- [Instructor] We're told Martha has a fixed budget of $20,
and she spends it all on two goods, good X and good Y.
The price of X is $4 per unit,
and the price of Y is $2 per unit.
The table below shows the total benefit,
measured in dollars, Martha receives
from the consumption of each good.
All right, we see that here, this is total benefit,
not marginal benefit.
What is Martha's marginal benefit
of the fifth unit of good X?
So just to answer this question, let's see,
she has a total benefit of $40 when she has four of X.
And then when she goes to the fifth,
her total benefit is an incremental $1.
So she goes from $40 to $41.
The marginal benefit of that fifth one is that extra dollar.
So we added a dollar of total benefit,
so that's the marginal benefit.
So it is $1.
Calculate the total consumer surplus
if Martha consumes five units of X.
Show your work.
Well, the consumer surplus is going to be the benefit,
benefit
minus the cost, which is going to be equal to,
well, when she has five units of X,
her total benefit is $41.
So I'll write that here, $41.
And then what's her cost of five units of X?
Well, X costs $4 per unit.
So five times four is $20, so her cost is going to be $20.
So her consumer surplus
is going to be equal to $21.
Martha is currently consuming four units of X
and two units of Y.
Use marginal analysis to explain
why this combination is not optimal for Martha.
So pause this video, and see if you can answer that.
All right, well, let's just think about
what the marginal benefit is
from every incremental unit of X or Y,
and then let's think about the marginal benefit per dollar.
So I'm gonna make an extra column here.
Let's call this marginal benefit of X,
and then let's call this marginal benefit of Y.
I'm doing it over this table just for the sake of space.
And so the marginal benefit
of this first one is going to be 16.
We went from zero to 16.
The second one, we go from 16 to 28, so it's $12.
And then to go from 28 to 36 is eight.
To go from 36 to 40 is four extra dollars of benefit.
And to go from 40 to 41, we already talked about that,
that's $1 of marginal benefit.
If we talk about Y, well, the first unit,
you get $10 of benefit.
The next one is the total benefit increase eight, by $8,
so that's the marginal benefit of the next unit.
The next one, to go from 18 to 24, six,
to go from 24 to 28 is four more,
and then 28 to 30 is two more.
And then we could use this information to think about
marginal benefit of X
per price of X,
and we could also think about marginal benefit of Y
per price of Y.
And so, let's see,
and we're gonna start with the first units.
And so for this first unit,
if you take the marginal benefit of X
divided by the cost of a unit of X,
$16 divided by $4 is going to get four.
12 divided by four is three.
Eight divided four is two.
Four divided by four is one.
And one divided by four is 0.25.
And then for Y, the cost of Y is $2 per unit.
So the marginal benefit per dollar
of this first unit right over here is $10 divided by $2,
which is five.
Eight divided by two is four.
Six divided by two is three.
Four divided two is two.
And then two divided by two is one.
And so let's just think about
what would be an optimal combination for Martha.
When she's thinking about spending her first few dollars,
she'll get a higher marginal benefit per dollar
by going with Y, so she's going to start here.
And then after that, her second unit of Y has the same
marginal benefit per dollar as her first unit of X.
So she's indifferent.
She would do these in some order.
So maybe she could do this one
and then move on to that one,
or, though, it could go in the other order.
And so far, she's only spent,
let's see, $2, $2 is $4 plus another $4,
she's only spent $8, and she has a budget of 20.
And then after that, her next incremental
unit of either X or Y,
the marginal benefit per dollar is the same.
So just thinking about whether this
could be an optimal combination,
she's already bought two units of Y.
Let's just give the benefit of the doubt here.
Let's say that she goes for the X,
so she buys this one here.
But once she has two units of both,
and she hasn't spent all of her money,
she spent $8 here plus another $4.
She has $8 to spend.
The next rational thing for her to do,
her marginal benefit per dollar for that third incremental Y
is higher than the third incremental X.
So it would be optimal for her to buy a third Y.
But here, we see that she only has two units of Y,
so that's why we know it's not an optimal combination.
So we could say once
she has
two of each,
the
marginal benefit of Y
per price of Y
is greater than the marginal benefit of X
per price of X for
the third unit,
so she will buy
more than two Ys,
let me write, let me scroll down a little bit, buy more than
two Ys.
All right, the next they say is
what is Martha's optimal combination of goods of X and Y?
Well, we've already started that conversation up here.
She would buy this Y, and so far she spent $8 plus $6,
so she has another $6 to spend.
And then now her next incremental unit, she's indifferent,
so maybe she buys another Y.
$8 plus $8, this is $16, so she has $4 left.
And so then she would buy this,
and she has spent all of her money, $12 here, $8 here.
So she would buy three Xs and four Ys.
So I would say three Xs
and four Ys.
All right part e, indicate whether each of the following
will cause the optimal quantity of good Y
to increase, decrease, or stay the same.
So look at these and pause this video,
and see if you can answer those.
So the price of good Y doubles.
Well, if the price of good Y doubles,
then the marginal benefit per price of Y will go down.
So she will buy less of Y, so it would decrease.
She would get less bang for her buck on Y,
so she would buy less of Y.
Martha's income falls to $10 with no price changes.
Well, if we go through the exercise we just did,
her budget would run out much faster,
and so she would definitely decrease
the number of Ys she would buy.
So the Ys would decrease.
Martha's income doubles, and the price of both goods double.
Well, in that case, things would stay the same,
stay the same.
Because once again,
she could buy that exact same combination.
It would just cost twice as much,
but then her budget is now twice as much.
So things would stay the same.
She would buy the same quantities of both Xs and Ys,
and they're just asking about Ys.
And we're done.