<blockquote>
  <p>$\frac{dy}{dx}$ is written as a function of a single variable right?</p>
</blockquote>

<p>I don't think so. Your shorthand notation should be standing for</p>

<p>$$\frac{dy}{dx} = -\frac{U_x(x, y)}{U_y(x,y} $$</p>

<p>and hence is a function of both variables ${x,y}$. That makes sense, as your MRS could depend on the locus you're at.</p>

CC BY-SA 3.0
2015-04-10T13:48:18.550Z
Okay. I buy what you are saying about the RHS being multivariable. But the LHS is clearly a single variable notation.....right?
Stan Shunpike
CC BY-SA 3.0
2015-04-10T13:58:26.640Z
@StanShunpike yes, the shorthand  is abuse of notation. When introduced, one should be clear that.
FooBar
CC BY-SA 3.0
2015-04-10T14:00:49.153Z
But how can we abuse it like than and still consider them differentials? Like that just seems like it wouldn't make any sense. Is there a way to write it that doesn't involve this abuse of notation?
Stan Shunpike
CC BY-SA 3.0
2015-04-10T12:22:50.503Z
2015-04-10T12:22:50.503Z
FooBar
<p>We have the <em>function</em></p>

<p>$$ U = U(x,y)$$</p>

<p>from which we create the <em>equation</em></p>

<p>$$x^*,y^*:  U(x^*,y^*) = \bar U$$</p>

<p>with $\bar U$ being fixed a constant, and so $d\bar U = 0$.</p>

<p>Then</p>

<p>$$dU(x^*,y^*) = d\bar U = 0 \implies U_x(x^*,y^*)dx^*+U_y(x^*,y^*)dy^* = 0$$</p>

<p>$$\implies \frac {dy^*}{dx^*} = -\frac {U_x(x^*,y^*)}{U_y(x^*,y^*)}$$</p>

<p>Leaving aside the (mathematical) "Civil War of differentials", there is no abuse of notation here, only simplification for reasons of compactness. Why? $x^*$ and $y^*$ are <em>solutions to the equation</em>, not variables free to vary independently.So in reality we have $y^* = h(x^*,\bar U)$ and</p>

<p>$$\implies \frac {dh(x^*,\bar U)}{dx^*} = -\frac {U_x(x^*,h(x^*,\bar U))}{U_y(x^*,h(x^*,\bar U))}$$</p>

<p><strong>NOTE:</strong> these kinds of notation simplifications appear in all disciplines that use mathematics. They are not always beneficial though: for example M. Caputo in <a href="https://books.google.gr/books?id=XZ2yYSVKWJkC&amp;pg=PA582&amp;lpg=PA582&amp;dq=Caputo%20dynamic%20optimization%20book&amp;source=bl&amp;ots=FpgQhEcvm0&amp;sig=8l4D_Ln0ESLzY9MeIoiZytaOcLE&amp;hl=en&amp;sa=X&amp;ei=QQdrVbfuFMelsgGvlYP4Bw&amp;ved=0CD8Q6AEwBg#v=onepage&amp;q=Caputo%20dynamic%20optimization%20book&amp;f=false" rel="nofollow">Foundations of Dynamic Economic Analysis</a> more than most writes explicitly all dependencies. This makes notation cumbersome and confusing at first, but once you get the hang of it, you are never lost.</p>

CC BY-SA 3.0
2015-05-31T13:10:51.470Z
2015-05-31T13:20:05.367Z
2015-05-31T13:20:05.367Z
Alecos Papadopoulos
<p>Suppose I have $U(x,y)$ and a level set of indifference curves. Suppose the value of $U$ along a given curve is $\bar{U}$. We know $dU = 0$. We compute total derivative, rearrange, and now have </p>

<p>$$\frac{dy}{dx} = -\frac{U_x}{U_y}$$ </p>

<p><strong>My Confusion and Question:</strong></p>

<p>$\frac{dy}{dx}$ is written as a function of a single variable right? But when we talk about $MU$, we always talk about it as a multivariable function. I thought $MRS$ was the ordinal utility analog of the cardinal utility $MU$. So why is $MRS$ a single variable function? </p>

<microeconomics><consumer-theory>
Is Marginal Rate of Substitution a multivariable function?
Stan Shunpike
# Is Marginal Rate of Substitution a multivariable function?

(Asked by: username_0 on 2015-04-10)

Suppose I have $U(x,y)$ and a level set of indifference curves. Suppose the value of $U$ along a given curve is $\bar{U}$. We know $dU = 0$. We compute total derivative, rearrange, and now have

$$\frac{dy}{dx} = -\frac{U\_x}{U\_y}$$

**My Confusion and Question:**

$\frac{dy}{dx}$ is written as a function of a single variable right? But when we talk about $MU$, we always talk about it as a multivariable function. I thought $MRS$ was the ordinal utility analog of the cardinal utility $MU$. So why is $MRS$ a single variable function?

**Tags:** microeconomics, consumer-theory

---

## 2 answers

### Answer by username_1 on 2015-04-10. Score: 1

> $\frac{dy}{dx}$ is written as a function of a single variable right?

I don't think so. Your shorthand notation should be standing for

$$\frac{dy}{dx} = -\frac{U\_x(x, y)}{U\_y(x,y} $$

and hence is a function of both variables ${x,y}$. That makes sense, as your MRS could depend on the locus you're at.

> **Comments:**
> **username_0**: Okay. I buy what you are saying about the RHS being multivariable. But the LHS is clearly a single variable notation.....right?  
> **username_1**: @StanShunpike yes, the shorthand  is abuse of notation. When introduced, one should be clear that.  
> **username_0**: But how can we abuse it like than and still consider them differentials? Like that just seems like it wouldn't make any sense. Is there a way to write it that doesn't involve this abuse of notation?  

---

### Answer by username_2 on 2015-05-31. Score: 1

We have the *function*

$$ U = U(x,y)$$

from which we create the *equation*

$$x^\*,y^\*: U(x^\*,y^\*) = \bar U$$

with $\bar U$ being fixed a constant, and so $d\bar U = 0$.

Then

$$dU(x^\*,y^\*) = d\bar U = 0 \implies U\_x(x^\*,y^\*)dx^\*+U\_y(x^\*,y^\*)dy^\* = 0$$

$$\implies \frac {dy^\*}{dx^\*} = -\frac {U\_x(x^\*,y^\*)}{U\_y(x^\*,y^\*)}$$

Leaving aside the (mathematical) "Civil War of differentials", there is no abuse of notation here, only simplification for reasons of compactness. Why? $x^\*$ and $y^\*$ are *solutions to the equation*, not variables free to vary independently.So in reality we have $y^\* = h(x^\*,\bar U)$ and

$$\implies \frac {dh(x^\*,\bar U)}{dx^\*} = -\frac {U\_x(x^\*,h(x^\*,\bar U))}{U\_y(x^\*,h(x^\*,\bar U))}$$

**NOTE:** these kinds of notation simplifications appear in all disciplines that use mathematics. They are not always beneficial though: for example M. Caputo in [Foundations of Dynamic Economic Analysis](https://books.google.gr/books?id=XZ2yYSVKWJkC&pg=PA582&lpg=PA582&dq=Caputo%20dynamic%20optimization%20book&source=bl&ots=FpgQhEcvm0&sig=8l4D_Ln0ESLzY9MeIoiZytaOcLE&hl=en&sa=X&ei=QQdrVbfuFMelsgGvlYP4Bw&ved=0CD8Q6AEwBg#v=onepage&q=Caputo%20dynamic%20optimization%20book&f=false) more than most writes explicitly all dependencies. This makes notation cumbersome and confusing at first, but once you get the hang of it, you are never lost.


---

