<p>So here $C$ is a function with the argument $Y-T$, like $C=f(Y-T)$</p>

<p>So differentiate the equation: $dY=C' * (dY-dT)$</p>

<p>The result immediately follows like in the book. </p>

<p>The reason you don't do the partial derivative is that C has only one argument - while Y and T are in there, it only takes in 1 input. </p>

CC BY-SA 3.0
2016-09-05T03:01:10.877Z
Ah, so you are saying treat Y-T as one argument (which totally makes sense, because we could just rename the difference as something else and we would have one argument). Cool! So just to clarify we could think of C' being dC/d(Y-T) then, right? Thanks for your help!
C_Dot_James
CC BY-SA 3.0
2016-09-05T03:13:02.237Z
Also, does this also imply you can distribute the "d" operator in calculus? I have never come across this before. If you know anything that explains this, that would be great.
C_Dot_James
CC BY-SA 3.0
2016-09-05T03:39:30.597Z
So you definitely do not distribute it, C'=dC/d(Y-T) , but since we are totally differentiating this, we need to chain rule out the inside arguments so I multiply by the derivative of the inside.
VCG
CC BY-SA 3.0
2016-09-05T03:56:06.257Z
I think I understand. It is really weird seeing a function that is of the form f(x-y) instead of f(x,y), and that is really throwing me off.
C_Dot_James
CC BY-SA 3.0
2016-09-05T11:48:21.730Z
Ya as you said, just think of Y-T as one thing on the inside, and totally differentiate it on the outside. But having multiple things inside 1 argument is something you do come across a decent amount in econ.
VCG
CC BY-SA 3.0
2016-09-05T15:02:42.370Z
Sweet. If you like my answer, approving it would be nice.
VCG
CC BY-SA 3.0
2016-09-05T15:41:05.963Z
For sure! (words to fill character requirement)
C_Dot_James
CC BY-SA 3.0
2016-09-05T02:28:07.590Z
2016-09-05T02:28:07.590Z
VCG
<p>It looks like I am making a calculus mistake here, and I am really banging my head against the wall.</p>

<p>Here is my work:</p>

<p>[img]<a href="https://i.sstatic.net/P6Cyl.jpg" rel="nofollow noreferrer">https://i.sstatic.net/P6Cyl.jpg</a></p>

<p>I know I should be getting $dY/dT=-(MPC)/(1-MPC)$, where $MPC=dC/dY$
If this is at all helpful here is a picture of page 295 in 7th edition of Macroeconomics By Mankiw that briefly describes the process. </p>

<p>[img]<a href="https://i.sstatic.net/gLsLJ.jpg" rel="nofollow noreferrer">https://i.sstatic.net/gLsLJ.jpg</a></p>

<p>It is confusing to me that he uses $C'$ for both terms on right-hand side when $C$ is composed of two arguments, so I decided to use Leibniz notation to see what is going on.</p>

<p>Thanks for help! 
Chris</p>

CC BY-SA 3.0
2016-09-05T01:01:09.107Z
Imgur link isn't working for me.
123
CC BY-SA 3.0
2016-09-05T01:26:24.153Z
Thanks for letting me know. I think it is fixed. I guess you have to "BBCode" link for Stackexchange.
C_Dot_James
CC BY-SA 3.0
2016-09-05T02:30:36.340Z
Is C a function so C=f(Y-T) or is C supposed to be multiplier by Y-T?
VCG
CC BY-SA 3.0
2016-09-05T02:42:28.557Z
Good question: I am convinced it is C=f(Y-T) in the text. Page 290 of Mankiw calls it a consumption function and it also says that C depends on (disposable income). Also Mankiw tends to use this type of notation. Like to model investment as a function of the interest rate he writes I=I(r).
C_Dot_James
<macroeconomics>
Deriving Simple Tax Multiplier with Calculus (Macro Ugrad)
C_Dot_James
# Deriving Simple Tax Multiplier with Calculus (Macro Ugrad)

(Asked by: username_0 on 2016-09-04)

It looks like I am making a calculus mistake here, and I am really banging my head against the wall.

Here is my work:

[img]<https://i.sstatic.net/P6Cyl.jpg>

I know I should be getting $dY/dT=-(MPC)/(1-MPC)$, where $MPC=dC/dY$
If this is at all helpful here is a picture of page 295 in 7th edition of Macroeconomics By Mankiw that briefly describes the process.

[img]<https://i.sstatic.net/gLsLJ.jpg>

It is confusing to me that he uses $C'$ for both terms on right-hand side when $C$ is composed of two arguments, so I decided to use Leibniz notation to see what is going on.

Thanks for help!
Chris

**Tags:** macroeconomics

> **Comments:**
> **username_1**: Imgur link isn't working for me.  
> **username_0**: Thanks for letting me know. I think it is fixed. I guess you have to "BBCode" link for Stackexchange.  
> **username_2**: Is C a function so C=f(Y-T) or is C supposed to be multiplier by Y-T?  
> **username_0**: Good question: I am convinced it is C=f(Y-T) in the text. Page 290 of Mankiw calls it a consumption function and it also says that C depends on (disposable income). Also Mankiw tends to use this type of notation. Like to model investment as a function of the interest rate he writes I=I(r).  

---

## 1 answers

### Answer by username_2 on 2016-09-05. Score: 0 | Accepted answer

So here $C$ is a function with the argument $Y-T$, like $C=f(Y-T)$

So differentiate the equation: $dY=C' \* (dY-dT)$

The result immediately follows like in the book.

The reason you don't do the partial derivative is that C has only one argument - while Y and T are in there, it only takes in 1 input.

> **Comments:**
> **username_0**: Ah, so you are saying treat Y-T as one argument (which totally makes sense, because we could just rename the difference as something else and we would have one argument). Cool! So just to clarify we could think of C' being dC/d(Y-T) then, right? Thanks for your help!  
> **username_0**: Also, does this also imply you can distribute the "d" operator in calculus? I have never come across this before. If you know anything that explains this, that would be great.  
> **username_2**: So you definitely do not distribute it, C'=dC/d(Y-T) , but since we are totally differentiating this, we need to chain rule out the inside arguments so I multiply by the derivative of the inside.  
> **username_0**: I think I understand. It is really weird seeing a function that is of the form f(x-y) instead of f(x,y), and that is really throwing me off.  
> **username_2**: Ya as you said, just think of Y-T as one thing on the inside, and totally differentiate it on the outside. But having multiple things inside 1 argument is something you do come across a decent amount in econ.  
> **username_2**: Sweet. If you like my answer, approving it would be nice.  
> **username_0**: For sure! (words to fill character requirement)  

---

