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### Course: Microeconomics > Unit 5

Lesson 2: Utility maximization using marginal utility per dollar spent# Marginal utility free response example

In this video, walk through the solution to a question on the 2012 AP Microeconomics exam applying the concepts of marginal utility and utility maximization.

## Want to join the conversation?

- I thought total utility was the culmination of all individual units of utility because the marginal utility shows the incremental increase/decrease. So why isnt it 10+18+24=32?(3 votes)
- From the author:You are right that total utility is the sum of all units of utility, but I think you are misunderstanding marginal utility here. When she consumes one train, she gets 10 units from that. When she increases consumption to two trains, she gets 8 units from the second train, not 18. Her
**total**utility from two trains is 18 because she gets 10 from the first, and an additional 8 from the second. Similarly, when she increases her consumption to three trains, she gets 10 from the first, 8 from the second, and 6 from the third.

It might be helpful to think of it incrementally. For example, "ok, I am consuming one toy train. What happens to my total utility if I increase to two trains? Well, the second train will provide me with 8 more utils, so that brings me up to a total of 18."(10 votes)

- When Sal comes to a point where the marginal utility per dollar of both goods are equal, he says that you can choose either at random. I may be overthinking this, but because a bagel is worth $2 (and I'm assuming you can't buy half a bagel), shouldn't you take the next 2 dollars into consideration? For example, when he gets to the point where marginal utility per dollar of both equals 4, he says that you could get either the toy car or the bagel. But if you choose the bagel, you get 4 utils per dollar for your next 2 dollars, whereas for the toy car, you would get 4 utils only for the first dollar, and 3 utils for the next one. So your marginal utility for the next 2 dollars would be 8 for the bagel, but only 7 for the toy car. In this situation, wouldn't you choose the bagel?

The final answer comes out the same, so I'm wondering if this matters at all, or if I'm just overcomplicating things for myself.(7 votes) - I understand how Sal figured out that Theresa spent 5$ on toys with the table, but I need someone to breakdown how he used the table to conclude she spent 6$ on Bagels.(3 votes)
- When Theresa actually buys a bagel she spends $2 so 2x3(bagels)=$6. The table just shows the MU per $1 (basically the MU per half a bagel).(1 vote)

- What happens when Theresa's income can't be split perfectly by the price of two products? For example, if she has $12, what will she do about that extra $1? Does she buy one more toy (affordable but has a lower MU/price than the MU/price of the next bagel)? Or does she saves the money?(3 votes)
- Theresa consumes toy cars? How come she doesn't have a tummy ache?

I get the bagel part, but who would eat CARS!?(2 votes) - In second question, if Theresa had a weekly income of $13, then she would have bought 5 toy cars and 8 bagels. In that situation MU per unit price would not be equal for the two. How to understand it?(2 votes)
- I thought it was asking for total utility. If MU= change in TU/change in Q, then wouldn't you use that to find TU, instead of simply adding the MUs(1 vote)
- You could make a different table for TU, then your answer would be the value corresponding to the number of toy cars.

When she consumes one toy car, she gets 10 units from that. When she increases consumption to two toy cars, she gets 8 units from the second toy, not 18. Her total utility from two toys is 18 because she gets 10 from the first, and an additional 8 from the second.

So you see, you can do it both ways.(1 vote)

- Why isn't the answer to this question 4 bagels and 3 cars?(1 vote)
- I think the confusion here is looking at total utility vs the (marginal) utility per dollar, which is what is used when calculating maximized utility given a certain budget, in this case $11. When we consider the price of bagels ($2), the utility per dollar is half of what their total utility is. Sal shows this new table at about4:56.(1 vote)

- If lets say the question doesn't give a number that works out like $11, and she has left over money (because let's say if she buys the item that gives her greatest marginal benefit makes her go over her budget, will she just not spend the money, or will she use that money to buy an item even if it doesn't give her the most (MU/P)/Bang for her buck. Thanks(1 vote)
- How can we incorporate the remaining budget and the possible choices? i.e. how can we explain the following?

A @2$: mu/p -> 7 - 6

B @1$: mu/p -> 7 - 5

budget=2$

in all of the videos, we never incorporated the remaining budget and the possible choices when mu/p is the same.(1 vote)

## Video transcript

- [Instructor] We are
told that Theresa consumes both bagels and toy cars and they tell us that the table above shows
Theresa's marginal utility from bagels and toy cars. And the first question is,
what is her total utility from purchasing three toy cars? So pause this video and
see if you can answer that. All right now lets work
through this together. So let's just make sure we
understand this table here. So this says that the first
bagel that Theresa consumes, she gets eight units of
marginal utility from that. Then the second bagel, she gets a little bit
less marginal utility. Some or her bagel craving
has already been satisfied by that first one. And then the third bagel,
the marginal utility goes down a little bit and
then that keeps happening for each incremental bagel. And on the toy side, we
see that that first toy, she gets a lotta marginal utility, 10, and then the next toy
gets a little bit less, and then you see that the
marginal utility for each incremental toy gets a
little bit lower and lower. So now let's answer the first question. What is her total utility from
purchasing three toy cars? Well that first toy car,
she gets utility of 10. Then that second toy car,
she gets a utility of eight. And then that third toy
car, her marginal utility for that incremental car is six. So the total utility is going
to be 10 plus eight plus six, which is what, 10 plus 14,
this is going to be equal to 24 units of marginal utility. All right now let's do part two. Theresa's weekly income is $11, the price of a bagel is $2, and the price of a toy car is $1. What quantity of bagels
and toy cars will maximize Theresa's utility if she
spends her entire weekly income on bagels and toy cars? Explain your answer
using marginal analysis. So once again, pause
this video and see if you can figure that out. All right now let's do this together. I'll scroll down a little
bit so I have some space. So the key thing is, is
when once we know the price of a bagel and the price
of a toy and we know the marginal utility for every
incremental bagel or toy, we can figure out our bang for our buck. We can figure out, what is
going to be the marginal utility per dollar from that incremental bagel and that incremental toy car? And so we can, let's just explain first. So Theresa, Theresa will maximize, maximize her marginal utility per incremental dollar, per dollar, or let me put it this way, per dollar, when making purchases on the margin, making maybe, making purchases on the margin. So her next incremental purchase, her next incremental, incremental purchase. So we can write over here bagel marginal utility per dollar. That'll be one row here. And then we could write car or toy marginal utility per dollar. And then we set up these
rows right over here. And then we could think about
it if she for the first one, for the second one, for the third one, for the fourth one, let's see we go up to six, fifth one, and then we go to our sixth one. So let's start with bagels. Bagels cost $2. So that first bagel, if
she gets eight marginal utility units, well that's going to be four marginal utility units per dollar, eight divided by two. So that's four units per dollar,
and then that second bagel, she gets seven units but it costs $2, so it's seven divided
by two units per dollar. So that's going to be 3.5. And then six divided by two is three, five divided by two is 2.5, four divided by two is two. So that fifth bagel, two marginal
utility units per dollar, and then that sixth bagel,
three divided by two is 1.5 marginal utility units per dollar. And now we can think about toys. Each toy is $1. So if she gets 10 marginal utility units from that first toy, it only cost her $1, so it's 10 utility units per dollar. So it's 10 there. It would be eight here. We're just dividing each of these by one. So six here, so this is gonna
be the same number again. Four, three, and two. So now that we set this
up, and let me scroll down a little bit so I have
a little bit more space. I have all the data I need. We can think about what
would be rational for her if we're thinking about
how she's gonna spend that $11 per week. Her first purchase, she's like wow, from the get go, that if I'm
picking between bagels and toys that first toy has a much
higher marginal utility per dollar than that first bagel. So she's going to start
here, and then she says, okay next do I wanna buy a bagel or a toy? But even that second toy, the
marginal utility per dollar is still higher than that first bagel. So then she'll buy a second toy. Then she'll think about it and
so far she's only spent $2, so we have a lot of money still left. Then she'll think about
okay, do I wanna spend that next incremental
amount on a toy or bagel? Well still, she gets more
marginal utility per dollar from the toy, so she'll spend that. She's spent $3 so far, $1, $2, $3. And now, when she thinks
about how to spend her next few dollars, she
says, well know I'm indifferent between bagels and toys. The marginal utility
per dollar is the same. So she might maybe spend
the next one on a toy and then right after that,
she'll go to bagels finally and buy a bagel. Well let's think about how
much she has spent so far. She's spent $4 on toys and $2 on bagels. So the order might look
something like this, and then she goes and
maybe buys her bagel. And now the marginal utility per dollar for that incremental
bagel is higher than for her next toy, and so
then she'll probably buy, she would buy another bagel right here, and let's see how much
money she has spent. Two bagels are $4 plus
she's spent $4 on toys because they're $1 each, so it's $8. So she still has $3 to spend. Now, her marginal utility
per dollar is neutral between bagels and toys,
between the incremental, the third bagel and that fifth toy, so she's indifferent between the two. So she could probably get both of them. So she might do something
like that, buy that, and then she could buy
that or she could do that in the other order. And let's see how much money she's spent. She has spent $5 on toys and she has spent $6 on bagels, and so she has spent her $11. And so to answer the first question, so she would buy, she would buy five toys, five toys, and three bagels, and three bagels based on this strategy of maximizing marginal utility per dollar for each incremental,
incremental purchase. Did I answer all of the questions? We said the quantity of bagels and toys that will maximize her marginal utility if she spends her weekly income,
and then we have explained using marginal analysis. Yep, we're looking good.