<p>The mathematical concepts we have to recall in order to answer your question  are that of <strong>level curves of a function</strong> and the so-called <strong>theorem of</strong> <strong>implicit functions</strong>, a very important theorem of mathematical analysis. To explain them in short  words isn't an easy task, I hope I can succeed in explaining it a little.</p>
<p>I  give below a short synthesis of these mathematical concept, to make clearer what I will say.</p>
<p><strong>Level curves of a function</strong></p>
<p>Through level curves  we  pass from a representation in three dimensions of a function to a representation in two dimensions: we will see that this  is the case of the usual representation of the indifference curves in the plane.</p>
<p>Consider a function <span class="math-container">$f$</span> of two real variables, <span class="math-container">$x_1,x_2$</span>  which assumes values in <span class="math-container">$R$</span>:</p>
<p><span class="math-container">$y=f(x_1,x_2)$</span></p>
<p>We want to know the pairs <span class="math-container">$x_1,x_2$</span> that make the <span class="math-container">$f$</span> assume a specific value, say <span class="math-container">$c$</span>, in <span class="math-container">$R$</span>. Formally we can write:</p>
<p><span class="math-container">$f(x_1,x_2)=c$</span></p>
<p>This last equation  describes all the pairs <span class="math-container">$(x_1,x_2)$</span>
that make the function <span class="math-container">$f$</span> equal to the constant <span class="math-container">$c$</span>. The <strong>set of these pairs</strong> is the level curve of the function <span class="math-container">$f$</span> corresponding to the constant <span class="math-container">$c$</span>.</p>
<p>Varying <span class="math-container">$c$</span> you have the whole map of level curves of the function <span class="math-container">$f$</span>.</p>
<p>From a geometrical  point of view, it is as if we 'cut' the 3-dimensional graphic of the <span class="math-container">$f$</span>
function with a plane parallel to <span class="math-container">$x_1,x_2$</span> plane, and project this 'cut' on the <span class="math-container">$x_1,x_2$</span> plane. More precisely, geometrically, a level curve is the orthogonal projection on the <span class="math-container">$x_1,x_2$</span> of the curves obtained  intersecting the plane <span class="math-container">$z=c$</span> and the graphic of the function <span class="math-container">$z=f(x_1, x_2)$</span>.</p>
<p>This way we pass from the 3-dimensional graphic of <span class="math-container">$f$</span>to the 2-dimensional graphic of the level curves.</p>
<p>For example, in the  pictures  below there are the 3-dimensional graphic of the function <span class="math-container">$z=1-x^2-y^2$</span>
and the 2-dimensional graphic of its level curves.</p>
<p><a href="https://i.sstatic.net/8Rhn7.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/8Rhn7.png" alt="enter image description here" /></a>
<a href="https://i.sstatic.net/vPCmz.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/vPCmz.png" alt="enter image description here" /></a></p>
<p>In the case of a utility function <span class="math-container">$U$</span> it is exactly the same, you 'cut' with planes, corresponding to various constant levels of utility, the utility function and project the level curves of <span class="math-container">$U$</span> on the <span class="math-container">$x_1, x_2$</span> plane.</p>
<p>This way you obtain, in two dimensions, the usual indifference curves. From a mathematical point they are exactly the level curves of the utility function, as you already said.</p>
<p><strong>Implicit function theorem</strong></p>
<p>Let's come back to the pairs of <span class="math-container">$_1,_2$</span>
that make  <span class="math-container">$f$</span> assume a specific value  <span class="math-container">$c$</span> in <span class="math-container">$R$</span>, that is to the equation</p>
<p><span class="math-container">$f(x_1,x_2)=c$</span>      or</p>
<p><span class="math-container">$f(x_1, x_2)- c=0$</span>.</p>
<p>The problem is if it is possible to have an explicit form of the function, that is if this equation can be explicitly solved for <span class="math-container">$x_1$</span> (or <span class="math-container">$x_2$</span>).</p>
<p>The answer is yes,  <strong>under certain assumptions</strong>.  Under certain assumptions, the equation  can be solved  for  <span class="math-container">$x_1$</span> (or <span class="math-container">$x_2$</span>) , and in this case it defines a function <span class="math-container">$g$</span> from <span class="math-container">$R$</span> to <span class="math-container">$R$</span>, that is a function</p>
<p><span class="math-container">$x_1=g(x_2)$</span></p>
<p>that gives explicitly all the pairs  <span class="math-container">$(x_1,x_2)$</span> that make the function <span class="math-container">$f$</span> equal to the constant <span class="math-container">$c$</span>.</p>
<p>In this case the equation <span class="math-container">$f(x_1, x_2)- c=0$</span> is said the <em>implicit form</em> of the function.</p>
<p>The <strong>implicit function theorem states the assumptions</strong> under which such a  function <span class="math-container">$g$</span> can be defined.</p>
<p>For an example of case in which such a <span class="math-container">$g$</span> function cannot be defined, let's come back to level curves. The point is that <strong>not any level curve represents a function</strong>, for examples the level curves of the picture  above are circumferences, and they do not represent a function, because the equation of a circumference  is not a function (it doesn't satisfy the definition of function).</p>
<p>(The implicit function theorem, however, states conditions under which such a  function can be <em>locally</em> defined).</p>
<p><strong>As for the derivative of <span class="math-container">$g$</span></strong> (for example the slope of the indifferences curve),  we resort again to the implicit function theorem.</p>
<p>The implicit function theorem not only states the conditions for the existence of <span class="math-container">$g$</span>, but <strong>gives us a formula to find its derivative.</strong> It gives the derivative of <span class="math-container">$g$</span>, <span class="math-container">$dx_1/dx_2$</span>, in terms of the partial derivatives of <span class="math-container">$f$</span>:</p>
<p><span class="math-container">$(dx_1/dx_2)=-(\frac{\partial f}{\partial x_2})/(\frac{\partial f}{\partial x_1})$</span></p>
<p>If you replace  <span class="math-container">$f$</span> with <span class="math-container">$U$</span>, you have the derivative of the indifference curve you wrote in your post.</p>
<p>Then you ask why this way we have a <strong>tangent</strong> in a plane parallel to <span class="math-container">$_1,_2$</span> .</p>
<p>Actually, you have not a 'tangent', you have a derivative of a function from <span class="math-container">$R$</span> to <span class="math-container">$R$</span>.</p>
<p>But you know that a derivative  of a function in a certain point geometrically represents the slope of the tangent to the function in that point.</p>
<p>So, knowing the derivative, you can graphically represent  the tangent to a level curve in a certain point in the plane <span class="math-container">$x_1,x_2$</span>, (or, if you prefer, you can 'lift' this tangent in the 3-dimensional space and see it in the plane that cuts the <span class="math-container">$f$</span> function where <span class="math-container">$f=c$</span>, parallel to the <span class="math-container">$(_1,_2)$</span> plane).</p>
<p>By the way, the formulas you wrote to find that derivative are a shortcut, a practical trick to find it,  but the rigorous foundation is the implicit function theorem.</p>

CC BY-SA 4.0
2022-07-26T19:31:13.483Z
2022-07-26T19:31:13.483Z
BakerStreet
<p>This isn't a perfect answer to your question, but I tend to show all my students pages 1-4 of Dixit's Optimization book. It describes exactly what the MRS is doing and why it must be tangent to the budget constraint. </p>

<p><a href="https://www.scribd.com/document/390573490/Dixit-Optimization-in-Economic-Theory-pdf" rel="nofollow noreferrer">https://www.scribd.com/document/390573490/Dixit-Optimization-in-Economic-Theory-pdf</a> (pages 1-3 here unless you sign in) </p>

CC BY-SA 4.0
2019-08-07T14:50:44.753Z
2019-08-07T14:50:44.753Z
Kyle Butts
<p><a href="https://i.sstatic.net/uKTvl.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/uKTvl.png" alt="enter image description here"></a></p>

<p>So I think I understand the math, but I don't understand the intuition on a 3d graph. Why does the ratio of partial derivatives give us a tangent line on a level curve(indifference curve)? My understanding is  <span class="math-container">$\frac{\partial U(x_1,x_2)}{\partial x_1}$</span> Will give a tangent line parallel to the <span class="math-container">$U(x_1,x_2), x_1$</span> plane and <span class="math-container">$\frac{\partial U(x_1,x_2)}{\partial x_2}$</span> Will give a tangent line parallel to the <span class="math-container">$U(x_1,x_2), x_2$</span> plane. So why does dividing them end up with a tangent line parallel to the <span class="math-container">$(x_1, x_2)$</span> plane?</p>

CC BY-SA 4.0
2019-08-06T04:33:22.853Z
Try looking up "implicit differentiation". This video might also help (I love this channel.) https://www.youtube.com/watch?v=qb40J4N1fa4
Art
<microeconomics>
What is the graphic intuition behind Marginal rate of subsitution
SLHep
# What is the graphic intuition behind Marginal rate of subsitution

(Asked by: username_0 on 2019-08-06)

[![enter image description here](https://i.sstatic.net/uKTvl.png)](https://i.sstatic.net/uKTvl.png)

So I think I understand the math, but I don't understand the intuition on a 3d graph. Why does the ratio of partial derivatives give us a tangent line on a level curve(indifference curve)? My understanding is $\frac{\partial U(x\_1,x\_2)}{\partial x\_1}$ Will give a tangent line parallel to the $U(x\_1,x\_2), x\_1$ plane and $\frac{\partial U(x\_1,x\_2)}{\partial x\_2}$ Will give a tangent line parallel to the $U(x\_1,x\_2), x\_2$ plane. So why does dividing them end up with a tangent line parallel to the $(x\_1, x\_2)$ plane?

**Tags:** microeconomics
> **Comments:**
> **username_1**: Try looking up "implicit differentiation". This video might also help (I love this channel.) https://www.youtube.com/watch?v=qb40J4N1fa4  

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## 2 answers

### Answer by username_2 on 2022-07-26. Score: 2

The mathematical concepts we have to recall in order to answer your question are that of **level curves of a function** and the so-called **theorem of** **implicit functions**, a very important theorem of mathematical analysis. To explain them in short words isn't an easy task, I hope I can succeed in explaining it a little.

I give below a short synthesis of these mathematical concept, to make clearer what I will say.

**Level curves of a function**

Through level curves we pass from a representation in three dimensions of a function to a representation in two dimensions: we will see that this is the case of the usual representation of the indifference curves in the plane.

Consider a function $f$ of two real variables, $x\_1,x\_2$ which assumes values in $R$:

$y=f(x\_1,x\_2)$

We want to know the pairs $x\_1,x\_2$ that make the $f$ assume a specific value, say $c$, in $R$. Formally we can write:

$f(x\_1,x\_2)=c$

This last equation describes all the pairs $(x\_1,x\_2)$
that make the function $f$ equal to the constant $c$. The **set of these pairs** is the level curve of the function $f$ corresponding to the constant $c$.

Varying $c$ you have the whole map of level curves of the function $f$.

From a geometrical point of view, it is as if we 'cut' the 3-dimensional graphic of the $f$
function with a plane parallel to $x\_1,x\_2$ plane, and project this 'cut' on the $x\_1,x\_2$ plane. More precisely, geometrically, a level curve is the orthogonal projection on the $x\_1,x\_2$ of the curves obtained intersecting the plane $z=c$ and the graphic of the function $z=f(x\_1, x\_2)$.

This way we pass from the 3-dimensional graphic of $f$to the 2-dimensional graphic of the level curves.

For example, in the pictures below there are the 3-dimensional graphic of the function $z=1-x^2-y^2$
and the 2-dimensional graphic of its level curves.

[![enter image description here](https://i.sstatic.net/8Rhn7.png)](https://i.sstatic.net/8Rhn7.png)
[![enter image description here](https://i.sstatic.net/vPCmz.png)](https://i.sstatic.net/vPCmz.png)

In the case of a utility function $U$ it is exactly the same, you 'cut' with planes, corresponding to various constant levels of utility, the utility function and project the level curves of $U$ on the $x\_1, x\_2$ plane.

This way you obtain, in two dimensions, the usual indifference curves. From a mathematical point they are exactly the level curves of the utility function, as you already said.

**Implicit function theorem**

Let's come back to the pairs of $\_1,\_2$
that make $f$ assume a specific value $c$ in $R$, that is to the equation

$f(x\_1,x\_2)=c$ or

$f(x\_1, x\_2)- c=0$.

The problem is if it is possible to have an explicit form of the function, that is if this equation can be explicitly solved for $x\_1$ (or $x\_2$).

The answer is yes, **under certain assumptions**. Under certain assumptions, the equation can be solved for $x\_1$ (or $x\_2$) , and in this case it defines a function $g$ from $R$ to $R$, that is a function

$x\_1=g(x\_2)$

that gives explicitly all the pairs $(x\_1,x\_2)$ that make the function $f$ equal to the constant $c$.

In this case the equation $f(x\_1, x\_2)- c=0$ is said the *implicit form* of the function.

The **implicit function theorem states the assumptions** under which such a function $g$ can be defined.

For an example of case in which such a $g$ function cannot be defined, let's come back to level curves. The point is that **not any level curve represents a function**, for examples the level curves of the picture above are circumferences, and they do not represent a function, because the equation of a circumference is not a function (it doesn't satisfy the definition of function).

(The implicit function theorem, however, states conditions under which such a function can be *locally* defined).

**As for the derivative of $g$** (for example the slope of the indifferences curve), we resort again to the implicit function theorem.

The implicit function theorem not only states the conditions for the existence of $g$, but **gives us a formula to find its derivative.** It gives the derivative of $g$, $dx\_1/dx\_2$, in terms of the partial derivatives of $f$:

$(dx\_1/dx\_2)=-(\frac{\partial f}{\partial x\_2})/(\frac{\partial f}{\partial x\_1})$

If you replace $f$ with $U$, you have the derivative of the indifference curve you wrote in your post.

Then you ask why this way we have a **tangent** in a plane parallel to $\_1,\_2$ .

Actually, you have not a 'tangent', you have a derivative of a function from $R$ to $R$.

But you know that a derivative of a function in a certain point geometrically represents the slope of the tangent to the function in that point.

So, knowing the derivative, you can graphically represent the tangent to a level curve in a certain point in the plane $x\_1,x\_2$, (or, if you prefer, you can 'lift' this tangent in the 3-dimensional space and see it in the plane that cuts the $f$ function where $f=c$, parallel to the $(\_1,\_2)$ plane).

By the way, the formulas you wrote to find that derivative are a shortcut, a practical trick to find it, but the rigorous foundation is the implicit function theorem.


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### Answer by username_3 on 2019-08-07. Score: 0

This isn't a perfect answer to your question, but I tend to show all my students pages 1-4 of Dixit's Optimization book. It describes exactly what the MRS is doing and why it must be tangent to the budget constraint.

<https://www.scribd.com/document/390573490/Dixit-Optimization-in-Economic-Theory-pdf> (pages 1-3 here unless you sign in)


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