# Marginal rate of substitution

This is a homework question.

Suppose a consumer has preferences over two goods that can be represented by the utility function $U(x,y) = 2\sqrt{x} + y$

The marginal rate of substitution of $x$ for $y$ in this case is $\frac{1}{ 2\sqrt{x}}$, which is the negative of the slope of the indifference curve. It's well defined only for $x > 0$

The question asks to plot the indifference curve with $x$ on the horizontal axis and y on the vertical axis, and indicate if the graph of indifference curve will intersect either or both axes.

I was thinking that since slope of the indifference curve tends to infinity as $x$ approaches $0$, therefore the indifference curve should not intersect the y-axis. However, the solution provided by the lecturer says "since it's possible to have positive utility when either $x$ or $y$ is zero, the indifference curve intersects both axes", and I kinda agree this statement as well...

So what should be the answer?

There is more than one indifference curve. There is one belonging to every utility level. So for any utility level $c$, the points $(x,y)$ that satisfy $$2\cdot \sqrt{x} + y = c$$ are an indifference curve.
For example let $c = 2$. Can you find a point $(x,y)$ that satisfies $$2\cdot \sqrt{x} + y = 2$$ and is on the $y$-axis or the $x$-axis?
• Now I see your problem. The MRS shows the slope of the indifference curve at a point, so if you were to draw a tangential line at that point, this would be its slope. At the exact point where an indifference curve reaches the $y$-axis, the tangential line would be vertical, since MRS$(x,y) = \frac{1}{\sqrt{x}}$. Technically a vertical line has no slope, it is 'infinitely' steep. But vertical lines still exist, and so do points that are both on an indifference curve and the $y$-axis. – Giskard Apr 30 '15 at 8:07
The slope does tend toward infinity but it does intersect. What's tricky about the question is that the "indifference curve" becomes a single point at $(y=0,x=0)$ for a utility value of $0$.