nm0757: er two three weeks is to er is to try and build up a framework a a 
model that will allow us to understand how consumers make choices and we've 
almost got to the end of the er what people would like to buy element er of the 
framework and then what we're going to try and do today when we've finished 
that off is to actually put it together to try and er create a framework a 
model that will allow us to understand the choices that actually made so we 
going to put together that availability set the things that people can choose 
and the part that looks at what they actually want to choose to actually look 
at the choices that are actually made and what we want to try and do is to put 
that framework together in such a way that we can use it not just in an 
academic sense to have some nice diagrams but to actually predict what people 
will choose er and to look at what happens when things change so if we can 
build this framework up we can try and understand well if prices change 
what's going to happen to the demand for a good is it going to go up is it 
going to go down and by how much likew-, wise with income for example if income 
changes how will demand change er and by how much so that's where we want to to 
end up now if you remember where we got to er at the end of last week was we we 
started at the end of last week hang on try and remember which of these light 
switches it is again where we have all of this this space these combinations of 
two goods X-one and X-two okay and if you remember we said well we can 
represent it in two dimensions but really we're looking at the choices between 
all the goods and services that consumers have available to them and we started 
saying well what the consumer can do we assume is compare any combination of 
these goods wherever it is and can say i prefer this point that combination of 
the two goods over this point and so on and that they can compare all the 
possible combinations of the things that are available 
to them okay and say i prefer this combination to that combination and they do 
that er they have these preferences they can say that they have strong 
preference i prefer A over B i'm indifferent i don't care which i have or weak 
preference i prefer A to B or i'm indifferent between the two and then what we 
we did was we said okay that's fine but what we want to try and do is to build 
up the model that will allow us to predict choices and to do that we have to 
make certain assumptions about the preferences that consumers have and we we 
had all these assumptions of completeness there are no black holes there are no 
er combinations of the goods that the consumer can't express a preference over 
transitivity that is that we have this rule that if you say i prefer A to B and 
i prefer B to C we can assume that they prefer A over C and we've we we said 
that often that doesn't hold but er at least logically it seems that a rational 
consumer might have transitive preferences 
we then had the idea of reflexivity that if a if a consumer couldn't compare a 
good or combination of goods to any other they could compare them to themselves 
and using that okay we said okay every single one of these points er we can 
express a preference over but there won't be any inconsistencies we then went 
on to say okay can we make any general rules about the direction of the 
consumer's preference and we had the idea of non-satiation which means that 
preferences go out in that direction I-E that within the realms of most of the 
choices that consumers make with lots and lots of goods and services available 
to them they will prefer more to less they prefer more of the goods to less of 
the goods and if that is the case their preferences will go out in that 
direction and hence we automatically can say for example that this point here 
is preferred to that point there simply because it has more of er both of er 
the goods er and therefore automatically the 
consumer will prefer more we then said okay if that's the case then we can 
start talking about indifference lines and because that is the case an 
indifference line or indifference curve must have a negative slope okay that is 
if we combi-, if we we draw a line that combines all of the combinations of 
these goods between which the consumer is indifferent it must be the case that 
if you get more of one good you have less of another and w-, the measure of 
that is the marginal rate of substitution okay the slope of the line so that's 
negative and it's a measure of the trade off between these two goods which 
reflects the relative preference that the consumer has for one good over 
another so if a consumer really likes X-one if they give up one unit of X-one 
they're going to require a lot of X-two to take its place okay for example then 
we said okay that's fine but looking at the way c-, they're looking at the 
sorts of preferences that consumers have what we tend to 
find is that this rate of trade off this marginal rate of substitution isn't 
constant okay and we said in fact that we would expect the indifference curve 
to be convex to the origin which means that the rate of trade off at this point 
here when you have a lot of X-two and not much X-one will be different to the 
rate of trade off here where you have a lot of X-one and not much X-two because 
we know that as you get more of a good its marginal utility declines and and 
vice versa so where we ended up last time was with basically the ability to 
represent sort of colour to represent the consumer's preferences between these 
two goods okay as a series of indifference curves like this that are negatively 
sloped convex to the origin the direction of the consumer's preferences is out 
here so we automatically know that indifference curve two is preferred to 
indifference curve one every single combination of the two goods on this 
indifference curve 
des-, is preferred to every single combination on indifference curve one and 
likewise indifference curve three every combination on that indifference curve 
is preferred to indifference curve two okay and so on so the least preferred 
point is this point here zero consumption 'cause the consumer gets no utility 
because they only get utility from consumption from the goods and the maximum 
is some infinite point up here okay er whoo-, whoops where they consume however 
much because at the moment they have no constraints okay because we haven't 
brought the availability set in er so this is what they would like to do okay 
now you may say okay that's totally unrealistic because eventually the consumer 
will grow satiated okay now that may be true with er two goods but remember if 
we try and represent this with all the goods available then we can have very 
small substitutions between all the different goods that are available and we 
would assume that normally within the realms of of balanced consumption of 
consuming a large number of different goods that the consumer 
will operate within the context of not having great satiation in most 
circumstances okay so that's where where we got to and that's fine we can so 
far wh-, if i could draw it do it in three dimensions we could do three goods 
okay but that is just a diagram it is it's just a representation a a 
theoretical model of er the consumer's preferences we can't use it at the 
moment to do anything except er to to try and think about the way in which 
choices are made what we need to do is to be able to represent this numerically 
we need to be able to represent these indifference curves in a numerical form 
'cause if we can do that if you remember the availability set can be 
represented er er mathematically because you have the budget line what we need 
to be able to do is to represent these algebraically mathematically er and then 
we can combine the two together and use that to actually predict the ch-, the 
co-, the choices the consumer would actually make if we can al-, also 
if we can represent it algebraically of course we're not constrained to three 
goods which is what we are in a three-dimensional diagram we can have however 
many goods that the consumer has available to them okay so we need to represent 
this in some other way and so the final assumption is that we can represent 
these indifference curves algebraically and that they are differentiable okay 
that is that if we we we can differentiate them so that we can derive an 
optimum okay we can derive a point er a choice that the consumer will make when 
we put all of all all of the information together the two sides so that's the 
final assumption that that we are we are going to make now we're not going to 
go into er a lot of the m-, of the mathematics okay it's it's the basic 
principle that really matters here okay now the complication we have is that we 
have said that utility is not measurable in a cardinal sense that is we cannot 
say for example that if a consumer er has a certain consumes a 
certain combination of goods their utility is level X and then if they consume 
another combination it's level Y okay we can't say there's a l-, level utility 
ten and another combination of goods gives you level of utility twenty we've 
said we can't do that we can't measure utility er and even if we could we 
couldn't combine it across individuals because it's totally individual er to 
that consumer that seems to fly in the face of saying well we want to represent 
those indifference curves mathematically algebraically and we want to be able 
to predict the choice they make that's because the reason for that conflict is 
that all of the mathematical functions that you will have seen to date are what 
called a-, are are are called cardinal functions that is functions where the 
numbers are measuring things so if we have a value of ten we can say that is 
half of a value of twenty okay now what we are going to do is to use a type of 
function that is called an ordinal 
function and an ordinal function is a function where the numbers only matter in 
terms of their relative position so if you have values of one and two the two 
is greater than one okay it doesn't matter how much but it's just greater okay 
so that all that matters in an ordinal function is order not relative position 
okay so what we want to do is we have our indifference curves like this and we 
want to be able to represent those indifference curves with a number okay 
algebraically and presume that we can represent that utility function as er U-X-
one-X-two that is utility the satisfaction they get is a function of the 
amounts of these goods which is what we've said and don't get they don't get 
satisfaction from anything else they just get it from consumption in in the 
context of the market and clearly the amount of the utility depends on the 
quantities of those two goods and that's exactly what this diagram says okay it 
says that er [cough] if we have this combination and this 
combination these two points here the level of utility's exactly the same 
although the quantities of these two goods is actually different because the 
consumer is indifferent between those two combinations and we've said that if 
we have a combination here the the level of utility is greater again the 
quantities of the two goods are different but the level of utility that the 
consumer gets is is greater because it's on a higher indifference curve and so 
what we want to do is to represent these these indifference curves [cough] as 
what is called a utility function a function that simply says the utility the 
consumer gets is a function mathematically of the quantity of X-one and the 
quantity of X-two okay the quantity the amount of the two goods one of these 
could be zero [cough] so if the consumer got no utility at all from the 
consumption of that good then simply the the expression relating to say X-one 
this good here would just be zero okay so that's possible er and if the 
consumer go-, got 
no utility at all from these two then it would be zero for both okay so we can 
represent that no no problem at all but we don't we're not going to represent 
these indifference curves with a function which assumes that we're measuring 
utility so we're going to use er what is termed a a an ordinal function [cough] 
okay so we're not going back to a world where yeah we can measure marginal 
utility et cetera we're still in a world of of indifference curves okay that's 
so that's that's fine okay [cough] the type of function we want is as is as 
follows if we assume that we have two consumption bundles X-one er and X-two 
okay and these are the utility functions that represent those two consumption 
bundles so X- one here is a combination of these two goods and X-two here is a 
combination of these two goods a different combination [cough] and what we want 
to do is to effectively so U-X-one okay could be associated with this 
indifference curve here and U-X-two could be associated with this indifference 
curve here okay for example what we want is to put numbers on these things so 
hence any combination of goods that lay on indifference curve two would have a 
greater number than any combination on indifference curve one okay now let's 
assume that we give we have a lar-, a larger value for U-X-one this combination 
than U-X-two what that would mean of course is that the consumer preferred this 
combination to that combination that is that X-one was preferred to X-two okay 
and that's what that would imply is that the consumer strictly preferred X-one 
over X-two the other possible scenario is that U -X-one is equal to U -X-two 
like if two goods were on the same indifference curve and that would assume 
that the consumer was indifferent okay so we want to represent these 
indifference curves in such a way that either if the consumer is indifferent 
the value we give the number we give is exactly the same okay if the consumer 
prefers one bundle of goods over another we will give 
a bigger number to the er bundle the consumer prefers what th-, th-, the the 
very important thing to get clear in your mind is it doesn't matter what the 
number is okay all that matters is that we give the same number to bundles of 
goods between which the consumer is indifferent and we give a bigger number to 
combinations of goods that the consumer prefers it doesn't matter how much 
bigger okay and that is the crucial difference if it mattered how much bigger 
the number was we would be using it as a real measure of utility we'd be saying 
that if it was a value of ten that would be half a value of twenty all we're 
saying is that if we give a value of ten that is less than twenty doesn't 
matter how much because we're not measuring utility we're just measuring order 
hence it is called an ordinal function okay so any combinations on one 
indifference curve will have the same number any higher indifference curve any 
combinations that are on a higher diss-, indifference 
curve will have a bigger number it doesn't matter how much bigger the number is 
okay so let's assume for example that we have er three no sorry four er 
consumption bundles okay these are four different combinations of those goods 
and we have a utility function U-X okay now if we assume that X-one that that 
this combination X-one and this combination X-two the consumer is totally 
indifferent between them they will have exactly the same number say five and 
five okay and let's assume that they prefer X-three over those two and it may 
then have a number of ten and they like X-four even more and that could have a 
number which is say twenty okay so X-one and X-two are indifferent they pref-, 
thi-, this consumer prefers X-three over two and one and prefers four over 
three two and one let's assume that we have let's think say another function 
that we rec-, we could we and we give another numerical representation to er to 
this consumer's preferences and let's say 
we have a value of er a thousand here again the numbers are the same hence the 
consumer's indifferent let's assume the number given to X-three is ten-thousand 
that indicates the consumer prefers X-three over two and one it does not say 
that they prefer the consu-, X-two more than when we had a value of ten okay 
all that matters is that that number is bigger than the numbers given to X-one 
and X-two it doesn't matter how much because we're not measuring utility all 
that matters is relative er positioning and so we could have a value of say a 
million given to X-four and again that doesn't tell us anything more than the 
values here it's exactly the same because all that matters is relative ordering 
hence it's an ordinal function the higher the number the greater the consumer's 
preference okay that's true but it doesn't say how much higher it is it just 
says it is higher so whenever a number is bigger than another number it means 
the consumer has greater 
preference they will go for that combination of goods rather than one to which 
we've allocated a lower number how much the number differs is irrelevant it 
could differ by you know one decimal point or it could differ by a million the 
information is exactly the same because what we're well what we're not saying 
is that this how the consumer chooses you know they they think in their head 
what is my utility function how much utility do i get from these combinations 
what we're saying is that we want to represent er these these ideas that we 
have about their preferences in a numerical way okay we don't want to measure 
utility because we assumed that we can't do that but what we do want to do is 
to say we want to give a bigger number to a higher indifference curve because 
of course what we want to do is to produce a model that says the consumer aims 
for the highest indifference curve okay they aim for the one which is furthest 
up okay we're not saying that we can 
measure their utility we're saying is that all they will do is to strive for 
the highest indifference curve okay and that is er that is all these are termed 
ordinal functions and they are very different to the sorts of functions that 
you see we can treat them in exactly the same way as any other mathematical 
function we can do everything to them that we can do to another mathematical 
function the only difference is the interpretation we give to these numbers all 
that matters is their relative position okay how much they are different does 
not matter simply because er we're not measuring their utility we're just 
measuring order okay 
do you do you understand that distinction between the two types of functions 
and and what those functions are representing yeah simply they're saying if we 
get more preference we have a bigger number doesn't matter how much that's 
irrelevant okay and it does take you know given you've been used to using other 
the other functions in in other ways it does take some sort of getting your 
head round but er okay that's th-, basically what what we want to do once we 
once we've said we can do that we can give a number to the preferences then we 
just have to decide what is the mathematical form okay what is the mathematical 
form and i'm just going to quickly give you some er examples [cough] okay er 
some some extreme examples okay th-, the normal case that we'd tend to find and 
then just some extreme examples so that you er you understand [cough] now 
you've been used to seeing er indifference curves like this okay 
fine they represent the sorts of things th-, the sort of assumptions that we've 
built up the sort of assumptions we built up last week and which is a convex 
marginal rate of substitution changes as the combinations of the good change 
tha-, tha-, it's negative okay so there's a trade off between the goods the 
consumer prefers a higher indifference curve to a lower one and so on okay if 
they're the assumptions we make then we want to be able to represent those 
types of preferences between the goods in some mathematical form and the normal 
form that is used there okay is er is a quadratic form of some kind okay that 
simply says that er and these here are your marginal rates of substitution 
alpha and beta okay so these are the marginal utilities of the goods the ratio 
of those will be the marginal rate of substitution okay so alpha is the 
marginal utility associated with X-one er and beta is the marginal rate of 
substitution associated with X-two and of course the 
ratio of those which is the slope of the indifference curve will give you the 
marginal rate of substitution okay and all we've said is okay this mathematical 
function okay will dem-, will will display mathematical properties like we have 
here okay I-E that represents that will allow us algebraically to represent 
that okay the alpha marginal utility of of X-one -beta marginal utility of X-
two the ratio between alpha and beta marginal rate of substitution which is a 
piece of information we may want and directly we can represent this 
algebraically we can get that marginal rate of substitution okay of course it 
may change according to which point on the indifference curve we are because 
we've said that's what happens so it will change according to the absolute 
amounts of X-one and X-two okay er but that would represent that and so that is 
the normal form that is used okay okay it's a bit er more difficult er 
mathematically to deal with because it's not linear 
but in fact if you take the logs of that it becomes linear anyway so so it's 
relatively easy to use but that is the sort of thing that we might use to 
represent the normal types of preferences that we assume consumers have okay so 
that's one form that we might use and then what we would do okay is X-one and X-
two will be the quantities of the goods and of course if there are ten goods 
there will be ten of these X-one X-two X-three up to X-ten and there'll be ten 
of these marginal utilities okay and then again there will be a marginal rate 
of substitution between all of the pairs of goods that the consumer has 
available to them so X-one and X-two there'll be a marginal rate of 
substitution between X -one and X-three and so on okay and that will simply be 
these ratios the ratios between these but all we're saying is that that 
mathematical form can represent that we're not saying that the consumer goes in 
the shop and gets out a calculator with that sort of 
mathematical function imprinted in it we're saying that that is a mathematical 
way of representing the preferences that we've said the consumer might have the 
model that we've we've actually built up okay [cough] an alternative form might 
be as follows these are linear indifference curves straight lines okay they're 
different to the previous ones because the other ones are convex they're curved 
these ones are straight lines how much or to what extent does the marginal rate 
of substitution change as you go up and down on the indifference curve like 
this go up go down 
sm0758: constant 
sf0759: constant 
nm0757: it's constant yeah so the marginal rate of substitution doesn't change 
it's constant okay it's still going to be that ratio al-, A alpha over beta 
okay because the alpha the coefficient of X-four represented marginal utility 
of X-one and the beta represented marginal utility of X-two so the marginal 
rate of substitution is still exactly the same and in every one of these it's 
exactly the same 
mathematically but in this case it doesn't change it's it's going to remain 
exactly constant what sort of goods demonstrate would have that sort of of er 
indifference curve what we're saying is that your margin utility okay doesn't 
change according to how much you have and the rate at which you substitute 
between the two goods remains exactly the same so if you alwa-, you will always 
require for example two units of X-one to replace one unit of X-two have you 
got any idea of the sorts of goods we might see that with how substitutable are 
these goods do you think very substitutable or not very substitutable very yeah 
these are perfect substitutes okay that doesn't mean that the marginal rate of 
substitution is one I-E one unit for one unit one unit of X-one for one unit of 
X-two but it does mean that the rate of substitution doesn't change at all it 
remains exactly the same hence they are perfect substitutes for one another and 
that remains exactly the the same 
however much you have and a function that would represent that is simply alpha-
X-one plus beta-X-two okay the marginal rate of substitution is again alpha 
over beta it's the ratio okay of of the two because alpha's the marginal 
utility of X-one beta's u-, marginal utility of X-two but in this case it 
remains it's it's a constant okay er however much X-one or X-two the consumer 
has these are perfect substitutes for one another okay these of course are easy 
to handle 'cause they're linear it's a very s-, simple function but again all 
we're saying is that if that was if we had two goods for which the consumer you 
know er regarded them as perfectly er substitutable then that that would 
represent that okay alpha-X-one plus beta-X-two and so that is the perfect 
substitute case which is one extreme away from the convex er indifference 
curves that we we saw before the other extreme i-, is as follows right-angled 
indifference curves have you any idea what sort of grid this 
might be because you may have seen demand curves that look exactly like this in 
the past 
sf0760: right and left shoe 
nm0757: sorry 
sf0760: [cough] left shoe and right shoe 
nm0757: left shift and right shift 
sf0760: left left left shoe and right shoe 
nm0757: oh left shoe and right shoe er not necessarily no mm maybe maybe what 
what do you mean by that 
sf0760: 
nm0757: okay yeah mayb-, yeah okay so what are you saying about the two goods 
sf0760: 
nm0757: mm 
sf0760: you mean like one or the other 
nm0757: okay in which case yeah can you think of any other examples these these 
goods are perfect complements yeah that is that they are consumed in certain 
combinations so in fact you're right left shoes right shoes er exactly the same 
er if someone likes er black coffee then they don't have any milk if they like 
very milky coffee then they they they consume a certain amount of milk in their 
coffee but the two cannot compensate for one another okay they can't compensate 
for one another so if you have so you're right if you have a left shoe then you 
have then have a right shoe that's fine but if you then have one extra right 
shoe that doesn't 
actually give you any more satisfaction because you need another left shoe 
presumably to consume with it or if you like a certain combination of of milk 
and and coffee in a drink if you add more milk then you don't get any extra 
satisfaction and that combination is that point there 
okay that combination says well in this case you require equal amounts of X-one 
and X-two so it could be left and right shoes for example okay er you have to 
consume them in that combination if you have more X-one okay which is going out 
along here you don't get any more satisfaction at all it remains exactly the 
same 'cause you don't go to another indifference curve and likewise if you have 
more X-two you don't get any more satisfaction you remain on the same 
indifference curve okay so the only way you get more satisfaction is to move up 
here on a line with the same combinations of X-one and X-two I-E you need more 
of both goods in exactly the same ratio to get more satisfaction if you get 
more 
of one and no more of the other you don't get any more satisfaction at all 
because you m-, you need to consume the two goods together okay now this is the 
other extreme of course perfect complements perfect substitutes it is however 
slightly er unrealistic i-, in in some senses because what we're saying is that 
if the consumer has this amount of the two goods so like milk and coffee so in 
this case the consumer would have white very milky coffee in the sense that 
they require the same quantities of both okay half and half what we're then 
saying is okay if this consumer was given coffee with more and more milk the 
amount of satisfaction they get doesn't go up it doesn't go down okay so they 
don't get any extra satisfaction but they don't get disutility either now that 
is probably unrealistic because what probably happens actually is that they 
require in this case same amount of milk the same amount of coffee but if they 
get more or less of one their actual utility 
goes down okay because the coffee gets more and more milk they don't like it 
hence their level of satisfaction goes down okay and we're assuming having the 
green lines here that that doesn't happen okay so they're perfect complements 
but if they get out of synch with one another they er you don't get disutility 
if you get disutility then in fact the indifference curve would simply be the 
points okay up here because the goods would have to be consumed in absolutely 
those quantities any other spoils everything okay so in certain cases extreme 
cases it could be that you have to consume them in absolute quantities in the 
same ratio or you got no utility at all okay [cough] and we're assuming here 
that that's not quite the extreme okay there is there are these combinations 
you have to consume them in but if you don't then er you don't get disutility 
but you don't get any more utility okay how we represent this algebraically is 
with a very strange function which is minimum alpha-X-one 
comma beta-X-two that is the again alpha is the marginal utility of X-one beta 
is the marginal utility of X-two the ratio between them will give you the 
marginal rate of substitution er but imagine utility is entirely determined by 
the good okay which is in the lowest amount okay so there may be excess amounts 
of one good so you might have say ten units of coffee and thirty units of milk 
but clearly you'd only use er ten units of milk because you'd need it to go in 
the coffee okay you'd you could use them in in equal amounts the excess amount 
of milk doesn't give any diss-, dissatisfaction but it doesn't give any 
satisfaction and so the amount of utility the consumer gets is entirely con-, 
er constrained by the amount of coffee the thing that's in short supply okay 
and that's the sort of function that would would represent that okay so what 
we're saying is that we can represent all types of of preference er perfect 
substitutes like that er perfect complements like that 
and the normal case with convex indifference curves like that and what we do is 
to represent these er types of preference okay mathematically and all we're 
saying is that if we look at the preferences that consumers have the way in 
which they trade off between the goods that are available to them and we 
observe that for some goods that have o-, er substitutes that is the consumer 
er doesn't really care which they have and will trade off between them at a 
rate that doesn't change or if we observe their perfect complements the other 
extreme these goods are always consumed in exactly the same quantities okay in 
a particular combination because that's how the two go together to produce 
something else and on their own they don't give any utillity or the middle case 
which is w-, w-, what we would normally er observe we think that in fact the 
goods are not perfect complements or per-, perfect substitutes there is 
substitutability between the two goods the amount 
of substitutability will depend on the slope so the flatter this indifference 
curve is this board's getting worse so if it was like that that would mean act-,
actually there's not that much s-, er er sorry there's a lot of 
substitutability between those goods okay 'cause it's it's getting towards flat 
it does change but not that much so we'd be erring in this direction here okay 
and maybe if we had indifference curves that were like that we'd now erring in 
this direction here towards perfect complements so the the slope of the 
indifference curve of course differs but we do have these two extremes perfect 
substitutes and perfect complements okay er we can represent these so if we 
observe these types of preference we can represent them in an algebraic way 
using functions that we've just looked at okay they're ordinal functions which 
means that all that matters is the higher the number the higher the 
indifference curve it means that if we have a number er which is higher for 
example 
we'd be on this indifference curve rather than that one perhaps we're not 
saying how much they differ we're not saying we can measure utility we are not 
saying that if the number is higher that means the consumer gets twice the 
amount of utility we're not doing that because we know we can't measure utility 
but also we don't need to if all we're interested in is how much of the goods 
the consumer consumes 'cause that's what we're interested in the choice the 
consumer makes which we assume is meas-, is driven by utility okay they want to 
maximize then the the degree to which they meet their needs and wants we 
measure that as this idea of utility or preference whatever we don't actually 
need to measure preference or utility in a c-, in a cardinal sense all we need 
to know is that the consumer does get the highest level of utility they 
possibly can okay and to do that we represent these preferences using these 
mathematical forms okay the mathematical form we use 
depends on the type of preferences we see if they're perfect substitutes then 
we use the type of function that we've just looked at okay er alpha-X-one plus 
alpha er plus beta-X-two if we observe they're perfect complements we use 
minimum alpha-X-one beta-X-two more normally we would er we would have 
preferences like in in the middle case they would differ in the amount of 
substitutability complementarity and that would be represented in the alpha and 
beta values but we would use a function which was er X-one -to-the-alpha X-two-
to-the-beta which is the normal form we would use those am-, those the values 
of alpha and beta represent the marginal utilities and therefore the ratio 
represents the marginal rate of substitution and that will differ according to 
all the goods that the consumer has and will differ between every consumer 
according to their preferences okay but all we're saying is that we can 
represent now these ideas that we've built up the ideas on preference 
and the the the properties that preferences have in a in a mathematical sense 
which means we can do it for any number of goods we want we're not constrained 
to three which we are here and we can now put it together and we can actually 
produce something that will allow us to measure the marginal rate of 
substitution which is a useful piece of information to predict the choices that 
a consumer will make in particular circumstances which is vaguely interesting 
most interesting we can put it together to u-, to build up a model that allows 
us to predict what happens if things change if this is what the consumer 
chooses now what happens if something a factor that influences their choices 
changes prices go up income goes down whatever that is very useful 'cause it 
allows us to derive elasticities for example and that is useful information 
somebody who wants to market a product wants to know if i increase my price ten 
per cent how much does demand change will my revenue go up or will my revenue 
go down what happens if my competitor changes their price reduces their price 
by five per cent how much is the demand for my product going to change so being 
able to represent all of this these notions in an algebraic sense allows us to 
to produce that sort of model okay and that's what we want to do okay let's 
take a a break there and then when we come back what we're going to do is to 
put the two sides together okay 
nm0757: 
this choice process can be broken down into two parts the things the the 
combinations of the goods that the consumer is able to buy and if you remember 
we said that that the first of all the availability part is determined by their 
budget line which is equal to P-one-X-one plus P-two -X-two er is smaller or 
equal to M so they cannot consume any combination of these goods which they 
can't afford the slope of that line is is equal to p-one over p-two the ratio 
of the prices the price of X-one over the price of X-two this is the same for 
every consumer the consumer is a minor part of the total market they double 
their consumption demand doesn't change price doesn't change that is the same 
for everyone the position of this line is determined by their income okay the 
higher their income the further out it is the lower their income the further in 
it is if there's zero income then the clearly they will be at that point there 
presuming there are no free goods there are no 
goods that don't cost anything okay we also had these non-negativities here and 
here it's not possible to consume negative amounts of goods and this one here 
we said X-two must be greater or equal to nought and here X-one must be greater 
or equal to normal so you can't consume anything less than zero which is at the 
origin that is the availability set what they can consume given the prices of 
the goods given their income prices change this changes and the slope of the 
budget line changes their income changes the slope remains the same and we just 
move in in and out that's what we looked at er a while ago now their 
availability set then the other part their preferences what do they want to 
consume okay and just like here we looked at what was available to them 
regardless of what they wanted to do what we've just done is to look at what 
they want to do regardless of what they can do okay regardless of what they 
they have available to them and we've said that that is 
driven consumption behaviour is induced motivated by your needs and wants 
consumers' needs and wants and we have the it's driven by the idea of 
preferences that is a c-, the consumer prefers a good that meets more of their 
needs and wants than one that pref-, that that meets less of their needs and 
wants so it's totally motivated behaviour and we can represent that through 
indifference curves and the direction of the consumer's preferences given non-
satiation is out in this direction so they want to get out as far as possible 
out here as they can these these preferences exhibit things like transitivity 
we have a convex er indifference curve which means that we have a rate of trade 
off between the good and the marginal rate of substitution which changes 
according to the amount of the goods that they consume so these are in this 
case some substitutability between the goods but they're not they're not 
perfect er complements okay but they're not perfect substitutes and that 
we can represent that these these preferences u-, using a utility function an 
ordinal function which is U -X-one-X-two and we can the mathematical form of 
that function will depend on what the preferences look like whether they're 
perfect substitutes perfect complements or like this in the normal type and 
we'll use the mathematical form which satisfies that which which has that that 
has those properties given that the consumer aims to consume as much of these 
goods as possible they're non-satiating we said the consumer and the idea of 
rationality we came up with in the first week is that the consumer aims to 
maximize their satisfaction okay aims to maximize the degree to which their 
needs and wants are satisfied and so what we're saying is on this diagram 
they've aimed for the highest indifference curve they want the one that's as 
high as possible because that's the one that gives them the highest level of 
utility or we want to maximize the value of their utility 
function 'cause the higher that is the higher the indifference curve they're on 
okay putting it together then we have what they want to do they want to aim to 
get that the the con-, th-, they want to consume as much X-one and X-two as 
possible and in the realm of all the goods as much of those goods as possible 
however many are available to them they want to maximize their utility they 
want to consume that combination of goods and services that gives them the 
greatest satisfaction meets as much of their needs and wants as possible but 
they're constrained in doing so because these goods are not free they have to 
pay for them and their ability to pay for them is constrained by their income 
okay the extent to which they're constrained depends on what their income is 
and how much the goods are but they are constrained in doing this and if we put 
those two together okay that's a consumer's budget line f-, for for an 
individual consumer okay and it will be the 
same slope for all consumers but the position will depend on how much income 
they have so all the consumers have the same level of income it'll be in the 
same position those that have more income it'll be further out those that have 
low income will be further in okay that's their availability set and we have 
these indifference curves which represent their preferences okay let's just 
call them U-one to U-three and what they will do is to choose that combination 
of goods that gives them the highest level of utility gives the maximum 
preferences given the resources that are available to them and that will occur 
at the point where the budget line just touches the highest indifference curve 
and that is the choice that the consumer will make so that's the quantity of X-
one and that's the quantity of X-two that the consumer will be able to consume 
okay so what we're saying is that given the the constraints on the choice 
choices that the consumer can make which is 
t-, t-, which are totally outside of their control they are economic 
constraints okay on er basically how much of these goods they can consume 
determined by the market prices and by their income and at any point in time 
those things are fixed of course someone can have influenced their income in 
the longer term work longer hours for example er can save at one point in time 
so they have more resources in the future and so on but at any point in time 
when the consumer chooses so when they're in the supermarket whatever then 
those are are fixed things the prices are fixed and their income is fixed and 
what we're saying is that how they decide between all the combinations of goods 
they can buy within those constraints is by thinking about how much all the 
different combinations will meet their needs and wants and they will be driven 
to choose that combination that provides the greatest utility that combination 
that meets their needs and wants most 
effectively we've represented that through a series of indifference curves and 
they're i i o-, we emphasize again we're not saying that the consumer goes into 
the supermarket with indifference curves or utility functions whatever what 
we're saying is we can represent how they make their choices in this way okay 
in this s-, sort of abstract model and putting the two sides together it will 
be the point at which the budget line just touches their highest indifference 
curve okay and that will be the choice they make that amount of X-one that 
amount of X-two okay if their income changes the choice will change if the 
price of any of the goods changes their choice will change if their preferences 
change then the choice may change okay so if there's an advertising campaign 
about one of the goods say X-one that shifts their preferences so they like X-
one more then their choice may change okay and we're going to look at those 
sorts of changes er next week er but those 
are the sorts of er changes that might go on and that would be reflected in a 
shift in the position of that point the choice that the consumer actually makes 
okay now we can represent that also er algebraically by just putting together 
these elements here okay because we've been able we've said to represent 
mathematically these the indifference curves through this utility function and 
the availability set through this budget constraint and through these non-
negativities so you can't consume negative amounts and so we can represent this 
mathematically it is simply that we want to maximize that utility function 
okay we want to be on the highest indifference curve possible that's how you 
want to think about it we want to maximize the utility the consumer gets and 
what we've done is to represent that through this utility function whatever its 
mathematical form and all that mathematical form does is to represent the way 
in which the consumer thinks about the goods the preferences the consumer has 
perfect complements perfect substitutes anything in between and that will 
depend on the individual one individual may regard products as perfect 
substitutes and another individual may not okay but all we're doing is 
representing how they see those goods in some mathematical sense and we'll 
maximize that okay and that would go to infinity if they weren't constrained 
but they are they're subject to two basic constraints the first is the budget 
line so they can maximize this freely until they hit this they can't spend more 
than they earn okay so 
they hit this budget line and secondly the non-negativities you can't consume 
less than zero of a good however much you hate it [cough] now that is obvious 
in real life but mathematically we have to allow for all eventualities because 
what we've now done is to say okay we can represent all these things that we 
can see go on and the assumptions we've made and what we can do is we can er 
now represent that this in this way because it's now mathematical we of course 
have to stop it doing stupid things okay when we run this and so we have to 
include those and that is the basic model that says what the consumer will do 
when faced with all of the goods and services that are available to them is 
they will select between those goods and services on the basis of their own 
personal preferences and we can represent those preferences in the mathematical 
sense okay and that mathematical representation will encapsulate will include 
all of the properties that we have observed okay in 
consumer preferences things like transitivity things like non-satiation things 
like the fact that the marginal rate of substitution diminishes er as or 
changes as the amount of the goods change because the marginal utility's 
changed all of those things that we can see okay er and which we know we think 
influence the way in which pe-, er people trade off the goods that they based 
we can represent that in within this and the mathematical form we use okay will 
depend on what we observed okay perfect substitutes we've one form perfect 
complements another form any other er type we use another form relative values 
of alpha and beta will again reflect those trade offs how much they like the 
two goods and how that rate of trade off changes as you get more and less of 
the goods okay so all of that can be represented and that's all that is doing 
is representing the way in which the consumer makes those choices again we're 
not saying that's how they make them we're 
saying that we can represent it in that way and they can do that they c-, they 
make their choices but they're constrained in so doing because well first of 
all the non-negativities which are fairly obvious but they're constrained by 
just economic facts of life the fact that they face these prices which are non-
zero so they have to pay for the goods and that's out of their control as an 
individual consumer okay and they also face their income which at any point in 
time when they make a choice is fixed okay it will depend on their money income 
from employment from other sources it will depend on their savings decisions in 
the past er access to credit and and all that but th-, at the point they make 
their choice will be fixed okay and so what we've done is to break down er the 
choices into these two parts and pu-, pulled together er the parts that we 
consider most important okay now of course this model er includes and there are 
certain elements of it include an awful lot of 
factors like utility preferences will include an awful lot of things that 
influence preferences like advertising access to information your own attitudes 
and beliefs about er the product about the world whatever and those will change 
over time and they will be different between individuals every individual will 
be different and hence this will change but at the point where the consumer 
stands in the supermarket or or whatever that is fixed okay at the point at 
which they make the choice the next time they make the choice that may be 
different that's okay but at the point that they make the choice that will be 
fixed just like this is fixed and we might be interested in knowing how this 
differs and the impact of changes in this okay and that's fine so we might be 
interested in looking at well if a consumer makes a choice now and then they 
make it next week after there's been an advertising campaign whatever we might 
be interested in knowing how that has 
changed okay that may be one of the variables that we want to consider but at 
the point they make the choice standing there that is fixed okay likewise these 
may change from week to week month to month or whatever but at the point where 
they're facing a choice at any point in time those are fixed and so what we've 
done is represent the choices that the consumer makes in this this algebraic 
sense what that allows us to do is to mathematically er look at the choices 
people make we can look at the prices people face the incomes they face we can 
represent their preferences through some utility function and we look at choice 
okay that combination of the two goods in this case that the consumer's likely 
to choose and we can then use that model to look at what happens if things 
change if the price of X-one doubles what's going to happen are they going to 
stop consuming X-one altogether or do they like X-one so much that they only 
reduce their consumption by a little 
bit and so on okay so that's what we can we can do with this we can also 
represent er the optimal point okay we can consider what happens at the point 
where they actually made their choices let me er quickly put this down again 
okay so that's the point at which they make their that's the optimal point okay 
and we can consider what are what what are the conditions er which must be 
satisfied for for the consumer to be at an optimum for them to actually be ma-, 
achieving the maximum level of satisfaction they can and there are two basic 
conditions first of all they must be using all of their income okay because 
we're assuming that we've included within our framework all of the things that 
give utility and remember this is a world where people get utility from 
consumption we can include if people like lots of money in the bank we can 
include that as well because we can include savings as a source of utility 
that's that's okay but what we have done is we've included 
within their utility function within the idea of our indifference curves 
everything that gives them satisfaction and hence at the end of the day they 
must allocate all of their incomings to to those things that give them 
satisfaction including maybe savings so first of all that must be the case 
their expenditure which is this that's their total expenditure must equal their 
money income okay that's the first thing that must be er must occur so that 
means they're going to be on their budget line okay providing we've included 
everything in the in here that gives them utility they must be on their budget 
line because they don't get any utility from not allocating their income that's 
the first er er thing we would we would see so their expenditure's not going to 
be less than their income it's going to be exactly equal to it of course it 
can't be more the second is that they are at this point here okay they're at 
that point there okay that means that the slope of the budget line is 
equal to the slope of the indifference curve okay 'cause at that point there 
they're going to be exactly the same the slope of their budget line and the 
slope of their indifference curve are going to be exactly the same and that's 
the second condition okay what is the slope of the budget line equal to 
sm0761: inverse price ratio 
nm0757: sorry 
sm0761: inverse price ratio 
nm0757: okay so it's the ratio of the prices yeah so the slope of that at any 
point is equal to p-one over p-two and it's exactly constant of course it's a 
straight line the prices do not depend on how much the the er consumer chooses 
so that one's linear that's a straight line what's the slope of the 
indifference curve what's the marginal rate of substitution 
sf0762: it's the ratio of marginal utility 
nm0757: okay it's the ratio of the marginal utility so the the slope of this is 
equal to marg utility of X-one over marg utility of X-two and at the optimum 
they must be equal okay the ratio of prices must be equal to the ratio of the 
marginal utility 
okay now you you know that that is equal to the marginal rate of substitution 
okay but in fact that is what is termed the marginal rate for substitution in 
consumption what it is is the rate at which the consumer wants to substitute 
between the goods given their preferences how they feel about the goods that is 
the rate at which they are willing to trade off between them okay or willing to 
trade off X-one and X-two or whatever goods we're dealing with okay so given 
their preferences that is the rate at which they are willing to trade them off 
and at this and it this is at that point there it depends on how much of the 
two goods because we this is convex to the origin so this differs this changes 
when diminishes as we move from the top to the bottom this is also the marginal 
rate of substitution this is the marginal rate of substitution in exchange it 
is the rate at which they are able to trade off the two goods and that is 
totally determined by their relative prices so the rate at 
which you are able to trade off one good for another depends on their relative 
price if one good is er costs you know a pound and another good costs fifty 
pence clearly then you can have you know two of the goods that cost fifty pence 
or one of the goods that cost a pound and that is totally determined by the 
prices out of your control okay but that is your your marginal rate of 
substitution all consumers' marginal rate of substitution in exchange is 
determined by the market place determined by the collective decisions of the 
suppliers and of all consumers but for you as an individual it's fixed because 
you are irrelevant within the market as a consumer and so at the optimal point 
we're saying that the rate at which the consumer is able to substitute between 
the goods determined by market prices is exactly equal to the rate at which 
they want to consu-, to to substitute between them which is determined by their 
ratio of marginal utilities okay so at the 
optimum it is the case that the marginal rate of substitution in exchange is 
exactly equal to the marginal rate of substitution in consumption the rate at 
which the consumers are able to substitute between the goods through market 
transactions is exactly equal to the rate at which they wish to substitute 
between them given their preferences okay the implication of that is that if we 
we order this we change this around okay what we find is that just rejigging 
this around we find that in fact that is equal to this expression here that at 
the optimum the ratio of the marginal utilities to the unit prices of the goods 
is exactly equal for every good that is the amount of utility you get for every 
unit of money spent is exactly the same for all of the goods okay so the 
marginal utility the amount of utility you get from consuming one extra unit of 
the good the ratio of that to the price that is how much it costs you to 
consume one extra unit of the good at the optimum is 
exactly equal for all of the goods you consume so the amount of utility for one 
unit of money is the same for X-one and for X-two can anyone remember what that 
is called you've done this in part one anyone remember this is termed the 
equimarginal principle okay if you remember you did it in part one with er 
looking at cardinal theory wi-, of consumer choice you said that what the d-, 
consumer does is they allocate their the goods and we assumed then we could 
measure it but what they did was they do it in such a way that the amount of 
income sorry the amount of utility they get for each of the for for the er unit 
amount of money they spend is the same for all the goods so it's not possible 
to allocate your money between the goods and get any extra utility so therefore 
you must be at the optimum okay you cannot reallocate your income in any way 
given the price of the goods and given the your preferences for those goods and 
achieve any extra utility this is called the 
equimarginal principle which you'll have met before so we end up in the s-, t-, 
in the same position okay as with other ideas but we're not measuring utility 
now we've represented it in a far more sophisticated model that we can actually 
applo-, employ to estimate things like elasticities to predict demand change in 
demand whatever but fundamentally it's based on the principle that consumers 
given their choices are driven by preferences that are individual to them and 
given they're constrained by their economic circumstances they will allocate 
their income in such a way given their preferences that they cannot achieve any 
extra utility they can't meet any more of their needs and wants by jigging 
around their er their their allocation of their income between the goods this 
is also equal to U-M the marginal utility of their money income if the consumer 
was given a fractional increase in their money income okay at the optimum it 
wouldn't matter which good 
they bought more of their extra marginal utility would be exactly the same 
because it has been equated across the groups ratio of the marginal utilities 
to their money their money to their prices it's exactly the same so if they 
were given a fractional increase in income it wouldn't matter which good they 
allocated it to simply because the ratio of marginal utility to prices is the 
same for every one of them and so that is at the margin the marginal utility of 
their money income of money okay so what we've done is to build up a model 
which we can represent okay diagrammatically for up to three goods but we've 
built it up taking account of all of the characteristics of consumer 
preferences and of their economic constraints okay that that you know really 
matter yes sure we have put a lot of things together like the indifference 
curve represents all of those things well includes all those things that 
influence the consumer's preferences every 
one of those is included within that and any one of those if it changes will 
change that indifference curve okay that's fine we've equally included the 
economic constraints okay and what we've now done is put them together to 
produce a model which we hope will allow us to okay on the f-, on the one hand 
predict the choice this consumer will make in this circumstance which is 
vaguely interesting what is more interesting is if we then use this to say okay 
what happens if things change can we use this given that we can now represent 
it er mathematically so we can actually represent it like this so we could 
given we can represent this mathematically we could apply this to data from the 
real world price data income data et cetera we so we can actually put real data 
into this when we've put it in an empirical model a-, and predict what will 
happen and what will happen if things change so now we can do that and to prove 
that this model er really does 
sort of encapsulate the way in which at least economists see the choices 
consumer makes the consumer makes again we're not saying this is how the 
consumer does make it we're sort of abstracting from the real world and trying 
to encapsulate how they make choices in a model that we can we can use that at 
that point they will spend all their income 'cause of another black holes that 
give utility and at that point they will maximize their utility they will not 
be able to reallocate their income in such a way that they can meet more of 
their needs and wants and so the rate at which they are able to substitute 
between the goods within the marketplace through through their market 
transactions is just equal to the rate at which they wish t-, to substitute 
between those goods and that gives us the principle the-, the marginal 
principle which says that the ratio of the marginal utility to price is the 
same for all of the goods so if the consumer's given a fractional 
increase in income it wouldn't matter which of the goods they allocated it to 
'cause it would be exactly the same okay so that is the basic the basic model 
that we've built up and it encapsulates all of the things that we we have said 
we think we observe in the real world so the ideas of diminishing marginal rate 
of substitution because of changes in er marginal utilities okay er the ideas 
of non-satiation transitivity all of those are encapsulated here 'cause we've 
built up the indifference curves and therefore are encapsulated here 'cause we 
can now represent them in a mathematical sense and the availability set which 
we we've said applies to the consumer the economic constraints they face so 
we've put all that together and we've produced this rather simple model the 
complication with it comes when you want to actually employ it in practice and 
we're not going to look at that in great detail because there you're getting 
into complex statistical 
procedures econometric procedures relating to the mathematical form that this 
takes okay and how we can put it together into something we can estimate what 
we're now going to do next week is to look at how we can use this okay we've 
said this is how we can model the choice they make okay I-E remember what we're 
saying is what we can observe in the real world is the prices the income and 
what people choose okay they're the things we can see we can see what people 
buy because we can actually collect that information we could do it on an 
individual basis 'cause we could ask people we could er ask them to give us 
their till receipts from the supermarket we could follow them around the shop 
we can gather that information in many ways we can do it on a collective basis 
of all consumers by looking at how much er how many bananas Sainsbury's and 
Tesco et cetera sell or at a national account level we can gather that 
information so we can look at what people choose I-E the X-one 
and X-two we can look at prices because again we can gather that information 
what the prices are in the shops and we can gather information on income so we 
can get all the variables income prices and choice we can get all of those and 
what we are go-, what we do is to use this model to make sense of those choices 
and when we've used this model to make sense of those choices I-E we use this 
model to understand why they've made the choices they have for example to 
discover what their preferences are we can then use that to say okay how much 
do we what do we think the consumption of bananas is going to be next year if 
we expect the prices to go up ten per cent and we expect people's incomes to go 
up say three four per cent what's the demand for bananas likely to be is it 
going to be higher is it going to be lower and by how much and that's useful so 
we're going to look at what happens if things change okay and what measures can 
we produce of those changes so things like elasticities er for example okay 
let's leave it there