<p>The idea here is that <span class="math-container">$m$</span> stands for money, which has some value from the possibility of spending it on other stuff.</p>
<p>Under the implicit assumption that <span class="math-container">$B(0)=0$</span>, that your instructor seems to make,
we get the equality <span class="math-container">$B(0)+m=B(x)+0$</span> if the consumer is indifferent between <span class="math-container">$0$</span> amount of the first good and <span class="math-container">$m$</span> amount of money, or <span class="math-container">$x$</span> amount of the first good and no money. Then <span class="math-container">$m=B(x)$</span>, so <span class="math-container">$B(x)$</span> is the amount of money that would compensate the consumer for not having the <span class="math-container">$x$</span> amount of the first good.</p>
<p>Money is here not referring to the total expenditure and the argument is not related to what a consumer would actually buy.</p>

CC BY-SA 4.0
2023-05-10T11:14:04.943Z
You said "() is the amount of money that would compensate the consumer for not having the  amount of the first good" But the professor gave the interpretation for $U(x,m)$. The statement "It's just the total amount of dollars that you would have to receive to be indifferent between consuming x and m and not consuming any x" is with reference to $U(x,m)$. Here's the video of the lecture for reference. At 8:40 he makes the statement. https://www.youtube.com/watch?v=WztFR_ESFrI
Adam Sorolla
CC BY-SA 4.0
2023-05-10T11:35:23.143Z
@AdamSorolla, yes, that's what he said, but it is obviously just an unfortunate (i.e., wrong) expression of what he most likely wanted to say, namely "It's just the total amount of dollars that you would have to receive to be indifferent between consuming x and m and *neither consuming any x nor any m*."
VARulle
CC BY-SA 4.0
2023-05-10T11:42:52.683Z
@AdamSorolla, PS: In the video, the speaker also provides an incorrect interpretation of $B(x)$ at 5:20. Since he explicitly mentions twice that $B(0)$ need *not* equal $0$, the WTP for $x$ units of the good is not $B(x)$, but $B(x)-B(0)$. The interpretation in his "Remark 1" is only valid for the normalization $B(0)=0$.
VARulle
CC BY-SA 4.0
2023-05-10T12:00:42.233Z
@VARulle Ah. That makes a whole lot more sense - "It's just the total amount of dollars that you would have to receive to be indifferent between consuming x and m and neither consuming any x nor any m" Thank you. The second point you made was actually listed as a correction for the lecture (i.e the B(x) interpretation only being valid for B(0) = 0)
Adam Sorolla
CC BY-SA 4.0
2023-05-10T12:14:20.160Z
@VARulle The interpretation I've taken for the second point is, for B(0)=0, B(x) is the maximum amount a consumer would pay for x units of good x. I reasoned as follows: If the total money a consumer has is split into two parts, one going towards m and the other towards buying good x, this only makes sense if B(x) >= just keeping that money and putting it towards m. So the limiting case would be where the benefit B(x) is equal to the money put to obtaining the good. So this is the maximum price, or "willingness to pay for good x"
Adam Sorolla
CC BY-SA 4.0
2023-05-10T12:54:06.677Z
@AdamSorolla, yes, the "WTP for consuming $x$ units of the good" is by definition the maximum of the set of prices such that you prefer buying to not buying, i.e. $WTP(x)=\max\{p:\, U(x,m-p)\ge U(0,m)\} = \max\{p:\, p\le B(x)-B(0)\}=B(x)-B(0)$. Thus, if you normalize $B(0)=0$, then $B(x)=WTP(x)$.
VARulle
CC BY-SA 4.0
2023-05-10T09:36:15.163Z
2023-05-10T09:36:15.163Z
Michael Greinecker
<p>I'm going through this principles of economics course, and at one point the lecturer gives the following interpretation of <span class="math-container">$U(x,m)$</span> where <span class="math-container">$U(x,m) = B(x)+m$</span> : &quot;It's just the total amount of dollars that you would have to receive to be indifferent between consuming x and m and not consuming any x&quot;, where m is a measure of all the other consumption in dollars.</p>
<p>I don't get this. Why wouldn't someone just have to give me the difference between B(x) and the the money needed to buy x units of the good. This would make up for the additional benefit I'm getting from purchasing B(x)</p>

<microeconomics>
Confused about this interpretation of U for.a quasilinear utility function
Adam Sorolla
# Confused about this interpretation of U for.a quasilinear utility function

(Asked by: username_0 on 2023-05-10)

I'm going through this principles of economics course, and at one point the lecturer gives the following interpretation of $U(x,m)$ where $U(x,m) = B(x)+m$ : "It's just the total amount of dollars that you would have to receive to be indifferent between consuming x and m and not consuming any x", where m is a measure of all the other consumption in dollars.

I don't get this. Why wouldn't someone just have to give me the difference between B(x) and the the money needed to buy x units of the good. This would make up for the additional benefit I'm getting from purchasing B(x)

**Tags:** microeconomics

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## 1 answers

### Answer by username_1 on 2023-05-10. Score: 1 | Accepted answer

The idea here is that $m$ stands for money, which has some value from the possibility of spending it on other stuff.

Under the implicit assumption that $B(0)=0$, that your instructor seems to make,
we get the equality $B(0)+m=B(x)+0$ if the consumer is indifferent between $0$ amount of the first good and $m$ amount of money, or $x$ amount of the first good and no money. Then $m=B(x)$, so $B(x)$ is the amount of money that would compensate the consumer for not having the $x$ amount of the first good.

Money is here not referring to the total expenditure and the argument is not related to what a consumer would actually buy.

> **Comments:**
> **username_0**: You said "() is the amount of money that would compensate the consumer for not having the  amount of the first good" But the professor gave the interpretation for $U(x,m)$. The statement "It's just the total amount of dollars that you would have to receive to be indifferent between consuming x and m and not consuming any x" is with reference to $U(x,m)$. Here's the video of the lecture for reference. At 8:40 he makes the statement. https://www.youtube.com/watch?v=WztFR_ESFrI  
> **username_2**: @AdamSorolla, yes, that's what he said, but it is obviously just an unfortunate (i.e., wrong) expression of what he most likely wanted to say, namely "It's just the total amount of dollars that you would have to receive to be indifferent between consuming x and m and *neither consuming any x nor any m*."  
> **username_2**: @AdamSorolla, PS: In the video, the speaker also provides an incorrect interpretation of $B(x)$ at 5:20. Since he explicitly mentions twice that $B(0)$ need *not* equal $0$, the WTP for $x$ units of the good is not $B(x)$, but $B(x)-B(0)$. The interpretation in his "Remark 1" is only valid for the normalization $B(0)=0$.  
> **username_0**: @VARulle Ah. That makes a whole lot more sense - "It's just the total amount of dollars that you would have to receive to be indifferent between consuming x and m and neither consuming any x nor any m" Thank you. The second point you made was actually listed as a correction for the lecture (i.e the B(x) interpretation only being valid for B(0) = 0)  
> **username_0**: @VARulle The interpretation I've taken for the second point is, for B(0)=0, B(x) is the maximum amount a consumer would pay for x units of good x. I reasoned as follows: If the total money a consumer has is split into two parts, one going towards m and the other towards buying good x, this only makes sense if B(x) >= just keeping that money and putting it towards m. So the limiting case would be where the benefit B(x) is equal to the money put to obtaining the good. So this is the maximum price, or "willingness to pay for good x"  
> **username_2**: @AdamSorolla, yes, the "WTP for consuming $x$ units of the good" is by definition the maximum of the set of prices such that you prefer buying to not buying, i.e. $WTP(x)=\max\{p:\, U(x,m-p)\ge U(0,m)\} = \max\{p:\, p\le B(x)-B(0)\}=B(x)-B(0)$. Thus, if you normalize $B(0)=0$, then $B(x)=WTP(x)$.  

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