hey there it's Zab again in this video I want to talk about the constrained optimization that consumers face if they have what we call a lay on TF utility function so this is a utility function that's named after the economists who first thought of the idea why C li lian TF and this is a utility function that is most commonly used when we're thinking about goods that are complements however just because you see a utility function like this doesn't mean that the goods have to be complements but in general if well if in all cases if goods are complements then this is likely to be the utility function that you would see so let's see what we have well this utility function says that our utility that we get by consuming some amount x1 of good 1 and some amount x2 of good 2 is going to be the minimum of a x1 be x2 and of course a and B here are both we're going to assume strictly positive real numbers so they are at least 0 now well you have this kind of utility function what you really care about is the smallest amount of either a x1 or B x2 because it's going to be the minimum of this or of this now if you're trying to figure out what the optimal consumption bundle is you're trying to figure out how much should the consumer buy of each good and of course they don't want to spend any more than they have to to get a desired or a given level of utility so in order to do this it makes the most sense for the consumer to set these two equal to each other so to set a x1 equal to B x2 because if one of them is larger than the other then that means you've bought too much of that good and you could cut back a little bit and buy more of the other and increase your utility so kind of optimally right this should be the case so optimally we're going to have a x1 equal to B x2 it of course that's what we're trying to find is the optimal consumption bundle so this is pretty easy this is a little easier than our video on the linear utility function so the first thing to do here is just solve for one of the goods in terms of the other so let's go ahead and solve for let's solve for x2 so if we solve for x2 we get that a over B x1 is equal to x2 all right now that's that's telling us something now the thing that we can do if we want to be able to figure out how much of good one we should buy is plug this into our budget constraint so remember the formula for the budget line so our budget constraint or our budget line is just going to be P 1 X 1 plus P 2 X 2 equals M because both of these are goods we like them both we know we're going to fully exhaust our income on this problem so we have just found that X 2 is equal to a over B x1 so we can rewrite our budget line this way so we get P 1 X 1 plus P 2 now we plug in our result for what we found X 2 to be so we get a over B x1 equals M and so now we can solve for x1 and exactly how much of x1 we want to by subject to they are given written only in terms of the parameters of the model right that is the price of good 1 the price of good 2 a and B and our income M so now we just want to solve for x1 so we can do this we can see that this is just X 1 times P 1 plus P 2 times a over B right this is exactly the same thing I've just factored out that x1 and that this is equal to em so now we can just divide in order to get our consumption bundle x1 SAR so x1 star the optimal consumption bundle is going to be our income M divided by p1 + and I really don't like that fraction being after the p2 a over B p2 and so there you have it that was relatively easy alright now we can do exactly the same thing trying to find the optimal amount of good - we'll start back here at this optimal condition and we will solve now for x1 so you see that x1 is going to be equal to B over a X - all right we just divided both sides by a and so we got that X 1 is equal to B over a X 2 now we can plug this into our budget constraint so let's move this up alright so hopefully that gives us enough room so we find that P 1 X 1 and now we've got X 1 written in terms of X 2 so we've got P 1 times B over a X 2 is equal to P 2 X 2 oops not equal to of course that's not what I meant I meant plus I was just checking to make sure you were following along and that's going to be equal to M all right so we can do the exact same thing here we factor out that X 2 and so we get that X 2 times B over a P 1 plus P 2 is equal to M and so we can come over and we can solve for the optimal amount of X 2 we get X 2 star is equal to M over B over a p1 plus b2 so that's the optimal amount of both Goods now we can rewrite this we can write it as we usually would that didn't work at all so we can rewrite our consumption bundle as X star equals now the optimal amount of good one so that's gonna be M over P one plus a over B p2 and then the optimal amount of good 2 is going to be M over B over a p1 plus p2 and so there we have it we have the consumers optimal bundle in the case of the Leone TF preferences now let's think about this in terms of indifference curves and a budget constraint so here we have x1 and x2 now we know that with Leone TF style preferences our indifference curves are going to look something like this alright so there's our indifference curves now imagine that the budget line looks something like this alright imagine that those points touch right there then this what we have just found is going to be the solution for this so this is gonna be our x1 star right this is this whole term here is gonna be x1 star and then this term is going to be X 2 star which is this quantity right so the only time that we would really have anything other than that would be if the price of one of the goods was 0 or if one of the goods prices were in then it could be that we would be okay consuming more so for instance if good to were free then our budget line could look something like this right but assuming that that's not the case then we don't really have to worry about that weird case so if we're dealing with strictly positive prices then that's not going to be a concern so really quickly let me give you kind of a bonus so related to the lay on TF utility function we could have a utility function that looks like this utility of X 1 X 2 is equal not to the minimum but to the maximum of a x1 B x2 now if we have this this is an incredibly easy function an incredibly easy one to solve because what we want to do then is we just want to maximize one of these things so which one should we choose to maximize well that's going to depend on the kind of bang for the buck and so the bang for the buck is going to be your marginal rate of substitution or well in this case we really don't have one know the marginal utility that you get from one of the goods right divided by the price that you would pay for it that's kind of your bang for your box so if for instance a over p1 the bang for the buck of Goodwin is greater than the bang for the buck for good to be over p2 then what should you do well you want to buy only Goodwin so your optimal consumption of Goodwin then in that case is going to be just all of your income spent on Goodwin if that's not the case right if on the other hand you have that a over p1 is less than B over P 2 that is the bang for the buck is better for good - then you should spend only I'm good too now you'll see that this is kind of a similar case your solution here is going to be a similar case to what you have in the linear utility preference there's a linear utility function so I'll leave you to that video to think about what this would look like but in this case essentially you're gonna want to set one of these to be your entire budget alright and you're gonna want to spend nothing on the other so it's just going to depend on which one gives you the better bang for the buck so of course well I'll go ahead and write it out so if a over p1 is greater than so let's actually write this out a little more formally so X star that is our optimal consumption bundle is going to be equal to so if the bang for the buck is bigger for Goodwin then you want to only buy Goodwin so we're gonna spend em over p1 here we're gonna buy 0 of good to if a over p1 is greater than B over p2 and of course if a over p1 is less than B over p2 then we want to buy none of good one and spend everything M over P to on good to so what do we get if these two are equal a over p1 the bang for the buck for Goodwin it is equal to B over B 1 B 2 the bang for the buck for good - well if that's the case then we really don't care right we would be happy with either M over P 1 comma 0 or 0 over 0 comma M over P - right so either one of those would be acceptable in this case right but you certainly want to divide up your consumption among those two bundles so that's really among those two good so that's really the the only difference between this and the ordinary case with linear utility function anyway just thought I'd show that so here you have now seen the consumers choice problem solved for Leon TF utility function and for the related function where we are trying to maximum Axum eyes either a X 1 or B X 2 so anyway hope this helped and we will see you in the next video thanks for watching