Marginal Rate of Substitution for perfect complements

Question

I have come across the following problem:

Determine the marginal rate of substitution MRS(x1, x2) at point (x1, x2) = (5,1) for the following function:

u(x1, x2) = min(x1, x2).

The solution is that the MRS is undefined at that point.

However, I don't understand why that is. With this utility function, we get an income expansion path that goes exactly 45 degrees from the origin, because the two goods are alway consumed in equal quantities. And as far as I know, the MRS of such a function (u(x1, x2) = min(αx1, βx2)) is only undefined at the exact angles of these curves, i.e. where x2 = (α/β)x1. In the problem at hand, however, we have x2 < (α/β)x1, i.e. 1 < 5. Shouldn't this mean that the MRS at the point (5, 1) is actually 0 and not undefined?

Answer 1

I think this is sort of a trick question.

First, you are completely right MRS is undefined at the kink - this is trivial MRS is the slope of indifference curve, which in this case is L shaped, and derivatives are not defined on the kink.

So this leaves us with two other parts of L shaped function. The vertical part and the horizontal part.

Over the horizontal portion of the indifference curve $\alpha x_1 > \beta x_2$ the MRS is given as:

$$MRS= \frac{U'_{x_1}}{U'_{x_2}} = \frac{0}{\beta}= 0 $$

So here the MRS is clearly defined.

However, at the vertical portion where $\alpha x_1<\beta x_2$ we will have a problem since:

$$MRS = \frac{U'_{x_1}}{U'_{x_2}} = \frac{\alpha}{0} = \infty | x_1 \wedge x_2 \geq 0 $$

but here on account of division by $0$ some people still say that the MRS is not defined.

However, the tricky part here is that choice of which good goes on $x$-axis and which good goes on $y$-axis is arbitrary. Usually people would put $x_1$ on $x$-axis and $x_2$ on $y$-axis but it is completely fine in principle to put $x_1$ on $y$-axis and $x_2$ on $x$-axis. In that case the result above would be exactly reversed.

If this is for an exam often people who draft exercises just make some simple answer key that might not cover all possible solutions and maybe the answer key included option where their role was reversed.

Answer 2

I was wrong before. I agree with you, it seems like it should be zero, and only undefined at the kink, where the derivatives of the utility functions don't exist.

Think of Leontief utility as CES utility where $\lim \rho \rightarrow \infty$. CES utility is: $$U(x_1,x_2, \rho) = (x_1^\rho + x_2^\rho) ^{1/\rho}$$ The MRS of of a CES utility function is: $$MRS = - (\frac{x_1}{x_2})^{\rho-1} $$ Taking the limit as $\rho \rightarrow \infty$: $$MRS = -(\frac{x_1}{x_2})^{\infty}$$ When $x_2 > x_1$ the MRS is negative is negative infinity. When $x_1 > x_2$ (as it is here), the MRS is 0. When $x_1 = x_2$, the Leontief utility function is not differentiable and this function doesn't exist.

Source:

Guoqiang Tian's Microeconomic Theory Lecture Notes (2013)