Okay, afternoon everyone. So hopefully
it's not news to you that there's
an exam tonight. Same format as
per usual, 30 multiple choice
questions. We're going to
go over a review of the exam content
in this class. And then from 5.30
to 7.30 in Wall, somewhere, the room's
on Brightspace. I'll be there
for two hours. I'm going over
probably practice questions and any
other questions people have. So please
feel free to attend. Otherwise, the exam's
in this building, I believe, room 200.
at IPM. So great yeah feel free to
interrupt me at any time ask questions this
is for you not me. So yeah we're going
over rationality and utility consumer optimization and
production process. So in economics
when we speak about rationality all we mean
is someone's preferences are both complete
and transitive. So that means that
people can have like not sensible preferences
but this can still be rational as long as it
satisfies completeness and transecutivity so
the key example here is like referring
less money to more as long as you're consistent
with that um and there's no cyclical
preferences that's going to be defined as
rational even though we'd all find that as a
weird preference to have so for completeness
we can take any two objects in the universe
that we care about and what we can say
depending on the binary relation either one of
three things are true, or that can all be true. So x is related to y, y is related to x, or both. So for every two
objects you need to be able to tick at
least one of those boxes, and if you
find one that you can't tick then it's
not complete. That's a counter example
and a violation. Transitivity on
the other hand says if x is related to
y and y is related to z, then X must
be related to Z. So completeness makes
sure that we have this consistent
ordering and a unique ordering of
preferences. So you can take any two objects
and you can compare them, so you can't
say I can't compare, you must be able
to compare them. And indifference
is allowed, so you can value two things
the exact same, incompleteness
is not, you can't say I
can't compare. Transitivity on the
other hand prevents what we call these cyclical
preferences and stops I guess this money pump
from getting up off the ground and a simple
example of transitivity is that if you prefer
apples at least as much as oranges and
oranges at least as much as bananas then you
must prefer apples at least as much as bananas
so given these first two claims this third
claim here must be true so if you remember
we have model preferences as binary relations
so there's two objects and we have this
relation between them. For preferences we have
what we call the weak preference, where x
is at least as good as y. We can have the
strict preference, which is x is strictly
preferred to y, and indifference where x is
indifferent to y. So here are some examples of
transitivity, some examples of things that
aren't transitive, we did this in class. And
for completeness, is taller than, is not
complete, because it doesn't take into account
the situation with two people the same
height so we went over that as well and this
is where these weak relations are really
important for completeness because it takes
into account all the situations which is simply
one person is taller than the other or
they're the same height so those are the two
situations in when we're talking about height
so at least as tall as takes into account
both those situations then we spoke about
this idea of expected value and all expected
value is at the end of the day is just an
average or what you'd expect to get on average
if you took a gamble played a lottery or
just made the decision with uncertainty so
the way you calculate expected value is simply
take each outcome and multiply it by the
probability of that outcome occurring and add it
all together so a simple example here is you
flip a coin if it lands on heads you win
five dollars if it lands on tails you lose one
dollar what do you expect to get on average
if you play this over and over and over and
over again well there's a 50 chance of heads
50 chance of tails so you take the first
outcome win five dollars multiply it by the
probability that it occurs 50 percent 0.5 then
add the second outcome which is lose one
dollar and multiply it by the probability
that occurs which is 50 as well and you end up
with two dollars so if you played this you know
um an infinite amount of times you'd expect
on on each click on average you'd earn two
dollars that's what the expected value is
saying but we don't use expected value so
much in our economics framework and the
example i gave to like um highlight this point
sorry is that a question What trouble do you more, completeness or
transitivity? I don't know if I
really understand the difference
between... Okay, great. Trying to
differentiate. Okay, I'll spend a few
extra minutes on that when we have time. So
the two car questions, one of the relations
was, is the same brand as? And the
other relation was, not the same brand as. So
these are different relations. And the
reason why I had both of these questions was
to highlight the point you can't just switch
them back and forth. you have one relation
so with completeness think about all the cars
that have ever existed there are only two
possible options if you pick two cars at
random they're either the same brand or not
the same brand there's like no other options
possible so for completeness remember
this check that we just talked about you will
either want to be able to say X is related
to Y Y is related to X or both so for the
binary relation is the same brand as this
is going to work for something like the
Toyota Corolla and the Toyota Prius. We can
say the Corolla is the same brand as the
Prius, but when you look at the other situation
where the cars aren't the same brand, you're
not going to be able to tick one of these.
So is the Toyota Corolla the same brand
as the Honda Odyssey? No. Is the Honda Odyssey
the same brand as the Toyota Corolla?
No. Are both true? No. So that's incomplete.
And just by thinking about that, not the
same brand as will also be incomplete, because
it can't account for the situation where
two cars are the same way so that's what you
need to think about with completeness you
need to think about what are the possible
options so with height either one person's
going to be taller than the other or
they're the same height there's no other possible
options so you can do both those checks
with completeness transitivity and this
is why i put that on the on the practice exam
transitivity is different from completeness
because we're saying if this is true
and this is true, then this must be true. So there's a lot of
situations where this doesn't get up off
the ground but that doesn't matter. So for
the same brand as if X is the Corolla and
the Y is the Odyssey, X, so the Toyota Corolla
is the same brand as the Honda Odyssey.
That just doesn't make any sense. So that's
not a check. We don't even get the idea of
proving transitivity off the ground. But
let's say we have the Corolla, a Toyota Prius,
and a Toyota Camry so we want to say the
Corolla is the same brand as the Prius
the Prius is the same brand as the Camry
therefore the Corolla must be the same brand
as the Camry and that's true so if you set up
any situation where this is true this
is true for the same brand then then it must
be true but and this is why I included
not the same brand as because you can see how
this will be violated so let's say X is the
the Toyota Corolla Y is the Honda Odyssey
and Z is the Prius what this means is the
Corolla is this is not the same brand as the
Odyssey that's true the Odyssey is not
the same brand as the Prius that's true
therefore the Corolla is not the same brand as
the Prius that's not true so So we can see
that the implication of transitivity here
is violated because the Corolla and the Prius
are the same brand. So that's why
transitivity doesn't work in that situation.
Similar to think about is in love
with. I think that's a really good example
of something that isn't transitivity.
It's possible that person A is in love with
B, and it's possible that B is in love
with C, but that doesn't imply that
A is in love with C. Does that make
sense? Great. Everyone on board? Alright, so we just
talked about expected value and the
reason why we don't use expected value
in our analysis is because there are
other things that people care about
when making decisions. And we drove
this point home, the St. Petersburg
paradox. So imagine a
gamble where you flip a coin as
many times as you want until it
lands on tails. And your payout is
$2 to the power of n, where tails occurs
on the nth toss. So the expected
value of this gamble, if you
remember, is actually infinite because
you can keep landing on heads forever
technically. However, when I
asked the class how much would you
be willing to pay for this, no one
said more than $10. So the expected value
is infinite, but no one wants to pay
anywhere close to that. And the reason
is people maximise utility, happiness,
and not money. So there are two
key things that expected value doesn't
take into account. The first is diminishing
marginal utility. And this is the
idea that the next dollar you earn is going
to be worth less to you than the previous
one. so as you have more and more of
something you still get happiness but
slightly less happiness than before and it's
the same idea that 500 will mean more
to someone with zero dollars than someone
with a million dollars so you can think
about it in that way and the second thing
is risk preferences we all have different
appetites to risk expected value assumes
we're all risk neutral but some people in
this class might not like risk and might
want to avoid it others might love risk and
want to actually gravitate towards it
so expected utility takes this into account
that expected value doesn't and i sorry
i don't have a slide for this but the only
difference between expected value and
expected utility is when you take each outcome
in expected utility you transform that
outcome by whatever the utility function
says so let's say the utility function equals
x squared what we would do is we take
each outcome, win $5 and before multiplying
it by the probability we transform it by
putting it in place of x, so 5 squared so
we get 25 times 0.5 and we do the same
thing here so 1 squared would be 1, so we end
up with minus 1 still it's important to
keep the minus outside of the bracket there
otherwise you'll get this being positive
and then you get the expected utility
of this which if my calculations are correct
it would be 12 I think. Someone can can check
up on that if they want but that's the
only difference. You're transforming the
outcome to account for what people's preferences
are. So if someone's risk loving it's
going to increase the expected utility
compared to the expected value and if they're
risk averse it's going to make it less than
the expected value. And finally what can
we actually do with utility functions what
can we say the utility function is just
assigning our preferences numbers so what we
can say is that one thing that has a high
number is preferred to something that has a
lower number that is all we can say we
can't make claims like um a is preferred
three times more than b was that just me my
imagination was that a noise just anyway and
And you can't say A gives you like seven
more utils than B. These cardinal statements
have no meaning. All you can do is put
them in a ranking. So like, DK, you're our
resident YouTuber, like tier lists are very
popular on YouTube. They're not saying tier
A is like seven more points than B. You're
just putting them in tiers. And it's the
same thing here. You're just putting your
preferences in tiers. So we have a bunch
of core axioms to set up expected
utility theory. We have completeness and transitivity, which
we've talked about, independence and
independence and irrelevant alternatives are very
similar. All they really say is, if you
prefer one thing to another, then making
the same change to both shouldn't change your
preference ordering. So, IARA is always easier to explain.
It's like, if you prefer X to Y, when you
have a choice between X and Y, if I give
you X, Y and Z, you shouldn't change
your preference from X to Y. So you should
still prefer extra and independence is just
the same thing with these lottery
combinations and we use an example in class think
you have an apple or a banana then you decrease
the the apple and the banana by 50% and
add 50% of a dollar to both nothing between
them is actually changed so your preference
should stay the same invariance simply
means you shouldn't be influenced by the way
things are described we are a lot this one
gets violated a lot but I think most importantly
are these auxiliary axioms because of
non-satiation and monotonicity so non
-satiation is you can't be satisfied you need more
money you need more utility there's always
a higher utility curve or indifference curve
you can get to and monotonicity is really
important because as we said before it's
rational to prefer less money to more as long
as your preferences are ordered correctly and
non-cyclical but we couldn't create
indifference curves if that was the case. So we add
these extra, I guess, implications to what
people can prefer, and monotonicity allows
us to set up this idea of more is better
than less, which is vital to the structured
indifference curves. Don't worry so much
about convexity, and we talked about diminishing
marginal utility as well. The more of
one thing you get, the less and less utility
you get each time. So there are four key
properties of indifference curves, and what
they try and tell us essentially is that
there are different possible bundles of
multiple goods that we can get and as the first
property says along the same indifference
curve these different bundles give us the
same level of pleasure so you could get 100
bananas and 20 apples or 50 of each and that
could give you the same pleasure it differs
depending on who you are and how you rank
these things the second thing is a higher curve
means high utility so as you move up
and to the right with your difference curves
you're better off and this is due to
monotonicity you always get at least more of one
thing and not lose anything else as you
move up and to the right third is indifference
curves cannot intersect as we
showed in the lecture slides and talked
about on friday for those who stuck
around this violates transitivity and
you can prove this finally they're convex
to the origin so we have this like kind
of bow shape like that if it was concave to
the origin that would look like that and
they're downward sloping and this is so we can
see that there's a diminishing marginal
rate of substitution and what the marginal
rate of substitution is is how much y do
you need to give up to get one more unit of x
and this slope is just the marginal utility
that you get from one more unit of x
divided by the marginal utility you get from
one unit of y and what this is essentially
saying is imagine you've got 10 units of y and
one unit of x. You have a lot of y, not
much x. So to get one more unit of x, you're
willing to give up a lot of y. So probably
like four units of y to get one more unit
of x. But as you keep moving along the
curve, and you get more and more x and less and
less y, you're going to be willing to give
up less y for that extra unit of x just
because of scarcity and diminishing marginal
utility. So you could have four units of x
and four units of y, and now you might only
be willing to give up 0.5 units of y, for
one more unit of X. So marginal utility and the marginal rate
of substitution is decreasing as you keep
moving down the slope. Okay, consumer optimization. So this kind of like
leads into consumer optimization because
the difference curve is like one part of
consumer optimization. If we didn't like
have any constraints, we'd just go up and
to the right forever. But that's not how
reality works. We have limited time, we have
limited money. and this creates some
scarcity and this doesn't allow us this makes
us have to choose between some amount of
x and y and the thing that does this for
us is the budget line so the budget line
when you're buying two goods is just your
income m is equal to the price of good x
times the amount of x you buy plus the price
of good y times the amount of y you buy and
you can rearrange this to get our actual equation
on the graph where y is on the left hand
side and x is on the right And as you can
see, the slope of X is minus the price of
X divided by the price of Y. This is really
important. This is one part of our consumer
optimization equation. So the budget line
is just all the possible bundles that
exhaust the income. If you're buying
on the budget line, you don't have any
income left. But the budget set is all
possible combinations of goods. Money
you have left over or money you
don't have left over. That entire triangle,
essentially. and the slope is the
market rate of substitution so the market rate
of substitution is pretty much if you
buy one less of a good how much of the other
can you buy so imagine x costs two dollars
and y costs one dollar if you have exhausted
your budget let's say and then you decide
to buy one less good of y which costs two
you've got two dollars left how much of x
can you buy if x is one dollar you've got
two dollars left you can buy two units of
x that's all that is. And we have certain
shifts in the budget line. So in income
change, if you get more or less income, the price
ratio doesn't change at all. So the slope
remains the same. You just shift parallel
outwards or inwards depending on what
happens to your income. And I should actually
mention this because it was a typo in my
lecture slides. The more income you have, if
you get more, it shifts your budget line up
and to the right. You have more choice than
you did before. But if you have less
income than before it brings it in closer to
the origin you have less choice than you have
before so hopefully that's intuitive enough
to account for my mistake and if there's
a price change what that's going to do is
let's say there's a price change in x it's
not going to affect the intercept on the
y-axis but it's going to change the intercept
on the x-axis so a price increase in x
will move the x-axis intercept on the x-axis
slope inwards towards the origin. What this
does is it changes your real income so
you have less. And if the price change is
that it decreases in x, then it's going to shift
it out. So the slope changes because the
intercept of either x or y is changing the
other remains cost. and as an example
here you can see if we had a budget of a hundred
dollars equals one dollar times x plus
five dollars times y we can calculate the
maximum you can buy of x and maximum of y
in this following way so if you buy no x
and spend it all on y then you just take the
income and divide it by the price of y 100
divided by 5 gives you 20 that's the
maximum amount of y you can buy and you can
do the same thing for x no y 100 divided
by one is 100 and the market rate of substitution
is just how much y can you buy if you
give up one unit of x so if you give up
one unit of x as you can see you have one
extra dollar but the y costs five dollars so
you can only get um one divided by five of
y so this px divided by py is the market
rate of substitution so our consumer
equilibrium we need two conditions for this to
occur the first is we want to be on the highest
possible indifference curve if we're at a
lowered curve and we can get up higher then
we're not maximizing and the second is we
need to be somewhere on the budget line if
you're not on the budget line if you remember
let's say where we're here you can consume
more of X in your budget set and remember more
is better than less so you have to be better
off and this idea is possible anywhere
within the budget set so you've got to be on
the budget line so as you can see here we've
got three indifference curves so what we have
infinite technically but let's say you're
consuming here at A you can do better by
moving to indifference curve 2 which is higher
and up to the right and consume anywhere
from b all the way to here which is in your set
and we can keep pushing it up and up and up
until we get to three this is the highest
we can go where point c on the indifference
curve is just touching the budget line it's
just touching the budget line if we moved it
up anymore we couldn't get the next indifference
curve within the budget set, so this
is the best we can do at C. And this occurs
where this line is a tangent, so just
touching the indifference curve, that is the marginal
rate of substitution, so the slope of the
indifference curve is the marginal rate
of substitution at that point, and that's going
to be the exact same as the budget line
or the market rate of substitution here. So
we covered this a fair bit in class and then
we have some simple changes so let's start
here if your income changes and if it's a
normal good if income increases the consumption
of that good will increase if it's inferior
it will decrease and vice versa and as we
know is substitutes and complements the simple
analysis is if two goods PY and PX sorry
Y and X are substitutes if the price of X
increases and their substitutes, there'll be
more consumption of Y. And if their complements
and the price of X increases, consumption
of Y will decrease. So this goes back
to what we talked about in the second
week, but it can be implied in this
analysis as well. Now, when income
changes, it just shifts the budget line,
like either outwards or inwards, the
slope remains the same. So there's only
an income effect. But when price changes,
not only does the ratio of prices change
between x and y but you're either relatively
poorer than before or relatively richer
because if you know the price of x goes from
let's say one dollar to a hundred dollars and
you only have a hundred dollars you can't buy
as much x as you did before you're gonna
have to make different trade-offs so this is
how we analyze it and break down the
substitution effect and the income effect so our
original graph is f sorry our original budget
line is f to g and as you can see the
tangential point with the indifference curve is
here at a here at a so we're consuming x zero
and y zero however what happens if the price
of x increases so if the price of x increases
as you can see it rotates the budget
line to the left and we have this new slope um
new intercept on the x-axis and now a new
slope so two things have happened here one we
have less real income than before and the
price ratio or the slope between x and y have
changed so to figure out the substitution
effect what we want to know is if real income
was the same as before just with these price
changes how would a person choose to consume
and the way we do this is we take the
new slope of the budget line and move it up
until it's tangential to the old indifference
curve remember that's where we were originally
consuming on the old indifference curve.
So if our real income remained constant, we
could still consume to have the same amount
of utility. So that's how we get Ji. It has
the same slope as FH, and here it's tangential
to the old indifference curve at B. So what
this tells us is, A to B is the
substitution effect. Holding income constant
and only changing the relative prices,
this person's going to consume less x and more
y, less x and more y. Then the second part
of this effect is the income effect and
that's moving from this ji curve all the way
back down to fh. They're the same slope so
the only thing that's changing from ji to fx
is moving it inwards so that's the loss of
real income essentially and as you can see
we decrease our x due to this income effect
but we also decrease y due to income
effect and the overall effect at our new
equilibrium c compared to a we're consuming less
x and more y so this tells us for y the
substitution effect dominates the income
effect the substitution effect increases the
amount of y we consume and the income effect
decreases the amount of y we consume but
overall we're consuming more y than before so
we say the substitution effect, dominates
the income effect. I know this
is a lot. Does anyone have any
questions? Need any clarification
on anything? Yeah. No, this is less
utility. It's down to the left. And that's
what's going to happen when you have an
increase in price. Because you're poorly
before, you just can't do as well as you
could have originally. Eric? Can you explain the
domination again? Yeah, sure. Yeah. So we can test to
see there's an income effect and also a
substitution effect. The substitution effect
is staying on the same indifference curve just
with the new slope. So the indifference
curve is this. With our old
budget line, you can see we'll
consume it at A. With our new
budget line, we move it up until
it's tangential to the old
indifference curve. And we're at this
point here. So this tells us the substitution
effect from A to B is that we consume
less X and more Y. And then the income
effect is just bringing this budget
line down until it reaches what the new
line should be. So that's just a
decrease in the shift leftwards. There's no
change in the slope. So at our new point
here where we're maximising, the
difference between B and C is purely due to
a change in income. And as you can see, there's less Y than at B. So when we
compare C and A overall, we're
like, okay, what happens at A?
What happens at C? At C, they're
consuming more Y. And the substitution
effect increases Y, but the income
effect decreases Y. So the substitution
effect must dominate the income effect
in this situation. However, for example, if, and this is in the graph, if Y was an inferior
good, if Y was an inferior good, and we
know it moves from A to B then we know
based on the idea that if your income decreases
with an inferior good consumption
increases then we would actually say both
effects would increase the consumption of Y
so you need to remember what happens if Y is
a normal good or if Y is an inferior good or
if Y is a complement or if Y is a substitute
these matter a lot then we applied
indifference curve analysis to two different
things the first is this idea of buy
one get one free and the way this
works is you have this original budget line a
to b here and currently this person is consuming
on this indifference curve one at c so they're
buying half a pizza or half a large pizza
and something out of y and by putting in
this promotion buy one get one free the the
key point here is it doesn't decrease price
by 50 percent you don't move from like this
budget line to this or something like that
and that's because for every unit up until
one you're not getting anything free so it's not
halving the price what this does is it keeps
from a to d the same slope but when you buy
at d and you get one pizza it shifts you
horizontally by how much you get free in this
case it's one pizza so it shifts everything
horizontally by one and then continues the
same slope down so it expands your budget
set if it was like buy one get five free then
at D you would move all the way out here and
then just continue down in the same way so
it expands a person's budget set which means
they might consume at a different place in
optimality as you can see here this person
will now purchase here at E so they will purchase
so they get two pieces which means they
buy one because they get the second one free
so that's what the the um is going on with
the budget line in the um buy one get one free
and finally we talked a lot about gift giving
so hopefully this is like embedded um in
your minds and the idea here is if a person's
originally consuming here at a this budget
line and you give them ten dollars as a gift
this is going to shift their budget line up
into the right so we know if they're rational
they'll get a bit more of x and a bit more
of y and then they'll consume here at c but
if you give them a cashing not a cash if
you give them an in-kind gift if you give them
a fruitcake that's worth ten dollars if
fruitcake's on our x-axis then as you can see it
moves them from a to b they don't have any
choice they just get moved to the right in
terms of their consumption so this gift isn't
bad i mean they're doing better than they
were before they're on a higher difference
curve than they were before the gift but this
is inefficient because if you gave them cash
then they could change their bundle so they'd
get up to a higher indifference curve here
on c so this is higher than here so this is
why we say um in-kind gifts are inefficient
and then obviously we talked about all the
reasons why we give gifts but that's the
the old school economic analysis and then we
spoke about gift cards. So with gift cards,
let's focus on the left at first. This
is a person's original budget line and then
you can give them a gift card that
they can only spend at one store to get
X or Y. So let's say they're only purchasing
Y at the moment, 0x. If you give them
a $10 gift card, this doesn't allow
them to actually buy any more Y.
They have to buy X with this $10 gift
card. They've already consumed the
optimal amount of Y. So what this does,
like the buy one, get one free, is it shifts
the budget set or the budget line
horizontally by how much the gift card is worth
and then just has the exact same slope all
the way down. So if you're originally
purchasing here, this person will then purchase
here, essentially. But when X is a normal
good, as you can see, it's going to
result in the efficient, optimal outcome for
the person receiving the gift. They're
able to adjust their consumption to
the gift card so they're able to consume the
highest point C. The issue occurs when
X is not a normal good. If X is an inferior
good, as you get more money, you want to get
less and less of X. So as you can see
here, if a person's originally consuming
here at A, if they get more money, so imagine
this is the full budget line here, as they
get richer, they'll consume now here at D
and purchase less X. but the gift card
constrains how much why they can actually buy
so you can see under this analysis they
still have to purchase the same amount of
eggs so a lot of stores and shops will
give out gift cards if they do sell like
inferior goods because they know it will
stop the the rot of them losing out on
sales essentially during the holiday season
okay finally we had this labor leisure
choice so there's a trade -off between leisure
and earnings how much you earn per hour
and your earnings just equaled the wage
what you get per hour times how many
hours you're able to work minus how many
leisure hours you work so it doesn't
have to be 24 the question could say
this person has 80 hours a week to
divide between leisure and labor then
the 24 will be 80. and the worker
equilibrium is just the exact same as what we
talked about before it's the tangency
of their their earning function and
their indifference curve so um I'm not
going to go too much into this example
actually, it's in the slides, but the key
idea here is there's also income and
substitution effects. So if someone's
working nine hours a day and
they get a raise, two things are
going to happen. First is the
substitution effect. This will mean that
they move more hours from leisure to work
because work is worth more now. But they're
going to be richer than before. So the income
effect is actually going to decrease the
amount of hours they work. So depending on
people's preferences, this could actually
result in them working less than they did
before, even though they have a higher wage.
So this is how the substitution income
effects can work together. Cost minimization and
other costs. So we have this production
function. You just take some amount of
capital, some amount of labor, you throw it
into this function, and it gives you some output.
And as we've seen, there are different
types of functions. So in the short run,
capital is usually fixed. Because you
can't adjust your capital, you can only
adjust your labour. So in the short run,
you're kind of constrained in how you can optimise
or minimise your costs. But in the long
run, everything is variable, so you can
make optimal decisions. We have a couple of measures of productivity. Total product
is just your total output,
given your inputs. The average product
of labour and the average product of
capital is just your output divided by how
much of each input you put in. and then
the marginal product of H1 is just how
much would your output change if you
increase labour by one unit or how much
would your output change if you increase
capital by one unit this is something
that troubled a lot of people in the
homework so how much should a manager mix
their inputs essentially and the basic idea
is you should hire workers up until
the value of the marginal product of
labour equals the wage So the marginal
product of labour is how much
output you get for increasing labour
by one unit. And the value of
the marginal product of labour is
how much can you sell those units
for, essentially. So that matters.
Essentially, how much revenue do you get
from one extra worker? And the wage is
how much does it cost to get
one extra worker? So if you're getting
more revenue than the cost of one extra worker,
then you're clearly not optimising
because you're leaving money on the table. You
can hire more workers and make more money.
And if the wage is higher than the earnings
you got from the last worker you're clearly
like not optimizing either. So we want to
be in this Goldilocks zone of the value
of the marginal product of labour equals
the wage rate and the same thing here for
capital. The value of the marginal product
of capital equals the rental rate on capital
which is just how much it costs for the
next unit of capital. So we had these
three algebraic functions. We had
linear which is just some parameter
times capital, some parameter times
labour equals output. this means
there's perfect substitution
between the two. So imagine A and
B are 1. Then it's just capital
plus labour. So it wouldn't
matter if you had 50 units of
capital or 50 units of labour, it
would equal 1. And the interesting
thing here is imagine capital costs $1 and labour costs $1 and then A is
2 and B is 1. Where would you
put your resources? if one unit of
labor one unit of capital costs the
same and you have 2k plus L as your
production function how would you divvy
up your resources yeah yeah you put all of
it in capital there's no difference in cost
so you can put one unit in labor and you
get one unit of output or you can put two,
one into capital and get two units of
output so with linear production functions
if you remember you can have these corner
solutions where you only have one of the
two things being used. The leontif production
function on the other hand is this
minimum function where you take the smallest
number in here so you're kind of constrained
by these fixed proportion there's no
substitution so as a result this one's pretty
straightforward to calculate just calculate
numbers and take out the smallest and
that's your output. Cobb-Douglas means
that there's some substitute ability
between inputs and this changes over
time. So our isoquants are our indifference
curves that have that convex
shape. This is due to like a Cobb
-Douglas type function. So our isoquants are
the indifference curves of production. So
on the same isoquant it produces the same
output. So you can have different combinations
of labour and capital and it will
produce the same output. That's what the
isoquant is telling in us and the marginal
rate of technical substitution tells
us how much of like essentially capital
do you need to give up to have one
more unit of labor to be able to produce
the same amount. So the rate of
substitution for L for K and this changes along
the curve as well exact same as diminishing
marginal utility. Then our budget line
for producers is the ISA cost where it's
your cost equals the wage rate times the
amount of labour you employ plus the
rental rate of capital multiplied by the
amount of capital you employ and if you do
the same transformation as we did before
you find that the slope of the isa
cost line is just the wage divided by the
rental rate on capital and this is just the
the the market rate of substitution between
um wage rate and and and rental rate on
capital and our cost minimization rule is we
want the marginal rate of technical substitution
equal to this slope or the the market
rate of substitution the exact same as before
do I have a I mean we have it in the no
I didn't have one we have one in the lecture
it just shows we have this same tangential
property same tangential property or
equivalently we want to maximize or minimize
with the marginal product of labor divided by
wage equals the marginal product of capital
divided by the rental rate of capital. And
the intuition here is imagine wage is $1
and rental rate is $1. You want to adjust your
inputs so the marginal products are both
equal to each other because if they don't,
you're over-investing in one and not investing
enough in the other. So if the marginal
product of capital is $20 and the
marginal product of labour is $10,
what that means is that last unit you
put into labour is giving you $10
units of output, whereas if you put it
into capital, you would have gotten $20 units
of output. So that's a misallocation of
resources if they're not equal to each
other and the the price just changes what the
calculation is essentially so that's the idea
behind it okay then we had different
cost curves and cost functions so we have
fixed cost and variable costs together that
equals the um total cost and then we have average
and marginal costs so you can see the first
thing is the average fixed cost fixed
cost is defined as something that doesn't
change with output so as you can see as output
increases the average fixed cost which is
fixed cost divided by quantity has to
decrease the numerator stays the same and the
denominator quantity is just increasing
increasing decreasing that number then as you
can see marginal costs have a special
relationship with average variable and average
total cost so as marginal cost is below the
average variable and average total cost, it
decreases it, and once it passes through,
that's the minimum point, and when it goes
above, it drags it up. This is just the
idea of averages and performance for the
next output. So, the example I gave, if
LeBron's averaging 30 points a game, and
then he drops 5 points on the next game,
this is going to decrease his average.
So the marginal cost is essentially the
cost of the next unit of output. When it's
below the average, it drags the average
down. When it's above the average, it
drags the average up. Then we talked about opportunity costs again. There were some
questions on the homework. You just
got to take into account the explicit
cost as well, like a wage foregone,
for example. And then sunk
costs, we've spoken about
that to death. So these are costs that
are already incurred and cannot be recovered
no matter what choice you make. So there's
no difference between following through or
doing something else. That cost is a cost
for both situations. Penultimately,
we talked about economies of scale
and diseconomies of scale. What economies
of scale mean, essentially, as your
output increases, the average cost
is decreasing. So as you can see,
as you have more output, the average
cost, or the long-run average
cost, is decreasing. And as we just
showed before, that's because the
marginal cost is decreasing. It's
dragging it down. So there are economies
of scale. The more you produce,
the cheaper it is. Then we hit this Q star
point with constant returns to scale.
So costs remain the same. and then in dis
-economies of scale as output increases
now marginal cost will be above the long-run
average cost curve and drags it up so
each extra unit you produce will be more
and more expensive. Finally how does the
long-run average cost curve relate to the
short-run average cost curve? So this curve
here is what we would produce if we were
acting optimally. So for example if we
wanted to produce this um much of q we would
produce here we can you know produce here
if we want it's just more expensive than
here and as you can see here with q1 q2 q3
the long run we want to produce it on the
line on the line we want to minimize but
in the short run we don't have flexibility
over everything our capital is fixed so in
the short run average cost curve um one
where k is one that means our capital is
fixed at one so what that means is there's
only one point where we actually optimise
or minimise our costs with our capital
according to the long-run average cost curve,
which is here at Q1. So if we wanted to
produce a Q2 in the short run, we can only move
our labour around, not our capital. So
we won't be at the optimal point in the
long run. We'd instead be up here. And it's only
in the long run that we can change our
capital allocation from K1 to K2 and get down
here. So as you can see with capital
equaling to 1, 2 or 3 in the short run, there's
only one point where it touches the long
run average cost curve so if you wanted to
produce at any other point in the short run
you're not actually going to be minimizing
your cost in a long run perspective and
that's just because you're constrained about
what you can change and finally if I remember
my slides correctly yeah we had these
multi-product cost functions economies of
scope simply mean if you have a factory that
produces like two goods together or if you
produce them at the same time it's actually
cheaper than having two separate factories
where you produce each good individually and
that's because you might have complementary
labor machines that you're able to use
for both goods etc and this idea of cost
complementarity is that you have two goods q1 and
q2 and if you hold q1 constant and you
don't change at all but you increase production
of q2 it's actually going to make q1 cheaper
it's going to make Q1 cheaper. And the
examples I gave were really bad, but it was
this idea of if you have a factory that
makes both steaks and handbags out of leather,
when you create a steak by killing a cow, you
also get the leather as well. So the more
steaks you create, the more leather you
have for handbags, so it's going to actually
be cheaper to make your handbags. That's
the idea of cost complementarity. And the
algebraic way to think about it is this function
here. So if you're producing q1 and q2
together you'll have some fixed costs plus the
complementarity of q1 and q2 and then q1
and q2 on their own so we take the marginal
cost of q1 so we take the derivative of q1
we end up with aq2 plus 2q1 aq2 plus 2q1 and
what this tells us is the marginal cost for
Well, good one doesn't just depend on good
one, 2Q1, but it also depends on AQ2, AQ2.
So if A is negative, as you produce more of
good two, it's going to decrease the marginal
cost of good one. So the heuristic
to keep in mind, if you see a cost
complementarity question, find A when these
two goods are multiplied together.
If A is negative, that means there's
going to be cost complementarity.
If A is not negative, there's
no complementarity. I mean, yeah, so if you
don't fully understand what's going on here,
just remember this heuristic. Look
where Q1 and Q2 are multiplied together,
where they interact, and if A is negative or
the number at the front is negative, there's
cost complementarity. All right, that's
all I have for you. Good luck for the
exam. For those of you who want to
come to a review session, I'll be
there just after 5.30. Yeah, Ivan. I said
a quick question for the cost
complementarity. You said if
you get Q1, Q2, yeah yeah as in as in this um good
question i don't want to make a definitive
claim but it doesn't matter and it doesn't
matter for this exam yeah yeah yeah yeah
why why like the second part have to keep
negative so much like that square that well
so yeah so if you yeah if you lose a dollar
yeah um it's still a loss in expected value
or utility so if you square minus one is
one it's going to be positive but then yeah
but then in math equation wise shouldn't you
include negative well this isn't this isn't
method this is um economic so it's like
you want to calculate your utility of
winning five dollars or losing a dollar so like
you don't get positive utility from losing a
dollar so that's why you need the minus on
the outside no worries hey DK what's up yeah
I'm good I have a question so the marginal
like like the name so like this line
right here yeah it's just marginal rate of
market right so both both yeah so both the
lines I mean are the market rate of substitution
and this is just a budget line and this
is the ISO cost line okay and then this
one is also the market rate and then the
slope is this it's like how much um if if let's
say labor costs two dollars and capital
costs one dollar if you reduce your labor by
one you'd have two dollars to spend on
on on capital so that that's what that trade
-off is telling you like if you're like at your
maximum cost and you reduce labor by one
how much can you spend on capital So it's
the exact same thing. How much can you
substitute between two goods with
the market prices? Okay, and then this
is the ISOQANT. ISOQANT, and then
that's the market rate of technical substitution,
or is that... Yeah, so, yeah,
exactly. So market rate of technical...
So that's telling you how much do you
have to change capital given your change
in labor by one unit to produce the same
amount of output. Okay, and then does this one have a name as well? Yeah, that's just
the marginal rate of substitution. Marginal,
okay, okay. yeah that's what i was confused
about just yeah just like those i know the
names are so bad and like they're also
because like this is like the same like if you
like right like this is this is like mrs
is mrs that's yeah i know i know it's a
nightmare it's an absolute nightmare i apologize
yeah yeah i got it all right cool see
you dk hey how's it going is it possible
to have something to be complete but not
translated yeah um the um i'm trying to think
what's a good example what was the question
is it possible yeah it's possible i'm just
trying to think nothing's coming to mind at
the moment i might have given an example
but yeah it can't be you know we can use google
here yeah yeah yes um what does
gemini say grayson it says uh rock
paper scissors it's oh yeah it's
cyclical yeah yeah it's that's example of a
complete yeah yeah so so rock paper scissors
better than um is is complete but it's not
um transitive because rock is better than paper
um so paper's better than rock rock is
better than scissors and by transitivity not
it doesn't have to be complete i don't think
oh no it would be yeah it would bit
yeah so intransitive preferences are possible
um uh defeats um like draws or defeats in in
sports would probably be complete and not
transitive as well yeah