<p>Just to integrate the very good answer of Nicolas Torres, maybe a picture is the better way to see the geometrical meaning of the condition MRS=relative prices.</p>
<p>Remember that relative prices are (the absolute value of) the slope of the budget line.</p>
<p>And that the MRS is the slope of the indifference curves.</p>
<p>The graphic representation can be as follows:</p>
<p><a href="https://i.sstatic.net/qK8dz.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/qK8dz.png" alt="enter image description here" /></a></p>
<p>The condition MRS=relative prices, therefore, says that the consumer achieves  her optimal choice, given her income,  when she chooses a bundle of the two goods for which the two slopes are equal, that is the <strong>indifference curve is tangent to the budget line.</strong></p>
<p>In the graph, at point <span class="math-container">$F$</span>.</p>
<p>From an intuitive point of view, this is clear: an indifference curve as <span class="math-container">$C$</span> cannot be reached, as the consumer has not enough income.</p>
<p>A curve like curve <span class="math-container">$A$</span>  is not convenient, as the consumer can reach a higher utility, with her income, on curve <span class="math-container">$B$</span>.</p>

CC BY-SA 4.0
2022-11-17T20:22:10.893Z
Thank you, and thanks to Nicolas too. Now I got it!
FuzzyOperator
CC BY-SA 4.0
2022-11-17T20:25:35.633Z
You are welcome!
BakerStreet
CC BY-SA 4.0
2022-11-17T18:04:20.497Z
2022-11-17T20:16:03.740Z
2022-11-17T20:16:03.740Z
BakerStreet
<p>We want to solve the optimization problem</p>
<ul>
<li><span class="math-container">$\max U(x,y)$</span> subject to <span class="math-container">$p_x x + p_y y = I$</span></li>
</ul>
<p>To set up the Lagrangian, we reorder the constraint so that it looks like “something <span class="math-container">$=0$</span>”:</p>
<ul>
<li><span class="math-container">$I - p_x x - p_y y = 0$</span></li>
</ul>
<p>The Lagrangian is</p>
<ul>
<li><span class="math-container">$\mathcal{L}(x,y,\lambda) = U(x,y) + \lambda (I - p_x x - p_y y)$</span></li>
</ul>
<p>Our first order conditions are</p>
<ul>
<li><p><span class="math-container">$\frac{\partial \mathcal{L}}{\partial x} = \frac{\partial U}{\partial x} - \lambda p_x = 0 \implies \lambda = \frac{MU_x}{p_x}$</span></p>
</li>
<li><p><span class="math-container">$\frac{\partial \mathcal{L}}{\partial y} = \frac{\partial U}{\partial y} - \lambda p_y = 0 \implies \lambda = \frac{MU_y}{p_y}$</span></p>
</li>
<li><p><span class="math-container">$\frac{\partial \mathcal{L}}{\partial \lambda} = I - p_x x - p_y y = 0 \implies p_x x + p_y y = I$</span></p>
</li>
</ul>
<p>Since <span class="math-container">$\lambda = \lambda$</span>, from the first two equations we have</p>
<p><span class="math-container">$\frac{MU_x}{p_x} = \frac{MU_y}{p_y} \implies MRS = \frac{MU_x}{MU_y} = \frac{p_x}{p_y}$</span></p>
<p>Note: <span class="math-container">$\frac{p_x}{p_y}$</span> is actually the absolute value of the slope of the budget constraint line.</p>

CC BY-SA 4.0
2022-11-17T15:58:14.537Z
Hi @Nicolas Torres, and thank you for taking the time.

What I am trying to understand, though, is why later on you use MRS = px/py (with new prices) in a system with the utility function. What is the first equation (MRS = px/py), in geometric terms? Is it a representation of the budget constraint equation? Or of its slope? Or is it something else?
FuzzyOperator
CC BY-SA 4.0
2022-11-17T03:19:49.740Z
2022-11-17T16:57:42.150Z
2022-11-17T16:57:42.150Z
Nicolas Torres
<p>I'm trying to understand how to solve an exercise about income and substitution effects. I got the theory, I guess, but can't get the maths.</p>
<p>One of the steps would be to put in a system the equation MRS=relative prices with the utility function to calculate the coordinates of the substitution effect. Why do you do that, in plain terms?</p>

CC BY-SA 4.0
2022-11-16T15:29:10.683Z
Welcome to SE. Could you explain better your question? You are asking just why MRS= relative prices, as in the title, or you want help to solve a specific  exercise about income and substitution effects? In this case, you'd better post your exercise, to allow contributors to answer.
BakerStreet
CC BY-SA 4.0
2022-11-17T15:56:31.767Z
Hi @BakerStreet, sorry for not explaining my question better. I wanted to understand the reasons why that specific method is used, in general. More specifically, I'd like to know what it means from a geometric point of view.
FuzzyOperator
CC BY-SA 4.0
2022-11-17T16:08:52.933Z
You are welcome. I said this just  to allow you to have the right answer you are looking for.
BakerStreet
<microeconomics><income-effect>
Income and substitution effect: why MRS=relative prices?
FuzzyOperator
# Income and substitution effect: why MRS=relative prices?

(Asked by: username_0 on 2022-11-16)

I'm trying to understand how to solve an exercise about income and substitution effects. I got the theory, I guess, but can't get the maths.

One of the steps would be to put in a system the equation MRS=relative prices with the utility function to calculate the coordinates of the substitution effect. Why do you do that, in plain terms?

**Tags:** microeconomics, income-effect

> **Comments:**
> **username_1**: Welcome to SE. Could you explain better your question? You are asking just why MRS= relative prices, as in the title, or you want help to solve a specific  exercise about income and substitution effects? In this case, you'd better post your exercise, to allow contributors to answer.  
> **username_0**: Hi @username_1, sorry for not explaining my question better. I wanted to understand the reasons why that specific method is used, in general. More specifically, I'd like to know what it means from a geometric point of view.  
> **username_1**: You are welcome. I said this just  to allow you to have the right answer you are looking for.  

---

## 2 answers

### Answer by username_1 on 2022-11-17. Score: 1 | Accepted answer

Just to integrate the very good answer of username_2 username_2, maybe a picture is the better way to see the geometrical meaning of the condition MRS=relative prices.

Remember that relative prices are (the absolute value of) the slope of the budget line.

And that the MRS is the slope of the indifference curves.

The graphic representation can be as follows:

[![enter image description here](https://i.sstatic.net/qK8dz.png)](https://i.sstatic.net/qK8dz.png)

The condition MRS=relative prices, therefore, says that the consumer achieves her optimal choice, given her income, when she chooses a bundle of the two goods for which the two slopes are equal, that is the **indifference curve is tangent to the budget line.**

In the graph, at point $F$.

From an intuitive point of view, this is clear: an indifference curve as $C$ cannot be reached, as the consumer has not enough income.

A curve like curve $A$ is not convenient, as the consumer can reach a higher utility, with her income, on curve $B$.

> **Comments:**
> **username_0**: Thank you, and thanks to username_2 too. Now I got it!  
> **username_1**: You are welcome!  

---

### Answer by username_2 on 2022-11-17. Score: 1

We want to solve the optimization problem

* $\max U(x,y)$ subject to $p\_x x + p\_y y = I$

To set up the Lagrangian, we reorder the constraint so that it looks like “something $=0$”:

* $I - p\_x x - p\_y y = 0$

The Lagrangian is

* $\mathcal{L}(x,y,\lambda) = U(x,y) + \lambda (I - p\_x x - p\_y y)$

Our first order conditions are

* $\frac{\partial \mathcal{L}}{\partial x} = \frac{\partial U}{\partial x} - \lambda p\_x = 0 \implies \lambda = \frac{MU\_x}{p\_x}$
* $\frac{\partial \mathcal{L}}{\partial y} = \frac{\partial U}{\partial y} - \lambda p\_y = 0 \implies \lambda = \frac{MU\_y}{p\_y}$
* $\frac{\partial \mathcal{L}}{\partial \lambda} = I - p\_x x - p\_y y = 0 \implies p\_x x + p\_y y = I$

Since $\lambda = \lambda$, from the first two equations we have

$\frac{MU\_x}{p\_x} = \frac{MU\_y}{p\_y} \implies MRS = \frac{MU\_x}{MU\_y} = \frac{p\_x}{p\_y}$

Note: $\frac{p\_x}{p\_y}$ is actually the absolute value of the slope of the budget constraint line.

> **Comments:**
> **username_0**: Hi @username_2, and thank you for taking the time.  What I am trying to understand, though, is why later on you use MRS = px/py (with new prices) in a system with the utility function. What is the first equation (MRS = px/py), in geometric terms? Is it a representation of the budget constraint equation? Or of its slope? Or is it something else?  

---

