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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?

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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.

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  • $\begingroup$ 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. $\endgroup$ – Oliv May 17 '15 at 10:58
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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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