throat]... Alright, so if we tr subtract... B from A... everything drops out apart from the following, so you have Y T minus [clears throat] lander Y T minus one plus B minus A, right equals alpha into Y minus lander... plus beta X T... okay. If you just rearrange that in terms of Y T... we get... so we just add Y T minus one to both sides and so we get Y T into alpha into one minus... plus... beta X T... plus lander Y T minus one... right, that's it.... Okay. So you too would assume quoick transformation... we've got... a geometric progression in here that goes off to infinity... alright... which is equivalent... to that, that little thing alright, we've got three parameters here... this one, that one and that one, alright, so we converted what is an infinite stream into a finite stream using this... er, quoick transformation.... Right, [clears throat]... so that's the digression over and done with... it's pretty fascinating stuff... okay, you'll you'll need that in the... in erm... in your Q M project... if you have to apply one to a particular problem.... Anyway [clears throat]... so... applying the quoick transformation back to er supply response, we apply this quoick transformation to equation seven... alright. We get the following supply response model... right, apply that quoick transformation to seven, yields S T... equals alpha gamma... plus beta gamma... P T minus one... plus, open brackets, one minus gamma, close brackets, beta S T minus one... alright, plus U T... minus open brackets, one minus gamma U T minus one... okay.... [clears throat]... Right, so this is [clears throat] if we incorporate... expectations... [clears throat]... in our model of supply response alright, so instead of saying that the the desired level of supply... instead of saying that, we say... it's actually this... right, if we actually incorporate expectations... specifically... alright, then we get a supply response model... alright, like this... and our hypothesis, y'know cos we don't know how expectations are formed, right, economics can't tell us anything... about how expectations are formed... are they rational, are they naive, are they adaptive, who knows, nobody knows... alright. But nevertheless it seems like a reasonable hypothesis the that that... er... expectations about future prices are revised in proportion, er the errors in expectations in the previous period, alright. If we incorporate that hypothesis we get... a supply response model of that form right, notice that we've only got observable variables, S T, P T minus one, S T minus one, we've only got observable variables in there... alright. So we can estimate this model although we've got a couple of problems that we'll come to later on, but nevertheless it's essentially estimatable.... Now [clears throat] the reason why we've gone through all these horrible equations... right, and looked at the adaptive expectations model and the partial adjustment model is because both of these two things are incorporated into single res erm... a single supply response model that was developed by Nerlove... Nur... Nerlove in nineteen fifty eight... old model but very very popular, right. This Nerlove supply response model that incorporates both adaptive expectations and the partial adjustment hypothesis... alright, has been applied... so that five hundred, right, different commodities throughout the world... right. So it's the, the most widely used supply response model in the entire universe, alright, and as a result of it's popularity... you have to learn it as well. Right, as you can imagine things might get a little bit nasty, we're incorporating... two er, two hypothesis and although very simple... produce algebra... that er, that's a bit nasty.... But never mind... never mind.... Now this Nerlove supply response model... alright... was introduced by Mark Nerlove in nineteen fifty eight... right in a heart breaking book called the dynamics... right of... supply... right erm estimation... of farmers... response to price... okay.... John Watkins University Press... I think there's a copy in the library... erm, you may want to have a little look at it, but [clears throat]... so what we're going do is we incorporate both these two hypothesis into a model supply response... okay. We'll er... do these just crack handle and then we'll show what information we can obtain alright from from this model and there's an example on a sheet of paper just to er... run it home... right. [clears throat] Okay. Let's start, start from the top... [clears throat]... let's suppose that the desired level of output... right, in any period... S star T... right, is simply... a function... of the expected value... of the price... so we're using our par our... well that's just a general, a general statement right. Decide supply... you subfunction beta of expected prices, right in T, supply in T is based on the expectation of what the price will be, in T... right. How are expectations formulated, right, it says nothing about how... expectations are formulated just says that... private supply will be dependent upon expected prices... alright. Well let's assume deducted expectations model... is a reasonable model of expectations formation... right... call that equation number one... right, so our adaptive expectations model is... E T minus one, so that's E T... right, minus E... T minus two... into P T minus one... right, so that equals... gamma... into... P T minus one, minus E T minus two... P T minus one... right so that's the same expression as we had before, right that's the our ad... our adapted expectations... model, that's, that's how we're assuming that expectations are generated... right, or revised... right [clears throat] so, we're incorporating expectations into this model but we also want to incorporate the fact that farmers take some time to respond... to changes in price... alright. Adjustment is not complete, it's partial... alright, so we're going to say [clears throat] that supply adjusts in the following manner, right, according to the partial adjustment... hypothesis... alright, so changes in actual supply... alright, will be delta... of... the difference between the desired level of output for T, right... and the actual level of... input T... minus one... alright, plus... er... alright, so that was just our partial adjustment model... that we looked at last week, right, we'll call that equation three... [clears throat]... okay... right. What we're going to do now... is combine both of these hypotheses... right... into... our supply response model, alright and we come out with the following.... [clears throat]... Now, it may look a little nasty but don't worry it simplifies... very easily.... So we combine those two hypothes hypotheses... right, we get S T equals alpha... delta... gamma... plus open brackets... open the second set of brackets... one minus delta... close brackets... open brackets, one minus gamma... close brackets... into S T minus one... minus... open brackets, one minus delta... one minus gamma... into S T minus two... right,... plus... beta, delta, gamma... into P T minus one... plus... E T... minus open brackets, one minus gamma... times E T minus one, right. U T was just that random error term... that we looked at. [clears throat]... Right. Now, looks pretty horrendous... alright, but if we... bum data on S T, S T minus one, S T minus two and P T minus one, all actual... variables, if we bung those into the... microfit and ask them to... form the regression, it would do, right, it would just... you'd get S T equals A plus B... that's T minus one... plus C... A T minus two... plus D... A T minus one... alright, and... B T [clears throat]... So before we we'll call that equation four... right, so although it looks nasty... alright... it's fairly straightforward, it's all, all we're doing is, we'd be asking the computer to regress S T on constant... like how you would yourself... T minus one... to like values... T minus two... and like prices... okay. [clears throat]... Right.... So it would actually be very straightforward to do this, although not... a couple of problems in this model that we ought to just say something about... alright, the specification of four does present a couple of problems... right. The first thing... and these are statistical problems that we won't spend too much time on them but just make sure they are... [clears throat]... our error term in this equation, right, is no longer a random error term... it's dependent on it's value in the previous period right, cos B T actually represents [clears throat] U T minus sumfunction of E T minus one... right. So... first problem... is one of serial correlation in this model... right, if we estimated this model and things were as we thought they were... this hypothes, these hypotheses were working... and farmers were behaving... erm, as we thought they might do, right, this model will automatically use serial correlation... right. So if you didn't get serial correlation and you estimated a model like this... you'd be a bit curious and wonder why... right. Secondly,... we've got erm...... [cough]... model [clears throat], we've got a dependent variable... a dependent variable, right, S T minus one... right, will be correlated... with the error term, right, so the dependent variable will be correlated with the errors... alright.... Because S T minus one... right... will be contained in the information up here.... Right. [clears throat]... Now because of these two problems... we violate... er a couple of assumptions... of ordinary lease squares... right, and as a result, the upshot of this... is that we get biased estimates... of our parameters A B C and D,... right, they will be biased... and also they won't... have the minimum variants property... right, they won't be the best estimators... you, the statistics you talk about blue estimates, best linear and biased... alright. So if for any one of those parameters, let's just say B... right, let's just say that's the true distribution of... the parameter B... what you'll... [clears throat] you estimate O L S and you'll get... alright, you'll get biased estimates, this is what you would get from the computer, alright... so we'd be biased cos the, the true value is over here but not only that, is that, instead of having a nice peak distribution, distribution would be quite flat, alright we'd... we wouldn't have the minimum variants property... alright.... So that there is a measure of the bias... and the flat distribution is a measure of er... erm... the fact that we don't have an estimator that's got this minimum variants property... right. So effectively what would happen all our T ratios right would be very small... cos the standard area would be very large, but don't worry too much about these ecometric problems... what I want you to do is to appreciate that... estimating a supply response model... right, can cause problems unless we do something about it.... Right, I think we'll, we'll leave it there for today, we'll finish off... [clears throat]... er... this tomorrow. No, no Er, just to appreciate, erm... what, what what it comprises, yes, so adaptive expectations what does that imply... partial adjustment, what does that imply. What can we get out of it and we'll do that tomorrow, we'll... derive y'know... estimates of... expectations, co-efficients and and elasticities... and it's those sorts of things that... you might be asked to in an exam... but, you won't need to derive anything in your exams Erm,... I haven't, probably better to see
[speaker001:] Is he in there now?
[speaker002:] He should be there now, yes he should. Right, so I'll see you all on er, on Wednesday.