<p>The marginal rate of substitution is the slope of the indifference curve multiplied by -1. Since the indifference curve is negatively sloped (as soon as the person prefers to consume the goods), the MRS is always <em>positive</em>. This has nothing to do with the convexity of the preferences.</p>

<p>If the indifference curve is convex without being strictly convex, then it is linear, i.e. it has constant slope. In that case, the marginal rate of substitution is constant (but still positive!).</p>

CC BY-SA 3.0
2015-05-17T09:55:35.750Z
2015-05-17T10:53:23.340Z
2015-05-17T10:53:23.340Z
Oliv
<p>If one of the goods is not beneficial and the other is then the slope of the indifference curve will be positive and hence the MRS negative. This is becuase
$$
MRS(x,y) = - \frac{MU_x(x,y)}{MU_y(x,y)},
$$
and if $x$ is beneficial then its marginal utility, $MU_x(x,y)$ is positive, whereas if $y$ is not beneficial then its marginal utility will be non-positive. (Zero if it is a neutral good.)</p>

<p>As @Oliv stated this has nothing to do with convexity.</p>

<p>An example: Let (y,t) denote the amount of money you have and the time it takes for you to get to work/school. You probably enjoy having as much money as possible. Because the same thing goes for free time you probably want to minimize the length of your commute. Two utility functions that could represent your preferences:
$$
U_1(y,t) = y^2 - t \hskip 20pt  U_2(y,t) = \sqrt{y} - t.
$$
Money is beneficial in both of these and travel time is not. But one represents convex preferences and the other concave.</p>

CC BY-SA 3.0
2015-05-17T10:58:00.323Z
This is true, thanks for the remark. In that case, I believe that the standard practice would be to make a change of variable and consider the good $-t$ on one axis (free time) instead of $t$ (time spent commuting) to keep the utility function increasing in both variables.
Oliv
CC BY-SA 3.0
2015-05-17T10:36:34.440Z
2015-05-17T15:39:36.953Z
2015-05-17T15:39:36.953Z
Giskard
<p>Strict convexity is defined as </p>

<blockquote>
  <p>Let $X$ be a convex set in a real vector space and let $f: X\rightarrow \Bbb{R}$ be a function. $f$ is called <em>strictly convex</em> if $\forall x_1 \neq x_2 \in X,$ and $\forall t \in (0, 1)$:$$f(tx_1+(1-t)x_2) &lt; t f(x_1)+(1-t)f(x_2)$$</p>
</blockquote>

<p>If this doesn't hold, does marginal rate of substitution still have to be negative? </p>

<consumer-theory>
If strict convexity of indifference curves isn't assumed, does MRS have to be negative?
Stan Shunpike
# If strict convexity of indifference curves isn't assumed, does MRS have to be negative?

(Asked by: username_0 on 2015-05-17)

Strict convexity is defined as

> Let $X$ be a convex set in a real vector space and let $f: X\rightarrow \Bbb{R}$ be a function. $f$ is called *strictly convex* if $\forall x\_1 \neq x\_2 \in X,$ and $\forall t \in (0, 1)$:$$f(tx\_1+(1-t)x\_2) < t f(x\_1)+(1-t)f(x\_2)$$

If this doesn't hold, does marginal rate of substitution still have to be negative?

**Tags:** consumer-theory

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## 2 answers

### Answer by username_1 on 2015-05-17. Score: 2

The marginal rate of substitution is the slope of the indifference curve multiplied by -1. Since the indifference curve is negatively sloped (as soon as the person prefers to consume the goods), the MRS is always *positive*. This has nothing to do with the convexity of the preferences.

If the indifference curve is convex without being strictly convex, then it is linear, i.e. it has constant slope. In that case, the marginal rate of substitution is constant (but still positive!).


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### Answer by username_2 on 2015-05-17. Score: 2

If one of the goods is not beneficial and the other is then the slope of the indifference curve will be positive and hence the MRS negative. This is becuase
$$
MRS(x,y) = - \frac{MU\_x(x,y)}{MU\_y(x,y)},
$$
and if $x$ is beneficial then its marginal utility, $MU\_x(x,y)$ is positive, whereas if $y$ is not beneficial then its marginal utility will be non-positive. (Zero if it is a neutral good.)

As @Oliv stated this has nothing to do with convexity.

An example: Let (y,t) denote the amount of money you have and the time it takes for you to get to work/school. You probably enjoy having as much money as possible. Because the same thing goes for free time you probably want to minimize the length of your commute. Two utility functions that could represent your preferences:
$$
U\_1(y,t) = y^2 - t \hskip 20pt U\_2(y,t) = \sqrt{y} - t.
$$
Money is beneficial in both of these and travel time is not. But one represents convex preferences and the other concave.

> **Comments:**
> **username_1**: This is true, thanks for the remark. In that case, I believe that the standard practice would be to make a change of variable and consider the good $-t$ on one axis (free time) instead of $t$ (time spent commuting) to keep the utility function increasing in both variables.  

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