<p>The tangency condition itself is the condition that the optimal point, for an interior optimum, is the point where the MRS must equal the slope of the budget line. So the second part of your question is true. It is at the tangent point where MRS equals the slope of the budget line. You've confused yourself with the second (sufficient) condition which, on the other hand, states that the utility function must be strictly quasi-concave in order for an interior optimum to exist.
Hope this clarifies. </p>

CC BY-SA 4.0
2019-07-23T07:27:59.260Z
Does strict quasi-concavity guarantee an interior solution? Why?
Giskard
CC BY-SA 4.0
2019-07-23T06:46:23.390Z
2019-07-23T06:46:23.390Z
Harper Lee
<p>Below is an example where the tangency (point <span class="math-container">$A$</span>) is not the optimum (assuming both <span class="math-container">$x$</span> and <span class="math-container">$y$</span> are "goods" so that more of either good is preferred to less). In fact, the optimum occurs at the corner (point <span class="math-container">$B$</span>) where the indifference curve is not tangent to the budget line. This happens because the preference over <span class="math-container">$x$</span> and <span class="math-container">$y$</span> is not <a href="https://en.wikipedia.org/wiki/Convex_preferences" rel="nofollow noreferrer">convex</a>, which is manifested by a set of indifference curves that is concave to the origin. </p>

<p><a href="https://i.sstatic.net/I6GrW.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/I6GrW.png" alt="enter image description here"></a></p>

<p>We normally assume that preference is convex, which generates indifference curves that are convex to the origin. Think of this as a taste for variety: if I'm indifferent between 2 can of pop and 2 slices of pizza, then I will be strictly better off if I get a combo of 1 pop and 1 pizza than having only one of the goods. </p>

<p>A non-convex preference represents the opposite taste for variety. Suppose <span class="math-container">$x$</span> is pop and <span class="math-container">$y$</span> is coffee, and I, as most people, would prefer to have them separately than combined. Then my preference over pop and coffee would be non-convex.</p>

CC BY-SA 4.0
2019-07-23T18:19:30.043Z
2019-07-23T18:50:57.627Z
2019-07-23T18:50:57.627Z
Herr K.
<p>I read that the tangency condition is not sufficient for optimality, and that one other condition is that the MRS must equal the slope of the budget line at an interior optimum. My confusion is that since MRS is the slope of an indifference curve, then wouldn't it automatically be equal to the slope of the budget line of the budget line is tangent to the indifference curve?</p>

CC BY-SA 4.0
2019-07-26T15:15:34.583Z
In this context, the "tangency condition" is that the MRS is equal to to slope of the budget line. That is, your two conditions are equivalent. However, there can still be a corner solution.
Bayesian
CC BY-SA 4.0
2019-07-23T06:23:33.110Z
1. Where did you read this? 2. What exactly do you mean by "tangency condition"?
Giskard
<optimization>
Optimal point and MRS
itssohardtocomeupwithusernames
# Optimal point and MRS

(Asked by: username_0 on 2019-07-22)

I read that the tangency condition is not sufficient for optimality, and that one other condition is that the MRS must equal the slope of the budget line at an interior optimum. My confusion is that since MRS is the slope of an indifference curve, then wouldn't it automatically be equal to the slope of the budget line of the budget line is tangent to the indifference curve?

**Tags:** optimization
> **Comments:**
> **username_2**: 1. Where did you read this? 2. What exactly do you mean by "tangency condition"?  
> **username_1**: In this context, the "tangency condition" is that the MRS is equal to to slope of the budget line. That is, your two conditions are equivalent. However, there can still be a corner solution.  

---

## 2 answers

### Answer by username_3 on 2019-07-23. Score: 0 | Accepted answer

The tangency condition itself is the condition that the optimal point, for an interior optimum, is the point where the MRS must equal the slope of the budget line. So the second part of your question is true. It is at the tangent point where MRS equals the slope of the budget line. You've confused yourself with the second (sufficient) condition which, on the other hand, states that the utility function must be strictly quasi-concave in order for an interior optimum to exist.
Hope this clarifies.

> **Comments:**
> **username_2**: Does strict quasi-concavity guarantee an interior solution? Why?  

---

### Answer by username_4 on 2019-07-23. Score: 0

Below is an example where the tangency (point $A$) is not the optimum (assuming both $x$ and $y$ are "goods" so that more of either good is preferred to less). In fact, the optimum occurs at the corner (point $B$) where the indifference curve is not tangent to the budget line. This happens because the preference over $x$ and $y$ is not [convex](https://en.wikipedia.org/wiki/Convex_preferences), which is manifested by a set of indifference curves that is concave to the origin.

[![enter image description here](https://i.sstatic.net/I6GrW.png)](https://i.sstatic.net/I6GrW.png)

We normally assume that preference is convex, which generates indifference curves that are convex to the origin. Think of this as a taste for variety: if I'm indifferent between 2 can of pop and 2 slices of pizza, then I will be strictly better off if I get a combo of 1 pop and 1 pizza than having only one of the goods.

A non-convex preference represents the opposite taste for variety. Suppose $x$ is pop and $y$ is coffee, and I, as most people, would prefer to have them separately than combined. Then my preference over pop and coffee would be non-convex.


---

